Efficient contact analysis method and modification optimization for hypoid gears based on substructure condensation and simplex method

This paper addresses the contact analysis problem of hypoid gear pairs and proposes an efficient numerical calculation method integrating substructure condensation with the simplex method for solving contact problems. The method employs substructure condensation technology to reduce the degrees of freedom of complex structures to the meshing interface, significantly decreasing the dimension of the system stiffness matrix while maintaining computational accuracy, thereby improving computational efficiency. The study utilizes eight-node hexahedral elements for gear finite element discretization, combined with shape function interpolation and isoparametric transformation techniques to accurately solve equivalent nodal forces and displacement responses in the contact region, constructing a flexibility matrix that characterizes the system’s elastic deformation properties. Based on this foundation, the simplex method is applied for contact iteration solving: first obtaining the relative angular deviation caused by elastic deformation under ideal uniform loading along a single contact line, then distributing the load torque according to the contact line length, and calculating the comprehensive angular deviation during multi-tooth synchronous meshing, thereby establishing the basis for tooth surface topological modification optimization design. Finally, by comparing the loaded transmission error curves before and after modification and validating against traditional finite element analysis results, the effectiveness of the method is demonstrated. Compared with conventional finite element methods, this approach improves computational efficiency by approximately 80% while maintaining calculation accuracy, providing a high-fidelity and high-efficiency numerical analysis tool for optimizing the contact fatigue performance of hypoid gears.