Nonlinear excitation and mesh characteristics model for spiral bevel gears

Accurate and rapid calculation of nonlinear excitation (NE) and loaded mesh characteristics (LMC) of gear pairs is a key to achieving rapid iterative optimization of gear system designs. Therefore, an analytical model (AM) was proposed for the accurate and numerically efficient calculation of NE and LMC for application to spiral bevel gears (SBG). First, a tooth contact analysis of an SBG was completed using Coons surface technology and mesh theory to determine the contact path and unloaded transmission error (UTE). Thus, an AM for the principal and relative normal curvatures of the SBG was derived using the Euler's formula. Based on the traditional calculation method for a Hertz contact ellipse, the relation between the relative curvatures was introduced to derive the formulas for the major and minor axes using the Muller's method, which could improve the programmability and avoid numerical instability of the program. Then, analytical models for the contact ellipse and pressure distribution on the contact tooth surfaces were derived using the Hertz model and geometric theory. Subsequently, based on a single tooth LMC, infinitesimal method, elasticity, and series stiffness model, a single mesh stiffness (SMS) model for SBGs was developed, which incorporates transverse tooth stiffness, transverse gear foundation stiffness, axial tooth stiffness, axial gear foundation stiffness, and Hertz nonlinear contact stiffness. Based on the SMS and UTE, compliance and load matrices were established, and the time-varying mesh stiffness, loaded transmission error, mesh force, and multi-tooth LMC models of the SBG were determined. Finally, the proposed method was validated against nonlinear FE simulations using examples. Compared to FE simulations, the proposed method requires significantly less computational effort and can be further extended to optimization or system analysis problems.