DYNAMIC CHARACTERISATION AND MODELLING OF DRY AND BOUNDARY LUBRICATED FRICTION FOR STABILISATION AND CONTROL PURPOSES

Arenb erg Do c t o raat ssc ho o l Wet ensc hap & Tec hno lo g ie Faculteit Ingenieurswetenschappen Departement Werktuigkunde

DYNAMIC CHARACTERISATION AND MODELLING OF DRY AND BOUNDARY LUBRICATED FRICTION FOR STABILISATION AND CONTROL PURPOSES

Thierry JANSSENS

Proefschrift voorgedragen tot het behalen van de graad van Doctor in de Ingenieurswetenschappen

2010D04

February 2010

KATHOLIEKE UNIVERSITEIT LEUVEN FACULTEIT INGENIEURSWETENSCHAPPEN DEPARTEMENT WERKTUIGKUNDE AFDELING PRODUCTIETECHNIEKEN, MACHINEBOUW EN AUTOMATISERING Celestijnenlaan 300B, 3001 Heverlee (Leuven), Belgium

DYNAMIC CHARACTERISATION AND MODELLING OF DRY AND BOUNDARY LUBRICATED FRICTION FOR STABILISATION AND CONTROL PURPOSES

Thierry JANSSENS

Jury: Prof. em. dr. ir. Y. D. Willems (voorzitter) Prof. dr. ir. F. Al-Bender (promotor) Prof. dr. ir. H. Van Brussel (promotor) Prof. dr. ir. J. Swevers Prof. dr. ir. J.-P. Celis Prof. dr. ir. P. Sas Prof. dr. ir. P. De Baets (Universiteit Gent) Prof. dr. ir. R. Dufour (Universit´ e de Lyon)

February 2010

Proefschrift voorgedragen tot het behalen van de graad van Doctor in de Ingenieurswetenschappen

c

Katholieke Universiteit Leuven Faculteit Toegepaste Wetenschappen Arenbergkasteel, B-3001 Heverlee (Leuven), Belgium

Alle rechten voorbehouden. Niets uit deze uitgave mag worden verveelvoudigd en/of openbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever. All rights reserved. No part of this publication may be reproduced in any form, by print, photoprint, microfilm or any other means without written permission from the publisher. D/2010/7515/10 ISBN 978-94-6018-172-6

Preface “There are those who look at things the way they are, and ask why... I dream of things that never were, and ask why not?” Robert Francis Kennedy

Hoe moet ik hier aan beginnen? Woorden zullen tekort schieten om uit te drukken wat hier nodig is. Toch zal ik een poging ondernemen. Ik draag dit werk op aan mijn beide grootmoeders, Maman Ju en Moeke, die ik gedurende mijn doctoraat verloren heb. Dit waren voor mij moeilijke tijden en ik had natuurlijk liever gehad dat ze er nog bij zouden zijn om deze volbrenging samen te delen. Dit onderzoek is begonnen met het schrijven van een proposal voor het Instituut voor de promotie van de Innovatie door Wetenschap en Technologie in Vlaanderen (I.W.T.-Vlaanderen). Bij de start van dit schrijven had ik eigenlijk geen flauw benul over wat ik moest schrijven. Uiteindelijk is dit allemaal goed gekomen door de juiste inzet en met de hulp van mijn promotoren, vandaar dat ik mijn dank betuig voor de financiering door I.W.T. als specialisatiebursaal. Mijn dank gaat natuurlijk ook uit naar mijn promotoren Prof. Farid Al-Bender en Prof. Hendrik Van Brussel, die me altijd hebben bijgestaan. Prof. Al-Bender die met zijn expertise me altijd heeft kunnen bijstaan en aan de hand van feedback op papers of op mijn onderzoek in het algemeen me binnen mijn traject in goede banen heeft geleid. Zo ook Prof. Van Brussel die voldoende tijd in me heeft ge¨ınvesteerd en die altijd klaar stond met goede raad. Zijn ervaring en eerlijkheid hebben me zeker veel bijgebracht. Ik dank ook mijn juryleden Prof. Jan Swevers, Prof. Jean-Pierre Celis, Prof. Paul Sas, Prof. Patrick De Baets en Prof. Régis Dufour voor de tijd die jullie hebben genomen om mijn doctoraatstekst kritisch door te nemen en te voorzien van commentaar. Jullie input heeft zeker voor een verbetering van mijn manuscript gezorgd. Prof. Régis Dufour, je veux vous remercier pour avoir accepté de devenir membre de mon jury, pour vos commentaires, vos

I

Voorwoord

contributions et pour votre cours intéressant. Ik mag de voorzitter van mijn jury, Prof. em. Yves Willems, niet vergeten. Dank u voor het voorzitten van mijn preliminaire en publieke doctoraatsverdediging. Ik zou ook graag de collega’s van de administratieve en technische diensten willen bedanken. Voor de administratieve hulp, dank u Luc Haine, Karin Dewit, Ann Letelier, Lieve Notré, Carine Coosemans en Regine Vanswijgenhoven. Voor de technische werkplaats, voor de productie van mijn testopstelling en voor de technische ondersteuning, dank u Dirk Bastiaensen, Eddy Smets, Frans ‘Franske’ De Kelver en Viggo Claeskens. Voor de dienst elektronica, voor de productie van meetapparatuur, voor het solderen van componenten, voor het mee bedenken van schakelingen, voor de bestellingen, ..., dank u Paul ‘Polleke’ Van Cauwenbergh, Bertram Van Soom, Luc De Simpelaere en Jean-Pierre Merckx voor het printen van posters en het fotograferen van testopstellingen. Voor de dienst informatica, voor het installeren van software, voor het herstellen van gecrashte systemen, voor het bestellen van computers en informaticabenodigdheden, dank u Jan Thielemans en Ronny Moreas. Iemand die ik zeker niet mag vergeten te bedanken is Prof. em. Paul Vanherck. Bedankt voor uw interesse en voor de vele gesprekken over verscheidene onderwerpen. Het was een ‘mooie’ ervaring en het was me een waar genoegen om zoveel van u te kunnen leren. I enjoyed working in such an international environment where I was able to meet and get to know many people from many nationalities with their specific cultural background and language. Many became very good friends and I learned a lot from them. Thank you all for these great four years and I hope we will stay in touch. Thank you. Dank u. Obrigado. Efcharisto. Merci. Terima kasih. Gracias. Grazie. Shukran. Spasibo. Xie xie ni. Motehshakeram. Two colleagues with which I had to pleasure to perform research, specifically in the domain of my PhD, are Kris De Moerlooze and Agusmian Partogi Ompusunggu. Thank you for the numerous discussions, for the friendly and constructive cooperation. Thanks to my colleagues, Tegoeh Tjahjowidodo, Hsiao-Wei Tang, Pauwel Goethals, Gorka Aguirre, Mauro Sette, Mohamed El-Said, Bert Willaert and Emmanuel Vander Poorten, with whom I shared an office and made it a pleasant environment. Special thanks go to Tegoeh who helped me with my first steps in my research, showed me around in the lab and was always there for a conversation or help on any subject. En ‘last but not least’ bedank ik mijn ouders voor hun steun en voor de kansen die ze mij gegeven hebben om dit allemaal waar te maken. Zonder jullie had ik hier natuurlijk niet gestaan. Dank u voor een liefdevolle thuis en voor het vertrouwen om mij vrij mijn eigen keuzes te laten maken en mijn eigen weg te laten gaan. Thierry Janssens Juni, 2009

II

Abstract For the realization of accurate actuation, friction is one of the most important phenomena which should be taken into account during design. To incorporate this phenomenon into the system simulation, friction models, which grasp the essence of the complex friction phenomena in a system in a set of equations, are a must. This research contributes to the development of such models, particularly models that can be used for a systematic design of automatic transmissions wet couplings, where (boundary and hydrodynamic) lubricated friction plays a crucial role. There is need for (1) a friction model that takes into account the underlying physical phenomena that form the basis of the frictional behaviour, for both dry and wet (lubricated) friction, (2) to derive an applicable, implementable model for the simulation and/or control of systems with friction elements, and (3) the development of control strategies, using (1) and (2) for the control and compensation of mechanical systems which contain those friction elements. The overall objective of the proposed research consists primarily in a theoretical modelling and experimental verification of the dynamic behaviour of the friction between surfaces for the case of dry and lubricated friction. Furthermore, the behaviour of a mechanical system where these surfaces form a crucial part, is examined using these models. The development of models based on the physical mechanisms that cause friction, however, always leads to a computationally intensive model, which cannot immediately be used for system simulation and control purposes. The development of a state model specifically for system simulation and control purposes that is less complex and useful for on-line applications will be the core of this work. Finally, the accuracy of this model is verified by means of experiments carried out on a transmission test setup. The research began with a literature study in order that the current state of the art in friction characterisation and modelling can be deduced. Research results related to physics-based (micro) models are critically and systematically analyzed with the aim of obtaining a suitable and satisfactory model.

III

Abstract

This model must contain the influences of the time dependence of the parameters as well as the interaction between the normal and tangential degrees of freedom. Parallel to this activity a test rig is designed and constructed to (mainly) qualify, quantify and correlate the wet friction behaviour with the model (and vice versa). The eXtended Generalised Maxwell-Slip (XGMS) model is developed for the compensation for friction in automatic transmissions. The extension is based on the Generalised Maxwell-Slip (GMS) model which is extended with a normal force dependence and a normal degree of freedom based on friction modelling and friction experiments. The normal load dependence is used to perform control on a wet multi-disk clutch using a developed adaptive torque controller. Keywords: dry friction, lubricated friction, characterisation, modelling, control, mechatronics

IV

Beknopte samenvatting Voor de realisatie van nauwkeurige aandrijvingen is wrijving één van de belangrijkste fenomenen die in rekening moeten gebracht worden tijdens het ontwerp. Om dit fenomeen te integreren in de systeemsimulatie zijn wrijvingsmodellen, die de essentie van het complexe wrijvingsfenomeen in een systeem van vergelijkingen vatten, een must. Dit onderzoek draagt bij tot de ontwikkeling van dergelijke modellen, meer bepaald modellen die kunnen gebruikt worden voor een systematisch ontwerp van automatische transmissies met natte koppelingen, waarbij (grens- en hydrodynamisch) gesmeerde wrijving een cruciale rol speelt. Er is nood aan (1) een wrijvingsmodel dat rekening houdt met de onderliggende fysische fenomenen die aan de basis van wrijving liggen, dit zowel voor droge als voor natte (gesmeerde) wrijving, (2) het afleiden van een werkbaar, implementeerbaar model voor de simulatie en/of de controle van systemen met wrijvingselementen, en (3) het ontwikkelen van controlestrategieën, met behulp van (1) en (2), voor de sturing en compensatie van mechanische systemen die wrijvingselementen bevatten. De algemene doelstelling van het voorgestelde onderzoek bestaat in de eerste plaats in een theoretische modellering en experimentele verificatie van het dynamisch gedrag van de wrijving tussen oppervlakken in het geval van droge en natte wrijving. Verder zal het gedrag van een mechanisch systeem waarvan deze oppervlakken deel uitmaken onderzocht worden met behulp van deze modellen. De ontwikkeling van modellen gebaseerd op de fysische mechanismen die wrijving veroorzaken leidt echter altijd tot een rekenintensief model dat niet onmiddellijk toepasbaar is voor systeemsimulatie en controledoeleinden. De ontwikkeling van een toestandsmodel specifiek voor systeemsimulatie en controledoeleinden dat minder complex en dat on-line is toe te passen vormt daarom de kern van dit project. Tenslotte, zal de nauwkeurigheid van dit model geverifieerd worden aan de hand van experimenten uit te voeren op een transmissielijn-testopstelling.

V

Beknopte samenvatting

Het project is begonnen met een literatuurstudie waaruit de huidige stand van kennis in wrijvingskarakterisering en modellering is af te leiden. Onderzoeksresultaten met betrekking tot fysicagebaseerde (micro)modellen worden kritisch en systematisch geanalyseerd met als doel een afdoend toestandsmodel te verkrijgen. Dit model moet de invloeden van de tijdsafhankelijkheid van de parameters zowel als de interactie tussen de tangentiële en de normale vrijheidsgraden bevatten. Parallel aan deze activiteit wordt een testopstelling ontwikkeld en gebouwd om (hoofdzakelijk) natte wrijvingsgedrag te kwalificeren, kwantificeren en correleren met het model (en vice versa). Het uitgebreide veralgemeende Maxwell-slip (XGMS) model is ontwikkeld voor de compensatie van de wrijving in automatische transmissies. De uitbreiding is gebaseerd op het veralgemeende Maxwell-slip (GMS) model dat wordt uitgebreid met een normale belastingsafhankelijkheid en een normale vrijheidsgraad op basis van wrijvingsmodellering en wrijvingsexperimenten. De normale belastingsafhankelijkheid wordt gebruikt om controle uit te voeren op een natte-plaat-koppeling met behulp van een ontwikkelde adaptieve koppelcontroller. Sleutelwoorden: droge wrijving, gesmeerde wrijving, karakterisering, modellering, controle, mechatronica

VI

List of abbreviations and symbols Abbreviations 1D 2D 3D ADC AFM ATF BL CAD DAC DCT DOF DOFs EDM EHL FE FEM FL FMTC FRF GMS HL ICP IO IT IWT

: : : : : : : : : : : : : : : : : : : : : : : : :

One-dimensional Two-dimensional Three-dimensional Analog to Digital Convertor Atomic Force Microscope Automatic Transmission Fluid Boundary Lubrication Computer Aided Design Digital to Analog Convertor Dual-Clutch Transmission Degree Of Freedom Degrees Of Freedom Electrical Discharge Machining Elasto-Hydrodynamic Lubrication Finite Element Finite Element Method Full Lubrication Flanders’ Mechatronics Technology Centre Frequency Response Function Generalized Maxwell-Slip Hydrodynamic Lubrication Integrated Circuit Piezo-electric Input Output Information Technology Institute for the Promotion of Innovation by Science and Technology in Flanders, Belgium

VII

List of abbreviations and symbols

LPV LTI ML MS MTM ODE PID PMA

: : : : : : : :

RMSE QFD RPM rps SAE XGMS

: : : : : :

Linear Parameter Varying Linear Time Invariant Mixed Lubrication Maxwell-Slip Metaalkunde en Toegepaste Materiaalkunde Ordinary Differential Equation Proportional Integral Derivative Production engineering, Machine design and Automation Root Mean Square Error Quality Function Deployment Revolutions Per Minute Revolutions Per Second Society of Automotive Engineers eXtended Generalized Maxwell-Slip

Latin symbols a aT A A Af ric Acyl b cd cef f ckf

: : : : : : : : : :

C C CN CT C1 C2 C3 C4 d Di Du

: : : : : : : : : : :

VIII

Contact patch radius Tangential acceleration Amplitude Linear system matrix Surface of friction contact Surface of cylinder Linear system vector Dynamometers damping Effective damping of dampers in series Non-linear damping component of friction material Constant Attraction parameter of (X)GMS model Normal attraction parameter of XGMS model Tangential attraction parameter of XGMS model Constant of integration Constant of integration Constant of integration Constant of integration Inner diameter of ring dynamometer Inner diameter of friction disk Outer diameter of friction disk

[m/s2 ]

[m2 ] [m2 ] [Ns/m] [Ns/m] [Ns/m]

[mm] [mm] [mm]


E E1 E2 E∗ f f (•) F Fa Fc Fn Fi Ft Fµ Ff Fs Fsi g(•) G G1 G2 G∗ h •

h ••

h h hthresh H H I1 Ief f j k k ka kb

: : : : : : : : : : : : : : : : : : : : : :

Young’s modulus Young’s modulus of material 1 Young’s modulus of material 2 Equivalent Young’s modulus Frequency Function of • Force Applied tangential force Coulomb friction force Normal force Output force of element i Tangential or friction force Friction force Friction force Static friction force Static friction force Function of • Shear modulus Shear modulus of material 1 Shear modulus of material 2 Equivalent shear modulus Dimensionless height of the back of a Rayleigh step bearing

:

Time derivative of height h

: Second time derivative of height h : Height of the back of a Rayleigh step bearing : Threshold on the bearing gap height : Dimensionless height of the front of a Rayleigh step bearing : Height of the front of a Rayleigh step bearing : 1st element of inertia matrix : Effective current √ : Imaginary unit = −1 : Stiffness : Stiffness vector : Shaft stiffness : Bellow coupling stiffness

[Pa] [Pa] [Pa] [Pa] [Hz] [N] [N] [N] [N] [N] [N] [N] [N] [N] [N] [Pa] [Pa] [Pa] []

[m]

[] [m] [kgm2 ] [A]

[N/m] [N/m] [N/m] [N/m]

IX


X

kef f kd kf ki kkf

: : : : :

kn kn1 kn2 kni kt kt1 kt2 kti L L

: : : : : : : : : :

m m1 M n N

: : : : :

N Nw p p1

: : : :

p2

:

pcontact

:

pcyl pm P Q r R R s

: : : : : : : :

Effective stiffness of stiffnesses in series Dynamometers stiffness Friction disk stiffness Linear spring constant of element i Non-linear stiffness component of friction material Normal stiffness Global normal stiffness Local normal stiffness Normal stiffness of element i Tangential stiffness Global tangential stiffness Local tangential stiffness Tangential stiffness of element i Length of front of Rayleigh step bearing Force constant for exponential decay of static friction force Mass of an asperity 1st element of mass matrix Mass of the Rayleigh step bearing Normal direction Amount of asperities, elasto-slide or Maxwell-Slip elements Amount of contacting pairs in a clutch Amount of unit steps Pressure Pressure in front part of a Rayleigh step bearing Pressure in back part of a Rayleigh step bearing Contact pressure (pressure behind piston at moment of contact) Oil pressure in the cylinder Maximum pressure Proportional control gain Tangential load Average radius of friction disk Distance Sphere radius Laplace variable

[N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [N/m] [m] [N] [kg] [kg] [kg]

[Pa] [Pa] [Pa] [Pa] [Pa] [Pa] [N] [m] [m] [m]


s s(v) S S S(v) t t t tr tx T T T0 Ts v vs vT V Vs w w W Wi Wn Wni x, y, z z z zb zi

: Stribeck effect function : Stribeck effect in function of sliding velocity : Surface : Stribeck behaviour function, consisting of the Stribeck and viscous effect : Stribeck behaviour in function of sliding velocity : Time : Tangential direction : Thickness of ring dynamometer : Rise time : Transition time : Torque : Temperature : Reference temperature : Static torque : Sliding velocity : Stribeck velocity : Tangential velocity : Sliding velocity : Stribeck velocity : Width of a unit step of the counterprofile : Width of ring dynamometer : Normal (pre)load : Maximum sliding force of element i : Lift-up force : Normal load for asperity i : Cartesian coordinates : Common displacement input : Normal displacement : Displacement of the main body : Initial height of asperity i

[m2 ]

[s] [mm] [s] [s] [Nm] [◦ C] [◦ C] [Nm] [m/s] [m/s] [m/s] [m/s] [m/s] [] [mm] [N] [N] [N] [N] [m]

XI


Greek symbols α αi

: :

β δ δn δt ∆h η

: : : : : :

Φ γ Λ Λthresh Λs Λ0

: : : : : :

µ µ0 µ∞ µmax µ(t) ν ω ω σ σv θ τ τ τt ξ ξi ζi ζi

: : : : : : : : : : : : : : : : : :

XII

Length ratio of Rayleigh step bearing Normalised maximum load beared by a Maxwell-slip element Ratio of Coulomb and static friction force Normal deformation Normal deformation Tangential deformation Step height of Rayleigh step bearing [] Coefficient of viscosity or dynamic viscosity (usual symbol is µ) Statistical distribution function Power of viscous velocity effect (shape factor) Sliding number Threshold on the sliding number Static sliding number Initial value of the sliding number, solution of the static Reynolds equation Friction coefficient Initial friction coefficient Limit value or asymptotic value of friction coefficient Maximum value of the friction coefficient Time dependent friction coefficient Poisson coefficient Rotational velocity [rad/s] Parameter to make time dimensionless [Hz] Squeeze film parameter Viscous velocity coefficient Angular position [rad] Shear stress [Pa] Force constant [N] Time constant [s] Position coordinate [m] Tangential deflection of the asperity State variable or the position of element i Normal defelection of asperity i


Miscellaneous symbols a A ¯ • ˆ • d L L−1

: : : : : : :

Vector (lower case, bold character) Matrix (upper case, bold character) Dimensional value of • Estimation of • Differential operator Laplace transform Inverse Laplace transform

XIII

XIV

Table of contents Preface

I

Abstract

III

Beknopte samenvatting

V


VII

Table of contents

I

XIV

Introduction

1

1 Introduction 1.1 Problem description . . . . . . . . . . . . . . . . . . 1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Framework of the research activities in the search unit . . . . . . . . . . . . . . . . . . . 1.2.2 Innovation . . . . . . . . . . . . . . . . . . . . 1.3 Research description . . . . . . . . . . . . . . . . . . 1.3.1 Autonomous torque controlled clutch . . . . . 1.3.2 Overview of the different tasks . . . . . . . . 1.4 Overview of main research aspects and challenges . . 1.5 Applications (in industry) . . . . . . . . . . . . . . . 1.6 Literature overview . . . . . . . . . . . . . . . . . . . 1.6.1 Friction characteristics . . . . . . . . . . . . . 1.6.2 Friction measurement . . . . . . . . . . . . . 1.6.3 Friction models . . . . . . . . . . . . . . . . . 1.6.3.1 Importance of friction modelling . . 1.6.3.2 Generic models . . . . . . . . . . . . 1.6.3.3 Control models . . . . . . . . . . . . 1.6.4 Clutch slip control . . . . . . . . . . . . . . . 1.7 Contributions to the state of the art . . . . . . . . .

XV

. . . . re. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

3 3 4

. . . . . . . . . . . . . . . .

5 5 5 8 9 12 13 14 14 19 22 23 23 25 28 29

Table of contents

1.8

II

Outline of the thesis . . . . . . . . . . . . . . . . . . . . .

Theoretical aspects

2 Physically based friction modelling 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2.2 The Generic model . . . . . . . . . . . . . . . . . . 2.2.1 Asperities . . . . . . . . . . . . . . . . . . . 2.2.2 Counter surface . . . . . . . . . . . . . . . . 2.2.3 Friction . . . . . . . . . . . . . . . . . . . . 2.2.4 Results . . . . . . . . . . . . . . . . . . . . 2.2.5 Conclusions . . . . . . . . . . . . . . . . . . 2.3 The Rayleigh step model . . . . . . . . . . . . . . . 2.3.1 Introduction . . . . . . . . . . . . . . . . . 2.3.2 Rayleigh step . . . . . . . . . . . . . . . . . 2.3.3 Implementation . . . . . . . . . . . . . . . . 2.3.4 Results . . . . . . . . . . . . . . . . . . . . 2.3.5 Conclusions . . . . . . . . . . . . . . . . . . 2.4 The Generic Rayleigh model . . . . . . . . . . . . . 2.4.1 Rayleigh step extension . . . . . . . . . . . 2.4.2 Implementation problems of the Rayleigh module within the Generic model . . . . . . 2.4.3 Conclusions . . . . . . . . . . . . . . . . . . 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . .

III

33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . step . . . . . . . . .

. . . . . . . . . . . . . . .

35 35 36 37 37 38 40 44 45 45 46 53 55 60 61 61

. . .

63 66 67

Experimental aspects

3 Tribometer for macroscopic measurements 3.1 Tribometer . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 System requirements . . . . . . . . . . . . . . . . . . . . . 3.5 Schematic/conceptual configuration and functional design 3.6 Physical design . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Comparison between different concepts . . . . . . . . . . . 3.8 Design and discussion of the developed tribometer . . . . 3.9 Measurement equipment and signal processing . . . . . . 3.9.1 Friction torque and normal force measurement . . 3.9.2 Rotational actuation . . . . . . . . . . . . . . . . . 3.9.3 Normal actuation . . . . . . . . . . . . . . . . . . . 3.9.4 Data acquisition . . . . . . . . . . . . . . . . . . . 3.10 Dynamic evaluation of the tribometer . . . . . . . . . . . 3.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .

XVI

31

69 71 71 73 74 74 75 77 77 80 81 81 81 82 82 82 89

Table of contents

4 Experimental results in the time domain 4.1 Friction characteristics . . . . . . . . . . . . . . . . . 4.1.1 Break-away force . . . . . . . . . . . . . . . . 4.1.2 Pre-sliding regime . . . . . . . . . . . . . . . 4.1.3 Friction lag in sliding regime . . . . . . . . . 4.1.4 Stribeck behaviour . . . . . . . . . . . . . . . 4.1.5 Stick-slip motion . . . . . . . . . . . . . . . . 4.2 Experimental characterisation of lubricated friction . 4.2.1 Lift-up and Stribeck behaviour considerations 4.2.2 Lift-up behaviour . . . . . . . . . . . . . . . . 4.2.3 Stribeck behaviour . . . . . . . . . . . . . . . 4.2.4 Parameter identification . . . . . . . . . . . . 4.3 Considerations on the lift-up effect . . . . . . . . . . 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . .

IV

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

Control aspects

5 Developed friction model for control 5.1 Background . . . . . . . . . . . . . . . . . . . 5.2 Friction Model . . . . . . . . . . . . . . . . . 5.3 Pre-sliding Friction Model . . . . . . . . . . . 5.4 Generalized Maxwell-Slip Model . . . . . . . 5.5 eXtended GMS Model . . . . . . . . . . . . . 5.6 Stribeck function . . . . . . . . . . . . . . . . 5.7 Comparison between Rayleigh step model and 5.8 Friction identification . . . . . . . . . . . . . 5.9 Friction compensation . . . . . . . . . . . . . 5.10 Conclusions . . . . . . . . . . . . . . . . . . .

91 91 92 93 94 96 97 98 98 99 101 102 104 106

109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XGMS . . . . . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

6 Modelling, identification and control of wet multi-disk clutches 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Formulation and analysis of frictional behaviour in multidisk clutches . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 System definition and preliminary assumptions . . 6.2.2 Frictional behaviour . . . . . . . . . . . . . . . . . 6.2.3 Used friction model . . . . . . . . . . . . . . . . . 6.3 Test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Identification results . . . . . . . . . . . . . . . . . . . . . 6.4.1 Temperature effect . . . . . . . . . . . . . . . . . . 6.4.2 Piston motion . . . . . . . . . . . . . . . . . . . . . 6.4.3 Stationary friction curve . . . . . . . . . . . . . . . 6.4.4 Dynamic behaviour . . . . . . . . . . . . . . . . . . 6.5 Controller design . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 112 113 113 117 120 120 122 122 125 126 127 127 129 130 133 136 136 137 140 141 143

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Table of contents

6.6

V

Discussion and conclusions . . . . . . . . . . . . . . . . . .

Conclusions

7 Conclusions and future developments 7.1 Main Contributions . . . . . . . . . . . 7.1.1 Theoretical contribution . . . . 7.1.2 Experimental contribution . . . 7.1.3 Control contribution . . . . . . 7.2 Future work . . . . . . . . . . . . . . . 7.3 Acknowledgements . . . . . . . . . . .

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Bibliography

VI

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149 149 150 150 151 152 152 154

Addenda

165

A Stick time in function of asperity stiffness

167

B Relation between the tangential and normal stiffness (Mindlin) B.1 Sphere on a flat surface . . . . . . . . . . . . . . . . . . . B.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . .

171 171 172

C ICP transducer DC measurement C.1 Introduction . . . . . . . . . . . . . . . . . . . . . C.2 Compensation . . . . . . . . . . . . . . . . . . . . C.3 Experiment . . . . . . . . . . . . . . . . . . . . . C.4 Pros and cons of PE sensor with charge amplifier C.4.1 Dirty environment . . . . . . . . . . . . . C.4.2 Cable length . . . . . . . . . . . . . . . . C.4.3 Range flexibility . . . . . . . . . . . . . . C.4.4 Durability . . . . . . . . . . . . . . . . . . C.4.5 Advantages and limitations . . . . . . . .

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175 175 176 177 180 180 180 181 181 181

D Maxwell-Slip parameter identification D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . D.2 Parameter identification . . . . . . . . . . . . . . . . . . . D.3 Asymmetric hysteresis conditions . . . . . . . . . . . . . .

183 183 183 188

E Normal degree of freedom of XGMS E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . E.2 Normal degree of freedom . . . . . . . . . . . . . . . . . .

191 191 191

Curriculum Vitae

197

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Table of contents

List of Publications Nederlandse samenvatting 1 Inleiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Probleemstelling . . . . . . . . . . . . . . . . . . . 1.2 Doelstelling . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Kadering van de onderzoeksactiviteiten van de onderzoekseenheid . . . . . . . . . 1.2.2 Innovatie . . . . . . . . . . . . . . . . . . 1.3 Onderzoeksbeschrijving . . . . . . . . . . . . . . . 1.3.1 Autonoom koppelgecontroleerde koppeling 1.3.2 Overzicht van de verschillende deeltaken van het project . . . . . . . . . . . . . . . 1.3.3 Relevantie voor de industrie en voor de onderzoeksgroep . . . . . . . . . . . . . . 1.4 Belangrijkste onderzoeksaspecten en uitdagingen . 1.5 Toepassingsmogelijkheden (in de industrie) . . . . 2 Theoretische bijdrage . . . . . . . . . . . . . . . . . . . . . 2.1 Wrijvingsmodel op basis van een Rayleigh stap . . 3 Experimentele bijdrage . . . . . . . . . . . . . . . . . . . . 3.1 Rotatieve tribometer . . . . . . . . . . . . . . . . . 4 Controle bijdrage . . . . . . . . . . . . . . . . . . . . . . . 4.1 Modellering en actieve controle van een gesmeerde koppeling . . . . . . . . . . . . . . . . . . . . . . . 5 Besluiten en toekomstige ontwikkelingen . . . . . . . . . . 5.1 Belangrijkste bijdragen . . . . . . . . . . . . . . . 5.1.1 Theoretische bijdrage . . . . . . . . . . . 5.1.2 Experimentele bijdrage . . . . . . . . . . 5.1.3 Controle bijdrage . . . . . . . . . . . . . 5.2 Toekomstig onderzoek . . . . . . . . . . . . . . . .

199 I I I II III III IV IX X XII XIII XV XVI XVI XIX XIX XXI XXI XXII XXII XXIII XXIII XXIV XXV

XIX

Table of contents

XX

Part I

Introduction

1

Chapter 1

Introduction “The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.” Albert Einstein

1.1

Problem description

Friction between two surfaces is always present in applications such as guideways in machines, brake and clutch surfaces, etc. .., where the surfaces of two objects interact and move relative to each other. The moving surface is always in contact with or supported by the rest of the mechanical system with a finite stiffness. The ‘friction coefficient’ µ, defined as the ratio of the tangential force and the normal force, is not constant, but depends on the two surfaces and conditions, such as the relative speed, surface topography, lubricant, etc. There is no theoretical method for predicting the friction coefficient. The highly non-linear nature of friction in mechanical systems can result in system errors, limit cycles, stick-slip and bad trajectory or following behaviour. These are reasons for friction instability with, as result, the occurrence of vibration. The effects of these vibrations are detrimental to the working behaviour for many reasons: the movement accuracy decreases, the wear of the contact surfaces increases, possible occurrence of self-induced vibrations in the whole structure that give rise to early failure by fatigue, wear or fretting. Moreover, the operator’s comfort of the control of a machine is affected by excessive vibration and noise. In this respect, friction is a major

3

1 Introduction

problem in applications where positioning and tracking behaviour is important. Friction models describe the essence of the complex phenomenon of friction in a set of equations. These can be used as important tools for system simulation and compensation, for example, in precision applications or in transmission. While a great deal of research in nano/micro tribology is done on the physical interaction of an asperity with a surface based on “Atomic Force Microscopy”, it is not an easy task to extrapolate this to a macroscopic surface. On the macroscopic level, some dynamic models are derived from experiments, but they are usually only applicable to specific cases for which they are set up and they can not explain the underlying physical phenomenon. There is need for (1) a friction model that takes into account the underlying physical phenomena that form the basis of the frictional behaviour, for both dry and wet (lubricated) friction. Validation of a model can be done by verifying the results with measurements from a tribometer. This tribometer should be able (i) to apply an arbitrarily chosen relative motion between two objects, (ii) to measure the relative motion between the two objects, (iii) to measure the friction force between the surfaces of the two objects, (iv) to apply and measure different normal forces between the two objects, and (v) to check the influence of different materials and conditions such as contact geometry and lubrication. (2) Deriving an applicable, implementable model for the simulation and/or control of systems with friction elements is also necessary, and finally (3) developing control strategies, using (1) and (2) for the control and compensation of mechanical systems which contain those friction elements is also a challenge.

1.2

Objective

The overall objective of the proposed research consists primarily in a theoretical modelling and experimental verification of the dynamic behaviour of the friction between surfaces for the case of lubricated friction in multi-disk clutches from an automatic transmission. Furthermore, the behaviour of a mechanical system where these surfaces form a crucial part, are examined using these models. The development of models based on the physical mechanisms that cause friction, however, always leads to a computationally intensive model which is not immediately applicable to system simulation and control purposes. The development of a state model specifically for system simulation and control purposes that is less complex and useful for on-line applications will be the core of this research. This control model should incorporate the influence of the normal force and the normal displacement on the friction behaviour. Finally, the previous activities should provide guidelines and tools for the development of control strategies as for friction compensation.

4

1.3 Research description

1.2.1

Framework of the research activities in the research unit

In recent years there has been extensive research into dry friction at the division PMA. Existing models were adapted and improved and new models were developed. The contribution of previous work in the field of dry sliding friction [1] is fourfold: (i) an experimental observation of friction dynamics, (ii) the development of a physically motivated friction model, (iii) the development of a friction model suitable for control purposes, and (iv) the validation of friction models through friction compensation. This previous work discusses a newly developed device for friction measurements, viz. a tribometer and discusses a new general model for the dynamics of dry sliding friction. Friction research is not limited only to tribology, but is also relevant in other domains. For control purposes, the understanding and control of friction is crucial in improving the performance and accuracy of mechanical systems. Compensation of friction effects is necessary for precise motion control, such as in precision mechanics, transmissions and actuators.

1.2.2

Innovation

Existing models will be used as a basis for establishing a dynamic friction model for (boundary) lubricated surfaces. Boundary lubricated friction occurs when the speed between the contacting bodies is not sufficient for the build-up of a hydrodynamic lubricating film such that asperity interaction still occurs. Recent (theoretical and experimental) research at the division PMA also demonstrated that a normal movement of the bodies always occurs when there is a tangential motion of the surfaces relative to each other. The intention of the investigation here is to expand the friction models such that they can also be used for boundary lubricated surfaces and characterize the influence of friction on the normal movement of the surfaces and vice versa. The influence of the normal load, which also affects the normal displacement, is also investigated. The introduction of time and position dependence of the parameters is also a relevant expansion. For determining the parameters of a model, a (reciprocating) test setup which is available in the division PMA, was initially used. With that setup parameter identification of the MaxwellSlip model was performed to investigate the relationship between normal load and friction force in pre-sliding frictional contacts [2]. The development of a second new rotational test setup is undertaken on the basis of the test needs specific to the lubricated contact problem.

1.3

Research description

The research aim is to develop a new dynamical model for friction in (boundary) lubricated surfaces starting from existing models and insights. The time

5

1 Introduction

Literature study Theoretical: Development of a micro-model

Experimental: Existing tribometer ⇓ New tribometer

Reduced (state) model

Test setup : Tranmission line with wet clutch

System simulation environment (Matlab)

Questions and observations from control

Models and guidelines for control

Figure 1.1: Overview of research activities.

and position dependence of the friction and the role of the normal-tangential coupling is also investigated, with as purpose the use of the research results in system control and compensation. The adaptation of this new dynamic friction model into a simplified model suitable for control and motion guidance is also needed. An important aspect here is the refinement of an existing tribometer or developing a new test setup. The test setup will be adapted or redesigned and constructed to validate the friction model. A good mechanical design can never entirely cancel friction and its influence. Moreover, a certain amount of friction is needed in many cases to provide some damping to the system. To remedy this it is possible to suppress the unwanted friction effects in a mechatronic manner, i.e. by a “holistic” approach, i.e. a mechanical, an electrical and an IT approach. To this end, there are a multitude of techniques available in several disciplines. The main building blocks, necessary for a successful design of an efficient system, are reliable models of the subsystems that are part of the overall system. Friction elements (bearings, couplings, etc.) are examples of such elements which show a complex behaviour and are difficult to model. Figure 1.1 gives an overview of the various activities that were performed in this research. The gray area covers the activities that ensure a transition from existing models and test setups to new models and setups. The orange blocks are the main part of the research and the blue block contains the ques-

6


tions and needs coming from control which may affect the modelling. The investigation starts with a thorough literature study. This allows an understanding of and insight in the state of the art regarding friction. The theoretical part of the research is as follows. The existing generic model [1, 3], developed at PMA for dry friction, forms the starting point for the development of a friction model for (boundary and hydrodynamic) lubricated surfaces. The basic components of this model are the friction mechanism and the contact scenario. The elements of the friction mechanism are creep, adhesion and deformation. The interaction of the contact is due to the asperities and surface topography. Within this framework adaptations and improvements of this model are necessary. A friction model that takes into account the position dependence of the friction, the time dependence of the friction coefficient or of the parameters in general, the normal force dependence of the friction and which allows a normal degree of freedom should be implemented. Adjustments of this model to ensure that it takes into account the lubricated friction mechanisms are also necessary. The interaction of the surfaces must take the lubrication into account. This can be achieved e.g. by an elementary contact modelled as a Rayleigh-step. The further derivation of this model uses the dynamic Reynolds equation, see Section 2.3. After the development of all these extensions the simulation can be achieved, which will be carried out in Matlab. The modelling and simulation has to take into account questions and observations that come forward from experiments. The experimental part starts with an existing test setup. This setup is discussed in the literature overview in Section 1.6. A necessary extension to examine (boundary and hydrodynamic) lubricated friction, is to make a system that can provide a controlled flow of a lubricant between the two interacting opposing surfaces. A further extension is to develop a new test setup that uses a rotational motion instead of a linear. The possibility to apply a high and constant speed is a requirement to simulate the rotational contact in a transmission. In the current tribometer this is not possible and it also has a limited stroke. The orange blocks of Figure 1.1, which, as just discussed, contain a theoretical and an experimental part, constitute the core of the research. The issues encountered and observations made during control experiments are guidelines for modelling and other experiments to be carried out.

7

1 Introduction

The study of boundary lubricated friction mainly focuses on wet clutches. The friction behaviour of wet clutches, used in applications with automatic transmissions, has a strong influence on the dynamic behaviour of the machine or vehicle, including the transmission itself. Self-excited vibration can increase the wear of the clutch, and thus shorten its life. An economic need is the reduction or limitation of the cost that is associated with the life of the coupling (repair, replacement) and the oil. This is achieved by proper selection of materials and oil that prevent the occurrence of vibration and reduces wear. Because designers tend to over-dimension clutches, a decrease of the friction coefficient does not put a constraint on the life time of the clutch [4]. This on the other hand brings an additional cost. A better friction model including wear, validated for different frictional materials can extend the lifetime of clutches and makes the selection of appropriate materials easier and more accurate. This can be done by applying an appropriate control based on this friction model. There is not only an economic advantage but an appropriate control can improve the behaviour and thus the driving comfort. Wet paper-based friction materials are commonly used in automatic transmissions of vehicles in which they serve as a clutch and blocking mechanism. Shock reduction during the gear switching process, to prevent vibration during the process of slip-controlled locking mechanism, and the creation of a higher friction coefficient µ are simultaneous requirements. Notwithstanding the fact that these requirements generally form a trade-off, they are achieved by obtaining a positive slope of the relationship between the friction coefficient and sliding speed and an automatic transmission fluid (ATF) standard which ensures that the dynamic friction force is larger than the static friction force in SAE#2 experiments. In previous research (in collaboration with Spicer Off-Highway) the aim was to compare the friction and wear behaviour of wet friction material on steel on a large (SAE#2) and small (coupon on disk) scale [4]. It appeared that for any large-scale test, self-excited vibrations (vibrations of the clutch) occurred at the end of the test. The fundamental cause of these vibrations can be explained as follows. Some researchers attributed this to stick-slip [5], where others claim it is due to the negative slope relationship between the friction force and velocity [6]. Those claims are related. Armstrong-Hélouvry [7] investigated stick-slip arising from the Stribeck effect. This Stribeck effect describes the negative slope relationship between the friction force and velocity. More about this is discussed in the literature overview in Section 1.6.1.

1.3.1

Autonomous torque controlled clutch

The control of the transferred torque in a coupling is a difficult undertaking. The desired torque results in the actual torque after transfer. The various components in the control system are software, DAC, solenoid, amplifier, transmission, hydraulic drive, valve ... At various places disturbances affect

8


the system. Some of these disturbances originate from hysteresis, stick-slip, pressure drop, temperature changes, variation in friction coefficient, etc. ... The assumption of a constant friction coefficient is now commonplace and usually the whole transmission is considered. An error occurs if a difference between the desired and actual torque is present. The largest contribution to the error is the uncertainty of the plates of the coupling being in contact, because the distance between the plates is not known because it is not measured. An obvious way to solve this problem is to take precautions such that as much as possible components of the system are linearized. Alternatives are using open-loop feedforward [8, 9], adaptive (feedforward) control [10, 11] and an autonomous torque controlled clutch. With feedforward, the inverse system model is often used. If a good estimate of the disturbance parameters and a good model are available, a one-to-one relationship between the desired and actual torque can in theory be obtained. This implementation of the controller is often done based on a look-up table. Some parameters, such as wear and degradation of the oil, are very difficult to determine. An extension of this method is the use of a better estimator for the disturbance parameters on the basis of the error, viz. the torque difference. In this case, we speak of learning or adaptive feedforward. Armstrong-Hélouvry et al. give a very good and exhaustive survey of friction models and friction compensation techniques in [12]. Another approach, instead of considering and controlling the entire transmission, is to isolate an autonomous controlled torque coupling. It is, as it were, a transition to an ‘intelligent’ system, viz. the coupling on its own. The friction coefficient may not be assumed constant, but it must be function of speed, pressure, temperature, oil, materials, etc. .. It is necessary to have an internal control such that a desired output is obtained for a given input. Externally, the transmission can be seen as a linear subsystem. This makes the implementation of the coupling in the transmission easier.

1.3.2

Overview of the different tasks

• As a starting point the existing model (generic model) will be reimplemented to form a basis for extension. – The model can be extended with a normal degree of freedom, normal to the contact surface, such that the normal-tangential coupling can be taken into account. Current models assume a constant normal load and constant normal distance. Within friction systems, there is always a variation of the normal distance, e.g. due to a variation in normal load. This implies that the further extension of the Maxwell Slip model must take into account this changing normal distance. A sudden increase in normal load causes

9

1 Introduction

the creation of a certain amount of new interacting asperities. If the normal degree of freedom in a system is added, the interaction between the mass and stiffness of the asperities also have to be taken into consideration. Adding an extra degree of freedom makes the problem more complicated. In analogy to to the generalized Maxwell-slip model, a simplified model appropriate for control purposes has to be deduced. This can be achieved once a generic model, based on the physical mechanism that takes into account the normal displacement and normal force, exists. – The model can be extended with the time dependence of the parameters. Friction has a certain running in time, viz. a transitional period during which parameters such as roughness, temperature, degradation of lubrication, etc ..., can change. Phenomena with a small time constant are e.g. pressure modulation and vibration, phenomena with medium time constant are temperature change and oil flow and phenomena with long time constant include material degradation and wear. Fish and Lloyd [13] showed that during running in the friction and wear vary considerably in function of time. If the initial roughness of the sliding surfaces is well chosen, running in will eventually reach a steady-state. At that moment, the surfaces are flatter and the amount of wear is low and constant. An inappropriate choice of the roughness, however, can lead to a faster degradation of the sliding surfaces. Experiments show that the surface micro-geometry is one of the main factors that deter¨ mine the life time of mechanical components. Ostvik and Christensen [14] noted that running-in consists in reducing the height of the highest asperities, and increases the number of contacting asperities and the load capacity of the surface. Wear during run in depends both on the height and the shape of the roughness or asperity distribution and the friction and lubrication also depend on these parameters. Due to wear, the lubrication can evolve from boundary lubrication to full film lubrication. – The model can be extended with the position dependence of the parameters. The normal load may vary depending on the position inaccuracies. Geometric variations can also affect the friction. This phenomenon can also be intrinsic to the system itself, no system is perfect. These variations may have a direct influence on the friction. – The model can be extended from dry friction to (boundary) lubricated friction. Wet, boundary lubricated friction occurs as the second dynamic regime, after the pre-sliding regime. In this regime the speed is not high enough to ensure the build-up of a liquid layer due to which asperity interaction between the two surfaces still happens. For lubricated friction it is necessary to take into

10


account the behaviour of the lubricant, so the interaction of the lubricated surfaces. A phenomenon that occurs with opposing lubricated interacting surfaces is the action of a squeeze film. • The derivation of a simpler, less complex friction model (similar to the Generalized Maxwell-Slip Model) useful for on-line control applications is a second challenge. The complexity of a model should not be too big if it must be suitable for control purposes. For the practical applicability of a friction model, easy parameter identification plays a key role. The parameters should be relatively easily determinable. • As a starting point the existing tribometer should be checked if it is suitable for this specific research. – Experiments have to be performed such that its shortcomings for this research become clear, modifications can easily be made and take into account the lubrication and the normal displacement. – The development of a new test setup or a new tribometer has to be considered which has a rotational motion for which it is possible to impose a controlled supply of lubricant. • The next step is the development of a test setup that represents the actual test case better than the previous setup, viz. a transmission line with a wet clutch on which simulation and control can be performed. The questions and observations from control theory, as stated above, form guidelines for modelling and experiments. Applying control can happen in different ways, viz. in a passive, semi-active and active way [15]. Passive methods include changing the stiffness, damping and the mass of a system without adding power. In a mass-spring-damper system, these parameters affect the friction dynamics. An adjustable suspension can be regarded as a semi-active method of control. Another example of a semi-active method is to apply a dither signal. Dither is a high frequency signal superimposed on the control signal [16]. In this way the stiction problem can be avoided. Dither is commonly used in valve technology for friction compensation [17, 18]. For active control, many possibilities exist. Here active elements such as actuators are applied. The use of sensors, for the measurement of the different states of a system, is indispensable. Feedforward and closed-loop systems, i.e. the use of feedback, are suitable. This requires the development of appropriate control schemes. Linear methods of control have been frequently discussed in the literature. This would not constitute difficulties. Since the issue discussed here has a strong non-linear character, non-linear control techniques will be applied. However, these are less addressed in the literature.

11

1 Introduction

1.4

Overview of main research aspects and challenges

Literature Study The investigation started with an increasingly thorough literature study. This allows an understanding of the situation, the state of the art with regard to friction. In this way, a good knowledge about the existing friction models could be obtained. Important is their accuracy and their applicability and their adaptability or potential improvement. A literature review can also initiate the development of new ideas and the design of new experiments which can be used for validation. The literature overview is presented in Section 1.6. Include the position dependence of the friction In addition to the displacement dependence of the friction in the pre-sliding regime, the friction also depends on the position. This may be due to a poor design of a system or a variation of normal force. Study and develop the role of the tangential-normal coupling Because of the surface roughness, tangential motion induces a normal motion and possibly also a rotation effect. The normal displacement is often called the ‘lift-up’ effect. This is due to the fact that the bearing capacity of the asperities increases with increasing sliding speed. It is similar to the effect that occurs in hydrodynamic lubrication due to pressure build-up. This coupling has an influence on the friction behaviour. Transition from dry to wet friction, i.e. friction in (boundary) lubricated surfaces A generic model on asperity level already exists for dry friction. This model has to be adapted or further developed such that it can be applied in boundary lubricated friction. For lubricated friction, the stationary force in function of the speed can be represented by the Stribeck curve. In boundary lubricated friction, the speed between the contacting bodies is not large enough for buildup of a liquid film such that asperity interaction between the two surfaces still occurs. This results in the process of shear. The friction characteristics of a wet clutch must be identified and modelled. Design and build a new tribometer Establish a new test setup for experiments and validation of models. The development of a tribometer for dry and (boundary) lubricated friction with a large speed range, which is rotational and with which the phenomena of the position dependence and the normal-tangential coupling can be investigated. Transmission line The application of the friction model in the control of a wet clutch of a machine

12

1.5 Applications (in industry)

with automatic transmission (AT) is the final aspect. Here, the test setup consists of a transmission line with a wet clutch. Control The control problem is not a real main research aspect but it is certainly worth mentioning. After the modelling phase comes the design of a controller. At this stage the development of (a) control scheme(’s) will be addressed. This should take into account the highly non-linear nature of this problem.

1.5

Applications (in industry)

There are several areas where friction plays an important role. The domain of structural dynamics studies the vibrations of systems where friction or damping in the joints has a significant impact on the dynamics of structures, see e.g. [19]. Geomechanics also belongs to the domain of friction research. In biomechanics, the knowledge of the friction in joints is important in the development of prostheses. The knowledge of the dynamic behaviour of boundary lubricated friction is to be used in various applications. Boundary lubricated friction not only applies to automatic transmissions. Wet clutches form the main application in this research, but this does not mean that other applications should not be taken into account. The demand from industry for modelling lubricated friction for this research comes from Dana (Spicer Off-Highway) with as application wet-plate clutches. This research joins the activities that FMTC (Flanders’ Mechatronics Technology Centre) performs as part of a project called Strategic Basic Research, in which the control of wet-plate clutches is investigated in detail. The industry in general has more need for improved friction models and active control methods, where mechatronics fits in, for ever more precise high-performance motion systems. This research can, in a more generic form, be extended in order that obtained results can be materialized in a later stage for various applications with boundary lubricated contact surfaces such as guideways, robots, machine tools and conveyors, boundary lubricated bearings, etc. ... Robots, for example, often have a gravitational force and position dependent friction. The design and control of boundary lubricated friction has the following applications: Lubricated clutches The friction behaviour of wet couplings for applications with automatic transmissions has a strong influence on the dynamic behaviour of the machine or vehicle, including the transmission itself. This automatic coupling can be found mainly in tractors and in general in off-road vehicles (bulldozers, forklifts ...). A friction model and a good control of the friction dynamics can greatly improve control of the switching behaviour of the transmission and a larger specific load of the transmission.

13

1 Introduction

Motion control In most applications where motion control is used, friction is a dominant factor that affects the performance. The control of the friction behaviour with a control system is in this case also applicable. Medical applications Haptic interface and feedback of surgical robots, for example in non invasive procedures, can be used so that the surgeon is not influenced by friction and can perform a procedure more accurately and more easily. The development of knee and hip prostheses with a better understanding of friction so that they have a longer life span and a better, more natural, behaviour. Boundary lubricated applications Applications with boundary lubricated contact surfaces such as guideways, robots, machine tools and conveyors, boundary lubricated bearings, etc. Friction in robotic systems is a source of imperfections in the tracking of an imposed path leading to steady-state errors and delays (“lag”) on the path to follow. Design and life cycle An understanding of friction in the domain of design and life cycle design is also relevant (e.g. in cars, there are more than 2000 tribological contacts [20]). A better understanding of friction can lead to a better design, a longer lifetime (e.g. bearings, gearboxes, etc.) and lower energy consumption.

1.6

Literature overview

This section gives a general overview of the relevant elements for this research of the state of the art about friction measurement, characterisation, modelling and control. The rest of the literature study is spread over the different chapters and is cited where necessary.

1.6.1

Friction characteristics

In this section the main friction characteristics or friction effects that are discussed in this dissertation are explained. The different frictional mechanisms that cause the following characteristics are normal creep, adhesion and deformation and in lubricated conditions pressure build-up and squeeze film are also involved. Creep is the tendency of a solid material to slowly move or deform permanently under the influence of stresses. In the developed generic model in [1, 3], the ‘creep’ mechanism is assumed to arise not only from material creep/relaxation but also from general interlocking of asperities in time. Adhesion is the molecular attraction exerted between the surfaces of bodies in contact. Here with adhesion we mean tangential surface forces arising from a variety of sources, when the two surfaces are in a certain proximity of each

14

1.6 Literature overview

other, thus not only metallic adhesion is considered, but any short or long range forces. Pressure build-up is the generation of a bearing force or film due to the relative tangential motion of two bodies with a fluid in between. Squeeze film is the generation of a bearing force or film due to the relative normal motion of two bodies with a fluid in between. In general friction can be defined as the resistance for sliding one object in reference to another. Lubricated friction contains a third body, which is the viscous film between the first two bodies. Several micro- and macroscopic properties will influence the frictional behaviour. The most important are: -

surface geometrie (micro- and macroscopic) the effective contact area elastic and plastic deformation of the asperities local contact temperature dynamics of nearby connected structures lubrication wear ...

Friction can be divided in two main regimes, the pre-sliding regime and the sliding or gross sliding regime. The pre-sliding regime is defined as the region in which the adhesive forces at asperity contacts are dominant such that the friction force appears to be a function of the displacement [12, 21, 22]. In this regime the global contact is not sliding yet but partial slip can occur, which means part of the contacting asperities are slipping. The transition to the sliding or gross sliding regime occurs when all the asperities start slipping. In this regime the global contact is sliding, total slip occurs and the friction force is not dependent on the displacement anymore but becomes dependent on the sliding velocity. In the pre-sliding regime a typical hysteresis occurs when applying a reciprocating motion and the hysteresis force is function of displacement. A hysteresis behaviour with non-local memory is defined as an input-output relationship for which the output at any time instant not only depends on the output at some time instant in the past and the input since then, but also on the past extreme values of the input or the output [23]. This means hysteretic friction is not a unique function of the displacement (the input of the system) but depends on the previous history of the displacement, i.e. the hysteresis has a non-local memory property. The force necessary to initiate total slip or gross sliding, can be determined. The maximum force which occurs in this initiation is called the break-away force [24, 25]. The break-away force is here defined as the maximum friction force during the transition from pre-sliding (stick) to gross sliding regime. One

15

1 Introduction

may use this definition of break-away force, as being equivalent to the static friction. But due to the dependence on the application rate, several values for the static friction can be found for one and the same normal load. For the lowest acceleration the highest maximal force is attained and this maximum decreases with increasing acceleration [24, 26]. Similar results are obtained by Kato et al. [27]. The change of the break-away force is a result of the contact time dependent friction coefficient [28] which on its turn is a result of normal creep and adhesion. The longer a body stands still, the more normal creep can occur in the contact due to which the adhesion force increases and also the force necessary to initiate motion due to the deformation of the asperities. In the gross sliding regime the friction force is dependent on the velocity. As soon as the break-away force is exceeded and the object starts to slide, the friction force generally drops to a lower value. It was found by Stribeck that the velocity dependence is continuous [29]. The typical drop (rise) in the Stribeck curve, which represents the friction force in function of velocity, for positive (negative) velocities is called Stribeck effect. In lubricated multiplate clutches, the objective is to always have a lubricated surface. Therefore we will assume that the contact surface always contains a lubrication film. Nevertheless, at very high pressure values, the frictional behaviour will more and more tend to dry friction properties. The main difference between these two is the forming of a lubrication film, causing hydrodynamic effects, like pressure build-up and squeeze film effect. This film will also mainly cause a drop of the friction coefficient. On the other hand, the frictional behaviour will change in function of sliding speed or position. In Figure 1.2 (a) there are three main regimes as a function of sliding speed, as described in [1, 30]. This figure is similar to the more extensive figure used in Section 4.3, viz. Figure 4.17. In the boundary lubrication regime, for low speeds, the system is not capable of building up a lubrication film. Therefore, the properties of the contact surfaces determine the global friction behaviour. At high speed, a lubrication film will be formed, and the viscosity of this film determine the behaviour. This is the full fluid lubrication regime. The transition between these two regimes has mixed properties. In Section 4.3 this is discussed in more detail. The stationary friction curve from Figure 1.2 (a) can be divided in two parts (Equation 1.1). s(v) is the weakening curve, which contains the friction from asperity interaction and lowers as a function of speed. As mentioned before this is called the Stribeck effect. The second part, f (v), is the strengthening curve, better known as a viscous effect curve [31]. F (v) = s(v) + f (v)

(1.1)

The Stribeck curve drops with increasing speed and in lubricated conditions this is a consequence of the hydrodynamic pressure build up. Hereby the con-

16

mixed lubrication

boundary lubrication

friction force


full fluid lubrication speed

friction force

(a) Lubrication regimes

acceleration

deceleration

stationary friction curve speed

(b) Stribeck curve and dynamic behaviour

Figure 1.2: Properties of lubricated friction. tact time between the asperities and the amount of contacting asperities will drop together with the friction force, related to this interaction. The viscous component arises as a function of stationary speed, because of the tangential viscous forces in the lubrication film. This is often considered proportional to the sliding speed [1]. It is important to understand that the Stribeck curve is a collection of points, which represent the friction force for a constant and stationary speed. Therefore, this is called a stationary friction curve (Figure 1.2 (b)). In a friction contact, when applying a reciprocating motion, we can see a higher friction force during acceleration compared to the friction force during deceleration. This phenomenon is called friction lag or hysteresis in velocity in the sliding regime. Lampaert et al. [24], Kappagantu and Feeny [32]

17

1 Introduction

and Hess and Soom [33] also measured a friction lag behaviour for oscillatory rubbing contacts. [24] and [32] examined friction lag in dry conditions as opposed to [33] who examined friction-velocity characteristics of line contacts operating under unsteady sliding velocities in the mixed, elastohydrodynamic and hydrodynamic lubrication regimes. So if the speed changes dynamically, the friction force will deviate from the earlier mentioned stationary curve, see again Figure 1.2 (b). Berger [25] gives a detailed overview of friction models and also a discussion of all known friction phenomena. He shows that experiments indicate a functional dependence upon a large variety of parameters, including sliding speed, acceleration, critical sliding distance, temperature, normal load, humidity, surface preparation, and, of course, material combination. The normal-tangential coupling is one of those dependenties. In addition to position/velocity/acceleration dependent friction, as mentioned before, a normaltangential coupling in the system dynamics has been examined as a contributor to friction related problems. The first profound impact in this area was made by Tolstoi [34], who completed delicate experiments which measured both the in-plane (tangential) motion as well as the out-of-plane (normal) motion of a slider against a counter surface. He constructed a test rig to determine the influence of small normal vibrations (including impacts) upon break-away behaviour. He noticed two key points: i) tangential slip events were invariably accompanied by simultaneous upward normal motion, and ii) a normal contact resonance condition could be observed under which apparent friction was reduced. The argument here is that in order for friction to change, the real area of contact must change, and therefore the mean normal separation of the surfaces must also change. Sakamoto [35] used a pin-on-disk configuration to carefully examine normal separation effects in sliding contacts; he emphasizes the slip portion only of the stick-slip cycle. Clockwise friction-velocity loops are observed, which reflect the friction lag phenomenon, and the variation in friction is interpreted as a change in the real area of contact during sliding (as inferred from contact resistance measurements). In lubricated conditions Sugimura et al. [36] examined the unsteady-state EHL film thickness behaviour under constant accelerations/decelerations. They have shown that the film thickness becomes thinner during acceleration and thicker during deceleration than the steady-state film thickness, that the difference between the steady-state and unsteady-state film thicknesses becomes larger at greater accelerations/decelerations, and that the difference is larger during deceleration than during acceleration. In this way they show that the lift-up or the change in lift-up is both function of the velocity and the acceleration or deceleration, due to the pressure build-up and squeeze film. Armstrong-Hélouvry [7] investigated stick-slip arising from Stribeck friction. He mentions that machines at low velocity in general exhibit stick-slip motion. Experimental work has mapped out the structure of Stribeck friction,

18


a non-linear low-velocity friction effect that contributes to and perhaps dominates stick-slip. Using dimensional analysis, one of the properties he examined was the minimum velocity below which stick-slip will occur, viz. the critical velocity or the Stribeck velocity. In [1] the stick-slip behaviour is explained as follows: at the beginning the actuated body is at standstill and the friction force increases with the spring force until break-away occurs. At break-away the friction force is maximal and the body starts to move. The increase of velocity results in a decrease of friction force due to the Stribeck effect and a decrease of the spring force (because the real velocity is higher than the desired velocity, the spring elongation decreases). The velocity increases until the friction force equals the spring force. From that moment on, there will be a deceleration of the mass, resulting in a higher force, where the spring force decreases as long as the real velocity is higher than the desired velocity. The deceleration becomes higher until the body is in the pre-sliding regime (stick). The term stick-slip has first been used by Bowden and Leben [37]. Blok [38], analysing experimental results of [37], defines stick-slip motion as a periodic cycle of alternating motion and arrest being the consequence of a lower friction force at higher sliding velocities. As opposed to the main reason for stick-slip to occur, Gao et al. [39] use a dynamic analysis to show that the rate of increase of the static friction coefficient with sticking time is a crucial parameter and that stick-slip may occur even if the dynamic friction force increases with speed.

1.6.2

Friction measurement

In a device for measuring friction, two specimens are placed together under a known normal load. One of them is brought to slide relative to the other and the tangential frictional resistance is then measured. There are several ways to measure the friction force, but the inclined-plane test is probably the simplest test, see Figure 1.3.

W.sinθ W

W.cosθ

θ

Figure 1.3: Friction measurement with inclined-plane test. With this setup, at the moment the upper body starts to slide, the friction

19

1 Introduction

coefficient µ can be calculated as the friction force F divided by the normal contact force Wn : µ=

W.sinθ F = = tgθ Wn W.cosθ

This method measures only the static friction coefficient and is therefore not applicable in cases where the variation of the friction force at continuous sliding has to be measured. If the friction and wear should be measured over a given period, alternative more complex experiments are needed. The test geometries for these experiments can be divided into two groups: non-conformal and conformal geometries as depicted in Figure 1.4.

Figure 1.4: Friction measurement using conformal and non-conformal geometries. Tribometers can be used to investigate rolling or sliding friction in dry or lubricated conditions. Here the focus is put on lubricated friction tribometers. An overview of possible tribometers is given after which one tribometer for dry friction is discussed more thoroughly to explain the necessary features a tribometer should contain. This is done here since all friction testing devices or tribometers have some typical main features in common. They provide a way to (i) fix two materials or bodies for which friction data is desired, (ii) to apply an arbitrary displacement or velocity signal in a controlled way, (iii) to apply and measure a normal load on the frictional contact, and (iv) to determine the tangential or friction force [40]. Typically for lubricated friction

20


experiments, SAE#2 setups are used. The Society of Automotive Engineers (SAE) standard describes a SAE No. 2 or a SAE#2 Friction Test Machine as follows. The SAE No. 2 Friction Test Machine is used to evaluate the friction characteristics of automatic transmission plate clutches with automotive transmission fluids. It can also be used to conduct durability tests on wet friction systems. There are SAE#2 and other test setups in which the overall behaviour of wet-plate clutches and friction behaviour in general can be measured or evaluated [4, 5, 24, 33, 41–47]. In [42–47] friction characteristics of paper-based friction material are investigated, more specifically friction and real contact area measurement, engagement behaviour and influence of surface topography. Holgerson [45] designed a wet clutch apparatus to allow measurements of the characteristics of engagement. The experiments conducted used a paper based wet clutch material taken directly from a standard automatic transmission. To measure brake torque and normal force two different piezoelectric load cells are used. The load cells take up all the transmitted force from the clutch plate. Gao [43] employed a bench test set-up to simulate the friction characteristics of a paper-based friction material operating against a steel plate. The friction force is measured by using a strain gage arm. Three pieces of friction materials are assembled on the upper plate and are spaced 120 degrees apart. The measured friction force is divided by the normal load to determine the coefficient of friction. The simulation is done both for dry friction and for lubrication with transmission fluid. In these experiments all similar setups are used, but the tangential-normal coupling and the relative normal displacement of the friction disks was never investigated, only in [44] the contact pressure and the film thickness based on electrical contact resistance was measured. As mentioned before, some tribometers are used to investigate rolling friction, some sliding friction, in dry or in lubricated conditions, but the main subdivision of tribometers is based on the type of displacement which can be applied. The two basic movements, which are used, are: (i) a reciprocating linear movement (e.g. in a sledge configuration) and (ii) a rotational movement (e.g. in a pin-on-disk configuration) [40]. The existing tribometer developed by Lampaert et al. at division PMA, which has been used in previous research, looks as depicted in Figure 1.5 [1, 24]. On the picture of the tribometer, the following components can be discerned: (1) frame, (2) support (unit), (3) force sensor, (4) elastic connection, (5) friction block, (6) driven block, (7 ) moving mirror, (8) fixed mirror, (9) stinger, (10) Lorenz actuator (11) linear guideway, (12) Plexiglas, (13) air bearing, (14) normal load (15) rotation point (16) lever. The instrument can roughly be divided into three main parts: the driving portion (parts 6, 7, 9 and 10), the friction part (parts 2 – 5 and 12) and the load part (parts 13 – 16). The different parts are decoupled as much as possible: the driving part and the friction part are only linked through the friction interface, and the load part and the friction part are completely decoupled by using an air bearing which ensures that the entire tangential force goes true the force sensor.

21

1 Introduction

16

12 cm

2

3

4

1

14 3 13 5

15 12 6 7

8

9

10

11

Figure 1.5: Linear reciprocating tribometer.

The development of a test setup specific for the investigation of the friction in lubricated multi-disk clutches in automatic transmissions, where the normal degree of freedom and the influence of the lubricant can be studied, is discussed in Chapter 3. A dedicated test setup has been developed and built to perform the friction control on a multi-disk clutch, see Chapter 6, Section 6.3.

1.6.3

Friction models

For the last few centuries the friction phenomenon has been studied and modelled in various ways. The most famous and probably most widely used model is Coulomb’s, which states that the friction force is equal to the so called Coulomb friction force multiplied by the sign of the speed of motion. This section describes friction models, which consist out of two main categories, viz. generic or physically based friction models and control models. Of particular interest here is “the Generic model” which is a physical asperity based friction force model [3]. The control models are split up in static and dynamic friction models. The most famous state control models include the Dahl model [48], the Bristle model [49], the LuGre model [50], the elasto-plastic model [51, 52], the Leuven model [22], the generalised Maxwell-Slip model [53]. These are called state models because they are described using a state equation with an internal state, e.g. the average deflection of the asperities. A thorough survey of models can be found in [12] and also in [54]. Here first the importance of friction modelling is addressed, then generic physically motivated friction models will be discussed, putting focus on the generic friction model developed by Lampaert and Al-Bender et al. [1, 3], after which some models useful for control are discussed.

22


The complexity and purpose of available friction models varies in many ways. The first distinction is the structure. A generic model is looking for phenomenological mechanisms on a microscopic scale, in search for a macroscopic behaviour. On the other hand, a model for control purposes, has to be able to run in real-time. Therefore the calculation can only last for a limited time (calculation time ≤ simulation time).

1.6.3.1

Importance of friction modelling

The objective of modelling friction is to reduce uncertainties about the behaviour of a system. Modelling and identifying the non-linear friction behaviour is an essential step towards understanding the principles behind the friction phenomena. In control the required model complexity depends on the desired control performance of the system. Controller designers benefit from having more detailed knowledge about the system to be modelled, to obtain the best available controller. Next to understanding friction behaviour, friction models are useful for mathematical analysis, integrated mechatronic design and offline simulations. Simulations using friction models play also an important role in understanding tribological processes. They allow controlled numerical experiments where the structural components of the system and sliding conditions can be varied at will such that their effect on friction, lubrication and wear can be explored. They can also be used in the integrated design of the structure and controller of new systems. 1.6.3.2

Generic models

Generic models are models, which focus on the physical friction properties. These models often require a lot of calculation, and are therefore only applicable in off-line use. The high calculation load is directly dependent on the number of modelled asperities, because for every asperity, a state has to be calculated individually. The aim of this type of models is to gain insight in frictional behaviour as a physical phenomenon. The generic model used in this thesis as basis for further development has been developed by Lampaert and Al-Bender et al. [1, 3]. Despite its simplicity, this model is able to simulate all experimentally observed properties and facets of low velocity friction dynamics, such as no friction model could do before. In [1] a literature survey on physically motivated friction models is given in a division of classes according to the scale of approach. The different scales are: (i) atomic/molecular scale (common in nanotechnology), (ii) asperity scale (common in the control domain), and (iii) tectonic plate scale (common in geophysics). An historical overview is also given in [3]. One reference certainly worth mentioning is Bowden and Tabor [55], which discusses most occurring mechanisms like

23

1 Introduction

normal load

force

(i)

(ii)

(iii)

z

(iia)

(iib)

x

(iii)

kn

kt 2

spring force

(A)

spring extension

(C)

kt 2 ζ ξ

w λw

αw

(B)

Figure 1.6: Life cycle and deformation of one asperity in contact (From [3]).

adhesion, gross plastic deformation, ploughing, junction growth, the rising static friction with dwell or rest time (creep), ... Schematically the generic model further discussed here looks as follows, see Figure 1.6. Figure 1.6 (A) shows the general contact between two objects. Figure 1.6 (B) shows the contact cycle of one asperity interaction: (i) no contact, (ii) contact, (iii) loss of contact. Figure 1.6 (C) shows the spring force behaviour as a function of the spring extension during the life cycle of an asperity (iia) –during stick, (iib) x at slip, and (iii) when loosing contact [3]. The author of this generic dry friction model [1] describes its limitations as follows. The model assumes a constant separation between the two sliding objects. However, if a normal degree of freedom is added to the system, the slider dynamics (inertia) will influence the friction force dynamics. A general feature of the normal degree of freedom is that as the normal oscillations increase, the average separation of the surfaces increases. The model should be extended to lubricated friction. At low velocities the lubricant is not able to build-up a fluid film by hydrodynamic effects, yielding that the dominating mechanism at low velocities will be solid-solid or dry friction. Due to the lubrication the model has to be extended for higher velocities taking the

24


hydrodynamic effect into account. When the normal degree of freedom is incorporated into the model a lift-up of the moving object will be possible, resulting in a smooth transition from solid-solid effects (i.e. dry sliding effects) to purely hydrodynamic effects. A generic model is often a good aid to develop a new control model. Recently a novel generic model is developed and discussed by De Moerlooze et al. [56]. In this novel approach, the presence of the creep phenomenon is intrinsically occurring due to the incorporated normal degree of freedom and the elasto-plastic behaviour of the asperities. Where the old implementation requires the a-priori knowledge of a time dependent coefficient of friction to be able to simulate the creep behaviour, the novel implementation has this quality without any presumption of the evolution of the coefficient of friction. Another point where the novel model distinguishes itself from the old version is a more rigorous implementation of the elasto-plastic material behaviour. In this novel model the springs are implemented as hysteretic dissipative elements which represent the elasto-plastic material behaviour. 1.6.3.3

Control models

Models for on-line implementation require a very short calculation time and are therefore built on a simplified interpretation of the physical phenomenon. The degree of simplification differs from application to application, as a function of the required model precision. There are two major levels of modelling: static and dynamic friction modelling. Static friction models These classic friction models give a static link between the friction force Ff and the sliding speed v [57]. The best known and maybe simplest friction model is probably the Coulomb friction model. This model has 2 parameters: the static friction force FS and the sliding friction force FC , also called the Coulomb friction force. It is logic to assume that this latter is smaller then the static friction force. FC is assumed to be independent of the sliding speed vrel . This gives the following function description, with Fa the applied tangential force:   sign(v)FC Fa Ff (v) =  sign(F )F a S

if |v| > 0 if v = 0 and |Fa | < FS if v = 0 and |Fa | ≥ FS

(1.2)

In lubricated contacts, it is often assumed to have a viscous part in the sliding friction force. At higher speeds, the total friction force will become greater then the static friction force. This viscous part is often assumed to be linear

25

1 Introduction

with the sliding speed. This gives the following function description:   sign(v) (FC + Cv |v|) if |v| > 0 Fa if v = 0 and Fa < FS Ff (v) =  F if v = 0 and Fa ≥ FS S

(1.3)

At a large speed range, the previous models describe a general behaviour, but in a lower range of sliding speed the linear assumption of the viscous friction is less accurate. Therefore, a general description for the static friction curve is given by the Stribeck friction model plus a viscous friction part, described by Figure 1.7 and given by eqn. 1.4. This is one of the many functions one can use with a typical exponential part and a part proportional to velocity. VS Ff (v) = sign(v) FS 1 + (β − 1) exp − + Cv |v|γ (1.4) v FS is the static friction force, β the ratio FC /FS , VS , called the Stribeck velocity, indicates at which speed the transition to viscous friction occurs and finally Cv determines the size of the viscous friction, with γ a shaping factor. 2

FS Friction Force

F

C

1

0

−FC −FS −2 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Sliding Speed

Figure 1.7: Graph of Ff for the Stribeck and the viscous model. These different static descriptions can be the result of a curve-fitting method on measurements of the friction force at steady state sliding speeds. At a speed, equal to zero, the friction force is (faulty) assumed to be equal to the applied force, because the friction force is then dependent on the relative position between the two bodies. This gives a so called hysteretic behaviour. Dynamic friction models As an extension to static models, dynamic models should provide a more complete modelling of the frictional behaviour. Sources for this modelling are [31] and [58]. The best known dynamic control model is probably the LuGre model [50]. Some models that were derived before the LuGre model are the Dahl model [48] and the Bristle model [49]. Some other models that were derived after the LuGre model are the elasto-plastic model [51, 52], the Leuven model [22] and the GMS model [53]. A comparison between the LuGre and GMS model in the friction compensation in an electro-mechanical

26


ζ1

k1

W1 ζi

ki

Fi

Fi

Wi

Wi

z

ki

Wi

z- ζi ζN

-Wi

z -Wi

kN

WN

Figure 1.8: Left: N elasto-slide elements; Right: characteristic of one element. system can be found in [9, 59]. The ability to estimate the friction behaviour in pre-sliding regime is the superiority of the GMS model, while it does not lose its ability to estimate the friction in sliding regime. Therefore the GMS model can capture the friction behaviour for any working range of displacement and velocity. In all of the validation cases, the GMS model yields the best results. This can be understood since this accommodates the hysteresis behaviour with non-local memory for the pre-sliding regime friction, while also modelling the gross sliding. The transition to a reduced control model, that is from the generic to the Generalized Maxwell-Slip (GMS) model [53], is based on the Maxwell-Slip model. The implementation of hysteresis in the Generalized Maxwell-Slip model is shown in Figure 1.8. A Maxwell-Slip model is a way to simulate the pre-sliding behaviour, using a combination of Maxwell-Slip elements. One such an element contains a spring, connected to a Coulomb friction block. This was combined with the dynamic behaviour in the gross sliding regime in the Generalised Maxwell Slip model (GMS-model) [31]. The idea is to put N elasto-slide elements in parallel with each other. Each element has a common input z and output Fi and each element is characterised by a maximum sliding force Wi , a linear spring constant ki and a state variable ζi . The state variable ζi describes the position of the element i. The characteristic of an element is shown right in the figure. The GMS model uses this implementation as a basis in the pre-sliding regime with another law for the sliding phase, in the form of a state (differential) equation [1, 53]. The pre-sliding regime is defined as the region in which the friction force appears to be a function of the displacement. The relationship between the displacement and the friction force is observed as an hysteresis [22]. The main property of the dynamic gross sliding behaviour is called ‘friction-lag’. Acceleration or deceleration produce a deviation around the Stribeck-curve. The magnitude of this deviation indicates the change (or

27

1 Introduction

derivative) of the friction force. This results in eqn. 1.5, with v the sliding speed. This expression is only valid in the gross sliding regime. dFf ric dt

Ff

Ff ric = sign(v)C 1 − s(v) VS with e.g. s(v) = FS 1 + (β − 1) exp − v = Ff ric + f (v)

(1.5)

f (v) is an additional friction force, mostly this is the viscous friction force, which has an instantaneous effect. The additional parameter C is called the attraction parameter, which determines the rate of convergence of the friction force Ff to the stationary friction force s(v). The result of the model gives a behaviour with dynamic friction as described in Figure 1.9.

Figure 1.9: Result of friction-lag modelling. Higher acceleration gives other results. dashed = reference, solid = . . . × 16 and dotted = . . . × 100. [31] The discussed friction models appropriate for control assume a constant normal force. Therefore, the GMS friction model is only valid for sliding friction if the normal force on the sliding object is constant or the variation is negligible compared to the pre-loading of the bearing. Therefore the GMS model is extended in order to take the influence of the normal force and therefore also the influence of normal displacement is taken into account. The lubrication effect or viscous effect is also investigated.

1.6.4

Clutch slip control

The main focus in this research goes to the torque transmission in transmission lines with multi-disk clutches where the technique of power shifting is used.

28

1.7 Contributions to the state of the art

Power shifting in transmission means using high torque friction clutches to engage gears, designed to shift while transmitting power without completely disengaging all clutches. It is indispensable for off-road machines to have a drive train that can deliver continuous power while shifting. In [60] the aim was to perform slip control enabled by model based control design for the control in the slipping phase of a multi-disk wet clutch. The model of the system from current, to the proportional valve, to slip, in the clutch, was calculated as a series connection of 3 models, G(s) = H(s).F (s).D(s), with s the Laplace operator. H(s) is a LinearParameter-Varying (LPV) model of the hydraulic circuit from current to pressure in the clutch, depending on the output pressure of the valve. F (s) is a model of the friction plates from the pressure in the clutch to the transmitted torque in the clutch. The relationship in this model is simply taken proportional to the friction coefficient and the dimensions of the friction plates and the friction coefficient is taken constant even though it depends on the temperature, the slip, the pressure and the life-time of the clutch. It is mentioned in that work that this behaviour should be further investigated and this is preformed in the research presented here. As last part of the model, D(s) is a model of the drive-line dynamics from the torque transmitted through the clutch to the slip between the friction plates in the clutch. Again here this part of the model is roughly approximated by a first order model. Finally, an LPV model G(s, p) is calculated of the complete model from current to slip by interpolating between the set of local LTI models. In [61] the control of the shift of a dual-clutch transmission (DCT) is discussed. Here the clutch is also modelled and controlled using a model based approach. The clutch torque, or friction torque, is modelled as a function of the relative angular velocity and the hydraulic pressure in the clutch piston in terms of a look-up table. The clutch has three operation states and the torque transmitted in each state is described by an equation for the slipping, closed and open condition. The friction torque is also here taken proportional with a constant coefficient of friction. It is clear that the friction models used in the previous discussed literature are a strong simplification of the actual friction behaviour and that a better more refined model can drastically reduce uncertainties about the behaviour of a system and can increase performance. That is why model based friction or torque control is performed in this research.

1.7

Contributions to the state of the art

In this section the main contributions of this work are discussed. This work focusses on the lubricated friction force interaction between the separator disk

29

1 Introduction

and friction disk of a multi-disk clutch from an automatic transmission. The aim is to model the friction behaviour and to develop a friction model for control that can eliminate, compensate or control the non-linear friction behaviour in order to have a good torque transmission in a multi-disk clutch. The main contributions of this work are: • A theoretical contribution is the development of the Generic Rayleigh model, based on an existing generic physically based friction model that is able to describe all the observed types of friction behaviour. This existing generic model was developed for dry sliding friction for constant separation between the two sliding objects. Here this model is extended to lubricated friction with the use of the Rayleigh step model and extended with the addition of a normal degree of freedom. In the Rayleigh step model this normal degree of freedom is also incorporated such that the normal-tangential coupling of the friction behaviour is taken into account. • A new rotational tribometer has been developed and build that enables us to measure different types of frictional behaviour on one and the same setup for dry and lubricated friction. The main focus here goes to lubricated friction between a pair of disks from a multi-disk clutch. An added functional requirement to the newly developed rotational tribometer is the ability to measure the normal displacement associated with friction, namely the lift-up effect. With this setup the normal load dependence of the friction force can also be investigated. The experimental results coming from this tribometer can be used to validate physically motivated friction models or to formulate or validate empirically motivated friction models. A second test setup was developed and build for the main purpose here, i.e. the torque control in function of hydraulic pressure, sliding velocity and lubrication oil temperature of an actual multi-disk wet clutch coming from the gearbox of an automatic transmission line. • The Generic Rayleigh model is a computationally expensive friction model and the identification of its parameters based on macroscopic friction measurements would be extremely difficult to achieve. Therefore, a new friction model appropriate for control has been developed based on the existing GMS model. The new model is called the eXtended Generalised Maxwell-Slip (XGMS) model and it is developed for friction torque control in automatic transmissions. The extension is based on the Generalised Maxwell-Slip (GMS) model which is extended with a normal force dependence and a normal degree of freedom based on friction modelling and friction experiments. The normal load dependence is used to perform control on a wet multi-disk clutch using a developed adaptive torque controller and for this the second developed setup has

30

1.8 Outline of the thesis

been used. Its parameters a relatively easy to identify and only a set of a few amount of elements is needed to represent the measured types of friction behaviour accurately.

1.8

Outline of the thesis

The dissertation consists of 6 parts and is organized as follows. Figure 1.10 illustrates the structure and the interdependence of the chapters. They are represented in two columns, one related to the theoretical aspect and one to the experimental aspect. There is also a control part that is both theoretical and experimental. Part I: Introduction Part I comprises an introductory chapter which includes the problem description and the objective of this work. It also gives an overview of the state of the art about friction measurement, characterisation, modelling and control. Part II: Theoretical aspects Part II involves friction modelling. Chapter 2 describes the Generic model, the Rayleigh step model and the joined Generic-Rayleigh friction model. Part III: Experimental aspects The design and development of a new tribometer are discussed in Chapter 3 and its experimental results, namely the investigation of the friction behaviour in Chapter 4. Part IV: Control aspects The basis for the formulation of a control model and its extension are described in Chapter 5. The application of the friction model on a test setup which represents a transmission line are discussed in Chapter 6. Part V: Conclusions Part V concludes the dissertation, stating the main conclusions of the research, the main achievements and some recommendations for future research. Part VI: Addenda Part VI comprises the author’s publication list and curriculum vitae. Furthermore, some appendices are included.

31

32

Figure 1.10: Outline of the thesis and chapter interdependence.

1 Introduction

Part II

Theoretical aspects

33

Chapter 2

Physically based friction modelling “The formulation of a problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.” Albert Einstein

2.1

Introduction

There has been a growing interest in the modelling of friction and its dynamics over the last couple of decades. This interest is shared by various disciplines, ranging from geophysics to engineering, and is stimulated principally by one underlying motion, viz. to be able to understand, model and simulate mechanical systems that comprise friction elements. Such modelling may be for the purpose of forecasting certain phenomena such as landslides and earthquakes or optimizing/stabilising processes, such as moving mass-damper elements or wet clutches or transmissions. Friction may be dry or lubricated; however the qualitative behaviour of its dynamics appears to be the same, see [3]. In that reference, a generic model at asperity level for dry, wearless friction was formulated in the literature, where the asperity contacts were modelled by a phenomenological creep/adhesion/deformation model. Here the analysis is extended to the case of lubricated contacts. In this way the influence of the lubricant can be eval-

35

2 Physically based friction modelling

uated and used for the possible extensions including the shape of the Stribeck curve, the relation between friction force or friction coefficient and the normal load and the viscous behaviour. This chapter described this physically based friction modelling. It consists of two models, viz. the Generic dry friction model and the Rayleigh step lubricated friction model. These models are developed separately and are joined to create an asperity based friction model which includes dry friction and all the lubricated friction regimes.

2.2

The Generic model

The basic ingredients for this model are a certain amount of asperities and a counter surface consisting of a profile with unit height steps. The logical steps taken to arrive at such a representation are exhaustively explained in [1, 3]. Based on this model the friction behaviour, resulting from a relative motion between the two contacting surfaces, is described. A schematic representation of the friction contact looks as depicted in Figure 2.1.

asperity base

main body kn kt/2

kt/2

asperity tip

counter surface

Figure 2.1: N = 10 asperities, Nw = 5 unit steps The main body can be represented by asperities, each with its own stiffness, mass and length. The counter surface of these asperities is a rigid surface. Only the asperities can deform. This representation allows determining whether or not an asperity is active, i.e. in contact. Figure 2.2 depicts the life cycle of one asperity, where it is assumed that the upper surface is moving from left to right with respect to the fixed lower surface. The characteristics of an asperity are lumped into one point (•), namely the asperity tip, for simplicity of treatment. This point (Figure 2.2) is initially moving freely (i), until it touches the lower rigid surface (ii), after sticking to and then slipping over the lower profile it breaks completely loose from the lower profile (iii). In case (ii) the asperity is said to be in an active state, for the other cases the asperity is said to be inactive.

36

2.2 The Generic model

(i)

(ii)

(iii)

spring force

spring extension

zb

h ζ zi

ξ

counter surface

Figure 2.2: The life cycle of one asperity contact

2.2.1

Asperities

The asperities are characterised by their amount N , by their tangential position and height distribution. These are generated with a certain statistical distribution with a certain average and standard deviation. Each asperity has its tangential stiffness kti and normal stiffness kni . The tangential stiffness is also generated with a certain statistical distribution. In the initial generic model the values for the normal and tangential stiffness are independently chosen from a distribution function Φ. In its normalised formulation the normalised tangential stiffness is proportional with the ratio of the tangential and normal stiffness. In the implementation here the normal and tangential stiffness are coupled. Based on the following equation, the normal stiffness can be obtained. (2 − ν) kti (2.1) kni = 2 (1 − ν) In this equation ν represents the Poisson coefficient. This relation between the tangential and normal stiffness results from the theory of Mindlin [62], which is valid when the asperity is in full stick, see also Appendix B.

2.2.2

Counter surface

The profile of the counter surface consists of an amount Nw , of unit steps each with a certain width w. The width of each unit step is on its turn also generated with a statistical distribution with a certain average and standard deviation. The average length determines the filling degree of the total profile length. E.g. if the average value is 70 then about 70% of the total length of the profile has value one and 30% is a pit or recess.

37


As position input for the relative motion between the asperities and counter surface a certain trajectory is applied to the main body because the counter surface is considered fixed. This could be a sinusoidal position input with certain amplitude A and frequency. In this way an oscillation around the initial position could occur.

2.2.3

Friction

To determine the friction force between one asperity and one unit step an adapted Maxwell-Slip (MS) model is used. This adapted MS model consists of an asperity with mass and a contact-time-dependent friction coefficient. This model is only used when the asperity is active or in contact with a unit step. [1, 3] describe the life cycle of one asperity as depicted in Figure 2.2 as follows. From the moment the asperity becomes active, it will begin to follow the profile of the counter surface, by deforming normally and tangentially; resulting in a normal and tangential force. When the asperity tip has fully traversed the bottom surface, it will break loose, vibrate (tangentially and normally) and thereby dissipate (part of) its “elastic” and “inertial” energy, by internal hysteresis, until it comes to rest or comes in contact with the next bottom asperity. In the model that will follow, we assume that all of the asperity energy is lost. This assumption is evidently valid for low sliding speeds and relatively large separation of consecutive asperities. For the adapted MS model two states can be distinguished, namely stick and slip. As initial state the asperity is considered being in stick. This assumption can be made because at the initial condition the asperities are standing still. In stick the maximum tangential force an asperity can sustain, which means before slipping, equals the adhesion force. Here, the local friction coefficient µ(t) is function of the contact time, owing to normal creep. While different appropriate relations are available [28], here we have chosen an exponential saturating function [27]: t µ (t) = µ0 + (µ∞ − µ0 ) · 1 − e− τt (2.2) In this equation µ0 is the initial value for the friction coefficient, µ∞ the limit value or asymptotic value for the friction coefficient and τt is a parameter which determines the rate of change from µ0 to µ∞ . Each asperity has its own time dependent friction coefficient. As mentioned earlier, other time dependencies could also be used [1, 3]. To evaluate the friction force with the MS model the state of each asperity is checked, i.e. if it is active or in contact with a unit step or not. The unit steps act as some kind of switching function between contact and no contact. As already mentioned earlier, in contact, two conditions can be discerned,

38


namely stick and slip. While sticking the stick time has to be taken into account to be able to determine the time dependent friction coefficient. The friction force Ft for each asperity is equal to the previous friction force, so at time step t − 1, plus the respective tangential stiffness multiplied by the elongation. The time is discrete, so t represents the index of the time instance. This means, in stick, the asperity behaves as a linear spring in tangential direction. The equation for this looks as follows: Ft (t, i) = Ft (t − 1, i) + kti · ∆x = kti · ξ

(2.3)

The i represents the ith asperity. ξ is the tangential deflection of the asperity. After each evaluation it has to be checked if the friction force is larger than the maximal allowed friction force in the contact, which is function of the normal load and the time dependent friction coefficient. This can be made clear with Figure 2.3. 1.8 1.6 1.4

F[ ]

1.2 1 0.8 0.6 0.4 0.2 0 0

500

1000

1500

2000

2500

3000

3500

x[]

Figure 2.3: Force in function of displacement (linear) and stick-time-dependent friction force (curved) If the friction force becomes larger than the maximal allowed friction force in the contact, then the state has to change from stick to slip. For a more exhaustive discussion, see Appendix A where the stick time in function of the asperity stiffness is discussed. The maximum allowed friction force for each asperity is also function of the normal load it is bearing. The normal load for each asperity can be calculated as follows: Wni = (1 − zi − zb ) · kni 1 − zi − z b = ζ i Wni = ζi · kni

(2.4)

In here 1 is the dimensionless height of a unit step, zi is the initial height of the asperity i as a result of the statistical distribution, zb is the displacement of the main body in the normal direction and kni is the normal stiffness. ζi is the normal deflection of one asperity when the asperity is active, see Figure

39


2.2. When zb becomes to big due to an increase in the bearing capacity or due to a decrease in normal load, ζi can become negative and no contact for that specific asperity is possible anymore. As a recapitulation, from the moment the asperity becomes active, it begins to follow the profile of the lower surface, by deforming normally through ζ and tangentially through ξ, resulting in normal and tangential forces. The normal force is given by Fn (t) = kn ζ (t) (2.5) while the tangential force in stick is given by Ft (t) = kt ξ (t)

(2.6)

where kn and kt equal the normal and tangential stiffness of the equivalent asperity, respectively. The maximum tangential force Fµ (t) an asperity can sustain before slipping equals the adhesion force, that is, Fµ (t) = µ (t) kn ζ (t)

(2.7)

where the local friction coefficient µ(t) is a function of adhesion or the contact time, owing to normal creep. At slip the friction force for each asperity is equal to its respective maximal allowed friction force. At direction reversal or stand still the state of an asperity goes from slip to stick. In both cases the speed crosses or becomes zero. A last component of the friction force is the result of the dissipation of the inertial energy due to the vibration of the asperity. The inertial energy resulting from impact of the mass m of the asperity equals: Wm =

mv 2 2

(2.8)

Here we consider this energy being totaly lost at impact, which means there is no energy recuperation, and the tangential force for this inertial component can be calculated as: Fm =

Wm mv 2 = (1 + α) w 2 (1 + α) w

(2.9)

With w the width of a profile step and αw the width of the following recess. It is clear that the higher the mass m or the higher the velocity v the higher this force will be, resulting in a strengthening effect.

2.2.4

Results

As a preliminary overview of the output of the model a sinusoidal position input is applied to the main body with amplitude A = 0.5 [ ] and 1.25 periods as shown in the Figure 2.4.

40


Input position 0.5 0.4

Displacement [ ]

0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 0

1

2

3

4

5

6

7

8

Time [ ]

Figure 2.4: Input signal of the model The output which result from the model is hysteresis, namely the friction force Ff in function of the displacement x and in function of the velocity v which can show the friction lag. In this result the friction lag is not very clear due to the signal noise. The friction force Ff in function of the displacement x and in function of the velocity v can be seen in the two following plots in Figure 2.5. For this simulation N = 250 asperities and Nw = 100 unit steps with a filling degree of 50 %.

Ff

Ff

Hysteresis

x

v

Figure 2.5: Representation of the friction force Ff i.f.o. the displacement x or hysteresis (left) and i.f.o. the velocity v or friction lag (right) As can be seen in Figure 2.5 for the friction force in function of the velocity, for this particular set of parameters there is no clearly visible Stribeck effect. As means of better visualisation, a fit of a possible Stribeck curve is shown. This Stribeck effect is related to the time dependency of the friction coefficient. A more apparent effect is the viscous effect which can be seen as a rising friction force with increasing speed. This effect is a result of the influence of the asperity mass m. This phenomenon is called velocity strengthening, i.e. for high speeds the friction force increases with the velocity [1]. The same

41


behaviour is measured for lubricated sliding friction, where it is attributed to the viscosity of the lubricant. The force necessary to initiate total slip, or gross sliding, can be determined. The maximum force which occurs in this initiation is called the break-away force [24, 25]. This is another resulting phenomenon of the generic model. The break-away force is here defined as the maximum friction force during the transition from pre-sliding to gross sliding regime. One may use this definition of break-away force, as being equivalent to the static friction force. But as can be seen in Figure 2.6, due to the dependence on the application rate, several values for the static friction force can be found for one and the same normal load. This result is obtained with a different counter profile, due to which no velocity strengthening is observed, but the effect on the break-away force is clear. For the lowest applied rate the highest maximal force is attained and this maximum decreases with increasing rate [24, 26].

Figure 2.6: Break-away force for different applied rates A third phenomenon is the transition from stick to slip at asperity level. A representation of this phenomenon can give insight in the model and the behaviour at asperity scale. Figure 2.7 shows the stick time as a colour map for each asperity at each time step of the simulation. It is clear that stick occurs more at motion reversal, as observed as a rising stick time for most asperities at reversal. This is logic because the motion is slowing down and the main body is instantaneously standing still. The periodic behaviour of the input can be recognised in this stick-slip representation. It has to be mentioned that stick can only occur when an asperity is active and as can be seen in the figure some asperities are not in contact at motion reversal and that is why they appear not to stick.

42

Time step

Stick time


Asperity

Figure 2.7: Stick time for each asperity To check the stick time dependency of the friction coefficient, the counter surface can be made totally flat, without pits or recess, then rerun the simulation and look at the resulting hysteresis. In this hysteresis an overshoot in the friction force would be expected because of the time dependent friction coefficient, which can be seen in Figure 2.8. This matches with the adapted MS model where the maximum allowed friction force, before slip occurs, is time dependent.

Ff

Hysteresis

x

Figure 2.8: Result from the adapted MS model with a flat counter surface and N = 250

43


2.2.5

Conclusions

This section presented the re-implementation of an analytical model at asperity scale which can reproduce most of the dry friction force dynamics. It is based on different well known physical friction mechanisms: deformation of asperities, adhesion theory, normal creep, ... Here the model was implemented in a force based way instead of an energy based method as it was in the original model [1, 3]. The contribution to the generic model is that the normal degree of freedom is added as opposed to the original model where the normal distance was fixed. In this way a change in normal load can affect the gap distance and on its turn also the friction force. It will be used as a basis for the extension to a friction model including lubricated friction conditions.

44

2.3 The Rayleigh step model

2.3

The Rayleigh step model

We consider a 2-D (infinitely wide) Rayleigh step bearing and formulate the time dependent Reynolds equation [63]. The solution of this equation, considering cavitation [64], yields the normal force and the traction force as function of the bearing sliding number Λ (proportional to velocity) and the squeeze number σ (proportional to normal velocity, or gap rate of change). The case of σ = 0 with constant bearing load, corresponding to steady state sliding, yields that part of the Stribeck curve, which reveals velocity weakening and velocity strengthening (i.e. excluding only the Boundary Lubrication (BL) part). The dynamic sliding case (σ 6= 0) yields a behaviour that is determined by the inertia of the slider and the dynamics of the (imposed) sliding velocity. Restriction to positive, periodic velocities (and low slider inertia’s) reveals the ‘friction lag’ phenomenon very clearly and in a manner that is consistent with reported experimental observation (both for lubricated and dry friction). Consequently it is our intention to use the elementary Rayleigh step as building block for a general model to simulate the dynamics of normal and tangential forces in lubricated rough contacts.

2.3.1

Introduction

Most models are only used to describe dry friction force behaviour and no direct link is made to lubricated or hydrodynamic friction and especially to boundary lubricated friction. In previous research a way to implement wet friction is to add a viscous term or to force the friction force to follow a certain Stribeck function. A novel way is described here, to link dry friction with wet friction to extend “the generic dry friction model” and incorporate a wet friction model. This wet friction model consists of the solution of the 2D Reynolds equation for a Rayleigh step bearing. The suitability of the Reynolds equation is described in [65]. This extension module allows us to include the different lubrication regimes. Elastohydrodynamic lubrication (EHL) is the type of hydrodynamic friction in which the elastic deformation of the contacting surfaces cannot be neglected [66]. This deformation is often calculated using FEM in the case of non-conformal contacts. With conforming contacts the elastic deformation can be taken into account at asperity level. The lubrication regimes are commonly divided into: Boundary Lubrication (BL), Mixed Lubrication (ML) and Full Lubrication or Hydrodynamic Lubrication (FL, HL) [30]. In the BL regime the lubricant’s hydrodynamic action is negligible and the load is carried directly by the surface asperities, similar to dry friction contact. In de ML regime the load is carried by the lubricant’s hydrodynamical action and/or directly by surface asperities. When the hydrodynamic action of the lubricant fully separates the surfaces and the load is carried totally by the lubricant film the contact enters FL regime. In terms of the friction coefficient

45


the Stribeck effect together with the hydrodynamic or viscous effect can give a good visualization of the different regimes, combined in the Stribeck curve. The surface topography has an influence on the occurrence of the different regimes in function of the velocity of the moving surfaces but also on the rate of wear [13]. The generic model [3], which treats friction as a dynamic process that evolves in the contact, forms a good basis for the continuation of this study. It is an effective and comprehensive model for the evolution of the friction force, as a function of the states of the system, namely time and displacement (or equivalently, the displacement and its time derivatives), in the totality of a rubbing contact. Since this model is only concerning dry friction, an extended model which also takes care of lubricated and especially boundary lubricated friction may be useful for the study of wet clutches in automatic transmissions. In automatic transmissions, paper-based wet friction materials are used for clutch and lock-up mechanisms [4, 42, 43]. Shock reduction and shudder prevention in a slip controlled process of a clutch and realization of higher friction coefficient (µ) are required simultaneously. Although these demands are a trade-off generally, understanding the friction characteristics through modelling can lead to the control and reduction of dynamic oscillations.

2.3.2

Rayleigh step

The extension of “the generic dry friction model” to lubricated friction can be made by a Rayleigh step module. This module simulates a 2-D Rayleigh step bearing, moving over a lubricated surface, by using the Reynolds equation. The tip of each asperity is replaced by two back-to-back Rayleigh steps. which are solved independently. Figure 2.9 illustrates a Rayleigh step bearing with normalisation parameters. Each elementary contact or asperity of the generic

x1

1

0

0

L H=

x

H Δh

x2 pm

αL

α

Δh

V (t ) or Λ(t )

h=

h Δh

Figure 2.9: Rayleigh step bearing and its parameters

46


friction model can be implemented as an elementary Rayleigh step which can reproduce the characteristics of lubricated or hydrodynamic friction. Thus one model can simulate or be used for control of dry friction or asperity contact, boundary lubricated friction which is a combination of asperity contact and lubrication, as well as fully lubricated friction otherwise known as hydrodynamic friction. In this section the behaviour of a Rayleigh step, which undergoes a motion with a certain positive velocity or a tangential oscillation around a velocity offset, will be described. This offset prevents the velocity V or the proportional sliding number Λ of the bearing to become negative. The two forces which are the result of this model are the bearing or the lift up force Wn as a result of the pressure in the bearing film and the friction force, key to an adequate friction model. This model can be extended by connecting several Rayleigh steps back to back to simulate a convex asperity. This enable negative displacements towards the lowest part of the Rayleigh step bearing, unlike a positive displacement where the relative displacement of the plate is towards the highest. In order to simplify solving the differential equation, eqn. 2.10, and to interpret the result not only for a specific bearing, use of a normalised equation is advised. The normalised Reynolds equation can be expressed in terms of the tangential sliding number Λ and the normal squeeze film parameter σ. The normalised 2-D Reynolds equation can be derived as follows: The dynamic Reynolds equation for 1D flow is ¯ 3 ¯ x) ¯ x) x) ∂p ∂ h(¯ ∂ h(¯ ∂ h(¯ = 6V (t) + 12 ∂x ¯ η ∂x ¯ ∂x ¯ ∂ t¯

(2.10)

Writing all the derivatives w.r.t. the position x ¯ at the left hand side of the equation gives ¯ 3 ¯ x) ¯ x) x) ∂p V (t¯) h(¯ ∂ h(¯ ∂ h(¯ − = (2.11) ∂x ¯ 12η ∂ x ¯ 2 ∂ t¯ ¯ x) x ¯ h(¯ With x = , h(x) = and t = ω t¯ being the dimensionless position, L ∆h height and time respectively this results in the normalised 2-D Reynolds equation. The boundary condition for the pressure is p = 0, which means that at the beginning and the end of the Rayleigh step the pressure p = 0. ∂ ∂h(x) 3 ∂p h(x) − Λ (t) h(x) = σ (2.12) ∂x ∂x ∂t In this equation x, h(x) and t are dimensionless. In the steady state case the derivative of h(x) to the dimensionless time t is equal to 0. A condition for

47


this to be true is that the bearing profile cannot deform, which is considered here to be the case. h(x) is defined as: h(x) =

H; 0 ≤ x ≤ 1 h; 1 ≤ x ≤ 1 + α

The two squeeze film parameters used in eqn. 2.12 are: Λ (t) =

6ηV (t) L ∆h2

(2.13)

12ηL2 ω (2.14) ∆h2 They both have the units of pressure [kg/m.s2 ], since here the pressure in the Reynolds equation is not normalised. σ=

The parameter η, namely the viscosity, which is used in the parameters Λ and σ must not been given a priori. This makes the solution more general. The horizontal dimensions are made dimensionless by dividing them by L [m], the vertical by dividing them by ∆h [m]. The time is made dimensionless by means of the parameter ω [Hz], viz. t = t¯ω. The module of this Rayleigh step results the lift up force Wn and the friction force Ff . If the friction force is plotted in function of the parameter Λ(t), together with the Stribeck curve, then the phenomenon of friction lag is clearly seen, as there is a higher friction force at acceleration than at deceleration (see also Figure 2.12, 2.13, 2.15 and 2.19). The initial value of Λ(t) can be fixed by means of the solution of the steady state Reynolds equation. This ensures that the transient does not last too long and therefore sufficient periods of the movement occur in sequence for a certain simulation time. The analytical solution of the Reynolds equation for a Rayleigh step can be obtained as follows: • Steady state normalised Reynolds equation ∂ 3 ∂p h(x) − Λ (t) h(x) = 0 ∂x ∂x

(2.15)

Region with height H: integration of eqn. 2.15 w.r.t. x, yields H3

48

∂p − Λ (t) H = C1 ∂x

(2.16)


∂p C1 + Λ H = ∂x H3 C1 + Λ H p1 (x) = x H3

(2.17) (2.18)

Region with height h: integration of eqn. 2.15 w.r.t. x, yields h3

∂p − Λ (t) h = C2 ∂x

C2 + Λ h ∂p = ∂x h3 C2 + Λ h p2 (x) = x + C3 h3

(2.19) (2.20) (2.21)

The mass flow at the passage of H to h must be conserved. This can be written as follows, taking the flow part of eqn. 2.12: − H3

∂p ∂p + Λ (t) H = −h3 + Λ (t) h ∂x ∂x

(2.22)

Thus from (2.14), (2.17) and (2.20) it follows that C1 = C2 . This equation can be rewritten in the following form because the pressure is linear in both zones: pm H 3 pm − Λ H = −h3 − Λ h, (2.23) α out of which the maximum pressure pm can be determined. pm =

α Λ (H − h) h3 + α H 3

(2.24)

In the region with height h for x2 = 0: p(x) = pm and therefore C3 = pm . For x2 = α : p(x) = 0 so C2 and C1 can be determined. C2 + Λ h α + pm p2 (α) = 0 = h3 Λ h3 C1 = C2 = − Λ h + 3 h + αH3

(2.25)

(2.26)

The bearing force or lift up force can be ascertained from these results. It is equal to the integral of the pressure over the total length of the bearing multiplied by the unit width. Since the steady state case is of concern here, there is an easier way to determine the bearing force. The

49


pressure distribution forms a triangle of which the surface is equal to the maximum value pm divided by two times the dimensionless length of the bearing. Therefore: Wn =

1 (1 + α) α Λ (H − h) pm (1 + α) = 2 2 (h3 + α H 3 )

(2.27)

This equation can be simplified when h → 0 : H → 1 and

h H

3 →0

(2.28)

Resulting in: (1 + α) Λ (2.29) 2 In this expression α can be neglected, because for h = 0 the bearing is in contact with the counter surface such that this region has no bearing capacity anymore due to the lubricant. Wn =

By taking the integral of the shear stress τ one can obtain the friction force Ff . Here only the viscous shear force on the bottom surface is calculated. If the viscous shear force on the top surface would be calculated a different result would be obtained. This difference is a result of the horizontal component of the hydrodynamic film pressure acting on the step surface of the bearing. If h is not dimensionless and denoted as h then the shear stress on the bottom surface can be written as: τ =η

V h ∂p − 2 ∂x h

ZL

Ff

Zα L H ∂p V h ∂p V − dx + η − dx 2 ∂x 2 ∂x H h 0 0 pm 1 α = h − H + ηV L + 2 H h ∆h pm ηV L 1 α = − + + 2 ∆h H h ∆h pm ∆h Λ (t) 1 α = − + + 2 6 H h

(2.30)

=

η

(2.31)

(2.32) (2.33) (2.34)

In the last two expressions H and h are dimensionless and pm is as determined before.

50


• Dynamic normalised Reynolds equation ∂ ∂h 3 ∂p h − Λ (t) h = σ ∂x ∂x ∂t

(2.35)

Region with height H σ ∂h ∂2p = 3 ∂x2 H ∂t

(2.36)

∂p σ ∂h = 3 x + C1 ∂x H ∂t

(2.37)

σ ∂h x2 + C1 x + C2 H 3 ∂t 2

(2.38)

∂2p σ ∂h = 3 ∂x2 h ∂t

(2.39)

∂p σ ∂h = 3 x + C3 ∂x h ∂t

(2.40)

p1 (x) = Region with height h

p2 (x) =

σ ∂h x2 + C3 x + C4 h3 ∂t 2

(2.41)

For x1 = 0 : p1 (x1 ) = 0 so that C2 = 0

(2.42)

For x2 = α : p2 (x2 ) = 0, therefore C4 = −αC3 − C3 = −

σ ∂h α2 h3 ∂t 2

C4 ασ ∂h − 3 α 2h ∂t

(2.43) (2.44)

Pressure continuity at the boundary from H to h implies that p1 (x1 = 1) = p2 (x2 = 0). The following conclusion may be drawn: C4 =

σ ∂h + C1 2H 3 ∂t

(2.45)

An extra condition to determine the unknown values C1 , C3 and C4 stipulates that the mass flow must be equal throughout the bearing. This can be represented with the same equation as used in the steady

51


state case. After solving the system of three equations, (2.41), (2.42) and (2.43), with three unknown variables the following result is obtained: " • σh 1 αΛ − C1 = 3 h + αH3 2

h H

!#

3 + 2α + α

2

(2.46)

•

•

C1 ασ h σh − − C3 = − 3 2αH α 2h3

(2.47)

•

σh C4 = + C1 2H 3 •

In these equations h =

(2.48)

∂H ∂h = ∂t ∂t

Now every unknown variable of the pressure equation is determined such that the bearing or lift up force can be determined. Z1 Wn =

Zα p1 (x1 ) dx1 +

p2 (x2 ) dx2

0

( 0 #!) • " 3 h 1+α 1 σh 2 + 2α + α αΛ − h 3 2 2 H α+ H • • # σh 1 α α3 σ h + + − 2 3 2 12h3 (2.49) The previous theory about the Rayleigh step model is developed in a similar way as performed by Al-Bender [67]. In [68] an alternative derivation of the steady state Reynolds equation for a stepped pad, also known as a Rayleigh step bearing, can also be found. 1 = 3 H

"

An observation which must be made is the fact that cavitation in the previous equation is not taken into account. It is assumed here that the region of cavitation is at p ≤ 0, i.e. that the pressure cannot become negative. The pressure profile can be as shown in Figure 2.10. To take cavitation into account, the integration boundaries for the calculation of the bearing force must be adapted to arrive at the next equation. 0 Z1 Zx2 Wn = p1 (x1 ) dx1 + p2 (x2 ) dx2 (2.50) x01

52

0


p pm

α x 0

x’1

x’2

Figure 2.10: Parabolic pressure profile with negative part, viz. cavitation This procedure is applicable because the pressure distribution is parabolic and therefore cannot become positive between the borders 0 and x01 , and x02 and α. For the friction force, the same expression as obtained with the solution of the steady state Reynolds equation is used, but then with pm = C4 . Therefore: ∆h C4 ∆h Λ (t) 1 α Ff = − + + (2.51) 2 6 H h In eqn. 2.51 cavitation is not explicitly taken into account, but as with the calculation of the bearing force Wn , here the integration boundaries could also be adjusted. As mentioned above, the initial value of Λ can be stipulated such that the transient of the solution does not last too long. This can be done as follows. Λ0 =

2 Wn (1 + α)

(2.52)

Following this procedure, for the dynamic case Λ(t) can oscillate around Λ0 , namely around the solution of the steady state equation. The expres 3 h →0 sion for Λ0 , however, only applies for h → 0 : H → 1 and H

2.3.3

Implementation

For the implementation, the use of blocks or modules facilitates the overview. The main block contains the solution of the Reynolds equation for a Rayleigh step. This block forms a function needed to solve the differential equation. For the final solution of the total problem, the use of an ODE solver with high enough order is advised. This solver should use a variable time step which is beneficial for the calculation speed. A small sketch of the link between the different blocks (modules) is shown in Figure 2.11. Rayleighstepdimlesssol, the main part of the code, gives all the parameters and the initial conditions

53


Rayleighstepdimlesssol - parameters - initialisation

ODE solver Rayleighstepdimlessode - State space representation

Rayleighstepdimless - Solution Reynolds eqn.

Rayleighstepdimlessres - Results - Visual representation

Figure 2.11: Flowchart of the Rayleigh step model and uses the ODE solver. The ODE solver calls the module Rayleighstepdimlessode, which in its turn receives the bearing force and the friction force from Rayleighstepdimless. Rayleighstepdimless contains the analytical solution of the Reynolds equation for a Rayleigh step. Rayleighstepdimlessres ensures the reproduction of the results such as the evolution of the height h of the bearing, the bearing force Wn in function of time, the friction force Ff in function of time and the parameter Λ. The integration to the height h can be solved in two ways by the use of a different order in the solution. •

• A first manner consists of rewriting the bearing force Wn (h, h, Λ(t)) • into an expression of the form h (h, Wn , Λ (t)). Integrating this expression, the ODE function immediately determines the evolution of the height h. In this expression a certain value W can be imposed on Wn . That value is considered as a certain preload of the bearing. • A second way is to work with the calculated Wn , with a higher order as before, viz. acceleration instead of velocity. In this way adding a mass, namely the mass of the bearing, is a simple task. As a result one obtains the following expression. ••

M h +W = Wn (t)

(2.53)

(Wn (t) − W ) M

(2.54)

or ••

h =

54


Integrating this acceleration twice by means of the ODE function gives the height h.

2.3.4

Results

The influence of the variation of the parameters is analysed by means of following plots. For the steady state case the friction force Ff in function of Λ visualises the Stribeck curve. The arrows in the following graphs reflect the rising evolution of the respective parameters. For better visualisation the x-axis is rescaled by dividing Λ by the preload W . Λ/W is similar to the Sommerfeld number, which is a bearing characteristic number often used in the design of hydrodynamic bearings. The influence of the amplitude A of Λ, being the input of the model, on the friction force can be seen in Figure 2.12. Ff(Λs/W) and Ff(Λt/W) Stribeck A = 0.1 A = 0.01 A = 0.001

0.5

Ff

0.4

0.3

0.2

0.1

0 0.8

0.805

0.81

0.815

0.82

0.825

Λ/W

0.83 0.835

0.84 0.845

0.85

Figure 2.12: The influence of the amplitude A [kg/m.s2 ] of Λ on the friction force There is a higher acceleration and deceleration for a higher amplitude A as seen in the more expressed friction lag phenomenon. The influence of the frequency f of Λ on the friction force can be seen in Figure 2.13. It can obviously be seen that the higher the frequency f the more the orientation of the loop changes from vertical to horizontal and the more the friction lag is expressed. The influence of the mass M of the Rayleigh step bearing can be seen in Figure 2.14.

55


Stribeck f = 0.25 f = 0.5 f=1 f=2 f=5 f = 10

0.5

Ff

0.4

0.3

0.2

0.1

0 0.8

0.802 0.804 0.806 0.808 0.81 0.812 0.814 0.816 0.818 0.82

Λ/W

Figure 2.13: The influence of the frequency f of Λ on the friction force 1.6 Stribeck M = 1e−2 M = 1e−1 M=1 M=2 M=5 M = 6.5

1.4 1.2

Ff

1 0.8 0.6 0.4 0.2 0 0.8

0.805

0.81

0.815

Λ/W

0.82

0.825

Figure 2.14: The influence of the mass M of the Rayleigh step bearing on the friction force We can clearly see that the lower the value of the mass the closer the oscillation lies to the Stribeck curve, due to the friction lag. The larger the mass M , the larger the difference between friction force either belonging to acceleration or deceleration, so the more the friction lag is expressed. The influence of the preload W on the Rayleigh step bearing can be seen in Figure 2.15. As in Figures 2.12 - 2.13 - 2.14, the influence of the preload W on the Rayleigh step bearing can be plotted in an easy to understand way (see Figure 2.15). It is clear that the friction force is normal load dependent. The higher

56


Stribeck W = 0.05

1

W = 0.25 0.8 W = 0.5 W = 0.75

Ff

0.6

W=1 W=2

0.4

0.2

0 0.8

0.805

0.81

0.815

0.82

Λ/W

0.825

0.83

0.835

Figure 2.15: The influence of the preload W on the Rayleigh step bearing on the friction force the normal load the higher the friction force. When the Stribeck curves for the different normal loads are plotted as the friction coefficient one common Stribeck curve is obtained, see Figure 2.16. Stribeck W = 0.05 Stribeck W = 0.25 Stribeck W = 0.5 Stribeck W = 0.75 Stribeck W = 1 Stribeck W = 2

1

0.8

µ

0.6

0.4

0.2

0 0.8

0.81

0.82

0.83

Λ/W

0.84

0.85

Figure 2.16: The influence of the preload W on Stribeck curve plot as the friction coefficient In the BL and ML regime this dependency is non-linear, as will be discussed further in Section 4.2 where this is shown based on experimental results. When Λ is not normalised we see that its minimum value changes with a change of the preload W . This influence can be seen in Figure 2.17. This can be explained by the way the initial value has been determined, like explained before. The influence of the frequency ω on the friction force Ff can be seen in

57


Stribeck W = 0.05 W = 0.25

1

0.8

W = 0.5 W = 0.75

0.6

Ff

W=1 W=2

0.4

0.2

0

0

0.2

0.4

0.6

0.8

Λ

1

1.2

1.4

1.6

Figure 2.17: The influence of the preload W on the Rayleigh step bearing on the friction force without normalization of the x-axis the next figure, Figure 2.18. The lower the parameter ω the closer the friction curve gets to the steady state solution, namely the Stribeck curve. This is the expression of friction lag.

Stribeck ω = 1e−6 ω = 1e−7 ω = 1e−8 ω = 1e−9 ω = 1e−10 ω = 1e−11

1

Ff

0.8

0.6

0.4

0.2

0 0.8

0.805

0.81

Λ/W

0.815

0.82

Figure 2.18: The influence of the frequency ω on the friction force Ff As mentioned earlier, this model can be extended by connecting several Rayleigh steps as back to back to better represent a surface and to enable negative displacements towards the lowest part of the Rayleigh step bearing. In this way reciprocating motion can be allowed. The resulting dynamic friction force of such an applied motion can be observed in Figure 2.19 for multiple frequencies.

58


−3

x 10 8 6 4

Ff

2 0 −2 Stribeck Stribeck f = 0.5 f=1 f=2 f=5 f = 10

−4 −6 −8 −5

−4

−3

−2

−1

0 Λ/W

1

2

3

4

5

Figure 2.19: Result of two Rayleigh steps connected back to back The lift-up effect of the Rayleigh step model has also been analysed. The lift-up is the variation in normal displacement of the bearing due to a variation in pressure in the bearing film as a result of the applied motion. For this a sinusoidal velocity input was applied and the resulting lift-up height was plotted against the tangential displacement in Figure 2.20. This results in the typical butterfly curves. Theoretical simulations typically give perfectly symmetric curves, on the other hand for experimental results the curves tend to be asymmetric due to the profile of the contact as described in Section 4.3.

i

Figure 2.20: Lift-up butterfly effect for two Rayleigh steps connected back to back for different input frequencies

59


2.3.5

Conclusions

The behaviour of the Rayleigh step module is one which represents lubricated friction in a good way because it can be used as a representation of a convex asperity. It has been derived and simulated, taking into account the effect of parameter variations, and it has been shown that it can be used as a general model to simulate the dynamics of normal and tangential forces in lubricated contacts. Using it as an extension of the generic friction model can result in a model that represent boundary and mixed lubricated friction. Similar results can be obtained by extending the generic friction model with a module containing the analytical solution of a plain inclined bearing. A next step could be to use the solution of the Reynolds equation for a sphere on a plane, which lies closer to the shape of the top of asperities. Pressure dependence of viscosity like in EHL and temperature dependence of viscosity [66, 69], by using Roelands equation, could also give a more exact behaviour, and the results would lie closer to reality. On the other hand, both the complexity and the simulation time of the model would drastically increase. An extension in this direction could also be investigated.

60

2.4 The Generic Rayleigh model

2.4 2.4.1

The Generic Rayleigh model Rayleigh step extension

The extension of the generic model to a friction model that takes lubricated friction into account can be based on a Rayleigh step module. This module can be added in the slip-part of the model discussed in Section 2.2. Here, each asperity can be represented as two back-to-back connected Rayleigh steps, see Figure 2.21. It is considered here that the pad is symmetric and that the pad cannot tilt. Generic

Generic Rayleigh

kni kti/2

kni kti/2

kti/2

kti/2

Figure 2.21: Representation of an asperity for the Generic model and for the Generic Rayleigh model In the slip condition two situations can be distinguished, namely dry slip and wet slip. Dry slip occurs when the Rayleigh step makes contact with a profile step and wet slip occurs when an oil film exists between the Rayleigh step and a profile step. The life cycle of an asperity in contact is usually as follows: initially stick, then dry slip followed by wet slip during which loss of contact occurs. The transition form dry slip to wet slip occurs when there is enough pressure build up and the asperity lifts up or loses contact. The structure of the code is shown in the following flowchart in Figure 2.22. Also here, for the implementation, the use of blocks or modules facilitates the overview, as represented in the flowchart. Generic Rayleigh main, the main part of the code, gives all the parameters and the initial conditions. Those parameters and conditions are the input variables to generate the position input signal, containing amplitude, frequency, total simulation time, sample frequency. It also contains the parameters to create the counter profile by sending them to Profile Rayleigh. Profile Rayleigh generates a square wave with a given amount of profile steps, filling percentage and standard deviation of the step length. The other inputs contains the amount of asperities, the distribution of asperities both in normal and tangential direction,

61


Generic_Rayleigh_main

Profile_Rayleigh

for loop over time

Generic_Rayleigh_Euler

for loop over asperities to determine contact No

Yes if contact No

if stick

Ff =0 or dynamic component (inertia)

Yes

Ff =k.x Rayleighstepode_Generic_Euler if stick=0

Rayleighstep_Generic

Ff

t

Figure 2.22: Flowchart for the Generic Rayleigh model

the normal load and the time dependant friction coefficient. All these inputs are used in Generic Rayleigh Euler for the actual calculation of the friction force over time, in a for loop. In Generic Rayleigh Euler, first a for loop over the asperities is run to determine which of them are in contact. If the

62


asperity is in contact his stick or slip condition has to be checked and if it is slipping the Rayleigh step solution is calculated, represented by Rayleighstepode Generic Euler and Rayleighstep Generic. Rayleighstep Generic contains the analytical solution of the Rayleigh step as determined in Section 2.3. The flowchart of the Rayleigh step model is represented in Figure 2.11.

2.4.2

Implementation problems of the Rayleigh step module within the Generic model

There is a problem related to the implementation of the Rayleigh step module within the Generic model. This problem arises in the calculation of the tangential sliding number Λ and the normal squeeze film parameter σ based on the speed derived from the input position or motion input of the generic model.

Λ (t) =

6ηV (t) L ∆h2

(2.55)

12ηL2 ω (2.56) ∆h2 In the initial Rayleigh step model the calculation of Λ happens on the basis of the solution of the steady state normalized Reynolds Equation such that an oscillation can occur around an equilibrium. The solution for the lift-up force in the steady state case for a height h going to zero is as follows: σ=

(1 + α) ·Λ 2 For a given load W , Λ can therefore be calculated. W =

(2.57)

If L and ∆h, and in this case also η are given, then using the speed V , Λ can also be calculated on the basis of the expression above but the result can be many orders larger or smaller than the result obtained from the steady state calculation, i.e. the solution is not at equilibrium. This means that the friction force and lift-up force on there turn can be many orders too large or too small. The initial assignment of L and ∆h is thus crucial. For a certain load W there is a certain minimum Λ > 0 for which the Stribeck function goes to infinity due to the asymptotic character of the solution of the Reynolds equation for a Rayleigh step bearing, see Figure 2.23. This part of the solution should be avoided because otherwise the friction force goes to infinity. This can be achieved by a minimum allowed speed which acts as a threshold underneath which dry friction occurs. The transition of the friction force at stick to the friction force at dry friction at slip

63


should occur smoothly. The same applies to the transition from dry slip to wet or lubricated slip. This transition is marked with a circled line in Figure 2.23. This region represents the BL part.

Fs

Λ

Figure 2.23: Stribeck curve for a Rayleigh step with dry-wet transition A threshold in velocity or on the horizontal sliding number Λ can be applied to overcome the previous mentioned implementation problem. Λthresh is determined based on the time dependent friction coefficient, see eqn. 2.2. The threshold is determined such that it can never result in a friction force or friction coefficient bigger then µ∞ , being the limit value for the time dependent friction coefficient. At this threshold different heights h can occur depending on the conditions of the model. So the condition to switch from the Generic model to the Rayleigh step model for each asperity in contact looks as follows: if

Λ Λ

≥ Λthresh < Λthresh

Rayleigh step model Generic model

An alternative is to apply a threshold on the height h to overcome the previous mentioned implementation problem. At this threshold different velocities or sliding numbers Λ can occur depending on the conditions of the model. So in this case the condition to switch from the Generic model to the Rayleigh step model for each asperity in contact looks as follows: if

h ≥ hthresh h < hthresh

Rayleigh step model Generic model

Both solutions take the normal load W into account. In case of the first solution, where the velocity is considered, the threshold is function of the steady state sliding number Λs which on its turn is function of the normal load. In

64


case of the second solution, where the height is considered, the higher the normal load, the higher the velocity should be, before enough pressure build-up occurs to overcome the load and the lift-up reaches the threshold. Within the Rayleigh step model a similar threshold is used because for a too small height h the problem becomes numerically to difficult. The condition here looks as follows: if

h h

≥ hthresh < hthresh

Rayleigh step model Coulomb friction (dry contact)

To ensure the transition, there must be a connection between the time dependence of the friction force and the Stribeck function as represented in Figure 2.24. The transition between the friction forces from both must be done based on the friction force that occurs at dry slip.

Figure 2.24: Left: Time dependent friction force in function of the normal load W and linear tangential spring characteristic; Right: Stribeck curve in function of the normal load W Three zones can be distinguished in Figure 2.25, marked with three black bold lines. The first is the dry stick zone which is not function of speed but of stick time and displacement. The second is the dry slip zone where a drop occurs in the friction force that ensures the transition in the friction force. This fall is a result of the time dependent friction coefficient. The peak value here represents the break-away force. The third zone is the wet slip zone represented by the Stribeck curve. It follows that the time dependence of the friction coefficient and the Stribeck function cannot be chosen independently from each other. This coupling is represented in Figure 2.24. Some results for low speed which corresponds with boundary lubrication (or dry friction), where the speed is not large enough for build up of an oil

65


Fs

Stribeck curve

Λ

Figure 2.25: Left: Inverse rescaled Stribeck curve, Fs normalised; Right: Stribeck curve

Ff

x

Ff

Ff

film and therefore the Rayleigh step stays in contact, can be observed in Figure 2.26 on the left. These results are obtained by applying a reciprocating motion to the main body. On the right of this figure similar results are shown for higher speeds which reflect the hydrodynamic effect. This figure includes only the friction force in function of speed. It can be compared with Figure 2.19

v

v

Figure 2.26: Hysteresis for low speed in function of position and speed and for high speed only in function of speed

2.4.3

Conclusions

The behaviour of the Rayleigh step module is one which represents lubricated friction in a good way. The Rayleigh step model as a stand alone model is not able to represent BL friction due to its asymptotic behaviour. Use of it as an extension of the generic friction model gives us a model that is able to

66

2.5 Conclusions

represent dry friction and all the lubrication regimes, going from boundary lubricated friction to hydrodynamic lubricated friction.

2.5

Conclusions

This chapter presented the re-implementation of an analytical model at asperity scale which can reproduce most of the dry friction force dynamics. The model is implemented in a force based way and it is used as a basis for the extension to a friction model including lubricated friction conditions. These lubricated friction conditions are modelled as a Rayleigh step module, as it can be used as a representation of a convex asperity. It has been derived and simulated, taking into account the effect of parameter variations. It has been shown that it can be used as a general model to simulate the dynamics of normal and tangential forces in lubricated contacts considering cavitation. The Rayleigh step model as a stand alone model is not able to represent BL friction due to its asymptotic behaviour. The application of it as an extension of the generic friction model gives us a model that is able to represent dry friction and all the lubrication regimes, going from boundary to hydrodynamic lubricated friction. As a combined model it is used to evaluate the influence of the lubricant and it is used deriving a control model, including the shape of the Stribeck curve, the relation between friction force or friction coefficient and the normal load and the viscous behaviour. The results, together with experimental results, are used to develop a new friction control model based on the GMS model, see Chapter 5.

67

68

Part III

Experimental aspects

69

Chapter 3

Tribometer for macroscopic measurements “It is better to create than to learn! Creating is the essence of life.” Julius Caesar

3.1

Tribometer

The characterisation of friction depends on the accuracy of the experimental measurements, and thus the used test setup. Design, qualification and quantification of the used setup are important in the interpretation of the measurement results, which are used to derive the friction characteristics. Owing to its relevance in machines, friction has been measured a numerous amount of times, for many centuries, and it still did not divulge all its secrets. Some measurements are performed on the used machines subject of investigation, and some on specially developed devices, viz. tribometers. The 1989 edition of the Oxford English Dictionary defines a tribometer as an instrument for estimating sliding friction. Thus, the definition of a tribometer is applicable to a broad range of measurement setups. Leonardo da Vinci (1452–1519) was the first to propose an experimental setup for measuring friction, see Figure 3.1. He measured friction by determining the mass needed for which the block would start moving. At the moment the block starts moving, the friction coefficient is the ratio of the mass of the block and the mass hanging at the

71

3 Tribometer for macroscopic measurements

cord. The tangential force is applied by gravitation and the use of some kind of pulley. The principle of estimating friction at macroscopic level did not change much during the following half millennium; only the precision of measurement has evolved a great deal.

Figure 3.1: Leonardo da Vinci’s sketch of his apparatus for friction experiments (From [70]). Friction measurements on machines and on some tribometers may be very specific, valid only for those configurations on which they are produced, and may have no general convenience and no general practical use. Frictional behaviour is generally a property of the system and not only of the friction materials in contact [71]. Therefore no single test can reproduce all types of frictional situations. Nevertheless, all friction testing devices, that is tribometers, have several features in common, related to their functional requirements: they provide a way to (i) support two bodies for which friction data is desired, (ii) to apply an arbitrary relative displacement or velocity signal in a controllable way to the contacting bodies, (iii) to apply and measure a load normal to the contact, and (iv) to measure the tangential friction force. The main subdivision of tribometers is based on the type of displacement which can be applied. The two basic movements, which are used, are: (i) a reciprocating linear movement (e.g. in a sledge configuration) and (ii) a rotational movement (e.g. in a pin-on-disk configuration) [40]. The tribometer described in this chapter is of the second type. The main advantages of this type are that it allows high relative velocities and that it can be used when constant sliding velocities are desired. Compared to a linear tribometer, it has an infinite periodic stroke. More specifically, the discussed tribometer is from the disk-on-disk type. It is intended to represent a single contact surface of the type present in a wet clutch of an automatic transmission. Thus it can be categorised as a test rig which can be used for both dry and lubricated friction measurement. As is usual in experimental friction measurements, only two quantities need to be measured accurately, at constant normal load. These are the friction force and the relative, tangential displacement or velocity between the two objects [40]. As simple as this may seem, it is not always straightforward to measure these quantities dynamically. In many tribological experiments,

72

3.2 Background

the friction measurement is complicated because of the vibrational characteristics of the transducer and the structural mechanical system [71]. The static calibration factor of the transducer no longer provides the relation between the strain and friction force when the dynamics of the mechanical force transducer is excited. The ambiguity is that friction and system dynamics cannot be decoupled because the act of measuring friction involves the use of a sensing element with finite compliance. Moreover, inertia effects of the transducer and of the structural mechanical system must be taken into account. The aim of this chapter is to dynamically characterise a newly developed rotational tribometer for macroscopic friction measurement under dry and/or lubricated friction conditions. The experimental results can be used to validate physically motivated friction models or to find or validate empirically motivated friction models, such as those used in control applications. In the case of lubricated conditions the control can be applied in automatic power shifting transmissions. The structure of the rest of this chapter is as follows. Section 3.2 to 3.7 contain the objective, the system requirements, the conceptual design and the selection of the test setup. Section 3.8 discusses the design and the development of the tribometer. Section 3.9 and 3.10 contain the measurement equipment and signal processing and the dynamic evaluation of the tribometer. It discusses the dynamic characterisation, putting focus on the measurement possibilities concerning range, frequency, accuracy, . . .

3.2

Background

There are SAE#2 and other test setups in which the overall behaviour of wetplate clutches and friction behaviour in general can be measured or evaluated [4, 5, 33, 42–47]. Some try to imitate the behaviour of a typical wet clutch of an automatic transmission. The typical range of the input parameters of an automatic transmission are as follows: sliding speed: 0 to 50 m/s, oil pressure between the friction disks: 0 to 2 MPa, drive torque: 0 to 40 Nm, inertia torque per surface 0 - 0.01 kg/m2 . The properties and possibilities of most setups are such that an automatic transmission is approximated as much as possible. It is however not possible to determine the friction behaviour with fine enough detail. Thus there is a need for a dedicated test setup to accurately measure the physical mechanism that lies behind the friction behaviour. The results of these measurements can be compared to a model based on the physical phenomena that occur in a wet-plate clutch.

73


3.3

Objective

The main objective is to simulate a wet-plate clutch with two plates that can rotate and axially move relative to each other and study its hydrodynamic characteristics and friction behaviour, which means developing a test setup with the following features/possibilities: To be able to apply in a controlled way: • Motor torque • Motor speed • Inertial torque • Normal load (static and dynamic) To enable the measurement of: • Normal load • Transmitted torque • Relative speed • Temperature • Gap between plates (lift-up)

3.4

System requirements

The different system requirements can be listed as follows: • Size of test samples (friction disks from Dana, Spicer Off-Highway) minimum inner diameter Di = 110 mm, maximum outer diameter Du = 190 mm, full rings or three sliding patches. • Forces and torques: maximum normal force Fn = 1290 to 2000 N, maximum torque T = 15 to 27.1 Nm. Note that the maximum torque and the maximum normal force are coupled. A friction coefficient of µ = 1 is used to determine this coupling. The actual friction coefficient is smaller such that a higher normal load can be applied to achieve the corresponding maximum torque. • Speed: 144 rpm for boundary lubricated friction and mixed friction, higher speed for hydrodynamic friction. Note that the friction condition is function of the normal load, which means that the friction condition is not only determined by the speed but also by the contact pressure.

74

3.5 Schematic/conceptual configuration and functional design

• Phenomena to be measured: measure and determine the relationships between e.g. relative speed and relative displacement at a constant normal force and relative velocity and normal force at a constant relative distance, etc., measurement of stick-slip and torsional vibrations. • Measurement resolution and dynamic range: Normal force: 1/3675 of the full range, Torque: 1/1000 of the full range, dynamic range: 100 Hz. Thus, all eigenfrequencies in the test setup must exceed this value otherwise the results are not representative anymore.

3.5

Schematic/conceptual configuration and functional design

For the configuration and functional design two main questions should be asked: (i) What are the possibilities? How can these possibilities be represented by sketches or block diagrams, and which are the various configurations? (ii) How would the previous requirements be met? This determines the functional outline. Different configurations with the different required functions are represented in the block diagrams of Figure 3.2.

Actuator Axial Force

Actuator Axial Force

Coupling

Coupling

Actuator Radial Torque Speed

Coupling Friction part

Actuator Radial Torque Speed

Measurement unit

Coupling

Coupling

Friction part

Measurement unit Coupling Friction part

Measurement unit Block diagram 1

Actuator Radial Torque Speed Block diagram 2

Actuator Axial Force Block diagram 3

Figure 3.2: Block diagrams representing different configurations

75


The various components of these diagrams should fulfill their function in a certain way and in what follows various alternatives are listed. These can later be developed in a physical design. • Actuator, axial force: static load based on a mass suspended on a lever or applied by a pneumatic actuator, using a shaker for the dynamic load (Philips 0.5 A –> 17.85 N) attached to the environment (ground), possibly with existing shaker holder. • Actuator, radial, torque, speed: electric motor: Dynaserv Direct Drive or Torque Motor (ETEL), attached to the environment with a degree of freedom for installation (alignment) and for removing the test pieces. • Friction part: test pieces mounted on friction disks that fit together and which contain the automatic transmission fluid (ATF), static part connected to the measurement unit. • Measurement unit: Measurement table with three ring dynamometers, with 8 strain gauges each, which can measure the normal and tangential load independently, the whole mounted as 6 or 2 full bridges (depending on the behaviour or non equal sensitivity of the dynamometers or poor alignment of the test setup) attached to the environment with a degree of freedom for alignment. An alternative to the three ring dynamometers is the use of two S-beam force sensors, see Figure 3.3, for the measurement of the transferred torque and a force cell for the measurement of the normal load. This alternative is not chosen because of the difficulty of alignement.

Friction part

Force cell

Figure 3.3: S-beam force sensor and force cell • Coupling to axial actuator, must ensure the transition from axial to radial motion by means of a bearing guiding the axial force, coupling with spherical ball bearing or spherical air bearing to avoid misalignment. • Coupling to radial actuator, must ensure the transfer of torque and allow a certain misalignment or with a degree of freedom in the coupling itself (bellow coupling with ball joint) or by a degree of freedom in the engine (Torque motor without bearings).

76

3.6 Physical design

3.6

Physical design

The physical design consists of the discussed concepts being implemented with real components (CAD). On the basis of a CAD program (Solid Edge), the previous conceptual configurations and their block diagrams and the proposed components lead to the physical design. These result in four different concepts, see Figure 3.6. Concept 1 and 2 are based on block diagram 1, concept 3 and 4 are based on block diagram 2. From the above concepts, it is clear that there is no concept design based on block diagram 3. This is because locating the axial actuator at the bottom of the test setup induces many problems, such as the difficulty to measure the normal load dynamically without influence of the inertial effect of the measurement unit.

3.7

Comparison between different concepts

To compose a QFD (Quality Function Deployment) there is a need for representative parameters that can be used to compare the different concepts. The various parameters that can be used are the cost, ease to assemble and disassemble or ergonomics, the total weight, the volume or the compactness and the dynamic behaviour. The volume consist the own volume which is the sum of the volume of all components, and the surrounding volume. The price of a test setup should not be very high because it is not a commercial product and the only considerations are the measurements that may lead to the verification of physical models. The ability to mount and dismount the setup is also important, especially regarding the replacement of the friction components. Replacing the friction components should not be too complex, so that it is not too time consuming and that it can be achieved with a few manipulations. The weight should be such that the setup still remains manageable and possible to move. The volume of the test setup should not be too large such that it can be mounted on a concrete block in the lab. The dynamic behaviour is probably the most important parameter. Each dynamic disturbance from the setup itself must be avoided as much as possible. The eigenfrequencies of the system (structure) should be as large as possible and certainly exceed the intended dynamic (measurement) range of the friction behaviour (this does not mean exceeding the bandwidth of the sensor).The comparison between different concepts is represented in Table 3.1 and the choice is made to continue with concept 1, certainly with respect to the replacement of the friction disks.

77


Axial actuator Axial actuator

Leaf spring

Radial actuator


Radial actuator

Spherical air bearing


(a) Concept 1

(b) Concept 2

Axial actuator Axial actuator

Leaf spring

Leaf spring


Spherical air bearing

Friction measurement Spherical air bearing

Radial actuator

Radial actuator

(c) Concept 3

(d) Concept 4

Figure 3.4: Concepts of the rotational test setup

78

3.7 Comparison between different concepts

Table 3.1: Table of QFD Mount and Weight Volume dismount [kg] [m3 ]

QFD

Cost [A C]

Concept 1

±5500

Remove the shaker and load Lift motor + upper friction component

±175

Own 0.0278 Surr. 0.0751

Eigenmodes frame (stiffness frame) Oscillations spherical ball bearing Oscillations shaft Misalignment of coupling

Concept 2

±5500 + price of ETEL Torque Motor

Remove the shaker and load Remove stator + leaf springs Lift spherical bearing + upper friction component

±156

Own 0.0283 Surr. 0.1218

Eigenmodes frame (stiffness frame) Torque Motor without bearing supported by spherical air bearing

Concept 3

±5500

Remove the shaker and load Remove stator + leaf springs Lift upper friction component and shaft

±184

Own 0.03354 Surr. 0.1431

Eigenmodes frame (stiffness frame) Oscillations spherical air bearing Oscillations shaft Misalignment of coupling

Concept 4

±5500 + price of ETEL Torque Motor

Remove the shaker and load Remove stator + leaf springs Lift upper friction component and rotor

± 165

Own 0.03547 Surr. 0.1310

Eigenmodes frame (stiffness frame) Torque Motor without bearing supported by spherical air bearing and spherical roller bearing

Dynamic haviour

be-

79


3.8

Design and discussion of the developed tribometer

The design of a setup must take the functional requirements and specifications into account. These are specific for each type of machine or system and, consequently, also for each individual part. The main purpose of this tribometer is to measure the friction force (dry and/or lubricated) occurring between two contacting disks from an automatic transmission with a paper based friction material. Other material and contact combinations are generally also possible. This chapter focuses on the measurement of the friction force as a function of the relative tangential motion and normal load for a given specific combination of materials, although this does not necessarily exclude the possibility to use other materials and configurations. Figure 3.5 shows an overview picture of the newly developed rotational tribometer.

2 4 Disk (1)

6 8 10

Disk (2) w

12

t d

14

1 3 5 7 9 11 13 15

Figure 3.5: Right: General overview of the rotational tribometer; Left: Cross-section to illustrate the friction disks and ring dynamometers In this picture the following parts can be discerned: (1) shaker (dynamic load), (2) shaker holder, (3) frame, (4) vertical guide ways, (5) cantilever, (6) static load with cantilever, (7) axial bearing, (8) Direct Drive actuator, (9) ball spline bearing, (10) shaft, (11) bellow coupling, (12) tub containing the two friction disks, (13) capacitive displacement sensors (3 off), (14) three ring dynamometers for force and torque measurement, (15) base plate. The normal force is applied with the shaker and the torque is applied with the Direct

80

3.9 Measurement equipment and signal processing

Drive motor, both through the shaft, which is connected to the upper disk via the bellow coupling. The lower disk is connected to the ring dynamometers. An added functional requirement to the newly developed rotational tribometer is the ability to measure the normal displacement associated with friction, namely the lift-up effect. This is a relevant effect, certainly when it concerns the normal dynamic effect on the friction behaviour [34, 72]. This is a feature, which is absent on most previously developed tribometers, and so they do not allow the measurement of this lift-up effect.

3.9

Measurement equipment and signal processing

To characterise the friction behaviour between disk 1 and disk 2, see Figure 3.5, the friction torque, normal force, relative displacement and velocity should be measured dynamically as accurately as possible. This is carried out as follows.

3.9.1

Friction torque and normal force measurement

For the torque and the normal force, a 6 DOF measuring table is designed based on ring dynamometers comprising strain gauges. Three identical ring dynamometers were designed, following the design rules of the state of the art [73], and manufactured by wire EDM. The main dimensions are: a width w of 25.5 mm, a thickness t of 8 mm and an inner diameter d of 72 mm, see Figure 3.5. The specifications for one ring dynamometer are as follows. For the normal DOF, the maximum allowable force is about 431 N and the resolution is 1/3675 of the full scale. For the tangential DOF, the corresponding values are 58.8 N (equivalent to 5 Nm) and 1/1000 of the full scale respectively. In the design, a safety factor of 2 for the maximal allowable stress is used. This is done to make sure the dynamometers do not deform plastically. Each ring dynamometer is equipped with two full bridges of strain gauges such that each sensor can measure the normal and the tangential force independently. As a bridge amplifier a VISHAY BA 660 strain gauge amplifier with auto-zero is used. The BA 660 is a miniature amplifier, designed to provide signal conditioning for full bridge strain gauge sensors.

3.9.2

Rotational actuation

For the rotational actuation, a Direct Drive Servo Actuator from Dynaserv YOKOGAWA Precision is used with a maximum torque of 15 Nm and a maximum speed of 2.4 rps The motor has a built-in encoder which is used for the position and velocity measurement. An encoder resolution of 163840

81


pulses/revolution is used. This actuator is driven by a proportional gain (P) position or velocity controller.

3.9.3

Normal actuation

For the normal actuation, a Philips PR 9270 shaker is used, which can generate a force of 35.7 × Ief f [N], with Ief f the effective current in amperes, with a nominal amplitude of 0.5 amperes and a peak value of 1 amperes. The shaker has a full stroke of 8 mm. This actuator is driven in open loop and has a bandwidth of 0 – 10000 Hz.

3.9.4

Data acquisition

For the data acquisition, a CLP1103 dSPACE system is used. The friction torque measurement is connected to the first three ADC channels which allow a resolution of 16 bit. The other channels have a resolution of 12 bit. The other ADC inputs contain three channels for the normal force, three channels for the normal displacement coming from the LION Precision capacitive sensors, one encoder connection and six digital I/O to reset the torque and the normal force (auto-zero). Two DAC channels are used, one for the motor and one for the shaker.

3.10

Dynamic evaluation of the tribometer

The tribometer system dynamics and its interaction with the friction behaviour play a crucial role in determining the friction force measurement ranges, in regard to amplitude and frequency. Thus, a dynamic identification must be performed in the tangential and in normal directions to obtain the allowable measurement bandwidth. The structural dynamic behaviour of the setup has been optimized during the design phase, the structural resonances being placed at as highest as possible frequencies. The use of a weak P control in a position loop was necessary to get rid of a constant drift due to the velocity control of the motor drive. The results of the identification, the dynamic as well as the frictional, are described further below by several frequency response functions (FRF’s). In the tangential excitation case, the torque measured by the dynamometers is considered as being the input and the rotation angle of the motor encoder is considered as being the output. In the normal excitation case, the applied signal to the shaker is considered as being the input, and the normal position of the dynamometric table is considered as being the output. Figure 3.6 shows a linear schematic representation of the tribometer to illustrate the considered inputs and outputs used for the dynamic identification. The following figures show the measured FRF’s using a multisine excitation with different amplitudes, random phase, and a bandwidth of 500 Hz. For each measurement, 4 different multisines, each with

82

3.10 Dynamic evaluation of the tribometer

different phase, were used and, for each of them 10 periods were taken. By taking the average of the time signal of the 10 measurements, random noise is filtered out. In Figure 3.7, 3.8 and 3.9, the tangential dynamic response is shown given the measured transmitted torque as input and the angular position of the motor coming from the encoder as output.

θ output

θ

F input

3

x3, m3

ka kb m

kf

M

ka kb

θ 2, I2

x2, m2

m

θ 1 , I1

x1, m1

M

x output

kd

T input

kd (a)

kf

(b)

Figure 3.6: (a): considered input torque T and output rotation angle θ for tangential identification, (b): considered input force F and output deflection x for normal identification, (ka , kb , kf and kd are the stiffness of the shaft, bellow coupling, friction disk and dynamometers respectively), kf is represented as a variable spring/damper function of the amplitude and frequency of the deflection and of the normal load In Figure 3.6(a) the torsional stiffness is considered and in Figure 3.6(b) the axial stiffness. The two stiffnesses in series, in case (a) and (b) respectively, namely ka and kb , can be represented by their combined stiffness. This stiffness is equal to kef f =

1 1 + ka kb

−1 ,

representing the torsional and axial stiffness for Figure 3.6(a) and 3.6(b) respectively. kf represents the non-linear stiffness and damping of the friction material. It is function of the amplitude of the rotation or deflection and the amplitude of the normal force. In the following equations both components are kkf and ckf . The system equations can be written as follows:

83


• In tangential direction: cd + ckf I1 0 θ¨1 · ¨ + 0 I2 −ckf θ2 +

kd + kkf −kkf

−kkf kkf + kef f

−ckf ckf + cef f

θ˙1 · ˙ θ2

θ1 0 · = θ2 kef f θ3

with T = −kd θ1 so, T θ3

=

−kd θ1 θ3

=

(ckf s+kkf )kef f kd 2 (ckf s+kkf ) −(I1 s2 +ckf s+(kd +kkf ))(I2 s2 +ckf s+(kef f +kd ))

• In normal direction: 

m1  0 0

0 m2 0 

   0 cd + ckf x ¨1 0 · x ¨2 + −ckf 0 m3 x ¨3

kd + kkf +  −kkf 0

−kkf kkf + kef f −kef f

−ckf ckf + cef f −cef f

  0 x˙ 1 −cef f · x˙ 2  cef f x˙ 3

     0 x1 0 −kef f  ·  x2  =  0  kef f x3 F

so, X1 (ckf s+kkf )kef f = 2 F (m3 s2 +kef f ) (m2 s2 +ckf s+(kkf +kef f ))(m1 s2 +ckf s+(kef f +kd ))−(ckf s+kkf )

(

(

−k2 m1 s2 +ckf s+ kef f +kd ef f

))

The damping which is not taken into account in Figure 3.6 and which is not taken into account in the transfer functions is marked in grey. This tangential identification has been performed for three different amplitudes of the multisine excitation, namely for A = 0.015 V, A = 0.03 V and A = 0.06 V, and for different normal preloads.

84

Amplitude [rad/Nm (in dB)]


-40 -50 -60 -70 own weight 280 N 560 N 840 N 1120 N 1400 N

-80 -90

-100 -110 0 10

1

2

10

10

3

10

600

Phase [°]

400 200 0 -200 0

1

10

2

10

10

3

10

Freq [Hz]


Figure 3.7: Tangential FRF for A = 0.015 V -30 -40 -50 -60 -70

own weight 280 N 560 N 840 N 1120 N 1400 N

-80 -90

-100 -110 0 10

1

2

10

10

3

10

Phase [°]

600 400 200 0 -200 0

10

1

2

10

10

3

10

Freq [Hz]

Figure 3.8: Tangential FRF for A = 0.03 V As can be seen in the FRF’s the tangential dynamic behaviour is relatively linear when high normal loads are applied. When the normal load is not high enough, that is, at a normal load which consists only of the weight of the driving parts (6, 7, 10, 11 and the upper disk, which amount to about 58 N), a non-linear effect can be observed. The slope of the amplitude at low frequencies goes from 0 dB/decade towards -40 dB/decade for increasing amplitudes

85



0 -20 -40 -60

own weight 280 N 560 N 840 N 1120 N 1400 N

-80

-100 -120 0 10

1

2

10

10

3

10

Phase [°]

600 400 200 0 -200 0

10

1

2

10

10

3

10

Freq [Hz]

Figure 3.9: Tangential FRF for A = 0.06 V

of excitation, corresponding to a mass-line. This behaviour, which arises from the hysteresis characteristics of the frictional contact, is treated exhaustively in [74]. In particular, the irregularities observed in the FRF pertaining to the lowest preload arise from the sensitivity of the response to small amplitude variations; a jump-like phenomenon [74]. The hysteresis nature of the contact will become apparent in the section on friction identification further below. In the three previous plots a dashed vertical line indicates the frequency at which the first tangential resonance frequency lies; i.e. somewhere around 170 Hz. The normal dynamic response has also been identified, in a similar way to the tangential response. In Figure 3.10, the normal dynamic response is shown, where the applied signal to the shaker is the input and the normal force, measured in the dynamometers by strain gauges, is the output. This output force is proportional to the normal displacement in the dynamometers. Also in this plot a dashed vertical line indicates the frequency at which the first normal resonance frequency, pertaining to lowest preload, is lying, namely at about 186 Hz. One can observe a shift in the first eigenfrequency to the right and a change in global dynamic response with rising preload, seen as a merge of the consecutive poles and zeros. This behaviour is due to the increase in the stiffness of the contact, it being essentially a Hertzian contact, with increased loading. At low preloads, one can also observe multiple eigenfrequencies, whereas with high preload there are only two, one at the shifted first eigenfrequency and one at about 288 Hz, due to the merge.

86


Amplitude [μm/N (in dB)]

60 50 40 30 20 10 0

-10 0 10

own weight 100 N 200 N 400 N 800 N 1200 N 1

2

10

10

3

10

0

Phase [°]

-100 -200 -300 -400 -500 -600 -700

0

10

1

10

2

Freq [Hz]

10

3

10

Figure 3.10: Normal FRF (deflection of dynamometers/shaker force) In order to ascertain the non-linear, friction hysteresis effect on the FRF, the case of lowest preload is further examined, presently with increasing amplitude of excitation. Figure 3.11 shows the FRF for this identification. The amplitude of the FRF increases with amplitude of excitation, to approach a virtual mass-line. This identification was also performed with a stepped sine, as in [74], and resulted in a similar response, see Figure 3.12. Compared to the four preceding FRF’s, this one is much noisier. This is due to the lower signal to noise ratio, which is in turn dictated by the relatively low stiffness in the contact. Another reason for this is the jumplike phenomenon of the non-linear effect as mentioned before. This FRF was performed in a non-preloaded situation, with the only normal load the own weight of the friction plate and the driving ball spline and shaft. As mentioned earlier the slope of the amplitude at low frequencies goes from 0 dB/decade to -40 dB/decade for increasing amplitudes of excitation. The bounds, within which this measured FRF will lie, are the mass-line (for high excitation) and the mass-spring curve (for low excitation). The occurrence of an anti resonance before the resonance in the tangential FRF’s should also be mentioned. This result can be due to a particular component in the setup which behaves as an energy sink at a frequency lower then the first eigenfrequency. Modal analysis bases on hammer excitation revealed some extra modes

87



0 -20

A = 0.01 V A = 0.02 V A = 0.03 V A = 0.04 V A = 0.05 V A = 0.06 V

A

1

-40 -60

-80

-100 -120 0 10

1

2

10

3

10

10

Phase [°]

600 400 200

XXX XXX X z X

0 -200 0

A 1

10

2

10

3

10

10

Freq [Hz]


Figure 3.11: Non-linear tangential FRF (effect of friction)

0 −20 −40

A

−60 −80 −100 −120 0 10

1

10

2

10

2

10

10

3

200

Phase [º]

150 100 50 0 −50 −100 0 10

A 1

10

10

3

Freq [Hz]

Figure 3.12: Non-linear tangential FRF (effect of friction) with stepped sine

within this bandwidth. This hammer excitation was performed on the measuring table composed of the lower friction disk and the three ring dynamometers. Most of these identified modes are tilting modes and the main normal and tangential mode where also clearly visible. The torsional mode of the measuring table has a frequency of about 222 Hz as compared to 170 Hz

88

3.11 Conclusions

when the friction surfaces are in contact and preloaded. A dynamic identification is performed in the tangential and in normal directions to obtain the allowable measurement bandwidth. Based on the previous results and FRF’s this can be determined up to a bandwidth of 100 Hz. This bandwidth is used to filter the experimental results in time domain.

3.11

Conclusions

This chapter described the new developed rotational tribometer which can be used for macroscopic friction measurement and identification. The functional requirements of the tribometer are: (i) accurate displacement and friction force measurement, (ii) normal loading and measurement possibility, both statically and dynamically, and (iii) the possibility of applying arbitrary displacement signals over a large range of magnitude and frequency. These functionalities are decoupled as much as possible based on the principles of precision engineering. A dynamic identification is performed to obtain the measurement bandwidth, which is determined to be 100 Hz. The structural and friction dynamics are uncoupled as much as possible such that the structural behaviour does not contaminate the friction identification. With this rotational tribometer experiments are performed for friction identification, which are used for control purposes. The tribometer allows experiments in a large range of displacements and velocities, thus allowing various friction characteristics, such as break-away force, pre-sliding hysteresis, friction lag in the sliding regime, stick-slip and limit-cycle oscillations, the Stribeck and the lift-up behaviour to be measured for one and the same configuration, and under dry or lubricated friction conditions. Such experimental results are used to validate physically motivated friction models, and to establish or validate empirically motivated friction models, such as used in control applications, see Chapter 4.

89

90

Chapter 4

Experimental results in the time domain “It is the weight, not numbers (sic) of experiments that is to be regarded.” Isaac Newton

This Chapter describes the experimental results in the time domain which consist of the break-away force, hysteresis function of the position in the pre-sliding regime, friction lag in the sliding regime, stick slip or dynamic oscillations, the Stribeck and the lift-up behaviour.

4.1

Friction characteristics

The tribometer allows experiments in a large range of displacements/velocities, thus allowing numerous friction characteristics to be observed on one and the same configuration and under dry and/or lubricated friction conditions. Typical friction characteristics are: break-away (or the static friction) force, hysteresis as a function of the position in the pre-sliding regime, friction lag in the sliding regime, stick-slip or dynamic oscillations, and Stribeck behaviour as described in literature [24, 29]. These experimental results can be used to validate physically motivated friction models or to formulate or validate empirically motivated friction models, which are often used for control purposes, e.g. to compensate for friction in the form of stick-slip, dynamic oscillations or other self induced vibrations such as shudder [6].

91

4 Experimental results in the time domain

The friction force as a function of the relative displacement and normal load is given for a specific combination of materials under investigation, namely a paper based clutch disk on a steel counter disk. Here the dynamic friction force for different combinations of materials is not relevant because here paper-based friction material on a steel separator disk is investigated. This does not necessarily exclude the possibility of using different materials and configurations. The results discussed in the next sections are all obtained after running-in the contact, to ensure as uniform subsequent behaviour as possible. It should be mentioned here that since a rotational tribometer is used, position dependent friction, such as periodic behaviour due to the profile of the disks, may easily occur in the measurements. For the experiments in lubricated conditions the TEXTRAN THD PREMIUM transmission oil is used. It is a combination of carefully refined basic oils to which special combinations of additives are added. They provide a high viscosity index, good resistance against corrosion, oxidation and foam generation as well as anti wear and high pressure properties. The kinematic viscosity for this oil is 56 and 9.3 [mm2 /s (cSt)] at 40◦ C and 100◦ C respectively. The control of frictional forces has been traditionally approached by chemical means, usually by supplementing base lubricants with friction modifier additives as can be seen in the following experimental results.

4.1.1

Break-away force

The force necessary to initiate total slip, or gross sliding, can be determined. The maximum force which occurs in this initiation is called the break-away force [24, 25]. In the case of a rotational tribometer the motion is circular and then we talk about the break-away torque. The experiment to measure the break-away torque is performed for different accelerations, from stick, under a normal load of 280 N. These different accelerations are applied to the system, and these result in different increasing torques due to stick in the friction contact. The break-away torque is here defined as the maximum friction torque during the transition from pre-sliding (stick) to gross sliding regime. Similar results where perceived as an output from the generic model, see Section 2.2.4. This is due to the stick time dependent friction coefficient, owing to normal creep. One may use this definition of break-away force, as being equivalent to the static friction. But as can be seen in Figure 4.1, due to the dependence on the application rate, several values for the static friction can be found for one and the same normal load. For the lowest acceleration the highest maximal torque is attained and this maximum decreases with increasing acceleration [24, 26]. The maxima can be compared with the friction torque at zero velocity with the same applied load, as can be seen in Figure 4.6 on the left.

92

4.1 Friction characteristics

Figure 4.1: Break-away torque for different applied accelerations with a normal load of 280 N

4.1.2

Pre-sliding regime

The pre-sliding regime is defined as the region in which the adhesive forces at asperity contacts are dominant such that the friction force appears to be a function of the displacement [12, 21, 22]. To investigate the friction behaviour in the pre-sliding regime, a desired position signal is applied to the system using a proportional controller. In dry conditions this desired trajectory consists of a saw tooth position in function of time, with maximum amplitude of about 4 × 10−4 rad, having two reversal points with smaller amplitude to investigate the non-local memory effect, see Figure 4.2. The amplitude of the signal is chosen such that the actual displacement remains within the pre-sliding regime in the case of dry friction. Applied position signal

-4

5

x 10

4 3

θ [rad]

2 1 0 -1 -2 -3 -4 -5 0

0.5

1

1.5

2

2.5

3

3.5

4

time [s]

Figure 4.2: Applied position signal used to investigate the pre-sliding behaviour and the non-local memory effect The resultant friction force, from this pre-sliding experiment, in function

93


Figure 4.3: Hysteresis in pre-sliding regime with non-local memory effect; Left: Dry friction, Right: Lubricated friction of displacement, can be seen in Figure 4.3. A hysteresis behaviour with nonlocal memory is defined as an input-output relationship for which the output at any time instant not only depends on the output at some time instant in the past and the input since then, but also on the past extreme values of the input or the output [23]. This can be perceived in the two inner loops in the hysteresis in Figure 4.3, see also [74] for details of the hysteresis function construction and modelling.

4.1.3

Friction lag in sliding regime

To investigate the behaviour in the transition from pre-sliding to sliding and vice-versa, a sinusoidal position command trajectory is applied to the system in dry condition, with an amplitude of 0.06 rad for three different frequencies. The measured velocity is not a pure sinusoidal signal due to the effect of friction, as can be seen at velocity reversal in Figure 4.4. This demonstrates the paradox of a tribometer: the controller which tries to impose the desired displacement signal is limited due to non-linear disturbances. The main disturbance signal comes from the friction interaction. We can see a higher friction force during acceleration compared to the friction force during deceleration. This phenomenon is called friction lag or hysteresis in the sliding regime. The Stribeck curve (see section 4.1.4), should intersect the loop formed by the friction force belonging to increasing and decreasing velocity during sliding. The measurement of this Stribeck curve is not trivial due to the stick-slip property of friction (see section 4.1.5). [24] and [32] also measured a friction lag behaviour for oscillatory rubbing contacts. For the lubricated condition similar measurements are performed. The results can be seen in Figure 4.5 on the left. In here the friction lag phenomenon

94


Hysteresis

0.5 0

Zoom in of friction lag

-0.5 -1 -0.08

0.95 -0.06

-0.04

-0.02

1

Tf [Nm]

0

0.02

0.04

0.06

0.08

θ [rad] Friction lag

Tf [Nm]

Tf [Nm]

1

0.9 0.85 0.8 0.75

0.5

0.2

f = 3 Hz f = 2 Hz f = 1 Hz

0 -0.5 -1 -1.5

-1

-0.5

0

0.5

1

0.4

0.6

0.8

1

1.2

w [rad/s]

1.5

w [rad/s]

Figure 4.4: Left: Hysteresis and corresponding friction lag for dry friction: f = 1 Hz, f = 2 Hz and f = 3 Hz, Right: zoom in of friction lag

is not really visible but the velocity strengthening is clearly visible. Therefore another test was performed by applying a sinusoidal velocity input around a certain velocity offset, to have an oscillation only for positive velocities [33]. The resulting friction torque can be seen in Figure 4.5 on the right. As opposed to a higher friction torque for acceleration than for deceleration, in this result the respective forces cross each other. Similar results are obtained for different normal preloads and different frequencies. This phenomenon may be related to the normal dynamic effect of the friction behaviour and must be further investigated because it is an uncommon friction lag result.

0.56

0

0.54

-0.06

-0.04

-0.02

0

0.02

θ [rad] Friction lag

0.5

0.04

0.06

0.08

Tlf [Nm]

Tlf [Nm]

0.58

-0.5 -0.08

Tlf [Nm]

Friction lag

Hysteresis, Lubricated

0.5

0.52 0.5

Decelerating Accelerating

0.48 0.46 0.44

f = 3 Hz f = 2 Hz f = 1 Hz

0

0.42 0.4

-0.5 -1.5

-1

-0.5

0

w [rad/s]

0.5

1

1.5

0.38 0

0.1

0.2

0.3

0.4

w [rad/s]

0.5

0.6

0.7

Figure 4.5: Left: Hysteresis and corresponding friction lag for lubricated friction: f = 1 Hz, f = 2 Hz and f = 3 Hz, Right: Uncommon friction lag result, lowest normal load, input freq. 9 Hz

95


4.1.4

Stribeck behaviour

As soon as the break-away force is exceeded and the object starts to slide, the friction force generally drops to a lower value. It was found by Stribeck that the velocity dependence is continuous [29]. The typical drop (rise) in the Stribeck curve, which represents the friction force in function of velocity, for positive (negative) velocities is called Stribeck effect. Figure 4.6 shows the Stribeck curve for a velocity range up to 0.6 rad/s in dry condition, for three different applied loads. For all the dry experiments the Stribeck effect is clearly visible and the viscous effect can also be seen as the rising part of the Stribeck curve, see Figure 4.6. The Stribeck curve is the combination of the Stribeck effect and the viscous effect. For the lowest applied load the position dependence of the friction force strongly affects the shape of the Stribeck curve and perceived as a fluctuation in the friction force. In case of lubricated conditions, for the specific used ATF, there is no apparent Stribeck effect, the dynamic friction force rises with rising velocity, see Figure 4.6 on the right. The shape of the Stribeck curve has an influence on the anti-shudder performance of wet clutches. A positive torque-velocity relationship or velocity strengthening is advantageous because it acts as positive damping; see also next section about stick-slip.

4 7 3.5 6 560 N 280 N +-58 N

3

Tlf [Nm]

Tf [Nm]

5

4

3

2 1.5

2

1

1

0 0

2.5

0.5

0.1

0.2

0.3

0.4

ω [rad/s]

0.5

0.6

0.7

0 0

1

2

3

ω [rad/s]

4

5

6

Figure 4.6: Stribeck curve for different loads, Left: dry condition, Right: lubricated condition with three same applied normal loads As can be seen in Figure 4.6, there is substantial decrease in the transmitted torque due to the addition of the lubricant, which indicates a lower friction coefficient. Another change is the drastic effect on the dynamic friction force, instead of having a drop, there is a strong rise in friction torque for increasing velocity. Similar results are found by [47] in lubricated conditions. This shows the strengthening effect of the dynamic friction torque. This effect is a result of the additives in the automatic transmission fluid (ATF) [6].

96


4.1.5

Stick-slip motion

To measure stick-slip motion a constant desired velocity is applied to the system. A necessary condition to induce stick-slip is a decreasing friction force for a rising velocity. That is, the dynamic friction force is lower than the static friction force. This phenomenon is known as the Stribeck effect, see section 4.1.4. Stick-slip can be described as follows. The critical velocity is defined as the maximum driving velocity below which stick–slip will occur [7]. When a constant driving velocity, lower than the critical velocity, is applied to the upper disk, the disk will initially stick and the friction force will build up until it reaches a maximum equal to the break-away force. Once this maximum is reached, the disk will start slipping and due to the decreasing friction force it will accelerate. After this acceleration the system will decelerate until it reaches again zero velocity due to the actuating stiffness. At this moment, the cycle starts again. If the moving body never reaches zero velocity, it will never stick but it can still exhibit an oscillatory behaviour around the desired velocity. This motion is referred to as dynamic oscillations or limit cycle hunting, but in literature this is also often called stick-slip motion [33, 42].

Figure 4.7 shows the stick-slip behaviour for two different applied loads and five command velocities (ω1 = 4, ω2 = 5.98, ω3 = 9.44, ω4 = 13.74, ω5 = 20 [×10−3 rad/s]). In these figures the velocity is rising from top to bottom and as can be seen in Figure 4.7 on the left, the velocity apparently reaches a value higher than the critical velocity because the stick-slip phenomenon does not occur anymore. For the same applied velocity at 87 N stick-slip still occurs. Stick−Slip: 75 N

Stick−Slip: 87 N

1.5

2 1.5

1

0

2

4

6

8

1

10

1.5

0

2

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6

8

10

0

2

4

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8

10

0

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8

10

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1.5 0

2

4

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8

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1.5

1

0

2

4

6

8

10

1.5

1

Tf [Nm]

Tf [Nm]

1

ω2

2 1.5 1

ω3 ω ↑

2

ω4

1.5 1

0

2

4

6

8

10

1.5

1 2

ω5

1.5 1

ω1

2

0

2

4

time [s]

6

8

10

1

time [s]

?

Figure 4.7: Stick-Slip with 75 N and 87 N normal force We may conclude, in dry condition, that the higher the normal load, the higher the critical velocity for which stick-slip still occurs. This is supported by the fact that the velocity corresponding to the minimum point in the

97


Stribeck curve shifts up with increasing preload, see Figure 4.6. For this velocity effect see also [75]. Because of the necessary condition to induce stick-slip, explained earlier, stick-slip will not occur in the case of lubricated conditions for this specific used oil. This is because the system is positively damped. On the other hand dynamic oscillations can still occur as limit-cycle behaviour as a result of interaction of the friction, the dynamics of the system due to the profile of the friction disks and the control action.

4.2 4.2.1

Experimental characterisation of lubricated friction Lift-up and Stribeck behaviour considerations

The newly developed rotational tribometer [76] has an added functional requirement, which is the ability to measure the normal displacement due to friction or the lift-up effect. This lift-up effect is a relevant dynamic friction characteristic both for dry and for lubricated friction. This is a feature which is absent in most previously developed tribometers, which do not allow the measurement of this friction effect. This phenomenon manifests itself as a butterfly shaped plot of the normal displacement as function of the periodic tangential displacement. The measurement of the normal degree of freedom, or the lift-up effect, is performed by three capacitive sensors, secured to the lower disk, and positioned at the same radius separated by an angle of 120, see Figure 4.8.

3 clamping holes

Lower friction disk

Metallic ring Upper friction disk

Figure 4.8: Lower friction disk with clamping holes for the capacitive sensors and detailed view The measurement counter surface is part of the upper disk and is in the form of a metallic ring. In this way the relative displacement between the two disks is measured, by a special averaging out of the three sensor readings. The

98

4.2 Experimental characterisation of lubricated friction

friction material has a stiffening behaviour in function of the normal load. The velocity effect on the lift-up is also investigated for a set of constant velocities at different applied loads. The friction force w.r.t. these constant applied velocities gives us the Stribeck behaviour. The described tribometer is used to investigate these effects.

4.2.2

Lift-up behaviour

Figure 4.9 shows the behaviour of the friction material under five applied loads (about 50, 280, 560, 840, 1120 N) and four applied velocity signals with their respective relative displacements. What should be mentioned is that the absolute reference for the relative displacement is unknown; this means these measurements all contain a common displacement offset which is most probably not equal to 0. 52 48 44 -6

-4

-2

0

2

4

6

-8 -10 -12

-6

-4

-2

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2

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6

-10 -12 -14

-6

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-2

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4

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-12 -14 -16

Normal load [N]

Normal rel. displacement [µm]

0 -5

-6

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280 270 570 560

840 830

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0

2

4

6 1110

-16

1100

-18 -6

-4

-2

0

q [rad]

2

4

6

1090

q [rad]

Figure 4.9: Normal relative displacement and normal load for different experiments The input consists of a stepped saw velocity signal of which each steady state velocity lasts 5 seconds, see Figure 4.10. The effect of the velocity on the lift-up can be seen when the full range of the angle θ of the experiment is shown as compared to the previous figure in which only two revolution where shown, see Figure 4.11. A decrease in absolute value of the normal displacement with higher velocities is perceived. This means that the higher the velocity, the higher the lift-up, as if the stiffness of the contact increases with increasing velocity. This can be explained by the pressure build-up in the oil. The bearing capacity increases due to an increasing pressure. As an example the main trend (upper and lower limit) for one experiment at the highest load is marked with two black lines in Figure 4.11. Another clear trend is the periodic displacement due to the shape of the two contacting disks. This effect is much more accentuated than the velocity effect, which has a peak to peak value of about 3.5 µm as compared to a lift-up effect of about 1 µm. When the average normal

99


5

w [rad/s]

6,4

3,2 1,6 0,8

[rad]

Figure 4.10: Different stepped saw velocity signal inputs

Normal rel. displacement [μm]

0 -2 -4 -6 -8 -10 -12 -14 -16 -18 -40

-20

0

20

40

60

80

θ [rad]

Figure 4.11: Normal relative displacement in function of full range of rotation angle θ relative displacement is plot against the average normal load for the different experiments the stiffening effect of the friction contact due to the increasing load becomes apparent, see Figure 4.12. Neglecting the hydrodynamic effect, this could be represented by a Hertzian contact. Figure 4.13 shows the stiffness in function of the rotation angle θ where the stiffening effect is again clearly visible. A zoom in on this figure show the position dependent stiffness as well as the velocity effect on stiffness. This can be sharply discerned for all loads except for 280 N where the velocity effect is less obvious. The velocity not only has a stiffening effect in the normal direction, due to which the lift-up effect occurs, but also has a positive damping effect in the tangential direction. This positive damping effect is resulting from the specific ATF used for these experiments. Based on Figure 4.14 it is clear that the friction coefficient decreases with

100


1200

Normal force [N]

1000

800

600

400

200

0 2

4

6

8

10

12

14

16

18

Normal rel. displacement [μm]

Figure 4.12: Stiffening effect due to rising normal load

Figure 4.13: Left: Velocity stiffening (envelope); Right: Position dependent stiffness

increasing normal load from about 0.25 at 50 N to about 0.11 at 1100 N. This decrease is non-linear; it has some kind of exponential decay with saturation at high load, see eqn. 4.2. The Stribeck curve, also in Figure 4.14, is obtained by plotting the friction coefficient or the friction torque in function of the angular velocity ω.

4.2.3

Stribeck behaviour

An appropriate function for the Stribeck behaviour is proposed and used for parameter identification on the performed experiments. In this function the non-linear effect of the friction torque or friction coefficient is considered, as well as the Stribeck effect [29] and the viscous effect. The used equation is of the following form:

101


Z [rad/s]

Figure 4.14: Left: Exponential decay of the friction coefficient in function of rising normal load; Right: Stribeck curve for three different applied normal loads (positive damping)







   1 −vs    γ S (v, Fsi ) = sign (v) . Fsi . 1 − + |v| .σv   exp | {z }   a1 |v|   | {z } Viscous effect

(4.1)

Stribeck effect

with Fsi = Wn µmax

L 1 exp − 1− a2 Wn

(4.2)

Eqn. 4.1 represents for the Stribeck behaviour which is function of the velocity v and the static friction force Fsi . The choice of this function is discussed in Chapter 5, Section 5.6. The first term is the Stribeck effect and the second term is the viscous effect. a1 , vs , γ and σv determine the minimum attainable value of the friction force, vs is the Stribeck velocity which determines the rate of decay of the Stribeck effect and γ and σ v determine the viscous effect. In eqn. 4.2 µmax is the maximum value of the friction coefficient, a2 determines the minimum attainable value of the friction coefficient, L determines the rate of decay and Wn is the normal load.

4.2.4

Parameter identification

The friction torque has been measured for different velocity ranges and each for five different loads, namely at about 50 N, 280 N, 560 N, 840 N and 1120 N. The averaged steady state velocities with their respective steady state torque are shown in Figure 4.15. A fit based on the Stribeck behaviour model, see eqn. 4.1, is also shown. The fit is based on parameter identification based on a technique discussed in Section 5.8. The Sommerfeld number is not used

102


here to more clearly see the effect of the normal load. An investigation on a better representation of the Stribeck curve, formulated in terms of more accessible parameters, can be found in [77]. The parameter used there for the abscissa is the shear rate, which has the units of an angular velocity ω [rad/s] and for a linear motion this gives a velocity v [m/s or µm/s]. Since the experiments are performed each time at a constant load for the five velocity ranges respectively, they should represent one and the same Stribeck curve. This also means the five fits should lie on top of each other for each load and can be seen as one fit on the average Stribeck curve for each load. This is achieved by joining the datasets for the different velocity ranges and fit them as one; see Figure 4.15 on the right.

Figure 4.15: Left: Stribeck curve for five different applied normal loads; Right: Stribeck behaviour and fit for the joined datasets A similar fit has been performed on the experiments of [78] (see Figure 4.16), such that the results can be compared. As a first comparison the friction coefficient can be plotted and as can be seen it also has an exponential decay in function of a rising normal load, see Figure 4.16 on the left. The friction coefficient is calculated as follows: µ=

Ts W ·r·N

(4.3)

With Ts the static torque, W the normal load, r the average radius of the friction disk and N the amount of contacting pairs in the clutch. For the fit of the Stribeck behaviour, which can be seen in Figure 4.16 on the right, two parameters are kept constant, namely vs = 11.52 [rad/s] and γ= 1/3 [ ]. The largest visible difference is the lower friction torque at high speed for the experimental results from the rotational tribometer especially at high normal load. This could be due to the lower amount of oil interacting between the friction contacts due to the orientation and because of the lack of additional oil supply in the contact. This means the contact could more easily run short on oil. Another way of comparing experimental results from different

103


30

0.1148 0.1146

20

0.1142

Tlf [Nm]

Friction coefficient []

25

0.1144

0.114 0.1138 0.1136

15

10

0.1134 5

0.1132 0.113

500

1000

1500

Normal load [N]

2000

0 0

5

10

15

20

25

ω [rad/s]

Figure 4.16: Left: Exponential decay of the friction coefficient in function of rising normal load; Right: Stribeck curve, experimental data and fit (as in Figure 4.14) test setups is to convert the torque data into the friction coefficient, converting the torque into a force taking the radius of the friction disk into account and dividing it by the normal load, see eqn. 4.3. The experimental results show that at low normal loads and at low velocity the Stribeck behaviour has a positive damping effect. This positive damping or the rising friction force for increasing velocity does not exclude stick-lip behaviour [39]. When the normal load is increased and also the velocity range is increased the Stribeck effect becomes more apparent [4]. This could be explained by the fact that the contact behaves more and more as a dry contact because of the shift from hydrodynamic, to mixed, to boundary lubrication, so in a way from lubricated to dry conditions. This effect can also be explained by the viscosity-pressure relation of oil. The higher the pressure the more the oil behaves as a solid.

4.3

Considerations on the lift-up effect

One characteristic of sliding friction is that the apparent load carrying capacity of the contact increases with the sliding speed. This is true for both dry and lubricated friction. In dry conditions this is so because, owing to decreasing adhesion influence with speed, a single asperity will be more active, so more in contact, per unit displacement, or more “available” for load carrying. This situation may be compared to trying to walk on the surface of a sticky quicksand: if you cannot free your feet in time to take the next step you will just sink down [3]. In lubricated conditions a pressure build up can be observed with rising sliding speed due to the hydrodynamic effect of the fluid. In this configuration at high speed or low load the contact can be seen as a hydrodynamic bearing. The lubrication regimes are commonly divided into: Boundary Lubrication (BL), Mixed Lubrication (ML) and Full Lubrication or Hydrodynamic Lubrication (FL, HL) [30]. In the BL regime the

104

4.3 Considerations on the lift-up effect

lubricant’s hydrodynamic action is negligible and the load is carried directly by the surface asperities, similar to dry friction contact. In de ML regime the load is carried by the lubricant’s hydrodynamic action and/or directly by surface asperities. When the hydrodynamic action of the lubricant fully separates the surfaces and the load is carried totally by the lubricant film the contact enters FL regime. In terms of the friction coefficient the Stribeck effect together with the hydrodynamic or viscous effect can give a good visualization of the different regimes, combined in the Stribeck curve, see Figure 4.17.

μ

Fn v

body 1 lubricant

h h→0 Boundary Lubrication

h≈R Mixed Lubrication

–Significant –Intermittent contact –Micro EHL contact of surfaces –Surface films and additives –An area of intense research –Not perfectly understood

body 2 h >> R surface roughness R Full or Hydrodynamic Lubrication

–Full separation of surfaces –Reynolds Equation –Fully understood

v

Figure 4.17: Stribeck curve and lubrication regimes with some characteristics (adapted from [7]) Consequently, when the normal load is constant, for a given slider, the number of contacting asperities will decrease with increasing sliding speed, i.e., the slider will lift up. This behaviour has already been observed by several researchers [34, 79]. Several explanations have been offered in the past, notably (i) Amontons’ classical, but controversial hypothesis that the asperities on the mating surfaces have to surmount each other (the mean asperity slope representing then the coefficient of friction), and (ii) Tolstoi’s questionable hypothesis that the effect be owing to induced normal vibrations coupled

105


with the non-linear asperity stiffness behaviour. Another explanation of the phenomenon could be given, namely that it is a direct consequence of adhesion, deformation, creep and pressure build up, which in fact resolves the contradiction of the first view without excluding the second. In this paragraph, experimental evidence of the increasing bearing capacity which results in a lift-up effect for lubricated friction is presented. Lubricated, sliding rubbing experiments where performed that show a relative normal displacement associated with the tangential motion. In particular, with periodic tangential input, the normal motion describes regular, hysteresis, butterfly like curves (similar in nature to those found in piezo-electric [80, 81] and magnetic materials [82]). This paragraph explores the basic behaviour of the lift-up effect and those butterfly curves experimentally. The results which are described are obtained from experiments performed to test friction lag without velocity offset, using position control, for six different normal loads, for six different frequencies and with an amplitude of 0.06 rad on a rotational tribometer [83]. The results here are only shown for the highest frequency applied, viz. 9 Hz, because at this frequency the effect was most obvious. The profile of the contacting bodies has to be taken into account, because it has an influence on the shape of the butterfly curves especially on its inclination. For a perfectly flat surface a symmetric butterfly would be expected. The shape of the profile is apparent in the shape of the butterfly curves, as can be perceived in Figure 4.18. A butterfly loop can clearly be observed especially at the lowest load. These experimental results show a lift-up behaviour for which a physical cause was discussed [84]. This lift-up effect was already discussed as a result of the Rayleigh step model in Section 2.3.4. It is a good reason to take the effect of this normal degree of freedom or this lift-up effect into account, also in the development of a control model. This will be more thoroughly discussed in Section E.2.

4.4

Conclusions

Friction and system dynamics cannot be decoupled because the act of measuring friction involves the use of a sensing element with finite compliance. Moreover, inertia effects of the transducer and of the structural mechanical system must be taken into account, they are never infinitely stiff. It appears that friction between two contacting bodies is not strictly a function of the materials in contact and the contact conditions, but also of the measurement approach. After the dynamic evaluation of the developed tribometer, different friction experiments in time domain are carried out. This setup allows us to investigate different friction phenomena, such as break-away, the pre-sliding

106

4.4 Conclusions

52 N

Fit for pos. vel. Fit for neg. vel. Average fit

-0.3 -0.4

Lift-up [μm]

0

-2

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-4

-0.7

Lift-up [μm]

280 N

-0.8

-6

-0.9 -1 -0.06 -0.04 -0.02

-8 560 N

0 0.02 θ [rad]

-4.8

Lift-up [μm]

-10 840 N -12 1120 N -14 1400 N -16 -0.1

0.04

0.06

Fit for pos. vel. Fit for neg. vel. Average fit

-4.9 -5 -5.1 -5.2 -5.3 -5.4 -5.5

-0.05

0

θ [rad]

0.05

0.1

-0.06

-0.04 -0.02

0

θ [rad]

0.02

0.04

0.06

Figure 4.18: Lift-up effect for different normal loads and butterfly curve for the two lowest loads at 9 Hz and sliding behaviour, the transition between regimes, stick slip motion and Stribeck behaviour. The comparison between the friction phenomena in dry and lubricated conditions shows a great difference. This difference is due to the lubricant and its additives which induce a higher dynamic friction coefficient. The results of these experiments can be used as a validation for friction models. These results are qualitatively consistent with the results obtained from the friction models developed and discussed in Chapter 2. These theoretical and experimental results can give insight in the development of a adequate control model which takes the different friction phenomena discussed here into account.

107

108

Part IV

Control aspects

109

Chapter 5

Developed friction model for control The Butterfly Effect: “The fluttering of a butterfly’s wings can effect climate changes on the other side of the planet.” Paul Erlich

5.1

Background

Being a complex non-linear phenomenon, friction is perhaps the most difficult part to identify and compensate for in a mechanical system. Hence an extended friction control model is introduced to be used in the compensation for friction in automatic clutches. In reality in an automatic transmission the friction is not compensated, as it is in accurate position control, but controlled such that it can be used in a optimal way, because it is the driving principle like in all friction drives. The extension is based on the Generalised Maxwell-Slip (GMS) model which is extended with a normal degree of freedom as explained in Chapter 2. Here the GMS model should not be confused with the extended linear stop operator (ELSO) model discussed in [85] which, when the number of operators goes to infinity, is also called the Generalised Maxwell-Slip Model [86]. The original formulation of this model was done in the mid 1800s by J.C. Maxwell.

111

5 Developed friction model for control

5.2

Friction Model

Modelling friction in mechanical systems usually considers only classical friction models, such as Coulomb and viscous friction. In fact, friction never starts or stops abruptly as implied by the Coulomb model, and (micro) presliding displacements are actually observed [12, 22]. In the pre-sliding region (i.e. the small displacement range just after velocity reversal) the friction force appears as a displacement dependent function, namely a hysteresis function with non-local memory, which can be simulated by the Maxwell-Slip model. As an existing model and basis for extension, the GMS model was introduced by Al-Bender et al. in order to model the friction not only in the pre-sliding regime but also in the gross sliding regime [31]. The extended control model here will be called the eXtended Generalized Maxwell-Slip model. The eXtended Generalized Maxwell-Slip (XGMS) model allows a varying normal load and adds a normal degree of freedom to the existing GMS model.

5.3

Pre-sliding Friction Model

The behaviour of hysteresis with non-local memory in the pre-sliding regime friction can be modelled by a parallel connection of N Maxwell-Slip or Jenkins elements [87], see Figure 5.1. ζ1

k1

W1 Fi ζi

Fi

Wi ki

Wi

z

ki

Wi

z- ζi ζN

-Wi

z -Wi

kN

WN

Figure 5.1: Left: N elasto-slide elements; Right: characteristic of one element. 1. if the friction block is sticking, the element acts like a linear spring: dFi = ki v dt 2. if the friction block is sliding (Fi = Wi ): dFi =0 dt

112

5.4 Generalized Maxwell-Slip Model

5.4

Generalized Maxwell-Slip Model

The GMS model uses the Maxwell-Slip formulation but adds sliding dynamics to the slip phase. This model is able to capture the friction behaviour in the pre-sliding regime, while effectively modelling the (gross) sliding regime. Lampaert et. al. [53] have used the merged hysteretic with non-local memory property of friction force in pre-sliding regime, represented by Maxwell-Slip elements, with the original Leuven model. The model equation in sliding regime was derived from a physically motivated generic friction model [3]. The friction force is given as the summation of the outputs of N elementary state models. The friction behaviour of each elementary model can be written as: 1. if an element in the model is sticking: dFi = ki v dt 2. if an element in the model is sliding: Fi dFi = sign (v) .C αi − dt s (v) Once an element is sliding, it remains sliding until the direction of motion is reversed or its velocity becomes zero. The GMS model tries to capture the friction behaviour from the generic friction model developed at K.U.Leuven, division PMA and has been used for compensation by Tjahjowidodo [9], Jamaludin et al. [88] or as its basic MS model by Vo Minh et al. [89, 90].

5.5

eXtended GMS Model

The XGMS model uses the same formulation as the GMS model with the viscous effect in the Stribeck function but adds the dependency of the normal force. Rice et al. [91] also proposed a general state-rate law where the normal force variations are taken into account. In the following the Stribeck force refers to the global friction force in function of the relative steady state velocity in the contact surface. As with the GMS model, the Stribeck function S (v,Fsi ) should be identified before the model can be used for compensation. Here Fsi represents the sum of the maximum static friction forces of each of the elementary blocks. An extra parameter can be added to the exponential term to control the ratio of the static force to the minimum asymptotically attainable force. The basic formulation for the normal force dependent Stribeck function used is (see also eqn. 4.1 in Section 4.2.3):

113


S (v, Fsi ) = sign (v) . Fsi . 1 − exp

−vs |v|

γ

+ |v| .σv

S (v, Fsi ) = s (v, Fsi ) + f (v, Fsi ) The state equation, when an element is sliding, is the same as the one used in the GMS model, viz. dFi Fi = sign (v) .C αi − dt s (v, Fsi ) For solving this state equation the viscous effect has to be omitted from the Stribeck behaviour and added separately, because it has an instantaneous effect as opposed to the Stribeck effect, which induces a lag, called friction lag. The total friction force, as output of the model, is: Ff =

N X

γ

Fi + sign (v) . |v| .σv

i=1

σv is also function of the normal load and in general one can write σv = f (Wn ). In the Stribeck function given above this dependency is written as Fsi σv . For constant velocities, i.e. when no dynamic sliding is present Ff = S (v, Fsi ) In the Stribeck function we have three terms, viz. a static term, a Stribeck term which represents the velocity weakening and a viscous term which repγ resents the velocity strengthening. The viscous term is following a |v| behaviour, this is based on the model for the lubricated friction derived as a Rayleigh step, see Section 2.3. This is discussed later at the end of Section 5.6. This formulation can represent different kinds of Stribeck behaviour which are also measured experimentally for different material combinations and lubrications [33, 83, 88, 92, 93]. As formulated in the GMS model there is a dependency between the αi parameter and the maximal allowed friction force Wi in each element. Wi αi = PN i=1

Wi

In the newly formulated Stribeck behaviour this dependency is extended into the normal force Wn = Fsi /µ, which can vary in time. Fsi =

N X

Wi

i=1

So any change in the normal force Wn = Fsi /µ will induce a change in each elementary maximum allowed force Wi . In general one can write that

114

5.5 eXtended GMS Model

Fsi = f (Wn ). Based on experimental work this dependency is non-linear as described in [2, 94]. There should be a decreasing proportionality with a rising normal force. The Fsi component in the Stribeck function should have a non-linear effect, as it appears the friction coefficient drops with rising normal load, in a saturating way. For this a function f (Wn ) (eqn. 4.2) has been proposed in Section 4.2 based on experimental data. Figure 5.2 and 5.3 give an overview of different simulated Stribeck shapes.

Figure 5.2: Stribeck curve in function of parameters vs and σv In Figure 5.2 σv varies between 0.01 and 1, vs varies between 0.25 and 0.01. When vs = ∞ and σv = 0 this results in Coulomb friction, viz. a constant friction force in function of velocity. For a certain given vs , σv and varying Fsi the Stribeck function is rescaling like shown in Figure 5.2 and 5.3. In Figure 5.3 Fsi varies between 0.1 and 2.

Figure 5.3: Stribeck curve in function of normal force as Fsi = f (Wn ) For arbitrary values for the given variables the friction force Ff in function of velocity v for N elementary blocks looks as in Figure 5.4 for a sinusoidal position input.

115


15

10

Ff

5

0

-5

-10

-15 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

v

Figure 5.4: Ff in function of velocity v for N = 10 The friction force in the XGMS model can, in a similar way as in the GMS model, be given as the summation of the outputs of N elementary state models. In this case the friction behaviour of each elementary model can be written as: 1. if an element in the model is sticking: dFT i = ki v dt 2. if an element in the model is sliding: dFT i = sign (v) .CT dt

Fi αi − s (v, Wn )

The only difference here with the GMS model is the normal load dependence as discussed before. The addition of a normal degree of freedom to the XGMS model is discussed in Appendix E. Its effect on the friction force and its link with the friction lag effect is also discussed. The generic model requires a lot of calculation, and is therefore only applicable in off-line use. The high calculation load is directly dependent on the number of modelled asperities, because for every asperity, a state has to be calculated individually. It has to be checked if an asperity is in contact or not, and when it is in contact if it is sticking or slipping, and this has to be calculated for each asperity at every time step. In order to have an realistic frictional behaviour enough asperities should be used in the simulation. In [1] 5000 asperities where used in the simulation. The derived model appropriate for control, viz. the XGMS model, is based on the Generic Rayleigh friction

116

5.6 Stribeck function

model and tries to incorporate as many friction characteristics as possible with as few parameters as possible. The developed model is called the eXtended Generalised Maxwell-Slip (XGMS) model because it is an extension of the GMS model. In this control model only a few amount of elements are necessary to simulate the friction behaviour and that’s why it is applicable for online use. In [9], Tjahjowidodo et al. evaluated the effect of the number of elements needed. The identification utilising the GMS model with 10 Maxwell-Slip elements was conducted, as a comparison with the initial amount of 4. Although this gives a slightly better result, the improvement is not significant. Increasing the Maxwell-Slip by 6 elements means adding two times six (=12) parameters in the optimisation process, which is a high price for the slight improvement. The same conclusion can be made for the XGMS model due to the similar used structure. For the sliding regime in principle only one element is necessary, because it captures the whole dynamic macroscopic friction behaviour in the differential equation with the use of the Stribeck effect and the addition of the viscous effect.

5.6

Stribeck function

The validation of the proposed basic Stribeck function is done based on experimental data from different sources. The experimental data represent the Stribeck behaviour and/or hydrodynamic behaviour for different test rigs, viz. the friction force measured for multiple constant velocities over a wide range. The basic formulation for the normal force dependent Stribeck function can be written as (again see also eqn. 4.1 in Section 4.2.3):

1 S (v, Fsi ) = sign (v) . Fsi (Wn ). 1 − exp a

−vs |v|

γ

+ |v| .σv

1 In here the factor is added to the exponential term to control the ratio of a the static force to the minimum asymptotically attainable force, as explained before. The power γ in the hydrodynamic (viscous) behaviour can be determined based on the Rayleigh step model. In the following figures the Stribeck function is fit to the experimental data using the Nelder-Mead algorithm with a minimisation function. In this case the minimisation function is the RMSE (root mean square error). v u N u1 X 2 RM SE = t · (Ff unct − Fexp ) N i=1

117


In this equation Ff unct represent the friction force at a given certain velocity for the Stribeck function and Fexp represent the experimental data for the same respective velocities. The first experimental data is coming from Zhang et al. [93]. Figure 5.5 shows the experimentally observed curves using different frictional material configurations and under different lubrication conditions, together with their respective fits. In Figure 5.5 on the right, the fit is less good because less data points were available for these two experiments. 0.75

0.17

0.7

0.16

0.65

0.15

0.6

0.5

0.13

μ []

μ []

0.14

exp. 1 exp. 2 fit 1 fit 2

0.55

0.45

0.12 0.11

0.4

0.1

0.35

0.09

0.3

0.08

0.25 0

exp. 1 exp. 2 fit 1 fit 2

5

10

15

20

25

30

35

40

45

50

0

5

10

v [mm/s]

15

20

25

v [mm/s]

Figure 5.5: Stribeck data (from [93]) and fit The second experimental data are coming from measurements performed on a X-Y table from [88, 92]. Figure 5.6 shows the data together with the fit. The fit of experimental results performed on the newly developed tribometer described in Chapter 3 and on the actual clutch used for control described in Section 6.3 are already discussed in Section 4.2.4. In those measurements the normal load effect on the friction force or on the friction coefficient was investigated. Stribeck for the X table 180 170 160

Ff [N]

150 140 130 120 110 100

exp. data model fit

90 80 0

5

10

15

20

25

30

35

40

45

v [mm/s]

Figure 5.6: Stribeck data (from [88, 92]) and fit

118

5.6 Stribeck function

The next fits, in Figure 5.7, are used to validate the basic expression for the Stribeck function based on the Rayleigh step module. In these plots the velocity and the friction force are represented in dimensionless form. The first fit, in Figure 5.7 on the left, is for a relatively small velocity range. The second fit, in Figure 5.7 on the right, is for a much larger velocity range to check the hydrodynamic behaviour and to see which power must be used in γ the |v| behaviour as explained later. These fits give relatively good results as can be seen in the Figure 5.7, viz. the Rayleigh step model result and the fits can hardly be distinguished. Stribeck Rayleigh step module + fit

0.5

1

Stribeck Rayleigh model fit

Stribeck Rayleigh step module + fit Stribeck Rayleigh model fit

0.9

0.45 0.8 0.4

Ff []

Ff []

0.7 0.35

0.6 0.5

0.3 0.4 0.25 0.3 0.2 0

0.5

1

1.5

2

2.5

v []

3

3.5

4

4.5

0.2 0

5

10

15

20

25

30

v []

Figure 5.7: Stribeck data and fit on Rayleigh step model for small and big velocity range As mentioned before the order of the hydrodynamic term of the Stribeck function can be determined based on the Rayleigh step module. This can be done because the static friction force in function of velocity is a closed form expression and looks as follows. For a more exhaustive explanation of this model see Section 2.3. 1 ∆h Λ α 2W Fs = + − 2L 3 Hs hs (1 + α) 13 (1 + α) Λ with Hs = and hs = Hs − 1 2W In this expression we can distinguish three terms, from which the first two are function of Λ (proportional to velocity) and the last one is a constant as shown in Figure 5.8. The fit of the hydrodynamic behaviour is done with a function of the 1 (1 + α) 1.7 form Ff it = σv Λ − . This is done because of the asymptotic 2W 2W behaviour of the static friction of the Rayleigh step at Λ = . Asymp(1 + α) totic here means the solution goes to infinity at the value for Λ mentioned

119


0.8

Rayleigh step module static friction force 1st term 2d term 3d term Ff

0.7 0.6

Hydrodynamic fit

F []

0.5 0.4 0.3 0.2 0.1 0 -0.1 0

5

10

15

20

25

30

35

40

45

50

Λ []

Figure 5.8: Rayleigh Stribeck curve and its different components before and the solution is not valid for lower values because hs becomes negative, which is impossible. Analytically the best fit for full velocity range, so 2 for Λ going to ∞, can be achieved with a power . But it seems, for the 3 Rayleigh step model, the power in the hydrodynamic behaviour has to be chosen in function of the velocity range to fit.

5.7

Comparison between Rayleigh step model and XGMS

A qualitative comparison for the friction lag phenomenon can be seen in Figure 5.9. The left one shows some results of the Rayleigh step module which also has some dynamic mass effect (normal degree of freedom). The right figure shows some results from the XGMS model for an arbitrary Stribeck function for several frequencies. After the extension with the normal force dependence into the XGMS model its effect on the normal degree of freedom should also be investigated. This effect results into a lift-up effect and behaves as an internal phenomenon, viz. as an internal dynamic effect. The normal force taken into account can be seen as an external phenomenon, e.g. as a disturbance, viz. as an external dynamic effect. The normal degree of freedom in sliding friction is discussed in Appendix E.

5.8

Friction identification

As mentioned before, the Stribeck function S (v,Fsi ) and the other parameters used in the XGMS model should be identified before the model can be used for compensation. The Maxwell-Slip parameters can be identified as explained in Appendix D. For this, optimization will be done by using non-linear regression

120

5.8 Friction identification

Stribeck f = 0.25 f = 0.5 f=1 f=2 f=5 f = 10

0.5

0.3

1.5

Ff

Ff

0.4

Stribeck 10 Hz 5 Hz 2 Hz 1 Hz 0.5 Hz

2

1 0.2

0.5 0.1

0 0.8

0.802 0.804 0.806 0.808 0.81 0.812 0.814 0.816 0.818 0.82

Λ/W

0

0

1

2

3

4

5

v

Figure 5.9: Qualitative comparison of Rayleigh step model and XGMS model friction lag

with the Nelder-Mead Downhill Simplex Algorithm as used by Tjahjowidodo T. [95]. The Nelder-Mead Algorithm is a heuristic search algorithm. It is a simplex method or downhill simplex method and it is a commonly used non-linear optimization algorithm. It is due to Nelder & Mead (1965) [96] and is a numerical method for minimising an objective function in many-dimensional space. It is a direct search method of optimization that works moderately well for stochastic problems. The method uses the concept of a simplex, which is a polytope of N + 1 vertices in N dimensions; a line segment on a line, a triangle on a plane, a tetrahedron in three-dimensional space and so forth. The method approximately finds a locally optimal solution to a problem with N variables when the objective function varies smoothly. For example, a suspension bridge engineer has to choose how thick each strut, cable and pier must be. Clearly these all link together, but it is not easy to visualise the impact of changing any specific element. The engineer can use the Nelder-Mead method to generate trial designs which are then tested on a large computer model. As each run of the simulation is expensive it is important to make good decisions about where to look. Nelder-Mead generates a new test position by extrapolating the behaviour of the objective function measured at each test point arranged as a simplex. The algorithm then chooses to replace one of these test points with the new test point and so the algorithm progresses. An alternative for the optimization using non-linear regression with the Nelder-Mead Downhill Simplex Algorithm could be “A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization” developed by Burk et al. [97].

121


5.9

Friction compensation

In this section, position control, with the incorporation of friction compensation using friction models in feedforward, is described. For this purpose, the inertial force and the friction force, as modelled in the previous section, are included in the feedforward loop. The control scheme is depicted in Figure 5.10. The feedback loop, which is required to track setpoint changes and to suppress unmeasured disturbances, is chosen to be a PID controller.

Figure 5.10: Friction compensation control scheme for position control Results of such friction compensation in position control of a DC motor can be found in [9, 59], where the friction models used are eg. the Coulomb friction model, the LuGre model, the GMS model, ... An added feedback arrow can take parameter variations into account, leading to adaptive control, where the parameters can change online for example because of a change in normal load as in the XGMS model. This shows how the XGMS model can be used in position control, while in Chapter 6 it is applied to torque control, see Figure 6.16. These two examples shows that the XGMS, similar to the GMS model, is a general applicable friction model, i.e. it can be used in position and force control, even though the model is not invertible.

5.10

Conclusions

In this chapter an extended friction control model is introduced to be used in the compensation for friction in automatic clutches. The extension is based on the Generalised Maxwell-Slip (GMS) model which is extended with a normal force dependence and a normal degree of freedom. It is shown that the tangential and normal dynamics in a frictional contact are coupled. The normal dynamic effect is called the lift-up effect. This coupling is taken into account

122

5.10 Conclusions

by adding a new differential equation to the sliding part of the existing GMS model and this model is named the eXtended Generalised Maxwell-Slip model (XGMS). The relation between the normal degree of freedom or the lift-up and the friction lag phenomenon was also discussed. The normal load dependence is used in Chapter 6 to perform control on a wet multi-disk clutch.

123

124

Chapter 6

Modelling, identification and control of wet multi-disk clutches “No student knows his subject: the most he knows is where and how to find out the things he does not know.” Woodrow Wilson

During a gear shift in automatic transmissions of heavy all-terrain vehicles, the transferred torque is shifted from one clutch to another. This shifting is realized by simultaneously opening and closing the two clutches, resulting in a gear shift without any torque interruption. This chapter contains the analysis of the frictional behaviour of multiplate wet clutches and the development of a torque controller, which is to be implemented in power shifting automatic transmissions. The controller is developed and tested on a test-rig in a lab environment with one multi-plate wet clutch. After an analysis of the influence parameters, a dynamic friction model for the clutch is developed. After identification of the model, the simulation is validated with measurements on the test rig. In the next fase, a controller is developed, using the identified model and an estimated inverse model. This implementation will be a basis for the further development of this torque controller in industrial applications.

125

6 Modelling, identification and control of wet multi-disk clutches

The rise time and regime error of the controller, which indicate the performance, are discussed in detail. The result show the performance of the controller, where the regime error is 3% and the rise time 0.25 s for the used trajectory.

6.1

Introduction

Applications with multi-plate clutches are commonly used in off-road vehicles. The control of these clutches can be achieved, based on several different strategies. As the industrial applications evolve, also the requirements of these controllers demand better efficiency and comfort performance. Clutch control commonly uses the differential speed of the clutch. In this chapter a torque-based control strategy is developed and the feasible proporties are explored. The gross sliding properties of multi-disk clutches constitute an important aspect for the control strategies in the drive-line off heavy off-road vehicles. This is because these clutches are integrated in the transmission to change gear without any interruption in the torque going to the wheels. In automatic transmissions with multi-disk clutches, there are always two clutches acting during a gear shift. For an up-shift, the clutch of the lowest gear will open when the pressure on the plates of the clutch from second gear is increasing. As a result, torque is still transmitted to the out-going axle when both clutches are slipping. During such a gear change the friction torque of the clutch is mainly dependent on the normal pressure on the plates. This means that a good knowledge of the friction force is required in order to maintain smooth gear changes. Therefore, in this chapter friction modelling is used to predict the friction force of one multi-disk clutch. The closing behaviour of these clutches contains several fases when starting from an open clutch to a totally closed clutch. The friction coefficient undergoes a change of properties over these fases, as stated by [45]. This means that the interaction between the friction material and the oil is very important, as mentioned by [98]. This interaction will determine the character of the slipping and closing behaviour, studied in [99, 100]. The temperature of the oil and friction material can vary a lot during an engagement of the clutch. The change in viscosity influences the frictional behaviour. The effect on the friction characteristic is not only due to the viscosity of the oil, but also due to the thermal coefficient of expansion and the interaction between the friction material and the additives in the oil.

126

6.2 Formulation and analysis of frictional behaviour in multi-disk clutches

The friction models, used here, are mostly used in very accurate positioning mechatronic systems. The aim here is to have a good compensation for the non-lineair behaviour during sliding. The goal is to control the friction by varying the normal force on the friction contact. This means that the application is different, but the base models will be the same. In positioning systems there is often a change in moving direction, which causes a transition between the sliding and pre-sliding regime. For the modelling of the clutch friction behaviour, only the gross sliding regime is of our interest. Various groups of models are available. Static friction models are useful [57], but more complex and dynamic models have a greater flexibility, which usually resolves in a more accurate friction torque prediction. The implementation of different models is discussed in this text, also as a possible implementation in the torque control of a wet multi-disk clutch. In this chapter friction modelling is used to control the closing behaviour of a clutch. The structure is as follows. The second section gives a description of the frictional contacts together with the investigated models. Section 6.3 gives more details about the test rig that was used for the experiments, and describes the different experiments that are executed. The identification of the system and the different experiments are described in Section 6.4. The application of the different torque control strategies are discussed in Section 6.5 and Section 6.6 will conclude this chapter with the important conclusions.

6.2

Formulation and analysis of frictional behaviour in multi-disk clutches

As mentioned earlier, this section gives a description of the frictional contacts within the clutch, which are essential for the closing behaviour. Additionally to that, the different models, used to simulate this behaviour, are described.

6.2.1

System definition and preliminary assumptions

In general, an automatic transmission has two main functionalities. The first is the connection with the combustion engine, which can be achieved by a torque converter or a clutch. The second functionality is the selection of the gear, which will provide the right gear ratio for the current vehicle speed. In heavy off-road vehicles it is also important to have a constant traction of the wheels, therefore a concept of ‘power-shifting’ is used. The aim of ‘power-shifting’ is to maintain the torque going through the transmission during a gear shift. This can be explained, using Figure 6.1. Before a gear shift (e.g. 1st to 2nd ), the first clutch is closed and the second

127


Figure 6.1: Principle of power-shifting. gear clutch is fully opened. After this gear shift the second gear clutch will be fully closed and the first one open. ‘Power-shifting’ means they will start slipping simultaneously and that the first clutch is smoothly opened, while the second clutch is closing. This is accomplished by controlling the hydraulic pressure on the pistons of both clutches. It is clear that the total torque going through the transmission is fully dependent on the closing and opening behaviour. This is why our attention will be fully focused on the characterization of this behaviour and how to use this knowledge in following control strategies. According to this, the boundaries for the system can be set. The observation of the torque going through a single clutch, as a function of the hydraulic pressure, slipping speed and the oil properties will be the first intention of this chapter. Figure 6.2 show a sectional plane of the hydraulic multi-disk clutch, used for the test setup. The clutch contains the following main parts, see Figure 6.2. This configuration can transfer torque as a function of the pressure pN between the friction plates and the separation plates. This pressure is realized by applying an hydraulic pressure pcyl in the cylinder. If no pressure is applied, the spring will push the piston in its rest position. This means there is a certain amount of play x between the piston and the plates. To calculate the normal pressure on the plates, which is necessary for the friction coefficient calculation, the following equation can be used: pN = with:

128

pcyl Acyl − kx Af ric

(6.1)


Figure 6.2: Sectional plane of a multi-disk clutch. -

6.2.2

pcyl the oil pressure in the cylinder, x the play between piston and plates, k the repositioning spring constant, Acyl the surface of the piston, Af ric the total friction surface between the plates.

Frictional behaviour

The friction material on the external toothed plates is a combination of organic fibres (cotton or cellulose) and a thermoset. The fabrication of this material is very similar to the process to make paper, hence the name of ‘paper-based friction material’ [4, 101, 102]. On the plates, in the surface of this material, grooves are present which provide lubrication and cooling of the contact surface in a closed clutch. The most important properties of this material are its high compressive strength, a good thermal conductivity and a stable friction coefficient. It is also a porous material for better internal cooling. The topography of the friction surface depends on the chemical composition of the friction material, in combination with the used automatic transmission fluid (ATF). The goal in the design of the friction material is to always have a positive derivative of the friction coefficient as a function of slipping speed [103]. This is a preferable property in automatisch transmissions, to have better behaviour regarding shudder and oscillations, as is described by T. Kugimiya in [6].

129


This behaviour can change if another oil is used, this means that every friction material should be used in combination with the right ATF. This fluid is an hydraulic oil with additives for better lubrication of other components in the transmission. The base oil mainly determines the viscosity, which influences the hydrodynamic friction. Additives in the oil mostly have there effect in the lubrication regime with the formation of tribolayers, as described in [6, 100]. These layers introduce a lower friction coefficient and a protection against extreme conditions for the friction surface and base oil. The closing behaviour of the plates is often tested. During this test, the plate pressure is increase from zero, while the slip speed will drop down because of this braking action. Hereby the speed and pressure vary rapidly during the tests. Because, in this chapter, the aim is to investigate the use of friction models to predict the transferred torque, more basic tests will be performed. Therefore we refer to [45, 99] for more information on the dynamic closing behaviour of wet multi-disk clutches.

6.2.3

Used friction model

The intention is to use friction modelling to predict the torque going through the clutch. This means that the friction in function of plate pressure and slip speed has to be determined. Over history, a lot of friction models have been developed, each for its specific purpose. For this section, [1] was used as a basic reference. An overview of friction models is given in the literature overview in Chapter 1. In this application, hysteretic position dependent behaviour does not fall in the scope of the velocity range. Therefore, it is not discussed in this chapter. Since our interest goes to the gross sliding regime, only this part will be discussed and reworked for the wet multi-plate clutch application. Proposed solution The GMS-model is chosen as a reference for the solution, suited for this application and is extended into the XGMS model as described in Chapter 5 to include the normal load dependence. Only the gross sliding regime is the area of interest here, as clutch control does not require hysteretic behaviour modelling. In Figure 6.3 (a), a scheme of a multi-plate clutch is given, together with the most important variables of the system. The friction torque is dependent on the hydraulic pressure p, the temperature of the lubrication oil Toil and the sliding speed ωs = ω2 − ω1 . There are clearly two kinds of dynamics in this system. The first will be called the tangential dynamics, which refer to friction dynamics in the contact

130


surface, related to the sliding velocity. The second dynamic is caused by the change in hydraulic pressure behind the piston and causes a change in normal force and will be referred to as the normal dynamics. The tangential and the normal dynamics in the model have to be decoupled. To do this, a normalized friction curve will be used in the GMS-model, which is nested in the global model. In this case, a normalized friction curve, is a Stribeck function with a static friction torque of 1 Nm (eqn. 6.2). To result in the total friction force Mout , the normalized friction force is multiplied by the actual Coulomb friction force Mc , which is dependent on the hydraulic pressure and represents the normal dynamics. Additionally to this, the viscous friction, assumed to have a direct effect, is dependent on the hydraulic pressure and sliding velocity. This calculation is given in the scheme of Figure 6.3 (b).

(a) Physical representation

(b) Extended friction model (XGMS) at reference temperature

Figure 6.3: Schematic representation of the multi-plate clutch. There are three main calculation blocks in this scheme: GMS-model contains the characterization of the friction curve, with parameters, which can depend on pressure and which represents the normal load dependant Stribeck effect part of the XGMS model.

131


Coulomb friction block contains the hydraulic pressure, which represents the normal dynamics. This block is also dependent on the friction surface and friction radius of the clutch. Viscous friction This block represents the viscous friction as a function of pressure and sliding velocity, which means it represents the normal load dependant viscous effect part of the XGMS model. The friction force in the XGMS model can, in a similar way as in the GMS model, be given as the summation of the outputs of N elementary state models. The expression, presented as a block diagram in Figure 6.3 (b) forms part of the total XGMS model. The part in black, see eqns. 6.2, represents the normal load dependent part that is used in the further developed torque controller. The friction behaviour of each elementary model can be written as: 1. if an element in the model is sticking: dMf ric = ki ω dt dWn = still to be determined from lift-up in pre-sliding dt 2. if an element in the model is sliding: dMf ric dt

Mout

Mf ric = sign(ω)C 1 − s(ω, p) with Stribeck effect VS s(ω, p) = MC (p) 1 + (β(p) − 1) exp − ω with Viscous effect γ f (ω, p) = sign (ω) Cv (p) |ω| and combined Stribeck behaviour S(ω, p) = s(ω, p) + f (ω, p) = Mf ric + f (ω, p)

(6.2)

dWn Wn = sign (ω) .Cn 1 − dt Fn (ω, α) Hereby, the global structure of a friction model with normal an tangential dynamics is proposed, based on the XGMS friction model. This global behaviour will be used to simulate the friction model in multi-plate clutches. Therefore, an identification of such a system is required and will be handled in Section 6.4. First the system will be described, together with the designed and developed test rig in Section 6.3.

132

6.3 Test setup

6.3

Test setup

The previous section clearly defines the goal of this research. As a validation of the theory, an experimental analysis will be performed. The required test environment will be described in this section. The main objective is to characterize the frictional behaviour of a multiplate wet clutch as a function of hydraulic pressure, sliding velocity and lubrication oil temperature. The used test setup provides the right environment for these measurement. A scheme is given in Figure 6.4.

Figure 6.4: Scheme of the test setup. The hydraulic pump (1) provides the pressure for the lubrication and hydraulic pressure circuit. The main pressure is 2.5 MPa, and provides enough oil flow for lubrication and clutch control. A safety valve (3) protects the proportional valve (4), which is connected to the high pressure circuit of the clutch. Before entering the clutch, the hydraulic pressure is measured (7), and send to the data acquisition system (dSPACE). The overload flow of the safety valve is used for the lubrication of the clutch (6). A recovery pump (9) recuperates the lubrication oil to the main reservoir (10). The clutch itself (20) is on one side connected to the environment through a (fixed) torque measurement, while the output shaft is connected to an AC-motor (14) with external motor drive, provided with controls for speed and torque. The torque measurement is discussed more exhaustively in Appendix C. The clutch, used here, only contains one friction plate, i.e. 2 frictional contacts instead of 10, to lower its torque capacity. The most important component of the test setup is the wet clutch (20). To be able to mount it, connect it to the AC-motor for actuation and to

133


connect it to the torque measurement unit a frame was designed. Figure 6.5 show a CAD drawing with the main components. A more detailed view is shown in Figure 6.6 as a draft with the different views and with the main dimensions. The torque measurement unit consist of two identical 208C03 PCB Piezotronics piezoelectric force sensors. Coupling

Ingoing shaft

Wet clutch Outgoing shaft

Frame

Torque measurement unit

Figure 6.5: CAD drawing of the wet clutch and its frame. A picture of the actual setup can be seen in Figure 6.7 with its different components: (1) Low pressure or lubrication oil connection, (2) Pressure sensor, (3) Force sensors, (4) High pressure or actuation oil connection, (5) Clamp, (6) Supply connector, (7) Output shaft, (8) Wet clutch, (9) Oil receptacle, (10) Input shaft, (11) Base clamps, (12) Flexible coupling, (13) Base, (14) AC-motor. Measured signals for data acquisition: - lubrication oil temperature Toil - clutch feedthrough torque Mout - rotational slip speed ωs - delivered AC-motor torque Mm - oil pressure in the cylinder pcyl Signals sent through the data acquisition system to control the test rig: - control voltage Vvalve for the V/A converter of the proportional valve - desired rotational velocity ωd of the AC-motor - maximum motor torque Mmax

134

6.3 Test setup

650

A

180

95

180

A

SECTION A-A

Figure 6.6: Draft with different views and main dimensions.

Figure 6.7: Picture of the test setup with its different components.

135


First, a proper pressure regulator was made to control the hydraulic pressure in the clutch. This is achieved with a feedback loop, using the measured oil pressure pcyl . Figure 6.15 shows the general dynamic behaviour of the hydraulic system. This is important information when the feasibility of the clutch controller is discussed. The available bandwidth (9.8 Hz at -3 dB) provides a restriction for the clutch torque controller capacity. This system is built to do an exhaustive identification on the clutch system, followed by a step by step controller development for the clutch torque capacity. These aspects are handled in the following sections.

6.4

Identification results

In this section, the dynamic as well as the static properties of the multi-plate clutch will be linked to the modelling, mentioned earlier. The different parts of the developed friction model from Section 6.2.3 will be discussed step by step.

6.4.1

Temperature effect

To show to what extent the oil affects the friction torque, this friction torque is measured at constant speed. In function of time, the disks and the oil between them will heat up, due to which the total friction torque decreases. Figure 6.8 (a) shows this effect for different pressures. In Figure 6.8 (b) these measurements are converted into the coefficient of friction µ in function of the temperature. The temperature mentioned here is the bulk temperature measured in the oil as close as possible to the contacting friction disks. This friction coefficient has a logarithmic evolution in function of temperature, as proposed in eqn. 6.3. The results of the estimation are shown in Table 6.1. µ = Alog(T − T0 ) + B

(6.3)

Table 6.1: Fit of A, B and T0 for the temperature effect based on eqn. 6.3 A[−] B[−] T0 [◦ C] Value

−0.0145

0.143

30.0

This temperature dependence also appears with varying speed. An experiment at constant pressure at acceleration of the ingoing shaft does not necessarily have the same frictional torque than at deceleration. The temperature dependence is, among other reasons, the cause of this, as shown in Figure 6.9. Marklund et al. in [100] come to the same conclusion after performing similar experiments. Holgerson and Lundberg [46] also investigated

136

6.4 Identification results

22

0.136 3.2 bar 3.4 bar 3.6 bar

0.134 Friction coefficient [−]

Torque [Nm]

20 18 16 14

0.132 0.13 0.128 0.126 0.124

12 40

60

80 100 120 Temperature [°C]

140

0.122 50

160

100 Temperature [°C]

150

Figure 6.8: Temperature dependence of the friction torque and friction coefficient.

the temperature influence on the engagement characteristics of a paper-based wet clutch. What is striking is that the friction torque at 1.05 rad/s (= 10 rpm) is about the same, despite the temperature difference. It can be deduced that the temperature mostly affects the viscous term. Therefore the asperity interactions are virtually not influenced. This becomes visible at low speeds, because there the viscous term is small.

Figure 6.9: Influence of the oil temperature on the friction curve at 3.4 bar.

6.4.2

Piston motion

The main parameter which influences the friction torque, is the pressure p. The pressure determines the position of the piston and the normal force on the friction disks. The position of the piston is easily determined, if only the stiffness of the configuration is taken into account. The first of these stiffnesses, is the overall stiffness of the conical spring washers, while the second

137


stiffness refers to the axial stiffness of the disk stack. Figure 6.10 shows a diagram of the situation, while the right figure indicates the cylinder pressure in function of the piston position x. In the figure, the transition is marked where the disk stack and the piston make contact. The respective pressure at contact is named the contact pressure pcontact .

friction

Pressure [-]

cylinder

pcyl

pcontact

disk spring washers

x

friction plate stack

Position [-]

Figure 6.10: Schematic representation of the clutch, Left: scheme w.r.t. the piston position, Right: Cylinder pressure in function of the piston position. Because the stiffness of the disk stack is much higher than that of the conical spring washers, it can be assumed that the disk stack is infinitely stiff. Because in this case, the conical spring washers bear no additional force after the disks are in contact. This allows to determine the normal pressure pN in the disks in function of the pressure in the cylinder pcyl . When the contact pressure pcontact is known, eqn. 5.3 gives the normal pressure pN and friction coefficient µ as a result. µ=

Ff ric pN Af ric

with pN = max(0,

Acyl (pcyl − pcontact )) Af ric

Ff ric =

Mf ric rf ric Nf

Ff ric is in this case unknown, but should be approximated by Mf ric divided by the average distance from the friction surface Af ric to the axis. This distance rf ric is here approximated by the average radius of the friction surface. The factor Nf is the number of friction contacts (2 per internal toothed disk). The values for this coupling can be found in Table 6.2.

138


Table 6.2: Dimensions of piston surface Acyl and friction disk rf ric Acyl [m2 ] −2

Value

1.131e

douter [m]

dinner [m]

rf ric [m]

0.133

0.099

0.058

5

25

4 20 3 2 0

20

40

60

80

100

30

Torque [Nm]

Torque [Nm]

Pressure [bar]

In the experiment, shown in Figure 6.11, a linear relationship between friction torque and normal force is visible. However, it is clear that a direction dependence is present, due to which the friction force is different from the average linear trend as can be perceived as hysteresis. To explain this phenomenon, three alternatives are proposed.

15

10

20 5

10 0 0

20

40

60

Time [s]

80

100

0 2

p

2.5

contact

3

3.5

4

4.5

Pressure [bar]

Figure 6.11: Friction torque Mf ric in function of pressure at a constant rotation speed of 200 rpm. The reason for this possibly lies in the interaction of the sealing rings with the housing or drum and piston, which provide a non-linear behaviour in the movement of the piston. Because of this the normal force between the disks becomes direction dependent, and its effect on the friction torque is displayed in Figure 6.11. An alternative explanation lies in the quantity of oil which is located between the plates. As already discussed, the presence of oil introduces an hydrodynamic effects. In the discussed experiment, the quantity of oil which is located between the plates differs between the closing and opening movement. During the closing movement an oil film is initially present between the disks before they come into contact. If the pressure further increases, there will be less and less oil between the contacting surface and in the friction material. If then the normal force drops, in the opposite motion for the same pressure, less lubrication of the contact surfaces will occur. Both between the asperities and in the porous friction material less oil will be available. This lower level of lubrication will introduce more asperity interactions, which in turn, entails in a greater friction force. A final possible

139


explanation is the hydrodynamic effect. According to Berger et al. [98], these hydrodynamic effects only occur at much higher pressure variations then were applied in the proposed experiment. Table 6.3: Result of a first order fit of the friction torque i.f.o. pressure Mf ric aM [Nm/(105 Pa)] bM [Nm] pcontact [105 Pa] µ[−] Value

−34, 7

14.8

2.35

0.1128

The linear fit, seen right in Figure 6.11, indicates a way to determine the contact pressure pcontact . The average of the linear fit for 25 different measurements is shown in Table 6.3. The average friction coefficient can be approximated. The resulting value corresponds to those found by Mäki [99]. Following equations have been used: Mf ric pcontact µ

6.4.3

= aM p + bM = −bM /aM aM = rf ric Acyl Nf

(6.4)

Stationary friction curve

The first step is to identify the static frictional torque as a function slipping velocity and hydraulic pressure. The temperature of the lubrication oil was at a regime value when these tests were executed. The general temperature of the components were about 70 ◦ C. 30 4.2 bar 4 bar

25

3.6 bar

20

3.4 bar 15

3.2 bar

10

2.9 bar

3 bar

Torque [Nm]

Torque [Nm]

3.8 bar

2.8 bar 5

2.7 bar fit

0 0

50

100

150

200

250

Pressure [bar]

Rotational Speed [rpm]

Rotational Speed [rpm]

Figure 6.12: Static friction curve as a function of pressure p and sliding velocity ωs . Parameter identification Five different parameters (Mc , Cv , β, VS and γ) determine the shape of the

140


frictional curve. For the measurements given in Figure 6.12, following values were calculated: γ VS

= =

1/3 11.52

[−] [rad/s]

(6.5)

As expected, there is a linear dependency of pressure for the static friction 20 Mc [Nm]

1

Cv [Nm/(rad/s)1/3]

0 2.5 6

0.6

3

3.5

4

0.4

4

0.2

2 0 2.5

beta [−]

0.8

10

0 2.5 3 3.5 4 Pressure [bar]

3 3.5 4 Pressure [bar]

Figure 6.13: Variation of parameters (Mc , β and Cv ) as a function of piston pressure p. torque Mc and the viscous factor Cv . The variation of β, fitted on eqn. 6.6, is a consequence of the flatter shape of the friction curve at higher pressures. The exact values can be found in Table 6.4 and 6.5. From the 5 different Table 6.4: Result of a 1st order fit of Mc and Cv Mc [∗ =Nm] Cv [*=Nm/(rad/s)1/3 ] a [∗/(105 P a)]

10.8

2.67

b [∗]

−25.1

−6.49

parameters, it can be said that Mc and Cv are directly dependent on the geometry of the clutch. The important values for the geometry are the friction surface Af and the friction radius rf . p − p1 β = min 1, β∞ + (1 − β∞ ) exp − (6.6) p0

6.4.4

Dynamic behaviour

As the normal and tangential dynamics are understood to be independent, they will be handled separately.

141


Table 6.5: Result for the fit of β β∞ [−] p1 [105 Pa] p0 [105 Pa] 0.704

2.71

0.158

18

16

Pressure [bar]

15 0

Velocity [rpm]

17.5

17

3.6 3.4 3.2 3 2.8 0

17 0.5

1

1.5

2

Torque [Nm]

Torque [Nm]

Tangential dynamics These dynamics are set by the attraction parameter C in the model. As this effect is, relative to other dynamics, small, the identification is only basic to have an idea of the tangential dynamics. The results of this study are given in Figure 6.14.

Measurement Desired 0.5

1

1.5

16.5

16

2

200 15.5 100

0 0

Accelerate Decelerate Model fit 0.5

1

1.5

2

Time [s]

15 0

50

100

150

200

Velocity [rpm]

Figure 6.14: Experiment for attraction parameter C at 3.4 bar. The velocity varies at 5 Hz and an amplitude of 60 rpm, with a bias of 100 rpm.

C = 69.0

[s−1 ]

(6.7)

Normal dynamics The most important behaviour, dependent on the hydraulic pressure in the cylinder, is the normal pressure on the plates. This is because this will be the system variable which will be controlled in the further developed torque controller. In Figure 6.15 the dynamic behaviour of the hydraulic system is shown together with the dynamics of the total system (pd = desired pressure, pm = measured pressure). Both measurements are executed at a pressure where the piston applies a pressure on the friction plates. The figure clearly show the link between both systems.

142

6.5 Controller design

! "#$

Figure 6.15: Identification of normal dynamics of the multi-plate (Mf ric /pd ) clutch and pressure regulator (pm /pd ).

6.5

Controller design

In this section the developed controller for the torque capacity of the clutch is explained. This controller will use the introduced friction model together with the identified parameters. This will be a controller, without knowing the actual transmitted torque, because the main intention is to develop a controller without having to use a torque measurement.

ω

pcyl

Toil

XGMS-Model

M sim Md

Feedforward Controller

p ff − +

Feedback Controller

++

pd

System

M out

Figure 6.16: Block diagram of the torque controller with the developed friction model. As Figure 6.16 shows, the controller is built up from two main parts. The first part, a feedforward model, contains a simplified inverse model of the

143


system, while the second part tries to minimize the steady state error by using the full developed model in the feedback part of the controller. Msim represents the simulated friction torque of the actual transmitted torque Mout that occurs in the system. Msim is also equal to Mf from eqn. 6.9. The structure of both parts of the controller will be discussed shortly. Feedforward part With a linearisation of the XGMS model it is possible to have a feedforward part on the controller. This linearisation assumes a Coulomb friction behaviour with an added viscous part, as shown in eqn. 6.8. The parameters for this calculation are taken from the sensitivity from Figure 6.13 based on the parameter identification of the XGMS model and the contact pressure pcontact , which compensates the free play between the friction plates and the separation plates. This will result in a faster reaction time of the total controller. The values for the parameters aM c and aCv can be found in Table 6.4 and the value for pcontact can be found in Table 6.3. pf f (Md ) =

Md + pcontact aM c + aCv |ωs |1/3

[105 Pa]

(6.8)

Feedback part Using the identified XGMS model, it is possible to implement a simulated or model based adaptive feedback loop. The model is used as a friction torque estimator. The model will be used in combination with a PI-controller, used on the proportional valve, to have a more accurate steady state error without realizing an unstable behaviour. As a recapitulation, the normal load dependent part of the XGMS model used here, looks as eqn. 6.9.

dMf ric dt

Mf

Mf ric = sign(ω)C 1 − s(ω) VS with s(ω) = MC (p) 1 + (β(p) − 1) exp − ω γ = Mf ric + Cv (p).sign (ω) . |ω|

(6.9)

Summary In Figure 6.17 a summary is given on the performance of the different parts in the controller. The controller is tested in the regime where the clutch is already filled and the piston applies pressure against the friction plates. The results are given in Table 6.6. The rise time tr is defined as the time necessary for the response to reach 90% of the desired trajectory for the first time. The

144

6.6 Discussion and conclusions

regime error is defined as the average error of the response from the moment the rise time is reached.

!

Figure 6.17: Following behaviour of the different controllers.

Table 6.6: Following behaviour of the different controllers Controller Regime error [Nm] tr [s] with feedforward

1.00

0.30

with XGMS model

0.50

0.76

combination

0.30

0.25

The developed controller, based on the identified behaviour of the proportional valve and the multi-plate clutch, is to be applied in the regime where the friction plates are already applied to the separation plates. The structure of the controller was built up in different steps, which were described earlier. The regime behaviour also indicates the accuracy of the identified model. In the operation of this clutch, several regimes can be identified. Before the piston can apply a normal force on the friction plates, a filling fase should precede. From the time that the clutch is filled, then the clutch is in a regime where the friction model is identified and tested. This regime is then active until the clutch is drained, meaning that the normal force on the plates is removed. To have a good filling and draining control of the clutch, the position of the piston during filling and draining should be modelled.

6.6

Discussion and conclusions

In this experimental study, the global frictional behaviour of a multi-plate clutch has been characterized. General characteristics, like the stationary friction curve are revealed.

145


Starting from available friction models, an extension of the Generalized Maxwell Slip model is defined. This extension, named the XGMS model, introduces the influence of the normal force on frictional torque going through the friction plates. This is combined with the tangential dynamics from the GMS-model in such a manner that both dynamics are independent, as measurements show. The identification of the friction properties are achieved afterwards. As a result, the parameters of the model are set, to simulate the frictional torque as a function of the hydraulic pressure and slipping velocity. The accuracy that was reached with the identified model is shown in the steady state error of the further developed torque controller. In the last part of this chapter, a torque controller is developed, based on the friction model of the clutch. The goal of this controller is to be used in applications where the transmitted torque is not measured. The performance and robustness of this controller is indicated by the performed measurements. The results of this chapter can also be found in [104]. The focus for future development can be searched in different areas. Since this article discusses situations of an engaged clutch, it can be an added value to investigate the piston position during the filling fase. Closing test can be performed on a SAE#2 test bench for more general results. These aspects can give an added value in the further development of this controller in industrial applications.

146

Part V

Conclusions

147

Chapter 7

Conclusions and future developments This chapter summarizes the main achievements and conclusions and provides some recommendations for future research.

7.1

Main Contributions

This dissertation mainly deals with the modelling and experimental investigation of lubricated friction in mechanical systems, more specifically with the torque transmission in lubricated clutches of heavy off road vehicles. First an existing model, viz. the Generic model, was re-implemented to form the basis for extension to lubricated friction. Second a lubricated friction model was developed, based on a Rayleigh step bearing, as an extent of the friction behaviour analysis in lubricated contacts, to be used as a stand alone model and a submodel of the Generic model. The work also contributes to the experimental measurement of dry and lubricated friction, to the understanding of the physical behaviour behind lubricated friction and to the problem of modelling and compensation of lubricated friction of mechanical systems, especially lubricated clutches.

149

7 Conclusions and future developments

7.1.1

Theoretical contribution

The Generic physically motivated dry friction model is re-implemented in a force based instead of a energy based way. This model contains different friction mechanisms: deformation of asperities, adhesion, normal creep, mass effect of asperities, ... Based on this model the friction behaviour, resulting from a relative motion between two contacting surfaces, is described. A normal degree of freedom is added to the system due to which the friction behaviour is influenced by the main body or slider inertia. The incorporation of this normal degree of freedom allows a lift-up of the moving body resulting in a possible transition from dry to fully lubricated or hydrodynamic friction. This last friction behaviour is described by a new developed model, based on a Rayleigh step bearing, which takes the lubricated friction conditions into account. In lubricated conditions a pressure build up can be observed with rising sliding speed due to the hydrodynamic effect of the fluid. In this configuration at high speed or low load the contact can be seen as a hydrodynamic bearing. The lubrication regimes that can occur are Boundary Lubrication (BL), Mixed Lubrication (ML) and Full Lubrication or Hydrodynamic Lubrication (FL, HL).

7.1.2

Experimental contribution

A rotational tribometer is developed which can be used for macroscopic friction measurement and identification. Its functional requirements are: (i) accurate displacement and friction force measurement, (ii) normal loading and measurement possibility, both statically and dynamically, and (iii) the possibility of applying arbitrary displacement signals over a large range of magnitude and frequency. These functionalities are decoupled as much as possible based on the principles of precision engineering. A dynamic identification of the setup is performed in the tangential and in normal directions to obtain the allowable measurement bandwidth. The structural dynamic behaviour of the setup has been optimized during the design phase, the structural resonances being placed at as highest as possible frequencies. This setup allows us to perform experiments for friction identification, which can be used for control purposes. The tribometer allows experiments in a large range of displacements and velocities, thus allowing various friction characteristics, such as break-away force, pre-sliding hysteresis, friction lag in the sliding regime, stick-slip and limit-cycle oscillations, and the Stribeck behaviour to be measured for one and the same configuration, and under dry or lubricated friction conditions [76, 83]. Such experimental results have been used to validate physically motivated friction models, or to establish or validate empirically motivated friction models, such as used in control applications.

150

7.1 Main Contributions

An added functional requirement to the newly developed rotational tribometer is the ability to measure the normal displacement associated with friction, namely the lift-up effect. This is a relevant effect, certainly when it concerns the normal dynamic influence on the friction behaviour. This is a feature, which is absent on most previously developed tribometers, so that they do not allow the measurement of this normal displacement or lift-up effect. By integrating capacitive sensors to measure the distance between the friction disks, it is possible to measure the normal displacement and its influence on the frictional behaviour.

7.1.3

Control contribution

As an existing model and basis for extension, the GMS model was introduced in order to model the friction not only in the pre-sliding regime, but also in the gross sliding regime. The eXtended Generalized Maxwell-Slip (XGMS) model allows a varying normal load and adds a normal degree of freedom to the existing GMS model. The normal load dependency is taken into account through the use of a new appropriate formulation for the normal force dependent Stribeck function. This Stribeck function is validated with different experimental data sets. In addition to this, an adequate function for the friction coefficient in function of the normal load is proposed, which takes its non-linear effect into account. For sliding friction one can observe a typical drop in the friction coefficient for an increasing normal load. The normal degree of freedom is incorporated by an extra differential equation in the sliding part of the model which expresses the changing bearing or normal load capacity of the contact due to the tangential motion and which results in the lift-up effect. The normal load dependent XGMS model is implemented after dynamic identification of the different influence parameters of the clutch sliding friction behaviour. To be able to apply this model, an SAE#2 without flywheel with control facility has been developed. The objective is to be able to control the friction torque by varying the normal force on the friction contact. The model estimates the friction torque which is dependent on the regime of the clutch. A feedforward controller has been developed based on the dynamic friction model as well as an approximate inverse model of the system. The control of the closing behaviour together with the control of the torque transmission, performed in lab conditions, forms the basis for research in industrial applications. The obtained results show that the control strategy, based on the proposed friction modelling, is effective in following a desired torque trajectory. The output torque has a steady state error of 3% and a rise time of 0.25 s, which are adequate for the intended application. It has been shown that the approach of adaptive control is applicable on the XGMS model, when the parameters of the friction model are changing during operation. The main varying parameters here are the normal load or the applied pressure, which is

151

7 Conclusions and future developments

the most important control parameter, the relative velocity and the oil temperature. An overview of the contributions is represented in Table 7.1 with the three main categories: theoretical, experimental and control contributions.

7.2

Future work

• An additional parameter to incorporate in the adaptive friction based control model could be the degradation of the friction material which also influences the friction behaviour due to a change in Stribeck behaviour. This is a long term effect. This would be a way to take wear of the friction disks into account. Part of the wear effect is already discussed by Ompusunggu et al. [105]. • The connection between the physical behaviour of the lift-up effect and extra differential equation, introduced in the XGMS model, has to be analysed. A function, similar to the Stribeck function, that described the normal-tangential relation should be found experimentally such that parameter identification can be performed. • In the modelling, identification and control of wet multi-disk clutches the closing behaviour of the piston should be further investigated. The reason for the non-linear behaviour in the motion of the piston lies in the interaction of the sealing rings with the housing or drum. A more accurate closing behaviour could improve the performance. Position control of the piston could reduce the peak torque which arises at contact. For this the filling phase and the approach of the piston to the friction disks should be modelled. This could also give a more decisive answer to the reason for the non-linear closing behaviour. • An alternative way to perform control could be the use of fractional derivatives. The influence of the non-linear friction on the frequency response function could be investigated in this way. The linear and non-linear components could be separated and a transfer function, in the form of fractional derivatives in function of the amplitude, could represent this non-linear effect and used as a basis for control.

7.3

Acknowledgements

This research is funded by the IWT, the Institute for the Promotion of Innovation by Science and Technology in Flanders, Belgium, grant SB-53043.

152

7.3 Acknowledgements

Table 7.1: Contributions with their related developments and benefits Development Benefit/contribution

Theoretical Contribution

Generic physically motivated dry friction model

incorporation of normal degree of freedom: allows lift-up

Rayleigh step friction model/module

takes the lubricated friction conditions into account

Generic Rayleigh model

for macroscopic friction measurement and identification

Experimental Rotational Contribution tribometer

contains different friction mechanisms: deformation of asperities, adhesion, normal creep, mass effect of asperities

pressure build up can be observed with rising sliding speed: resulting in lift-up

general model to simulate the dynamics of normal and tangential forces in lubricated rough contacts for macroscopic friction measurement and identification dynamic identification in tangential and normal directions to obtain the allowable measurement bandwidth: 100 Hz ability to measure the normal displacement associated with friction, lift-up effect: this feature is absent on most previously developed tribometers

Control Contribution

eXtended Generalized Maxwell-Slip (XGMS) model

allows a varying normal load and adds a normal degree of freedom to the existing GMS model normal load dependency is taken into account through a new appropriate formulation for the normal force dependent Stribeck function the normal degree of freedom is incorporated by an extra differential equation in gross sliding which results in the lift-up effect

SAE#2 without flywheel

developed with control facility to apply a torque controller normal load dependent XGMS model implemented after parameter identification the model estimates the friction torque (function of applied pressure, relative velocity and oil temperature)

153

154

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164

Part VI

Addenda

165

Appendix A

Stick time in function of asperity stiffness To study the behaviour of stick in the generic model, the introduction of the basic implementation of the stiffness is necessary. An asperity that is in contact with a counter surface, which starts to move, will initially be in stick. After a certain displacement it will go into slip. In function of position this can be represented as a hysteresis. The tangential stiffness can be represented as a linear spring and the maximal allowed friction force in the contact as a time dependency. This behaviour can be represented in the following figure.

1.8 1.6 1.4

F[ ]

1.2 1 0.8 0.6 0.4 0.2 0 0

500

1000

1500

2000

2500

3000

3500

x[]

Figure A.1: Force in function of displacement (linear) and stick time dependent friction force (curved) In this figure the black line represents the tangential force resulting from the linear spring and the curve represents the time dependency of the maximal allowed friction force in contact. From the moment the black curve crosses

167


the curved one the motion goes from stick to slip. • The derivation of these functions is as follows: By using the adhesion theory we can assume a time dependency of the friction coefficient. It can be implemented by using an exponential behaviour like:

t µ (t) = µ0 + (µ∞ − µ0 ) · 1 − e− τt In this equation µ0 is the initial value for the friction coefficient, µ∞ the limit value or asymptotic value for the friction coefficient and τt is a parameter which determines the rate of going from µ0 to µ∞ . The tangential stiffness kt is a constant (linear spring) such that the tangential force can be represented as: kt = cst. Ft = kt · x Using the next equation gives the normal stiffness. kn =

(2 − ν) kt 2 (1 − ν)

In this equation, ν is the Poisson coefficient for a certain material. This relation between the tangential and normal stiffness results from the theory of Mindlin [62] . As a convenience we can change this equation into: kn = c · kt with c=

(2 − ν) 2 (1 − ν)

For a certain constant normal displacement z, the normal force will result in Fn = z · kn = z · c · kt

168

This normal force combined with the time dependent friction coefficient results into the maximal allowed friction force. Fµ = Fn · µ (t) For this linear stiffness the point in which the motion goes from stick to slip occurs when Ft = Fµ . So this point is equal to x = c · z · µ (tx ). tx represents the time corresponding with the transition position x. In the rest of the text it will be shown that this point of transition is the same for whatever behaviour is given to the stiffness if we have a constant normal displacement z for the asperity in contact. As an example a non-linear softening spring with an exponential behaviour will be used. • Now the derivation is as follows: The tangential stiffness kt (x) is an exponential function of position (nonlinear spring) such that the tangential force can be represented as:

−kt kt (x) = kZt · e( τ ·x)

Ft (x) =

( −kτ t ·x) kt (x) dx = τ · 1 − e

kt is the initial stiffness, at x = 0. The integration constant is found by the knowledge of the initial condition of the force, Ft (0) = 0. For the normal stiffness we have kn (x) = c · kt (x) Now the normal force in function of the tangential displacement x is ( −kτ t ·x) Fn (x) = c · τ · 1 − e Since we have a normal constant displacement z it is not possible to integrate the normal stiffness over this normal displacement. To find the normal force for this normal displacement we need to convert the normal stiffness to a point to point linear representation of this stiffness, so we need to find the stiffness for each displacement x. This can be done as follows

169


Fn (x) kn0 (x) = = x Fn0 (x) = kn0 (x) · z

( −kτ t ·x) c·τ · 1−e x

An explanation of the previous conversion can best be done by a figure.

2

k1(x)

1

k2(x)

0

k2(x) 0

500

1000

1500

2000

2500

3000

3500

Figure A.2: Stiffness and force In this figure the full curve, which represents the force in function of the displacement x, viz. F (x), can be calculated in two different ways: Z F (x) =

k2 (x) dx

and F (x) = k1 (x) · x So we can also write that R k1 (x) =

k2 (x) dx x

Out of these results we can find the maximal allowed friction force in contact, viz.

Fµ = Fn0 (x) · µ (t) If we now calculate the transition point we find the same result as with the linear spring, viz. x = c · z · µ (tx ). tx represents again the time corresponding with the transition position x.

170

Appendix B

Relation between the tangential and normal stiffness (Mindlin) B.1

Sphere on a flat surface

Taken a sphere with radius R compressed on a flat surface with a preload W, the contacting surface will be a circle with radius a (see Figure 1.8) and the sphere will undergo deformation δ. Derivation of thea relation between the tangential and normal

stiffness of a sphere on a flat surface (Mindlin) Taken a sphere with radius R compressed on a flat surface with a preload W, the contacting surface will be a circle with radius a (see Figure) and the sphere will undergo a deformation δ.

W R r

R

Q

r

u

u-δ

a Figure B.1: Sphere in contact with a flat surface and its parameters

171

B Relation between the tangential and normal stiffness (Mindlin)

B.2

Derivation

The following variables can be determined based on the geometry of the sphere. The distance between the flat surface and the sphere can be expressed as: p u = R − R2 − r 2 And for small r this gives r2 2R2

u=

The radius of the contacting patch a is equal to r a=

3

3W R 4E ∗

1 = E∗

with

1 − ν12 1 − ν22 + E1 E2

For a spherical contact the normal deformation is expressed by δ=

a2 R

So using the previous expression for a, so taking into account the preload W and the radius of the sphere R one gets the following expression δ=

9W 2 16E ∗2 R

13 or W =

4 ∗ 1 3 E R2 δ2 3

Based on this deformation the normal stiffness can be determined. For the local normal stiffness: kn2 =

1 1 dW = 2E ∗ R 2 δ 2 dδ

kn1 =

1 1 4 W = E∗R 2 δ 2 δ 3

global normal stiffnes:

When a tangential force Q is applied the sphere will also deform in tangential direction equal to " 2 # 3µW Q 3 δt = 1− 1− 16aG∗ µW

172

with

1 2 − ν1 2 − ν2 = + ∗ G G1 G2

B.2 Derivation

Q = 1 the sphere gets into gross sliding, at that moment the tangential µW deformation becomes: If

3µW 16aG∗ Based on this deformation the normal stiffness can be determined. For the local tangential stiffness: δt =

kt2

1 dQ Q 3 ∗ = = 8G a 1 − dδ µW

Using the normal stiffness and the tangential stiffness the ratio can be obtained kn2 = kt2

1

1

2E ∗ R 2 δ 2 13 Q 8G∗ a 1 − µW

Replacing d and a with there respective expression one gets kn2 = kt2

E∗ 4G∗ 1 −

Q µW

13

This relationship can be reduced for a Hertzian contact where the footprint (a) is in full stick for two the same materials (E = E1 = E2 , G = G1 = G2 ) kn2 E∗ = kt2 4G∗

2E 1 − ν2 2G G∗ = 2−ν E G= 2 (1 + ν) E ∗ G = (2 − ν) (1 + ν) 2−ν = 2 (1 − ν)

with E ∗ =

kn2 kt2

This expression has also been derived by Mindlin [62].

173

174

Appendix C

ICP transducer DC measurement C.1

Introduction

The use of an Integrated Circuit Piezo-electric (ICP) force transducer has one main drawback, viz. the occurrence of leakage of the capacitor. This means measuring a DC signal is not possible or is only possible for a short time depending on the discharge time constant. In general, if the frequency component of the input is low w.r.t. the rate of leakage of the ICP transducer the output signal does not represent the measured value anymore. So e.g. if a linearly slowly varying input is applied than the output signal of the ICP transducer will not follow a line but will decrease because of the leakage. With a DC component in the input, e.g. a constant load, the output signal will have an exponential decay to zero. This shows that an ICP transducer together with its power-unit behaves as a high-pass filter. For high frequency components the output voltage follows the input and for low frequency and DC components the signal tends to go to zero as the charge leaks faster than the effect of the input. To have a better idea about the scheme of the ICP transducer and its power unit, see Figure C.1. ranging capacitor

piezo or quartz element

bias resistor

2 to 20mA

mosfet ic amplifier

current reg. diode

C 18-24V DC power

R

o u t p u t

Figure C.1: Typical ICP transducer and basic scheme of power unit

175

C ICP transducer DC measurement

C.2

Compensation

To overcome this problem and broaden the frequency measuring range determining the transfer function of the ICP transducer and its power-unit is crucial because its inverse will be used. This can be done on sight by looking at the decay rate of the leakage. Applying a constant input or DC input the output has an exponentially decreasing behaviour which looks like: O0 .e−a.t With O0 the initial DC value and a function of σ the decay time constant. a can be determined as follows:

O0 O0.e-a.t

0.02 O0 t

σ

Figure C.2:

−a =

1 ln (0.02) σ

We are looking for a high-pass filter which represents the ICP transducer together with its power-unit. A basic first order high-pass filter looks like: FH =

τs τs + 1

Now applying a constant signal input O0 gives the exponential behaviour in time domain. τ τ 1 −1 and L O0 FH .O0 . = O0 = O0 .e−a.t s τs + 1 τs + 1 Now the inverse of this transfer function can be used to filter the output signal to regenerate the real measured signal without the leakage for low frequencies. s+a s

176

with a =

1 τ

C.3 Experiment

In this inverse transfer function there is an integrator which has to be taken into account. A slight offset of the initial output of the power unit in unloaded condition will give a linear drift. To overcome this problem the initial output voltage can be zeroed such that the integration gives a zero output. After zeroing the measurement can begin.

C.3

Experiment

To demonstrate this idea a PCB FORCE SENSOR model number 208A03 was used. In Figure C.3 the effect of an applied constant force can be seen, in this case an exponential drift to the initial output value which is very close to zero. From this exponential behaviour the parameter a can be determined. 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 0

2

4

6

8

10 12 time [s]

14

16

18

20

Figure C.3: Exponential drift of the piezo cell In Figure C.4 the effect of the zeroing of the initial output value on the filtered and measured value can be seen. As explained before, an offset w.r.t. zero gives a linear drift in the filtered signal because of the integrator. Zeroing reduces this drift to a minimum. Here the unity is [V/10] coming from a factor 100 of the amplifier and a factor 1/10 of the dSPACE.

Zeroin

0.02 0

-0.02 0

2

4

6

8

10 time [s]

12

14

16

18

20

2

4

6

8

10 time [s]

12

14

16

18

20

0.12

0.1 0.08 0

Figure C.4: Effect of zeroing on the output signal

177


This slight offset from zero is due to the amplifier in the power unit. With a perfect amplifier without zero error this problem would not occur. The initial output and the adapted output can be compared in Figure C.5. In Figure C.5 on the left, four of the same loads are applied consecutively and then taken off one by one. As can be seen in de that plot (lower plot) the applied load or the DC offset drifts away. A similar result can be seen in Figure C.5 on the right, where all the four loads are applied at the same time and then taken off one by one. What also can be noted is that the values for the different loads are the same for the different plots, which shows the filter does not affect linearity nor has memory effect. 0.5

0.7 0.6

0.4

0.5 0.3 0.4 0.2

0.3

0.1

0.2 0.1

0 0 -0.1

-0.2 0

-0.1

2

4

6

8

10 time [s]

12

14

16

18

20

-0.2 0

2

4

6

8

10 time [s]

12

14

16

18

20

Figure C.5: Effect of the applied load Recalculating the load out of the voltage signal by the use of the sensitivity mentioned in the data sheet only gives an error of 6 g on 500 g, which represents the weight of one load. This error is not measured in function of time but is instantaneous so no drift is taken in to account. In Figure C.6 the dynamic behaviour of the force sensor with and without filter can be seen. In that plot (lower signal) the offset drifts away almost immediately whereas the upper signal keeps oscillating around the offset without drift. The AC component for both signals is the same so AC and DC measurements are compatible. Another point is the temperature sensitivity of the ICP transducer. In the following figure, Figure C.7, the first change is coming from a load of 4 kg, the second and the third are coming from blowing on the sensor and the last change comes from gently holding the sensor. As can be seen, the effect of temperature change is relatively large and the change is not only due to the possible change of load because of blowing on or holding the sensor. For this the change is too pronounced. The biggest change corresponds to about one fourth of the initial displacement which is 1 kg. To really only test the

178

C.3 Experiment

1

0.8

0.6

0.4

0.2

0

-0.2

-0.4 0

2

4

6

8

10 time [s]

12

14

16

18

20

Figure C.6: AC input with certain DC term temperature effect radiation without touching the sensor should be used.

0.2

0

-0.2

-0.4

-0.6

-0.8

-1

-1.2 0

20

40

60

80

100 120 time [s]

140

160

180

200

Figure C.7: Temperature effect on ICP transducer It was mentioned above that there is about an error of 1.2% on the measured load. This was with the amplifier of the power-unit set on x100. It should be checked how much the filtered system still drifts because of a nonperfect zero of the amplifier output. In Figure C.8 this was done with the power-unit set on x10. After about 15 seconds a load of 2 kg was applied. The instantaneous error of the measured load was 0.55%, after about 65 seconds the error was 3.5%. In the other zones where the load was decreased the drift was neglectable since the time for which drift could occur was only 40 seconds or less and the zero offset of the amplifier was less. It should be mentioned that the drift of the DC signal is function of the accuracy of the zeroing and also of the zero error of the amplifier of the power-unit, see Figure C.4.

179


0.05

0.04

0.03

[V]

0.02

0.01

0

-0.01

-0.02 0

20

40

60

80

100 120 time [s]

140

160

180

200

Figure C.8: Plot to determine the error in function of time

C.4 C.4.1

Pros and cons of PE sensor with charge amplifier Dirty environment

With any type of measurement system, every component in the measurement chain (the sensor, cable, and amplifier) should be kept clean and dry in order to achieve the desired performance characteristics. If water gets into the cable connector, for example, a fault condition will develop. However, PE accelerometers are especially sensitive to external contamination due to their high impedance nature. This requires more care in daily maintenance.

C.4.2

Cable length

Signal-to-noise ratio in the PE based system is a function of the amount of capacitance (length of cable) between the sensor and the charge amplifier. In applications where long cables are needed between the charge amplifier and the accelerometer, two points require consideration: a) With a typical charge amplifier, the addition of 6,000 pF of cable capacitance will increase the noise floor of the amplifier by about 4 times. Although the relationship between cable length and amplifier noise is not linear, the noise increase from about less than 15 m of cable is typically considered negligible. b) With a very long cable run, the difference in cost between noise treated cables and ordinary coaxial cables may be significant.

180

C.4 Pros and cons of PE sensor with charge amplifier

C.4.3

Range flexibility

One of long-standing advantages of PE based system is its ability to change its usable dynamic range. In a typical shock and vibration measurement application, the range of measurement is unknown. During the equipment selection/setup process, the test engineer usually estimates the required maximum range based on past experiences, and supplements that with a safety factor when choosing the range of the accelerometer. With a high impedance PE transducer and an external charge amplifier, the engineer can manipulate the full scale range adjustment on the amplifier, and the test may continue without requiring a substitution. This is due to the fact that most PE accelerometers are capable of greater than 120 dB of dynamic range. Using an external charge amplifier gives the user the flexibility to customize the operating range in any given test.

C.4.4

Durability

With a PE accelerometer the charge amplifier can be sitting outside of the test chamber at ambient condition, minimizing unnecessary stress on the amplifier.

C.4.5

Advantages and limitations

Sensor Type: PE (Piezoelectric) Advantages: 1. Adjustable full scale output through range changes in charge amplifier 2. High temperature operation to 700 ◦ C available for special purpose devices 3. Interchangeable with existing charge systems with no system compatibility issues 4. Simple design, few parts, durable 5. Charge converter electronics is usually at ambient condition, away from test environment, minimizing necessary stress Limitations: 1. A lot of care/attention is required to install and maintain high impedance systems at peak operating condition 2. Special low noise cable required to minimize tribo-electric noise (generated by cable motion) 3. Capacitive loading from long cable run results in noise floor increase 4. External charge amplifier is required for operation

181

182

Appendix D

Maxwell-Slip parameter identification D.1

Introduction

The following paragraph describes a way to find the parameters for N Maxwell-Slip elements which can be used to model a measured hysteretic behaviour. This solution uses a linear algebra technique to find the solution of Ak = b.

D.2

Parameter identification

Consider a measured hysteretic behaviour as represented in Figure D.1. x is the generalised input and h the generalised output. The outer loop is the contour and one half of this curve can be used to determine the parameters of the Maxwell-Slip elements. This explanation will be given for the upper left part of the curve. If the hysteresis is symmetric the solution of the upper left side can also be used for the lower right side. If the hysteresis is asymmetric the same technique can be performed a second time. The hysteresis curve, see Figure D.1, can be converted to the virgin curve as follows, based on the Masing rules. Dividing one half of the curve by 2 in the two dimensions results in the virgin curve, see Figure D.2.

183

D Maxwell-Slip parameter identification

h

x

Figure D.1: Example of possible measured hysteresis

h h hm

hvirg (x) x ≥ 0 = f (x) , with f (x) = −hvirg (−x) x ≤ 0 x − Am = hm + 2f (D.1) , 2 = h (Am ) calculated with the formula of h before reversal point

h

A/2 = Am

virgin curve

B/2

B x

A

Figure D.2: Virgin curve from hysteresis Now that the virgin curve is determined, the 2xN parameters for N Maxwell-Slip elements can be determined. This is achieved by selecting a

184

D.2 Parameter identification

distribution of points in the x direction, which is considered being the input. Before continuing the behaviour of one Maxwell-Slip element is described. One such element is determined by two parameters, one determines its stiffness while the other determines the maximal allowed or saturation force (or output of the element in general). This is shown in following figure, Figure D.3. N of such element can be used as a model to fit an hysteretic behaviour.

h Wi Wi

ki

ki

x

x

Maxwell-Slip element

-Wi

Figure D.3: One Maxwell-Slip element and its behaviour

For a certain choice of the amount of elements N and the position of the break points x1 , x2 , x3 , . . . , xN the corresponding break heights h1 , h2 , h3 , . . . , hN can be determined. The pairs (xi , hi ) in addition to the origin (0, 0) determine the points on the virgin curve which are a piece-wise linear representation of the non-linear hysteresis shape, see Figure D.4.

Those different heights can be expressed in function of the displacement and the parameters of the Maxwell-Slip elements, i.e. ki ’s and Wi ’s.

185


h hN hN-1

virgin curve . :

h3 h2 h1 . .. x1

x2 x3

x

xN-1 xN

Figure D.4: The virgin curve

h1

= k1 · x1 + k2 · x1 + k3 · x1 + ... + kN −1 · x1 + kN · x1 =

N X

ki · x1 = W1 +

i=1

h2

1 X

Wi +

N X

i=1

i=2

2 X

Wi +

N X

i=1

2 X

Wi +

i=1

N X

ki · x2

i=3

ki · x3 =

i=3

3 X

Wi +

i=1

N X

ki · xN −1

i=4

... = W1 + W2 + W3 + ... + kN −1 · xN −1 + kN · xN −1 =

N −2 X

Wi +

N X

ki · xN −1 =

i=N −1

i=1

hN

ki · x2 =

= W1 + W2 + k3 · x3 + ... + kN −1 · x3 + kN · x3 =

hN −1

ki · x1

i=2

= W1 + k2 · x2 + k3 · x2 + ... + kN −1 · x2 + kN · x2 =

h3

N X

N −1 X i=1

Wi +

N X

(D.2)

ki · xN −1

i=N

= W1 + W2 + W3 + ... + WN −1 + kN · xN =

N −1 X i=1

Wi +

N X i=N

ki · xN =

N X

Wi

i=1

These equations can be rewritten in one matrix equation of the form Ak = b with:

186

D.2 Parameter identification



x1 x1 x1 .. .

   A=    x 1 x1

x1 x2 x2 .. .

x1 x2 x3 .. .

··· ··· ··· .. .

x2 x2

x3 x3

· · · xN −1 · · · xN −1



k1 k2 k3 .. .



h1 h2 h3 .. .



   k=    k N −1 kN



x1 x2 x3 .. .

   b=    h N −1 hN

x1 x2 x3 .. . xN −1 xN

      

       

(D.3)

(D.4)

      

(D.5)

Solving this linear system gives k = A/b. Element-wise multiplication of k with x gives W .



W1 W2 W3 .. .

   W =    W N −1 WN





k1 x1 k2 x2 k3 x3 .. .

      =       k N −1 xN −1 kN xN

       

(D.6)

A typical phenomenon for the Maxwell-Slip model is that it tends to saturate for high input amplitude, at the point all elements are slipping. To overcome this problem additional elements can be used which do not saturate, such that they behave as a linear spring. These additional elements tilt the orientation of the hysteresis, due to the increase of the global stiffness, such that the output cannot saturate anymore. It should also be mentioned that the choice of the break points xi is done randomly or they can be chosen equidistant. This choice can be optimised such that the difference between

187


the fit and the actual hysteresis is minimal. A good choice for the distribution can be obtained when the distance between the break points are taken inversely proportional to the second derivative of the virgin curve.

D.3

Asymmetric hysteresis conditions

In this paragraph some requirements for the use of MS to model asymmetric hysteresis will be discussed. Since two sets of parameters are used for each side of the hysteresis loop, so based on the virgin curve of the upper left part and the lower right part, some conditions have to be met, e.g. to overcome drift due to the model. The first condition is that the sum of the stiffnesses of each set must be equal to each other, otherwise drift will occur. This can be proven as follows. Right after motion reversal all the elements of a MS model will be in the stick condition. When a small reciprocating motion is applied with an amplitude such that no element starts to slip the model should behave as a linear spring. If the sum of the stiffnesses of both models is not equal the output will have a different slope or total stiffness when moving in one direction compared to moving in the opposite direction. The model with the biggest total stiffness will determine the direction of drift of the output, see Figure D.5.

Drift down because k sum1 ≺ k sum 2

ksum1

ksum2

x

Figure D.5: Drift due to different sum of stifnesses for small stroke A second condition is that after reversal of motion, if the motion is kept on going in one direction, the output should go through the maximal achieved

188

D.3 Asymmetric hysteresis conditions

output reached before. This condition is explained by a two element, so N = 2, hysteresis shown in Figure D.6.

h hmax

-x2

(1)

-x1 x1

x2

x

(2)

-hmax

Figure D.6: Asymmetric two element hysteresis Consider the state at reversal in point (2). Returning from this point on, so back to the right, the output should pass through point (1). The state before reversal of the two elements is: • 1st element is in slip • 2nd element is in stick The deflection of the second element is x2 − x(2) and x2 = x(1) . The output h at (2) can be calculated as: h(2)

= h(1) − Wd1 − x(1) − x(2) kd2 = h(1) − Wu1 − x(1) − x(2) kd2

(D.7)

Based on this expression the new deflection of element 2 right after reversal can be calculated such that the hysteresis loop will go though point (1), which is the maximal achieved output reached before. Such a condition should also be written for a 3 element hysteresis and so on up to an N element hysteresis loop. It is clear that such a formulation for N elements for any possible breakpoint is difficult or maybe even impossible to find. An alternative, as a solution to overcome this problem, could be to make the hysteresis symmetric by multiplying it with a function, apply the MS model and make it again asymmetric.

189

190

Appendix E

Normal degree of freedom of XGMS E.1

Introduction

In this appendix the addition of a normal degree of freedom to the XGMS model is discussed. Its effect on the friction force and link with the friction lag effect is also discussed. The function characterising lift-up remains unknown and should experimentally be investigated. After this is performed parameter identification can be achieved. This can be the topic of future work.

E.2

Normal degree of freedom

As can be seen based on experimental results from Section 4.3 and based on simulations of the Rayleigh step model from the end of Section 2.3.4 the normal degree of freedom in friction has an important influence and can be perceived as the lift-up effect. This effect could be implemented in the XGMS model in order to get an improved control friction model. As described earlier the extension based on the static normal force is not complete. The normal displacement associated with friction, namely the lift-up effect is a relevant effect, certainly when it concerns the normal dynamic effect on the friction behaviour [34]. The implementation of this normal dynamic effect and the normal degree of freedom which is a result of it can be performed in a similar way as the friction lag phenomenon in the GMS model, i.e. based on the same mathematical formulation. The normal distance between two contacting sliding bodies is function of the relative speed and the change of this distance is function of the acceleration [36, 84, 106]. This dynamic effect is perceived as the lift-up

191

E Normal degree of freedom of XGMS

effect or the butterfly effect, because the shape of this behaviour resembles a butterfly. The friction lag behaviour uses a static curve around which it evolves because of the dynamic behaviour. In that case this static curve is the Stribeck curve. In the case of the lift-up effect a similar static curve could be used, representing the static displacement for different velocities. Related to this curve one can think of a curve which represents the change in normal force due to an acceleration or deceleration. The formulation for this normal force generation due to the tangential motion could be as follows: dFN i FN i = sign (v) .C αi − dt FN (v) With FN (v) a yet to be determined function. In comparison to the Stribeck function which is a point symmetric function this new function has to be a line symmetric function such that the resulting force is not hysteretic but has a butterfly shape. A certain shape for this function is used for a proof of concept and can be seen in Figure E.1. For the Rayleigh step bearing, for a given constant bearing film gap, the lift-up force or bearing force can be expressed in function of velocity. Similar to this, for a given constant preload, the lift-up height can be expressed in function of velocity. It is clear that gap height or normal distance, normal force and tangential velocity are related.

25

20

FN

15

10

5

0 -4

-3

-2

-1

0

1

2

3

4

v

Figure E.1: Lift-up force in function of velocity v When a sinusoidal position input is applied, the resulting lift-up force looks as depicted in Figure E.2. As with the friction force in the GMS model the change in lift-up force becomes zero at motion reversal or at velocity zero crossing. This force can be used to simulate the normal motion which is a result of the tangential motion of a mass on a frictional surface. This simulation is achieved as follows. Consider a mass M which is preloaded by its own mass and a normal load Wn on the non-linear stiffness of the contact with a certain damping, thus behaving

192

E.2 Normal degree of freedom

2

0

FN

-2

-4

-6

-8

-10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

xT

Figure E.2: Lift-up force in function of position x as a hysteresis. This stiffness and damping represents the behaviour of the lubricant and the asperities. The generated force FN due to the tangential motion acts as a disturbance on the preload G = M.g + Wn which represents the static normal load. This disturbance will make the mass move in normal direction. The resulting normal motion in function of the tangential motion will also have a butterfly behaviour, see Figure E.3.

0.091 0.0905

hN

0.09 0.0895 0.089 0.0885 0.088 0.0875 -1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

xT

Figure E.3: Lift-up in function of position x The contact lift-up hysteresis for the non-linear stiffness with damping looks in this case as shown in Figure E.4. The shape of the lift-up force function is very important in the final result and should be identified in a similar way as the Stribeck curve is identified. It could also be that the function has to be defined in function of acceleration instead of only in function of velocity. In what follows the used function is a 3D function being the change of normal force in function of the tangential velocity and the tangential acceleration as:

193


2

0

FN

-2

-4

-6

-8

-10

0.0875

0.088

0.0885

0.089

0.0895

0.09

0.0905

0.091

hN

Figure E.4: Contact lift-up hysteresis

dFN dFN • = f (vT , aT ) or = f vT , vT dt dt This function has to be integrated to get FN and used in the time-lag expression. The physical behaviour of the lift-up could be explained with a thought experiment. The height of a moving body on a surface in function of speed has to be a saturating function as shown in Figure E.5. This velocity dependence was experimentally investigated and discussed in Section 4.2.2.

1

Lift-up hN in function of tangential velocity vT

0.9 0.8 0.7

hN

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

8

9

10

vT

Figure E.5: Lift-up in function of tangential steady state velocity vT For each constant velocity a constant height is reached, so at constant speed there is no force variation due to the motion, the body is at equilibrium. At very high speed any acceleration has no influence on the height, so the change in force has to be zero. At low speed or at stand still the change in generated normal force is function of the acceleration. So accelerating from low speed or from high speed results in a different change of height. Based on

194 10

E.2 Normal degree of freedom

this thought experiment the function could look as follows, see Figure E.6.

Figure E.6: Derivative of the lift-up force in function of the tangential velocity vT and the tangential acceleration aT This has to be investigated in more detail. Results of lift-up experiments can give more insight in the actual behaviour. The resulting normal force FN can be seen as a disturbance and can be coupled with the existing slip equation via the Stribeck function because it is normal load dependent. Previous research obtained similar results. In [106], the LuGre model [50] is modified taking the dynamics of lubricant film formation into consideration. The lubricant film dynamics is approximated by a first-order lag element and its time constant is varied among the acceleration, deceleration and dwell periods. Experiments show the dynamic behaviour of the friction of a hydraulic cylinder. It was shown that the lubricant film thickness change lags behind the velocity change. Sugimura et al. [36] examined the unsteady-state EHL film thickness behaviour under constant accelerations/decelerations. They have shown that the film thickness becomes thinner during acceleration and thicker during deceleration than the steady-state film thickness, that the difference between the steady-state and unsteady-state film thicknesses becomes larger at greater accelerations/decelerations, and that the difference is larger during deceleration than during acceleration. In this way they show that the lift-up or the change in lift-up is both function of the velocity and the acceleration or deceleration. They also get a similar lift-up curve in function of steady state velocity as shown in Figure E.5. The resultant lift-up or butterfly behaviour for the previously explained function looks like the curves in Figure E.7 for different sinusoidal tangential displacements xT . The signal with the highest amplitude, which also means the highest attained velocity vT , reaches the highest lift-up. These results are directly related to the friction lag phenomenon, which [3] Tjahjowidodo T., Report: "Generalized Maxwell-Slip Model for friction dynamics 195 identification" [4] Nelder J. A. and Mead R., "A Simplex Method for Function Minimization." Comput. J. 7, 308-313, 1965

11


0.082 0.0818 0.0816

hN

0.0814 0.0812 0.081 0.0808 0.0806 0.0804 0.0802 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

xT

Figure E.7: Butterfly lift-up curves generates a higher friction force with acceleration and a lower friction force with deceleration. At acceleration the film thickness decreases due to which more asperity interaction can occur. On the other hand at deceleration the film thickness increases due to which less asperity interaction can occur. The friction force in the XGMS model can, in a similar way as in the GMS model, be given as the summation of the outputs of N elementary state models. In this case the friction behaviour of each elementary model can be written as: 1. if an element in the model is sticking: dFT i = ki v dt dFN i = still to be determined from lift-up in pre-sliding dt 2. if an element in the model is sliding:

196

dFT i = sign (v) .CT dt

dFN i = sign (v) .CN dt

Fi αi − s (v, FN i ) αi −

FN i FN (v, a)

Curriculum Vitae Personal data Thierry Vincent Marie Corneil Janssens Address: Vloetgrachtstraat 15 B-3078 Meerbeek (Kortenberg), Belgium E-mail: [email protected] Place and date of birth: Leuven, June 21st 1979 Nationality: Belgian Education • 2005-2009: Ph.D. student at the Department of Mechanical Engineering, K.U.Leuven; funded by the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT), scholarship (20062009) • July 2005: Master in Engineering Sciences at K.U.Leuven, Department of Mechanical Engineering, Division of Production, Mechatronics and Automation Graduated (with distinction)with Master thesis: Modelling of aerostatic foil bearings under the supervision of prof. Farid Al-Bender • 1999-2005: Student Mechanical-Electrotechnical Engineer, Faculty of Engineering Sciences, K.U.Leuven • 1998-1999: Preparation year in mathematics at K.U.Leuven, Preparation Institute, Certificate of Dutch for students of the Preparation Institute at ILT (Instituut Levende Talen) • 1993-1998: Economics-Modern Languages at Don Bosco Haacht College Additional certificate on the knowledge of management (participated twice in mini enterprise, Vlaamse Jonge Ondernemingen)

197

198

List of publications Full papers in proceedings of international conferences [1] T. Janssens, F. Al-Bender and H. Van Brussel, “Experimental characterisation of dry and lubricated friction on a newly developed rotational tribometer for macroscopic measurements”, Proceedings of the International Conference on Noise and Vibration Engineering (ISMA2008), Leuven (Belgium), pp. 857-869, September 15-17, 2008. [2] T. Janssens, F. Al-Bender and H. Van Brussel, “Comparison between experimental characterisation of dry and lubicated friction”, Proceedings of the XIII International symposium on Dynamic Problems of Mechanics (DINAME2009), Angra dos Reis (Brazil), 10 pp., March 2-6, 2009. [3] K. Laurijssen, T. Janssens, K. De Moerlooze, F. Al-Bender, H. Van Brussel, “Modelling, Identification and Control of Wet Multi-Disk Clutches”, Proceedings of the 2th European Conference on Tribology (EcoTrib2009), Pisa (Italy), 6 pp., June 7-10, 2009. [4] T. Janssens, F. Al-Bender and H. Van Brussel, “Experimental Characterisation of Lubricated Friction: Lift-Up and Stribeck Behaviour Considerations”, Proceedings of the 2th European Conference on Tribology (EcoTrib2009), Pisa (Italy), 6 pp., June 7-10, 2009. [5] T. Janssens and F. Al-Bender, “Considerations on the Lift-up effect in Sliding Friction”, Proceedings of the World Tribology Congress (WTC2009), Kyoto (Japan), 1 pp., September 6-11, 2009. [6] A.P. Ompusunggu, T. Janssens, F. Al-Bender, P. Sas, H. Van Brussel and S. Vandenplas, “Contact Stiffness Characteristics of a Paper-Based Wet Clutch at Different Degradation Levels”, Proceedings of the 17th International Colloquium Tribology (TAE2010), Stuttgart/Ostfildern (Germany), 14 pp., January 19-21, 2010.

199

List of publications

Papers as a result of Master Theses 1. K. De Moerlooze, Investigation into the relation between the normal load and the friction force in pre-rolling and pre-sliding contacts in dry friction, Katholieke Universiteit Leuven, Faculteit Ingenieurswetenschappen, Departement Werktuigkunde, 2006 [1] K. De Moerlooze and F. Al-Bender, Experimental Investigation into the Tractive Prerolling Behavior of Balls in V-Grooved Tracks, Advances in Tribology, vol. 2008, Article ID 561280, 10 pages, 2008. doi:10.1155/2008/561280 [2] F. Al-Bender and K. De Moerlooze, A Model of the Transient Behavior of Tractive Rolling Contacts, Advances in Tribology, vol. 2008, Article ID 214894, 17 pages, 2008. doi:10.1155/2008/214894 [3] F. Al-Bender and K. De Moerlooze, A theoretical analysis on the relation between normal load and friction force in pre-sliding frictional contacts Manuscript Submitted to Wear, 2009 [4] K. De Moerlooze and F. Al-Bender, An experimental investigation into the relationship between normal load and friction force in presliding frictional contacts Manuscript Submitted to Wear, 2009 2. K. Laurijssen and L. Van Houdt, Modellering, Identificatie en actieve controle van gesmeerde lamellenplaatkoppelingen, Katholieke Universiteit Leuven, Faculteit Ingenieurswetenschappen, Departement Werktuigkunde, 2008 [1] K. Laurijssen, T. Janssens, K. De Moerlooze, F. Al-Bender, H. Van Brussel, “Modelling, Identification and Control of Wet MultiDisk Clutches”, Proceedings of the 2th European Conference on Tribology (EcoTrib2009), Pisa (Italy), 6 pp., June 7-10, 2009.

200

Nederlandse Samenvatting Dynamische Karakterisering en Modellering van Droge en Grensgesmeerde Wrijving voor Stabilisatie- en Controledoeleinden 1 1.1

Inleiding Probleemstelling

Wrijving tussen twee oppervlakken komt altijd voor in toepassingen zoals geleidingen van machines, rem- en koppelingsoppervlakken, enz..., waar deze twee oppervlakken interageren en t.o.v. elkaar bewegen. Het bewegend oppervlak is steeds met een eindige stijfheid verbonden aan of ondersteund door de rest van het mechanisch systeem. De ‘wrijvingscoëfficiënt’ µ is geen constante, maar is afhankelijk van de twee oppervlakken en van voorwaarden, zoals de relatieve snelheid, de oppervlaktetopografie, het smeermiddel enz... Er bestaat geen theoretische methode voor het voorspellen van de wrijvingscoëfficiënt. Het sterk niet-lineaire karakter van wrijving kan bij mechanische systemen resulteren in regimefouten, limietcycli, stick-slip en slecht volggedrag. Dit zijn redenen van wrijvingsinstabiliteit met als resultaat het optreden van trillingen. De gevolgen van deze trillingen zijn om vele redenen nefast: de bewegingsnauwkeurigheid neemt af, de slijtage van de contactvlakken neemt toe, mogelijkerwijze ontstaan in de ganse structuur zelfopgewekte trillingen die tot vervroegd falen aanleiding geven door vermoeiing, slijtage of fretting. Bovendien neemt het bedieningscomfort van een

I

Nederlandse samenvatting

machine af door overmatige trillingen en lawaai. In dit opzicht vormt wrijving een groot probleem voor toepassingen waar positioneernauwkeurigheid en volggedrag belangrijk is. Wrijvingsmodellen kunnen de essentie van het complexe wrijvingsfenomeen in een systeem van vergelijkingen vatten. Deze kunnen als belangrijk hulpmiddel voor systeemsimulatie en -compensatie gebruikt worden, bijvoorbeeld in precisietoepassingen of in overbrengingen en aandrijvingen. Hoewel er reeds heel wat onderzoek in de nano/micro tribologie is gedaan rond de fysische interactie van een ruwheidstop op een oppervlak a.d.h.v. “Atomic Force Microscopy”, is het geen eenvoudige taak om dit te extrapoleren naar een macroscopisch oppervlak. Op macroscopisch vlak bestaan er wel dynamische modellen verkregen uit experimenten, maar deze zijn meestal enkel van toepassing voor de specifieke gevallen waarvoor ze zijn opgesteld en ze kunnen het onderliggende fysische fenomeen niet verklaren. Er is bijgevolg nood aan (1) een model dat rekening houdt met het onderliggende fysische fenomeen dat de wrijving veroorzaakt, dit zowel voor droge als voor natte (hydrodynamisch- of grensgesmeerde) wrijving. Validatie van een model kan gebeuren door de resultaten ervan te controleren met meetresultaten afkomstig van een tribometer. Deze tribometer moet in staat zijn (i) een arbitrair gekozen relatieve beweging tussen de twee objecten op te leggen, (ii) de relatieve verplaatsing tussen de twee voorwerpen nauwkeurig te meten, (iii) de wrijvingskracht tussen de twee voorwerpen nauwkeurig te meten, (iv) verschillende normale krachten tussen beide voorwerpen op te leggen en te meten, en (v) de invloed van verschillende materialen en randvoorwaarden zoals contactgeometrie en smering na te gaan. (2) Het afleiden van een werkbaar, implementeerbaar model voor de simulatie van systemen met wrijvingselementen is ook noodzakelijk en tot slot is er nood aan (3) het ontwikkelen van controletools, vertrekkende van (1) en (2), voor de sturing en compensatie van mechanische systemen.

1.2

Doelstelling

De algemene doelstelling van het voorgestelde onderzoek bestaat in de eerste plaats in een theoretische modellering en experimentele verificatie van het dynamisch gedrag van de wrijving tussen oppervlakken in het geval van droge en natte (of gesmeerde) wrijving. Verder zal het gedrag van een mechanisch systeem waarvan deze oppervlakken deel uitmaken onderzocht worden met behulp van deze modellen. Men zal verwachten dat de ontwikkeling van een nieuw model, gebaseerd op de fysische mechanismen die wrijving veroorzaken, zal leiden tot een rekenintensief model dat niet onmiddellijk toepasbaar is voor (online) controledoeleinden. De ontwikkeling/afleiding van een ‘gereduceerd’ model, dat minder complex en dat on-line is toe te passen, geschikt voor systeemsimulatie en controledoeleinden is dus noodzakelijk. Tenslotte, dient

II


de voorgaande activiteit richtlijnen en “tools” te leveren zowel voor het ontwikkelen van controlestrategieën als voor wrijvingscompensatie.

1.2.1

Kadering van de onderzoeksactiviteiten van de onderzoekseenheid

De voorbije jaren is er op de afdeling PMA reeds veel onderzoek gedaan naar droge wrijving. Hiervoor werden zowel bestaande modellen aangepast en verbeterd als nieuwe modellen ontwikkeld. De bijdrage van voorgaand werk op het gebied van droge glijdende wrijving [1] is viervoudig: (i) een experimentele observatie van wrijvingsdynamica, (ii) de ontwikkeling van een fysisch gemotiveerd wrijvingsmodel, (iii) de ontwikkeling van een wrijvingsmodel toepasbaar voor controledoeleinden, en (iv) de validatie van wrijvingsmodellen door wrijvingscompensatie. Dit voorgaand werk bespreekt een nieuw ontwikkeld apparaat voor wrijvingsmetingen, namelijk een tribometer en het bespreekt een nieuw algemeen model voor de dynamica van droge glijdende wrijving. Wrijvingsonderzoek beperkt zich niet enkel tot de tribologie, maar is ook relevant in andere domeinen. Voor controledoeleinden is het begrijpen en controleren van wrijving cruciaal bij het verbeteren van de performantie en de nauwkeurigheid van mechanische systemen. Compensatie van de wrijving is noodzakelijk voor een nauwkeurige beweging, bijvoorbeeld in de precisiemechanica, bij overbrengingen en aandrijvingen.

1.2.2

Innovatie

Bestaande modellen zullen als uitgangspunt gebruikt worden voor het opstellen van een dynamisch wrijvingsmodel voor (grens)gesmeerde oppervlakken. Grensgesmeerde wrijving treedt op wanneer de snelheid niet voldoende groot is voor de opbouw van een hydrodynamische smeerfilm waardoor er nog steeds een ruwheidstopinteractie is. Tevens heeft recent (theoretisch) onderzoek op de afdeling PMA aangetoond dat er altijd een normale beweging van de oppervlakken optreedt bij tangentiële beweging van de oppervlakken relatief t.o.v. elkaar. De bedoeling is om in deze fase van het onderzoek de bestaande wrijvingsmodellen uit te breiden zodat zij ook kunnen gebruikt worden voor grensgesmeerde oppervlakken en om de invloed van de wrijving op de normale beweging van de oppervlakken te karakteriseren. Het invoeren van de tijds- en positieafhankelijkheid van de parameters is ook een belangrijke uitbreiding. Voor de bepaling van de parameters van het model zal in eerste instantie gebruik worden gemaakt van een bij afdeling PMA beschikbare (heen- en weergaande) proefstand. Het ontwikkelen van een tweede, nieuwe rotatieve proefstand zal ondernomen worden op basis van de testnoden specifiek aan het gesmeerde-contactprobleem.

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1.3

Onderzoeksbeschrijving

Het onderzoek beoogt de ontwikkeling van een nieuw dynamisch wrijvingsmodel voor wrijving bij (grens)gesmeerde oppervlakken vertrekkend van bestaande modellen en inzichten. De tijds- en positie-afhankelijkheid van de wrijving en de rol van de normale-tangentiële koppeling zullen hierbij aan bod komen, met het oog op de toepasbaarheid van de projectresultaten in systeemcontrole en -compensatie. De aanpassing van dit nieuw dynamisch wrijvingsmodel naar een eenvoudiger model toepasbaar voor regeling en sturing is dus ook nodig. Een belangrijk aspect hierbij is de verfijning van een bestaande tribometer of de ontwikkeling van een nieuwe testopstelling. De testopstelling zal aangepast of herontworpen en gebouwd worden om het wrijvingsmodel te kunnen valideren. Een goed mechanisch ontwerp kan wrijving en dus haar invloed nooit volledig uitschakelen. Bovendien is een zekere hoeveelheid wrijving in veel gevallen nodig om enige demping te voorzien in het systeem. Als remedie hiervoor is het mogelijk om op een mechatronische wijze, d.w.z. door een “holistische” aanpak, mechanisch, elektrisch en IT, de ongewenste wrijvingseffecten te onderdrukken. Hiertoe bestaan er een veelheid aan beschikbare technieken in verscheidene disciplines. De belangrijkste bouwblokken, nodig voor een geslaagd ontwerp van een performant systeem, zijn betrouwbare modellen van de subsystemen die deel uitmaken van het totale systeem. Wrijvingselementen (lagers, koppelingen, enz. . . ) zijn voorbeelden van dergelijke elementen die zowel een complex gedrag vertonen als moeilijk te modelleren zijn. Figuur 1 geeft een overzicht van de verschillende activiteiten die zullen doorlopen worden in dit project. Het grijze gedeelte omvat de activiteiten die voor een overgang zorgen van reeds bestaande modellen en opstellingen naar nieuwe modellen en opstellingen. De oranje blokken vormen het hoofdgedeelte van het project en het blauwe blok vormt de vragen en de noden vanuit de controle die de modellering kunnen be¨ınvloeden. Het onderzoek start steeds met een grondige literatuurstudie. Deze laat toe om een inzicht te krijgen in de stand van zaken, de ‘state of the art’ wat betreft wrijving. Het theoretische gedeelte van het project verloopt als volgt. Het bestaande generisch model [1, 3], ontwikkeld op PMA voor droge wrijving, zal het vertrekpunt vormen voor de ontwikkeling van een wrijvingsmodel voor (hydrodynamisch- en grens) gesmeerde oppervlakken. De basiscomponenten van dit model zijn het wrijvingsmechanisme en het contactscenario.

IV


Literatuurstudie Theoretisch: Ontwikkeling van micro-model

Experimenteel: Bestaande tribometer ⇓ Nieuwe tribometer

Gereduceerd (toestands)model

Testopstelling : Tranmissielijn met natte koppeling

Systeemsimulatieomgeving (Matlab)

Vraagstukken en observaties vanuit controle

Modellen en richtlijnen voor controle

Figuur 1: Overzicht van de projectactiviteiten.

De elementen van het wrijvingsmechanisme zijn kruip, adhesie en vervorming. De interactie van het contact is te wijten aan de ruwheidstoppen en de oppervlaktetopografie. Schematisch ziet het model eruit als volgt (zie Figuur 2): In Figuur 2 (A) is het algemeen contact tussen de twee voorwerpen te zien. Figuur 2 (B) laat de levenscyclus zien van één ruwheidstopcontact: (i) geen contact, (ii) contact, (iii) verlies van contact. Figuur 2 (C) toont het gedrag van de veerkracht i.f.v. de vervorming van de veer tijdens de levenscyclus van een ruwheidstop (iia) –tijdens slip, (iib) x tijdens slip, en (iii) bij contactverlies [1]. De overgang naar een gereduceerd controlemodel, dus van het generisch naar het Veralgemeend Maxwell-Slip (GMS) model [53], gebeurt aan de hand van het Maxwell-Slip model. De implementatie van hysteresis in het Veralgemeend Maxwell-Slip model is weergegeven in Figuur 3. Het idee hierbij is om N elasto-glij elementen in parallel met elkaar te plaatsen. Elk element heeft een gezamenlijke input z en een output Fi en elk element is gekarakteriseerd door een maximum kracht Wi , een lineaire veerconstante ki en een toestandsvariabele ζ i . De toestandsvariabele ζ i beschrijft de positie van het element i. De karakteristiek van één element is rechts in de figuur weergegeven. Het GMS model gebruikt deze implementatie als basis in het

V


normal load

force

(i)

(ii)

(iii)

z

(iia)

(iib)

x

(iii)

kn

kt 2

spring force

(A)

spring extension

(C)

kt 2 ζ ξ

w λw

αw

(B)

Figure 2. Figure A shows a general contact between two objects. Figure B shows the life cycle of one asperity contact: (i) no contact, (ii) contact, Figuur 2: the Levenscyclus en vervorming van eenspring ruwheidstop in contact. (iii) loss of contact. Figure C shows spring force behaviour as a function of the extension during a life cycle of an asperity contact: (ii a)—during stick, (ii b) x during slip, and (iii) when loosing contact. ζ1

k1

W1 Fi ζi

Fi

Wi ki

Wi

z

ki

Wi

z- ζi ζN

-Wi

z -Wi

kN

WN

Figuur 3: Links: N elasto-glij elementen; Rechts: karakteristiek van één element. pre-glijden regime samen met een andere wet voor de glij-fase, in de vorm van een toestands(differentiaal)vergelijking [1, 53]. Aanpassingen en verbeteringen zullen hier echter nodig zijn. Een wrijvingsmodel dat rekening houdt met de positieafhankelijkheid van de wrijving, met de tijdsafhankelijkheid van de topografie of met de parameters in het algemeen, is een belangrijke uitbreiding. Aan dit model zijn ook aanpassingen nodig die ervoor zorgen dat er rekening kan gehouden worden met

VI


de gesmeerde wrijvingsmechanismen. De interactie van de oppervlakken moet rekening houden met de smering. Dit kan bijvoorbeeld gebeuren door een elementair contact te modelleren als een Rayleigh-stap. De verdere uitwerking hiervan kan gebruik maken van de dynamische Reynoldsvergelijking. De verdere uitbreiding van dit model maakt gebruik van de dynamische Reynoldsvergelijking, zie Sectie 2.3. Na het uitwerken van al deze uitbreidingen kan het simuleren van het systeem beginnen, wat in Matlab zal gebeuren. De modellering en de simulatie zullen rekening moeten houden met vragen en observaties die vanuit controle experimenten naar voor gekomen zijn. Het experimentele gedeelte start met de reeds bestaande testopstelling. De bestaande tribometer die in voorgaand onderzoek gebruikt is, ziet eruit als volgt [1, 24]:

16

12 cm

2

3

4

1

14 3 13 5

15 12 6 7

8

9

10

11

Figuur 4: Lineaire tribometer. Op de foto van de tribometer zijn de volgende onderdelen te onderscheiden: (1) frame, (2) steun(stuk), (3) krachtsensor, (4) elastische verbinding, (5) wrijvingsblok, (6) aangedreven blok, (7) bewegende spiegel, (8) vaste spiegel, (9) stinger, (10) Lorenz actuator, (11) lineaire geleiding, (12) plexiglas, (13) luchtlager, (14) belasting, (15) rotatiepunt, (16) hefboom. Het instrument kan ruwweg in drie grote delen verdeeld worden: het aandrijvende gedeelte (onderdelen 6, 7, 9 en 10), het wrijvingsgedeelte (onderdelen 2, 3, 4, 5 en 12) en het belastingsgedeelte (onderdelen 13, 14, 15 en 16). De verschillende delen zijn zo goed als mogelijk ontkoppeld: het aandrijvende gedeelte en het wrijvingsgedeelte zijn enkel gekoppeld via de wrijvingsinterface, en het belastingsgedeelte en het wrijvingsgedeelte zijn volledig ontkoppeld door gebruik te maken van een luchtlager die ervoor zorgt dat heel de tangentiële kracht naar de krachtsensor gaat.

VII


Een noodzakelijke uitbreiding, voor het kunnen bestuderen van natte (hydrodynamisch en grensgesmeerde) wrijving, is het aanbrengen van een systeem dat kan zorgen voor een gecontroleerde toevoer van een smeermiddel tussen de twee interagerende oppervlakken. Een verdere uitbreiding is het ontwikkelen van een nieuwe testopstelling die gebruik maakt van een rotatieve beweging i.p.v. een lineaire beweging. De mogelijkheid om een hoge en constante snelheid op te leggen is hierbij een vereiste. Bij de huidige tribometer is dit niet mogelijk en deze heeft ook een beperkte slag. De oranje blokken van Figuur 1, die, zoals net besproken, een theoretisch en een experimenteel gedeelte bevatten, vormen de kern van het project. De vraagstukken en observaties uit controle experimenten vormen anderzijds richtlijnen voor de modellering en de uit te voeren experimenten. Het bestuderen van natte (grensgesmeerde) wrijving richt zich voornamelijk op natte koppelingen. Het wrijvingsgedrag van natte koppelingen voor toepassingen met automatische transmissies heeft een sterke invloed op het dynamisch gedrag van de machine of het voertuig, inclusief de overbrenging zelf. Zelfopgewekte trillingen kunnen de sleet van de koppeling verhogen en dus de levensduur verkorten. Een economische noodzaak is het beperken van de kost die samengaat met de levensduur van de koppeling (herstellen, vervangen) en de olie. Dit is te bekomen door een juiste selectie van de materialen om trillingen te voorkomen en sleet te beperken. Omdat ontwerpers de neiging hebben hun koppelingen te overdimensioneren, is een daling van de wrijvingscoëfficiënt geen beperking op de levensduur van de koppeling. Dit brengt wel een meerkost met zich mee. Een beter wrijvingsmodel, gevalideerd voor verschillende wrijvingsmaterialen, kan de levensduur van koppelingen verlengen en kan de selectie van de gepaste materialen vergemakkelijken en nauwkeuriger maken. Dit kan gebeuren door een gepaste controle toe te passen op basis van dit wrijvingsmodel. Er is niet alleen een economisch voordeel maar een gepaste controle kan het rijgedrag en dus ook het besturings- en rijgemak bevorderen. Natte wrijvingsmaterialen op basis van papier, komen vaak voor in automatische transmissies van voertuigen waar ze dienen als koppelingsen blokkeringsmechanisme. Schokvermindering tijdens het schakelproces ter voorkoming van trillingen tijdens het slipgecontroleerde proces van het blokkeringsmechanisme en het tot stand brengen van een hogere wrijvingscoëfficiënt µ zijn simultane vereisten. Niettegenstaande deze vereisten in het algemeen een trade-off vormen, is men hierin geslaagd door het bekomen van een positieve helling van het verband tussen de wrijvingscoëfficiënt en de glijsnelheid en door een ATF (automatic transmission fluid) standaard die ervoor zorgt dat de dynamische wrijving groter is dan de statische wrijving bij SAE#2 testen. In voorgaand onderzoek (in samenwerking met Spicer Off-Highway) was het doel om het wrijvings- en het slijtagegedrag

VIII


van nat wrijvingsmateriaal op staal op grote (SAE#2) en kleine (coupon op schijf) schaal te vergelijken. Hieruit bleek dat bij elke test op grote schaal zelfopgewekte trillingen (trillingen van de koppeling) optraden op het einde van de test. De fundamentele oorzaak van deze trillingen is nog niet bekend. Sommige onderzoekers wijten dit aan de stick-slip [5] , waar anderen het wijten aan het negatief hellend verband tussen de wrijvingskracht en de snelheid [6]. 1.3.1

Autonoom koppelgecontroleerde koppeling

Het controleren van het over te brengen koppel bij een koppeling is een moeilijke onderneming. Het gewenste koppel resulteert na overdracht in het effectieve koppel. De verschillende onderdelen in deze sturing zijn software, DAC, soleno¨ıde, versterker, overbrenging, hydraulische aandrijving, stuurklep, . . . Op verschillende plaatsen komen stoorsignalen het systeem be¨ınvloeden. Enkele stoorsignalen zijn hysteresis, stick-slip, drukverliezen, temperatuursveranderingen, variatie in de wrijvings-coëfficiënt, enz. . . Het veronderstellen van een constante wrijvingscoëfficiënt is tegenwoordig gebruikelijk en meestal wordt de gehele transmissie in beschouwing genomen. Een fout treedt op indien er een verschil is tussen het gewenste en het effectieve koppel. De grootste bijdrage tot de fout is de onzekerheid of de platen van de koppeling al dan niet contact maken. Een vanzelfsprekende manier om dit probleem op te lossen is het nemen van voorzorgen om zo veel mogelijk componenten van het systeem te linearizeren. Andere manieren zijn het gebruik maken van feedforward in open loop, adaptieve feedforward en een autonoom koppelgecontroleerde koppeling. Bij feedforward wordt vaak het inverse model van het systeemmodel toegepast. Hiermee kan men in theorie, indien een goede schatting van de storingsparameters en een goed model ter beschikking staan, een één-tot-één relatie tussen het gewenste en het effectieve koppel bekomen. Deze implementatie van de controller gebeurt vaak aan de hand van een look-up-table. Sommige parameters, zoals slijtage en degradatie van de olie, zijn echter zeer moeilijk te bepalen. Een uitbreiding van deze manier van werken is het gebruik maken van een betere schatter voor de storende parameters op basis van de fout, namelijk het koppelverschil. In dat geval spreekt men van een adaptieve of lerende feedforward. Een andere aanpak is, i.p.v. de gehele transmissie te beschouwen en deze te sturen, het afzonderen van een autonoom koppelgecontroleerde koppeling. Het is als het ware een overgang naar een ‘intelligent’ systeem, namelijk de koppeling op zich. De wrijvingscoëfficiënt mag men hierbij niet meer constant veronderstellen, maar deze moet in functie staan van snelheid, druk, temperatuur, olie, materialen, enz... Hierbij is dan wel nood aan een inwendige controle zodat een gewenste output bekomen wordt met een gegeven input. Uitwendig is de transmissie te aanzien als een lineair deelsysteem. Dit maakt de implementatie van de koppeling in de gehele transmissie eenvoudiger.

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1.3.2

Overzicht van de verschillende deeltaken van het project

Met als vertrekpunt het oude model (generiek model) • Uitbreiden met een normale vrijheidsgraad, dus rekening houden met de normale-tangentiële koppeling. Huidige modellen gaan uit van een constante normale belasting en constante normale afstand. Bij wrijvingsystemen is er echter altijd een variatie van de normale afstand. Daardoor zal een verdere uitbreiding van het Maxwell-Slip model rekening moeten houden met deze veranderende normale afstand. Een plotse stijging van de normaalbelasting zorgt voor het ontstaan van een aantal nieuwe interagerende ruwheidstoppen. Indien de normale vrijheidsgraad aan een systeem wordt toegevoegd, dan moet de interactie tussen de massa en de stijfheid van de ruwheidstoppen ook in beschouwing genomen worden. Het toevoegen van een extra vrijheidsgraad maakt het probleem ingewikkelder. Analoog aan het veralgemeend MaxwellSlip model kan, indien er een generisch model bestaat gebaseerd op het fysische mechanisme dat rekening houdt met de normale verplaatsing en de normale kracht, een vereenvoudigd model, gepast voor controledoeleinden, afgeleid worden. • Uitbreiden met de tijdsafhankelijkheid van de parameters. Wrijving heeft een zekere inlooptijd, een overgangsperiode tijdens dewelke parameters zoals ruwheid, temperatuur, degradatie van smering, enz. . . , kunnen wijzigen. Fenomenen met kleine tijdsconstante zijn bijvoorbeeld drukmodulatie en trillingen, fenomenen met middellange tijdsduur zijn bijvoorbeeld temperatuur en oliestroming en fenomenen met lange tijdsduur zijn bijvoorbeeld materiaaldegradatie en slijtage. Tijdens het inlopen variëren de wrijving en de sleet aanzienlijk in functie van de tijd. Als de initiële ruwheid van de glijdende oppervlakken goed gekozen is dan zal het inlopen uiteindelijk een “steady-state” bereiken. Op dat moment zijn de oppervlakken vlakker en de hoeveelheid sleet is laag en constant. Een ongepaste keuze van de ruwheid daarentegen kan leiden tot een snellere degradatie van de glijdende oppervlakken. Experimenten tonen aan dat de oppervlakte-microgeometrie één van de belangrijkste factoren is die de levensduur van mechanische componenten bepalen. ¨ Ostvik en Christensen [14] merkten op dat het inlopen bestaat uit het reduceren van de hoogte van de hoogste ruwheidstoppen, waardoor het aantal ruwheidstoppen die contact maken vergroten en waardoor de belastingscapaciteit van het oppervlak vergroot. Sleet tijdens het inlopen hangt zowel af van de hoogte als van de vorm van de ruwheidstopverdeling en de wrijving en de smering hangen af van deze parameters. De smering kan door de sleet overgaan van grenssmering naar volledige filmsmering.

X


• Uitbreiden met de positieafhankelijkheid van de parameters. De normaalbelasting kan variëren in functie van de positie door onnauwkeurigheden. Ook geometrische variaties kunnen de wrijving be¨ınvloeden. Dit fenomeen kan eveneens intrinsiek zijn aan het systeem zelf, geen enkel systeem is namelijk perfect. Deze variaties hebben een directe invloed op de wrijving. • Uitbreiding van droge wrijving naar (grens)gesmeerde wrijving. Bij natte wrijving komt grensgesmeerde wrijving voor als tweede dynamisch regime; na het pre-glijden regime. Hierbij is de snelheid niet hoog genoeg om te zorgen voor de opbouw van een vloeistoflaag waardoor er nog steeds een ruwheidstopinteractie is tussen beide oppervlakken. Bij gesmeerde wrijving is het noodzakelijk om rekening te houden met het smeermiddel, dus met de gesmeerde interactie van de oppervlakken. Een fenomeen dat voorkomt bij gesmeerde interagerende oppervlakken is bijvoorbeeld het optreden van een squeeze-film. Afleiden van een eenvoudiger, minder complex wrijvingsmodel (in de aard van het Veralgemeend Maxwell-Slip Model) bruikbaar voor on-line controletoepassingen. De complexiteit van een model mag namelijk niet te groot zijn indien het geschikt moet zijn voor controledoeleinden. Voor de praktische toepasbaarheid van een wrijvingsmodel speelt een gemakkelijke parameteridentificatie een belangrijke rol. De parameters zullen dus relatief gemakkelijk bepaalbaar moeten zijn. Met als vertrekpunt de reeds bestaande tribometer • Experimenten uitvoeren zodat de tekortkomingen duidelijk zijn voor het nieuwe project, aanpassingen aan de testopstelling eenvoudiger aan te brengen zijn en er rekening kan gehouden worden met de smering en de normaalverplaatsing. • Ontwikkelen van een nieuwe testopstelling, namelijk een nieuwe tribometer die een rotatieve beweging oplegt waarbij een gecontroleerde toevoer van een smeermiddel mogelijk is. Overgaan naar een testopstelling die het eigenlijke mechanisme, namelijk een transmissielijn met een natte koppeling, beter weergeeft dan de voorgaande testopstellingen, en waarmee simulatie kan gebeuren. De vraagstukken en observaties uit de controletheorie vormen, zoals reeds gezegd, richtlijnen voor de modellering en de experimenten. Toepassen van controle kan gebeuren op verschillende manieren, zowel passief, semi-actief als actief. Passieve methodes zijn bijvoorbeeld het ingrijpen op de stijfheid, de demping en de massa van een systeem. Bij een massa-veer-systeem kunnen deze parameters de wrijvingsdynamica be¨ınvloeden. Een regelbare ophanging kan aanzien worden als een semi-actieve wijze van controle. Een ander voorbeeld van een semi-actieve methode is het toepassen van een dithersignaal.

XI


Dither is een hoogfrequent signaal dat gesuperponeerd wordt op het stuursignaal. Hierdoor wordt het stictieprobleem vermeden. Voor actieve controle bestaan er veel mogelijkheden. Hierbij zijn actieve elementen zoals actuatoren van toepassing. Het gebruik van sensoren voor het opmeten van de verschillende toestanden van een systeem zijn dan onmisbaar. Feedforward en gesloten-lus-systemen, dus het gebruik maken van terugkoppeling, komen hier tot hun recht. Hiervoor is het ontwikkelen van gepaste controleschema’s nodig. Lineaire manieren van controle zijn reeds vaak besproken in de literatuur. Deze zouden dus geen problemen mogen vormen. Vermits de hier besproken problematiek een sterk niet-lineair karakter vertoont zullen nietlineaire controletechnieken van toepassing zijn. Deze komen echter beperkt aan bod in de literatuur. 1.3.3

Relevantie voor de industrie en voor de onderzoeksgroep

Dit project is een uitbreiding en aanvulling van het reeds verrichte werk en de vergaarde kennis in de industrie en aan de K.U.Leuven. Aan de K.U.Leuven, aan het departement Materiaalkunde (MTM), worden vooral materiaalkundige aspecten van de tribologie, meer bepaald tribochemie, onderzocht. Eveneens aan de K.U.Leuven, aan het departement Werktuigkunde binnen de afdeling PMA (Mechatronica), wordt onderzoek verricht naar optimale positienauwkeurigheid, de dynamica en trillingen van complexe machines. Een aspect dat hierbij ge¨ıntegreerd wordt, is de droge wrijving in lageringen en aandrijfsystemen. In de laatste 10 jaar is er heel wat kennis en ervaring ontwikkeld op het vlak van (droge) wrijvingsidentificatie, modellering en compensatie. Het deel van het onderzoek in het kader van dit project naar de modellering van wrijving in mechanische systemen, waarbij aandacht besteed wordt aan de niet-lineaire aspecten in systeemdynamica, komt voornamelijk aan bod bij het onderzoek van de onderzoeksgroep mechatronica (binnen de afdeling PMA) die grote ervaring bezit in systeemtheorie en dynamica van complexe machines (vb. robots en werktuigmachines) en in de uitvoering van dynamische experimenten. Twee doctoraatswerken die een beeld geven van het reeds verrichte werk zijn de doctoraatsthesissen van Tegoeh Tjahjowidodo [107], Vincent Lampaert [1], Wim Symens [108] en Tutuko Prajogo [21]. Bij de modellering van wrijving is er nood aan een uitbreiding met de positieafhankelijkheid van de wrijving voor het onderzoek en de modellering en controle van niet-lineariteiten van een harmonic-drive. De torsiestijfheid is namelijk afhankelijk van de precisie van montage en van onnauwkeurigheden die op hun beurt de wrijving be¨ınvloeden. Hoe groter deze fouten zijn hoe minder een model de realiteit en dus de positienauwkeurigheid zal kunnen weergeven [109, 110]. De contactwrijving bij de ontwikkeling van een ultrastijf piëzo-elektrisch positiesysteem met een stappende en resonate mode kan voor een lagere nauwkeurigheid zorgen. Een model van deze wrijving, kan

XII


alleen maar ten goede komen aan de positienauwkeurigheid [111]. De kennis van de wrijving bij het gebruik van verschillende afdichtingstechnieken en afdichtingsmaterialen bij de ontwikkeling van een lineaire hydraulische microactuator is niet van minder belang. Een gepast gebruik van een materiaal kan de wrijving in gunstige mate be¨ınvloeden [112]. Wrijving in industriële robots en bij machines met een koppeling kunnen nefast zijn voor de nauwkeurigheid en de gebruiksvriendelijkheid. De vraag uit de industrie naar modellering van gesmeerde wrijving komt voor dit project van Dana (Spicer Off-Highway) met als toepassing natteplaat koppelingen. Het project sluit verder aan bij activiteiten die FMTC (Flanders’ Mechatronics Technology Centre) uitvoert in het kader van een project Strategisch Basisonderzoek, waar de regeling van natte-plaat koppelingen nader wordt bekeken. De industrie heeft meer algemeen nood aan betere wrijvingsmodellen en actieve sturingsmethodes, waar de mechatronica haar intrede doet, voor de steeds nauwkeurigere hoge-performantie bewegende systemen. Dit onderzoek is in een meer generieke vorm uit te breiden zodat de verworven resultaten in een later stadium geconcretiseerd kunnen worden voor diverse toepassingen met grensgesmeerde contactoppervlakken zoals geleidingen van robots, werktuigmachines en transportbanden, grensgesmeerde glijlagers, enz... Robots hebben bijvoorbeeld vaak een gravitatiekracht- en positie-afhankelijke wrijving. Er zijn nog andere domeinen waar wrijving een belangrijke rol speelt. Het domein van de structuurdynamica bestudeert de trillingen van systemen waar de wrijving in de verbindingsstukken een grote invloed heeft op de dynamica van structuren. Ook de geomechanica behoort tot het domein van wrijvingsonderzoek. In de biomechanica kan de kennis van de wrijving in gewrichten belangrijk zijn bij de ontwikkeling van prothesen.

1.4

Belangrijkste onderzoeksaspecten en uitdagingen

Literatuurstudie: Het onderzoek start steeds met een grondige literatuurstudie. Deze laat toe om een inzicht te krijgen in de stand van zaken, de ‘state of the art’ wat betreft wrijving. Op deze manier kan een goede kennis verworven worden over de bestaande wrijvingsmodellen. Belangrijk hierbij is hun nauwkeurigheid en hun toepasbaarheid en hun mogelijke aanpasbaarheid of verbetering. Een literatuurstudie kan ook aanzet geven tot de ontwikkeling van nieuwe ideeën en het bedenken van nieuwe experimenten die gebruikt worden voor de validatie. Uitbreiden met de positieafhankelijkheid van de wrijving: Naast de verplaatsingsafhankelijkheid van de wrijving in het pre-glijden regime kan de wrijving ook afhankelijk zijn van de positie. Dit kan het gevolg

XIII


zijn van een slecht ontwerp van een systeem of van een variatie van de normaalkracht. Bestuderen en uitwerken van de rol van de normale-tangenti¨ elerotatie koppeling: Een tangentële verplaatsing heeft omwille van de oppervlakteruwheid ook een normale verplaatsing en zelfs een mogelijke rotatie tot gevolg. In het geval van de normale verplaatsing spreekt men van het ‘lift-up’-effect. Dit is te wijten aan het feit dat de draagkracht van de ruwheidstoppen toeneemt bij stijgende glijsnelheid. Het is te vergelijken met het effect dat optreedt bij hydrodynamische smering. Deze koppeling heeft ook een invloed op het wrijvingsgedrag. Overgang van droge naar natte wrijving, namelijk wrijving bij (grens)gesmeerde oppervlakken: Er bestaat reeds een generiek model op ruwheidstopniveau voor droge wrijving. Dit is aan te passen of verder te ontwikkelen voor het geval van grensgesmeerde wrijving. Bij gesmeerde wrijving kan de stationaire kracht i.f.v. de snelheid weergegeven worden via de Stribeck-curve. Bij grensgesmeerde wrijving is de snelheid nog niet groot genoeg om een vloeistoffilm op te bouwen waardoor er nog steeds een ruwheidstopinteractie van beide oppervlakken is. Dit resulteert in het proces van afglijden. De wrijvingskarakteristieken van een natte koppeling moeten ge¨ıdentificeerd en gemodelleerd worden. Nieuwe tribometer ontwerpen en bouwen: Tot stand brengen van een nieuwe testopstelling voor het vergaren van meetresultaten en het valideren van modellen. De ontwikkeling van een tribometer voor droge en (grens)gesmeerde wrijving met een groot snelheidsbereik, die rotatief is en waarmee de fenomenen van de positieafhankelijkheid en de normale-tangentiële koppeling te onderzoeken zijn. Transmissielijn: Wrijvingsmodel toepassen op een natte koppeling van een machine met automatische overbrenging. Hier zal de testopstelling bestaan uit een transmissielijn met een natte koppeling. Controle: Wat geen echte mijlpaal is in het project maar wel het vermelden waard is, is de controleproblematiek: Na de modelleringsfase volgt het ontwerp van een regelaar. De ontwikkeling van (een) controleschema(’s) komt hier aan bod. Hierbij moet rekening gehouden worden met het sterk niet-lineair karakter van deze problematiek.

XIV


1.5

Toepassingsmogelijkheden (in de industrie)

De kennis van het dynamisch gedrag van grensgesmeerde wrijving is te gebruiken in verschillende toepassingen. Grensgesmeerde wrijving komt niet alleen voor bij automatische transmissies. Natte koppelingen vormen de belangrijkste toepassing in dit project, maar dit wil niet zeggen dat andere toepassingen uit het oog worden verloren. Een model en de controle van grensgesmeerde wrijving heeft volgende toepassingsmogelijkheden: Natte koppelingen: Het wrijvingsgedrag van natte koppelingen voor toepassingen met automatische transmissies heeft een sterke invloed op het dynamisch gedrag van de machine of het voertuig, inclusief de overbrenging zelf. Deze automatische koppelingen kan men voornamelijk terugvinden bij tractoren, en in het algemeen bij “off-road” voertuigen (bulldozers, heftrucks, . . . ). Een wrijvingsmodel en een goede controle van de wrijving kunnen deze dynamica sterk verbeteren en zorgen voor een steeds betere beheersing van het schakelgedrag van de transmissie en een grotere specifieke belasting van de transmissie. In de meeste toepassingen, waar bewegingscontrole aangewend wordt, is wrijving een dominante factor die de performantie be¨ınvloed. De controle van deze wrijving met een regelsysteem is hier ook van toepassing. Medische toepassingen: Haptische interface en terugkoppeling van chirurgische robots, bijvoorbeeld bij niet invasieve ingrepen, zodat de chirurg niet wordt be¨ınvloed door de wrijving en nauwkeurig te werk kan gaan, kan een ingreep vergemakkelijken. Ontwikkeling van knie- en heupprotheses met een beter begrip van de wrijving zodat ze een langere levensduur hebben en een beter, natuurlijker gedrag vertonen. Toepassingen met grensgesmeerde contactoppervlakken zoals geleidingen van robots, werktuigmachines en transportbanden, grensgesmeerde glijlagers, etc. Wrijving in roboticasystemen is een bron van onvolmaaktheden in het volgen van een opgelegd pad die leiden tot “steady-state” fouten en een achterstand (“lag”) op het te volgen pad. Een begrip van wrijving in het domein van design en levencyclus ontwerp is ook relevant (vb. in auto’s zijn er meer dan 2000 tribologische contacten [20]). Een beter begrip van de wrijving kan hier leiden tot een beter ontwerp, een langere levensduur (vb. van lagers, tandwielkasten, enz.) en een lager energieverbruik.

XV


2

Theoretische bijdrage

2.1

Wrijvingsmodel op basis van een Rayleigh stap

De uitbreiding van het generisch wrijvingsmodel voor droge wrijving [3] naar gesmeerde wrijving kan gebeuren aan de hand van een Rayleigh stap module. Deze module simuleert een Rayleigh stap aan de hand van de Reynoldsvergelijking. Elk elementair contact, elke ruwheidstop, van het generisch model kan ge¨ımplementeerd worden als een elementaire Rayleigh stap die het gesmeerd of hydrodynamisch karakter kan weergeven. Op deze manier kan er aan de hand van één model zowel droge (ruwheidstopcontact), als grensgesmeerde (combinatie van ruwheidstopcontact en smering), als volledig gesmeerde (hydrodynamische) wrijving gesimuleerd en eventueel gecontroleerd worden. In dit gedeelte komt de beschrijving van het gedrag van een Rayleigh stap voor, die onderhevig is aan een tangentiële oscillatie rond een zekere offset. Deze offset zorgt ervoor dat de snelheid V of de dimensieloze snelheid λ van het lager niet negatief kan worden. Twee krachten die uit dit model voortvloeien zijn de ’lift up’ kracht, die het gevolg is van de drukopbouw onder het lager ten gevolge van de beweging, en de wrijvingskracht die zeer belangrijk is om een adequaat wrijvingsmodel te bekomen. Een uitbreiding kan bestaan uit het koppelen van twee rug aan rug liggende Rayleigh stappen zodat een negatieve relatieve verplaatsing toegelaten is. Om het oplossen van de differentiaalvergelijking te vereenvoudigen en om het resultaat algemeen, en dus niet voor een specifiek lager, te kunnen interpreteren is de overgang naar een genormalizeerde vergelijking aangeraden. De Reynoldsvergelijking kan genormalizeerd opgelost worden door het invoeren van een horizontale glijparameter Λ en een verticale squeeze film parameter σ. Na herschrijven ziet de Reynoldsvergelijking er dan uit als volgt: ∂h ∂ 3 ∂p − Λ (t) h = σ (1) h ∂x ∂x ∂t In deze vergelijking zijn x, h en t dimensieloos. In het statisch geval is de afgeleide van h naar de dimensielose tijd t gelijk aan 0. Deze twee film parameters zien eruit als volgt: Λ (t) =

σ=

XVI

6ηV (t) L ∆h2 12ηL2 ω ∆h2

(2)

(3)


Ze hebben beide de eenheid van druk [kg/m.s2 ], vermits de druk in de Reynoldsvergelijking niet genormaliseerd is. Ter verduidelijking van een Rayleigh stap is volgende schets gegeven (zie Figuur 5).

x1

1

0

0

L H=

x

H Δh

x2 pm

αL

α

Δh

V (t ) or Λ(t )

h=

h Δh

Figuur 5: Schets van een Rayleigh stap lager met enkele parameters De parameter η, namelijk de viscositeit, die voorkomt in de parameters Λ en σ moet men niet a priori meegegeven. De horizontale dimensies worden dimensieloos gemaakt door ze te delen door L [m] en de verticale door ze te delen door ∆h [m]. De tijd is dimensieloos gemaakt aan de hand van de parameter ω [Hz]. De module van deze Rayleigh stap geeft na berekening met de nodige inputs als resultaat de ‘lift up’ kracht Wn en de wrijvingskracht Ff . Als de wrijvingskracht in functie van de parameter Λ(t) geplot word samen met de Stribeck curve, dan is het fenomeen van frictional lag duidelijk te zien, namelijk in het algemeen een hogere wrijvingskracht bij versnelling dan bij vertraging. De beginwaarde voor Λ(t) kan men bepalen aan de hand van de oplossing van de statische Reynoldsvergelijking. Dit zorgt ervoor dat het overgangsverschijnsel niet te lang duurt en er dus voldoende periodes van de beweging in regimetoestand voorkomen. De oplossing voor een Rayleigh stap bestaat uit twee onderdelen, namelijk uit de oplossing van de statische vergelijking en deze van de dynamische vergelijking. De ‘lift-up’ kracht Wn en de wrijvingskracht Ff die hieruit voortkomen zijn: • Voor de statische genormaliseerde Reynoldsvergelijking Wn =

1 (1 + α) α Λ (H − h) pm (1 + α) = 2 2 (h3 + α H 3 )

(4)

XVII


p pm

α x 0

x’1

x’2

Figuur 6: Parabolisch drukprofiel met negatief gedeelte te wijten aan cavitatie Hierin is pm de maximale druk die onder het lager voorkomt.

Ff = −

∆h pm ∆h Λ (t) + 2 6

α 1 + H h

(5)

In deze uitdrukking zijn H en h dimensieloos. • Voor de dynamische genormaliseerde Reynoldsvergelijking ( #!) • " 3 h 1+α 1 σh 2 αΛ − + 2α + α h 3 2 2 H α+ H • • # α3 σ h σh 1 α + − + 2 3 2 12h3 (6)

1 Wn = 3 H

"

Een opmerking die hierbij gemaakt moet worden is het feit dat er in voorgaande vergelijking geen rekening gehouden is met cavitatie. Het drukprofiel kan er namelijk uitzien als volgt, zie Figuur 6. Om rekening te houden met cavitatie moeten de integratiegrenzen voor de berekening van de ‘lift up’ kracht aangepast worden. Deze manier van werken is toepasbaar omdat het drukverloop parabolisch is en dus niet meer positief kan worden tussen de grenzen 0 en x01 , en x02 en α respectievelijk. Voor de wrijvingskracht heeft men dezelfde uitdrukking als bij de oplossing van de statische Reynoldsvergelijking, maar dan met pm = C4 . C4 is één van de intergartieconstanten. Men heeft dus: ∆h Λ (t) 1 α ∆h C4 + + (7) Ff = − 2 6 H h Voor een meer uitgebreide uitwerking, zie Sectie 2.3 in Hoofdstuk 2.

XVIII


3

Experimentele bijdrage

3.1

Rotatieve tribometer

Er bestaan SAE#2 en andere testopstellingen waarin het globale gedrag van een natte-plaat-koppeling kan gemeten/geëvalueerd worden. Deze proberen het gedrag van een typische automatische transmissie na te bootsen. Het typische bereik van de invoerparameters van een automatische transmissie zijn als volgt: glijsnelheid: 0 - 50 m/s, oliedruk: 0 - 2 MPa, aandrijfkoppel: 0 - 40 Nm, inertie per koppeloppervlak 0 - 0.01 kg/m2 . De eigenschappen, mogelijkheden van de meeste opstellingen zijn zodanig dat ze een automatische transmissie zo goed als mogelijk benaderen. Het is echter niet mogelijk om het wrijvingsgedrag in voldoende fijn detail te meten. Daardoor is er nood aan een ‘dedicated’ testopstelling waarmee het fysische mechanisme dat achter het wrijvingsgedrag schuilt nauwkeurig te kunnen opmeten. De resultaten van deze metingen kunnen in een later stadium vergeleken worden met een model dat gebaseerd is op de fysische fenomenen die optreden in een natteplaat-koppeling. Het doel van deze testopstelling bestaat uit het simuleren van een natteplaat-koppeling aan de hand van twee platen die t.o.v. elkaar kunnen roteren en axiaal bewegen, het bestuderen van het hydrodynamisch karakter en het wrijvingsgedrag van een natte-plaat-koppeling, wat neer komt op het ontwikkelen van een testopstelling met de volgende eigenschappen/mogelijkheden: Opmeten van: • Normale belasting • Overgedragen koppel • Relatieve snelheid • Temperatuur • Spleethoogte Aanleggen van: • Motorkoppel • Motorsnelheid • Inertiekoppel • Normale belasting (statisch en dynamisch)

XIX


Het ontwerp van een testopstelling moet de functionele vereisten en specificaties in acht nemen. Deze zijn specifiek voor elk type machine of systeem en bijgevolg ook voor elk afzonderlijk onderdeel. Het belangrijkste doel van onze tribometer is het meten van de wrijvingskracht (droge en/of gesmeerd) voorkomende in het contact tussen de twee schijven van een automatische transmissie met papiergebaseerd wrijvingsmateriaal. Andere materialen en contact configuraties zijn in het algemeen ook mogelijk. Dit hoofdstuk richt zich op de meting van de wrijvingskracht als functie van de relatieve tangentiële beweging en normale belasting voor een bepaalde specifieke combinatie van materialen, hoewel dit niet noodzakelijkerwijs het gebruik van andere materialen en configuraties uitsluit. Figuur 7 toont een overzichtsfoto van de nieuw ontwikkeld rotatieve tribometer. Op deze foto kunnen de volgende onderdelen worden onderscheiden: (1) shaker (dynamische belasting), (2) shakerhouder, (3) frame, (4) verticale geleiding, (5) hefboom, (6) statische belasting met hefboom, (7) aslagering, (8) Direct Drive actuator, (9) ball spline lager, (10) as (11) balgkoppeling, (12) kuip met de twee wrijvingsschijven (13) drie capacitieve afstandssensoren, (14) drie ringdynamometers voor krachten koppelmeting, (15) basisplaat. De normale kracht wordt aangelegd met de shaker en het koppel is aangelegd met de Direct Drive motor, beide via de as, die is verbonden met de bovenste schijf via de balgkoppeling. De lagere schijf is verbonden met de ringdynamometers.

2 4 Disk (1)

6 8 10

Disk (2) w

12

t d

14

1 3 5 7 9 11 13 15

Figuur 7: Rechts: Algemeen overzicht van de rotatieve tribometer; Links: Sectie ter illustratie van de wrijvingsschijven en de ringdynamometers Een toegevoegde functionele vereiste voor de nieuw ontwikkelde rotatieve

XX


tribometer is de mogelijkheid voor het meten van de normale verplaatsing die gerelateerd is aan wrijving, namelijk het lift-up effect. Dit is een relevant effect, zeker wanneer het gaat om de normale dynamische effect op de wrijvingsgedrag [34, 72]. Dit is een functie, die afwezig is op de meeste eerder ontwikkelde tribometers, zodat zij de meting van dit lift-up effect niet kunnen verwezenlijken. Een dynamische identificatie werd uitgevoerd in de tangentiële en in de normale richting om de bruikbare meetbandbreedte te kunnen bepalen. Op basis van de resultaten van deze dynamische identificatie en de bekomen FRF’s kan deze bepaald worden op een bandbreedte van 100 Hz. Voor een uitgebreide beschrijving zien Sectie 3.10. Deze bandbreedte wordt gebruikt voor het filteren van de experimentele resultaten in de tijd domein. De nieuw ontwikkelde rotatieve tribometer kan worden gebruikt voor macroscopische wrijvingsmetingen en identificatie. De functionele vereisten van de tribometer zijn: (i) nauwkeurige meten van de verplaatsing en de wrijvingskracht, (ii) normale belasting aanleggen en opmeten, zowel statisch als dynamisch, en (iii) de mogelijkheid van toepassing van willekeurige verplaatsingssignalen over een groot bereik van amplitude en frequentie. Deze functies worden zoveel mogelijk ontkoppeld gebaseerd op de beginselen van precisie-engineering. Met deze rotatieve tribometer kunnen experimenten worden uitgevoerd voor wrijvingsidentificatie, die kunnen worden gebruikt voor controledoeleinden. De tribometer kan proeven in een groot scala van verplaatsingen en snelheden uitvoeren, waardoor verschillende wrijvingskarakteristieken, zoals break-away kracht, pre-sliding hysteresis, friction lag in het glijregime, stick-slip en limiet-cyclus oscillaties, het Stribeck en het lift-up gedrag te meten voor één en dezelfde configuratie, en onder droge of gesmeerde wrijvingsomstandigheden. Dergelijke experimentele resultaten kunnen worden gebruikt voor het valideren van fysisch gemotiveerde wrijvingsmodellen, of voor het vaststellen of valideren van empirisch gemotiveerde wrijvingsmodellen, zoals gebruikt in controletoepassingen.

4 4.1

Controle bijdrage Modellering en actieve controle van een gesmeerde koppeling

Toepassingen met natte-plaat-koppelingen worden vaak gebruikt in off-road voertuigen. De controle van deze koppelingen kunnen worden gerealiseerd, gebaseerd op verschillende strategieën. Vermits de industriële toepassingen evolueren, veranderen ook de eisen van de controllers en stijgt de vraag naar een betere efficiëntie en comfort prestaties. Koppelcontrole kan worden

XXI


gebaseerd op het verschil in snelheid van de koppeling. Deze paragraaf beschrijft de ontwikkeling van een koppelgebaseerde controlestrategie en verkent haar haalbaarheid. De gross-glij-eigenschappen van een natte-plaat-koppeling vormen een belangrijk aspect voor de controle-strategieën van de aandrijflijn uit zware off-road voertuigen. Dit is omdat deze koppelingen zijn ge¨ıntegreerd in de transmissie van voertuigen om het koppel naar de wielen te veranderen zonder onderbreking. In de automatische transmissies met natte-plaat-koppelingen, zijn er altijd twee koppelingen actief tijdens het schakelen. Voor het opschakelen wordt de koppeling van de laagste versnelling geopend terwijl de druk op de platen van de koppeling van de tweede versnelling steeds groter wordt. Als gevolg is er steeds koppel aan de uitgaande as wanneer beide koppelingen aan het slippen zijn. Tijdens een dergelijke versnelling veranderd het wrijvingskoppel van de koppeling dat in hoofdzaak afhankelijk is van de normale druk op de platen. Dit betekent dat een goede kennis van de wrijvingsgedrag is vereist om soepele versnellingsveranderingen te kunnen handhaven. Daarom wordt wrijvingsmodellering gebruikt om de wrijvingskracht van een natte-plaat-koppeling te voorspellen. Voor een uitgebreide uitwerking, zie Hoofdstuk 6.

5 5.1

Besluiten en toekomstige ontwikkelingen Belangrijkste bijdragen

Dit hoofdstuk bevat de belangrijkste bijdragen en besluiten en geeft enkele aanbevelingen voor toekomstig onderzoek. Dit proefschrift behandelt voornamelijk de modellering en het experimentele onderzoek van gesmeerd wrijving in mechanische systemen, meer specifiek de koppeloverdracht in gesmeerd koppelingen van zware off-road voertuigen. Eerst werd een bestaand model opnieuw ge¨ımplementeerd, namelijk het generisch model, om de basis te vormen voor de uitbreiding met gesmeerd wrijving. Dan werd een gesmeerd wrijvingsmodel ontwikkeld, gebaseerd op een Rayleigh stap lager, als uitbreiding van de wrijvingsgedraganalyse in gesmeerde contacten, om te worden gebruikt als een stand-alone model en als submodel van het generisch model. Het werk draagt ook bij aan de experimentele meting van droge en gesmeerde wrijving, om tot een beter begrip van het fysische gedrag achter gesmeerde wrijving te komen en voor het probleem van modellering en de compensatie of controle van gesmeerde wrijving in mechanische systemen, met name gesmeerd koppelingen.

XXII


5.1.1

Theoretische bijdrage

Het generisch fysisch gemotiveerd droge wrijvingsmodel is opnieuw ge¨ımplementeerd op een krachtgebaseerde in plaats van een energiegebaseerde manier. Dit model bevat verschillende wrijvingsmechanismen: vervorming van de ruwheidstoppen, adhesie, normale kruip, massa effect van de ruwheidstoppen, ... Op basis van dit model wordt het wrijvingsgedrag beschreven, als gevolg van een relatieve beweging tussen twee contactmakende oppervlakken. Een normale vrijheidsgraad wordt toegevoegd aan het systeem, dewelke het wrijvingsgedrag be¨ınvloed, met name door de inertie van het bewegende lichaam. Het opnemen van deze normale vrijheidsgraad laat een lift-up van het bewegende lichaam toe met als gevolg een mogelijke overgang van droge naar volledig gesmeerd of hydrodynamische wrijving. Dit laatste wrijvingsgedrag wordt beschreven door een nieuw ontwikkelde model, gebaseerd op een Rayleigh stap lager, die de gesmeerd wrijvingsomstandigheden in acht neemt. In gesmeerd omstandigeheden kan een drukopbouwd worden waargenomen met stijgende glijsnelheid te wijten aan het hydrodynamisch effect van de vloeistof. In deze configuratie kan het contact bij hoge snelheid of lage belasting aanzien worden als een hydrodynamisch lager. De smeringsregimes die kunnen optreden zijn Grenssmering (Boundary Lubrication: BL), Gemengde Smering (Mixed Lubrication: ML) en Volledige Smering of Hydrodynamische Smering (Full Lubrication: FL, Hydrodynamic Lubrication: HL) [30]. 5.1.2

Experimentele bijdrage

Een nieuw rotatieve tribometer is ontwikkelde die kan worden gebruikt voor macroscopische wrijvingsmetingen en identificatie. De functionele vereisten van de tribometer zijn: (i) nauwkeurige meten van de verplaatsing en de wrijvingskracht, (ii) normale belasting aanleggen en opmeten, zowel statisch als dynamisch, en (iii) de mogelijkheid van toepassing van willekeurige verplaatsingssignalen over een groot bereik van amplitude en frequentie. Deze functies worden zoveel mogelijk ontkoppeld gebaseerd op de beginselen van precisie-engineering. Een dynamische identificatie van de opstelling is uitgevoerd in de tangentiële en in de normale richting om de toelaatbare meetbandbreedte te achterhalen. Het structurele dynamische gedrag van de setup is geoptimaliseerd tijdens de ontwerpfase, met name door de structurele resonanties op zo hoog mogelijke frequenties te plaatsen.

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Met deze rotatieve tribometer kunnen experimenten worden uitgevoerd voor wrijvingsidentificatie, die kunnen worden gebruikt voor controledoeleinden. De tribometer kan proeven in een groot scala van verplaatsingen en snelheden uitvoeren, waardoor verschillende wrijvingskarakteristieken, zoals break-away kracht, pre-sliding hysteresis, friction lag in het glijregime, stickslip en limiet-cyclus oscillaties, het Stribeck en het lift-up gedrag te meten voor één en dezelfde configuratie, en onder droge of gesmeerde wrijvingsomstandigheden [76, 83]. Dergelijke experimentele resultaten kunnen worden gebruikt voor het valideren van fysisch gemotiveerde wrijvingsmodellen, of voor het vaststellen of valideren van empirisch gemotiveerde wrijvingsmodellen, zoals gebruikt in controletoepassingen. Een toegevoegde functionele vereiste voor de nieuw ontwikkelde rotatieve tribometer is de mogelijkheid voor het meten van de normale verplaatsing in die gekoppeld is met wrijving, namelijk het lift-up effect. Dit is een relevant effect, zeker wanneer het gaat om het normale dynamische effect op het wrijvingsgedrag [34, 72]. Dit is een functie, die afwezig is op de meeste eerder ontwikkelde tribometers, zodat zij de meting van dit lift-up effect niet kunnen verwezenlijken. Door de integratie van capacitieve sensoren voor het meten van de afstand tussen de wrijvingsschijven, is het mogelijk om de normale verplaatsing en haar invloed op het wrijvingsgedrag na te gaan.

5.1.3

Controle bijdrage

Als een bestaand model en als basis voor uitbreiding, is het GMS-model ingevoerd om de wrijving te modelleren, niet alleen in de pre-glij-regime, maar ook in de gros-glij-regime. Het eXtended Generalized Maxwell-slip (XGMS) model laat een wisselend normale belasting toe of voegt een normale vrijheidsgraad toe aan het bestaande GMS model. De normale belastingsafhankelijkheid wordt in acht genomen door het gebruik van een nieuwe geschikte formulering voor de normale belastingsafhankelijke Stribeckfunctie. Deze Stribeckfunctie is gevalideerd met verschillende experimentele datasets. Daarnaast is een passende functie voor de wrijvingscoëfficiënt in functie van de normale belasting voorgesteld, waarin de niet-lineaire effecten vervat zitten. Voor glijdende wrijving kan men een typische daling in de wrijvingscoëfficiënt voor een verhoging van de normale belasting waarnemen. Een normale vrijheidsgraad wordt opgenomen door een extra toestandsvergelijking die de veranderende draag- of normale belastingscapaciteit van het contact te wijten aan de tangentiële beweging beschrijft en die resulteert in het lift-up effect. Het normale belastingsafhankelijke XGMS model wordt ge¨ımplementeerd na dynamische identificatie van de invloed van verschillende parameters op het gedrag van de koppeling. Het doel is om het wrijvingskoppel te controleren door variatie van de normale belasting op het wrijvingscontact. Het model

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schat het wrijvingskoppel die afhankelijk is van het regime van de koppeling. Een feedforward controller is ontwikkeld gebaseerd op het dynamische wrijvingsmodel alsmede een benaderend inverse model van het systeem. De controle van het sluitingsgedrag samen met de controle van de koppeloverdracht, uitgevoerd in lab-omstandigheden, vormt de basis voor het onderzoek in industriële toepassingen. De verkregen resultaten tonen aan dat de controlestrategie, gebaseerd op de voorgestelde wrijvingsmodellering, effectief is in het volgen van een gewenst koppeltraject. De output heeft een koppel steadystate fout van 3 % en een stijgtijd van 0,25 s, die toereikend is voor de beoogde toepassing. Het is aangetoond dat de aanpak van adaptieve controle toepasbaar is op de XGMS model, wanneer de parameters van het wrijvingsmodel veranderen tijdens gebruik. De belangrijkste variërende parameters hier zijn de normale belasting of de aangelegde druk, wat de belangrijkste controleparameter is, de relatieve snelheid en de olie-temperatuur.

5.2

Toekomstig onderzoek

• Een extra parameter, op te nemen in het adaptief wrijvingsgebaseerde controlemodel, zou kunnen volgen uit de sleet van het wrijvingsmateriaal dat ook het wrijvingsgedrag benvloedt te wijten aan een verandering in Stribeck gedrag. Dit is een lange termijn effect. Dit zou een manier zijn om de slijtage van de wrijvingsschijven in rekening te brengen. • De verbinding tussen het fysische gedrag van het lift-up effect en de derde toestandsvergelijking, ge¨ıntroduceerd in de XGMS model, moet worden geanalyseerd. Een functie, vergelijkbaar met de Stribeck functie, die het normale tangentiale verband beschrijft, dient experimenteel te worden gevonden zodanig dat parameteridentificatie kan worden uitgevoerd. • In de modellering, identificatie en controle van de gesmeerde plaatkoppeling moet het sluitingsgedrag van de zuiger verder onderzocht worden. De reden voor het niet-lineaire gedrag in de beweging van de zuiger ligt in de interactie van de dichtingsringen met de behuizing of trommel. Een nauwkeuriger sluitingsgedrag kan de prestaties verbeteren. Positiecontrole van de zuiger kan het maximumkoppel dat ontstaat bij contact verminderen. Voor dit moet de vulfase en het contact van de zuiger met de wrijvingsschijven worden gemodelleerd. Dit zou ook een meer beslissende antwoord kunnen geven voor de reden van het niet-lineair gedrag van de sluiting. • Een alternatieve manier om controle uit te voeren kan bestaan uit het gebruik van fractionele afgeleiden. De invloed van de niet-lineaire wrijving

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op de frequentieresponsiefunctie kan op deze manier worden onderzocht. De lineaire en niet-lineaire componenten kunnen worden gescheiden en een transfertfunctie, in de vorm van fractionele afgeleiden in functie van de amplitude, kan dit niet-lineaire effect weergeven en gebruikt worden als basis voor controle.

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DYNAMIC CHARACTERISATION AND MODELLING OF DRY AND BOUNDARY LUBRICATED FRICTION FOR STABILISATION AND CONTROL PURPOSES

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