# Solution to Nonlinear Vibration of Spur Gear Drive System

The experimental research shows that the gear transmission system shows obvious non-linear vibration characteristics, and the traditional linear model can not reliably reflect the vibration characteristics of gears. Therefore, the theory and method of non-linear dynamics must be used to study this non-linear vibration characteristics.

Due to the complexity of the nonlinear dynamic system itself, the solutions of the system often vary from problem to problem, so far there is no suitable universal solution.At present, the solution method of nonlinear dynamic equation and the research on non-linear vibration characteristics have become the main research contents of domestic and external gear drive system dynamics.

At present, analytical method and numerical method are the main methods to solve the non-linear dynamic problems of gear transmission system.The application range of classical analytical method is very limited. Only a few special periodic and quasi-periodic solutions of weakly nonlinear systems with less than one degree of freedom can be calculated and conclusions about stability can be obtained.

As the degree of freedom increases in the dynamic model of gear transmission system, the non-linear factors considered become more and more complex. The non-linear dynamic equation is a complex high-dimensional non-linear system, and its solution is more complex than that of traditional linear system and low-dimensional non-linear system. It is generally impossible to study directly by analytical method and can only be solved by numerical method.

The prominent function of numerical method in nonlinear vibration is to find new phenomena, which has become the prominent feature of modern development of nonlinear vibration.The advantage of the numerical method is that the dynamic response of the system can be obtained. The solution not only includes the main harmonic response, but also the super-harmonic, sub-harmonic, combined harmonic response and chaos response.

The numerical method is mainly to solve the non-linear differential equation by numerical value in order to obtain the motion law of the non-linear system under the specific parameter conditions and initial conditions. Its mathematical basis is the numerical solution of the initial value problem of the ordinary differential equation, mainly including iteration method, variational method, finite element method, etc.

Due to the incomplete mathematical tools for dealing with non-linear vibration, numerical methods play a very important or even irreplaceable role.Generally, the following problems need to be considered when choosing numerical method to solve the dynamic equation of gear transmission system: the solution accuracy of the algorithm, the stability of the algorithm, the convergence of the algorithm, the calculation efficiency of the algorithm and the rigidity of the dynamic equation.

With the continuous development of high-precision numerical calculation methods, the analysis methods for non-linear vibration characteristics of gear transmission system have been developed from the initial time-domain and frequency-domain methods to the combination of non-linear dynamic analysis methods. The transient, steady-state and chaos characteristics of gear transmission system have been comprehensively studied from many aspects.In the field of nonlinear dynamics, the most commonly used nonlinear dynamic analysis methods include qualitative analysis methods such as phase space, phase plane, section and global bifurcation diagram, and quantitative analysis methods such as fractal dimension and index.

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