An efficient multibody dynamic model of three-dimensional meshing contacts in helical gear-shaft system and its solution

The dynamics of helical gear-shaft systems are characterized by three-dimensional (3D) meshing contacts that have significant variations in the location and size of the contact area, resulting in noise that is unavoidably transmitted to the gearbox through the shaft. Accurate and efficient predictions of the dynamic behaviors of helical gear and shaft are indispensable in reliable and cost-effective gearbox design. Available analytical methods, though computationally feasible, cannot consider multi-point contacts and uneven tooth-load distribution. In contrast, the finite element (FE) method provides a high-fidelity approach to compute the dynamic behaviors of a general gear-shaft system at high expenses of computation. This paper aims to establish a high-efficiency multibody dynamic model for 3D contacts in helical gear-shaft systems, in which the helical gear is pertinently represented under the framework of Arbitrary Lagrangian Eulerian formulation and the shaft is discretized by 3D Timoshenko beam elements. The computational efficiency is greatly improved through the following four steps. First, the low-frequency approximation technique is adopted to reduce the degrees of freedom (DOFs) resulting from the fixed boundary normal modes. Second, under the framework of ALE formulation, only the FE nodes of three meshing tooth-faces are defined as boundary nodes. Then, the dynamic equations and Jacobian matrix are simplified by ignoring the inertial forces associated with deformation. Finally, a two-step algorithm is adopted to accelerate the contact detection process. The accuracy and efficiency of the proposed method are demonstrated through five numerical tests with correlation to commercial nonlinear finite element software.