Analytical solution of spur gear mesh using linear model

The paper is concerned with vibration problem in a spur gearing. A one-degree-of-freedom (1 DOF) linear and periodic system including mesh stiffness and manufacturing error is considered. The parabolic function is used to describe the stiffness of tooth pair in a single pair contact. The approach using the periodic Green's function (PGF) in the form of truncated Fourier series is applied to find an analytical periodic solution in a steady state. Moreover, the method enables to find the borders of (in)stability by means of the real eigenvalues of the so called system matrix. The presented approach enables to solve the problem even in the case when periodic stiffness is implicit function of fluctuation parameter. The stability diagrams are presented and the validation of their correctness is performed by the Floquet method. The dynamic system behaviour in a steady state is also investigated and the periodic solution obtained by the presented analytical method is compared with the results given by the Runge-Kutta continuation. A very good agreements are achieved in all cases.