0
Research Papers: Design of Direct Contact Systems

Optimization of Profile Modifications With Regard to Dynamic Tooth Loads in Single and Double-Helical Planetary Gears With Flexible Ring-Gears

[+] Author and Article Information
M. Chapron

R&T Department,
Hispano-Suiza,
18 Boulevard Louis-Seguin,
Colombes Cédex 92707, France
e-mail: Matthieu.Chapron@insa-lyon.fr

P. Velex

Université de Lyon,
INSA de Lyon,
LaMCoS,
UMR CNRS 5259,
Bâtiment Jean d'Alembert,
20 Avenue Albert Einstein,
Villeurbanne Cédex 69621, France
e-mail: Philippe.Velex@insa-lyon.fr

J. Bruyère

Université de Lyon,
INSA de Lyon,
LaMCoS,
UMR CNRS 5259,
Bâtiment Jean d'Alembert,
20 Avenue Albert Einstein,
Villeurbanne Cédex 69621, France
e-mail: Jerome.Bruyere@insa-lyon.fr

S. Becquerelle

R&T Department,
Hispano-Suiza,
18 Boulevard Louis-Seguin,
Colombes Cédex 92707, France
e-mail: Samuel.Becquerelle@hispano-suiza-sa.com

1Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 13, 2015; final manuscript received October 16, 2015; published online November 19, 2015. Assoc. Editor: Hai Xu.

J. Mech. Des 138(2), 023301 (Nov 19, 2015) (11 pages) Paper No: MD-15-1489; doi: 10.1115/1.4031939 History: Received July 13, 2015; Revised October 16, 2015

This paper is mostly aimed at analyzing optimum profile modifications (PMs) in planetary gears (PGTs) with regard to dynamic mesh forces. To this end, a dynamic model is presented based on 3D two-node gear elements connected to deformable ring-gears discretized into beam elements. Double-helical gears are simulated as two gear elements of opposite hands which are linked by shaft elements. Symmetric tip relief on external and internal gear meshes are introduced as time-varying normal deviations along the lines of contact and time-varying mesh stiffness functions are deduced from Wrinckler foundation models. The equations of motion are solved by coupling a Newmark time-step integration scheme and a contact algorithm to account for possible partial or total contact losses. Symmetric linear PMs for helical and double-helical PGTs are optimized by using a genetic algorithm with the objective of minimizing dynamic tooth loads over a speed range. Finally, the sensitivity of these optimum PMs to speed and load is analyzed.

Copyright © 2016 by ASME
Topics: Gears , Stress , Optimization
Your Session has timed out. Please sign back in to continue.

References

Velex, P. , and Ajmi, M. , 2006, “ On the Modeling of Excitations in Geared Systems by Transmission Errors,” J. Sound Vib., 290(3–5), pp. 882–909. [CrossRef]
Velex, P. , and Ajmi, M. , 2007, “ Dynamic Tooth Loads and Quasi-Static Transmission Errors in Helical Gears—Approximate Dynamic Factor Formulae,” Mech. Mach. Theory, 42(11), pp. 1512–1526. [CrossRef]
Munro, R. G. , 1990, “ A Review of the Theory and Measurement of Gear Transmission Error,” First International Conference on Gearbox Noise and Vibration, Cambridge, MA, pp. 3–10.
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, 213(6), pp. 551–562. [CrossRef]
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, 213(6), pp. 563–568. [CrossRef]
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,” Seventh ASME International Power Transmission and Gearing Conference, San Diego, CA, pp. 365–372.
Vinayak, H. , and Houser, D. R. , 1992, “ A Comparison of Analytical Prediction With Experimental Measurements of Transmission Error of Misaligned Loaded Gears,” Sixth ASME International Power Transmission and Gearing Conference, Phoenix, AZ, pp. 11–19.
Velex, P. , Bruyère, J. , and Houser, D. R. , 2011, “ Some Analytical Results on Transmission Errors in Narrow-Faced Spur and Helical Gears: Influence of Profile Modifications,” ASME J. Mech. Des., 113(3), p. 031010. [CrossRef]
Bruyère, J. , and Velex, P. , 2013, “ Derivation of Optimum Profile Modifications in Narrow-Faced Spur and Helical Gears Using a Perturbation Method,” ASME J. Mech. Des., 135(7), p. 071009. [CrossRef]
Bruyère, J. , Gu, X. , and Velex, P. , 2015, “ On the Analytical Definition of Profile Modifications Minimizing Transmission Error Variations in Narrow-Faced Spur and Helical Gears,” Mech. Mach. Theory, 92, pp. 257–272. [CrossRef]
Abousleiman, V. , and Velex, P. , 2006, “ A Hybrid 3D Finite Element/Lumped Parameter Model for Quasi-Static and Dynamic Analyses of Planetary/Epicyclic Gear Sets,” Mech. Mach. Theory, 41(6), pp. 725–748. [CrossRef]
Bahk, C. , and Parker, R. G. , 2013, “ Analytical Investigation of Tooth Profile Modification Effects on Planetary Gear Dynamics,” Mech. Mach. Theory, 70, pp. 298–319. [CrossRef]
Wu, X. , and Parker, R. G. , 2006, “ Vibration of Rings on a General Elastic Foundation,” J. Sound Vib., 295(1), pp. 194–213. [CrossRef]
Wu, X. , and Parker, R. G. , 2008, “ Modal Properties of Planetary Gears With an Elastic Continuum Ring Gear,” ASME J. Appl. Mech., 75(3), p. 031014. [CrossRef]
Parker, R. G. , and Wu, X. , 2010, “ Vibration Modes of Planetary Gears With Unequally Spaced Planets and an Elastic Ring Gear,” J. Sound Vib., 329(11), pp. 2265–2275. [CrossRef]
Kahraman, A. , Kharazi, A. A. , and Umrani, M. , 2003, “ A Deformable Body Dynamic Analysis of Planetary Gears With Thin Rims,” J. Sound Vib., 262(3), pp. 752–768. [CrossRef]
Kahraman, A. , and Vijayakar, S. , 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. [CrossRef]
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. [CrossRef]
Jauregui, J. , and Gonzalez, O. , 1999, “ Modeling Axial Vibrations in Herringbone Gears,” Proceedings of ASME Design Engineering Technical Conference, Las Vegas, NV, Sept. 12–15, DETC99/VIB-8109.
Ajmi, M. , and Velex, P. , 2001, “ A Model for Simulating the Quasi-Static and Dynamic Behavior of Double Helical Gears,” The JSME International Conference on Motion and Power Transmission, pp. 132–137.
Kang, M. R. , and Kahraman, A. , 2015, “ An Experimental and Theoretical Study of the Dynamic Behavior of Double-Helical Gear Sets,” J. Sound Vib., 350, pp. 11–29. [CrossRef]
Sondkar, P. , and Kahraman, A. , 2013, “ A Dynamic Model of a Double-Helical Planetary Gear Set,” Mech. Mach. Theory, 70, pp. 157–174. [CrossRef]
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. [CrossRef]
Gu, X. , and Velex, P. , 2012, “ A Dynamic Model to Study the Influence of Planet Position Errors in Planetary Gears,” J. Sound Vib., 331(20), pp. 4554–4574. [CrossRef]
Weber, C. , and Banaschek, K. , 1953, Formänderung und Profilrücknahme bei Gerad-und Schrägverzahnten Antriebstechnik, Vol. 11, Vieweg, Braunschweig, Germany.
Lundberg, G. , 1939, “ Elastische Berührung Zweier Halbraüme,” Forsch. Geb. Ingenieurwes., 10(5), pp. 201–211. [CrossRef]
Botman, M. , and Toda, A. , 1979, “ Planet Indexing in Planetary Gears for Minimum Vibrations,” ASME Paper No 79-DET-73.
Ma, P. , and Botman, M. , 1985, “ Load Sharing in a Planetary Gear Stage in the Presence of Gear Errors and Misalignment,” ASME J. Mech. Trans. Autom. Des., 107(1), pp. 4–10. [CrossRef]
Velex, P. , and Flamand, L. , 1996, “ Dynamic Response of Planetary Trains to Mesh Parametric Excitations,” ASME J. Mech. Des., 118(1), pp. 7–14. [CrossRef]
Bonori, G. , Barbieri, M. , and Pellicano, F. , 2008, “ Optimum Profile Modifications of Spur Gears by Means of Genetic Algorithms,” J. Sound Vib., 313, pp. 603–616. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Internal gear element—planet/flexible ring-gear connection

Grahic Jump Location
Fig. 2

External gear element definition

Grahic Jump Location
Fig. 3

Potential contact points with respect to beam nodes

Grahic Jump Location
Fig. 4

Deformed (thin line) and undeformed (thick line) PGT, (a) and (b) in two perpendicular planes (amplification factor = 100, carrier not represented)

Grahic Jump Location
Fig. 5

Deformed (thin line) and undeformed (thick line) double-helical PGT (a), (b) in two perpendicular planes and (c) zoom on the left helix (upper, amplification factor = 100, carrier not represented)

Grahic Jump Location
Fig. 6

Influence of a flexible ring-gear on the dynamic factor of (a) external and (b) internal gear meshes—single helical PGT

Grahic Jump Location
Fig. 7

Influence of a flexible ring-gear on the dynamic factor of (a) external and (b) internal gear meshes—double-helical PGT (results on left and right helices are identical)

Grahic Jump Location
Fig. 8

PM definition on planet teeth

Grahic Jump Location
Fig. 10

Optimum PMs for a double-helical PGT set with a deformable ring-gear (● for a flexible ring-gear, thick line and line for the modified or not Master Curve)

Grahic Jump Location
Fig. 9

Optimum PMs for a single helical PGT with and without flexible ring-gears (◆, ● for a rigid or flexible ring-gear, thick line and line for the modified or not Master Curve)

Grahic Jump Location
Fig. 11

Influence of long, medium and short PMs on the dynamic factor of (a) external and (b) internal gear meshes

Grahic Jump Location
Fig. 12

Influence of long, medium and short PMs on the dynamic factor of (a) external and (b) internal gear meshes (results for left and right helices are identical)

Grahic Jump Location
Fig. 13

Influence of different optimum PM combinations (external/internal) on the dynamic factor of (a) external and (b) internal gear meshes

Grahic Jump Location
Fig. 14

Influence of optimum PM combinations (external/internal) on the dynamic factor of (a) external and (b) internal gear meshes (results for left and right helices are identical)

Grahic Jump Location
Fig. 15

Influence of torque on the mesh dynamic responses of external gear meshes with (a) long, (b) medium, and (c) short optimum reliefs (helical)

Grahic Jump Location
Fig. 16

Influence of torque on the mesh dynamic responses of internal gear meshes with (a) long, (b) medium, and (c) short optimum reliefs (helical)

Grahic Jump Location
Fig. 17

Influence of torque on the mesh dynamic responses of external gear meshes with (a) long, (b) medium, and (c) short optimum reliefs (double-helical)

Grahic Jump Location
Fig. 18

Influence of torque on the mesh dynamic responses of internal gear meshes with (a) long, (b) medium, and (c) short optimum reliefs (double-helical)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In