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TECHNICAL BRIEFS

# Energy Conserving Equations of Motion for Gear Systems

[+] Author and Article Information
Sejoong Oh

Senior Engineer General Motors Corporation, Warren, MI

Karl Grosh, James R. Barber

Associate ProfessorProfessor Department of Mechanical Engineering,  The University of Michigan, Ann Arbor, MI

J. Vib. Acoust 127(2), 208-212 (Apr 21, 2004) (5 pages) doi:10.1115/1.1891815 History: Received June 12, 2003; Revised April 21, 2004

## Abstract

A system of two meshing gears exhibits a stiffness that varies with the number of teeth in instantaneous contact and the location of the corresponding contact points. A classical Newtonian statement of the equations of motion leads to a solution that contradicts the fundamental principle of mechanics that the change in total energy in the system is equal to the work done by the external forces, unless the deformation of the teeth is taken into account in defining the direction of the instantaneous tooth interaction force. This paradox is avoided by using a Lagrange’s equations to derive the equations of motion, thus ensuring conservation of energy. This introduces nonlinear terms that are absent in the classical equations of motion. In particular, the step change in stiffness associated with the introduction of an additional tooth to contact implies a step change in strain energy and hence a corresponding step change in kinetic energy and rotational speed. The effect of these additional terms is examined by dynamic simulation, using a system of two involute spur gears as an example. It is shown that the two systems of equations give similar predictions at high rotational speeds, but they differ considerably at lower speeds. The results have implications for gear design, particularly for low speed gear sets.

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Copyright © 2005 by American Society of Mechanical Engineers
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## Figures

Figure 1

Dynamic model for two different gears: T0 is the constant applied torque, N1,N2 are number of teeth of gears 1 and 2, respectively; θ1 is the rotation angle of gear 1 and θ2 is a rotation angle of gear 2; k(θ) is the meshing stiffness

Figure 2

Limiting process occurring during the change in the number of teeth in contact

Figure 3

Total energy (Utotal) comparison between the energy conserving formulation (with the jump) and the rigid kinematic Newtonian equations (without the jump) using balanced torques

Figure 4

Power spectral density of the temporal response: Predictions in (a) and (d) are made from the equations with the jump, (b) and (e) are made from the rigid kinematic Newtonian equations (without the jump), and (c) and (f) are made from the classical model. (a)–(c) are power spectral densities for an initial angular velocity of the driving gear corresponding to 600RPM, (N1Ω1∕ωd=0.1). (d)–(f) are power spectral density for an initial angular velocity of the driving gear corresponding to 4500RPM(N1Ω∕ωd=0.75)

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