Discrete Fourier transform of nonlinear function of gear clearance in reducer variable speed integrated gear

In the system of equations Si, the coefficient Zi of the steady-state response q is the unknown quantity to be solved. In addition, f (q) is a function of Q, and its harmonic coefficients fi are functions of the unknown quantity Zi. Before solving, FI is expressed by Zi.

Firstly, the formula is discretized so that:

Where m and m are integers, m ∈ [0, M-1].

The discrete sequence GM of the gap nonlinear function f (q) is obtained

Then, the discrete Fourier transform is used to obtain the coefficients fi: F (q)

By substituting the formula into the equation set Si = 0, we can obtain the unknown variable Zi (I = 1,2 (10R + 5).

Newton method is widely used in the field of numerical calculation of nonlinear equations, which is the basis of many algorithms in this field. However, there are many shortcomings in this method, such as the partial derivative of equations must be calculated in each iteration. The Broyden method adopted in this paper is one of the two iterative methods for solving nonlinear equations, which is developed on the basis of Newton method’s defects. It is a variant form of Newton’s method. Considering that the quasi Newton method can avoid the inversion of Jacobi matrix of the system of equations in each iteration process, and the calculation efficiency is high. In this paper, Broyden’s method of quasi Newton’s method is used to solve the nonlinear algebraic equations Si = 0 according to a given set of initial values of Zi, and the steady-state periodic solution of Q is obtained.

In order to more clearly express the solution process of the periodic steady-state solution of the reducer variable speed integrated gear based on the harmonic balance method and Broyden method of quasi Newton method, the solution process is shown in the form of flow chart to assist understanding.

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