This work develops a hybrid analytical-computational (HAC) method for nonlinear dynamic response in spur gear pairs. The formulation adopts a contact model developed in (Eritenel, T., and Parker, R. G., 2013, “Nonlinear Vibration of Gears With Tooth Surface Modifications,” ASME J. Vib. Acoust., 135(5), p. 051005) where the dynamic force at the mating gear teeth is determined from precalculated static results based on the instantaneous mesh deflection and position in the mesh cycle. The HAC method merges this calculation of the contact force based on an underlying finite element static analysis into a numerical integration of an analytical vibration model. The gear translational and rotational vibrations are calculated from a lumped-parameter analytical model where the crucial dynamic mesh force is calculated using a force-deflection function (FDF) that is generated from a series of static finite element analyses performed before the dynamic calculations. Incomplete tooth contact and partial contact loss are captured by the static finite element analyses and included in the FDF, as are tooth modifications. In contrast to typical lumped-parameter models elastic deformations of the gear teeth, including the tooth root strains and contact stresses, are calculated. Accelerating gears and transient situations can be analyzed. Comparisons with finite element calculations and available experiments validate the HAC model in predicting the dynamic response of spur gear pairs, including for resonant gear speeds when high amplitude vibrations are excited and contact loss occurs. The HAC model is five orders of magnitude faster than the underlying finite element code with almost no loss of accuracy.
Issue Section:
Research Papers
References
1.
Özgüven
, H. N.
, and Houser
, D. R.
, 1988
, “Mathematical Models Used in Gear Dynamics—A Review
,” J. Sound Vib.
, 121
(3
), pp. 383
–411
.2.
Velex
, P.
, and Maatar
, M.
, 1996
, “A Mathematical Model for Analyzing the Influence of Shape Deviations and Mounting Errors on Gear Dynamic Behaviour
,” J. Sound Vib.
, 191
(5
), pp. 629
–660
.3.
Wang
, J.
, Li
, R.
, and Peng
, X.
, 2003
, “Survey of Nonlinear Vibration of Gear Transmission Systems
,” ASME Appl. Mech. Rev.
, 56
(3
), pp. 309
–329
.4.
Gregory
, R. W.
, Harris
, S. L.
, and Munro
, R. G.
, 1963
, “Dynamic Behaviour of Spur Gears
,” Proc. Inst. Mech. Eng.
, 178
(1
), pp. 207
–226
.5.
Blankenship
, G. W.
, and Singh
, R.
, 1995
, “A New Gear Mesh Interface Dynamic Model to Predict Multi-Dimensional Force Coupling and Excitation
,” Mech. Mach. Theory
, 30
(1
), pp. 43
–57
.6.
Blankenship
, G. W.
, and Singh
, R.
, 1995
, “Dynamic Force Transmissibility in Helical Gear Pairs
,” Mech. Mach. Theory
, 30
(3
), pp. 323
–339
.7.
Velex
, P.
, and Ajmi
, M.
, 2006
, “On the Modelling of Excitations in Geared Systems by Transmission Errors
,” J. Sound Vib.
, 290
(3–5
), pp. 882
–909
.8.
Eritenel
, T.
, and Parker
, R. G.
, 2012
, “Three-Dimensional Nonlinear Vibration of Gear Pairs
,” J. Sound Vib.
, 331
(15
), pp. 3628
–3648
.9.
Eritenel
, T.
, and Parker
, R. G.
, 2012
, “An Investigation of Tooth Mesh Nonlinearity and Partial Contact Loss in Gear Pairs Using a Lumped-Parameter Model
,” Mech. Mach. Theory
, 56
(1
), pp. 28
–51
.10.
Eritenel
, T.
, and Parker
, R. G.
, 2013
, “Nonlinear Vibration of Gears With Tooth Surface Modifications
,” ASME J. Vib. Acoust.
, 135
(5
), p. 051005
.11.
Liu
, G.
, and Parker
, R. G.
, 2008
, “Dynamic Modeling and Analysis of Tooth Profile Modification for Multimesh Gear Vibration
,” ASME J. Mech. Des.
, 130
(12
), p. 121402
.12.
Ichimaru
, K.
, and Hirano
, F.
, 1974
, “Dynamic Behavior of Heavy-Loaded Spur Gears
,” ASME J. Eng. Ind.
, 96
(2
), pp. 373
–381
.13.
Lee
, C.
, Oswald
, F. B.
, Townsend
, D. P.
, and Lin
, H. H.
, 1991
, “Influence of Linear Profile Modification and Loading Conditions on the Dynamic Tooth Load and Stress of High-Contact-Ratio Spur Gears
,” ASME J. Mech. Des.
, 113
(4
), pp. 473
–480
.14.
Lin
, H. H.
, Oswald
, F.
, Townsend
, D.
, and Lee
, C.
, 1993
, “Computer-Aided Design of High-Contact-Ratio Gears for Minimum Dynamic Load and Stress
,” ASME J. Mech. Des.
, 115
(1
), pp. 171
–178
.15.
Kahraman
, A.
, and Blankenship
, G. W.
, 1997
, “Experiments on Nonlinear Dynamic Behavior of an Oscillator With Clearance and Periodically Time-Varying Parameters
,” ASME J. Appl. Mech.
, 64
(1
), pp. 217
–226
.16.
Ambarisha
, V. K.
, and Parker
, R. G.
, 2007
, “Nonlinear Dynamics of Planetary Gears Using Analytical and Finite Element Models
,” J. Sound Vib.
, 302
(3
), pp. 577
–595
.17.
Chabert
, G.
, Tran
, T. D.
, and Mathis
, R.
, 1974
, “An Evaluation of Stresses and Deflection of Spur Gear Teeth Under Strain
,” ASME J. Eng. Ind.
, 96
(1
), pp. 85
–93
.18.
Oda
, S.
, Nagamura
, K.
, and Aoki
, K.
, 1981
, “Stress Analysis of Thin Rim Spur Gears by Finite Element Method
,” Bull. JSME
, 24
(193
), pp. 1273
–1280
.19.
Oda
, S.
, Miyachika
, K.
, Koide
, T.
, and Mizune
, M.
, 1986
, “Stress Analysis of Thin-Rimmed Spur Gears by Boundary Element Method
,” Bull. JSME
, 29
(248
), pp. 593
–599
.20.
Bibel
, G. D.
, Reddy
, S.
, Handschuh
, R.
, and Savage
, M.
, 1994
, “Effects of Rim Thickness on Spur Gear Bending Stress
,” ASME J. Mech. Des.
, 116
(4
), pp. 1157
–1162
.21.
Baud
, S.
, and Velex
, P.
, 2002
, “Static and Dynamic Tooth Loading in Spur and Helical Geared Systems-Experiments and Model Validation
,” ASME J. Mech. Des.
, 124
(2
), pp. 334
–346
.22.
Wang
, J.
, and Howard
, I.
, 2005
, “Finite Element Analysis of High Contact Ratio Spur Gears in Mesh
,” ASME J. Tribol.
, 127
(3
), pp. 469
–483
.23.
Kawalec
, A.
, Ceglarek
, D.
, and Wiktor
, J.
, 2006
, “Comparative Analysis of Tooth-Root Strength Using ISO and AGMA Standards in Spur and Helical Gears With Fem-Based Verification
,” ASME J. Mech. Des.
, 128
(5
), pp. 1141
–1158
.24.
Thirumurugan
, R.
, and Muthuveerappan
, G.
, 2010
, “Maximum Fillet Stress Analysis Based on Load Sharing in Normal Contact Ratio Spur Gear Drives
,” Mech. Based Des. Struct. Mach.
, 38
(2
), pp. 204
–226
.25.
Palmer
, D.
, and Fish
, M.
, 2010
, “Evaluation of Methods for Calculating Effects of Tip Relief on Transmission Error, Noise and Stress in Loaded Spur Gears
,” Am. Gear Manuf. Assoc. Fall Tech. Meeting
, 2010
, pp. 112
–126
.http://geartechnology.com/issues/0112x/palmer.pdf26.
Li
, N.
, Li
, W.
, Liu
, N.
, and Liu
, H. G.
, 2011
, “Analytical Method on Contact Stress of Helical Gear With Asymmetric Involutes
,” Adv. Mater. Res.
, 321
, pp. 157
–160
.27.
Li
, S.
, 2012
, “Contact Stress and Root Stress Analyses of Thin-Rimmed Spur Gears With Inclined Webs
,” ASME J. Mech. Des.
, 134
(5
), p. 051001
.28.
Vijayakar
, S. M.
, Busby
, H. R.
, and Houser
, D. R.
, 1988
, “Linearization of Multibody Frictional Contact Problems
,” Comput. Struct.
, 29
(4
), pp. 569
–576
.29.
Vijayakar
, S. M.
, Busby
, H. R.
, and Wilcox
, L.
, 1989
, “Finite Element Analysis of Three-Dimensional Conformal Contact With Friction
,” Comput. Struct.
, 33
(1
), pp. 49
–61
.30.
Vijayakar
, S. M.
, 1991
, “A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem
,” Int. J. Numer. Methods Eng.
, 31
(3
), pp. 525
–545
.31.
Parker
, R. G.
, Vijayakar
, S. M.
, and Imajo
, T.
, 2000
, “Non-Linear Dynamic Response of a Spur Gear Pair: Modelling and Experimental Comparisons
,” J. Sound Vib.
, 237
(3
), pp. 435
–455
.32.
Parker
, R. G.
, Agashe
, V.
, and Vijayakar
, S. M.
, 2000
, “Dynamic Response of a Planetary Gear System Using a Finite Element/Contact Mechanics Model
,” ASME J. Mech. Des.
, 122
(3
), pp. 304
–310
.33.
Tamminana
, V. K.
, Kahraman
, A.
, and Vijayakar
, S.
, 2007
, “A Study of the Relationship Between the Dynamic Factors and the Dynamic Transmission Error of Spur Gear Pairs
,” ASME J. Mech. Des.
, 129
(1
), pp. 75
–84
.34.
Cooley
, C. G.
, Parker
, R. G.
, and Vijayakar
, S. M.
, 2011
, “A Frequency Domain Finite Element Approach for Three-Dimensional Gear Dynamics
,” ASME J. Vib. Acoust.
, 133
(4
), p. 041004
.35.
Dai
, X.
, Cooley
, C. G.
, and Parker
, R. G.
, 2016
, “Dynamic Tooth Root Strains and Experimental Correlations in Spur Gear Pairs
,” Mech. Mach. Theory
, 101
, pp. 60
–74
.36.
Kahraman
, A.
, and Vijayakar
, S. M.
, 2001
, “Effect of Internal Gear Flexibility on the Quasi-Static Behavior of a Planetary Gear Set
,” ASME J. Mech. Des.
, 123
(3
), pp. 408
–415
.37.
Singh
, A.
, Kahraman
, A.
, and Ligata
, H.
, 2008
, “Internal Gear Strains and Load Sharing in Planetary Transmissions: Model and Experiments
,” ASME J. Mech. Des.
, 130
(7
), p. 072602
.38.
Ambarisha
, V. K.
, and Parker
, R. G.
, 2006
, “Suppression of Planet Mode Response in Planetary Gear Dynamics Through Mesh Phasing
,” ASME J. Vib. Acoust.
, 128
(2
), pp. 133
–142
.39.
Kahraman
, A.
, Ligata
, H.
, and Singh
, A.
, 2010
, “Influence of Ring Gear Rim Thickness on Planetary Gear Set Behavior
,” ASME J. Mech. Des.
, 132
(2
), p. 021002
.40.
Guo
, Y.
, and Parker
, R. G.
, 2012
, “Stiffness Matrix Calculation of Rolling Element Bearings Using a Finite Element/Contact Mechanics Model
,” Mech. Mach. Theory
, 51
, pp. 32
–45
.41.
Kahraman
, A.
, and Blankenship
, G. W.
, 1999
, “Effect of Involute Contact Ratio on Spur Gear Dynamics
,” ASME J. Mech. Des.
, 121
(1
), pp. 112
–118
.42.
Kahraman
, A.
, and Blankenship
, G. W.
, 1999
, “Effect of Involute Tip Relief on Dynamic Response of Spur Gear Pairs
,” ASME J. Mech. Des.
, 121
(2
), pp. 313
–315
.43.
Harris
, S. L.
, 1957
, “Dynamic Loads on the Teeth of Spur Gears
,” Proc. Inst. Mech. Eng.
, 172
(1
), pp. 87
–100
.44.
Chong
, T.-H.
, Suzuki
, T.
, Aida
, T.
, and Fujio
, H.
, 1982
, “Bending Stresses of Internal Spur Gear
,” Bull. JSME
, 25
(202
), pp. 679
–686
.45.
Wang
, M.-J.
, 2003
, “A New Photoelastic Investigation of the Dynamic Bending Stress of Spur Gears
,” ASME J. Mech. Des.
, 125
(2
), pp. 365
–372
.46.
Houser
, D. R.
, and Seireg
, A.
, 1970
, “An Experimental Investigation of Dynamic Factors in Spur and Helical Gears
,” ASME J. Eng. Ind.
, 92
(2
), pp. 495
–503
.47.
Botman
, M.
, 1980
, “Vibration Measurement on Planetary Gears of Aircraft Turbine Engines
,” J. Aircr.
, 17
(5
), pp. 351
–357
.48.
Krantz
, T. L.
, 1991
, “Gear Tooth Stress Measurements of Two Helicopter Planetary Stages
,” National Aeronautics and Space Administration, Glenn Research Center, Cleveland, OH, Technical Report No. 91-C-038
.https://ntrs.nasa.gov/search.jsp?R=1992001731249.
Kahraman
, A.
, 1999
, “Static Load Sharing Characteristics of Transmission Planetary Gear Sets: Model and Experiment
,” SAE
Paper No. 1999–01–1050.50.
Rao
, S. B.
, Schwanger
, V.
, McPherson
, D. R.
, and Rudd
, C.
, 2005
, “Measurement and Validation of Dynamic Bending Stresses in Spur Gear Teeth
,” ASME
Paper No. DETC2005-84419.51.
Hotait
, M.
, and Kahraman
, A.
, 2008
, “Experiments on Root Stresses of Helical Gears With Lead Crown and Misalignments
,” ASME J. Mech. Des.
, 130
(7
), p. 074502
.52.
Hotait
, M.
, and Kahraman
, A.
, 2013
, “Experiments on the Relationship Between the Dynamic Transmission Error and the Dynamic Stress Factor of Spur Gear Pairs
,” Mech. Mach. Theory
, 70
, pp. 116
–128
.53.
Kahraman
, A.
, and Singh
, R.
, 1990
, “Non-Linear Dynamics of a Spur Gear Pair
,” J. Sound Vib.
, 142
(1
), pp. 49
–75
.54.
Vinayak
, H.
, and Singh
, R.
, 1998
, “Multi-Body Dynamics and Modal Analysis of Compliant Gear Bodies
,” J. Sound Vib.
, 210
(2
), pp. 171
–214
.55.
Vinayak
, H.
, Singh
, R.
, and Padmanabhan
, C.
, 1995
, “Linear Dynamic Analysis of Multi-Mesh Transmissions Containing External, Rigid Gears
,” J. Sound Vib.
, 185
(1
), pp. 1
–32
.56.
Sainte-Marie
, N.
, Velex
, P.
, Roulois
, G.
, and Caillet
, J.
, 2017
, “A Study on the Correlation Between Dynamic Transmission Error and Dynamic Tooth Loads in Spur and Helical Gears
,” ASME J. Vib. Acoust.
, 139
(1
), p. 011001
.http://vibrationacoustics.asmedigitalcollection.asme.org/article.aspx?articleid=255185557.
Cooley
, C. G.
, Liu
, C.
, Dai
, X.
, and Parker
, R. G.
, 2016
, “Gear Tooth Mesh Stiffness: A Comparison of Calculation Approaches
,” Mech. Mach. Theory
, 105
, pp. 540
–553
.58.
Benton
, M.
, and Seireg
, A.
, 1978
, “Simulation of Resonances and Instability Conditions in Pinion-Gear Systems
,” ASME J. Mech. Des.
, 100
(1
), pp. 26
–32
.59.
Benton
, M.
, and Seireg
, A.
, 1981
, “Factors Influencing Instability and Resonances in Geared Systems
,” ASME J. Mech. Des.
, 103
(2
), pp. 372
–378
.60.
Kahraman
, A.
, and Singh
, R.
, 1991
, “Interactions Between Time-Varying Mesh Stiffness and Clearance Non-Linearities in a Geared System
,” J. Sound Vib.
, 146
(1
), pp. 135
–156
.61.
Cai
, Y.
, 1995
, “Simulation on the Rotational Vibration of Helical Gears in Consideration of the Tooth Separation Phenomenon
,” ASME J. Mech. Des.
, 117
(3
), pp. 460
–469
.62.
Kahraman
, A.
, and Blankenship
, G. W.
, 1996
, “Interactions Between Commensurate Parametric and Forcing Excitations in a System With Clearance
,” J. Sound Vib.
, 194
(3
), pp. 317
–336
.63.
Theodossiades
, S.
, and Natsiavas
, S.
, 2000
, “Non-Linear Dynamics of Gear-Pair Systems With Periodic Stiffness and Backlash
,” J. Sound Vib.
, 229
(2
), pp. 287
–310
.64.
Palermo
, A.
, Mundo
, D.
, Hadjit
, R.
, and Desmet
, W.
, 2012
, “Effects of Variable Loads and Misalignments on Gear Noise and Vibration Through Multibody Simulation Based on Transmission Error
,” International Conference on Noise and Vibration Engineering and 4th International Conference on Uncertainty in Structural Dynamics
, Leuven, Belgium, Sept. 17–19, pp. 3995
–4006
.65.
Palermo
, A.
, Mundo
, D.
, Hadjit
, R.
, and Desmet
, W.
, 2013
, “Multibody Element for Spur and Helical Gear Meshing Based on Detailed Three-Dimensional Contact Calculations
,” Mech. Mach. Theory
, 62
, pp. 13
–30
.66.
Korta
, J.
, Palermo
, A.
, Mundo
, D.
, and Shweiki
, S.
, 2015
, “Combining Finite Element and Multibody Modeling Techniques for Time-Efficient Simulation of Nonlinear Gear Dynamics
,” The Seventh International Conference on Advances in System Simulation
Barcelona, Spain, Nov. 16–21, pp. 94–99.Copyright © 2019 by ASME
You do not currently have access to this content.
