A critical analysis on friction force models in dynamical systems
Abstract
As sociedades atuais têm enfrentado bastantes desafios no que diz respeito à economia de recursos naturais, reciclagem e reutilização destes mesmos recursos, assim como na procura por novas fontes de energia mais sustentáveis. O contacto e movimento relativo entre as superfícies de componentes mecânicos são responsáveis pela dissipação de percentagens consideráveis de energia sob a forma de ruído, calor, entre outras, contribuindo para um desgaste acelerado e falha prematura destes mesmos componentes. A correta modelação e previsão do comportamento dinâmico de componentes e mecanismos dos mais variados tipos de máquinas e equipamentos recorrendo a ferramentas computacionais é crucial para o correto controlo do seu funcionamento e prolongamento da sua vida útil, possibilitando a economia de recursos naturais e energéticos que de outra forma seriam gastos na manufatura de componentes de substituição. Deste modo, este trabalho centra-se, principalmente, na análise crítica de modelos de força de atrito teóricos, comparando o desempenho computacional destes na previsão do comportamento dinâmico de sistemas mecânicos de complexidade crescente na presença de fenómenos dissipativos relacionados com as forças de atrito que se desenvolvem entre os diferentes tipos de contacto. Depois da análise dos conceitos fundamentais e teóricos associados aos fenómenos de atrito são analisadas as formulações dos modelos de atrito estáticos e dinâmicos para contactos secos, comummente aplicados para a simulação destes fenómenos dissipativos em problemas que se inserem no quadro da dinâmica de sistemas multicorpo. O projeto de uma bancada simples e versátil para a determinação de dados relativos a forças de atrito é apresentado com o objetivo de calibrar e prever os parâmetros numéricos associados aos diferentes modelos de atrito existentes na literatura e que melhor permitem representar as propriedades estruturais, físicas e cinemáticas de um determinado sistema mecânico. O estudo da influência dos modelos de força de atrito apresentado neste trabalho culmina na análise de um sistema multicorpo, onde é feita uma análise qualitativa e quantitativa dos efeitos destes fenómenos dissipativos no comportamento dinâmico da suspensão de um carro do tipo Formula Student.
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Universidade do Minho Escola de Engenharia Universidade do Minho Escola de Engenharia Francisco Miguel Gonçalves Novais A critical analysis on friction force models in dynamical systems dezembro de 2024 A critical analysis on friction force models in dynamical systems Francisco Miguel Gonçalves Novais UMinho | 2024
Universidade do Minho Escola de Engenharia Francisco Miguel Gonçalves Novais A critical analysis on friction force models in dynamical systems Dissertação de Mestrado Mestrado em Engenharia Mecânica Área de especialização em Conceção e Construção Mecânica Trabalho efetuado sob a orientação do Professor Doutor João Paulo Flores Fernandes Professor Doutor Pedro Filipe Lima Marques dezembro de 2024
ii DIREITOS DE AUTOR E CONDIÇÕES DE UTILIZAÇÃO DO TRABALHO POR TERCEIROS Este é um trabalho académico que pode ser utilizado por terceiros desde que respeitadas as regras e boas práticas internacionalmente aceites, no que concerne aos direitos de autor e direitos conexos. Assim, o presente trabalho pode ser utilizado nos termos previstos na licença abaixo indicada. Caso o utilizador necessite de permissão para poder fazer um uso do trabalho em condições não previstas no licenciamento indicado, deverá contactar o autor, através do RepositóriUM da Universidade do Minho. Licença concedida aos utilizadores deste trabalho Atribuição-NãoComercial CC BY-NC https://creativecommons.org/licenses/by-nc/4.0/
iii ACKNOWLEDGEMENTS The execution of this thesis involved a significant amount of work, dedication and resilience from myself to carry out all the tasks that I set to accomplish. However, all these efforts would have been in vain without the help of several individuals. In the first place, I would like to express my most sincere and profound gratitude to my supervisors, Professor Paulo Flores and Professor Filipe Marques. To Professor Paulo Flores for the invitation to work with him and for sharing his expertise and knowledge with me. To Professor Filipe Marques for his expertise and patience in the countless long meetings and laboratory work. Thank you for your friendship and carefulness throughout this work and for believing in my potential, even when I felt that I was not good enough. It was an honor to work with the best from the best and I hope to one day reach your level. A special acknowledgment to Engineer Filipe Marques from the Mechanical Engineering Department workshops for the multiple suggestions to improve the experimental apparatus design and for all the lessons taught about manufacturing techniques. I would like also to thank all the individuals from Laboratório de Ensaio de Materiais for letting me use the required equipment for the experimental apparatus and for all the tips given. I want to give a special thanks to all my friends from high school (where some of them were met way before that) and from the “Roubocopo” group for all the laughs and good moments throughout the years. I am grateful for your efforts to uplift my spirits during difficult times. To my girlfriend, Marta, a huge acknowledgement for her unwavering love, support and belief in my capacities. Also, thank you for all your patience and understanding. To my brother, Fernando, for his true friendship and support, even during the moments when his actions test my patience. The final and most profound acknowledgment is reserved for my parents, to whom I dedicate this work. I am eternally grateful for their unwavering support and love throughout my life. For the countless opportunities they have provided me, the sacrifices they have made, and for their constant encouragement to become a better person, both professionally and personally. Thank you for the constant desire to see me succeed and achieve the best.
iv STATEMENT OF INTEGRITY Declaro ter atuado com integridade na elaboração do presente trabalho académico e confirmo que não recorri à prática de plágio, nem a qualquer forma de utilização indevida ou falsificação de informações ou resultados em nenhuma das etapas conducente à sua elaboração. Mais declaro que conheço e que respeitei o Código de Conduta Ética da Universidade do Minho.
v RESUMO A critical analysis on friction force models in dynamical systems As sociedades atuais têm enfrentado bastantes desafios no que diz respeito à economia de recursos naturais, reciclagem e reutilização destes mesmos recursos, assim como na procura por novas fontes de energia mais sustentáveis. O contacto e movimento relativo entre as superfícies de componentes mecânicos são responsáveis pela dissipação de percentagens consideráveis de energia sob a forma de ruído, calor, entre outras, contribuindo para um desgaste acelerado e falha prematura destes mesmos componentes. A correta modelação e previsão do comportamento dinâmico de componentes e mecanismos dos mais variados tipos de máquinas e equipamentos recorrendo a ferramentas computacionais é crucial para o correto controlo do seu funcionamento e prolongamento da sua vida útil, possibilitando a economia de recursos naturais e energéticos que de outra forma seriam gastos na manufatura de componentes de substituição. Deste modo, este trabalho centra-se, principalmente, na análise crítica de modelos de força de atrito teóricos, comparando o desempenho computacional destes na previsão do comportamento dinâmico de sistemas mecânicos de complexidade crescente na presença de fenómenos dissipativos relacionados com as forças de atrito que se desenvolvem entre os diferentes tipos de contacto. Depois da análise dos conceitos fundamentais e teóricos associados aos fenómenos de atrito são analisadas as formulações dos modelos de atrito estáticos e dinâmicos para contactos secos, comummente aplicados para a simulação destes fenómenos dissipativos em problemas que se inserem no quadro da dinâmica de sistemas multicorpo. O projeto de uma bancada simples e versátil para a determinação de dados relativos a forças de atrito é apresentado com o objetivo de calibrar e prever os parâmetros numéricos associados aos diferentes modelos de atrito existentes na literatura e que melhor permitem representar as propriedades estruturais, físicas e cinemáticas de um determinado sistema mecânico. O estudo da influência dos modelos de força de atrito apresentado neste trabalho culmina na análise de um sistema multicorpo, onde é feita uma análise qualitativa e quantitativa dos efeitos destes fenómenos dissipativos no comportamento dinâmico da suspensão de um carro do tipo Formula Student. Palavras-chave: Bancada Experimental, Dinâmica de Sistemas Multicorpo, Modelos de Atrito, Simulação Computacional
vi ABSTRACT A critical analysis on friction force models in dynamical systems Current societies have been facing considerable challenges regarding the management of natural resources, recycling, and the search for new and more sustainable energy sources. The contact and relative movement between the surfaces of mechanical components are responsible for the dissipation of significant amounts of energy, which is wasted in the form of noise, heat, among others, contributing to accelerated wear and premature failure of these components. Proper modeling and prediction of the dynamic behavior of components and mechanisms in various types of machines and equipment utilizing computational resources is crucial for controlling their operation and extending their lifespan, enabling the conservation of natural and energy resources that would otherwise be spent on manufacturing replacement components. Thus, this work focuses primarily on the critical analysis of theoretical friction force models, comparing their computational performance in predicting the dynamic behavior of mechanical systems of increasing complexity in the presence of dissipative phenomena related to the friction forces developed between different contact types. After analyzing the fundamental and theoretical concepts associated with friction phenomena, static and dynamic friction models for dry contacts which are commonly applied for simulating these dissipative phenomena in problems within the framework of multibody systems dynamics are examined in detail. The design of a simple and versatile experimental apparatus for determining data related to friction forces is presented with the aim of calibrating and predicting the numerical parameters associated with different friction models found in the literature, which best represent the structural, physical, and kinematic properties of a specific mechanical system. The study of the influence of friction force models presented in this work culminates in the analysis of a multibody system, where a qualitative and quantitative analysis of the effects of these dissipative phenomena on the dynamic behavior of the suspension system of a Formula Student car is carried out. Keywords: Computational Analysis, Experimental Apparatus, Friction Models, Multibody Dynamics
vii TABLE OF CONTENTS Acknowledgements ........................................................................................................... iii Resumo .............................................................................................................................. v Abstract .............................................................................................................................vi Table of Contents .............................................................................................................. vii List of Abbreviations ........................................................................................................ xiii List of Figures .................................................................................................................. xiv List of Symbols ................................................................................................................. xxi List of Tables ................................................................................................................ xxxii 1. Introduction ................................................................................................................ 1 1.1 Motivation ............................................................................................................................... 2 1.2 Scope and Objectives .............................................................................................................. 4 1.3 State of the Art of Friction Force Models ................................................................................. 5 1.4 Organization of the Dissertation .............................................................................................. 9 1.5 Contributions of this Work ..................................................................................................... 11 2. Literature Review ...................................................................................................... 12 2.1 Multibody Dynamics .............................................................................................................. 12 2.2 Experimental Apparatus to Study Friction Phenomena .......................................................... 17 2.3 Influence of Friction on Vehicle Suspension Systems ............................................................ 19 3. A Brief History about Friction .................................................................................... 23 3.1 Friction in the Early Human Civilizations ............................................................................... 23 3.2 Friction Studies by Leonardo da Vinci ................................................................................... 24 3.3 Friction Studies by Guillaume Amontons ............................................................................... 26 3.4 Friction Studies by Leonhard Euler ....................................................................................... 30 3.5 Friction Studies by Charles August Coulomb ......................................................................... 31
xiv LIST OF FIGURES Figure 1.1 – Final energy consumption by sector in the European Union in 2022 (Adapted from Eurostat, 2024) ..................................................................................................................................................... 2 Figure 2.1 – Abstract representation of a generalized multibody system. ............................................. 12 Figure 3.1 – Representation of an Egyptian individual using a bow drill (Dowson, 1998). .................... 23 Figure 3.2 – Transportation of an Egyptian colossus from the tomb of Tehuti-Hetep, El-Bersheh (Dowson, 1998). .................................................................................................................................................. 24 Figure 3.3 – Leonardo Da Vinci’s friction experimental apparatus: (a) Sliding blocks in a horizontal surface disposed in various configurations (MacCurdy, 1955); (b) Experimental apparatus used to measure friction using a weight connected to a sliding block (Pitenis et al., 2014). ............................................ 25 Figure 3.4 – Experimental apparatus developed by Guillaume Amontons to measure friction. A specimen BB is sliding along the test surface AA while being pressed downwards by a CCC flexible body ( flèche ). The spring connected to the scale is used to measure the friction force (Hutchings, 2021). ................ 26 Figure 3.5 – a) Seventeenth century workshop with workers polishing glass; b) Components of the workbench used to polish glass (Hutchings, 2021). ............................................................................. 27 Figure 3.6 – Guillaume Amonton’s experiment using a set of specimen blocks A rubbing against a set of specimen blocks B while being pressed down by the weight body C (Serrano et al., 2015). ................ 29 Figure 3.7 – Amontons’ theory on friction being described by the spring behaviour of asperities. (Desplanques, 2014). .......................................................................................................................... 29 Figure 3.8 – Schematic representation of surface roughness as a set of triangular-shaped asperities in a saw tooth pattern by Euler. Each side of the triangular shape exhibits an inclination, α, in relation to the horizontal plane (Besson, 2013). ......................................................................................................... 30 Figure 3.9 – Adapted Euler’s inclined plane schematic (Dowson, 1998).............................................. 31 Figure 3.10 – Coulomb’s experimental apparatus for the study of sliding friction. Adapted from Coulomb (1821).................................................................................................................................................. 32 Figure 3.11 – Interlocking phenomena between asperities surfaces. Adapted from (Coulomb, 1821). 34 Figure 3.12 – Coulomb’s experimental apparatus for the study of rolling friction (Coulomb, 1821). .... 34 Figure 4.1 - Contact between asperities in the interface of two surfaces. Contact only occurs in the dots, which explains the difference between the real area and the apparent area of contact. ....................... 39 Figure 4.2 – Schematic representation of sectioned asperities, corresponding to the real area of contact, and the apparent area of contact. ........................................................................................................ 39
xv Figure 4.3 – Simple representation of the contact and plastic deformation of asperities (a) initial static position (b) deformation of asperities due to the existence of a tangential velocity, t v . Adapted from Green (1955). ...................................................................................................................................... 40 Figure 4.4 – Asperities of a rough surface in contact with a smooth surface to the applied load W. An increase in the value of W leads to the deformation of the contacting asperities and more asperities come into contact with the smooth surface, increasing the total real contact area, A. ................................... 41 Figure 4.5 – Forces acting on a block placed in an inclined plane. ...................................................... 42 Figure 4.6 – Inclined plane positioned at different angles of inclination. In the bottom right scheme, the block starts sliding along the plane. ..................................................................................................... 43 Figure 4.7 – Graph of the variation of friction force between static and kinetic states as a function of the applied external force f . ..................................................................................................................... 44 Figure 4.8 – Graphical representation of the Coulomb dry friction force model as a function of tangential velocity. ................................................................................................................................................ 45 Figure 4.9 – Macroscopic sliding motion of a block and zoom exhibiting the microscopic phenomenon that explains sliding friction as elastic and plastic deformation of asperities. ........................................ 46 Figure 4.10 – Rolling friction of a disc in a plane a) static situation and pressure distribution b) kinetic situation due to the application of the force f. ...................................................................................... 47 Figure 5.1 – Graphical representation of the Coulomb dry friction model exhibiting the static friction (stiction) phenomena and the Coulomb friction force level for non-null values of tangential relative velocity, t v . ...................................................................................................................................................... 52 Figure 5.2 – Graphical representation of the Coulomb friction model considering viscous effect. ........ 53 Figure 5.3 – Graphical representation of the Coulomb friction model considering the Stribeck effect. . 54 Figure 5.4 – Coulomb friction force model with viscous and Stribeck effects. ...................................... 56 Figure 5.5 – Friction force, f f , variation as a function of time , t , associated with the stick-slip phenomena for a) low velocities, b) high relative velocities. .................................................................. 57 Figure 5.6 – Pre-sliding displacement and association with the micro-slip and macro-slip phenomena.58 Figure 5.7 – Graphical representation of the frictional lag phenomena. ............................................... 59 Figure 5.8 – Theoretical schematization of the non-reversibility of friction force. In the direction from 1 to 2, the system is in an acceleration phase, while from 3 to 4 is in a deceleration phase. ...................... 61 Figure 6.1 – Static friction force models divided by models with stiction and without stiction categories. ............................................................................................................................................................ 64
xvi Figure 6.2 – Linear regularization of the Coulomb friction model proposed by Bernard (1974). .......... 65 Figure 6.3 – Graphical representation of the Threlfall regularization strategy for the Coulomb friction model. .................................................................................................................................................. 66 Figure 6.4 – Rooney and Deravi’s static friction model graphical representation. ................................. 68 Figure 6.5 – Graphical representation of Ambrósio’s regularization approach to Coulomb friction model. ............................................................................................................................................................ 69 Figure 6.6 – Graphical representation of the Karnopp static friction force model. ................................ 70 Figure 6.7 – Graphical representation of the Bengisu and Akay static friction force model. ................. 71 Figure 6.8 – Andersson et al. static friction force model graphical representation. ............................... 75 Figure 6.9 – Graphical representation of Awrejcewicz et al. static friction force model. ........................ 78 Figure 6.10 – Graphical representation of the linear static friction force model with stiction. ............... 81 Figure 6.11 – Schematic representation of the one degree of freedom benchmark problem. .............. 83 Figure 6.12 – Main window of the computational code created in MATLAB to study the benchmark problem. .............................................................................................................................................. 84 Figure 6.13 – Friction force models selection window of the MATLAB code created. ........................... 84 Figure 6.14 – Static friction force models without stiction window from the MATLAB computational code to define the specific parameters of each model. ................................................................................. 85 Figure 6.15 – Graphical results of the benchmark problem for the static friction force models without stiction a) Position [m] vs Time [s] graph b) Relative velocity [m/s] vs Time [s] graph c) Acceleration [m/s2] vs Time [s] graph d) Friction force [N] vs Time [s] graph using the ode45 solver. ..................... 87 Figure 6.16 – Static friction force models with stiction window from the MATLAB computational code to define the specific parameters of each model. ..................................................................................... 90 Figure 6.17 – Graphical results of the benchmark problem for the static friction force models with stiction a) Position [m] vs Time [s] graph b) Relative velocity [m/s] vs Time [s] graph c) Acceleration [m/s2] vs Time [s] graph d) Friction force [N] vs Time [s] graph using the ode45 solver. .................................... 92 Figure 7.1 – Analyzed dynamic friction force models. .......................................................................... 97 Figure 7.2 – Analogy of bristle elastic and plastic deformation with material’s behavior a) brittle material b) ductile material. Adapted from Dahl (1968). .................................................................................... 99 Figure 7.3 – Schematic representation of bristle deflection and location of the bond between asperities. .......................................................................................................................................................... 101
xvii Figure 7.4 – Physical interpretation of bristle interaction between surfaces and bristle deflection, z, which is influenced by the bristle stiffness and damping, represented by the spring and the damper elements, respectively. ....................................................................................................................................... 104 Figure 7.5 – a) Schematic representation of an asperity with d being the distance between reference planes of contacting surfaces and h the relative height to a different reference plane b) Asperity’s deformation, δ, due to tangential relative motion. .............................................................................. 106 Figure 7.6 – Physical representation of the elasto-plastic dynamic friction model where the displacement x of the sliding body is divided into an elastic displacement portion z and a plastic displacement w. 109 Figure 7.7 – Physical interpretation of the Generalized Maxwell Slip friction model as the parallel interaction of N elementary single state friction models. .................................................................... 111 Figure 7.8 – Contact between to different bodies (Body 1 and Body 2) in two different instants, t and tt+ . .............................................................................................................................................. 116 Figure 7.9 – a) Representation of maximum static (𝑧s max), and kinetic (𝑧k max) deflections for a given acting force 𝐹 in a certain time instant t and b) for another time instant given by 𝑡 + ∆𝑡, where the bristle deflection shows an expansion and rotation around the common tangential plane. ........................... 116 Figure 7.10 – Dynamic friction force models window from the MATLAB computational code to define the specific parameters of each model..................................................................................................... 119 Figure 7.11 – Graphical results of the benchmark problem for the dynamic friction force models a) Position [m] vs Time [s] graph b) Relative velocity [m/s] vs Time [s] graph c) Acceleration [m/s2] vs Time [s] graph d) Friction force [N] vs Time [s] graph using the ode45 solver. ........................................... 121 Figure 7.12 – Zoomed region of the friction force vs time graph. ....................................................... 122 Figure 7.13 – Graph of friction force [N] as a function of relative velocity [m/s] for dynamic friction models. .......................................................................................................................................................... 123 Figure 7.14 – Graph of friction force [N] as a function of displacement [m] for dynamic friction models. .......................................................................................................................................................... 124 Figure 8.1 – CAD model of the experimental workbench designed in SolidWorks. ............................. 128 Figure 8.2 – Real-world manufactured workbench for friction force data acquisition. ......................... 129 Figure 8.3 – Free body diagram of the slider during a forward motion cycle. ..................................... 129 Figure 8.4 – Electronic equipment utilized to acquire analog force data from the load cell. ............... 130 Figure 8.5 – Illumination LED and camera utilized for the slider’s tracking procedure. ...................... 131 Figure 8.6 – Diagram of the different data sources, equipments and used softwares to obtain friction force data. .......................................................................................................................................... 132
xviii Figure 8.7 – Displacement profile of the slider for the conducted dynamic tests. ............................... 133 Figure 8.8 – Average friction force-time experimental data obtained from the reciprocating motion test. .......................................................................................................................................................... 134 Figure 8.9 – Average friction force-displacement experimental data obtained from the reciprocating motion test. ........................................................................................................................................ 134 Figure 8.10 – Average friction force-relative velocity results obtained from the reciprocating motion test. .......................................................................................................................................................... 135 Figure 8.11 – Comparison of the literature and optimized parameters of the LuGre friction force model relatively to the experimental data as a function of a) displacement b) relative velocity. ..................... 141 Figure 9.1 – Complete CAD model of the FSUM prototype car of 2023. ............................................ 145 Figure 9.2 – Subsystems of the FSUM car multibody model a) chassis b) steering system................ 145 Figure 9.3 – Suspension subsystems of the FSUM car multibody model a) front left b) rear left. ....... 146 Figure 9.4 – Different stages of rotation for the Bryant angles convention. a) Reference global system; b) First rotation c) Second rotation d) Third rotation. .............................................................................. 148 Figure 9.5 – Flowchart of the determination of Euler parameters from Bryant angles convention. ..... 150 Figure 9.6 – Real-world kinematic joints used to connect the different bodies of the front left suspension subsystem.......................................................................................................................................... 151 Figure 9.7 – Representation of the multibody dynamics computational approach to solve the over constrained problem in a double wishbone A-arm type suspension a) Spherical joints in both A-arm mounting points to the chassis that generate an over constrained model b) Revolute joint that solves the over constrained problem while being kinematically equivalent. ......................................................... 152 Figure 9.8 – Replacement of real-world joints with kinematically equivalent joints in the shock absorber a) Real world revolute joints on both ends of the SA b) Universal joints replacement on both ends of the SA. ..................................................................................................................................................... 153 Figure 9.9 – Real-world kinematic joints used to connect the different bodies of the left rear suspension. .......................................................................................................................................................... 153 Figure 9.10 – Real-world kinematic joints connecting the different bodies that compose the steering subsystem.......................................................................................................................................... 154 Figure 9.11 – Degrees of freedom associated to the car's chassis – 3 translations and 3 rotations. .. 155 Figure 9.12 – Tire testing machine (Adapted from (Calspan, 2019). ................................................. 156 Figure 9.13 – SAE tire forces and moments axis system. .................................................................. 158 Figure 9.14 – Inputs and outputs of the nonlinear empirical Magic Formula Tire Model. ................... 159
xix Figure 9.15 – DNM RCP-2S shock absorber utilized for the suspension subsystem of the FSUM car. 162 Figure 9.16 – Section view of a downhill bicycle shock absorber. The arrows indicate the critical contact zones where the relative motion between surfaces generates friction................................................. 162 Figure 9.17 – Generic schematic of a kinematic cylindrical joint and allowed degrees of freedom. .... 163 Figure 9.18 – Cylindrical joint with reaction forces and moments determined from the Lagrange multipliers vector................................................................................................................................ 165 Figure 9.19 – Schematic section view of a shock absorber with the possible contact points considered. .......................................................................................................................................................... 167 Figure 9.20 – Schematic of the SA projected in the ajnj plane and the resultant reaction forces and moments............................................................................................................................................ 168 Figure 9.21 – Schematic of the SA projected in the mjaj plane and the corresponding reaction forces and moments............................................................................................................................................ 169 Figure 9.22 – Front and rear springs curves considering bumpstops. ................................................ 175 Figure 9.23 – Track layout and dimensions for severe lane-change maneuver according to ISO 3888-2. .......................................................................................................................................................... 177 Figure 9.24 – a) Steering wheel input angles defined for the severe lane-change maneuver at the initial speed of 72 km/h b) Trajectory of the center of gravity of the car during the maneuver. ................... 178 Figure 9.25 – Preliminary analysis of the FSUM car behavior during the obstacle avoidance maneuver a) Tire vertical forces on all four wheels b) Spring deformation on all four shock absorbers of the car. .. 178 Figure 9.26 – Comparison of the shock absorber force for the front left wheel of the FSUM car considering an ideal no friction scenario and considering friction forces calculated by different models................ 180 Figure 9.27 – Detailed views of the total force acting on the shock absorber during the evasive maneuver for a a) rebounding phase b) compression phase. ............................................................................. 180 Figure 9.28 – Total friction force acting on the front left shock absorber during the evasive maneuver. .......................................................................................................................................................... 181 Figure 9.29 – Vertical displacement of 40 mm applied to both front and rear wheels on the four-poster rig with a rad phase delay. ............................................................................................................. 183 Figure 9.30 – Tire modeling as a set of three spring-damper elements for the four-poster rig simulation. .......................................................................................................................................................... 184 Figure 9.31 – Total shock absorber force for the front left wheel during the four-poster rig test. ........ 185 Figure 9.32 – Detailed views of the shock absorbers motion during the four-poster rig simulation a) Rebounding phase b) Compression phase. ........................................................................................ 186
xx Figure 9.33 – Total friction force verified in the front left shock absorber during the four-poster rig simulations. ....................................................................................................................................... 187 Figure 9.34 – Total friction force as function of the relative tangential velocity between the shock absorber bodies during the four-poster rig test. ................................................................................................. 188
xxi LIST OF SYMBOLS Chapter 2 Symbol SI Unit Description DoF n - Number of Degrees of Freedom b n - Number of Bodies i q - Generalized coordinate vector for i-index body i r - Translational coordinates of i-index body in global coordinates i p - Euler parameters vector of i-index body rad Angle of rotation e u - Unit vector i v - Generalized velocities vector for i-index body i r - Vector of translational velocities i r - Vector of translational accelerations i - Vector of angular velocities i - Vector of angular accelerations ( ) ,tq - Set of holonomic constraints t s Time variable - Velocity constraint equations in matrix form D - Jacobian matrix - Right-hand side of the velocities constraint equations - Acceleration constraint equations i v - Generalized accelerations vector - Right-hand side of the acceleration constraint equations - Mass matrix g - Generalized forces and gyroscopic moments - Vector of Lagrange multipliers - Velocity feedback control parameter of Baumgarte stabilization method
xxii - Position feedback control parameter of Baumgarte stabilization method Chapter 3 Symbol SI Unit Description rad Inclination angle f N External force g f N Gravity force Chapter 4 Symbol SI Unit Description i f N Intrinsic frictional force a f N Force required to shear adhered junctions d f N Force associated with deformation energy t v m/s Relative tangential velocity A m2 Total real area of contact W N Normal load f N External force g f N Gravity force n f N Normal force f f N Friction force m kg Mass of the block g m/s2 Gravity acceleration rad Inclination angle - Friction coefficient k - Kinetic friction coefficient s - Static friction coeffic e f N Resultant of external forces applied C f N Coulomb friction force magnitude r M N·m Rolling resistance moment
xxiii Chapter 5 Symbol SI Unit Description k - Kinetic friction coefficient s - Static friction coefficient s f N Static friction force magnitude C f N Coulomb friction force magnitude t v m/s Relative tangential velocity v f N Viscous friction force v - Non-linear velocity dependent variable ( ) t fv - Arbitrary velocity dependent friction function S v m/s Stribeck velocity - Constant factor defined to control the shape of the Stribeck curve - Factor dependent on the geometry of contacting surfaces - Specific curve fitting parameter - Specific curve fitting parameter t s Time k N/m Constant slope of the elastic range of friction force-displacement relationship s m Displacement t s Small time interval fu f N Friction force behavior for the upper curve acceleration phase fl f N Friction force behavior for the lower curve deceleration phase a - Parameter to adjust the acceleration phase slope b - Parameter to adjust the deceleration phase slope
xxx crit c N·s/m Critical damping t m kg Tire mass i a - Longitudinal unit vector j m - Unit vector j n - Unit vector d - Vector defined between i P and j P points - Set of constraint equations g - Vector of external generalized forces M - Mass matrix v - Acceleration vector T D - Transpose of the Jacobian matrix - Vector of Lagrance multipliers ai , mj , sP i , sP j , d - Skew-symmetric matrices of the respective vectors 1 2 3 4 , , , - Reaction forces ( ) i f f - Generalized friction forces vector obtained from reaction forces 1C P - Point contact 1 2C P - Point contact 2 d - Norm of vector d ii P - Auxiliar point in i-index body jj P - Auxiliar point in j-index body ii d m Immutable dimension on i-index body jj d m Immutable dimension on j-index body t v m/s Tangential relative velocity N CP f N Normal force in contact point f CP f N Friction force in contact point f CP n N·m Friction torque in contact point
xxxi s k N/m Spring stiffness pl f N Preload bs k N/m Bumstop stiffness m Spring deformation d c N·s/m Damping coefficient m/s Time rate of spring deformation - Baumgarte stabilization method parameters s k - Vector of spring stiffnesses - Vector of springs deformations d c - Vector of damping constants - Vector of time-rate of spring deformation T i i i - Direction of actuation of spring-damper forces in local coordinates
xxxii LIST OF TABLES Table 1.1 – Energy expended to overcome frictional forces (Adapted from (Holmberg & Erdemir, 2017). .............................................................................................................................................................. 3 Table 2.1 – Quantification of friction force in suspension systems. ...................................................... 22 Table 4.1 – Summary of the most relevant friction laws. Adapted from Mo et al. (2009). .................... 38 Table 4.2 – Factors influencing frictional behaviour (Adapted from Blau (2001)) ................................. 48 Table 6.1 – Benchmark problem simulation parameters. .................................................................... 83 Table 6.2 – Static friction force models without stiction specific parameters. ....................................... 85 Table 6.3 – Solver and average simulation time for the static friction force models without stiction. .... 89 Table 6.4 - Static friction force models with stiction specific parameters. ............................................. 90 Table 6.5 – Solver and average simulation time for the static friction force models with stiction. ......... 94 Table 7.1 – Dynamic friction force models specific parameters. ........................................................ 119 Table 7.2 – Solver and average simulation time for the dynamic friction force models. ..................... 125 Table 8.1 – Optimized parameters fitted to the LuGre model by the MATLAB genetic algorithm. ....... 139 Table 8.2 – Comparison of the percent difference between literature values and optimization fitted values for the LuGre friction force model....................................................................................................... 139 Table 9.1 – Calculation of the total DoF of the FSUM car multibody model........................................ 155 Table 9.2 – Tire structural and geometric input data ......................................................................... 160 Table 9.3 – Static and kinetic friction coefficients for contact points 1 and 2 of the shock absorber (Axsom, 2023; Giesbers, 2012). ..................................................................................................................... 171 Table 9.4 – Front and rear shock absorbers and bump stops parameters. ........................................ 175 Table 9.5 – Solver and stabilization method options for the equations of motion. .............................. 176 Table 9.6 – Parameters for the friction force models utilized in the FSUM car shock absorbers. ....... 179 Table 9.7 – Comparison of the simulation time for the evasive maneuver for the ideal non-friction scenario and the different simulations considering static and dynamic friction force models. Values inside brackets correspond to quantities normalized to the lowest value. ................................................................... 182 Table 9.8 – Comparison of the simulation time and number of function evaluations for the four-poster rig test. Values inside brackets correspond to quantities normalized to the lowest value. ........................ 188
1 1. INTRODUCTION The rapid advancement in computational power over recent decades has transformed the design and validation processes for mechanical systems. The growing market competitiveness demands optimized product development while minimizing costs, resources, and time. These constraints render traditional trial-and-error procedures impractical due to their inefficiencies. Additionally, the increasing complexity of mechanical systems has exacerbated the challenges of performing purely analytical analyses. Consequently, Computer-Aided Engineering (CAE) has emerged as an essential tool in modern engineering practices, facilitating enhanced reliability and efficiency across the product development cycle. CAE tools have become indispensable for their ability to perform simulations and analyses that encompass multidisciplinary perspectives, enabling engineers to achieve higher performance levels in system design. This shift from experimental or trial-based methods to computational approaches reflects the demand for integrated and realistic simulations that consider all factors significantly impacting the performance of a product. These methodologies have become crucial across all engineering domains, particularly in fields like multibody system dynamics and its application to areas such as automotive, robotics, aeronautical and aerospace, which are characterized by their substantial economic and technological relevance. As the field advances, it increasingly necessitates the incorporation of complex phenomena such as friction, which can significantly influence dynamic behavior. Frictional forces play a pivotal role in determining system performance, yet accurately modeling these forces remains a challenge due to their nonlinear and complex nature. Understanding the influence of frictional phenomena is therefore crucial for achieving realistic and reliable simulations in dynamical systems. This work aims to explore and evaluate various friction force models employed in the simulation of mechanical systems. By critically analyzing their applicability, limitations, and impact on system dynamics, it is possible to enhance the fidelity of dynamic simulations and address the growing demands of modern engineering challenges.
2 1.1 Motivation The primary aim of this study is to investigate the impact of friction modeling on the dynamic behavior of mechanical systems. Friction is commonly described as a dissipative force that acts as a resistance between two contacting surfaces that exhibit a relative tangential motion between them. The increasing evolution and development of modern societies highly relies in industrial activities that can only operate when transportation and power generation sectors exist alongside. These sectors involve the movement of people and a wide variety of materials, utilizing diverse machines and mechanical systems composed of numerous moving components with interacting surfaces. The smooth, reliable, and durable operation of such machines is critically dependent on the effective control of friction at these interacting surfaces (Holmberg & Erdemir, 2017). The dissipative nature of friction forces indicates that these forces lead to a loss of energy that in the majority of cases is not recoverable. Figure 1.1 shows the final energy consumption data by sector in the European Union in 2022 from Eurostat (2024). Figure 1.1 – Final energy consumption by sector in the European Union in 2022 (Adapted from Eurostat, 2024) Holmberg & Erdemir (2017) estimated from the energy consumed in a paper mill and in the mining industry that the overall energy consumption to overcome friction in industry is roughly 20% of the total supplied energy. In the transportation sector, the effects of friction in energy consumption were calculated with great detail for the passenger vehicles, trucks and buses, reaching a value of around 32% of total energy usage just to overcome friction. Although there was not enough available data about the aviation and marine sectors, it was estimated that the share of energy wasted to overcome friction is about 10% 4% 25% 31% 13% 27% Final Energy Consumption by Sector (EU, 2022) Other Industry Transport Services Households
3 and 20%, respectively, which leads to an estimate 30% of total supplied energy consumption to overcome friction in the transportation sector (Holmberg & Erdemir, 2017). In what concerns the services and households category, there is a wide range of energy conversion and utilization technologies, such as HVAC systems, lighting, household appliances and business equipment. The mechanical systems which need energy and thus friction can impact their performance are mainly ventilation systems, fans, pumps, etc., where an estimate of 10% of the total energy is used to overcome friction (Holmberg & Erdemir, 2017). In average, considering the economic sectors explored, a total of 20% of the global energy use is utilized to overcome friction. Table 1.1 summarizes the percentage of total energy that is used to overcome friction in the economic sectors considered. Table 1.1 – Energy expended to overcome frictional forces (Adapted from (Holmberg & Erdemir, 2017). Economic sector Percentage of total energy use to overcome friction [%] Industry 20 Transportation 30 Services and Households 10 Average 20 Currently, significant efforts are focused on developing more energy-efficient vehicles and machines, driven not only by economic considerations, but also by the need to meet CO2 emission reduction targets set by the Kyoto Protocol. In certain countries, these efforts are further motivated by the desire to avoid substantial government-imposed financial penalties (Holmberg & Erdemir, 2017). Although friction is commonly seen as an undesirable phenomenon, there are several examples of its utility role in everyday life. Some examples are related to the friction forces generated in the contact patch of automobile tires with the ground, the high friction forces needed in cars disk brakes, the high friction forces associated with clutch-type transmissions, the friction between the soles of shoes and the ground, or the friction force that allows a robot gripper to pick and place objects. On the other hand, there are several applications where friction is undesirable and should be minimized such as bearings and other machine elements, since its dissipative effects cause temperature rise and other phenomena that can lead to excessive wear and ultimately to the premature and/or unexpected failure of mechanical components and systems.
4 The latest technological advancements that can reduce friction are associated with the use of new synthetic lubricants, which contain anti-wear and anti-friction additives, that perform better than traditional mineral oil lubricants and new materials and new surface treatments that reduce friction between contacting surfaces, as well as the introduction of super-hard and self-lubricating particles within the material’s structure using additive manufacturing techniques (Holmberg & Erdemir, 2017). A key technology that allows to better comprehend, model and predict frictional behavior between contacting bodies is the use of computational methods. The utilization of multiscale integrated material modelling and simulation based on sophisticated computer codes, which can integrate various methods such as finite element analysis and multibody dynamics, permits the prediction and correction of the dynamical behavior of mechanical systems considering friction phenomena. By simulating the behavior of a system in a virtual environment, it is possible to reduce monetary expenses both in designing, building and testing prototypes, as well as during the system’s useful life, since the final design should be optimized in terms of energy consumption to take into account friction forces, and by knowing how many more cycles the component/system has left until failure, considering friction phenomena. Computational modeling of friction phenomena has paramount importance in the design of control systems, because using a suitable friction force model, the system’s controller can compensate for the effects of friction without requiring high gain control loops (Canudas De Wit and Lischinsky, 1995). To summarize, the presented work is motivated by identifying friction phenomena and the influence that friction forces have in the dynamic behavior of mechanical systems through computational and experimental approaches in dry contacts. 1.2 Scope and Objectives The main scope of this work centers on the critical analysis and evaluation of friction force models in dynamic mechanical systems. Friction, a nonlinear and multifaceted phenomenon, plays a pivotal role in influencing the behavior of mechanical systems. Accurately modeling and understanding friction is essential for developing simulations that reflect real-world dynamics, enabling engineers to optimize system performance, durability, and efficiency. This work delves into the theoretical foundations, practical implementations, and experimental validations of various friction models, aiming to bridge the gap between theoretical insights and their practical applications.
5 In order to explore the different dimensions in which the critical analysis of friction force models is divided, it is of paramount importance to establish a division between the tasks required to analyze the impact of the dissipative effects of friction in the dynamic behavior of mechanical systems in a systematic manner. The subsequent topics indicate the objectives that must be fulfilled: • present a brief review of the most important contributions throughout the centuries to the scientific knowledge of friction; • present and explore in detail the different phenomena associated with friction; • review the most utilized dry friction force models in literature in the context of multibody dynamics simulations; • develop an intuitive and simple interface to test different static and dynamic friction force models within the context of a simple one degree of freedom benchmark problem; • design and manufacture a simple and versatile experimental apparatus to obtain friction force data; • develop an optimization routine in MATLAB to predict and calibrate friction force model parameters; • develop a rigid multibody model of the 2023 Formula Student of University of Minho (FSUM) car prototype, comprising the modeling of tire-ground contact utilizing the Magic Formula (MF) tire model calibrated with experimental data and subject the obtained model in a variety of dynamic test to assess the influence of incorporating friction phenomena in the behavior of the suspension subsystem. Throughout the different chapters and sections of this work is possible to verify the accomplishments of the aforementioned topics. 1.3 State of the Art of Friction Force Models Friction is a force that opposes the relative motion between two surfaces in contact. It encompasses various phenomena, including dry friction, which occurs between unlubricated surfaces, and lubricated friction, which involves surfaces with a lubricating layer. Each of these frictional interactions is characterized by distinct mechanisms and behaviors that influence the resistance to motion. The importance of studying friction phenomena was recognized several centuries ago, being the first works developed by Da Vinci, followed by Amontons and Coulomb (Dowson, 1998). The Coulomb friction model is a foundational model for describing friction and serves as the basis for many subsequent
6 friction models. It is a relatively simple approach, requiring only a single parameter, the coefficient of friction, to characterize the frictional force. However, the Coulomb model has significant limitations, particularly its inability to capture many complex frictional phenomena. One of its main problems is the discontinuity in the friction force at null relative velocity between the contacting bodies. This discontinuity poses a major challenge in dynamic simulations, as it introduces numerical instabilities. Further research into frictional behavior has suggested that friction forces are greater when an object is at rest (Morin, 1833; Rabinowicz, 1951). This led to the differentiation of two distinct friction coefficients: one for static friction and another for kinetic friction. These variations in frictional force contribute to the occurrence of the stick-slip phenomenon (Rabinowicz, 1957, 1961). The continuous progress made to better understand and characterize friction led to the identification of several complex phenomena such as viscous and Stribeck effects, pre-sliding displacement, micro-slip and frictional lag (Andersson et al., 2007; Olsson et al., 1998). Several friction models have been proposed in literature. They can be divided into multiple categories based on the capability of modeling certain phenomena, complexity or simply by chronological order of appearance. The most common criterion used to differentiate friction models consists of dynamic and static models (Andersson et al., 2007a; Marques et al., 2016; Olsson et al., 1998; Pennestrì et al., 2016). Usually, static friction models are simpler in terms of mathematical formulations and do not have the ability to capture complex phenomena, being sufficiently accurate in describing the behavior of friction forces at steady state. Dynamic friction models are capable of modeling complex phenomena due to the use of extra state variables, which although increases the accuracy of the friction force results, normally produces more computational effort and less efficient simulation times. Different works are found in literature regarding the different phenomena associated with friction and review works concatenating the state of the art static and dynamic friction force models. Olsson et al. (1998) presented a review of the recognized complex friction phenomena and the friction models of interest for automatic control. Regarding friction phenomena, static friction (stiction) is identified as being the friction force level reached before a system initiates macroscopic motion, being the breakaway force the level of force required to leave the static state. The microscopic motion verified between the fully static state and the breakaway force level is described as pre-sliding displacement (Olsson et al., 1998). The frictional lag phenomenon is associated with the hysteresis relationship verified between friction force and velocity where for decreasing velocities the friction force is lower than for increasing velocities. This highly non-linear phenomenon is one of the main reasons that dynamic friction forces are required to describe the physical behavior of contacting surfaces (Olsson et al., 1998). The static friction force models
7 presented are mainly based on the Coulomb friction model considering viscous and Stribeck effects as well as stiction. The only static regularized approaches presented for friction modeling are the Karnopp and the Armstrong friction models. For the dynamic category, Dahl, Bristle Model, Reset Integrator, Bliman and Sorine and LuGre models are presented in a comprehensive way. The LuGre friction force model is further analyzed by comparing the numeric results with experimental data obtained from a laboratory test rig. Andersson et al. (2007) presented various friction models tailored for different sliding contact conditions, including dry, boundary, and mixed lubrication scenarios. The models analyzed include the Coulomb friction model, which is widely used but often inadequate for accurately representing friction behavior; the viscous friction model, which simplifies simulations but may not reflect real conditions; and the Stribeck friction model, which accounts for lubrication effects and varying friction with sliding speed. Additionally, the paper introduces combined models, such as the Coulomb and viscous friction model, which aim to mitigate the limitations of individual models. The Dankowicz and Dahl models are also discussed, focusing on their applicability in control engineering and their representation of friction as a function of displacement. The authors emphasize the importance of considering micro-slip phenomena and the stochastic nature of friction, suggesting that friction can be modeled as a stochastic process influenced by the interactions of surface asperities. Regarding the experimental apparatus, the authors utilized a one-degree-of-freedom (1 DoF) dynamic system to study the behavior of different friction models through numerical simulations conducted in MATLAB. Marques et al. (2016) provides a very complete review and comparison of various friction force models for multibody mechanical systems in a dry sliding regime. The authors categorize friction models into two primary groups: static models, which describe steady-state friction behaviors, and dynamic models, which incorporate additional state variables to capture time-dependent friction effects. Complex phenomena such as stick-slip, viscous friction, and the Stribeck effect are also described in a comprehensive manner within the friction force models that consider them. A comparative analysis of the computational efficiency of the different friction force models was conducted for the ode45 and ode15s solvers of MATLAB. The basic Coulomb model serves as a foundation, describing friction as a force proportional to the normal force that acts in the opposite direction of the relative motion between contacting bodies. Due to its limitations, variants such as models with viscous and Stribeck components were developed to address speed-dependent and transitional friction phenomena. Dynamic models, like the LuGre model,
14 and i p represents the vector of Euler parameters for body i, which can be written in the following form T T 0 1 2 3 cos cos 22 ie e e e e == pu (2.4) where represents the angle through which the body must rotate about a unique axis, given by the unit vector e u to align its reference frame with the global reference frame. Although using Euler parameters requires four coordinates to describe the orientation of the body, the system retains only three independent coordinates due to the constraint that i p remains a unit vector. Regarding generalized velocities and accelerations, the problem of singular positions does not exist. Thus, the generalized velocities of a rigid body can be written as T i i i =vr (2.5) in which i r is the vector of the translational velocities and i is the vector of angular velocities of body i. By differentiating Equation (2.5), the generalized accelerations vector is obtained being defined as T i i i =vr (2.6) where i r is the vector of the translational accelerations and i is the vector of angular accelerations. The degree of accuracy wanted for a certain dynamic analysis dictates the type of joints modeled to simulate the connections between the different bodies of a multibody system that constrain their relative motion. Two main categories of joints can be distinguished in the framework of multibody dynamics: ideal joints and imperfect joints. The ideal joints are simply kinematic relations defined between the connected bodies that resort on algebraic equations that constrain the motion through vectors, coordinates of points, etc. The imperfect joints are usually based on contact-impact events and can model more complex phenomena such as clearance, friction and wear. Despite the enhanced realistic behavior of imperfect joints, their computational efficiency is normally lower compared to the simpler kinematic joints. Ideal joints are mathematically defined by kinematic constraint equations, where the number of independent equations represents the number of constrained degrees of freedom. Kinematic constraints are classified as holonomic or nonholonomic, depending on whether they can be integrated to form a position constraint or not, respectively (Flores, 2015). Additionally, constraints are categorized as
15 rheonomic if they explicitly depend on time or scleronomic if not (Flores, 2015). Therefore, in a constrained multibody system, the set of holonomic constraint equations can be expressed as follows ( ) ,t=q0 (2.7) where q denotes the vector of the generalized coordinates of the system and t is the time variable. The velocities constraint equations are obtained from differentiating Equation (2.7) in order to time and are represented in matrix form as −=Dv 0 (2.8) in which D is the Jacobian matrix of the constraint equations, v represents the vector of the generalized velocities of the system and represents the right-hand side of the velocities constraint equations. The second derivative of Equation (2.7) with respect to time yields the constraint equations at acceleration level defined as =−Dv 0 (2.9) where v is the vector of generalized accelerations and is the right-hand side vector of the acceleration constraint equations. In this work, the equations of motion of the multibody system are derived according to NewtonEuler’s methodology. The Newton-Euler equations establish a relation between the motion of the center of mass of the rigid bodies and the external and reaction forces and moments of the kinematic constraints, which are formulated as T Mv +D = g (2.10) where M represents the mass matrix of the system which encompasses both the mass and the inertia tensor of the system’s bodies and the vector g denotes the vector of the external generalized forces and gyroscopic moments. The term T D includes the reaction forces and moments due to the kinematic joints in the global coordinate system, in which denotes the vector of the Lagrange multipliers, where its size is equal to the number of holonomic constraints, with each element representing the reaction force or moment associated with a specific kinematic constraint. In dynamic analysis, a unique solution is obtained when the algebraic constraint equations at the acceleration level are considered simultaneously with the differential equations of motion (Flores, 2015). The matrix form of the dynamic equations of motion of a multibody system can be obtained by introducing
16 Equation (2.9) in Equation (2.10) which results in the following system of differential algebraic equations (DAE) T = vg MD D0 (2.11) that is then solved for accelerations vector, v , and Lagrange multipliers vector, . At each integration time step, the acceleration vector, v , along with the velocity vector, v , is integrated to compute the system’s velocities and positions for the subsequent time step. This process is iteratively repeated until the final analysis time is reached (Flores, 2015). Solving Equation (2.11) determines the accelerations of the system at each time step, but does not ensure that the position and velocity constraints are satisfied, since the constraints at both the position and velocity levels are not explicitly included in the equations of motion. Consequently, errors introduced through numerical integration can lead to violations of both position and velocity constraint equations. To address constraint violations, several methods are available, categorized into three main approaches: constraint stabilization techniques, coordinate partitioning methods, and direct correction formulations, which are well detailed and analyzed in literature (Marques et al., 2017). In this work, the Baumgarte stabilization technique (Baumgarte, 1972) was introduced in the dynamic equations of motion in order to keep the violation of the constraints under acceptable values. It should be noted that the constraint stabilization approaches do not fully eliminate the violations but allows to keep their values under control. In this constraint stabilization approach, the acceleration constraint equations are modified to include two additional terms that provide feedback control to correct violations of the position and velocity constraints. The system of equations of Equation (2.11) can then be rewritten as T 2 = vg MD D0 − − (2.12) in which and are positive values that represent the feedback control parameters of the velocity and position constrain violations, respectively. The numerical resolution of Equation (2.12) should be carefully addressed. The multibody dynamics equations of motion are differential and algebraic equations (DAE), but it is useful to convert them to ordinary differential equations (ODE), since in the later form it is possible to solve the equations through numerical integration algorithms that are rather simple to implement (Flores, 2015). The most
17 widely used numerical integration methods in the literature include the Euler method, Runge-Kutta methods, and Adams predictor-corrector methods (Flores, 2015), which are not the most accurate nor efficient due to their fixed integration time-step nature. However, MATLAB and other programming languages have ODE solvers already integrated that can utilize variable time-step integrators, which results in improved accuracy and efficiency of the solution of highly non-linear problems. 2.2 Experimental Apparatus to Study Friction Phenomena In the middle centuries of the previous millennium, many scientists were interested in understanding the dissipative phenomena associated with parts that exhibited relative motion between them, which led to wear and premature failure of mechanical components. Leonardo da Vinci constructed simple wooden devices with different weights to investigate friction forces, driven by his interest in optimizing mechanical systems like gears and pulleys. Guillaume Amontons developed various sliding apparatuses using blocks, plates and ropes to measure frictional resistance, aiming to address mechanical inefficiencies in machines. Charles-Augustin Coulomb designed more sophisticated apparatuses to analyze friction, motivated by the need to improve naval and structural engineering. These tools were built to systematically measure and quantify friction for solving real-world mechanical challenges. Experimental apparatuses have been pivotal to understanding friction phenomena, providing controlled environments to observe and quantify frictional effects. From hysteretic behaviors to stick-slip dynamics and non-reversible friction characteristics, these setups have allowed researchers to validate theoretical models and refine them. There are apparatuses built specifically to study certain isolated friction phenomena. The work by Awrejcewicz & Olejnik (2007), for example, consists of a two-degree-of-freedom mechanical system with a feedback-reinforced laboratory rig, where a block moving on a belt coupled with springs and sensors allows to measure both friction force and displacement in order to explore stick-slip dynamics and develop a friction model accounting for the behavior at both positive and negative tangential velocities. Wojewoda et al. (2008) built a setup designed to study hysteretic behaviors in dry friction, focusing on pre-sliding displacement and non-reversible friction characteristics. The apparatus captures phenomena such as the Stribeck effect and varying breakaway forces in order to investigate hysteresis during transitions from static to dynamic friction and validate friction models through experimental results compared with numerical simulations.
18 The phenomenon of non-reversibility is another complex phenomenon associated with friction, that due to its complexity and stochastic nature, has led to the construction of experimental apparatus dedicated to capture this specific phenomenon. Wiercigroch et al. (1999) built a Coulomb oscillator equipped with a pneumatic actuator to control the normal force, allowing precise measurement of friction dynamics and non-reversibility of friction force during acceleration and deceleration phases. This apparatus allowed to investigate non-reversible friction characteristics and validate theoretical friction models through dynamic experimentation. Guo et al. (2008) conducted similar studies utilizing a reciprocating motion test rig with a vibration table and laser displacement sensors to measure frictional forces during sinusoidal and rectangular wave motions with the goal of modeling and identifying nonreversible friction characteristics, including hysteresis in pre-sliding and sliding regimes, and to analyze the effects of relative velocity and acceleration in the overall dynamic response of the system. Friction phenomena identification and modeling are important to develop energy efficient and high precision mechanical and electromechanical systems. Thus, it is of paramount importance to also build tailored made experimental rigs to test real-world systems in order to correctly predict and compensate for the dissipative effects of friction forces. The work of Kebairi et al. (2015) describes an apparatus for accurately measuring the influence of friction in modeling actuator dynamics, incorporating static and LuGre friction models, and evaluating their performance under various operating conditions. The experimental rig includes a DC motor, a two-stage spur gear system, a helical spring, and position sensors to measure friction torque and actuator displacement, simulating real-world engine conditions that affect the operation of a Bosch GPA-S electromechanical actuator used in Diesel engines to control air intake. The incorporation of friction phenomena in automation and mechatronic systems, such as pneumatic and hydraulic cylinders or robots, is crucial for ensuring precision and reliability in dynamic control systems. Friction, as a dissipative phenomenon, significantly influences system behavior, especially at low speeds or during transitions between static and dynamic states. Accurate identification and modeling of friction allow control systems to mitigate its negative effects, enhancing performance in terms of stability, accuracy, and energy efficiency. Felix & Silveira (2018) built a low-cost rig using a Raspberry Pi controller and pneumatic instrumentation to measure friction forces in real time. The setup calculates friction based on pressures in actuator chambers and piston dynamics to analyze the static and dynamic friction behaviors of pneumatic actuators, enabling precise parameter identification for friction compensation in industrial positioning systems. Wakasawa et al. (2018) performed tests in a pneumatic cylinder driven by a stepper motor, equipped with load cells, and pressure and velocity sensors to isolate and measure friction
19 contributions from rod and piston seals under varying lubrication conditions. Feng et al. (2019) performed an identification procedure of nonlinear friction parameters using an improved Stribeck model and implemented friction compensation, improving trajectory accuracy and mitigating low-velocity issues in electro-hydraulic systems. The real-world application consisted of a robotic excavator arm equipped with hydraulic cylinders, pressure sensors, and displacement sensors to measure friction forces during controlled movements at constant velocities. Qian et al. (2022) studied the impact of piston speed, pressure, and pressure differences on friction in pneumatic cylinders utilizing a friction test rig featuring two pneumatic cylinders mounted face-to-face with their chamber pressures balanced to eliminate pressure effects on force measurements, enabling precise and isolated evaluation of friction forces. The experimental test rig built and tested in the context of this work focuses specifically on the parameter estimation of numerical friction force models, allowing a better understanding of the influence of parameter selection in the dynamical behavior of mechanical systems. 2.3 Influence of Friction on Vehicle Suspension Systems Friction is modeled in suspension systems through various formulations such as finite element analysis, vibrations and multibody dynamics. In highly complex and realistic simulations, multiple formulations can be employed in a single model, by combining finite element analysis and multibody dynamics, for example. The influence of friction in the dynamical behavior of suspension systems can be explored by two different perspectives: comfort and NVH (Noise Vibration and Harshness) requirements, usually of capital importance for regular passenger vehicles or dynamic performance, for sports and racing cars. Primarily, friction forces affect the vertical dynamics of a vehicle. The increased sales of electric vehicles in recent years brought particular interest in studying the influence of friction in suspension systems, because the substitution of the internal combustion engine, that produces higher noise and vibration levels compared to an electric or hybrid powertrain, leads to an increased perception of the noise generated by suspension components, reducing the overall NVH quality of a certain vehicle. The topic of friction influence analysis in shock absorber's performance is not a novelty in literature. Several works explore the critical role of friction in shock absorbers and suspension systems, focusing on its effect on vehicle dynamics and ride comfort. Lizarraga et al. (2008) provide insights into the dual role of friction in suspension systems and its influence on both comfort and energy dissipation. Their research addresses limitations in existing friction
20 models by proposing a new formulation that combines the Stribeck effect with viscous friction, improving the understanding of shock absorber behavior. A specialized hydraulic testing device was developed to isolate and measure friction, minimizing hydraulic forces' influence. Experimental data confirmed the presence of the Stribeck effect, highlighting stick-slip phenomena and asymmetrical friction characteristics during rebound and compression phases. Benini et al. (2017) explored the impact of friction in suspension systems for racing cars, finding that friction significantly affects the vehicle’s vertical dynamics, especially at low frequencies near the car's natural pitch frequency. Using a Formula 4 car tested on a Four Poster Rig, the study highlights that friction, particularly in the front suspension, acts as an additional damping element, influencing vehicle behavior during dynamic testing. Friction’s role in vertical response and transient maneuvers indicates that ignoring it in simulations can result in an inaccurate assessment of the overall vehicle dynamic behavior. Körner & Mayer (2020) focus on the friction characteristics of dampers in relation to driving comfort and vehicle dynamics. The study reveals that frictional forces at damper contact surfaces, between seals made from rubber and other materials, significantly influence the damper’s response to excitation and the initial breakaway force at the start of movement. The research demonstrated that internal pressure and relative velocity impact friction. Unlike pure Coulomb friction, damper friction exhibits elastohydrodynamic characteristics, influenced by movement direction and surface conditioning. Although frictional forces are generally lower than hydraulic damping forces, the authors stated that understanding them is critical for damper optimization and comfort enhancement. Herzog & Augsburg (2021) presented a two-part study on modeling friction in shock absorbers. The first part develops a dynamic friction model within a finite element framework, focusing on unwanted static friction generated by seals and guides. Using a hybrid model, the researchers integrated physical and dynamic friction models, applying the LuGre model to simulate contact friction. This model, validated through high accuracy simulations in Ansys Mechanical, accurately represents shock absorber friction dynamics, enhancing shock absorber design efficiency by reducing physical testing needs. The second part (Herzog & Augsburg, 2021) validates the friction model using an experimental setup with two specialized test rigs, which measure friction at key contact points in shock absorbers, such as the rod guide assembly and piston/tube interfaces. The data obtained confirmed the simulation model's accuracy in predicting friction across various components, though some deviations were noted in the floating piston/tube interface due to the complexity of seal deformation modeling. The validated model effectively aids shock absorber design and optimization.
21 Lv et al. (2021) examined the kinematics and compliance (K&C) characteristics of double wishbone air suspensions (DWAS), addressing limitations in traditional models that often overlook friction and joint clearances. This research incorporates a nonlinear dynamic model using ADAMS/VIEW, integrating flexible components and friction forces to enhance suspension simulation accuracy. Experimental validation using a K&C test rig demonstrated that including friction and joint clearances in the model significantly improved predictions of suspension performance, accurately replicating force-related and displacement-related behaviors observed in real-world conditions. The findings underscore the importance of these factors in suspension design, since they have substantial impacts on kinematics, though they affect compliance less. Deubel et al. (2024) investigated the importance of accurately modeling friction in shock absorbers, noting that friction characteristics are often treated as quasi-static, despite being influenced by dynamic conditions. Using modified valve-free shock absorbers, the researchers observed dynamic friction properties, which revealed that dynamic friction forces significantly exceed quasi-static friction forces, especially at lower velocities (up to 60 mm/s) and under substantial side forces. The study found that side force and velocity should be considered in friction analysis, as conventional models underestimate total shock absorber forces by neglecting dynamic friction as a key component. This understanding can lead to improvements in active suspension control and vehicle simulations by allowing better parametrization of shock absorber models. A common problem across the various studies on the friction role in the dynamic behavior of suspension systems is the general way friction is studied within these systems, where real world road events are not considered nor a clear quantification of the friction effect is made (Deubel & Prokop, 2024). Another notable issue is the limited availability of studies that provide accurate, reliable data on friction levels within vehicle suspension systems. This scarcity applies both to overall suspension friction and to friction in specific components. Furthermore, there is a lack of comprehensive research on how various parameters, such as temperature and amplitude, may influence suspension friction (Deubel & Prokop, 2024). The limited number of studies in this area may be attributed to several challenges: (i) the need for extensive measurements to achieve accurate and reproducible friction data, (ii) the significant effort required to model friction with precision, both in terms of capturing all the phenomena associated with friction, as well as the required computational power, and (iii) the inherent difficulties in practically validating how variations in friction affect vehicle dynamics (Deubel & Prokop, 2024). Table 2.1 shows a summary of the quantification of friction force influence on suspension systems.
22 Table 2.1 – Quantification of friction force in suspension systems. Authors Data Source Friction Force Values Kölsch (1994) Information obtained from Deubel & Prokop (2024) paper. 25 N Manger (1995) Information obtained from Deubel & Prokop (2024) paper. 30 – 45 N Lizarraga et al. (2008) Experimental test of a shock absorber in a dedicated test rig and numerical simulation using the Coulomb friction model modified to account for the Stribeck effect. Numerical Simulation Compression: 32 N Rebounding: 23 N Experimental Data Compression: 22 N Rebounding: 15 N Fujimoto et al. (2018) MATLAB/Simulink numerical data comparison with experimental data 20 – 150 N, depending on the type of shock absorber and damping level Wegener (2020) List of front and rear axles data for various vehicles. No information about test conditions. 75 – 350 N Körner & Mayer (2020) Experimental data obtained from dedicated testbench. Asymmetrical behavior of the shock absorber, ranging from 20 to 70 N for compression and 10 to 40 N for rebounding, depending on excitation frequency and pressure. Herzog & Augsburg (2021) FEA analysis in Ansys and experimental data from dedicated test rig. Values ranging from 35 N for compression to 30 N for rebounding. Deubel & Prokop (2024) Analytical formulation based on kinematic constraint forces and experimental data from full-scale four-poster rig 62 – 134 N for different combinations of displacement and wheel lateral forces. In this work, a model of the Formula Student of University of Minho will be used to assess the influence of friction forces in the dynamic behavior of suspension systems and its influence on the overall vehicle dynamics.
23 3. A BRIEF HISTORY ABOUT FRICTION Friction represents a challenging phenomenon within the dynamic behavior of mechanical systems since ancient times. Early civilizations, such as the Egyptians, developed ingenious methods to manage friction while moving massive stones and statues. The Renaissance marked a turning point with Leonardo da Vinci’s pioneering experiments, laying the foundation for the study of friction. Following the first scientific methodology to study friction presented by Da Vinci, Guillaume Amontons formalized the first laws of friction, followed by the critical refinements of Leonhard Euler and Charles-Augustin de Coulomb. This chapter traces the evolution of friction’s understanding, from the practical applications of each historic era to the outcome scientific knowledge of the last century. 3.1 Friction in the Early Human Civilizations It is known that throughout the ages the importance of friction and resistance of motion has no doubt been recognized. Evidence from early civilizations, such as rock art and engravings, indicate an awareness of this phenomena’s influence on everyday tasks. The Early Human Civilizations, more specifically the Sumerian and Egyptian civilizations, have developed tools in which the friction phenomena were beneficial and of great use such as the so-called drills that employed alternating rotary motion and were developed to produce fire and holes (Dowson, 1998). Figure 3.1. shows the specific example of an Egyptian individual using a bow drill. Figure 3.1 – Representation of an Egyptian individual using a bow drill (Dowson, 1998).
30 3.4 Friction Studies by Leonhard Euler The first mathematical studies of friction were attributed to Leonhard Euler in the 18th century. Euler (1750b) treated the aspects concerning the equilibrium state and uniform motion of bodies, developing mathematical formulations to describe these phenomena. The first mathematical approach to friction proposed by Euler was the interlocking asperity theory, where the surfaces of contacting bodies were represented by a set of triangular-shaped asperities in a sawtooth pattern, which interlock perfectly and are inclined at an angle, , in relation to the horizontal plane (Besson, 2013). Figure 3.8 shows a schematic representation of the sawtooth pattern constituted by small triangular-shaped asperities that originate the roughness of a surface. Figure 3.8 – Schematic representation of surface roughness as a set of triangular-shaped asperities in a saw tooth pattern by Euler. Each side of the triangular shape exhibits an inclination, α, in relation to the horizontal plane (Besson, 2013). The block placed in an inclined plane, represented in Figure 3.9, is in equilibrium state when the external applied force, f , required to move the block up the plane and the frictional force acting on the block, ff, exhibit the following equality relation f g g sin cosf f f f = = (3.1) Utilizing the tangent trigonometric function, it is possible to rewrite Equation (3.1) in a more simple and convenient form sin tan cos = = (3.2) which leads to the conclusion that Euler demonstrated the dependency relation of the friction coefficient, , with the tangent of the inclined plane angle, 𝛼. Moreover, it can also be attributed to Euler the introduction of the greek letter, 𝜇, into the friction study area to represent the coefficient of friction.
31 Another important contribution of Euler to the scientific knowledge of friction is the distinction made between static friction and kinetic friction. The distinction between static and kinetic friction was also made using the inclined plane apparatus represented in Figure 3.9. Figure 3.9 – Adapted Euler’s inclined plane schematic (Dowson, 1998). Although Euler did not present a justification that allowed to explain the loss of mechanical energy due to friction, since that problem did not belong to the scientific paradigm of the 18th century, demonstrations were made and led to the conclusion that if the plane was inclined until reaching a limit slope where the static equilibrium was verified, then a very small angle increase would result in a disturbance of the equilibrium state of the block and it would exhibit a fast linear motion down the plane. This observation contradicted Euler’s first hypothesis that a small angle increase would result in the block’s motion with a very small velocity, concluding that the kinetic friction, associated with the motion of a body, must assume a smaller value in comparison to the static friction (Dowson, 1998). 3.5 Friction Studies by Charles August Coulomb Charles August Coulomb (1736 – 1806) is undoubtedly a pioneer researcher in the scientific area of friction. As described by Dowson (1998), Coulomb’s studies on friction are a good example of highquality scientific work prompted by practical problems. The base for Coulomb’s friction works relies on the developments by Amontons, although Coulomb demonstrated through several experiments that some of his conclusions contradicted the ones formulated by Amontons. Coulomb studied the sliding and rolling types of friction, proposing experimental validation of their behavior in practical large-scale applications, mainly related to pulleys, capstans, and the launching of ships on slipways (Dowson, 1998).
32 Figure 3.10 shows the apparatus developed by Coulomb to study the sliding friction between plane surfaces, which is composed of a solid wooden table, serving as the lower surface of the contacting pair, represented by (a), and by a sledge, represented by (b), that can have rails of various widths and materials attached. Figure 3.10 – Coulomb’s experimental apparatus for the study of sliding friction. Adapted from Coulomb (1821). The influence of the normal force in the friction force that acts in the reverse direction of the sledge sliding motion was studied by placing different weights on the sledge of Figure 3.10 (d) schematic. The sliding force was exerted by placing a weight along the length of a bar which is connected to a pulley that pushed the sledge over the table through a rope. The data obtained with this experiment was for various combinations of tribological pairs of materials such as oak, green oak, guaiac wood, fir and elm for the woods, and iron and copper for what concerns the metallic materials. Dry and lubricated conditions were analyzed using lubricants such as axle grease, tallow, soot, water and olive oil, both in smooth and rough surfaces, for pressures ranging up to approximately 30.4 MPa and maximum sliding velocities of about 18 km/h. The static state of motion time was also an important variable in Coulomb’s theories of friction and his experimental data showed stationary times ranging from 0.5 s up to 4 days (Besson, 2013; Dowson, 1998).
33 Coulomb investigated the influence of four different types of factors that can have a significant impact on friction: (i) the nature of the materials in contact as well as the effect of dry and lubricated conditions, (ii) the area of contact, (iii) the normal load supported by the surfaces, which Coulomb described as pressure, and (iv) the duration of the contact between surfaces (Besson, 2013; Dowson, 1998). The main contributions to friction knowledge resulting from the experimental findings of Coulomb can then be exposed: (i) the friction exhibits an initial rise but soon reaches a maximum for tribological contacts of wood sliding on wood under dry conditions, being the force of friction essentially proportional to the load; (ii) the friction force for wood sliding on wood is essentially proportional to the load at any speed, but the kinetic friction force, normally associated to the steady state sliding motion, has a lower magnitude compared to the static friction force, especially those seen after long periods of rest; (iii) the friction force for metals sliding over metals without lubricant is proportional to the load and there is no difference between the magnitude of both static and kinetic friction; (iv) for the metal-wood tribological pair under dry conditions, the static friction rises at a small rate, taking some hours or even days to reach its limit, while for metal-metal pairs this limit is reached almost immediately, and for wood-wood pairs this static friction rise takes up a short minute time. It was also verified that for wood-wood and metalmetal pairs, the kinetic friction force magnitude under dry conditions is almost independent of the tangential velocity, while for wood-metal contacts, the friction force magnitude increases with the increase of tangential velocity between contacting surfaces (Dowson, 1998). The contributions by Coulomb to the development of friction did not only encompass the practical effects of friction, but also delved into the theoretical explanation of friction phenomena. According to Coulomb, friction could have two possible causes, one of them being the interlocking of asperities on surfaces, as represented in Figure 3.11, that can only be separated by bending, breaking or rising above other, or by adhesion/cohesion phenomena verified between the surfaces in contact (Besson, 2013). Coulomb verified that only the interlocking of surface asperities was responsible for the friction phenomena, while cohesion represents a small influence, since friction is proportional to the normal load and independent of the contact area. The cohesion represents an inevitable increase of friction because there are multiple contact points between surfaces, but the magnitude of the referred contacts, although not being null, can be negligible for a macroscopic analysis (Besson, 2013). The asperities of contacting surfaces, after the application of a tangential force, deform until a critical point where the opposing asperities of the adjacent surface slip out of the interlocking mesh and sliding motion initiates, as it is represented in the Figure 3.11 (c) scheme. Since the deformation of asperities is assumed to be constant in the sliding phase, the friction force is also constant, fact that led
34 a third law of friction to be attributed to Coulomb where is stated that “kinetic friction is independent of the sliding velocity” (Dowson, 1998). Figure 3.11 – Interlocking phenomena between asperities surfaces. Adapted from (Coulomb, 1821). Another important aspect of Coulomb’s friction discoveries is correlated to experimental apparatuses built to study the governing phenomena of rolling friction. Figure 3.12 shows a workbench projected to study the rolling friction phenomenon, where Coulomb stated that rolling friction was dependent on the normal reaction force, although presenting a magnitude lower that sliding friction and that the resistance to rolling was inversely proportional to the radius of the roller (Dowson, 1998). Figure 3.12 – Coulomb’s experimental apparatus for the study of rolling friction (Coulomb, 1821). It is clear that Coulomb was one, if not the most influential researcher in the study of friction, paving the way for the subsequent research findings in the 20th century by introducing and developing the concepts of surface roughness, asperity deformation, adhesion and breaking of asperities, which are arguably the most important concepts in the scientific domain of friction.
35 3.6 Other Studies on Friction Following the essential contributions made to the knowledge of friction right after the Industrial Revolution, that remain unchanged or suffered small improvements until present, in this section are briefly presented new interpretations of the friction phenomenon, primarily associated to wear effects. The initial significant theoretical inquiries into the mechanisms of wear were conducted by Holm, Burwell and Strang, Archard, and Archard and Hirst (Besson, 2013). These researchers presented a coherent theoretical analysis, introducing models to describe and explain wear and providing formulas for calculation of the wear rate as the volume of material removed per unit length of work. More specifically, Holm proposed an adhesive mechanism at the atomic level, suggesting that bodies in contact only made contact at the peaks of the surface irregularities (asperities). This concept, later expanded upon by Bowden and Tabor, involved Holm assuming that with each approach of two atoms between the contacting asperities, existed a constant probability of an atom being detached from the surface of either of the two bodies (Besson, 2013). The comprehensive synthesis of research on friction, lubrication, and wear can be found in the collective works of Bowden and Tabor, encompassed in their two volumes published in 1950 and 1964 (Bowden & Tabor, 1950, 1964). Despite some knowledge gaps or incomplete aspects, their contributions established the foundation for the modern study of friction and served as a primary reference in the field. Bowden and Tabor provided extensive experimental results, particularly focused on metals and wood, covering fundamental aspects of the subject such as surface structure, topography, sliding and rolling friction, lubrication, and wear (Dowson, 1998). One of the most significant contributions of Bowden and Tabor was the introduction of the concept of the real area of contact, comprising numerous small regions known as asperities or junctions of contact. These authors demonstrated the strong dependence of friction force between sliding surfaces on the real contact area and developed an interpretation of Amontons' laws based on an adhesion model called the adhesive (plastic) junction model, which remains a fundamental reference in the friction domain. To support their model, Bowden and Tabor delved into the topography of surfaces and asperities, especially in metals (Besson, 2013). Additionally, Bowden and Tabor explored friction from the perspective of purely elastic sliding processes. In such cases, they applied the theory developed by Hertz in 1881 concerning the deformations of contact between two elastic solids under load. The Hertz contact theory, still considered valid today, forms the basis for calculating asperities' deformations in friction models.
36 3.7 Summary and Discussion This chapter presented an overview of the first scientific contributions to the knowledge of friction phenomena. Although there were no scientific methodologies at the time, early civilizations such as the Egyptians recognized the existence of dissipative effects between contacting surfaces when sliding large objects in the ground, leading to the use of ancient lubricants to smooth these processes. Motivated by improving his designs and comprehend the physical phenomena that governed his designs, Leonardo Da Vinci can be considered the first scientist to propose a research methodology to study friction. The experimental apparatuses of Da Vinci, despite its simplicity, allowed him to being the first to study the influence of apparent contact area upon frictional resistance, creating the embryogenesis of the well-known friction law that specifies that the apparent area of contact does not influence the friction response. Guillaume Amontons proposed and built more sophisticated experimental workbenches, utilizing mechanisms inspired by industrial tools of that era, such as polishing glass tables. By testing different pairs of contacting materials and the addition of coats of pork fat that acted as lubricants, Amontons was able to formulate two of the most widely recognized laws of friction: (i) the force of friction is directly proportional to the applied load and (ii) the force of friction is independent of the apparent area of contact. Moreover, Amontons was also responsible for associating friction with the deformation of asperities, which constitutes nowadays the fundamental basis of many friction force models. The theoretical works of Leonhard Euler were mainly associated with the proposal of the interlocking asperity theory to explain the frictional behavior between contacting surfaces. The main contribution of Euler to friction’s scientific knowledge is associated with the distinction of static and kinetic friction coefficients, through studies based on inclined planes. The introduction of the Greek letter to refer to the friction coefficient is also attributed to this author. Charles August Coulomb is probably the most widely recognized and associated name with friction, creating many experimental apparatuses based on large scale real-world applications. Coulomb’s works relies on the developments by Amontons, although contradicting some theories formulated by the latter. Although the third fundamental law of friction is attributed to Coulomb, which states that the kinetic friction is independent of the sliding velocity, the first two fundamental laws are associated with Amontons. The fundamental laws of friction should be referred to as Amontons-Coulomb friction laws instead of only attributing credit to the latter, as it is done by many authors. The nature of contacting materials, dry or lubricated conditions, contact area, normal load, and contact duration significantly influence friction were also studied by Coulomb, as well as other key variables include surface roughness,
37 asperity deformation, adhesion, and the breaking of asperities, forming the fundamental concepts in the scientific study of friction. To conclude, the studies performed in the past millennium paved the way to a solid scientific research methodology associated with friction, leading to numerous mathematical and phenomenological models that can accurately predict the dynamic behavior of mechanical systems considering the dissipative effects of friction.
38 4. FRICTION – CONCEPT, DISTINCTION AND INFLUENCING FACTORS Friction is a phenomenon which is observed in a great variety of situations. This subject is studied in the context of tribology, a word that derives from the Greek root tribos , which means “rubbing”, and the suffix -logy that represents the “study or science of”, and encompasses the study of lubricants, lubrication, friction, wear and bearings (Dowson, 1998). Since this science covers a wide range of scientific areas, it can be described as an interdisciplinary subject that requires the expertise study of physicists, chemists, and mechanical engineers, as well as materials scientists and metallurgists (Hutchings & Shipway, 2017). In a more common approach, friction can be universally described as the resistance to relative tangential motion which occurs between two interacting surfaces. There are several laws that attempt to describe the friction phenomenon, being the most relevant ones exposed in Table 4.1. Table 4.1 – Summary of the most relevant friction laws. Adapted from Mo et al. (2009). Nature of the friction theory Friction law Macroscale Amonton’s law Bowden and Tabor Single asperity Non-adhesive – based on Hertz model Adhesive Multi-asperity picture of nanoscale contact Non-adhesive Adhesive Since the main objective of this work is to make a critical analysis of the friction force models in dynamical systems, the description and analysis will be focused on the macroscale friction concept laws. The most utilized and accepted macroscale theory to explain friction is related to the existence of asperities representing the topography of a certain surface, known as the roughness of a surface. Bowden and Tabor (1950, 1964) made valuable contributions to tribology science that are still relevant and used today, even if more advanced theories exist nowadays. These authors were responsible for the introduction of the concept of real area of contact, composed of numerous small regions, known as asperities or contact junctions, and demonstrated that the frictional force between sliding surfaces heavily relied on this real contact area. Figure 4.1 exhibits the real area of contact between two surfaces characterized by the black dots.
39 Figure 4.1 - Contact between asperities in the interface of two surfaces. Contact only occurs in the dots, which explains the difference between the real area and the apparent area of contact. A section view of the graphical representation of the contact between asperities is presented in Figure 4.2 exhibiting the physical difference between the real and apparent area of contact. Figure 4.2 – Schematic representation of sectioned asperities, corresponding to the real area of contact, and the apparent area of contact. It is then intuitive to understand that the real area of contact is a relatively small fraction of the apparent area of contact. Bowden & Tabor (1950) determined for various loads the ratio between the apparent area of contact and the real area of contact in steel-on-steel tribological pairs through electrical conductivity methods, concluding that the real contact of area is 10-5 to 10-2 times smaller than the apparent area. These authors developed an interpretation of Amontons' laws based on an adhesion model called the adhesive (or plastic) junction model, which remains a fundamental reference till present days. In this Real area of contact Apparent area of contact
46 lead to numerical instabilities in dynamic simulations of mechanical systems. This uncertainty in the friction force value in the nearby region of the null value of tangential velocity requires regularization techniques that prevent numerical instabilities from occurring during simulations. 4.1.3 Sliding Friction The sliding friction is related to the resistance that an object offers when it slips over another. This type of dynamic friction is directly related to the theory that states that the surfaces are composed of asperities with different lengths and heights. The movement is then characterized by the relative motion between the asperities of both surfaces in a relatively large area of contact and the resistance to motion, i.e., the friction force, results from the shearing of junctions formed as a result of adhesion phenomena at the interface (Bowden & Tabor, 1950; Greenwood et al., 1961; Persson, 1998). In Figure 4.9 is possible to observe the classical example of sliding friction materialized by a sliding block and it is also represented a zoom-in of the shearing deformation of asperities in the interface of both surfaces in contact. Figure 4.9 – Macroscopic sliding motion of a block and zoom exhibiting the microscopic phenomenon that explains sliding friction as elastic and plastic deformation of asperities. 4.1.4 Rolling Friction Rolling friction exists when two contacting bodies exhibit rolling motion between them. This type of friction can be described through two different situations that can be verified in Figure 4.10: a) a rolling disc in a stationary state on a plane surface where the contact pressure translates into a normal resultant force, fn, which is normal to the contacting surface and aligned with the weight acting on the rolling body; b) when the disc starts to roll due to an external force, f, the contact pressure distribution follows the deformation disc-plane, which results in a translation of the contact point by a distance, d, in relation to fg fn f ff
47 the center of gravity of the disc. This displacement of the contact point originates a moment, r M , named rolling resistance moment, that opposes to the rolling movement of the disc and is directly related to the deformation of the contacting bodies. a) b) Figure 4.10 – Rolling friction of a disc in a plane a) static situation and pressure distribution b) kinetic situation due to the application of the force f. Moreover, this surface contact that is generated due to the elastic deformation of the contact surfaces of both bodies can be explained by hysteresis losses. Tabor (1952, 1955) and Atack (1955, 1958) demonstrated that the front portion of the contact of a sphere with the plane translates into elastic work, due to compression stresses, that would be equally compensated by the rear portion of the circle of contact when it recovers elastically and moves the sphere forward, due to tension stresses on the rear of the sphere. That is not verified in reality due to the hysteresis phenomenon, so the force required to move the sphere along the plane is non-null, which then allows to conclude that exists a resistance force acting on the contact surface (Greenwood et al. 1961). According to Dahl (1968), tension and compression properties principally govern the friction process. The rolling form of motion is more complex than sliding motion and most of the knowledge that exists nowadays is related to Reynolds (1876), Hertz (1886) and Heatcote (1921) (Doménech et al. 1987). One fact that confirms this statement was made by Reynolds where whatever materials that compose the rolling and plane bodies, there will always be slipping motion in the contact surface, although the magnitude of it might be too minor to be measured, so rolling motion then combines rolling and sliding motions which makes it complex to characterize (Reynolds, 1875). p fn fg f ff d fg p f fn Mr
48 4.2 Factors influencing frictional behavior The resistance to sliding primarily manifests in the regions near and between solid surfaces. Identifying the specific attributes of contact conditions and materials that contribute significantly to friction force is a significant challenge in developing friction tests and analytical models. Friction models incorporate various approaches, including geometric considerations such as surface roughness and asperity interlocking, mechanical properties-based variables involving shear properties of solids and substances between surfaces, being the most common situation the existence of lubricants, fluid dynamics principles, electrostatic forces between surface atoms, chemical compatibility, surface temperature, and other variables regarding the dynamics of the system in analysis, such as relative sliding velocity, load or time of contact, for example. Given the diversity of these approaches, the number of potential variables for use in predictive friction models can be substantial. Table 4.2 exhibits a summary of the most common factors influencing frictional behavior accordingly to Blau (2001). Table 4.2 – Factors influencing frictional behaviour (Adapted from Blau (2001)) Category Factor Contact geometry Conformity of the components: macroscale mating of shapes Surface roughness: microscale features – asperity shape, size distribution Fluid properties and flow Lubrication regime; boundary, mixed, elastohydrodynamic, hydrodynamic Lubricant film thickness, pressure, temperature, shear stress and viscosity Oxidation and acidification of lubricants Relative motion Unidirectional or reciprocating motion Motion variation – accelerations, pauses, stick-slip Applied forces Magnitude and variation of the normal force Third-bodies Particles entrained in the lubricant and wear particles External contaminants Temperature Thermal effects on material properties: thermoelastic instabilities, ductility increase Thermal effects on lubricant properties Stiffness and vibrations Contact compliance Damping of frictional or external vibrations
49 Due to the numerous potential factors influencing friction, it is essential to pinpoint the key variables relevant to each specific case. This allows for the selection of appropriate test methods or simulations tailored to each particular case study (Blau, 2001). In the context of this work, the effect of each variable presented in Table 4.2 will not be analyzed in detail, since it is not the main focus of the critical and practical analysis of friction force models in dynamical systems conducted in this work. 4.3 Summary and Discussion In this chapter, a summarized presentation of the main aspects of friction was elaborated, focusing primarily on the concept, distinction and influencing factors that affect the relative motion between contacting bodies. The distinction between static and kinetic friction coefficients was made considering the practical and intuitive example of the inclined plane, similarly to what has been done in the works of Euler. The conducted analysis allowed to recognize the existence of higher friction forces at static states of motion, compared to the lower and constant value of kinetic friction force associated with sliding motion. Although the exposition of these concepts was made using the inclined plane as a practical example, other not so intuitive examples, such as the pressure cone theory (Flores et al., 2023), could be utilized to explain the differences between static and friction coefficients. The Coulomb friction force, considered the predecessor of all modern friction force models, was analyzed both in terms of its general graphical representation and mathematical representation. The simplicity of this model is associated with the fact that only needs one input parameter to be fully defined – the kinetic friction coefficient, k . Despite its rudimentary nature, implementing this friction model in computational models is complex due to the non-differentiable behavior near the null value of relative velocity, which can cause numerical instabilities. Regularization techniques, which will be further discussed in the next chapters, are required to ensure stable dynamic simulations. Both sliding and rolling friction phenomena were also analyzed in a simple and effective manner. In one hand, sliding friction is mainly associated with the theory that surfaces consist of asperities of varying sizes, with movement arising from their relative motion across a large contact area. Friction force originates from the shearing of junctions formed by adhesion at the interface. On the other hand, rolling friction is more complex than sliding friction due to hysteresis, where elastic recovery at the rear region of the contact area does not fully compensate for compression at the front. This results in a non-zero
50 force associated to the motion of rolling objects. Additionally, rolling motion inherently involves minor slipping at the contact surface, combining rolling and sliding motions, which complicates its characterization. A summary of actors influencing frictional behavior was also gathered in this chapter, ranging from contact geometry to fluid properties and flow associated with lubricants, relative motion between bodies, applied forces, existence of third-bodies, temperature and stiffness and vibrations, although the effects of each variable are not presented in the context of this work.
51 5. FRICTION PHENOMENA Friction is an inherently non-linear phenomenon, shaped by a range of complex sub-phenomena that influence the interactions between contacting surfaces. While many of these sub-phenomena are well understood and supported by mathematical formulations, incorporating them into comprehensive friction force models remains a challenge. This complexity arises from the diversity of materials, lubrication regimes, kinematic variables like relative velocity, and stochastic factors that impact surface interactions. Moreover, characterizing each phenomenon in terms of a numeric value attributed to a model parameter is also a challenging task. Classical models, such as Coulomb, offer simplified approximations to determine the magnitude of the dissipative effects of friction, but fail to capture critical aspects like static friction exceeding the kinetic friction force levels or the intricate transitions between motion states. Over time, advancements in tribology have uncovered new frictional behaviors, allowing for more detailed and realistic representations of surface contact phenomena. The evolution of computational power has further enabled the accurate simulation of these phenomena, improving predictions of the behavior of dynamic mechanical systems. This chapter provides a structured exploration of key friction phenomena, progressing in order of increasing complexity. Topics include stiction, breakaway force and dwell-time, viscous effects, the Stribeck effect, stick-slip motion, pre-sliding displacement and micro-slip, frictional lag, and nonreversibility. Together, these phenomena form the foundation for understanding the variables and physical assumptions materialized in the mathematical formulations of friction force models. 5.1 Static friction (stiction), breakaway force and dwell-time It is intuitive to understand that the force needed to maintain the movement of a certain object is lower than the force required to initiate the motion. With this fact in mind and analyzing the graphical representation of the Coulomb friction force model, it can be contradictive in the first place, since an increasing value of relative tangential velocity between surfaces does not translate into a reduction in friction force, due to the fact that this friction model does not consider any differentiation between the static friction force level and the Coulomb (kinetic) friction force level. Considering the previously mentioned phenomenon, several studies have been developed to explain it. Morin (1833) introduced the idea of a friction force associated with a static state that is higher than the Coulomb friction and Rabinowicz (1951) was one of the most influential scientists in the study
52 of the nature of static and kinetic coefficients of friction (Morin, 1833; Rabinowicz, 1951). Through several experiments, the latter author was able to address the transition between sticking and sliding motion and, consequently, between static and kinetic friction as a function of the displacement of a block sliding along an inclined plane. Due to adhesive bonds that are generated between the asperities of both surfaces in contact and the shear forces developed in the initial phase of motion to overcome the resistance offered by these same asperities, it is possible to explain why the static coefficient of friction, denoted as s , is higher than the kinetic coefficient of friction, k . This means that in the diagram of friction force in function of the tangential relative velocity, there is a static friction force s f , higher than the Coulomb friction force, C f , that now represents unequivocally the kinetic friction force according to Figure 5.1. Figure 5.1 – Graphical representation of the Coulomb dry friction model exhibiting the static friction (stiction) phenomena and the Coulomb friction force level for non-null values of tangential relative velocity, t v . The introduction of the stiction phenomenon in the Coulomb friction force model leads to the following formulation ( ) C t t f e s e t sgn( ) if 0 min , sgn( ) if 0 f v v ff f f v == (5.1) where s f represents the static friction force level (stiction). Associated with this stiction phenomenon is commonly introduced the breakaway force concept. The breakaway force is associated with the maximum friction force value, thus corresponding to the static ff vt fs –fs fC –fC
53 friction force level. Due to the high nonlinearities associated with this phenomenon, the measurement of it is quite challenging and complex, which can explain the inaccuracy sometimes verified in control applications during acceleration and deceleration phases of motion. These inaccuracies arise from the dependency of the static friction on the time of contact, which is referred to as dwell-time dependency. The transition from null relative velocities until reaching the breakaway force level during the stick phase occurs in a finite but continuous time interval, that is often difficult to capture and model. Only highly complex dynamic models such as the one proposed by Gonthier et al. (2004) are capable of modeling and predicting this phenomenon. 5.2 Viscous effect In the early 19th century, with the increasing interest and development of hydrodynamics theories, the viscous effect was introduced in the Coulomb friction model due to the influence of lubricants in the surface’s contact. This behavior is materialized by the fluid lubricant layer that exists between the two rubbing surfaces. In this case, the viscous behavior is considered to be a linear function of the tangential relative velocity as represented in Figure 5.2 (Olsson et al., 1998). Figure 5.2 – Graphical representation of the Coulomb friction model considering viscous effect. The linear increasing behavior of the friction force due to viscous effects indicates that the lubrication regime is the full-film lubrication type. The Coulomb friction force model formulation considering the viscous effect component can then be written as ff vt fC –fC
54 ( ) ( ) C t v t t f e C e t sgn( ) if 0 min , sgn if 0 f v f v v ff f f v + == (5.2) in which v f is a viscous friction coefficient related to the viscosity of the fluid. Although the relation between friction force and velocity is considered to be linear, to have a better fit to experimental data, this effect can be described by a non-linear dependence on velocity using an extra variable, v , which depends on the geometry of the application (Olsson et al., 1998), making the second parcel of the first branch of Equation (5.2) equal to v t t sgn( ) v f v v (5.3) 5.3 Stribeck effect The viscous effect mentioned in the previous section described the relationship between friction force and relative tangential velocity in the presence of a lubricated contact between the surfaces as a linear relation, but this phenomenon only captures the behavior of a full-film lubrication situation (Olsson et al., 1998). To be able to capture multiple stages of the lubrication regime related to different tangential relative velocities, the Stribeck effect was added to the Coulomb dry friction model (Stribeck, 1902). This phenomenon is responsible for ensuring a continuous decrease from static to kinetic friction as can be observed in Figure 5.3. Figure 5.3 – Graphical representation of the Coulomb friction model considering the Stribeck effect. ff vt fs –fs fC –fC
55 This continuous drop is characterized by a decreasing friction force with increasing relative tangential velocity, t v , at low velocity, and is responsible for capturing the transition from a boundary lubrication regime to a mixed-film lubrication, being described numerically by the following formulation ( ) tt f e S e t ( ) if 0 min , sgn( ) if 0 f v v ff f f v == (5.4) where t ()fv is an arbitrary velocity dependent friction function. Many authors in literature proposed mathematical expressions to represent the Stribeck effect as a form of a velocity dependent friction function. Tustin (1947) utilized an exponential expression to establish a relationship between the tangential velocity, t v , and the Stribeck velocity, S v ( ) t S f C s C 1 v v f f f f e− = + − − (5.5) Bo & Pavelescu (1982) also proposed an exponential function to represent the Stribeck effect where the total friction force is determined by the following function ( ) ( ) t t C s C v f v f f f e − = + − (5.6) where is a constant factor defined to control the shape of the Stribeck curve and is a factor dependent on the geometry of the contacting surfaces. Hess and Soom (1990) defined a mathematical expression to numerically describe the Stribeck curve, typically known as Lorentzian model, and that is formulated as ( ) sC t C v t 2 t S 1 ff f v f f v v v − = + + + (5.7) where vt fv is a term used to describe the viscous effect phenomena. Popp & Stelter (1990) described the decreasing characteristic of the friction force associated with the Stribeck effect using the following formulation 2 sC t s t t () 1 ff f v f v v − = + + + (5.8)
62 5.8 Summary and Discussion In this chapter, a complete and detailed presentation and analysis of the principal phenomena associated with friction that are considered in the mathematical formulations of various friction force models was conducted. Each phenomenon, including stiction, breakaway force, dwell-time, viscous effects, the Stribeck effect, stick-slip motion, pre-sliding displacement and micro-slip, frictional lag, and non-reversibility, was explored to highlight its unique contributions to the overall frictional behavior between contacting surfaces. Understanding these phenomena is essential for advancing the accuracy of friction force models. Each contributes to the complex dynamics observed in real-world mechanical systems, yet their nonlinear and interdependent nature poses significant challenges for integration into multibody simulations. Accurate modeling requires not only a thorough comprehension of these mechanisms but also robust mathematical formulations that can replicate their effects under varying conditions, such as changes in relative velocity, material properties, and lubrication regimes. The necessity of incorporating these phenomena into friction models becomes evident when considering the precision required in engineering applications. Failure to adequately represent these behaviors can lead to discrepancies between simulated and observed performance, potentially compromising the reliability of various systems. With advancements in computational capabilities, it is now possible to incorporate these phenomena into increasingly sophisticated models. Consequently, such enhanced models are indispensable for optimizing system performance, improving durability, and reducing energy losses across a broad range of engineering applications. In conclusion, the phenomena discussed in this chapter underscore the complexity of friction phenomena and the necessity of accurately predicting their effects on the dynamic behavior of mechanical systems.
63 6. STATIC FRICTION MODELS The Coulomb friction model is characterized by a simple graphical shape where the friction force, given by the product between the kinetic friction coefficient, k , and the normal reaction, n f , assumes a positive value for a certain direction of relative tangential velocity and a symmetric value of friction force when the system in analysis changes its direction. This model can be defined as an empirical model since its graphical representation derived simply from experiments and observations made by Coulomb. Despite its simplicity at first sight, given by the fact that its implementation only requires one parameter, the kinetic friction coefficient, k , the Coulomb friction model becomes a very complex problem to solve when trying to implement it in computational applications. The observed challenge stems from the discontinuity associated with the frictional force value corresponding to a relative velocity of null magnitude. When the value of relative tangential velocity is null, this friction force model cannot be described by any type of mathematical function since it violates one of the basic principles for a given set of data being described by a function – there is a value of the domain, in this case the null value of tangential velocity, that has multiple and undetermined correspondent range values, in this case friction force values, which means that the model is non-differentiable in the null relative velocity region. Considering this computational implementation problem, several strategies were developed throughout the years to surpass this numerical instability and “smooth” the transition near the null value of tangential velocity so that friction can be described by a mathematical continuous function. The models analyzed in this section, which can be fitted in the static category, have a common variable in common – the tangential velocity, vt. This relative velocity developed in the interface between contacting bodies is very important in the context of this type of models since most of their formulations are made as a function of the aforementioned velocity. The existence of a relatively small number of parameters needed to define these models, limits their capacity of modelling complex phenomena related to physical interaction between the surfaces The static friction force models studied are divided into two main categories as the Figure 6.1 diagram indicates: static without stiction, where the maximum value of friction force is the kinetic one, also recognized as the Coulomb friction force level, C f , and the static with stiction models, that as the name suggests, are capable of modelling the maximum value of friction force as the static friction force level, s f .
64 Figure 6.1 – Static friction force models divided by models with stiction and without stiction categories. It is important to note that the models in each category are presented in an ascending order of appearance, with the oldest models being presented first until reaching the most recent formulations. Furthermore, one of the main objectives of this work is to investigate the computational performance of these friction models, so only regularized approaches to static friction models are studied and simulated through a benchmark problem. 6.1 Static friction force models without stiction 6.1.1 Bernard model (1974) Bernard (1974) proposed one of the simplest regularization approaches to the Coulomb friction force model by considering a linear transition from a negative value of Coulomb friction force, C f− , to a positive value, C f+ . Figure 6.2 shows the smoothened regularization modeled by Bernard. Static friction force models without Stiction with Stiction Bernard (1974) Rooney and Deravi (1982) Ambrósio (2003) Karnopp (1985) Bengisu and Akay (1994) Armstrong-Hélouvry et al. (1994) Hollars (1994) Andersson et al. (2007) Wojewoda et al. (2007) Awrejcewicz et al. (2008) Specker et al. (2014) Linear with stiction (Marques et al.) (2016) Brown and McPhee (2016) Threlfall (1978)
65 Figure 6.2 – Linear regularization of the Coulomb friction model proposed by Bernard (1974). In mathematical terms, this model can be described by the following system of equations C l t t f C l t t min( ,1) if 0 max( , 1) if 0 f k v v ff k v v =− (6.1) where fC is the Coulomb friction force magnitude, l k represents a coefficient defined to determine the speed of transition from negative values to positive values of friction force, which in practical terms represents the linear slope of this friction force model and vt is the relative tangential velocity between the contacting surfaces. As a clear advantage of this model can be referred the continuous transition between positive and negative values of friction force using a simple (linear) mathematical function that has small computational cost in comparison with other regularization strategies, but the disadvantages related to it are the sharp slope change at the points where the friction force reaches its maximum value, namely in the transition between the linear slope and the Coulomb friction force level, and the fact of not being capable of capture the stiction phenomenon. Despite some of the graphical representations of this model exhibiting a velocity tolerance, which is the value of tangential velocity that marks the transition between sticking and sliding phases, the mathematical formulation does not consider any type of equation to establish this transition, being fully defined by the transition constant slope. ff vt fC –fC
66 6.1.2 Threlfall model (1978) As it was previously stated in the Bernard (1974) model section, one of its main disadvantages is the sharp transition between the linear slope and the constant slope associated with the steady-state friction force value, which in this specific case, is represented by the Coulomb friction force level, C f . To overcome this problem, a curve shape defined by an exponential function was defined by Threlfall (1978), which led the friction force to be mathematically expressed as ( ) t 0 3 f C t 1 sgn v v f f e v − =− (6.2) where the constant 3 is responsible for ensuring that at t0 vv= the friction force is approximately equal to fC 0.95ff (6.3) being 0 v the velocity at which friction is velocity independent (Threlfall, 1978). When the dynamic system in analysis reaches the 0 v value, the friction force assumes the steady state magnitude, which in the case of this model represents the Coulomb friction force value. Figure 6.3 exhibits the graphical representation of the Threlfall model. Figure 6.3 – Graphical representation of the Threlfall regularization strategy for the Coulomb friction model. ff vt fC –fC v0 -v0
67 The value of the parameter 0 v is selected by the model user, however, it should not be very small, i.e., close to zero, because this would result in a behavior similar to the Coulomb friction model, thus introducing a discontinuity near the value of null tangential velocity. Regarding this model, Marques et al. (2016) proposed an alternative version of it represented formulated as ( ) t 0 3 f C t 1 1sgn 1 v v e f f v e − − − = − (6.4) Besides reaching the Coulomb friction force, the modification of the model ensures that the transition between the raising sticking phase and the sliding phase is made in a smoother and continuous manner without yielding numerical instabilities. 6.1.3 Rooney and Deravi model (1982) Another relatively simple approach, in mathematical terms, of a smoothened Coulomb friction model was proposed by Rooney and Deravi (1982). In this case, a trigonometric function, more specifically a hyperbolical tangent function, tanh , is used, being the friction force defined as t C t 0 f0 C t 0 tanh if if v f v v fv f v v = (6.5) where 0 v represents the velocity tolerance. This parameter defines by itself the width of the region in which the friction force is dependent on the tangential velocity. Figure 6.4 exhibits the graphical representation of the Rooney and Deravi regularization approach to the Coulomb friction force model.
68 Figure 6.4 – Rooney and Deravi’s static friction model graphical representation. It is possible to notice that the Rooney and Deravi’s model considers that for tangential velocities superior to the velocity tolerance ( t0 vv ), the friction force is equal to the Coulomb friction force, as it is demonstrated by the second branch of Equation (6.5), not having a static threshold friction force to represent the stiction phenomenon. The positive aspects of considering this friction force model are related to the simple computational implementation, the need of only defining two parameters, the kinetic friction coefficient, k , and velocity tolerance, 0 v , and its very suitable for high-speed cyclic motions which makes it very applicable to automotive (engine, transmission and drivetrain’s moving parts) and industry applications (Rooney and Deravi, 1982). Downsides to the use of this friction model can be related to the inexperience or lack of sensitivity of the user in defining an adequate value of the velocity tolerance, 0 v , that provides both precise results and minimizes the computational time. 6.1.4 Ambrósio model (2003) The friction force model proposed by Ambrósio (2003) for application in multibody contact problems characterizes itself as a form of regularization of the Coulomb friction model based in a linear relation between the symmetric maximum values of friction force, C f− and C +f , respectively. The formulation of this model is made using the following piecewise function ff vt fC –fC v0 -v0
69 t0 t0 f 0 t 1 10 C t t 1 0 if if sgn( ) if vv vv f v v v vv f v v v − = − (6.6) where 0 v and 1 v are velocity tolerances that establish the beginning of the stiction phase where the friction force assumes a non-null value and the transition from the sticking phase to the sliding state, respectively. This model prevents the friction force from changing direction near the null value of relative tangential velocity, but as a disadvantage, it also considers a null value of friction force for small sliding velocities. Also, it is not capable of modelling the stiction phenomenon, reason why is inserted in the without stiction static friction force models category. Figure 6.5 represents the graphical aspect of this friction model. Figure 6.5 – Graphical representation of Ambrósio’s regularization approach to Coulomb friction model. Another disadvantage that arise from the use of this model is related to the sharp transitions verified between the thresholds 0 v and 1 v . ff fC -fC -v1-v0 v0v1vt
70 6.2 Static friction force models with stiction 6.2.1 Karnopp model (1985) In order to reduce the probability of existence of numerical instabilities near the value of null tangential relative velocity, Karnopp (1985) proposed a model where the friction force, f f , is a function of the tangential velocity, t v , for what is considered the steady state of the dynamical system in analysis. In the nearby region of null tangential velocity, a small velocity range is defined by the conjunction of conditions 0 t 0 v v v− , where 0 v represents a velocity threshold. Since this model is a regularization of the Coulomb friction model, then it can be expressed as ( ) ( ) ( ) t t 0 f e s e t 0 if min , sgn if f v v v ff f f v v = (6.7) where ( ) t fv is a function of tangential velocity, that in practical terms, represents the value of Coulomb friction force, ( ) tC f v f= , for values of tangential velocity outside the range defined by the velocity threshold 0 v . Figure 6.6 presents a graphical representation of this friction model. Figure 6.6 – Graphical representation of the Karnopp static friction force model. Inside the hatched region of the Karnopp graphical representation, which is delimited by the velocity threshold value and its symmetrical value, 0 v and 0 v− , respectively, the tangential velocity is ff fsf (vt) = fC -f (vt) = -fC-fs v0 -v0 vt
71 considered to be null, letting the system change its state and its corresponding response from the kinetic friction force value to a static (stiction) friction force value, s f . Thus, the friction force is determined as (i) the value needed to keep the system at zero velocity, that is, the static friction force level, s f , or (ii) the breakaway force threshold that marks the transition to the kinetic friction force value, C f , which is associated with the sliding phase. 6.2.2 Bengisu and Akay (1994) As the Karnopp model allowed the modelling of the stiction phenomenon, a similar strategy was adopted by Bengisu and Akay (1994) in order to model the Stribeck effect, with the purpose of being able to correctly and more realistically represent the friction phenomena in practical applications. Mathematically, the proposed model can be described by the following function tt f C t r sgn( )(1 )[1 ( 1) ] vv f f v e f e −− = − + − (6.8) where and are parameters that can be either related to the surface roughness or the bonding ability of the surfaces and r f represents the ratio of the asymptotic value of the curve to its maximum, that is, the ratio between the static and kinetic friction force values sn r kn f ff = (6.9) Figure 6.7 exhibits the graphical representation of this dry friction model. Figure 6.7 – Graphical representation of the Bengisu and Akay static friction force model. vt fs ff fC -fC -fs v0 -v0
78 ( ) ( ) t s t e t 0 e s s t t 0 e s e t f t e s e t 0 e s e t t t t 0 ( sgn( )) if (2 1) sgn( ) if 0 ,sgn( ) if 0 sgn( ) if A v f v f v v f f A f v v v f f f v f v f f f v v f f f v f v v v v − + + − = (6.22) where 2 tt 2 00 32 vv Avv =− (6.23) and s f represents the maximum value of static friction force and 0 v being a velocity tolerance similarly to the ones defined in previously analyzed models to mark the transition between sticking and sliding phases. It is verifiable that for values of tangential velocity below the threshold defined by the velocity tolerance, 0 v , the friction force is not only dependent of the tangential velocity, but also dependent of the external resultant force, e f , as it was stated in the beginning of this section. Figure 6.9 shows the graphical representation of this model in which the black lines represent the possible behaviors of this friction model for the minimum and maximum values of external resultant force. Figure 6.9 – Graphical representation of Awrejcewicz et al. static friction force model. vt fs ff fe -fs v0 -v0 f (vt) f (vt) Maximum curve for f (vt) Minimum curve for f (vt)
79 6.2.8 Specker et al. model (2014) The static friction model proposed by Specker et al. (2014), alongside the Coulomb friction, is capable of modeling the viscous and Stribeck effects through continuous functions, which makes it ideal for computational applications due to its capabilities of avoiding numerical instabilities. The Coulomb friction in the analyzed model is described by the following formulation t C k n 0 tanh v ff v = (6.24) where k is the kinetic friction coefficient, n f is the normal force, t v is the tangential velocity and 0 v is a transition velocity that marks the shift from sticking to sliding phase. This velocity is one of the arguments of the hyperbolic tangent function used to regularize the Coulomb friction force model, thus being responsible for the shape of this function. In what concerns the Stribeck effect, this model considers that it can be modeled as 2 t Sp 11 2 Sp t s C v Sp 0 Sp tanh v v vv f f f c v e vv − = − − (6.25) where s f is the static friction force level, C f is the Coulomb friction force level, Sp v is the Stribeck peak velocity that defines the friction force decay due to the Stribeck effect, and v c is a damping friction coefficient. The final term of this friction force model is related to the viscous effect and is expressed as v v t f c v= (6.26) where the meaning of each variable has already been described before. The complete formulation of this friction force model can be obtained by summing the Coulomb, Stribeck and viscous effect terms as follows 2 t Sp 11 2 Sp tt f k n s C v Sp v t 0 0 Sp Viscous effect Coulomb Stribeckeffect tanh tanh v v v vv f f f f c v e c v v v v − = + − − + (6.27)
80 Like other friction models, the disadvantage of the Specker et al. model resides in the definition of the parameter values, since it can only be made in a precise way via experimental setups. Nonetheless, the transition velocity and the Stribeck peak velocity can be approximated as S 0 Sp 42v v v (6.28) being S v the Stribeck velocity. An interesting fact of this model is that the authors performed an adaptation of it to a dynamic friction model by introducing a linear parameter-varying (LPV) system in a form of a first-order lowpass filter, which allowed to capture hysteresis and memory effects that are complex phenomena normally only modelled by dynamic friction force models (Specker et al. 2014). 6.2.9 Linear static with stiction model (2016) Another regularization approach to the Coulomb friction force model discontinuity at null relative tangential velocity is the linear friction model with stiction (Marques et al., 2016). In practical terms, this model considers a peak of friction force corresponding to the value of static friction which occurs at the velocity threshold, 0 v , which marks the transition where the friction force is independent of the tangential velocity, t v . The transition from the static friction force level, s f , to kinetic friction force, C f , verifiable in the Coulomb friction force with stiction, is also characterized by another numerical discontinuity, since this transition occurs for the null value of tangential velocity. To overcome this problem, the strategy followed was to consider a linear function between 0 v and a second velocity threshold named 1 v , that allows a continuous transition from static to kinetic friction force level. The model can be mathematically expressed by the following system of equations (Marques et al., 2016) ( ) t s t t 0 0 t0 f s s C t 0 t 1 10 C t t 1 sgn( ) if sgn( ) if sgn( ) if vf v v v v vv f f f f v v v v vv f v v v − = − − − (6.29)
81 Figure 6.10 shows the graphical representation of the linear friction model with stiction. Figure 6.10 – Graphical representation of the linear static friction force model with stiction. An overview of the presented model allows to conclude that its mathematical formulation is very simple to implement in a computational code and the type of equations used (linear) lead to very low computational costs. The disadvantages shown by this model are the lack of capability of modelling more complex phenomena such as Stribeck and viscous effects, and the sharp transitions between the different branches of the model’s mathematical formulation, that could lead to numerical instabilities. 6.2.10 Brown and McPhee model (2016) Brown and McPhee (2016) proposed a velocity-based friction force model capable of capturing static, dynamic and viscous behaviors. These three different behaviors are represented by three noncoupled terms which can be expressed as ( ) t tn 0 f C s C v t 2 2 0 nt t Coulomb Viscous effect steady state dynamic friction 0 Static friction (stiction) tanh 4 tanh 4 13 44 v vf v f f f f v vf v v = + − + + (6.30) where the first term represents the steady state dynamic friction, which corresponds to the Coulomb friction force level, the second term the static friction (stiction) and the third one the viscous friction. This non-coupled approach is very useful and simple to implement when certain friction phenomena, for ff vt fs fC v1 v0 -v1-v0 -fC -fs
82 example the viscous effect, is not a variable of interest for the analysis of a certain system, which is done by removing the corresponding term from Equation (6.30). In what regards the meaning of the variables, C f represents the Coulomb friction force level, t v is the tangential velocity, 0 v is a threshold or tolerance velocity that marks the transition from stick to slip phase, which occurs for the maximum value of static friction force, s f , v is the viscous coefficient of friction, n f is the normal contact force and nt f is a transition normal force which is used to detect when the viscous friction effects should be accounted. When n nt ff , the viscous friction term has a significant impact on the friction force value. Without considering viscous effects, the third term of Equation (6.30) can be neglected and the friction force can be simply calculated as ( ) t t0 f C s C 2 2 0 t 0 tanh 4 13 44 v vv f f f f vv v = + − + (6.31) This friction force model reveals a simple approach to regularize the Coulomb friction model, through the use of a hyperbolic tangent function and is able to capture stiction and viscous effect in an intuitive to implement model. 6.3 Benchmark problem – static friction force models To evaluate the behavior and computational efficiency of the static friction force models presented and analyzed in the previous sections, it was necessary to select a dynamic problem with significant relevance and well documented in literature, that is, a benchmark problem. This simple system is widely used in literature to validate friction force models, and it was first proposed by Rabinowicz (1956). Figure 6.11 exhibits the one degree of freedom system composed by a block with mass m which is connected to a fixed reference by a spring with stiffness k and is placed on a conveyor belt that moves with a linear speed b v .
83 Figure 6.11 – Schematic representation of the one degree of freedom benchmark problem. Using the equations of dynamics it is possible to describe the mass movement of this mechanical system through the following differential equation 2 f 2 dx m kx f dt = − − (6.32) where f f represents the tangential friction force. The simulation parameters chosen to study the performance, behavior and efficiency of the static friction force models analyzed in this section are presented in Table 6.1, being particularly based on the ones used by Gonthier et al. (2004) and Marques et al. (2016). Table 6.1 – Benchmark problem simulation parameters. Parameter Symbol Value Parameter Symbol Value Mass of the block m 1 kg Initial velocity 0 v 0.1 m/s Gravity acceleration g 9.81 m/s2 Time-step - 0.0001 s Conveyor belt speed b v 0.1 m/s Simulation time - 40 s Spring stiffness constant k 2 N/m ODE Solver - ode45, ode15s, ode23s, ode78 Kinetic friction coefficient μk 0.1 Solver Relative Tolerance, RelTol - 1e-8 Static friction coefficient μs 0.15 Solver Absolute Tolerance, AbsTol - 1e-8 Initial position x0 0 m These parameters were then introduced in the interface shown in Figure 6.12 which is an integrant part of a computational code dedicatedly programmed in MATLAB to study the behavior of a variety of friction force models in an intuitive way. k m vb x
84 Figure 6.12 – Main window of the computational code created in MATLAB to study the benchmark problem. After introducing all the parameters in the main window and hitting the Next button, the user is taken to another window, presented in Figure 6.13, where it is possible to choose the models that want to be studied. Figure 6.13 – Friction force models selection window of the MATLAB code created.
85 In the next sections a more detailed study of the static friction force models with and without stiction will be made using the programmed MATLAB code. 6.3.1 Performance analysis of static friction force models without stiction After selecting all the static friction force models without stiction inserted in the MATLAB code, it is necessary to define the values of certain specific parameters of each model. Table 6.2 displays the various parameters considered for this category of models. Table 6.2 – Static friction force models without stiction specific parameters. Parameter Symbol Value Transition coefficient (Bernard) kl 2000 Velocity threshold (Threlfall, Rooney and Deravi, Threlfall modified) v0 0.001 m/s Velocity threshold (Ambrósio) v0 0.0001 m/s Velocity threshold (Ambrósio) v1 0.001 m/s The previous parameters were then defined in the static friction force without stiction window in each respective box as visible in Figure 6.14. Figure 6.14 – Static friction force models without stiction window from the MATLAB computational code to define the specific parameters of each model.
86 After clicking the Solve button, it is possible to obtain the graphs that quantify the behavior of the system in analysis in terms of position, relative velocity, acceleration, and friction as functions of the simulation time and which are represented in Figure 6.15 from a) to d). a) b)
87 c) d) Figure 6.15 – Graphical results of the benchmark problem for the static friction force models without stiction a) Position [m] vs Time [s] graph b) Relative velocity [m/s] vs Time [s] graph c) Acceleration [m/s2] vs Time [s] graph d) Friction force [N] vs Time [s] graph using the ode45 solver. A general overview of all the graphs allows to conclude that all the static models without stiction have remarkably similar behaviors. Figure 6.15 a) exhibits an initial linear slope that corresponds to the
94 Although not represented in Table 6.5, the average value of the simulation times of all the solvers yields the Awrejcewicz friction model as the most efficient model in terms of computational elapsed time, while the Andersson et al. model is the least efficient. These results show once again that the ode23s is not a very reliable solver, since that for the Karnopp friction force model, the problem did not converge to a solution. Table 6.5 – Solver and average simulation time for the static friction force models with stiction. Model Solver Average Simulation Time [s] Model Solver Average Simulation Time [s] Karnopp ode45 1.4239 Specker ode45 1.5924 ode15s 1.9092 ode15s 2.2854 ode23s DNC* ode23s 1.4331 ode78 2.2380 ode78 2.4739 Bengisu and Akay ode45 1.7400 Linear with stiction ode45 1.7217 ode15s 2.4848 ode15s 2.3644 ode23s 1.3448 ode23s 1.1405 ode78 2.6984 ode78 2.6100 Andersson ode45 2.6665 Hollars ode45 1.6364 ode15s 2.3516 ode15s 2.2919 ode23s 1.2756 ode23s 1.2626 ode78 4.2416 ode78 2.5369 Awrejcewicz ode45 1.4188 Brown and McPhee ode45 1.8278 ode15s 2.2201 ode15s 2.5609 ode23s 1.1813 ode23s 1.4607 ode78 2.1865 ode78 2.8658 *DNC – Did Not Converge
95 6.4 Summary and Discussion This chapter has presented a comprehensive analysis of 14 static friction force models, comprising 4 static models without stiction and 10 static models with stiction. Each model represents a regularized alternative to the Coulomb friction model, that is, they do not exhibit a numerical discontinuity near the value of null tangential velocity. The analyses focused primarily on the computational performance, physical realism, and ability of these models to incorporate phenomena such as viscous effects and the Stribeck curve. The static friction models without stiction demonstrated remarkable computational efficiency, with behaviors that are largely consistent across all models. However, their inability to simulate stiction phenomena limits their applicability in scenarios where a realistic representation of static friction is required. These models reach only the kinetic friction force level during simulations, as evident from their friction force vs. time responses. While the absence of stiction simplifies their implementation and improves computational performance, it also constrains their use in simulations demanding higher fidelity. In contrast, the static friction models with stiction offer enhanced capabilities to represent friction phenomena more realistically. These models successfully simulate the stiction phase, achieving the expected maximum static friction force level in line with theoretical expectations. The distinction between sticking and sliding phases was evident, with the sliding phase exhibiting faster transitions compared to the oscillatory stick-slip behavior observed in static models without stiction. This improved realism, however, comes at the cost of increased computational complexity, which is still relatively low. The computational efficiency of the models was evaluated using a benchmark dynamic problem based on the 1 DoF problem proposed by Rabinowicz, for which an intuitive graphical interface was created in MATLAB. For static models without stiction, the Threlfall modified model emerged as the most computationally efficient, achieving an average simulation time of 1.2801 s using the ode23s solver. However, the Coulomb friction model, while simple in concept, exhibited the least computational efficiency and failed to converge under certain conditions, underscoring its limitations in numerical applications due to instabilities in the non-differentiable region of null relative velocity. Among static models with stiction, the Linear with stiction model demonstrated the shortest average simulation time of 1.1405 s with the ode23s solver. Conversely, the Andersson et al. model was the least computationally efficient, with an average time of 4.2416 s using the ode78 solver. The results also indicated that the reliability of solvers like ode23s can vary significantly, as evidenced by nonconvergence in some scenarios.
96 The performance discrepancies among solvers, as observed in this study, further underscore the importance of selecting appropriate numerical solvers for friction simulations. The results suggest that while ode23s often yields faster computations, its reliability can be problematic, necessitating careful consideration of solver choice based on the specific friction model and dynamical system in analysis. The findings of this chapter reinforce the importance of tailoring static friction force models to the specific needs of a given application. Static friction models without stiction are well-suited for scenarios requiring high computational efficiency and moderate fidelity, while static models with stiction provide a better representation of real-world friction phenomena, by being able of representing the static friction force phenomenon, having similar computational demands.
97 7. DYNAMIC FRICTION MODELS Static friction models are not capable of capturing certain complex phenomena, such as hysteresis or frictional lag. With this problem in mind, a more complex and precise category of friction models is introduced – the dynamic friction force models. Dynamic friction models not only depend on the tangential relative velocity between contacting surfaces but also incorporate a state variable, often represented by bristle deflection, as most dynamic models are formulated at the asperity level. A state variable can be defined as a mathematical variable that captures the condition or configuration of a certain dynamic system in a certain instant of time. This variable is then explicitly updated using an update expression at the beginning or end of each time step of the numerical simulation. The use of a state variable enables the mathematical formulation of dynamic friction models to be described by continuous functions that not only correctly describe the friction phenomenon at the asperity level, but also eliminate the numerical instabilities normally verified in the proximity region of the null value of tangential velocity. Despite its enhanced precision in predicting the friction force acting on the contact interface between bodies that exhibit relative motion, their increased computational cost and complexity make some of these models hardly usable for practical engineering tasks. Another major problematic associated with these asperity level models is the difficulty to choose parameter values to model the physical behavior of surface bristles, which often requires calibration procedures utilizing experimental data. In this section, similarly to what has been done for the static models, the dynamic friction force models analyzed appear in chronological order of appearance, being the studied ones presented in the schematic of Figure 7.1. Figure 7.1 – Analyzed dynamic friction force models. Dahl (1968) Liang et al. (2012) Dynamic friction force models Bristle (1991) Reset Integrator (1991) LuGre (1995) Dankowicz (1999) Leuven (2000) Elasto-Plastic (2000) GMS (2004) Gonthier et al. (2004) Bliman and Sorine (1993)
98 7.1 Dynamic friction force models analyzed – formulations 7.1.1 Dahl model (1968) A new mathematical model of solid friction was formulated and presented by Dahl (1968) with the main objective of describing friction for both sliding and rolling friction that can be used in simulations of dynamic systems involving mechanical elements that are subjected to friction. The Coulomb friction concept was introduced based on experimental data and it is widely accepted as a physical macro-phenomenon. In this model, a new theory emerged by establishing a comparison between frictional behavior and the quasi-static properties of materials. When an external force is applied to a material, due to the inertial properties of it, a resistance to the deformation is offered by the material. In frictional terms, this resistance to deformation can be thought of as the friction force that develops at the interface between the bodies and prevents the tangential relative motion between them. With that being said, stiction, the maximum value of static friction, can be directly correlated with the ultimate tensile stress, UTS , of a stress-stain curve, since from this point onwards, the applied force will lead to a continuum plastic deformation of the bristles between both contacting surfaces until Coulomb friction force is reached. This kinetic friction force level can be directly correlated with the rupture point of a material in a stress-strain curve where the contacting surfaces will exhibit relative tangential motion, that is, will start to slide (Dahl 1968). It was found that this analogy to deformation of materials can be distinct when brittle and ductile materials are in contact due to differences between the stress-strain curves of these types of materials. Thus, for materials that have a brittle behavior, the stiction (static friction) and Coulomb friction are indistinguishable, as it can be observed in Figure 7.2 a) and for ductile materials, the ultimate stress point will correspond to the maximum static friction value (stiction) and Coulomb friction force will be correlated with rupture point as it is represented in Figure 7.2 b) and as it was stated in the previous paragraph.
99 a) b) Figure 7.2 – Analogy of bristle elastic and plastic deformation with material’s behavior a) brittle material b) ductile material. Adapted from Dahl (1968). In what concerns the mathematical formulation of this model, Dahl (1968) described this deformation and hysteretic behavior of materials as dependent of both the relative displacement and velocity of the contacting surfaces, being the stress strain curve defined by a differential equation as f f f 0t CC 1 sgn 1 sgn( ) df f f v dx f f = − − (7.1) where f f corresponds to the friction force, x is the displacement, 0 denotes the stiffness of the bristles and is a parameter that defines the shape of the material curve, being dependent of the material and usually varying between 0 and 1. It is possible to notice that Equation (7.1) is a function of displacement, but according to Dahl (1968) it can be differentiated in order to time, leading to ( ) f f f f f t 0 t t CC 1 sgn 1 sgn df df dx df f f v v v dt dx dt dx f f = = = − − (7.2) In the case of considering the parameter being equal to 1, which is the most common case, then Equation (7.2) can be written as ( ) ff 0 t t C 1 sgn df f vv dt f =− (7.3) The Dahl model is considered to be a dynamic friction model, so it must have a state variable which describes the dynamic behavior of the system in analysis. As this formulation considers the Rupture Stiction and Coulomb Friction Levels Rupture Coulomb Friction Level Ultimate Stress Stiction Level
100 existence of bristles, then the state variable is the bristle deflection, represented by the parameter z, and assuming that these entities exhibit a linear spring behavior that can be described as f0 fz = (7.4) then the rate of bristle deflection is defined as 0 tt C 1 sgn( ) dz z vv dt f =− (7.5) which when the system reaches the steady state yields f C t sgn( )f f v= (7.6) being Equation (7.6) simply the Coulomb friction force model. The analyzed approach allows to model both pre-sliding displacement as well as Coulomb friction, through the introduction of a state variable related to the bristle’s deflection, but the main drawbacks of this model are its incapability of modelling the Stribeck effect and to capture stiction, which led to the formulation of new friction models based on the Dahl model, such as the Bliman and Sorine (1993, 1991) and the LuGre model (Canudas De Wit et al., 1995). 7.1.2 Bristle model (1991) The continuous search for a friction force model able to realistically model the physical phenomena and simultaneously exhibit good computational performance was the main goal of Haessig and Friedland (1991). These authors stated that friction force should be defined as a function of displacement rather than tangential velocity, so the model proposed must be able to describe the sticking phase of friction as a function of displacement and during the slipping phase the friction force could be described as a function of tangential velocity. The friction between surfaces is then given by the interaction between a N number of bristles in which each bristle contributes with an infinitesimal fraction of the total friction force. With that being said, the total friction force can be expressed as ( ) f i i i 1 N i f x b = =− (7.7) where i is the stiffness of each bristle, i b is the reference position related to the initial bond of the bristles and i x is the is the relative position of the bristles.
101 Figure 7.3 illustrates the physical interaction between two bristles and the corresponding deflection of the lower one. Figure 7.3 – Schematic representation of bristle deflection and location of the bond between asperities. The difference between the last two variables indicated before ( i b and i x ) represents the relative displacement that should be used as a state variable to describe the sticking phase of the relative motion between surfaces. When the strain at a certain bristle exceeds its limit, ( ) i i i xb = − , the rupture of that bond occurs, and a new bond is formed in a new location, which is calculated from the previous one. The bond between a rigid bristle and an elastic bristle is destroyed when a critical deflection point is reached and then a new bond is randomly created between the rigid bristle and another different elastic bristle. The new bristle location can be determined as ' i i uniform( )sgn( ) i bb = + (7.8) where i b is the reference bond location and the new bristle range. Although the bristle model was designed for accuracy, this comes at a cost, that in this case is the computational efficiency. Despite considering a finite relatively small number of bristles, that should be less than 50 as stated by the authors (Haessig & Friedland, 1991), because of the spacing between bristles and the discontinuities related to the snapping of bristle bonds, this model becomes x xibi
102 computationally inefficient, requiring very small integration steps to provide precise results. This is the main reason why this model is not commonly used. 7.1.3 Reset Integrator model (1991) The same authors of the bristle model exposed in the previous section proposed a new approach to reduce the computational time requirements, while still being capable of accurately representing and modelling the stick-slip friction phenomena. Haessig and Friedland (1991) formulated a model that does not consider the use of multiple bristles to accurately represent the bonding effect of stick-slip phenomena in the sticking phase, but instead considers this stick phase of the friction force as a function of position for a single bristle. The input of the integrator used in this model is the strain rate of the bristle deformation, that can assume the following values t0 t t 0 0 if 0 if 0 z v z z dz v z v z z dt = (7.9) where t v is the tangential relative velocity, z is the bristle deflection and 0 z the pre-determined maximum value of the bristle deflection. To determine the correspondent friction force of this model it is necessary to use two different equations: one to represent the sticking phase until reaching the breakaway force and another to describe the system’s behavior during the sliding phase as it is expressed below 0 t 1 t 0 f 0 t t 0 ( )(1 ) if ( ) sgn( ) if dz v a z v z fdt v z z v z + + = (7.10) where 0 is the bristle stiffness, a is a coefficient related to the stiction phenomena, normally representing the ratio between the stiction load, s f , and the sliding steady-state load, C f , and 1 is a damping coefficient used to eliminate unwanted oscillations in the sticking phase and can be defined as 10 0.707 m = (7.11) where m represents the mass of the sliding body.
103 In conclusion, this model requires significantly less computational time than the previously proposed model (Bristle model, 1991), since it exhibits a smooth and continuous behavior, so the small time steps associated with breaking the bristles is not a mandatory requirement. 7.1.4 Bliman and Sorine model (1993) The friction models developed by Bliman and Sorine (1993; 1991) are dynamical models based on the works of Rabinowicz (1951; 1957; 1961). The models presented in this section fulfil both dependence on the sign function of the tangential velocity, t sgn( )v , and the space variable s, which results in the integral form t t 0()s v d= (7.12) to replace the time variable, t, with the space variable, s, through a variable transformation process. Considering that s x is the absolute relative displacement of the contacting bodies, then the model can be expressed as a linear system of equations s ss f s s dx Ax Bv ds f Cx Dv =+ =+ (7.13) This model depends on the sign function of the tangential velocity, t sgn( )v , which in this case is mathematically represented by the variable s v , being ( ) st sgnvv= The first model analyzed is a first-order model, characterized by its reduced complexity, which uses the following variables 1f f 110A B f C D = − = = = which results in the model form f f f 1 f t t t f1 df df ds df f f v v v dt ds dt ds f = = = − (7.14) This formulation is remarkably similar to the Dahl model, where 1 C1 f 1 f ff = = =
110 in which f ss t ()Fv is the friction force value in steady state, i is an integer coefficient and t ( , )zv is a piecewise continuous function used to capture stiction, since it allows the elastic displacement to be developed until reaching the breakaway force. This function depends on the regime (elastic or plastic) of the displacement as it is represented t ba t ba ss t t t ss t t 0 if sgn( ) sgn( ) 0 1 if sgn( ) sgn( ) ( ) ( , ) 1 if sgn( ) sgn( ) ( ) 0 if sgn( ) sgn( ) v z z z v z z z z v zv v z z z v vz = = == (7.33) where ba z is the breakaway bristle deflection and ss t ()zv is the steady-state bristle deflection, which corresponds to the maximum value of deflection, ss t max ()z v z= . The relation ( ) ba ss t 0.7 z zv can be used to determine the value of the breakaway deflection, ba z . In the specific case of small vibrations, the t ( , )zv function for the ba ss t ()z z z v interval can be equal to ss t ba t ss t ba () 11 2 ( , ) sin 2 ( ) 2 z v z z zv z v z + − = + − (7.34) The use of the sine function is explained by the fact of being very useful to study the transition behavior from static to kinetic friction force levels in friction force models. Despite its accuracy, this model still presents some problems related to the determination of the mathematical description of 𝛼(𝑧, 𝑣𝑡) function, since its shape cannot be intuitively chosen due to the inexistence of a physical relation with the friction phenomena. 7.1.9 Generalized Maxwell Slip model (2003) The Generalized Maxwell Slip Model (GMS), proposed by Lampaert et al. (2003), is a friction force model based in physical properties of interacting surfaces which simulates the contact physics at the asperity levels. This model can predict pre-sliding regime, Stribeck effect, frictional lag, breakaway force and stiction behaviors.
111 There are two main categories of phenomena introduced in this model: (i) friction mechanics and (ii) the asperity contact scenario. In what concerns the friction mechanics, it is possible to evaluate and analyze phenomenological mechanisms such as normal creep of contacting asperities, adhesion raising and hysteresis losses due to asperity deformation. Regarding asperity contact, this category can be divided into three different phases: (i) the asperity of one contact surface is free, thus not contacting or suffering any kind of bending solicitation from the other surface; (ii) the asperity of one surface is in contact with the opposed surface which leads to a deformation that can be expressed as a function of the relative displacement between the interacting surfaces; (iii) the surfaces are no longer in contact and all the deformation energy is dissipated. This analysis leads to the representation of friction as the parallel interaction of 𝑁 elementary single state friction models as it is represented in Figure 7.7. Figure 7.7 – Physical interpretation of the Generalized Maxwell Slip friction model as the parallel interaction of N elementary single state friction models. Each one of the elements of this model is described by the same dynamical model, which is materialized by the bending behavior of bristles as springs, where the displacement and velocity applied to each one of the elementary models is the same. The sticking and sliding phases of each one of the elementary friction models is evaluated to describe the general response of the system in analysis. (…) (…) x z1 zi k1 ki kN zN
112 The GMS model is based on three friction properties: (i) the Stribeck curve for constant velocities, (ii) the hysteresis function with non-local memory in the pre-sliding regime, and (iii) the frictional memory in the sliding regime (Lampaert et al. (2003)). Al-Bender et al. (2005) stated that if an individual friction element i is sticking, then the bristle deflection rate can be expressed as i t dz v dt = (7.35) where t v is the tangential velocity. This regime maintains until the deflection of the bristle equals the value of the Stribeck velocity in sliding, ( ) i i t z s v= . For the slipping phase of an individual friction element i, the bristle deflection rate can be described as ( ) ii ti it sgn( ) 1 dz z vC dt s v =− (7.36) where i C is an attraction parameter which represents a gain, similar to the proportional term of a PID controller, that determines how fast i z converges to i s . Equation (7.36) is valid until the relative velocity of the elementary model reaches a null value. Since the sticking and slipping equations of the state variable z are written for each individual friction element i, then to determine the total friction force acting on a system is necessary to sum all the outputs of the N elementary friction models as ( ) ( ) i f i i i t i1 Ndz f k z t f v dt = = + + (7.37) where i k is the stiffness of each elementary model spring, i is a damping coefficient and ( ) t fv is a viscous function which is defined to model the viscous component. This function is referred to as velocity strengthening, since it is proportional to the tangential velocity. It is possible to notice that each element is characterized by four different parameters: a stiffness i k , a damping coefficient i , an attraction parameter i C and a Stribeck velocity function i s . The definition of i s introduces one more unknown parameter i , which is the reciprocal of the Stribeck velocity for element i. To solve this problem without introducing additional unknown parameters nor
113 sacrificing the essence of the problem, a i v variable must be defined which acts as a scale parameter for all the elemental friction models existing on the system. Although being very precise and faithful to experimental data, this dynamic friction force model has the disadvantage of requiring a lot of computational resources, needing around 1000 equivalent asperities to be modeled to obtain smooth responses. Thus, its main applications can be associated to the validation of simpler friction force models which are more suitable for practical control and multibody applications. 7.1.10 Gonthier et al. model (2004) The Gonthier model, proposed by Gonthier et al. (2004), is a regularized three-dimensional contact force model with asymmetric damping and dwell-time dependent friction. Since the focus of this work is to describe the friction force models in a comprehensive way, only the tangential component of this model will be studied in detail. In this model, the friction force is also defined as a function of bristle deflection, similarly to the approach proposed in the LuGre model by Canudas De Wit et al. (1995), being expressed as br 0 1 dz fz dt =+ (7.38) where 0 is the stiffness of the bristles, z the bristle deflection, 1 a damping coefficient and dz dt the rate of bristle deflection. The novelty introduced with this model is directly related to the bristle state of deflection. This deflection was then divided into two components: a static rate of deflection defined for the sticking phase st dz dt , and a dynamic rate of deflection defined for the sliding phase sl dz dt . The general formulation of the deflection rate is presented below st sl (1 ) dz dz dz ss dt dt dt = + − (7.39) where s is a sticking state function defined as 2 t S v v se − = (7.40) in which t v represents the tangential relative velocity and S v the Stribeck velocity.
114 It is then intuitive to understand that when t v tends to zero, the deflection rate becomes equal to the static deflection rate, st dz dz dt dt = , and when s tends to zero, the deflection rate is practically equal to the dynamic deflection rate, sl dz dz dt dt . Moreover, in the sticking regime, the bristle deflection will correspond to the tangential velocity between the interacting surfaces, st t dz v dt = , and in the sliding regime the bristle deflection will be defined in terms of the Coulomb friction force C f , since it represents the steady state of the system in analysis. The Coulomb friction force opposes the external force applied to a system and can be defined as ( ) C C 0 t 0 dir ,f v v=f (7.41) where fC represents the Coulomb friction force level and ( ) 0 t 0 dir ,vv is a piecewise function of both tangential velocity t v and a small velocity tolerance 0 v which returns a unit vector along the t v direction expressed as ( ) t t0 t 3 0 t 0 tt t t0 0 0 0 if dir , 31 if 22 vvv v vv vv vvv v v v = − (7.42) The velocity tolerance, 0 v , is used to reduce the discontinuities in velocity direction and should be at least one tenth of S v , being commonly considered to be equal to one hundredth of S v . Combining Equation (7.39) with the assumption that sl dz dz dt dt results in a dynamic bristle deflection equal to 0 C 11 1 sl dz fz dt =− (7.43) which leads to a general bristle deflection equation formulated as 22 tt SS 0 tC 11 1 1 vv vv dz e v e f z dt −− = + − − (7.44)
115 To correctly model the transition existent in the stick-slip motion due to the frictional lag associated with the dwell-time, it is necessary to include a new state variable dw s . This variable is a function defined by two time constats that allow the correct modelling of dwell-time effect on the maximum static friction force and can be expressed as ( ) ( ) dw dw dw dw dw dw br 1if 0 1if 0 s s s s s s s s s − − = − − (7.45) where dw is the dwell-time dynamics time constant and br is the bristle dynamics time constant which can be defined as 1 br 0 = . The maximum static friction (stiction) force can be defined as ( ) max C s C dw f f f f s= + − (7.46) Considering that the friction force applied by the bristle br f should be equal to the Coulomb friction force C f when the contacting bodies are sliding, br C ff= , then the friction force can be modelled as br 2 t br max fbr max 2 t br max br if if f v f f fff v f f f + =+ (7.47) where 2 is a viscous damping coefficient. This friction model is then characterized by eight (or seven, if it is considered that the velocity tolerance, v0, is defined as a function of the Stribeck velocity, vS) different parameters: k , s , 0 , 1 , 2 , 0 v , S v and dw . 7.1.11 Liang et al. (2012) The Liang et al. (2012) bristle model is an extension of the bristle model proposed by Haessig and Friedland (1991) to three dimensional multibody systems that can be used to describe the tangential contact between general rigid or near-rigid body contact dynamics applications.
116 This model consists in treating the contact between interacting surfaces as multiple individual contacts, materialized by bristles. In Figure 7.8 is represented the contact between two different bodies, Body 1 and Body 2 in two different time instants, t and tt+ . Figure 7.8 – Contact between to different bodies (Body 1 and Body 2) in two different instants, t and tt+ . For a time instant, t , the contact point between both bodies is the point 1 P . If the system is in a sticking phase, then for the instant tt+ the two bodies remain in the same relative position, so the point of contact between bodies 1 and 2 is still 1 P . On the other hand, if one body moves relatively to the other, then the contact point will change from 1 P to 2 P . Then, the motion of these bristles can be described as a spring like behavior that exhibits stretch and rotation around the common tangential plane, being the friction force computed as the average displacement of these elements, as depicted in Figure 7.9 a) and b). Figure 7.9 – a) Representation of maximum static (𝑧s max), and kinetic (𝑧k max) deflections for a given acting force 𝐹 in a certain time instant t and b) for another time instant given by 𝑡 + ∆𝑡, where the bristle deflection shows an expansion and rotation around the common tangential plane. Body 2 Body 1 at instant Body 1 at instant Tangential plane at instant Tangential plane at instant a) b)
117 The main advantages of this model are that it is a standalone model, so it can be easily implemented and integrated into an existing general dynamics solver for multibody dynamical systems, and it is able to describe both static and dynamic friction in the same model and simulate usual frictional phenomena (Liang et al. 2012). In what concerns each contact point, the three-dimensional friction force can be described by Hooke’s law in the form of f0 fz =− (7.48) where 0 is the bristle stiffness and z is a three-dimensional vector that represents the average bristle deflection or displacement, which can be expressed as ( ) 0 0 t max t max max t ( ) if ( ) () ( ) if ( ) () t t z t v t dt z z t zvt z t z z t vt + = (7.49) where 0 t is the initial time instant corresponding with the beginning of contact, t is the current time of contact, t()vt represents the three-dimensional vector of tangential velocity and t()vt is simply the magnitude of tangential velocity. The maximum bristle deflection in a given time instant, max ()zt , can be defined as C max t 0 0 max s max t 0 0 () ( ) if () () ( ) if k s ft z t v v zt ft z t v v = = = (7.50) in which C f is the Coulomb friction force acting in each instant, 0 v is a velocity threshold that marks the transition between sticking and sliding phases and s f is the static friction force acting in each instant. It is also important to notice that this definition of the maximum bristle deflection makes a distinction between values for the maximum static deflection smax (t)z , and maximum kinetic deflection k max ()zt , as it is visible in Figure 7.9. The model formulation and the corresponding illustration presented in Figure 7.9 a) and b) show that this model approximates the bristle behavior to the one exhibited by a linear spring. Thus, this can lead to numerical instabilities in the transition between stick and slip state of motion due to excessive
118 stiffness and/or oscillatory behavior of the system. To prevent the noise generated by high frequency friction force variation, that can lead to numerical discontinuities, it is introduced a bristle damping coefficient 1 which leads the friction force mathematical expression to be equal to f 0 1 dz fz dt = − − (7.51) This friction force model reduces to the Coulomb friction force model when the friction is in sliding regime and while the system in analysis is in the sticking phase, the friction force magnitude is below the one corresponding to the stiction threshold. To fully define this friction force model it is necessary to specify the values of five parameters: the kinetic coefficient of friction k , the static coefficient of friction s , the velocity threshold 0 v , the bristle stiffness 0 , which should be defined, by rule of thumb, with a value of one order of magnitude (101) higher than the largest dominant stiffness of the system in analysis, and the bristle damping coefficient 1 . 7.2 Benchmark problem – dynamic friction models After presenting the most commonly utilized dynamic friction force models in literature, it becomes imperative to assess the computational behavior and efficiency of this category of models. The MATLAB computational program briefly described in the previous chapter about static friction force models, was also adapted to study the one degree of freedom dynamic system of the sliding block using dynamic friction force models. The selection of this type of model implies the definition of the specific parameters of each model. Table 7.1 displays the various parameters considered for the dynamic category of friction force models.
119 Table 7.1 – Dynamic friction force models specific parameters. Parameter Symbol Value Parameter Symbol Value Initial bristle deflection (common to all models) 0 z 0 m Stictionsteady state ratio (Reset Integrator) a 0.52 Stiffness coefficient (common to all models) 0 105 N/m Integer power (Elasto-Plastic) i 1 Damping coefficient (LuGre, Reset Integrator, ElastoPlastic, Gonthier, Liang) 1 √105 N⋅s/m Initial dwell-time (Gonthier) dw s 0 Viscous coefficient (LuGre, Elasto-Plastic, Gonthier) 2 0.1 N⋅s/m Dwell-time dynamics constant (Gonthier) dw 2 Stribeck velocity (LuGre, Elasto-Plastic, Gonthier) S v 0.001 m/s Velocity threshold (Liang et al.) 0 v 0.001 m/s The previously specified parameters were then introduced in the respective interface window of the MATLAB program shown in Figure 7.10. Figure 7.10 – Dynamic friction force models window from the MATLAB computational code to define the specific parameters of each model.