Impact and in non-linear mechanical systems with

combined - contact

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

By

Sriram Sundar, B. E.

Graduate Program in Mechanical

The Ohio State University

2014

Dissertation Committee:

Prof. Rajendra Singh, Advisor

Prof. Dennis A Guenther

Prof. Ahmet Kahraman

Prof. Vishnu Baba Sundaresan

Dr. Jason T Dreyer

Copyright by

Sriram Sundar

2014

ABSTRACT

This research is motivated by the need to have a better understanding of the non- linear contact of systems with combined rolling-sliding contact such as cam- follower mechanism, gears and drum . Such systems, in which the dominant elements involved in the sliding contact are rotating, have unique interaction among contact , siding friction and . Prior models used in the literature are highly simplified and do not use formulation hence the dynamics of the system are not well understood. The main objective of this research is to gain a fundamental understanding of the non-linearities and contact dynamics of such systems, for which a cam-follower mechanism is used as an example case. Specifically, the non- linearities, impact damping and coefficient of friction are analyzed in this study. The problem is examined using a combination of analytical, experimental, and numerical methods.

First, the various non-linearities (kinematic, dry friction, and contact) of the cam- follower system with combined rolling-sliding contact are investigated using the Hertzian contact theory for both line and point contacts. Alternate impact damping formulations are assessed and the results are successfully compared with experimental results as available in the literature. The applicability of the coefficient of restitution model is also critically analyzed. Second, a new dynamic experiment is designed and instrumented to

ii precisely acquire the impact events. A new -domain based technique is adopted to accurately calculate the system response by minimizing the errors associated with numerical integration. The impact damping is considered in a generalized form as a product of damping coefficient, indentation raised to the of damping index, and the time derivative of the indentation displacement. A new signal processing procedure is developed (in conjunction with a contact mechanics model) to estimate the impact damping parameters (damping coefficient and index) from the measurements by comparing (on the basis of three residues) them to the results from the contact mechanics model. Also few unresolved issues regarding the impact damping model are addressed using the experimental results. Third, the coefficient of friction under lubrication is estimated using the same experimental setup (operating under sliding conditions). A signal processing technique based on complex-valued Fourier amplitudes of the measured and is proposed to estimate the coefficient of friction. An empirical relationship for the coefficient of friction is suggested for different surface roughnesses based on a prior model under lubrication. Possible sources of errors in the estimation procedure are identified and quantified.

Some of the major contributions of this research are as follows. First, impact damping model was determined experimentally and related unresolved issues were addressed. Second, coefficient of friction for a cam-follower system with point contact under lubricated condition was estimated. Finally, better understandings of the effect of non-linearities and shortcomings of coefficient of restitution formulation are obtained.

iii

Dedication

To the lotus feet of my spiritual master

His Holiness Sri Rangaraamaanuja Mahaadesikan

iv

ACKNOWLEDGEMENTS

First, I would like to thank my advisor, Prof. Rajendra Singh, for his patience and guidance throughout my graduate study. His tremendous experience and knowledge has been helped me overcome the difficulties faced during this process. I also would like to express my deepest appreciation to Dr. Jason Dreyer for his extremely valuable support in the experimental and many technical discussions. I would like to thank my committee members, Prof. Guenther, Prof. Kahraman and Prof. Sundaresan for their time to review my work. I also would like to thank Caterina Runyon-Spears for her careful reviews of this work and all the members of and Dynamics Lab for their providing with an amicable atmosphere over the past four years. Special thanks to

Laihang for helping me record the experimental data. I would like to thank the Vertical

Lift Consortium, Inc., Smart Vehicles Concept Center (www.SmartVehicleCenter.org) and the National Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) for partially supporting this research.

I am most grateful to my parents, brother, fiancée and other family members for their constant faith, support and patience. I would like to thank all my friends especially,

Adarsh, Ranjit, Sriram, Saivageethi and Darshan who made my graduate life, away from home, a memorable one. Also a special thanks to Sriram’s mom, for her mother-like care during all her visits in these four years.

v

VITA

December 25, 1985……………………………… Born - Chennai, India

2003……………………………………………… B. E. Anna University, Chennai, India

2009 – Present…………………………………… University Fellow/ SVC Fellow Graduate Research Associate The Ohio State University Columbus, OH

PUBLICATIONS

1. S. Sundar, J. T. Dreyer and R. Singh, Rotational sliding contact dynamics in a non- linear cam-follower system as excited by a periodic , Journal of Sound and , (2013).

2. S. Sundar, J. T. Dreyer and R. Singh, Effect of the tooth surface waviness on the dynamics and structure-borne noise of a spur gear pair, SAE Technical Paper 2013-01- 1877, 2013, SAE Noise and Vibration Conference.

FIELDS OF STUDY

Major Field: Mechanical Engineering

Main Study Areas: Mechanical , Nonlinear Dynamics, Sliding Contact Systems, Contact Mechanics.

vi

TABLE OF CONTENTS

Page ABSTRACT……………………………………………………………...……………… ii

DEDICATION…………………………………………………………………………... iv

ACKNOWLEDGEMENTS……………………………………………………...………. v

VITA……………………………………………………………………………..…….... vi

LIST OF TABLES ...... xi

LIST OF FIGURES ...... xiii

LIST OF SYMBOLS ...... xix

CHAPTER 1: INTRODUCTION...... 1

1.1 Motivation ...... 1

1.2 Literature review...... 2

1.3 Problem formulation...... 4

References for Chapter 1 ...... 10

CHAPTER 2: ROTATIONAL SLIDING CONTACT DYNAMICS IN A NON-LINEAR

CAM-FOLLOWER SYSTEM AS EXCITED BY A PERIODIC MOTION………..… 16

2.1 Introduction ...... 16

2.2 Problem formulation………………………………………………………... 17

2.3 Analytical model……………………………………………………….…… 20

2.3.1 Relationship between the coordinate systems……………….……. 20 vii

2.3.2 ………………………………………………. 21

2.3.3 Static equilibrium and linearized natural ……………… 24

2.3.4 Contact damping and dry friction models……………………….... 25

2.4 Examination of the contact non-linearity and alternate damping models…... 27

2.5 Assessment of the coefficient of restitution ( ξ) concept……………………. 33

2.6 Study of the line and point contact models in the sliding contact regime….. 38

2.7 Analysis of the friction non-linearity……………………………………….. 40

2.7.1 Effect of direction………………………………………………… 40

2.7.2 Dynamic bearing and friction forces……………………………… 40

2.8 Study of kinematic non-linearity…………………………………………… 47

2.9 Conclusion………………………………………………………………….. 49

References for Chapter 2 ...... 53

CHAPTER 3: ESTIMATION OF IMPACT DAMPING PARAMETERS FROM TIME- DOMAIN MEASUREMENTS ON A MECHANICAL SYSTEM……………...…….. 58

3.1 Introduction…………………………………………………………...…….. 58

3.2 Problem formulation……………………………………………………...… 59

3.3 Design of the laboratory experiment and instrumentation………………….. 61

3.4 Analytical model……………………………………………………………. 62

3.4.1 Kinematics of the system……………………………………….… 62

3.4.2 Non-contact regime……………………………………………..… 65

3.4.3 Contact regime…………………………………………………..... 66

3.5 Estimation of the impact damping parameters ( β and n)…………………… 68 viii

3.5.1 Time-domain based technique to estimate the system response….. 68

3.5.2 Signal processing procedure to estimate β and n…………………. 70

3.6 Error and sensitivity analyses on the estimation procedure……………….... 72

3.6.1 Error analysis…………………………………...………………… 72

3.6.2 Sensitivity analysis……………………………………..…………. 76

3.7 Estimation of the impact damping from the measurements………………… 79

3.8 Equivalent coefficient of restitution………………………………………… 84

3.8.1 Governing equation………………………………………..……… 84

3.8.2 Estimation of the equivalent ξ model……………………...……… 86

3.8.3 Justification of the estimated impact damping parameters……….. 90

3.9 Conclusion………………………………………………………………….. 91

References for Chapter 3……………………………………………………….. 93

CHAPTER 4: ESTIMATION OF COEFFICIENT OF FRICTION FOR A MECHANICAL SYSTEM WITH COMBINED ROLLING-SLIDING CONTACT USING VIBRATION MEASUREMENTS………………………………………..…… 98

4.1 Introduction………………………………………………………………..... 98

4.2 Problem formulation……………………………………………………...… 99

4.3 Contact mechanics model……………………………………………...….. 102

4.4 Experiment for the determination of ………………………………….…. 107

4.5 Identification of system parameters…………………………………..…… 108

4.5.1 Identification of geometrical parameters…………………..…..... 108 ix

4.5.2 Identification of the modal damping ratio………………………. 113

4.6 Signal processing technique to estimate ……………………………….... 114

4.7 Experimental results and friction model…………………………………... 119

4.8 Potential sources of error in the estimation of …………………...……… 123

4.9 Conclusion…………………………………………………………...……. 130

References for Chapter 4 ...... 132

CHAPTER 5: CONCLUSION……………………………………………………...… 137

5.1 Summary ………………………………………………………………..… 137

5.2 Contributions …………………………………………………………...… 138

5.3 Future work ……………………………………………………………..… 139

References for Chapter 5 ...... 141

BIBLIOGRAPHY…………………………………………………………………..…. 142

x

LIST OF TABLES

Table Page

3.1 Comparison of average residues per impact ( Λ1, Λ2 and Λ3) using two simulations

S S −2.5 S1 S 2 (S1 and S2) with β1= β 2 = 24.7 GNsm and n= n = 1.5 ………………….. 75

3.2 Comparison of normalized average residues per impact ( Λ1, Λ 2 and Λ 3 ) using

S −2.5 S2 simulation S2 ( β 2 = 24.7 GNsm and n =1.5 ) with e/rc = 0.2 and c = 16 Hz.

a) For different values of β S1 in the proximity of β S2 with constant value of

nS1= n S 2 .b) For different values of nS1 in the proximity of nS2 with constant value

of βS1= β S 2 ………………………...………………………………………...…. 78

3.3 Error in the estimation of ξ model using time histories from simulation S3

(γ S3 = 0.8s/m ) ………………………………………………………………...….. 88

4.1 Comparison of measurements and predictions (from the contact mechanics model)

with = 0.51 and e/a = 0.116 at the harmonics of c = 11.55 Hz…………....… 122

4.2 Error in the estimation of for the mechanical system with a circular cam for

different values of e at c = 11.55 Hz………………………………………..…. 127

4.3 Error in the estimation of for the mechanical system with circular cam for

different cam with e/a = 0.1…………………………………………...… 128

xi

4.4 Error in the estimation of for the mechanical system with an elliptic cam at c =

8.33 Hz and e = 0.1 a…………………………………………………………..... 130

xii

LIST OF FIGURES

Figure Page

1.1 Analytical model of typical cam -follower system with contact mechanics

formulation …………………………………………………………………... ….. … 6

1.2 Cam-follower experiment designed to study the contact mechanics. a) Isometric

view of the cam -follower experiment built using a lathe; b) Closer view of the

contact……………………………………………………………………………… 7

2. 1 Cam-follower system in the general state where a non -linear contact stiffness

model, kλ(ψi(t)), is employed ……………………………………….. ……...……. . 19

2.2 of the follower in the sliding contact regime ……………... … 22

2.3 Normalized d ry friction models (equations are given in Section 2. 3.4). Key: ,

Coulomb friction (Model I); , Smooth ened Coulomb friction (Model II) ... 28

r r r r 2.4 Comparison of αrms and αp at lower speeds. (a) αrms ; (b) α p . Key: , contact

mechanics formulation with damping model A ; , damping model B; , damping

model C; , damping model D; , damping model E; , experimental result

from literature [8]; , prior analytical result from literature [2.8] ….……...... 31

2.5 Comparison of predicted α r (t ) using different damping models with experimental

data at c = 155rpm. Key: , contact mechanics formulation with damping model

xiii

A; , damping model B; , damping model C; , damping model D; , damping

model E; , prior experimental result from literature [2.8] ……. …………. .. 32

r 2.6 Comparison of experimental and analytical results for α at c = 155rpm . (a) Time

domain comparison; (b) Frequency domain comparison. Key: , analytical

contact mechanics formulation with damping model D; , experimental result

from Alzate et al. [2.8] …………………. ……………………………. .………….. 33

r 2.7 Map of α p vs. c at lower speeds. Key: , predictions from contact mechanics

model; , experimental results from Alzate et al. [2.8] ; , prior analytical results

from Alzate et al. [2.8]; , prediction based on approximate ba lance

technique (given in Section 2. 4.2) with ξ = 0.05; , ξ = 0.2; , ξ = 0.4;

, ξ = 0.6 ……………………………………………………………….. … 37

r 2.8 Map of α p vs. c over a broad range of speeds. Key: , predictions from contact

mechanics formulation; , experimental results from Alzate et al. [2.8]; , prior

analytical results from Alzate et al. [2.8]; , prediction based on approximate

energy balance technique (given in Section 2. 4.2) with ξ = 0.2; , ξ = 0.6;

, ξ = 1. …………………………………………………………………... 38

2.9 Identification of contact domains based on ks - c mapping at a constant cam

with e = 0.1 rc……………………………………………………... …………... ….. 42

D D 2.10 Comparison of αɺɺ spectra (with µm = 0.3, ζ = 0.01 and β = 4.25 s/m). (a) Spectra

showing harmonics of c; (b) Spectra showing natural frequency of the system .

Key: , de-energizing system with line contact ( lλ = 0.0016m, c = 300 rpm );

xiv

, self-energizing system with line contact ( lλ = 0.0016m, c = -300 rpm);

, de-energizing system with point contact ( c = 300 rpm)……….……….. 43

2.11 Comparison of Fn(t) for different direction of cam rotation with line contact ( lλ =

D D 0.0016m, µm = 0.3, ζ = 0.01 and β = 4.25 s/m). Key: , de-energizing (c =

300 rpm); , self-energizing (c = -300 rpm)…………………………….… 44

2.12 Comparison of relative sliding vr(t) for two dry friction models of Fig. 2.3

D D (with c = 50 rpm, e = 0.7 rc , ζ = 0.01 and β = 4.25 s/m). Key: , Coulomb

friction; , Smoothened Coulomb friction…………………….….………… 45

2.13 Comparison of forces for two dry friction models of Fig. 2.3 (with c = 50 rpm, e =

D D 0.7 rc, ζ = 0.01 and β = 4.25 s/m). (a) Nx(t); (b) Ff(t). Key: , Coulomb friction;

, Smoothened Coulomb friction……………………………………...……. 46

2.14 Comparison of relative sliding velocity vr(t) for two dry friction models of Fig. 2.3

D D (with ωc = 40 rpm, e = 0.7 rc, ζ = 0.01 and β = 4.25 s/m). Key: , Coulomb

friction; , Smoothened Coulomb friction………….………………………. 47

2.15 Comparison of forces for two dry friction models of Fig. 2.3 (with ωc = 40 rpm, e =

D D 0.7 rc, ζ = 0.01 and β = 4.25 s/m). (a) Ff(t); (b) Nx(t). Key: , Coulomb friction;

, Smoothened Coulomb friction……………………………………..…….. 51

ɺɺ D D 2.16 Comparison of α spectra (with µm = 0.3, c = 300 rpm, ζ = 0.01 and β = 4.25

s/m). (a) Spectra showing harmonics of c; (b) Spectra showing natural frequency

of the system. Key: , Non-linear system; , Linear system…….……. 52

3.1 Cam-follower experiment designed to determine impact damping parameters…. 61

3.2 Analytical contact mechanics model of the experiment shown in Fig. 3.1…….... 64

xv

3.3 Regimes of contact and impact for the system (with parameters given in section

3.6.1) via c vs. e/rc. Key: , Operational points (with periodic impacts) selected

for the purpose of error analyses……………………………………………..….. 74

3.4 Comparison of hysteresis loops for single impacts during simulation S2 (

S −2.5 S2 S S S1 S 2 β 2 = 24.7 GNsm and n =1.5 ) and simulation S1 ( β1= β 2 and n= n )

given e/rc = 0.2 and c = 16 Hz. Key: , Simulation S1; , Simulation

S2………………………………………………………………………………… 76

3.5 Time histories of the measured forces and acceleration with e/rc = 0.13 and c =

14 Hz (with other parameters given in section 3.6.1). a) Normalized reaction force

along eˆ x ; b) Normalized reaction force along eˆy ; c) of the

follower………………………………………………………………….………. 81

3.6 Sample measured forces and acceleration during the contact sub-event from a

single impact from measurements shown in Fig. 3.5. a) Reaction forces; b)

Angular acceleration. Key: , Normalized reaction force along eˆ x ; ,

Normalized reaction force along eˆy …………………………………….………. 82

3.7 Comparison of the hysteresis loops from measured data of Fig. 3.6 and simulation

S1 (using the impact damping model selected based on minimization of Λ1). Key:

S -2.5 , Measured; , Simulation S1 (with β 1 = 49.3 GNsm and

nS1 =1.5 )………………………………………………………………..……….. 83

3.8 Comparison of the hysteresis loops from measured data of Fig. 3.6 and simulation

S1 (using viscous damping model selected based on minimization of Λ1) Key:

xvi

S1 , Measured; , Simulation S1 with viscous damping ( n = 0 and

β S1 = 1.47 kNs/m )………………………………………………………….....… 85

i ɺ b 3.9 Variation in estimated ξ (during different impacts) with ψ i given e/rc = 0.10 and

S c = 18 Hz. Key: , Simulation S 3 (γ 3 = 0.8s/m ) ; , Estimated ξ model

with γ = 0.799 s/m (using least square curve-fitting technique)…………..…….. 89

i ɺ b 3.10 Variation in estimated ξ (during different impacts) with ψ i for the experimental

data of Fig. 3.5. Key: , Experimental data for different impact; ,

Estimated ξ model with γ = 0.758 s/m (using least square curve-fitting technique).

…………………………………………………………………………………… 90

4.1 Example case: A mechanical system with an elliptic cam and follower supported by

a lumped spring ( ks)………………………………………………………..……. 101

4.2 Free-body diagram of the follower; refer to Fig. 4.1 for the two coordinate

systems……………………………………………………………………..……. 106

4.3 Mechanical system experiment used to determine the coefficient of friction ( ) at

the cam-follower interface………………….……………………………..…….. 110

4.4 Classification of response regimes of the mechanical system with a circular cam in

terms of c vs. e/a map with the parameters of section 4.5. Key: ,

Operational range of the experiment…………………………….………………. 112

4.5 Slide-to-roll ratio for the cam-follower system with e/a = 0.12 and c = 11.55 Hz

and other parameters of section 4.5…………………………...……………...…. 113

xvii

4.6 experiment to determine the viscous damping ratio associated with the

lubricated contact regime. a) Experimental setup; b) Top view of the dowel pin

arrangement showing the three point contacts. Key: , contact point…….…….. 115

4.7 Relative accelerance spectra in the vicinity of the system . Key: ,

dry (unlubricated); , lubricated with AGMA 4EP oil; , lubricated with ISO 32

oil ………………………………………………………………………….. ……. 116

4.8 Estimated for different Rm values and comparison with prior values (including the

range) for the dry friction regime [4.13] . Key: , With AGMA 4EP oil; , With

ISO 32 oil; , dry contact - iron pin with steel disk [4.13] ; , dry contact - copper

pin with steel disk [4.13] …………………………………………………….. ….. 121

4.9 Comparison of the modified Benedict -Kelley model from the results of Fig. 4.8

with friction values reported in the literature [4.16, 4.33 - 35] . Key: , Model

for AGMA 4EP oil ( EHL regime ); , Model for ISO 32 oil ( mixed

lubrication regime ), , Shon et al. [4.16] ; , , Xu and Kahraman [4.33] ;

Grunberg and Campbell [4.34] ; , Furey [4.35] ………………………….. …… 124

4.10 Classification of response regimes of a mechanical system with an elliptic cam in

terms of a c – b/a map with e = 0.1 a and other parameter values given in section

4.5. Key: , Operational range of simulation ………………………... …. 129

xviii

LIST OF SYMBOLS

List of symbols for Chapter 1 c Damping F Dynamic force k Stiffness

α of the follower δ Indentation

κ Arbitrary constant

Coefficient of friction η Arbitrary constant

Θ Angular displacement of the cam

Subscripts k Stiffness s Spring

λ Contact

Operators

, First and second derivative with respect to time List of symbols for Chapter 2 c damping d Length from cam pivot point to the point where follower spring base

xix

E Cam pivot point e Eccentricity or runout

(eˆx ,eˆy ) Fixed co-ordinate system along vertical and horizontal directions f Frequency

F Dynamic force h Non-linear function (for Jacobian method)

G Center of point

I of

J Jacobian matrix

(iˆ, ˆj) Moving co-ordinate system (with ̂ being parallel to the follower) k Translational stiffness

L Length l Length of line contact

N Bearing reaction force n Impact damping index

O Contact stiffness location

P Follower pivot point

Q Origin of ̂, ̂ coordinate system r Radius of the cam t Time

T ɵ u Velocity of the contact point along i v Sliding velocity

V w width

Y Young’s modulus

xx

α Rotation of the follower from the horizontal in both the coordinate systems

β Impact damping factor

ξ Coefficient of restitution

Change

χ Moment arm of the normal force on the follower about the pivot point.

Ξ State vector

ˆ ˆ (ψi, ψ j ) Translational displacement variables for the cam in the (i, j) coordinate system

ε Generalized state space variable

ω of

Θ Angular displacement variable of the cam in the ( ,) coordinate system

θ Angular displacement variable of the cam in the ( ̂,̂) coordinate system

Coefficient of friction

σ Regularizing factor for smoothing (hyperbolic tangent) function

ν Poisson’s ratio

ζ Damping ratio

ϑ Natural frequency Subscripts b Follower c Cam e Equivalent f Friction l Linearized m Static n Normal

λ Denotes contact parameters

xxi p Peak to peak r Relative rms root-mean-square s Spring x, y horizontal and vertical directions Superscripts a After impact b Before impact c Out of contact

0 Zero displacement state i In contact

P Denotes moment (or) about the follower pivot point r Residual u uncompressed

* Static equilibrium point

A-E Damping model numbers

I,II ,.. Friction model numbers Operators

, First and second derivative with respect to time

( ) Normalized

δ( ) Small increment sgn Signum function List of symbols for Chapter 3 c damping d Length from cam pivot point to the point where follower spring base

E Cam pivot point

xxii e Eccentricity or runout

(eˆx ,eˆy ) Fixed co-ordinate system along vertical and horizontal directions f Frequency

F Dynamic force h Non-linear function (for Jacobian method)

G Center of gravity point

I Mass moment of inertia

J Jacobian matrix

(iˆ, ˆj) Moving co-ordinate system (with ̂ being parallel to the follower) k Translational stiffness

L Length

N Bearing reaction force

O Contact stiffness location

P Follower pivot point

Q Origin of ̂, ̂ coordinate system r Radius of the cam

S Simulation t Time v Sliding velocity w width

Y Young’s modulus

α Rotation of the follower from the horizontal in both the coordinate systems

β Impact damping factor

γ Velocity factor in COR

κ Arbitrary constant

χ Moment arm of the normal force on the follower about the pivot point.

xxiii

Ξ Function to output time of return of follower

ˆ ˆ (ψi, ψ j ) Translational displacement variables for the cam in the (i, j) coordinate system

ξ Coefficient of restitution

Angular velocity

Λ Residue

Θ Angular displacement variable of the cam in the ( ,) coordinate system

Coefficient of friction

ν Poisson’s ratio

ζ Damping ratio

ϑ Natural frequency Subscripts

1 Trial values

2 Known values a End of contact event c Cam b Follower d Dowel pin e End of the impact cycle f Friction l Linearized m maximum

λ Denotes contact parameters r Relative s Spring x, y horizontal and vertical directions Superscripts

xxiv

0 Zero displacement state

S1 Simulation S1

S2 Simulation S2 i experimental impact event

K known values

T Trial values

P Denotes moment (or) moment of Inertia about P u uncompressed

* Static equilibrium point Operators

, First and second derivative with respect to time

( ) Normalized sgn Signum function List of symbols for Chapter 4

A Semi-major axis point a Semi-major axis of the elliptic cam

B Semi-minor axis point b Semi-minor axis of the elliptic cam c Translational viscous damping

C Arbitrary constants for B-K model

D Arbitrary point on the cam circumference d Length from cam pivot point to the point where follower spring is attached to the ground

E Cam pivot point e Eccentricity

(eˆx ,eˆy ) Fixed co-ordinate system along vertical and horizontal directions f Frequency

xxv

F Dynamic force g Acceleration due to gravity

G Center of gravity point

I Mass moment of inertia

J Jacobian matrix

̂, ̂ Moving co-ordinate system (with ̂ being parallel to the follower) k Translational stiffness

L Length l Length of line contact m Mass

N Bearing reaction force

O Contact stiffness location

P Pivot point p Hertizian pressure

Q Origin of ̂, ̂ coordinate system q Arc length of the ellipse

R Roughness

S Scoring s slope t Time u Velocity of the contact point v Sliding velocity w width

Y Young’s modulus of the material

α Rotation of the follower from the horizontal in both the coordinate systems

ˆ ˆ (ψi, ψ j ) Translational displacement variables for the cam in the (i, j) coordinate system

χ Moment arm of the normal force on the follower about the pivot point. xxvi

Ξ State space vector

β Impact damping factor

γ parameter of the ellipse in canonical form

ε Generalized state space variable

Angular velocity

ρ Radius of curvature

ς Fourier amplitude of trigonometric functions of α

ω Angular frequency

Θ Angular displacement variable of the cam in the (eˆx ,eˆy ) coordinate system

ˆ ˆ θ Angular displacement variable of the cam in the (i, j) coordinate system Coefficient of friction

ν Poisson’s ratio

ζ Damping ratio

η dynamic

ϑ Natural frequency Subscripts a Average b Follower c Cam d dowel pin e Entrainment f Friction h Hertzian l Linearized n Normal

λ Denotes contact parameters

xxvii p Peak to peak r Relative rms root-mean-square s Spring x, y horizontal and vertical directions Superscripts d DC term k kinematically calculated e Equivalent

0 Zero displacement state

P Denotes moment (or) moment of Inertia about P r Reconstructed u uncompressed

* Static equilibrium point Operators

, First and second derivative with respect to time

( ) Normalized sgn Signum function List of symbols for Chapter 5

F Dynamic force δ Indentation

κ Arbitrary constant

Coefficient of friction ξ Coefficient of restitution Subscripts

xxviii k Stiffness

λ Contact

Operators

, First and second derivative with respect to time

xxix

CHAPTER 1

INTRODUCTION

1.1 Motivation

Cam-follower systems, gears and drum-brakes are widely used in vehicles and machineries. The dynamics of such systems significantly differ from the translational sliding contacts due to the unique non-linear interaction of contact mechanics and sliding friction in the source regime with the kinematics of system. The knowledge of the contact dynamics of these systems is limited and its effect on the response of the system is not well understood. For better understating of the dynamics, a fundamental investigation of the system with combined rolling-sliding contact is required. In scientific literature, simpler systems are often investigated (as it aids in more controlled research) to understand the dynamics of similar systems; thus, a cam-follower system is selected for this research.

The dynamics of cam-follower systems have traditionally been described by lumped parameter, linear system theory for the follower with motion input from the cam, as reported by Chen [1.1] in a literature survey (1977). Alzate et al. [1.2] used the coefficient of restitution concept to model the contact between the cam and follower.

Such coefficient of restitution type models usually have several deficiencies as stated by

1

Gilardi and Sharf [1.3]. Overall, the contact stiffness and damping non-linearities of a cam-follower system are yet to be rigorously studied. Also the effect of friction and its non-linearity has been neglected in the cam-follower models [1.4 - 11] since there is no motion along the direction of friction. Since friction plays a significant role in the dynamics of such systems under sliding contacts [1.12 - 14], the value of the coefficient of friction ( ) must be accurately estimated. The methodology adopted to estimate in prior experimental studies (specific to translational sliding contact) [1.15 - 18] cannot be directly employed for a system with combined rolling-sliding contact system, since the kinematics at the contact is different. Furthermore, impacts commonly occur in cam- follower systems with [1.1, 1.2, 1.10] at high cam speeds, affecting the dynamic response. Hence the impact is a very important phenomenon to be analyzed.

Therefore, one of the primary motivations for this research is the need to understand the non-linearities of combined rolling-sliding contact cam-follower system

(only in the context of a single degree-of-freedom system). Hence, the proposed formulation would include kinematic, friction and contact non-linearities. Next, having a precise model for impact damping is mandatory to achieve accurate prediction of the dynamics of impacting systems. Finally, there is a need to have an experiment to estimate

for combined rolling-sliding contact systems.

1.2 Literature review

The sliding and/or rolling contacts are of interest in many mechanical systems such as pin-disk models [1.19 - 1.22], geared transmission systems [1.23 - 1.25], and bearings [1.26]. However, the dynamics of the sliding contact is sometimes studied using

2 simple translating systems [1.27 - 29]. In the case of combined rolling-sliding contact models, investigators have employed piecewise linear systems to study the loss of contact in a cam-follower system [1.4 - 6], and some studies [1.7 - 9] have examined the stability issues. Hence the non-linear dynamics of the combined rolling-sliding contact systems is examined in this research using contact mechanics principles [1.30].

Contact mechanics formulation (with impact damping model) has been employed by few researchers [1.33, 1.34] for analyzing systems undergoing impacts. The widely used contact force formulation [1.33, 1.35] is of the following form where the force due to contact damping is proportional to force due to stiffness,

ɺ Fλ = F k (1 + κδ ) . (1.1)

Here, Fλ is the contact force (with λ representing contact parameter), Fk is the contact stiffness force, δ is the indentation distance and κ is an arbitrary constant. However, other

1/ 4 ɺ models such as Fλ = F k + ηδ δ (where η is a constant) also have been used [1.34] to represent the contact force during impacts. Hence a more generalized formulation for the

n ɺ contact force of the form Fλ = F k + βδ δ should be analyzed experimentally.

Furthermore, among the experimental work done in rotational systems to determine , investigators have analyzed a pin-disk apparatus [1.36, 1.37], two rotating circular plates

[1.38], and a radially loaded disk-roller system [1.39, 1.40]. However, none of the previous combined rolling-sliding contact experiments rely on vibration measurements.

Hence there is a need to develop an experimental method to determine for combined rolling-sliding contact systems with vibration measurements.

3

Based on the available literature on the dynamics of cam-follower system, some of the major unresolved issues are as follows, a. Is coefficient of restitution model applicable to such a system during impacts? b. What are effects of different non-linearities on the dynamics? c. Is the contact damping force proportional to contact stiffness force during impact? d. Is the equivalent viscous damping model appropriate for impacts? e. Can the coefficient of friction be estimated from the vibration measurements of reaction forces and acceleration? f. What is the generalized friction model for combined rolling-sliding contact systems under lubrication?

1.3 Problem formulation

Fig. 1.1 shows a typical single degree-of-freedom (SDOF) cam-follower system

(when the cam and follower are not in contact) which is considered for analysis in this research. The circular cam rotates about the fixed pivot, which is at a distance from the geometric center of the cam. The angular displacement of the cam is given by Θ(t), which is also the motion input to the system. The follower consisting of a long bar of square cross-section is hinged at one of its end to a frictionless pivot. The angle α(t) made by the follower with the horizontal line in the clockwise direction is the only generalized coordinate. The follower is supported by a linear follower spring (ks). The contact mechanics between the cam and follower is represented by means of non-linear contact stiffness (kλ) and damping (cλ) terms. The coefficient of friction between the cam and follower is given by time-varying (t). During the operation, the system can be in either

4 the sliding contact regime or the non-contact regime at a given instant based on the cam speed, and hence the system should be studied on both contact and non-contact regimes.

The experiment designed for this study is shown in Fig 1.2. The reaction forces at the follower pivot and the acceleration of the follower are measured from this experiment.

The scope of this study is restricted to the following:

i) A single degree-of-freedom cam-follower system with combined rolling-

sliding contact having contact, friction and kinematic non-linearities.

ii) Cam with elliptical profile is analyzed using the analytical model, while only

a circular cam is studied experimentally.

iii) Though line and point contacts are studied analytically, only point contact is

taken up for experimental studies.

iv) Dynamics of the system is examined with constant cam speed.

v) The angular velocity of the cam is restricted to 1500 rpm (25 Hz) in the

experiments, much below the natural frequency of the system (≈1400 Hz for

point contact).

vi) The non-linear dynamics of the system is investigated only under stable and

deterministic conditions.

vii) Variation in the surfaces of the cam and follower due to ageing is not

considered.

The major assumptions of this study are as follows,

1. The bearings at the pivots of the cam and follower are frictionless and rigid,

allowing only rotation without any translation.

5

2. The axes of rotation of the cam and the follower do not change under any load.

3. The cam and follower are elastic bodies, and their contact follows Hertzian

contact theory.

4. Kelvin-Voigt model is used to represent the contact.

5. The moment of the follower is negligible.

6. The angular velocity of the cam is constant and unaffected by the frictional

load.

Cam pivot Cam

Follower Follower pivot

Follower spring

Fig. 1.1. Example case: Cam-follower system with contact mechanics formulation.

6

a) Lathe

Rigid fixture

b) Tri-axial Roller Cam load cell bearings Follower

Point contact Accelerometer Spring

Fig. 1.2. Cam-follower experiment designed to study the contact mechanics. a)

Isometric view of the cam-follower experiment built using a lathe; b) Closer view of the

contact.

7

The specific objectives of this dissertation are outlined along with sub-objectives, to resolve the major issues state above. The objectives are organized to parallel Chapters 2 to 4.

Objective 1: Study the non-linear dynamics of the cam-follower system with combined rolling-sliding contact (Addressed in Chapter 2).

(1a) Develop a contact mechanics model for the cam-follower system with

combined rolling-sliding contact.

(1b) Examine the applicability of different viscous and impact damping models

and the coefficient of restitution concept by comparing the predictions with

the experimental results reported by Alzate et al. [1.2].

(1c) Study the effects of contact and friction non-linearities in the sliding contact

regime.

(1d) Analyze the effect of kinematic non-linearity of the system by comparing it

with a linearized model.

Objective 2: Determine the impact damping parameters (β and m) for the mechanical system using time-domain measurements (Addressed in Chapter 3).

(2a) Design a controlled cam-follower experiment with lubricated point contact to

directly measure forces and motion under periodic impacts.

(2b) Propose and evaluate time-domain based signal processing techniques to

determine β and m from the measured data.

8

(2c) Verify if the contact damping force proportional to contact stiffness force

during impact.

(2d) Analyze the applicability of the viscous damping model to impacting

conditions.

Objective 3: Estimate the coefficient of friction for a mechanical system with combined rolling-sliding contact using vibration measurements under lubrication (Addressed in

Chapter 4).

(3a) Develop a contact mechanics model for a mechanical system with a

combined rolling-sliding contact to design a suitable experiment and to

predict the dynamic response.

(3b) Design a controlled laboratory experiment for the cam-follower system to

measure dynamic forces and acceleration.

(3c) Propose a signal processing technique to estimate using Fourier amplitudes

of measured forces and acceleration

(3d) Suggest an empirical formula for and compare the estimated values with

the literature.

9

References for Chapter 1

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and Machine Theory 12 (3) (1977) 201–224

[1.2] R. Alzate, M. di Bernardo, U. Montanaro, and S. Santini, Experimental and

numerical verification of bifurcations and chaos in cam-follower impacting

systems. Nonlinear Dynamics 50 (3) (2007) 409–429

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and Machine Theory 37 (2002) 1213–1239.

[1.4] T.L. Dresner, P. Barkan, New methods for the dynamic analysis of flexible

single-input and multi-input cam-follower systems. ASME Journal of Mechanical

Design 117 (1995) 150.

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Engineering for Industry 86 (1964) 343.

[1.6] R.L. Norton, Cam Design and Manufacturing Handbook, Industrial Press Inc.,

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Machine Theory 42 (9) (2007) 1238–1250.

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of discontinuity-induced bifurcations in cam-follower systems. Journal of Applied

Dynamical Systems 7 (2007) 18–38.

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[1.9] H. S. Yan, M. C. Tsai, M. H. Hsu, An experimental study of the effects of cam

speeds on cam-follower systems. Mechanism and Machine Theory 31 (4) (1996)

397–412.

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numerical investigation of coexistence, novel bifurcations, and chaos in a cam-

follower system. SIAM Journal on Applied Dynamical Systems 8 (2009) 592–

623.

[1.11] M.P. Koster, Vibrations of Cam Mechanisms: Consequences of Their Design,

Macmillan, 1974.

[1.12] C.A. Brockley, R. Cameron, A.F. Potter, Friction-induced vibration, Journal of

Tribology 89 (2) (1967) 101–107.

[1.13] A.H. Dweib, A.F. D’Souza, Self-excited vibrations induced by dry friction, part 1:

Experimental study, Journal of Sound and Vibration 137 (2) (1990) 163–175.

[1.14] A.F. D’Souza, A.H. Dweib, Self-excited vibrations induced by dry friction, part 2:

Stability and limit-cycle analysis, Journal of Sound and Vibration 137 (2) (1990)

177–190.

[1.15] B.N.J. Persson, Sliding friction, Surface Science Reports 33 (3) (1999) 83–119.

[1.16] H.D. Espinosa, A. Patanella, M. Fischer, A novel dynamic friction experiment

using a modified kolsky bar apparatus, Experimental Mechanics 40 (2) (2000)

138–153.

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[1.17] E.R. Hoskins, J.C. Jaeger, K.J. Rosengren, A medium-scale direct friction

experiment, International Journal of Rock Mechanics and Mining 5 (2)

(1968) 143–152.

[1.18] K. Worden, C.X. Wong, U. Parlitz, A. Hornstein, D. Engster, T. Tjahjowidodo, F.

Al-Bender, D.D. Rizos, S.D. Fassois, Identification of pre-sliding and sliding

friction dynamics: Grey box and black-box models, Mechanical Systems and

Signal Processing 21 (1) (2007) 514–534.

[1.19] V. Aronov, A.F. D’Souza, S. Kalpakjian, I. Shareef, Interactions among friction,

, and system stiffness. Part 2. Vibrations induced by dry friction. ASME

Journal of Lubrication Technology 106 (1984) 59–64.

[1.20] Y. Masayuki, N. Mikio, A fundamental study on frictional noise (1st report, the

generating mechanism of rubbing noise and squeal noise). Bulletin of JSME 22

(1979) 1665–1671.

[1.21] Y. Masayuki, N. Mikio, A fundamental study on frictional noise (3rd report, the

influence of periodic surface roughness on frictional noise). Bulletin of JSME 24

(1981) 1470–1476.

[1.22] Y. Masayuki, N. Mikio, A fundamental study on frictional noise (5th report, the

influence of random surface roughness on frictional noise). Bulletin of JSME 25

(1982) 827–833.

[1.23] S. He, R. Gunda, R. Singh, Effect of sliding friction on the dynamics of spur gear

pair with realistic time-varying stiffness. Journal of Sound and Vibration 301

(2007) 927–949.

12

[1.24] S. Kim, R. Singh, Gear surface roughness induced noise prediction based on a

linear time-varying model with sliding friction. Journal of Vibration and Control

13 (2007) 1045–1063.

[1.25] Y. Michlin, V. Myunster, Determination of power losses in gear transmissions

with rolling and sliding friction incorporated. Mechanism and Machine Theory 37

(2002) 167–174.

[1.26] M.N. Sahinkaya, A.-H.G. Abulrub, P.S. Keogh, C.R. Burrows, Multiple sliding

and rolling contact dynamics for a flexible rotor/magnetic bearing system.

IEEE/ASME Transactions on , 12 (2007) 179 –189.

[1.27] B. L. Stoimenov, S. Maruyama, K. Adachi, and K. Kato, The roughness effect on

the frequency of frictional sound. International 40 (4) (2007) 659–664.

[1.28] M. Othman, A. Elkholy, Surface-roughness measurement using dry friction noise.

Experimental Mechanics 30 (3) (1990) 309–312.

[1.29] M. Othman, A. Elkholy, A. Seireg, Experimental investigation of frictional noise

and surface-roughness characteristics. Experimental Mechanics 30 (4) (1990)

328–331.

[1.30] G.G. Gray, K.L. Johnson, The dynamic response of elastic bodies in rolling

contact to random roughness of their surfaces. Journal of Sound and Vibration 22

(1972) 323–342.

[1.31] P.R. Kraus, V. Kumar, Compliant contact models for collisions, in:

IEEE International Conference on and Automation, 1997, pp. 1382 –

1387 vol.2.

13

[1.32] O. Ma, Contact dynamics modelling for the simulation of the space station

manipulators handling payloads, in: IEEE International Conference on Robotics

and Automation, 1995, pp. 1252 –1258 vol.2.

[1.33] C. Padmanabhan, R. Singh, Dynamics of a piecewise non-linear system subject to

dual harmonic excitation using parametric continuation. Journal of Sound and

Vibration 184 (1995) 767–799.

[1.34] D. Zhang, W.J. Whiten, The calculation of contact forces between particles using

spring and damping models. Powder Technology 88 (1996) 59–64.

[1.35] M.A. Veluswami, F.R.E. Crossley, G. Horvay, Multiple impacts of a ball between

two plates - part 2: mathematical modelling, Journal of Engineering for Industry

97 (1975) 828–835.

[1.36] S.C. Lim, M.F. Ashby, J.H. Brunton, The effects of sliding conditions on the dry

friction of metals, Acta Metallurgica 37 (3) (1989) 767–772.

[1.37] M. Wakuda, Y. Yamauchi, S. Kanzaki, Y. Yasuda, Effect of surface texturing on

friction reduction between ceramic and steel materials under lubricated sliding

contact, Wear 254 (3 - 4) (2003) 356–363.

[1.38] S. Mentzelopoulou, B. Friedland, Experimental evaluation of friction estimation

and compensation techniques, Proceedings of the American Control Conference,

Baltimore, MD, USA, June 29 – July 1, 1994, 3132-3136.

[1.39] S. Shon, A. Kahraman, K. LaBerge, B. Dykas, D. Stringer, Influence of surface

roughness on traction and scuffing performance of lubricated contacts for

aerospace and automotive gearing, Proceedings of the ASME/STLE International

14

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61212.

[1.40] G.H. Benedict, B.W. Kelley, Instantaneous coefficients of gear tooth friction,

ASLE Transactions 4 (1) (1961) 59–70.

15

CHAPTER 2

ROTATIONAL SLIDING CONTACT DYNAMICS IN A NON-LINEAR CAM-

FOLLOWER SYSTEM AS EXCITED BY A PERIODIC MOTION

2.1 Introduction

The dynamics of cam-follower systems have traditionally been described by lumped parameter, linear system theory for the follower with motion input from the cam, as reported by Chen [2.1] in a literature survey (1977). More recent investigators have employed piecewise linear system models to study the loss of contact in a cam-follower system [2.2 - 4], and some studies [2.5 - 7] have examined the stability issues. In particular, Alzate et al. [2.8, 2.9] and Koster [2.10] studied bifurcations in a non-linear cam-follower system, though they did not focus on the kinematic non-linearity. Alzate et al. [2.8] used the coefficient of restitution concept to model the contact between the cam and follower. Such coefficient of restitution type models usually have several deficiencies as stated by Gilardi and Sharf [2.11]. Overall, the contact stiffness and damping non- linearities of a cam-follower system are yet to be rigorously studied. Also the effect of dry friction non-linearities has been neglected in the cam-follower models [2.1 - 10] since there is no motion along the direction of friction. The chief goal of this paper is, therefore, to overcome the void in the literature and study the combined rolling-sliding

16 contact dynamics (only in the context of a single degree-of-freedom system) given a periodic motion by the cam rotating about a fixed pivot. The proposed formulation would include kinematic, friction and contact non-linearities.

The sliding and/or rolling contacts are of interest in many mechanical systems such as pin-disk models [2.12 - 15], geared transmission systems [2.16 - 18], and bearings [2.19]. However, the dynamics of the sliding contact is sometimes studied using simple translating systems [2.20 - 26]. The rolling contacts have been investigated by

Remington using a lumped system model [2.27] and experiments [2.28]. Gray and

Johnson [2.29] have analyzed the rolling contact problem using a simple vibration model that included the contact mechanics concept. This paper will examine only the combined rolling-sliding contact and utilize some of the contact mechanics principles employed in other mechanical system [2.30, 2.31].

2.2 Problem formulation

Fig. 2.1 shows a single degree-of-freedom (SDOF) cam-follower system in the general position, when the cam and follower are not in contact. The fixed orthogonal coordinate system (eˆ ,eˆ ) describes the horizontal and vertical directions, with its origin x y at E. The circular cam of radius, rc, is considered; it rotates about the fixed pivot E, which is at a distance, e, from the geometric center of the cam ( Gc). The angular movement of the cam is given by Θ(t), the angle made by Gc E with the horizontal line in the counter- clockwise direction; it is also the motion input to the system. The follower is described by a rectangular bar (with width, wb), which is pivoted at its center of gravity to a fixed pivot

P which is at a distance, dy, above the ground. The angle α(t) made by the follower with

17 the horizontal line in the clockwise direction is the only generalized coordinate. The follower is supported by a linear follower spring ( ks) at a distance, dx, from P. The contact between the cam and follower is represented by means of non-linear contact stiffness

(kλ(ψi(t))) and damping ( cλ(ψi(t))) terms. Contact points in the follower and cam are Ob and Oc, respectively. During the operation, the system can be in either the sliding contact regime or the non-contact regime at a given instant, which is determined by the sign of

ψi(t). The coefficient of friction between the cam and follower is given by time-varying 0 (t) . When the follower just touches the cam (QO c = 0 ) for a given Θ , the state of the

0 0 0 system is denoted as the 0-state. This 0-state (where Q , Ob and Oc are coincident) is used to define the geometry of the cam-follower system and to derive the relationship between the fixed coordinates and moving coordinates (iˆ, ˆj) as attached to the follower at Q. This system is similar to the cam-follower experiment that has been studied by

Alzate et al. [2.8]; the results will be discussed later in section 2.4.

The cam-follower system, as discussed in this thesis includes kinematic (from the geometry of the system), dry friction, and contact non-linearities. The friction non- linearity arises due to the dependence of the friction force, Ff(t), on the magnitude as well as on the direction of the of sliding, vr(t). The contact non-linearity is from the non-linear Hertzian point contact model, the non-linear contact damping model

(function of the displacement and velocity of contact points), and the discontinuity during the contact. The key assumptions in the proposed formulation include the following: (i) the bearings at the pivots of the cam and follower are frictionless and rigid, allowing only rotation without any translation; (ii) cam and follower are elastic bodies, and their 18 surfaces are smooth; (iii) the contact force between the cam and follower follows the

Hertzian theory [2.32]; and (iv) the bending moment of the follower is negligible.

Follower Cam

Follower spring

Fig. 2.1. Cam-follower system in the general state where a non-linear contact stiffness

model, kλ(ψi(t)), is employed

The objectives of this chapter are as follows: (a) Develop a contact mechanics model for the cam-follower system with combined rolling-sliding contact; (b) Examine the applicability of different viscous and impact damping models and the coefficient of restitution concept by comparing the predictions with the experimental results reported by Alzate et al. [2.8]; (c) Study the effects of contact and friction non-linearities in the

19 sliding contact regime; and (d) Analyze the effect of kinematic non-linearity of the system by comparing it with a linearized model. Since all the non-linearities are inter- related with each other, the dynamic system is very complex even with a single degree- of-freedom formulation.

2.3 Analytical model

2.3.1 Relationship between the coordinate systems

In Fig. 2.1, QO is represented by ψ()tiˆ+ ψ () tj ˆ in the moving coordinate c i j system, and ψi(t) and ψj(t) are used to calculate the contact force and the moment imparted by the cam on the follower, respectively. A non-negative value of ψi(t) indicates that the cam and follower are not in contact. When ψi(t) is negative, the system is in the sliding contact regime with the magnitude of ψi(t) representing the deflection of the contact spring. At any instant, ψi(t) and ψj(t) can be calculated for a given α(t) and Θ(t) from the system geometry as shown below. From Fig. 2.1 the vectors are calculated as follows: PO= PE + EG + GO , (2.1) b ccb wb  w b  POtb =χα( )cos() ( t ) + sin()() α ( te ) ˆx +− χα ( t )sin ( t ) + cos() α ( te )  ˆ y , 2  2 

(2.2)

EGc=− ecos( Θ ( tee )) ˆ x − sin( Θ ( te )) ˆ y , (2.3)

GOcbci=−+( rψ()sin t) ( α () ter) ˆ xci −+( ψ ()cos t) ( α () te) ˆ y . (2.4)

20

0 0 0 Here, χ(t) = χ -ψj(t), where χ(t) and χ are the components of PO b and PO b respectively along ɵj . The constant vector PE is evaluated based on the 0-state as follows, where α0 is the angle of the follower at the 0-state:

 00wb  0 0  PE=χcos() α ++ rc  sin()() α +Θ ee cos  ˆx  2   (2.5)  00wb  0 0  +− χsin() α ++rc  cos()() α +Θ e sin eˆy .  2  

Using Eqs. (2.2) to (2.5) in Eq. (2.1) and rearranging, ψi (t) and ψj(t) are evaluated as,

0 0 wb  0 ψχααi ()t= sin()() tr −++c  () cos()() αα t −− 1 2  (2.6) +et[sin()α ( ) +Θ−0 sin() α ( tt ) +Θ ( ) ],

w  0 0  b 0 ψχj ()t=− 1cos()() αα tr −++  c  sin()() αα t − 2  (2.7) 0  −etcos()α ( ) +Θ− cos() α ( tt ) +Θ ( )  .

Next, differentiate Eqs. (2.6) and (2.7) with respect to time to yield the following:

w  ɺ0 0 ɺb 0 ɺ ψχαααi ()t= cos()() ttr −−+ ()c  sin()() ααα tt − () 2  (2.8)  0 ɺ ɺ ɺ  +ecos()α () t +Θ α ()cos t −() α () tttt +Θ ()() α () +Θ (),

w  ɺ0 0 ɺb 0 ɺ ψχαααj ()t= sin()() ttr −++ ()c  cos()() ααα tt − () 2  (2.9)  0 ɺ ɺ ɺ  +etsin()α () +Θ− αα sin() () tttt +Θ ()() α () +Θ ().

2.3.2 Equations of motion

Fig. 2.2 shows the free body diagram of the follower in the sliding contact regime.

The moment balancing about P yields the following equation of motion for the follower 21

P in the sliding contact regime, where Ib is the mass moment of inertia of the follower about P:

P ɺɺ Ibα() t=− Ftd sxn () + Ft ()()0.5 χ t − Ftw fb () . (2.10)

Fig. 2.2. Free body diagram of the follower in the sliding contact regime

u The elastic force ( Fs(t)) from the follower spring is given by the following, where Ls is the un-deflected length of the follower spring:

u  FtkLddsssyx( )= −+ tan(α ( t )) − 0.5 w b sec( α ( t ))  . (2.11)

The normal contact force ( Fn(t)) is given by

Ft()= − kψψ () t () tc − ψψ () tɺ (). t (2.12) nλ( ii) λ ( ii)

22

The Hertzian theory [2.32] for line contact is used to define kλ(ψi(t)) as follows, where lλ is the length of line contact, and Y is the equivalent Young’s modulus (with subscript e denoting equivalent):

π kλ()ψ i( t )= Yl e λ . (2.13) 4

The equivalent Ye of the two materials in contact is calculated based on the Hertzian theory [2.32] as well:

1 2 2  − 1−νc 1 − ν b Ye = +  . (2.14) Yc Y b 

Here, ν is the Poisson’s ratio of the material, and the subscripts b and c represent the follower and cam, respectively. The force Ff(t) due to sliding friction exerted on the

follower by the cam is Ftf()= µ () tFt n () where two models for time varying (t) are utilized (discussed later in this section). The equation of motion for the system in the non- contact regime is derived below, similar to the sliding contact regime, but now with Fs(t)

= 0 and Fn(t) = 0.

IPαɺɺ () t= − Ftd (). (2.15) b s x Eqs. (2.10) and (2.15) are numerically solved using MATLAB’s [2.33] ODE solver for stiff problems (which uses simultaneous first and fifth order Runge-Kutta formulations) for a given initial value of α(t). These results were found not to differ significantly from the results from the slower but accurate fourth and fifth order Runge-Kutta formulations for some test cases. One must, however, keep track of the condition for switching

23 between the contact regimes (‘event detection’ feature of MATLAB [2.33] is used) based on the value of ψi(t) as discussed earlier.

2.3.3 Static equilibrium and linearized natural frequency

The force Fs(t) is assumed to be sufficiently large at the static equilibrium point to maintain the cam-follower contact. The equations for the static equilibrium point (given

0 by superscript *) are derived for Θ(t) = Θ by replacing α(t), ψi(t), and ψj(t) with the corresponding values at the static equilibrium point (α*, ψi*, and ψj*, respectively), and forcing all time derivative terms to zero in the Eqs. (2.6), (2.7) and (2.10), as follows:

0 0 wb 0  w b χααsin() *−++rc cos() αα * −−+−=  r c ψ i * 0, (2.16) 2  2

w  0 0  b 0 χ1− cos() αα * −++  rc  sin() ααψ * −−=j * 0, (2.17) 2 

u  π 0 −−+kdLddsxsyxtan()()α *0.5sec − w b α *  + Yl eiλ ψχψ *() −= j * 0. (2.18) 4

Simultaneous Eqs. (2.16) to (2.18) are numerically solved to find α*, ψi*, and ψj*. The system is then linearized about the static equilibrium point. Writing the linearized equation of motion of the system in the sliding contact regime in state space form as

Ξɺ = h(Ξ) , where

ɺ T T T Ξ =()()αα(),()tt = εε1 (), tt 2 () , h()Ξ= ( h1 (), Ξ h 2 ()). Ξ (2.19 a, b)

The state space equations are derived below from Eq. (2.10) as,

ɺ t t h ε1()= ε 2 () = 1 (), Ξ

[−Ftd () + Ft ()()0.5χ t − wFt ()] ɺ sxn bf ε2 (t )=P = h 2 ( Ξ ). (2.20 a, b) Ib 24

The Jacobian matrix ( J) at the static equilibrium point is

* ∂h  J=i  ; i , j = 1,2. (2.21) ∂ε j 

The following partial derivatives are calculated to evaluate J at the static equilibrium point:

* * ∂h  ∂h  1  = 0; 1  =1; ∂ε1  ∂ε2 

* ∂F ∂ F ∂χ ∂F  * s n f   −dx +χ + F n − 0.5 w b  ∂h2 ∂∂αα ∂ α ∂ α   = P  ; ∂ε1  Ib   

* ∂F ∂ F ∂χ ∂F  * s n f   −dx +χ + F n − 0.5 w b  ∂h2 ∂∂ααɺɺ ∂ α ɺ ∂ α ɺ   = P  . (2.22 a- d) ∂ε 2  Ib   

The linearized natural frequency ( ϑ) is calculated as follows where the operator, Im, yields the imaginary part of the operand, and the operator, Eig , gives the eigenvalues of a square matrix:

0.5 ϑ = Im ()Eig[] J (2.23)

2.3.4 Contact damping and dry friction models

Five different damping models are utilized for cλ(ψi(t)), as described in Eq. (2.12), to examine the dissipation of energy by impact and other mechanisms. First is the pure viscous damping formulation (denoted as damping model A) where the damping

A coefficient cλ (ψ i ( t ) ) is time-invariant and is calculated as follows from Eqs. (2.10) and

25

(2.12) using the linearized modal viscous damping ratio ζ (with a superscript representing

0 * the damping model) and χ* = χ − ψ j :

A P A 2ζ ϑ Ib cλ ()ψ i ( t )= 2 . (2.24) ()χ *

Two pure impact damping formulations (denoted as damping models B and C) are analyzed next; these models are suggested by Padmanabhan et al. [2.34] and Zhang et al.

[2.35], respectively; here β is the impact damping factor:

cBψ( t )= β B kt ψ ( ); (2.25) λ( i) λ i

0.25 cCψ() t= β C ψ (). t (2.26) λ ( i) i

The dissipation of energy in the system might be through impact from the point of contact (at t = t a) to the point of maximum deformation of the contact spring (at t = t b and

ɺ where ψ i = 0 ), and then through the material until the follower goes out of contact of the cam (at t = t c). Therefore, combined viscous-impact damping formulations (as denoted damping models D and E) are proposed as follows:

 D a b βkλ ψ i ( t ); ttt< < D  D P cλ ()ψ i ( t )= 2ζ ϑ Ib b c ; (2.27 a, b)  2 ; t< t < t  ()χ *

βE ψ (t )0.25 ; ttt a< < b  i cE t E P λ ()ψ i ( )= 2ζ ϑ Ib b c . (2.28 a, b)  2 ; t< t < t  ()χ *

The time-varying coefficient of friction between the cam and follower is described using two well known dry friction formulations: Coulomb friction (model I) and Smoothened

26

Coulomb friction (model II). Fig. 2.3 shows these as a function of the vr(t) between the cam and follower; here, vr(t) is given by

ɺ ɺ ɺ vtr()=ψ j () tre −+( c sin( α () t +Θ () t))( α () t +Θ () t ) . Model I (with a maximum value of

I m) is given by µ()t= µ m sgn( v r () t ) and has a sharp discontinuity at vr(t) = 0, which is smoothened by model II using a regularizing factor ( σ) for the hyperbolic tangent

II function as: µ()t= µm tanh( σ v r () t ).

2.4 Examination of the contact non-linearity and alternate damping models

The proposed contact mechanics model is used to represent the physics of the cam-follower experimental system as reported by Alzate et al. [2.8]. In the prior experiment [2.8], the follower (along with its spring) is above the cam, unlike in Fig. 2.1 where the follower (and its spring) is placed below the cam. The follower is pivoted at its center of gravity, and the magnitude of Fs(t) is assumed to be the same in tension or compression. Consequentially, the proposed contact mechanics model is representative of the cam-follower experiment [2.8], and hence calculations can be compared with the reported measurements.

27

1

0.5

0

-0.5

-1 -1 -0.5 0 0.5 1 v r

Fig. 2.3. Normalized dry friction models (equations are given in Section 2.3.4). Key:

, Coulomb friction (Model I); , Smoothened Coulomb friction (Model II).

The results are viewed in terms of the residual response ( αr(t)) for a given constant of the cam ( c), where the measurements are available at 110,

135, 143, 148, 150, 155, and 159 rpm. Mathematically, αr(t) is given by αr(t) = α(t) -

αi(t), where αi(t) is the response assuming the follower to be in contact with the cam, and

αi(t) is calculated from the kinematics of Fig. 2.1 as follows:

0.5  4 22 2   2   PGc+ PG cc PG −+ PG ccb( r 0.5 w ) ++ PG ccb ( r 0.5 w )  ( x x y x ) y α i (t )= cos−1   .  2 2  PGc+ PG c  x y    (2.29)

28

Here, PG c and PG c , the magnitudes PG c along x and y directions, respectively, are x y

r functions of Θ(t). At any c, α (t) is predicted using the contact mechanics model with

0 the parameter values given in [2.8], and the input is Θ(t) = Θ +ct with the static equilibrium point as the initial condition. The contact stiffness is evaluated using the

Hertzian theory [2.32] for a steel cam and follower (with Yc = 200 GPa, Yb = 200 GPa, νc

= 0.3, and νb = 0.3) and by assuming line contact of length lλ as 16 mm. For several measurement cases, the cam-follower system impacts and stays in the non-contact regime for most of the time. Consequentially, friction model I is used with m = 0.3 since the dry friction does not play a major role in the response. Alternate damping models are first

r used to predict the root-mean-square of residual response ( αrms ) using the contact mechanics model, and then appropriate damping parameter(s) are identified for each model based on the best correlation with the measurements [2.8]. The following damping parameters are identified where superscripts denote the model type: ζA = 0.125, βB = 3.25 s/m, βC = 0.325 MNs/m 1.25 ζD = 0.119, βD = 4.25 s/m, ζE = 0.096 and βE = 0.325

MNs/m 1.25 .

r Fig. 2.4 shows the variation in αrms and the peak-to-peak of the residual

r amplitude ( α p ) for five damping models (using these identified parameters), along with the digitized experimental and analytical results of Alzate et al. [2.8]. Fig. 2.5 compares predicted normalized residual responses ( α r (t ) ) with different damping models along with experimental data [8] in the time domain. The cam-follower system in [2.8] is observed to go into a chaotic state beyond 155 rpm, and it is accompanied by increased

29 amplitude. From Fig. 2.4 and Fig. 2.5 it is inferred that damping models A and C are still periodic (at c = 155 rpm), while damping model E seems to yield chaotic even before 155 rpm. Only two damping models (B and D) start to behave chaotically for c >

155 rpm; this is similar to the previous experiment [2.8]. Hence, these two damping models are deemed more applicable to the cam-follower system during impacts. The pure viscous damping model and the impact damping model suggested by Zhang et al. [2.35] do not seem to predict the physics for the current example. Even though the applicability of contact damping models are specific to a given contact element or mechanical system, damping models A, C or E should be suitable for other impacting systems.

The combined damping model D (with ζD = 0.119 and βD = 4.25 s/m) is utilized for further analyses. Fig. 2.6 shows sample time domain and frequency domain

r comparisons of the normalized residual responses ( α ) at c = 155 rpm; the predictions from the contact mechanics model is compared with the experimental result [2.8]. The normalized time scale ( t ) is calculated based on the period required for one revolution of the cam; the normalized frequency scale ( f ) is calculated based on the speed of the cam, and α r (t ) is calculated based on the time average of αr(t) for one revolution of the cam.

A good correlation between the previous experiment [2.8] and the proposed formulation is observed both in time and frequency domains. Using the inverse kinematics [2.36], the

c needed for the follower to lose contact is predicted as 130 rpm from the contact mechanics model, which closely matches the measured value of 125 rpm as reported by

Alzate et al. [2.8]. These comparisons validate the contact mechanics model.

30

0.04 (a) 0.035

0.03

0.025

0.02

0.015

0.01

0.005

0 110 115 120 125 130 135 140 145 150 155 160 c [rpm] (b) 0.1 0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0 110 115 120 125 130 135 140 145 150 155 160 [rpm] c

r r r r Fig. 2.4. Comparison of αrms and α p at lower speeds. (a) αrms ; (b) α p . Key: , contact mechanics formulation with damping model A; , damping model B; , damping model

C; , damping model D; , damping model E; , experimental result from

Alzate et al. [2.8] ; , prior a nalytical result from Alzate et al. [2.8]. 31

6

5

4

3

2

1

0 0 0.5 1 1.5 2 2.5 3

t

Fig. 2.5. Comparison of predicted α r (t ) using different damping models with experimental data at c = 155rpm. Key: , contact mechanics formulation with damping

model A ; , damping model B; , damping model C; , damping model D; ,

damping model E; , prior experimental result from literature [2.8].

32

4 (a)

3

2

1

0 0 0.5 1 1.5 2 2.5 3 t (b) 2

1.5

1

0.5

0 1 2 3 4 5 f

r Fig. 2.6. Comparison of experimental and analytical results for α at c = 155rpm. (a)

Time domain comparison; (b) Frequency domain comparison. Key: , analytical contact mechanics formulation with damping model D; , experimental result from

Alzate et al. [2.8].

33

2.5 Assessment of the coefficient of restitution ( ξ) concept

Alzate et al. [2.8] assumed the coefficient of restitution, ξ, to be a function of c.

They empirically determined ξ for various c using a linear interpolation for which no reference was cited. Also, they employed a very small range of c with very few measured data points to estimate ξ as a function of c. According to Goldsmith [2.37] and Stronge [2.38], ξ should decrease as c or the velocity of impact increases.

Nevertheless, Alzate et al. [2.8] have estimated ξ to increase with c. This contradiction suggests that the ξ estimation by Alzate et al. [2.8] is not accurate, and thus this issue is examined next.

An approximate energy balance technique has been developed next using the

r coefficient of restitution concept in order to predict α p at different c. The velocity of

the contact point ( Ob or Oc) along the direction of iˆ is represented by u; the subscripts b and c denote the contact point in follower and the cam, respectively, while the superscripts a and b denote the after and before impact, respectively. The following assumptions are made in this technique: (i) the follower is at rest before impact, and the cam and follower impact at the static equilibrium point; (ii) the velocities

a b uc and uc are equal to ec (the maximum possible velocity of Oc along the direction of

ɵ i ) during impact; and (iii) ξ is constant for all values of c. From the definition of ξ the following relationship is derived as,

a a ub− u c ξ = b b . (2.30) uc− u b

34

b b Based on the assumption made, ub = 0. Using this value of ub and the assumptions in

a Eq. (2.30), ub is calculated as,

a ub=(1 +ξ ) e Ω c . (2.31)

The maximum angular velocity of the follower after impact is (1+ξ )e Ω c / χ * .

Therefore, the kinetic energy ( T) of the follower after impact is,

P 2 Tb=0.5 I b( (1 +ξ ) e Ω c / χ *) . (2.32)

This kinetic energy, Tb, is equal to an increase in the potential energy (V) of the system essentially deflecting the follower spring. The increase in the potential energy of the system is given by,

u α p kLdd− +tan()()α − 0.5 w sec α   ssyx b   ∆V = ∫ d α. (2.33) 2   α* dxsec()()()α− 0.5 w b sec α tan α  

r By equating Tb and V (from the Eqs. (2.32) and (2.33), respectively) , αp, and hence α p

r , are calculated at each c. Fig. 2.7 shows a map of α p vs. c for the approximate energy balance technique (with ξ = 0.05, 0.2, 0.4 and 0.6). Results from the contact mechanics model and the literature [2.8] are also given. It is inferred that below 160 rpm, ξ should

r be less than 0.05 for successfully predicting α p with the approximate energy balance technique. In contrast, Alzate et al. [2.8] have used much higher values of ξ say from 0.37

(c = 130 rpm) to 0.6 ( c = 155 rpm). This clearly suggests an ambiguity in using the coefficient of restitution method for predicting the non-linear the response of the system.

35

The study using the contact mechanics model is extended to higher speeds in Fig. 2.8 where the proposed model, the previous experiment [2.8], and the approximate energy balance technique (with ξ = 0.2, 0.6 and 1) are compared. Also from Fig. 2.8, it is inferred that Alzate et al. [2.8] have analyzed a very small range of c in their study. As

r r observed, α p is very low for c < 160 rpm. With an increase in speed, α p increases and saturates at 1.36 rad, which corresponds to α = 90 o. Unlike the contact mechanics model, the approximate energy balance technique using the coefficient of restitution concept yields only a global (but a smooth) trend for different values of ξ. This supports the claim made by Gilardi and Sharf [2.11] that the coefficient of restitution model has inherent problems.

36

0.16

0.14

0.12 ξ = 0.6

0.1

0.08 ξ = 0.4

0.06 ξ = 0.2

0.04 ξ = 0.05

0.02

0 0 20 40 60 80 100 120 140 160

c [rpm]

r Fig. 2.7. Map of α p vs. c at lower speeds. Key: , predictions from contact mechanics model; , experimental results from Alzate et al. [2.8]; , prior analytical results from

Alzate et al. [2.8]; , prediction based on approximate energy balance technique

(given in Section 2.4.2) with ξ = 0.05; , ξ = 0.2; , ξ = 0.4; , ξ = 0.6.

37

1.4

1.36 rad 1.2 ξ = 1

1 ξ = 0.6

0.8

0.6

0.4

ξ = 0.2 0.2

0 100 200 300 400 500 600 700 800 900 [rpm] c

r Fig. 2.8. Map of α p vs. c over a broad range of speeds. Key: , predictions from

contact mechanics formulation; , experimental results from Alzate et al. [2.8]; ,

prior analytical results from Alzate et al. [2.8]; , prediction based on approximate energy balance technique (given in Section 2.4.2) with ξ = 0.2; , ξ = 0.6;

, ξ = 1.

38

2.6 Study of the line and point contact models in the sliding contact regime

The speed, c, at which the follower would lose contact with the cam, is calculated for different values of ks using inverse kinematics [2.36]. Fig. 2.9 shows the sliding contact and impacting regimes in terms of ks - c map given e = 0.1 rc for a constant speed of the cam. Higher c is needed to lose the contact with the increase in ks.

For further analysis in the sliding contact regime ks = 2000 N/m and c = 300 rpm are chosen. The non-linear analyses in the sliding contact regime, as reported in this study, cannot be obviously performed using the coefficient of restitution model developed in

[2.8].

Recall that a line contact (with lλ =16 mm) is assumed between the cam and follower in the previous section. However, this contact can be approximated as a non- linear Hertzian point contact [2.32] for small lλ as,

4 0.5 kλ ()ψi() t= Yr eci() ψ (). t (2.34) 3

To have a meaningful comparison of the line and point contact stiffness models, lλ has been reduced to 1.6 mm. Only the Coulomb friction (with µm = 0.3) and combined damping model D (with ζD = 0.01 and βD = 4.25 s/m) are utilized in the comparison of the

αɺɺ spectra of both line and point contact models as shown in Fig. 2.10 for e = 0.1 rc. The

αɺɺ spectra with the two contact models do not differ in the lower shown in

Fig. 2.10 (a) which are dominated by the harmonics of c. However, the spectra differ at the resonant peaks shown in Fig. 2.10 (b). For the line contact, the frequency is 2182 Hz but for the point contact the frequency is 1331 Hz, mainly because the linearized contact

39 stiffness at the static equilibrium point for line and point contacts are 138 MN/m and 34.2

MN/m respectively.

2.7 Analysis of the friction non-linearity

2.7.1 Effect of direction

The dynamic effects of friction non-linearity are studied for the cam-follower system next since such issues are of importance in mechanical systems [2.39-41]. The system is a self-energizing type when vr(t) > 0 (the cam rotates clockwise), as the friction force tends to increase the normal force; and the system is a de-energizing type when vr(t)

< 0 (the cam rotates counter-clockwise). The comparison of the αɺɺ spectra for the two different systems with the same cam speed (300 rpm) with the Coulomb friction model is shown in Fig. 2.10. Due to different directions of Ff(t), ϑ is 2182 Hz for the de-energizing system and 2125 Hz for the self-energizing system, which can be identified by the peaks in the spectrum in Fig. 2.10 (b). The harmonics of c (in the lower frequency range) of both systems match well, as seen in Fig. 2.10 (a). The effect of change in the direction of rotation is more pronounced in the Fn(t) time history that is displayed in Fig. 2.11. The self-energizing system has a higher mean component of Fn(t) compared to the de- energizing system; this is because the Ff(t) increases the Fn(t) in the self-energizing system which in turn increases Ff(t).

2.7.2 Dynamic bearing and friction forces

The friction models are found to more significantly affect the dynamic forces than the displacement or the acceleration. As such, the response of the system with the alternate friction models is found not to vary significantly as long as vr(t) stays in the

40 same direction. Consequentially, a change in direction in vr(t) is introduced (twice per revolution of the cam) by increasing e to 0.7 rc as observed from Fig. 2.12; the system is verified to be in the sliding contact regime at 50 rpm, using the inverse kinematics [2.36].

The force time histories Ff(t) and Nx(t) (dynamic bearing force along horizontal direction) are shown in Fig. 2.13 (a) and (b), respectively, for the alternate dry friction models. Note that Nx(t) is calculated from Fig. 2.2 as,

NtFtx()= f ()cos(α () tFt) + n ()sin( α () t ) (2.35)

As seen from Fig. 2.13 the forces Ff(t) and Nx(t) with friction model I (with µm = 0.3) are discontinuous during the change in direction of vr(t) followed by some high frequency . This discontinuity is smoothened by using a small value of σ = 10 for friction model II, but the result of the friction model II should be close to that of the friction model I for a high value of σ.

Next, it is assumed that the cam oscillates with a particular frequency ( ωc); the motion of the cam is described by Θ(t) = Θ(0) +0.5 Θp sin( ωct), where Θp is the peak-to- peak value of Θ(t). Fig. 2.14 shows that vr(t) < 0 for nearly half the period of oscillation

(π/ωc), and vr(t) > 0 for the other part of the period; here Θp = π rad, ωc = 40 rpm, and e =

0.7 rc. Hence, the system acts as self-energizing and de-energizing types on a cyclic basis.

Fig. 2.15 shows the periodic profiles of the forces Ff(t) and Nx(t) where a discontinuity is observed for friction model I when vr(t) changes its sign; this discontinuity is smoothened using the friction model II. When | vr(t)| >> 0, the forces Ff(t) and Nx(t) predicted by model I and II are very close.

41

1500

Impacting regime

1000 [rpm] c

500 Sliding contact regime

0 0 500 1000 1500 2000 2500 3000 3500 4000

ks [N/m]

Fig. 2.9. Identification of contact domains based on ks - c mapping at a constant cam

speed with e = 0.1 rc.

42

2 (a) 10

10 0

10 -2

10 -4

10 -6

0 5 10 15 20 25 30 f [Hz] (b) 10 -3

10 -4

10 -5

10 -6

10 -7

1000 1500 2000 2500 f [Hz]

D D Fig. 2.10. Comparison of αɺɺ spectra (with µm = 0.3, ζ = 0.01 and β = 4.25 s/m). (a)

Spectra showing harmonics of c; (b) Spectra showing natural frequency of the system.

Key: , de-energizing system with line contact ( lλ = 0.0016m, c = 300 rpm);

, self-energizing system with line contact ( lλ = 0.0016m, c = -300 rpm); , de-

energizing system with point contact ( c = 300 rpm). 43

46

45

44

43 [N] 42 ) t ( n

F 41

40

39

38

37 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Fig. 2.11. Comparison of Fn(t) for different direction of cam rotation with line contact ( lλ

D D = 0.0016m, µm = 0.3, ζ = 0.01 and β = 4.25 s/m). Key: , de-energizing (c =

300 rpm); , self-energizing (c = -300 rpm).

44

0.1

0

-0.1

-0.2 ) [m/s] t ( r

v -0.3

-0.4

-0.5

-0.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Fig. 2.12. Comparison of relative sliding velocity vr(t) for two dry friction models of Fig.

D D 2.3 (with c = 50 rpm, e = 0.7 rc , ζ = 0.01 and β = 4.25 s/m). Key: , Coulomb

friction; , Smoothened Coulomb friction.

45

15 (a) 10

5

0 ) [N] t ( f

F -5

-10

-15 0 0.5 1 1.5 2 t (b) 20

15

10 ) [N] t ( x

N 5

0

-5 0 0.5 1 1.5 2 t

Fig. 2.13. Comparison of forces for two dry friction models of Fig. 2.3 (with c = 50

D D rpm, e = 0.7 rc, ζ = 0.01 and β = 4.25 s/m). (a) Nx(t); (b) Ff(t). Key: , Coulomb

friction; , Smoothened Coulomb friction. 46

1.5

1

0.5

0 ) [m/s] t ( r v

-0.5

-1

-1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t

Fig. 2.14. Comparison of relative sliding velocity vr(t) for two dry friction models of Fig.

D D 2.3 (with ωc = 40 rpm, e = 0.7 rc, ζ = 0.01 and β = 4.25 s/m). Key: , Coulomb

friction; , Smoothened Coulomb friction.

2.8 Study of kinematic non-linearity

The system is linearized next by assuming small angles and by carrying out a perturbation analysis about the static equilibrium point with the following limitations: (i) the system must always be in the sliding contact regime, and (ii) (t) must be time- invariant, hence the Coulomb friction model (where the magnitude of (t) is constant for 47 all vr(t)) must be used, and the sign of vr(t) must be time-invariant. By implementing the above limitations, the friction non-linearity and the contact non-linearity (in the equations for the sliding contact regime as discussed in section 2.3.2) have been eliminated resulting in only the kinematic non-linearity of the system. The linearized equation of motion for this particular system with line contact is derived below by replacing α(t) = α*

+ δα (t) in Eq. (2.10), where δα (t) is the perturbation of α(t) about α*:

 u dx  Ls− d y +()sin(2α *) + 2 δ ( α )  1+ cos(2α *) P ɺɺ   Ibδ() α ( t ) + kd s x  0.5 wb  −()2cos(α *) + 2sin( αδα *) ( )   1+ cos(2α *)   χ0cos( αα *− 0 ) +e cos( α * +Θ 0 )   δ( α ) +  0   −(rc + 0.5 w b )sin(α * − α )   +k()χ* − 0.5 µ w   λ b  0 0 0   χααsin( *−++ ) (rc 0.5 w b )cos( αα * − ) −1   ()  0  +esin(α * +Θ− ) e sin() α * +Θ () t    χαα0cos( *−−+ 0 ) (r 0.5 w )sin( αα * − 0 )   +cχ* − 0.5 µ w  δα (ɺ )  c b   = 0. λ ()b 0  +ecos(α * +Θ )   (2.36)

Rearrange Eq. (2.36) and write it in the standard form as follows:

IPδαɺɺ() tC+ δα () ɺ + K δα () = Ft (). (2.37) b( ) l l l

Here, the effective damping coefficient ( Cl), stiffness ( Kl), and time-varying forcing function ( Fl(t)) are given as follows:

χαα0cos( *−−+ 0 ) (r 0.5 w )sin( αα * − 0 )  C= c()χ* − 0.5 µ w  c b  , lλ b  0  +e cos(α * +Θ ) 

0 0 0  ks d x[2− w b sin(α *) ] χcos( αα *− ) +e cos( α * +Θ ) K= + k()χ* − 0.5 µ w   , l λ b 0 1+ cos(2α *) −(rc + 0.5 w b )sin(α * − α ) 

48

u dxsin(2α *)− w b cos( α *)  Ftl( ) =− kd sxs L −+ d y  1+ cos(2α *)  * 0  (rc+ 0.5 w b )() cos(α −−+ α ) 1 (2.38 a- c)   −kχ0 − 0.5 µ w 0   . λ ()b 0 0 sin(α *+Θ ) χsin( α *− α ) + e   −sin()α * +Θ (t )  

The αɺɺ spectra of the linearized and the non-linear systems are compared in Fig.

2.15. Observe that the linear system has only the fundamental harmonic of c while the non-linear system exhibits the super-harmonics of c as shown in Fig. 2.16(a). The linear system has a single peak at the natural frequency of the system, but the non-linear system displays side bands associated with this peak as seen in Fig. 2.16 (b). Therefore, the linear system approximation does not accurately predict the results of the system with only kinematic non-linearity.

2.9 Conclusion

The non-linear dynamics of the cam-follower system have been analyzed in this study. First, a contact mechanics formulation for a cam-follower system with combined rolling-sliding contact has been developed, and the predictions with combined viscous- impact damping models are successfully compared with the experimental results reported by Alzate et al. [8]. Second, the accuracy of the coefficient of restitution concept is analyzed using the approximate energy balance technique, and the estimated value of the coefficient of restitution by Alzate et al. [2.8] is found to be much lower than the estimates of this study; this suggests that there is some ambiguity in employing this concept for the cam-follower system. Third, the effect of friction non-linearity on the dynamic forces is studied using discontinuous and smoothened dry friction models.

49

Finally, a linearized system is found to be inadequate in representing the system with only kinematic non-linearity.

Several contributions emerge from this study over the current literature on the cam-follower dynamics [2.1-10]. The new contact mechanics formulation successfully predicts the dynamics of the cam-follower system with combined rolling-sliding contact in both impacting regime and sliding contact regimes. A better understanding of the applicability of different damping models and the inaccuracies of the coefficient of restitution model to the cam-follower system during impacts is obtained. Even though the applicability of contact damping models are specific to the cam-follower system, this thesis provides better insights into the damping mechanisms of a family of mechanical systems. This analysis also yields a better understanding of the roles of the friction and kinematic non-linearities in the sliding contact regime. The chief limitation of the chapter is the utilization of a single degree-of-freedom model that assumes the cam is rigidly pivoted about its center of rotation. This deficiency should be removed in future study with a higher degree-of-freedom system. Also, semi-analytical solutions can be sought.

50

(a) 15

10

5

0 ) [N] t ( f -5 F

-10

-15 0 0.5 1 1.5 2 t (b) 30

20

10 ) [N] t ( x N 0

-10 0 0.5 1 1.5 2 t

Fig. 2.15. Comparison of forces for two dry friction models of Fig. 2.3 (with ωc = 40

D D rpm, e = 0.7 rc, ζ = 0.01 and β = 4.25 s/m). (a) Ff(t) ; (b) Nx(t) . Key: , Coulomb

friction; , Smoothened Coulomb friction. 51

10 2 (a)

10 0

10 -2

10 -4

10 -6

0 5 10 15 20 25 30 f [Hz] -3 (b) 10

10 -4

10 -5

10 -6

10 -7 1000 1500 2000 2500 f [Hz]

ɺɺ D D Fig. 2.16. Comparison of α spectra (with µm = 0.3, c = 300 rpm, ζ = 0.01 and β =

4.25 s/m). (a) Spectra showing harmonics of c; (b) Spectra showing natural frequency

of the system. Key: , Non-linear system; , Linear system.

52

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(1979) 1665–1671.

[2.14] Y. Masayuki, N. Mikio, A fundamental study on frictional noise (3rd report, the

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pair with realistic time-varying stiffness. Journal of Sound and Vibration 301

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[2.17] S. Kim, R. Singh, Gear surface roughness induced noise prediction based on a

linear time-varying model with sliding friction. Journal of Vibration and Control

13 (2007) 1045–1063.

[2.18] Y. Michlin, V. Myunster, Determination of power losses in gear transmissions

with rolling and sliding friction incorporated. Mechanism and Machine Theory 37

(2002) 167–174.

[2.19] M.N. Sahinkaya, A.-H.G. Abulrub, P.S. Keogh, C.R. Burrows, Multiple sliding

and rolling contact dynamics for a flexible rotor/magnetic bearing system.

IEEE/ASME Transactions on Mechatronics, 12 (2007) 179 –189.

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the frequency of frictional sound. Tribology International 40 (4) (2007) 659–664.

[2.21] M. Othman, A. Elkholy, Surface-roughness measurement using dry friction noise.

Experimental Mechanics 30 (3) (1990) 309–312.

[2.22] M. Othman, A. Elkholy, A. Seireg, Experimental investigation of frictional noise

and surface-roughness characteristics. Experimental Mechanics 30 (4) (1990)

328–331.

[2.23] A. Le Bot, E. Bou Chakra, Measurement of friction noise versus contact area of

rough surfaces weakly loaded. Tribology Letters 37 (2010) 273–281.

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[2.24] A. Le Bot, E. Bou-Chakra, G. Michon, Dissipation of vibration in rough contact.

Tribology Letters 41 (2011) 47–53.

[2.25] H. Ben Abdelounis, H. Zahouani, A. Le Bot, J. Perret-Liaudet, M.B. Tkaya,

Numerical simulation of friction noise. Wear 271 (2011) 621–624.

[2.26] A. Soom, J.-W. Chen, Simulation of random surface roughness-induced contact

vibrations at Hertzian contacts during steady sliding. ASME Journal of Tribology

108 (1986) 123–127.

[2.27] P.J. Remington, Wheel/rail rolling noise, I: Theoretical analysis, Journal of the

Acoustical Society of America 81 (1987) 1805–1823.

[2.28] P.J. Remington, Wheel/rail rolling noise, II: Validation of the theory, The Journal

of the Acoustical Society of America 81 (1987) 1824–1832.

[2.29] G.G. Gray, K.L. Johnson, The dynamic response of elastic bodies in rolling

contact to random roughness of their surfaces. Journal of Sound and Vibration 22

(1972) 323–342.

[2.30] P.R. Kraus, V. Kumar, Compliant contact models for rigid body collisions, in:

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1387 vol.2.

[2.31] O. Ma, Contact dynamics modelling for the simulation of the space station

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[2.32] K.L. Johnson, Contact Mechanics, Cambridge University Press, Cambridge,

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56

[2.33] MATLAB User’s Guide, The Math Works Inc., Natick, MA, 1993.

[2.34] C. Padmanabhan, R. Singh, Dynamics of a piecewise non-linear system subject to

dual harmonic excitation using parametric continuation. Journal of Sound and

Vibration 184 (1995) 767–799.

[2.35] D. Zhang, W.J. Whiten, The calculation of contact forces between particles using

spring and damping models. Powder Technology 88 (1996) 59–64.

[2.36] F.Y. Chen, Mechanics and Design of Cam Mechanisms, Pergamon Press, 1982.

[2.37] W. Goldsmith, Impact: The Theory and Physical Behavior of Colliding Solids,

Dover Publications, New York, 2001.

[2.38] W.J. Stronge, Impact Mechanics, Cambridge University Press, New York, 2000.

[2.39] F.-J. Elmer, Nonlinear dynamics of dry friction. Journal of Physics A:

Mathematical and General 30 (1997) 6057–6063.

[2.40] M. Vaishya, R. Singh, Sliding friction-induced non-linearity and parametric

effects in gear dynamics. Journal of Sound and Vibration 248 (2001) 671–694.

[2.41] R.I. Leine, D.H. van Campen, A. de Kraker, L. van den Steen, Stick-Slip

vibrations induced by alternate friction models. Nonlinear Dynamics 16 (1998)

41–54.

57

CHAPTER 3

ESTIMATION OF IMPACT DAMPING PARAMETERS FROM TIME-DOMAIN

MEASUREMENTS ON A MECHANICAL SYSTEM

3.1 Introduction

Periodic impacts commonly occur in mechanical systems having clearance or backlash; these include geared systems [3.1 - 5], cam-follower mechanisms [3.6 - 9], and four-bar linkages [3.10 - 12]. There is a significant body of literature on such impacting systems employing linear system method [3.13, 3.14], non-linear analysis [3.15], stability investigations [3.16 - 18] and energy dissipation analyses [3.19, 3.20]. However, only a few researchers [3.15, 3.21, 3.22] have utilized contact mechanics formulation (with an impact damping model) for such systems. The commonly used contact force formulation

[3.21, 3.23 - 25] is of the form,

ɺ Fλ = F k (1 + κδ ) . (3.1)

Here, Fλ is the contact force (with λ representing a contact parameter), Fk is the contact stiffness force, δ is the indentation displacement and κ is an arbitrary constant.

1 / 4 ɺ Additionally, alternate formulation such as Fλ = F k + η δ δ (where η is a constant) also has been used [3.22] to represent the contact force during impacts. Overall, there is a clear need to experimentally determine the most appropriate impact damping model, but

58 the direct measurement of the contact force during impact is a challenge. Therefore, the main goal of this chapter is to propose a new method that would combine time domain measurements and analytical predictions to estimate damping parameters.

3.2 Problem formulation

A generalized model for the contact force is proposed below where n is the damping index and β is the impact damping coefficient,

n ɺ Fλ = F k + βδ δ . (3.2)

Hertzian contact theory [3.26] could be used to find Fk since it gives a reasonable estimate of the elastic force as suggested by Veluswami et al. [3.23]. The values of β and n could then be experimentally determined, though only a limited number of researchers

[3.27] have conducted experimental studies using an impact damping model.

Nevertheless, the following key questions remain unanswered: a) Is eq. (3.2) in a general form, with experimentally estimated values of β and n, consistent with eq. (3.1), which has been used for impacting systems without much experimental corroboration? b) Could the hysteresis loop and the contact force be utilized to estimate β and n? c) What is the relative significance of the numerical values of β and n? d) How could one justify the numerical values of β and n, given the literature for a typical system? e) Is the equivalent viscous damping model appropriate for this problem? The scope of this study is accordingly formulated to address the above mentioned questions, though this work is restricted to impacts with point contacts between two objects made of steel under lubricated conditions. The key objectives of this chapter are defined as follows: (i)

Design a controlled cam-follower experiment with lubricated point contact to measure

59 forces and motion (in time domain) under periodic impacts; (ii) Propose an analogous analytical model for the experiment with contact mechanics formulation; (iii) Develop and evaluate a signal processing procedure to experimentally determine β and n without directly measuring the contact force; and (iv) Determine an equivalent coefficient of restitution model from the same experiment and then justify the estimated value(s) of β using the relationship suggested by Hunt and Crossley [3.24].

The cam-follower system proposed for this study is shown in Fig. 3.1, which is a representative experiment for impacting systems. The system consists of a cylindrical steel cam rotating about an axis not passing through its centroid but parallel to the axis if the cylinder. The follower consists of a long bar of square cross-section attached to a thin cylindrical steel dowel pin as shown in Fig. 3.1, and pivoted at its end by a pair of roller

bearings. The follower is supported along the vertical direction ( eˆ y ) by a coil spring which is always compressed, thereby forcing it towards the rotating cam. The main assumptions regarding the experiment are: a) The axes of rotation of the cam and the follower remain unchanged at any load; b) The bearings at the follower pivot are frictionless; and c) The angular velocity of the cam ( c, subscript c denoting the cam) is constant and unaffected by the impact loads. The following conditions are to be considered in designing the cam-follower experiment to achieve a good estimate of β and n. First, the sliding friction force at the contact during impacts should not interfere with the impact dynamics of the system. Second, the effect of flexural vibrations of the follower caused by impacts should not affect the measured force and acceleration. Third, the responses have to be accurately measured during the impacts which take place within

60 a very short time interval. Finally, the follower must impact with the cam periodically at the rate of once per cam revolution; the need to have this particular condition is explained later in section 3.5.

Axis of Cam rotation

Tri-axial load cell Frictionless Point of impact bearings (lubricated) Dowel pin

Follower

Damping Accelerometer material Spring

Rigid fixture

Fig. 3.1. Cam-follower experiment designed to determine impact damping parameters.

3.3 Design of the laboratory experiment and instrumentation

The cam-follower experiment is carefully designed based on the requirements stated in section 3.2 for accurate estimation. First, a point contact is achieved between the cam (of radius rc, subscript c denoting the cam) and the dowel pin (of radius rd) of the follower, since two cylindrical surfaces (with axes perpendicular to each other) are in contact. Second, the coefficient of friction ( ) is minimized by having smooth contacting surfaces (with average roughness of 0.2 microns). Moreover, the contact is constantly lubricated with gear oil (AGMA 4EP with dynamic viscosity of 0.034 kg m -1 s-1 [3.28], 61

[3.29]), hence is taken to be as low as 0.2. Third, the flexural vibrations in the follower

(of width wb, subscript b denoting the bar) are minimized using a damping material (Sika damp 620 [3.30]). Fourth, a tri-axial force transducer (PCB 260A01 [3.31]) located at the

follower hinge measures the reaction forces along eˆ x (horizontal) and eˆ y directions, while the shock accelerometer (PCB 350B02 [3.32]) attached to the follower near the contact point (at a distance of la from follower pivot) measures its tangential acceleration.

Both these transducers (capable of accurately recording the impacts) are simultaneously sampled at a very high frequency of 204800 Hz using the LMS Scadas III [3.33] data acquisition system. Finally, the contact is maintained when c is low; however, the follower (of length lb) starts losing contact and making impacts as c is increased considerably. As observed by Alzate et al. [3.8], the system quickly goes into a chaotic state once c is increased beyond a certain limit. Hence a variable speed is used to carefully control c, so that the system does a periodic impact of exactly once per revolution of the cam. A digital tachometer (Neiko tools, USA) is used to accurately measure c. The variation in c during the impacts is found to less than 2.5% of the mean value, which is the same even when there is no load on the cam (follower is not in contact with the cam).

3.4 Analytical model

3.4.1 Kinematics of the system

Fig. 3.2 shows a sketch of the analytical model with a contact mechanics formulation for the experiment. The cylindrical cam rotates about E which is at a distance e from the centroid ( Gd). The linear stiffness of the coil spring supporting the bar is ks and

62 it is grounded at a distance dy below the bearing pivot P and at a distance of dx from it

along eˆ x . The angle made by the follower with eˆ x is given by α(t) measured in the clockwise direction, while the angular displacement of the cam in the counter-clockwise direction is given by Θ(t). The instantaneous points of contact in the cam and the follower are given by Oc and Ob, respectively. A moving coordinate system (iˆ , ˆj ) attached to the follower is defined with its origin at Q where iˆ is orthogonal to the follower. The vector ˆ ˆ ˆ ˆ QO c in the (i , j ) coordinate system is given by ψii+ ψ j j as shown in Fig. 3.2. The 0- state is defined with Θ0 and α0 (with superscript 0 representing the 0-state), as discussed by Sundar et al. [3.9] for a similar system. The contact mechanics is represented by point contact stiffness ( kλ) and impact damping ( cλ) elements. The chief assumptions in the analytical formulation are as follows: (1) The friction does not affect the impact mechanism and it follows a Coulomb friction model; (2) The coil spring supporting the follower is linear; and (3) The bending moment in the follower is negligible, while the amplitude of flexural vibrations are negligibly small compared to the amplitude of angular oscillations due to impact;

At the 0-state, α0 and Θ0 is defined as the following,

 4 22 2 2   PE+ PE PE − PE()0.5 wb +++ 2 r d r c e  x xy x     +PE()0.5 wb +++ 2 r d r c e α 0= cos− 1  y  , (3.3)  2 2  PE+ PE  x y       

Θ0 =π − α 0 . (3.4) 2 63

Fig. 3.2. Analytical contact mechanics model of the experiment shown in Fig. 3.1.

64

Here PE and PE are the magnitudes of PE along eˆ x and eˆ y , respectively. Using x y the procedure discussed by Sundar et al. [3.9], the moving coordinates ( ψi(t) and ψj(t)) and their time derivates are calculated using the following equations:

0 0 0 ψχααi (t )= sin( ( t ) −+++) () rrwc 2 d 0.5 b ( cos( αα ( t ) −−) 1 ) (3.5) +et[sin()α ( ) +Θ−0 sin() α ( tt ) +Θ ( ) ],

0 0  0 ψχααj (t )=− 1 cos( ( t ) −+++)  () rrwtc 2 d 0.5 b sin( αα ( ) − ) (3.6) 0  −etcos()α ( ) +Θ− cos() α ( tt ) +Θ ( )  ,

ɺ0 0 ɺ0 ɺ ψχαααi (t )= cos( ( t ) −) ( trrw ) −++( c 2 d 0.5 b ) sin( ααα ( t ) − ) ( t ) (3.7)  0 ɺ ɺ ɺ  +ecos()α () t +Θ α ()cos t −() α () tttt +Θ ()() α () +Θ (),

ɺ0 0 ɺ0 ɺ ψχαααj ()t= sin( () t −) () trrw +++( c 2 d 0.5 b ) cos( ααα () t − ) () t (3.8)  0 ɺ ɺ ɺ  +etsin()α () +Θ− αα sin() () tttt +Θ ()() α () +Θ ().

0 Here, χ= χ()t + ψ j () t , where χ(t) is the moment of the Fλ(t) about P and is given by the ˆ instantaneous magnitude of PO b along j .

3.4.2 Non-contact regime

When the instantaneous value of ψi(t) > 0 the follower is not in contact with the cam, the equation of motion depends only on the dynamics of the follower and coil

P spring. It is given by the following, where I b is the moment of inertia of the follower

(along with the damping material) about P, mb is the mass of the follower (along with the damping material), lb is the distance from the center of gravity of the follower ( Gb) from

P:

65

P ɺɺ Ibα() t= mgl bg cos( α () t) − Ftd sx (). (3.9)

u Here, Fs(t) is the elastic force from the coil spring which is given as follows, where Ls is the undeflected length of the spring,

u  FtkLddsssyx( )= −+ tan(α ( t )) + 0.5 w b sec( α ( t ))  . (3.10)

The eq. (3.9) is solved numerically for a given value of c, as long as ψi(t) > 0. The system goes to a contact regime once ψi(t) goes less than 0 and the response of the system has to be calculated using the contact mechanics formulation, which is discussed next.

3.4.3 Contact regime

Using the contact mechanics formulation, the response of the system is calculated using Hertzian contact theory [3.26]. The contact stiffness for a point contact is calculated as,

4 e e 0.5 ktλ ()ψi()= Y() ρ ψ i (). t (3.11) 3

Here, Y is the Young’s modulus (with superscript e denoting equivalent) given by the following, where ν is the Poisson’s ratio,

−1 1−ν2 1 − ν 2  Y e =c + b  . (3.12) Yc Y b 

The equivalent radius of curvature at the contact (ρe) is given by,

e −1 − 1  −1 ρ =()rc + () r d  . (3.13)

The impact damping is defined as,

n cλ ()ψi() t= β ψ i (). t (3.14)

66

The total contact force is given by,

ɺ Ftλ()= − k λ(ψψii () t) () tc − λ ( ψψ ii () t) (). t (3.15)

Taking the moment balance of the forces acting on the follower about P, the equation of motion in the contact regime is calculated as,

P ɺɺ Ibα() tmgl= bg cos( α () t) −+− FtdFttFt sx ()λ ()() χ f ()0.5( w bd + 2 r ) . (3.16)

Here, Ff(t) is the friction force given by

Ftf()= µ Ftλ ()sgn( vt r ().) (3.17)

Here, vr(t) is the relative sliding velocity at the contact point given by

ɺ  ɺ ɺ vtr()=ψ j () tre −+ c sin( α () t +Θ () t)  ( α () t +Θ (). t ) (3.18)

Note that from eq. (3.16), Ff(t) does not significantly affect the dynamics of the system as its moment arm (0.5 wb + 2 rd) is very small compared to χ(t) and moreover is very low. The system response is computed in the non-contact regime by solving eq.

(3.9), while in the contact regime by solving eq. (3.16). The system constantly switches between these two regimes when c is greater than a certain value, depending on the system. At the very beginning of the simulation α(0)= α * and αɺ(0)= 0 are used as initial conditions, where α* is the value of α(t) at the static equilibrium point (with superscript *). This is evaluated using eqs. (3.5), (3.6) and (3.16) in the Jacobian matrix method as discussed by Sundar et al. [3.9]. After this the initial conditions for each regime is taken from the final state of the previous regime.

67

3.5 Estimation of the impact damping parameters ( β and n)

3.5.1 Time-domain based technique to estimate the system response

The reaction forces along eˆ x and eˆ y (Nx and Ny, respectively, as shown in Fig.

3.2) and αɺɺ (t ) (calculated by dividing the measured tangential acceleration of the follower by la) are the experimental data measured in the time-domain. The geometrical parameters and inertia are obtained directly from the experiment. The entire procedure is performed in the time-domain as impacts excite a wide range of frequencies (including the natural frequencies of the cam-follower system and flexural vibrations of the follower), and hence the frequency domain data cannot be used directly. Some of the important numerical issues for which care needs to be taken in this estimation process are as follows. First, the measured αɺɺ (t ) is numerically integrated obtain αɺ(t ) , which is numerically integrated again to obtain the system response ( α(t ) ). The integration process does not give the DC component of the signal, moreover numerical integration process has inherent errors [3.34] (truncation and round-off) associated with it. Also since integration has to be performed twice, these errors would have a cumulative effect on

α(t). Thus, higher accuracies can be achieved by having a shorter time resolution ( τ) and a smaller length of integration vector. Second, the magnitude of indentation ( ψi(t)) during contact is very small compared to the maximum value of α(t) in the non-contact regime and the time of impact is very short. Hence it is very difficult to accurately estimate ψi(t) during impacts from the experimental data.

The following technique is adopted to minimize the errors due to the numerical integration and to estimate the DC component of α(t). The impact damping estimation

68 procedure discussed later in section 3.5.2 can be employed even without adopting this technique, if α(t) can be accurately calculated from the experiment. The measured time- domain data (forces and acceleration) have many impacts, but each impact is considered an independent event for the purpose of analysis. Furthermore, each impact event is divided into two sub-events, namely contact and non-contact. The contact sub-event begins when the cam and the follower are just in contact ( ti = 0, superscript i represents

i experimental data for an impact event) with ψ i = 0 and ends when the follower looses

i i contact with the cam ( t= t a ). Then the non-contact sub-event starts and lasts until the

i i follower next comes into contact with the cam ( t= t e ). The experimental angular acceleration for each impact event is measured as αɺɺ i(t i ) and its time-average

αɺɺ it i should be 0 since the impacts are periodic (as stated in section 3.3). ( ( ) ti )

However, generally this is not the case as the measured data has some errors, and hence

αɺɺ it i is subtracted from αɺɺ i(t i ) . Then αɺɺ i(t i ) is integrated numerically using the ( ) ti

Runge-Kutta method to get αɺ i(t i ) . Though αɺ it i should be 0, it is not due to the ( ) ti errors of the numerical integration technique. This is eliminated by subtracting αɺ it i ( ) ti from αɺ i(t i ) . The resultant signal is again integrated numerically to get α i(t i ) . The fact that the follower impacts exactly once per revolution of the cam, is used again to calculate the DC component of α i(t i ) . For a given system, the time period of each

i impact ( te ) depends on its state at the end of the contact sub-event. Writing it

69 mathematically as follows where Ξ is a function that gives the time required by the follower to return to its initial position based on initial conditions,

i iiɺ ii te= Ξ (α( t a ), α ( t a ).) (3.19)

ɺ i i Equation (3.9) is solved iteratively with known α (ta ) and different values of α as initial conditions to evaluate Ξ. The time required for the follower to come back to same initial position is calculated for each α. The value of α for which the calculated time matches

i i i i with te , is chosen α (ta ) since ta ≈ 0 (due to extremely short time of contact sub-event).

The DC component of α i(t i ) (calculated by numerical integration) is adjusted so that

ii ii . The angle of the cam at ti = 0 i is calculated from eq. (3.5) by α()ti i = α () t a (Θ (0) ) t= t a

i replacing α(t) with α (0) , forcing ψi(t) = 0 and solving for Θ(t). Thus the time-history of

Θ can be calculated as,

ii i i Θ()t =Θ (0) +Ω c t . (3.20)

3.5.2 Signal processing procedure to estimate β and n

It is not possible to have a direct method to estimate the impact damping parameters, because of their inter-relationship with the measured acceleration and reaction forces. Hence they are identified using an indirect method of comparing the experimental data for each impact with the results from analytical model with trial values of β and n which is defined as simulation S1 (where S1 represents simulation with trial

i i i i values). To aid in the comparison process, the Fλ ( t ) during contact ( 0

iiɺɺ iiii i iiii  ii FtmltNtλ ()=bgα () + x ()sin()( α t) ++− NtFtmg ysb () ()  cos( α (). t ) (3.21)

70

i i i Here Fs ( t ) is calculated from eq. (3.10) by replacing α(t) with α (t ) . Also the maximum

i amplitude of response (α m ) is calculated as the following where ‘max’ is a function that returns the maximum value of a set of inputs,

i i i αm = max( α (t ).) (3.22)

The results of simulation S1 are obtained for each impact event by solving the equations of motion of the contact sub-event (eq.(3.16)) with α i (0) and αɺ i (0) as initial conditions from the experimental data of the impact event, followed by the equation of

S1 i motion in the non-contact sub-event (eq. (3.9)) until t= t e (superscript S1 represents results with simulation S1 with trial values of β and n for the corresponding impact event).

Simulation S1 is conducted using different trial values of β and n for each impact event to

S1 S 1 compare with the experimental results. Similar to the experimental results Fλ ( t ) ,

S1 S 1 S1 ψ i (t ) and α m are estimated for each simulation. The following residues are defined to compare the experimental results with that of simulation S1:

S1 i αm− α m Λ1 = i , α m

i te 2 i S1 ∫()Fλ() t− F λ (). t dt Λ = 0 , 2 i te i 2 ∫()Fλ ( t ) . dt 0

i i i S1 S 1 S 1 ∫Fdλ(ψψii ).( ) − ∫ F λ ( ψψ ii ). d ( ) Λ = . (3.23 a - c) 3 i i i ∫ Fλ (ψi ). d () ψ i

71

The maximum response of the system after the impact is the criterion considered for the first residue ( Λ1). The second residue ( Λ2) is based on the root mean square difference in the contact forces from the experiment and the simulated data in time-domain, while the third residue ( Λ3) uses the area under the hysteresis loop. The appropriate numerical values for β and n are identified based on the impact damping model which gives the minimum value of the average of the residues of all impact events. The estimation procedure can be based on either of the residues. Hence the accuracy of the estimation process using these residues will be discussed in section 3.6.

3.6 Error and sensitivity analyses on the estimation procedure

3.6.1 Error analysis

Before employing the procedure discussed in section 3.5, the robustness (using error analysis) and accuracy (using sensitivity) of the procedure have to be critically studied. The robustness can be better understood by the error in the residues for an ideal case, since an indirect method of comparison is being employed for the estimation process. Simulation S2 (where S2 represents simulation with known values) is defined as solving the analytical model for each impact event with given initial conditions using

S S2 known values of damping parameters ( β 2 and n , superscript S2 represents simulation

S2). The force and acceleration time-histories from simulation S2 are used instead of the experimental data in the procedure discussed in section 3.5 and the residues are

S S S1 S 2 calculated using the simulation S1 with the β1= β 2 and n= n . Ideally all three residues should be 0, but that is not the case due to the approximations in the estimation procedure.

72

The following parameters for the cam-follower experiment are selected to have an accurate estimation of impact parameters: mb = 0.257 kg, rc = 17.5 mm, Ib = 3300 kg-

2 mm , lg = 173 mm, la = 63 mm, lb = 86 mm, wb = 12.7 mm, rd = 3.2 mm, ks = 3319 N/m,

u Ls = 53 mm, dx = 35 mm and dy = 58 mm. The relative positions of the pivot points of the cam and the follower are given by PE=89mm eˆx + 27mm e ˆ y . The material properties for a steel cam and a steel follower are Yc = Yb = 210 GPa and νc = νb = 0.3. Inverse kinematic analysis discussed by Sundar et al. [3.9] is employed for the current system with the given parameters to obtain the regions on impact on an e/rc vs. c map as shown in Fig. 3.3. Table 3.1 gives the average residue per impact, calculated using simulation S1

S S −2.5 S1 S 2 and S2 with β1= β 2 = 24.7 GNsm and n= n = 1.5 for different values for e/rc and

c in the impact regime (shown in Fig. 3.3). As observed, the residues for all cases are very low, which shows that the estimation procedure is very robust for these examples.

Also, it can be inferred that as e/rc increases, Λ1 reduces and reaches a minimum at e/rc =

0.2 and starts increasing again. With the increase in e/rc the signal to noise ratio increases; hence Λ1 reduces, but for very high values of e/rc the system operates close to a chaotic state. The noise here might be from experimental measurements or from the numerical error (in solving equations of motion). Similar trends are not observed in the cases of Λ2 and Λ3. As observed, the values of Λ1 are the lowest, followed by Λ3, with Λ2 being the highest. Fig. 3.4 compares sample hysteresis loops of simulations S1 and S2, for

i i * i i * a case with e/rc = 0.2, where Fλ= F λ/ F λ and ψi= ψ i/ ψ i . It is observed that for this case, a point contact the maximum value of Fλ during impact is about 3 orders of

73

* magnitude greater than Fλ while the maximum value of ψi during impact is only about 2

* two orders of magnitude greater than ψ i . Also even for an ideal case there is not a very good match in the hysteresis loop for low values of indentation. The relative accuracies of the residues in estimating the damping parameters should not be decided from the Λ values in Table 3.1, but should be decided from the sensitivity of these residues to variation in β and n. Hence it is analyzed next.

40

35

30 Impacting regime

25 [Hz] c

20

15

10 Contact regime

5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

e/rc

Fig. 3.3. Regimes of contact and impact for the system (with parameters given in section

3.6.1) via c vs. e/rc. Key: , Operational points (with periodic impacts) selected for the

purpose of error analyses.

74

e/rc c (Hz) Average Λ1 Average Λ2 Average Λ3

0.05 23 0.0056 0.0849 0.0024

0.10 18 0.0361 0.1032 0.0033

0.15 17 4.75 x 10 -7 0.0703 0.0041

0.20 16 0.78 x 10 -7 0.0728 0.0039

0.25 15 1.65 x 10 -7 0.0834 0.0043

Table 3.1 Comparison of average residues per impact ( Λ1, Λ2 and Λ3) using two

S1 S 2 −2.5 S1 S 2 simulations ( S1 and S2) with β= β = 24.7GNsm and n= n = 1.5 .

75

1000

800

600

400

200

0 0 10 20 30 40 50 60 70 i ψ i

Fig. 3.4. Comparison of hysteresis loops for single impacts during simulation S2 (

S2 −2.5 S2 S1 S 2 S1 S 2 β = 24.7GNsm and n =1.5 ) and simulation S1 ( β= β and n= n ) given

e/rc = 0.2 and c = 16 Hz. Key: , Simulation S1; , Simulation S2.

3.6.2 Sensitivity analysis

Table 3.2a gives the values of normalized residues ( Λ ) for simulation S2 (

S −2.5 S2 S1 S 2 β 2 = 24.7 GNsm and n =1.5 ) and simulation S1 with n= n with different values of β S1 in the close proximity of β S2 . The residues are normalized based on its value when βS1= β S 2 . The sensitivity of the residues to a change in β can be understood from

this table. It can be easily inferred that Λ 1 has a very high sensitivity even to a very small

change in the value of β, compared to that of Λ 2 and Λ 3 . Also ideally Λ should be

S1 S 2 lowest for β= β , but that is not the case with Λ 2 and Λ 3 , which may lead to an

76 incorrect estimation of β. A similar analysis is performed to study the sensitivity of residues to a change in n and the result is shown Table 3.2b. Compared to the other two

residues Λ 1 is more sensitive to changes in n. Also unlike Λ 1 , the lowest value does not

S1 S 2 occur at n= n for Λ 2 and Λ 3 . Thus Λ1 is more accurate than other residues and hence it will be used in the estimation of the impact damping model for the experimental system. Note that the estimation procedure using Λ1 uses only the measured acceleration and not the forces.

The reasons for the inaccuracies of Λ2 and Λ3 are as follows. The residue Λ2 is inaccurate because it is based on the calculated Fλ during the impact which is for a very short time period. Since Λ2 uses the integration of Fλ over time even a small error in the estimation of Fλ is magnified in the residue calculation. The residue Λ3 is based on hysteresis loop where Fλ is plotted against ψi which is very small in magnitude (< 50 m) during contact compared to the magnitude of ψi (~ 20 mm) during non-contact.

Moreover, ψi(t) is calculated using α(t) which has some error caused by numerical integration. Hence, even a small error in the estimation of ψi results in a very high error in the hysteresis loop. Thus, the sampling frequency should be much higher (than the one which is used in this experiment) to use Λ2 and Λ3 in the estimation procedure.

77

βS1= β S 2 Normalized S S S S S S S S β1= 0.98 β 2 β1= 0.99 β 2 (Ideal β1=1.01 β 2 β1=1.02 β 2 Residue case)

Λ 1 1.17 x 10 5 0.582 x 10 5 1 0.574 x 10 5 1.14 x 10 5

Λ 2 0.94 0.95 1 1.05 1.06

Λ3 0.95 0.96 1 1.05 1.06

S S a) For different values of β 1 in the proximity of β 2 with constant value of nS1= n S 2 .

Normalized nS1= n S 2 nS1= 0.98 n S 2 nS1= 0.99 n S 2 nS1=1.01 n S 2 nS1=1.02 n S 2 Residue (Ideal case)

Λ 1 17.24 x 10 5 8.94 x 10 5 1 9.42 x 10 5 19.2 x 10 5

Λ 2 2.49 1.67 1 0.76 1.16

Λ3 2.07 1.52 1 0.49 0.0086

S S b) For different values of nS1 in the proximity of mS2 with constant value of β1= β 2 .

Table 3.2 Comparison of normalized average residues per impact ( Λ1, Λ 2 and Λ 3 ) using

S2 −2.5 S2 simulation S2 ( β = 24.7GNsm and n =1.5 ) with e/rc = 0.2 and c = 16 Hz.

78

3.7 Estimation of the impact damping from the measurements

The procedure discussed in section 3.5.2 had been employed to estimate the impact damping parameters for the experimental system shown in Fig. 3.1. The experiment was conducted for a given value of e/rc and the c is slowly increased until a one impact per revolution of the cam is achieved. The distinct impacts are visible in the

* time histories of the normalized measured reaction forces ( Ntx()= Nt x ()/ F λ ,

* Nty()= Nt y ()/ F λ ) and measured acceleration shown in Fig. 3.5 from which T is obtained and its relationship with 1/ c is verified. Fig. 3.6 shows normalized reaction forces and acceleration data measured during the contact sub-event of a sample impact from the experiment with e/rc = 0.13 and c = 14.05 Hz (with measured T = 0.0712 s), with all other parameters being the same as given in section 3.6.1, where the normalized

i i i time is given by t= t/ t a . The values of the impact damping parameters identified

-2.55 using minimization of Λ1 are β = 92.6 GNsm and n = 1.55 (Note that the unit of β

n ɺ depends on the numerical value of n, since β ψi ψ i should have the units of force).

The estimated value of n from the experimental data agrees closely with n = 1.5 which

ɺ would fit the contact force formulation of Fλ = F k (1 + κδ ) , which is used by many researchers [3.21, 3.23 - 25]. Hence the same procedure is followed by forcing n = 1.5 and β is evaluated as 49.3 GNsm -2.5 . Fig. 3.7 shows a comparison of the contact forces of sample impact from the experimental data and the simulation S1 (with

S β S1 = 49.3 GNsm -2.5 and n 1 =1.5 ). As it can be inferred, though the damping parameters have been selected based on the αm there is a good match in the shape and

79 peak value of the contact force. Also it is very important to note that for a very small change in the value of n (about 3%) there is a very significant change in the estimated value of β of about 46%. Hence n is more critical than β in the impact damping formulation. Instead, if n is taken as 1.45, β is estimated as 29.3 GNsm -2.45 . The experiment was repeated with different values of e/rc and c and the same estimation procedure had been followed to obtain β and n values.

80

30 a) 20

10

0

-10

-20

-30 0 0.5 1 1.5 2 2.5 3 b) 40 30 20 10 0 -10 -20 0 0.5 1 1.5 2 2.5 3 50 c) 40 ]

2 30 20 10 rad/s

k 0 [ -10 -20 -30 -40 0 0.5 1 1.5 2 2.5 3 t [s]

Fig. 3.5. Time histories of the measured forces and acceleration with e/rc = 0.13 and c

= 14 Hz (with other parameters given in section 3.6.1). a) Normalized reaction force

along eˆx ; b) Normalized reaction force along eˆy ; c) Angular acceleration of the follower. 81

a) 40 30

20

10

0

-10

-20 0 0.2 0.4 0.6 0.8 1 b) 50

40 ] 2 30 rad/s k

[ 20

10

0

-10 0 0.2 0.4 0.6 0.8 1 ti

Fig. 3.6. Sample measured forces and acceleration during the contact sub-event from a

single impact from measurements shown in Fig. 3.5. a) Reaction forces; b) Angular

acceleration. Key: , Normalized reaction force along eˆx ; , Normalized

reaction force along eˆy . 82

250

200

150

100

50

0 0 0.2 0.4 0.6 0.8 1 i t

Fig. 3.7. Comparison of the contact forces (in the contact sub-event) from measured data

of Fig. 3.6 and simulation S1 (using the impact damping model selected based on

minimization of Λ1). Key: , Measured; , Simulation S1 (with

S -2.5 β 1 = 49.3 GNsm and nS1 =1.5 )

83

Since the estimated values of β and n were consistent, the repeatability of the experiment and the accuracy of the estimation procedure are validated. Also, Sundar et al. [3.9] estimated the values of κ for a similar system under dry conditions ranging from 3.25 s/m to 4.25 s/m which are comparable to κ = 3.8 s/m to 6.5 s/m (for n = 1.45 to 1.5, respectively) for the current lubricated system.

To check if a viscous damping model can be used to represent an impact event, n is forced to 0 and the same procedure is followed to obtain β = 1.47 kNs/m. A sample

S1 hysteresis loop comparing the experimental data and simulation S1 (with n = 0 and

β S1 = 1.47 kNs/m ) for the same is shown in Fig. 3.8. As it can be easily inferred there is a very clear variation in the shape of the hysteresis loops. Furthermore, Fλ goes to a negative value (tensile force) for part of the loop which is impractical, hence a viscous damping model is not a good approximation to model impacts.

3.8 Equivalent coefficient of restitution

3.8.1 Governing equation

The equivalent ξ model is determined from the same measurements and it is used to justify the impact damping parameters. Though researchers [3.9, 35] have assumed ξ = constant, a more general model of the form is considered for the current analysis as given by the following,

ξ=1 − γ va (3.24)

84

300

250

200

150

100

50

0

-50 0 5 10 15 20 25 30 35 i ψ i

Fig. 3.8. Comparison of the hysteresis loops from measured data of Fig. 3.6 and

simulation S1 (using the viscous damping model selected based on minimization of Λ1).

S1 Key: , Measured; , Simulation S1 with viscous damping ( n = 0 and

β S1 =1.47 kNs/m ).

Here, ξ decreases with the velocity of approach ( va) at a constant rate of γ as suggested by

Hunt and Crossley [3.24]. Instead of using eq. (3.16) in the contact regime (for the contact mechanics formulation), the state of the system after the impact (with superscript a) is calculated for ξ formulation using the state of the system before impact (with superscript b). From the definition of ξ, 85

ψɺ a ξ = − i (3.25) ɺ b ψ i

ɺ b Since v a= ψ i , as per eq. (3.24),

ɺ b ξ=1 − γ ψ i (3.26)

From eqs. (3.25) and (3.26) the velocity of separation is calculated as

ɺa ɺ b ɺ b ψi= − ψ i(1 − γψ i ) . The state of the system before impact is obtained from the response

a of the system in the non-contact regime when ψi(t) = 0. Rearranging the eq. (3.7) αɺ is calculated as

ɺ a a a ɺ ψi +ecos ( α +ΘΘ) αɺ a = (3.27) 0a 0 a 0 χααcos()−−+()rc 0.5 w b + 2 r d sin () αα − a0 a a  +e cos()()α +Θ− cos α +Θ 

Mathematically, in ξ formulation the system is in contact just for a single instant and hence αa and Θa are approximated by their corresponding values before impact.

3.8.2 Estimation of the equivalent ξ model

Unlike the estimation of impact damping, a direct method can be employed to estimate ξ. Since the state of the system just before impact (with superscript b) is needed for this purpose, it is calculated using a numerical backward difference technique as given by the following equations,

αbi= ατ( −= ) α i (0) − τα ɺ i (0),

αɺbi= ατ ɺ( −= ) α ɺ i (0) − τα ɺɺ i (0),

Θ=Θ−bi(τ ) =Θ i (0) −Θ τ ɺ i (0). (3.28 a - c)

86

ɺ b ɺ Velocity of approach ( ψ i ) is calculated from eq. (3.7) by replacing α(t ) , α(t ) and Θ(t)

b ɺ b b ɺ a with α , α and Θ , respectively, while the velocity of separation (ψ i ) is the value of

ɺ i ɺ i i ψ i (t ) when α (t ) reaches maximum. Using eq. (3.25), ξ is estimated for each impact.

i ɺ a From ξ and (ψ i ) , γ is identified using least square curve-fitting technique.

Analysis similar to the one discussed in section 3.6.1 has been performed to evaluate the error in the estimation of ξ. Force and acceleration time histories from

S simulation S 3, which is an analytical model with known γ 3 = 0.8s/m (other parameters are same as given in section 3.6.1) is taken as reference. The ξi is estimated for each impact for cases with different e/rc. The estimated γ and the % error associated with its estimation is given in Table 3.3. It is inferred from the magnitude of errors that this

i procedure yields very accurate γ for all the cases of e/rc. Fig. 3.9 shows the variation of ξ

i with v a on a normalized basis (on a scale of 0 to 1), for a sample case with e/rc = 0.1 and

i i c = 18 Hz. The normalized ξ and v a are calculated as follows, where ‘min’ is a function that returns the minimum value of a set of inputs,

ξi − min( ξ ) ξ i = , (3.29) max()()ξ− min ξ

i i va− min ( v a ) va = . (3.30) max()()va− min v a

Fig. 3.9 also shows the ξ model estimated using the least square curve-fitting technique with γ = 0.799 s/m.

87

For the same measured data (shown in Fig. 3.5), ξi is estimated for each impact using the procedure (discussed in section 3.8.2) and the results are shown on a normalized basis (using eqs. (3.29) and (3.30)) in Fig. 3.10. The least square curve- fitting technique was applied to estimate γ as 0.758 s/m using data the measured data and line of fit is also shown in Fig. 3.10. The average of the estimated ξ for all the impacts is 0.75 with velocity of approach of 0.34 m/s which is similar to the value of ξ =

0.65 reported by Seifried et al. [3.35] for va of 0.3 m/s, under similar impacting conditions.

⌢ Estimated γ γ− γ r e/rc c % error = r ×100 (Hz) (s/m) γ

0.05 23 0.8 0

0.10 18 0.799 0.13

0.15 17 0.8 0

0.20 16 0.76 5.0

0.25 15 0.8 0

Table 3.3 Error in the estimation of ξ model using time histories from simulation S 3 (γ S3 = 0.8s/m ) .

88

1

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 i va

i ɺ b Fig. 3.9. Variation in estimated ξ (during different impacts) given ψi with e/rc = 0.10

S3 and c = 18 Hz. Key: , Simulation S 3 (γ = 0.8s/m ) ; , Estimated ξ model

with γ = 0.799 s/m (using least square curve-fitting technique).

89

1

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 i va

i ɺ b Fig. 3.10. Variation in estimated ξ (during different impacts) with ψi for the experimental data of Fig. 3.5. Key: , Experimental data for different impact; ,

Estimated ξ model with γ = 0.758 s/m (using least square curve-fitting technique).

3.8.3 Justification of the estimated impact damping parameters

The relationship between coefficient of restitution and impact damping formulations has been derived by Hunt and Crossley [3.24] for vibroimpacts. Using this

90 formula the equivalent β for the estimated γ (0.758 s/m) is obtained as 9.1 GNsm -2.5 , which is comparable to the values estimated experimentally in section 3.7 (49.3 GNsm -2.5 for n = 1.5 and 29.3 GNsm -2.45 for n = 1.45). Some reasons for the difference between the

β values are: a) Hunt and Crossley [3.24] derived the relationship for vibroimpacts under dry condition assuming pure metal to metal contact, but the experiments were under lubricated condition; b) Even a small change in n value affects β significantly; and c) the

ξ concept has inherent drawback [3.9, 3.23] and hence the experimental estimation of γ might have some inaccuracies. Taking these points into consideration, the estimated values for impact damping parameters are justified on an order of magnitude basis.

3.9 Conclusion

The major contributions of this experimental study are as follows. First, a new cam-follower experiment has been designed for periodic impacts, and instrumentation and data acquisition parameters have been carefully chosen to accurately measure forces and motion during impacts which takes place within a very short time interval. Second, a novel time-domain based technique to estimate α(t) is developed, which minimizes the effect of errors associated with the numerical integration. Also a new signal processing procedure using only the measured acceleration has been developed to estimate impact damping parameters. Third, a better understanding of the impact damping model is obtained, and the following issues (stated in section 3.2) have been resolved: a) The contact mechanics model given by eq. (3.1) is an acceptable formulation for impacting systems; b) The residue using αm is more accurate than other residues (using hysteresis loop and contact forces) for the purpose of damping parameter estimation; c) The impact

91 damping model is more sensitive to the damping index ( n) than the damping coefficient

(β); d) The estimated value of n is successfully compared with those reported in the literature, while the value of β is justified on an order of magnitude basis; and e) The viscous damping model is not appropriate for impacting systems. Since the proposed signal processing procedure (using Λ1) does not require force measurements, it could be extended to other mechanical systems. The chief limitation of this study is the indirect estimation of impact damping model; thus future work may be directed towards development of a direct method, possibly for a line contact.

92

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[3.32] PCB piezotronics Inc., sensors for acceleration, shock, vibration and acoustic

measurements: Product catalog (Model 350B02). ( http://www.pcb.com ;

Accessed 15 Aug 2013), Depew, NY, 2004.

[3.33] LMS instruments. LMS SCADAS III data acquisition front-end,

(http://www.lmsintl.com , Accessed 15 Aug 2013), Breda, The Netherlands,

2008.

[3.34] P.J. Davis, P. Rabinowitz, Methods of Numerical Integration, Courier Dover

Publications 2007.

96

[3.35] R. Seifried, W. Schiehlen, P. Eberhard, Numerical and experimental evaluation

of the coefficient of restitution for repeated impacts. International Journal of

Impact Engineering 32 (2005) 508–524.

97

CHAPTER 4

ESTIMATION OF COEFFICIENT OF FRICTION FOR A MECHANICAL

SYSTEM WITH COMBINED ROLLING-SLIDING CONTACT USING

VIBRATION MEASUREMENTS

4.1 Introduction

Friction plays a significant role in the dynamics of mechanical systems under sliding contacts [4.1 - 4.7]. The friction force is often modeled using the Coulomb formulation, though the analyst must judiciously select the value of the coefficient of friction ( ). In many prior experimental studies, is found from a simple translational sliding contact system as summarized by Persson [4.8]. For instance, Espinosa et al. [4.9] used a modified Kolsky bar apparatus, while Hoskins et al. [4.10] used sliding block of rocks to estimate the normal and friction forces. Furthermore, the translational sliding experiments were employed by Worden et al. [4.11] to estimate the dependence of friction forces on displacement and velocity, and then by Schwingshackl et al. [4.12] to model the non-linear friction interface. Such experiments (specific to translational sliding contact) cannot be directly employed for a system with combined rolling-sliding contact to estimate , since the kinematics at the contact is different. Accordingly, several investigators have conducted sliding contact experiments using a pin-disk apparatus

98

[4.13, 4.14], two rotating circular plates [4.15], and a radially loaded disk-roller system

[4.16, 4.17]. Also, Kang and Kim [4.18] determined the Coulomb friction in sight stabilization equipment using and angular displacement characteristics, while

Povey and Paniagua [4.19] estimated the bearing friction for a turbomachinery application. Further, Radzimovsky et al. [4.20] conducted experiments on gears to determine the instantaneous over a mesh cycle. However, none of the previous combined rolling-sliding contact experiments rely on vibration measurements. Hence there is a need to develop a combined rolling-sliding contact experiment to determine for a mechanical system with vibration measurements under certain conditions.

Some researchers have experimentally studied cam-follower mechanisms [4.21,

4.22] from the stability and bifurcation perspective under impacting conditions. In contrast, a cam-follower mechanism with combined rolling-sliding contact (with no impacts) is used to experimentally determine in this study. Since cannot be directly measured from vibration experiments, an analogous contact mechanics model [4.23] is developed to aid the process. The goal is to vary the surface roughness, slide-to-roll ratio, lubrication film thickness, contact pressure and velocities at contact (sliding and entrainment). The proposed system could then be utilized to simulate the contact conditions seen in drum brakes and geared systems.

4.2 Problem formulation

Fig. 4.1 shows the mechanical system with an elliptic cam (with semi-major and minor axes as a and b, respectively). The cam is pivoted at E along its major axis with a radial runout, e, from its centroid ( Gc, with subscript c denoting cam). The angle made by

99 the end point of the major axis ( A) with the horizontal axis ( eˆx ) is Θ(t), which is an excitation to the system (where t represents the time). The equation of the elliptic cam is given by the following, where r is the radial distance from Gc to any point on the circumference of the cam, and is the polar angle of that point,

ab r ()∆ = . (4.1) 2 2 asin()()∆  + b cos ∆ 

The cam is in a point contact (at Oc) with the follower (at Ob, with subscript b denoting follower), which consists of a thin cylindrical dowel pin (of radius rd) attached to a bar

(of length lb) of square cross-section (of width wb). The center of gravity of the follower lies at Gb at a distance of lg from the pivot point P (using roller bearings) which is at dy distance about the ground. The follower is supported by a linear spring ( ks) along the

vertical direction ( eˆy ), which is at a distance of dx from P as shown in Fig. 4.1. The

angular motion of the follower is given by α(t) in the clockwise direction from the eˆx axis; it is also the only dynamic degree-of-freedom of the system. The contact mechanics at O between the cam and the follower is represented by non-linear contact stiffness ( kλ)

ˆ ˆ and viscous damping ( cλ) elements. A coordinate system (i , j ) attached to the follower is defined with its origin at Q where iˆ is orthogonal to the follower. The angle subtended by

GOc c from eˆx is given by φ(t), which is used in the following equation to calculate the

O(t) for the contact point Oc as,

∆O(t ) = mod(ϕ ( t ) −Θ ( t ),2 π ) . (4.2)

100

Here, “mod” is the modulus function defined as: mod(xy , )= xy − .floor( xy / ) , if y ≠ 0. ˆ ˆ ˆ ˆ The vector QO c is represented in the (i , j ) coordinate system by ψii+ ψ j j . When the instantaneous value of ψi(t) is negative, that would ensure that the cam and the follower are in contact.

Fig. 4.1 Example case: A mechanical system with an elliptic cam and follower

supported by a lumped spring ( ks).

101

The scope of the current study is restricted to an estimation of under a mixed lubrication and elastohydrodynamic lubrication (EHL) regimes. The key assumptions in the proposed system are as follows: (i) The bearings at the follower pivot are frictionless and rigid; (ii) the surfaces of the cam and follower have no other irregularities with the exception of random surface roughness; (iii) the sliding friction between cam and the follower can be described by the Coulomb friction model; (iv) the contact force can be represented by the Hertzian point contact model [4.23]; and (v) the bending moment of the follower is negligible. The specific objectives of this study are: (1) Develop a contact mechanics model for a mechanical system with a combined rolling-sliding contact to design a suitable experiment and to predict the dynamic response; (2) Design a controlled laboratory experiment for the cam-follower system to measure dynamic forces and acceleration; and (3) Propose a signal processing technique to estimate using Fourier amplitudes of measured forces and acceleration an empirical formula for will be suggested and potential sources of errors will be identified.

4.3 Contact mechanics model

The 0-state of the system (represented by superscript 0) is defined as the state 0 0 when QO c =0 and the major axis is parallel to the follower ( α = −Θ ). In the 0-state

0 0 0 Q , Ob and Oc are coincident. From the geometry of the system, α and the magnitude of ˆ PO b along j (χ) are calculated in the 0-state as,

102

 4 22 2 2   PE+ PEPE − PE()0.5 wbd +++ 2 rb PE() 0.5 w bd ++ 2 rb  0− 1 x x y x y α = cos  2 2  ,  PE+ PE   x y  (4.3) χ0=PEcos( α 0 ) − PE sin( α 0 ) − e . (4.4) x y

Here, PE and PE represent the magnitudes of PE along eˆ x and eˆ y , respectively. x y

The instantaneous values of the moving coordinates ψi(t) and ψj(t) are determined from

α(t), Θ(t) and the system geometry using the following vector equation,

PE= POb + O bc O + O c E . (4.5)

Employing the vector polygon procedure discussed by Sundar et al. [4.24], the equations for ψi(t) and ψj(t) are obtained as,

0 0 0 ψχααi ()t=+( e) sin()( t −+∆++) ( rtwr( o ()0.5) b 2cos() d ) ( αα t − ) (4.6) +∆rt()()()()o ()sinα () tte +− ϕ () sin α () tt +Θ− () 0.5 wrb + 2 d ,

0 0 0 0 ψχχααj ()t=−+( et) cos()( −+∆++) ( rtwr( o ()0.5) b 2sin() d ) ( αα t − ) (4.7) −∆rt()()()o ()cosϕ () tte ++ α () cos α () tt +Θ ().

ɺ ɺ Differentiating Eqs. (4.6) and (4.7) with respect to time, ψ i (t ) and ψ j (t ) are obtained as follows:

ɺ0 0 ɺ 0 ɺ ψχi ()t=+( e) cos()( ααα t −) () trtwr −∆++( ( o ()0.5) b 2sin() d ) ( ααα t − ) () t ɺ ɺ ɺ +∆rt()()()()()o ()cosαϕαϕ () ttttrt + () () ++∆ ()o ()sin αϕ () tt + () −ecos()α () tttt +Θ ()() α ɺ () +Θ ɺ (),

(4.8)

103

ɺ0 0 ɺ 0 ɺ ψχj ()t=+( e) sin()( ααα t −) () trtwr +∆++( ( o ()0.5) b 2cos() d ) ( ααα t − ) () t ɺ ɺ ɺ +∆rt()()()()()o ()sinϕαϕα () ttttrt + () () +−∆ ()o ()cos ϕα () tt + () −esin()α () t +Θ () tt() α ɺ () +Θ ɺ (). t

(4.9)

Here,

0.5abba2− 2 sin2( ∆ () t) ϕɺ () t −Θ ɺ () t  ɺ ( ) o   r()∆o () t = . (4.10) 2 2 1.5 asin∆ () t + b cos ∆ () t  ()()o()() o 

The angle φ(t) corresponding to the contact point Oc is determined at every instant for a given α(t) and Θ(t) by locating the point on the elliptic profile of the cam which is tangential to the follower. Hence the slope of the follower, sb(t) = tan( -α(t)), should be

c equal to the slope of cam at Oc ( sO ( t ) ) which is calculated as follows,

2   c −1 b sO () t= tan Θ+ () t tan  − 2   (4.11) atan()ϕ ( t )− Θ ( t )  

b c Equating s (t) and sO ( t ) and rearranging, φ(t) is calculated by the following:

2  −1 b ϕ(t )= Θ ( t ) − tan 2  (4.12) atan()α ( t )−Θ ( t ) 

The equation of motion of the follower when it is in contact with the cam is derived by balancing the moments (from Fig. 4.2) about P as,

P ɺɺ Ibα() tmgl= bg cos( α () t) −+− FtdFttFt sxn () ()() χ f ()0.5( w bd + 2 r ) . (4.13)

P Here, Ib is the moment of inertia of the follower about P, mb is the mass of the follower, g is the acceleration due to gravity, and χ(t) is the moment arm of the contact force about

104

u the pivot P. The elastic force from the spring, Fs(t), is given by the following, where Ls is the original length of the follower spring:

FtkLdd( )=u −+ tanα ( t ) + 0.5 w sec α ( t )  . (4.14) sssyx( ) b ( ) 

The normal force ( Fn(t)) arising from the point contact with the cam is given by,

Ft()= − kψ () t ψ () tct − ψ ɺ (). nλ( ii) λ i (4.15)

The non-linear contact stiffness is defined for a point contact based on the Hertzian contact theory [4.23] as,

4 e e 0.5 kλ ()ψi() t= Y() ρ ()(). tt ψ i (4.16) 3

Here, Y is the Young’s modulus (with superscript e denoting equivalent) in accordance with the Hertzian contact theory given by the following, where ν is the Poisson’s ratio,

−1 1−ν2 1 − ν 2  Y e =c + b  . (4.17) Y Y c b 

The equivalent radius of curvature at the contact (ρe(t)) and the radius of curvature of the elliptical cam at Oc (ρc(o(t))) are given by,

−1 −1 ρe ()t= ρ ∆ () t + () r −1  , (4.18) ()c() o d 

2 2 1.5 asinγ∆ ( tb )  + cos γ ∆ ( t )   ()()o ()() o   ρc ()∆0 ()t = . (4.19) ab

105

Fig. 4.2 Free-body diagram of the follower; refer to Fig. 4.1 for the two coordinate

systems.

The viscous contact damping is given by the following expression, where ζ is the modal damping ratio which will be experimentally found under lubricated conditions, as explained later in section 4.4,

P 2ζϑ Ib cλ = . (4.20) ()χ * 2

Here, ϑ is the linearized natural frequency of the system, and χ* is value of χ(t) at the static equilibrium point (discussed in later in this section). The friction force is given as, 106

Ftf()= µ Ft n ()sgn( vt r ().) (4.21)

Here the relative sliding velocity, vr(t) is given by,

ɺ   ɺ ɺ vtr()=−∆ψ j () trt ( ()sin) ( ϕα () tte ++ ()) sin( α () tt +Θ ()) ( α () tt +Θ ().) (4.22)

The static equilibrium point is used as the initial condition while numerically solving Eq. (4.13). In Eqs. (4.6), (4.7) and (4.13), α(t), ψi(t), and ψj(t) are replaced with their corresponding values at the static equilibrium point (with superscript *), and all time-derivative terms are set to zero and solved. Using the method of Jacobian matrix as discussed by Sundar et al. [4.24], ϑ is then calculated at the static equilibrium point.

4.4 Experiment for the determination of

Since the measured time domain signals are bound to have significant noise, a frequency domain based signal processing technique is preferred for the estimation of .

Accordingly, measured forces and acceleration must not be affected by discontinuities and system . Design criteria for the experimental system can be given by the following. First, the follower must always be in contact with the cam, as a loss in contact would generate impulses in force and acceleration signals. Second, vr(t) should not change direction during the operation, as that would induce a sudden change in the direction of the Ff(t), thereby making the measured forces discontinuous. Furthermore, the variation in vr(t) should be minimal. Third, the cam should rotate with a constant speed ( c) in order to accurately measure the spectral contents of forces and acceleration.

Fourth, at least the first five harmonics of c should lie in the stiffness controlled regime.

Fifth, the experiment should permit a mixed lubrication and EHL regimes. Finally, a variation in the slide-to-roll ratio should be possible in the experiment. 107

Fig. 4.3 shows the schematic of a cam-follower experiment having a hollow cylindrical cam of outer radius, a, driven by the output shaft of an electric motor. The radial runout between the center of the rotation (axis of the shaft) and the centroid of the cam can be easily varied. A point contact is obtained, as the cam and the dowel pin have cylindrical surfaces with their axes oriented orthogonal to each other. The contact is continuously lubricated using either a heavy gear oil (AGMA 4EP) [4.25, 4.26] or a light hydraulic oil (ISO 32) [4.25, 4.26]. The follower is hinged at one of its ends with two frictionless rolling element bearings and is supported by a coil spring. A tri-axial force transducer (PCB 260A01 [4.27]) located at the follower hinge measures the reaction

forces, Nx(t) and Ny(t), along eˆx and eˆ y , respectively. An accelerometer (PCB 356A15

[4.28, 4.29]) located at the end of the follower measures its tangential acceleration. These are dynamic transducers with a very high frequency bandwidth [4.27, 4.29]. Both force and acceleration signals are simultaneously sampled.

4.5 Identification of system parameters

4.5.1 Identification of geometrical parameters

The following parameters for the cam-follower system are carefully chosen to satisfy the design constraints stated in section 4.4: mb=0.21 kg, a = b = 17.5 mm, Ib =

2 u 2020 kg-mm , lg = 179 mm. lb = 89 mm, wb = 12.7 mm, rd = 3.2 mm, ks = 2954 N/m, Ls

= 57 mm, dx = 40 mm and dy = 61 mm. The relative positions of the pivot points of the cam and the follower are given by PE=86mm eˆx + 24mm e ˆ y . The averaged surface roughness ( R) and root-mean-square roughness ( Rrms ) of the cam and follower surfaces are measured using an optical profilometer. For the precision ground surfaces used in the 108 experiment Rc = 0.29 m and Rb = 0.25 m, while for sand-blasted surfaces Rc = 0.36 m and Rb = 0.89 m. The key parameters that dictate a loss of contact between the follower and the cam and the sign reversal in vr(t) are e and c. Inverse kinematics [4.30] is employed, as explained below, to predict a range of values for these two parameters over which the system neither has a loss of contact nor a sign reversal in vr(t). For a given value of e and c, the angle of the follower (assuming it is just in contact) with the cam

(αk(t)) is kinematically calculated for different values of Θ(t) in the range [0, 2 π]

(superscript k represents values calculated using the inverse kinematics). By setting ψi = 0 in Eq. (4.6), αk(t) is calculated using the following equation,

0k 0 k 0 (χαα+e)sin( ( t ) −+∆++) ( rtwr( o ( )) 0.5 b 2 d ) cos( αα ( t ) − ) (4.23) k kk k +∆rt()o ()sin()()α () tte +− ϕ () sin α () tt +Θ− ()() 0.5 wrb += 2 d 0.

k Here, r (o(t)) is obtained using Eqs. (4.1) and (4.2) as,

ab rk ()∆() t = . (4.24) o 2 2 kk  kk  asin()()ϕα ( tt )− ( )  + b cos ϕα ( tt ) − ( ) 

Equations (4.23) and (4.24) are solved along with Eq. (4.12) after replacing α(t) with

k k k k k k α (t), to get r (o(t)), α (t) and φ (t). Then, differentiating α (t) with respect to t, αɺ (t ) and αɺɺ k (t ) are obtained. The normal force is estimated (as stated below) from the moment balance about P and by neglecting the moment due to Ff(t) in comparison with the moment due to Fn(t) because of system geometry,

IPkkαɺɺ () tFtdmgl+ () − cos α k () t k b s xbb ( ) Fn () t = . (4.25) χαα0cosk ()t−−++ 0 () bwr 0.5 2 sin ααk () t − 0   () b d ()  k k k   +rt() ∆o ()cos()α () tt − ϕ ()  109

Housing for electric motor

Output shaft of Tri-axial Cam the electric motor load cell

Lubricated Frictionless interface bearings Dowel pin Accelerometer

Follower Spring

Rigid fixture

Fig. 4.3 Mechanical system experiment used to determine the coefficient of friction ( ) at

the cam-follower interface.

k k Here, Fs ( t ) is calculated from Eq. (4.14) corresponding to α (t). If the minimum value of

k Fn ( t ) calculated from Eq. (4.25) is negative, it would indicate that the follower would lose contact with the cam during the steady-state operation. Similarly, the relative

k velocity ( vr ( t ) ) is kinematically calculated to check for any sign reversal from Eqs. (4.9) and (4.22) by replacing α(t) and φ(t) with αk(t) and φk(t), respectively. The procedure mentioned above is repeated for different values of e and c to calculate the c - e/a map as shown in Fig. 4.4; the regimes with and without loss of contact and reversal in the sliding velocity direction are clearly marked. All experiments are conducted in the e/a 110 range from 0.05 to 0.15, and c is varied only between 10.1 Hz and 11.7 Hz; thus the system is well within the contact regime (as shown) with a constant sgn( vr(t)) = -1 and minimal variation in vr(t). With these parameters, the linearized natural frequency of the system is found to be 1040 Hz for a steel cam and a steel follower ( Yc = Yb = 200 GPa; νc

= νb = 0.3); thus the first five harmonics of c lie in the stiffness controlled regime. Also, the lubrication regime is identified based on the “lambda ratio” ( Λ), which is the ratio of minimum lubrication film thickness [4.31] to the composite surface roughness

0.5 R2+ R 2 . With AGMA 4EP oil [4.25, 4.26] (with dynamic viscosity, η = 0.034 (( rms, c rms , b ) ) kg m -1 s-1, pressure viscosity coefficient = 20x10 -9 m2/N at 60° C) the system lies in the

EHL regime as Λ is approximately between 2 and 5. The system lies in the mixed lubrication regime when ISO 32 oil [4.25, 4.26] (with η = 0.012 kg m -1 s-1, pressure viscosity coefficient = 18x10 -9 m2/N at 60° C) is used, as lower values of Λ (0.7 to 1.5) is utilized. Since the temperature at the contact is higher than the ambient (due to continuous sliding), it is assumed that the interfacial oil operates at 60° C. Furthermore, the slide-to-roll ratio which is given by the ratio of vr(t) (as given in Eq. (4.22)) and entrainment velocity ( ve(t) as defined below), varies between 0.75 and 1.25 for e/a =

0.116 and c = 11.55 Hz as shown in Fig. 4.5. Here,

ɺ   ɺ ɺ vte()=+∆ψ j () trt ( ()sin) ( ϕα () tte ++ ()) sin( α () tt +Θ ()) ( α () tt +Θ ().) (4.26)

The slide-to-roll ratio ( vtr() vt e () ) could be easily changed by altering the geometry, such as PE , a, b and e.

111

No sign Sign reversal

30 reversal of vr(t) of vr(t)

25 [Hz]

c 20 In-contact Loss of contact regime regime 15

10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 e/a

Fig. 4.4 Classification of response regimes of the mechanical system with a circular cam

in terms of c vs. e/a map with the parameters of section 4.5. Key: , Operational

range of the experiment.

112

1.3

1.2

1.1

1 Slide-to-roll ratio Slide-to-roll 0.9

0.8

0 0.2 0.4 0.6 0.8 1

t [s]

Fig. 4.5 Slide-to-roll ratio for the cam-follower system with e/a = 0.12 and c = 11.55 Hz

and other parameters of section 4.5.

4.5.2 Identification of the modal damping ratio

The modal damping ratio under lubrication depends on the oil viscosity and the materials in contact; hence it is determined experimentally using the half-power bandwidth method with both lubricants. The experimental setup consists of two

(m1 = 1.4 kg and m2 = 1.8 kg) connected by three identical point contacts which are lubricated as shown in Fig. 4.6. These point contacts are obtained by placing three dowel 113 pins ( rd = 3.2 mm) attached to m1 in one direction and two more dowel pins attached to m2 in the orthogonal direction, as shown. The system is placed on a compliant base

(foam), and two accelerometers are attached to each mass. An impulse excitation is imparted to the system in the vertical direction with an impact hammer. The response accelerance spectrum of each mass along the vertical direction is then found by averaging signals from two accelerometers. Impact tests are conducted with two lubricants. Fig. 4.7 shows the relative accelerance spectra (between m1 and m2), focusing on the system resonance (~ 1000 Hz). As observed, there is a reduction in the amplitude and the natural frequency with lubrication. For a single point contact, the damping ratio ( ζ) with unlubricated, ISO 32 oil and AGMA 4EP oil conditions are found to be 1.8%, 1.9% and

4.1%, respectively. Note that the damping for ISO 32 oil is very close to the dry case. It is assumed that these values of ζ are also valid for the running cam-follower experiment.

4.6 Signal processing technique to estimate

The is estimated from measured reaction forces ( Nx(t) and Ny(t)) along the eˆx and ˆ directions, respectively, and the tangential acceleration ( ɺɺ ) of the follower at e y lbα ( t ) its free end. By dividing the measured tangential acceleration by lb, αɺɺ (t ) is obtained, and then numerically integrating it twice w.r.t. time, the time-varying component of α(t) is computed, while the integration constant ( αd) is obtained from the time-averaged value of

k α (t). The instantaneous elastic force Fs(t) is calculated from α(t) using Eq. (4.14). From

Fig. 4.2, Nx(t) and Ny(t) are evaluated as follows,

ɺɺ NtFtxn()= ()sin(α () tFt) + f ()cos( α () tmlt) − bg αα ()sin( (), t ) (4.27)

114

NtFty()= n ()cos(α () tFt) − f ()sin( α () t ) (4.28) ɺɺ +mgmlbbg −α( t )cos() α ( t ) − Ft s ( ).

a) Impact hammer

Triaxial Dowel pins accelerometer

b)

Dowel pins

Fig. 4.6 Impulse experiment to determine the viscous damping ratio associated with the

lubricated contact regime. a) Experimental setup; b) Top view of the dowel pin

arrangement showing the three point contacts. Key: , contact point.

115

15 ] 2

s 10 - [m/N

5 accelerance Relative

1 850 900 950 1000 1050 1100

Frequency [Hz]

Fig. 4.7 Relative accelerance spectra in the vicinit y of the system resonance. Key:

, dry (unlubricated); , lubricated with AGMA 4EP oil; , lubricated with ISO

32 oil.

116

Rearrange Eqs. (4.27) and (4.28) to yield the friction and normal forces as,

  FtNtfx()= ()cos(α () t) +−− mgNtFt bys () ()sin  ( α (), t ) (4.29)

ɺɺ   FtmltNtnbg()=α () + x ()sin( α () t) ++− NtFtmg ysb () ()  cos( α (). t ) (4.30)

Since the dynamic force transducer used does not measure the DC component, a technique to estimate is proposed that utilizes complex-valued Fourier amplitudes while maintaining the phase relationship among the measured signals. First, the measured Nx(t),

Ny(t) and αɺɺ (t ) are converted to the frequency domain using the fast Fourier transform

(FFT) algorithm. Then, the harmonic reaction forces are reconstructed (with superscript r) using only their DC components (with superscript d) and the fundamental harmonic component of c (with superscript 1) as,

r d 1 r d 1 NtNNx( )= xx + cos() Ω c t , NtNNy( )= yy + cos() Ω c t . (4.31 a, b)

1 1 In the above equation, Nx and Ny (where, ~ represents a complex-valued signal) are

d d known from measurements while Nx and N y are unknown. Similarly, the following

harmonic signals have also been reconstructed as the following where, ςs (t )= sin( α ( t ) )

and ςc (t )= cos( α ( t ) ) :

r d 1 r d 1 ςs(t )= ς ss + ς cos() Ω c t , ςc(t )= ς cc + ς cos() Ω c t ,

r d 1 r d 1 FtFFn( )= nn + cos() Ω c t , FtFFf( )= ff + cos() Ω c t ,

r d 1 FtFFs( )= ss + cos() Ω c t . (4.32 a-e)

Since αɺɺ (t ) does not have a DC component, it is written as,

117

ɺɺr ɺɺ 1 α(t )= α cos() Ω c t . (4.33)

Substituting these reconstructed harmonic signals in Eqs. (4.29) & (4.30) and

r rearranging, the following DC components and first harmonic components of Ff ( t ) and

r Fn ( t ) are found as,

FNd= ddςς − N dd +−+ F dd ς mg ς d +0.5 NNF11 ςςς −− 11 11 , (4.34) f xc ys{ ss bs xc ysss }

FN1=−+ddςς 1 N 111 N ςς dd −+ N mgF ςςς 111 −− dd F , f xc ys{ xc ys bs ss ss } (4.35)

ɺɺ d dd d  0.5 mlbbα+ F sc ς − mg bc ς d dd dd   Fn= N xsς + N yc ς +   , (4.36) +0.5 N11ς + N 11 ς + F 11 ς xs yc sc 

FN11111=++ddςςςς N N dd +− N mgF ςςς 111 +++ dd F0.5 ml α ɺɺ 1 . n xs yc{ xs yc bc sc sc bb } (4.37)

From Eq. (4.21) and since sgn( vr(t)) = -1 is a constant, the following relationships can be derived:

d d Ff= − µ F n , (4.38)

1 1 Ff= µ F n . (4.39)

r r Since Ff ( t ) and Ff ( t ) are exactly out-of-phase,

1 1 ∠Ff =−∠ F n . (4.40)

Substituting Eqs. (4.34) to (4.37) into Eqs. (4.38) to (4.40), three non-linear equations

d d with three unknowns ( , Nx and Ny ) are obtained. These equations are numerically solved

118 to estimate . In order to computationally validate this technique, predicted forces and acceleration from the contact mechanics model with e/a = 0.3, c = 11.55 Hz and a known = 0.3 are used. The signal processing technique (with 9460 Hz sampling frequency and frequency resolution of 1.15 Hz) yields an estimate of as 0.302, which is

d d about 99.3% accurate. This method also accurately estimates Nx and Ny as -2.26 N and -

12.88 N, respectively, compared to the known values of -2.26 N and -12.7 N, respectively.

4.7 Experimental results and friction model

Spectral tests are conducted under lubricated conditions with different surface roughness levels at the contact. Care is taken during the experiments to record the steady state force and acceleration measurements only after the initial transients have sufficiently decayed. Using the measured data in the signal processing technique ⌢ discussed in section 4.6, the µ (estimated value of ) is indentified for various values of ⌢ mean surface roughness Rm=0.5*( R c + R b ) as shown in Fig. 4.8. This µ is compared with the values reported in the literature [4.13] for dry friction contact. A higher range is ⌢ observed in the case of a pure dry friction regime in comparison with µ for the ⌢ lubricated contact. Also, observe that µ with ISO 32 lubricant (with a low Λ value) is similar to the dry friction contact case [4.13].

He et al. [4.32] used the Benedict-Kelly friction model [4.17] to develop an empirical relationship between and Rm, but this was specific to a line contact in gears.

Hence that relationship is generalized for both point and line contacts as the following, where < > t is the time-average operator, 119

C p  µ = 1 logλ  . (4.41) C− R vt() vt () 2  2 m η rt e t 

Here, C1 and C2 are the arbitrary constants and pλ is the time-averaged Hertzian contact pressure given by,

3 F( t ) p = n . (4.42) λ 2e ()()t t π ρ ψ i t

With a non-linear curve-fitting technique, the constants of Eq. (4.41) are found from the experimental results for each lubricant: C1 = 0.0288 m and C2 = 2.03 m for AGMA

4EP oil, and C1 = 0.0509 m and C2 = 1.6512 m for ISO 32 oil.

The measured force and acceleration spectra are compared with the contact ⌢ mechanics model (with estimated µ ) in Table 4.4 for a typical case with e/a = 0.116, c ⌢ = 11.55 Hz and µ = 0.51. The contact mechanics model successfully predicts the forces and acceleration at the first three harmonics of c, which are dominant compared with the higher harmonics.

120

1.2

1

0.8

0.6

0.4

0.2

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Rm [m]

Fig. 4.8 Estimated for different Rm values and comparison with prior values (including

the range) for the dry friction regime [13]. Key: , With AGMA 4EP oil; , With ISO

32 oil; , dry contact - iron pin with steel disk [2.13] ; , dry contact - copper pin with

steel disk [2.13] .

121

N (N) N (N) αɺɺ (rad/s 2) x y Harmonic of c Measured Predicted Measured Predicted Measured Predicted

1 122.7 122.6 0.99 0.99 2.48 2.47

2 3.2 5.5 0.08 0.07 0.07 0.07

3 0.2 1.3 0.02 0.003 0.02 0.003

Table 4.4 Comparison of measurements and predictions (from the contact mechanics model) with = 0.51 and e/a = 0.116 at the harmonics of c = 11.55 Hz.

The normalized coefficient of friction ( µ ) for the empirical model of Eq. (4.41)

is defined as,

µ µ = . (4.43) p  λ  log 10 vt() vt () 2  η rt e t 

From Fig. 4.9 it is observed that µ monotonically increases with Rm. Also µ is lower

with AGMA 4EP (EHL regime) as compared to ISO 32 oil (mixed lubrication regime).

Fig. 4.9 compares some results of prior friction experiments [4.16, 4.33 - 35] in terms of

selected µ values which are calculated based on certain assumptions given a lack of

pertinent data. For instance, Shon et al. [4.16], Xu & Kahraman [4.33] and Furey [4.35]

conducted experiments under EHL regime, and hence their µ values are very low.

Conversely, Grunberg and Campbell [4.34] conducted experiments under poorly

lubricated conditions (mixed lubrication regime). It can be easily inferred that µ

122 decreases as Λ increases. There are some differences in the µ values from (Eq. (4.41)) and the ones reported in the literature; these may be attributed to different lubrication regimes as well as potential sources of error in the estimation process which is discussed next.

4.8 Potential sources of error in the estimation of

Some of the common measurement errors which are difficult to minimize include the following. First, a variation in the frictional load torque on the cam causes small variations in c during the experiment. This in turn introduces inaccuracy in the harmonic contents of the measured forces and acceleration, thereby affecting the estimated . Second, a small error in the angular alignment ( κ) of a force transducer could measure Ny(t) cos( κ) instead of the actual Ny(t). From the static analysis it is found that for κ = 5°, the estimate has only a 0.5 % error. Third, if the follower spring is oriented at an angle of σ (from the vertical in the clockwise direction), the elastic force Fs(t) will be as follows as opposed to the one given by Eq. (4.14),

kcosα ( t ) s ( ) u  Fts( )= Ldd syx −+ tan()()α ( tw ) + 0.5 b sec α ( t )  . (4.44) cos()α (t ) − σ

123

0.06

0.05

0.04

0.03

0.02

0.01

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Rm [m]

Fig. 4.9 Comparison of the modified Benedict-Kelley model from the results of Fig. 4.8 with friction values reported in the literature [4.16, 4.33 - 35]. Key: , Model for

AGMA 4EP oil (EHL regime) ; , Model for ISO 32 oil (mixed lubrication

regime), , Shon et al. [4.16]; , Xu and Kahraman [4.33]; , Grunberg and

Campbell [4.34]; , Furey [4.35].

124

Also the reaction forces will have to be calculated from the following instead of using the expressions of Eqs. (4.27) and (4.28),

NtFtx()= n ()sin(α () tFt) + f ()cos( α () t ) (4.45) ɺɺ −0.5mltb bα ( )sin()() α ( t ) − Ft s ( )sin σ ,

NtFtyn()= ()cos(α () t) − Ft f ()sin( α () t) + mg b (4.46) ɺɺ −0.5mltb bα ( )cos()() α ( t ) − Ft s ( )cos σ .

Based on the static force balance, the error in the estimation of is about 9% with only σ

= 1°, which is very significant.

The estimation of involves some error-prone numerical methods [4.36]. For instance, bias errors [4.37] might be caused in the computation of the spectral contents of forces and acceleration due to a coarse frequency resolution (constrained by the length of the measured time domain signal) and the usage of Hanning window. Furthermore, equations (4.38) to (4.40) are solved using the Levenberg-Marquardt algorithm which has limited accuracy as dictated by its relative and absolute tolerance values [4.38].

The error in is simulated for the system with a circular cam for different values of e under a constant c = 11.55 Hz. Using the predicted force and acceleration responses from the contact mechanics model (with known = 0.3) in signal processing technique, ⌢ µ is calculated and the results are given in Table 4.5. For a very low value of e/a, the ⌢ error in µ is high because the amplitude of αɺɺ (and the reaction forces) at the first harmonic of c is very small. As the amplitude of αɺɺ at the fundamental harmonic of c increases, the error reduces and reaches a minimum at e/a = 0.3 (error = 0.67%). Beyond e/a = 0.3, the error again starts increasing since the amplitude of αɺɺ at the second

125 harmonic of c becomes significant compared with that of the first. Next the error is calculated for different c with a constant e. The error monotonically decreases (as observed from Table 4.6) with an increase in c; this is because the amplitude of αɺɺ (and the reaction forces) at the first harmonic of c increases, while the amplitude ratio of the second harmonic to the fundamental harmonic is a constant.

A similar analysis is done for the system with an elliptic cam given e/a = 0.1, for

0.5 b2  different values of eccentricity ∈=1 − 2  with known = 0.3 (other parameters ( a )  remaining the same as in section 4.5). Fig. 4.10 gives a map of c - b/a, showing different regimes that are obtained using the inverse kinematics procedure of section 4.5.

Comparison of Fig. 4.10 with Fig. 4.6 suggests that ϵ for an elliptic cam provides a similar motion input as e does for a circular cam. Care is taken so that the system lies in the regime without a loss of contact and no direction reversal of the vr(t). Table 4.7 shows ⌢ the µ values for an elliptic cam for different ϵ at c = 8.33 Hz. Only a small variation in the error is observed. However, an increase in the ϵ increases the acceleration amplitude at the second harmonic of c due to a change in the type of motion input to the system.

Overall, it is inferred that can be satisfactorily estimated even for a system with an elliptic cam.

126

αɺɺ (rad/s2) ⌢ Estimated µ – known µ e/a % error = ×100 At the first At the second ⌢ known µ µ harmonic of c harmonic of c

0.05 52.5 1.01 0.284 5.3

0.10 105.0 4.02 0.286 4.7

0.15 157.5 9.05 0.288 4.0

0.20 210.0 16.1 0.292 2.8

0.25 262.5 25.1 0.296 1.3

0.3 315.1 36.2 0.302 0.7

0.35 367.6 49.3 0.31 3.1

0.4 420.1 64.4 0.32 6.0

0.45 472.7 81.5 0.33 9.4

Table 4.5 Error in the estimation of for the mechanical system with a circular cam for different values of e at c = 11.55 Hz.

127

ɺɺ 2 α [rad/s ] Estimated ⌢ c µ – known µ % error = ×100 At the first At the second ⌢ [Hz] µ known µ

harmonic of c harmonic of c

2 3.24 0.135 0.28 6.7

5 20.26 0.84 0.281 6.3

8 51.77 2.15 0.283 5.7

11 97.9 4.1 0.285 5.0

14 158.5 6.6 0.288 4.0

17 233.8 9.7 0.292 2.7

21 356.7 14.9 0.297 1.0

Table 4.6 Error in the estimation of for the mechanical system with circular cam for different cam speeds with e/a = 0.1.

128

24

22 Loss of contact regime 20 In-contact 18 regime No sign Sign reversal

reversal of vr(t) of vr(t) [Hz] 16 c

14

12

10

8 0 0.2 0.4 0.6 0.8 1 ϵ

Fig. 4.10 Classification of response regimes of a mechanical system with an elliptic cam in terms of a c – b/a map with e = 0.1 a and other parameter values given in section 4.5.

Key: , Operational range of the simulation.

129

αɺɺ (rad/s 2) ⌢ Estimated µ – known µ ϵ % error = ×100 At the first At the second ⌢ µ known µ

harmonic of c harmonic of c

0 52.2 2.18 0.283 5.7

0.31 52.1 52.7 0.281 6.2

0.44 52.1 104.9 0.281 6.2

0.53 52.05 156.8 0.284 5.5

0.6 52.0 208.4 0.288 4.2

0.66 51.95 259.7 0.294 2.2

Table 4.7 Error in the estimation of for the mechanical system with an elliptic cam at c = 8.33 Hz and e = 0.1 a.

4.9 Conclusion

The major contributions of these analytical and experimental studies are as follows. First, a new vibration experiment has been designed to estimate for a mechanical system with combined rolling-sliding contact under lubrication. This experiment permits the contact pressure, “lambda ratio”, contact velocity (sliding and entrainment), lubrication regime and surface roughness to be changed while satisfying the design constraints. Thus, the same experiment can be used to estimate for similar 130 combined rolling-sliding contact systems such as gears and drum brakes. Second, an improved contact mechanics model for a mechanical system with an elliptic cam and follower is formulated that successfully predicts the system responses, as theory and experiment match well. This mathematical model yields a better understanding of the system dynamics as well as the accuracy of the estimation procedure. Third, an improved signal processing method is proposed to calculate using the complex-valued

Fourier amplitudes of measured forces and acceleration. The DC components of the measured signals are also estimated by this method (along with ) by numerically solving a set of nonlinear equations. The chief limitation of this study is related to the angular ⌢ alignment of the follower spring. Also the error in µ is controlled by the choice of system geometry and cam speed; in particular the speed should be fairly low in order to avoid impacting conditions.

131

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136

CHAPTER 5

CONCLUSION

5.1 Summary

This study examines the non-linear dynamics and contact mechanics of system with combined rolling-sliding contact using a cam-follower mechanism. The contact dynamics especially coefficient of friction and impact damping are estimated using analytical, experimental and numerical methods.

In chapter 2, the non-linearities of the cam-follower system have been analyzed in this study. A contact mechanics formulation for a cam-follower system with combined rolling-sliding contact has been developed, and the predictions with combined viscous-impact damping models are successfully compared with the experimental results reported by Alzate et al. [5.1]. The inaccuracy of the coefficient of restitution (ξ) concept is analyzed using the approximate energy balance technique. The effect of friction non- linearity on the dynamic forces is studied using discontinuous and smoothened dry friction models. Finally, a linearized system is found to be inadequate in representing the system with only kinematic non-linearity.

In chapter 3, the parameters of impact damping model for a point contact under lubricated condition are estimated from time-domain measurements. An experiment is 137 designed and instrumented to measure force and acceleration in a system during periodic impacts. A new time-domain based technique is developed to accurately calculate the system response. An indirect signal processing estimation procedure (using residue minimization) is developed to get the damping parameters using an analogous contact mechanics model. Three different residues were defined and their accuracies were analyzed. The estimated values of damping parameters were justified using literature

[5.2] and equivalent ξ model.

In chapter 4, a new experimental method to estimate the coefficient of friction ( ) under lubricated condition for a system with combined rolling-sliding contact is developed. The cam-follower experiment is designed to have continuous contact. A contact mechanics model with an elliptic cam is developed from which the experimental parameters satisfying the design constraints are obtained. A new signal processing technique had been developed to estimate from the Fourier amplitudes of the measured forces and acceleration. This technique has a very good numerical accuracy, as inferred from the error analysis conducted. An empirical relation to get is derived based on

Benedict-Kelley model [5.3]. The different types of errors in the estimation process are analyzed and it is found that, even a minor misalignment in the angle of the follower spring causes large error in estimation.

5.2 Contributions

In this dissertation, several contributions emerge that are related to the improved understanding of the non-linear contact dynamics of systems with combined rolling-sliding contact. Some of the major contributions are as follows. First, the impact 138 damping model is estimated using the measurements of periodic impact events with point contact, with the help of a new signal processing procedure (minimizing the inherent errors associated with the numerical integration). Also some of the major issues regarding impact damping model (stated in section 3.2) have been resolved. Furthermore, the

ɺ applicability of Fk (1+κδ ) formulation and the inaccuracy of viscous damping for an impacting system have been experimentally verified.

Second, the coefficient of friction has been determined for the cam-follower system with combined rolling-sliding contact from the measured forces and acceleration

(without the DC component) using a new frequency-domain based technique. A generalized model to predict for a given surface roughness under lubricated condition

(with point contact) is proposed. Also the major sources of error in the estimation process have been quantified. Some of the other contributions from this research are the following. This research yields better understanding of the inaccuracies of coefficient of restitution formulation during impacts and the roles of the friction and kinematic non- linearities in the sliding contact regime.

5.3 Future work

There are several paths to further extend the examination of the non-linear dynamics and contact mechanics of systems with combined rolling-sliding contact. Each path should be independent and build upon the knowledge gained from this research:

1. Improve the analytical model by developing a higher degrees-of-freedom

(DOF) system for a cam-follower system by relaxing the rigid pivot

139

assumption. The improved model will more accurately represent the real

system under higher loads.

2. Analyze the non-linear dynamics of system with different scenarios of cam

motion like, constant acceleration, constant deceleration, oscillating speeds

and friction torque dependent speed variation.

3. Seek semi-analytical solutions to the non-linear differential equations of motion

to achieve improved accuracy in the prediction of system response.

4. Perform similar experimental study with line contacts to widen the knowledge

of the contact mechanics of such system. A very high degree of precision is

required to obtain a line contact experimentally.

5. Develop a direct method to obtain the impact damping parameters with higher

accuracy, instead of using indirect method discussed in this research.

6. Examine the non-linear dynamics of cam-follower system with two

kinematically liked followers in contact with the cam. A 3 DOF system can be

developed for this purpose with a rotational DOF each for the followers and

the cam. Friction induced vibrations like stick-slip and sprag-slip can be

experimentally analyzed using this system, which is a simplified model of a

drum .

140

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Transactions 4 (1) (1961) 59–70.

141

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