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The torsional stiffness of involute spur gears

J Wang and I Howard*
Department of Mechanical Engineering, Curtin University of Technology, Perth, Western Australia, Australia
Abstract: This paper presents the results of a detailed analysis of torsional stiffness of a pair of involute spur gears in mesh using finite element methods. Adaptive meshing has been employed within a commercial finite element program to reveal the detailed behaviour in the change over region from single- to double-tooth contact zones and vice versa. Analysis of past gear tooth stiffness models are presented including single- and multitooth models of the individual and combined torsional mesh stiffness. The gear body stiffness has been shown to be a major component of the total mesh stiffness, and a revised method for predicting the combined torsional mesh stiffness is presented. It is further shown that the mesh stiffness and load sharing ratios will be a function of applied load.
Keywords: gear, FEA, adaptive mesh, individual torsional stiffness, combined torsional mesh stiffness, handover region
Gears are one of the most critical components in industrial rotating machinery. In recent years, many different procedures have been developed to model the behaviour of gears in mesh. Examples of this can be seen in references [1] to [33]. One of the many factors that can be investigated is the torsional mesh stiffness variation as the gear teeth rotate through the mesh cycle. For prediction of the torsional mesh stiffness, finite element analysis (FEA) modelling in particular can encompass three major stages: analysis with partial teeth models, analysis with single tooth gear models and analysis of multiteeth gears over a complete mesh cycle.
With recent hardware and FEA software advances, it is now possible to predict accurately the combined torsional mesh stiffness of two spur gears (multiteeth) in mesh, where one of the gear hubs is restrained from rotating, with the other gear hub having an input torque load. The combined torsional mesh stiffness of two gears in mesh is calculated here at each selected position in the mesh cycle, and the overall FEA solution shows that the combined torsional mesh stiffness varies periodically with the meshing position as the teeth rotate within the mesh cycle. In particular,
the modelling of the combined mesh stiffness in the handover region has shown that the handover point is a function of the applied load, which has further implications for the mesh stiffness and load-sharing ratios.
In order to understand the combined torsional mesh stiffness, the variations of the individual torsional stiffness for each of the gears in the mesh cycle have to be studied. However, to predict the individual torsional stiffness for one of the gears in mesh is a rather complex procedure, due to the non-linear contact. The actual position of the contact(s) is usually unknown until the solution for both gears in mesh is completed.

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