# Theoretical research and solution method of knock model of automobile transmission gear

The phenomenon of gear knocking in automobile transmission belongs to the NVH dynamics problem of transmission system. The research idea is shown in Figure 4. The phenomenon of gear knocking is abstracted as a mechanical model, which is transformed into a solvable mathematical model by using the relevant dynamics principle, and the phenomenon of gear knocking is reproduced by using the simulation model.

The study of gear knocking belongs to the category of nonlinear vibration. For the practical nonlinear vibration in engineering, in addition to the experimental method, the commonly used theoretical research includes qualitative and quantitative research methods. The former is based on the response of nonlinear model to analyze the nonlinear system of gear from the perspective of stability, bifurcation and chaos, The latter focuses on the response solution and natural mode calculation of dynamic vibration simulation model.

#### 1. Qualitative research method of gear knock model

The qualitative analysis method of gear system is based on the differential equation of motion of gear system, which studies the properties of solution directly to judge the state of motion. The phase plane method is the most intuitive qualitative evaluation method, which uses the phase trajectory to describe the characteristics of motion stability, bifurcation and chaos of the system.

For the qualitative study of the gear system of automobile transmission, it starts from the study of single-stage gear pair. Theodosiades  uses the Poincare mapping method to analyze the influence of dimensionalized frequency on the motion stability of the system. The results show that when the equivalent dimensionalized frequency takes a certain value, it can be seen from the Poincare diagram of dimensionalized displacement difference and speed difference that characterize the motion stability of the system, The system has chaotic motion. Zhang suohuai et al  established a single-stage gear system knock model, and studied the impact of external dynamic excitation amplitude on gear knock by using the phase plane method. The results show that when the excitation amplitude changes, the gear system will generate one cycle, two cycle, multi cycle and quasi cycle motion. The phase diagram of the gear system has complex phase trajectory, For example, it is not easy to observe the motion stability of the gear system by direct observation method when the chaotic motion occurs in the gear system. The method of Poincare section, frequency division sampling and the calculation of the maximum Lyapunov exponent are often used to judge the stability of the system.

#### 2. Quantitative research method of gear knock model

The quantitative research methods of nonlinear system include analytical method, approximate analytical method, numerical method and semi numerical semi analytical incremental harmonic balance method.

2.1 analytical and approximate analytical solutions

The analytic method is to find the analytic solution of the nonlinear differential equation accurately and get the motion law of the nonlinear system. The solution process usually involves the introduction and research of non elementary functions (such as elliptic functions), etc. for the gear system with more nonlinear factors, this method is of high difficulty. In the actual research, the approximate analytical method is often used, For example, Kahraman  Based on the “vibration impact” model of single-stage gear system, the solution process of harmonic balance method is derived, and the solutions of the excitation terms and equations of the gear system are all expanded into Fourier series, The approximate solution of the model is obtained by using the self phase equilibrium of harmonic components of the system force and inertia force. The nonlinear characteristics of the system, such as multi value solution and amplitude jump, are obtained by using the amplitude frequency characteristic results.

2.2 numerical solution method

For example, the single-step Runge Kutta method, various improved algorithms and multi-step Adams method, as well as the central difference method, houbert method, Wilson – θ method and Newmark – β method, which are commonly used to solve the linear vibration system with multiple degrees of freedom, are commonly used to solve the ordinary differential equations.

For the gear system of automobile transmission, high-dimensional non-linear mathematical models considering various factors such as gear tooth lubrication and backlash are often established, which are difficult to be solved by analytical method, and the numerical method is most widely used. For example, Wang  applies the high-precision explicit one-step method Runge Kutta method, Two kinds of solvers, ode15 and ode23, were used to solve the knock model of a five gear manual transmission. The time domain results of each gear movement were obtained by numerical method, and the knock strength of each gear pair was analyzed and compared.

2.3 incremental harmonic balance method

Incremental harmonic balance method (IHBM) is a semi analytical and semi numerical method for solving the equations of motion of nonlinear systems. It can be applied to both weak nonlinear systems and strong nonlinear systems. It can also be applied to solving the knock system of gears. For example, Yang Shaopu et al.  established a gear system model considering the time-varying meshing stiffness and backlash of gears, By using the incremental harmonic balance method, the approximate solution with any accuracy can be obtained. The influence of damping ratio and external excitation amplitude on the amplitude frequency characteristics of the system, as well as the bifurcation characteristics of the system, are analyzed. It is shown that the incremental harmonic balance method is feasible to solve the nonlinear vibration. Li Yinggang et al  established a single degree of freedom gear pair system dynamics model considering elastic damping and external dynamic excitation, The incremental harmonic balance method is used to effectively prove the nonlinear dynamic behaviors of the gear system, such as parametric resonance, multivalued solution and amplitude jump, under the external dynamic excitation.

As the same as the analytical and approximate analytical methods, the incremental harmonic balance method is more difficult to solve when the factors such as gear surface lubrication and bearing characteristics are taken into account. This method is also suitable for solving the low degree of freedom gear knock model. 