Impact damping and friction in non-linear mechanical systems with
combined rolling-sliding 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 Engineering
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 dynamics of systems with combined rolling-sliding contact such as cam- follower mechanism, gears and drum brakes. Such systems, in which the dominant elements involved in the sliding contact are rotating, have unique interaction among contact mechanics, siding friction and kinematics. Prior models used in the literature are highly simplified and do not use contact mechanics 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 time-domain based technique is adopted to accurately calculate the system response by minimizing the errors associated with numerical integration. The impact damping force is considered in a generalized form as a product of damping coefficient, indentation displacement raised to the power 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 forces and acceleration 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 work 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 Acoustics 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 Science 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. Mechanical Engineering 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 motion, Journal of Sound and Vibration, (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 Vibrations, Nonlinear Dynamics, Sliding Contact Systems, Contact Mechanics.
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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 Equations of motion………………………………………………. 21
2.3.3 Static equilibrium and linearized natural frequency……………… 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
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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 speeds with e/a = 0.1…………………………………………...… 128
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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
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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 Free body diagram 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
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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 energy 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 speed
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 );
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, 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 velocity 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
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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) Angular acceleration 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:
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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 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
4.7 Relative accelerance spectra in the vicinity of the system resonance. 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 Angular displacement of the follower δ Indentation
κ Arbitrary constant