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FRICTION FORCE EXCITATIONS IN SPUR AND HELICAL INVOLUTE PARALLEL AXIS GEARING
Dissertation
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
David Hochmann, B.S, M.S.
The Ohio State University
1997
Dissertation Committee;
Professor Donald R. Houser, Adviser Approved By Professor Gary L. Kinzel
Professor Si C. Lee Adviser Professor Chia-Hsiang Menq Department of Mechanical Engineering T3MI Number; 9813272
UMI Microform 9813272 Copyright 1998, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 Copyright by David Hochmann 1997 ABSTRACT
Current thought is that the main sources of dynamic excitation in spur
and helical gearing occurs along the line-of-action and are due to time varying
tooth stiffness and static transmission. This study examines friction forces as a
potential dynamic excitation source in the gear mesh of involute parallel axis
spur and helical gearing. The friction forces act in a direction perpendicular to
the line-of-action and occur at the tooth pair contact point.
To support the claim that friction force is a potential dynamic excitation
source, experimental evidence in the form of shaft motions measured near the
support bearings in the line-of-action and off line-of-action directions is
presented. These experimental results show that the off line-of-action motion
is of the same order of magnitude as the line-of-action motion and at times the off line-of-action motion at gear mesh frequency is several times larger than the line-of-action motion. The shaft motion data is gathered from the published literature and performed experiments. The motions are converted to forces through the support bearing stiffness matrix. Other potential causes of the large off line-of-action shaft motion such as bearing cross coupling phenomena, reduced bearing stiffness, and bearing clearances are examined. The measurement of the shaft motion in the off line-of-action direction is related to the net friction force. The net friction force is the sum of all friction forces on the individual tooth pairs in contact at each instant. A methodology to m easure the individual friction forces for spur gears with a contact ratio less than or equal to two is introduced. The measurement scheme is based on the measurement of the net friction force and of the instantaneous power lost at the gear mesh.
A general dynamic model incorporating off line-of-action friction force is introduced. The developed model typically forms nonlinear and linear time varying equations dependent upon the assumed friction force model. From the general model, a gear system supported on rigid bearings is examined in detail both under static and dynamic operating conditions. Under dynamic operating conditions, friction force models based on both a Dry friction and elastohydrodynamic fluid film theory are studied. The only extemal input function examined is a time invariant input torque. A constant input torque highlights the effect of the off line-of-action friction force on the output torque and the normal mesh force. The linear time varying and non-linear equations developed typically are solved using numerical techniques. For a certain class of linear time varying problems incorporating an off line-of-action friction force, an exact analytical solution is presented.
Ill ACKNOWLEDGMENTS
I would like to express my appreciation to my adviser, Dr. Donald R.
Houser for his advice, guidance, and patience throughout the course of this research. I thank the other members of my dissertation committee, Dr. Gary L.
Kinzel, Dr. Si C. Lee and Chia-Hsiang Menq for there suggestions and comments. I would like to thank the sponsors of the Gear Dynamics and Gear
Noise Research Laboratory. I would also like to thank Fred Oswald for his assistance in the experiments performed at NASA Lewis Research Center,
Cleveland Ohio. I would further like to thank the Army Research Office
(URI Grant DAA03-92-G-0120), Project Monitor; Dr. T.L. Doligalski for the funding support for this research.
I am grateful to the many students of the Gear Lab over the years. I also thank Jim Sorenson for his assistance with the test rigs.
Finally, I would like to thank my Mom, Dad, and my Sister.
IV VITA
April 5, 1967 Bom - Prague, Czechoslovakia.
May, 1990 Bachelor of Science in Mechanical Engineering, The University of Texas at San Antonio, San Antonio, Texas.
September, 1990-Present Graduate Research Associate, Gear Dynamics and Gear Noise Research Lab, The Ohio State University, Columbus, Ohio.
August, 1992 Master of Science in Mechanical Engineering, The Ohio State University, Columbus Ohio.
PUBLICATIONS
1. D. Hochmann, D.R. Houser and J. Thomas, "Transmission Error and Load Distribution Analysis of a Spur and Double Helical Gear Pair Used in a Split Path Helicopter Transmission Design", Proceedings of the American Helicopter Society, 1991.
2. D. Hochmann and D.R. Houser, "An Experimental Test Stand for Measurements on Loaded Parallel Axis Gears", Proceedings of the Intemational Gearing Conference, 1994. FIELDS OF STUDY
Major Field: Mechanical Engineering
Studies in Vibrations, Signal Processing and Measurements
VI TABLE OF CONTENTS
Abstract...... ii
Acknowledgments ...... iv
Vita...... V
List of Tables ...... xi
List of Figures ...... xiii
Nomenclature ...... xxiii
Chapters:
1. Introduction ...... 1
1.1 Introduction ...... 1 1.2 Scope and Objective ...... 4 1.3 Organization ...... 5
2. Experimental and Analytical Evidence of Off Line-of-Action Friction Force as a Potential Dynamic Excitation Source ...... 8
2.1 Introduction ...... 8 2.2 Literature Review...... 10 2.3 Experimental Evidence ...... 10 2.3.1 Experimental Results of Umezawa ...... 10 2.3.2 NASA Gearbox Test Rig ...... 14 2.3.2.1 NASA Gearbox Test Rig Description ...... 14 2.3.2.2 NASA Experiment Description ...... 17 2.3.2.S NASA Experimental Results ...... 19 2.3.2.4 NASA Result Summary ...... 22 2.3.3 Back-to-Back Gearbox Test Rig ...... 25 2.3.3.1 Back-to-Back Gearbox Test Rig Description ...... 25 2.3.3.2 Back-to-Back Experiment Description ...... 27 2.3.3.3 Back-to-Back Experimental Results...... 29 2.3.5.4 Back-to-Back Results Summary ...... 34 2.3.4 Experimental Results Conclusion ...... 34 vii 2.4 Possible Off Line-of-ActIon Shaft Motion Explanations ...... 34 2.4.1 Friction Force in the Off Line-of-Action Direction ...... 35 2.4.2 A Rotordynamic Phenomenon ...... 37 2.4.3 Shaft Support Bearings ...... 38 2.4.3.1 Bearing Cross Coupling ...... 39 2.4.3.2 Bearing Reduced Stiffness in the Off Line-Of-Action Direction ...... 40 2.4.5.3 Bearing Radial Clearance ...... 43 2.4.3.3.1 Analytical Model Incorporating Radial Clearance...... 43 2.4.3.3.2 Experimental Results Examining Radial Clearance In A Tapered Roller ...... 45 2.5 Analytical Evidence ...... 46 2.5.1 Gearbox System Dynamics ...... 46 2.5.1.1 Three Degree of Freedom Gear Model ...... 46 2.5.1.2 Dynamic Finite Element Model of the Back-to-Back Gear Tester...... 51 2.5.1.2.1 Rotational Lumped Parameter Model ...... 51 2.5.1.2.2 Translational/Bending Finite Element Model .53 2.5.1.2.3 Back-to-Back Tester Dynamic Finite Element/Lumped Parameter Model ...... 55 2.5.1.3 Experimental Inertance and Analytical Inertance Comparisons ...... 62 2.5.1.4 Section Conclusion ...... 66 2.5.2 Energy Loss and Friction Force To Normal Mesh Force Ratio 67 2.5.2.1 Friction Force to Normal Force Ratio Derivation ...... 67 2.5.2.2 Friction Force to Normal Force Ratio R esults ...... 70 2.5.2.3 Section Conclusion ...... 74 2.6 Chapter Conclusion ...... 74
3. The Experimental Measurement of the Off Line-of-Action Friction Forces in the G ear Mesh of Parallel Axis Involute Spur G e ars ...... 77
3.1 Introduction ...... 77 3.2 Literature Review...... 79 3.3 New Measurement Methodology Introduction ...... 83 3.3.1 New Measurement Method Theory...... 84 3.3.2 Numerical Test of Friction Force Measurement Schem e ...... 86 3.3.3 Error Examination of Measurement Methodology ...... 93 3.4 Experimental Setup ...... 94 3.4.1 Powerless Measurement ...... 96 3.4.1.1 Torque Measurement ...... 98 3.4.1.2 Angular Velocity Measurement ...... 104 3.4.1.2.1 Mean Angular Velocity Measurement ...... 105 3.4.1.2.2 Time-Varying Angular Velocity Measurement...... 105 v iii 3.4.1.3 Power Flow Computation ...... 107 3.4.2 Force Measurement ...... 108 3.4.2.1 Displacement Measurement ...... 108 3.4.2.2 Bearing Stiffness Matrix ...... 109 3.4.2.3 System Dynamic and Gearbox Geometry Effects Ill 3.4.3 Absolute Position Measurement ...... 115 3.4.4 Experimental Data Reduction Flow Chart ...... 117 3.5 Experimental Results ...... 117 3.6 Concluding Remarks ...... 129
4. Extemal Involute Spur and Helical Parallel Axis G ear Model Incorporating Off Line-Of-Action Friction F orces ...... 131
4.1 Introduction ...... 131 4.2 Literature Review...... 133 4.3 Twelve Degree-Of-Freedom Extemal Spur and Helical Involute Gear Mesh Parallel Axis Model ...... 138 4.3.1 Forces at the Gear Mesh Interface ...... 140 4.3.1.1 Forces Acting on Pinion at the Gear Mesh Interface.. 140 4.3.1.2 Forces Acting on Gear at Mesh Interface ...... 142 4.3.2 Forces at the Pinion and Gear Axis of Rotation ...... 144 4.3.2.1 Gear Mesh Forces and Gear Mesh Moments about the Pinion Axis of Rotation ...... 145 4.3.2.2 Gear Mesh Forces and Gear Mesh Moments about the G ear Axis of Rotation ...... 147 4.3.3 A Dynamic Gear Model Incorporating Friction Force ...... 149 4.4 Normal Force Distribution and Friction Force Distribution Models ..152 4.4.1 Normal Force Distribution ...... 153 4.4.2 Friction Force Distribution ...... 155 4.4.2.1 Determination of the Friction Force Direction ...... 156 4.4.2.2 Fluid Film Model ...... 157 4.4.3 Dynamic G ear Model With an Assumed Normal And Friction Force Distribution ...... 160 4.5 Spur and Helical Gear Mesh Supported on Rigid Bearings...... 164 4.5.1 Reduced Model Formulation ...... 165 4.5.2 Static Examination of Gear Model ...... 170 4.5.2.1 Static Normal Mesh Force, Output Torque, and Friction Force ...... 170 4.5.2.2 Output Torque Considerations ...... 180 4.5.2.3 Power Loss Due to Off Line-Of-Action Friction ...... 190 4.6 Numerical Solution of Gear Model Supported on Rigid Bearings.... 193 4.7 Analytical Solution of Gear Model Supported on Rigid ...... 211 4.7.1 Analytical Solution of the Vector Form of the Meissner Equation ...... 213 4.7.2 Vector Form of the Meissner Equation applied to Off Line-Of- Aotion Friction Force in a Gear M esh ...... 219 ix 4.7.3 Analytical Solution Summary ...... 234 4.8 Non-Linear Elastohydrodynamic Friction Force Model Solution 235 4.9 Concluding Rem arks...... 243
5. Contributions and Recommendations ...... 247
5.1 Contributions ...... 247 5.2 Future Research...... 249
A ppendix ...... 253
Bibliography ...... 260 LIST OF TABLES
Table Page
Table 2.1 NASA gear noise test rig spur gear pair summary ...... 15
Table 2.2 Summary for selected NASA test rig results presenting gear mesh frequency linear magnitudes ...... 20
Table 2.3 Back-to-Back test rig gear pair summary ...... 26
Table 2.4 Computed tapered roller bearing stiffness terms kj.j for mean F io a= 0 Ibf, mean axial preload Fz = 50 Ibf ...... 42
Table 2.5 Computed tapered roller bearing stiffness terms k,., for mean F io a= 50 Ibf, mean axial preload Fz = 50 Ibf ...... 42
Table 2.6 Computed tapered roller bearing stiffness terms kj, j for mean Fioa= 1000 Ibf, mean axial preload Fz = 50 Ibf ...... 42
Table 2.7 System parameters used for the 3 DOF off line-of-action and line-of-action model ...... 50
Table 2.8 Element properties used in the back-to-back dynamic finite element model ...... 58
Table 2.9 System parameters used in dynamic finite element model of back-to-back gear tester ...... 58
Table 3.1 Relevant gears parameters for numerical test ...... 87
Table 3.2 Relevant spur gears parameters for experimental test 119
Table 3.3 Comparison of measured mean pinion and gear torque at various pinion shaft speeds ...... 119
XI Table 4.1 Baseline gear parameters used for the linear time varying numerical solution ...... 196
Table 4.2 Natural frequencies and mode shapes for undamped and damped frictionless c ase ...... 197
Table 4.3 Operating conditions examined in numerical study ...... 197
Table 4.4 Baseline gear parameters used for comparison between numerical and analytical linear time varying numerical solution ...... 221
Table 4.5 Gear parameters used for the non-linear fluid film lubrication model ...... 236
XII LIST OF FIGURES
Figure Page
Figure 1.1 Line-of-Action and Off Line-of-Action Definition ...... 3
Figure 2.1 Test rig used by Umezawa ...... 12
Figure 2.2 Sliaft measurement locations ...... 12
Figure 2.3 Vibration mode of gear system, 1235 Hz mesh frequency ..... 13
Figure 2.4 Vibration mode of gear system, 1445 Hz mesh frequency ..... 13
Figure 2.5 NASA Lewis research test rig layout ...... 15
Figure 2.6 NASA Lewis research gearbox details ...... 16
Figure 2.7 NASA gearbox measurement plane location ...... 18
Figure 2.8 NASA gearbox displacement probe locations ...... 18
Figure 2.9 NASA test rig results for output speed= 2000 rpm, varying torque ...... 23
Figure 2.10 NASA test rig results for output torque = 1000 Ibf-in, varying speed ...... 23
Figure 2.11 NASA test rig results showing an overlay of shaft displacement measurements in the off line-of-action direction for two separate shaft revolutions in the time domain, output speed=1000 rpm, output load=1000 Ibf-in ...... 24
Figure 2.12 NASA test rig results showing the off line-of-action shaft displacement in the frequency domain, output speed=1000 rpm. output load=1000 Ibf-in ...... 24
x iii Figure 2.13 Back-to-Back Gear Test Rig ...... 26
Figure 2.14 Back-to-Back coordinate system ...... 28
Figure 2.15 Tracking of the first mesh frequency (25th shaft order) for various output shaft loads ...... 31
Figure 2.16 X-Axis shaft displacement, no input torque preload ...... 32
Figure 2.17 Y-Axis shaft displacement, no input torque preload ...... 32
Figure 2.18 X-Axis shaft displacement, 10000 Ibf-in output torque preload ...... 33
Figure 2.19 Y-Axis shaft displacement, 10000 Ibf-in output torque preload ...... 33
Figure 2.20 Translational bearing reactions for a mean load applied in the line-of-action direction ...... 44
Figure 2.21 Simple off line-of-action and line-of-action dynamic model ....47
Figure 2.22 Reduced order off line-of-action and line-of-action dynamic m o d e l ...... 47
Figure 2.23 Force transmissibility comparison between the off line-of-action and the line-of-action directions using the 3 dof model ...... 50
Figure 2.24 Schematic of line-of-action dynamic FE model, rotational lumped parameter model, and the off line-of-action dynamic FE model ...... 52
Figure 2.25 Force transmissibility between measurement gear mesh and off line-of-action and line-of-action pinion bearing ...... 60
Figure 2.26 Force transmissibility between measurement gear mesh and off line-of-action and line-of-action pinion bearing ...... 61
Figure 2.27 Force transmissibility for off line-of-action and line-of-action directions for the pinion shaft bearing between the measurement side mesh and the slave gearbox ...... 62
Figure 2.28 Experimental inertance measurement for off line-of-action and line-of-action test case ...... 64
XIV Figure 2.29 Coherence function for 5000 Ibf-in off line-of-action test case presented in Figure 2.28 ...... 64
Figure 2.30 Inertance comparison in the line-of-action direction between experimental results and dynamic finite element model 65
Figure 2.31 Inertance comparison in the off line-of-action direction between experimental results and dynamic finite element model 65
Figure 2.32 Friction force to normal force ratio geometric assignments.... 68
Figure 2.33 Friction force to normal mesh force ratio per unit efficiency, rp = 2.5, various — ...... 72
Figure 2.34 Friction force to normal mesh force ratio per unit efficiency, rp = 2.5, various — ...... 73 *g
Figure 2.35 Friction force to normal mesh force ratio per unit efficiency, —= 0.5, various rp ...... 73 “"g
Figure 3.1 Normal and friction forces via strain gage m easurem ents 81
Figure 3.2 Plot of the friction coefficient, |i, versus position along line-of-action, p,=0.4s^ ...... 87
Figure 3.3 Analytically computed instantaneous power loss and net friction force, ff, n=0.05, T,n=1 Ibf-in, Qin=1 rad /s ...... 89
Figure 3.4 Analytically computed individual tooth forces, p=0.05, Tin=1 Ibf-in, Oin=1 rad/s ...... 90
Figure 3.5 Inverted individual tooth forces from Pioss and ff, n=0.05, Tin=1 Ibf-in, Qin=1 rad/s ...... 90
Figure 3.6 Analytically computed power loss and net friction force, ff, 0.4s^, Tin=1 Ibf-in, 0^=1 rad/s ...... 91
Figure 3.7 Analytically computed individual tooth forces , ji=0.4s^, Tin=1 Ibf-in, Qin=1 rad/s ...... 92
XV Figure 3.8 Inverted Individual tooth forces from Pioss and ff, ji=0.4s^, Tin=1 Ibf-ln, Qin=1 rad /s ...... 92
Figure 3.9 Schematic of back-to-back gear tester presenting location of commercial torque sensor ...... 95
Figure 3.10 Schematic of the Instrumented back-to-back gear test rig test side gear box ...... 95
Figure 3.11 Power loss measurement excluding bearing losses ...... 97
Figure 3.12 Calibration curve for the semi-conductor strain gage bridge mounted on pinion shaft ...... 102
Figure 3.13 Calibration curve for the semi-conductor strain gage bridge mounted on gear shaft ...... 102
Figure 3.14 Dynamic torque signal of commercial torque sensing unit, Tin=5000 Ibf-ln, Qjn=1080 rpm ...... 103
Figure 3.15 Dynamic torque signal constructed torque sensing bridge, Tin=5000 Ibf-ln, Qin=1080 rpm ...... 103
Figure 3.16 Comparison of dynamic torque signal between commercial torque sensing unit and constructed torque sensing bridge, Tin=5000 Ibf-ln, ^n=1080 rpm ...... 104
Figure 3.17 Time varying angular velocity measurement scheme 106
Figure 3.18 Force transmissibility magnitude between gear mesh and the pinion bearing nearest the slave gearbox In the off llne-of-actlon direction using the FEA m odel...... 114
Figure 3.19 Force transmissibility phase between gear mesh and the pinion bearing nearest the slave Gearbox In the off llne-of-actlon direction using the FEA model ...... 114
Figure 3.20 Force transmissibility phase between gear mesh and the pinion bearing nearest the slave gearbox In the off llne-of-actlon direction using the FEA model ...... 115
Figure 3.21 Schematic of absolute position determination ...... 116
Figure 3.22 Flow chart of friction force computation procedure ...... 118
XVI Figure 3.23 Pinion shaft torque sensor T 1, torque harmonics 0-peak amplitude, Tin=2500 Ibf-in, Qin= 1080 rpm ...... 124
Figure 3.24 Pinion shaft torque sensor, torque harmonics around mesh frequency for location T 1, 0-peak amplitude, Tin=2500 Ibf-in, Oin= 1080 rpm ...... 124
Figure 3.25 Pinion shaft torque sensor T2, torque harmonics 0-peak amplitude, T,n=2500 Ibf-in, 0;n= 1080 rpm ...... 125
Figure 3.26 Pinion shaft torque sensor T3, torque harmonics 0-peak amplitude, Tjn=2500 Ibf-in, 1080 rpm ...... 125
Figure 3.27 Pinion shaft time domain torque measurement, T 1, Tin=2500 Ibf-in, Qin= 1080 rpm ...... 126
Figure 3.28 Pinion shaft angular velocity harmonics, 0-peak amplitude, Tin=2500 Ibf-in, n,n= 1080 rpm (113 rad/s) ...... 126
Figure 3.29 Gear mesh power loss computed from experimental measurements to 5000 Hz, 0-peak amplitude, Tjn=2500 Ibf-in, Oin= 1080 rpm ...... 127
Figure 3.30 Inverse Fourier Transform of gear mesh power loss using mean component and 10 mesh frequency harmonics, nin= 1080 rpm ...... 127
Figure 3.31 Net friction and computed individual tooth friction forces, koioa=1-6 X 10® Ibf/in, Tn=2500 Ibf-in, Qjn= 1080 rpm ...... 128
Figure 3.32 Friction force on tooth pair 1 and tooth pair 2 using the load distribution computed by LDP, Tin=2500 Ibf-in, p.=0.05 ...... 128
Figure 4 .1 G ear model development flow c h art ...... 132
Figure 4.2 Transverse view of gear mesh ( xy plane ) ...... 139
Figure 4.3 Plane-of-Action view of the gear mesh zone ( xz plane) 139
Figure 4.4 Normal, n;, and friction, fi, force distributions (force/length) acting on the pinion at gear pair mesh interface ...... 141
Figure 4.5 Normal, n-,, and friction, f„ force distributions (force/length) acting on the gear at gear pair mesh interface ...... 143
XVII Figure 4.6 Force summed about pinion axis of rotation ...... 146
Figure 4.7 Force summed about gear axis of rotation ...... 148
Figure 4.8 Actual tooth length and projected transverse tooth length ... 160
Figure 4.9 Schematic of reduced model on rigid bearings ...... 166
Figure 4.10 Schematic of reduced model on rigid bearings with gear element attached to a massive load inertia through torsional spring element, kg ...... 168
Figure 4.11 Normal mesh force per unit input torque for |i'=0.01, 0.05, 0.1, contact ratio= 1 , rp=2.5, rg=5, Y= 0°, LPSTC=-0.5, Zb=0.5, Za=-O.S...... 173
Figure 4.12 Normal mesh force per unit input torque for different helix angles, p'= 0.05, rp=2.5, rg=5, single tooth pair in contact.... 173
Figure 4.13 Normal mesh force per unit input torque for contact ratio= 1, 1.25, 1.5 , fi = 0.05, rp=2.5, rg=5, 'P= 0 , bp= 1, LPSTC=-0.5, Zb=0.5, frictionless ratio=0.4 ...... 174
Figure 4.14 Line-of-Action geometry used for normal mesh force, input/output torque ratios and friction force examples presented in Figure 4.13, Figure 4.17 and Figure 4.20 ...... 174
Figure 4.15 Output to input torque ratio, Tout/Tin, for various friction coefficient, contact ratio= 1 , rp=2.5, rg=5, 'F= 0°, bp=1, LPSTC=-0.5, Zb=0.5, Za=~0.5, frictionless Tout/Tin = 2 ...... 176
Figure 4.16 Output to input torque ratio,Tout/Tin, for various helix angles, fi'= 0.05, rp=2.5, rg=5, single tooth in contact, frictionless Tout/Tin=2 ...... 177
Figure 4.17 Output to input torque ratio, Tout/Tin, for various contact ratios, fi = 0.05, rp=2.5, rg=5, Y= 0 , LPSTC=-0.5, Zb=0.5, frictionless Tout/Tm=2 ...... 177
Figure 4.18 Off line-of-action friction force acting on pinion element,Qpy, per unit input torque. Tin, for fi"=0.01, 0.05, 0.1, contact ratio= 1, rp=2.5, rg=5, 'F= 0 , LPSTG=-0.5, Zb=0.5, Za=-0.5 ...... 179
XVIII Figure 4.19 Off line-of-action friction force acting on pinion element,Qpy, per unit input torque, Tin, for various helix angles, |i = 0.05, rp=2.5, rg=5, single tooth pair in contact ...... 179
Figure 4.20 Off line-of-action friction force acting on pinion element,Qpy, per unit input torque. Tin, for contact ratio= 1, 1.25, 1.5, p. =0.05, rp=2.5, rg=5, 'F=0 , bp=1, LPSTC=-0.5, Zb—0.5 ...... 180
Figure 4.21 Critical friction coefficient during approach action, -1 < a < 0 ...... 184
Figure 4.22 Recess action output torque to input torque ratio as 0 < a <1 ...... 186
Figure 4.23 Gear mesh forces, input/output torques and reaction forces acting on pinion and gear bodies in approach action ...... 187
Figure 4.24 Gear mesh forces, input/output torques and reaction forces acting on pinion and gear bodies recess in action ...... 188
Figure 4.25 Dynamic mesh force, fm, fundamental parametric excitation frequency, cOp, at off resonant conditions, Tin=1000 Ibf-in, frictionless normal mesh force=425 Ibf ...... 201
Figure 4.26 Dynamic mesh force, fm, fundamental parametric excitation frequency exciting the first damped natural frequency of oscillation, cOp =03^, Tn= 1000 Ibf-in, frictionless normal mesh force=425 Ibf ...... 202
Figure 4.27 Frequency spectrum of the dynamic mesh force, fm, for the results presented in Figure 4.26 ...... 202
Figure 4.28 Dynamic mesh force, fm, fundamental parametric excitation frequency exciting the second damped natural frequency of oscillation ...... 203
Figure 4.29 Dynamic output torque. Tout, fundamental parametric excitation frequency, cOp, at off resonant conditions, Tin=1000 Ibf-in, frictionless output torque=1880 Ibf-in ...... 205
Figure 4.30 Frequency spectrum of the dynamic output torque. Tout, for the case of C0p =8 Hz presented in Figure 4.29 ...... 205
XIX Figure 4.31 Dynamic output torque, Tout, fundamental parametric excitation frequency exciting the first damped natural frequency of oscillation, cOp = 0)^1 > Tin=1000 Ibf-in, frictionless output torque=1880 Ibf-in ...... 206
Figure 4.32 Dynamic output torque, Tout, fundamental parametric excitation frequency exciting the second damped natural frequency of oscillation, o)p = C 0p2 > Cdd=C0e=C, frictionless output torque=1880 Ibf-in ...... 206
Figure 4.33 Off line-of-action dynamic friction force, Qpy, acting on the pinion body with cOp at off resonant conditions, Tin=1000 Ibf-in...... 209
Figure 4.34 Off line-of-action dynamic friction force, Qpy, acting on the pinion body with excitation frequency exciting the first damped natural frequency of oscillation, cOp = Wp,, Tin= 1000 Ibf-in...... 210
Figure 4.35 Off line-of-action dynamic friction force, Qpy, acting on the pinion body with excitation frequency exciting the second damped natural frequency of oscillation, cOp = 00^2 - for various Tm, and C d d = C e e = C ...... 211
Figure 4.36 Spand Eq versus position along line-of-action, contact ratio=1, bp=0.59 ...... 213
Figure 4.37 Normal mesh force comparison between analytical (top) and numerical (bottom) solution of the Meissner equation, (Op = 8 H z, Tin=1000 Ibf-in, |i’=0.05, bp=0.59, contact ratio= 1, frictionless normal mesh force=425 Ibf 224
Figure 4.38 Expansion of Figure 4.37; normal mesh force overlay comparison between numerical and analytical solution of the Meissner equation, cOp = 8 Hz, Tn=1000 Ibf-in, |i’=0.05, bp=0.59, contact ratio=1, frictionless normal mesh force=425 Ibf ...... 225
Figure 4.39 Output torque overlay comparison between numerical and analytical solution of the Meissner equation, cOp = 8 Hz, Tin=1000 Ibf-in, p'=0.05, bp=0.59, contact ratio=1, frictionless output torque=1880 Ibf-in ...... 225
XX Figure 4.40 Normal mesh force overlay comparison between numerical and analytical solution of the Meissner equation, COp = 2964 Hz, Tjn=1000 Ibf-ln, p =0.05, bp=0.59, contact ratlo= 1, frictionless normal mesh force=425 Ibf ...... 226
Figure 4.41 Output torque overlay comparison between numerical and analytical solution of the Meissner equation, cOp = 2964 Hz, Tin=1000 Ibf-ln, |i*=0.05, bp=0.59, contact ratlo= 1, frictionless output torque=1880 Ibf-ln ...... 226
Figure 4.42 Waterfall plot of dynamic mesh force. Tin = 1000 Ibf-ln, 11=0.05, bp=0.59, contact ratlo=1, frictionless mean mesh force= 425 Ibf ...... 230
Figure 4.43 Waterfall plot of dynamic output torque, Tn=1000 Ibf-ln, (1=0.05, bp=0.59, contact ratlo=1, frictionless mean output torque=1880 Ibf-ln ...... 231
Figure 4.44 Comparison of steady state dynamic output torque between numerically solution (top) and exact analytical solution (bottom) where cop=o)di, Oin=1000 rpm, Tm=1000 Ibf-ln ...... 232
Figure 4.45 Waterfall plot of off llne-of-actlon friction force, Qin=96-9600 rpm, Tin=1000 Ibf-ln, (i =0.05, bp=0.59, contact ratlo=1, frictionless mean output torque=1880 Ibf-ln ...... 233
Figure 4.46 Off llne-of-actlon friction force harmonic content versus Input speed contour plot, Tm=1000 Ibf-ln, ^i’=0.05, bp=0.59, contact ratlo=1 ...... 233
Figure 4.47 Instantaneous friction coefficient, |i', h=1.15hn •mini Tin=1000 Ibf-ln, bp=0.59, contact ratlo = 1 ...... 239
Figure 4.48 Dynamic normal mesh force, h=1 .IShmin, T,n=1000 Ibf-ln, bp=0.59, contact ratlo=1, frictionless normal mesh force = 425 Ibf ...... 239
Figure 4.49 Dynamic output torque, h=1.15hmin, Tm=1000 Ibf-ln, bp=0.59, contact ratlo=1, frictionless output torque=1 880 Ibf-ln...... 240
Figure 4.50 Dynamic off llne-of-actlon friction force, h=1.15hmin, Tin=1000 Ibf-ln, bp=0.59, contact ratlo=1 ...... 240
XXI Figure 4.51 Instantaneous viscosity, h=1 .IShmin, Tjn=1000 Ibf-in, bp=0.S9, contact ratio=1 ...... 241
Figure 4.52 Minimum fluid film thickness, hmin. Tin=1000 Ibf-in, bp=0.59, contact ratio=1 ...... 241
Figure 4.53 Mean contact pressure, h=1.15hmin. T,n=1000 Ibf-in, bp=0.59, contact ratio=1 ...... 242
Figure A. 1 Modified instantaneous friction coefficient, h=1.15hmin. o)p=1690 Hz ( pinion speed = 4056 rpm ), 7^=1000 Ibf-in, bp= 0.59, contact ratio = 1 ...... 257
Figure A. 2 Dynamic normal mesh force using modified fluid film model, (Op =1690 Hz ( pinion speed = 4056 rpm ), T,n=1000 Ibf-in, friction less normal mesh force=425 Ibf ...... 258
Figure A. 3 Dynamic output torque. Tout, using modified fluid film model, (Op =1690 Hz ( pinion speed = 4056 rpm ), Tm=1000 Ibf-in, friction less output torque =1880 Ibf-in ...... 258
Figure A. 4 Dynamic off line-of-action friction force, using modified fluid film model, o)p=1690 Hz ( pinion speed = 4056 rpm ), Tin=1000 Ibf-in...... 259
XXII Nomenclature
Chapter 2 ioa line-of-action oloa off line-of-action spl sound pressure level 4) operating pressure angle pinion base radius gear base radius X, Y,Z coordinate system L normal mesh force f mean ’m average normal mesh force Afm small time varying normal mesh force f, off line-of-action friction force friction coefficient, ratio of friction force to normal mesh force V tooth pair relative sliding velocity i.j indices Rloa rotation about line-of-action axis Roloa rotation about off line-of-action axis R z rotation about the axis of rotation km gear mesh stiffness k„. pinion shaft support translational bearing stiffness along i**^ axis gi gear shaft support translational bearing stiffness along i^ axis e(t) time varying static transmission error 6p perturbation angular motion of pinion about axis of rotation 0 perturbation angular motion of pinion about axis of rotation for reduced order model force input along the off line-of-action direction for the reduced order model base radius kx.ky translational bearing stiffness in the x and y direction for the reduced order model moment of inertia of gear body
XXIII fx force input along the off line-of-action direction for the reduced order model s Laplace variable and linear position along the line of action fbx bearing reaction the along line-of-action direction for the reduced order model fby bearing reaction along the off line-of-action direction for the reduced order model em(t) static transmission error on the measurement side gearbox of the FEA model es(t) static transmission error on the slave side gearbox of the FEA model kmm gear mesh stiffness on the measurement side gearbox of the FEA model kms gear mesh stiffness on the slave side gearbox of the FEA model ffm friction force input on the measurement side gearbox of the FEA model ffs friction force input on the slave side gearbox of the FEA model [mje beam element m ass matrix [K ]e unmodified beam element stiffness matrix Wg beam element displacement vector Wi translational degree of freedom for node i 0i rotational degree of freedom of node i L (x) hermite cubics shape function vector Wx displacement vector for torsional and line-of-action motion Wyp displacement vector for off line-of-action motion of pinion shaft Wyg displacement vector for off line-of-action motion of gear shaft f^ line-of-action and torsion input vector off line-of-action force input vector for the pinion shaft off line-of-action force input vector for the gear shaft Tin pinion input torque Tout gear output torque 9g perturbation angular motion of gear about axis of rotation cp angle between gear center line and the line-of-action, ^ - (p D linear displacement, rp9p+rg0g AEin input energy AEout output energy AEioss energy loss, AEm-AEout St, S2 integration limits along the line-of-action
XXIV Chapter 3
Ioa line-of-action oloa off line-of-action ff net off line-of-action friction force fi, fa off line-of-action friction force acting on tooth contact pair 1 and 2 , respectively Vi relative sliding speed for tooth pair i P mesh power loss at the gear mesh bp base pitch Z a start of tooth pair contact LPSTC lowest point of single tooth pair contact Zb end of tooth pair contact Tin input torque input angular velocity kcomputed experimentally measure or computed off line-of-action bearing stiffness k exact off line-of-action bearing stiffness Ak deviation between exact and computed off line-of-action bearing stiffness r net total gearbox power loss P ' beanng power loss due to the bearings
Plub power loss due to lubricant churning and slinging away from the mesh
windage power loss due to aerodynamic effects ^in, instantaneous input angular velocity Qout instantaneous out angular velocity Tin input torque to the gearbox Tout output torque of the gearbox T1 pinion torque measured on loaded side of the shaft T2 pinion torque measured on unloaded side of the shaft T3 gear torque measured on loaded side of the shaft T4 gear torque measured on unloaded side of the shaft Tk instantaneous torque at location k -rm ean 'k mean torque at location k Tl i^ Fourier harmonic of instantaneous torque at location k strain gage bridge torque sensitivity for fully active four gage Wheatstone bridge mounted on a round shaft, mV/(V Ibf-in) G gage factor of individual strain gage d diameter of shaft V Poisson ratio E modulus of elasticity
XXV m instantaneous angular velocity ç^mean mean angular velocity i^ Fourier harmonic of instantaneous velocity m ean n in mean input angular velocity m ean mean output angular velocity nout 3l , &2 linear accelerations measured with translational accelerometer r radius to accelerometer circle H.O.T. higher order term fx bearing reaction force in the line-of-action (X axis) direction fv bearing reaction force in the off line-of-action (Y axis)direction fz bearing reaction force in the axial (Z axis) direction Mx bearing reaction moment about the X axis My bearing reaction moment about the Y axis Mz bearing reaction moment about the Z axis X. Y ,Z perturbation linear displacement along the line-of-action, off line- of-action, and axial directions, respectively 0X, 0Y, 0Z perturbation angular displacement about the line-of-action, off line-of-action, and axial directions, respectively bearing stiffness matrix term coupling directions i and j. i=X,Y,Z, 8 x, 0Y, 0Zi j= X,Y,Z, 0x, 0Y. 0z k y.Y off line-of-action bearing stiffness term kY.Z bearing stiffness coupling between off line-of-action direction and axial direction 'Y.Bv bearing stiffness coupling between off line-of-action direction and rotation about the line-of-action axis. ffaeanng (cû) off line-of-action bearing force in frequency domain fmesh(w) off line-of-action mesh force in frequency domain G(cû) transfer function between off line-of-action forces in the gear mesh and the off line-of-action bearing ; zeta, damping ratio CMM coordinate measuring machine □FT discrete Fourier Transform IDFT inverse discrete Fourier Transform product © summation
XXVI Chapter 4
Tin pinion input torque Tout gear output torque ^in mean angular speed of pinion body mean angular speed of gear body
Za start of contact LPSTC lowest point of single/integer(contact ratio) tooth pair contact PR pitch point HPSTO highest point of single/integer(contact ratio) tooth pair contact Zb end of contact bp base pitch Aj position of tooth pair i relative to the reference tooth pair, Aj = integer t independent variable, time s independent variable, position along the line-of-action Si independent variable, position of tooth pair i along line-of- action (parallel to x axis) Y helix angle w position along face width (parallel to z axis) wOi starting position of tooth pair i along gear mesh face width, wOi XXVII resultant moment vector due to the normal force distribution acting on the gear element at the Ith tooth pair resultant moment vector due to the friction force distribution _ ' acting on the gear element at the ith tooth pair forces and moments column vector acting on the pinion element computed at the pinion element’s axis of rotation due forces and moments at the mesh Interface. Qpk scalar element of Qp representing the force acting In the direction on the pinion element computed at the pinion elements axis of rotation due to forces at the mesh Interface, k=x,y,z _ QpMk scalar element of Qp representing the moment acting about the k^ axis on the pinion element computed at the pinion elements axis of rotation due to forces and moments at the mesh Interface, k=x,y,z Rp reaction forces and moments column vector acting on the pinion element Ppi^ scalar element of Rp representing the reaction force In the k*^ direction for the pinion body, k=x,y,z scalar element of Rp representing the reaction moment about the k^ axis for the pinion body, k=x,y,z — forces and moments column vector acting on the gear element G computed at the gear element’s axis of rotation due forces and moments at the mesh interface. Qck scalar element of Qq representing the force acting In the k‘^ direction on the gear element computed at the pinion elements axis of rotation due to forces at the mesh Interface, k=x,y,z QcMk scalar element of Qq representing the moment acting about the k‘^ axis on the gear element computed at the pinion elements axis of rotation due to forces and moments at the mesh Interface, k=x,y,z Rq reaction forces and moments column vector acting on the gear element Rok scalar element of Rq representing the reaction force in the k“^ direction for the gear body, k=x,y,z Mgr scalar element of Rg representing the reaction moment about the k'*^ axis for the gear body, k=x,y,z Uj gear body displacement column vector for the body Xj scalar element of U, representing the translational displacement of the body In the x direction XXVlll Yi scalar element of Uj representing the translational displacement of the body in the y direction Zj scalar element of U, representing the translational displacement of the body in the z direction 0xj scalar element of Uj representing the angular displacement of the body about the x axis 0yi scalar element of Uj representing the angular displacement of the body about the y axis 0zj scalar element of U, representing the angular perturbation displacement of the body about the z axis [k ] bearing stiffness matrix for the body [m | mass matrix for the body fm normal gear mesh force km instantaneous gear mesh stiffness 5 transverse gear mesh deformation 7 S load sharing factor for the i^ tooth pair on contact |i signed coefficient of friction, ratio of off line-of-action friction force to normal mesh force |i absolute value of signed coefficient of friction p direction friction force acts, P= -1 or +1, |i=p.’p Vi sliding speed for tooth pair i T shear stress v instantaneous viscosity b 1/2 of Hertzian contact width R equivalent radius E’ equivalent modulus of elasticity hm mean fluid film thickness hmin minimum fluid film thickness ^ ^1 + tan"('F) related to Qp, when the normal load distribution model used in this research is applied, Qp = %k,^0 related to Q q , when the normal load distribution model used in this research is applied, Q q = ep effect of the off line-of-action friction force on the torsional motion of the pinion body £g effect of the off line-of-action friction force on the torsional motion of the gear body D line-of-action translation variable ks output shaft torsional stiffness XXIX Cd.d , Ce.e. viscous damping terms between dependent variables D and 0 Co.e. Ce.D X state space column vector Ross It gear mesh power loss computed using the input/output torques and input/output angular speeds Ross Ip gear mesh power loss computed using the off line-of-action friction force and the gear mesh sliding speed o)di first damped natural frequency of oscillation C0d2 second damped natural frequency of oscillation Q)p fundamental parametric excitation frequency, equal to the fundamental gear mesh frequency XXX Chapter 1 Introduction 1.1 Introduction In an ideal gear box configuration, the efficiency would equal unity and no vibration would exist. Unfortunately the efficiency of the gear mesh is not unity, though it can approach unity. The lost energy is dissipated as heat, sound, and vibration, though they are not independent mechanisms. The energy lost as heat is usually dissipated in the lubricant and the environment. The energy lost as sound can lead to pressure levels which can damage hearing, create dangerous working conditions, cause annoyance and allow for detection, but generally have no direct detrimental effect on the operation of the gear box. The energy lost in vibration is transmitted to components external to the gear mesh. These vibrations can excite natural frequencies and generate sound via housing vibration and under extreme conditions can lead to premature failure of the gearbox or 1 JL attached machinery [Houser, 1990]. As mentioned above, the energy mechanisms are not independent since sound is generated from a vibrating source, but appears as different manifestations. At the gear mesh there are several excitations which include transmission error, tooth impact, tooth stiffness variation, frictional forces, and moments of forces. Transmission error, tooth impact and stiffness variation have been examined in previous investigations [Ozguven, 1988] and deal in forces and motion in a direction normal to the tooth. Tooth friction acts in the direction parallel to the transverse tooth surface (perpendicular to the line-of-action) which for the purpose of this research is defined as the off line-of-action direction This research is motivated by and based upon observations by Hochmann [1992] and Yoshida [1993], in which measured gear shaft motion mesh frequency vibration amplitudes decomposed into line-of-action motion and off line-of-action components showed vibration in the off line-of-action to be generally larger than the vibration amplitude in the line-of-action direction. The off line-of-action is tangent to the point of an gear mesh contact surface while the line-of-action is normal to the gear mesh contact surface as presented in Figure 1.1. Measurements done by Hochmann were performed in the static operating region, while measurements performed by Yoshida were in the dynamic operating region. Figure 1.1 Line-of-ActIon and Off Llne-of-Actlon Definition The effect of gear mesh friction as a vibration excitation has generally been ignored [Smith, 1983], whereas the effect of friction on efficiency has been considered by Tso [1961], Yada [1972], Radzimovsky [1973] and Chiu [1975]. 1.2 Scope and Objective The primary focus of this research is to examine the friction force at the gear mesh as a potential dynamics excitation source of involute parallel axis spur and helical gears. Specific objectives of this research are: 1. Gather experimental data to further show that the large off line-of-action motion observed in earlier research by Hochmann and Yoshida is not an isolated phenomenon. 2. Relate the off line-of-action motion to off line-of-action forces at the gear mesh interface. 3. Develop an analytical model of the gear mesh incorporating the effect of the off line-of-action friction forces. 4. Experimental measurement of the individual tooth friction forces in the off line-of-action direction for a spur gear pair with a contact ratio less than or equal to two. 1.3 Organization The outlined objectives of the previous section are grouped Into a chapter format. Given below Is a brief description of the subsequent chapters: Chapter 2 Further experimental data of the off llne-of-actlon and llne-of- actlon motion Is presented. The experimental data consists of measurements performed on the Ohio State University Gear Noise Gear Dynamics Research Laboratory Back-to-Back Gear Test Stand and the NASA Lewis Research Center Gear Noise Test Rig. Another set of experimental data Is obtained from the technical literature. The off llne-of-actlon and llne-of- actlon motion Is related to the force through support bearing stiffnesses. Next, two analytical approaches are used to further support the Idea of friction force at the gear mesh as a dynamic excitation source. The first approach compares the force transmlsslblllty between the gear mesh and the support bearing In the llne-of-actlon and off llne-of-actlon directions. The second approach uses energy concepts to show that high friction forces are needed to generate typical efficiencies reported for the gear mesh. Chapters A measurement methodology is presented to compute the Instantaneous individual tooth friction forces for a spur gear pair with a contact ratio less than or equal to two. The method is based upon the measurement of the power loss across the gear mesh and the measurement of the net friction force in the off line-of-action direction. The method is shown to be valid, both in a general analytical sense and through several numerical examples. Finally, the method is applied to the Gear Noise and Gear Dynamics Research Laboratory Back-to-Back Tester. Both the experimental results and experimental setup are presented. Chapter 4 A six degree-of-freedom model is proposed for each parallel axis helical gear blank element. The gear blank elements are coupled through a generalized force distribution in the line-of- action (normal force load distribution) and the off line-of-action (friction force load distribution) directions. A normal force and friction force load distribution model is introduced for the generalized force distributions. The friction force model is based upon a friction coefficient concept, where the friction coefficient may be a function of the dependent and independent variables resulting in linear time varying or non linear equations. A gear system mounted on rigid bearings is 6 derived based upon the developed general model. The rigid bearing model is examined under static and dynamic conditions, first for a linear time varying form which implies the friction coefficient is a function of the independent variable and then for a non-linear form which represents a simple elastohydrodynamic fluid film lubrication situation. Chapter 5 The conclusions and contributions based on this research are outlined. Recommendations on research issues which have not been examined in this dissertation, but which need to be explored for greater understanding are also discussed. Chapter 2 Experimental and Analytical Evidence of Off Line-of-Action Friction Force as a Potential Dynamic Excitation Source 2.1 Introduction This chapter introduces evidence to support the claim that friction force at the gear mesh is a potential excitation source. Both experimental and analytical evidence are examined. The experimental evidence consists of shaft displacement measurements gathered near the gear shaft support bearing. The results show that the motion in the off line-of-action direction can be several times larger than motion in the line-of-action direction. Evidence of this nature is produced from three different experiments. The first experimental displacement results are from independent research performed by Umezawa [1996]. The next set of results present displacement measurements taken from the NASA Lewis Research Center gear noise test 8 rig. The last set of experimental results come from the Ohio State University Gear Dynamics Gear Noise Laboratory Back-To-Back Gear test rig. The measured data is in the form of shaft displacements which are related to the force by the bearing's stiffness. Therefore, the support bearing is examined to estimate the bearing contribution to the large off line-of-action shaft displacement. Bearing cross coupling, reduced bearing stiffness and bearing radial clearance are examined. Friction force as an excitation force is examined analytically using two different concepts. The first concept compares the force transmissibility between the gear mesh and the bearing support in the line-of-action and off line-of-action directions. First, a simple 3 degree-of-freedom model is studied. Then the force transmissibility concept is extended into a dynamic finite element model representing the Gear Lab’s Back-to-Back Tester. The results show that the off line-of-action force transmissibility can be orders of magnitude larger than the line-of-action force transmissibility at frequencies below the first few shaft natural frequencies. The next analytical study examines the relationship between the gear mesh energy efficiency and the friction force to normal mesh force ratio. These results show that even for the normally high efficiencies found in spur and helical gearing, there is a significant friction force in the off line-of-action direction. 2.2 Literature Review Gear mesh friction force as a dynamic excitation has been neglected [Smith, 1983]. Therefore the body of literature covering the topic is virtually non-existent. Research by Hochmann [1992] and Yoshida [1993], provided the motivation for this research. Experimental measurements demonstrated large off line-of-action shaft vibratory motion as compared to the vibratory motion in the line-of-action. Other relevant published research include the gear shaft motion studies by Umezawa [1996], which is discussed in a section of this chapter. Another important relevant topic is the roller element bearing research performed by Lim [1989]. Matson [1995] developed the torsional lumped param eter model which is coupled to a dynamic finite element formulation of the Ohio State University back-to-back gear tester. 2.3 Experimental Evidence 2.3.1 Experimental Results of Umezawa This section presents the results of the independent research performed and published by Umezawa [1996]. The study shows that shaft motion in the off line-of-action is of the sam e order of magnitude as motion in the line-of-action direction. Umezawa offered no opinion on the possible cause of the large off line-of-action motion that he observed. 1 0 The test rig used by Umezawa is a power absorbing design equipped with a driving motor and a power absorbing brake dynamometer. The bearing pedestal (bearing type 6207ZZ) was designed to be rigid in the radial and thrust directions. Figure 2.1 shows a simple schematic of the experimental test rig used by Umezawa. Figure 2.2 shows the laser measurement locations on the gear shaft. Displacements are measured at several locations on each shaft. Measurement points U and V correspond to points on the shoulder of the pinion and gear blanks, respectively. The next two measurement points are located near the bearings in the gearbox and the two outermost measurement points are outside of the gearbox. The laser measurements are taken in the horizontal and vertical planes. A normal load of 561 Ibf (2.5x10^ N) was applied and tested under forced lubrication conditions. Figure 2.3 and Figure 2.4 present measurement results of the test rig’s shaft translational motion under several operating conditions. The orbit plots show that motion in the off line-of-action direction generally has a larger mesh frequency amplitude than the corresponding line-of-action displacement. The motion of the driving gear shown in Figure 2.4, is almost entirely in the off line-of-action direction. 11 (1) Slip ring (2) Diaphragm coupling Motor a 8 i - Dynamo meter 3 ; Du— Figure 2.1 Test rig used by Umezawa [1996] Gear n t i Measurement Points with Laser Doppler Velodmeter am Figure 2.2 Shaft measurement locations, [Umezawa, 1996] 1 2 Bearing Figure 2.3 Vibration mode of gear system, 1235 Hz mesh frequency, [Umezawa, 1996] Figure 2.4 Vibration mode of gear system, 1445 Hz mesh frequency, [Umezawa, 1996] 1 3 2.3.2 NASA Gearbox Test Rig This section presents the results from experimental research performed on the NASA Lewis gear noise test rig at the NASA Lewis Research Center, Cleveland Ohio. Rebbechi [1996;1991] and Oswald [1995, 1996] are recent publications of experimental research performed on the NASA gear test rig. The experiment for this research was performed on a gearbox designed specifically for research to independently verify results gathered at The Ohio State University Gear Dynamics Gear Noise Research Laboratory. 2.3.2.1 NASA Gearbox Test Rig Description The NASA Lewis test rig is a single gear mesh design with a 200 hp (150 kW) driving motor and a power absorbing eddy-current dynamometer. The gearbox can operate at speeds up to 6000 rpm and was built to carry out fundamental studies of gear noise and dynamic behavior of gear systems. Table 2.1 presents a short summary of the gear pair design parameters used in the experiment. Figure 2.5 presents the test rig layout and Figure 2.6 shows details of the gearbox. The gear shafts are supported with a deep grove thrust bearing at one end and a roller bearing at the other end. 1 4 Pinion Gear Teeth 20 36 Base Diameter 2.349 inch 4.229 inch Outside Diameter 2.750 inch 4.750 inch Face Width 0.25 inch Center Distance 3.5 inch Normal Diametral Pitch 8 Normal Pressure Angle 20 degree Table 2.1 NASA gear noise test rig spur gear pair summary Dynamomeiar Optional \ Test output tiyvheel -, oeaftnx I Optional input llywheei-\ Figure 2.5 NASA Lewis research test rig layout, [Oswald, 1995] 1 5 o Figure 2.6 NASA Lewis research gearbox details, [Oswald, 1995] 1 6 2.3.2 2 NASA Experiment Description The experiment measured the motion of the gear shaft using two orthogonal non-contact inductive displacement transducers (Electro Corporation, Model 4937) placed near the gear shaft bearing farthest from the dynamometer. This location was selected based on spatial constraints within the gearbox. The eddy current probes have a nominal sensitivity of 400 volt/inch and an operating range of 25,000 micro-inches. Figure 2.7 locates the measurement plane inside of the NASA gearbox while Figure 2.8 defines the probe measurement coordinate system. The displacement probes are fixed to an L-shaped bracket which is attached to a cantilever beam below an I-beam mounted across the top of the gearbox. Accelerometers are attached next to each probe on the L- shaped bracket to measure the motion of the bracket in space. The measured displacement and acceleration are transformed from the measured horizontal and vertical coordinate system to the off line-of- action and line-of-action coordinate system. The line-of-action coordinate system is parallel to the gear line-of-action which is the tangent line connecting the two gears as pictured in Figure 2.8. The off line-of-action coordinate system is perpendicular to the line-of-action coordinate system. 1 7 V A Figure 2.7 NASA gearbox measurement plane location pinion shaft gear shaft \ \ Horizontal Probe Vertical Probe LIne-of-Actlon Figure 2.8 NASA gearbox displacement probe locations 1 8 2.3 2.3 NASA Experimental Results Table 2.2 presents the frequency domain gear mesh linear magnitude for two output shaft test speeds and for various output torque levels. Presented are the off line-of-action and the line-of-action probe displacements and bracket displacements and the vector combination of the probe and bracket displacements. The bracket displacements, as measured with the accelerometers mounted parallel to the displacement probes, is significant only for the 1000 rpm speed case. It is insignificant for the other test speeds, as seen in Table 2.2 for the 2000 rpm results. This could suggest a possible probe bracket resonance condition at the 1000 rpm operating speed. The phase between the shaft motion and the bracket motion is approximately zero, resulting in an increased shaft motion harmonic amplitude. The off line-of-action and line-of-action shaft displacements shown in Table 2.2 are of the same order of magnitude. 1 9 test torque, oloa, probe loa, probe oloa, loa, oloa, loa, output disp disp brk. acc. brk acc. probe & probe & acc acc rpm Ibf-in micro-in micro-in micro-in micro-in micro-in micro-in 1000 200 20 57 15 6 31 64 1000 400 84 110 45 30 103 133 1000 600 103 93 51 35 147 115 1000 800 97 114 74 36 166 133 1000 1000 111 140 59 38 170 151 1000 1200 150 174 60 39 179 189 2000 200 20 36 1 0.1 2000 400 21 25 0.5 0.6 2000 600 32 23 0.2 2.5 2000 800 48 24 0.5 4 2000 1000 46 22 1.4 5 2000 1200 51 8 1.9 1.2 Table 2.2 Summary for selected NASA test rig results presenting gear mesh frequency linear magnitudes Figure 2.9 and Figure 2.10 graphically summarize the gear mesh frequency zero-to-peak, linear amplitude for the off line-of-action displacement, the line-of-action displacement and a sound pressure level reading. Also presented on Figure 2.9, the predicted static transmission error in terms of linear displacement using the Load Distribution Program (LDP) [Gear Dynamics and Gear Noise Research Lab, no date] is presented. The results are in a logarithmic scale with displacement 2 0 referenced to 1.0 |iin and the sound pressure level referenced to the minimum sound level recorded. To compute the absolute sound pressure level referenced to 20 pPa, 47.6 dB must be added to the sound data in Figure 2.9 and Figure 2.10. Figure 2.9 shows results with the output speed fixed at 2000 rpm and the output torque varied from 200 Ibf-in to 1200 Ibf-in.. The off line-of-action and line-of-action displacements are in the same order of magnitude. Above 400 Ibf-in output torque the off line-of-action displacement is larger than the line-of-action displacement. Figure 2.9 also presents mesh frequency sound pressure level at the various test conditions. Since both the off line-of-action and line-of-action displacements occur at mesh frequency, the total sound pressure level is a combination of the off line-of-action and line-of-action generated sound, but at an unknown amount. The sound pressure level generally appears to decrease when either off line-of-action or line-of-action displacement decreases and vice versa. This is also observed in Figure 2.10. Figure 2.10 presents results with the output torque fixed at 1000 Ibf-in and the output speed varied. In Figure 2.10, the off line-of-action displacement is larger than line-of-action displacement. The mesh frequency harmonic amplitude decreases until the displacements reach a minimum at 3000 rpm and then increases as speeds increase. The sound pressure level also follows this trend. As noted earlier, at 1000 rpm, a possible resonance 2 1 condition exists. Therefore the decrease in amplitude as excitation frequency leaves the resonance at 1000 rpm and starts to increase after 3000 rpm, where the mesh excitation is perhaps approaching the next resonance, is explained. Figure 2.11 shows an overlay of two different shaft revolutions in the time domain of the off line-of-action shaft motion for a gear shaft speed of 1000 rpm and output torque of 1200 Ibf-in. Superimposed on the once per revolution component is the mesh frequency component of 600 Hz. Figure 2.12 presents in the frequency domain the data of Figure 2.11. Clearly, the 600 Hz fundamental mesh frequency is the dominant component. 2.3.2 4 NASA Result Summary In general, results from the NASA Gear Noise Test Rig show that at gear mesh frequencies the off line-of-action motion is of the same order of magnitude as the line-of-action shaft motion. Also, at many conditions, off line-of-action motion can be up to several times the magnitude of the line-of- action shaft motion at mesh frequencies. 2 2 s 50 K 45 r 40 Q. a: 35 = 30 ô 25 2 20 - * oloa. displacement - a - loa. displacement —K- LDPTE • vector sum of oloa & loa 200 300 500 600 700 800400 900 1000 1100 1200 output side torque. In-lbf Figure 2.9 NASA test rig results for output speed= 2000 rpm, varying torque, displacement dBref = Ipin, absolute SPL =SPL+47.6 dB 50 • - ♦ - • oloa. displacement - « - loa. displacement 40 T3 35 vector sum of oloa & loa 15 1000 1500 2000 2500 3000 3500 4000 Output Speed, rpm Figure 2.10 NASA test rig results for output torque = 1000 Ibf-in, varying speed, displacement dBref = 1 pin, absolute SPL =SPL+47.6 dB 2 3 off line-of-action motion. 1000 rpm. 1200 Ibf-in 2.5 I I 0.5 ! I -0.5 -1.5 50100 150 200 250 300 350 400 Gear Shiaft Rotation, degree Figure 2.11 NASA test rig results showing an overlay of shaft displacement measurements in the off line-of-action direction for two separate shaft revolutions in the time domain, output speed=1000 rpm, output load=1000 Ibf-in _ , o'* off ine-of-action shall motion. 1000 rpm. 1200 bl-in 1.4 — mesh frequency 0.6 0.6 2 mesh frequency 0.4 0.2 100 1000 1500 2000 2500 frequency, Hz Figure 2.12 NASA test rig results showing the off line-of-action shaft displacement in the frequency domain, output speed=1000 rpm, output load=1000 Ibf-in 2 4 2.3.3 Back-to-Back Gearbox Test Rig This section presents results from the Ohio State University Back-to- Back Gear Test Stand. Similar to the previous experiment, shaft motion at gear mesh frequencies In the off line-of-action and line-of-action directions near the gear shaft bearings are examined. Large off line-of-action motion is obsen/ed. 2.3.3.1 Back-to-Back Gearbox Test Rig Description The back-to-back test rig [Matson, 1995] is a power recirculating design. Two gear boxes are attached together as shown schematically in Figure 2.13. Torque is applied using a rotary hydraulic actuator and a direct current motor spins the rig through a belt drive system. The test rig gear boxes are a standard industrial type with the gear shafts being supported by tapered roller bearings. Table gives the basic measurement side gear geometry. 2 5 Pinion Gear Teeth 25 47 Base Diameter 4.698 inch 4.229 inch Outside Diameter 5,608 inch 10.192 inch Face Width 1.25 inch Center Distance 7.5 inch Normal Diametral Pitch 5 Operating Pressure Angle 25 degree Table 2.3 Back-to-Back test rig gear pair summary Drive Motor C o m m e r c ia l D riv e T o rq u e B e lt — .S e n s o r Pinion Shaft G e a r S h a ft T o rq u e ^ T e s t G e a r Actuator Slave Gear B ox B ox Figure 2.13 Back-to-Back Gear Test Rig 2 6 X 2.S.3.2 Back-to-Back Experiment Description In the experiment, the shaft motion near the bearings was measured using two perpendicular non-contact eddy-current displacement transducers. Similar to the arrangement of Figure 2 .8 , the probes are mounted in a horizontal and vertical fashion. The results are transformed to the off line-of-action and line-of-action coordinate system [Hochmann, 1992]. The probes are mounted in an L-bracket that is attached to the gear box housing using strong permanent magnets [Yoshida, 1993]. Figure 2.14 defines a coordinate system used in the back-to-back tester for shaft displacement measurements. Coordinate axes X and Y correspond respectively, to the horizontal and vertical coordinate system. For the test gears, the operating pressure angle is 25 degrees and is defined by (j) in Figure 2.14. For a qualitative examination, the off line-of- action motion is approximately equal to the horizontal shaft motion (X axis) and the line-of-action motion is approximately equal to the vertical shaft motion (Y axis). 2 7 V Figure 2.14 Back-to-Back coordinate system 2 8 2.3.3.3 Back-to-Back Experimental Results The results are presented in the form of speed sweeps starting from the 100 rpm and increasing to 1200-1500 rpm depending upon specific case. The measurement are presented in the orders domain which is analogous to the frequency domain except the independent variable is spatial instead of temporal. Figure 2.15 shows the 25th order of shaft vibration (fundamental mesh frequency) for pinion shaft speeds between 100 and 1500 rpm at 25 rpm increments. Three load cases are presented for the X axis direction shaft displacement motion which is approximately the off line-of-action direction. The actuator pressures correspond to no-load, 5000 Ibf-in, and 10000 Ibf-in on the output side, respectively. Figure 2.15 shows that the off line-of-action harmonic motion generally increases as the normal load increases. For each actuator pressure, there are two independent runs showing data repeatability. Several probable natural frequencies are also observed. These occur at pinion speeds of 600, 1000 and 1250 rpm, respectively. Figure 2.16, Figure 2.17, Figure 2.18, and Figure 2.19 present orders domain waterfall results for no-load ‘X’, no-load ‘Y’, 10,000 Ibf-in X' and 10,000 Ibf-in ‘Y’ motion, respectively. These results range from 100 rpm through 1200 rpm in 50 rpm increments. Again, the line-of action harmonic motion is significantly smaller than the corresponding off line-of-action motion. 2 9 In addition to the large 1st mesh order amplitude, other significant shaft motions occur at higher order mesh harmonics corresponding to the 50th, 75th, 100th, etc. shaft orders. These higher harmonics are capable of exciting higher natural frequencies at a given speed. Two additional natural frequencies are readily seen in the 10,000 Ibf-in case and but are more difficult to detect in the no-load case. The dominant natural frequency is at approximately 1350 Hz. This frequency is excited by the 3rd, 4th, 5th, 6th, 7th mesh orders at approximately 1000, 800, 650, 550, and 450 rpm, respectively. Another smaller resonance at approximately 625 Hz is excited by the 2nd, 3rd, and 4th mesh orders at 750 rpm, 500 rpm, and 375 rpm, respectively. These resonances appear to be excited in both the on line-of- action and off line-of-action directions, but not at the identical frequency. For example, in Figure 2.16, at the 100th order, the peak amplitude occurs at 800 rpm while in Figure 2.17, at the 100th order, the peak amplitude occurs at 850 rpm. This corresponds to about 85 Hz difference. This can possibly be attributed to degenerate modes that vary due to the spring stiffness in the tooth. This observation will be examined in more detail later in this chapter. 3 0 X 10’® First mesh order amplitude for X axis shaft motion Ü 10000 Ibf-in Ç no load 5 0 0 1 0 0 0 1 5 0 0 rpm 208 Hz 416 Hz 6 2 5 H z Pinion Shaft Speed / Mesh Frequency Figure 2.15 Tracking of the first mesh frequency (25th shaft order) for various output shaft loads 3 1 '"o« RPW 180 ISO '<0 120 <00 8 0 SO <0 20 0 P'«Rency(o,aei5) ‘displacement, no « '"put torque preload 1200 npM 180 160 140 120 100 80 SO 40 20 0 Shaft dfepiao no input torque Pn^oad 3 2 *'«0800.300 ®s'Df»s*/a P<» ^ent, 10000 ibf., '°mue preload ’"^“"«Owidresre Pmamncyforewi FW reajsy.^^ "^"««splacereent, loooo in torque prefoad 3 3 2.3.5.4 Back-to-Back Results Summary In general, results from the Back-to-Back Gear Test rig show that at gear mesh frequencies off line-of-action motion is of the same order of magnitude as line-of-action shaft motion. At many conditions, off line-of- action motion can be up to several times the magnitude of the line-of-action shaft motion at gear mesh frequencies. 2.3.4 Experimental Results Conclusion The published experimental work by Umezawa [1996], the experimental data gathered at NASA Lewis Research Center on the gear noise test rig, and the experimental data gathered on the back-to-back gear test rig all show that motion at gear mesh frequencies in the off line-of-action direction, is of the same order of magnitude as motion in the line-of-action direction. Furthermore, for many conditions the off line-of-action motion is larger. 2.4 Possible Off Line-cf-Action Shaft Motion Explanations These results could suggest a force acting in the off line-of-action direction is responsible for this motion. A possibility would be the gear friction forces that occurs as the meshing gear teeth slide with respect to each other. However, other altemative explanations of the large off line-of- 3 4 action motion will be examined in this dissertation. Other possible explanations to be considered in this chapter include rotordynamic phenomena, the shaft support bearings, and system dynamics properties. 2.4.1 Friction Force in the Off Line-of-Action Direction The large amplitude of motion in the off line-of-action direction as compared to the on line-of-action direction may possibly be explained as follows; In the line-of-action direction the force transmitted across the mesh is a combination of the mean load, f™®", plus a time varying load, Af,^(t), at mesh frequency and its harmonics. Eqn. 2.1 presents this relation. The time varying loads are typically attributed to profile errors, profile modifications, tooth impact, and mesh stiffness variations. Eqn. 2.1 f Jt)=fr'’+ Af Jt) The friction force may be represented as being proportional to the normal load, where the proportionality variable is the coefficient of friction irrespective of the friction mechanism as presented in Eqn. 2.2. The friction force acts along the off line-of-action in the direction resisting the sliding motion. Therefore the friction force due to the transmitted normal load in the gear tooth mesh can be represented by Eqn. 2.3. Eqn. 2.2 f, 3 5 -V Eqn. 2.3 f, = + jiAf J VI In gear design, one design philosophy is to minimize the line-of-action vibration by minimizing the force variation in the line-of-action as the tooth mesh proceeds through the mesh cycle. This is done by profile modifications and by attempting to maintain integer profile or face contact ratios such that mesh stiffness remains through the mesh cycle. These adjustments minimize the time varying components of transmission error and mesh stiffness which in tum minimize vibration transmitted to the bearing along the line-of-action. This implies that Afm(t) will be very small as compared to fmean sjp^e, typlcally, f » Afm, Eqn. 2.3 may be approximated as Eqn. 2.4. -V Eqn. 2.4 f, = IV The ratio of surface sliding speed to the absolute value of tooth sliding speed is either, plus one and negative one, depending upon the position of contact relative to the pitch point. Since for loaded gears, f™®" » Aft, the friction force can be of the same order as the time varying normal load which leads to the line-of-action vibration. This could occur even though the friction coefficient in a gear mesh is small and the gear mesh efficiency is high. Since the friction force changes direction at the pitch point, once per tooth 3 6 contact, the friction force changes direction once per tooth contact generating a time varying excitation at the gear mesh frequency and its harmonics. Though, explained in the context of a gear with a contact ratio of unity, the analogy can be extended to contact ratios greater than one by summing the friction force at each point of contact along the mesh. 2.4.2 A Rotordynamic Phenomenon A possible cause of the observed shaft motion could potentially be attributed to a rotordynamic phenomenon. The large off line-of-action motion phenomenon was observed consistently at gear mesh frequencies on different test rigs for different gear geometries. It is unlikely to get a rotor dynamics phenomenon consistently showing up at gear mesh frequencies. Secondly, the observed experimental data show that the off line-of-action amplitude is a function of load and speed while rotordynamic phenomena are typically speed driven. In conclusion, for this research, rotordynamics phenomena will be neglected since the types of speed excited phenomena have not been observed. 3 7 2.4.3 Shaft Support Bearings The shaft support bearings are another potential cause for the large off line-of-action motion observations. There are three distinct reasons this could occur; first, bearing radial clearance; second, reduced stiffness in the off line-of-action direction; and third, bearing cross coupling. The observed shaft motion in the off line-of-action direction is a measured displacement, related to force via the bearing stiffness and inertia. This implies large motion equates to a large force, but it is not always the case. If the bearing stiffness is near zero, due to radial bearing clearance, small forces can create large displacements. These topics are addressed in the following sections. Bearing cross coupling suggests that forces or moments in the line- of-action direction cause reaction forces in the off line-of-action directions through the internal operation of the bearing. This topic will also be examined in a following section. This discussion on bearings is based upon work and equations developed by Lim[1989]. Lim’s work examined vibration transmission through rolling element bearings (ball and roller element bearings) in geared rotor systems. As part of Lim’s work, a six degree-of-freedom bearing stiffness matrix was developed. The stiffness matrix is developed about the mean bearing deflected position, taking into account properties such as 3 8 radial clearance, number of rolling elements, and type of rolling element. The stiffness matrix is computed by solving a set of simultaneous non-linear algebraic equations. 2.4.3.1 Bearing Cross Coupling The general 6 x 6 stiffness matrix proposed by Lim potentially couples the off line-of-action and line-of-action directions through the rolling bearing elements. Lim’s research concluded that, in general, certain stiffness elements of the generalized bearing stiffness matrix are dominant. These terms have magnitudes significantly larger than other non-dominant terms, therefore defining the static response of the bearing. In Table 2.4 through Table 2.6 the results of the analysis of the bearings stiffness matrix are presented for the tapered roller element bearing in the back-to-back tester. The axis loa, oloa, Z represent translation about the line-of-action, the off line-of-action, and the shaft rotation axis, respectively. The Rioa, Roioa, and Rz represent rotation about their respective axes. Lim’s model assumed free motion about the shaft axis of rotation (Z axis), therefore row 6 and column 6 are always equal to zero. In Table 2.4 through Table 2.6 the dominant terms, like kioa,ioa and koioa.oioa, are of the order 10® and 10^ while non-dominant terms, like kioa.oioa and koioa,ioa. are of the order 10‘^°. In these results the dominant stiffness terms do not form a cross coupling situation between the off line-of-action 3 9 direction and the line-of-action direction via the bearing stiffness matrix. Similar results are predicted by Lim’s model for ball bearing elements. For further details, the Ph.D. dissertation by Lim contains parametric studies for both ball and roller element bearings. 2.4.S.2 Bearing Reduced Stiffness in the Off Line-Of-Action Direction The potential off line-of-action force excitation is related to the off line- of-action displacement through the bearing stiffness terms. Cross coupling has been eliminated as a potential cause of the observed large off line-of- action displacements. Another possible cause of the off line-of-action displacements is reduced bearing stiffness in the off line-of-action direction as compared to the line-of-action direction. Table 2.4 presents the bearing stiffness matrix values for a tapered bearing that is unloaded in the radial direction and has a 50 Ibf preload in the axial direction. Table 2.5 presents the tapered bearing stiffness with equal axial and line-of-action radial bearing load of 50 Ibf. The application of the minimal radial load nearly doubles the bearing stiffness term, kioa,ioa. as compared to the same stiffness term in Table 2.4 while no change occurs in the off line-of-action stiffness term, koioa.oioa. Finally, 2.6 presents the bearing stiffness terms with a 1000 Ibf line-of-action radial load. Examining the results presented in Table 2.4 to Table 2.6 shows that the off line-of-action translational bearing stiffness term, koioa.oioa, is less than the line-of-action translational stiffness, k)oa,ioa. but only by approximately 4 0 one-half. The observed experimental data presented in the previous sections show that off llne-of-actlon motion can be several times the magnitude of the llne-of-action motion. Therefore, reduced bearing stiffness does play a minor role In the large off llne-of-actlon motions observed In the experimental data. However, reduced stiffness does not explain the entire picture. Even with the off llne- of-actlon stiffness term being approximately one-half the llne-of-actlon stiffness term, the off llne-of-actlon gear mesh frequency force excitation may be of the same order of magnitude as the llne-of-actlon gear mesh force excitation. Similar results are predicted by Lim’s model for ball bearing elements. 4 1 i.i loa oloa Z Rio« Roioa Rz loa 0.14E+07 -0.35E-10 -0.51 E+06 0.15E-10 -0.62E+06 O.OOE+OO oloa -0.35E-10 0.12E+07 0.73E-10 0.56E+06 -0.15E-10 O.OOE+00 z -0.51 E+06 0.73E-10 0.22E+06 0.21 E-10 0.22E+06 O.OOE+OO Rloa 0.15E-10 0.56E+06 0.21 E-10 0.24E+06 -0.70E-11 O.OOE+OO Roioa -0.62E+06 -0.15E-10 0.22E+06 -0.70E-11 0.27E+06 O.OOE+OO Rz O.OOE+00 O.OOE+00 O.OOE+00 O.OOE+00 O.OOE+00 O.OOE+OO Table 2.4 Computed tapered roller bearing stiffness terms ki.j for mean Fioa= 0 Ibf, mean axial preload Fz == 50 Ibf, units: Ibf/in and Ibf-in as appropriate loa oloa Z R|oa Roloa Rz loa 0.23E+07 0.30E-10 -0.24E+06 0.15E-10 -0.10E+07 O.OOE+OO oloa 0.30E-10 0.13E+07 0.84E-10 0.59E+06 -0.15E-10 O.OOE+OO z -0.24E+06 0.84E-10 0.30E+06 0.40E-10 0.10E+06 O.OOE+OO Rloa 0.15E-10 0.59E+06 0.40E-10 0.26E+06 -0.13E-10 O.OOE+OO Roloa -0.10E+07 -0.15E-10 0.10E+06 -0.13E-10 0.44E+06 O.OOE+OO Rz O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO Table 2.5 Computed tapered roller bearing stiffness terms kj,, for mean Fioa= 50 Ibf, mean axial preload Fz = 50 Ibf, units: Ibf/in and Ibf-in as appropriate i|j. . loa oloa Z j R|oa Roloa Rz loa 0.31 E+07 0.62E-10 0.28E+05 0.28E-10 -0.13E+07 O.OOE+OO oloa 0.62E-10 0.17E+07 -0.20E-10 0.75E+06 -0.28E-10 O.OOE+OO z 0.28E+05 -0.20E-10 0.39E+06 0.24E-10 -0.12E+05 O.OOE+OO Rloa 0.28E-10 0.75E+06 0.24E-10 0.33E+06 -0.18E-10 O.OOE+OO Roloa -0.13E+07 -0.28E-10 -0.12E+05 -0.18E-10 0.59E+06 O.OOE+OO Rz O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO O.OOE+OO Table 2.6 Computed tapered roller bearing stiffness terms kj, j for mean Fioa= 1000 Ibf, mean axial preload Fz = 50 Ibf, units: Ibf/in and Ibf-in as appropriate 4 2 2.4.S.3 Bearing Radial Clearance Another possible cause of the large off line-of-action motion is bearing radial clearance introduced in design or in manufacturing. The radial clearance can be viewed as a small region of zero bearing stiffness. Under general operating conditions, the mean normal load in the line-of-action deforms the bearings which forms the new equilibrium operating position, therefore removing any bearing clearance. In the off line-of-action direction, the mean load is approximately zero, suggesting that the large off line-of- action is merely a small friction force moving the shaft back and forth over this region of zero stiffness. This option will be examined analytically through Lim’s work, and qualitatively through results from the back-to-back gear tester. 2.4.3.3.1 Analytical Model Incorporating Radial Clearance Lim’s non-linear algebraic equations for the bearing stiffness matrix include the effects of radial bearing clearance. Figure 2.20 shows the translational reaction forces due to a mean load applied in the line-of-action direction. The sum of the reaction forces in the line-of-action direction is equal in magnitude to the applied mean load, Roa- The off line-of-action reaction components sum to zero. Though the off line-of-action net reaction force is zero, individual rolling elements are deformed in the off line-of-action 4 3 direction, forming the off line-of-action stiffness terms observed in Table 2.4 through Table 2.6. This process also eliminates the off line-of-action clearance, even though the mean load in the off line-of-action direction is zero. 0 0 0 0 Figure 2.20 Translational bearing reactions for a mean load applied in the line-of-action direction 4 4 2.4.3.S.2 Experimental Results Examining Radial Clearance In A Tapered Roller Radial clearance can also be eliminated as a cause for large off line- of-action motion by examining the tapered roller bearings used in the back- to-back gear tester. An axial preload is applied to the tapered roller bearing through the application of shims. By increasing the shim thickness, the axial preload is reduced. Decreasing the shim thickness increases the bearing preload. This axial preload is uniform about the radial bearing direction, eliminating the radial bearing clearance in all radial directions. Yoshida[1993], performed an experiment measuring horizontal (X axis) and vertical (Y axis) gear mesh frequency motion. The coordinate system used is defined in Figure 2.14. Yoshida’s results show that at large shim thickness, the off line-of-action motion can be affected by radial clearance, but quickly reduces as shim thickness is decreased. The back-to-back experimental results, presented in Section 2.S.3.3, were gathered with a minimum shim thickness, therefore minimizing the radial clearance effect. The conclusion is that an external force, such as sliding friction. Is causing the most significant portion of the off line-of-action gear mesh frequency motion. Since there is nothing particularly unique about the back- to-back test gears, it can be surmised that this friction force is also causing the large off line-of-action motion presented in the results by Umezawa and the NASA gear noise test rig results. 4 5 2.5 Analytical Evidence 2.5.1 Gearbox System Dynamics The force transmissibility between the gear mesh and the support bearing in the off line-of-action and line-of-action directions will be examined. Two approaches will be used. First, a simple three degree of freedom model, representing one half of the line-of-action gear system is examined. The second approach is a dynamic finite element model of the back-to-back gear test rig. The force transmissibility will be examined between the on and off line-of-action forces at the gear mesh and the on and off line-of-action bearing forces. Experimentally measured inertance for a pinion shaft bearing on the back-to-back gear test rig will be compared against the inertance computed from the developed finite element model at the same bearing location. 2.5.1.1 Three Degree of Freedom Gear Model Figure 2.21 graphically presents a simple gear system dynamic model (Nakada [1956], Utagawa [1958], Gregory [1963], Ozguven [1988]). This six degree of freedom model can be reduced to a three degree of freedom model by assuming the gear mass, gear inertia and gear bearing stiffnesses are infinite. This reduces the system to one-half of the original gear system. The new model is presented in Figure 2.22. 4 6 y "y Figure 2.21 Simple off line-of-action and line-of-action dynamic model \ Figure 2.22 Reduced order off line-of-action and line-of-action dynamic model 4 7 The dynamic equations for the reduced gear system of Figure 2.22 are shown in Eqn. 2.5. Defining the input for the line-of-action component in terms of force results in Eqn. 2.6 which is substituted into Eqn. 2.5 to obtain Eqn. 2.7. Solving Eqn. 2.7 using Laplace Transforms, and writing the off line-of-action and line-of-action force transmissibility results in Eqn. 2.8 and Eqn. 2.9. lg 8 + k y 8 -Hk,^rx = rk,^e(t) Eqn. 2.5 mx + (k„,+kjx+k,„r 8 = k,„e(t) mÿ + kyy = fy(t) Eqn. 2.6 f,(t) = k^e(t) lg 8 4-k^r"8 + k,^rx = rfXt) Eqn. 2.7 mx + (k,„+k Jx+k,„r 8 = f,<(t) mÿ + kyy = fy(t) Eqn. 2.8 L(s) ^ k,x(s) ______^k,s=______fx(s) fx(s) (s^ -(a + b )^ )(s ^ -(a -b )^ ) (lgkm+rnk„r"+lgk„ cL — ——————— 2lgm ^ ^ VOgkq, + m ky+ lgkj" -4(lgm)(k,k,^r^ 2lgm 4 8 Eqn.2.9 yîL.M!). lis) fJs) ms^+k, Table 2.7 presents the system parameters used in the force transmissibility computation. The system parameters defined in Table 2.7 represent an extremely reduced form of the back-to-back tester described in section 2.3.3. The results of Eqn. 2.8 and Eqn. 2.9 are presented in Figure 2.23. The two directions do not behave similarly at frequencies below the first shaft translational mode. In this frequency range, the off line-of-action direction has a significantly larger force transmissibility as compared to the line-of-action direction. This occurs because in the line-of-action direction, the force input acts about a moment arm of radius, r, supported by a bearing that allows free rotation but resists translational motion. This introduces a zero into the line-of-action equation of motion (Eqn. 2.8). Also in. Figure 2.23 the line-of-action and off line-of-action shaft mode natural frequencies (=1500 Hz) are separated due to the presence of the line-of-action gear mesh stiffness and the different off line-of-action and line-of-action bearing stiffnesses. 4 9 system parameter units magnitudes km Ibf/in 2.9 X 10® kx Ibf/in 2.5 X 10® ky Ibf/in 1.6 X 10® Ibf-in-s^ 0.06 m Ibf-s^/in 0.019 r in 2.34 Table 2.7 System parameters used for the 3 DOF off line-of-action and line- of-action model 10 102 1 10 oloa ,0 10 1 E 10 •2 10 loa -5 10 ■€ 10 0 500 1000 1500 2000 2500 3000 3500 4000 frequency, Hz Figure 2.23 Force transmissibility comparison between the off line-of-action and the line-of-action directions using the 3 dof model 5 0 2.5.1.2 Dynamic Finite Element Model of the Back-to-Back Gear Tester The simple 3 DOF model representing one-half of a generic gear system is extended in this section. A combined linear time-invariant dynamic finite element formulation and lumped parameter representation of the back- to-back gear tester is developed. The axial rotation of the back-to-back tester is modeled using a 14 DOF lumped parameter model that has been taken from Matson[1995]. The on and off line-of-action translation and bending are modeled using dynamic finite elements. Figure 2.24 presents a schematic of the back-to-back gear tester model broken into the line-of- action motions, the torsional motion, and the off line-of-action motions. In Figure 2.24 the individual elements are defined by a ‘e’ preceding the number and the nodal DOF are represented by a number. 2.5.1.2.1 Rotational Lumped Parameter Model The rotational model consists of 14 degrees of freedom that is based on analysis by Matson[1995]. The translational components of the rotational model are removed, reducing the model to 11 degrees of freedom. The rotational model is coupled to the translational models developed using finite element methods. Complete development of the rotational lumped parameter model can be found in the masters thesis by Matson. Nodal degrees of freedom 57 through 67 are assigned to the lumped parameter rotational model (See Figure 2.24). 5 1 of Action TrGnstaxicn/Benai.nQ Dynamic Finite Element Mcoet *^px ^ ► el3 *10 elS *16 ► *17 *18? 29JO 31.02 3134 I 37.30 30MO < l« 2 , k.'ns "T- I ‘ -y- 3 I «y * 4^46 47^49 1 4 9 JC i 5 1 ^ S3jS4 s y 6 •2! *22 Rotational Lunpeo °arooete^ Mcdei I I 65 i j 64 I 'I 63 I P:r„cn iLU! L I—J L_J l— i i LI 11 S ^ a rt > ^on * -'c S i =e1_(t) =ei<7-t) C l rP kF i I' Gear ii^% !j 5 ij3 l!Lj 2iJ' Sha-t □ rf Line o f Action Transla'Cion/Benoi.'^g Dynanic finite Element Model ■ k py W py. ' "py hoy " * 5 * 6 - # # ♦ 13 1 4 5.6 , 9.10 1U2 1114 k * I iffs iTm ! : i i 6 i 7 .:a 19.20 ■ !X2< Jia s 27.20 .10 I > .1! rllb rg y sy, j_^9^ Measurement Slave Gearbox Gearbox Figure 2.24 Schematic of line-of-action dynamic FE model, rotational lumped parameter model, and the off line-of-action dynamic FE model. Arabic numeral alone defines the nodal degree-of-freedom and an ‘e’ followed by an Arabic numeral defines the element number. 5 2 2.5.1.2.2 Translational/Bending Finite Element Model The translation and bending motion of the back-to-back gear tester are modeled through the use of finite element techniques. As mentioned previously, the complete model consists of 3 decoupled systems. The 3 systems are the rotational model coupled to the line-of-action translation/bending model, the off line-of-action pinion side translational/bending model, and the off line-of-action gear side translational/bending model. Before the line-of-action translational/bending model is coupled to the rotational model, the off line-of-action pinion model is identical to the line-of-action pinion model, only representing motion perpendicular to the off line-of-action direction. Nodal degrees of freedom 1 through 28 are assigned to the off line-of-action models, with 1 through 14 assigned to the pinion shaft and 15 through 28 assigned to the gear shaft. Degrees of freedom 29 through 56 are assigned to the line-of-action models, with 29 through 42 assigned to the pinion shaft and 43 through 56 assigned to the gear shaft. Odd numbered degrees of freedom represent shaft translations and even numbers represent shaft bending (See Figure 2.24). The dynamic finite elements for the translation/bending components of the back-to-back tester model are modeled using a beam element for the stiffness matrix and a consistent mass matrix formulation [Meirovitch, 1986]. 5 3 and Eqn. 2.11 present the formulation for the mass and stiffness matrices for an individual beam element and Eqn. 2.13 presents the selected shape functions known as the Hermite cubics. Eqn. 2.10 [m]^ = Jm(x){L(x)}{L(x)}'’dx Eqn. 2.11 Wi Eqn. 2.12 8 i w i+1 i+1 r x f 1 -3 + 2 z ^x ' fx> X^ — 2 + h h h Eqn. 2.13 L(x) = V / z ( x l r x i + Vh h The individual mass and stiffness elements are assembled to form the complete mass and stiffness matrix. The assembled stiffness matrix represents a shaft floating in space whereas the actual shaft is attached to the ground through concentrated bearing stiffness terms. This is accomplished by adding the bearing stiffness to the appropriate diagonal 5 4 stiffness term as shown In Eqn. 2.14. This technique also is used to couple the pinion and gear shaft line-of-action translational and bending model to the rotational bending model through the gear mesh spring stiffness, as shown in Eqn. 2.15. The two equations implement reciprocity. Eqn. 2.14 ii =kj] ii +k ‘ ■'bearing kij =Kj+k,, Eqn. 2.15 k ji= k ;+ k ,. 2.5.1.2.3 Back-to-Back Tester Dynamic Finite Element/Lumped Parameter Model Eqn. 2.16, Eqn. 2.17 and Eqn. 2.18 represent the formulation for the rotation/line-of-action translation model, off line-of-action pinion shaft model and the off line-of-action gear shaft model, respectively. The damping matrix [C] is assumed to be 5 percent in the modal domain and is transformed to the physical coordinate system through the modal matrix computed from the mass and stiffness matrices. Eqn. 2.16 contains 39 degrees of freedom, Eqn. 2.17 contains 14 degrees of freedom and Eqn. 2.18 contains 14 degrees of freedom. The complete model consists of 3 sets of decoupled systems. The first system consists of the rotational model and the line-of- action translation/bending model coupled together through the stiffness 5 5 matrix via the gear mesh spring stiffness. The second and third systems consist of the off line-of-action translational/bending model for the pinion shaft and the gear shaft respectively. The equations are solved numerically in the frequency domain to determine force transmissibility functions between the gear mesh and the rolling element bearings. Eqn. 2.22 through Eqn. 2.24 define the force excitation vector for the rotation/line-of-action translation model, off line-of- action pinion shaft model and off line-of-action gear shaft model, respectively. Table 2.8 defines the element parameters used to formulate the finite elem ents for element 1 through 12. Elements 13 through 24 are identical to elements 1 through 12. Elements 1, 2 , 7 and 8 represent the measurement side gearbox and a step change in elements properties occur at the gear as shown in Table 2.8. Table 2.9 presents the system properties used in the dynamic finite element model. Similar, to the 3 degree-of- freedom model developed in section 2.5.1.1, the parameters represent a more detailed model of the back-to-back gear tester. Eqn. 2.16 Mlw^+[cl,w^+[K^ w^ =f^ Eqn. 2.17 |)vi]yp Wyp + w^^ + K L ^yp = fy yp Eqn. 2.18 [M^gW^g+ [C ]^w ^+ [K]^Wyg =fy vg Eqn. 2.19 w^ = [wgg, 830 , •••• Wgg, 8 gg, 857 , .... 8 gy 5 6 Eqn. 2 .2 0 =[w„ w , ,, Q,J Eqn. 2 .2 1 Wyg=[w,5, w„, Q^J ~ [^29 ^30 ^56 T” where, f - f '31 “ 'mm ^39 - ^ms ^45 ~ ~^mm Eqn. 2.22 fga =-L f — r f 57 'pm 'm m ^58 ~ ~*"gm^mm ^62 “ ~*"gs^ms ^63 ” ^ps^ms fj = 0 Otherwise Eqn. 2.23 fyp=[0 0 0 0 0 0 0 0 0 0 f„ 0 O f Eqn. 2 .2 4 fyg=(o 0 ^ 0 0 0 0 0 0 0 0 f,, 0 Of 5 7 element number moment of area mass per length element length units in^ Ibf s^/in^ inch 1 1.92 0.003, 0S(< 2.5 3.16 1 continued 30.7 0.014, 2.5 Table 2.8 Element properties used in the back-to-back dynamic finite element model system units magnitude parameter kox Ibf/in 2.5 X 10* kov Ibf/in 1.6 X 10* 2.5 lO*’ kgx Ibf/in X Ibf/in 1.6x10* kgy kmm Ibf/in 2.9 X 10* kms Ibf/in 9.0x10* inch 2.34 r . inch 4.41 *"gm inch 2.33 inch 4.38 Tgs Table 2.9 System parameters used in dynamic finite element model of back- to-back gear tester 5 8 Figure 2.25, Figure 2.26, and Figure 2.27 present force transmissibility results from the dynamic finite element/lumped parameter formulation. Figure 2.25 corresponds to the force transmissibility between the measurement side gear mesh and the measurement side pinion shaft bearing nearest to the slave gearbox. Figure 2.26 show the force transmissibility between the measurement side gear mesh and the measurement side pinion shaft bearing farthest from the slave gearbox. Similar to the 3 degree of freedom model presented earlier, the off line-of- action force transmissibility is significantly higher than the line-of-action force transmissibility for frequencies up to 1500 Hz and are of the same order of magnitude between 1500 Hz and 2500 Hz. Figure 2.27 presents force transmissibility results for the measurement side gearbox pinion shaft bearing nearest to the slave and an input excitation applied in the off line-of- action and line-of-action direction at the same bearing. Since the excitation is applied at the bearing a static reaction force is created for a static input. A similar result will be compared to experimental results gather at the same location in section 2.5.1.3. 5 9 10 oloa loa £1 0 u- 10 0 500 1000 1500 2000 2500 3000 3500 4000 frequency, Hz Figure 2.25 Force transmissibility between measurement gear mesh and off line-of-action (nodal DOF 5) and line-of-action (nodal DOF 33) pinion bearing, 5% assumed modal damping ratio 6 0 10 oloa 10 @10 \ A / l o a 10 10 10 500 1000 1500 2000 2500 3000 3500 4000 frequency, Hz Figure 2.26 Force transmissibility between measurement gear mesh and off line-of-action (nodal DOF 1) and line-of-action (nodal DOF 29) pinion bearing, 5% assum ed modal damping ratio 6 1 1 2.5 o> oloa 0.5 500 1000 1500 2000 2500 frequency, Hz Figure 2.27 Force transmissibility for off line-of-action and line-of-action directions for the pinion shaft bearing between the measurement side mesh and the slave gearbox (DOF 5 and 33), 5% assumed modal damping ratio 2.5.1.3 Experimental Inertance and Analytical Inertance Comparisons Analytically expressing the force transmissibility relationship between the gear mesh and the shaft support bearings is an easy task. Experimentally measuring the force transmissibility between the gear mesh and the bearing is difficult. Therefore, in this section, experimental inertance results from the back-to-back gear test rig are compared to the analytical inertance trends predicted by the dynamic finite element/lumped parameter model developed in Section 2.5.1.2.3. 62 The experiment consists of impulse testing near the pinion shaft bearing nearest the slave gearbox. This bearing location corresponds to the nodal degrees-of-freedom 5 and 33, as presented in Figure 2.24. Using a modal hammer, an impulse is applied in the line-of-action direction and off line-of-action direction. The corresponding acceleration in that same direction is measured using an accelerometer. Figure 2.28 present the inertance for a 5000 Ibf-in input preload torque cases. In Figure 2.28, the off line-of-action and line-of-action inertance demonstrates the separating natural frequency peak phenomena discussed in the previous section, particularly at 1500 Hz. The separated peaks at 1500 Hz in Figure 2.28 compare with the 3 DOF model results of Figure 2.23. Figure 2.29 shows the coherence computed for the off line-of-action test case presented in Figure 2.28. Figure 2.30 and Figure 2.31 present the comparison between the experimentally measured inertance and the inertance computed using the dynamic finite element/lumped parameter model, developed in Section 2.5.1.2. The results in Figure 2.30 and Figure 2.31 demonstrate a favorable trend between the experimental inertance measurement and the analytical inertance predictions based on the dynamic finite element/lumped parameter model. This favorable comparison between the experimental and analytical inertance taken with the driving point and response point at the pinion bearing brings credence to the results of the analytical model’s predictions between the gear mesh and the pinion bearing. 6 3 0 .3 — 500C Ibf-in loa 5000 Ibfnn oloa 0.2 - u 0.15 - - - J 0.05 - 0 500 1000 1500 2000 2500 frequency. Hz Figure 2.28 Experimental inertance measurement for off line-of-action and line-of-action test case with a input preload torque of 5000 Ibf-in 1.00 0.90 - - Coherence 0.80 - 0.70 - g 0.60 5I 0.50 ■ 0.40 0.30 0.20 0.10 0.00 0 500 1000 1500 2000 2500 Hz Figure 2.29 Coherence function for 5000 Ibf-in off line-of-action test case presented in Figure 2.28 6 4 0 .2 5 S 0.15 - 0.1 - - r ijf o I ii\P ' g 0 0.05 11- - - 0 üaSl 0 5001000 1500 2000 2500 frequency, Hz Figure 2.30 Inertance comparison in the line-of-action direction between experimental results and dynamic finite element model, 5% modal damping assumed for FEA model 0.25 5000 Ibf-in oloa 0.2 FEA. oloa I 0.15 S c g 0.1 0.05 ■ 0 500 1000 1500 2000 2500 frequency, Hz Figure 2.31 Inertance comparison in the off line-of-action direction between experimental results and dynamic finite element model, 5% modal damping assumed for FEA model 6 5 2.5.1.4 Section Conclusion The simple 3 degree of freedom analytical model and the dynamic finite element model of the back-to-back gear tester predict the force transmissibility between the gear mesh and the shaft support bearings is significant larger at frequencies lower than the first several shaft natural frequencies in the off line-of-action direction. This can be explained by the presence of a zero in the line-of-action response equations. This suggests that friction forces in the off line-of-action direction are readily transmitted to the bearing across a broad frequency range. These forces are then transmitted to the gearbox housing and ultimately result in radiated sound. Experimental inertance measurement near the gearbox bearing are compared to analytically determined inertance for the same gearing using the developed dynamic finite element model. 6 6 2.5.2 Energy Loss and Friction Force To Normal Mesh Force Ratio In this section, a friction force to normal mesh force ratio is determined as a function of energy loss efficiency. The derivation is independent of the friction force mechanism, assuming a generic friction force acting in the off line-of-action direction. This problem can be thought of as the inverse to the standard efficiency problem. Instead of calculating an efficiency, the efficiency is assumed and the required friction force is determined for that efficiency. The derivation will show that even for the high efficiencies found in the spur gearing, there can exist comparatively large friction forces. 2.5.2.1 Friction Force to Normal Force Ratio Derivation Figure 2.32 defines the important gear geometry for the friction force to normal force ratio computation. Important assumptions include point contact at the mesh, gear kinematics follows theoretical involute motion, gear mesh operates in a quasi-static condition, and a contact ratio of one is maintained. It is further assum ed that the limits of integration for the efficiency computation do not cross the pitch point. A simplification is the normal m esh force is constant throughout the computation. The friction force generating mechanism is not assumed and left as a generic force. 6 7 V c V Figure 2.32 Friction force to normal force ratio geometric assignments Eqn. 2.25 gives the static equilibrium conditions for the gear mesh in terms of the linear mesh deflection, D and the output torque, Tout- The linear mesh deflection is defined as D=rp6p+rg0g. Solving Eqn. 2.25 results in a solution for D, Eqn. 2.26, and Tout, Eqn. 2.27, in terms of the friction force, position along the line of action, and some gear geometry parameters. Defining the efficiency as the input energy minus output energy divided by the input energy, efficiency can be shown to be equal to Eqn. 2.28, noting that — = 0, implies no energy loss. Dividing Eqn. 2.27 by the input torque, Tn, results in Eqn. 2.29. The mesh force, fm, is the gear mesh stiffness times the linear mesh deflection, D. Substituting the linear mesh deflection into the mesh force equation results in Eqn. 2.30 This equation is written in terms of the input torque and a position varying term containing 6 8 the friction force. The mesh force which is assumed constant through the mesh cycle, is approximated as the input torque divided by the pinion gear base radius. Substituting the approximate mesh force into Eqn. 2.29 results in Eqn. 2.31. Eqn. 2.31 defines the input torque to output torque ratio as a constant plus a position varying term that contains the friction force to normal mesh force ratio. Substituting Eqn. 2.31 into the efficiency equation, Eqn. 2.28, and integrating results in Eqn. 2.32, the efficiency of the gear mesh is calculated over the integration limits, Si and Sg. Rearranging Eqn. 2.32 provides Eqn. 2.33, which computes the friction force to normal force ratio per unit efficiency. kmCp 0 [ D ] fTi„+(rp/tan((p) + s)f/ Eqn. 2.25 ^ToutJ 1 -(-rg/tan((p) + s)f, _ rp/tan(cp)+s\ Eqn. 2.26 D = ■ + Cpkm rpkm r Eqn. 2.27 =^Tin + s^^^f, Eqn. 2.28 - AE,„ r„ r„ + r„ p p 6 9 L = kn,D Eqn. 2.30 f, f =Ii!L " - r , Eqn. 2.31 = in *p f L^m / AE|o„- kA ,+ #:+ S ,)f f, 'I Eqn. 2.32 AE:„ 2r_ v^m y ft V AEL 2r„ Eqn. 2.33 AE V A loss 2.S.2.2 Friction Force to Normal Force Ratio Results The results of evaluating Eqn. 2.33 is plotted in Figure 2.33 in a normalized per unit efficiency form for a gear mesh with a pinion base radius of 2.5 inches and various gear ratios. Efficiency is defined a s energy in minus energy out divided by energy in, as shown in Eqn. 2.32. From Figure 2.33 it can be observed that as the contact point approaches the pitch point, Si+S2=0, the friction force required to generate a desired efficiency rapidly increases, approaching infinity at the pitch point. At the pitch point the gear mesh relative motion turns to pure rolling and under a point contact condition no energy is lost. By definition, the efficiency must be greater than or equal to zero to avoid the gear drive becoming a perpetual motion 7 0 machine. This constraint can be observed in Figure 2.33 by the sign change in the ratio of friction force to normal force. The sign of normal mesh force will not change as a function of position because the teeth in mesh must remain in contact. Therefore, the sign or direction of the friction force changes as the contact point crosses the pitch point. Figure 2.34 is an expansion of the Si+Sa < 0 plane of Figure 2.33. From Figure 2.34, the required friction force to achieve a defined efficiency decreases at an amount defined by the pinion’s base radius and gear’s base radius. This corresponds to the fact that as the point of mesh contact moves away from the pitch point, the sliding velocity increases which, for a constant friction force magnitude, increases the energy loss. Also, from Figure 2.34, for a given pinion base radius, the gear mesh efficiency decreases (increasing AE„ ■loss ratio) as a gear ratio increases. AEin Dudley’s Gear Handbook [Townsend, 1991] states that an approximate gear loss at the mesh for intemal and external spur gears is between 0.5 and 3 percent. From Figure 2.34, for a gear ratio of unity, the levels of efficiencies can result in friction force to normal m esh force ratios ,at S2-hsi= -0.5, ranging from 0.025 to 0.15. Figure 2.35 is a similar plot to Figure 2.34, except the gear ratio is set at a reduction ratio of — = 0.5 with the pinion base radius varied. 7 1 The results presented In this section show that for the high operating efficiencies found in spur gearing (low — ratios), that the off line-of- action friction force may be of sufficient magnitude as to represent a significant dynamic excitation source. 1000 ■= 0.25 750 -- •= 0.5 500 -- — = 2 250 -- -250 -- -500 - - -750 - - -1000 - 0.6 -0.4 - 0.2 0 0.2 0.4 O.f S, +Sj Figure 2.33 Friction force to normal mesh force ratio per unit efficiency, r„ = 2.5, various — rg 7 2 i. 4 0 30 - 25 -- = 4 4' 20 •• to -- 5 ■ — -0.5 -0.45 -0.35 -0.3 -0.2 -0.15 -0.1 s , f s. Figure 2.34 Friction force to normal mesh force ratio per unit efficiency, r„ = 2.5, various — 30 Tp = 0.25 25 •• Tp = 0.5 •p = 1.0 *9 s 1.5 20 •• ^»2 Î ^ 15 - - 10 - 5 -" -0.5-0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 S, -t-Sj Figure 2.35 Friction force to normal mesh force ratio per unit efficiency, —=0.5, various r„ 7 3 2.S.2.3 Section Conclusion This section computes a friction force to normal force ratio based upon an the energy efficiency at the gear mesh. This computation is independent of the mechanism causing the friction force and is based upon energy concepts. The results support a conclusion that the high efficiencies found in involute spur gearing does not necessarily imply low friction forces. Assuming a 3 percent energy loss in the mesh, the friction force to normal mesh force ratio for a unity gear ratio pair can be 15 percent, away from the pitch point. The friction force to normal force ratio increases as the gear mesh contact point approaches the pitch point. The friction force, which changes sign at the pitch point, therefore can represent a significant excitation source. 2.6 Chapter Conclusion This chapter presented evidence in several forms supporting the idea that the friction force in the off line-of-action direction can be a potential and possibly significant source of dynamic excitation. The evidence comes in several forms including the presentation of experimental results from several sources and the results of analytical models. Experimental results by Umezawa [1996], measurements taken on the Gear Noise Test Rig at NASA Lewis Research Center, and measurements gathered on the Back-to-Back Gear Tester at the Gear Dynamic and Gear Noise Research Laboratory show large motions in the off 7 4 il line-of-action direction. The amplitudes of these motions are of the same order of magnitude as or larger than measured motions in the line-of-action direction near shaft support bearings. The measured parameter of displacement is related to force through the bearing stiffness. This measurement indirectly supports the large friction force hypothesis. For example, reduced bearing stiffness in the off line-of-action direction could account for large displacements with small off line-of-action forces. Therefore, the next sections examine in detail possible causes for large off line-of-action motion that result from the shaft support bearing. The examination of the bearings was based upon the research by Lim[1989] in ball and roller bearing elements. Topics examined included bearing cross coupling, bearing reduced stiffness in the off line-of-action direction and bearing radial clearance. These potential sources of off line-of- action motion were eliminated as general causes of the large off line-of- action motion. Lastly, two different analytical concepts which support the concept of friction force as a dynamic excitation in the gear mesh, were developed. The first concept involves two dynamic models of the gear system in the off line- of-action and line-of-action directions. The first model is a simplified model representing one-half of the gear mesh. The other model is a dynamic finite element model of the back-to-back gear tester. The results of these models suggest that the force transmissibility between the gear mesh and the 7 5 bearings in the off line-of-action is significantly larger than the force transmissibility in the line-of-action direction. The other concept examines the ratio of off line-of-action friction force to the normal mesh force for a given energy efficiency. The results show that for even the high efficiencies typically found in involute spur gearing, there can exist large off line-of- action friction forces. 7 6 Chapter 3 The Experimental Measurement of the Off Line-of-Actlon Friction Forces in the Gear Mesh of Parallel Axis Involute Spur Gears 3.1 Introduction The friction force in the gear mesh has been examined as a potential dynamic excitation source in Chapter 2. This chapter introduces a different experimental measurement technique for the determination of the instantaneous friction force at the gear mesh for the individual teeth on an involute spur gear pair having a contact ratio of less than or equal to two. The measurement of the instantaneous friction force at each tooth may potentially lead to a greater understanding of the friction force mechanism under various gear geometries, operating conditions and lubricants. This greater understanding can be employed to minimize the effect of friction force excitation at the source. 7 7 The measurement of the friction force experimentally can be divided into two general measuring technique categories and two operating conditions. The first group of techniques measure away from the gear mesh and the second group takes measurements near the gear mesh. The two operating conditions relate to high and low speed running conditions. At low speeds the lubricating regime is likely boundary or mixed lubrication while at high speeds a elastohydrodynamic lubrication regime likely exists. The results of the measurements can then be grouped into two categories, time or spatially changing information and mean information. Time varying information can be expressed as a time varying friction coefficient while mean data can be expressed as an average friction coefficient over some mesh cycles. The various methods used by previous researchers for these types of measurements are further examined in the literature review of this chapter. This chapter introduces a measurement methodology capable of measuring the instantaneous friction force on the individual tooth pairs in contact for an involute spur gear pair with a contact ratio between one and two. 7 8 3.2 Literature Review Radzimovsky [1973,1974] and Mirarefi [1974] determined a friction coefficient a s a function of roll angle based on a coulomb friction model of a spur gear involute mesh. The speeds were low in order to maintain a boundary or extreme boundary lubrication regime. The gear mesh efficiency was computed from the power loss as m easured from the input torque to a back-to-back test stand. The losses due to the bearings were determined and the remaining power loss was attributed to gear mesh losses. Approximating the path of the contact point over a small time interval and using the coulomb friction model, the calculated energy loss was equated to the m easured energy loss, to determine an "apparent" coefficient of friction. This coefficient of friction contained the effects of sliding resistance and rolling resistance. The gear pairs examined approached a contact ratio of one but during two tooth pair contact, the friction coefficient combined the effects of both teeth into an equivalent value, from the power loss point of view. Yada[1972] experimentally determined the gear losses of a spur gear pair at the mesh by monitoring the temperature of the oil bath. Losses due to churning and bearings were examined. A gear loss coefficient, similar to efficiency, was introduced. The gear loss coefficient was computed over a long period of time representative of many mesh cycles. Therefore, this 7 9 procedure does not represent what is happening at the mesh from the instantaneous point of view. Yada concluded that speed, followed by machining method influenced efficiency significantly, whereas torque had only secondary effects on gear pair efficiency. Rebbechi[1991,1996] took strain gage measurements in the tooth roots to compute the normal and tangential forces at the mesh interface of an involute spur gear pair. Gages were mounted in the compressive and tensile side on each tooth for the maximum number of teeth in contact. Using a tooth force influence coefficient matrix concept relating tooth strain to gear mesh force, the normal and tangential tooth loads can be compute instantaneously for each particular tooth in contact. The 1991 paper presented the friction force on a normalized scale, while the 1996 paper presented the friction force in physical units. The forces were examined in both the static and dynamic senses. Close agreement occurred for the normal load in the static case and good agreement for the normal load in the dynamic case with theoretical normal load values. The tangential loads showed trends such as direction change at the pitch point. A typical result of Rebbechi’s is presented in Figure 3.1. 8 0 Pitch Pt Pitch Pt 445 tooth 1 tooth 2 3550 -1 I z o Vi o Maximum static normaE force — % U 2660 - -445 o 63 fZ 1 7 7 0 - o Z 890 - r T ~r T T i 30 22 14 10 1“ I I I —r "I 30 26 22 18 14 10 Roll angle, deg Figure 3.1 Normal and friction forces via strain gage m easurements [Rebbechi,1996] 8 1 Benedict [1961] performed experiments using a disk machine to m easure the friction coefficient. The machine consists of two disks with one disk rotating slower such that both rolling and sliding velocity can be achieved. Benedict found the friction coefficient to increase with load and decrease with rolling velocity, sliding velocity, and oil viscosity, respectively. An empirical formula for the coefficient of friction as a function of viscosity, sliding velocity, rolling velocity, and normal load was presented. O’Donoghue [1966] developed an empirical relationship between the friction coefficient and the parameters: speed, viscosity and surface finish. The friction force is assum ed proportional to the normal load with the friction coefficient being the proportionality constant. The study was performed on an Amsler machine which is a disk machine to simulate the rolling and sliding velocity found in gear contact. Film thickness was measured via a voltage discharge device, implying a hydrodynamic or elastohydrodynamic lubrication condition. Ishida [1967] experimentally examined the noise generated by sliding motion through a two disk machine simulating the approach and recess action during the gear mesh cycle. The measurement conditions included boundary lubrication regime and low operating speeds, such as to maintain surface contact of the disks. The noise was classified into surface asperity generated friction noise and pitch circle impulse noise caused by the change 8 2 in friction direction at the pitch point. Ishida concluded that friction noise that is transmitted mainly as sound conducted, can account for 5 to 15 percent of the total gear noise. 3.3 New Measurement Methodology Introduction Currently, the involute curve for spur and helical gear tooth profiles is used almost exclusively in power transmission. The involute has many properties that facilitate its use as a gear tooth profile. One involute-curve property is for a given center distance between involute gears, the common tangent to the base circles form the line-of-action. Theoretical tooth contact always occurs along this line-of-action for both single and multiple tooth contact [Buckingham, 1963]. The unit normal at the points of contact is always parallel to the line-of-action so that the unit vector tangent to the point of contact is perpendicular to the line-of-action. The line-of-action is known from kinematic parameters, therefore, the theoretical directions of the friction force and normal force are known. The tooth reaction forces are transferred through the bearings to the housing. By monitoring the forces transmitted through the bearings, the net sum of forces acting along the line-of-action and the off line-of-action direction at the gear mesh should be computable. For a gear with a contact ratio of unity, the net force is equal to the vector sum of the normal and tangential force at the mesh but for a contact ratio greater than one and up 8 3 to two, during a portion of the engagement cycle, two teeth will be in contact and the force measured at the bearings will be the sum of these forces. The mean normal force will remain approximately constant since the direction the normal force acts does not change during the engagement cycle. In the friction force direction, the direction changes depending upon the position in the mesh cycle, with the individual tooth friction forces adding or subtracting to create the net friction force. Since the m easurem ent at the bearings measures only the net force, additional information will be required to break the net friction force into its individual components during two tooth engagements. This additional information can be gathered by monitoring the power loss of the gear mesh. 3.3.1 New Measurement Method Theory The total friction force at the mesh in the off line-of-action direction is the sum of all friction forces acting on the individual teeth at any position of contact. For a spur gear pair with a contact ratio less than or equal to two, the net friction force is the sum of the friction force acting on the individual tooth pairs as represented in Eqn. 3.1. Assuming point or line contact at the gear mesh, the power loss due to the sliding of the teeth in the mesh cycle can be written as Eqn. 3.2. Writing Eqn. 3.1 and Eqn. 3.2 in matrix form results in Eqn. 3.3 and rearranging results in Eqn. 3.4. Eqn. 3.4 suggests. 8 4 that by measuring the net friction force at the gear mesh and the instantaneous power lost at the gear mesh, the individual friction forces on the two teeth can be reconstructed. Eqn. 3.1 f, =fi+fz V, Eqn. 3.1a fi = i=1,2 Eqn. 3.2 ■ 1 1 Eqn. 3.3 f, mesh. V , V , 1 Eqn. 3.4 Va -1 f, [fsj (V z -V j .-V, 1. m esh . The transformation of equation Eqn. 3.3 to Eqn. 3.4 depends upon inverting the matrix in Eqn. 3.3. Eqn. 3.5 takes the determinant of the matrix from Eqn. 3.4. Noting, from Eqn. 3.5, the determinant is simply the difference between the relative sliding velocities of tooth pairs in contact. It can be shown, that the matrix in Eqn. 3.3 is linearly independent for all single or double tooth contact combinations except for a single case. For single tooth contact at the theoretical pitch point, where the sliding velocities, Vi=0 and V2=0, the matrix of Eqn. 3.3 is singular. For single tooth pair contact at the pitch point, Eqn. 3.1 which is reduced f, =f,, can be used to find the friction force. 8 5 Eqn. 3.4 is derived under the following assumptions: first the gear mesh contact is assumed to be point or line contact and secondly, the gear pair follows the theoretical kinematics for the involute tooth geometry. The first assumption introduces an indeterminate solution at the pitch point, where pure rolling exists. Therefore at this point any friction force present does not produced any lost power. -1 Eqn. 3.5 y, = V2-V, #0 -V, 1 3.3.2 Numerical Test of Friction Force Measurement Scheme This section numerically tests for two spur gear cases the friction force measurement scheme presented earlier in this chapter. The first case is for a constant friction coefficient of n=0.05 and the second for a position varying p.(s)=0.4s^, where s is the position along the line of action. The position varying |i(s) is plotted in Figure 3.2. The test case has a contact ratio of 1.41 and during two tooth pair contact, the load sharing is assumed equal. Table 3.1 presents the relevant gear parameters for this test case. 86 0 .0 8 0.07 -- 0.06 -- 0.05 4 0.04 •c 0.03 f 0.02 -0.43 -0.33 -0.13 0.07 0.17 0.27-0.03 0.37 s, position along line-of-action Figure 3.2 Plot of the friction coefficient, |i, versus position along line-of- action, |i=0.4s^ param eter value pinion base radius, inch 2.25 gear base radius, inch 4.12 face width, inch 1.25 base pitch, bp, inch 0.59 lowest point of tooth contact, Za, inch -0.429 lowest point of single tooth contact, LPSTC, inch -0.183 highest point of tooth contact, Zb, inch 0.407 input torgue, Ibf-in 1 input angular velocity, rad/s 1 nominal single tooth contact normal tooth force, Ibf 0.444 Table 3.1 Relevant gears parameters for numerical test 8 7 Figure 3.3, Figure 3.4 and Figure 3.5 present results for the test case |i=0.05. Figure 3.3 shows the net mesh power loss and the net friction force. The net mesh power loss and the net friction force represent the measured experimental parameters. From Figure 3.3, the percent difference between the peak-to-peak friction force to nominal normal mesh force is 9.9%. Figure 3.4 and Figure 3.5 present a comparison between the analytically computed tooth friction forces and the tooth friction forces computed from the net friction force and the net mesh power loss. The results from Figure 3.4 and Figure 3.5 show virtually identical agreement. Figure 3.6, Figure 3.7 and Figure 3.8 present the same results as Figure 3.3 through Figure 3.5 respectively, for the case of p=0.4s^. From Figure 3.6, the peak-to-peak friction force is approximately 5.4% of the nominal normal mesh force. Again, Figure 3.7 and Figure 3.8 show virtually identical results between the computed individual tooth friction forces and the inverted friction forces computed from the analytically obtained net friction force and net mesh power loss. These results show, that if the net mesh instantaneous power loss and the net friction force are known along the line-of-action, the individual tooth friction forces can be computed. 8 8 Ibf-in/s —o— fj , Ibf o% —I I 0.01 s. cO _ ? i -02 - 0.1 0.1 02 0.3 0.4 LL C O - 0.01 Z Li. - 0.02 — Sngle Tooth Pair Contact Two Tooth Pair Contact -0.03 i s, position along line-of-action Figure 3.3 Analytically computed instantaneous power loss and net friction force, ff, |i=0.05, Tm=1 Ibf-in, Q,n=1 rad/s 89 Calculated Friction Forces. Tin= 1 lbf*in. Qins 1 rad/s 0.03 T ■fl c a lc . Ibf 0.02 f2 calc. Ibf 0.01 •0.1 0.1 0.2 0.3 0.4 •0.01 - 0.02 •0.03 s. position along line-of-action Figure 3.4 Analytically computed individual tooth forces, |i=0.05, Tin=1 Ibf-in, Qin=1 rad/s Inverted Friction Forces. Tin= t Ibf-in. Qin= t rad/s 0.03 - -It invert. Ibf 0.02 12 invert. Ibf O.Ot S S. i - £O -0 .2 -O.t 0.1 0.2 0.3 0 . 4 -O.Ot -0.02 - -0.03 - s. position along line-of-action Figure 3.5 Inverted individual tooth forces from Pioss and ff, ji=0.05, Tin=1 Ibf-in, Qin=1 rad/s 9 0 0.015 . Ibf-in/s 0.01 I 0.005 - O0I 9 % - 0.2 - 0.1 0.2 0.4 LL 0C 1 -0.005 - -0.01 T -0.015 -L position along line-of-action Figure 3.6 Analytically computed power loss and net friction force, ff, |i= 0.4s^, Tin=1 Ibf-in, % =1 rad/s 9 1 Calculated Fnction Forces. Tin= 1 Ibf-in. Qin= 1 rad/s 0.02 M calc, ibf 0.015 12 c a lc , ibl 0.01 s £ 0.005 -L I •0.2 - 0.1 0.2 0.3 0.4 -0.005 - •0.01 •0.015 - s. position along line-of-action Figure 3.7 Analytically computed Individual tooth forces , |i=0.4s^, Tin=1 Ibf-in, C2in=1 rad/s Inverted Fnction Forces. Tin= 1 Ibf-in. Qin= 1 rad/s 0.02 - 0.015 0.01 I 0.005 - -0.1 0.2 0.3 0.4 -0.005 - -0.01 -0.015 - s. position along line-of-action Figure 3.8 Inverted individual tooth forces from Pioss and ft, |i=0.4s^, Tin=1 Ibf-in, Qin=1 rad/s 9 2 3.3.3 Error Exammation of Measurement Methodology This section will examine the bearing stiffness effect on the computed individual friction tooth forces assuming the net friction force in the off line- of-action direction is computed from a measured bearing deflection and bearing stiffness, as represented in Eqn. 3.6. In Eqn. 3.6, kcomputed is the determined off line-of-action bearing stiffness which is a combination of the actual bearing stiffness, k, and any error, Ak, in the bearing stiffness determination. Substituting Eqn. 3.6 into Eqn. 3.4 results in Eqn. 3.7. Expanding Eqn. 3.7 into the individual tooth friction forces results in Eqn. 3.8 and Eqn. 3.9. During single tooth pair contact the sliding velocity conditions are: Vi=V and V2=0. For tooth pair 1 under single pair contact, the computed friction force is independent of the assumed bearing stiffness as can be seen by substituting the sliding velocity conditions into Eqn. 3.8. During single tooth pair contact the friction force on tooth pair 2 which must be equal to zero since only the first pair is contact. Eqn. 3.9 can produce a check of the experimental measurement since during single tooth pair contact, f 2=0 . During double tooth pair contact, an error proportional to deviation from the actual off line-of-action bearing stiffness is introduced into each friction force term. Depending on whether the off line-of-action bearing 9 3 stiffness Is over or under estimated, one friction force term is overestimated and the other tooth pair is underestimated since the sum of the individual friction force terms must equal to the net friction force, Eqn. 3.1. Eqn. 3.6 f, = = (k + Ak)y 1 'V , -1" 'Va -1" Aky] Eqn. 3.7 I ky 1+ 1 (V g-V J -V , 1. [Pmeshl (V g-V J -V, 1. 0 I yVz mesh AkY V, Eqn. 3.8 f, =k / V^-V, k(v,-vj\k J Eqn. 3.9 f = k -V lV , Pmesh V, Va-V, k(Vg-Vj AVa-V, 3.4 Experimental Setup This section describes the experimental setup used to implement the individual friction force measurement experiment presented in the earlier sections of this chapter. The experiment was performed on the Ohio State University Gear Noise Gear Dynamics Back-to-Back Gear Test Rig, shown schematically in Figure 3.9. The experiment is broken up into several measurement groups: 1) gear mesh powerless measurement, 2) net friction force measurement, and 3) absolute position measurement. Figure 3.10 presents a schematic of the instrumented Back-to-Back gear test rig test side gear box. 9 4 Drive Motor C o m m e rc ia l D riv e T o rq u e B elt — .S e n s o r Pinion Shaft G e a r S h a ft T e s t G e a r Actuator Slave Gear B ox B ox Figure 3.9 Schematic of back-to-back gear tester presenting location of commercial torque sensor Figure 3.10 Schematic of the Instrumented back-to-back gear test rig test side gear box 9 5 3.4.1 Powerless Measurement The net power loss in a gearbox can be grouped Into four contributing losses, Eqn. 3.10. These losses are the mesh losses, bearing losses, lubricant churning and slinging losses, and gear windage losses [Townsend, 1991]. For friction force calculations, the power loss at the mesh, which typically must be extracted from the measured net power loss. Is of Interest. Eqn. 3.10 =Pn,esh +Pbearing +P|ub + Pondage = Tn^in +T^ut^out The net power loss In the gearbox Includes losses that can be attributed to the bearings. Radzlmovsky [1973] strain gauged the bearing elements to calculate the bearing power loss component. Bearing losses may also be eliminated by measuring the torque on either side of the respective gears between the gear and the bearing with the setup presented In Figure 3.11. Eqn. 3.10 represents the net power loss In the gear box, noting that îîout Is the opposite sign of ^n. Eqn. 3.11 computes the gearbox losses excluding the support bearings. In equation Eqn. 3.11, the sign of T2 is opposite the sign of T1, reducing the potential transmitted power. Also, T4 Is the opposite sign of T3 but Is subtracted since the goal Is the power loss at the mesh. The power loss component due to the resisting torque, T4 has been transmitted by the mesh, therefore It Is not considered a mesh loss. 9 6 A;in T 2 T1 T:, in To Slave G earbox -*• ^out T 4 T 3 out |T1 I > |T 2| |T3| > |T4l Figure 3.11 Power loss measurement excluding bearing losses 9 7 Eqn. 3.11 Pmesh + P,ub + Pwindage = (T1 + T2)Qi„ + (T3 - T The measurement of the power losses such as churning and slinging due only to lubrication away from the mesh is extremely difficult. Likewise, the measurement of the power loss due only to gear windage is also extremely difficult. It m ay be best to minimize and neglect the power losses due to the lubricant and gear windage, rather than attempt to measure them. The gear windage power loss is most prevalent with high speed gears that operate with pitch line velocities of 10,000 ft/min [Townsend, 1991]. The gear windage losses, if applicable, can be estimated from works similar to Dawson [1984]. 3.4.1.1 Torque Measurement The back-to-back test rig schematic of Figure 3.9 shows a commercially installed torque sensor, Eaton/Lebow 2048-20K, mounted between the test and slave gearbox on the pinion shaft. Unfortunately, this placement is not ideal for measuring pinion torque for computing power loss at the gear mesh. Therefore, an alternative torque measuring scheme must be used with the constraint of minimal test rig modification. This constraint was imposed for two reasons, first the friction force measuring technique is unproven in practical application, so major test rig modifications were undesirable. Secondly the back-to-back test rig is a general research rig, so 9 8 modifying the test stand specifically for friction force measurements was not practical. Based upon the imposed constraint, an ideal solution to measure the torque at each location is through the use of strain gages. At the maximum 5000 Ibf-in pinion side mean torque capacity of the back-to-back test rig, the principal strains on the pinion shaft and gear shaft are of the order of 50 microstrain. Since Eqn. 3.12 requires the measurement both the mean torque and the time varying components, special attention must used in strain gage selection. To get the maximum signal output, a Wheatstone Bridge configuration with 4 active gages was selected [Measurement Group, 1982]. For a pure shear stress loading condition, the sensitivity for a circular shaft can be computed using Eqn. 3.13. The static sensitivity, 8, is defined as the output voltage per volt excitation per Ibf-in. Commonly used strain gages are of a foil design with a nominal gage factor, G, of two. Using foil design strain gages would result in static sensitivities of approximately 3x10*® mV/(V Ibf-in) and 2x1 O'® mV/(V Ibf-in) for the pinion shaft and gear shaft respectively. This is extremely low when compared to the commercial torque sensor (Eaton/Lebow Model 1248-20K) installed on the back-to-back tester that has a static sensitivity of 1x10^ mV/(V Ibf-in). An alternative choice is to use semi-conductor strain gages. Semi-conductor strain gages have a gage factor of the order of 150. With these gages, static sensitivities of approximately 2x10*® mV/(V Ibf-in) can be achieve for the pinion and gear shaft, respectively. The semi 99 i conductor strain gage’s static sensitivity is approximately 65 times greater than the foil gage setup and 20 times greater than the commercial unit. Based upon the static sensitivity, the semi-conductor strain gage should be the obvious choice, but semi-conductor strain gages are temperature sensitive with the gage factor and resistance changing on the order of 10% per 100 degree Celsius change [Micron Catalog Specifications]. Eqn. 3.12 T,(t) = Tr"+YT'(t) k = 1,2,3.4 i Eqn. 3.13 S = ^^= ^0 + ^) V«T Tcd^ E For the back-to-back test rig application, the change in operating temperature is small for the short test periods, and deemed insignificant. Another temperature consideration, is the local heating of the strain gage due to the excitation current. To overcome this problem, the gages are allowed to preheat to a steady state temperature prior to any calibration or testing. The semi-conductors gages are deemed, “stabilized” when the D.C. signal drift ceases, allowing calibration and testing. This procedure does force excitation specific calibration and operation. Ultimately, the semi-conductor strain gage was selected based upon its superior static sensitivity. The selected gages, SS-060-033-500P-S4, were supplied and mounted by Micron Instrument in Simi Valley, California. This strain gage is a bar design, with a 540 ohm nominal resistance and 140 nominal gage factor and is supplied in a matched set of four. 1 0 0 The strain gage bridges are calibrated using a static moment arm technique with the calibration results presented for two bridges in Figure 3.12 and Figure 3.13. The resulting static sensitivities are 2.1x10’^ mV/(V Ibf- in) and 9.6x10"^ mV/(V Ibf-in) for the pinion shaft bridge and the gear shaft bridge, respectively. Figure 3.14 and Figure 3.15 compares the measured torque signal content from the commercial torque sensor and the constructed semi-conductor torque sensor, respectively for an nominal input torque preload of 5000 Ibf-in. The commercial torque sensor location shown in Figure 3.9 corresponds to the torque measurement point Tin presented in Figure 3.11 while the constructed torque sensor corresponds to location T1. The two sensors show reasonable agreement in content, but because the sensors are not at the same location and are separated by a pinion bearing, it is difficult to establish whether the cause of the differences is due to the sensor construction or due to sensor location. Figure 3.16 compares the sensitivity due to the sign change of the normal bending stress due to shaft rotation, between the commercial and constructed unit. At shaft speed, Qin=1080 rpm and fshaft= 18 Hz, the constructed unit output is an order of magnitude larger than the commercial unit. This is likely related to the strain gauge misalignment and the location of the commercial and constructed torque sensors. 1 0 1 1000 2000 3000 4000 5000 6000 7000 - 0.2 PA-T2 PA-T3 -0.3 PA-T4 PA-T5 4 -0.5 -0.7 •0.8 -0.9 Torque. Itwn Figure 3.12 Calibration curve for the semi-conductor strain gage bridge mounted on pinion shaft, excitation voltage=7 V, output gain=11 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0.2 -0.2 GA-T2 GA-T3 GA-T5 -0.8 r - 1.2 Torque. Ib-in Figure 3.13 Calibration curve for the semi-conductor strain gage bridge mounted on gear shaft, excitation voltage=10 V, output gain=11 1 0 2 COtrmenjal. Ibt-m œ 25 - ,2 20 - 100 300 500 900 1100 1300 1500 Frequency, Hz Figure 3.14 Dynamic torque signal of commercial torque sensing unit, Tin=5000 Ibf-in, Q,n=1080 rpm Constructea Ibf-in 40 r -£ 30 m 25 - g 20 100 300 500 900700 1100 1300 1500 Frequency, Hz Figure 3.15 Dynamic torque signal constructed torque sensing bridge, Tin=5000 Ibf-in, ^ = 1 0 8 0 rpm 1 0 3 Compariaan Batwaan Commarical Torqua Sanaor and Conatructad Torqua Sanaor 160 -Commercial. Ibf-in 140 -Conairucied. Ibt-m 120 100 + o 80 40 20 0 A 10 20 30 40 50 60 70 80 90 100 Frequency. Hz Figure 3.16 Comparison of dynamic torque signal between commercial torque sensing unit and constructed torque sensing bridge, Tin=5000 Ibf-in, Qin=1080 rpm 3.4.1.2 Angular Velocity Measurement To get the power loss measurement, besides the torque measurement, the instantaneous angular velocities of both the pinion and gear shaft are required. The instantaneous angular velocity can be divided into the mean angular velocity and the time varying harmonic terms represented by Eqn. 3.14. Eqn. 3.14 1 0 4 3.4.1.2.1 Mean Angular Velocity Measurement The mean angular velocity of the pinion shaft is measured through the use of a once per revolution trigger. The experimental data is recorded on a Sony PC208A digital tape recorder with a sampling rate of 12,000 samples/second. Eqn. 3.15 is the formula used to compute the mean input angular velocity, in radians per second, where N is the number of counted once per revolution triggers and K is the total number of samples between the first and last trigger. The pinion and gear are mechanically interlocked, so the gear mean angular velocity and may be compute by the ratio of the pinion base radius to the gear base radius as shown in Eqn. 3.16. Eqn. 3.15 =271^-°^—j Eqn. 3.16 ""g 3.4.1.2.2 Time-Varying Angular Velocity Measurement The time varying angular velocity component is a small perturbation about a shaft's respective mean angular velocity and is caused by elastic deformations of the teeth in contact, tooth profile modifications, manufacturing errors and small variations in the input angular velocity. The perturbations are not generally related by the gear ratio and must be measured independently. 1 0 5 The method selected to measure the time varying angular velocity uses translational accelerometers mounted tangentially about the shaft’s axis of rotation at some radial distance. The measurement scheme is displayed in Figure 3.17. The scheme measures the time varying rotational acceleration which must be integrated to obtain time varying angular velocity. Eqn. 3.17 shows the functional relationship between the measured translational acceleration and the rotational angular velocity. Additional reference material on this measurement technique can be found in Matson [1995], Bolze [1996], and Liauwnardi [1997]. Eqn. 3.17 Figure 3.17 Time varying angular velocity measurement scheme 1 0 6 ll 3.4.1.3 Power Flow Computation To compute the power loss at the gear mesh, the power flow through the pinion shaft and gear shaft must be computed. Eqn. 3.18 and Eqn. 3.19 describe the instantaneous angular velocity and the instantaneous torque in terms of a mean component and time varying components. The time varying components are the Fourier harmonics of the angular velocity and torque. The power flow is the product of the instantaneous angular velocity and the instantaneous torque as depicted in Eqn. 3.20. Substituting Eqn. 3.18 and Eqn. 3.19 into Eqn. 3.20 results in Eqn. 3.21. This equation for power presents the idea that the power flow is the sum of the products of the mean values, the product of mean values and the Fourier harmonic terms, and products of the Fourier harmonic terms. Noting the Fourier harmonic terms are small perturbations about the mean value, the product of the harmonic terms is of a higher order (H.O.T.) and is neglected resulting in Eqn. 3.22. This assumption simplifies the power flow computation by eliminating the negligible convolution component from the computation and is valid in both the time and frequency domains. Eqn. 3.18 £l(t) (0 i Eqn. 3.19 T(t) = T " * " + X T (t) 1 0 7 Eqn. 3.20 P(t) = T(t)A(t) Eqn. 3.21 P(t) = I ' Eqn. 3.22 P(t) = +Q"’"^"£Tj +KO.T. > i 3.4.2 Force Measurement In addition to the power loss measurement, another ingredient is the net friction force in the gear mesh. The direct measurement of the net friction force is extremely difficult, therefore, an indirect measurement is performed which is related to the net friction force in the gear mesh. For this experiment, the force transmitted through the shaft support bearings is measured and related to the net friction force through the gearbox geometry and system dynamics. The transmitted force through the bearing is related to the bearing displacement and the bearing stiffness. The gearbox geometry and dynamic effects are modeled through a dynamic finite element model of the back-to-back test rig. 3.4.2.1 Displacement Measurement The displacement of the bearing in the off line-of-action direction is measured through the use of eddy current non-contact displacement probes. The probes measure the shaft displacement near the bearing in the plane perpendicular to the shaft’s axis of rotation. The probes used in this 1 0 8 research are manufactured by Electro Corporation, probe model 4937. The probe is calibrated to a nominal sensitivity of 400 V/inch which provides a 0.0025 inch operating range. The probe is capable of measuring both the static and dynamic component of shaft displacement. Capacitance style non-contact displacements probes which have potentially higher nominal sensitivities (=2000 V/inch) were also used in the early phases of this research. They were discarded because of the large changes in sensitivity related to the surface curvature of the shafts. The measurement of the shaft displacement, rather than the bearing deflection directly, introduces some potential errors such as shaft eccentricity and local material conductivity variation [Electro Corp., 1993]. The errors will mostly be restricted to once per revolution and harmonics. These errors can be minimized by measuring the shaft rotation under an unloaded condition and subtracting these results from the loaded m easurem ent [Hochmann,1992]. Once the shaft deflection in the off line-of- action direction is known, the force can be determined through the application of the bearing stiffness matrix. S.4.2.2 Bearing Stiffness Matrix The bearing force is related to the bearing deflection through the bearing stiffness matrix. A general linear time invariant 6 degree-of-freedom matrix model developed by Lim[1989] is presented in general form in Eqn. 3.23. Lim concluded that there are certain “dominant” stiffness terms in Eqn. 1 09 3.23 and these were discussed in Chapter 2. Since the main goal here is the bearing force in the off line-of-action, that equation may be extracted from the generalized bearing stiffness formulation. The off line-of-action bearing force equation with only the dominant stiffness terms included is presented in Eqn. 3.24,. The Y axis is parallel to the off line-of-action, the X axis is parallel to the line-of-action direction and Z axis is parallel to the axis of rotation. fx ' ''X.X "X.Y "X.Z 'X .0X ■'X.0Y r x i Y fY ^Y.X "Y.Y "Y.Z Eqn. 3.24 fv — fgioa — k vv'Y.Y Y + kvYZ' yZ + ky a"Y.8.'^x 0 In this research, the gear test pair is a spur gear design, the bearing translation along the shaft axis of rotation can be neglected. Measuring 8% can be approximated through a finite difference approach by measuring the line-of-action displacement on both sides of the respective bearing. For this research, 6%, is neglected and left to be studies in future research, therefore reducing Eqn. 3.24 to Eqn. 3.25. Once the bearing force in the off line-of- 1 1 0 action direction is known, the force must be related to the net friction force in the gear mesh. This may be accomplished through the analysis of the gearbox geometry and system dynamics. Eqn. 3.25 fmoa — ^ y .y '^ S.4.2.3 System Dynamic and Gearbox Geometry Effects The measurement point near the bearing is located away from the contact region of the gear mesh. The gear mesh forces are transmitted from the contact point or line through the gear teeth, the gear blank, the gear shaft and finally through the bearing to the gearbox housing to ground. As the speed and excitation frequencies increase, the transmitted force is modified by the system dynamics through amplitude attenuation and phase changes. The system dynamics matrix can be formulated either analytically or experimentally. An analytically developed system matrix would consider dynamic effects from the gear tooth to the bearing, but the mechanical parameters, such as stiffness and damping, must be estimated. The uncertainties in the mechanical parameters and the analytical models are always considerations when applying inverse solving techniques. An experimentally formulated system dynamics matrix, based on system 1 1 1 identification techniques inherently accounts for the questionable mechanical parameters. When the main interest is the forces in the gear mesh, an ideal experiment would place the excitation point at the gear mesh and measure the response at the bearing. A potentially close approximation would be to place the excitation point on the gear shaft next to the gear body and then measure the response at the bearings. Matson [1995] took the latter approach in his modal examination of the back-to-back gear test stand. A unique feature of the back-to-back tester is that preloading the gear teeth is inherent to the operation of the test rig. Once the bearing force is determined, the force can be transformed to a more useful reference system such as the line-of-action and off line-of- action coordinate system. For this experiment, the system dynamics of the back-to-back gear test rig are examined using the dynamic finite element model developed in Chapter 2. Eqn. 3.26 presents the relation between the off line-of-action friction force at the gear mesh and the off line-of-action force transmitted through the bearing where G(co) is a complex function. Eqn. 3.27 presents the compensation operation to compute the net friction force from the indirect force measurement at the bearings. Eqn. 3.26 f^eanog (û j) = G(cû)fmesh (® ) 1 1 2 Eqn. 3.27 = Figure 3.18 plots the calculated force transmlssibllity magnitude between the gear m esh and measurement bearing support in the off line-of- action direction for various modal damping ratios. Figure 3.19 represents the phase data. As can be expected from the mass, damper, and spring system, the force transmissibility at the natural frequencies is controlled by the damping ratio, while away from the natural frequencies, the damping affects the force transmissibility at higher frequencies through the phase angle. For a modal damping ratio, Ç=0.05, Figure 3.20 shows that the phase lag is less than 5 degrees at frequencies less than 500 Hz except when the frequency is near the first natural frequency. The gearbox geometry affects the load sharing of the bearings and is also considered in the finite element model. As ©->0, the static load sharing of the gearbox is revealed. The net friction force at the mesh can be computed from a single bearing measurement or from an average of multiple bearing m easurements. 1 1 3 100 —a— zetasO.01 ze(a=0.05 -«*--zeta=0.l -o-zeta=0.2 zeta=0.4 I” 0.1 500 1000 1500 2000 2500 Hz Figure 3.18 Force transmissibility magnitude between gear mesh and the pinion bearing nearest the slave gearbox in the off line-of-action direction using the FEA model 180 pi 150 —o—zeta=0.01 —a— Z6ta=0.05 w zeta=0.1 —o—zeta=0.2 zeta=0.4 2500 Figure 3.19 Force transmissibility phase between gear mesh and the pinion bearing nearest the slave Gearbox in the off line-of-action direction using the FEA model 1 1 4 -o—zeta=0.01 O) -ù—zeta=0.05 40 zeta=0.l ■o—zeta=0^ — zeta=0.4 30 li. 0 50 100 150200 250 300 350 400 500450 Hz Figure 3.20 Force transmissibility phase between gear mesh and the pinion bearing nearest the slave gearbox in the off line-of-action direction using the FEA model up to 500 Hz 3.4.3 Absolute Position Measurement The absolute position of contact in the gear mesh must be known in order to compute the instantaneous sliding velocity in the gear mesh. The instantaneous sliding velocity is required to compute individual tooth friction forces. For the measurements on the back-to-back tester, this position was determined by measuring the shaft keyway planes and determining the locations these planes intersect the teeth. These measurements were performed on a coordinate measuring machine [Sheffield Measurement Division, 1987], [Giddings and Lewis, 1993]. Figure 3.21 presents a schematic of the absolute position measuring scheme. To compensate for 1 1 5 il possible errors In shaft keyway edge detection during the actual experiment, both the left and right keyway flanks are determined and the center keyway plane is used as the reference position. The displacement probed is placed tangent to the gear pair center line. A possible source of significant error is the placement of the displacement probe. The probe must lie on the line intersecting the pinion and gear shaft’s center of rotation. This error may take two forms: static and dynamic errors. Static errors result from offset or angular errors between the probe axis and the gear pair center line. Dynamic errors result from instantaneous changes in the axis of rotation for either the pinion or gear shaft. For this research the probe was aligned by eye. Therefore static error is the most likely cause of possible error. \ Figure 3.21 Schematic of absolute position determination on pinion gear 1 1 6 Jtl 3.4.4 Experimental Data Reduction Flow Chart Figure 3.22 presents the flow chart that represents the data reduction process used for the friction force measurement experiment. The strain gage bridge used to measure torque location T4 (Figure 3.11) failed after calibration and is not included in the final result computation. The conversion to the frequency domain was performed to extract the once per revolution component which was believed to be predominately caused by strain gage misalignment in the torque sensing bridges. Test speeds and data ensemble lengths were adjusted to minimize the effects of converting to the frequency domain and back to the time domain. 3.5 Experimental Results This section presents results performed on the back-to-back gear tester to measure the individual tooth friction forces based upon the techniques presented earlier in this chapter. Table 3.2 presents some of the relevant gear parameters of the test gear pair. Table 3.3 summarizes the mean torque on the input pinion shaft and the output gear shaft. At no-load, the difference between the input torque and the output torque is significant. Under load, the percent difference is less than three percent. In order to provide a reference for harmonic comparison. Table 3.3 also presents the mean input power. 1 1 7 pinion shaft gear shaft pinion shaft Torque Torque Torque Torque angular velocity angular velocity Shaft Deflection once per rev Signal Signal Signal Signal in oloa direction trigger based on based on T1 T2 T3 T4 rotating acc. rotating acc. compute m ean pinion Extract speed Mean & Mesh Harmonics oioa X bearing i (ÇK stiffness aosolute position rstaranca Extract Mean & Mesh Harmonics 1 V2 -1 (V2-V1) -Vl 1 gear mast! Figure 3.22 Flow chart of friction force computation procedure 1 1 8 Pinion Gear Teeth 25 47 Base Diameter, inch 4.698 4.229 Outside Diameter, inch 5.608 10.192 Face Width, inch 1.25 Center Distance, inch 7.5 Normal Diametral Pitch 5 Operating Pressure Angle, degree 25 base pitch, inch/degree 0.59/14.4° lowest point of tooth contact, inch -0.429 lowest point of single tooth contact, inch -0.183 highest point of tooth contact, inch 0.407 off line-of-action bearing stiffness, Ibf/in 1.6 xIO * Table 3.2 Relevant spur gears parameters for experimental test Pinion Shaft, Measured Measured T 1 ^ - T 3 Power In, Ibf-in/s, R P M / Mesh Pinion Torque, Gear Torque, 1 hp=6600 Ibf-in/s — ^------Id 00 Frequency, Hz T1,lbf-in T3, ibf-in T 1 ^ n, 216 rpm / 90 Hz 13 129 -427% 216 rpm / 90 Hz 2595 4906 -0.56% 58,980 216 rpm / 90 Hz 4188 7813 0.76% 94,730 540 rpm / 225 Hz 4 8 -6.38% 540 rpm / 225 Hz 1985 3631 2.70% 112,300 540 rpm / 225 Hz 4252 7817 2.21% 240,400 1080 rpm / 450 Hz 1 77 -3995% 1080 rpm / 450 Hz 2539 4712 1.28% 287,200 1080 rpm / 450 Hz 3957 7228 2.83% 447,500 Table 3.3 Comparison of measured mean pinion and gear torque at various pinion shaft speeds 1 1 9 Figure 3.23 presents the frequency spectrum of the pinion shaft torque sensor, T1, for the first four mesh harmonics. This figure shows a "ghost" noise frequency at 679 Hz. Figure 3.24 presents a frequency magnification about the fundamental mesh frequency. The spectrum of Figure 3.24 shows both pinion shaft and gear shaft sidebands. Figure 3.25 and Figure 3.26 present the frequency spectra for the strain gage bridges at torque locations T2 and T3, respectively. Figure 3.25 and Figure 3.26 have sidebands of the same order as the mesh harmonic components. Attempts to explain the modulation as a result of strain gage mounting error causing once per revolution bending strains that modulate gear mesh frequency torque signal were unsuccessful [Measurement Group, 1983]. Models of misaligned strain gages identified the once per revolution bending strains, but the once per revolution bending components failed to modulate higher frequency torque fluctuations. Figure 3.27 shows two shaft revolutions of the pinion shaft torque time domain signal. The once per pinion shaft revolution component has visibly high frequency oscillations at gear mesh frequency. Figure 3.28 presents the pinion shaft angular velocity time varying components in a frequency spectrum. The first several mesh frequency components are visible. A 679 Hz "ghost" component in Figure 3.28 is also present in the T1 pinion shaft torque spectra. Figure 3.23. 1 2 0 Using the measured torque and angular velocity measurements, the power loss at the gear mesh can be computed using Eqn. 3.22. Figure 3.29 is the frequency spectrum of the computed gear mesh power loss. The mean power loss is computed from the input power flow and the torque efficiency from Table 3.3. For the presented case, the mean power loss is (287,200 lbf-in/s)(1.28%)=3700 Ibf-in/s. From the power loss frequency spectrum, a time domain power loss is determined via the inverse discrete Fourier transform. Figure 3.30 is the time domain gear mesh power loss computed from the gear mesh power loss frequency spectrum using only the mean power loss and the first ten gear mesh harmonics for a nominal input torques of 2500 Ibf-in and 4000 Ibf-in, respectively. On Figure 3.30, the pitch point noted on the abscissa corresponds to the point of minimum power loss for each case. For the tested gear set the pitch point Is crossed in the single tooth pair contact regime. Therefore, the gear mesh is at a point “nearest’ to pure rolling where the ideal power loss is zero. Due to fluid film formulations and elastic deformation of the tooth surface, there is no instant of pure rolling during the mesh cycle but near the pitch point a minimum power loss point could be expected and is demonstrated for the two separate loading cases. Both cases have a minimum power loss near the pitch point. For the case at the nominal input torque of 2500 Ibf-in, the gear mesh power loss approaches zero near the pitch point. The 4000 Ibf-in case has a minimum of -7500 Ibf-in/s at the pitch point. 1 2 1 The minimum energy loss of the high torque case is more than 1/2 of the mean power loss of (-2.83%)(447,500 lbf-in/s)=12664 Ibf-in/s (See Table 3.3). Since there is only a single tooth pair in contact at the pitch point a minimum closer to zero would be anticipated. The large minimum power loss for the 4000 Ibf-in case may be attributed to the addition of numbers of different orders of magnitude. In the 4000 Ibf-in case, the mean power flow of 447,000 Ibf-in/s is much larger than the fundamental m esh frequency power loss harmonic of 6,250 Ibf-in/s. Any error, such as in the mean power loss, could create a significant offset of the same order as the gear mesh harmonics power loss magnitudes. For the test conditions, Dm = 1080 rpm and Tin = 2500 Ibf-in, the power loss reaches a near zero minimum in the pitch point region. Using this test case, the inverted friction forces, f1 and f2, are presented in Figure 3.31 with an assumed off line-of-action bearing stiffness of 1.6 x 10® Ibf/in. The net friction force, fnet, as computed from the bearing displacement is also superimposed in Figure 3.31. Along the abscissa of Figure 3.31 important mesh positions along the line-of-action are labeled starting with the lowest point of single tooth contact (LPSTC), leading to the highest point of single tooth contact (HPSTC), and finally to the highest point of tooth contact (Zb). During single tooth contact, f1 should be non-zero, while f2 should equal 1 2 2 zero. From Figure 3.31 both f1 and f2 are nonzero. In the double tooth contact regime, the computed friction forces are not acting In opposite directions as could be anticipated from the gear geometry. The inconclusive results of f1 and f2 In Figure 3.31 may stem from the experimental data. From Figure 3.30, the time domain gear mesh power loss or from the net friction force of Figure 3.31, no significant change can be detected In the transition from single to double tooth contact. Figure 3.32 presents f1, f2 and fnet based upon a constant friction coefficient of 0.05 and the load sharing computed by the Load Distribution Program (LDP) [Gear Dynamics Gear Noise Research Laboratory, no date]. Comparing fnet from Figure 3.31 and Figure 3.32 shows fnet from Figure 3.31 lagging 90 degrees to fnet In Figure 3.32. Therefore, an Improved system dynamic model may be needed. Also the net force amplitude of 300 Ibf Is high compared to the normal mesh force of 1000 Ibf, giving a peak friction force to normal force ratio of 0.3 as compared to typical values between 0.05 and 0.1, suggesting Improved bearing stiffness estimation Is required. 1 2 3 45 Mesh 40 Frequency 35 30 tz 25 679 Hz (D 3 20 F S 15 10- 5- 4 500 1000 1500 2000 Frequency, Hz Figure 3.23 Pinion shaft torque sensor T1, torque harmonics 0-peak amplitude, Tin=2500 Ibf-in, A;n= 1080 rpm 45 40 Mesh Frequency . Gear Shaft . Pinion Shaft Sideband Sideband 20 O’ 440420 460 480 500 Frequency, Hz Figure 3.24 Pinion shaft torque sensor, torque harmonics around mesh frequency for location T1, 0-peak amplitude, Tin=2500 Ibf-in, Qm= 1080 rpm 1 2 4 Mesh L. Frequency 3rd £ (D 3 2nd E 4th 5 0 5001000 1500 2000 Frequency, Hz Figure 3.25 Pinion shaft torque sensor T2, torque harmonies 0-peak amplitude, Tin=2500 ibf-in, Un= 1080 rpm 25 Mesh 4 7 Frequency Mesh Frequency £ É 3 CT 4th . 1000 1500 Frequency, Hz Figure 3.26 Pinion shaft torque sensor T3, torque harmonics 0-peak amplitude, Tjn=2500 Ibf-in, Qjn= 1080 rpm 1 2 5 -1800 -2000 Pinion Shaft Revolution -2200 - -2 4 0 0 o -2 6 0 0 .£ -2 8 0 0 -3 0 0 0 -3 2 0 0 roll angle, degrees Figure 3.27 Pinion shaft time domain torque measurement, T1, Tin=2500 Ibf- in, Qin= 1080 rpm 0 .0 6 3rd 0 .0 5 M esh 2nd F req u en cy 0 .0 4 o 0 .0 3 679 Hz 0.02 4th O) 0.01 5 0 0 1000 1 5 0 0 2000 Frequency, Hz Figure 3.28 Pinion shaft angular velocity harmonics, 0-peak amplitude, Tin=2500 Ibf-in, Qin= 1080 rpm (113 rad/s) 1 2 6 4 0 0 0 350 0 Mesh Frequency 300 0 2 0 0 0 3000 4 0 0 0 5 0 0 0 Frequency, Hz Figure 3.29 Gear mesh power loss computed from experimental measurements to 5000 Hz, 0-peak amplitude, Tin=2500 Ibf-in, Qin= 1080 rpm X 10 computed gear mesh power loss T1=2500 Ibf-in T1=4000 Ibf-in 10 20 30 40 Pitch pinion shaft revolution, degree Point Figure 3.30 Inverse Fourier Transform of gear mesh power loss using mean component and 10 mesh frequency harmonics, Qin= 1080 rpm 1 2 7 JLl (D U f1= tooth pair 1 f2= tooth pair 2 f,=f1+f2 double pair single pair region region -0.2 -0.1 0 0.1 0.2 0.3 0.4 LPSTC=-0.19 Pitch Point=0 HPSTC=0.16 Zb=0.4 Position Along LOA, Inch Figure 3.31 Net friction and computed individual tooth friction forces, koioa=1-6 X 10® Ibf/in, Tin=2500 Ibf-in, Q,n= 1080 rpm LPSTC 40 20 -- S PP Zb -20 -- -40 HPSTC -60 - 0.1 0 0.1 0.2 0.3 0.4 Position Along LOA. Inch Figure 3.32 Friction force on tooth pair 1 and tooth pair 2 using the load distribution computed by LDP, Tin=2500 Ibf-in, p=0.05 1 2 8 3.6 Concluding Remarks This chapter has presented a measurement methodology to compute the Individual tooth friction forces during single or double tooth contact by measuring the net friction force at the gear mesh and the net gear mesh power loss. The method assumes that the gear kinematics follow involute theory and line contact between the mating teeth is maintained. Numerical examples analytically demonstrated the method. The net friction force was determined by measuring the bearing deflection in the off line-of-action direction and relating the displacement to force through the bearing stiffness. The power loss was computed by measuring the input and output torque using semi-conductor strain gage bridges and the input and output instantaneous angular velocities. The torque measurements and the instantaneous angular velocity harmonic component results demonstrate the ability to perform the required measurement. The input and output measured torque for loaded cases are within 3 percent agreement, which can be related to power loss and experimental uncertainty. The computed power loss as shown in section 3.5, repeatedly demonstrated a minimum power loss near the gear mesh pitch point. The computation of the tooth friction forces from the net bearing force and the gear mesh powerless produced inconclusive results. During single tooth pair contact, two tooth friction forces were computed. While during 1 2 9 double tooth pair contact, the friction forces had the sam e sign while on opposite side of the pitch point. The time domain power loss and bearing forces did not agree in form with the numerically computed friction and power loss results for the same gear pair, particularly when the pair are in double tooth contact. Techniques, to improve and verify the gathered measurements, particularly the torque and net friction force components must be developed to further develop the proposed experimental method. 1 3 0 Chapter 4 External Involute Spur and Helical Parallel Axis Gear Model Incorporating Off Line-Of-Action Friction Forces 4.1 Introduction Many studies have dealt with gear vibration from the analytical modeling point of view. However, literally all of these studies dealt with pure torsional vibration or coupled torsional and translational line-of-action motion. Force excitations in the off line-of-action direction, which is the direction frictional forces act, has not been extensively examined. This chapter of the dissertation will present the development of a general analytical model capable of representing the off line-of-action excitation and the resulting off line-of-action vibration. Figure 4.1 presents a flowchart for the model development in this chapter. First, a general model for the pinion and gear elements is introduced with each element possessing 6 degrees-of-freedom. 1 3 1 6 DO F model for Pinion element 6 DDF model for Gear element Mesh Forces and Moment computed from generalized normal force and friction force distributions. Sections: 4.3.1-4.3.3 Introduce normal force distribution model based on elastic coupling through the gear teeth. Section: 4.4.1 Introduce friction force distribution models. 1 ) Model based on known friction coefficient. Section- 4.4.2 2) Model based on a simole elastohydrodynamic theory. Section: 4.4.2.2 Introduce normal and friction force models into the general twelve degree of freedom gear model, Section : 4.4.3 Formulate Reduced Gear Pair Model on Rigid Bearings, Section: 4.5.1 Examine LTV Reduced Examine non-linear Examine Static Reduced Model. Reduced Model with Model. Elastohydrodynamic Numerically: Section: 4.6 lubrication. Sections :4.5.2.1-4.5.2.3 Analytically: Section: 4.7 Section: 4.8 Figure 4.1 Gear model development flow chart 1 3 2 The elements are coupled through generalized normal force and friction force distributions. Next, specific normal force and friction force models are introduced into the generalized load distributions and substituted into the general gear model. The general model is further reduced by assuming the gear elements are supported on rigid bearings. The reduced gear model is finally examined under quasi-static and dynamic operating conditions. 4.2 Literature Review Work by Gregory [1963-1964], Sato [1979], Neriya [1984], Iwatsubo [1984], Furukawa [1991], Hao [1991], Ozguven [1988, 1991], Cai [1992] and Kahraman [1992] are examples of dynamic modeling that has dealt with the coupled torsional and line-of-action motion issues. The coupling occurs through force and dynamic coupling modes. Force coupling occurs due to the load at gear mesh interface and the dynamic coupling due to mass and geometric eccentricities. Different solution methods examined by these authors include finite element analysis, time domain simulation, modal expansion, and linear constant coefficient solutions. Parameters such as mesh stiffness, tooth profile modifications, mass unbalance, and geometric eccentricities were examined by these authors. The friction force at the mesh in the off line-of-action direction was neglected in all of these works. 1 3 3 Blankenship [1992] developed a 6 degree-of-freedom model for the pinion and gear elements with the purpose of analyzing force coupling and modulation phenomena. Off line-of-action friction forces were neglected in this research. Viscous energy loss mechanisms were used in the models developed by Blankenship. Lund [1977], Mitchell [1975], lida [1985, 1986], Neriya [1985], and Vinayak [1995] have examined the coupling of gear trains with three or more shafts in the gear train. In general, the line-of-actions for the different gear meshes are not co-linear so there are both force and dynamic coupling of the line-of-action and off line-of-action motion. The coupling phenomena are the sam e as the single mesh case but due to the multiple directions the line- of-action forces, there is coupling in the off line-of-action and on line-of- action directions. The parameters and solution methodologies used by these authors are similar to the single mesh case. They conclude that the angle between the lines-of-action of the various meshes affects the natural frequencies and mode shapes of the gear mesh system. As with the single gear m esh system, gear tooth damping or friction in the off line-of-action direction were neglected by the authors. Friction has usually been dealt with the power loss concept by computing efficiency or power loss due to the friction force, whether considering boundary, mixed, or elastohydrodynamic lubrication. Work of this type has been performed by Tso [1961], Chiu [1975], and Yousif [1981]. 1 3 4 Tso [1961] analytically studied friction loss in spur gears. Tso assum ed all energy loss at the gear mesh is com posed of the product of the friction force and the sliding velocity. The friction force followed a constant friction coefficient model. An empirical tooth deflection relationship was applied to allow the normal tooth load to be computed for two tooth pair contact regimes. Tso concluded that from an energy loss point of view, equal load distribution on the teeth is a sufficient assumption. Chiu[1975] decomposed the energy loss of the gear mesh into two distinct components. The first component consisted of the sliding friction force, with the friction force being proportional to the normal tooth load. Equal load sharing was assumed during multi-tooth contact and Chiu used the empirical friction coefficient developed by O’Donoghue [1966]. The second component consists of the pumping force required to pump or draw the lubricant from the extemal low pressure region to the high pressure contact region. Chiu concluded that at high speed, the pumping loss can be com parable to the sliding loss. In the work by Yousif [1981] the friction force was theoretically computed as a function of position along the line-of-action. Equal load sharing was considered during two tooth pair contact. The regime was assumed to be elastohydrodynamic with the viscosity and film thickness determined from work by Dowson [1977]. The friction force was calculated using Newton’s Law of Viscosity. 1 3 5 Work that has considered the effect of friction from the vibration and noise point of view has been performed by Ishida [1967], Buckens [1980], Ikeda [1981], Rebbechi[1981,1983] and lida [1985]. Buckens [1980] developed a 8 degree of freedom model gear model with rotors using the Lagrangian equation. The constraint equation written in terms of velocity components is non-holonomous which required the introduction of a Ferrer’s multiplier to take into account the constraint equation. The change of pressure angle due to center distance variation was introduced but neglected in the analysis. The model includes damping at the mesh in the sliding direction but had a Rayleigh’s dissipation function form. This corresponds to viscous damping in the profile direction and is proportional to velocity squared while the Coulumb model would be proportional to velocity. The translational motion is coupled with the profile direction friction force, but this is due to the coordinate system in which the equations were derived. By the application of a coordinate transform the equations would decouple. Ikeda [1981] examined the effect of tooth friction on the torsional vibration of the gears. The friction force effect on torsional vibration was determined experimentally by measuring transmission error and torque variation. The torsional motion was separated into two effects: geometric and frictional. The friction was assumed to act along the line-of-action direction. Ikeda used a torsional model to extract the components from the 1 3 6 measured transmission error and torque variation. Ikeda employed a single degree of freedom torsional model with rigid teeth, and applied constraints such that the teeth always remained in contact. The model assumed the friction force was speed independent. Rebbechi [1981,1983] developed a 4 degree of freedom torsional model which was reduced to a three degree of freedom model through the use of an equation which imposed a constraint of maintained tooth contact. It included friction in the off line-of-action direction and allowed the common normal of the teeth to vary based on the tooth motion. The gear teeth and gear blank were separate elements. The teeth were assumed to be rigid but were allowed to pivot about a torsional spring near the base circle of the gear blank. lida [1985] examined the vibration due to friction in the off line-of- action with a single degree of freedom model, lida examined the vibration characteristic from several levels. They include a numerically integrated model which includes a friction coefficient as a function of sliding speed and a simple load sharing model for two tooth contact. A simplified numerical model was then examined assuming a constant friction coefficient and contact ratio of unity. Finally the model was further simplified by introducing the equivalent damping concept to reduce the problem to the standard second order mass-damper-spring system. 1 3 7 4.3 Twelve Degree-Of-Freedom External Spur and Helical Involute Gear Mesh Parallel Axis Model in this section a générai tweive degree-of-freedom modei including the effect of off iine-of-action friction is developed for the pinion and gear bodies. Each gear blank possesses six degrees-of-freedom for the three translation and three rotation components. Figure 4.2 presents a view along the axis of rotation of the gear system geometry. Figure 4.3 presents a view of the plane of action for spur and helical involute gearing. The model incorporates several assumptions in the model development. The important assumptions are: gear mesh contact can be described by the kinematics of involute theory, line contact is assumed at the gear teeth, the description perturbations are about the gear system mean angular velocity, and the gear mesh geometrical description is time invariant. Each of these assumptions and others are introduced at the appropriate section of the model development. 1 3 8 Off Line- / of-Action / Pinion \ Line-of- Action not to scale. for illustrative use Gear only Figure 4.2 Transverse view of gear mesh ( xy plane ) not to scale, for illustrative use only Figure 4.3 Piane-of-Action view of the gear mesh zone ( xz plane) 139 4.3.1 Forces at the Gear Mesh Interface This section determines the forces acting on the pinion and gear blanks at the mesh interface for ith tooth in contact for any position along the gear mesh cycle. The force distribution and moment distribution acting at the gear tooth mesh interface are summed about the point defined by the intersection of the ith tooth contact line and the x axis. In absolute terms, the point is defined by the point, Si=Sref+Ajbp, on the x axis. The summing of the distributions about a point implies that dynamic effects due to the gear tooth are neglected. Other important assumptions introduced in this section include: line contact is assumed at the gear teeth interface and gear mesh contact can be described by the kinematics of involute theory. 4.3.1.1 Forces Acting on Pinion at the Gear Mesh Interface Figure 4.4 shows the plane of action for spur and helical involute gearing with the gear mesh force distribution acting on the pinion. In Figure 4.4, the normal force distribution and friction force distribution are shown in the positive sense. Summing the normal and friction force distributions about Si, results in the force and moment vectors defined by Eqn. 4.1 through Eqn. 4.4. Eqn. 4.1 solves for the force vector, Np,, computed from the normal force distribution and represents the familiar transverse normal force along the X axis and axial force due to any helix angle along the z axis. Eqn. 4.2 1 4 0 solves for the force vector, Fpj, computed from the friction force distribution and acts perpendicular to the plane of action. Eqn. 4.3 solves for the moment vector, Mp^, due to the normal force distribution acting along the y axis. Eqn. 4.4 solves for the moment vector, Mp^, due to the friction force distribution and acts along a vector normal to the tooth contact line. friction force distribution, f, acts perpendicular to tfie plane of action not to scale, normal force for illustrative use distribution, n, only acts in the plane of action Figure 4.4 Normal, ni, and friction, f,, force distributions (force/length) acting on the pinion at gear pair mesh interface 1 4 1 7 I------ô— - X ^ tan(Y)z Eqn. 4.1 Np, = J nj-y/l + tan^(^)dw wOi ■y/l + tan^('P) -y/l + tan^('P) ______Eqn. 4.2 Fp, = Jf|.Jl + tan^(Y)dw[y] wOj ______Eqn. 4.3 Mp„j = J-Hj ^1 + tan^ (Y)wdw[y] wOj Wlj -X tan(Y)z Eqn. 4.4 Mpfi = Jnjw7l + tan^('P)dw wO, _Vl + tan"('F) ^1 + tan"M 4.3.1.2 Forces Acting on Gear at Mesh Interface Figure 4.5 shows the plane of action for spur and helical involute gearing with the gear mesh force distribution acting on the gear. In Figure 4.5, the normal force distribution and friction force distribution are shown in the positive sense. Summing the normal and friction force distributions about Si results in the force and moment vectors defined by Eqn. 4.5 through Eqn. 4.8. These vectors are equal and opposite to the forces acting on the pinion. Eqn. 4.5 solves for the force vector, Ng,, computed from the normal force distribution and represents the familiar transverse normal force along the x axis and axial force due to the helix angle along the z axis. Eqn. 4.6 solves for the force vector, Fg,, computed from the friction force distribution and 1 4 2 acts perpendicular to the plane of action. Eqn. 4.7 solves for the moment vector, Meni, due to the normal force distribution acting along the y axis. Eqn. 4.8 solves for the moment vector, Mg,^, due to the friction force distribution and acts along a vector normal to the tooth contact line. friction force distribution, f, acts perpendicular to the plane of action \ not to scale, \ \ normal force for illustrative use : distribution, n, only acts in the plane of action Figure 4.5 Normal, nj, and friction, f|, force distributions (force/length) acting on the gear at gear pair mesh interface 1 4 3 - t a n ( Y ) z Eqn. 4.5 = Jn|^1 + tan^(Y)dw = -Nc wO, ^1 + tan^(Y) i/l + tan^(Y) wl. ______Eqn. 4.6 Fgj = - Jfj^ 1 + tan^('P)dw[y] = -Fp, wQ, «Ji ______Eqn. 4.7 Ms„j = j n,^1 + tan^('P)wdw[y] = -Mp^j wOi **'i _____ X - ta n (Y )z Eqn. 4.8 = jn|W-\/l + tan^(Y)dw f — ■ + 7 - = -Md wO, y i + tan^W ^j^ + \an^{'i') 4.3.2 Forces at the Pinion and Gear Axis of Rotation This section transforms the mesh Interface forces and moments determined In section 4.3.1 to the respective pinion shaft and gear shaft axes of rotation. The forces are moved to the Intersection of the XY plane and the respective pinion and gear axis of rotation which Is parallel to the z axis. This transformation Introduces an additional assumption that the gear blank does not attenuate the magnitude or change the phase of the forces and moments. This Imposes either of two constraints, the gear blank behaves as a rigid body and/or the frequency content of the forces and moments are In the static operating region of the elastic gear blank. 1 4 4 Also introduced in this section is a generalized reaction force vector at the pinion and gear axis of rotation from the gear blank support structure. This support could represent a rigid ground, flexible shafts, or elastic bearings. 4.3.2.1 Gear Mesh Forces and Gear Mesh Moments about the Pinion Axis of Rotation Figure 4.6 shows the pinion body, generalized reactions, and mesh interface forces for the ith line-of-contact. In Figure 4.6 the generalized reactions at the pinion are defined in the positive direction. Eqn. 4.9 presents the mesh interface force vector, transformed to the pinion axis of rotation for all teeth in contact at any given position along the line-of-action. Eqn. 4.10 introduces the generalized reaction force vector, Rp, for the pinion element. Summing Eqn. 4.9 and Eqn. 4.10 and the input torque vector results in Eqn. 4.11, the sum of forces and moment about the intersection of the XY plane and the pinion axis of rotation. 1 4 5 Figure 4.6 Force summed about pinion axis of rotation jn ,d w ' wO, w1* % j fi-y/l + tan^('P)dw Qp. ' wO, wl, Qpy ^ jrii tan(Y)dw Qpz wO, Eqn. 4.9 QpMx J "j tan(Y)dw - J fjVl + tan^('P)wdw QpMy wO, ' wO, QpM2 E J 7 T + S' k t a n M d w - % J n,^1 + tan^(Y)wdw wo,vtan((p) y Wl, Wl, -rp% j n^dw + 2 j f +s, +w tan(Y) f,^1 + tan^(Y)dw ‘ wO, ' wo, V i3n((p) y 1 4 6 R,Px Ipy R, Eqn. 4.10 = Pz Px Py PzJ 0 f z p . l 0 0 Eqn. 4.11 + + Qp I K 0 SM, 0 P .T n . 4.3 2.2 Gear Mesh Forces and Gear Mesh Moments about the Gear Axis of Rotation Figure 4.7 shows the gear blank, generalized reactions, and mesh interface forces for the ith line-of-contact. In Figure 4.7, the generalized reactions at the gear are defined in the positive direction. Eqn. 4.12 presents the mesh interface force, Qoy Qo, Qow. Qa^v transformed to the gear axis of rotation for all teeth in contact at any given position along the line-of-action. Eqn. 4.13 introduces the generalized reaction vector, Rg, for the gear element. Summing Eqn. 4.12 and Eqn. 4.13 and the output torque vector results in Eqn. 4.14, the sum of forces and moment about the intersection of the XY plane and the gear axis of rotation. 147 \,' - Figure 4.7 Force summed about gear axis of rotation ' wO. wl. ______ Qc i wO, Wl, Q Gy - Z Jn, tan(Y)dw Eqn. 4.12 Q Gz ‘ wO, wl, ______r,Z j "i tan(Y)dw + ^ Jf,Vl + tan^(M')wdw ^GMy ' wO, ' wO, wl, Qqmz - I f -Si n, tan('F)dw + Z Vl + *3n^(Y)wdw wO. wl, -''g Z f "idw + ^ J — -Si -w tan('P)t^1 + tan"(Y)dw wO, > wO, V 1 4 8 G x 'Gy RGz Eqn. 4.13 Rq = MGx M,Gy M,G z. ' 0 ■ f S F . ' 0 IF, SF. 0 Eqn. 4.14 + R q + Q g S m , 0 SM, 0 G ."Hjut. 4.3.3 A Dynamic Gear Model Incorporating Friction Force This section takes the sum of forces about the respective pinion and gear axes of rotation from section 4.3.2, and Newton’s Law of Motion for the gear system to formulate a dynamic model incorporating friction. Also in this section, the generalized reaction vector is cast in the form of bearing reactions and a mass matrix is introduced to complete the dynamic model formulation. This model introduces the additional assumptions: the described perturbations are about the gear system mean angular velocity, and the gear mesh geometrical description is time invariant. The mean input angular speed is Qfn and the mean output angular speed is Qout- These mean speeds are constrained by Eqn. 4.15. 149 Eqn. 4.15 ^out - —"T^in Eqn. 4.16 introduces a generalized displacement vector of the blank centroid for the pinion, Up, and gear, Ug, respectively. The generalized displacement vector is defined about the unloaded position of the gears. Eqn. 4.17 presents the general reaction vector cast into the form of the bearing stiffness matrix, [K]j, proposed by Lim[1989]. Eqn. 4.19 presents Eqn. 4.17 in compact notation. The negative sign in Eqn. 4.17 and Eqn. 4.19 represents the fact that a positive reaction force needs a negative displacement. Eqn. 4.16 j = P,G P is the pinion blank, G is the gear blank 150 ^X.Y k x j kx.8x k x.9. 0 k y x ^Y.Y k y z ky,8x ky .9 . 0 k & x k&Y k z .z kz,8y 0 Eqn. 4.17 Rj = — kz.8x U k ex X k8x.Z k8x.e. 0 *^9y.x ^ 8 ,. Y k e ,.z k8,.8x k e ,,8 . 0 0 0 0 0 0 0 -ii k x .x kx.Y k x .z kx.8x kx.8. 0 k y .x ky.Y k y ,z ky.ex ky .8 . 0 0 Eqn. 4.18 [K], = k z .x kz.Y k ^ z kz.8x k z .8 . ^8x.X kex.Y k8x.Z *^ex.ex kex.9. 0 key.x kSy.Y k8,,Z *^8,.8x k e ,.8 . 0 0 0 0 0 0 0 Eqn. 4.19 Rj = -[K].Uj The inertial matrix, [m Jj , is defined by Eqn. 4.20 for the pinion and gear blanks (j=P,G) about the respective blank’s axis of rotation. The matrix terms, m,, is the m ass of the jth blank. Ixj, Ivj, and Izj are the moments of inertia about the x , y , and the z axis, respectively for the jth blank. Applying Newton’s Law of Motion, Eqn. 4.11, Eqn. 4.14, Eqn. 4.19, and Eqn. 4.20 results in Eqn. 4.21 and Eqn. 4.22. Eqn. 4.21 and Eqn. 4.22 describe the motion of the pinion and gear bodies in terms of bearing reaction forces, input and output torque, and the mesh forces at the gear mesh interface. 1 5 1 m ^ 0 0 0 0 0 0 m j 0 0 0 0 0 0 m j 0 0 0 [Ml = 0 0 0 0 0 Eqn. 4.20 •xi 0 0 0 0 'vi 0 0 0 0 0 0 *Z) . D ^ P is the pinion blank, G is the ' = gear blank 0 0 0 Eqn. 4.21 0 0 J i n 0 0 0 Eqn. 4.22 [M]3Ü o +[K],U3 = + Q r 0 0 ^out . 4.4 Normal Force Distribution and Friction Force Distribution Models Up to this section, the normal force distribution, nj, and the friction force distribution, fj, have been left in a general functional form with the only requirement being that they are represented as forces per unit length. This is an ideal form when computing the normal force or friction force distributions from external computer programs such as the Load Distribution 152 Program (LDP) [Gear Noise Gear Dynamics Research Laboratory] or the Contact Analysis Program Package (CAPP) [Vijayakarj. In this section, the normal load distribution and the friction force distribution functions are defined explicitly in terms of the normal mesh force, a known load sharing factor, and the ratio of friction force to normal mesh force. 4.4.1 Normal Force Distribution The generality of normal force distribution is reduced by defining the normal force distribution in terms of the load sharing factor, 7s, and the normal mesh force, fm as presented in Eqn. 4.23. Eqn. 4.24 defines the normal mesh force in terms of the gear pair mesh stiffness, km, the helix angle, and the deformation across the gear mesh, Ô. The deformation across the gear mesh is a scalar value which depends upon the structure of the gear system model. For a gear system supported on rigid bearings, Ô can take the form of Eqn. 4.25, while a gear system supported on elastic bearings in the line-of-action or x direction could take the form of Eqn. 4.25a. Another from of Ô incorporates axial motion of the helical gear blanks in the z direction as presented in Eqn. 4.25b. The load sharing factor, with units 1/length, is a function of the position along the gear mesh line-of-action and position along the face width subject to the constraint of Eqn. 4.26 is known apriori. The load sharing 1 5 3 factor distributes the normal mesh force over the individual teeth in contact at every position along the line of action. In this research, the load sharing factor is given by Eqn. 4.27. The load sharing factor defined by Eqn. 4.27 m eans that tooth i is uniformly loaded across the face width with a load equal to the normal mesh force times the length of tooth i divided by the total length of all teeth in contact at that instant. Inserting Eqn. 4.24 and Eqn. 4.27 into Eqn. 4.23 results in Eqn. 4.28, the normal load distribution used in the remainder of this research. Eqn. 4.23 Eqn. 4.24 f„ = k^Ô^I + tan^W Eqn. 4.25 5 = rp6,p + rg0,Q Eqn. 4.25a 5 = (rp0,p +rg0,e)+(xQ -x ,) (z g -Z p)tan('F) Eqn. 4.25b 5 = (rp0,p +rg0,e)+(xe -X p ) + - .Jl + tan^(Y) wii Eqn. 4.26 ^ jy,(s,,w)-\/l + tan^(Y)dw = 1 ' wOi Eqn. 4.27 Yi(Si) = w i i % J V‘l+ tan^('F)dw i w O i 1 5 4 Eqn. 4.28 n,{s,) =------^ ------^ j^ l + tan^Mdw % jc ' wOi ' wOj 4.4.2 Friction Force Distribution The friction force distribution is defined by Eqn. 4.29. This form is recognizable as Coulomb or dry friction, but in the context of this research, it is defined as the ratio of friction force to normal force, irrespective of whether the mechanism is dry, boundary, mixed, or full film lubricant regime. This friction model form has been used for gear efficiency studies by Tso[1961], Chui [1961], and Martin [1972,1980]. The friction coefficient, p, is a real number and may be a function of dependent and independent variables. The sign of p. determines the direction the friction force is acting. Eqn. 4.30 defines p as the product of the absolute value of p and |3 where P=±1 depending upon the direction the friction force is acting. fi = pn. Eqn. 4.29 = fiYiL = PYiVl + tan ^ M k ^ 0 Eqn. 4.30 p = |p|p = p ‘p 155 4.4.2.1 Determination of the Friction Force Direction For parallel axis involute spur and helical gears supported by rigid bearings, the sliding speed defined by the points Sj and w in the plane of action is given by Eqn. 4.31. Assuming the pinion and gear axis of rotation are parallel to the z axis the sliding speed vector lies in the XY plane. If the point of contact is in approach action, Vj < 0, then p=+1. If the point of contact is in recess action, Vi > 0, then p=-1. Eqn. 4.31 y = —— — (s, + wtan('P))Qjn At V|=0, the pitch point, the sliding velocity is zero, but the sliding acceleration is positive. Therefore, the friction force is chosen to act in a direction impeding potential movement. Accordingly, at the pitch point, p=-1. This is done because the gear system model represents the motion perturbation about the mean operating angular velocity. Assuming the gears are supported by rigid bearings, p is simply a function of the independent variable, Sj. If the sliding velocity at the gear mesh incorporates motion from the gear elements, which could occur for gear bodies supported by elastic bearings in the off line-of-action direction, is a function of both the independent variable, s„ and the dependent variables, ÿp and y^, making p non-linear. Eqn. 4.32 describes such a situation. 1 5 6 Eqn. 4.32 V, = + w tan(Y))0,, + (ÿ^ - ÿp ) For the situation described by Eqn. 4.32, if V, < 0, then p =+1 and if Vj> 0, then P=-1. For this research, it is assumed that the sliding velocity due to the gear mesh kinematics described by Eqn. 4.31 is much greater than any relative velocities introduced from the dependent variables, such as , ÿp and Ÿq . This is generally valid when contact is not at the pitch point, but when near the pitch point this assumption may fail. This situation is left for further research. Therefore, for this research, p is only a function of the independent variable. 4.4.2.2 Fluid Film Model The friction force model for this research is defined by Eqn. 4.29 and Eqn. 4.30. In this section, a fluid film model based on works performed by Dowson [1966] and later by Gu [1972] are cast into the form defined by Eqn. 4.29. The quasi-static fluid film model approximates the involute profile by assuming equivalent cylinders for the pinion and gear at each point of contact. Eqn. 4.33 presents Newton’s Law of Viscosity, which is defined in terms of shear stress, x. To convert the shear stress to the force per unit length needed to model the friction force distribution, Eqn. 4.33 is multiplied 157 by the Hertzian contact width, 2b, given by Gu [1972], and presented in Eqn. 4.34. Next, the rate of change of sliding speed with respect to the film thickness is approximated by the gear mesh sliding speed as defined by Eqn. 4.31 and the fluid film thickness as given by Eqn. 4.35. Dowson [1966] empirically developed a minimum film thickness, hmm, relationship given in Eqn. 4.36 from experimental data. The mean fluid film thickness, hm, is approximately 1.1-1.2 times the minimum fluid film thickness, hmm [Dowson, 1966]. Combining Eqn. 4.33, Eqn. 4.34, Eqn. 4.35, and Eqn. 4.36 results in Eqn. 4.37. The negative sign on the relative sliding velocity implies that the friction force resists the sliding motion. Finally, to cast Eqn. 4.37, into the defined friction model, Eqn. 4.37, must be divided by the normal load, resulting in Eqn. 4.38. The viscosity, v, is defined by Dowson as Eqn. 4.39 and by Gu as Eqn. 4.39a. It is noted that the main difference between the two equations is the formulation by Gu includes temperature dependent viscosity. Another formulation similar to this approach is the gear efficiency work with fluid film lubrication by Martin [1980]. This work is based upon the approximate sliding friction coefficient equation developed by Trachman [1976]. For this research, the viscosity will be assumed independent of temperature and will follow the viscosity model of Dowson presented in Eqn. 4.39. Temperature effects are left for future research. 1 5 8 dV Eqn. 4.33 T = V dh. ofSRn, Eqn. 4.34 2b = \ e'k ) (rptan4) + s,)(rgtan(|)-Si) Eqn. 4.34a R = (rp + rjtan * Eqn. 4.34b E’= 2 Eqn. 4.35 dh ~ h„ Eqn. 4.36 R ■ Si(cos({))(rg-rp) Eqn. 4.36a sin(t) + E’R Eqn. 4.36b W = Hj E’R Eqn. 4.37 f; =2vb V / f 2vb Eqn. 4.38 Hi = — = n, n. Eqn. 4.39 V = Vp6 2b Eqn. 4.39a V = Vp6 159 4.4.3 Dynamic Gear Model With an Assumed Normal And Friction Force Distribution This section takes the dynamic gear model derived in section 4.3.3 and presented as Eqn. 4.21 for the pinion body and Eqn. 4.22 for the gear body and introduces the normal force distribution and friction force distribution discussed in sections 4.4.1 and 4.4.2. The normal force and friction force distribution are reproduced as Eqn. 4.40 and Eqn. 4.41, respectively. Eqn. 4.42 relates to actual tooth length due to the helix angle to the projected transverse tooth length. For a spur gear, \j/=0, the actual tooth length and the projected transverse tooth length are equal, implying Eqn. 4.42 equals one. The actual tooth length and the transverse projected tooth length is presented in Figure 4.8. w 1i Projected Transverse Tooth Length, Actual Tooth wlj - wOj L e n g th , h(w1j - wO|) w 0 Figure 4.8 Actual tooth length and projected transverse tooth length 1 6 0 Substituting Eqn. 4.40, Eqn. 4.41, and Eqn. 4.42 into Eqn. 4.9 results in Eqn. 4.43 Qp in terms of km, 5, and is the mesh interface force defined in terms of the normal force and friction force models developed in Sections 4.4.1 and 4.4.2. H.=[Np, Np^ Np, Np^, Np^^ Np„^f is defined by Eqn. 4.44 and relates the forces and moments in the gear mesh to the forces acting at the pinion element’s axis of rotation. Similarly, substituting Eqn. 4.40, Eqn. 4.41, and Eqn. 4.42 into Eqn. 4.12 results in Eqn. 4.45, Qq in term s of km, 6, and 1%. N^f is defined by Eqn. 4.46 and relates the forces and moments in the gear mesh to the forces acting at the gear element's axis of rotation, and Nq are functions only of the independent variable and may be determined apriori. Introducing Eqn. 4.43 and Eqn. 4.45 into Eqn. 4.21 and Eqn. 4.22 results in Eqn. 4.47 and Eqn. 4.48, the equation of motion for the pinion and gear bodies, respectively, in terms of Ô, the deformation across the gear m esh, and the input and output torques. Due to the assumption that the point of contact is known from kinematics, time, t, and the position along the line of action, s, are related through Eqn. 4.49. 1 6 1 Eqn. 4.40 ni=7ihk^S Eqn. 4.41 f. =nn| =n7,hkm6 Eqn. 4.42 h = .^1 + tan^('P) Eqn. 4.43 Q p = i^ k ,5 wl, jYihdw ' wO, Wl, X J^Yih^dw Px ' wO, Wl, Py X JY i tan(Y)hdw Eqn. 4.44 Pz ' wO, wl, wl, PMx -rpX J Yi tan('P)hdw-X j^iYih^wdw PMy ' wO, i wO, Wl, / J. \ Wl, PMz -? wO,i KV ' " " ' r wl. wl, , wO, Eqn. 4.45 Q q = N^k^S 1 6 2 Z Jïihdw ' wOj JtiYih'dw Gx ' wOj wii Gy “ X J7i tan('P)hdw Gz ' wOi Eqn. 4.46 wl, wl, GMx rg% j" Yi tanM hdw + 2 JnYih^wdw GMy ' w O , I w O , w l , w l . G M zJ -IJ -^ " ^ -SijYi tan(¥)hdw + Ç jYih^wdw I w O , w l , w l , - r , S J Yihdw +X J Sj - wtan(Y) HYjh^dw ' wOj ' w O , tan(cp) 01 0 0 Eqn. 4.47 [M]^ + [K], = 0 0 0 0 0 Eqn. 4.48 [M]^ + [K]^ - l^k^S = 0 0 ^OUI fr \ / Eqn. 4.49 s = b g w , -in t b„ b„ \\ V yj 163 4.5 Spur and Helical Gear Mesh Supported on Rigid Bearings This section examines a reduced model of Eqn. 4.47 and Eqn. 4.48, where the support bearing is assumed rigid. This reduced model will be expresses in two forms, simultaneous second order differential equations and state variable form. Next, the equations are studied in the static form to examine if the model is consistent, meaning, the model does not create spontaneous energy when using physically realizable gear parameters. The static model is also examined to determine if the power flow loss computed from the input and output torque and angular speeds equals the power loss computed from the gear mesh friction force and gear mesh sliding velocity. In both of the static studies, the answer is positive meaning the static model behaves sensibly for all physically realizable gear parameters. Next, the dynamic model is examined, when p.' is only a function of the independent variable. In general, the linear time-varying problem, requires a numerical solution. However, one goal of this research is to investigate an analytically tractable linear time varying solution approximating the actual system. The analytical solution is compared with a numerical solution of the same system. Then, the dynamic model is examined employing the non-linear fluid film model developed earlier in section 4.4.2.2. 1 6 4 4.5.1 Reduced Model Formulation This section reduces the developed equations of motion described by Eqn. 4.47 and Eqn. 4.48 by assuming the shaft support bearing is rigid. A schematic of the reduced model is presented in Figure 4.9. With the rigid bearing assumption Eqn. 4.47 and Eqn. 4.48 reduce to Eqn. 4.50 and Eqn. 4.51, respectively. From the model kinematics, Ô can be defined as 8 ^fpGzP +fgôzG - e(f) - This equation also includes the effect of the static transmission error. Noting Eqn. 4.52 and the substitutions defined by Eqn. 4.53 and Eqn. 4.54, equations Eqn. 4.50 and Eqn. 4.51 become +Eqn. 4.55 and Eqn. 4.56, respectively. Examination of +Eqn. 4.55 and Eqn. 4.56, reveals a semi-definite system which can be transformed to a positive definite system by applying the coordinate transformation defined a s D = rp0^p-i-rge^Q and D = rp0^p+rg0,Q. The positive definite system that results is given by Eqn. 4.57. From the friction point of view, Eqn. 4.57, is rather uninteresting. The torques, -^T’n +— , *ZP IzG are independent variables specified as a differential amount across the gear mesh. A more interesting form would be to specify the input torque and compute the output torque based upon the friction in the gear mesh. 1 6 5 y Figure 4.9 Schematic of reduced model on rigid bearings ( w). \ \ Egn. 4.50 Izp®zp liY.h^dw kn.5 = T„ wl, wl, Eqn. 4.51 Izg ^zg ~ -^-s,-w tan( 4 ') Ly.h^dw -rgX J Yihdw + Xr^wo,Uan((p) J ) J wl, Eqn. 4.52 ^ j Yjhdw = 1 by definition ' wOi wl. Eqn. 4.53 Ep == ][] j + S| + wtan(Y) \iy, {s„ w)h"dw i wo,Uan((p) ) wi, / r Eqn. 4.54 = X 7~~/' x ~^i -wtan(Y) HYi(Si,w)h^dw i woAtan((p)J 1 6 6 +Eqn. 4.55 l^pë^p +(r. -6p)k^(r_8zp +r,02e) = T,„ +(r. -£p)k^e(t) Eqn. 4.56 l^sëzG +(<'g -eG)km(fp02P +r,0zs) = T<,^ +(r„ -es)k^e(t) D+ ^(fp-ep) + ^(rg-ee) kmD = V 2P 'ZG Eqn. 4.57 \ r^ T n + 7 ^ Tout + 7 ^ ( f p -ep)+^(fg - £ q) kn,e(t) 'z p 'zG v 'z p 'ZG The output torque is defined as = -ks0zQ, which physically represents a massive load inertia attached to the output gear through the torsional spring stiffness,30 kg. This gear configuration is presented in Figure 4.10. Analytically, the configuration presented in Figure 4.10 can be obtained by noting T^u, = -k^G^g, Eqn. 4.58 and Eqn. 4.58a and substituting into +Eqn. 4.55 and Eqn. 4.56. This results in the matrix formulation represented by Eqn. 4.59. Also introduced in Eqn. 4.59 is a viscous damping matrix. Another useful form of Eqn. 4.59 can be the state space representation. Eqn. 4.59 written in state space form is given by Eqn. 4.60, where the matrices and vectors are defined in Eqn. 4.61 through Eqn. 4.65. 167 Figure 4.10 Schematic of reduced model on rigid bearings with gear element attached to a massive load Inertia through torsional spring element, kg. Eqn. 4.58 TpQzp - D rgOzG Eqn. 4.58a rp0zp=D-rgÔz3 1 — (fp-ep) p-Ep^m Izp Izp T.(t)+- e(t) 0 g “E q )<„, Eqn. 4.60 ^ = - [ a EÎ b L>^+[f L ü ) 1 6 8 Eqn. 4.61 X = ZG D ZG. 1 -fg C&D C&D 0 «'9 C&8 C&8 Eqn. 4.62 [A l.= 0 0 -1 0 0 0 0 -1 0 0 rp -E p k 0 'ZP Eqn. 4.63 [b L = 0 0 Cg 'ZG 'z G 1 0 0 0 0 1 0 0 rp o W 1 ) "Epjk^ •z p *ZP Tn / Eqn. 4.64 F L = 0 j— k ""Egjk^ VG 0 0 0 0 Eqn. 4.65 169 4.5.2 Static Examination of Gear Model This section quasi-statlcally examines the model developed In the section 4.5.1. First, the normal mesh force and output torque to Input torque relationship Is developed. Next, from the output/input torque relationship, It Is shown that the developed gear model loses output torque capacity for any physically realizable friction coefficient. The Ideal torque relationship, Tinrg=Toutrp, only occurs when contact Is at the pitch point or when |i=0. Finally, It Is shown that the power loss computed from the torque and angular speeds equals the power loss computed from the sliding speed and the friction force. 4.5.2.1 Static Normal Mesh Force, Output Torque, and Friction Force Neglecting the dynamics and rearranging +Eqn. 4.55 and Eqn. 4.56 results In Eqn. 4.66. Solving for D and Tout, results In Eqn. 4.67 and Eqn. 4.69, respectively. The coordinate D can be transformed to the coordinate 5, for direct computation of the transverse mesh force, as shown In Eqn. 4.68. The normal mesh force Is simply the transverse mesh force multiplied by h as presented In Eqn. 4.68a. When |i=0 both £p, Eqn. 4.53, and eg , Eqn. 4.54, are equal to zero, therefore reducing Eqn. 4.67 and Eqn. 4.69 to the mesh force and output torque predicted by standard m eans that neglect friction In the gear mesh. 1 7 0 (rp-ep)kn, O' D fTin+(rp-£p)k^e(t)| Eqn. 4.66 {l'g-£G)km o u t, [ (rg-EG)k^e(t) J T Eqn. 4.67 D = ■e(t) (fp-ep)kn Eqn. 4.68 5k_ (fp - Ep ) Eqn. 4.68a (rp-Ep) (l'a ~Eq ) Eqn. 4.69 Tn (""p - Ep ) Figure 4.11 through Figure 4.20 present the normal force to input torque ratio, output torque to input torque ratio, and the friction force to input torque ratio under quasi-static operating conditions. The effects of the friction coefficient, the helix angle, and contact ratios on the normal force, output torque and the friction force are presented. The position along the line-of-action defined as the coordinate along the x axis as defined in Figure 4.3 with the start of tooth pair contact Za shifted to zero. Figure 4.11, Figure 4.12 and Figure 4.13 examines the normal mesh force to input torque ratio for various friction coefficients, various helix angles and various contact ratios, respectively. Figure 4.11 varies the friction coefficient, but assum es the friction coefficient constant over the mesh cycle. Note, in approach action, the friction force tends to increase the mesh force, while in recess the friction force tends to decrease the mesh force as compared to the ideal friction less case of fi=0. 171 Figure 4.12 presents the normal force to input torque ratio for various helix angles for a single tooth pair in contact. The effect of the helix angle on the mean normal mesh force is apparent. Increasing the helix angle increases the magnitude of the mean normal mesh force. The helix angle also smoothens the transition from approach action to recess action. For a helix angle of zero the transition is a step change. By increasing the helix angle from zero, the transition from approach action to recess action transforms from a step change to a ramp change with the magnitude of the slope decreasing as the helix angle increases. Figure 4.13 compares the effect of profile contact ratio of a spur gear mesh on the normal force to input torque ratio. Figure 4.14 presents the line- of-action configuration used in Figure 4.13. In Figure 4.13 during two tooth pair contact, one tooth is in approach action and one tooth is in recess action. The position varying ramp component cancels, but the constant remainder is still lower than the frictionless normal mesh force to input torque ratio of 0.4. 172 0.42- H=0.1 0.415 ■ 0.41 ■ (1=0.05 0.405 ■ (1 =0.01 (1 = 0 .0 — 0 .4 - 0.395 0.39 0.385 0.38 0.37|L 5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Za position along the line-of-action Zb Figure 4.11 Normal mesh force per unit input torque for n'=0.01, 0.05, 0.1, contact ratio= 1 , rp=2.5, rg=5, 0 , LPSTC=-0.5, Zb=0.5, Za=-0.5 0.44 0.43 0.42 0.41 - 0.4 0.39 •0.4 -0.3 -0.2 -0.1 0.1 0.2 0.4 0.50.3 position along the line-of-action Figure 4.12 Normal mesh force per unit input torque for different helix angles, |i'= 0.05, rp=2.5, rg=5, single tooth pair in contact 1 7 3 0.41 0.405 0.4 CR=1.S 0.395 CR=1.25 0.39 - CR=1 -0.4 -0.3 - 0.2 -0.1 0.2 0.30.1 0.4 0.5 LPSTC position along the line-of-actionline-of- Zb Figure 4.13 Normal mesh force per unit input torque for contact ratio= 1, 1.25, 1.5 , fi = 0.05, rp=2.5, rg=5, 'F= 0 , bp=1, LPSTC=-0.5, Zb=0.5, frictionless ratio=0.4. (See Figure 4.14 for line-of-action geometry) CR=1 Zb =-0.5 PP=0 Zb=0.5 LPSTC HPSTC CR=1.25 [- Zb=-0.75 LPSTC PP=0 HPSTC Zh=0.5 CR=1.5 |- Z b = -1 .0 LPSTC PP=0 Zb=0.5 HPSTC Figure 4.14 Line-of-Action geometry used for normal mesh force, input/output torque ratios and friction force examples presented in Figure 4.13, Figure 4.17 and Figure 4.20 1 7 4 Figure 4.15 through Figure 4.17 examine the output torque to input torque ratio for various friction coefficients, various helix angles and various contact ratios, respectively. The power loss is related to the output to input torque ratio and the gear ratio. The lower torque ratio compared to the ideal case, the lower the operating efficiency. Figure 4.15 examines the effect of friction coefficient on the torque ratio. The friction coefficient is assum ed constant over the entire mesh cycle. One intuitively obvious result is that an increase in the friction coefficient increases the power loss. The power loss is manifested through a decrease in the output torque for a constant input torque. Also, the torque ratio is not symmetric about the pitch point, even for a gear ratio of unity, where the pinion base radius equals the gear base radius. Figure 4.16 presents the effect of helix angle on the torque ratio when only a single tooth pair is in contact. For spur gearing, at the pitch point the developed model predicts a torque ratio equal to the frictionless case for any given friction coefficient. This results because contact at the pitch point is in a pure rolling condition. In helical contact, only one point along each diagonal contact line may be in pure rolling contact at any instant, therefore reducing the torque ratio. The reduced torque output effect increases both in amount and in percent of overall contact along the line-of-action as the helix angle increases for a fixed face width. 1 7 5 Figure 4.17 presents the effect of contact ratio on the input torque to output torque ratio. As for the normal mesh force examples, when two tooth pairs are in contact and one pair is in approach and the other is In recess, the position varying component cancels leaving a constant component. Unlike the normal force component that approaches the ideal frictionless case, the torque ratio decreases significantly. |1 = 0 .0 1 1.99 1.98 1.96 1.95 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 Za position along line-of-actionline-of- action Zb Figure 4.15 Output to input torque ratio, Tout/Tm, for various friction coefficient, contact ratio= 1 , rp=2.5, rg=5, Y= 0°, bp=1, LPSTC=-O.S, 2b=0.S, Za=-0.5, frictionless Tout/Tm = 2 1 7 6 4* = 0' 1.995 S '= 5" Y = 1 0 ' Y = 2 0 ' 1.99 1.985 1.98 1.975 1.97 1.96! •0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 position along line-of-actionline-of- action Figure 4.16 Output to input torque ratio,Tout/Tin, for various helix angles, \i 0.05, rp=2.5, rg=5, single tooth in contact, frictionless Tout/T,n=2 1.995 1.99 ^ 1.985 CR=1.5 CR=1 1.98 1.975 CR=1.25 1.97 -0.5 -0.3-0.4 - 0.2 - 0.1 0 0.2 0.3 0.4 0.50.1 LPSTC position along line-of-action Zb Figure 4.17 Output to input torque ratio, Tout/Tin, for various contact ratios, p*= 0.05, rp=2.5, rg=5, 4^= 0°, LPSTC=-0.5, Zb=0.5, frictionless Tout/Tin=2. (See Figure 4.14 for line-of-action geometry) 1 7 7 Figure 4.18 through Figure 4.20 examine the friction force acting on the pinion support bearing, Eqn. 4.43, to input torque ratio for various friction coefficients, various helix angles and various contact ratios, respectively. The friction force acting on the gear element is equal and opposite to the friction force acting on the pinion element. The sign change at the pitch point causes a direction change in the friction force. The friction force sign change results in a potential dynamic mesh excitation source. Dynamic studies are examined in later sections. Figure 4.18, presents the results of various friction coefficients on the friction force. Figure 4.19 shows the effects of the helix angle for a single tooth pair in contact on the friction force. Increasing the helix angle increases the peak-to-peak amplitude. This effect results from the increased normal mesh force due to the increased helix angle. Similar to the normal mesh force case. Figure 4.12, the transition from approach action to recess action is transformed from a step change to a ramp change. In Figure 4.20 the effects of contact ratio on the off line-of- action friction force are shown. During double tooth pair contact, when one tooth pair is in approach and the other in recess, the friction forces cancel entirely. This friction force cancellation comes at a significantly larger power loss as shown in Figure 4.17. 1 7 8 0.05 0.04 0.03 H=0.05 0.02 0.01 H = 0.0 -0.01 -0.02 -0.03 -0.( -0.4 -0.3 - 0.2 -0.1 0.2 0.3 0.4 0.5 position along the line-of-action Figure 4.18 Off line-of-action friction force acting on pinion element,Qpy, per unit input torque, Tin, for p.'=0.01, 0.05, 0.1, contact ratio= 1, rp=2.5, rg=5, Y= 0 , LPSTC=-0.5, Zb=O.S, Za=-0.5 0.025 Ÿ = O ' 0.015 T = 10 Y = 20 0.005 -0.005 -0.015 - 0.02 - 0.2 - 0.1 0 0.1 0.2 position along the line-of-action Figure 4.19 Off line-of-action friction force acting on pinion element,Qpy, per unit input torque, Tm, for various helix angles, p'= 0.05, rp=2.5, rg=5, single tooth pair in contact 1 7 9 0 .025 0.02 0.015 0.01 0.005 CR=1.5 -0.005 CB=1.25 - 0.01 -0.015 CR=1 - 0.1 -0.4 -0.3 - 0.2 - 0.1 0.1 0.2 0.3 0.4 0.5 position along the line-of-actionline-of- ZbLPSTC Figure 4.20 Off line-of-action friction force acting on pinion element,Qpy, per unit input torque, Tin, for contact ratio= 1, 1.25, 1.5, p'=0.05, rp=2.5, rg=5, Y=0°, bp=1, LPSTC=-0.5, Zb=0.5 (See Figure 4.14 for line-of-action geometry) 4.S.2.2 Output Torque Considerations Figure 4.11 through Figure 4.20 examines the normal mesh force, output torque, and off line-of-action friction force for various gear parameters such as friction coefficient, helix angle, and contact ratio. Eqn. 4.69 relates the output torque to the input torque when friction is included in the gear mesh. This section examines the output torque for the extreme cases, meaning the off line-of-action friction force is so large the output torque is 1 8 0 reduced to zero or on the other extreme, the output torque is equal to the frictionless case. Eqn. 4.70 presents these constraints on the input/output torque. The upper constraint, ^ - ; r , being the output torque may never be in *P larger than the case of p=0, for this would violate the Laws of Thermodynamics. The lower constraint, Lss->o, is a limiting case, where Tn output torque goes to zero, and no output work may be achieved. Any resistance on the output side would stop the output shaft’s rotation, therefore locking up the gearbox. The next section examines the static model considering the constraint defined by Eqn. 4.70, and shows that for any physically realizable friction coefficient, Eqn. 4.70 is not violated. Eqn. 4.70 o < ^ < ^ in Applying the lower limit set by Eqn. 4.70 to Eqn. 4.69 results in ^ = 1^” =0 which implies that r^ =E q . Substituting Eqn. 4.54 and Eqn. 4.30 for £q results in Eqn. 4.71. Assuming p' is a time or position invariant, p ’ can be solved for by rearranging Eqn. 4.71 into Eqn. 4.72. Noting in Eqn. 4.72 that Sj -wtan('P) > 0 must be greater than or equal to zero tan((p) from geometric considerations, 7j(Si,w)>0, and h >0, the sign of Eqn. 4.72 is determined by the sign of p (See Sections 4.4.2 and 4.4.2.1). In 1 8 1 approach action, Si+wtan(\(r) <0, P=+1 and at the pitch point and in recess action, Sj+wtan(\|r) >0, p=-1. Each of these conditions will be separately examined. '"r' f r ^ Eqn. 4.71 r = % j , -S; -w tan (Y ) |i'pYi(Si,w)h^dw i jo,Uan((p) Eqn. 4.72 p ’ = ------wi, / r ^ E f ^-S j-w tan M yXSi,w)h"pdw i vJ^o,Uan((p) Substituting p=+1 into Eqn. 4.72 results in Eqn. 4.73 which defines the critical value of p ' during approach action for the condition J°a.=o. Since Tn equation Eqn. 4.73 is simply a summation, over the number of tooth pairs in contact, of the integral across the face width for each individual tooth pair and a helical gear at any cross section along the z axis is a spur gear, we can examine a spur gear with a single tooth pair in contact. For this case Eqn. 4.73 reduces to Eqn. 4.74. At p '= p ’r, the output torque. Tout, is zero given any input torque, Tn. Therefore, no work may be performed by the output shaft. At p’>p„, the friction force has become so large that. Tout must be in the opposite direction of Tn, to hold the gear mesh in equilibrium. At Sj=0, the pitch point, p ’r=tan((p), which for a 25 degree pressure angle is equal to approximately 2, but as the point of contact approaches the pinion, the critical value decreases, reaching a minimum at the pinion base circle. This is shown in Figure 4.21. 1 8 2 Eqn. 4.73 p.*, = ------ Eqn. 4.74 n;, = ^ -S, tan(cp) Figure 4.21 presents a normalized plot of the critical friction coefficient, , for approach action gear tooth pair contact. The abscissa, a, of Figure 4.21 is the normalized position of along the line-of-action for the approach portion of the gear mesh cycle. The actual position along the line-of-action is r„ defined by s = a- p tan((p) ’ Dudley’s Gear Handbook [Townsend, 1991] briefly mentions this gear locking phenomenon in terms of speed increasers. As shown in Figure 4.21, the locking phenomenon is most apparent for speed increasers where the ratio of the gear base radius to the pinion base radius, ^ , is less than unity. The locking phenomenon can potentially occur for speed reduction gearing which has a — ratio greater than unity, but at higher critical friction *p coefficient levels as compared to the speed increaser case. 183 0.9 0.8 0.7 = 4 0.6 = 2 = 1.5 0.5 0.4 = 0.75 i =0.5 0.3 -0.9 - 0.8 -0.7 -0.6 -0.5 -0.4 -0.3 - 0.2 - 0.1 Pinion Nonnalized position along line-of-action. a Pitch Base Radius Point Figure 4.21 Critical friction coefficient during approach action, -1 < a < 0 Next, Eqn. 4.72 will be examined in the recess portion of the gear tooth pair contact. At the pitch point and during recess action of the gear mesh, Si-f-wtan(v|/) >0 and p=-1. Letting p=-1, Eqn. 4.72 becomes Eqn. 4.75. Eqn. 4.75 defines the critical value of |i’ during the recess portion of the mesh cycle. Noting that during recess action. — (Sj -f-wtan(Y))>0, tan( 7 i > 0 and h >1, then the denominator of Eqn. 4.75 is greater than or equal to zero. Therefore, during recess action is negative. Since by definition, |i'> 0, a negative friction coefficient condition cannot physically be reached. 184 This means that at the pitch point and during the recess portion of the gear mesh, Tout will be greater than zero. Examining a single spur gear tooth pair contact line in recess action, the output torque may be written as Eqn. 4.76. Allowing p'^+oo, the most extreme friction coefficient, results in Eqn. 4.77. Eqn. 4.77 presents the output torque to input torque ratio, , when the friction coefficient is -h » during recess action. -r„ Eqn. 4.75 l^a - " — S: tan((p) Eqn. 4.76 ""g ~^G T„ ■"p - E p Tp + tan((p) — Si tan(cp) Eqn. 4.77 * out T„ —^ + s,l tan((p) ) Plotting Eqn. 4.77 results in Figure 4.22. Figure 4.22 shows that in recess, the effects of high friction coefficient values on the output is minimal at the pitch point and approaches zero at the intersection of the line-of-action and the gear base circle. 1 8 5 3.5 2.5 out =0.75 = 0.5 1.5 0.5 0.1 0.2 0.4 0.5 0.6 0.7 0.8 0.90.3 Pitch Normalized position along line-of-action, a Gear Point Base Radius Figure 4.22 Recess action output torque to input torque ratio as p -><», 0 < a < 1 The abscissa, a, of Figure 4.21 is the normalized position of along the line- of-action for the approach portion of the gear mesh cycle. The actual position along the line-of-action is defined by s = a — tan((p) The results of the limiting case = 0 can be explained in terms of "in the effect of the friction force on the gear m esh normal force. This is graphically presented in Figure 4.23 for approach action and in Figure 4.24 for recess action. Figure 4.23 and Figure 4.24 are exploded views of Figure 4.2. In Figure 4.23 the friction forces, Fpj and F^j, act on the pinion and gear 1 8 6 bodies In a direction that tends to increase the normal mesh force. The increase in normal force then increases the friction force. Conversely, in Figure 4.24 the friction forces, Fpj and Fgj, act on the pinion and gear bodies in a direction that tends to decrease the normal mesh force which reduces the friction force. Line-of-Action not to scale, for illustrative use only Figure 4.23 Gear mesh forces, input/output torques and reaction forces acting on pinion and gear bodies in approach action 1 8 7 Line-of-Action not to scale, for illustrative use only Figure 4.24 Gear mesh forces, input/output torques and reaction forces acting on pinion and gear bodies recess in action Next, the case when - ^ = — , upper limit of Eqn. 4.70 is examined. I in Substituting the upper limit of Eqn. 4.70 into Eqn. 4.69 results in Eqn. 4.78. Simplification of Eqn. 4.78 reduces to Eqn. 4.79. Expanding Ep, Eqn. 4.53, and Eg , Eqn. 4.54, in Eqn. 4.79 in terms of a spur gear with a single tooth pair in contact, results in Eqn. 4.80. There are two conditions in which Eqn. 4.80 is valid. The first condition is the trivial condition of p=0. The second condition is given by Eqn. 4.81, which is satisfied only at Sj=0, i.e. the gear mesh pitch point. 1 8 8 Eqn. 4.78 ^ ^ ^ Tin r, (r.-£ p ) Eqn. 4.79 r £p - r £q = 0 Eqn. 4.80 + S; -r„ -Si H = 0 tan(cp) tan((p) // Eqn. 4.81 ■ + Si -r„ — Si = 0 tan(cp) tan((p) Since the model assumes line contact and at the pitch point, motion at the contact point consists of pure rolling, no energy is lost at the m esh. In reality, contact at the mesh interface is over a finite area. Therefore in real gearing there is no instant of pure rolling but a minimum energy loss point can be anticipated at the pitch point. At any position other than the pitch point for p >0, the output torque ratio to input torque ratio is always less than the ideal frictionless case of = — In conclusion, this section has shown that the model follows the physical constraints presented Eqn. 4.70 for any physically realizable friction coefficient. 189 4.5.2 3 Power Loss Due to Off Line-Of-Action Friction This section compares the computed power loss due to off line-of- action friction forces at gear mesh through two methods. The first technique computes the power loss using the input/output torque and input/output angular speeds. The second method computes the power loss from the friction force, Fpj , and sliding velocity, Vi, at the gear mesh. As expected, it can be shown that the two losses are identical for the developed quasi-static gear model. Eqn. 4.82 is the computed power loss in terms of the torques and the angular speeds. Note that Eqn. 4.82 is the power loss for both spur and helical gears of arbitrary contact ratio and is directly related to the output to input torque ratio as determined by Eqn. 4.69. Combining Eqn. 4.69, Eqn. 4.82, and Eqn. 4.83, one gets Eqn. 4.84. This equation is a generalization of the power loss calculation developed by Martin [1972,1980]. Pioss It =P|n - Pout Eqn. 4.82 =1^0,, ^ T n — Tn^in 1 _ _2HL_2!il V T n ^ i n J Eqn. 4.83 ^in 1 9 0 Eqn. 4.84 F^bsslT = Pin I (rp-ep)r. 9 / The total power loss due the friction force and sliding speed is the inner product of the friction force, Fp;, and the sliding velocity, Vj y, at each tooth pair summed over the number of tooth pairs in contact. This product is given in Eqn. 4.85. The negative sign compensates for the sign difference of the friction force and sliding speed during approach and recess action. Combining the normal force distribution model discussed in Section 4.4.1 with Eqn. 4.2 results in Eqn. 4.86. Eqn. 4.87 restates the gear mesh sliding speed for tooth pair i with the gear pair supported on rigid bearings, as discussed in Section 4.4.2.1. Enq. 4.88 is a rearranged form of Eqn. 4.68, the transverse mesh force. Eqn. 4.89 results when Eqn. 4.85, Eqn. 4.86, Eqn. 4.87 and Enq. 4.88 are combined. This approach is a generalization of the power loss computation performed by Tso [1961]. Eqn. 4.85 Piossjp = Rn ~Pom = ' YŸ) i Wl, N ^ Wl, \ Eqn. 4.86 Ppi — JpYihdw |f,^y= JpYihdw |<„,0hy Eqn. 4.87 % = +wtan(Y))Q;, T O P Enq. 4.88 ""p -Gp Tp-Sp 191 wl, + ' g ) Eqn. 4.89 P,loss Ip JnYj(Sj +wtan('F))h^dw ''.('•p-Ep) ' V«°i By energy conservation, the energy loss measured through torque must equal the energy lost due to the friction force. This implies that Pioss |t =Pioss|p shown in Eqn. 4.90 where the following simplifying notation s' =Sj + wtan(Y) is introduced. Multiplying Eqn. 4.90 by r^(rp -Sp) results in Eqn. 4.91. Inserting the Sp, Eqn. 4.53, and Eg, Eqn. 4.54 into Eqn. 4.91 results in Eqn. 4.92. Algebraically reducing Eqn. 4.92 gives Eqn. 4.93. The left hand side of Eqn. 4.93 equals the right hand side of Eqn. 4.93 proving that Eqn. 4.82, P|oss |t-. and Eqn. 4.85, P,pss|p, are identically equal. ^ (""g-Eg) i;, ^ Wl, \ Eqn. 4.90 Pj^ j pYjS'h^dw (Cp-Ep) r, 9 / \ w Q , j ' ' w l , Eqn. 4.91 r^Eg - r^Ep = -(r^ + r j ^ J^iY|S Vdw l^wO, ' ' w l , r ^ "l —^ - s h^dw ■ + s h^dw r p l tan((p) J I JV y : tan(cp) J Eqn. 4.92 \wO, - k + rj% J^YiS'h^dw ' I wO, wl. wl, Eqn. 4.93 ~(rp+rg)X jnYiS'h^dw = ~('’p+''9)X jnYiS'h^dw ' VwOi 192 This section has proven that the power loss determined through the torque and angular speeds equals the power loss determined using the off line-of-action friction force and the sliding speed for the proposed model in quasi-static conditions. This section has extended power loss computation techniques of previous authors such as Martin and Tso to include helical and spur gears of arbitrary contact ratios. Another contribution is that works of the discussed authors and most previous attempts at power loss computations would not pass the P,oss |t =Pioss|p test. This can be explained by the fact that the previous attempts simplified the problem by assuming that the normal force remained constant throughout the mesh cycle. This research allows the off line-of-action friction force to affect the line-of-action normal force throughout the mesh cycle. 4.6 Numerical Solution of Gear Model Supported on Rigid Bearings This section numerically examines the rigid bearing gear model incorporating friction force at the gear mesh in a linear time varying form that was presented by Eqn. 4.60 and Figure 4.10. This implies all gear parameters are known as a function of the independent variable. The gear parameters km, kg. and all viscous damping terms are assumed to be time invariant. These param eters are left as time invariant parameters to emphasize the effect of off line-of-action friction force on the normal mesh force and the output torque. For the spur gear case, the gear mesh stiffness, 1 9 3 km, is the mean mesh stiffness. In practice, the time varying gear mesh stiffness can represent a significant dynamic excitation source, particularly in spur gearing. Besides the dependent variables, only £p and change with time, and are known apriori as computed by Eqn. 4.53 and Eqn. 4.54 The numerical solution of the linear time varying form of Eqn. 4.60 is accomplished using a 4th/5th order Runge-Kutta adaptive step size algorithm [Mathworks, 1992]. Table 4.1 defines the baseline parameters used in the gear model. The parameters represent the gear pair used in the back-to-back gear test rig for the experimental part of this research. Viscous damping is introduced to eliminate the homogenous part of the solution leaving the steady state component. Static transmission error is neglected and Tin is assumed constant to emphasize the parametric effect of the friction force a s a mesh excitation. This reduces Eqn. 4.65 to Eqn. 4.94. Eqn. 4.94 u = in Table 4.2 lists the natural frequencies, cù^ and cOnj, damped natural frequencies of oscillation, cOj,, and baseline viscous damping values, modal damping ratios, and the mode shapes for the gear parameters in Table 4.1, determined by assuming zero off line-of-action mesh friction force. Neglecting off line-of-action friction force reduces the problem to a linear time invariant system which implies Ep and Eq are equal to zero. The 194 lower natural frequency mode corresponds primarily to out of phase motion of the pinion and gear body. The second natural frequency mode corresponds primarily to in phase motion of the pinion and gear resulting in large deflections of the mesh spring. Based upon the time invariant natural frequencies, Table 4.3 list the operating conditions that are numerically examined with the Runge-Kutta method. Table 4.3 presents the input speed, in rpm, the corresponding parametric excitation frequency, cop, and the period of the parametric excitation. The parametric excitation frequency is equal to the gear mesh frequency and varies the stiffness term of the differential equation. The conditions chosen respectively correspond to operating well below the first natural frequency, excitation of the first damped frequency of oscillation, operation between the first and second natural frequencies, excitation of the second damped natural frequency, and operating well above the second natural frequency. 1 9 5 Units Pinion Gear Teeth 25 47 Base Diameter inch 4.698 4.229 Outside Diameter inch 5.608 10.192 Face Width inch 1.25 Center Distance inch 7.5 Normal Diametral Pitch 1/inch 5 Operating Pressure Angle degree 25 Face width inch 1.25 Helix Angle degree 0 b o inch 0.59 z. inch -0.429 LPSTC inch -0.183 Zb inch 0.407 Contact Ratio 1.41 la. Ibf-in-s*^ 0.06 0.72 rT ii, Ibf-s'^/in 0.02 0.07 k m Ibf/in 2 .9 x 1 0 '’ k . Ibf-in 6 .5 x 1 0 '’ n 0.05 C d . d Ibf-s/in 500 C e ,0 Ibf-s/rad 500 C o . 9 Ibf-s/rad 0 Cb.D Ibf-s/in 0 Table 4.1 Baseline gear parameters used for the linear time varying numerical solution (See Figure 4.10) 1 9 6 no damping baseline damping mode 1 mode 2 mode 1 mode 2 natural freq., coni, ct)n2 419 Hz 2966 Hz 419 Hz 2966 Hz damped natural freq. of N/AN/A 0)^1=418 Hz Q)jJ2=2964 Hz oscillation, • damping ratio 0 0 0.05 0.04 modal amplitude, D 1 1 -(0.09+i0.08) 0.28+i0.96 modal amplitude, 0zg -8.49 0.053 0.70+i0.72 0.0l4-i0.05 Table 4.2 Natural frequencies and mode shapes for undamped and damped friction less case rpm parametric excitation periodic frequency period, sec. comment CÛD, Hz 19 8 1.25 X 1 o ' c O p « 0 ) ( ji 1000 418 2 .3 9 x 1 0 ^ ^ d l 4056 1690 5.92 X lO'^ Ü)tjT<(üpCQ)(j2 7143 2964 3.37 X 10*^ Ü)p=Û)jj2 71438 29640 3.37 X 10'^ 0)p»CÙ^2 Table 4.3 Operating conditions examined in numerical study 1 9 7 Figure 4.25 through Figure 4.35 present the dynamic normal mesh force, the dynamic output torque, and the dynamic friction force for various operating conditions. The results are grouped into operating conditions away from natural frequencies and operating conditions near natural frequencies. In each case, the friction coefficient was assumed to be 0.05 and it was assumed to remain constant along the entire line-of-action. The input torque is a constant input of 1000 Ibf-in. The independent variable, time is transformed to position along the line-of-action through Eqn. 4.95. Eqn. 4.95 s = (a,„rp)t The position along the line-of-action is defined as the coordinate along the x axis as defined in Figure 4.3 with the lowest point of single tooth contact shifted to zero. The presented results are for times after the initial transients due to the selected initial conditions have decayed away. The results appear periodic over the few mesh cycles presented, but no detailed study was performed to assure absolute periodicity. Unlike linear time invariant problems, in linear time varying problems there is no theory stating that the dependent variable’s period of oscillation must be equal to the excitation’s period of oscillation. Examining the longer periods of oscillation is left for further research. 1 9 8 Figure 4.25 presents the dynamic mesh force, when the operating conditions are away from any predicted natural frequencies. In the static case, cOp =8 Hz, the response appears similar in structure to the normal force for a contact ratio of 1.25 presented in Figure 4.13 with a superimposed damped high frequency component. The high frequency component is dominated by oscillations at the second natural frequency. The transient is due to the step change in friction force as the contact crosses the pitch point and at the position where the contact changes from single to multiple tooth pair contact. The high speed case, o)p= 29640 Hz, is simply a case of the fundamental parametric excitation frequency being too high for the gear components to follow, resulting in a flat line response. Figure 4.26 presents the dynamic mesh force when the first damped frequency of oscillation is excited by the fundamental parametric excitation frequency, oOp =418 Hz. The response at this speed deals primarily with out of phase motion of the pinion and gear elements and does not primarily affect the mesh force as can be seen by comparing the relative amplitudes of Figure 4.25 and Figure 4.26. The parametric excitation frequency, o)p = 418 Hz, does not excite only the 1st natural frequency, as can be seen by the 7 cycles per base pitch frequency component superimposed on Figure 4.26. Figure 4.27 presents a frequency spectrum for the results in 199 Figure 4.26. The seventh harmonic of the fundamental parametric excitation frequency, is extremely close to the 2nd damped natural frequency of oscillation, 7c0p =0.989cOp2- Figure 4.28 is the dynamic mesh force when the fundamental parametric excitation frequency, cOp, is at the second damped natural frequency of oscillation, Figure 4.28 compares the effect of an increased input torque, Tin=3000 Ibf-in and an increase in the viscous damping, Cd,d=C=1500 Ibf-s/in and Ce.e=C=1500 Ibf-s/rad. For Tjn= 1000 Ibf-in, the damping affects the steady state amplitude. When the input torque is increased to 3000 Ibf-in, and the increased damping is used, the model predicts loss of contact as observed by the negative normal load. The model developed in this research does not model the loss of contact phenomenon once the tooth pair has separated but does predict operating conditions where loss of contact occurs. For the 3000 Ibf-in case, neglecting tooth friction results in a static normal load of 1277 Ibf. The use of the model developed in this research for loss of contact prediction is left for further research. 2 0 0 460 one two %=8 Hz tooth tooth ► 0 ^ =1690 Hz 450 pair pair ' ûj^ =29640 Hz PP HPSTC s o x: (DU3 E 420 CO o c 410 400 390 0.2 0.4 0.6 0.8 1.2 1.4 LPSTC position alongalong the line-of-action,action inch Figure 4.25 Dynamic mesh force, fm, fundamental parametric excitation frequency, cOp, at off resonant conditions, T,n=1000 Ibf-in, frictionless normal mesh force=425 Ibf 2 0 1 4 6 0 B 420 1 1.5 2 2.5 position along the line-of-action. inch Figure 4.26 Dynamic mesh force, fm, fundamental parametric excitation frequency exciting the first damped natural frequency of oscillation, cOp = co^i, Tin= 1000 Ibf-in, frictionless normal mesh force=425 Ibf normal mesh force for first frequency mode condition 16 14 7(ùp = 0.989 (0^2 « 12 i (QE 10 ; 8 2cû. (0 6 cI 4 2 0 0 500 1000 1500 2000 2500 3000 3500 4000 frequency, Hz Figure 4.27 Frequency spectrum of the dynamic mesh force, fm, for the results presented in Figure 4.26. 2 0 2 3500 T-=1000 IbWn. C=500 3000 T«=1000 Ibf-in. C=1500 Tr=3000 Ibf-in. C=1500 2500 _ 2000 M 1500 I 1000 -500 -1000, 0.5 1 1.5 2 2.5 position along the line-of-action. inch Figure 4.28 Dynamic mesh force, fm, fundamental parametric excitation frequency exciting the second damped natural frequency of oscillation Figure 4.29 through Figure 4.32 present the dynamic output torque, Tout, at three different parametric excitation conditions. The presented dynamic output torques are computed based upon only a constant input torque. The time varying output torque oscillations are a potential torsional exciter for rotating equipment attached to the output side of the gear mesh. The output torque oscillation due to the off line-of-action friction force as a dynamic excitation source is left for further research. Figure 4.29 presents the dynamic output torque when the fundamental parametric excitation frequency, cOp, is away from any system resonant frequencies. The near static case, C0p= 8 Hz , appears similar to 2 0 3 the static output torque case, computed for the contact ratio of 1.25, as presented in Figure 4.17, but has the first natural frequency transients superimposed. Again, the extremely high speed case, o)p= 29640 Hz, is a flat line, but with an output torque that is lower than the frictionless ideal case. Figure 4.30 presents the frequency spectrum of the output torque for the case of cOp = 8 Hz. In Figure 4.30 the fundamental frequency, cOp, is the dominate frequency component with the first natural frequency transient response also being apparent. Figure 4.31 presents the dynamic output torque when the fundamental parametric excitation frequency equals the first damped frequency of oscillation, cOp = 0)^,. The amplitude of the output torque is significantly larger than the results for the cases where the fundamental parametric excitation frequency was at off resonant conditions. Figure 4.32 presents the dynamic output torque when the fundamental parametric excitation frequency is at the second damped natural frequency of oscillation. The output torque amplitude is slightly larger than the amplitudes of Figure 4.29, where cOp is at off resonant conditions, but significantly less than the amplitudes shown in Figure 4.31, where û)p = copi. This can be explained due to the fact the first natural frequency mode effects mainly the output torque, Tp^, = -k^e^Q whereas the second natural frequency mode affects mainly the normal mesh force. 2 0 4 1890 ca, =8 Hz % = 1690 Hz 0 4 =29640 Hz 5 1870 a- 1865 0.6 0.8 1 1.2 1.4 position along the line-of-action. inch Figure 4.29 Dynamic output torque, Tout, fundamental parametric excitation frequency, cOp, at off resonant conditions, Tin=1000 Ibf-in, frictionless output torque=1880 Ibf-in C .Q Q. E o(0 E o 3 Q. O3 100 200 300 400 500 frequency, HzHz Figure 4.30 Frequency spectrum of the dynamic output torque. Tout, for the case of (Op =8 Hz presented in Figure 4.29 2 0 5 2200 2100 2000 Ç I 1900 g 1800 1700 1600, 0.5 1 1.5 2 Z5 position along the line-of-action. inch Figure 4.31 Dynamic output torque, Tout, fundamental parametric excitation frequency exciting the first damped natural frequency of oscillation, cOp = , Tin=1000 Ibf-in, frictionless output torque=1880 Ibf-in 1920 1910 1900 1890 à0 1880 1 o 1870 ■3 t o I860 1850 1840 Tin=1GC0 Ibf-in, 0=500 Tio=1000 Ibf-in. 0=1500 1830, 0.5 1 1.5 2 2.5 position along the line-of-action. inch Figure 4.32 Dynamic output torque. Tout, fundamental parametric excitation frequency exciting the second damped natural frequency of oscillation, (Dp = (0^2 ' Cdd=Cee=C, frictionless output torque=1 8 8 0 Ibf-in 2 0 6 Figure 4.33 through Figure 4.35 present the dynamic off line-of-action friction force at three different parametric excitation conditions. The first condition is when the fundamental parametric excitation frequency is not near any system resonances. The second condition is when the fundamental parametric excitation frequency is on the first damped natural frequency of oscillation, co^,, and the third condition is when the fundamental parametric excitation frequency is on the second damped natural frequency of oscillation, co^g- The presented dynamic off line-of-action friction forces are computed based upon only a time invariant input torque. The time varying friction force is a potential dynamic source that can be transmitted through the gear blank, the gear shaft, through the support bearings and finally to the housing. The friction force is assum ed to be the normal mesh force times the friction coefficient, p., as discussed in section 4.4.2. As will be seen in Figure 4.33 and Figure 4.34, the friction force can be closely approximated by the static normal force times the friction coefficient, without concem for the normal force dynamics. When the fundamental parametric excitation frequency corresponds with the second natural frequency, the normal force dynamics must be considered as shown in Figure 4.35. In Figure 4.33 through Figure 4.35, during double tooth pair contact, the friction forces cancel out to zero. The friction force cancellation during double tooth pair contact results are based upon, one tooth pair is in approach and the other tooth pair is in recess action, the teeth have equal load sharing and 2 0 7 the friction coefficient is assumed constant throughout the mesh cycle. Figure 4.35 shows the significant effect that the gear dynamics have on the friction force when the parametric excitation frequency excites the gear mesh frequency, 0)^,2. Static transmission error effects on the developed model are included in the model derivation but have been neglected in the presented results and should be incorporated in future research. By incorporating static transmission error effects, guidelines suggesting when the off line-of-action friction force becom es a significant dynamic excitation source could be developed. The off line-of-action friction force results presented in Figure 4.33 through Figure 4.35 represent idealized cases. To compare the results of the model developed in this research against experimental results would require more accurate load sharing models and friction coefficient models. The presented off line-of-action friction forces in this research change direction at the pitch point. These results compare with the experimental friction force measurements performed by Rebbechi [1991,1996]. Incorporating more accurate load sharing models and a time variant friction coefficient for the comparison with experimental results is left for future research. 2 0 8 = 8 Hz one two (üj, = 1690 Hz tooth tooth ► pair pair = 29640 Hz 8 o c o u -10 -15 -20 -25 0.2 0.4 0.6 0.8 1.4 position along the line-of-action, inch Figure 4.33 Off line-of-action dynamic friction force, Qpy, acting on the pinion body with cOp at off resonant conditions, T^=1000 Ibf-in 2 0 9 25 20 one two tooth tooth pair pair _o (D 5£ o c o -10 -15 -20 -25 0.5 2.5 position along the line-of-action. Inch Figure 4.34 Off line-of-action dynamic friction force, Qpy, acting on the pinion body with excitation frequency exciting the first dam ped natural frequency of oscillation, cOp =oidi> Tin= 1000 Ibf-in 2 1 0 200 Tio=1000 Ibf-in, C=500 150 Tin=1000 Ibf-in, C=1500 T«=3000 Ibf-in, C=1500 100 one two tooth tooth pair pair 50 i o c o u -50 -100 -150 -200 0.5 2.5 position along the line-of-action, inch Figure 4.35 Off line-of-action dynamic friction force, Qpy, acting on the pinion body with excitation frequency exciting the second damped natural frequency of oscillation, cOp = 00^2 . for various Tin, and Cdd=Cee=C 4.7 Analytical Solution of Gear Model Supported on Rigid Section 4.6 examined the numerical solution of the linear time varying system incorporating friction force at the gear mesh. For most general linear time varying systems numerical solutions are required [Richards, 1983]. Certain time varying forms lend themselves to exact analytical solution. One such form is the Meissner equation presented as Eqn. 4.96 [Richards, 2 1 1 1983]. The terms Ai and Aa, where A 1M 2, periodically repeat on the period T. The Meissner equation form can approximate the linear time varying system developed in section 4.5 for a contact ratio of unity. The approximation takes the exact functions of Sp and £q , which relates the effect of the off line-of-action friction on the gear system, and casts them in the form of the Meissner equation. Figure 4.36 presents the exact and the used approximation of Ep and Eq , for the gear parameters listed In Table 4.4. The approximated Ep and Eq form two linear time invariant pieces that are periodic over the base pitch, bp, describing the Meissner equation. The next section introduces an exact solution to a periodically forced vector form of the Meissner equation which may solve the problem of off line-of-action friction forces acting on a gear system for gear systems of contact ratio of unity. Using the presented solution methodology, an exact solution may be developed to solve gear friction problems of contact ratio greater than unity. This entails extending the solution for the parametric time varying component. Ai, to cases where i>2. This extension is left for future research. 2 1 2 0.15 Approach Action Ep exact Ep approximate 0.1 Eq exact Eq approximate 0.05 Recess Action -0.05 - 0.1 - 0.1 - 0.2 - 0.1 0.1 0.2 0.3 position along the line-of-action■action Figure 4.36 Epand Eq versus position along line-of-action, contact ratio= 1, bp=0.59 4.7.1 Analytical Solution of the Vector Form of the Meissner Equation The solution of the vector form of the Meissner equation is presented in this section. A solution to a general system of first order linear time varying differential equations, Eqn. 4.97, is presented as an overview. More information about the general solution presented can be found in Zadeh[1963] or Richards[1983]. The technique of variation of param eters can be used to form the solution of Eqn. 4.97. A general solution of the form given by Eqn. 4.98, is assumed, where di(t) are differentiable weighting functions and x^{t) are 2 1 3 fundamental or basis solutions to the homogeneous form of Eqn. 4.97. Forming a matrix out of the fundamental solution vectors, >q(t), results in Eqn. 4.100, which is called the Wronskian matrix. Substituting Eqn. 4.98 into Eqn. 4.97 and integrating, results in Eqn. 4.101 the solution for the weighting function vector, d(t). Introducing Eqn. 4.102 into Eqn. 4.101 results in Eqn. 4.103. Eqn. 4.102 implies that for a system of n simultaneous first order differential equations, the Wronskian must contain n linearly independent fundamental solution vectors of length, n. Finally, defining the state transition matrix in Eqn. 4.104, Eqn. 4.103 can be transformed into Eqn. 4.105. Therefore, the solution of Eqn. 4.97 can be reduced to finding the Wronskian matrix, satisfying Eqn. 4.97. Eqn. 4.97 x(t) = [A(t)]x(t) + f(t) Eqn. 4.98 x(t) = % d;(t)^(t) = [W(t)]d(t) i=i Eqn. 4.99 X;(t) = Xni(t) Xii(t) Xln(t)‘ Eqn. 4.100 [W(t)] = [x,(t) ••• x„(t)] = Xn,(t) Xnn(t) Eqn. 4.101 d(t) = d(0) + 1[W(x)]'’f(x)dx Eqn. 4.102 d(t) = [W(t)] x(t) 2 1 4 I Eqn. 4.103 x(t) = [W{t)IW{ 0)]''x (0) +j[W{t)IW(T)]''f(t)dt Eqn. 4.104 1 -1 / t ' Eqn. 4.105 x(t) = [ The linear time varying solution presented in Eqn. 4.105 will be applied to the solution of the vector form of the forced Meissner equation. Books and published literature deal mainly with the homogeneous forms of linear time varying systems and on topics of stability. The key to finding the forced solutions to linear time varying problems is finding the Wronskian matrix of the homogenous form [Richards, 1983]. The stability of the linear time varying system is also extensively covered. The Wronskian matrix for tractable forms of linear time varying systems and discussions on stability can be obtained in Richards[1983], Zadeh[1963] and Pipes[1953]. Timoshenko[1937] derives a forced linear time varying formulation for the vibrations in a side rod drive system of electric locomotives but only examines the stability of the homogeneous system. Richards[1983] presents a forced response to a sinusoidal excitation. The solution is limited in the sense that the state transitions matrix , |o(ti,to)], can be composed of only one Wronskian matrix, [o(ti,to)]= [W(tjIW(to)]"\ A major idea to the solution of periodic linear time varying problems is that the state transitions 2 1 5 matrix can be constructed piecewise from any number of different Wronskian matrices, [(t2,to)]=[W 2(t2)IW 2(ti)]'’[W,(ti)IW,(to)]‘’ where t2 Two important observations emerge from Eqn. 4.106. The first observation is that the steady state solution contains a large number of frequency components whose relative amplitudes depend on the participation factor. Second, the steady state solution is aperiodic unless cop and coi are commensurate. Eqn. 4.106 x„ = £ ^ gV^(üJi+(n+k)ûn,> n=-«« k=- Limiting [A(t)] to periodic time varying coefficients that satisfy Eqn. 4.107 reduces Eqn. 4.97 to a form of the Hill equation. Eqn. 4.108 is introduced for the independent variable, time. [A(t)] is further defined by Eqn. 4.109, and f(t) is limited to periodic functions of period T as described by Eqn. 4.110 and Eqn. 4.110a. Eqn. 4.107 [A(t + T)] = [A(t)] t = mT + x m = 0,1,2... Eqn. 4.108 ^ 0 216 E q n .4 ,0 9 [A(t)] = [A,(x)] %T<% Eqn. 4.110 f(t + T)=f(t) f(t) = û,(%) 0 A recursive solution to the nonhomogeneous problem is presented in Eqn. 4.111. Eqn. 4.111 piecewise solves the problem by using the last solution from the previous section as the initial condition for the next section. Starting with the initial conditions at t=0, (m=0 and %=0), an exact solution for any time t can be determined. Defining Eqn. 4.112 and Eqn. 4.113, Eqn. 4.114 and Eqn. 4.115 present solutions to these “pseudo” initial conditions, x(mT)and x((m + ^)T), for each individual piecewise section. Though shown here for two piecewise sections, this process can be extended to any number of finite pieces within the fundamental period T. The recursive solution for the “pseudo” initial conditions, Eqn. 4.114 and Eqn. 4.115, has an explicit solution presented in Eqn. 4.117 and Eqn. 4.118 where Ki and Ka are defined by Eqn. 4.116. 0 < % < ^T x(mT +%) = [ / Z-4T x(mT + x) = [‘t>2(x.^T); x((m + Ç)T)+ + \ 0 2 1 7 ST Eqn. 4 . 1 1 2 X, = J[ TIM) Eqn. 4.113 Xj = J [^>2 (t,0 )] U2 (T + 4T)dx x(mT)=x(0) m = 0 Eqn. 4.114 x(mT)= [ Eqn. 4.115 x((m + 4)T)= [<(>, feT,0)](x(mT)+ X, ) K, =[<6,ftT,0)] Eqn. 4.116 K,=[4.,(T,|T)] x(mT)=x(0) m = 0 Eqn. 4.117 m . \ \ x(mT)=(KjK,ri<(0)+ 2 K K J X, + £(K;K,)i (<,X. m >1 Eqn. 4.118 x((m + = (k.KgrK,x(0) + 1"% (K,K 2) )(,X , +f]x . I . 1=1 . For the periodic linear time varying vector form of the Meissner equation, the matrix [A(t)j is divided into two linear time invariant pieces, [A(t)] = [A, ] for 0 < X < and [A(t)] = [A 2] for < % < T . The key to the solution of the linear time varying problem is the determination of the state transition matrix which is built up from the Wronskian matrix. For the vector form of the Meissner equation, the Wronskian matrix is easily constructed for each piecewise section with the respective eigenvectors, E^, and the 2 1 8 eigenvalues, h, that are computed from the time invariant matrix [A,]. Eqn. 4.119 presents the eigenvector/eigenvalue problem for each piecewise section and Eqn. 4.120 presents the construction of the Wronskian for that respective section. 'k, 0 0 0 _ 0 X, 0 0 Eqn. 4.119 [A,][Ê, •• E„] = [E, Ep 0 0 ■. 0 0 0 0 Xp e^’' 0 0 0 ' 0 e^= 0 0 Eqn. 4.120 [W|(t)] = |E, 0 0 0 0 0 0 e^"' 4.7.2 Vector Form of the Meissner Equation applied to Off Line-Of- Action Friction Force in a Gear Mesh This section analytically examines the rigid bearing gear model incorporating friction force at the gear mesh in a linear time varying form as presented by Eqn. 4.60 and schematically as Figure 4.10. This implies that all gear parameters are known as a function of the independent variable, time. The gear parameters km, kg. and all viscous damping terms are time invariant. These parameters are left as time invariant parameters to emphasize the effect of off line-of-action friction force on the normal mesh force and the output torque. For the examined spur gear case the mean 2 1 9 mesh stiffness between single and double tooth pair contact is used. In practice, the time varying gear mesh stiffness can represent a significant dynamic excitation source, particularly in spur gearing. Besides the dependent variables, only Sp and Eq change with time. These variables are known apriori in Eqn. 4.53 and Eqn. 4.54 Table 4.4 presents the gear system parameters used for the examination of the linear time varying system developed in Section 4.5.1 and presented schematically as Figure 4.10. The parameters are virtually identical to the parameters used in the numerical case examined in section 4.6 except the parameters, Za and Zb are adjusted to create a gear pair with a contact ratio of one. A contact ratio of one was selected to match the vector form of the Meissner equation developed in Section 4.7.1. By extending the linear time varying solution to contain more time invariant pieces (See Section 4.7.1) gear problems with contact ratios greater than one or position varying friction coefficients may be solved. The mass, stiffness values, damping values and friction coefficient are identical to the test case examined in section 4.6. This also means that the linear time invariant system, with Ep and Eq equal to zero, natural frequencies and mode shapes are also identical and listed in Table 4.2. 2 2 0 Units Pinion Gear Base Diameter inch 4.698 4.229 Outside Diameter inch 5.608 10.192 Face Width inch 1.25 Operating Pressure Angle degree 25 Face width inch 1.25 Helix Angle degree 0 bo . inch 0.59 Za inch -0.295 PP 0 Zf inch 0.295 Contact Ratio 1.0 Ep (approach,recess) 0.04,-0.07 Eq (approach, recess) 0.12,-0.09 Iz,, i=P,G Ibf-in-s^ 0.06 0.72 m,, i=P,G Ibf-s^/in 0.02 0.07 Kn Ibf/in 2.9 X 10" ki Ibf-in 6.5 X 10*” F 0.05 Coo ibf-s/in 500 Ce.8 Ibf-s/rad 500 Co.e Ibf-s/rad 0 Ce.o ibf-s/in 0 Table 4.4 Baseline gear parameters used for comparison between numerical and analytical linear time varying numerical solution 2 2 1 Using the parameters of Table 4.4 and the equations developed in section 4.5.1, the linear time varying problem incorporating gear friction can be approximated (See Figure 4.36) in the form of the Meissner equation and may be solved using the results of section 4.7.1. The state matrices [a J andlAg] are expressed in terms of the friction force gear model parameters [AJsg and [B^. The state space model matrices [A]gs and |B]gs were developed in Section 4.5.1 and are evaluated using the approximate Ep and Eq values presented in Figure 4.36. Eqn. 4.123 and Eqn. 4.124 relate the parametric period, T, and the transition point between the respective piecewise components, to the gear model parameters. The reciprocal of T is the gear mesh frequency, while ^ represents the normalized location at which the tooth contact pair crosses the pitch point. Eqn. 4.121 [Ai ]= [a J^[b L 0 Eqn. 4.122 [Ag]= [ A ÇT < t < T recess action Zk-Z= Eqn. 4.123 T = -b Tp^in PP-Z Eqn. 4.124 ^ = a 2 2 2 Figure 4.37 through Figure 4.41 present comparisons between the analytical solution results and the numerical solution results for the gear system modeled as the Meissner equation. The position along the line-of- action is defined as the x axis coordinate as defined in Figure 4.3. All the figures start at t= 0 , to show agreement in the homogeneous part of the solution which is dependent upon the initial conditions. Figure 4.37 presents a side by side comparison of the mesh force for the gear mesh operating in the static region, £^m=19 rpm. Unlike the side by side comparison of Figure 4.37, the comparison plots of Figure 4.38 through Figure 4.41 presents overlayed numerical and analytical data. Figure 4.38 expands in on the results presented in Figure 4.37 at the time t=0*. Figure 4.39 presents the output torque for the case of Qin=19 rpm. Figure 4.40 and Figure 4.41 present the overlay comparison for dynamic mesh force and the dynamic output torque at an input speed of 7143 rpm. This speed corresponds to the fundamental parametric excitation frequency, cop, exciting second damped frequency of oscillation, C 0d2. as computed from the frictionless linear time invariant model. In the presented sample runs, the agreement between the analytical and numerical solution is virtual identical. 2 2 3 460 4 5 0 Pitch Point Pitch Point 4 4 0 I 4 1 0 g 400 3 90 0.2 0.4 0.6 0.8 position along the line-of-actionaction 460 4 5 0 Pitch Point Pitch Point 4 4 0 ÔS 430 1I 420 oI 410 c 400 390 0.2 0.4 0.6 0.8 position along the line-of-actionaction Figure 4.37 Normal mesh force comparison between analytical (top) and numerical (bottom) solution of the Meissner equation, cOp = 8 Hz, Tjn=1000 Ibf-in, p =0.05, bp=0.59, contact ratio=1, frictionless normal mesh force=425 Ibf 2 2 4 445 440 5 435 OS s E 430 g 425 420. 0.01 0.02 0.03 0.04 0.05 position along the line-of-actionline-of- Figure 4.38 Expansion of Figure 4.37; normal mesh force overlay comparison between numerical and analytical solution of the Meissner equation, oOp = 8 Hz, Tfn=1000 Ibf-in, |i'=0.05, bp=0.59, contact ratio=1, frictionless normal mesh force=425 Ibf 1885 1880 1875 Ç 1870 B “ 1865 3 Q. 1855 E Pitch Point 1850 1845 1840 0.1 0.2 0.3 0.4 0.5 0.6 position along the line-of-action■action Figure 4.39 Output torque overlay comparison between numerical and analytical solution of the Meissner equation, o)p = 8 Hz, Tin=1000 Ibf-in, p.'=0.05, bp=0.59, contact ratio=1, frictionless output torque=1880 Ibf-in 2 2 5 900 800 700 x> 600 <0 500 i j= 400 3 E 300 1§ 200 o 100 -100 25 position along the line-of-action Figure 4.40 Normal mesh force overlay comparison between numerical and analytical solution of the Meissner equation, ojp = 2964 Hz, Tin=1000 Ibf-in, p. =0.05, bp=0.59, contact ratio=1, frictionless normal mesh force=425 Ibf 1940 1920 1900 S -@ 1880 F 2 I860 I g 1840 1820 1800. 25 position along the line-of-action Figure 4.41 Output torque overlay comparison between numerical and analytical solution of the Meissner equation, cOp = 2964 Hz, T,n=1000 Ibf-in, p.*=0.05, bp=0.59, contact ratio=1, frictionless output torque=1880 Ibf-in 2 2 6 JL The Runge-Kutta algorithm [Mathworks, 1992] used in this research for the numerical solution has an adaptive step size for accuracy and therefore it is difficult to get uniformly spaced solution steps. The analytical solution has no such limitation and lends itself to standard frequency domain analysis such as the Discrete Fourier Transform (DFT). Another major advantage is in computation time of the analytical solution numerical evaluation time versus the numerical solution, particularly since the steady state solution is desired. The closed form solution may potentially be studied in the frequency domain analytically by applying the Fourier Transform, Fourier Series, or some potential joint time-frequency analysis. In this research, the closed form solution is evaluated numerically at various input speeds to get a data ensemble that is converted to the frequency domain by the application of the DFT. The sampling parameters include a sampling rate of 24 KHz and an ensemble of 2400 sampled points. This gives a frequency resolution of 10 Hz. The input speed range is from 96 rpm to 9600 rpm in 96 rpm increments. This speed combination and increment result in the minimum mesh frequency of 40 Hz to a maximum mesh frequency of 4000 Hz in 40 Hz increments. Figure 4.42, Figure 4.43, Figure 4.45 present waterfall plots of normal mesh force, output torque, and off line-of-action friction force, respectively. 2 2 7 In Figure 4.42, when the fundamental parametric excitation frequency is in the vicinity of the second natural frequency. The harmonic normal force amplitude is near the mean normal force of 425 Ibf, creating a potential loss of contact situation. For the output torque waterfall plot, Figure 4.43, the maximum frequency component amplitude is approximately 40 Ibf-in, but occurs when the fundamental parametric excitation frequency is in the vicinity of the second natural frequency. This is contrast to the numerical simulation performed in Section 4.6. For the numerical simulation the maximum output torque harmonic amplitude occurred when the fundamental parametric excitation frequency is in the vicinity of the first natural frequency (See Figure 4.29 through Figure 4.32). Figure 4.44 presents a comparison between the steady state solutions for the Meissner formulation computed both numerically and analytically for the case of cùp=cùdi showing matching results. The small harmonic output torque amplitude at cùp=o)di is a result of the approximated ep and eq functions (See Figure 4.36) used to fit the model to the Meissner equation. The approximated ep and eq functions are created by selecting one point along the line-of-action from the approach section and one point from the recess section of the exact functions. This approximation fixes the output torque to the output torque based upon the selected approximate ep and eq. Since the output torque is approximately symmetric about the pitch point (see Figure 4.17) the nominal output torque change over the mesh cycle is decreased. The decrease in nominal output torque 2 2 8 can be seen in Figure 4.39 and compared to the actual nominal output shape presented in Figure 4.17. in effect, the current analytical formulation requires the user to select an approximated ep and eq approximation that either accurately models the normal mesh force or output torque, while sacrificing accuracy of the other dependent variable output. This limitation can be reduced by increasing the number of time varying steps in the approximated ep and eg functions. Figure 4.46 presents the waterfall plot of the off line-of-action friction force. Due to the nature of the off line-of-action friction force waveforms (See Figure 4.33 through Figure 4.35), the frequency spectrum contains an infinite number of harmonic components which at high frequencies become aliased. Therefore, the waterfall plot presented in Figure 4.46 has been cleaned up by setting any harmonic amplitude less than 4 Ibf equal to zero. The one large aliased component occurs at ScOp is not removed by the cleanup operation due to its large amplitude. Figure 4.46 presents a contour view of Figure 4.45 out to the Nyquist sampling rate of 12000 Hz showing the 5oùp aliased component. Examining Figure 4.45 reveals a constant fundamental harmonic amplitude over the entire speed range even when cûp=û)d 2- At cûp=cûd2, the normal mesh force harmonic amplitude is approximately 425 Ibf which results in a off line-of-action friction force amplitude of (0.05)(425 lbf)=21 Ibf. The effect of the increased normal mesh force at Q)p=c 0d2 on the off line-of-action friction force output occurs at 2cop. 2 2 9 (0 200 \ c 100\ 10000 input speed, rpm 0 \oOO 2000 3000 4000 5000 6000 frequency, Hz Figure 4.42 Waterfall plot of dynamic mesh force, Tm = 1000 Ibf-in, |i*=0.05, bp=0.59, contact ratio=1, frictionless mean mesh force= 425 Ibf 2 3 0 0000 ^ , _ k input speed, rpm 0 ',000 '2000 3000 4000 5000 6000 frequency, Hz Figure 4.43 Waterfall plot of dynamic output torque, Tin=1000 Ibf-in, |i’=0.05, bp=0.59, contact ratio=1, frictionless mean output torque=1880 Ibf-in 231 1890 1880 c 1870 2. 1860 3 - 1850 1840 1830 position along the line-of-action, inch 1890 1880 I c 1870 g. 1860 Q. 1850 1840 i i r- 1830 2 3 4 5 position along the line-of-action, inch Figure 4.44 Comparison of steady state dynamic output torque between numerically solution (top) and exact analytical solution (bottom) where û)p=cùdi, Qin=1000 rpm, T,n=1000 Ibf-in 2 3 2 s 20, Aliased 5ûjp 10000 input speed, rpm % 2000 3000 4000 5000 6000 7000 frequency, Hz Figure 4.45 Waterfall plot of off line-of-action friction force, Qin=96-9600 rpm, Tin=1000 Ibf-in, p'=0.05, bp=0.59, contact ratio=1, frictionless mean output torque=1 880 Ibf-in 9000 Aliased 5(0. 8000 7000 e-E 6000 ■g 5000 0 g- 4000 1 3000 2000 1000 5(4, 2000 4000 6000 8000 10000 12000 frequency, Hz Figure 4.46 Off line-of-action friction force harmonic content versus input speed contour plot, Tjn=1000 Ibf-in, |i*=0.05, bp=0.59, contact ratio =1 2 3 3 4.7.3 Analytical Solution Summary In this section, the dynamic model Incorporating the effects of llne-of- actlon friction force has been solved analytically through an approximation of the Ep and eg functions (See Figure 4.36 and Eqn. 4.54 and +Eqn. 4.55). A limitation of the approximation Is that the approximate ep and Eg functions must be selected either to accurately model the normal mesh force or the output torque at the expense of the other output. By Increasing the number of time Invariant sections to the approximate ep and eg functions, both the normal mesh force and the output torque results can be simultaneous determined. The potential advantages of the analytical solution over the numerical solution Include computational speed and an explicit solution which can be examined. The explicit solution may be transformed Into frequency domains directly with out the worry of aliasing. Another potential advantage of the explicit solution may be when attempting optimization techniques to minimize off llne-of-action friction force or maximize operating efficiency. For all cases In this section and in this research the input torque is a constant time invariant Input. In linear time Invariant theory once the initial condition transients have damped out, a constant Input results in a constant output. The models developed in this research show that off line-of-action friction force can be a source of dynamic excitation not only In the off llne-of- 2 3 4 action direction, but may also cause line-of-action excitation due to its effect on the normal mesh force and in the torsional direction through the effect on the output torque. 4.8 Non-Linear Elastohydrodynamic Friction Force Model Solution This section examines the dynamic gear model incorporating the friction force model developed in section 4.5.1 with the non-linear fluid film friction coefficient model constructed in section 4.4.2.2. The solution is generated numerically employing a 4th/5th order Runge-Kutta numerical algorithm [Mathworks, 1992]. Table 4.5 lists the gear mesh parameters and the lubricating fluid parameters. The lubrication properties are taken from Gu[1972]. The natural frequencies and mode shapes of the linear time invariant system are given by Table 4.2. The gear parameters km, kg, and all viscous damping terms are time invariant. Viscous damping is introduced to eliminate the homogenous part of the solution, leaving the steady state component. The parameters are left as time invariant parameters to emphasize the effect of off line-of-action friction force on the normal mesh force and the output torque. For the spur gear case examined the gear mesh stiffness, km, is the mean mesh stiffness between single and double tooth pair contact. In practice, the time varying gear mesh stiffness can represent a significant dynamic excitation source, particularly in spur 2 3 5 gearing. Static transmission error is neglected and Tin is assumed constant, to emphasize the parametric effect of the friction force as a mesh excitation. This reduces Eqn. 4.65 to Eqn. 4.125. Eqn. 4.125 u = 0 Units Pinion Gear Base Diameter inch 4.698 4.229 Face Width inch 1.25 Operating Pressure Angle degree 25 Face width inch 1.25 Helix Angle degree 0 bp inch 0.59 Za inch -0.295 LPSTC inch -0.295 PP inch 0 Zb inch 0.295 Contact Ratio 1.0 Iz. Ibf-in-s^ 0.06 0.72 rrii, Ibf-s^/in 0.02 0.07 km Ibf/in 2.9 X 10" ks Ibf-in 6.5x10" Vo Ibf-s/in^ 3.7x lO'" a in^/lbf 1.2x10"' Cod Ibf-s/in 500 Ce.e Ibf-s/in 500 Co.e Ibf-s/in 0 Ce.o Ibf-s/in 0 Table 4.5 G ear parameters used for the non-linear fluid film lubrication model 2 3 6 1 Figure 4.47 through Figure 4.53 present the numerical results for two input speed cases, Q,n=19 rpm/(0p=8 Hz and fl,n=4056 rpm/cop=1 690 Hz. The input speed of 19 rpm corresponds to a mesh frequency well below the first natural frequency and the input speed of 4056 rpm corresponds to a mesh frequency between the first and second natural frequencies. All results are steady state occurring after the transients due to initial conditions have died out. The independent variable of time is transformed to position along the line-of-action using Eqn. 4.95. The position along the line-of-action is defined as the coordinate along the x axis as defined in Figure 4.3 with the start of contact, Za, shifted to zero. Figure 4.47 presents the instantaneous friction coefficient, |i', when operating with a fluid film. The friction coefficient is not symmetric about the pitch point and at the pitch point has a friction coefficient of zero. The computed friction coefficient shows the same non-symmetry as reported by Radzimovsky[1973,1974] in the experimental measurement of the instantaneous friction coefficient. The value of |i'=0 at the pitch point comes from the zero sliding velocity and the elastohydrodynamic formulation (See Section 4.4.2.2). 2 3 7 Figure 4.48, Figure 4.49 and Figure 4.50 present the dynamic mesh force, output torque, and the off line-of-action friction force, respectively. The step change at the pitch point for the normal mesh force and the off line-of- action friction force observed in earlier presented results when a constant friction coefficient was assumed has been transformed to a smooth transition. Figure 4.51, Figure 4.52 and Figure 4.53 present the instantaneous fluid viscosity, the minimum fluid film thickness, and the mean contact pressure, respectively. The minimum fluid film thickness and the mean contact pressure trends agree with the results by Gu[1972]. The fluid film thickness increases as the contact point moves from approach action to recess action while the contact pressure decreases. Gu[1972] computed fluid film parameters employing a more sophisticated viscosity model and incorporated temperature, but assumed static operating conditions and a constant normal mesh force. The developed model in this research is isothermal but allows a time varying normal force and incorporates dynamic effects. 2 3 8 0 . 3 “ ■ —— (Qj a 8 H z 0.25 AcOon (Ü, = 1690 Hz 0.2 I I a1 0.15 J 0.1 0.05 0.2 0.4 0.6 0.8 1.2 1.4 position alongalong the line-of-actlon.-action. Inch Figure 4.47 Instantaneous friction coefficient, p.", h=1.15hmin, Tin=1000 Ibf-in, bp=0.59, contact ratio=1 500 480 460 S 440 o I E 420 CO 2 400 380 CO, = 8 Hz (0, =1690 Hz 360 0.2 0.4 0.6 0.8 1.2 1.4 position along the line-of-action.-action. inch Figure 4.48 Dynamic normal mesh force, h=1.15hmin, Tin=1000 Ibf-in, bp=0.59, contact ratio= 1, frictionless normal mesh force = 425 Ibf 2 3 9 1885 S 1870 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 position along the line-of-action. inch Figure 4 .4 9 Dynamic output torque, h=1.15hmin, Tjn=1000 Ibf-in, b p = 0 .5 9 , contact ratlo=1, frictionless output torque=1 8 8 0 ibf-in 120 . * ApproaOi, Rocess Action Action 100 — 01 , = 8 Hz 0 1 , = 1690 Hz 80 60 40 8O § 20 •ë -20 -40 -60, 0.2 0.4 0.6 0.8 1.2 1.4 position along the line-of-action,■action, inch Figure 4.50 Dynamic off line-of-action friction force, h=1.15hmin, Tin=1000 Ibf- in, bp=0.59, contact ratio=1 2 4 0 xIO Recess Action — — (üy = 8 Hz oij = 1690 Hz I0) 0.2 0.4 0.6 0.8 1 1.2 position along the line-of-action.line-of-action. inch Figure 4.51 Instantaneous viscosity, h=1.15hmin, Tm=1000 Ibf-in, bp=0.59, contact ratio=1 X 10 3.5 3 - 2.5 - Approacn^ I Recess ^ Action Action “ = 8 Hz m, = 1690 Hz I 1.5h I 1 0.5 - 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 position along the line-of-action. Inch Figure 4.52 Minimum fluid film thickness, hmin, Tin=1000 Ibf-in, bp=0.59, contact ratio=1 2 4 1 X 10 4.7, Appreacti, Recass Acton Acton 4.6| 4.5 & CL I 4.2 I V' 3.9 3.8 0.2 0.4 0.6 0.8 1 1.2 position along the line-of-action, inch Figure 4 .5 3 Mean contact pressure, h=1.15hmin, Tin=1000 Ibf-in, b p = 0 .5 9 , contact ratio=1 This section has demonstrated the capability of the developed model to incorporate not only time invariant coefficient models examined in a previous section of this research but also non-linear friction coefficient formulations typically associated with fluid film lubrication. Future research should include incorporating more sophisticated fluid film models which typically include lubricant and bulk material temperatures. The inclusion of temperature requires the addition of ordinary differential or partial differential equations to model the heat generation and flow between the lubricant and the bulk material. The heat transfer equations have not been developed in this research. Work by Gu[1972] and Wang[1981] incorporate temperature effects in fluid film problems related to spur gearing. 242 4.9 Concluding Remarks This chapter has developed a dynamic gear model for parallel axis involute spur and helical gearing incorporating the effect of the off line-of- action friction force at the gear mesh interface. The model is based upon generalized normal load and friction load distributions. A gear model supported on rigid bearing that is connected to a load with a massive inertia is examined in detail using a constant friction coefficient and a simple fluid film model. The gear mesh stiffness (a potential dynamic excitation source particularly in spur gearing) is assumed time invariant and static transmission error is incorporated in the model but neglected. This approach is used to demonstrate the effect of off line-of-action friction force on the normal mesh force and the output torque. For all cases in this research the input torque is a constant time invariant input. In linear time invariant theory, a constant input once the initial condition transients have damped out results in a constant output. The models developed in this research show that off line-of-action friction force may be a source of dynamic excitation not only in the off line-of-action direction, but may also cause line-of-action excitation through the affect on the normal mesh force and in the torsional direction through the effect on the output torque. 2 4 3 Under static operating conditions the model is examined for the effects of various time invariant friction coefficients, various helix angles, and various contact ratios on the normal mesh force, the output torque and the off line-of-action friction forces. In approach action, the friction force tends to increase the mesh force, while in recess action the friction force tends to decrease the mesh force. The output torque compared to the frictionless output torque, decreases as the distance of the contact point to the pitch point increases. The output torque also decreases as the number of teeth in contact increase. Furthermore the output torque decreases as the helix angle increases, particularly near the point of pure rolling. The results from this work can be used to computed the gear mesh efficiency for a spur or helical gear pair of arbitrary contact ratio and helix angle. Under dynamic conditions, a gear mesh representative of that used in the experimental work performed on the back-to-back gear tester is examined under various operating conditions. The model was examined with an assum ed time invariant friction coefficient and with a simple isothermal fluid film lubrication model. The dynamic gear models resulted in linear time varying and non-linear models representative of the constant friction coefficient model and the fluid film model, respectively. The linear time varying model was examined both numerically and analytically. An analytical solution to the forced vector form Meissner equation was 2 4 4 introduced. This vector form of the Meissner equation can be used to approximate the problem of off line-of-action friction force in the gear mesh system with a contact ratio of unity. By expanding the analytical solution to incorporate more time invariant piecewise steps, contact ratios greater than unity can be solved. From the dynamic results, these models suggest that the off line-of- action friction force may be accurately approximated by using the static normal load. This approximation appears to be valid when operating at any condition away from a natural frequency whose mode significantly affects the normal mesh force or when the inputs (such as static transmission error or input torque harmonics) or parametric terms (such as time varying mesh stiffness or off line-of-action friction forces) cause a significant increase in the normal mesh force. This implies the “better” the gear pair in terms of line-of-action excitations, the more significant the off line-of-action friction force modeling potentially becomes. This is a preliminary observation base upon a single gear geometry and limited parameter changes. A single gear configuration allowed for the comparison and examination of the gear system under all potential operating conditions. This includes operating conditions well below the first natural frequency, parametric frequency excitation of the first natural frequency, operating conditions between the first and second natural frequency, parametric frequency excitation of the second natural frequency, and at an operating condition well above the 2 4 5 second natural frequency. Other geometry’s would simply modify the location of the operating conditions. Parametric studies of other gear geometries and studies of the developed model including more features such as added input and output inertia and translational components (increasing the number of degrees of freedom) is left for further research. 2 4 6 Chapter 5 Contributions and Recommendations 5.1 Contributions This thesis has studies the concept gear mesh friction force from a dynamics perspective. In particular, the following major contributions emerge. 1. Friction force has been shown to be a potential dynamic excitation source. This conclusion is based upon experimental data and analytical arguments. The experimental results come from both the published literature and experiments performed specifically for this research. The analytical arguments are based upon energy and force transmissibility concepts. 2 4 7 2. A new dynamic gear model that incorporates off line-of-action friction forces for involute parallel axis spur or helical gears of arbitrary contact ratio has been developed. The dynamic gear model is capable of handling dry friction and fluid film model formulations. The gear model results in linear time varying or non-linear formulations. 3. The dynamic gear model studied under quasi-static conditions results in a formulation for computing the gear mesh efficiency for a spur or helical gear pair of arbitrary contact ratio. This is a significant improvement over current gear mesh efficiency computational techniques that are available in the technical literature. 4. An explicit analytical solution for the nonhomogeneous Meissner equation is developed. The forcing term is limited to functions which are periodic, with a fundamental period equal to the period of the time varying parameter. For gear related problems, this restriction does not present a significant problem, since the time varying parameters have a fundamental displacement period equal to the base pitch of the gear pair. Therefore, excitation sources such as static transmission error can be easily incorporated, since the transmission error also has a fundamental period equal to the base pitch. 2 4 8 5. A methodology Is introduced for the m easurem ent of the off line-of- action friction force on the individual gear teeth for an involute spur gear pair with a contact ratio less than or equal to two. The power loss m easurem ents were at a minimum at the pitch point, but the individual tooth friction force results were inconclusive. 5.2 Future Research This dissertation has identified several major issues that merit further investigation. Some of these issues are listed below: 1. The off line-of-action displacement is related to the off line-of-action force through the analytical bearing stiffness matrix model proposed by Lim[1989]. Experimental verification of the bearing model is desirable to further strengthen the conclusions of Chapter 2. 2. The off line-of-action shaft displacement measurements performed on the Gear Noise Gear Dynamics Laboratory Back-to-Back Tester and the NASA Lewis Gear Noise Tester used only involute spur gears. The off line-of-action shaft displacement measurements should be extended to helical gears. 2 4 9 3. For this research, the orbital motion of the shafts were measured using inductance probes with a nominal sensitivity of 400 volts per inch. By employing capacitance probes, nominal sensitivities of 2500 volts per inch can be achieved. In-situ calibration is required due to the capacitance probe’s sensitivity to surface curvature. The surface curvature sensitivity requires implementing special mounting fixtures. The higher probe sensitivity may allow detection of smaller changes, such as additional tooth contact pair engagement, missed by the inductance probes. 4. The results of Chapter 4 suggest that the dynamic effects in the line- of-action direction have minimal effect on the off line-of-action friction force for off resonant speeds. This implies that an estimate of the friction force time signal can be determined by using the static normal load rather than the dynamic normal load. At resonant conditions, the dynamic normal load may be significantly larger than the static normal load. This conclusion should be further studied through a series of parametric studies. 5. A criterion should be developed that gives the gear designer conditions, such as gear geometry, operating conditions, or transmission error levels, an idea when the friction force excitation 2 5 0 could represent a significant source of dynamic excitation. Conditions such as low transmission error gear pairs or operating conditions such that the gear mesh frequency is excited, could suggest friction force should be considered. 6. The analytical solution to the Meissner equation should be extended to an arbitrary number of linear time invariant piecewise steps. This would allow a closer approximation of the exact Sp and Zq functions defined in chapter 4. This extended formulation would also enable analytical modeling of the gear system containing off line-of-action friction forces to contact ratios greater than unity. 7. Reinterpreting one the forms of the friction force model derived in chapter 4 and reproduced here as Eqn. 5.1, in terms of time varying mesh stiffness, the linear time varying analytical solution may be used to study the effects of time varying mesh stiffness on the gear system's line-of-action parameters such as dynamic mesh forces and dynamic transmission error. D + r^(fp -ep) + T^(fg - Eg) k„,D = Eqn. 5.1 V 'ZP 'ZG r r ^ -£p) + r ^ ( r g -Eq) V e ( t) 'ZP 'zG V'ZP *ZG 2 5 1 8. The developed analytical solution for the linear time varying model may have practical uses in the prediction of operating conditions where loss of tooth pair contact could be anticipated. Excitation sources that may be examined include input torque pulsations, time varying mesh stiffness and off line-of-action friction force. 2 5 2 APPENDIX Modified Fluid Film Model 2 5 3 In Chapter 4.8 of this thesis a simple isothermal elastohydrodynmaic lubrication model based on work by Dowson [1966] was incorporated into the friction model. For the test case of cü^= 1690 Hz (pinion input speed of 4056 rpm), the simple model of Dowson predicted large friction coefficient values at positions away from the pitch point ( Figure 4.47). This appendix modifies the computed friction coefficient used in Chapter 4.8 by introducing a limiting value to the friction coefficient. Qualitatively the limiting value is introduced in the following manner; if the computed friction coefficient is greater than the limiting value, than the friction coefficient equals the limiting value. Quantitatively, the limiting value is expressed as A. 1. Chapter 4.8 simply assumes p. = Pcomputed for all values of jicomputed • Using the limiting value follows the results from the elastohydrodynamic model used in the power loss computation study for spur gears by Wu [1991]. A. 1 M-compuied = friction coefficient based on model by Dowson [1966] If M-compuled ^ M-litniting then n = Iiiimrnng else p = Pcmpuwd The gear geometry and physical parameters are identical to the values used in section 4.8. Figure A. 1 presents the instantaneous friction coefficent determined using a limiting value of |i,in,jting= 0.05. Approximately 2 5 4 60 percent of the mesh cycle, centered about the pitch point, has a friction coefficient based upon the model by Dowson [1966]. The remaining 40 percent of the mesh cycle, near the start and end of gear tooth pair contact, is based upon the limiting friction coefficient of 0.05. Figure A. 2 presents the dynamic normal mesh force results using the modified friction coefficient. Comparing Figure A. 2 to Figure 4.48 reveals a very similar time trace with only a difference in the peak-to-peak amplitude of the normal mesh force. This suggests that in regions near and above the first natural frequency, the magnitude of the friction coefficient is more important than the waveform trace of the friction coefficient over the mesh cycle. Figure A. 3 presents the dynamic output torque using the modified friction coefficient. Similar to the dynamic normal force comparison, comparing Figure A. 3 and Figure 4.49 reveals a similar time trace between the modified and unmodified friction coefficients. The peak-to-peak amplitude is lower in Figure A. 3 as compared to the peak-to peak amplitude in Figure 4.49. Figure A. 3 also has a increase in the mean output torque due to the limiting value on the friction coefficient. Figure A. 4 presents the off line-of-action friction force for the modified friction coefficient case. As with the unmodified friction force case, Figure 4.50, the time trace of the friction force is determined mainly by the time trace of the friction coefficient. The waveform shape of the normal force 2 5 5 is of secondary Importance, at non-resonant conditions, since the friction force amplitude Is driven by the mean normal load and the position of tooth pair contact. The results of this appendix suggest that at frequencies grater than the first natural frequency the normal mesh force and the output torque waveform shape are driven by system dynamic properties rather than the friction coefficient waveform shape. W hereas, the peak-to-peak amplitudes of normal mesh force and the output torque appear to be driven by the magnitude of the friction coefficient. The off line-of-action friction force is primarily determined by the waveform shape of the friction coefficient. 2 5 6 0.05 0.045 0.04 0.035 0.03 S ÿ 0.025 0.02 0.015 0.01 0.005 Recess Action 0.2 0.4 0.6 0.8 1 1.2 1.6 position along the line-of-action. inch Figure A. 1 Modified instantaneous friction coefficient, h=1.15hmin, cop =1690 Hz ( pinion speed =4056 rpm), Tjn=1000 Ibf-in, bp= 0.59, contact ratio =1 257 450 445 440 435 g 430 425 I 420 £ 415 410 405 Approach Rnzcss 4 - Acuon ■ ■4— Actjoo 400 0.2 0.4 0.6 0.8 1 1.2 position along the line-of-action. inch Figure A. 2 Dynamic normal mesh force using modified fluid film model, o)p=1690 Hz ( pinion speed = 4056 rpm ), Tin=1000 Ibf-in, frictionless normal mesh force= 425 Ibf 1875 1874 1873 1872 I E. 1871 1870 1869 Approach Recess *■ Action - ■ Action- 1868 0.2 0.4 0.6 0.8 1.2 1.4 position along the line-of-aciion. inch Figure A. 3 Dynamic output torque. Tout, using modified fluid film model, (Op =1690 Hz ( pinion speed = 4056 rpm ), Tn=1000 Ibf-in, frictionless output torque =1880 Ibf-in 2 5 8 i i 25 20 15 10 £ 5 0 - Approach Recess .1 Action - Action ■ •5 -10 -15 -20 -25 0.2 0.4 0.6 0.8 position along the line-of-action. inch Figure A. 4 Dynamic off line-of-action friction force, using modified fluid film model, o)p=1690 Hz ( pinion speed = 4056 rpm ), T^=1000 Ibf-in 259 BIBLIOGRAPHY Benedict, G.H. and Kelley, B.W., "Instantaneous Coefficients of G ear Tooth Friction", ASLE Trans., Vol. 4, 1961, pp. 59-70. Bolze, V.M., “Static and Dynamic Transmission Error Measurements and Predictions and Their Relation to Measured Noise for Several Gear Sets”, MSc. Thesis, Ohio State University, Columbus, Ohio, 1996. Blankenship, G.W., “Analysis of Force Coupling and Modulation Phenomena in Geared Rotor Systems”, Ph.D. Dissertation. Ohio State University, Columbus, Ohio, 1992. Buckens, F., "On the Damped Coupled Torsional and Flexural Vibrations of Gear-Connected Parallel Shafts", ASME Paper 80-C2/DET-6. Buckingham, E., Analytical Mechanics of G ears. Dover Publications, New York, 1963. Cai, Y. and Hayashi, T.,"The Linear Approximated Equation of Vibration For A Pair of Spur Gears: Theory and Experiment", ASME Intemational Power Transmission and Gearing Conference , DE-Vol.43-2, 1992, pp. 521-528. Chiu, Y.P., "Approximate Calculation of Power Loss in Involute Gears", ASME Paper 75-PTG-2. Dawson, F.H., “Windage Loss in Large High-Speed Gears”, Proc., Inst of Mech. Eng., Part A-Power and Process Engineering, 1984, pp. 51-59. Dowson, D., and Higginson, G.R., “A Theory of Involute Gear Lubrication”, Institute of Petroleum Gear Lubrication Symposium, London, 1964, pp. 8-15. Dowson, □., and Higginson, G.R., Elasto-Hvdrodvnamic Lubrication. Pergamon Press, Oxford, 1977. Electro Corporation Catalog, Catalog #94APC, October 1993. 2 6 0 Furukawa, T., "Vibration Analysis of Gear and Shaft System by Modal Method", JSME International Conference on Motion and Power Transmissions, 1991, pp. 123-127. Gear Dynamics Gear Noise Research Laboratory (no date). Load Distribution Program, [Computer Program], Gear Dynamics Gear Noise Research Laboratory, Department of Mechanical Engineering, Ohio State University, Columbus, Ohio. Giddings and Lewis, “Directlnspecf, Manual # 56040013, February 1993. Gregory, R.W., Harris, R.L., and Munro, R.G., "Dynamic Behavior of Spur G ears”, Proceeding of the Institution of Mechanical Engineers, Vol. 178, 1963-1964, pp. 207-226. Gregory, R.W., Harris, R.L., and Munro, R.G., Torsional Motion of a Pair of Spur Gears”, Proceeding of the institution of Mechanical Engineers, Vol. 178, 1963-1964, pp. 166-173. Hao, J., and Yao, W., "Non-Linear Vibration of Spur Gears ( Study on Mesh - Impulse)", JSME Intemational Conference on Motion and Power Transmissions, 1991, pp. 5-9. Hochmann, D., "An Improved Loaded Single Flank Static Transmission Error Tester", MSc Thesis, Ohio State University, Columbus, Ohio, 1992. Houser. D.R., “Gear Noise Sources and Their Prediction Using Mathematical Models”, Chapter 16 of Gear Design, Manufacturing and inspection Manual, Society of Automotive Engineers, 1990. Hutchings, I.M., Triboloov: Friction and Wear of Engineering Materials. Edward Arnold, London, 1992. lida, H., Tamura, A., and Yamato, H., "Dynamic Characteristics of a Gear Train System with Softly Supported Shafts", Bulletin of JSME, Vol. 29, No. 252, June 1986. lida, H., Tamura, A., and Yamada, Y., "Vibrational Characteristics of Friction Between Gear Teeth", Bulletin of JSME, Vol. 28, No 241, July 1985, pp.1512-1519. 2 6 1 lida. H., Tamura, A., and Oonshi, M., “Coupled Torsional-flexural Vibration of a Shaft in a Geared System", Bulletin of JSME, Vol. 28, No. 245, Nov. 1985, pp. 2694-2698. Ikeda, H., and Muto, E., "The Simulations of the G ear Vibration Based on the Actual Transmission Errors and Tooth Frictions", Intemational Symposium on Gearing and Power Transmission, Tokyo, 1981, c-6. Ishida, K., "On the Friction Noise and Pitch Circle Impulse Noise of Gear", JSME Semi-lntemational Symposium, Tokyo, Sept. 1967, Paper 318. Iwatsubo, T., Arii, S., and Kawai, R., "The Coupled Lateral Torsional Vibration of a Geared Rotor System", Proceedings of the Third Intemational Conference on Vibrations in Rotating Machinery (Institution of Mechanical Engineers), 1984, pp. 59-66. Joule, D., Hinduja, S., Ashton J.N., Thermal Analysis of a Spur Gearbox Part 1: Steady State Finite Element Analysis, Proc. Inst, of Mechanical Engineers Part 0, Vol. 202, No. C64, 1988, pp. 245-256. Kahraman, A., Ozguven, H.N, Houser, D.R., and Zarajsek, J.J., "Dynamic Analysis of Geared Rotors by Finite Elements", Joumal of Mechanical Design, TRANS. ASME, Vol. 114, Sept. 1992, pp. 507-513. Liauwnardi, L., “Plastic Gear Transmission Error Measurement and Predictions”, MSc. Thesis, The Ohio State University, Columbus, Ohio, 1997. Lim, T.K., "Vibration Transmission Through Rolling Element Bearings in G eared Rotor Systems", Ph.D. Dissertation. Ohio State University, Columbus, Ohio, 1989. Lund, J.W., "Critical Speeds, Stability and Response of a Geared Train of Rotors", ASME Paper 77-DET-30. Math Works Inc., Matlab Reference Guide. August 1992. Martin, K.F. ,"Theoretical Efficiency of Straight Involute Spur Gear Teeth", ASME Paper 72-Mech-66. Martin, K.F. ,”The Efficiency of Involute Spur Gears", ASME Paper 80- C2/DET-16. 2 6 2 Matson, G.A.,”A Dynamic Study of a Back-to-Back Gear Test Stand”, MSc Thesis, Ohio State University, Columbus, Ohio, 1995. Measurement Group, “Errors Due to Wheatstone Bridge Nonlinearity”, Tech Note, TN-507, 1982. Measurement Group, “Errors due to Misalignment of Strain Gages”, Tech Note, TN-511, 1983. Meirovitch, L., Elements of Vibration Analysis. McGraw-Hill Inc., New York, 1986. Mirarefi, A., and Radzimovsky, E., "Effects of Torsional Vibrations on the Efficiency and the Coefficient of Friction in G ear Transmission Systems", ASME Paper 74-DET-98. Mitchell, L.D., and Mellen, D.M., "Torsional-Lateral Coupling in a Geared High-Speed Rotor System", ASME Paper 75-DET-75. Nakada, T., and Utagawa, M., “The Dynamic Loads on Gears Caused by the Varying Elasticity of the Mating Teeth of Spur G ears”, Proceedings of the Sixth Japanese National congress on Applied Mechanics, pp.493-497, 1956. Neriya, S.V., Bhat, R.B., and Sankar, T.S.,"Effect of Coupled Torsional- Flexural Vibration of a Geared Shaft System on the Dynamic Tooth Load", The Shock and Vibration Bulletin, Part 3, June 1984, pp.67-75. Neriya, S.V., Bhat, R.B., and Sankar, T.S., "Vibration of a Geared Train of Rotors with Torsional-Flexural Coupling", ASME Paper 85-DET-124. O' Donoghue, J.P., and Cameron, A., "Friction and Temperature in Rolling Sliding Contacts", ASLE Trans., Vol. 9, 1966, pp.186-194. Oswald, F.B., and Townsend, D.P., “Influence of Tooth Profile Modification on Spur Gear Dynamic Tooth Strain", NASA TM-106952, 1995. Oswald, F.B., Townsend, D.P., Rebbechi, B., and Lin, H.H., “Dynamic Forces in spur Gears - Measurement, Prediction and Code Validation”, NASA TM-107223, 1996. 2 6 3 Ôzgûven, H.N., and Houser, D.R., "Mathematical Models Used in Gear Dynamics-A Review”, Joumal of Sound and Vibration, Vol. 121, No. 3, March 1988, pp. 383-411. Ôzgûven, H.N., “Assessment of Some Recently Developed Mathematical Models in Gear Dynamics”, Proc. 8th World Congress Theory of Machines and Mechanisms, Prague, Czechoslovakia, 1991. Pipes, L.A., “Maxtrix Solution of Equations of the Mathieu-Hill Type”, Joumal of Applied Physics, Vol. 24, No. 7, July 1953, pp. 902-910. Radzimovsky, E., and Mirarefi, A., “Dynamic Behavior of Gear Systems and Variation of Coefficient of Friction and Efficiency During the Engagement Cycle", ASME Paper 74-DET-86. Radzimovsky, E. I., and Mirarefi, A., and Broom W.E., “Instantaneous Efficiency and Coefficient of Friction of an Involute G ear Drive", Joumal of Engineering for Industry, TRANS. ASME, Vol. 5, No. 4, Series B, Nov. 1973, pp. 1131-1138. Rebbechi, B., and Crisp, J.D.C., "A New Formulation of Spur-Gear Vibration", Proc. Intemational Symposium on Gearing and Power Transmission, Tokyo, Vol. II, Paper c-11, 1981, pp. 61-66. Rebbechi, B., and Crisp, J.D.C., "The Kinetics of the Contact Point, and Oscillatory Mechanisms in Resilient Spur Gears", Proc. of the Sixth World Congress on Theory of Machines and Mechanisms, New Delhi, 1983, pp. 802-806. Rebbechi, B., Oswald, F.B., and Townsend, D.P., "Dynamic Measurements of Gear Tooth Friction and Load", NASA TM-103281, 1991. Rebbechi, B., Oswald, F.B., and Townsend, D.P., "Measurement of Gear Tooth Dynamic Friction”, ASME Intemational Power Transmission and Gearing Conference , DE-Vol. 88, 1996. Richards, J.A., Analvsis of Periodicallv Time-Varvino Svstems. Springer- Verlag, Berlin, 1983. 2 6 4 Sato, K., Kamada, O., and Takatsu, N., “Dynamical Distinctive Phenomena in Gear Systems ', Bulletin of JSME, Vol. 22, No. 174, Dec. 1979, pp. 1840- 1847. Sheffield Measurement Division, ‘The Function Library FLB Reference Manual”, Manual # 58004140, October 1987. Smith, J.D., Gears and Their Vibration: A Basic Approach to Understanding G ear Noise. M. Decker Inc., New York, 1983. Timoshenko, S., Vibration Problems in Engineering. D. Van Nostrand Company Inc., New York, 1937. Townsend, D.P. editor, Dudlev’s Gear Handbook 2nd. Edition. McGraw-Hill Inc., New York, 1991. Trachman, E.G.,"A Simplified Technique for Predicting Traction in Elastohydrodynamic Contact”, ASLE Trans. , Vol. 21-1, 1976, pp. 53-62. Tso, L.N. and Prowell, R.W., "A Study of Friction Loss for Spur G ear Teeth", ASME Paper 61-Wa-85. Umezawa, K., Houjoh, H., and Matsumura, S., “Experimental Precise Identification of Dynamic Behavior of a Helical Gear System”, Proc. of Intemational Conference on Gears, Dresden, Germany, 1996, pp. 779-792. Utagawa, M., "Dynamic Load on Spur Gear Teeth”, Bulletin of the Japanese Society of Mechanical Engineers, Vol. 1, 1958, pp.397-403. Vinayak, H., "Multi-body Dynamics and Modal Analysis Approaches for Multi-Mesh Transmissions with Compliant Gear Bodies", Ph.D. Dissertation. Ohio State University, Columbus, Ohio, 1995. Vijayakar, S.M. (no date). Contact Analysis Program Package, [Computer Program ], Advanced Numerical Solutions, Columbus, Ohio. Wang, K.L. and Cheng, H.S., “A Numerical Solution to the Dynamic Load, Film Thickness, and Surface Temperature in Spur Gears, Part I Analysis”, ASME Joumal of Mechanical Design, Vol. 103, January 1981, pp. 177-187. 2 6 5 Wang, K.L. and Cheng, H.S., “A Numerical Solution to the Dynamic Load, Film Thickness, and Surface Temperature in Spur Gears, Part II Results”, ASME Joumal of Mechanical Design, Vol. 103, January 1981, pp. 188-194. Wu, S. and Cheng, H.S., “A Friction Model of Partial-EHL Contacts and its Application to Power Loss in Spur Gears”, STLE Tribology Trans., Vol. 34, No. 3, pp. 398-407. Yada, T., “The Measurement of the Gear Mesh Friction Losses", ASME Paper 72-PTG-35. Yoshida, T., “Gear System Static and Dynamic Analysis by Theoretical and Experimental Approach", Report, Gear Dynamics and Gear Noise Research Laboratory, Ohio State University, Columbus, Ohio, 1993. Yousif, A.E. and Sadeck, K.S.H., "Traction-Slip Characteristics Applied to Gear Tooth", Intemational Symposium on Gearing and Power Transmission, Tokyo, b-16, 1981. Zadeh, L.A. and Desoer ,C.A., Linear System Theory The State Space Approach. McGraw-Hill Book Co., New York, 1963. 2 6 6