The dynamic equation of bevel gear is a differential equation, which belongs to the initial value problem. The traditional methods to solve this kind of equation include perturbation method, phase plane method, Ritz average method, modal analysis method, state space method and numerical method. According to different actual situations, the above methods have certain applications, and can achieve good results. For example, the perturbation method is very suitable for solving small parameter nonlinear problems, and satisfactory results can be obtained. In 1992, Tang hengzhen gave the solution method of perturbation method, and obtained the calculation formula and method of steady-state response of four degree of freedom dynamic model of bevel gear pair by using direct expansion method.
In solving the steady-state response of bevel gear transmission system, Ritz method is more appropriate. This method is a quantitative method originally proposed by K. klotter. Later, M.C. Benton used this method to solve the steady-state response of dynamic systems with time-varying parameters in 1978. The phase plane method can be used to solve the dynamic stability and steady-state response of bevel gear system with sometimes variable meshing stiffness. The main advantage of this method is that it does not need to be numerically integrated in the whole cycle, but only in the middle part. Therefore, it has outstanding advantages in solving the analytical solution of complex bevel gear system when the vibration period cannot be determined in advance. The modal analysis method is to change the coordinates of the differential equation of the dynamic system and diagonalize the coefficient matrices [M], [C] and [k] of the equation, so as to decouple the original equation to obtain an independent and decoupled modal equation, and then solve the dynamic response of the multi degree of freedom system by solving the single degree of freedom dynamic system. Then the modal coordinates are transformed into physical coordinates by linear transformation, and the solution of the dynamic system equation is obtained.
For the state space method, a set of state vectors is used to represent the state of the system. As long as the set of variables at the initial time and the subsequent inputs are known, the behavior of the system at any time can be determined. For a steady system, the initial time is usually taken as zero. The state space method has considerable advantages in solving multi degree of freedom time-varying dynamic systems. Numerical methods are mainly calculated by iteration, and the most representative is Runge Kutta method. The Runge Kutta method is used to solve the differential equations of the dynamic system, Δ T is the iteration step size, which is determined by T+ Δ Calculate the value of time from the value of time t. At present, computer technology has developed to a high level, so numerical method has been widely used. For large-scale dynamic systems with multiple degrees of freedom, the numerical method has obvious advantages. In theory, the numerical method is suitable for solving any type of nonlinear dynamic equations, especially the strong nonlinear equations considering time-varying characteristics. Under the given initial conditions, the transient solution and steady-state order of the system can be obtained at the same time, which is very convenient for solving the dynamic equation of bevel gear transmission system.