EFFECT OF SLIDING FRICTION ON SPUR AND HELICAL GEAR
DYNAMICS AND VIBRO-ACOUSTICS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate School
of The Ohio State University
By
Song He, B.S., M.S.
*****
The Ohio State University
2008
Dissertation Committee:
Approved by
Dr. Rajendra Singh, Advisor
Dr. Ahmet Kahraman
____________________________
Dr. Ahmet Selamet
Adviser
Dr. Marcelo Dapino
Graduate Program in Mechanical Engineering
ABSTRACT
This study examines the salient effects of sliding friction on spur and helical gear
dynamics and associated vibro-acoustic sources. First, new dynamic formulations are
developed for spur and helical gear pairs based on a periodic description of the contact
point and realistic mesh stiffness. Difficulty encountered in the existing discontinuous
models is overcome by characterizing a smoother transition during the contact. Frictional
forces and moments now appear as either excitations or periodically-varying parameters,
since the frictional force changes direction at the pitch point/line. These result in a class
of periodic ordinary differential equations with multiple and interacting coefficients,
which characterize the effect of sliding friction in spur or helical gear dynamics.
Predictions (based on multi-degree-of-freedom analytical models) match well with a
benchmark finite element/contact mechanics code and/or experimental results.
Second, new analytical solutions are constructed which provide an efficient
evaluation of the frictional effect as well as a more plausible explanation of dynamic
interactions in multiple directions. Both single- and multi-term harmonic balance
methods are utilized to predict dynamic mesh loads, friction forces and pinion/gear
displacements. Such semi-analytical solutions explain the presence of higher harmonics
in gear noise and vibration due to exponential modulations of the periodic stiffness,
ii
dynamic transmission error and sliding friction. This knowledge also analytically reveals
the effect of the tooth profile modification in spur gears on the dynamic transmission
error, under the influence of sliding friction. Further, the Floquet theory is applied to
obtain closed-form solutions of the dynamic response for a helical gear pair, where the
effect of sliding friction is quantified by an effective piecewise stiffness function.
Analytical predictions, under both homogeneous and forced conditions, are validated
using numerical simulations. The matrix-based methodology is found to be
computationally efficient while leading to a better assessment of the dynamic stability.
Third, an improved source-path-receiver vibro-acoustic model is developed to
quantify the effect of sliding friction on structure-borne noise. Interfacial bearing forces
are predicted for the spur gear source sub-system given two gear whine excitations (static
transmission error and sliding friction). Next, a computational model of the gearbox, with
embedded bearing stiffness matrices, is developed to characterize the motilities of
structural paths. Radiated sound pressure is then estimated by using two numerical
techniques (the Rayleigh integral method and a substitute source technique). Predicted
pressures match well with measured noise data over a range of operating torques. In
particular, the proposed vibro-acoustic model quantifies the contribution of sliding
friction, which could be significant when the transmission error is minimized through
tooth modifications.
iii
DEDICATION
Dedicated to my parents and wife
iv
ACKNOWLEDGMENT
I would like to express my sincere appreciation to my advisor, Professor Rajendra
Singh, for his time, guidance, and support over the years both in my academic research
and personal life. His intellectual insight and encouragement had a huge impact on my
professional growth. I would also like to thank Professor Ahmet Kahraman, Professor
Ahmet Selamet and Professor Marcelo Dapino for their services on the doctoral
committee and for providing constructive suggestions.
I sincerely thank Dr. Todd Rook for providing valuable suggestions. I greatly
appreciate Dr. Rajendra Gunda for granting access to the Calyx software and for offering
insightful comments. I gracefully acknowledge the experimental work conducted by
Vivake Asnani and Fred Oswald, as well as the collaboration with Allison Lake. Dr.
Chengwu Duan is thanked for helping me in both my research and personal life. All
colleagues in the Acoustics and Dynamic Laboratory are acknowledged for their
encouragement and friendship. I thank Professor Goran Pavić, Professor Jean-Louis
Guyader and Corinne Lotto for their advice and kind help during my stay in INSA Lyon.
The financial support from the Army Research Office, EU’s Marie Curie
Fellowship and OSU Presidential Fellowship is gracefully appreciated.
Finally, I would like to thank my parents and my wife, Lihua, for their love and
encouragement throughout my pursuit for the doctoral degree.
v
VITA
October 24, 1979………………………..…..….Born – Jiangsu, China
2002………………………..………………..….B.S. Instrumentation Engineering
Shanghai Jiao Tong University
Shanghai, China
2004………………………..………………..….M.S. Mechanical Engineering
The Ohio State University
2004 - 2006………………..………………..….Graduate Teaching and Research
Associate, Mechanical Engineering
The Ohio State University
2007………………………..………………..….Marie Curie Fellow (EU)
National Institute of Applied Science
Lyon, France
2007 - 2008………………………..……………Presidential Fellow (Graduate School)
The Ohio State University
PUBLICATIONS
Research Publications
1.
He, S., Gunda, R., and Singh, R., 2007, “Effect of Sliding Friction on the Dynamics
of Spur Gear Pair with Realistic Time-Varying Stiffness,” Journal of Sound and
Vibration, 301, pp. 927-949.
2.
He, S., Gunda, R., and Singh, R., 2007, “Inclusion of Sliding Friction in Contact
Dynamics Model for Helical Gears, ASME Journal of Mechanical Design, 129(1),
pp. 48-57.
vi
3.
He, S., Cho, S., and Singh, R., 2008, “Prediction of Dynamic Friction Forces in Spur
Gears using Alternate Sliding Friction Formulations,” Journal of Sound and
Vibration, 309(3-5), pp. 843-851.
FIELDS OF STUDY
Major Field: Mechanical Engineering
Dynamics of Mechanical Systems
Vibro-Acoustics and Noise Control
vii
TABLE OF CONTENTS
Page
Abstract ............................................................................................................................... ii
Dedication .......................................................................................................................... iv
Acknowledgment ................................................................................................................ v
Vita..................................................................................................................................... vi
List of Tables .................................................................................................................... xii
List of Figures .................................................................................................................. xiii
List of Symbols ................................................................................................................. xx
Chapter 1 Introduction ........................................................................................................ 1
1.1 Motivation..................................................................................................................... 1
1.2 Literature Review.......................................................................................................... 4
1.3 Problem Formulation .................................................................................................... 8
1.3.1 Key Research Issues ........................................................................................ 8
1.3.2 Scope, Assumptions and Objectives .............................................................. 11
References for Chapter 1 .................................................................................................. 14
Chatper 2 Spur Gear Dynamics with Sliding Friction and Realistic Mesh Stiffness ....... 18
2.1 Introduction................................................................................................................. 18
2.2 Problem Formulation .................................................................................................. 20
2.2.1 Objectives and Assumptions.......................................................................... 20
2.2.2 Timing of Key Meshing Events..................................................................... 23
2.2.3 Calculation of Realistic Time-Varying Tooth Stiffness Functions................ 24
2.3 Analytical Multi-Degree-of-Freedom Dynamic Model.............................................. 27
2.3.1 Shaft and Bearing Stiffness Models............................................................... 27
2.3.2 Dynamic Mesh and Friction Forces............................................................... 28
2.3.3 MDOF Model................................................................................................. 32
2.4 Analytical SDOF Torsional Model............................................................................. 33
viii
2.5 Effect of Sliding Friction in Example I....................................................................... 35
2.5.1 Validation of Example I Model using the FE/CM Code ............................... 35
2.5.2 Effect of Sliding Friction ............................................................................... 40
2.5.3 MDOF System Resonances ........................................................................... 42
2.6 Effect of Sliding Friction in Example II ..................................................................... 43
2.6.1 Empirical Coefficient of Friction................................................................... 43
2.6.2 Effect of Tip Relief on STE and k(t).............................................................. 45
2.6.3 Phase Relationship between Normal Load and Friction Force Excitations... 50
2.6.4 Prediction of the Dynamic Responses ........................................................... 53
2.7 Experimental Validation of Example II Models......................................................... 56
2.8 Conclusion .................................................................................................................. 65
Chapter 3 Prediction of Dynamic Friction Forces Using Alternate Formulations ........... 68
3.1 Introduction................................................................................................................. 68
3.2 MDOF Spur Gear Model ............................................................................................ 69
3.3 Spur Gear Model with Alternate Sliding Friction Models ......................................... 74
3.3.1 Model I: Coulomb Model .............................................................................. 74
3.3.2 Model II: Benedict and Kelley Model ........................................................... 75
3.3.3 Model III: Formulation Suggested by Xu et al. ............................................. 76
3.3.4 Model IV: Smoothened Coulomb Model ...................................................... 78
3.3.5 Model V: Composite Friction Model............................................................. 78
3.4 Comparison of Sliding Friction Models ..................................................................... 80
3.5 Validation and Conclusion.......................................................................................... 85
References for Chapter 3 .................................................................................................. 89
Chapter 4 Construction of Semi-Analytical Solutions to Spur Gear Dynamics............... 91
4.1 Introduction................................................................................................................. 91
4.2 Problem Formulation .................................................................................................. 96
4.3 Semi-Analytical Solutions to the SDOF Spur Gear Dynamic Formulation ............... 97
4.3.1 Direct Application of Multi-Term Harmonic Balance (MHBM) .................. 99
ix
4.3.2 Semi-Analytical Solutions Based on One-Term HBM................................ 100
4.3.3 Iterative MHBM Algorithm......................................................................... 103
4.4 Analysis of Sub-Harmonic Response and Dynamic Instability................................ 107
4.5 Semi-Analytical Solutions to 6DOF Spur Gear Dynamic Formulation ................... 110
4.6 Conclusion ................................................................................................................ 116
References for Chapter 4 ................................................................................................ 122
Chapter 5 Effect of Sliding Friction on the Vibro-Acoustics of Spur Gear System....... 124
5.1 Introduction............................................................................................................... 124
5.2 Source Sub-System Model........................................................................................ 126
5.3 Structural Path with Friction Contribution ............................................................... 130
5.3.1 Bearing and Housing Models....................................................................... 130
5.3.2 Experimental Studies and Validation of Structural Model .......................... 132
5.3.3 Comparison of Structural Paths in LOA and OLOA Directions ................. 138
5.4 Prediction of Noise Radiation and Contribution of Friction..................................... 140
5.4.1 Prediction using Rayleigh Integral Technique............................................. 140
5.4.2 Prediction using Substitute Source Method................................................. 141
5.4.3 Prediction vs. Noise Measurements............................................................. 144
5.5 Conclusion ................................................................................................................ 148
References for Chapter 5 ................................................................................................ 150
Chapter 6 Inclusion of Sliding Friction in Helical Gear Dynamics................................ 152
6.1 Introduction............................................................................................................... 152
6.2 Problem Formulation ................................................................................................ 153
6.3 Mesh Forces and Moments with Sliding Friction..................................................... 155
6.4 Shaft and Bearing Models......................................................................................... 169
6.5 Twelve DOF Helical Gear Pair Model ..................................................................... 173
6.6 Role of Sliding Friction Illustrated by an Example .................................................. 176
6.7 Conclusion ................................................................................................................ 183
x
References for Chapter 6 ................................................................................................ 184
Chapter 7 Analysis of Helical Gear Dynamics using Floquet Theory............................ 186
7.1 Introduction............................................................................................................... 186
7.2 Linear Time-Varying Formulation ........................................................................... 187
7.3 Analytical Solutions by Floquet Theory ................................................................... 199
7.3.1 Response to Initial Conditions ..................................................................... 201
7.3.2 Forced Periodic Response............................................................................ 206
7.4 Conclusion ................................................................................................................ 214
References for Chapter 7 ................................................................................................ 216
Chapter 8 Conclusion...................................................................................................... 218
8.1 Summary ................................................................................................................... 218
8.2 Contributions............................................................................................................. 221
8.3 Future Work .............................................................................................................. 223
References for Chapter 8 ................................................................................................ 224
Bibliography ................................................................................................................... 225
xi
LIST OF TABLES
Table
Page
2.1
Parameters of Example I: NASA-ART spur gear pair (non-unity ratio) .............. 21
2.2
Parameters of Example II-A and II-B: NASA spur gear pair (unity ratio). Gear
pair with the perfect involute profile is designated as II-A case and the one with
tip relief is designated as II-B case ....................................................................... 22
2.3
Averaged coefficient of friction predicted over a range of operating conditions for
Example II by using Benedict and Kelly’s empirical equation [2.14].................. 44
5.1
Parameters of the example case: NASA spur gear pair with unity ratio (with long
tip relief).............................................................................................................. 128
5.2
Comparison of measured natural frequencies and finite element predictions .... 133
7.1
Relationship between Contact Zones and Contact Regions for the NASA-ART
helical gear pair................................................................................................... 197
xii
LIST OF FIGURES
Figure
Page
1.1
(a) Schematic of the spur gear contact, where LOA and OLOA represent the lineof-action direction and off line-of-action direction, respectively. (b) Directions of
the sliding velocity (V), normal mesh load and friction force in spur gears........... 3
1.2
Comparison of periodic mesh stiffness functions for a spur gear pair ................... 5
1.3
MDOF helical gear dynamic model (proposed in Chapter 6) and its contact
mechanics with sliding friction............................................................................. 10
1.4
Block diagram for the vibro-acoustics of a simplified geared system with two
excitations at the gear mesh (as proposed in Chapter 5)....................................... 14
2.1
Snap shot of contact pattern (at t = 0) in the spur gear pair of Example I. ........... 24
2.2
Tooth mesh stiffness functions of Example I calculated by using the FE/CM code
(in the “static” mode). (a) Individual and combined stiffness functions. (b)
Comparison of the combined stiffness functions.................................................. 26
2.3
Schematic of the bearing-shaft model................................................................... 28
2.4
Normal and friction forces of analytical (MDOF) spur gear system model. ........ 31
2.5
Validation of the analytical (MDOF) model by using the FE/CM code (in the
“dynamic” mode). Here, results for Example I are given in terms of δ (t ) and its
spectral contents ∆ ( f ) with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are
for µ = 0 and (c-d) are for µ = 0.2. .................................................................... 36
2.6
Validation of the analytical (MDOF) model by using the FE/CM code (in the
“dynamic” mode). Here, results for Example I are given in terms of FpBx (t ) and
its spectral contents FpBx ( f ) with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b)
are for µ = 0 and (c-d) are for µ = 0.2. .............................................................. 37
xiii
2.7
Validation of the analytical (MDOF) model by using the FE/CM code (in the
“dynamic” mode). Here, results for Example I are given in terms of FpBy (t ) and
its spectral contents FpBy ( f ) with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b)
are for µ = 0 and (c-d) are for µ = 0.2 ............................................................... 39
2.8
Effect of µ on δ (t ) based on the linear time-varying SDOF model for Example I
at Tp = 2000 lb-in. Here, tc = 1 s ........................................................................... 41
2.9
Coefficient of friction µ as a function of the roll angle for Example II, as
predicted by using Benedict & Kelley’s empirical equation [2.14]. Here, oil
temperature is 104 deg F and T = 500 lb-in.......................................................... 45
2.10
Mesh harmonics of the static transmission error (STE) calculated by using the
FE/CM code (in the “static” mode) for Example II: (a) gear pair with perfect
involute profile (II-A); (b) gear pair with tip relief (II-B) .................................... 46
2.11
Tooth stiffness functions of a single mesh tooth pair for Example II: (a) gear pair
with perfect involute profile (II-A); (b) gear pair with tip relief (II-B) ................ 48
2.12
Combined tooth stiffness functions for Example II: (a) gear pair with perfect
involute profile (II-A); (b) gear pair with tip relief (II-B) .................................... 49
2.13
Dynamic loads predicted for Example II at 500 lb-in, 4875 RPM and 140 °F with
tc = 0.44 ms: (a) Normal loads of gear pair with perfect involute profile (II-A); (b)
normal loads of gear par with tip relief (II-B); (c) friction forces of gear pair with
perfect involute profile (II-A); (d) friction forces of gear pair with tip relief (II-B).
............................................................................................................................... 51
2.14
Dynamic shaft displacements predicted for Example II at 500 lb-in, 4875 RPM
and 140 °F with tc = 0.44 ms and fm = 2275 Hz: (a) x p (t ) ; (b) X p ( f ) ; (c) y p (t )
and (d) Yp ( f ) ....................................................................................................... 54
2.15
Dynamic transmission error predicted for Example II at 500 lb-in, 4875 RPM and
140 °F with tc = 0.44 ms and fm = 2275 Hz: (a) δ (t ) ; (b) ∆ ( f ) ........................... 55
2.16
Sensors inside the NASA gearbox (for Example II-B)......................................... 57
2.17
Mesh harmonic amplitudes of X p as a function of the mean torque at 140 °F. All
values are normalized with respect to the amplitude of Yp at the first mesh
harmonic ............................................................................................................... 59
xiv
2.18
Mesh harmonic amplitudes of y p as a function of the mean torque at 140 °F for
Example II-B. All values are normalized with respect to the amplitude of y p at the
first mesh harmonic............................................................................................... 60
2.19
Predicted dynamic transmission errors (DTE) for Example II over a range of
torque at 140 °F: (a) gear pair with perfect involute profile (II-A); (b) with tip
relief (II-B). All values are normalized with respect to the amplitude of δ (II-A) at
the first mesh harmonic with 100 lb-in ................................................................. 62
2.20
Mesh harmonic amplitudes of x p as a function of temperature at 500 lb-in for
Example II-B. All values are normalized with respect to the amplitude of y p at the
first mesh harmonic............................................................................................... 63
2.21
Mesh harmonic amplitudes of y p as a function of temperature at 500 lb-in for
Example II-B. All values are normalized with respect to the amplitude of y p at the
first mesh harmonic............................................................................................... 64
3.1
(a) Snap shot of contact pattern (at t = 0) in the spur gear pair; (b) MDOF spur
gear pair system; here k (t ) is in the LOA direction............................................. 71
3.2
(a) Comparison of Model II [3.7] and Model III [3.8] given Tp = 22.6 N-m (200
lb-in) and Ω p = 1000 RPM. (b) Averaged magnitude of the coefficient of friction
predicted as a function of speed using the composite Model V with Tp = 22.6 Nm (200 lb-in). Here, tc is one mesh cycle............................................................. 81
3.3
Comparison of normalized friction models. Note that curve between 0 ≤ t / tc < 1
is for pair # 1; and the curve between 1 ≤ t / tc < 2 is for pair # 0......................... 83
3.4
Combined normal load and friction force time histories as predicted using
alternate friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM.
............................................................................................................................... 84
3.5
Predicted LOA and OLOA displacements using alternate friction models given
Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM ................................................. 86
3.6
Predicted dynamic transmission error (DTE) using alternate friction models given
Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM ................................................. 87
xv
3.7
Validation of the normal load and sliding friction force predictions: (a) at Tp =
79.1 N-m (700 lb-in) and Ω p = 800 RPM; (b) at Tp = 79.1 N-m (700 lb-in) and
Ω p = 4000 RPM.................................................................................................... 88
4.1
Realistic mesh stiffness functions of the spur gear pair example (with tip relief)
given Tp = 550 lb-in. (b) Periodic frictional functions......................................... 92
4.2
(a) Normal (mesh) and friction forces of 6DOF analytical spur gear system model.
(b) Snap shot of contact pattern (at t = 0) for the example spur gear pair. ........... 93
4.3
Semi-analytical vs. numerical solutions for the SDOF model, expressed by Eq.
(4.3), given Tp = 550 lb-in, Ω p = 500 RPM, µ = 0.04. (a) Time domain responses;
(b) mesh harmonics in frequency domain........................................................... 102
4.4
Semi-analytical vs. numerical solutions for the SDOF model as a function of
pinion speed with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4.....
............................................................................................................................. 106
4.5
Normalized determinant of the sub-harmonic matrix K u as a function of ωn / Ω NS
with µ = 0.04: (a) Tp = 100 lb-in.; (b) Tp = 550 lb-in. ................................. 109
4.6
(a) 6DOF spur gear pair model and its subset of unity gear pair (3DOF model)
used to study the natural frequency distribution; (b) Natural frequencies ΩN as a
function of the stiffness ratio K B / km ........................................................ 111
4.7
Semi-analytical vs. numerical solutions for the 6DOF model as a function of Ω p
with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4 .................... 113
4.8
Semi-analytical vs. numerical solutions of the LOA displacement xp for the 6DOF
model as a function of Ω p with KB/km = 100, µ = 0.04 (a) Mesh order n = 1; (b) n
= 2; (c) n = 3; (d) n = 4 ....................................................................................... 117
4.9
Semi-analytical vs. numerical solutions of the OLOA displacement yp for the
6DOF model as a function of Ω p with KB/km = 100, µ = 0.04. (a) Mesh order n =
1; (b) n = 2; (c) n = 3; (d) n = 4........................................................................... 118
xvi
4.10
Semi-analytical vs. numerical solutions of the LOA displacement xp for the 6DOF
model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n = 1; (b)
n = 2; (c) n = 3; (d) n = 4 .................................................................................... 120
4.11
Semi-analytical vs. numerical solutions of the OLOA displacement yp for the
6DOF model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n =
1; (b) n = 2; (c) n = 3; (d) n = 4........................................................................... 121
5.1
Block diagram for the vibro-acoustics of a simplified geared system with two
excitations at the gear mesh. ............................................................................... 125
5.2
Bearing forces predicted under varying Tp given Ωp = 4875 RPM and 140 °F. (a):
LOA bearing force; (b) OLOA bearing force. Key: m is the mesh frequency index
............................................................................................................................. 129
5.3
(a) Schematic of NASA gearbox; (b) Finite element model of NASA gearbox
with embedded bearing stiffness matrices .......................................................... 131
5.4
Comparison of the gearbox mode shape at 2962 Hz: (a) modal experiment result
[5.12]; (b) finite element prediction.................................................................... 134
5.5
(a) Experiment used to measure structural transfer functions; (b) comparison of
transfer function magnitudes from gear mesh to the sensor on top plate. .......... 137
5.6
Magnitudes of the combined transfer mobilities in two directions calculated at the
sensor location on the top plate........................................................................... 139
5.7
Comparison of normal surface velocity magnitudes and substitute source strength
vectors under Tp = 500 lb-in and Ωp = 4875 RPM. (a) Line 1: interpolated surface
velocity on top plate; (b) Line 2: simplified 2D gearbox model with 15 substitute
source points; (c) Line 3: substitute source strengths in complex plane for 2D
gearbox. Column 1: mesh frequency index m = 1; Column 2: m = 2; Column 3: m
= 3. ...................................................................................................................... 145
5.8
Normalized sound pressure (ref 1.0 Pa) predicted at the microphone 6 inch above
the top plate under varying torque Tp given Ωp = 4875 RPM and 140 °F .......... 147
5.9
Overall sound pressure levels (ref: 2e-5 Pa) and their contributions predicted at
the microphone 6 in above the top plate under Ωp = 4875 RPM and 140 °F. (a) Tp
= 500 lb-in (optimal load for minimum transmission error); (b) Tp = 800 lb-in. 149
xvii
6.1
Schematic of the helical gear pair system........................................................... 156
6.2
Tooth stiffness function calculated using Eq. (6.1) based on the FE/CM code
[6.11]................................................................................................................... 157
6.3
Contact zones at the beginning of a mesh cycle. (a) In the helical gear pair; (b)
contact zones within contact plane. Key: PP’ is the pitch line; AA’ is the face
width W; AD is the length of contact zone Z ...................................................... 166
6.4
Predicted tooth stiffness functions ...................................................................... 169
6.5
Schematic of the bearing-shaft model. Here, the shaft and bearing stiffness
elements are assumed to be in series to each other. Only pure rotational or
translational stiffness elements are shown. Coupling stiffness terms K xθ y , K yθ x are
not shown ............................................................................................................ 170
6.6
Time and frequency domain responses of translational pinion displacements
u xp , u yp ,u zp at Tp = 2000 lb-in and Ω p = 1000 RPM. All displacements are
normalized with respect to 39.37 µinch (1 µm).................................................. 178
6.7
Time and frequency domain responses of pinion bearing forces FSB , xp , FSB , yp and
FSB , zp at Tp = 2000 lb-in and Ω p = 1000 RPM. All forces are normalized with
respect to 1 lb ...................................................................................................... 179
6.8
Time and frequency domain responses of composite displacements δ x , δ y , δ z and
velocity δz at Tp = 2000 lb-in and Ω p = 1000 RPM. All motions are normalized
with respect to 39.37µinch (1µm) or 39.37µinch (1µm/s) ................................. 181
7.1
Schematic of the helical gear pair system........................................................... 189
7.2
Contact zones at the beginning of a mesh cycle within the contact plane. Key: PP’
is the pitch line; AA’ is the face width W; AD is the length of contact zone Z .. 191
7.3
Individual effective stiffness Ke,i(t) along the contact zone, where Tmesh is one
mesh cycle ............................................................................................ 196
7.4
Piece-wise effective stiffness function defined in six regions within one mesh
cycle with µ = 0.4 .................................................................................. 198
xviii
7.5
(a) Effective stiffness and (b) homogeneous responses predictions within two
mesh cycles given x0 = 2×10-6 in., v0 = 20 in./s at Ωp = 1000 RPM ................ 204
7.6
Predictions of damped homogeneous responses within two mesh cycles given x0 =
2×10-6 in., v0 = 20 in./s, µ = 0.2 at Ωp = 1000 RPM ..................................... 206
7.7
Predictions of (undamped) forced periodic responses within two mesh cycles
given x0 = 2×10-6 in., v0 = 20 in./s, Tp = 2000 lb-in, µ = 0.2 and Ωp = 1000 RPM
........................................................................................................... 210
7.8
Steady state forced periodic responses given x0 = 2×10-6 in., v0 = 20 in./s, Tp =
2000 lb-in., µ = 0.1 and Ωp = 1000 RPM: (a) DTE vs. time; (b) DTE spectra... 212
7.9
Predicted mesh harmonics of (undamped) forced periodic responses as a function
of µ given x0 = 2×10-6 in., v0 = 20 in./s, Tp = 2000 lb-in and Ωp = 1000 RPM: (a)
DTE; (b) slope of DTE ....................................................................................... 213
7.10
Mapping of eigenvalue κ (absolute value) maxima as a function of the ratio of
time-varying mesh frequency fmesh(t) to the system natural frequency fn ......... 215
xix
LIST OF SYMBOLS
List of Symbols for Chapter 1
O
P
t
V
x
y
z
ε
pinion/gear center location
pitch point
time (s)
contact point speed (in./s)
line-of-action direction
off line-of-action direction
axial direction
unloaded static transmission error (µin.)
Subscripts
1
2
pinion
gear
Abbreviations
ART
DOF
DTE
EHL
LOA
MDOF
OLOA
SDOF
STE
Advanced Rotorcraft Transmission
degree-of-freedom
dynamic transmission error
elasto-hydrodynamic lubrication
line-of-action
multi-degree-of-freedom
off line-of-action
single degree-of-freedom
static transmission error
List of Symbols for Chapter 2
a, b
CR
c
E
F
f
I
i, j
shaft distance (in)
surface roughness constant
viscous damping (lb-s/in)
Young’s modulus (psi)
force (lb)
frequency (Hz)
area moment of inertia (in4)
indices of gear tooth
xx
J
K
K
k
L
N
n
R
r
T
t
Ve
Vs
Wn
X
XΓ
x
Y
y
∆
δ
ε
γ
λ
µ
υ0
θ
σ
Ω
ζ
polar moment of inertia (lb-s2-in)
stiffness matrix (lb/in)
stiffness (lb/in)
tooth mesh stiffness (lb/in)
geometric length (in)
normal contact force (lb)
mesh index
tooth surface roughness (in)
radius (in)
torque (lb-in)
time (s)
entraining velocity (in/s)
sliding velocity (in/s)
normal load per unit length of face width (lb/in)
moment arm (in)
load sharing factor
motion variable along LOA axis (in)
frequency spectrum of motion along OLOA axis (in)
motion variable along OLOA axis (in)
frequency spectrum of dynamic transmission error (in)
dynamic transmission error (in)
static transmission error (in)
nominal roll angle (rad)
base pitch (in)
coefficient of friction
dynamic viscosity (lb-s/in2)
vibratory angular displacement (rad)
contact ratio
angular speed (rad/s)
viscous damping ratio
Subscripts
0, 1… n
avg
B
b
c
e
f
g
i
m
P
p
indices of meshing teeth/modes
average
bearing
base
(mesh) cycle
effective
friction
gear
index of gear tooth
mesh
pitch point
pinion
xxi
S
x
y
z
shaft
LOA direction
OLOA direction
axial direction
Superscripts
.
..
first derivative with respect to time
second derivative with respect to time
Abbreviations
ART
DOF
DTE
FE/CM
HPSTC
LPSTC
LTV
LOA
MDOF
OLOA
SDOF
Advanced Rotorcraft Transmission
degree-of-freedom
dynamic transmission error
finite element/contact mechanics
highest point of single tooth contact
lowest point of single tooth contact
linear time-varying
line-of-action
multi-degree-of-freedom
off line-of-action
single degree-of-freedom
Operators
ceil
floor
mod
sgn
ceiling function
floor function
modulus function
sign function
List of Symbols for Chapter 3
b
bH
c
E
F
G
H
J
K
k
L
m
N
empirical coefficient
semi-width of Hertzian contact band
viscous damping (lb-s/in)
Young’s modulus (GPa)
force (lb)
dimensionless material parameter
dimensionless central film thickness
polar moment of inertia (lb-s2-in)
stiffness (lb/in)
tooth mesh stiffness (lb/in)
geometric length (in)
mass (lb⋅s2/in)
normal contact force (lb)
xxii
Ph
r
S
SR
T
t
U
V, v
W
wn
X
x
y
Z
α
ηM
Λ
λ
µ
ν
θ
ρ
Φ
σ
Ω
ζ
maximum Hertzian pressure (GPa)
radius (in)
surface roughness (µm)
slide-to-roll ratio
torque (lb-in)
time (s)
speed parameter
velocity (m/s)
load parameter
normal load per unit length of face width (N/mm)
moment arm (in)
motion variable along LOA axis (in)
motion variable along OLOA axis (in)
face width (mm)
pressure angle (rad)
dynamic viscosity (N-s/mm2)
film parameter
base pitch (in)
coefficient of friction
Poisson’s ratio
vibratory angular displacement (rad)
profile radii of curvature (mm)
regularizing factor
contact ratio
angular speed (rad/s)
viscous damping ratio
Subscripts
0, 1… n
avg
B
b
C
c
comp
e
f
g
i
p
r
S
s
X
indices of meshing teeth
average
bearing
base
Coulomb friction
(mesh) cycle
composite
entraining component
friction
gear
index of gear tooth
pinion
rolling component
smoothened friction model
sliding component
Xu and Kahraman model
xxiii
x
y
LOA direction
OLOA direction
Superscripts
.
..
’
first derivative with respect to time
second derivative with respect to time
effective value
Abbreviations
DOF
DTE
EHL
HPSTC
LPSTC
LTV
LOA
MDOF
OLOA
degree-of-freedom
dynamic transmission error
elasto-hydrodynamic lubrication
highest point of single tooth contact
lowest point of single tooth contact
linear time-varying
line-of-action
multi-degree-of-freedom
off line-of-action
Operators
floor
mod
sgn
floor function
modulus function
sign function
List of Symbols for Chapter 4
A, B
C
c
D
E
F
F
f
i, j
J
K
K
k
L
M
m
N
harmonic balance coefficients
damping parameter (lb-s/in)
viscous damping (lb-s/in)
Fourier differentiation matrix
gear constant
harmonic balance matrix
force (lb)
frictional function
indices
polar moment of inertia (lb-s2-in)
harmonic balance matrix
stiffness (lb/in)
tooth mesh stiffness (lb/in)
geometric length (in)
(friction) torque (lb-in) or mass (lb⋅s2/in)
mass (lb⋅s2/in)
normal contact force (lb) or harmonic order
xxiv
n
R
r
S
T
t
X
∆
δ
ε
ϑ
λ
µ
θ
σ
τ
υ
Ω
ω
ζ
mesh index
base radius (in)
radius (in)
index
(normalized) period
time (s)
moment arm (in)
Fourier coefficient vector
dynamic transmission error (in)
static transmission error (in)
angle (rad)
base pitch (in)
coefficient of friction
vibratory angular displacement (rad)
contact ratio
dimensionless time
sub-harmonic index
angular speed (rad/s)
mesh frequency (rad)
viscous damping ratio
Subscripts
0, 1… n
B
b
e
f
h
g
i
k
P
p
u
x
y
δ
indices of meshing teeth
bearing
base
effective
friction
super harmonic matrix
gear
index of gear tooth
stiffness coefficient
pitch point
pinion
sub-harmonic matrix
LOA direction
OLOA direction
dynamic transmission error coefficient
Superscripts
.
..
’
”
^
first derivative with respect to time
second derivative with respect to time
first derivative with respect to dimensionless time
second derivative with respect to dimensionless time
differential operator
xxv
nominal value
~
iterative harmonic balance parameter
+
pseudo-inverse
Abbreviations
DFT
DOF
DTE
FFT
LOA
LTV
MDOF
MHBM
OLOA
SDOF
discrete Fourier transform
degree-of-freedom
dynamic transmission error
fast Fourier transform
line-of-action
linear time-varying
multi-degree-of-freedom
multi-term harmonic balance method
off line-of-action
single degree-of-freedom
Operators
floor
mod
sgn
floor function
modulus function
sign function
||
matrix determinant
List of Symbols for Chapter 5
e
F
f
H
Hv
I
i, j
J
L
K
k
k(ω)
m
N
n
P
Q
error
force (lb)
frequency (Hz)
transfer function
Hankel function
identity matrix
indices of gear tooth
polar moment of inertia (lb-s2-in)
geometric length (in)
stiffness (lb/in)
tooth mesh stiffness (lb/in)
wave number
mass (lb⋅s2/in)
normal contact force (lb)
natural (frequency)
sound pressure (Pa)
source strength (Pa-in2)
xxvi
r
S
T
t
V
w
X
x
Y
y
α
λ
µ
θ
ρ
σ
Ω
ω
Ξ
ζ
radius (in)
surface area (in2)
torque (lb-in)
time (s)
velocity (in/s)
weighting function
moment arm (in)
motion variable along LOA axis (in)
mobility (in/s/lb)
motion variable along OLOA axis (in)
angle (rad)
base pitch (in)
coefficient of friction
vibratory angular displacement (rad)
air density (lb⋅s2/in4)
contact ratio
angular speed (rad/s)
angular frequency (rad)
velocity error matrix
viscous damping ratio
Subscripts
B
b
e
f
g
i
m
P
p
S
x
y
bearing
base
effective parameter
friction
gear
index of gear tooth
mean component
path
pinion
shaft or source
LOA direction
OLOA direction
Superscripts
.
..
~G
-1
*
Abbreviations
first derivative with respect to time
second derivative with respect to time
complex value
vector
matrix inverse
complex conjugate
xxvii
DOF
LOA
MIMO
OLOA
RMS
STE
degree-of-freedom
line-of-action
multi-input multi-output
off line-of-action
mean square root
static transmission error
Operators
floor
mod
sgn
||
floor function
modulus function
sign function
absolute value
List of Symbols for Chapter 6
E
e
F
K
k
I
J
L
l
M
m
N
N
r
T
Tmesh
t
u
W
v
x
z
β
∆
µ
λ
δ
φ
σ
Young’s modulus (psi)
unit vector along axis
force (lb)
tooth mesh stiffness (lb/in)
tooth mesh stiffness density (lb/in2)
area moment of inertia (in4)
polar moment of inertia (lb-s2-in)
length of contact line (in)
variable along contact line (in)
moment (lb-in)
mass (lb⋅s2/in)
normal contact force (lb)
mesh index
radius (in)
torque (lb-in)
mesh period (s)
time (s)
translational motion (in)
face width (in)
velocity of contact point (in/s)
LOA coordinate of contact point (in)
axial coordinate of contact point (in)
helical angle
deformation of contact point (in)
coefficient of friction
base pitch (in)
dynamic transmission error (in)
pressure angle (deg)
contact ratio
xxviii
Θ
θ
Ω
ζ
(static) angular deflection (rad)
vibratory angular displacement (rad)
angular speed (rad/s)
viscous damping ratio
Subscripts
0, 1… n
A
b
c
g
h
i
l
P
p
S
s
V
x
y
z
indices of meshing teeth
(shaft) cross section area (in2)
base
contact point
gear
(coordinate) upper limit
index of gear tooth
(coordinate) lower limit
pitch point
pinion
shaft
sliding component
viscous component
LOA direction
OLOA direction
axial direction
Superscripts
.
..
−
<-1>
T
first derivative with respect to time
second derivative with respect to time
mean component
matrix inverse
matrix transverse
Abbreviations
ART
DOF
FE/CM
LOA
LTV
MDOF
OLOA
Advanced Rotorcraft Transmission
degree-of-freedom
finite element/contact mechanics
line-of-action
linear time-varying
multi-degree-of-freedom
off line-of-action
Operators
×
ceil
floor
cross product
ceiling function
floor function
xxix
mod
sgn
modulus function
sign function
List of Symbols for Chapter 7
C
e
F
f
G
H
J
K
k
L
M
m
r
T
t
v
W
X
x
Z
z
β
δ
ε
Φ
φ
γ
κ
λ
µ
Π
θ
σ
τ
Ω
ζ
viscous damping coefficient (lb-s/in)
unit vector along axis
force (lb)
frequency (Hz)
state matrix
transition matrix
polar moment of inertia (lb-s2-in)
tooth mesh stiffness (lb/in)
tooth mesh stiffness (lb/in)
length of contact line (in)
moment (lb-in)
mass (lb⋅s2/in)
radius (in)
torque (lb-in)
time (s)
velocity (in/s)
face width (in)
state space response
LOA coordinate of contact point (in)
contact zone
axial coordinate of contact point (in)
helical angle
dynamic transmission error (in)
static transmission error (in)
state transition matrix
pressure angle (deg)
basis solution
eigenvalue
base pitch (in)
coefficient of friction
Wronskian matrix
vibratory angular displacement (rad)
contact ratio
integration variable
angular speed (rad/s)
viscous damping ratio
Subscripts
xxx
0, 1… n
c
e
g
m
p
x
y
z
indices of meshing teeth
contact point
effective parameter
gear
mesh (frequency)
pinion
LOA direction
OLOA direction
axial direction
Superscripts
.
..
−
first derivative with respect to time
second derivative with respect to time
time average
Abbreviations
DOF
DTE
FE/CM
LOA
LTV
OLOA
SDOF
degree-of-freedom
dynamic transmission error
finite element/contact mechanics
line-of-action
linear time-varying
off line-of-action
single-degree-of-freedom
Operators
ceil
floor
LommelS1
mod
sgn
||
ceiling function
floor function
Lommel function
modulus function
sign function
absolute value
List of Symbols for Chapter 8
µ
coefficient of friction
Abbreviations
DOF
DTE
LTV
LOA
MDOF
OLOA
degree-of-freedom
dynamic transmission error
linear time-varying
line-of-action
multi-degree-of-freedom
off line-of-action
xxxi
SDOF
single-degree-of-freedom
xxxii
CHAPTER 1
INTRODUCTION
1.1
Motivation
Spur and helical gears are widely used in vehicles and mechanical devices to
transmit large torques while maintaining a constant input-to-output speed ratio. One
remaining challenge for modern gear engineering is the reduction of gear noise in ground
and air vehicles such as heavy duty trucks and helicopters. Typically, steady state gear
(whine) noise is generated by several sources [1.1-1.2]. Virtually all of the prior
researchers [1.1-1.6] have assumed that the main source is static transmission error (STE),
which is defined as the derivation from the ideal (kinematic) tooth profile induced by
manufacturing errors and elastic deformations. Accordingly, design engineers tend to
reduce STE, via improved manufacturing processes and tooth modifications [1.7]. Yet, at
higher torque loads, noise levels are still relatively high though STE might be somewhat
minimal (say at the design loads). In other cases, the trend in sound pressure levels does
not necessarily match the STE vs. torque curves [1.2]. Typical examples include
experimental data on the Advanced Rotorcraft Transmission (ART) gears tested by
1
NASA Glenn and OSU [1.8-1.10]. These suggest that additional vibro-acoustic sources
must be considered.
The relative speed between V2 and V1 of two meshing gear teeth (with centers at
O1 and O2), as depicted in Figure 1.1(a), changes direction at the pitch point P during
each contact event, thus providing additional periodic excitations normal to the direction
of contact, as shown in Figure 1.1(b). Certain unique characteristics of the gear tooth
sliding make it a potentially dominant factor, despite the somewhat lower magnitudes of
friction force. First, due to the reversal in the direction during meshing action, friction is
associated with a large oscillatory component, which causes both higher magnitudes as
well as higher bandwidth in dynamic responses. Furthermore, friction is more significant
at higher torque and lower speeds. In reality, frictional source mechanism is associated
with
surface
roughness,
lubrication
regime
properties,
time-varying
friction
forces/torques and mesh interface dynamics. These lead to interesting gear dynamic
phenomena, such as super-harmonic response, unstable regimes, sub-harmonic resonance
and angular modulation [1.11-1.15]. Clearly, the diverse effects of friction can only be
analyzed by adopting an intra-disciplinary approach, wherein the principles of meshing
kinematics, contact and tribological characteristics, dynamics and noise propagation
mechanisms are integrated into a cohesive model.
2
(a)
Sliding
Velocity
V2 – V1
Normal Load
Friction
Force
(b)
Figure 1.1 (a) Schematic of the spur gear contact, where LOA and OLOA represent the
line-of-action direction and off line-of-action direction, respectively. (b) Directions of the
sliding velocity (V), normal mesh load and friction force in spur gears.
3
Historically, the friction between gear teeth and its cyclic nature have been either
ignored or incorporated as an equivalent viscous damping term [1.1-1.2]. Such an
approach is clearly inadequate since viscous damping is essentially a passive
characteristic and it cannot act as the external excitation to the governing system. Neither
does it consider the dynamic effects in the off-line-of-action (OLOA) direction. Hence,
there is a definite need for new or improved models that could predict the dynamic and
vibro-acoustic responses of a geared system and clarify the role of sliding friction. This is
the salient focus of this study.
1.2
Literature Review
In a series of recent articles, Vaishya and Singh [1.13-1.15] have provided an
extensive review of prior work. They developed a spur gear pair model with sliding
friction and rectangular mesh stiffness by assuming that load is equally shared among all
the teeth in contact, as shown in Figure 1.2. They also solved the SDOF system equations
in terms of the dynamic transmission error (DTE) by using the Floquet theory and the
harmonic balance method [1.13-1.15]. While the assumption of equal load sharing yields
simplified expressions and analytically tractable solutions, it may not lead to a realistic
model (as shown in Figure 1.2 and then Chapter 2). Houser et al. [1.16-1.17]
experimentally demonstrated that the friction forces play a pivotal role in determining the
load transmitted to the bearings and housing in the OLOA direction; this effect is more
pronounced at higher torque and lower speed conditions.
4
Tooth stiffness (lb/in)
Figure 1.2 Comparison of periodic mesh stiffness functions for a spur gear pair. Key:
, realistic load sharing (proposed in Chapter 2);
, equal load sharing assumed by
Vaishya and Singh [1.13].
Velex and Cahouet [1.18] described an iterative procedure to evaluate the effects
of sliding friction, tooth shape deviations and time-varying mesh stiffness in spur and
helical gears and compared simulated bearing forces with measurements. They reported
significant oscillatory bearing forces at lower speeds that are induced by the reversal of
friction excitation with alternating tooth sliding direction. In a subsequent study, Velex
and Sainsot [1.19] analytically found that the Coulomb friction should be viewed as a
non-negligible excitation source to error-less spur and helical gear pairs, especially for
translational vibrations and in the case of high contact ratio gears. However, their work
was confined to a study of excitations and the effects of tooth modifications were not
5
considered. Lundvall et al. [1.20] considered profile modifications and manufacturing
errors in a multi-degree-of-freedom (MDOF) spur gear model and examined the effect of
sliding friction on the angular dynamic motions. By utilizing a numerical method, they
reported that the profile modification has less influence on the dynamic transmission
error when frictional effects are included. However, incorporation of the time-varying
sliding friction and the realistic mesh stiffness functions into an analytical (MDOF)
formulation and their dynamic interactions remain unsolved.
In all of the work mentioned above and related literature [1.8-1.20], the sliding
friction phenomenon has been typically formulated by assuming the Coulomb
formulation with a constant coefficient of friction for modeling convenience. This is
partially related to the difficulty associated with the measurement of friction force in a
gear mesh. In reality, tribological conditions change continuously due to varying mesh
properties, dynamic fluctuations and lubricant film thickness as the gears roll through a
full cycle [1.21-1.26]. Thus, coefficient of friction varies instantaneously with the spatial
position of each tooth and the direction of friction force changes at the pitch point.
Alternate tribological theories, such as elasto-hydrodynamic lubrication (EHL), boundary
lubrication or mixed regime, have been employed to explain the sliding friction under
varying operating conditions [1.21-1.23]. For instance, Benedict and Kelley [1.21]
proposed an empirical dynamic friction coefficient under mixed lubrication regime based
on measurements on a roller test machine. Xu et al. [1.22-1.23] recently proposed yet
another friction formula that is obtained by using a non-Newtonian, thermal EHL
formulation. Duan and Singh [1.27] developed a smoothened Coulomb model for dry
friction in torsional dampers; it could be applied to gears to obtain a smooth transition at
6
the pitch point. Hamrock and Dawson [1.28] suggested an empirical equation to predict
the minimum film thickness for two disks in line contact. They calculated the film
parameter, which could lead to a composite, mixed lubrication model for gears. Rebbechi
et al. [1.8] have successfully used root strains to compute friction force under dynamic
conditions. Recently, Vaishya and Houser [1.9] have shown that quasi-static
measurement of friction force is possible by using the technique of digital filtering to
eliminate the dynamic effects. However, no comprehensive work could be found which
critically evaluates the existing lubrication theories in the framework of an actual gear
mesh. Also, no prior work has incorporated the time-varying coefficient of friction into
MDOF gear dynamics or examined its effect.
Sliding friction at gear teeth also manifests as a noise source, as contended by the
dynamic tests by Houser et al. [1.16]. Borner and Houser [1.17] predicted the dynamic
forces due to friction and qualitatively discussed the radiated sound from the housing.
Most studies on gearbox system dynamics [1.2] have relied on a combination of detailed
finite element, boundary element and semi-analytical methods. Van Roosmalen [1.29]
formulated a gearbox model including analytical formulations for the vibration at the
gears due to tooth deflections and the vibration transfer through the bearings. Lim et al.
[1.30-1.33] developed a lumped parameter model with a rigid casing and a finite element
model with a flexible casing for a simple geared system. However, finite and boundary
element methods often require extensive computational time. Over the last four decades,
some simplified lumped parameter models have been developed though few have
incorporated the torsional and translational motions in both the line-of-action (LOA) and
OLOA directions. Steyer [1.34] examined a single mesh geared system with 6 DOF. By
7
assuming the housing mass is much larger than the gears and shafts, an impedance
mismatch was created with a rigid boundary condition at the bearing location. Thus, the
internal geared system was modeled separately and analytical expressions were presented
for a unity gear pair in terms of the resulting force transmissibility curve. Kartik [1.35]
developed a frequency-response based model to predict noise radiation from gearbox
housings with a multi-mesh gear set. His work showed that the bearing and mesh
stiffness significantly affects the sound pressure in the high frequency range while the
casing stiffness controls the response in the range below 4 kHz. However, the transfer
function relating the bearing forces to the equivalent force at the housing panel was based
on limited experiments. Overall, the above mentioned system models fall short of
providing a complete vibro-acoustic model.
1.3
Problem Formulation
1.3.1
Key Research Issues
Governing equations for gear dynamics should lead to a class of damped
inhomogeneous periodic differential equations [1.36-1.38] with multiple interacting
coefficients [1.13-1.20]. Although similar equations may also be found in a variety of
disciplines such as communication networks [1.36] and electrical circuits [1.39], the gear
friction problems, however, significantly differ from existing models such as the classical
Hill’s equations [1.36] in several ways. First, unlike classic friction problems in most
mechanical systems, the direction of gear friction is normal rather than in the direction of
8
nominal motions. Second, the frictional forces and moments emerge on both sides of the
governing equations as either excitations or periodically-varying parameters. Also, the
periodic damping should capture not only the kinematic effects but also the energy
dissipation due to sliding friction. Third, the periodic mesh stiffness is not confined to a
rectangular wave assumed by Manish and Singh [1.13-1.15], or a simple sinusoid as in
the Mathieu’s equation [1.36]. Instead, they should describe realistic, yet continuous,
profiles of Figure 1.2 resulting from a detailed finite element/contact mechanics analysis
[1.40]. Lastly, the stiffness and viscous damping terms incorporate combined (but phase
correlated) contributions from all (yet changing) tooth pairs in contact.
Historically, such periodic differential equations are seldom investigated and
limited prior research efforts, as reported in the literature review [1.1-1.20], are based
largely on numerical integration and the Fast Fourier Transform algorithm. Consequently,
there is a clear need for closed form analytical (say by using the Floquet theory) and
semi-analytical (say by using the multi-term harmonic balance method) solutions to the
dynamic responses of spur and helical gear pairs under the influence of sliding friction.
Recently, Velex and Ajmi [1.41] implemented a harmonic analysis to approximate the
dynamic factors in helical gears (based on tooth loads and quasi-static transmission
errors). Their work, however, does not describe the multi-dimensional system dynamics
or include the frictional effect, which may lead to “multiplicative” terms as described
earlier. The parametric friction force excitation may have an influence on the stability of
the homogenous system.
Further, for a satisfactory understanding of dynamic behaviors of gears, a higher
number of degrees-of-freedom are required for analysis, such as the MDOF helical gear
9
model of Figure 1.3. This is essential for additional phenomena like friction force,
torsional-flexural coupling, shaft wobble and axial shuttling, which are yet to be fully
understood. Also, to represent practical geared systems, a generalized model is required
that incorporates different gear design configurations, lubrication conditions and meshing
parameters. Existing solution methodology [1.29-1.35] has to be improved to compute
the dynamic response of the entire gearbox, for a combined excitation of transmission
error, sliding friction, mean torque and other sources. Subsequently, the relative
contribution of various parameters and the resulting noise characteristics need to be
understood. This requires an improved source-path-receiver model for the entire gearbox
system that incorporates competing noise sources.
ε (t )
Figure 1.3 MDOF helical gear dynamic model (proposed in Chapter 6) and its contact
mechanics with sliding friction.
10
1.3.2
Scope, Assumptions and Objectives
Chief goal of this research is to improve the earlier work by Vaishya and Singh
[1.13-1.15] by developing improved mathematical models and proposing new analytical
solutions that will enhance our understanding of the influence of friction on gear
dynamics and vibro-acoustic behavior. Many dynamic phenomena that emerge due to
interactions between parametric variations (time-varying mesh stiffness and viscous
damping) and sliding friction will be predicted, along with a better understanding of the
relative contributions of transmission error versus sliding friction noise to the gear whine
noise. The specific objectives of this study are therefore as follows:
ƒ
Extend Vaishya and Singh’s work [1.13-1.15] by developing improved MDOF
dynamic system for a spur gear pair that incorporates realistic time-varying mesh
stiffness functions, accurate representations of sliding friction and load sharing
between meshing tooth pairs. (Chapter 2)
ƒ
Comparatively evaluate alternate sliding friction models [1.21-1.28] and predict
the interfacial friction forces and motions in the OLOA direction. Also, validate
dynamic system models and analytical solutions by comparing predictions to
numerical solutions, the benchmark finite element/contact mechanics code as well
as measurements. (Chapters 2 and 3)
ƒ
Propose a semi-analytical algorithm based on both single- and multi-term
harmonic balance methods to quickly construct frequency responses of multidimensional spur gear dynamics with sliding friction. This should provide new
insights into the dynamic interactions between parametric excitations. (Chapter 4)
11
ƒ
Propose a refined source-path-receiver model that characterizes the structural
paths in two directions and develop analytical tools to efficiently predict the
whine noise radiated from gearbox panels and quantify the contribution of sliding
friction to the overall whine noise. Analytical predictions of the structural transfer
function and noise radiation will be compared with measurements. (Chapter 5)
ƒ
Propose a new three-dimensional formulation for helical gears to characterize the
dynamics associated with the contact plane including the reversal at the pitch line
due to sliding friction. A 12 DOF model will be developed which includes the
rotational and translational motions along the LOA, OLOA and axial directions as
well as the bearing/shaft compliances. (Chapter 6)
ƒ
Develop improved closed form solutions for the linear time-varying helical gear
system in terms of the dynamic transmission error under the effect of sliding
friction by using the Floquet theory. (Chapter 7)
Scope and assumptions include the following: For the internal spur and helical
gear pair sub-systems, the pinion and gear are modeled as rigid disks. The elastic
deformations of the shaft and bearings are modeled using lumped elements which are
connected to a rigid casing. Also, vibratory angular motions are small in comparison to
the mean motion, and the mean load is assumed to be high such that the dynamic load is
not sufficient to cause tooth separations [1.42]. If these assumptions are not made, the
system model would have implicit non-linearities. Consequently, the position of the line
of contact and relative sliding velocity depend only on the nominal angular motions; this
leads to a linear time-varying system formulation. Note that different mesh stiffness
schemes are assumed for spur and helical gears: For the spur gear analysis, the realistic
12
and continuous mesh stiffness is considered based on an accurate finite element/contact
mechanics analysis code [1.40]; Thus, the time-varying stiffness is indeed an effective
function which may also include the effect of profile modifications. For the helical gear
analysis, however, only those gears with perfect involute profiles are considered and the
mesh stiffness per unit length along the contact line (or stiffness density) is assumed to be
constant [1.19]. This is equivalent to the equal load sharing assumption by Vishya and
Singh [1.13-1.15]. Such limitation may be further examined in future work.
For the structure-borne whine noise model of the gearbox system, a source-pathreceiver model of Figure 1.4 is used. All the assumptions as mentioned above are
embedded in the modeling of the internal gear pair sub-system. The unloaded static
transmission error and sliding friction are considered as the two main excitations to the
system; these are assumed to be most dominant in the LOA and OLOA directions,
respectively. Hence, only corresponding structural paths in these two directions are
considered by neglecting the moment transfer in the bearing matrices. Also, by assuming
the housing mass is much larger than the gears and shafts, an impedance mismatch is
created with a rigid boundary condition at the bearing location. Thus, the internal geared
system could be modeled separately and its resulting force response provides force
excitations to the structural paths. Finally, for the NASA gearbox used as the case study,
the box plate is assumed to be the main radiator due to its relatively high mobility as well
as the way the gearbox was assembled.
Finally, it is worthwhile to mention that all chapters of this thesis are written in a
self-contained manner in terms of formulation, literature review, methods and results.
13
SOURCE
Transmission
error
Sliding
friction
LOA bearing forces
6 DOF linear-timevarying spur gear
pair model + shafts
OLOA bearing forces
RECEIVER
Sound
pressure
Coupling at
bearings
PATH
Radiation
model
Housing
velocity
Housing structure
model
Figure 1.4 Block diagram for the vibro-acoustics of a simplified geared system with two
excitations at the gear mesh (as proposed in Chapter 5).
References for Chapter 1
[1.1] Ozguven, H. N., and Houser, D. R., 1988, “Mathematical Models Used in Gear
Dynamics - a Review,” Journal of Sound and Vibration, 121, pp. 383-411.
[1.2] Lim, T. C., and Singh, R., 1989, “A Review of Gear Housing Dynamics and
Acoustic Literature,” NASA-Technical Memorandum, 89-C-009.
[1.3] Oswald, F. B., Seybert, A. F., Wu, T. W., and Atherton, W., 1992, “Comparison of
Analysis and Experiment for Gearbox Noise,” Proceedings of the International Power
Transmission and Gearing Conference, Phoenix, pp. 675-679.
[1.4] Baud, S., and Velex, P., 2002, “Static and Dynamic Tooth Loading in Spur and
Helical Geared Systems-Experiments and Model Validation,” American Society of
Mechanical Engineers, 124, pp. 334-346.
[1.5] Comparin R. J., and Singh, R., 1990, “An Analytical Study of Automotive Neutral
Gear Rattle,” ASME Journal of Mechanical Design, 112, pp. 237-245.
[1.6] Mark, W. D., 1978, “Analysis of the Vibratory Excitation of Gear Systems: Basic
Theory,” Journal of Acoustical Society of America, 63(5), pp. 1409-1430.
14
[1.7] Munro, R. G., 1990, “Optimum Profile Relief and Transmission Error in Spur
Gears,” Proceedings of IMechE, Cambridge, England, 9-11 Apr., pp. 35-42.
[1.8] Rebbechi, B. and Oswald, F. B., 1991, “Dynamic Measurements of Gear Tooth
Friction and Load,” NASA-Technical Memorandum, 103281.
[1.9] Vaishya, M., and Houser, D. R., 1999, “Modeling and Measurement of Sliding
Friction for Gear Analysis,” American Gear Manufacturer Association Technical Paper,
99FTMS1, pp. 1-12.
[1.10] Schachinger, T., 2004, “The Effects of Isolated Transmission Error, Force
Shuttling, and Frictional Excitations on Gear Noise and Vibration,” MS Thesis, The Ohio
State University.
[1.11] Blankenship, G. W., and Singh, R., 1995, “A New Gear Mesh Interface Dynamic
Model to Predict Multi-Dimensional Force Coupling and Excitation,” Mechanism and
Machine Theory Journal, 30(1), pp. 43-57.
[1.12] Padmanabhan, C., and Singh, R., 1995, “Analysis of Periodically Excited NonLinear Systems by a Parametric Continuation Technique,” Journal of Sound and
Vibration, 184(1), pp. 35-58.
[1.13] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration,
243(3), pp. 525-545.
[1.14] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and
Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671694.
[1.15] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear
Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.
[1.16] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of
Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.
[1.17] Borner, J., and Houser, D. R., 1996, “Friction and Bending Moments as Gear
Noise Excitations,” SAE Paper # 961816.
[1.18] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on
the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of
Mechanical Design, 122(4), pp. 515-522.
15
[1.19] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction
Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp. 641658.
[1.20] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body
Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3),
pp. 479-499.
[1.21] Benedict, G. H., and Kelley B. W., 1961, “Instantaneous Coefficients of Gear
Tooth Friction,” Transactions of the American Society of Lubrication Engineers, 4, pp.
59-70.
[1.22] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, “Prediction of
Mechanical Efficiency of Parallel-Axis Gear Pairs,” ASME Journal of Mechanical
Design, 129 (1), pp. 58-68.
[1.23] Xu, H., 2005, “Development of a Generalized Mechanical Efficiency Prediction
Methodology,” PhD dissertation, The Ohio State University.
[1.24] Seireg, A. A., 1998, Friction and Lubrication in Mechanical Design, Marcel
Dekker, Inc., New York.
[1.25] Baranov, V. M., Kudryavtsev, E. M., and Sarychev, G. A., 1997, “Modeling of the
Parameters of Acoustic Emission under Sliding Friction of Solids,” Wear, 202, pp. 125133.
[1.26] Drozdov, Y. N., and Gavrikov, Y. A., 1968, “Friction and Scoring under the
Conditions of Simultaneous Rolling and Sliding of Bodies,” Wear, 11, pp. 291-302.
[1.27] Duan, C., and Singh, R., 2005, “Super-Harmonics in a Torsional System with Dry
Friction Path Subject to Harmonic Excitation under a Mean Torque”, Journal of Sound
and Vibration, 285(2005), pp. 803-834.
[1.28] Hamrock, B. J., and Dowson, D., 1977, “Isothermal Elastohydrodynamic
Lubrication of Point Contacts, Part III – Fully Flooded Results,” Journal of Lubrication
Technology, 99(2), pp. 264-276.
[1.29] Van Roosmalen, A., 1994, “Design Tools for Low Noise Gear Transmissions,”
PhD Dissertation, Eindhoven University of Technology.
[1.30] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling
Element Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and
Vibration, 139(2), pp. 179-199.
16
[1.31] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling
Element Bearings. Part II: System Studies,” Journal of Sound and Vibration, 139(2), pp.
201-225.
[1.32] Lim, T. C., and Singh R., 1991, “Statistical Energy Analysis of a Gearbox with
Emphasis on the Bearing Path,” Noise Control Engineering Journal, 37(2), pp. 63-69.
[1.33] Lim, T. C., and Singh, R., 1991, “Vibration Transmission Through Rolling
Element Bearings. Part III: Geared Rotor System Studies,” Journal of Sound and
Vibration, 151(1), pp. 31-54.
[1.34] Steyer, G., 1987, “Influence of Gear Train Dynamics on Gear Noise,” NOISECON 87 proceedings, pp. 53-58.
[1.35] Kartik, V., 2003, “Analytical Prediction of Load Distribution and Transmission
Error for Multiple-Mesh Gear-Trains and Dynamic Studies in Gear Noise and Vibration,”
MS Thesis, The Ohio State University.
[1.36] Richards, J. A., 1983, Analysis of Periodically Time-varying Systems, New York,
Springer.
[1.37] Jordan, D. W., and Smith, P., 2004, Nonlinear Ordinary Differential Equations, 3rd
Edition, Oxford University Press.
[1.38] Thomsen, J. J., 2003, Vibrations and Stability, 2nd Edition, Springer.
[1.39] Kenneth S. K., Jacob K. W. and Alberto S-V, 1990, “Steady-State Methods for
Simulating Analog and Microwave Circuits,” Kluwer Academic Publishers, Boston.
[1.40] External2D (CALYX software),
www.ansol.www, ANSOL Inc., Hilliard, OH.
2003,
“Helical3D
User’s
Manual,”
[1.41] Velex, P., and Ajmi, M., 2007, “Dynamic Tooth Loads and Quasi-Static
Transmission Errors in Helical Gears – Approximate Dynamic Factor Formulae,”
Mechanism and Machine Theory Journal, 42(11), pp. 1512-1526.
[1.42] Blankenship, G. W., and Kahraman, A., 1995, “Steady State Forced Response of a
Mechanical Oscillator with Combined Parametric Excitation and Clearance Type Nonlinearity,” Journal of Sound and Vibration, 185(5), pp. 743-765.
17
CHAPTER 2
SPUR GEAR DYNAMICS WITH SLIDING FRICTION AND REALISTIC MESH
STIFFNESS
2.1
Introduction
In a series of recent articles, Vaishya and Singh [2.1-2.3] developed a spur gear
pair model with periodic tooth stiffness variations and sliding friction based on the
assumption that load is equally shared among all the teeth in contact. Using the simplified
rectangular pulse shaped variation in mesh stiffness, they solved the single-degree-offreedom (SDOF) system equations in terms of the dynamic transmission error (DTE)
using the Floquet theory and the harmonic balance method [2.1-2.3]. While the
assumption of equal load sharing yields simplified expressions and analytically tractable
solutions, it may not lead to a realistic model. This chapter aims to overcome this
deficiency by employing realistic time-varying tooth stiffness functions and the sliding
friction over a range of operational conditions. New linear time-varying (LTV)
formulation will be extended to include multi-degree-of-freedom (MDOF) system
dynamics for a spur gear pair.
18
Vaishya and Singh [2.1-2.3] have already provided an extensive review of prior
work. In addition, Houser et al. [2.4] experimentally demonstrated that the friction forces
play a pivotal role in determining the load transmitted to the bearings and housing in the
off-line-of-action (OLOA) direction; this effect is more pronounced at higher torque and
lower speed conditions. Velex and Cahouet [2.5] described an iterative procedure to
evaluate the effects of sliding friction, tooth shape deviations and time-varying mesh
stiffness in spur and helical gears and compared simulated bearing forces with
measurements. They reported significant oscillatory bearing forces at lower speeds that
are induced by the reversal of friction excitation with alternating tooth sliding direction.
In a subsequent study, Velex and Sainsot [2.6] analytically found that the Coulomb
friction should be viewed as a non-negligible excitation source to error-less spur and
helical gear pairs, especially for translational vibrations and in the case of high contact
ratio gears. However, their work was confined to a study of excitations and the effects of
tooth modifications were not considered. Lundvall et al. [2.7] considered profile
modifications and manufacturing errors in a MDOF spur gear model and examined the
effect of sliding friction on the angular dynamic motions. By utilizing a numerical
method, they reported that the profile modification has less influence on the dynamic
transmission error when frictional effects are included. Nevertheless, two key questions
remain unresolved: How to concurrently incorporate the time-varying sliding friction and
the realistic mesh stiffness functions into an analytical (MDOF) formulation? How to
quantify dynamic interactions between sliding friction and mesh stiffness terms
especially when tip relief is provided to the gears? This chapter will address these issues.
19
2.2
Problem Formulation
2.2.1
Objectives and Assumptions
Chief objective of this chapter is to propose a new method of incorporating the
sliding friction and realistic time-varying stiffness into an analytical MDOF spur gear
model and to evaluate their interactions. Key assumptions are: (i) pinion and gear are
modeled as rigid disks; (ii) shaft-bearings stiffness in the line-of-action (LOA) and
OLOA directions are modeled as lumped elements which are connected to a rigid casing;
(iii) vibratory angular motions are small in comparison to the mean motion; and (iv)
Coulomb friction is assumed with a constant coefficient of friction µ . If assumption (iii)
is not made, the system model would have implicit non-linearities. Consequently, the
position of the line of contact and relative sliding velocity depend only on the nominal
angular motions.
An accurate finite element/contact mechanics (FE/CM) analysis code [2.8] will be
employed, in the “static” mode, to compute the mesh stiffness at every time instant under
a range of loading conditions. Here, the time-varying stiffness is calculated as an
effective function which may also include the effect of profile modifications. The
realistic mesh stiffness is then incorporated into the LTV spur gear model with the
contributions of sliding friction. The MDOF formulation should describe both the LOA
and OLOA dynamics; a simplified SDOF model will also be derived that describes the
vibratory motion in the torsional direction. Proposed methods will be illustrated via two
spur gear examples (designated as I and II) whose parameters are listed in Table 2.1 and
Table 2.2. The MDOF model of Example I will be validated by using the FE/CM code
20
[2.8] in the “dynamic” mode. Issues related to tip relief will be examined in Example II in
the presence of sliding friction. Finally, experimental results of Example II will be used
to further validate our method.
Parameter/property
Pinion
Gear
Number of teeth
25
31
Diametral pitch, in
8
8
Pressure angle, deg
25
25
Outside diameter, in
3.372
4.117
Root diameter, in
2.811
3.561
Face width, in
1.250
1.250
Tooth thickness, in
0.196
0.196
Gear mass, lb⋅s ⋅in
6.72E-03
1.04E-02
Polar moment of inertia, lb⋅s2⋅in
8.48E-03
2.00E-02
-1
2
-1
Bearing stiffness (LOA and OLOA), lb/in
20E6
Center distance, in
3.5
Profile contact ratio
1.43
Elastic modulus, psi
30E6
Density, lb⋅s-2⋅in-4
7.30E-04
Poisson’s ratio
0.3
Table 2.1 Parameters of Example I: NASA-ART spur gear pair (non-unity ratio)
21
Parameter/property
Pinion/Gear
Number of teeth
28
Diametral pitch, in-1
8
Pressure angle, deg
20
Outside diameter, in
3.738
Root diameter, in
3.139
Face width, in
0.25
Tooth thickness, in
0.191
Roll angle where the tip modification
starts (for II-B), deg
24.5
Straight tip modification (for II-B), in
7E-04
Center distance, in
3.5
Profile contact ratio
1.63
Elastic modulus, psi
30E6
Density, lb⋅s-2⋅in-4
7.30E-04
Poisson’s ratio
0.3
Range of temperatures, °F
104, 122, 140, 158, 176
Range of input torques, lb⋅in
500, 600, 700, 800, 900
Table 2.2 Parameters of Example II-A and II-B: NASA spur gear pair (unity ratio). Gear
pair with the perfect involute profile is designated as II-A case and the one with tip relief
is designated as II-B case
22
2.2.2
Timing of Key Meshing Events
Analytical formulations for a spur gear pair are derived via Example I (NASA-
ART spur gear pair) with parameters of Table 2.1. For a generic spur gear pair with noninteger contact ratio σ , n = ceil( σ ) meshing tooth pairs need to be considered, where the
“ceil” function rounds the σ element to the nearest integer towards a higher value.
Consequently, two meshing tooth pairs need to be modeled for Example I ( σ = 1.43).
First, transitions in key meshing events within a mesh cycle need to be determined
from the undeformed gear geometry for the construction of the stiffness function. Figure
2.1 is a snapshot for Example I at the beginning of the mesh cycle (t = 0). At that time,
pair #1 (defined as the tooth pair rolling along line AC) just comes into mesh at point A
and pair #0 (defined as the tooth pair rolling along line CD) is in contact at point C,
which is the highest point of single tooth contact (HPSTC). As the gears roll, when pair
#1 approaches the lowest point of single tooth contact (LPSTC) of point B at t = tB, pair
#0 leaves contact. At t = t P , pair #1 passes through the pitch point P, and the relative
sliding velocity of the pinion with respect to the gear is reversed, resulting in a reversal of
the friction force. This should provide an impulse excitation to the system. Finally, pair
#1 goes through point C at t = tc, completing one mesh cycle (tc). These key events are
defined below, where Ω p is the nominal pinion speed, rbp is the base radius of the pinion,
length LAC is equal to one base pitch λ.
tc =
λ
Ω p rbp
,
tp
tb LAB
,
=
tc
λ
tc
23
=
LAP
λ
.
(2.1)
Ωg
Ωp
Figure 2.1 Snap shot of contact pattern (at t = 0) in the spur gear pair of Example I.
2.2.3
Calculation of Realistic Time-Varying Tooth Stiffness Functions
The realistic time-varying stiffness functions are calculated using a FE/CM code,
External2D [2.8]. An input torque Tp is applied to the pinion rotating at Ω p , and the
mean braking torque Tg on the gear and its angular velocity Ω g obey the basic gear
kinematics. Superposed on the nominal motions are oscillatory components denoted as
θ p and θ g for the pinion and gear, respectively. The normal contact forces N0(t), N1(t)
and pinion deflection θ p (t ) are then computed by performing a static analysis using
FE/CM software [2.8]. The stiffness function of the ith meshing tooth pair for a generic
24
spur gear pair is given by Eq. (2.2), where the “floor” function rounds the contact ratio σ
to the nearest integer towards a lower value, i.e. floor(σ ) = 1 for Example I.
ki (t ) =
Ni (t )
, i = 0, 1, ... , n = floor(σ ).
rbpθ p (t )
(2.2)
The stiffness function k(t) for a single tooth pair rolling through the entire
meshing process is obtained by following the contact tooth pair for n = ceil( σ ) number
of mesh cycles. Due to the periodicity of the system, expanded stiffness function ki (t ) of
the ith meshing tooth pair is calculated at any time instant t as:
ki (t ) = k [ (n − i)tc + mod(t , tc )] , i = 0, 1, ... , n = floor(σ ).
(2.3)
Here, “mod” is the modulus function defined as:
mod( x, y ) = x − y ⋅ floor( x / y ), if y ≠ 0.
(2.4)
For Example I, calculated k0 (t ) , k1 (t ) functions and their combined stiffness are
shown in Figure 2.2(a). Note that k0 (t ) and k1 (t ) are, in fact, different portions of k (t ) as
described in Eq. (2.3). Figure 2.2(b) compares the continuous k (t ) of the realistic load
sharing model against the rectangular pulse shaped discontinuous k (t ) based on the equal
load sharing formulation proposed earlier by Vaishya and Singh [2.1-2.3].
25
Figure 2.2 Tooth mesh stiffness functions of Example I calculated by using the FE/CM
code (in the “static” mode). (a) Individual and combined stiffness functions. Key:
,
total stiffness;
stiffness of pair #0;
, stiffness of pair #1. (b) Comparison of the
combined stiffness functions. Key:
, realistic load sharing;
, equal load sharing
as assumed by Vaishya and Singh [2.1-2.3].
26
2.3
Analytical Multi-Degree-of-Freedom Dynamic Model
2.3.1
Shaft and Bearing Stiffness Models
Next, we develop a generic spur gear pair model with 6 DOFs including rotational
motions ( θ p and θ g ), LOA translations ( x p and xg ) and OLOA translations ( y p and yg ).
The governing equations are derived in the subsequent sections. First, a simplified shaft
model, as shown in Figure 2.3, is developed based on the Euler’s beam theory [2.9].
Corresponding to the 6 DOFs mentioned above, only the diagonal term in the shaft
stiffness matrix needs to be determined as follows, where E is the Young’s modulus,
I = π rs4 / 4 is the area moment of inertia for the shaft, and a and b are the distances from
pinion/gear to the bearings.
K Sx = K Sy = 3EI
a+b ⎡
2
a − b ) + ab ⎤ , K Sθ z = 0 .
3 3 ⎣(
⎦
ab
(2.5)
The rolling element bearings are modeled using the bearing stiffness matrix K Bm
formulation (of dimension 6) as proposed by Lim and Singh [2.10, 2.11]. Assume that
each shaft is supported by two identical axially pre-loaded ball bearings with a mean
axial displacement; the mean driving load Tm generates a mean radial force Fxm in the
LOA direction and a moment M ym around the OLOA direction. The time-varying
friction force and torque are not included in the mean loads.
27
K Bx / 2
a
K By / 2
x
b
θz
y
K Bx / 2
K By / 2
z
Figure 2.3 Schematic of the bearing-shaft model.
Corresponding to the 6 DOFs considered in our spur gear model, only two
significant coefficients, K Bxx and K Byy , are considered for K Bm [2.10, 2.11]. The
combined bearing-shaft stiffness ( K Bx and K By in the LOA and OLOA directions) are
derived by assuming that the bearing and shaft stiffness elements act in parallel.
2.3.2
Dynamic Mesh and Friction Forces
Figure 2.4 shows the mean torque and internal reaction forces acting on the pinion
for Example I. For the sake of clarity, forces on the gear are not shown, which are equal
in magnitude but opposite in direction to the pinion forces. Based on the Coulomb
28
friction law, the magnitude of friction force ( F f ) is proportional to the nominal tooth
load (N) as F f = µ N where µ is constant. The direction of F f is determined by the
calculation of nominal relative sliding velocity, which results in the LTV system
formulation. Denote X pi (t ) as the moment arm on the pinion for the friction force acting
on the ith meshing tooth pair
X pi (t ) = LXA + ( n − i )λ + mod(Ω p rbp t , λ ), i = 0, 1, ... , n = floor(σ ).
(2.6)
The corresponding moment arm for the friction force on the gear is
X gi (t ) = LYC + iλ − mod(Ω g rbg t , λ ), i = 0, 1, ... , n = floor(σ ).
(2.7)
Assume time-varying mesh (viscous) damping coefficient and relate it to ki (t ) by
a time-invariant damping ratio ζ m as follows, where J e = J p J g / ( J p rbg2 + J g rbp2 )
ci (t ) = 2ζ mi ki (t ) ⋅ J e ,
i = 0, 1, ... , n = floor(σ ).
(2.8)
The normal forces acting on the pinion are
N pi (t ) = N gi (t ) = ki (t ) ⎡⎣ rbpθ p (t ) − rbgθ g (t ) − ε p (t ) + x p (t ) − xg (t ) ⎤⎦ +
ci (t ) ⎡⎣ rbpθp (t ) − rbgθg (t ) − ε p (t ) + x p (t ) − x g (t ) ⎤⎦ , i = 0, 1, ... , n = floor(σ ).
29
(2.9)
Here ε P (t ) is the profile error component of the static transmission error (STE), and xp(t)
and xg(t) denote the translational bearing displacements of pinion and gear, respectively.
For a generic spur gear pair whose jth meshing pair passes through the pitch point within
the mesh cycle, the friction forces in the ith meshing pair are derived as follows
⎧ µ N pi (t ),
i = 0, 1, ... , j − 1,
⎪⎪
Fpfi (t ) = ⎨ µ N pi (t ) sgn ⎡⎣ mod(Ω p rbp t , λ ) + (n − i )λ − LAP ⎤⎦ , i = j ,
⎪
i = j , j + 1, ... , n = floor(σ ),
⎪⎩− µ N pi (t ),
(2.10a)
⎧ µ N gi (t ),
i = 0, 1, ... , j − 1,
⎪⎪
Fgfi (t ) = ⎨ µ N gi (t )sgn ⎣⎡ mod(Ω g rbg t , λ ) + (n − i )λ − LAP ⎦⎤ , i = j ,
⎪
i = j , j + 1, ... , n = floor(σ ).
⎪⎩− µ N gi (t ),
(2.10b)
Consequently, the friction forces for Example I of Figure 2.4 are given as:
Fpf 0 (t ) = µ N p 0 (t ) ,
(2.10c)
Fpf 1 (t ) = µ N p1 (t ) sgn ⎡⎣ mod(Ω p rbp t , λ ) − LAP ⎤⎦ ,
(2.10d)
Fgf 0 (t ) = µ N g 0 (t ) ,
(2.10e)
Fgf 1 (t ) = µ N g1 (t ) sgn ⎡⎣ mod(Ω g rbg t , λ ) − LAP ⎤⎦ .
(2.10f)
30
Figure 2.4 Normal and friction forces of analytical (MDOF) spur gear system model.
31
2.3.3
MDOF Model
The governing equations for the torsional DOFs are
J pθp (t ) = Tp +
n = floor(σ )
J gθg (t ) = −Tg +
∑
i =0
X pi (t ) Fpfi (t ) −
n = floor(σ )
∑
i =0
n = floor(σ )
X gi (t ) Fgfi (t ) +
∑
rbp N pi (t ) ,
i =0
(2.11)
n = floor(σ )
∑
rbg N gi (t )
i =0
(2.12)
.
The governing equations of the translational DOFs in the LOA direction are
m p x p (t ) + 2ζ pBx K pBx m p x p (t ) + K pBx xg (t ) +
mg xg (t ) + 2ζ gBx K gBx mg x g (t ) + K gBx xg (t ) +
n = floor(σ )
∑
N pi (t ) = 0 ,
(2.13)
N gi (t ) = 0 .
(2.14)
i =0
n = floor(σ )
∑
i =0
Here, K pBx and K gBx are the effective shaft-bearing stiffness in the LOA direction, and
ζ pBx and ζ gBx are their damping ratios. Similarly, the governing equations of the
translational DOFs in the OLOA direction are
m p y p (t ) + 2ζ pB y K pBy m p y p (t ) + K pBy y p (t ) −
mg y g (t ) + 2ζ gB y K gBy mg y g (t ) + K gBy yg (t ) −
32
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
Fpfi (t ) = 0 ,
(2.15)
Fgfi (t ) = 0 .
(2.16)
The composite DTE, which is the relative dynamic displacement of pinion and
gear along the LOA direction, is defined as
δ (t ) = rbpθ p (t ) − rbgθ g (t ) + x p (t ) − xg (t ) .
(2.17)
Finally, the dynamic bearing forces are as:
2.4
FpBx (t ) = − K pBx x p (t ) − 2ζ p Bx K pBx m p x p (t ) ,
(2.18a)
FpBy (t ) = − K pBy y p (t ) − 2ζ p Bb K pBy m p y p (t ) ,
(2.18b)
FgBx (t ) = − K gBx xg (t ) − 2ζ gBb K gBx mg x g (t ) ,
(2.18c)
FgBy (t ) = − K gBy y g (t ) − 2ζ g Bb K gBy mg y g (t ) .
(2.18d)
Analytical SDOF Torsional Model
When only the torsional DOFs of the spur gear pair are of interest, a simplified
but equivalent SDOF model can be derived by assuming that the shaft-bearings stiffness
is much higher than the mesh stiffness. After eliminating θp(t) and θg(t) in terms of the
DTE, δ (t ) = rbpθ p (t ) − rbgθ g (t ) , the governing SDOF model is obtained for a generic spur
gear pair whose jth meshing pair passes through the pitch point within the mesh cycle:
33
J eδ(t ) +
µ
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
⎡⎣ci (t )δ (t ) + ki (t )δ (t ) ⎤⎦ +
⎧sgn ⎡ mod(Ω p rbp t , λ ) + (n − j )λ − LAP ⎤ ⋅
⎫
⎣
⎦
⎪⎪
⎪⎪
⎡ X pj (t ) J g rbp + X gj (t ) J p rbg ⎤ ⎬ =
⎨
⎥⎪
⎪ ⎡⎣ci (t )δ(t ) + ki (t )δ (t ) ⎤⎦ ⋅ ⎢
J p rbg2 + J g rbp2
⎢⎣
⎥⎦ ⎭⎪
⎩⎪
n = floor(σ )
Te
⎡⎣ci (t )ε p (t ) + ki (t )ε p (t ) ⎤⎦ +
+
∑
J p rbg2 + J g rbp2
i =0
µ
n = floor(σ )
∑
i =0
.
(2.19)
⎧sgn ⎡ mod(Ω p rbp t , λ ) + (n − j )λ − LAP ⎤ ⋅
⎫
⎣
⎦
⎪⎪
⎪⎪
⎡ X pj (t ) J g rbp + X gj (t ) J p rbg ⎤ ⎬
⎨
⎥⎪
⎪ ⎡⎣ci (t )ε p (t ) + ki (t )ε p (t ) ⎤⎦ ⋅ ⎢
J p rbg2 + J g rbp2
⎣⎢
⎦⎥ ⎭⎪
⎩⎪
Here the effective polar moment of inertia J e is consistent with that defined in Eq. (2.8)
and the effective torque is Te = Tp ⋅ J g ⋅ rbp + Tg ⋅ J p ⋅ rbg . The dynamic response δ (t ) is
controlled by three excitations: (i) time-varying Te , (ii) ε p (t ) and its derivative ε p (t ) and
(iii) sliding friction. For Example I, the governing Eq. (2.19) could be simplified as
J eδ(t ) + [ c1 (t ) + c0 (t )] δ(t ) + [ k1 (t ) + k0 (t )]δ (t ) +
⎡ X p1 (t ) J g rbp + X g1 (t ) J p rbg ⎤
⎥ ⋅ sgn ⎡⎣ mod(Ω p rbpt , λ ) − LAP ⎤⎦ +
J p rbg2 + J g rbp2
⎣⎢
⎦⎥
µ ⎡⎣c1 (t )δ(t ) + k1 (t )δ (t ) ⎤⎦ ⎢
µ ⎡⎣c0 (t )δ(t ) + k0 (t )δ (t ) ⎤⎦
=
(X
p0
(t ) J g rbp + X g 0 (t ) J p rbg )
J p rbg2 + J g rbp2
Te
+ [ c1 (t ) + c0 (t )] ε p (t ) + [ k1 (t ) + k0 (t )] ε p (t ) +
2
J p rbg + J g rbp2
⎡ X p1 (t ) J g rbp + X g1 (t ) J p rbg ⎤
⎥ ⋅ sgn ⎡⎣ mod(Ω p rbpt , λ ) − LAP ⎤⎦ +
J p rbg2 + J g rbp2
⎢⎣
⎥⎦
µ ⎡⎣c1 (t )ε p (t ) + k1 (t )ε p (t ) ⎤⎦ ⎢
µ ⎡⎣c0 (t )ε p (t ) + k0 (t )ε p (t ) ⎤⎦
34
(X
p0
(t ) J g rbp + X g 0 (t ) J p rbg )
J p rbg2 + J g rbp2
(2.20)
2.5
Effect of Sliding Friction in Example I
2.5.1
Validation of Example I Model using the FE/CM Code
The governing equations of either SDOF or MDOF system models are
numerically integrated by using a 4th-5th order Runge-Kutta algorithm with fixed time
step. The ε p (t ) and ε p (t ) components are neglected, i.e. no manufacturing errors other
than specified profile modifications are considered. Concurrently, the dynamic responses
are independently calculated by running the FE/CM code [2.8] using the Newmark
method. Predicted and computed results are compared with good correlations in terms of
the DTE, and LOA and OLOA forces, as shown in Figure 2.5 to Figure 2.7. Note that
time domain comparisons include both transient and steady state responses but the
frequency domain results report only the steady state responses. Figure 2.5 shows that the
sliding friction introduces additional DTE oscillations when the contact teeth pass
through the pitch point. Figure 2.6 illustrates that the sliding friction enhances the
dynamic bearing forces in the LOA direction, especially at the second mesh harmonic.
This is because the moments associated with Fpfi (t ) and Fgfi (t ) are coupled with the
moments of Npi(t) and Ngi(t).
35
(f) (in)
(t) (in)
(f) (in)
(t) (in)
Figure 2.5 Validation of the analytical (MDOF) model by using the FE/CM code (in the
“dynamic” mode). Here, results for Example I are given in terms of δ (t ) and its spectral
contents ∆ ( f ) with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ = 0 and
(c-d) are for µ = 0.2. Key:
, Analytical (MDOF) model;
, FE/CM code (in t
domain); o, FE/CM code (in f domain).
36
x 103
x 103
0.2
3
2
0.1
1
0
0
0
0.5
(a)
1
1.5
2
2
(b)
Normalized time t / t c
x
0
2.5
4
6
8
10
8
10
Mesh order n
103
x 103
0.2
3
2
0.1
1
0
0
(c)
0.5
1
1.5
Normalized time t / t c
2
0
2.5
0
(d)
2
4
6
Mesh order n
Figure 2.6 Validation of the analytical (MDOF) model by using the FE/CM code (in the
“dynamic” mode). Here, results for Example I are given in terms of FpBx (t ) and its
spectral contents FpBx ( f ) with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ
, Analytical (MDOF) model;
, FE/CM code
= 0 and (c-d) are for µ = 0.2. Key:
(in t domain); o, FE/CM code (in f domain).
37
Further, the normal loads mainly excite the vibration in the LOA direction, as
illustrated by Eqs. (2.9), (2.11) and (2.13). The scales of the bearing forces of Figure
2.7(a-b) for µ = 0 case are the same as those of Figure 2.7(c-d) for the sake of comparison.
The bearing forces predicted by the MDOF model for µ = 0 case approach zero (within
the numerical error range). This is consistent with the mathematical description of Eqs.
(2.15-2.16). Larger deviations at this point are observed in Figure 2.7(a-b) for the FE/CM
analysis. Figure 2.7shows that the OLOA dynamics are more significantly influenced by
the sliding friction when compared with the LOA results of Figure 2.6. In order to
accurately predict the higher mesh harmonics, refined time steps (say more than 100
increments per mesh cycle) are needed. Consequently, the FE/CM analysis tends to
generate an extremely large data file that demands significant computing time and postprocessing work. Meanwhile, the lumped model allows much finer time resolution while
being computationally more efficient (by at least two orders of magnitude when
compared with the FE/CM). Hence, the lumped model could be effectively used to
conduct parametric design studies.
38
x 103
x 103
0.2
0
F pBy (f) (lb)
F pBy (t) (lb)
5
-5
0.1
-10
0
0
0.5
(a)
1
1.5
2
0
2.5
2
(b)
Normalized time t / tc
4
6
8
10
8
10
Mesh order n
x 103
x 103
0.2
0
F pBy (f) (lb)
F pBy (t) (lb)
5
-5
0.1
-10
0
0
(c)
0.5
1
1.5
2
2.5
0
(d)
Normalized time t / t c
2
4
6
Mesh order n
Figure 2.7 Validation of the analytical (MDOF) model by using the FE/CM code (in the
“dynamic” mode). Here, results for Example I are given in terms of FpBy (t ) and its
spectral contents FpBy ( f ) with tc = 2.4 ms and fm = 416.7 Hz. Sub-figures (a-b) are for µ
, Analytical (MDOF) model;
, FE/CM code
= 0 and (c-d) are for µ = 0.2. Key:
(in t domain); o, FE/CM code (in f domain).
39
2.5.2
Effect of Sliding Friction
Figure 2.8 shows the calculated DTE without any friction is almost identical to
the STE at a very low speed ( Ω p = 2.4 rpm). However, the sliding friction changes the
shape of the DTE curve. During the time interval t ∈ [0, tP ] , the friction torque on the
pinion opposes the normal load torque as shown in Figure 2.4, resulting in a higher value
of the normal load that is needed to maintain the static equilibrium. Also, friction
increases the peak-peak value of the DTE as compared with the STE. For the remainder
of the mesh cycle t ∈ [tP , tc ] , friction torque acts in the same direction as the normal load
torque. Thus a small value of normal load is sufficient to maintain the static equilibrium.
Detailed parametric studies show that the amplitude of second mesh harmonic increases
with the effect of sliding friction.
40
Figure 2.8 Effect of µ on δ (t ) based on the linear time-varying SDOF model for
Example I at Tp = 2000 lb-in. Here, tc = 1 s. Key:
41
, µ = 0;
, µ = 0.1.
2.5.3
MDOF System Resonances
For Example I, the nominal bearing stiffness K pBx = K pBy = K gBx = K gBy = 20 ×106
lb/in are much higher than the averaged mesh stiffness km . The couplings between the
rotational and translational DOFs in the LOA direction are examined by using a
simplified 3 DOF model as suggested by Kahraman and Singh [2.12]. Note that the DTE
is defined here as δ = rbpθ xp − rbgθ xg , and the undamped equations of motion are
⎡ me
⎢
⎢0
⎢
⎣0
0
mp
0
⎡
0 ⎤ ⎧ δ ⎫ ⎢ km
⎥⎪ ⎪ ⎢
x p ⎬ + km
0 ⎥ ⎨ ⎢
mg ⎥⎦ ⎩⎪ xg ⎭⎪ ⎢ − k
⎣ m
(k
km
m
+ K pBx )
−km
⎤
⎥ ⎧ δ ⎫ ⎧0 ⎫
⎥ ⎪⎨ x p ⎪⎬ = ⎨⎪0⎬⎪ .
− km
⎥⎪ ⎪ ⎪ ⎪
( km + K gBx )⎦⎥ ⎩ xg ⎭ ⎩0⎭
− km
(2.21)
Here, the effective mass is defined as me = J p J g / ( rg2 J p + rp2 J g ) . The eigensolutions of
Eq. (2.21) yield three natural frequencies: Two coupled transverse-torsional modes ( f1
and f3 ) and one purely transverse mode ( f 2 ); numerical values are: f1 = 5,130 Hz, f 2 =
8,473 Hz and f3 = 11,780 Hz. Predictions of Eq. (2.21) match well with the numerical
simulations using the formulations of section 2.3 (though these results are not shown
here). A comparative study verifies that one natural frequency of the MDOF model shifts
away from that of the SDOF model (6,716 Hz) due to the torsional-translational coupling
effects. In the OLOA direction, simulation shows that only one resonance is present at
f pBy =
1
2π
K pBy / m p = 9,748 Hz, which is dictated by the bearing-shaft stiffness.
42
2.6
Effect of Sliding Friction in Example II
Next, the proposed model is applied to Example II with the parameters of Table
2.2. The chief goal is to examine the effects of tip relief and sliding friction. Further,
analogous experiments were conducted at the NASA Glenn Research Center Gear Noise
Rig [2.13]. Comparisons with measurements will be given in section 2.7.
2.6.1
Empirical Coefficient of Friction
The coefficient of friction varies as the gears travel through mesh, due to
constantly changing lubrication conditions between the contact teeth. An empirical
equation for the prediction of the dynamic friction variable, µ , under mixed lubrication
has been suggested by Benedict and Kelley [2.14] based on a curve-fit of friction
measurements on a roller test machine. Rebbechi et al. [2.15] verified this formulation by
measuring the dynamic friction forces on the teeth of a spur gear pair. Their
measurements seem to be in good agreement with the Benedict and Kelley equation
except at the meshing positions close to the pitch point. This empirical equation, when
modified to account for the average gear tooth surface roughness ( Ravg ), is
⎛ 3.17 ×108 X Γ (γ )Wn ⎞
44.5
.
⎟ , CRavg =
2
44.5 − Ravg
⎝ ν o Vs (γ )Ve (γ ) ⎠
µ (γ ) = 0.0127 CRavg log10 ⎜
43
(2.22a,b)
where C Ravg is the surface roughness constant, Wn is the normal load per unit length of
face width, and υo is the dynamic viscosity of the lubricant. Here Vs (γ ) is the sliding
velocity, defined as the difference in the tangential velocities of the pinion and gear, and
Ve (γ ) is the entraining velocity, defined as the addition of the tangential velocities, for
roll angle γ along the LOA. Further, Ravg in our case was measured with a profilometer
using a standard method [2.13]. Lastly, X Γ (γ ) is the load sharing factor as a function of
roll angle, and it was assumed based on the ideal profile of smooth meshing gears. Figure
2.9 shows µ as a function of roll angle calculated using Eq. (2.22). Since µ was
assumed to be a constant earlier, an averaged value is found by taking an average over
the roll angles between 19.8 and 21.8 degrees. Table 2.3 lists the µ values that were
computed at each mean torque and oil temperature for Example II (with Ravg = 0.132
µ m).
Torque (lb-in)
Temperature (°F)
500
600
700
800
900
104
0.032
0.033
0.034
0.035
0.036
122
0.034
0.036
0.037
0.037
0.038
140
0.036
0.037
0.038
0.039
0.040
158
0.038
0.040
0.041
0.041
0.042
176
0.040
0.041
0.042
0.043
0.044
Table 2.3 Averaged coefficient of friction µ predicted over a range of operating
conditions for Example II by using Benedict and Kelly’s empirical equation [2.14]
44
Figure 2.9 Coefficient of friction µ as a function of the roll angle for Example II, as
predicted by using Benedict & Kelley’s empirical equation [2.14]. Here, oil temperature
is 104 deg F and T = 500 lb-in. Key P: Pitch point at 20.85 deg.
2.6.2
Effect of Tip Relief on STE and k(t)
The STE is calculated as a function of mean torque for both the perfect involute
gear pair (designated as II-A) and then one with tip relief (designated as II-B) using
FE/CM code. Figure 2.10 compares the amplitudes of STE spectra at mesh harmonics for
both cases. (In this and following figures, predictions are shown as continuous lines for
the sake of clarity though they are calculated only at discrete torque points.)
45
x 10-4
2.5
Transmission error (in)
2
1.5
1
0.5
0
100
500
700
900
Torque (lb-in)
Transmission error (in)
(a)
300
Figure 2.10 Mesh harmonics of the static transmission error (STE) calculated by using
the FE/CM code (in the “static” mode) for Example II: (a) gear pair with perfect involute
profile (II-A); (b) gear pair with tip relief (II-B). Key:
, n = 1;
, n = 2;
,n=
3.
46
The first two mesh harmonics are most significantly affected by the tip relief and
they are minimal at the “optimal” mean torque around 500 lb-in. For both the II-A and IIB cases, typical k (t ) functions of a single meshing tooth over two complete mesh cycles
are calculated using Eq. (2.2) for various mean torques, as shown in Figure 2.1. Note that
k (t ) is defined as the effective stiffness since it incorporates the effect of profile
modification such as the linear tip relief (II-B). Observe that although the maximum
stiffness remains the same, application of the tip relief significantly changes the stiffness
profile. For the perfect involute profile (II-A), steep slopes are observed in the vicinities
near the single or two teeth contact regimes, and a smooth transition is observed in
between these steep regimes. Also, k (t ) is found to be insensitive to a variation in the
mean torque. However, with tip relief, an almost constant slope is found throughout the
transition profile between single and two teeth contact regimes. Moreover, a smaller
profile contact ratio (around 1.1 at 100 lb-in) is observed for the tip relief case when
compared with around 1.6 (at all loads) for the perfect involute pair. The realistic k (t )
function is then incorporated into the lumped MDOF dynamic model.
Figure 2.12 shows the combined k (t ) with contributions of both meshing tooth
pairs over two mesh cycles for Example II. Observe that the profile of case II-A is
insensitive to a variation in the mean torque, but the profile of case II-B shows a
minimum around 500 lb-in. Frequency domain analysis reveals that the first two mesh
harmonics are most significantly affected by the linear tip modification. Overall, it is
evident that significant changes take place in the STE, tooth load distribution and mesh
stiffness function due to the profile modification (tip relief), which may be explained by
an avoidance of the corner contact at an “optimized” mean torque.
47
Figure 2.11 Tooth stiffness functions of a single mesh tooth pair for Example II: (a) gear
pair with perfect involute profile (II-A); (b) gear pair with tip relief (II-B). Key:
, 100
lb-in;
, 500 lb-in;
, 900 lb-in.
48
Figure 2.12 Combined tooth stiffness functions for Example II: (a) gear pair with perfect
involute profile (II-A); (b) gear pair with tip relief (II-B). Key:
, 100 lb-in;
, 500
lb-in;
, 900 lb-in.
49
2.6.3
Phase Relationship between Normal Load and Friction Force Excitations
Using the 6DOF spur gear model with parameters consistent with the
experimental conditions, dynamic studies are conducted for Example II. First, a mean
torque of 500 lb-in is used corresponding to the “optimal” case with minimal STE.
Equations (2.13-2.16) show that the normal loads
∑N
i
and friction forces
∑F
fi
excite
the LOA and OLOA dynamics, respectively. The force profile of a single tooth pair
undergoing the entire meshing process is obtained by tooth pairs #0 and #1 for two
continuous meshing cycles as shown in Figure 2.13(a-b) and (c-d) for II-A and II-B cases
respectively. Observe that the peak-to-peak magnitude of combined pinion normal load
∑N
pi
is minimized for the tip relief gear due to reduced STE at 500 lb-in. However, the
combine pinion friction force ∑ Ffpi with tip relief has a higher peak-to-peak magnitude
when compared with the perfect involute gear. This implies that the tip relief amplifies
∑F
fi
in the OLOA direction while minimizing
∑N
i
in the LOA direction. Such
contradictory effects are examined next using the phase relationship between N pi and
F fpi .
50
Figure 2.13 Dynamic loads predicted for Example II at 500 lb-in, 4875 RPM and 140 °F
with tc = 0.44 ms: (a) Normal loads of gear pair with perfect involute profile (II-A); (b)
normal loads of gear par with tip relief (II-B); (c) friction forces of gear pair with perfect
involute profile (II-A); (d) friction forces of gear pair with tip relief (II-B). Key:
,
combined;
, tooth pair #0;
, tooth pair #1.
51
At points A, B, C and D, corner contacts are observed for N pi of the perfect
involute gear, corresponding to the time instants when meshing tooth pairs come into or
out of contact. These introduce discontinuous points in the slope of the
∑N
pi
profile.
Note that N p1 and N p 2 between A and B (or C and D) are in phase with each other,
which should amplify the peak-to-peak variation of ∑ N pi . For the Ffpi profile of Figure
2.13(c), an abrupt change in the direction is observed at the pitch point P in addition to
the corner contacts. Unlike N pi , the profiles of Ffp1 and Ffp 2 of Figure 2.13(c) between A
and B (or C and D) are out of phase with each other. This should minimize the peak-topeak variation of ∑ Ffpi . When tip relief is applied in Figure 2.13(b), corner contacts of
N pi are reduced and smoother transitions are observed at points A, B, C and D. Unlike
the perfect involute gear, N p1 and N p 2 are now out of phase with each other between A
and B (or C and D), which reduces the peak-to-peak variation of
∑N
pi
. However, the
profiles of Ffp1 and Ffp 2 of Figure 2.13(d) are in phase with each other in the same region,
which amplifies the variation of
∑F
fpi
. The out of phase relationship between N pi and
F fpi explains why the tip relief (designed to minimize the STE) tends to increase the
friction force excitations. This relationship is mathematically embedded in Eq. (2.10) and
graphically illustrated in Figure 2.13, where N p1 and N p 2 are in phase while Ffp1 and
F fp 2 are out of phase. Consequently, a compromise would be needed to simultaneously
address the dynamic responses in both the LOA and OLOA directions.
52
2.6.4
Prediction of the Dynamic Responses
Dynamic responses including x p (t ) , y p (t ) , Fpbx (t ) , Fpby (t ) and DTE δ (t ) are
predicted by numerically integrating the governing equations. Predictions from both
perfect and tip relief gears are compared to examine the effect of profile modification in
the presence of sliding friction. Figure 2.14 shows that the normalized x p (t ) at 500 lb-in
is much smaller (over 90% reduction) when the tip relief is applied. This is because that
the STE is the most dominant excitation in the LOA direction and it is minimized at 500
lb-in when the tip relief is applied. An alternate explanation is that the peak-to-peak
variation of
∑N
pi
is minimized with the tip relief as shown in Figure 2.13.
In the OLOA direction, more significant oscillations are observed for y p (t ) due
to increased
∑N
pi
∑F
fpi
excitations with tip relief. Despite that the vibratory components of
are larger than that of ∑ Ffpi , predicted y p (t ) is actually higher than x p (t ) . This
shows the necessity of including sliding friction when other excitations such as the STE
are minimized. Note that a phase difference is present in simulated y p (t ) before and after
the tip relief is applied. Predicted pinion bearing forces are not shown here since they
depict the same features as the displacement responses of Figure 2.14.
Figure 2.15 shows the DTE predictions, as defined by Eq. (2.17), with and
without the tip relief. Similarity between Figure 2.14(a-b) and Figure 2.15(a-b) suggests
that the relative LOA displacement plays a dominant role in the DTE responses. However,
this conclusion is somewhat case specific as the DTE results depend on the mesh stiffness,
bearing stiffness, and gear geometry.
53
Figure 2.14 Dynamic shaft displacements predicted for Example II at 500 lb-in, 4875
RPM and 140 °F with tc = 0.44 ms and fm = 2275 Hz: (a) x p (t ) ; (b) X p ( f ) ; (c) y p (t ) and
, gear pair with perfect involute profile in t domain (II-A); o, with
(d) Yp ( f ) . Key:
perfect involute profile in f domain (II-A);
, with tip relief (II-B).
54
Dynamic transmission error (in)
Figure 2.15 Dynamic transmission error predicted for Example II at 500 lb-in, 4875 RPM
and 140 °F with tc = 0.44 ms and fm = 2275 Hz: (a) δ (t ) ; (b) ∆( f ) . Key:
, gear pair
with perfect involute profile in t domain (II-A); o, with perfect involute profile in f
domain (II-A);
, with tip relief (II-B).
55
2.7
Experimental Validation of Example II Models
Experiments corresponding to Example II-B were conducted at the NASA Glenn
Research Center (Gear Noise Rig) to validate the MDOF spur gear pair model and to
establish the relative influence of friction force excitation on the system. Figure 2.16
shows the inside of the gearbox, where a bracket was built to hold two shaft displacement
probes one inch away from the center of the gear in the LOA and OLOA directions [2.13].
The probes face a steel collar that was machined to fit around the output shaft with
minimal eccentricity. Accelerometers were mounted on the bracket, so the motion of the
displacement probes could be subtracted from the measurements, if necessary. A
thermocouple was installed inside the gearbox to measure the temperature of the oil
flinging off the gears as they enter into mesh. The thermocouple position was chosen to
be consistent with Benedict and Kelley’s [2.14] experiment. A common shaft speed of
4875 rpm is used in all tests so that the first five harmonics of the gear mesh frequency
(2275, 4550, 6825, 9100, and 11375 Hz) do not excite system resonances. Data of shaft
displacement in the LOA and OLOA directions are collected from the proximity sensors
under oil inlet temperatures over the range of temperatures (104, 122, 140, 158, and 176
°F). At each temperature the torque is varied from 500 to 900 lb-in increments of 100 lbin.
56
Thermocouple
OLOA
LOA
Proximity Probes
Bracket
Accelerometer
Figure 2.16 Sensors inside the NASA gearbox (for Example II-B).
Parametric studies are conducted to examine the dynamic responses under varying
operational conditions of temperature and nominal torque. Benedict and Kelly’s [2.14]
friction model is used to calculate the empirical µ as given in Table 2.3 and the realistic
k (t ) calculated using FE/CM under varying torques are incorporated into the dynamic
model. Since the precise parameters of the experimental system are not known [2.13],
both simulated and measured data are normalized with respect to the amplitude of their
first mesh harmonic of the OLOA displacement (which is then designed as 100%). This
57
facilitates the comparison of trends and allows simulations and measurements to be
viewed in the same graphs from 0 to 100%.
Figure 2.17 compares the first five mesh harmonics of the LOA displacement as a
function of mean torque. (In this and other figures, predictions are shown as continuous
lines for the sake of clarity though they are calculated only at discrete points like the
measurements.) It was observed that overall simulation trend matches well with the
experiment. Magnitudes of the first two mesh harmonics are most dominant and they
have minimum values around the optimized load due to the linear tip relief. Figure 2.17
also shows predicted first two harmonics for the prefect involute gear (II-A). Compared
with the tip relief gear, they increase monotonically with the mean torque and have much
higher values than the tip relief gear around the “optimal” torque.
Figure 2.18 compares the first five mesh harmonics of the OLOA displacement,
on a normalized basis, as a function of mean torque. The overall simulation trend again
matches well with the experiment. However, unlike the LOA responses, the first
harmonic of OLOA displacement grows monotonically with an increase in the mean
torque. This is because the friction forces increase almost proportionally with normal
loads as predicted by the Coulomb law, but the frictional contribution of each meshing
tooth pair tends to be in phase with each other for the tip relief gear (II-B). Thus it should
amplify the combined friction force excitation in the OLOA direction. Consequently it is
not reducing the OLOA direction responses induced by the sliding friction, even though
the profile modification can be efficiently used to minimize gear vibrations in the LOA
direction.
58
Figure 2.17 Mesh harmonic amplitudes of X p as a function of the mean torque at 140 °F.
All values are normalized with respect to the amplitude of Yp at the first mesh harmonic.
Key:
, n = 1 (prediction of II-B);
, n = 2 (prediction of II-B);
, n = 3
(prediction of II-B);
, n = 1 (prediction of II-A);
, n = 2 (prediction of II-A);
, n = 1 (measurement of II-B); , n = 2 (measurement of II-B); O, n = 3 (measurement
of II-B).
59
Figure 2.18 Mesh harmonic amplitudes of Yp as a function of the mean torque at 140 °F
for Example II-B. All values are normalized with respect to the amplitude of Yp at the
first mesh harmonic. Key:
, n = 1 (prediction of II-B);
, n = 2 (prediction of II-B);
, n = 3 (prediction of II-B);
, n = 1 (measurement of II-B); , n = 2
(measurement of II-B); O, n = 3 (measurement of II-B).
60
Figure 2.19 compares first five mesh harmonics of the normalized DTE for II-A
and II-B cases over a range mean torques. Observe that the DTE spectral trends are very
similar to the STE spectral trends of Figure 2.10. For example, the harmonic amplitudes
of the perfect involute gear grow monotonically with mean torque while the harmonic
amplitudes of the tip relief gear have minimum values around the “optimal” torque. Also,
the DTE spectra show a dominant second harmonic, whose magnitude is comparable to
that at the first harmonic. In some cases for the tip relief gear the second harmonic
becomes the most dominant component especially when the mean torque is lower than
350 lb-in.
Finally, Figure 2.20 compares the first five mesh harmonics of the normalized
LOA displacement as a function of operational temperature. The changes in temperature
are converted into variation in µ of Table 2.3. Compared with the OLOA motions, both
predictions and measurements in the LOA direction give almost identical results at all
temperatures. Figure 2.21 shows the first five mesh harmonics of the OLOA
displacement with a change in temperature. The first harmonic varies quite significantly
even though the changes in µ are relatively small. Consequently, the OLOA dynamics
tends to be much more sensitive to a variation in µ as compared with the LOA motions.
Measured data of Figure 2.21 show some variations due to the experimental errors [2.13].
61
Figure 2.19 Predicted dynamic transmission errors (DTE) for Example II over a range of
torque at 140 °F: (a) gear pair with perfect involute profile (II-A); (b) with tip relief (II-B).
All values are normalized with respect to the amplitude of δ (II-A) at the first mesh
harmonic with 100 lb-in. Key:
, n = 1 (II-B);
, n = 2 (II-B);
, n = 3 (II-B).
62
Figure 2.20 Mesh harmonic amplitudes of X p as a function of temperature at 500 lb-in
for Example II-B. All values are normalized with respect to the amplitude of Yp at the
first mesh harmonic. Key:
, n = 1 (prediction of II-B);
, n = 2 (prediction of II-B);
, n = 3 (prediction of II-B);
, n = 1 (measurement of II-B); , n = 2
(measurement of II-B); O, n = 3 (measurement of II-B).
63
100
80
60
40
20
0
95
115
135
155
175
Temperature (deg F)
Figure 2.21 Mesh harmonic amplitudes of Yp as a function of temperature at 500 lb-in for
Example II-B. All values are normalized with respect to the amplitude of Yp at the first
mesh harmonic. Key:
, n = 1 (prediction of II-B);
, n = 2 (prediction of II-B);
, n = 3 (prediction of II-B);
, n = 1 (measurement of II-B); , n = 2
(measurement of II-B); O, n = 3 (measurement of II-B).
64
2.8
Conclusion
Chief contribution of this study is the development of a new multi-degree of
freedom, linear time-varying model. This formulation overcomes the deficiency of
Vaishya and Singh’s work [2.1-2.3] by employing realistic tooth stiffness functions and
the sliding friction over a range of operational conditions. Refinements include: (1) an
accurate representation of tooth contact and spatial variation in tooth mesh stiffness based
on a FE/CM code in the “static” mode; (2) Coulomb friction model for sliding resistance
with empirical coefficient of friction as a function of operation conditions; (3) a better
representation of the coupling between the LOA and OLOA directions including
torsional and translational degrees of freedom. Numerical solutions of the MDOF model
yield the dynamic transmission error and vibratory motions in the LOA and OLOA
directions. The new model has been successfully validated first by using the FE/CM code
while running in the “dynamic” mode and then by analogous experiments. Since the
lumped model is more computationally efficient when compared with the FE/CM
analysis, it could be quickly used to study the effect of a large number of parameters.
One of the main effects of sliding friction is the enhancement of the DTE
magnitude at the second gear mesh harmonic. A key question whether the sliding friction
is indeed the source of the OLOA motions and forces is then answered by our model. The
bearing forces in the LOA direction are influenced by the normal tooth loads, but the
sliding frictional forces primarily excite the OLOA motions. Finally, effect of the profile
modification on the dynamic transmission error has been analytically examined under the
influence of frictional effects. For instance, the tip relief introduces an amplification in
the OLOA motions and forces due to an out of phase relationship between the normal
65
load and friction forces. This knowledge should be of significant utility to the designers.
Future modeling work should examine the effects of other profile modifications and find
the conditions for minimal dynamic responses when both STE and friction excitations are
simultaneously present. Also, the model could be further refined by incorporating
alternate friction formulations.
References for Chapter 2
[2.1] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration,
243(3), pp. 525-545.
[2.2] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and
Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671694.
[2.3] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear
Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.
[2.4] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of
Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.
[2.5] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on
the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of
Mechanical Design, 122(4), pp. 515-522.
[2.6] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations
in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp. 641-658.
[2.7] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body
Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3),
pp. 479-499.
[2.8] External2D (CALYX software), 2003,
www.ansol.www, ANSOL Inc., Hilliard, OH.
66
“Helical3D
User’s
Manual,”
[2.9] Vinayak, H., Singh, R., and Padmanabhan, C., 1995, “Linear Dynamic Analysis of
Multi-Mesh Transmissions Containing External, Rigid Gears,” Journal of Sound and
Vibration, 185(1), pp. 1-32.
[2.10] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling
Element Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and
Vibration, 139(2), pp. 179-199.
[2.11] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling
Element Bearings. Part II: System Studies,” Journal of Sound and Vibration, 139(2), pp.
201-225.
[2.12] Kahraman, A., and Singh, R., 1991, “Error Associated with a Reduced Order
Linear Model of Spur Gear Pair,” Journal of Sound and Vibration, 149(3), pp. 495-498.
[2.13] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared
Systems,” Technical Report submitted to the Army Research Office, grant number
DAAD19-02-1-0334.
[2.14] Benedict, G. H., and Kelley B. W., 1961, “Instantaneous Coefficients of Gear
Tooth Friction,” Transactions of the American Society of Lubrication Engineers, 4, pp.
59-70.
[2.15] Rebbechi, B., Oswald, F. B., and Townsend, D. P., 1996, “Measurement of Gear
Tooth Dynamic Friction,” ASME Power Transmission and Gearing Conference
proceedings, DE-Vol. 88, pp. 355-363.
67
CHAPTER 3
PREDICTION OF DYNAMIC FRICTION FORCES USING ALTERNATE
FORMULATIONS
3.1
Introduction
Gear dynamic researchers [3.1-3.6] have typically modeled sliding friction
phenomenon by assuming Coulomb formulation with a constant coefficient (µ) of friction
(it is designated as Model I in this chapter). In reality, tribological conditions change
continuously due to varying mesh properties and lubricant film thickness as the gears roll
through a full cycle [3.7-3.10]. Thus, µ varies instantaneously with the spatial position of
each tooth and the direction of friction force changes at the pitch point. Alternate
tribological theories, such as elasto-hydrodynamic lubrication (EHL), boundary
lubrication or mixed regime, have been employed to explain the interfacial friction in
gears [3.7-3.10]. For instance, Benedict and Kelley [3.7] proposed an empirical dynamic
friction coefficient (designated as Model II) under mixed lubrication regime based on
measurements on a roller test machine. Xu et al. [3.8, 3.9] recently proposed yet another
friction formula (designated as Model III) that is obtained by using a non-Newtonian,
thermal EHL formulation. Duan and Singh [3.11] developed a smoothened Coulomb
68
model for dry friction in torsional dampers; it could be applied to gears to obtain a
smooth transition at the pitch point and we designate this as Model IV. Hamrock and
Dawson [3.10] suggested an empirical equation to predict the minimum film thickness
for two disks in line contact. They calculated the film parameter Λ, which could lead to a
composite, mixed lubrication model for gears (designated as Model V). Overall, no prior
work has incorporated either the time-varying µ (t ) or Models II to V, into multi-degreeof-freedom (MDOF) gear dynamics. To overcome this void in the literature, specific
objectives of this chapter are established as follows: 1. Propose an improved MDOF spur
gear pair model with time-varying coefficient of friction, µ (t ) , given realistic mesh
stiffness profiles of Chapter 2; 2. Comparatively evaluate alternate sliding friction models
and predict the interfacial friction forces and motions in the off-line-of-action (OLOA)
direction; and 3. Validate one particular model (III) by comparing predictions to the
benchmark gear friction force measurements made by Rebbechi et al. [3.12].
3.2
MDOF Spur Gear Model
Transitions in key meshing events within a mesh cycle are determined from the
undeformed gear geometry. Figure 3.1(a) is a snapshot for the example gear set (with a
contact ratio σ of about 1.6) at the beginning (t = 0) of the mesh cycle (tc). At that time,
pair # 1 (defined as the tooth pair rolling along line AC) just comes into mesh at point A
and pair # 0 (defined as the tooth pair rolling along line CD) is in contact at point C,
which is the highest point of single tooth contact (HPSTC). When pair # 1 approaches the
69
lowest point of single tooth contact (LPSTC) at point B, pair # 0 leaves contact. Further,
when pair #1 passes through the pitch point P, the relative sliding velocity of the pinion
with respect to the gear is reversed, resulting in a reversal of the friction force. Beyond
point C, pair # 1 will be re-defined as pair # 0 and the incoming meshing tooth pair at
point A will be re-defined as pair # 1, resulting in a linear-time-varying (LTV)
formulation. The spur gear system model is shown in Figure 3.1(b) and key assumptions
for the dynamic analysis include the following: (i) pinion and gear are rigid disks; (ii)
shaft-bearings stiffness elements in the line-of-action (LOA) and OLOA directions are
modeled as lumped springs which are connected to a rigid casing; (iii) vibratory angular
motions are small in comparison to the kinematic motion. Overall, we obtain a LTV
system formulation, as explained in Chapter 2 with a constant µ . Refinements to the
MDOF model of Figure 3.1(b) with time-varying sliding friction µ (t ) are proposed as
follows. The governing equations for the torsional motions θ p (t ) and θ g (t ) are as follows:
J pθp (t ) = Tp +
n = floor(σ )
J gθg (t ) = −Tg +
∑
i =0
X pi (t ) ⋅ Fpfi (t ) −
n = floor(σ )
∑
i =0
n = floor(σ )
X gi (t ) ⋅ Fgfi (t ) +
70
∑
i=0
rbp ⋅ N pi (t ) ,
n = floor(σ )
∑
i=0
rbg ⋅ N gi (t ) .
(3.1)
(3.2)
(a)
−Tg , −Ω g
J g , mg
K gBx
k (t )
K pBy
K gBy
J p , mp
K pBx
x
Ω p , Tp
y
θ
(b)
Figure 3.1 (a) Snap shot of contact pattern (at t = 0) in the spur gear pair; (b) MDOF spur
gear pair system; here k (t ) is in the LOA direction.
71
Here, the “floor” function rounds off the contact ratio σ to the nearest integer
(towards a lower value); J p and J g are the polar moments of inertia for the pinion and
gear; Tp and Tg are the external and braking torques; N pi (t ) and N gi (t ) are the normal
loads defined as follows:
N pi (t ) = N gi (t ) = ki (t ) ⎡⎣ rbp ⋅ θ p (t ) − rbg ⋅ θ g (t ) + x p (t ) − xg (t ) ⎤⎦ +
ci (t ) ⎡⎣ rbp ⋅ θp (t ) − rbg ⋅ θg (t ) + x p (t ) − x g (t ) ⎤⎦ , i = 0, 1, ... , n = floor(σ ).
(3.3)
where ki (t ) and ci (t ) are the time-varying realistic mesh stiffness and viscous damping
profiles; rbp and rbg are the base radii of the pinion and gear; x p (t ) and xg (t ) denote the
translational displacements (in the LOA direction) at the bearings. The sliding (interfacial)
friction forces Fpfi (t ) and Fgfi (t ) of the ith meshing pair are derived as follows; note that
five alternate µ (t ) models will be described later.
Fpfi (t ) = µ (t ) N pi (t ),
Fgfi (t ) = µ (t ) N gi (t ), i = 0, ... , n.
(3.4a,b)
The frictional moment arms X pi (t ) and X gi (t ) acting on the ith tooth pair are:
X pi (t ) = LXA + ( n − i )λ + mod(Ω p rbp t , λ ), i = 0, ... , n ,
(3.5a)
X gi (t ) = LYC + iλ − mod(Ω g rbg t , λ ), i = 0, ... , n .
(3.5b)
72
where “mod” is the modulus function defined as: mod( x, y ) = x − y ⋅ floor( x / y ), if y ≠ 0 ;
“sgn” is the sign function; Ω p and Ω g are the nominal operational speeds (in rad/s); and λ
is the base pitch. Refer to Figure 3.1(a) for length LAP . The governing equations for the
translational motions x p (t ) and xg (t ) in the LOA direction are:
m p x p (t ) + 2ζ pBx K pBx ⋅ m p ⋅ x p (t ) + K pBx x p (t ) +
mg xg (t ) + 2ζ gBx K gBx ⋅ mg ⋅ x g (t ) + K gBx xg (t ) +
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
N pi (t ) = 0 ,
N gi (t ) = 0 .
(3.6)
(3.7)
Here, m p and mg are the masses of the pinion and gear; K pBx and K gBx are the
effective shaft-bearing stiffness values in the LOA direction, and ζ pBx and ζ gBx are their
damping ratios. Likewise, the governing equations for the translational motions y p (t )
and y g (t ) in the OLOA direction are written as:
m p y p (t ) + 2ζ pB y K pBy ⋅ m p ⋅ y p (t ) + K pBy y p (t ) −
mg y g (t ) + 2ζ gB y K gBy ⋅ mg ⋅ y g (t ) + K gBy yg (t ) −
73
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
Fpfi (t ) = 0 ,
(3.8)
Fgfi (t ) = 0 .
(3.9)
3.3
Spur Gear Model with Alternate Sliding Friction Models
Following a similar modeling strategy of Chapter 2, we obtain a LTV system
formulation. Refinements to the MDOF model with time-varying sliding friction µ (t ) are
proposed as follows. The sliding (interfacial) friction forces Fpfi (t ) and Fgfi (t ) of the ith
meshing pair are
Fpfi (t ) = µ (t ) N pi (t ),
Fgfi (t ) = µ (t ) N gi (t ), i = 0, ... , n.
(3.10a,b)
Five alternate µ (t ) models are described as follows:
3.3.1
Model I: Coulomb Model
The Coulomb friction model with time-varying (periodic) coefficient of
friction µCi (t ) for the ith meshing tooth pair is derived as follows, where µavg is the
magnitude of the time-average.
µCi (t ) = µavg ⋅ sgn ⎡⎣ mod(Ω p rbp t , λ ) + (n − i)λ − LAP ⎤⎦ , i = 0, ... , n.
74
(3.11)
3.3.2
Model II: Benedict and Kelley Model
The instantaneous profile radii of curvature (mm) ρ (t ) of ith meshing tooth are:
ρ pi (t ) = LXA + (n − i )λ + mod(Ω p rbp t , λ ) , i = 0, ... , n .
(3.12a)
ρ gi (t ) = LXY − ρ pi (t ), i = 0, ... , n .
(3.12b)
The rolling (tangential) velocities vr (t ) (m/s) of ith meshing tooth pair are:
vrpi (t ) =
Ω p ρ pi (t )
1000
,
vrgi (t ) =
Ω g ρ gi (t )
1000
,
i = 0, ... , n .
(3.13a,b)
The sliding velocity vs (t ) and the entraining velocity ve (t ) (m/s) of ith meshing
tooth pair are:
vsi (t ) = vrpi (t ) − vrgi (t ) ,
vei (t ) = vrpi (t ) + vrgi (t ) , i = 0, ... , n .
(3.14a,b)
The unit normal load (N/mm) is wn = Tp / ( Z ⋅ rwp cos α ) , where α is the pressure
angle, Z is the face width (mm), Tp is torque (N-mm) and rwp is the operating pitch
radius of pinion (mm). Our µ (t ) prediction for the ith meshing tooth pair is based on the
Benedict and Kelley model [3.7], though it is modified to incorporate a reversal in the
direction of friction force at the pitch point. Here, S avg = 0.5 ( S ap + S ag ) is the averaged
75
surface roughness ( µ m ), and η M is the dynamic viscosity of the oil entering the gear
contact.
µ Bi (t ) =
⎡ 29700wn ⎤
0.0127 ×1.13
⋅ log10 ⎢
⎥ ⋅ sgn ⎡⎣ mod(Ω p rbpt , λ ) + (n − i )λ − LAP ⎤⎦ ,
2
1.13 − Savg
⎣η M vsi (t )vei (t ) ⎦
i = 0, ... , n .
3.3.3
(3.15)
Model III: Formulation Suggested by Xu et al.
The composite relative radius of curvature ρ r (t ) (mm) of ith meshing tooth pair is:
ρ ri (t ) =
The
effective
ρ pi (t ) ρ gi (t )
,
ρ pi (t ) + ρ gi (t )
modulus
of
elasticity
i = 0, ... , n
(GPa)
of
(3.16)
mating
surfaces
is
⎡1 −ν p2 1 −ν g2 ⎤
E′ = 2 / ⎢
+
⎥ , where E and ν are the Young’s modulus and Poisson’s ratio,
Eg ⎥⎦
⎢⎣ E p
respectively. The maximum Hertzian pressure (GPa) for the ith meshing tooth pair is:
Phi (t ) =
wn E ′
,
2000πρ ri (t )
76
i = 0, ... , n .
(3.17)
Define the dimensionless slide-to-roll ratio SR(t ) and oil entraining velocity
Ve (t ) (m/s) of ith meshing tooth pair as:
SRi (t ) =
2vsi (t )
,
vei (t )
Vei (t ) =
vei (t )
,
2
i = 0, ... , n .
(3.18a,b)
The empirical sliding friction expression (for the ith meshing tooth pair), as
proposed by Xu et al. based on non-Newtonian, thermal EHL theory [3.8-3.9], is
modified in our work to incorporate a reversal in the direction of the friction force at the
pitch point:
f SR ( t ), P
µ Xi (t ) = e (
i
hi ( t ),η M
, Savg
)
Phib2 SRi (t ) 3 Veib6 (t )η Mb7 Rib8 (t ) ⋅ sgn ⎡⎣ mod(Ω p rbp t , λ ) + ( n − i )λ − LAP ⎤⎦ ,
b
f ( SRi (t ), Phi (t ),η M , Savg ) = b1 + b4 SRi (t ) Phi (t ) log10 (η M ) + b5 e
− SRi ( t ) Phi ( t )log10 (η M )
+ b9 e
i = 0, ... , n .
(3.19a,b)
S avg
,
Xu [3.9] suggested the following empirical coefficients (in consistent units) for
the above formula: b1 = −8.916465 , b2 = 1.03303 , b3 = 1.036077 , b4 = −0.354068 ,
b5 = 2.812084 , b6 = −0.100601 , b7 = 0.752755 , b8 = −0.390958 , and b9 = 0.620305 .
77
3.3.4
Model IV: Smoothened Coulomb Model
Xu [3.9] conducted a series of friction measurements on a ball-on-disk test
machine and measured the µ (t ) values as a function of SR; these results resemble the
smoothening function reported by Duan and Singh [3.11] near the pitch point (SR = 0)
especially at very low speeds (boundary lubrication conditions).
By denoting the
periodic displacement of ith meshing tooth pair as xi (t ) = mod(Ω p rbp t , λ ) + (n − i )λ − LAP , a
smoothening function could be used in place of the discontinuous Coulomb friction of
Chapter 2. The arc-tangent type function is proposed as follows though one could also
use other functions [3.11]:
µ Si (t ) =
2 µavg
π
arctan [ Φ ⋅ xi (t )] + xi (t ) ⋅
2µavgσ
π ⎡⎣1 + Φ 2 xi2 (t ) ⎤⎦
, i = 0, ... , n
(3.20)
In the above, the regularizing factor Φ is adjusted to suit the need of smoothening
requirement. A higher value of Φ corresponds to a steeper slope at the pitch point. In our
work, Φ = 50 is used for a comparative study.
3.3.5
Model V: Composite Friction Model
Alternate theories (Models I to IV) seem to be applicable over specific operational
conditions. This necessitates a judicious selection of an appropriate lubrication regime as
indicated by the film parameter, Λ, that is defined as the ratio of minimum lubrication
78
2
2
film thickness and composite surface roughness Rcomp = Rrms
, g + Rrms , p measured with a
filter cutoff wave length Lx , where Rrms is the rms gear-tooth surface roughness [3.13].
The film parameter for rotorcraft gears usually lies between 1 and 10. In the mixed
lubrication regime the films are sufficiently thin to yield partial asperity contact, while in
the EHL regime the lubrication film completely separates the gear surfaces. Accordingly,
a composite friction model is proposed as follows:
⎧ µC (t )
⎪ µ (t )
⎪ B
µ (t ) = ⎨
⎪ µ X (t )
⎪⎩ µ S (t )
simplified Coulomb model, computationally efficient (Model I)
1<Λ <4, mixed lubrication, (Model II)
4 ≤ Λ <10, EHL lubrication, (Model III)
low Ω p , high Tp , Λ < 1, boundary lubrication (Model IV)
(3.21)
Application of Model II, III or IV would, of course, depend on the operational and
tribological conditions though Model I could be easily utilized for computationally
efficient dynamic simulations. Note that the magnitude µavg of Model I or IV should be
determined separately. For instance, the averaged coefficient based on Model II was used
in Chapter 2. Also, the critical Λ value between different lubrication regimes must be
carefully chosen. The film thickness calculation employs the following equation
developed by Hamrock and Dowson [3.10, 3.13], based on a large number of numerical
solutions that predict the minimum film thickness for two disks in line contact. Here, G is
the dimensionless material parameter, W is the load parameter, U is the speed parameter,
H is the dimensionless central film thickness, and bH is the semi-width of Hertzian
contact band:
79
Λ i (t ) =
H ci (t ) ρ r1 (t ) × 103
Rcomp
bH 1 (t ) =
LX
,
2bHi (t )
8wn ρ r1 (t )
,
π Er
i = 0, ... , n
(3.22a)
G = kη Ms Er ,
(3.22b-c)
G 0.56U i0.69 (t )
H ci (t ) = 3.06
,
Wi 0.10 (t )
U i (t ) =
3.4
η M vei (t )
× 10 −6 ,
2 Er ρ ri (t )
(3.22d)
Wi (t ) =
wn
Er ρ ri (t )
.
(3.22e.f)
Comparison of Sliding Friction Models
Figure 3.2(a) compares the magnitudes of µ (t ) as predicted by Model II and III
for the spur gear set of Chapter 2 given Tp = 22.6 N-m (200 lb-in) and Ω p = 1000 RPM.
The LTV formulations for meshing tooth pairs # 0 and 1 result in periodic profiles for
both models. Two major differences between these two models are: (1) The averaged
magnitude from Model II is much higher compared with that of Model III since friction
under mixed lubrication is generally higher than under EHL lubrication; and (2) while
Model III predicts nearly zero friction near the pitch point, Model II predicts the largest µ
value due to the entraining velocity term in the denominator.
80
0.1
0.09
0.08
0.07
µ
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.5
1
Normalized time t/tc
1.5
2
(a)
0.11
0.1
0.09
0.08
µ(t)
0.07
0.06
0.05
0.04
0.03
0.02
0
1000
2000
3000
4000
5000
Ω p (RPM)
(b)
Figure 3.2 (a) Comparison of Model II [3.7] and Model III [3.8] given Tp = 22.6 N-m
(200 lb-in) and Ω p = 1000 RPM. Key:
, pair # 1 with Model II;
, pair # 0 with
Model II;
, pair # 1 with Model III;
, pair # 0 with Model III; (b) Averaged
magnitude of the coefficient of friction predicted as a function of speed using the
composite Model V with Tp = 22.6 N-m (200 lb-in). Here, tc is one mesh cycle.
81
As explained by Xu [3.9], three different regions could be roughly defined on a µ
versus SR curve. When the sliding velocity is zero, there is no sliding friction, and only
rolling friction (though very small) exists. Thus, the µ value should be almost zero at the
pitch point. When the SR is increased from zero, µ first increases linearly with small
values of SR. This region is defined as the linear or isothermal region. When the SR is
increased slightly further, µ reaches a maximum value and then decreases as the SR value
is increased beyond that point. This region is referred to as non-linear or non-Newtonian
region. As the SR is increased further, the friction decreases in an almost linear fashion;
this is called as the thermal region. Model II seems to be valid only in the thermal region
[3.8, 3.9]. Figure 3.2(b) shows the averaged magnitude of µavg predicted as a function of
Ω p using the composite formulation (Model V) with Tp = 22.6 N-m (200 lb-in). An
abrupt change in magnitude is found around 2500 RPM corresponding to a transition
from the EHL to a mixed lubrication regime. Similar results could be obtained by plotting
the composite µ (t ) as a function of Tp . Though our composite model could be used to
predict µ (t ) over a large range of lubrication conditions, care must be exercised since the
calculation of Λ itself is based on an empirical equation [3.10].
Figure 3.3 compares four friction models on a normalized basis. The curves
between 0 ≤ t / tc < 1 are defined for pair # 1 and those between 1 ≤ t / tc < 2 are defined
for pair # 0. Discontinuities exist near the pitch point for Models I and II, and these might
serve as artificial excitations to the OLOA dynamics. On the other hand, smooth
transitions are observed for Models III and IV corresponding to the EHL lubrication
condition.
82
1.5
Normalized µ(t)
1
0.5
0
-0.5
-1
-1.5
0
0.5
1
1.5
Normalized time t/tc
Figure 3.3 Comparison of normalized friction models. Key:
, Model I (Coulomb
friction with discontinuity);
, Model II [3.7];
, Model III [3.8];
, Model
IV (smoothened Coulomb friction). Note that curve between 0 ≤ t / tc < 1 is for pair # 1;
and the curve between 1 ≤ t / tc < 2 is for pair # 0.
Figure 3.4 compares the combined normal loads and friction force time histories
as predicted by four friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875
RPM. Note that while Figure 3.3 illustrates µ (t ) for each meshing tooth pair the friction
forces of Figure 3.4 include the contributions from both (all) meshing tooth pairs. Though
alternate friction formulations dictate the dynamic friction force profiles, they have
negligible effect on the normal loads.
83
Np (N)
1400
1350
1300
0
0.5
1
1.5
2
1
1.5
2
100
Ffp (N)
50
0
-50
-100
0
0.5
Normalized time t/tc
Figure 3.4 Combined normal load and friction force time histories as predicted using
alternate friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key:
, Model I;
, Model II;
, Model III;
, Model IV.
84
3.5
Validation and Conclusion
Figure 3.5 compares the predicted LOA and OLOA displacements with alternate
friction models given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Note that the
differences between predicted motions are not significant though friction formulations
and friction force excitations differ. This implies that one could still employ the
simplified Coulomb formulation (Model I) in place of more realistic time-varying friction
models (Models II to IV). Similar trend is observed in Figure 3.6 for the dynamic
transmission errors (DTE), defined as δ (t ) = rbpθ p (t ) − rbgθ g (t ) + x p (t ) − xg (t ) . The most
significant variation induced by friction formulation is at the second harmonic, which
matches the results reported by Lundvall et al. [3.6].
Finally, predicted normal load and friction force time histories (with Model III)
are validated using the benchmark friction measurements made by Rebbechi et al. [3.12].
Results are shown in Figure 3.7. Based on the comparison, µ is found to be about 0.004
since it was not given in the experimental study. Here, we have made the periodic LTV
definitions of meshing tooth pairs # 0 and 1 to be consistent with those of measurements,
where meshing tooth pairs A and B are labeled in a continuous manner. Predictions
match well with measurements at both low ( Ω p = 800 RPM) and high ( Ω p = 4000 RPM)
speeds. Ongoing research focuses on the development of semi-analytical solutions given
a specific µ (t ) model and an examination of the interactions between tooth modifications
and sliding friction.
85
0.15
-59.8
0.05
-60
-60.2
0
0.1
xp (µm)
xp (µm)
-59.6
1
0
2
1
2
3
4
5
1
2
3
4
5
2
1
yp (µm)
yp (µm)
1
0
0.8
0.6
0.4
-1
0.2
-2
0
1
0
2
Normalized time t/tc
Mesh order n
Figure 3.5 Predicted LOA and OLOA displacements using alternate friction models given
Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key: in time domain:
, Model I;
, Model II;
, Model III;
, Model IV; in frequency (mesh order n)
,
domain:
, Model I; , Model II; , Model III; +, Model IV.
+
86
24
DTE ( µm)
23.5
23
22.5
22
21.5
0
0.5
1
1.5
2
Normalized time t/tc
DTE ( µm)
0.4
0.3
0.2
0.1
0
1
2
3
4
5
Mesh order n
Figure 3.6 Predicted dynamic transmission error (DTE) using alternate friction models
,
given Tp = 56.5 N-m (500 lb-in) and Ω p = 4875 RPM. Key: in time domain:
Model I;
, Model II;
, Model III;
, Model IV; in frequency (mesh order
n) domain:
, Model I; , Model II; , , Model III; +, Model IV.
+
87
2000
Np (N)
1500
1000
500
0
0
0.5
1
1.5
2
1
1.5
2
Ffp (N)
5
0
-5
0
0.5
Normalized time t/tc
(a)
2500
Np (N)
2000
1500
1000
500
0
0
0.5
0
0.5
1
1.5
2
1
1.5
Normalized time t/tc
2
4
Ffp (N)
2
0
-2
-4
-6
(b)
Figure 3.7 Validation of the normal load and sliding friction force predictions: (a) at Tp =
79.1 N-m (700 lb-in) and Ω p = 800 RPM; (b) at Tp = 79.1 N-m (700 lb-in) and Ω p =
4000 RPM. Key:
, prediction of tooth pair A with Model III;
, prediction of
tooth pair B with Model III; X, measurement of tooth pair A [3.12]; , , measurement of
tooth pair B [3.12].
88
References for Chapter 3
[3.1] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration,
243(3), pp. 525-545.
[3.2] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and
Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671694.
[3.3] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear
Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.
[3.4] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on
the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of
Mechanical Design, 122(4), pp. 515-522.
[3.5] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations
in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp 641-658.
[3.6] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body
Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3),
pp. 479-499.
[3.7] Benedict, G. H., and Kelley B. W., 1961, “Instantaneous Coefficients of Gear Tooth
Friction,” Transactions of the American Society of Lubrication Engineers, 4, pp. 59-70.
[3.8] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, “Prediction of
Mechanical Efficiency of Parallel-Axis Gear Pairs,” ASME Journal of Mechanical
Design, 129 (1), pp. 58-68.
[3.9] Xu, H., 2005, “Development of a Generalized Mechanical Efficiency Prediction
Methodology,” PhD dissertation, The Ohio State University.
[3.10] Hamrock, B. J., and Dowson, D., 1977, “Isothermal Elastohydrodynamic
Lubrication of Point Contacts, Part III – Fully Flooded Results,” Journal of Lubrication
Technology, 99(2), pp. 264-276.
[3.11] Duan, C., and Singh, R., 2005, “Super-Harmonics in a Torsional System with Dry
Friction Path Subject to Harmonic Excitation under a Mean Torque”, Journal of Sound
and Vibration, 285(2005), pp. 803-834.
[3.12] Rebbechi, B., Oswald, F. B., and Townsend, D. P., 1996, “Measurement of Gear
Tooth Dynamic Friction,” ASME Power Transmission and Gearing Conference
proceedings, DE-Vol. 88, pp. 355-363.
89
[3.13] AGMA Information Sheet 925-A03, 2003, “Effect of Lubrication on Gear Surface
Distress.”
90
CHAPTER 4
CONSTRUCTION OF SEMI-ANALYTICAL SOLUTIONS TO SPUR GEAR
DYNAMICS
4.1
Introduction
Periodic differential equations [4.1-4.3] are usually needed to describe the gear
dynamics [4.4-4.12] since significant variations in mesh stiffness k(t) and damping c(t)
are observed, within the fundamental period tc (one mesh cycle). Additionally, dynamic
friction force Ff(t) and torque Mf(t) also undergo periodic variations, with the same period
tc, due to changes in normal mesh loads and coefficient of friction µ, as well as a reversal
in the direction of Ff(t) at the pitch point [4.4-4.6], especially in spur and helical gears.
For the sake of illustration, typical ki(t) profiles and frictional functions fi(t) for the ith
meshing pair in spur gears are shown in Figure 4.1; derivations of fi(t) will be explained
later along with particulars of the example case. The fundamental nature of the linear
time-varying (LTV) system is illustrated in Figure 4.2(a); the system model is described
in Chapter 2.
91
ki(t) (lb/in)
f(t)
(a)
(b)
Figure 4.1(a) Realistic mesh stiffness functions of the spur gear pair example (with tip
relief) given Tp = 550 lb-in. Key:
, k0 (t ) ;
, k1 (t ) . (b) Periodic frictional
functions. Key:
, f 0 (t ) ;
, f1 (t ) ;
, f 2 (t ) .
92
(a)
Ωg
Ωp
(b)
Figure 4.2(a) Normal (mesh) and friction forces of 6DOF analytical spur gear system
model. (b) Snap shot of contact pattern (at t = 0) for the example spur gear pair.
93
The governing single degree-of-freedom (SDOF) equation in terms of dynamic
transmission error (DTE) δ (t ) = rbpθ p (t ) − rbgθ g (t ) is given below, where subscripts p and
g correspond to the pinion and gear, respectively; θ is the vibratory component of the
rotation; and rb is the base radius.
S
S
i =0
i =0
J eδ (t ) + J b ∑ ⎣⎡ci (t )δ (t ) + ki (t )δ (t ) ⎦⎤ + µ ∑ sgn ⎡⎣ mod(Ω p rbp t , λ ) + ( S − j )λ − LAP ⎤⎦
⋅ ⎡⎣ ci (t )δ (t ) + ki (t )δ (t ) ⎤⎦ ⎡⎣ X pi (t ) J g rbp + X gi (t ) J p rbg ⎤⎦ = Te + T (t )
Je = J p J g ,
J b = J g rbp2 + J p rbg2 ,
(4.1a)
,
Te = Tp J g rbp + Tg J p rbg ,
(4.1b-d)
X pi (t ) = LXA + ( S − i )λ + mod(Ω p rbp t , λ ), i = 0, ... , S = floor(σ ) ,
(4.1e)
X gi (t ) = LYC + iλ − mod(Ω g rbg t , λ ), i = 0, ... , S = floor(σ ) .
(4.1f)
Further, J is the moment of inertia; T and Ω are the nominal torque and rotation speed;
and λ is the base pitch. Tooth pairs #1 and #0 are defined as the pairs rolling along line
AC and CD in Figure 4.2(b), respectively. The jth tooth pair passes though the pitch point
P during the meshing event, and the reversal at P is characterized by the sign function
“sgn” with a constant coefficient of Coulomb friction µ [4.4]. The modulus function
(mod(x, y) = x − y⋅floor(x/y), if y ≠ 0) is used to describe the periodic friction force Ff(t)
and the moment arm X(t). The “floor” function rounds off the contact ratio σ to the
nearest integers towards a lower value, i.e. S = 1 for the example case. Finally, L
corresponds to the geometric length in Figure 4.2(b).
94
The chief goal of this chapter is to find semi-analytical solutions to Eq. (4.1) type
periodic systems which significantly differ from the classical Hill’s equation [4.1] in
several ways. First, the periodic ki (t ) is not confined to a rectangular wave assumed by
Manish and Singh [4.4-4.6], or a simple sinusoid as in the Mathieu’s equation [4.1].
Instead, Eq. (4.1) should describe realistic, yet continuous, profiles of Figure 4.1(a)
resulting from a detailed finite element/contact mechanics analysis [4.7]. Hence, multiple
harmonics of ki (t ) should be considered. Second, the periodic viscous ci (t ) term should
dissipate vibratory energy due to the sliding friction besides its kinematic effect. Third,
S
the ∑ δ i (t )ci (t ) and
i =1
S
∑ δ (t )k (t )
i =1
i
i
terms of Eq. (4.1) incorporate combined (but phase
correlated) contributions from all (yet changing) tooth pairs in contact. Consequently, the
relative phase between neighboring tooth pairs should play an important role in the
resulting response δ (t ) . Fourth, multiplicative effects between ki (t ) , ci (t ) , X i (t ) and
δ i (t ) should result in higher mesh harmonics, which poses difficulty in constructing
closed-form solutions. Lastly, T (t ) of Eq. (4.1) represents the time-varying component of
the forcing function due to unloaded (manufacturing) static transmission error ε (t ) . This
indicates that the frictional forces and moments reside on both sides of Eq. (4.1) as either
periodically-varying parameters or external excitations, thus posing further mathematical
complications.
95
4.2
Problem Formulation
Sliding friction has been found as a non-negligible excitation source in spur and
helical gear dynamics by Houser et al. [4.8], Velex and Cahouet [4.9], Velex and Sainsot
[4.10], and Lundvall et al. [4.11]. Earlier, Vaishya and Singh [4.4-4.6] developed a SDOF
spur gear model with rectangular k(t) and sliding friction profiles; they solved the δ(t)
response by using the Floquet theory [4.4] and multi-term harmonic balance method
(MHBM) [4.5]. Their work was recently refined and extended to helical gears in our
work (refer to Chapters 5 and 6) where closed-form solutions of δ(t) for a SDOF system
are derived under frictional excitations. While the equal load sharing assumption [4.4-4.6]
yields simplified expressions and analytically tractable solutions, they do not describe
realistic conditions. This particular deficiency has been partially overcame in Chapters 2
and 3 where we proposed a multi-degree-of-freedom (MDOF) model with realistic k(t)
and sliding friction functions. However, we utilized numerical integration and fast
Fourier transform (FFT) analysis methods in Chapters 2 and 3 that are often
computationally sensitive. Hence, a semi-analytical algorithm based on MHBM is highly
desirable for quick constructing frequency responses without any loss of generality.
Recently, Velex and Ajmi [4.12] implemented a similar harmonic analysis to
approximate the dynamic factors in helical gears (based on tooth loads and quasi-static
transmission errors). Their work, however, does not describe the multi-dimensional
system dynamics or include the frictional effect, which may lead to “multiplicative”
terms as described earlier.
96
The prime objective of this chapter is thus to extend the above mentioned
publications [4.4]. In particular, we intend to develop semi-analytical harmonic balance
solutions to the 6DOF spur gear model of Chapter 2 with realistic k(t) and sliding friction
functions. The example case used for this study is the unity ratio NASA spur gear (with
tip relief); refer to Table 2.2 of Chapter 2 for its parameters. Key assumptions include: (i)
the pinion and gear are rigid disks; (ii) vibratory motions are small in comparison to the
nominal motion; this would lead to a linear time-varying model; (iii) Coulomb friction is
assumed with a constant µ though sign is reversed at the pitch point; (iv) when the
torsional component is dominant over the translational component of δ(t) for the 6DOF
model of Chapter 2, the harmonic solutions of the SDOF system could be extended to
predict translational responses in the line-of-action (LOA) and off-line-of-action (OLOA)
directions. Note that semi-analytical method analyzes the 6DOF system as a 5DOF model
as it calculates the δ(t) and not absolute angular displacements θp(t) and θp(t); all 6
motion terms are determined in the numerical method.
4.3
Semi-Analytical Solutions to the SDOF Spur Gear Dynamic Formulation
Consider the example case with only the mean load Te, i.e. T(t) = 0 including
ε (t ) = 0 . Equation (4.1) can be rewritten over the mesh cycle 0 ≤ t ≤ tc as follows:
97
meδ (t ) + ⎡⎣c0 (t )δ (t ) + k0 (t )δ (t ) ⎤⎦ [1 + E1 + E3t ]
,
⎡
⎤
L
+ ⎣⎡c1 (t )δ (t ) + k1 (t )δ (t ) ⎦⎤ ⎢1 + ( E2 + E3t ) ⋅ sgn(t − AP ) ⎥ = Fe
Ω p rbp ⎥⎦
⎢⎣
(4.2a)
E1 = µ ⎡⎣( LXA + λ ) J g rbp + LYC J p rbg ⎤⎦ / J b ,
(4.2b)
E2 = µ ⎡⎣ LXA J g rbp + ( LYC + λ ) J p rbg ⎤⎦ / J b ,
(4.2c)
E3 = µΩ p rbp ( J g rbp − J p rbg ) / J b , me = J e / J b ,
Fe = Te / J b .
(4.2d-f)
Next express Eq. (4.2) in terms of the dimensionless time τ = t / tc , such that
δ ′(τ ) = dδ / dτ = tcδ (t ) and δ ′′(τ ) = d 2δ / d 2τ = tc2δ (t ) :
meδ ′′(τ ) + tc [ c0 (τ )δ ′(τ ) + tc k0 (τ )δ (τ ) ][1 + E1 + tc E3 f 0 (τ )]
,
(4.3a)
f 0 (τ ) = mod(τ ,1) , f1 (τ ) = sgn [ mod(τ ,1) − τ P ] = sgn [ f 0 (τ ) − τ P ] ,
(4.3b-c)
f 2 (τ ) = mod(τ ,1)sgn [ mod(τ ,1) − τ P ] = f 0 (τ ) f1 (τ ) , τ P = LAP / λ .
(4.3d-e)
+ tc [ c1 (τ )δ ′(τ ) + tc k1 (τ )δ (τ )][1 + E2 f1 (τ ) + E3 f 2 (τ )] = tc2 Fe
Each periodic function, ki (τ ) , ci (τ ) , f i (τ ) or δ (τ ) now has a period of T = 1; Figure
4.1(b) shows the typical f 0 (τ ) , f1 (τ ) and f 2 (τ ) functions, which describe the periodic
moment arm and sliding friction excitations for the example case.
98
4.3.1
Direct Application of Multi-Term Harmonic Balance (MHBM)
Define the Fourier series expansions of the periodic ki (τ ) and ci(t) in Eq. (4.3) up
to N mesh harmonics as follows, where ωn = 2π n (in rad/s) and n is the mesh order.
N
N
n =1
n =1
ki (τ ) = Aki 0 + ∑ Akin cos(ωnτ ) + ∑ Bkin sin(ωnτ ) .
(4.4a)
1
1
1
0
0
0
Aki 0 = ∫ ki (τ )dτ , Akin = 2∫ kin cos(ωnτ )dτ , Bkin = 2∫ kin sin(ωnτ )dτ .
N
N
n =1
n =1
ci (τ ) = 2ζ ki (τ ) I e = Aci 0 + ∑ Acin cos(ωnτ ) + ∑ Bcin sin(ωnτ ) .
1
1
1
0
0
0
Adi 0 = ∫ ci (τ )dτ , Acin = 2∫ cin cos(ωnτ )dτ , Bcin = 2∫ cin sin(ωnτ )dτ .
(4.4b-d)
(4.5a)
(4.5b-d)
The fi(τ) functions could be expanded explicitly as shown below:
f 0 (τ ) =
1 N 1
sin(ωnτ ) ,
−∑
2 n =1 nπ
N
4sin (ωnτ P )
n =1
ωn
f1 (τ ) = 1 − 2τ P − ∑
(4.6a)
N 4 ⎡ cos ( ω τ ) − 1⎤
n P
⎦ sin(ω τ ) , (4.6b)
cos(ωnτ ) + ∑ ⎣
n
n =1
ωn
⎧N
⎫
⎡⎣1 − ωnτ P sin (ωnτ P ) − cos (ωnτ P ) ⎤⎦ cos(ωnτ ) + ⎪
∑
⎪
1
4 ⎪ n=1
⎪
f 2 (τ ) = − τ P2 + 2 ⎨ N
⎬ . (4.6c)
ωn ⎪
2
⎡⎣ωnτ P cos (ωnτ P ) − sin (ωnτ P ) − 0.5ωn ⎤⎦ sin(ωnτ ) ⎪
⎪⎩∑
n =1
⎭⎪
99
Finally, assume that the periodic dynamic response δ (τ ) is of the following form:
N
N
n =1
n =1
δ (τ ) = Aδ 0 + ∑ Aδ n cos(ωnτ ) + ∑ Bδ n sin(ωnτ ) .
(4.7)
Substitute Fourier series expansions of Eqs. (4.4-4.7) into Eq. (4.3) and balance
the mean and harmonic coefficients of sin(ωnτ ) and cos(ωnτ ) . This converts the linear
periodic differential equation into easily solvable linear algebraic equations (as expressed
below) where K h is a square matrix of dimension (2N+1) consisting of known
coefficients of ki (τ ) , ci (τ ) and f i (τ ) . By calculating the inverse of K h , the 2N+1 Fourier
coefficients of δ (τ ) could be computed at any gear mesh harmonic (n).
⎡ Aδ 0 ⎤ ⎡ Fe ⎤
⎢A ⎥ ⎢0⎥
⎢ δ1 ⎥ ⎢ ⎥
⎢B ⎥ ⎢ 0 ⎥
⎡⎣K h ⎤⎦ ⎢ δ 1 ⎥ = ⎢ ⎥ .
⎢ ... ⎥ ⎢ ... ⎥
⎢ Aδ N ⎥ ⎢ 0 ⎥
⎢
⎥ ⎢ ⎥
⎢⎣ Bδ N ⎥⎦ ⎣ 0 ⎦
(4.8)
4.3.2 Semi-Analytical Solutions Based on One-Term HBM
Next, we construct one-term HBM [4.13] solutions to conceptually illustrate the
method. Set the harmonic order N = 1 (only the fundamental mesh, in addition to the
mean term) in Eqs. (4.4-4.8) and balance the harmonic terms in Eq. (4.3). This leads to a
100
K h matrix of dimension 3. Three of its typical coefficients are given as follows and the
rest could be found in a similar manner:
1 ⎛ tc Ak 11 Af 21 E3 + Bk 11 B f 11 E2 − (tc Bk 01 E3 / π ) + 2 Ak 00 + 2 Ak10 + Ak11 Af 11 E2 ⎞
K h11 = ⎜
⎟⎟
2 ⎜⎝ +2 Ak 00 E1 + tc Ak 00 E3 + 2 Ak 10 Af 10 E2 + 2tc Ak10 Af 20 E3 + tc Bk 11 B f 21 E3
⎠
K h 21 = Ak 01 + Ak 11 + A01 E1 + Ak10 A f 11 E2 + tc Ak 10 Af 21 E3 + Ak 11 Af 10 E2
+ tc Ak11 A f 20 E3 + ( tc Ak 01 E3 / 2 )
(4.9a)
(4.9b)
K h 31 = Bk 11 A f 10 E2 + Bk 11 + tc Bk11 Af 20 E3 + Ak10 B f 11 E2 + tc Ak10 B f 21 E3 + Bk 01 E1 + Bk 01
−
tc
π
Ak 00 E3 +
(4.9c)
tc Bk 01E3
2
The Fourier series coefficients of δ (τ ) are then obtained by inverting K h . Figure
4.3 shows that one-term HBM solution predicts the overall tendency (mean and first
harmonic) fairly well when compared with numerical simulations at Tp = 550 lb-in and
Ω p = 500 RPM. This confirms that the one-term HBM (and likewise the MHBM)
approach coverts the periodic differential Eq. (4.3) with multiple interacting coefficients
into simpler algebraic calculations that are computationally more efficient than numerical
integrations and subsequent FFT analyses. Thus, the semi-analytical solution provides an
effective design tool. Also, most coefficients of Eq. (4.9) show side-band effects that are
introduced by k(t) (or c(t)) and the fi(t) functions.
101
10.4
x 10-4
(in)
10
9.6
9.2
0
2
1
t/tc
(in)
(a)
(b)
Figure 4.3 Semi-analytical vs. numerical solutions for the SDOF model, expressed by Eq.
(4.3), given Tp = 550 lb-in, Ω p = 500 RPM, µ = 0.04. (a) Time domain responses; (b)
, , numerical simulations;
, ,
mesh harmonics in frequency domain. Key:
, , semi-analytical solutions using
semi-analytical solutions using one-term HBM;
5-term HBM.
102
4.3.3 Iterative MHBM Algorithm
When N ≤ 5, we can utilize a symbolic software [4.14] to balance multiple
harmonic terms and calculate K h . However, the computational cost involved with each
element of K h increases by N3 due to the triple multiplication of periodic coefficients in
Eq. (4.3). Consequently, for higher N (say >5), a direct computation of K h becomes
inefficient and thus inadvisable. Instead, we apply a matrix-based iterative MHBM
algorithm [4.15, 4.16]. First, define variables Ω and ϑ:
Ω = 2π / (υ tc ) , ϑ = Ωt ∈ [ 0, 2π ) , ϑ = mod (υϑ / 2π , 1) .
(4.10a-c)
where υ is the sub-harmonic index; also define the differential operator “^” as:
xˆ =
dx
= Ω −1 x ,
dϑ
x = Ωxˆ .
(4.11a,b)
Equation (4.3) is then converted into the following form:
ˆ
Ω 2 meδˆ (ϑ ) + ⎡⎣ Ωc0 (ϑ ) δˆ (ϑ ) + k0 (ϑ ) δ (ϑ ) ⎤⎦ ⎡⎣1 + E1 + E4ϑ ⎤⎦
+ ⎡⎣Ωc1 (ϑ ) δˆ (ϑ ) + k1 (ϑ ) δ (ϑ ) ⎤⎦ ⎡⎣1 + ( E2 + E4ϑ ) sgn ( λϑ / LAP − 1) ⎤⎦ = Fe
Or, express it more compactly as:
103
,
(4.12)
~
~
ˆ
Ω 2 meδˆ(ϑ ) + ΩC (ϑ )δˆ(ϑ ) + K (ϑ )δ (ϑ ) = Fe ,
(4.13a)
C (ϑ ) = c0 (ϑ )(1 + E1 + E4ϑ ) + c1 (ϑ ) ⎡⎣1 + ( E2 + E4ϑ ) sgn ( λϑ / LAP − 1) ⎤⎦ ,
(4.13b)
K (ϑ ) = k0 (ϑ )(1 + E1 + E4ϑ ) + k1 (ϑ ) ⎡⎣1 + ( E2 + E4ϑ ) sgn ( λϑ / LAP − 1) ⎤⎦ .
(4.13c)
For the MHBM, a discrete Fourier transform (DFT) matrix is formed as follows, where
ϑi = i 2π / M and M ≥ 2 N + 1 :
⎡1 sin (ϑ1 ) cos (ϑ1 ) … sin ( Nϑ1 ) cos ( Nϑ1 ) ⎤
⎢
⎥
1 sin (ϑ2 ) cos (ϑ2 ) … sin ( Nϑ2 ) cos ( Nϑ2 ) ⎥
⎢
,
F=
⎢
⎥
⎢
⎥
⎣⎢1 sin (ϑM ) cos (ϑM ) … sin ( NϑM ) cos ( NϑM ) ⎦⎥
⎧ δˆˆ ϑ ⎫
⎧ δˆ (ϑ1 ) ⎫
⎧ δ (ϑ1 ) ⎫
⎪ ( 1) ⎪
⎪
⎪
⎪
⎪
ˆ
⎪ δˆ (ϑ2 ) ⎪
ˆˆ ⎪⎪ δˆ (ϑ2 ) ⎪⎪
⎪ δ (ϑ2 ) ⎪
2
ˆ
,
,
≡
δ
δ ≡⎨
=
F
∆
δ
≡
=
FD
∆
⎨
⎬ = FD ∆
⎬
⎨
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪ˆ
⎪
⎪δ (ϑM ) ⎪
⎪δˆ ϑ ⎪
⎩
⎭
⎩ ( M )⎭
⎪⎩δˆ (ϑM ) ⎪⎭
(4.14a)
(4.14b-d)
Here, ∆ is a vector of 2 N + 1 Fourier coefficients; and the Fourier differentiation
matrix is given as:
104
⎡0
⎢0
⎢
⎢0
D=⎢
⎢
⎢0
⎢
⎣0
0 0 …
0 −1 …
1 0 …
0
0
0
0
0
0
… 0
… N
0
0
0 ⎤
0 ⎥⎥
0 ⎥
⎥,
0 ⎥
−N ⎥
⎥
0 ⎦
⎡0 0 0
⎢ 0 −1 0
⎢
⎢0 0 −1
2
D =⎢
⎢
⎢0 0 0
⎢
⎣0 0 0
…
…
…
0
0
0
0
… −N 2
…
0
0 ⎤
0 ⎥⎥
0 ⎥
⎥ (4.14e,f)
0 ⎥
0 ⎥
⎥
−N 2 ⎦
Applying the DFT to the equation of motion yields the following MHBM equations
+
where F is the Moore-Penrose or pseudo-inverse of the DFT matrix:
⎡Ω 2 me D2 + ΩF + CF + F + KF ⎤ ∆ = F ,
⎣
⎦
)
(4.15b)
C ≡ diag {C (ϑ1 ) C (ϑ2 ) … C (ϑM )} .
(4.15c)
K ≡ diag
({K (ϑ )
(4.15a)
1
K (ϑ2 ) … K (ϑM )} ,
(
)
Figure 4.3 shows that the five-term HBM solutions compare well with numerical
simulations. Likewise, an increase in N captures higher frequency components around the
10th mesh harmonic, as observed in the numerical simulations. The semi-analytical
solutions are efficiently used in Figure 4.4 for parametric studies of δ(t) at the gear mesh
harmonics over a range of Ω p ; and, these calculations are indeed achieved with reduced
computational cost.
105
Ωp
Ωp
Ωp
Ωp
Figure 4.4 Semi-analytical vs. numerical solutions for the SDOF model as a function of
pinion speed with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key: ,
numerical simulations;
, semi-analytical solutions using 5-term HBM.
106
4.4
Analysis of Sub-Harmonic Response and Dynamic Instability
Vaishya and Singh [4.5] examined the parametric instability of a spur gear pair
(with equal load sharing) via the sub-harmonic analysis. A similar approach is
implemented along with following improvements. First, since realistic rather than
rectangular k(t) profile is examined, system stability could now be evaluated as a function
of Tp with contributions from profile modifications. Second, the sub-harmonic matrix was
constructed earlier [4.5] as an external forcing function without the frictional effects. In
the proposed work, sliding friction is characterized as a parametric excitation and then its
effect on instability is examined. Re-expand δ (τ ) in terms of the first sub-harmonic term
at 0.5ω0 and higher sub-harmonics such as 1.5ω0 :
δ u (τ ) = Aδ 0.5 cos(ω0.5τ ) + Bδ 0.5 sin(ω0.5τ ) + Aδ 1.5 cos(ω1.5τ ) + Bδ 1.5 sin(ω1.5τ )
(4.16)
Substitute δ u (τ ) into Eq. (4.3) and expand coefficients at sub-harmonic frequencies. By
balancing harmonics at 0.5ω0 and 1.5ω0 , the following equation is obtained
⎡ Aδ 0.5 ⎤ ⎡ 0⎤
⎢ B ⎥ ⎢0⎥
⎡⎣K u ⎤⎦ ⎢ δ 0.5 ⎥ = ⎢ ⎥ .
4× 4 ⎢ A
⎥ ⎢0⎥
δ 1.5
⎢
⎥ ⎢ ⎥
⎣ Bδ 1.5 ⎦ ⎣ 0⎦
107
(4.17)
For a non-trivial (unstable) solution to the above homogeneous equation, matrix
K u must be singular. Such points correspond to period-doubling instability [4.5, 4.17]
and are found by computing the determinant K u as a function of the ratio of mesh
frequency ωn and the natural frequency Ω NS of the corresponding linear time-invariant
SDOF system, which could be estimated from I e and time-averaged mesh stiffness.
Figure 4.5 shows the normalized K u for the example case given Tp = 100 lb-in (light
load) and Tp = 550 lb-in (design load); each is calculated for four damping ratios ζ . The
zero-crossing points of the K u curve suggest the onset of instability [4.5]. Note that the
unstable zone depends on the value of Tp , which also influences k(t) and the “effective”
contact ratio. For instance, larger period-doubling unstable zone is observed under a light
load condition in Figure 4.5(a), as compared with the design loading condition in Figure
4.5(b). Further, instability could be effectively controlled by an increase in ζ. For
example, when Tp = 550 lb-in, 2% value is sufficient to stabilize the system under
period-doubling condition; however, about 8% is needed at Tp = 100 lb-in to achieve
stability under the same condition. Also, a variation in µ seems to have negligible
influence on K u ; however, in reality the energy dissipated by the sliding friction is
usually embedded in an equivalent ζ. Thus an increase in µ should enhance the stability.
108
|Ku|
(a)
(b)
Figure 4.5 Normalized determinant of the sub-harmonic matrix K u as a function of
ωn / Ω NS with µ = 0.04: (a) Tp = 100 lb-in.; (b) Tp = 550 lb-in. Key:
= 0.01;
,ζ = 0.05;
, ζ = 0.1.
109
, ζ = 0;
,ζ
4.5
Semi-Analytical Solutions to 6DOF Spur Gear Dynamic Formulation
When the excitation (mesh) frequencies do not coincide with any natural
frequency, the semi-analytical solution, δ (τ ) = rbpθ p (τ ) − rbgθ g (τ ) , of the SDOF model
could approximate the DTE, δ (τ ) = rbpθ p (τ ) − rbgθ g (τ ) + x p (τ ) − xg (τ ) , for a 6DOF spur
gear system of Chapter 2, where x(τ) and y(τ) are the bearing displacements in the LOA
and OLOA directions respectively. First, the distribution of natural frequencies is
examined by using a 6DOF linear time-invariant model of Figure 4.6(a). Since the OLOA
motions yp(τ) and yg(τ) are decoupled from other DOFs, we will focus on the coupling (in
terms of DTE) between transverse and torsional motions in the LOA direction. This leads
to a simplified 3DOF system model [4.18] in terms of xp(τ), xg(τ) and δ (t ) . Define the
following parameters for the example case: Mass of pinion (gear) m p = mg = M ; moment
of inertia J p = J g = J ; basic radius rbp = rbg = R ; the equivalent mass me = J /(2 R 2 ) ;
time-averaged mesh stiffness km (Tp=550 lb-in.); shaft-bearing stiffness K Bp = K Bg = K B .
The natural frequencies of the 3DOF system are found as follows [4.18].
[ km M + (2km + K B )me ] ± [ km M + (2km + K B )me ]
2
Ω
2
N 1, N 3
=
2Mme
Ω 2N 2 =
KB
.
M
110
− 4 Mme km K B
, (4.18a)
(4.18b)
θg
J,M
km
KB
KB
KB
J,M
x
KB
θp
y
θ
(a)
(b)
Figure 4.6 (a) 6DOF spur gear pair model and its subset of unity gear pair (3DOF model)
used to study the natural frequency distribution; (b) Natural frequencies ΩN as a function
, ΩSN of SDOF system (torsional only, in terms
of the stiffness ratio K B / km . Key:
of DTE);
, ΩN1 of 3DOF system;
, ΩN2 of 3DOF system;
, ΩN3 of 3DOF
system.
111
Mode 1 ( Ω N 1 ) and Mode 3 ( Ω N 3 ) correspond to the first and second coupled
transverse-torsional modes, respectively; and Mode 2 ( Ω N 2 ) is the purely transverse
mode. Also, the natural frequency of the corresponding SDOF torsional system can be
estimated as Ω NS = km / me . Figure 4.6(b) compares the natural frequencies ( Ω N ) of the
3DOF and SDOF models as a function of the stiffness ratio K B / km . For easy comparison
with the excitation (mesh) frequency ωn , which is determined by the nominal pinion RPM,
all natural frequencies are converted in Figure 4.6(b) from rad/s into RPM units. Observe
that Ω NS asymptotically approaches Ω N 3 (or Ω N 1 ) when K B / km < 1 (or K B / km > 10 ).
Moreover, ωn does not excite any resonance in Zone I ( ωn << Ω N 1 ) and Zone III
( ωn >> Ω N 3 ). Additionally, non-resonant Zone II could be found for both “soft”
( K B / km < 1 , Ω N 2 << ωn << Ω NS ) and “stiff” ( K B / km > 10 , Ω NS << ωn << Ω N 2 ) shaftbearing cases. In these non-resonant zones, the semi-analytical solution δ(t) of the SDOF
model could be extended to the 6DOF system. Figure 4.7 compares the 5-term HBM
prediction of δ(t) for the SDOF system with numerical simulation for a 6DOF model for
two limiting value of K B / km . When K B / km = 100 , prediction matches well with
numerical simulation. However, when K B / km = 0.37 (i.e. nominal case of Chapter 2),
good correlation is observed only away from system resonances Ω N in Zones I, II and III.
112
Ωp
Ωp
Ωp
Ωp
Figure 4.7 Semi-analytical vs. numerical solutions for the 6DOF model as a function of
Ω p with µ = 0.04. (a) Mesh order n = 1; (b) n = 2; (c) n = 3; (d) n = 4. Key:
,
predictions using five-term HBM; , numerical simulations with nominal KB (KB/km =
0.37); , numerical simulations with stiff KB (KB/km = 100).
113
The dynamic normal (mesh) loads N i (τ ) of the pinion and gear are equal in
magnitude but opposite in direction. To account for interactions between ki (τ ) and δ (τ )
as well as between ci (τ ) and δ (τ ) , Fourier series is expanded to find the N i (τ ) terms up
to 2N mesh harmonics.
2N
2N
n =1
n =1
2N
2N
n =1
n =1
N 0 (τ ) = k0 (τ )δ (τ ) + c0 (τ )δ (τ ) = AN 00 + ∑ AN 0 n cos(ωnτ ) + ∑ BN 0 n sin(ωnτ ) , (4.19a)
N1 (τ ) = k1 (τ )δ (τ ) + c1 (τ )δ (τ ) = AN 10 + ∑ AN 1n cos(ωnτ ) + ∑ BN 1n sin(ωnτ ) . (4.19b)
Similarly, the dynamic friction forces Ffi (τ ) are expanded in Eq. (4.20). Since
tooth pair #1 is associated with a periodic change of friction force at the pitch point,
Ff 1 (τ ) is expanded up to 3N mesh harmonics due to a multiplication of k1 (τ ) , δ (τ ) and
f1 (τ ) .
2N
2N
n =1
n =1
Ff 0 (τ ) = µ N 0 (τ ) = AF 00 + ∑ AF 0 n cos(ωnτ ) + ∑ BF 0 n sin(ωnτ ) ,
3N
3N
n =1
n =1
Ff 1 (τ ) = µ N1 (τ ) f1 (τ ) = AF 10 + ∑ AF 1n cos(ωnτ ) + ∑ BF 1n sin(ωnτ ) .
(4.20a)
(4.20b)
In the LOA (or x) direction, the transfer function (at frequency ω a) with Np(τ) as
input and x p (τ ) as output is found by using the corresponding linear time-invariant model
as follows, where K pBx and ζ pBx are the shaft-bearing stiffness and damping terms.
114
tc2
(ω ) =
.
Np
(−ω 2 m p + K pBx tc2 ) + j 2ωtcζ pBx K pBx m p
Xp
(4.21a)
Magnitude M pxn and phase α pxn at the nth mesh order, ωn = 2π nτ (rad/s), are
M px (ωn ) =
tc2
2
(−ωn2 m p + K pBx tc2 )2 + 4ωn2tc2ζ pBx
K pBx m p
α px (ωn ) = tan −1
2ωn tcζ pBx K pBx m p
ωn2 m p − K pBx tc2
,
.
(4.21b)
(4.21c)
Thus the pinion bearing displacement x p (τ ) in the LOA (or x) direction is expanded as
2N
x p (τ ) = M px (0)( AN 00 + AN 10 ) + ∑ M px (ωn )( AN 0 n + AN 1n ) cos ⎡⎣ωnτ + α px (ωn ) ⎤⎦
n =1
. (4.22)
2N
+ ∑ M px (ωn )( BN 0 n + BN 1n ) sin ⎡⎣ωnτ + α px (ωn ) ⎤⎦
n =1
In the OLOA (or y) direction, the magnitude and phase of the transfer function
from friction force Ff(τ) to pinion displacement y p (τ ) could be found at ωn as
M py (ωn ) =
tc2
2
(−ωn2 m p + K pBy tc2 ) 2 + 4ωn2tc2ζ pBy
K pBy m p
α py (ωn ) = tan −1
2ωn tcζ pBy K pBy m p
ωn2 m p − K pBy tc2
115
.
,
(4.23a)
(4.23b)
Thus the pinion displacement y p (τ ) in the OLOA (or y) direction is expanded as:
y p (τ ) = M py (0) AF 00 + M py (0) AF 10
2N
2N
+ ∑ M py (ωn ) AF 0 n cos ⎡⎣ωnτ + α py (ωn ) ⎤⎦ + ∑ M py (ωn ) BF 0 n sin ⎡⎣ωnτ + α py (ωn ) ⎤⎦ . (4.24)
n =1
n =1
3N
3N
n =1
n =1
+ ∑ M py (ωn ) AF 1n cos ⎡⎣ωnτ + α py (ωn ) ⎤⎦ + ∑ M py (ωn ) BF 1n sin ⎡⎣ωnτ + α py (ωn ) ⎤⎦
Equations (4.22, 4.24) confirm that multiplications between periodic coefficients
ki (τ ) (or ci (τ ) ), δ i (τ ) (or ci (τ ) ) and f i (τ ) lead to higher mesh harmonic components
which are commonly observed in spur gears [4.8].
4.6
Conclusion
Figure 4.8 compares the semi-analytical x p (τ ) with numerical prediction as a
function of Ω p with KB/km = 100. Good correlations are observed up to 16,000 RPM (in
Zone 1 with high KB/km) including the LOA shaft-bearing resonances at n = 3 and n = 4.
Likewise, the semi-analytical y p in the OLOA direction compares well with numerical
simulations in Figure 4.9, where the shaft-bearing resonances are also observed at n = 3
and n = 4.
116
Ωp
Ωp
Ωp
Ωp
Figure 4.8 Semi-analytical vs. numerical solutions of the LOA displacement xp for the
6DOF model as a function of Ω p with KB/km = 100, µ = 0.04 (a) Mesh order n = 1; (b) n
= 2; (c) n = 3; (d) n = 4. Key: , numerical simulations;
, predictions using five-term
HBM.
117
Ωp
Ωp
Ωp
Ωp
Figure 4.9 Semi-analytical vs. numerical solutions of the OLOA displacement yp for the
6DOF model as a function of Ω p with KB/km = 100, µ = 0.04. (a) Mesh order n = 1; (b) n
, predictions using five-term
= 2; (c) n = 3; (d) n = 4. Key: , numerical simulations;
HBM.
118
Although good correlations are usually expected for “stiff” shaft-bearings, it is
worthwhile to examine the case where KB is comparable or less than km. By using the
nominal parameters of Chapter 2 where KB/km = 0.37, semi-analytical predictions are
compared with numerical simulations in Figure 4.10 and Figure 4.11 for LOA and OLOA
responses, respectively. Observe that semi-analytical x p matches numerical simulations
only from system resonances Ω N , as explained earlier in Figure 4.6. Nonetheless, good
correlation is observed in the OLOA direction over the operating speed range in Figure
4.11 since the bearing resonance dictates the y p dynamics.
Overall, this chapter has successfully developed semi-analytical solutions to
periodic differential equations with time-varying parameters of spur gears including
realistic mesh stiffness and sliding friction. Proposed one-term and multi-term HBM
predictions compare well with numerical simulations; the computational efficiency is
achieved by converting the periodic differential equations into easily solvable algebraic
equations, while providing more insight into the dynamic behavior. Both super-and subharmonic analyses are successfully conducted to examine the higher mesh harmonics due
to multiplicative coefficients and the system stability, respectively. Finally, semianalytical solutions are developed for a 6DOF system model for the predictions of
(normal) mesh loads, friction forces and bearing displacements in the LOA and OLOA
directions, under non-resonant conditions. Methods of this work could be extended to
multi-mesh spur gear dynamics.
119
1.2
x 10-4
ΩN1
Z1
0.8
ΩN 2
x 10-5
2
Ω NS
(a)
Z2
(b)
Z3
1
ΩN 3
0.4
0
6
0
5
x 10-5
10
Ω p(RPM)
15 x103
0
0
5
x 10-5
10
Ω p(RPM)
(c)
15 x103
(d)
0.8
4
0.4
2
0
0
5
10
Ω p(RPM)
15 x103
0
0
5
10
Ω p (RPM)
15x103
Figure 4.10 Semi-analytical vs. numerical solutions of the LOA displacement xp for the
6DOF model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n = 1; (b) n
= 2; (c) n = 3; (d) n = 4. Key: , numerical simulations;
, predictions using five-term
HBM.
120
Ωp
Ωp
Ωp
Ωp
Figure 4.11 Semi-analytical vs. numerical solutions of the OLOA displacement yp for the
6DOF model as a function of Ω p with KB/km = 0.37, µ = 0.04. (a) Mesh order n = 1; (b) n
= 2; (c) n = 3; (d) n = 4. Key: , numerical simulations;
, predictions using five-term
HBM.
121
References for Chapter 4
[4.1] Richards, J. A., 1983, Analysis of Periodically Time-varying Systems, New York,
Springer.
[4.2] Jordan, D. W., and Smith, P., 2004, Nonlinear Ordinary Differential Equations, 3rd
Edition, Oxford University Press.
[4.3] Thomsen, J. J., 2003, Vibrations and Stability, 2nd Edition, Springer.
[4.4] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration,
243(3), pp. 525-545.
[4.5] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and
Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671694.
[4.6] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear
Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.
[4.7] External2D (CALYX software), 2003,
www.ansol.www, ANSOL Inc., Hilliard, OH.
“Helical3D
User’s
Manual,”
[4.8] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of
Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.
[4.9] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on
the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of
Mechanical Design, 122(4), pp. 515-522.
[4.10] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction
Excitations in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp 641658.
[4.11] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body
Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3),
pp. 479-499.
[4.12] Velex, P., and Ajmi, M., 2007, “Dynamic Tooth Loads and Quasi-Static
Transmission Errors in Helical Gears – Approximate Dynamic Factor Formulae,”
Mechanism and Machine Theory Journal, 42(11), pp. 1512-1526.
[4.13] Kim, T. C., Rook, T. E., and Singh, R., 2005, “Super- and Sub-Harmonic
Response Calculations for A Torsional System with Clearance Non-Linearity using
Harmonic Balance Method,” Journal of Sound and Vibration, 281(3-5), pp. 965-993.
122
[4.14] Maple 10 (symbolic software), 2005, Waterloo Maple Inc., Waterloo, Ontario.
[4.15] Padmanabhan, C., Barlow, R. C., Rook, T.E., and Singh, R., 1995, “Computational
Issues Associated with Gear Rattle Analysis,” ASME Journal of Mechanical Design, 117,
pp. 185-192.
[4.16] Duan, C., and Singh, R., 2005, “Super-Harmonics in a Torsional System with Dry
Friction Path Subject to Harmonic Excitation under a Mean Torque”, Journal of Sound
and Vibration, 285(2005), pp. 803-834.
[4.17] Den Hartog, J. P., 1956, Mechanical Vibrations, New York, Dover Publications.
[4.18] Kahraman, A., and Singh, R., 1991, “Error Associated with A Reduced Order
Linear Model of Spur Gear Pair,” Journal of Sound and Vibration, 149(3), pp. 495-498.
[4.19] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared
Systems,” Technical Report submitted to the Army Research Office, grant number
DAAD19-02-1-0334.
123
CHAPTER 5
EFFECT OF SLIDING FRICTION ON THE VIBRO-ACOUSTICS OF SPUR
GEAR SYSTEM
5.1
Introduction
Gears are known to be one of the major vibro-acoustic sources in many practical
systems including ground and air vehicles such as heavy-duty trucks and helicopters.
Typically, steady state gear (whine) noise is generated by several sources and the
reduction of gear noise is often challenging for most products. Virtually all prior
researchers [5.1-5.3] have assumed the main exciter to be the static transmission error
(STE) that is defined as the derivation from the ideal tooth profile induced by
manufacturing errors and elastic deformations. However, high precision gears are still
unacceptably noisy in practice. When the transmission error has been minimized (say via
modifying the tooth profile), the sliding friction remains as a potential contributor to gear
noise and vibration. Further, most prior research on gear friction [5.4-5.5] has been
confined to the dynamic analysis of the gear pair source sub-system and no attempt has
been made to examine the friction related structural path and noise radiation issues. To
124
fill in this void, the main objectives of this chapter are thus to: First, propose a refined
source-path-receiver model that characterizes the structural paths in two directions and,
second, propose analytical tools to efficiently predict the whine noise and quantify the
contribution of sliding friction to the overall whine noise. The system model is depicted
in Figure 5.1. The source sub-system includes the spur gear pair and shafts inside the
gearbox; these are characterized by a 6 degree-of-freedom (DOF) linear-time-varying
model of Chapter 2. The transmission error dominated bearing forces in the line-of-action
(LOA) direction and friction dictated bearing forces in the off line-of-action (OLOA)
direction are coupled and transmitted to the housing structure. Radiated sound pressure
p(ω) from gearbox panels (at gear mesh frequencies) are then received by microphone(s).
Analytical predictions of the structural transfer function and noise radiation will be
compared with measurements.
SOURCE
Transmission
error
Sliding
friction
LOA bearing forces
6 DOF linear-timevarying spur gear
pair model + shafts
OLOA bearing forces
RECEIVER
Sound
pressure
Coupling at
bearings
PATH
Housing
velocity
Radiation
model
Housing structure
model
Figure 5.1 Block diagram for the vibro-acoustics of a simplified geared system with two
excitations at the gear mesh.
125
5.2
Source Sub-System Model
The source sub-system is described by the 6DOF, linear time-varying spur gear
pair model of Chapter 2 that incorporates the sliding friction and realistic mesh stiffness,
which is calculated by an accurate finite element/contact mechanical code [5.6]. Rigid
bearing is assumed as boundary conditions due to the impedance mismatch at the
shaft/bearing interface. Overall, the system formulations are summarized as following.
The governing equations for the torsional motions θp(t) and θg(t) of pinion and gear are:
n
n
i =0
i =0
n
n
i =0
i =0
J pθp (t ) = Tp + ∑ X pi (t ) Fpfi (t ) − ∑ rbp N pi (t )
J gθg (t ) = −Tg + ∑ X gi (t ) Fgfi (t ) + ∑ rbg N gi (t )
(5.1)
(5.2)
where n = floor(σ) in which the “floor” function rounds off the contact ratio σ to the
nearest integer (towards a lower value); Jp and Jg are the polar moments of inertia of the
pinion and gear; Tp and Tg are the external and braking torques; rbp and rbg are base radii
of the pinion and gear; and, Npi(t) and Ngi(t) are the normal loads defined as follows:
N pi (t ) = N gi (t ) = ki (t ) ⎡⎣ rbpθ p (t ) − rbgθ g (t ) + x p (t ) − xg (t ) ⎤⎦
+ ci (t ) ⎡⎣ rbpθp (t ) − rbgθg (t ) + x p (t ) − x g (t ) ⎤⎦
126
(5.3)
where ki(t) and ci(t) are the realistic mesh stiffness and viscous damping profiles; xp(t)
and xg(t) denote the LOA displacements of pinion and gear centers. The sliding friction
forces Fpfi(t) and Fgfi(t) as well as their moment arms Xpi(t) and Xgi(t) of the ith meshing
pair are derived as:
Fpfi (t ) = µi (t ) N pi (t ) ,
Fgfi (t ) = µi (t ) N gi (t )
X pi (t ) = LXA + (n − i )λ + mod(Ω p rbp t , λ ),
X gi (t ) = LYC + iλ − mod(Ω g rbg t , λ )
(5.4a,b)
(5.5a)
(5.5b)
where the sliding friction is formulated by µi (t ) = µ0 sgn ⎡⎣ mod(Ω p rbp t , λ ) + ( n − i)λ − LAP ⎤⎦ ; λ
is the base pitch; “sgn” is the sign function; the modulus function mod(x, y) = x –
y ·floor(x/y), if y ≠ 0; Ωp and Ωg are the nominal speeds (in rad/s); and, LAP, LXA and LYC
are geometric length constants of Chapter 2. The governing equations for xp(t) and xg(t)
motions in the LOA direction are:
n
m p x p (t ) + 2ζ pSx K pSx m p x p (t ) + K pSx x p (t ) + ∑ N pi (t ) = 0
(5.6)
i =0
n
mg xg (t ) + 2ζ gSx K gSx mg x g (t ) + K gSx xg (t ) + ∑ N gi (t ) = 0
(5.7)
i =0
Here, mp and mg are the masses of the pinion and gear; KpSx and KgSx are the effective
shaft stiffness values in the LOA direction, and ζpSx and ζgSx are the damping ratios.
Likewise, the translational motions yp(t) and yg(t) in the OLOA direction are governed by:
127
n
m p y p (t ) + 2ζ pS y K pSy m p y p (t ) + K pSy y p (t ) − ∑ Fpfi (t ) = 0
(5.8)
i =0
n
mg y g (t ) + 2ζ gS y K gSy mg y g (t ) + K gSy y g (t ) − ∑ Fgfi (t ) = 0
(5.9)
i =0
Both LOA and OLOA bearing forces are predicted for the example case (unityratio NASA spur gear pair whose parameters are listed in Table 5.1) and compared in
Figure 5.2 at the first three gear mesh frequencies as a function of pinion torque Tp.
Observe that the friction dominated OLOA dynamic responses are less sensitive to a
variation in Tp.
Parameter/property
Pinion/Gear
Parameter/property
Pinion/Gear
Number of teeth
28
Face width, in
0.25
Diametral pitch, in-1
8
Tooth thickness, in
0.191
Pressure angle, º
20
Center distance, in
3.5
Outside diameter, in
3.738
Elastic modulus, psi
30×106
Root diameter, in
3.139
Shaft stiffness, lb/in
1.29×105
Table 5.1 Parameters of the example case: NASA spur gear pair with unity ratio (with
long tip relief)
128
6
5
m=1
m=2
m=3
|FpBx| (lb)
4
3
2
1
0
500
600
700
800
Torque (lb-in)
900
(a)
7
6
|FpBy| (lb)
5
4
3
2
1
0
500
600
700
800
Torque (lb-in)
900
(b)
Figure 5.2 Bearing forces predicted under varying Tp given Ωp = 4875 RPM and 140 °F.
(a): LOA bearing force; (b) OLOA bearing force. Key: m is the mesh frequency index.
129
5.3
Structural Path with Friction Contribution
5.3.1
Bearing and Housing Models
Predicted bearing forces by the source sub-system provide excitations to the
multi-input multi-output (MIMO) structural paths for the gearbox of Figure 5.3(a). Force
excitations are coupled at each bearing via a 6 by 6 stiffness matrix [K]Bm which is
calculated by using the algorithm proposed by Lim and Singh [5.7]. Nominal shaft loads
and bearing preloads are assumed to ensure a time-invariant [K]Bm. In order to focus on
the transmission error path and frictional path in the LOA and OLOA directions
respectively, [K]Bm is further simplified into a 2 by 2 matrix by neglecting the moment
transfer [5.8] and assuming that no axial force is generated by the spur gear sub-system.
Calculated nominal bearing stiffness [5.7] are KBx = KBy= 2.8×106 lb/in at mean operating
conditions; these are much larger than the shaft stiffness of 1.29×105 lb/in. This is
consistent with the impedance mismatch assumption made at the shaft/bearing interface.
130
(a)
(b)
Figure 5.3 (a) Schematic of NASA gearbox; (b) Finite element model of NASA gearbox
with embedded bearing stiffness matrices.
131
The implementation of [K]Bm into the finite element gearbox model of Figure
5.3(b) is given special attention [5.9-5.10]. At high mesh frequencies (say up to 5 kHz),
the dimensions of the bearings are comparable to the plate flexural wavelength. Hence
the holes may significantly alter the plate dynamics and such effects must be modeled
[5.10]. A rigid (with Young’s modulus 100 times higher than the casing steel) and massless (with density 1% of the casing steel) beam element is used to model the interface
from shaft to bearing. Its length is chosen to be very short to avoid the introduction of any
beam resonances in the frequency range of interest. The shaft beam element is connected
to the central bearing node though orthogonal foundation stiffness (KBx and KBy) in the
LOA and OLOA directions, respectively. The central node is then connected to the
circumferential bearing nodes by 12 rigid and mass-less beams (one at each rolling
element’s angular position) which form a star configuration, such that the displacement
of the plate around the bearing hole are equal to the “housing node” at the center.
5.3.2
Experimental Studies and Validation of Structural Model
The finite element model of Figure 5.3 is created by using I-DEAS for the NASA
gearbox with bearing holes, embedded stiffness matrices [K]Bm, stiffening plates as well
as clamped boundary conditions at four rigid mounts. Although the gear pair and shafts
are not included, it has been shown [5.7, 5.9] that an "empty" gearbox tends to describe
the dynamics of the entire gearbox system. Table 5.2 confirms that the natural
frequencies predicted by the finite element model correlate well with measurements
132
reported by Oswald et al. [5.11] despite minor modifications made to the gearbox. Mode
shape predictions also match well with modal tests, and Figure 5.4 gives a typical
comparison of structural mode at the 8th natural frequency (fn = 2962 Hz).
Method/mode index
Measurements [5.11]
Finite element predictions (Hz)
(Hz)
1
658
650
2
1049
988
3
1709
1859
4
2000
1940
5
2276
2328
6
2536
2566
7
2722
2762
8
2962
2962
Table 5.2 Comparison of measured natural frequencies and finite element predictions
133
(a)
(b)
Figure 5.4 Comparison of the gearbox mode shape at 2962 Hz: (a) modal experiment
result [5.12]; (b) finite element prediction.
134
In order to validate the structural paths, several transfer functions were measured
for the NASA gearbox by assuming that the quasi-static system response is similar to the
response under non-resonant rotating conditions. The gearbox was modified to allow
controlled excitations to be applied to the gear-mesh and measured. Brackets were
welded to the bedplate of the gear-rig to mount shakers in the LOA and OLOA directions
outside the gearbox, as shown in Figure 5.5(a). Stinger rods were connected from the
shakers through two small holes in the gearbox and attached to a collar on the input shaft.
Two mini accelerometers were fastened to a block behind the loaded gear tooth to
measure the LOA and OLOA mesh accelerations. Band-limited random noise signals
were then used as excitation signals and tests were done with only one shaker activated at
a time with a 600 lb-in static preload. Dynamic responses were measured to generate
vibro-acoustic transfer functions. Sensor # 1 of Figure 5.5(a) is a tri-axial accelerometer
mounted on the output shaft bearing cap to measure the LOA, OLOA, and axial
vibrations. Sensors #2 and #3 are unidirectional accelerometers mounted on the top and
back plates, respectively.
The transfer function of the combined source-path sub-systems is predicted as
following:
Y (ω )
H S − P (ω ) = H S (ω ) ⋅ H P (ω ) = H S (ω ) ⋅ plate
Ybearing (ω )
(5.10)
where H S (ω ) is the motion transmissibility from mesh excitation to translational bearing
responses (in LOA or OLOA directions) by using a 6DOF linear time-invariant spur gear
135
model [5.13]. Note that such a lumped model is insufficient to capture the bending and
flexural modes of the gear flanks and shafts. Here, Yplate (ω ) and Ybearing (ω ) are the
transfer and driving point mobilities for the (top) plate and the bearing; these are derived
from the finite element gearbox model by using the modal expansion method with 1%
structural damping for all modes. Figure 5.5(b) shows that the predicted motion
transmissibility from gear mesh to the top plate correlates reasonably well with
measurement given the complexity of the geared system. The highest frequency is chosen
such that the shortest wave-length is 4 times larger than the mesh dimension on the top
plate. Recall that the interactions between the shaft and bearings/casing were neglected in
our model by the impedance mismatch assumption. Consequently, a 10 dB empirical (but
uniformly applied) weighting function w is used to “tune” the magnitude of transfer
mobility prediction in Figure 5.5(b) for a better comparison. Further work is needed to
explain this shift.
136
(a)
40
|H| top plate (dB)
20
0
-20
-40
-60
500
1000
1500
2000
2500
3000
3500
Frequency (Hz)
(b)
Figure 5.5 (a) Experiment used to measure structural transfer functions; (b) comparison
of transfer function magnitudes from gear mesh to the sensor on top plate. Key:
,
measurements; , predictions,
.
137
5.3.3
Comparison of Structural Paths in LOA and OLOA Directions
First, assume that (i) the bearing forces predicted by the lumped source model
[5.6] are in phase at either bearing end for both the pinion and gear shafts; and (ii) the
bearing forces of pinion and gear are same in magnitude but opposite in directions due to
the symmetry of unity ratio gear pair. Second, the overall structural paths are derived for
the transmission error controlled LOA (or x) path and the friction dominated OLOA (or y)
path in terms of combined effective transfer mobilities Ye , x (ω ) and Ye , y (ω ) :
Ye , x (ω ) = ∑ wp , x ,nYp , x ,n (ω ) − ∑ wg , x , nYg , x ,n (ω )
(5.11a)
Ye , y (ω ) = ∑ wp , y ,nYp , y , n (ω ) − ∑ wg , y , nYg , y ,n (ω )
(5.11b)
n
n
n
n
where w is the empirical weighting function (10 dB, as discussed in the previous section);
and the subscript n is the index of the two ends of pinion/gear shafts. Figure 5.6 compares
the magnitudes of Ye , x (ω ) and Ye , y (ω ) at the sensor location on the top plate. Different
peaks are observed in the LOA and OLOA paths spectra. This implies that at certain
frequencies (e.g. 650 and 1700 Hz), the OLOA path (and thus the frictional effects) could
be dominant over the LOA path (and thus the transmission error effects) given
comparable force excitation levels. The proposed method thus provides a design tool to
quantify and evaluate the relative contribution of structural path due to sliding friction.
The top plate velocity distribution Vtop (ω ) could then be predicted by using Eq. (5.12),
where Fp , B , x (ω ) and Fp , B , y (ω ) are the pinion bearing forces predicted by the lumped
138
source model in the LOA and OLOA directions. Figure 5.7(a) shows the surface
interpolated velocity distributions on the top plate at three mesh harmonics (m =1, 2, 3)
given Tp = 500 lb-in and Ωp = 4875 RPM.
1
1
Vtop (ω ) = Fp , B , x (ω )Ye, x (ω ) + Fp , B , y (ω )Ye , y (ω )
2
2
(5.12)
Figure 5.6 Magnitudes of the combined transfer mobilities in two directions calculated at
the sensor location on the top plate. Key:
, mobility of the OLOA path;
,
mobility of the LOA path.
139
5.4
Prediction of Noise Radiation and Contribution of Friction
5.4.1
Prediction using Rayleigh Integral Technique
Since the rectangular top plate is the main radiator [5.14] of the gearbox due to its
relatively high mobility, Rayleigh integral [5.15] is used to approximate the sound
pressure radiation by assuming that the top plate is included in an infinite rigid baffle and
each elementary plate surface is an equivalent point source in the rigid wall. The sound
pressure amplitude is given as follows where ρ is the air density, Q i (ω ) = Vi (ω )∆Si is
the source strength of ith equivalent source with area ∆Si , k (ω ) is the wave number and
ri is the distance of ith source to the receiving point.
Q i (ω ) − jk (ω ) ri
jωρ
P (ω ) =
e
∑
2π i
ri
(5.13)
Compared with conventional boundary element analysis, Rayleigh integral
approximates sound pressure in a fraction of the required computation time [5.16]. Hence,
it is most suitable for parametric design studies. Although some researchers [5.16] have
pointed out that Rayleigh integral may give large errors for sound pressure prediction if
applied to strongly directional, three dimensional (3D) fields, such errors are not
significant here due to the flat (rather than curved) top plate and favorable surroundings
(such as rigid side plates and anechoic chamber).
140
5.4.2
Prediction using Substitute Source Method
As an alternative to Rayleigh integral, a newly developed algorithm based on the
substitute source approach [5.17] is used to compute radiated or diffracted sound field. It
is conducted by removing the gearbox and introducing acoustic sources within the
liberated space which yield the desired boundary conditions at the box surface (Neumann
problem). Solutions are obtained in terms of the locations and/or the strengths of the
substitute sources by minimizing the error function between original and estimated
particle velocity normal to the interface surface [5.17].
Since the surface velocity distributions of gearbox are essentially symmetric along
the center lines due to geometric symmetry, velocity distributions along the border lines
of EFGH plane in Figure 5.3(b) are chosen to simplify the 3D gearbox into a 2D radiation
model for simpler data representation as well as faster computation. Zero (negligible)
velocity distribution is assumed along lines EF, FG and HE since the microphone
(receiver) is positioned above the center of major radiator, i.e. the top plate. A 2D line
source uniformly pulsating with unit-length volume velocity Q′ is chosen as the
substitute source. Its radiation field is the same in any plane perpendicular to the source
line. Amplitudes of the sound pressure and radial velocity of such source are given by the
following, where H v(2) is the Hankel function of second kind and order v.
k (ω ) ρ c P (ω ) =
Q′(ω ) H 0(2) [k (ω )r ] ,
4
(5.14a)
k (ω ) (2)
Vr (ω ) = − j
Q′H1 [k (ω )r ]
4
(5.14b)
141
A “greedy search” algorithm [5.17] is used to search for “optimal” substitute
sources: First, a large number of candidate source positions within the vibrating body are
defined, e.g. at the vertices of a square grid. Second, a single position is first found which
allows the point source to produce the smallest deviation between the original and
estimated normal velocity of surface vibration. The estimation is then subtracted from the
original velocity to get a velocity residual. Third, among the rest of candidate points, a
new position is found which makes the second source acting at it, maximally reduce the
velocity residual of the first step. Once found, the source strengths of both sources are
adjusted for a best fit of the original surface velocity and a new residual velocity. Each
subsequent step defines a new optimum source position among the ones not already used.
The curve fitting of source strengths is done by minimizing the mean square root (RMS)
value of the velocity error. The vector of complex source strength Q ′ is related (as shown
below) to the vector V n of complex normal surface velocity at control points via the
G G
source-velocity transfer matrix T where rij = ri − rj and α ij is the angle between vector
G G
ri − r and the outer normal to the surface.
Q ′(ω ) = T −1 (ω )V n (ω ) ,
(5.15a)
k (ω ) (2)
T ij (ω ) = − j
H1 [k (ω )rij ]cos(α ij )
4
(5.15b)
To minimize the impact of an ill-conditioned matrix, the number of control points
is kept well above that of independent source points. Minimization of the RMS error
142
using pseudo-inverse yields the following, where the asterisk signifies the conjugate
transpose:
−1
Q ′(ω ) = ⎡⎣T (ω )* T (ω ) ⎤⎦ T (ω )*V n (ω )
(5.16)
The difference between synthesized and original surface normal velocities is:
∆V (ω ) = Ξ(ω )V n (ω ) ,
(5.17a)
−1
Ξ (ω ) = T (ω ) ⎡⎣T (ω )* T (ω ) ⎤⎦ T (ω )* − I (ω )
(5.17b)
where I (ω ) is the identity matrix. The matrix Ξ (ω ) appears as a velocity error matrix.
The RMS velocity error is normalized by dividing with the RMS value of original
velocity as:
V n (ω )* Ξ (ω )* Ξ (ω )V n (ω )
E RMS (ω )
eRMS (ω ) =
=
Vn , RMS (ω )
V n (ω )*V n (ω )
143
(5. 18)
5.4.3
Prediction vs. Noise Measurements
Figure 5.7(a) shows predictions of surface interpolated velocity distribution on the
top plate at the first three mesh harmonics under Tp = 500 lb-in and Ωp = 4875 RPM.
Note that predictions at high frequencies (e.g. mesh index m = 3) are less “reliable” due
to the limitation of element dimensions as compared the wave length. The symmetry of
surface velocity distribution leads to the simplification into a 2D gearbox model of Figure
5.7(b). To ensure necessary accuracy for the acoustic radiation, the selected central lines
of the 2D plane should capture the dominant structural modes of Figure 5.7(a). Also, the
structural wavelength along the central line should be higher than the acoustic
wavelength of interest to ensure the validity of the 2D approach.
The source points of Figure 5.7(b) are chosen from a mesh grid of candidate
points not too close to the boundary to prevent forming steep gradients of surface
pressure. Observe that only 15 substitute sources tend to predict well the surface
distribution of velocity magnitude at the gear mesh harmonics. Figure 5.7(c) illustrates
the predicted source strengths of the substitute sources in the complex plane for
evaluation of the acoustic source properties. A single dominant substitute source is
observed at the first mesh harmonic (monopole-like acoustic source); however, several
dominant substitute sources are present and these are more equally distributed in the
complex plane at the higher harmonics (multi-pole acoustic source).
144
(a)
12 6
15
8 1 9
2
4
3
14 3
14
1
2
9
10
9
13 8 11
15
7
5 12 4
13
7 8
10
5
11
15
3 4 14
10
6 7
13
1 5
12 6 2
11
(b)
90
120
90
5e-5
60
120
3e-5
30
150
150
90
1.5e-5
60
1e-5
120
30
5e-6
150
1.5e-5
60
1e-5
(c)
30
5e-6
1e-5
180
0
210
330
240
300
270
180
0
210
330
240
300
270
180
0
330
210
300
240
270
(c)
Figure 5.7 Comparison of normal surface velocity magnitudes and substitute source
strength vectors under Tp = 500 lb-in and Ωp = 4875 RPM. (a) Line 1: interpolated
surface velocity on top plate; (b) Line 2: simplified 2D gearbox model with 15 substitute
source points; Key:
, original surface velocity magnitude;
, surface velocity
magnitude by substitute sources; , locations of substitute sources. (c) Line 3: substitute
source strengths in complex plane for 2D gearbox. Column 1: mesh frequency index m =
1; Column 2: m = 2; Column 3: m = 3.
145
The simplification from 3D into 2D gearbox model requires examining surface
modes and a careful selection of representative plane, which poses additional limitations
to its application as a universal method. However, once applicable, the substitute source
method provides the following benefits. First, it enables an efficient evaluation of the
acoustic source characteristics for whine noise. Second, unlike the Rayleigh integral
which assumes that the top plate is part of an infinite rigid baffle, it takes the body shape
into account and thus reduces the errors especially in the low frequency range. This also
allows a straight forward synthesis of the radiation field for all the sources by using
simple superposition as diffraction on the sources does not take place. Finally, compared
with boundary element analysis, it does not suffer from the problems of singularities or
uniqueness of solution. Nonetheless, it is an approximate method.
Figure 5.8 compares sound pressure measured at the microphone 6 inch above the
top plate to predictions by using both the Rayleigh integral as well as the substitute
source method under varying pinion torque given Ωp = 4875 RPM and 140 °F.
Predictions correlate well with measurements in terms of trends and relative magnitudes
at first three gear mesh harmonics.
146
1.8
1.6
1.4
1.2
|P|
1
0.8
0.6
0.4
0.2
0
500
550
600
650
700
750
800
850
900
Torque (lb-in)
Figure 5.8 Normalized sound pressure (ref 1.0 Pa) predicted at the microphone 6 inch
above the top plate under varying torque Tp given Ωp = 4875 RPM and 140 °F. Key:
,
measurements; , Rayleigh integral predictions , substitute source predictions. Blue,
mesh frequency index m = 1; red, m = 2; black, m = 3.
147
5.5
Conclusion
A refined source-path-receiver model has been developed which characterizes the
sliding friction induced structural path and associated noise radiation. Proposed Rayleigh
integral method and substitute source technique are more efficient for calculating the
acoustic field than the usual boundary element technique and thus they provide rapid
design tools to quantify the frictional noise. Figure 5.9 compares the sound pressure level
predicted given Ωp = 4875 RPM and 140 °F under Tp = 500 lb-in (close to the “optimal”
load where transmission error is minimized) and under high torque with Tp = 800 lb-in.
At each gear mesh frequency, individual contributions of transmission error (via the LOA
path) and frictional effects (via the OLOA path) are compared to the overall whine noise.
Observe in Figure 5.9(a) that near the “optimal” load, friction induced noise is
comparable to the transmission error induced noise (especially for the first two mesh
harmonics); thus sliding friction should be considered as a significant contributor to
whine noise. However, at non-optimal torques in Figure 5.9(b), friction induced noise is
overwhelmed by the transmission error noise, thus sliding friction could be negligible
under such conditions. Further work is needed to extend the 2D gearbox into 3D model
by using multipole substitute source technique [5.18] and fully examine the friction
source, especially under varying lubrication conditions. Effects of the tooth surface finish
should be examined as well.
148
100
Overall
90
SPL (dB)
80
LOA
OLOA
70
60
50
40
30
1
2
3
Mesh index m
(a)
100
SPL (dB)
Overall
90
LOA
80
OLOA
70
60
50
40
30
1
2
3
Mesh index m
(b)
Figure 5.9 Overall sound pressure levels (ref: 2e-5 Pa) and their contributions predicted at
the microphone 6 in above the top plate under Ωp = 4875 RPM and 140 °F. (a) Tp = 500
lb-in (optimal load for minimum transmission error); (b) Tp = 800 lb-in.
149
References for Chapter 5
[5.1] Houser, D. R., 1994, “Comparison of Transmission Error Predictions with Noise
Measurements for Several Spur and Helical Gears,” NASA-Technical Memorandum,
106647, 30th AIAA Joint Propulsion Conference, Indianapolis, IN.
[5.2] Steyer, G., 1987, “Influence of Gear Train Dynamics on Gear Noise,” NOISE-CON
87 proceedings, pp. 53-58.
[5.3] Ajmi, M., and Velex, P., 2005, “A Model for Simulating the Quasi-Static and
Dynamic Behavior of Solid Wide-Faced Spur and Helical Gears,” Mechanism and
Machine Theory Journal, 40, pp. 173-190.
[5.4] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on
the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of
Mechanical Design, 122(4), pp. 515-522.
[5.5] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body
Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3),
pp. 479-499.
[5.6] External2D (CALYX software), 2003,
www.ansol.www, ANSOL Inc., Hilliard, OH.
“Helical3D
User’s
Manual,”
[5.7] Lim, T. C., and Singh, R., 1990, “Vibration Transmission Through Rolling Element
Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and Vibration, 139(2),
pp. 179-199.
[5.8] Rook T. E., and Singh, R., 1996, “Mobility Analysis of Structure-Borne Noise
Power Flow Through Bearings in Gearbox-Like Structures,” Noise Control Engineering
Journal, 44(2), pp. 69-78.
[5.9] Van Roosmalen, A., 1994, “Design Tools for Low Noise Gear Transmissions,” PhD
Dissertation, Eindhoven University of Technology.
[5.10] Rook, T. E., and Singh, R., 1998, “Structural Intensity Calculations for Compliant
Plate-Beam Structures Connected by Bearings,” Journal of Sound and Vibration, 211(3),
pp. 365-388.
[5.11] Oswald, F. B., Seybert, A. F., Wu, T. W., and Atherton, W., 1992, “Comparison of
Analysis and Experiment for Gearbox Noise,” Proceedings of the International Power
Transmission and Gearing Conference, Phoenix, pp. 675-679.
150
[5.12] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared
Systems,” Technical Report submitted to the Army Research Office, grant number
DAAD19-02-1-0334.
[5.13] Holub, A., 2005, “Mobility Analysis of a Spur Gear Pair and the Examination of
Sliding Friction,” MS Thesis, The Ohio State University.
[5.14] Jacobson, M. F., Singh, R., and Oswald, F. B., 1996, “Acoustic Radiation
Efficiency Models of a Simple Gearbox,” NASA Technical Memorandum, 107226.
[5.15] Cremer, L., and Heckl, M., 1973, Structure-Borne Sound, Chapter 6, Sound
Radiation from Structures, Springer-Verlag, New York.
[5.16] Herrin, D. W., Martinus, F., Wu, T. W., and Seybert, A. F., 2003, “A New Look at
the High Frequency Boundary Element and Rayleigh Integral Approximations,” SAE,
paper # 03NVC-114.
[5.17] Pavić, G., 2005, “An Engineering Technique for the Computation of Sound
Radiation by Vibrating Bodies using Substitute Sources,” Acta Acustica Journal, 91, pp.
1-16.
[5.18] Pavić, G., 2006, “A Technique for the Computation of Sound Radiation by
Vibrating Bodies Using Multipole Substitute Sources,” Acta Acustica Journal, 92, pp.
112-126.
151
CHAPTER 6
INCLUSION OF SLIDING FRICTION IN HELICAL GEAR DYNAMICS
6.1
Introduction
Sliding friction is believed to be one of the major sources of gear vibration and
noise especially under high toque and low speed conditions since a reversal in the sliding
velocity takes place along the pitch line. Yet few analytical contact dynamics
formulations that incorporate friction are available in the literature [6.1-6.8]. In a series of
recent articles, Vaishya and Singh [6.1-6.3] reviewed various modeling strategies that
have been historically adopted and then illustrated issues for spur gears by assuming
equal load sharing among the contact teeth. Further, Velex and Cahouet [6.4] considered
the effects of sliding friction in their models for spur and helical gears. They found that
the dynamic bearing forces, as caused by friction at lower speeds, can generate
significant time-varying excitations. Velex and Sainsot [6.5] examined friction
excitations in errorless spur and helical gear pairs, and reported that the friction appears
as a non-negligible excitation source especially for translating motions. Lundvall et al.
[6.6] proposed a multi-body model for spur gears and briefly discussed the role of profile
152
modification in the presence of sliding friction. In Chapter 2, we have developed a more
accurate model of the spur gears that includes realistic mesh stiffness and sliding friction,
while overcoming the deficiency of Vaishya and Singh’s work [6.1-6.3]. In particular,
this model shows that the tip relief could even amplify dynamic motions in some cases
due to interactions between mesh and friction forces. Based on the literature review, it is
clear that sliding friction has not been adequately considered in the dynamic models for
helical gears. Work presented in this chapter attempts to fill this void.
6.2
Problem Formulation
In a helical gear pair, the line-of-action (LOA) lies in the tangent plane of the base
cylinders, as defined by the base helix angle βb. Consequently, the moment arms for the
out of plane moments change constantly. This phenomenon introduces axial and friction
force shuttling effects [6.4]. Thus, the friction forces play a pivotal role in the loads
transmitted to the bearings, especially in the off-line-of-action (OLOA) direction.
Blankenship and Singh [6.9] have developed a three-dimensional representation of forces
and moments generated and transmitted via a gear mesh interface. This model is
formulated in terms of the spatially-varying mesh stiffness and transmission errors which
are assumed to be available from quasi-static elastic analyses. The vector formulation
leads to a multi-body analysis of geared systems and force coupling due to vibratory
changes in the contact plane is directly included. Similar strategy will be adopted here.
153
Objectives of this chapter include the following. First, propose a threedimensional formulation to characterize the dynamics associated with the contact plane
including the reversal at the pitch line due to sliding friction. Calculation of the contact
forces/moments will be illustrated via an example case (NASA-ART helical gear pair).
The tooth stiffness density along contact lines will then be calculated using a Finite
Element/Contact Mechanics (FE/CM) analysis code [6.10]. Second, develop a multidegree-of-freedom (MDOF) helical gear pair model (with 12 DOFs) which includes the
rotational and translational DOFs along the LOA, OLOA and axial directions as well as
the bearing/shaft compliances. Third, illustrate the effect of sliding friction including the
shuttling effect in the radial direction and coupling between the LOA and OLOA
directions. The following assumptions are made to derive the MDOF linear time-varying
system (LTV) model. 1. The position of the contact lines and relative sliding velocity
depend only on the mean angular motions of the gear pair. If this assumption is not made,
the system will have implicit non-linearities. 2. The mesh stiffness per unit length along
the contact line (or stiffness density k) is constant [6.7]. The constant k is estimated from
the geometrical calculation of total length of contact lines and the mesh stiffness, which
is computed using the finite element model [6.10]; this is equivalent to the equal load
sharing assumption in spur gears [6.1-6.3]. 3. Coulomb’s law with a constant coefficient
of friction (µ) is employed like previous researchers [6.1-6.3] though mixed lubrication
regime exists. 4. The elastic deformations of the shaft and bearings are modeled using
lumped parameter representations for their compliances. Also, it is assumed that the mean
load is high and the dynamic load is not sufficient to cause tooth separations [6.11]. Thus
vibro-impacts are not considered here.
154
6.3
Mesh Forces and Moments with Sliding Friction
The helical gear system is depicted in Figure 6.1. The pinion and gear are
modeled as rigid cylinders linked by a series of independent stiffness elements that
describe the contact plane tangent to the base cylinders. The normal direction at a point of
contact lies in the contact plane (due to the involute profile construction) and is
perpendicular to the line of contact. The pinion and gear dynamics are formulated in the
coordinate systems located at their respective centers as shown in Figure 6.1. The
nominal motions of the pinion and gear are given as −Ω p ez and Ω g ez = ez Ω p rbp / rbg ,
respectively. Here, z axis coincidences with the axial direction, e is the unit directional
vector and rbp and rbg are the base radii of pinion and gear. An (static) input torque Tp is
applied to the pinion and the (static) braking torque Tg on the gear obeys the basic gear
kinematics. Superposed on the kinematic motions are rotational vibratory motions
denoted by θ zp and θ zg for the pinion and gear. For the sake of illustration, analytical
formulations are demonstrated via the following example case with parameters of the
pinion (gear parameters are within the parenthesis): number of teeth 25 (31); outside
diameter 3.38 (4.13) in; pitch diameter 3.125 (3.875) in; root diameter 2.811 (3.561) in;
center distance 3.5 in; transverse diametral pitch 8 in-1; transverse pressure angle 25°;
helix angle 21.5°; face width 1.25 in; polar moment of inertia 8.33E-3 (1.64E-2) lb-s2-in;
mass 1.26E-2 (1.58E-2) lb-s2/in. Since the overall contact ratio is around 2.7, either two
or three tooth pairs are meshing with each other at any time instant.
155
θ
φ
θ
βb
Figure 6.1 Schematic of the helical gear pair system.
The three meshing tooth pairs within a mesh cycle are numbered as #0, #1 and #2,
respectively. Calculate the (static) contact loads N 0 (t ) , N1 (t ) , N 2 (t ) and (static) pinion
deflection Θ zp (t ) by performing a static analysis using the FE/CM formulation [6.10].
The stiffness K 0 (t ) , K1 (t ) and K 2 (t ) of the three meshing tooth pairs are calculated as
follows:
K i (t ) = N i (t ) / ⎡⎣ rbp Θ zp (t ) ⎤⎦ ,
156
i = 0,1, 2
(6.1)
x 10
5
Tooth stiffness (lb/in)
15
10
5
t/Tmesh
0
0
0.5
1.5
1
2
2.5
t/Tmesh
Figure 6.2 Tooth stiffness function calculated using Eq. (6.1) based on the FE/CM code
[6.11]. Key:
, tooth pair #0;
, tooth pair #1;
, tooth pair #2;
, combined
tooth pair stiffness.
For our example, the single tooth pair stiffness function K(t) are obtained by
following one tooth pair for three complete mesh cycles, as shown in Figure 6.2, where
Tmesh is the period of one mesh cycle. Observe that the stiffness profile has a trapezoidal
shape; some discontinuities exist where corner contacts take place. The overall stiffness
function, defined as a combination of all meshing tooth pairs, also follows a trapezoidal
pattern. Meanwhile, the sum of the lengths of contact lines can be calculated by using
either the FE/CM code [6.10, 6.12] or a simplified approximation method based on gear
geometry [6.13]. Since the total length of contact lines and combined tooth stiffness
157
follow a similar pattern, we can assume a constant mesh stiffness density (k) along the
contact lines. Consequently, the time-averaged k is estimated as follows, where Li is the
length of the ith contact line.
2
2
i =0
i =0
k ≈ ∑ K i (t ) / ∑ Li
(6.2)
Denote the LOA, OLOA and axial axes as x, y and z axes respectively; the
dynamic motions of the pinion and gear centers consist of three translations (u) and three
rotations (θ) such that u p = {u xp , u yp , u zp , θ xp ,θ yp ,θ zp } , u g = {u xg , u yg , u zg , θ xg ,θ yg ,θ zg } . For
T
T
a contact point with local coordinates ( x, rbp , z ) in the pinion coordinate system, its global
motion when considered as part of the pinion is derived as:
⎧u xp ⎫ ⎧θ xp ⎫
⎧u xp + zθ yp + rbp Ω p t − rbpθ zp ⎫
⎪ ⎪ ⎪ ⎪
⎪
⎪
u pc = ⎨u yp ⎬ + ⎨θ yp ⎬ × ( xex + rbp ey + zez ) = ⎨ u yp − zθ xp − xΩ pt + xθ zp ⎬
⎪u ⎪ ⎪θ ⎪
⎪
⎪
uzp + rbpθ xp − xθ yp
⎩ zp ⎭ ⎩ zp ⎭
⎩
⎭
(6.3a)
Denote xg as the x coordinate at the gear center in the pinion coordinate system,
the global motion of the contact point when it is considered as part of the gear is:
⎧uxg ⎫ ⎧θ xg ⎫
⎧
⎫
uxg + zθ yg + rbg Ω g t + rbgθ zg
⎪ ⎪ ⎪ ⎪
⎪
⎪
ugc = ⎨u yg ⎬ + ⎨θ yg ⎬ × ⎣⎡ −( xg − x)ex − rbg ey + zez ⎦⎤ = ⎨u yg − zθ xg − ( xg − x)Ω g t − ( xg − x)θ zp ⎬ (6.3b)
⎪u ⎪ ⎪θ ⎪
⎪
⎪
uzg − rbgθ xg + ( xg − x)θ yg
⎩ zg ⎭ ⎩ zg ⎭
⎩
⎭
158
The deformation of the mesh spring is ∆ mesh = (u pc − u gc ) ⋅ (− cos β b ex + sin β bez ) ,
which can be further simplified as
∆ mesh = − cos βb ⎡⎣(u xp − u xg ) + z (θ yp − θ yg ) − (rbpθ zp + rbgθ zg ) ⎤⎦ +
sin βb ⎡⎣(u zp − u zg ) + ( rbpθ xp + rbgθ xg ) − ( xθ yp + ( xg − x)θ yg ) ⎤⎦
(6.4)
The velocities of the contact point when considered as part of the pinion or gear
(ignoring the vibratory component) are derived as:
v pc = rbp Ω p ex − xΩ p ey ,
vgc = rbg Ω g ex − ( xg − x)Ω g ey
(6.5a,b)
The relative sliding velocity of the pinion with respect to the gear is:
vs = ⎡⎣ − xΩ p + ( xg − x)Ω g ⎤⎦ ey
(6.6)
From Eq. (6.6) it is clear that vs is in the positive y direction at the beginning of
contact (small x). Further, vs becomes zero at the pitch point xP and changes to the
negative y direction when x > xP . Hence, Eq. (6.6) can be used to determine the direction
of the friction forces. The elemental forces on the meshing tooth pairs of the pinion and
the gear are given in Eq. (6.7) assuming only the elastic effects, and the total mesh forces
are derived by integrating the elemental forces over the contact line as given below in Eq.
(6.8):
159
∆Fmesh , p = −∆Fmesh , g
Fmesh, p = − Fmesh, g
⎧ − k ∆ mesh cos βb ⎫
⎪
⎪
= ⎨ µ k ∆ mesh sgn( x − xP ) ⎬
⎪ k ∆ sin β
⎪
mesh
b
⎩
⎭
⎧
⎫
⎪ −k cos β b ∫ ∆ mesh dl ⎪
l
⎪
⎪
⎪
⎪
= ⎨ µ k ∫ ∆ mesh sgn( x − xP )dl ⎬
⎪ l
⎪
⎪ k sin β ∆ dl
⎪
b ∫ mesh
⎪
⎪
l
⎩
⎭
(6.7)
(6.8)
To facilitate the integration, the contact zone is divided into three regions as
shown in Figure 6.1, where xb and xe are defined as the lower and upper boundaries of
the contact zone in the LOA coordinate system of the pinion. In Region 1 (lightly shaded
area within the contact zone of Figure 6.1), the lower and upper limits of x are denoted
as xm = xb and x f ≥ xb , respectively. The z coordinate (on the line of contact) is written
as a function of the x coordinate as follows, where W is the face width.
z ( x) = 0.5W + ⎡⎣( x − x f ) / tan β b ⎤⎦
(6.9a)
In Region 2 (white area within the contact zone of Figure 6.1), xm ≥ xb and x f ≤ xe . The
z coordinate of the contact point is derived as:
z ( x) = 0.5W ( 2 x − x f − xm ) / ( x f − xm )
160
(6.9b)
In Region 3 (darkly shaded area within the contact zone of Figure 6.1), xm ≤ xe
and x f = xe . The z coordinate of a contact point is given as:
z ( x) = −0.5W + ⎡⎣( x − xm ) / tan β b ⎤⎦
(6.9c)
Consider the integral in equation (6.8):
⎡ − cos β b ⎡ (u xp − u xg ) + z (θ yp − θ yg ) − (rbpθ zp + rbgθ zg ) ⎤ + ⎤
⎣
⎦
⎥ dl (6.10a)
∆ = ∫ ∆ mesh dl = ∫ ⎢
⎥
⎤
−
+
+
−
+
−
(
u
u
)
(
r
θ
r
θ
)
(
x
θ
(
x
x
)
θ
)
l
l ⎢sin β b ⎡
zg
bp xp
bg xg
yp
e
yg ⎦ ⎦
⎣ zp
⎣
Recognizing that dz = dl ⋅ cos β b and dx = dl ⋅ sin β b the above integral yields:
∆ = −( z f − zm ) ⎡⎣(u xp − u xg ) − (rbpθ zp + rbgθ zg ) ⎤⎦ − 0.5( z f 2 − zm 2 )(θ yp − θ yg ) +
( x f − xm ) ⎣⎡(u zp − u zg ) + (rbpθ xp + rbgθ xg ) − xeθ yg ⎤⎦ − 0.5( x f 2 − xm 2 )(θ yp − θ yg )
(6.10b)
Compare Eqs. (6.8) and (6.10) and represent the contact forces in the LOA and
axial directions on the pinion as:
Fmesh, p , x = − k cos β b ∆ ,
Fmesh, p , z = k sin β b ∆
161
(6.11a,b)
Due to the sliding friction, the Fmesh, p , y involves a discontinuous sign function and
it needs to be evaluated separately for three different cases assuming a constant µ . Case 1:
Both limits of the contact line are less than xP , which implies the contact line on the
pinion is completely below the pitch cylinder (approach action) so that sgn( x − xP ) = −1 .
The friction force of the pinion is
Fmesh, p , y = − µk∆
(6.12)
Case 2: The contact line lies on either side of the pitch cylinder. The integral of friction
force need to be evaluated in two parts.
Fmesh, p , y = µk ( ∆ 2 − ∆1 )
(6.13a)
∆1 = −( z P − zm ) ⎡⎣(u xp − u xg ) − (rbpθ zp + rbgθ zg ) ⎤⎦ − 0.5( z P 2 − zm 2 )(θ yp − θ yg ) +
( xP − xm ) ⎡⎣(u zp − u zg ) + (rbpθ xp + rbgθ xg ) − xeθ yg ⎤⎦ − 0.5( xP2 − xm2 )(θ yp − θ yg )
∆ 2 = −( z f − z P ) ⎡⎣(u xp − u xg ) − (rbpθ zp + rbgθ zg ) ⎤⎦ − 0.5( z f 2 − z P 2 )(θ yp − θ yg ) +
( x f − xP ) ⎡⎣(u zp − u zg ) + (rbpθ xp + rbgθ xg ) − xeθ yg ⎤⎦ − 0.5( x 2f − xP2 )(θ yp − θ yg )
(6.13b)
(6.13c)
Case 3: Both limits are greater than xP , which implies the contact line on the pinion is
completely above the pitch cylinder (recess action). Consequently, sgn( x − xP ) = 1 and
the friction force on the pinion is:
162
Fmesh, p , y = µk∆
(6.14)
Hence, in summary the mesh forces are derived as:
Fmesh, p = − Fmesh, g
⎫
⎧
− k cos β b ∆
⎪
⎪
= ⎨− µk∆ or µk ( ∆ 2 − ∆1 ) or µk∆ ⎬
⎪
⎪
k sin β b ∆
⎩
⎭
(6.15)
The elemental moments on the pinion at a point on the contact line are derived as:
∆M mesh, p
⎧rbp sin β b − z µ sgn( x − xP ) ⎫
⎪
⎪
= k ∆ mesh ⎨− x sin β b − z cos β b
⎬
⎪ xµ sgn( x − x ) + r cos β ⎪
P
bp
b⎭
⎩
(6.16)
Integrating Eq. (6.16) over the contact line yields the total moments on the pinion
as:
M mesh, p = ∫ ∆M mesh, p dl
(6.17)
l
To facilitate the calculation of Eq. (6.17), define two integration operations over
the line of contact with lower and higher limits xl and xh , respectively:
163
xh
( ∆ ⋅ x ) ( xl , xh ) = ∫ x∆ mesh dl = −
xl
( xh + xl )( zh − zl )
( x 2 − xl2 )
c1 + c2 z f ) + h
c3
(
2
2
⎛ x − xf xf ⎞
⎛ x − xf xf ⎞
( x3 − x3 )
− h l c2 -( zh − z f ) 2 ⎜ h
+ ⎟ c2 + ( zl − z f ) 2 ⎜ l
+ ⎟ c2
3
2 ⎠
2 ⎠
⎝ 3
⎝ 3
xh
( ∆ ⋅ z ) ( xl , xh ) = ∫ z∆ mesh dl = (−c1 + c3 tan βb )
xl
− c2
x f tan β b
2
(6.18a)
( zh2 − zl2 )
( z3 − z3 )
− c2 h l
2
3
⎛ ( zh3 − zl3 ) z f ( zh2 − zl2 ) ⎞
2
2
2
( zh − zl ) − c2 tan βb ⎜
−
⎟⎟
⎜
3
2
⎝
⎠
c1 = (u xp − u xg ) − (rbpθ zp + rbgθ zg ) ,
c2 = (θ yp − θ yg )
c3 = (u zp − u zg ) + (rbpθ xp + rbgθ xg ) − xeθ yg
(6.18b)
(6.18c,d)
(6.18e)
Common integrals in the vector Eq. (6.17) are now evaluated as follows:
∫ x∆
mesh
dl = (∆ ⋅ x )( xm , x f ) ,
l
∫ z∆
mesh
dl = ( ∆ ⋅ z ) ( xm , x f )
(6.19a,b)
l
P
)∆ mesh dl = − ( ∆ ⋅ x ) ( xm , xP ) + ( ∆ ⋅ x ) ( xP , x f )
(6.19c)
P
)∆ mesh dl = − ( ∆ ⋅ z ) ( xm , xP ) + ( ∆ ⋅ z ) ( xP , x f )
(6.19d)
∫ x sgn( x − x
l
∫ z sgn( x − x
l
Consequently, the moments of the mesh force acting on the pinion are given as:
M mesh , p
⎧rbp ∆ sin β b + µ ( ∆ ⋅ z ) ( xm , xP ) − µ ( ∆ ⋅ z ) ( xP , x f ) ⎫
⎪
⎪
= k ⎨− sin βb ( ∆ ⋅ x ) ( xm , x f ) − ( ∆ ⋅ z ) ( xm , x f ) cos β b
⎬
⎪− µ ( ∆ ⋅ x ) ( x , x ) + µ ( ∆ ⋅ x ) ( x , x ) + r ∆ cos β ⎪
m
P
P
f
bp
b⎭
⎩
164
(6.20)
Similarly, the moments of the elemental mesh forces on the gear are given as:
∆M mesh , g
⎧rbg sin βb + z µ sgn( x − xP )
⎫
⎪
⎪
= k ∆ mesh ⎨ − xe sin βb + x sin β b + z cos β b
⎬
⎪ x µ sgn( x − x ) − xµ sgn( x − x ) + r cos β ⎪
P
P
bg
b⎭
⎩ g
(6.21)
The total moments due to the mesh forces on the gear are derived as:
M mesh, g
⎧rbg ∆ sin βb − µ ( ∆ ⋅ z ) ( xm , xP ) + µ ( ∆ ⋅ z ) ( xP , x f )
⎫
⎪
⎪
= k ⎨− xg ∆ sin β b + sin βb ( ∆ ⋅ x ) ( xm , x f ) + ( ∆ ⋅ z ) ( xm , x f ) cos β b
⎬ (6.22)
⎪ x (∆ − ∆ ) + µ ( ∆ ⋅ x ) ( x , x ) − µ ( ∆ ⋅ x ) ( x , x ) + r ∆ cos β ⎪
m
P
P
f
bg
b⎭
1
⎩ g 2
Note that Eqs. (6.15), (6.20) and (6.22) are formulated for a single tooth pair in
contact. For multiple tooth pairs in contact, the dynamics of all meshing tooth pair must
be considered. Consider an example to demonstrate the modeling strategy. Since the
contact ratio is around 2.7, three tooth pairs are considered in a single mesh cycle. For a
generic helical gear pair with contact ratio σ , n = ceil(σ ) number of meshing tooth
pairs need to be considered by following the same methodology, where the “ceil”
function rounds σ to the nearest integers towards infinity. Figure 6.3(a) shows the
contact plan of the example case within a helical gear pair, and Figure 6.3(b) illustrates a
zoomed-in snapshot of the contact zone at the beginning of a mesh cycle. At this instant,
pair #0 just comes into mesh at point A and pair #1 is in contact along line CI. Likewise
pair #2 contacts each other along line MN.
165
θzg
θzp
Figure 6.3 Contact zones at the beginning of a mesh cycle. (a) In the helical gear pair; (b)
contact zones within contact plane. Key: PP’ is the pitch line; AA’ is the face width W;
AD is the length of contact zone Z.
166
As the gears roll, the contact lines move diagonally across the contact zone. When
pair #0 reaches the pitch point P, the relative sliding velocity of the pinion with respect to
the gear starts to reverse, resulting in a reversal of friction force along the portion of
contact line surpass the pitch point. Once pair #0 reaches the CI line (and pair #1 reaches
MN line) at the end of the mesh circle, pair #0 becomes #1 (and pair #1 becomes #2)
corresponding to the start of the next mesh cycle. At any time instant t, the x coordinates
of the three pairs are projected along the AR line and denoted as x0 (t ) , x1 (t ) and x2 (t ) , as
defined below in Eq. (6.23). Here, λ is the base pitch, L represents the geometrical
distance,
and
“mod”
is
the
modulus
function
defined
as
mod( x, y ) = x − y ⋅ floor( x / y ), if y ≠ 0 .
x0 (t ) = mod(Ω p rbpt , λ ) + LT1A ,
x1 (t ) = mod(Ω p rbp t , λ ) + λ +LT1A
x2 (t ) = mod(Ω p rbp t , λ ) + 2λ +LT1A
(6.23a,b)
(6.23c)
To implement the integration algorithm, the contact regions are further divided
into eight contact zones as shown in Figure 6.3(b). Zones 1 and 2 correspond to pair #0
before and after reaching the pitch line; Zones 3-5 and Zones 6-8 correspond to pairs #1
and #2, respectively. The zone classifications and their corresponding integration limits
for the calculation of dynamic forces and moments are derived as following, where xm ,
x f , zm and z f denote the lower and upper limits along x and z axes.
167
Zone 1 (LT1A ≤ x0 (t ) < LT1P ): xm = LT1A , x f = x0 , z f = 0.5W ,
zm = 0.5W − ⎡⎣( x0 − LT1A ) / tan βb ⎤⎦
Zone 2 (LT1P ≤ x0 (t ) < LT1C ): xm = LT1A , x f = x0 , x p = LT1P , z f = 0.5W ,
zm = 0.5W − ⎡⎣( x0 − LT1A ) / tan βb ⎤⎦ , z p = 0.5W − ⎡⎣( x0 − LT1P ) / tan βb ⎤⎦
Zone 3 (LT1C ≤ x1 (t ) < LT1Q ): xm = LT1A , x f = x1 , x p = LT1P , z f = 0.5W ,
zm = 0.5W − ⎡⎣( x1 − LT1A ) / tan βb ⎤⎦ , z p = 0.5W − ⎡⎣( x1 − LT1P ) / tan βb ⎤⎦
Zone 4 (LT1Q ≤ x1 (t ) < LT1D ): xm = x1 − W ⋅ tan βb , x f = x1 , x p = LT1P , zm = −0.5W
z f = 0.5W , z p = 0.5W − ⎡⎣( x1 − LT1P ) / tan βb ⎤⎦
Zone 5 (LT1D ≤ x1 (t ) < LT1E ): xm = x1 − W tan βb , x f = LT1D , x p = LT1P , zm = −0.5W ,
z f = 0.5W − ⎡⎣( x1 − LT1D ) / tan βb ⎤⎦ , z p = 0.5W − ⎡⎣( x1 − LT1P ) / tan β b ⎤⎦
Zone 6 (LT1E ≤ x2 (t ) < LT1H ): xm = x2 − W ⋅ tan βb , x f = LT1D , x p = LT1P , zm = −0.5W ,
z f = 0.5W − ⎡⎣( x2 − LT1D ) / tan βb ⎤⎦ , z p = 0.5W − ⎡⎣ ( x1 − LT1P ) / tan β b ⎤⎦
Zone 7 (LT1H ≤ x2 (t ) < LT1G ): xm = x2 − W ⋅ tan βb , x f = LT1D , zm = −0.5W ,
z f = 0.5W − ⎡⎣( x2 − LT1D ) / tan β b ⎤⎦
Zone 8 (LT1G ≤ x2 (t ) < LT1R ): No definition
(6.24a)
(6.24b)
(6.24c)
(6.24d)
(6.24e)
(6.24f)
(6.24g)
(6.24h)
Figure 6.4 shows the analytical tooth stiffness functions of each meshing tooth
pair and the combined stiffness calculated for the example case by using the integration
algorithm. Observe that both magnitude and shape of the stiffness functions in Figure 6.4
correlate well with those in Figure 6.2 which are obtained using a detailed FE/CM code
[6.10]. Note that the stiffness functions in Figure 6.4 are defined similar to Eq. (6.23),
which is different from the definition of Eq. (6.1) corresponding to Figure 6.2.
168
Tooth Stiffness (lbf/in)
16
x 10 5
12
8
4
0
0
0.5
1
1.5
2
2.5
t/Tmesh
Figure 6.4 Predicted tooth stiffness functions. Key:
, tooth pair #0;
#1;
, tooth pair #2;
, combined tooth pair stiffness.
6.4
, tooth pair
Shaft and Bearing Models
Consider the simplified shaft model of Figure 6.5 where l1 and l2 are the
distances between the pinion/gear center to the bearing springs, E is the Young’s
modulus, I = π rs4 / 4 is the area moment of inertia and A is the cross sectional area
[6.14-6.15]. The shaft stiffness matrix [ K ]S corresponding to the displacement vector
[ x, y, z,θ x , θ y ,θ z ]T is:
169
[ K ]S
⎡ K Sxx
⎢
⎢
⎢
=⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
− K Sxθ y
K Syy
0
K Syθ x
0
K Szz
0
0
0
K Sθ xθ x
sym.
K Sθ yθ y
0⎤
⎥
0⎥
0 ⎥⎥
0⎥
⎥
0⎥
⎥
0⎦
(6.25)
2
3
where K Sxx = K Syy = 3EI ( l1 + l2 ) ⎡( l1 − l2 ) + l1l2 ⎤ / ( l1l2 ) is the bending stiffness, K Sθ xθ x =
⎣
⎦
(
)
K Sθ yθ y = 3EI ( l1 + l2 ) / ( l1l2 ) is the rocking stiffness, K Sxθ y = K Syθ x = 3EI l12 − l22 / ( l1l2 ) is
2
the rocking-bending coupled stiffness and K Szz = AE / ( l1 + l2 ) is the longitudinal stiffness.
Kθ xθ x / 2
K zz / 2
K xx / 2
l1
K yy / 2
Kθ xθ x / 2
l2
Kθ yθ y / 2
K xx / 2
K zz / 2
θx
K yy / 2
Kθ yθ y / 2
θy
Figure 6.5 Schematic of the bearing-shaft model. Here, the shaft and bearing stiffness
elements are assumed to be in series to each other. Only pure rotational or translational
stiffness elements are shown. Coupling stiffness terms K xθ y , K yθ x are not shown.
170
The rolling element bearings in Figure 6.5 are modeled by a stiffness matrix [ K ]B
of dimension six as proposed by Lim and Singh [6.16]. The mean shaft loads and bearing
preloads are assumed constant to ensure a time-invariant [ K ]B matrix. Assume that each
shaft is supported by two identical axially preloaded high precision deep groove ball
bearings with a mean axial displacement. The helical gear pair is driven by a mean load
Tm which also generates mean radial force FBx in the LOA direction, axial force FBz and
moment M ym around OLOA direction. The [ K ]B matrix for each bearing under mean
loads has significant coefficients K Bxx , K Byy , K Bzz , K Bθ xθ x , K Bθ yθ y , K Byθ x , K Bxθ y , K Bxz and
K Bxθ y . Note that the translational stiffness K Bxx , K Byy and K Bzz are highly sensitive to the
axial preload which is quantified by the mean axial displacement. Thus we obtain:
[ K ]B
⎡ K Bxx
⎢
⎢
⎢
=⎢
⎢
⎢
⎢
⎢
⎢⎣
0
K Bxz
0
K Bxθ y
K Byy
0
K Byθ x
0
K Bzz
0
K Bzθ y
K Bθ xθ x
0
sym.
K Bθ yθ y
0⎤
⎥
0⎥
0 ⎥⎥
0⎥
⎥
0⎥
⎥
0 ⎥⎦
(6.26)
The combined shaft-bearing stiffness matrix is derived as follows where <-1>
implies term by term inverse.
171
1>
1>
⎤⎦
+ [ K ]<−
[ K ]SB = ⎡⎣[ K ]<−
S
B
<−1>
⎡ K xx
⎢
⎢
⎢
=⎢
⎢
⎢
⎢
⎢
⎣
0
0
0
K xθ y
K yy
0
K yθ x
0
K zz
0
0
0
Kθ xθ x
sym.
Kθ y θ y
0⎤
⎥
0⎥
0⎥⎥
0⎥
⎥
0⎥
⎥
0⎦
(6.27)
The restoring forces due to the shaft/bearing stiffness cause forces and moments
at the centers of pinion and gear. Consider the two springs at both ends of the pinion shaft,
their corresponding forces on the pinion are as follows, where j is the bearing index:
2
∑F
j =1
SB , p , j
⎧ − K xx , p1 (u xp + l p1θ yp ) − K xθ y , p1θ yp − K xx , p 2 (u xp − l p 2θ yp ) − K xθ y , p 2θ yp ⎫
⎪
⎪
= ⎨− K yy , p1 (u yp − l p1θ xp ) − K yθ x , p1θ xp − K yy , p 2 (u yp + l p 2θ xp ) − K yθ x , p 2θ xp ⎬ (6.28)
⎪
⎪
− K zz , p1u zp − K zz , p 2u zp
⎩
⎭
The moments due to these bearing forces on the pinion are given as:
2
∑M
j =1
SB , p, j
⎧ l p1K yy , p1 (uyp − l p1θxp ) + l p1K yθx , p1θ xp + l p 2 K yy , p 2 (uyp + l p 2θ xp ) + l p 2 K yθx , p 2θ xp ⎫
⎪
⎪
= ⎨−l p1K xx, p1 (uxp + l p1θ yp ) − l p1K xθ y , p1θ yp − l p 2 K xx, p 2 (uxp − l p 2θ yp ) − l p 2 K xθ y , p 2θ yp ⎬ (6.29)
⎪
⎪
0
⎩
⎭
Similarly, the forces and moments due to the two bearings on the gear are:
172
2
∑F
j =1
SB , g , j
2
∑M
j =1
6.5
SB , g , j
⎧ − K xx , g1 (u xg + lg1θ yg ) − K xθ y , g1θ yg − K xx , g 2 (u xg − lg 2θ yg ) − K xθ y , g 2θ yg ⎫
⎪⎪
⎪⎪
= ⎨− K yy , g1 (u yg − lg1θ xg ) − K yθ x , g1θ xg − K yy , g 2 (u yg + l g 2θ xg ) − l g 2 K xθ y , g 2θ yg ⎬ (6.30)
⎪
⎪
− K zz , g1u zg − K zz , g 2u zg
⎪⎩
⎪⎭
⎧ lg1 K yy , g1 (u yg − lg1θ xg ) + lg1 K yθ x , g1θ xg + lg 2 K yy , g 2 (u yg + lg 2θ xg ) + lg 2 K yθ x , g 2θ xg ⎫
⎪
⎪
= ⎨ −lg1 K xx , g1 (u xg + l p1θ yg ) − l p1 K xθ y , p1θ yp − lg 2 K xx , g 2 (u xg − lg 2θ yg ) − lg 2 K xθ y , g 2θ yg ⎬ (6.31)
⎪
⎪
0
⎩
⎭
Twelve DOF Helical Gear Pair Model
First, the viscous damping matrix is derived from the modal properties of the
components by assuming modal damping ratios. In the 12 DOF model, the nominal
external load is treated as excitations and the parametric excitations of tooth stiffness
variation and friction effects are incorporated into a time-varying K matrix. Thus, a direct
implementation of modal damping ratio will result in complex-valued viscous damping
terms. Consequently, only the diagonal viscous damping terms (in the damping matrix)
correspond to the directions of motions are considered, i.e. the diagonal viscous damping
terms are assumed to be dominant over other coupling terms. More specifically:
(i) For the translational DOFs along x, y and z axes, the mesh damping force on pinion is:
2
∑F
j =1
V , p, j
(
(
(
)
)
)
⎧ −2 ζ xx , p1 K xx , p1m p + ζ xx , p 2 K xx , p 2 m p ⋅ u xp ⎫
⎪
⎪
⎪
⎪
= ⎨−2 ζ yy , p1 K yy , p1m p + ζ yy , p 2 K yy , p 2 m p ⋅ u yp ⎬
⎪
⎪
⎪ −2 ζ zz , p1 K zz , p1m p + ζ zz , p 2 K zz , p 2 m p ⋅ u zp ⎪
⎩
⎭
173
(6.32)
The mesh damping force on the gear is:
2
∑F
V ,g , j
j
(
(
(
)
)
)
⎧ −2 ζ xx , g1 K xx , g1mg + ζ xx , g 2 K xx , g 2 mg ⋅ u xg ⎫
⎪
⎪
⎪
⎪
= ⎨−2 ζ yy , g1 K yy , g1mg + ζ yy , g 2 K yy , g 2 mg ⋅ u yg ⎬
⎪
⎪
⎪ −2 ζ zz , g1 K zz , g1mg + ζ zz , g 2 K zz , g 2 mg ⋅ u zg ⎪
⎩
⎭
(6.33)
(ii) For the rotational DOFs along the x and y directions (rocking motions), the mesh
damping moments for the pinion and gear are:
M V ,θ x p = −2ζ θ xθ x , p Kθ xθ x , p J xp ⋅ θxp , M V ,θ y p = −2ζ θ yθ y , p Kθ yθ y , p J yp ⋅ θyp
(6.34a, b)
M V ,θ x g = −2ζ θ xθ x , g Kθ xθ x , g J xg ⋅ θxg , M V ,θ y g = −2ζ θ yθ y , g Kθ yθ y , g J yg ⋅ θyg
(6.35a, b)
(iii) For the rotational DOFs along the axial directions, the time-varying tooth mesh
damping is dominant. Thus the damping moments are:
M V ,θ z p = −2ζ θ z p J zp / rbp
M V ,θ z g = −2ζ θ z g J zp / rbg
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
K mesh,θ z p ,i (t ) ⋅ rrpθzp + rbgθzg
(
)
(6.36)
(
)
(6.37)
K mesh ,θ z g ,i (t ) ⋅ rrpθzp + rbgθzg
where i is the index for contact tooth pairs, and the floor function rounds the contact ratio
σ to the nearest integer towards minus infinity. Here, rrpθzp + rbgθzg is the relative
174
dynamic velocity along the LOA direction; K mesh ,θ z p (t ) and K mesh ,θ z g (t ) are the timevarying dynamic tooth stiffness functions derived earlier but considering only the
rotational DOF in the z direction, and they also incorporate the effect of sliding friction.
The equations of motion for the 12 DOF helical gear pair model are derived as
follows. The pinion equations in the three translational directions are:
m p uxp =
m p uyp =
m p uzp =
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
2
2
j =1
j =1
2
2
j =1
j =1
2
2
j =1
j =1
Fmesh, xp ,i + ∑ FSB, xp , j + ∑ FV , xp , j
Fmesh , yp ,i + ∑ FSB , yp , j + ∑ FV , yp , j
Fmesh, zp ,i + ∑ FSB , zp , j + ∑ FV , zp , j
(6.38a)
(6.38b)
(6.38c)
The pinion equations in the three rotational directions along x, y and z axes are:
J xpθxp =
J ypθyp =
J zpθzp =
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
2
M mesh,θ x p ,i + ∑ M SB ,θ x p , j + M V ,θ x p
(6.39a)
j =1
2
M mesh,θ y p ,i + ∑ M SB ,θ y p , j + M V ,θ y p
(6.39b)
M mesh ,θ z p ,i + M V ,θ z p + Tp
(6.39c)
j =1
The gear equations in the three translational directions are:
175
mg uxg =
mg uyg =
m p uzg =
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i=0
n = floor(σ )
∑
i =0
2
2
j =1
j =1
2
2
j =1
j =1
2
2
j =1
j =1
Fmesh, xg ,i + ∑ FSB, xg , j + ∑ FV , xg , j
Fmesh, yg ,i + ∑ FSB , yg , j + ∑ FV , yg , j
Fmesh, zg ,i + ∑ FSB , zg , j + ∑ FV , zg , j
(6.40a)
(6.40b)
(6.40c)
And, finally the gear equations in the three rotational directions along x, y and z
axes are:
J xgθxg =
J ygθyg =
J zgθzg =
6.6
n = floor(σ )
∑
i=0
n = floor(σ )
∑
i =0
n = floor(σ )
∑
i =0
2
M mesh ,θ x g ,i + ∑ M SB ,θ x g , j + M V ,θ x g
(6.41a)
j =1
2
M mesh ,θ y g ,i + ∑ M SB ,θ y g , j + M V ,θ y g
(6.41b)
M mesh,θ z g ,i + M V ,θ z g − Tg
(6.41c)
j =1
Role of Sliding Friction Illustrated by an Example
The governing equations are numerically solved for the example case. We will
examine the following variables for parametric studies: (i) translational displacements
u xp , u yp ,u zp , u xg , u yg ,u zg ; (ii) composite displacements δ z = rbpθ zp + rbgθ zg + u xp − u xg ,
δ y = rbpθ yp + rbgθ yg + u zp − uzg and δ x = rbpθ xp + rbgθ xg +u zp − uzg , which are the coupled
176
torsional-translational motions; (iii) dynamic bearing forces for the simplified case with
l p1 = l p 2
,
K xx , p1 = K xx , p 2 = 0.5 K xx , p
,
K yy , p1 = K yy , p 2 = 0.5 K yy , p
and
K zz , p1 = K zz , p 2 = 0.5 K zz , p . The combined bearing force in the LOA direction for the pinion
is: FSB , xp = − K xx , pu xp − 2ζ xp K xx , p m p . Other bearing forces are defined similarly. The role
of sliding friction is illustrated in Figure 6.6 by comparing normalized time and
frequency domain responses of u xp , u yp ,u zp (at Tp = 2000 lb-in and Ω p = 1000 RPM) for
µ = 0.01 and µ = 0.1 . Note that time t is normalized with respect to the mesh period Tmesh ,
and n is the harmonic number of the gear mesh frequency. Observe that the OLOA
vibratory motion u yp is most significantly affect by the sliding friction. Increasing µ
almost proportionally enhances the magnitude of u yp over the entire frequency range. It
is consistent with Eq. (6.8) which shows the magnitude of the friction force is
proportional to µ . The sliding friction has a moderate influence on u xp in the LOA
direction. An increase in µ significantly increases the amplitudes at n = 1 and 2 but the
higher harmonics remain almost unchanged. The axial displacements u xp are least
sensitive to µ except for the first two harmonics. Nevertheless, Eq. (6.15) shows that the
axial shuttling excitation is proportional to sin β b . Hence, it is implied that larger helical
angle β b will lead to increased shuttling excitations due to the time-varying mesh
stiffness. The bearing resonances are around n = 8 to 10 and they could be easily tuned by
varying bearing stiffness. Predicted gear displacements u xg , u yg ,u zg share essentially the
characteristics of pinion motions.
177
-3
8
x 10
1.82
4
1.78
1.74
4.5
5
5.5
6
0
6.5
1
2
3
4
5
1
x 10-3
2
3
4
5
1
2
3
4
5
0.06
0.1
0.04
0
0.02
-0.1
4.5
5
5.5
6
0
6.5
-1.3
6
4
-1.4
2
-1.5
4.5
5
5.5
6
0
6.5
t/Tmesh
Mesh order n
Figure 6.6 Time and frequency domain responses of translational pinion displacements
u xp , u yp ,u zp at Tp = 2000 lb-in and Ω p = 1000 RPM. All displacements are normalized
with respect to 39.37 µinch (1 µ m ). Key:
, µ = 0.01 ;
178
, µ = 0.1 .
3
-690
2
-700
1
-710
-720
4.5
5
5.5
6
6.5
0
100
30
50
20
0
10
-50
4.5
5
5.5
6
6.5
0
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1.5
290
1
280
0.5
270
260
4.5
5
5.5
6
6.5
t/Tmesh
0
Mesh order n
Figure 6.7 Time and frequency domain responses of pinion bearing forces FSB , xp , FSB , yp
and FSB , zp at Tp = 2000 lb-in and Ω p = 1000 RPM. All forces are normalized with respect
to 1 lb. Key:
, µ = 0.01 ;
, µ = 0.1 .
179
Figure 6.7 shows dynamic bearing forces in x, y and z directions for the pinion.
Characteristics similar to the displacement responses of Figure 6.6 are observed,
implying that the elastic force components dominate over the viscous forces (with 5%
damping ratios). Compared with the spur gear set [6.6], the bearing forces in the helical
gear pair are reduced by more than one order of magnitude due to the gradual approach
and recess motions. Figure 6.8 shows the composite displacements δ x , δ y , δ z around the x,
y and z axes. Observe that an increase in µ significantly increases the amplitudes at n = 1
and 2 for δ x and δ y , but has only minor influence on δ z . This observation is consistent
with the results reported by Velex and Cahouet [6.4]. Nevertheless, one has to note that
the amplitude of δ z is higher than those of δ x and δ y by at least two orders of magnitude.
To examine the effect of sliding friction, Figure 6.8 also shows the time and frequency
domain responses of δz , which is the relative torsional-translational velocity along the
LOA direction. It is seen that an increase in µ introduces additional oscillations when
tooth pairs pass across the pitch point. Despite that the first mesh harmonic dominates the
spectral contents of δ z or δz , a careful comparative study shows that the second harmonic
is most significantly amplified due to friction.
180
Figure 6.8 Time and frequency domain responses of composite displacements δ x , δ y , δ z
and velocity δ at Tp = 2000 lb-in and Ω = 1000 RPM. All motions are normalized with
z
p
respect to 39.37 µinch (1 µ m ) or 39.37 µ inch/s (1 µ m / s ). Key:
µ = 0.1 .
181
, µ = 0.01 ;
,
The effect of sliding friction could be better observed by varying µ from 0 to 0.3
and then by generating the spectral contents of u xp , u yp ,u zp , u xg , u yg ,u zg and δ x , δ y , δ z up to
15 harmonics (n) of the gear mesh frequency [6.17]. Though the resultant figures are not
included here due to space constraints, some observations are as follows. 1. In the LOA
direction, an in increase in µ significantly enhances amplitudes not only at n = 1 and 2 of
u xp and u xg (due to the reversal of friction force at pitch point), but also at higher
harmonics as well, say around the torsional-transverse mode. This implies that friction
force acts as a potential source in the LOA direction to excite resonances that are
controlled due to shaft/bearing compliances. 2. In the OLOA direction, an increase in µ
efficiently increases the amplitudes of u yp and u yg over the entire frequency range,
especially at n = 1 and 2 and at the torsional-transverse resonance. This clearly shows that
the OLOA dynamics are most significantly dictated by the friction effect. 3. The axial
vibratory u zp and u zg motions are high at the torsional-transverse resonance (controlled by
the axial bearing stiffness), but the resonant amplitude does not seem to depend much on
µ . Nonetheless, friction significantly increases the amplitudes at n = 1 and 2. This
indicates that the shuttling forces are relatively insensitive to the sliding friction when
compared with the LOA and OLOA dynamics. 4. For the composite torsional-transverse
displacements δ x and δ y , an increase in µ increases most harmonic amplitudes,
especially at the first two harmonics and at higher bearing stiffness controlled resonances.
5. An increase in µ has negligible influence on δ z except at the second mesh harmonic.
182
This observation is similar to that found in a spur gear pair of Chapter 2. However, the δ z
amplitude in the helical gear pair is not as significantly influenced by the sliding friction.
6.7
Conclusion
A new 12 DOF model for helical gears with sliding friction has been developed; it
includes rotational motions, translations along the LOA and OLOA directions and axial
shuttling motions. Key contributions include the following: Three-dimensional model has
been proposed that characterizes the contact plane dynamics and captures the reversal at
the pitch line due to sliding friction. Calculation of the contact forces and moments is
illustrated by using a sample helical gear pair. A refined method is also suggested to
estimate the tooth stiffness density function along the contact lines by using the FE/CM
analysis [6.11]. Among the 12 DOFs described above, the rotational (rocking) motions
around the LOA and OLOA directions and the axial motions are usually relatively
insignificant. Therefore, a simplified 6DOF model (with coordinates u xp , u yp ,u xg , u yg ,θ zp
and θ zg ) could be easily derived based on Eqs. (6.3) to (6.40) by neglecting the
u zp , u zg , θ xp , θ xg , θ yp and θ yg variables. Such a 6DOF model requires less computational
efforts though it should yield results comparable to those by the 12 DOF model. Future
work should also include validation of the proposed theory, say by running the FE/CM
code in the dynamic mode and by conducting analogous experiments on gears with
different friction conditions.
183
References for Chapter 6
[6.1] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear
Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.
[6.2] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration,
243(3), pp. 525-545.
[6.3] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and
Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671694.
[6.4] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on
the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of
Mechanical Design, 122(4), pp. 515-522.
[6.5] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations
in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp 641-658.
[6.6] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body
Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3),
pp. 479-499.
[6.7] Borner, J., and Houser, D. R., 1996, “Friction and Bending Moments as Gear Noise
Excitations,” SAE Transaction, 105(6), pp. 1669-1676.
[6.8] Houser, D. R., Vaishya M., and Sorenson J. D., 2001, “Vibro-Acoustic Effects of
Friction in Gears: An Experimental Investigation,” SAE Paper # 2001- 01-1516.
[6.9] Blankenship, G. W., and Singh, R., 1995, “A New Gear Mesh Interface Dynamic
Model to Predict Multi-Dimensional Force Coupling and Excitation,” Mechanism and
Machine Theory Journal, 30(1), pp. 43-57.
[6.10] Helical3D (CALYX software), 2003,
www.ansol.www, ANSOL Inc., Hilliard, OH.
“Helical3D
User’s
Manual,”
[6.11] Padmanabhan, C., Barlow, R. C., Rook, T. E., and Singh, R., 1995,
“Computational Issues Associated with Gear Rattle Analysis,” ASME Journal of
Mechanical Design, 117, pp. 185-192.
[6.12] Perret-Liaudet, J. and Sabot, J, 1992, “Dynamics of a Truck Gearbox,” Proceeding
of 5th Internation Power Transmission and Gearing Conference, Phoenix, 1, pp. 249-258.
184
[6.13] Houser, D. R. and Singh, R., 2004-2005, “Basic and Advanced Gear Noise Short
Courses,” The Ohio State University.
[6.14] Lim, T. C., and Singh, R., 1991, “Vibration Transmission through Rolling Element
Bearings. Part II: System Studies,” Journal of Sound and Vibration, 139(2), pp. 201-225.
[6.15] Vinayak, H., Singh, R., and Padmanabhan, C., 1995, “Linear Dynamic Analysis of
Multi-Mesh Transmissions Containing External, Rigid Gears,” Journal of Sound and
Vibration, 185(1), pp. 1-32.
[6.16] Lim, T. C., and Singh, R., 1990, “Vibration Transmission through Rolling Element
Bearings. Part I: Bearing Stiffness Formulation,” Journal of Sound and Vibration, 139(2),
179-199.
[6.17] Singh, R., 2005, “Dynamic Analysis of Sliding Friction in Rotorcraft Geared
Systems,” Technical Report submitted to the Army Research Office, grant number
DAAD19-02-1-0334.
[6.18] Umeyama, M., Kato, M., and Inoue, K., 1998, “Effects of Gear Dimensions and
Tooth Surface Modifications on the loaded Transmission Error of a Helical Gear Pair,”
ASME Journal of Mechanical Design, 120, pp. 119–125.
[6.19] Guilbault, R., Gosselin, C., and Cloutier, L., 2005, “Express Model for Load
Sharing and Stress Analysis in Helical Gears,” ASME Journal of Mechanical Design,
127(6), pp. 1161-1172.
[6.20] Guingand, M., de Vaujany, J. P., and Icard, Y., 2004, “Fast Three-Dimensional
Quasi-Static Analysis of Helical Gears Using the Finite Prism Method,” ASME Journal
of Mechanical Design, 126(6), pp. 1082-1088.
[6.21] Gagnon, P., Gosselin, C., and Cloutier, L., 1996, “Analysis of Spur, Helical and
Straight Level Gear Teeth Deflection by the Finite Strip Method,” ASME Journal of
Mechanical Design, 119(4), pp. 421–426.
185
CHAPTER 7
ANALYSIS OF HELICAL GEAR DYNAMICS USING FLOQUET THEORY
7.1
Introduction
This work is an extension of Chapter 6 where we proposed a 12 degrees-of-
freedom (DOF), linear time-varying (LTV) analytical model for helical gears that
characterizes the contact plane dynamics and captures the velocity reversal at the pitch
line due to sliding friction. Earlier, Velex and Cahouet [7.1] had found that the dynamic
bearing forces (as related to the sliding friction in helical or spur gears) can indeed
generate significant time-varying excitations at lower speeds. Velex and Sainsot [7.2]
have examined friction excitations in errorless spur and helical gear pairs, and reported
that the friction appears as a non-negligible excitation source especially for translating
motions. Lundvall et al. [7.3] proposed a multi-body model for spur gears and briefly
discussed the role of profile modification in the presence of sliding friction. Vaishya and
Singh [7.4-7.6] had illustrated frictional issues for spur gears by assuming equal load
sharing among the contact teeth. This assumption leads to a rectangular variation in
stiffness, which is a special case (zero helical angle) of a generic trapezoidal stiffness
186
profile for helical gears. To overcome this deficiency [7.4-7.6], in Chapter 2 we have
developed a more accurate model of the spur gears that incorporates realistic mesh
stiffness and sliding friction. Many of the models cited above are solved numerically and
thus there is a clear need for analytical (closed form) solutions to the dynamic response of
a helical gear pair under the influence of sliding friction. In fact, Vaishya and Singh [7.5]
had applied the Floquet theory to a simplified spur gear model to predict responses of
parametrically-excited system and to assess the system stability. This chapter will
enhance Vaishya and Singh’s work [7.5] by applying the Floquet theory to a helical gear
pair.
7.2
Linear Time-Varying Formulation
Chief objectives of this chapter are as follows. First, place emphasis on periodic
frictional effects at the gear tooth interface by ignoring other directional properties and
the auxiliary components of the gearbox. This will allow us to describe the single mesh
helical geared system as a simplified single degree-of-freedom (SDOF), linear timevarying (LTV) oscillator with piece-wise linear effective mesh stiffness; frictional
forces/moments will be formulated as parametric excitations. Second, derive closed form
solutions for the LTV system in terms of the dynamic transmission error (DTE) under
both homogeneous and forced conditions by using the Floquet theory. Third, validate the
proposed theory by using the numerical integration method. Key assumptions include: (1)
The vibratory motions are small compared with the kinematical motion, so that the
187
position of the contact lines and the relative sliding velocity depend only on the mean
angular motions of the gear pair. (2) The mesh stiffness per unit length along the contact
lines (i.e. stiffness density k) is constant; this is equivalent to the equal load sharing
assumption in spur gears [7.4-7.6]. (3) Coulomb’s law with a constant coefficient of
friction (µ) is employed [7.1-7.6] though mixed lubrication regimes exist [7.7]. (4) The
bearing stiffness is assumed to be much higher than the mesh stiffness and thus the
shaft/bearings could be simplified as rigid connections. Hence, only the torsional DOFs
are considered in terms of DTE. Also, it is assumed that the mean load is high such that
the dynamic load is insufficient to cause any tooth separations [7.8-7.9].
The helical geared system is depicted in Figure 7.1, where the pinion and gear are
modeled as rigid cylinders linked by a series of independent stiffness elements that
describe the contact plane tangent to the base cylinders. The pinion and gear dynamics
are formulated in the coordinate systems located at their respective centers; the nominal
motions are given as −Ω p ez and Ω g ez = ez Ω p rbp / rbg , respectively. Here, the z axis
coincidences with the axial direction, e is the unit directional vector, and rbp and rbg are
the base radii of pinion and gear. An (static) input torque Tp is applied to the pinion, and
the (static) braking torque Tg on the gear obeys the basic gear kinematics. Superposed on
the kinematic motions are rotational vibratory motions denoted by θzp and θzg for the
pinion and gear. Analytical formulations are demonstrated via the following example
case with parameters of the pinion (relevant gear parameters are within the parenthesis):
number of teeth 25 (31); outside diameter 3.38 (4.13) in.; pitch diameter 3.125 (3.875) in.;
root diameter 2.811 (3.56) in.; center distance 3.5 in.; transverse diametral pitch 8 in.-1;
transverse pressure angle 25°; helix angle βb = 21.5°; face width W = 1.25 in.; polar
188
moment of inertia Jpz = 8.33×10-3 (Jgz = 1.64×10-2) lb-s2in.; mass 1.26×10-2 (1.58×10-2)
lb-s2/in.-1. Since the overall contact ratio σc = 2.7, either two or three tooth pairs are in
contact at any time instant. The three meshing tooth pairs within one mesh cycle are
numbered as #0, #1 and #2, respectively. A constant mesh stiffness density (k) along the
contact lines could be estimated via a static analysis by using the finite elements/contact
mechanics (FE/CM) formulation [7.10-7.11].
θ
φ
θ
βb
Figure 7.1 Schematic of the helical gear pair system.
189
A
simplified
SDOF
model
could
be
derived
in
items
of
DTE
δ (t ) = rbpθ zp (t ) + rbgθ zg (t ) at the gear mesh by assuming rigid links at the shaft/bearings.
Note that the dynamic mesh forces oriented in other directions (such as the sliding force
in the OLOA direction) still need to be formulated for calculations of the dynamic
moments in the torsional direction. Hence, the effective torsional tooth stiffness should
have contributions from both the sliding friction and the time-varying elastic tooth
stiffness due to Hertzian contact.
For multiple tooth pairs in contact, n = ceil(σc) (n = 3 for the example case) pairs
of meshing teeth need to be formulated, where the “ceil” function rounds σc to the nearest
integers towards infinity. Figure 7.2 illustrates the snapshot at the beginning of a mesh
cycle. At this instant, pair #0 (defined as x0 (t ) = mod(Ω p rbpt , λ ) + LT1A , where λ is the base
pitch; L represents the geometrical distance; the modulus function is mod(x,
y)=x−y⋅floor(x/y) for y ≠ 0; and the “floor” function rounds x/y to the nearest integer
towards minus infinity) just comes into mesh at point A and pair #1 (defined
as x1 (t ) = mod(Ω p rbp t , λ ) + λ +LT1A ) is in contact along line CI. Likewise, pair #2 (defined
as x2 (t ) = mod(Ω p rbp t , λ ) + 2λ +LT1A ) contacts each other along line MN. As the gears roll,
the contact lines move diagonally across the contact zone. When pair #0 reaches the pitch
point P, the relative sliding velocity between pinion and gear starts to reverse, resulting in
a reversal of friction force along the portion of contact line surpass the pitch point. Once
pair #0 reaches line CI (and pair #1 reaches MN line) at the end of the mesh circle, pair
#0 becomes #1 (and pair #1 becomes #2) corresponding to the start of the next mesh
cycle. The dynamic tooth stiffness functions Kp,i(t) and Kg,i(t) are defined below where
190
δ (t ) is the dynamic transmission error, Mp,i(t) and Mg,i(t) are dynamic moments on the
pinion and gears, respectively:
K p ,i (t ) =
M p ,i (t )
rbpδ (t )
, K g ,i (t ) =
M g ,i (t )
rbg δ (t )
(i = 0,1, 2) .
(7.1a,b)
Figure 7.2 Contact zones at the beginning of a mesh cycle within the contact plane. Key:
PP’ is the pitch line; AA’ is the face width W; AD is the length of contact zone Z.
191
For the example case, Kp,i(t) and Kg,i(t) are explicitly derived for each meshing
tooth pair over eight contact zones (Zi) as shown in Figure 7.2. Also, refer to Chapter 6
for details. Zones 1 and 2 correspond to pair #0 before and after reaching the pitch line;
Zones 3, 4 and 5, and Zones 6, 7 and 8 correspond to pairs #1 and #2, respectively. The
stiffness of the two contact zones for the first pair (#0) are derived as follows, where xm,
xf, zm and zf denote the lower and upper limits along x and z axes, as shown in Figure 7.1.
⎧k
⎪r
⎪ bp
⎪
K p ,0 (t ) = ⎨
⎪k
⎪r
⎪ bp
⎩
⎧k
⎪r
⎪ bg
⎪
K g ,0 (t ) = ⎨
⎪ k
⎪r
⎪⎩ bg
⎡ µ
⎤
⎢ − 2 ( x f (t ) + xm ) + rbp cos βb ⎥ ( z f − zm (t ) )
⎣
⎦
Z1
⎡ µ
⎤
⎢ − 2 ( x p + xm )( z p (t ) − zm (t ) ) + rbp cos βb ( z f − zm (t ) ) ⎥
⎢
⎥ Z2
⎢ + µ ( x (t ) + x )( z − z (t ) )
⎥
p
f
p
⎢⎣ 2 f
⎥⎦
⎡µ
⎤
⎢⎣ 2 ( x f (t ) + xm − 2 xg ) + rbg cos β b ⎥⎦ ( z f − zm (t ) )
(7.2a)
Z1
(7.2b)
µ
⎡µ
⎤
⎢ 2 ( x p + xm )( z p − zm (t ) ) − 2 ( x f (t ) + x p )( z f − z p (t ) ) ⎥
⎢
⎥ Z2
⎢⎣ + xg µ ( z f + zm (t ) − 2 z p (t ) ) + rbg cos β b ( z f − zm (t ) ) ⎥⎦
The second meshing tooth pair (#1) is classified into Zones 3, 4 and 5 as shown in Figure
7.2. Its composite torsional stiffness could be derived for the pinion and gear as:
192
⎧
⎪k
⎪r
⎪ bp
⎪
⎪
⎪k
K p ,1 (t ) = ⎨
⎪ rbp
⎪
⎪
⎪k
⎪
⎪ rbp
⎩
µ
⎡ µ
⎤
⎢ − 2 ( x p + xm )( z p (t ) − zm (t ) ) + 2 ( x f (t ) + x p )( z f − z p (t ) ) ⎥
⎢
⎥
⎢⎣ + rbp cos β b ( z f − zm (t ) )
⎥⎦
µ
⎡ µ
⎤
⎢ − 2 ( x p + xm (t ) )( z p (t ) − zm ) + 2 ( x f (t ) + x p )( z f − z p (t ) ) ⎥
⎢
⎥
⎢⎣ + rbp cos β b ( z f − zm (t ) )
⎥⎦
µ
⎡ µ
⎤
⎢ − 2 ( x p + xm (t ) )( z p (t ) − zm ) + 2 ( x f + x p )( z f (t ) − z p (t ) ) ⎥
⎢
⎥
⎢⎣ + rbp cos βb ( z f (t ) − zm )
⎥⎦
⎧
⎪k
⎪r
⎪ bg
⎪
⎪
⎪k
K g ,1 (t ) = ⎨
⎪ rbg
⎪
⎪
⎪k
⎪
⎪ rbg
⎩
Z3
Z4 (7.3a)
Z5
µ
⎡µ
⎤
⎢ 2 ( x p + xm )( z p (t ) − zm (t ) ) − 2 ( x f (t ) + x p )( z f − z p (t ) ) ⎥
⎢
⎥
⎢⎣ + xg µ ( z f + zm (t ) − 2 z p (t ) ) + rbg cos β b ( z f − zm (t ) )
⎥⎦
µ
⎡µ
⎤
⎢ 2 ( x p + xm (t ) )( z p (t ) − zm ) − 2 ( x f (t ) + x p )( z f − z p (t ) ) ⎥
⎢
⎥
⎢⎣ + xg µ ( z f + zm − 2 z p (t ) ) + rbg cos β b ( z f − zm )
⎥⎦
Z3
Z4 (7.3b)
µ
⎡µ
⎤
⎢ 2 ( x p + xm (t ) )( z p (t ) − zm ) − 2 ( x f + x p )( z f (t ) − z p (t ) ) ⎥
⎢
⎥
⎢⎣ + xg µ ( z f (t ) + zm − 2 z p (t ) ) + rbg cos β b ( z f (t ) − zm )
⎥⎦
Z5
The third meshing tooth pair (#2) is divided into contact Zones 6, 7 and 8 as shown in
Figure 7.2. Its composite torsional stiffness could be derived for the pinion and gear as:
⎧
⎪
⎪
⎪
⎪
⎪
K p ,2 (t ) = ⎨
⎪
⎪
⎪
µ
⎡ µ
⎤
k ⎢ − 2 ( x p + xm (t ) )( z p (t ) − zm ) + 2 ( x f + x p )( z f (t ) − z p (t ) ) ⎥
⎥ Z6
rbp ⎢
⎢⎣ + rbp cos β b ( z f (t ) − zm )
⎥⎦
k ⎡µ
⎤
x f + xm (t ) ) + rbp cos β b ⎥ ( z f (t ) − zm )
Z7
(
⎢
rbp ⎣ 2
⎦
Z8
0
193
(7.4a)
⎧ ⎡µ
µ
⎤
⎪ k ⎢ 2 ( x p + xm (t ) )( z p (t ) − zm ) − 2 ( x f + x p )( z f (t ) − z p (t ) ) ⎥
⎪r ⎢
⎥ Z6
bg
⎪ ⎢⎣ + xg µ ( z f (t ) + zm − 2 z p (t ) ) + rbg cos β b ( z f (t ) − zm )
⎥⎦
⎪
⎪ k ⎡µ
⎤
K g ,2 (t ) = ⎨ ⎢ ( x f + xm (t ) ) + rbp cos β b ⎥ ( z f (t ) − zm )
Z7
r
2
⎣
⎦
bg
⎪
⎪ 0
Z8
⎪
⎪
(7.4b)
The undamped torsional equations for the pinion and gear are derived as follows:
2
J pzθzp (t ) + ∑ rbp K p ,i (t ) [δ (t ) − ε (t ) ] = Tp
(7.5)
i =0
2
J gzθzg (t ) + ∑ rbg K g ,i (t )δ (t ) = −Tg
(7.6)
i=0
Define DTE δ (t ) and reduce Eqs. (7.5) and (7.6) into one equation which
describes an equivalent translational definite system as follows, where ε(t) is the
unloaded static transmission error. A time-varying viscous damping coefficient Ce(t) is
also included given (assumed) damping ratio ζe.
meδ(t ) + Ce (t ) ⎡⎣δ(t ) − ε(t ) ⎤⎦ + K e (t ) [δ (t ) − ε (t ) ] = Fe
me =
J pz J gz
rbg2 J pz + rbp2 J gz
,
2
2
rbp2 J gz K p ,i (t ) + rbg2 J pz K g ,i (t )
i =0
i =0
rbg2 J pz + rbp2 J gz
K e (t ) = ∑ K e ,i (t ) = ∑
Ce ,i (t ) = 2ζ e me K e (t ) ,
(7.7a)
Fe =
194
rbpTp J gz − rbg Tg J pz
rbg2 J pz + rbp2 J gz
(7.7b-c)
(7.7d-e)
Here, me is the effective mass defined in the torsional-transverse direction and Fe is the
effective external force due to the nominal torques applied at the pinion and gear. The
periodically time-varying effective stiffness function Ke,i(t) of the ith meshing mesh pair is
piece-wise linear, and it incorporates contributions from both the mesh tooth stiffness and
the sliding friction. The frictional influence on Ke,i(t) is illustrated in Figure 7.3 over eight
contact zones, where a generic effective stiffness function is obtained by following a
single tooth pair for three complete mesh cycles since ceil(σc) = 3. When µ = 0 (no
friction), Ke,i(t) has a symmetric trapezoidal profile, when high sliding friction is
introduced with µ = 0.4, additional discontinuities in the slope emerge during the
transitions from Zone 1 to Zone 2, as well as from Zone 6 to Zone 7. These correspond to
the conditions when the contact line reaches or leaves the pitch line. Note that the
stiffness functions are “continuous” in a piece-wise manner due to the gradual
approaching and recess motions of the helical gear pair. Compared with the square-wave
shaped tooth stiffness function of a spur gear pair [7.4-7.6], this shape should be more
favorable as lower vibro-acoustic levels would be expected.
Given the piece-wise stiffness Ke(t) of Eq. (7.7), we denote j as the index for the
jth interval (with a constant slope) and define the generic periodic stiffness function Ke,j(t)
over m piece-wise intervals within one mesh cycle as follows:
K e , j (t ) = K e , j (t + T ) = K e , j −1 +
195
K e , j − K e , j −1
t j − t j −1
(t − t )
j −1
(7.8)
Ke,i (lb/in)
Figure 7.3 Individual effective stiffness Ke,i(t) along the contact zone, where Tmesh is one
mesh cycle. Key: ––, tooth pair #0 (µ = 0); ––, tooth pair #1 (µ = 0); ––, tooth pair #2 (µ
= 0); ...., tooth pair #0 (µ = 0.4); - - -, tooth pair #1 (µ = 0.4); - ⋅ -, tooth pair #2 (µ = 0.4).
For the example case with individual Ke,i(t) (i = 0, …, 2) of Figure 7.3, the
combined stiffness functions Ke,j(t) (j = 1, …, 6) are calculated over six contact regions
within one mesh cycle. Since the slope is constant within each region, only the stiffness
values Ke,j at the starting and ending time instants are needed with Ke,6 = Ke,0 due to the
periodicity. The time instants ti of each region within one period could be determined
196
based on Figure 7.2 as follows t0 = 0; t1 = (LEH/λ)T; t2 = (LCQ/λ)T; t3 = (LCD/λ)T; t4 =
(LAP/λ)T; t5 = (LEG/λ)T and t6 = T.
Table 7.1 lists the relationship between the six contact regions defined for the
combined stiffness functions Ke,j(t) and the eight contact zones defined for individual
meshing tooth pairs as given by Ke,i(t). Although the number of contact zones/regions
depends on gear geometry, the proposed modeling strategy could be easily applied to
other helical geared systems.
Contact
region
Contact zones of Figure 7.2
Pair #0
Pair #1
Pair #2
1
Z1
Z3
Z6
2
Z1
Z3
Z7
3
Z1
Z4
Z7
4
Z1
Z5
Z7
5
Z2
Z5
Z7
6
Z2
Z5
Z8
Table 7.1 Relationship between Contact Zones and Contact Regions for the NASA-ART
helical gear pair
197
Figure 7.4 compares the combined Ke,j(t) and individual Ke,i(t) functions over one
period. Observe that the profile of Ke,j(t) resembles those of individual Ke,i(t): Under zero
friction, Ke,j(t) follows a symmetric trapezoidal pattern, where 4 piece-wise intervals exist
within one mesh cycle. When the sliding friction is included, two additional
discontinuities in the slope are introduced at the transitions from Region 1 to Region 2, as
well as from Region 4 to Region 5. Hence, six piece-wise regions need to be analyzed for
one complete mesh cycle. Note that a high mean component exists for the combined
Ke,j(t), whose values are always positive (non-zero).
Figure 7.4 Piece-wise effective stiffness function defined in six regions within one mesh
cycle with µ = 0.4. Key: .... tooth pair #0, - - - tooth pair #1, - ⋅ - tooth pair #2, ––
combined stiffness function.
198
7.3
Analytical Solutions by Floquet Theory
Vaishya and Singh [7.5] suggested analytical solutions to a SDOF spur gear
system model with periodic square-wave stiffness. For a helical gear pair, Ke,i(t) varies
periodically in a trapezoidal pattern; hence, the spur gear could be regarded as a special
(limiting) case of the helical gear model where the slope of stiffness within each interval
is zero rather than an arbitrary constant. Assume ε(t) = 0 (perfect involute profile) and
Ce(t) = 0 (undamped condition), the parametrically excited system of Eq. (7.7) under a
mean load Fe is simplified as follows:
meδ(t ) + K e (t )δ (t ) = Fe
(7.9)
The Floquet theory [7.12] is then applied to find analytical solutions to both free
(including the case with a constant viscous damping Ce= 2ζ e me K e ) and forced
responses. For the example case, six contact regions need to be formulated with the
influence of sliding friction. Represent the governing equation in the state space form as:
X (t ) = G (t ) X (t ) + F (t ) , G (t + T ) = G (t )
(7.10a-b)
0 ≤ t < t1
⎧ G1 (t )
⎪
G (t ) = ⎨G j (t ) t j −1 ≤ t < t j (i = 2...5) ,
⎪
t5 ≤ t < T
⎩ G6 (t )
(7.10c)
⎧ 0 ⎫
⎧δ (t ) ⎫
X (t ) = ⎨ ⎬ , F (t ) = ⎨
⎬
⎩δ (t ) ⎭
⎩ Fe / me ⎭
199
(7.10d-e)
The solution over one complete mesh cycle T is written in the form of a state transition
matrix (Φ). For a piecewise periodic system, this matrix may further be decomposed into
Φj over each contact regions [7.5] in Eq. (7.11), where the functions are continuously
differentiable and analytical solutions to the homogeneous equation exist.
Φ (T , 0 ) = Φ (T , t5 ) ⋅⋅⋅Φ ( t2 , t1 ) Φ ( t1 , 0 )
(7.11)
Each Φ(tj, tj-1) is evaluated from the Wronskian matrix (Π) as:
Φ ( t j , t j −1 ) = Π (t j )Π −1 (t j −1 ), t j −1 ≤ t ≤ t j ,
(7.12a)
⎡γ γ ⎤
Π (t ) = ⎢ 1 2 ⎥
⎣γ1 γ2 ⎦
(7.12b)
Here, γ1 and γ2 are two basis solutions to the homogeneous equation X (t ) = G (t ) X (t ) .
Use the periodic property of Φ, Floquet theory extends solutions to future states of the
system that are apart by n mesh cycles. Thus, the state transition matrix Φ(nT , 0) over n
cycles and the resulting responses X(t) are given by:
Φ(nT , 0) = Φ (T , 0) n
(7.13)
t
X (t ) = Φ (t , 0) X (0) + ∫ Φ (t ,τ )F (τ )dτ ,
(7.14a)
X (t + nT ) = Φ n (T ,0 ) X (t )
(7.14b)
0
200
Equations (7.11-7.14) are of significant importance. First, they drastically reduce
the computational time since the results calculated for one mesh cycle can be easily
extended to other periods by using matrix multiplication, which is computationally
effective. Second, it allows an easier inversion of the matrix.
7.3.1
Response to Initial Conditions
Knowledge of the free response to initial conditions is important to assess the
dynamic stability property of the helical gear pair. Within each interval tj-1 ≤ t < tj, Eq.
(7.9) is rewritten in the homogeneous form as:
⎡
⎛
⎞⎤
2t
+ 1⎟ ⎥ δ (t ) = 0, t j −1 ≤ t ≤ t j (j = 1...6)
⎜ t j − t j −1 ⎟ ⎥
⎝
⎠⎦
δ(t ) + ⎢ a j − 2q j ⎜ −
⎢⎣
αj =
K j −1
me
aj = α j +
− β j t j −1 ,
β j ( t j − t j −1 )
2
βj =
,
1
me
⎛ K j − K j −1 ⎞
⎜⎜
⎟⎟
⎝ t j − t j −1 ⎠
qj =
β j ( t j − t j −1 )
4
(7.15a)
(7.15b-c)
(7.15d-e)
With a change of variable z j = α j + β j t , convert Eq. (7.15) into the Stoke’s
equation [7.12]:
d 2δ z j
δ =0
+
dz 2j β j2
201
(7.16)
A set of basis solutions are known over tj-1 ≤ t < tj:
γ 1 (t ) = z j J1/ 3 (σ j ),
γ 2 (t ) = z j J −1/ 3 (σ j )
(7.17a-b)
where σ j = 2 z 3/j 2 (3β j )−1 and J ±1/ 3 (σ j ) are the Bessel functions of the first kind of order
±1/3. Use the recurrence relation of Bessel functions to find the Wronskian matrix as:
⎡ z j J 1 3 (σ j )
Π j (t ) = ⎢
⎢ z j J −2 3 ( σ j )
⎣
Π −j 1 ( t ) = −
2π
3 3β j
z j J −1 3 ( σ j ) ⎤
⎥
− z j J 2 3 (σ j ) ⎥⎦
⎡ − z j J 2 3 ( σ j ) − z j J −1 3 ( σ j ) ⎤
⎢
⎥
⎢ − z j J −2 3 (σ j )
⎥
z
J
σ
(
)
j
13
j
⎣
⎦
(7.18a)
(7.18b)
Note than Eqs. (7.17-7.18) are valid only for the conditions with zj > 0. For the cases in
which zj are negative (or zero), the Wronskian matrices are derived in terms of the
modified Bessel functions of the first kind (or gamma functions), which could be treated
in a similar matter. However, for the SDOF helical gear model with a high positive mean
component, all zj have positive values so that Eqs. (7.17-7.18) hold. Also, for the
intervals with a negative slope βj (such as contact regions 4 and 5 of Figure 7.4), the
corresponding σj also has negative values, which lead to complex Πj(t) generated by the
Bessel functions in Eqs. (7.17-7.18). Nevertheless, due to the supplemental phase
relationship between Π(tj) and Π−1(tj-1), the state transition matrix Φ(tj-1, tj) = Π(tj)Π−1(tj-1)
still assumes real values within each interval tj-1 ≤ t < tj. Thus, responses to initial
202
conditions X (0) = {δ (0), δ(0)}T are derived in Eq. (7.19), where Φ(T, 0) is the discrete
transition matrix. Here, Φ(t−nT, 0) needs to be evaluated similar to Eq. (7.11) over the
last cycle.
X (t ) = Φ (t − nT , 0)Φ (T , 0 ) X (0), ( 0 ≤ t − nT < T )
n
(7.19)
Figure 7.5 compares the homogeneous responses given initial condition x0 =
2×10-6 in., v0 = 20 in./s at Ωp = 1000 RPM, as predicted by using the Floquet theory and
the numerical solution (based on the Runge Kutta scheme [7.13]). Since the numerical
solution completely overlaps with the Floquet theory prediction, only one pair of
comparative results are given with µ = 0.2 in Figure 7.5(b). Observe that increasing
sliding friction changes the slopes of the effective stiffness function Ke,j(t), while such
effect does not seem to be significant in the DTE response. This is because that the
undamped responses are dictated by the dynamic components at the system natural
(
)
frequency fn = K e / 2π me , where K e is the averaged stiffness. For the example case,
fn is found to be close to 9.5fm, where fm is the mesh frequency at Ωp = 1000 RPM. Side
bands around 8.5fm and 10.5 fm may also be present due to frequency modulation effects.
203
(a)
(b)
Figure 7.5(a) Effective stiffness and (b) homogeneous responses predictions within two
mesh cycles given x0 = 2×10-6 in., v0 = 20 in./s at Ωp = 1000 RPM. Key: ––, µ = 0
(Analytical solution by the Floquet theory); - ⋅ -, µ = 0.2 (Analytical); ...., µ = 0.2
(Numerical); - - - , µ = 0.4 (Analytical).
Damped
homogenous
response
could
also
be
derived
by
assuming
Ce ,i (t ) = 2ζ e me K e (t ) ≈ 2ζ e me K e = Ce 0 with a time-averaged viscous damping Ce0.
Thus, Eq. (7.7) is converted into constantly damped homogenous form as follows:
meδd (t ) + Ce 0δd (t ) + K e (t )δ d (t ) = 0
204
(7.20a)
By defining the transformation δ d (t ) = ψ (t )e
−
Ce 0 t
2
= ψ (t )e
−ζ e
Ke
t
me
, Eq. (7.20) is further
converted into the following expression:
⎛
Ke ⎞
meψ(t ) + ⎜1 − ζ e2
⎟ K e (t )ψ (t ) = 0
K e (t ) ⎠
⎝
(7.20b)
Since K e / K e (t ) ≈1, for small viscous damping (say ζe = 5%), its square value (2.5×10-3)
is negligible compared with 1. Hence, Eq. (7.20b) assumes the same form as the
undamped Eq. (7.15), so that it should have the same solution. This implies that for an
oscillator with small and constant viscous damping, the damped homogeneous response
could be calculated as follows, where δ(t) is the analytical solution to an undamped
system.
δ d (t ) = δ (t )e
−ζ e
Ke
t
me
(7.21)
Figure 7.6 shows that the analytical prediction of the damped homogeneous
response with a constant Ce 0 = 2ζ e me K e correlates well the numerical simulation with a
time-varying Ce(t) of Eq. (7.7e). Here, the Ke(t) profile is the same as illustrated in Figure
7.5(a) with µ = 0.2. This implies that Eq. (7.21) could be used to approximate the
homogeneous response with periodically varying stiffness and viscous damping
parameters.
205
Figure 7.6 Predictions of damped homogeneous responses within two mesh cycles given
x0 = 2×10-6 in., v0 = 20 in./s, µ = 0.2 at Ωp = 1000 RPM. Key: ––, Analytical solution by
the Floquet theory with Ce0; - ⋅ -, Numerical with Ce(t).
7.3.2
Forced Periodic Response
For the LTV system of Eq. (7.9), Φ could be applied to compute the response
under a periodic excitation. The tractability of the solution depends on both the
characteristics of the excitation and the nature of Φ. In general, this problem is solved by
expanding the forcing function as well as the time-varying parameters in terms of Fourier
coefficients. Clearly, this will lead to errors due to truncation of modes and also
206
significantly increase the computations [7.5]. For the example helical gear pair, three (or
six) piecewise linear segments need to be considered for each mesh cycle without (or
with) the influence of sliding friction. All integrals associated with the Floquet theory
could be analytically found under a mean torque excitation. However, as the number of
piecewise linear segments increases within the mesh cycle, such as for the realistic
stiffness profile, analytical solutions could become computationally expensive. The
forced response of this system is formulated as follows:
t
X (t ) = Π (t )Π −1 (t ) X (0) + ∫ Π (t )Π −1 (τ ) F (τ )dτ
(7.22)
0
Given the initial condition response Π (t )Π −1 (t ) X (0) has already been derived in
Eq.
(7.19),
only
the
forced
response
needs
to
be
derived
by
applying
Φ(t , 0) = Φ(t ,τ )Φ(τ , 0) for any τ:
nT
t
X ( t ) = Φ ( t , 0 ) ⎡ ∫ Φ −1 (τ , 0 ) F (τ ) dτ + ∫ Φ −1 (τ , 0 ) F (τ ) dτ ⎤
⎢⎣ 0
⎥⎦
nT
= Φ ( t , 0 ) ⎡⎣ H1 ( n ) + H 2 ( t ) ⎤⎦
(7.23)
The solution is found in two parts including the integral number of mesh cycles (H1) and
the last cycle (H2) [7.5]. For n complete cycles, the expression for H1(t) is found as:
n
H1 ( n ) = ∑
iT
∫
i =1 ( i −1) T
Φ i−1 (τ , 0 ) F (τ ) dτ
207
(7.24)
Define τ0 = τ + (i − 1)T and apply the Floquet theory such that:
⎧ ⎡Φ −1 (T , 0 ) ⎤ j −1 ⋅
⎫
⎦
⎪⎣
⎪
⎪⎡
t1
⎤⎪
−1
n ⎪ ⎢ Π1 ( t0 ) ∫ Π1 (τ ) F (τ 0 ) dτ +
⎥ ⎪⎪
0
⎪
H1 ( n ) = ∑ ⎨ ⎢
⎥⎬
t2
−1
−1
j =1 ⎪ ⎢ Π ( t ) Π ( t ) Π ( t )
Π
+
τ
τ
τ
F
d
(
)
(
)
⎥⎪
1 0
1
1
2 1 ∫t
2
0
1
⎪⎢
⎥⎪
⎪ ⎢Π t Π −1 t Π t Π −1 t Π t t3 Π −1 τ F τ dτ + ...⎥ ⎪
⎪⎩ ⎣⎢ 1 ( 0 ) 1 ( 1 ) 2 ( 1 ) 2 ( 2 ) 3 ( 2 ) ∫t2 3 ( ) ( 0 )
⎥⎦ ⎪⎭
(7.25)
For the last time cycle, the value of H2(t) depends upon the time instant t in the
whole mesh cycle. Hence, solutions are derived within each piecewise linear segment as
follows, where τ0 = τ + nT.
t − nT
n
H 2 ( t ) = ⎡⎣Φ −1 (T , 0 ) ⎤⎦ ⎡Π1 ( 0 ) ∫ Π1−1 (τ ) F (τ 0 ) dτ ⎤ ,
⎢⎣
⎥⎦
0
⎪
t − nT
⎪
Π1 ( 0 ) ∫ Π1−1 (τ ) F (τ 0 )dτ ;
t1
⎪
⎪
( t0 ≤ t − nT < t1 )
⎪
⎪ ⎡Π 0 t1 Π −1 τ F τ dτ + Π 0 Π −1 t Π t t − nT Π −1 τ F τ dτ ⎤ ;
( 0 ) ⎥⎦
1 ( ) 1 ( 1 ) 2 ( 1 ) ∫t
2 ( )
⎪ ⎢⎣ 1 ( ) ∫0 1 ( ) ( 0 )
1
(7.26)
n ⎪
= ⎡⎣ Φ −1 (T , 0 ) ⎤⎦ ⋅ ⎨
( t1 ≤ t − nT < t2 )
⎪
......
⎪
⎪
⎡Π ( 0 ) T Π −1 (τ ) F (τ ) dτ + ...
⎤
0
⎪
∫0 1
⎢ 1
⎥
⎪
t − nT
⎢
⎥;
−1
−1
−1
+Π
Π
Π
Π
⋅⋅⋅
Π
Π
t
t
t
τ
F
τ
d
τ
0
⎪
(
)
(
)
(
)
(
)
(
)
(
)
⎢ 1
⎥
j
j −1 ∫t
j
1
1
2 1
2
0
j −1
⎣
⎦
⎪
⎪
( t j −1 ≤ t − nT < t j , t6 = T )
⎩
208
All matrices in Eq. (7.26) have been analytically derived except for the
∫
0
0
Π −1 (τ ) F (τ ) dτ integral, which could be analytically found by using Eq. (7.27). Note
that the constant forcing function F (t ) = {0 Fe / me } could be taken out of the integral.
T
Here, the LommelS1 in Eq. (7.27) is the Lommel function [7.14].
∫ zJ
(σ )dt = − z J −1/ 3 (σ )
z J −1 3 (σ ) dt =
∫−
∫ zJ
∫
2/3
−2 / 3
(7.27a)
4σ
J −1/ 3 (σ ) L1 + σ J −4 / 3 (σ ) L2
3
(σ )dt = z J1/ 3 (σ )
z J 2 3 (σ ) dt = −
(7.27b)
(7.27c)
2σ
J1/ 3 (σ ) L3 − σ J −2 / 3 (σ ) L4
3
(7.27d)
4 2 32
L1 = LommelS1(−1, − , β z )
3 3
(7.27e)
1 2 3
L2 = LommelS1(0, − , β z 2 )
3 3
(7.27f)
2 2 3
L3 = LommelS1(−1, − , β z 2 )
3 3
(7.27g)
1 2 32
L4 = LommelS1(0, , β z )
3 3
(7.27h)
Analytical forced responses predicted by using Eqs. (7.22-7.27) compare well
with numerical results in Figure 7.7, given Ce(t) = 0, x0 = 2×10-6 in., v0 = 20 in./s, Tp =
2000 lb-in., µ = 0.2 and Ωp = 1000 RPM. Here, the Ke(t) profile is the same as Figure
209
7.5(a) with µ = 0.2; thus six piece-wise contact regions need to be considered within each
mesh cycle. Similar to the undamped homogeneous response, the forced responses are
also dominated by the dynamic component at the system natural frequency and some side
bands due to the frequency modulation effect. Such resonances, however, are efficiently
controlled by the viscous damping, which might have negligible effects on the mesh
harmonics. Consequently, if any mesh harmonic is away from the resonant frequency,
one can obtain the dynamic responses (which should share the same features as damped
responses) by filtering out the resonant components in frequency domain. In our example
case, a low pass filter is used since fn >> fmesh.
Figure 7.7 Predictions of (undamped) forced periodic responses within two mesh cycles
given x0 = 2×10-6 in., v0 = 20 in./s, Tp = 2000 lb-in, µ = 0.2 and Ωp = 1000 RPM. Key: ––,
Analytical solution by the Floquet theory; - ⋅ -, Numerical.
210
Analytical predictions of (undamped) forced responses are compared in Figure
7.8 with numerical simulations obtained from a viscously damped SDOF model as well
as a 6DOF model, which is similar to the 12DOF model of Chapter 6. Comparison of
steady-state time domain responses in Figure 7.8(a) shows that a viscous damping
coefficient of 5% tends to “remove” the dominant resonant components (as compared to
the mesh harmonic components) from the forced responses. Also, numerical simulations
of DTEs predicted from the SDOF model match well with those from the 6DOF model
despite some differences in the mean component. The steady-state time responses are
converted into frequency domain and Figure 7.8(b) (where the static term is cut off)
shows that the predictions at first five mesh harmonics of the undamped system match
very well with the spectra of viscously damped responses calculated by numerical
integration. This suggests that the analytical solution could be extended to examine the
damped dynamic response in frequency domain.
Figure 7.9(a) shows the predicted mesh harmonics of DTE as a function of µ.
Observe that an increase in µ has the most significant effect on the first two mesh
harmonics. Figure 7.9 (b) shows predicted DTE harmonics with respect to µ; these are
obtained by the finite difference method, i.e. dδ/dµ ≈ [δ(n)−δ(n−1)]/[µ(n) −µ (n−1)].
Observe that the second harmonic has the highest increasing rate followed by the first
harmonic. Moreover, since the amplitude at the second harmonic without friction (µ =0)
is much smaller than that of the first harmonic, it is implied that sliding friction has more
influences on the second harmonic. This is consistent with the results predicted by the 12
DOF formulation of Chapter 6.
211
(a)
(b)
Figure 7.8 Steady state forced periodic responses given x0 = 2×10-6 in., v0 = 20 in./s, Tp =
2000 lb-in., µ = 0.1 and Ωp = 1000 RPM: (a) DTE vs. time; (b) DTE spectra. Key:
, ,
undamped analytical prediction;
, , damped numerical simulation of SDOF system
with ζe = 5%;
, , damped numerical simulation of a 6DOF model with ζe = 5%
(with mean component compensated).
212
(a)
(b)
Figure 7.9 Predicted mesh harmonics of (undamped) forced periodic responses as a
function of µ given x0 = 2×10-6 in., v0 = 20 in./s, Tp = 2000 lb-in and Ωp = 1000 RPM: (a)
DTE; (b) slope of DTE. Key:
, n = 1;
, n = 2;
, n = 3;
, n = 4.
213
7.4
Conclusion
Work presented in this chapter extends the earlier work by Vaishya and Singh
[7.5] by applying the Floquet theory to a helical gear pair to examine the effect of sliding
friction on the DTE. In particular, the LTV formulations (with parametric excitations)
have been developed for a SDOF model, and the effect of sliding friction is quantified as
parametric excitations of effective mesh stiffness. The Floquet theory has been
successfully applied to obtain closed-form DTE solutions given periodic and piece-wise
linear stiffness functions. Responses to both initial conditions and forced periodic
functions, under a nominal preload, are derived. Analytical models have been validated
by comparing predictions with numerical simulations. Although the coefficient of friction
µ is assumed to be high for the example case for illustrative purpose, the same algorithm
could be implemented under realistic conditions when a smaller value of µ is expected.
Overall, the sliding friction has a marginal effect on the dynamic transmission error of
helical gears, as compared with spur gears, at least in the context of the torsional model.
Finally, parametric instability issues are briefly examined. Asymptotic stability of a
homogenous system can be determined from the discrete transition matrix over one
complete period of parametric changes. A sufficient condition for stability is that all the
eigenvalues κ of the Φ(T, 0) matrix have absolute values less than unity [7.12]. For the
sake of illustration, Figure 7.10 shows the mapping of maximum κ (absolute value) as a
function of the ratio of time-varying mesh frequency fmesh(t) to the system natural
frequency fn, without viscous damping. Observe that the most dominant unstable region
emerges when fmesh(t)/fn ≈ 2, such parametric instability is well explained by Den Hartog
214
[7.15]. Other unstable regions are found when the ratio of fmesh(t)/fn is close to 1, 2/3, 1/3,
etc. Also, an increase in µ tends to enhance the max { κ } value in the most dominant
unstable region around fmesh(t)/fn ≈ 2; in addition, it decreases the 1/3 peak while
enhancing the peak around 1. When the system operates near unstable regions, various
stability performances could be observed though these results are not shown here. For
instance, when fmesh(t)/fn is close to 2 (say at 18,000 RPM), long term stability
performance is observed. On the other hand, when fmesh(t)/fn falls within the unstable
κ
region (say at 19,000 RPM), the homogeneous response grows unbounded.
Figure 7.10 Mapping of eigenvalue κ (absolute value) maxima as a function of the ratio
of time-varying mesh frequency fmesh(t) to the system natural frequency fn. Key:
,µ=
0.01;
, µ = 0.1;
, µ = 0.2.
215
References for Chapter 7
[7.1] Velex, P., and Cahouet, V., 2000, “Experimental and Numerical Investigations on
the Influence of Tooth Friction in Spur and Helical Gear Dynamics,” ASME Journal of
Mechanical Design, 122(4), pp. 515-522.
[7.2] Velex, P., and Sainsot. P, 2002, “An Analytical Study of Tooth Friction Excitations
in Spur and Helical Gears,” Mechanism and Machine Theory, 37, pp. 641-658.
[7.3] Lundvall, O., Strömberg, N., and Klarbring, A., 2004, “A Flexible Multi-body
Approach for Frictional Contact in Spur Gears,” Journal of Sound and Vibration, 278(3),
pp. 479-499.
[7.4] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear
Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.
[7.5] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration,
243(3), pp. 525-545.
[7.6] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and
Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671694.
[7.7] Borner, J., and Houser, D. R., 1996, “Friction and Bending Moments as Gear Noise
Excitations,” SAE Transaction, 105(6), pp. 1669-1676.
[7.8] Padmanabhan, C., Barlow, R. C., Rook, T. E., and Singh, R., 1995, “Computational
Issues Associated with Gear Rattle Analysis,” ASME Journal of Mechanical Design, 117,
pp. 185-192.
[7.9] Xiao, D. Z., Gao Y., Wang, Z. Q., and Liu, D. M., 2005, “Conjugation Criterion for
Making Clearance of the Meshed Helical Surfaces,” ASME Journal of Mechanical
Design, 127(1), pp. 164-168.
[7.10] Helical3D (CALYX software), 2003,
www.ansol.www, ANSOL Inc., Hilliard, OH.
“Helical3D
User’s
Manual,”
[7.11] Tamminana, V. K., Kahraman, A., and Vijayakar, S., 2007, “A Study of the
Relationship between the Dynamic Factors and the Dynamic Transmission Error of Spur
Gear Pairs,” ASME Journal of Mechanical Design, 129(1), pp. 75-84.
[7.12] Richards, J. A., 1983, Analysis of Periodically time-varying Systems, New York,
Springer.
[7.13] Cartwright, J. H. E., and Piro, O., 1992, “The Dynamics of Runge-Kutta
Methods,” International Journal of Bifurcations Chaos, 2, pp. 427-449.
216
[7.14] Gray, A., and Mathews, G. B., 1966, A Treatise on Bessel Functions and Their
Applications to Physics, New York, Dover Publications.
[7.15] Den Hartog, J. P., 1956, Mechanical Vibrations, New York, Dover Publications.
[7.16] Abousleiman, V., Velex, P., and Becquerelle, S., 2007, “Modeling of Spur and
Helical Gear Planetary Drives with Flexible Ring Gears and Planet Carriers,” ASME
Journal of Mechanical Design, 129(1), pp. 95-106.
217
CHAPTER 8
CONCLUSION
8.1
Summary
A combination of analytical and numerical techniques has been utilized to fully
understand the influence of sliding friction on spur and helical gear dynamics with vibroacoustic sources. Many dynamic phenomena that emerge due to interactions between
parametric variations (time-varying mesh stiffness/damping) and sliding friction are
predicted and partially validated through experiments. A summary of the self-contained
chapters is described below.
In Chapter 2, a new multi-degree-of-freedom (MDOF), linear time-varying (LTV)
spur gear model has been formulated which overcomes the deficiency of Vaishya and
Singh’s work [8.1-8.3] by employing realistic tooth stiffness functions and the sliding
friction over a range of operational conditions. Refinements include: (1) an accurate
representation of tooth contact and spatial variation in tooth mesh stiffness based on a
finite element/contact mechanics code in the “static” mode; (2) Coulomb friction model
for sliding resistance with empirical coefficient of friction as a function of operation
218
conditions; (3) a better representation of the coupling between the LOA and OLOA
directions including torsional and translational degrees of freedom. Numerical solutions
of the MDOF model yield the dynamic transmission error (DTE) and vibratory motions
in the LOA and OLOA directions. The new model has been successfully validated first
by using the finite element code while running in the “dynamic” mode and then by
analogous experiments. Since the lumped model is more computationally efficient when
compared with the finite element analysis, it could be quickly used to study the effect of a
large number of parameters.
In Chapter 3, the MDOF spur gear pair model (initially proposed in Chapter 2)
has been improved with time-varying coefficient of friction, µ (t ) , given realistic mesh
stiffness profiles. Alternate sliding friction models have been comparatively evaluated
and the interfacial friction forces and motions in the OLOA direction were successfully
predicted. In particular, one model has been validated by comparing predictions to the
benchmark gear friction force measurements made by Rebbechi et al. [8.4].
In Chapter 4, semi-analytical solutions have been developed to periodic
differential equations with time-varying parameters of spur gears including realistic mesh
stiffness and sliding friction. Proposed one-term and multi-term harmonic balance
predictions compare well with numerical simulations; the computational efficiency is
achieved by converting the periodic differential equations into easily solvable algebraic
equations, while providing more insight into the dynamic behavior. Both super-and subharmonic analyses are successfully conducted to examine the higher mesh harmonics due
to multiplicative coefficients and the system stability, respectively. Semi-analytical
solutions are developed for a 6DOF system model for the predictions of (normal) mesh
219
loads, friction forces and bearing displacements in the LOA and OLOA directions, under
non-resonant conditions.
In Chapter 5, a refined source-path-receiver model has been developed which
characterizes the sliding friction induced structural path and associated noise radiation.
Proposed Rayleigh integral method and substitute source technique are more efficient for
calculating the acoustic field than the usual boundary element technique and thus they
provide rapid design tools to quantify the frictional noise. Individual contributions of
transmission error (via the LOA path) and frictional effects (via the OLOA path) are
compared to the overall whine noise at gear mesh frequencies.
In Chapter 6, a new 12 DOF model for helical gears with sliding friction has been
developed; it includes rotational motions, translations along the LOA and OLOA
directions and axial shuttling motions. Three-dimensional model has been proposed that
characterizes the contact plane dynamics and captures the reversal at the pitch line due to
sliding friction. Calculation of the contact forces and moments is illustrated by using a
sample helical gear pair. A refined method is also suggested to estimate the tooth
stiffness density function along the contact lines [8.5].
In Chapter 7, the Floquet theory has been applied to a helical gear pair to examine
the effect of sliding friction on the DTE. In particular, the LTV formulations (with
parametric excitations) have been developed for a SDOF model, and the effect of sliding
friction is quantified as parametric excitations of effective mesh stiffness. The Floquet
theory has been successfully applied to obtain closed-form DTE solutions given periodic
and piece-wise linear stiffness functions. Responses to both initial conditions and forced
periodic functions, under a nominal preload, are derived. Analytical models have been
220
validated by comparing predictions with numerical simulations. Although the coefficient
of friction µ is assumed to be high for the example case for illustrative purpose, the same
algorithm could be implemented under realistic conditions when a smaller value of µ is
expected. Parametric instability issues are briefly examined.
8.2
Contributions
Chief contribution of this research has been the development of new or refined
mathematical models and analysis techniques that enhance our understanding of the
influence of friction on the dynamics and vibro-acoustics of spur and helical gears. One
of the main effects of sliding friction is the enhancement of the DTE magnitude at the
second gear mesh harmonic. A key question whether the sliding friction is indeed the
source of the OLOA motions and forces (in spur gears) is then answered by this study.
The bearing forces in the LOA direction are influenced by the normal tooth loads, but the
sliding frictional forces primarily excite the OLOA motions. The effect of the profile
modification on the dynamic transmission error has been analytically examined under the
influence of frictional effects. For instance, the tip relief introduces an amplification in
the OLOA motions and forces due to an out of phase relationship between the normal
(mesh) load and friction forces. This knowledge should be of significant utility to the
designers.
New analytical solutions are constructed which provide an efficient evaluation of
the frictional effect as well as a more plausible explanation of dynamic interactions in
221
multiple directions. Both single- and multi-term harmonic balance methods are utilized to
predict dynamic mesh loads, friction forces and pinion/gear displacements. Such semianalytical solutions explain the presence of higher harmonics in gear noise and vibration
due to exponential modulations of the periodic stiffness, dynamic transmission error and
sliding friction. This knowledge also analytically reveals the effect of the tooth profile
modification in spur gears on the dynamic transmission error, under the influence of
sliding friction. Further, the Floquet theory is applied to obtain closed-form solutions of
the dynamic response for a helical gear pair, where the effect of sliding friction is
quantified by an effective piecewise stiffness function. Analytical predictions, under both
homogeneous and forced conditions, are validated using numerical simulations. The
matrix-based methodology is found to be computationally efficient while leading to a
better assessment of the dynamic stability.
Finally, the improved source-path-receiver model for friction-induced gear whine
noise reveals that near the “optimal” load, friction-induced noise is comparable to the
transmission error induced noise (especially for the first two mesh harmonics); thus
sliding friction should be considered as a significant contributor to whine noise. However,
at non-optimal torques, frictional noise is overwhelmed by the transmission error noise,
thus sliding friction could be negligible under such conditions. This confirms that the
sliding friction should be viewed as a potential contributor to structure-borne noise for
high precision, high power density geared systems.
222
8.3
Future Work
Directions for future work are suggested below (as an extension of this work).
1.
Develop coupling indices that would quantify the effect of sliding friction on the
gear dynamic and vibro-acoustic responses. For instance, the 6DOF spur gear
dynamics model (of Chapter 2) suggests that while a change in the coefficient of
sliding friction µ has a global effect, the OLOA dynamic displacements y p and
yg affect other degrees-of-freedom somewhat minimally. This implies a one-way
coupling effect and thus the system could be further divided into a 4DOF
“excitation” sub-system and a 2DOF “response” system for the OLOA dynamics.
This issue has been briefly discussed in Chapter 4 where the multi-term harmonic
balance solutions (based on the SDOF model) are extended to predict dynamics in
multiple directions due to the one-way coupling effect.
2.
Examine the effect of various modifications to spur gears and propose new
criteria which would simultaneously consider contributions from both static
transmission error and friction excitations. Also, conduct parametric design
studies to seek suitable profile modification schemes to minimize both combined
excitations over a range of operating loads. Use analytical models to examine
interactions between profile modifications and sliding friction. Further, it is
desirable to examine the mesh stiffness concept (associated with the Floquet
theory) and to quantify the role of sliding friction in terms of discontinuities and
generation of higher harmonics.
223
3.
Extend the closed form solutions and semi-analytical solutions (as proposed in
this work) to multi-mesh spur gear dynamics. Application of multi-term harmonic
balance method to multi-mesh gear systems with non-commensurable mesh
frequencies and/or backlash elements would lead to periodic, non-linear
formulations. Also, investigate friction-induced instabilities in the presence of
clearance non-linearities.
4.
Refine surface mechanics and tribological models and quantify the effects of
normal load, operating speed, lubricant viscosity, gear body materials on the
magnitude and dynamic characteristics of sliding friction forces. Also, incorporate
the effect of gear surface roughness (random components) and surface
undulations (periodic components) in gear vibration and structure-borne noise
models.
5.
Improve the system path model by including compliant shaft formulation and
moment transfer through the bearing matrix. Quantify the error induced by the
impedance mismatch assumed between the gear source sub-system and the
gearbox structural sub-system. Likewise, examine the limitations of the simplified
source-path-receiver network adopted in this work.
6.
Extend the simplified 2D sound radiation model [8.6] of the gearbox into a 3D
model which should require a more rigorous analysis by using monopoles (or
multipoles [8.7]) as the substitute sound sources. Examine the substitute sound
source property to quantify the frictional effect on the gear whine noise in terms
of acoustic source strength and directivity. Also, compare analytical predictions of
gear noise with the boundary-element solutions
224
7.
Describe other acoustic source mechanisms that may contribute to air-borne noise
induced by sliding friction between gear teeth. Potential acoustic sources may
include the oscillatory behavior of contact zone after release from sliding and
acoustic pumping within the cavities formed between mating teeth.
8.
Conduct new experimental work (over a range of tribological, thermal and
operating loads and speeds) to validate and refine analytical predictions. Apply
smart materials based sensors to measure in-situ forces and motions. Finally,
apply active and/or passive control methods to concurrently reduce dynamic mesh
and sliding friction force excitations.
References for Chapter 8
[8.1] Vaishya, M., and Singh, R., 2001, “Analysis of Periodically Varying Gear Mesh
Systems with Coulomb Friction Using Floquet Theory,” Journal of Sound and Vibration,
243(3), pp. 525-545.
[8.2] Vaishya, M., and Singh, R., 2001, “Sliding Friction-Induced Non-Linearity and
Parametric Effects in Gear Dynamics,” Journal of Sound and Vibration, 248(4), pp. 671694.
[8.3] Vaishya, M., and Singh, R., 2003, “Strategies for Modeling Friction in Gear
Dynamics,” ASME Journal of Mechanical Design, 125, pp. 383-393.
[8.4] Rebbechi, B., and Oswald, F. B., 1991, “Dynamic Measurements of Gear Tooth
Friction and Load,” NASA-Technical Memorandum, 103281.
[8.5] Helical3D (CALYX software), 2003, “Helical3D User’s Manual,” www.ansol.www,
ANSOL Inc., Hilliard, OH.
225
[8.6] Pavić, G., 2005, “An Engineering Technique for the Computation of Sound
Radiation by Vibrating Bodies using Substitute Sources,” Acta Acustica Journal, 91, pp.
1-16.
[8.7] Pavić, G., 2006, “A Technique for the Computation of Sound Radiation by
Vibrating Bodies Using Multipole Substitute Sources,” Acta Acustica Journal, 92, pp.
112-126.
226
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231
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233