Some theoretical developments are presented, which lead to approximate analytical results on quasi-static transmission errors valid for low and high contact ratio spur and helical gears. Based on a multidegree-of-freedom gear model, a unique scalar equation for transmission error is established. The role of profile relief is analyzed by using Fourier series and it is shown that transmission error fluctuations depend on a very limited number of parameters representative of gear geometry and profile relief definition. An original direct solution to the optimum relief minimizing transmission error fluctuations is presented, which is believed to be helpful for designers. The analytical results compare well with the numerical results provided by a variety of models and it is demonstrated that some general laws of evolution for transmission error fluctuations versus profile modifications can be established for spur and helical gears.
Issue Section:
Research Papers
Keywords:
Fourier series,
gears,
gear,
transmission error,
profile relief,
modifications,
analytical results
1.
Walker
, H.
, 1938, “Gear Tooth Deflection and Profile Modification
,” Engineer
, 166
, pp. 434
–436
.2.
Niemann
, G.
, and Baethge
, J.
, 1970, “Drehwegfehler, Zahnfederhärte und Geräusch bei Stirnrädern
,” VDI-Z
, 112
, pp. 205
–276
.3.
Gregory
, R. W.
, Harris
, S. L.
, and Munro
, R. G.
, 1963, “Dynamic Behavior of Spur Gears
,” Proc. Inst. Mech. Eng. Int. Conf. Gearing
, 178
, pp. 261
–266
.4.
Terauchi
, Y.
, Nadano
, H.
, and Nohara
, M.
, 1982, “On the Effect of the Tooth Profile Modification on the Dynamic Load and the Sound Level of a Spur Gear
,” Bull. JSME
0021-3764, 25
(207
), pp. 1474
–1481
.5.
Sato
, T.
, Umezawa
, K.
, and Ishikawa
, J.
, 1983, “Effects of Contact Ratio and Profile Correction on Gear Rotational Vibration
,” Bull. JSME
0021-3764, 26
(221
), pp. 2010
–2016
.6.
Smith
, J. D.
, 1983, Gears and Their Vibration
, Marcel Dekker
, New York/London
.7.
Tavakoli
, M.
, and Houser
, D. R.
, 1986, “Optimum Profile Modifications for the Minimization of Static Transmission Errors of Spur Gears
,” ASME J. Mech., Transm., Autom. Des.
0738-0666, 108
, pp. 86
–95
.8.
Houser
, D. R.
, 1991, “Gear Noise
,” Dudley’s Gear Handbook
, P. S.
Townsend
, ed., McGraw-Hill
, New York
, pp. 14.1
–14.32
, Chap. 14.9.
Lee
, C.
, Lin
, H. H.
, Oswald
, F. B.
, and Townsend
, D. P.
, 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.
0161-8458, 113
(4
), pp. 473
–480
.10.
Munro
, R. G.
, 1990, “A Review of the Theory and Measurement of Gear Transmission Error
,” Proceedings of the First International Conference on Gearbox Noise and Vibration
, Cambridge, pp. 3
–10
.11.
Munro
, R. G.
, Yildirim
, N.
, and Hall
, D. M.
, 1990, “Optimum Profile Relief and Transmission Error in Spur Gears
,” Proceedings of the First International Conference on Gearbox Noise and Vibration
, Cambridge, pp. 35
–41
.12.
Lin
, H. H.
, Oswald
, F. B.
, and Townsend
, D. P.
, 1994, “Dynamic Loading of Spur Gears With Linear or Parabolic Tooth Profile Modifications
,” Mech. Mach. Theory
0094-114X, 29
(8
), pp. 1115
–1129
.13.
Matsumura
, S.
, Umezawa
, K.
, and Houjoh
, H.
, 1996, “Performance Diagram of a Helical Gear Pair Having Tooth Surface Deviation During Transmission on Light Load
,” Proceedings of the Seventh ASME International Power Transmission and Gearing Conference
, San Diego, Oct. 6–9, pp. 161
–168
.14.
Velex
, P.
, and Maatar
, M.
, 1996, “A Mathematical Model for Analyzing the Influence of Shape Deviations and Mounting Errors on Gear Dynamics
,” J. Sound Vib.
0022-460X, 191
(5
), pp. 629
–660
.15.
Maatar
, M.
, and Velex
, P.
, 1997, “Quasi-Static and Dynamic Analysis of Narrow-Faced Helical Gears With Profile and Lead Modifications
,” ASME J. Mech. Des.
0161-8458, 119
(4
), pp. 474
–480
.16.
Umeyama
, M.
, Kato
, M.
, and Inoue
, K.
, 1998, “Effects of Gear Dimensions and Tooth Surface Modifications on the Loaded Transmission Error of a Helical Gear Pair
,” ASME J. Mech. Des.
0161-8458, 120
(1
), pp. 119
–125
.17.
Yildirim
, N.
, and Munro
, R. G.
, 1999, “A Systematic Approach to Profile Relief Design of Low and High Contact Ratio Spur Gears
,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
0954-4062, 213
(6
), pp. 551
–562
.18.
Yildirim
, N.
, and Munro
, R. G.
, 1999, “A New Type of Profile Relief for High Contact Ratio Spur Gears
,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
0954-4062, 213
(6
), pp. 563
–568
.19.
Kahraman
, A.
, and Blankenship
, G. W.
, 1999, “Effect of Involute Tip Relief on Dynamic Response of Spur Gears
,” ASME J. Mech. Des.
0161-8458, 121
(2
), pp. 313
–315
.20.
Velex
, P.
, and Ajmi
, M.
, 2006, “On the Modelling of Excitations in Geared Systems by Transmissions Errors
,” J. Sound Vib.
0022-460X, 290
(3–5
), pp. 882
–909
.21.
Mohamad
, E. N.
, Komori
, M.
, Murakami
, H.
, Kubo
, A.
, and Fang
, S.
, 2009, “Analysis of General Characteristics of Transmission Error of Gears With Convex Modification of Tooth Flank Form Considering Elastic Deformation Under Load
,” ASME J. Mech. Des.
0161-8458, 131
(6
), p. 061015
.22.
Houser
, D. R.
, Harianto
, J.
, Iyer
, N.
, Josephson
, J.
, and Chandrasekaren
, B.
, 2000, “A Multi-Variable Approach to Determining the ‘Best’ Gear Design
,” Proceedings of the Eighth ASME International Power Transmission and Gearing Conference
, Baltimore, Sep. 10–13.23.
Sundaresan
, S.
, Ishii
, K.
, and Houser
, D. R.
, 1991, “A Procedure Using Manufacturing Variance to Design Gears With Minimum Transmission Error
,” ASME J. Mech. Des.
0161-8458, 113
(3
), pp. 318
–325
.24.
Sundaresan
, S.
, Ishii
, K.
, and Houser
, D. R.
, 1991, “Design Optimization for Robustness Using Performance Simulation Programs
,” Proceedings of ASME Advances in Design Automation Conference
, DE-Vol. 32-1
, pp. 249
–256
.25.
Maatar
, M.
, and Velex
, P.
, 1996, “An Analytical Expression of the Time-Varying Contact Length in Perfect Cylindrical Gears—Some Possible Applications in Gear Dynamics
,” ASME J. Mech. Des.
0161-8458, 118
(4
), pp. 586
–589
.26.
Velex
, P.
, and Ville
, F.
, 2009, “An Analytical Approach to Tooth Friction Losses in Spur and Helical Gears—Influence of Profile Modifications
,” ASME J. Mech. Des.
0161-8458, 131
(10
), p. 101008
.27.
Velex
, P.
, Maatar
, M.
, and Octrue
, M.
, 1995, “Loaded Transmission Error Predictions and Measurements on Spur Gears With Profile Relief
,” Proceedings of IMechE, Gearbox Noise, Vibration and Diagnostics International Conference
, Londres, pp. 199
–208
.28.
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.
0161-8458, 124
(2
), pp. 334
–346
.29.
Ajmi
, M.
, and Velex
, P.
, 2002, “Multi-Criterion Design of Tooth Modifications on High-Contact-Ratio Spur Gears
,” VDI, Proceedings of the International Conference on Gear
, Münich, Vol. 2
, pp. 737
–749
.30.
Weber
, C.
, and Banaschek
, K.
, 1953, Formänderung und Profilrücknahme bei Gerad-und Schrägverzahnten Antriebstechnik
, Vieweg
, Braunschweig
, Vol. 11
.31.
Lundberg
, G.
, 1939, “Elastische Berührung zweier Halbraüme
,” Forsch. Geb. Ingenieurwes.
0367-2204, 10
(5
), pp. 201
–211
.32.
Conry
, T. F.
, and Seireg
, A.
, 1971, “A Mathematical Programming Method for Design for Elastic Bodies in Contact
,” ASME J. Appl. Mech.
0021-8936, 38
, pp. 1293
–1307
.33.
Conry
, T. F.
, and Seireg
, A.
, 1973, “A Mathematical Programming Technique for the Evaluation of Load Distribution and Optimal Modifications for Gear Systems
,” ASME J. Eng. Ind.
0022-0817, 95
, pp. 1115
–1122
.34.
Houser
, D. R.
, Bolze
, V. M.
, and Graber
, J. M.
, 1996, “Static and Dynamic Transmission Error Measurements and Predictions for Spur and Helical Gear Sets
,” Proceedings of the Seventh ASME International Power Transmission and Gearing Conference
, San Diego, pp. 365
–372
.35.
Vinayak
, H.
, and Houser
, D. R.
, 1992, “A Comparison of Analytical Prediction With Experimental Measurements of Transmission Error of Misaligned Loaded Gears
,” Proceedings of the Sixth ASME International Power Transmission and Gearing Conference
, Phoenix, pp. 11
–19
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
