Dynamic stability of interacting spur gears.

Abstract

Driven by an industry trend toward more stringent noise specifications, recent research has sought to develop more refined descriptions of gear system dynamics. This dissertation presents the re-examination of the basic governing equations of motion (EOM) for interacting spur gears, which was motivated by the need to develop more accurate models. In this dissertation, the kinematics of spur gears in conjugate action is determined, to obtain a mesh stiffness as a function of a mean rotation angle of gears. Then, the EOM are derived, including the nonlinear terms arising from the dependence of the mesh stiffness on the mean rotation angle. These terms ensure conservation of energy, which was neglected in previous studies. The effect of neglecting these terms is quantified, by comparing the new EOM and the classical EOM. A multi-degree-of-freedom system is discussed which may include, for instance, shafts, bearings, motors, etc, along with a discussion of backlash; however, the emphasis of the thesis is on an interacting gear pair. A normalization is performed to generalize the results. The allowable magnitude of the applied torque, for the gears not to yield, is determined. Next, the stability of a steady-state response is determined for both the new EOM and the classical EOM. The stability of the new EOM is obtained from a perturbation analysis, using the first-order-Jacobian-matrix analysis. Floquet Theory is used to determine the stability of the classical EOM. Parameter ranges in which the nonlinear term effects stability are determined. The effects of damping and the magnitude of the applied torque on the stability, combined with the nonlinear term, are shown. Finally, energy conservation of general systems with a kinematically varying stiffness is discussed, using simple examples. It is concluded that, unless the actual direction of the contact force is considered in the models for such systems, the models do not conserve energy.