# Research status of mathematical model of gear transmission dynamics

In the primary stage of gear transmission dynamics research, based on the lack of effective numerical methods to solve the gear transmission analysis model, some scholars always linearize the nonlinear factors in the gear transmission system with the help of various mathematical means to solve the practical problems in engineering. With the development of the vibration theory of nonlinear gear transmission, the nonlinear factors in the process of gear meshing are gradually taken into account in the study of the dynamic characteristics of gear transmission system. At present, the research on the nonlinear dynamic behavior of gear transmission has attracted more and more attention of gear scholars, including the research on the natural characteristics of the system and the vibration response caused by excitation; Research on the influence of structural parameters of gear, box and other parts of the system on the dynamic behavior of the system; Research on the influence of tooth profile modification on the vibration characteristics of the system, and verification of reducing the vibration and noise of the system; In order to obtain the gear transmission system with low noise and vibration and high reliability, the research idea of system structure and related parameters and optimization is put forward.

Nonlinear factors such as tooth meshing stiffness and tooth clearance will appear in gear transmission. In addition, there are machining errors, installation errors, deformation, tooth wear and other factors in gear transmission, which will affect the dynamic behavior of the system. Through theoretical analysis and experimental research, it is found that there will be obvious nonlinear dynamic characteristics in the process of gear transmission, such as vibration, noise and harmonic response. In serious cases, there will be howling noise, which will increase the dynamic load and vibration noise of gear transmission system.

In the early stage, the vibration model of gear transmission system was established by using the linear gear transmission dynamics theory, and the dynamic performance of gear transmission system was studied. However, the nonlinear factors in the meshing process of gear transmission, such as time-varying meshing stiffness and tooth side clearance, were not considered in the vibration model. The model was relatively simple, and the analysis of vibration equation was relatively easy. However, with the development of nonlinear dynamics theory of gear transmission, the vibration model of gear transmission may contain a variety of degrees of freedom. The analysis object can be either a single pair of gears, or a complete gear transmission system composed of transmission shaft and bearing. Since the linear dynamics theory of gear transmission has been developed more mature and can quickly analyze and solve the natural characteristics of the system, the linearized analysis model is still commonly used to analyze the natural characteristics of complex gear transmission system.

In the process of gear transmission, the number of teeth in meshing state is constantly changing, and the coincidence degree of gear transmission is generally not an integer, so the meshing stiffness of gear teeth is also constantly changing. The excitation caused by the time-varying meshing stiffness of gear transmission makes the differential equation of gear transmission vibration model have time-varying coefficients, so it is necessary to study the dynamic behavior of gear transmission according to the nonlinear vibration theory. Some scholars consider the influence of time-varying meshing stiffness on gear transmission, and give the corresponding dynamic vibration model of gear transmission, such as Hill equation. According to this kind of model, the influence of time-varying meshing stiffness on transmission dynamic behavior and stability can be analyzed.

In addition, the existence of machining error, installation error, tooth wear and other problems makes the gap between the teeth appear during gear transmission. If the gear transmission operates under heavy load, the tooth surface is always in contact, and there will be no gap or small gap between the teeth, so the tooth gap will not affect the dynamic behavior of the gear transmission. If the gear transmission system is under frequent starting and braking conditions, or works under small load but high speed, the gap between teeth will make the meshing teeth appear repeated contact and separation, resulting in the change of contact state, which will affect the dynamic contact load of teeth and the dynamic behavior of gear transmission. Li Runfang, Wang Jianjun considered the influence of meshing stiffness and tooth clearance on the dynamic performance of gear transmission, proposed the corresponding vibration analysis model, and studied the dynamic performance of gear transmission system through numerical calculation and experimental analysis.

According to whether nonlinear factors are considered in the dynamic analysis model of gear transmission system, the analysis model can be divided into linear time invariant system analysis model, linear time-varying system analysis model, nonlinear time invariant system analysis model and nonlinear time-varying system analysis model. The linear time invariant model ignores the time variation of meshing stiffness, assumes that the meshing stiffness of gear teeth is constant, and ignores the influence of factors such as backlash and error; The linear time-varying model expresses the meshing stiffness of gear teeth as a time-varying function; In the nonlinear time invariant model, the influence of nonlinear parameters such as gear transmission clearance is considered, but the meshing stiffness is considered not to change with time; The nonlinear time-varying analysis model includes two parameters: meshing stiffness and gear transmission clearance, and their effects on the vibration characteristics of the system are considered. According to the different degrees of freedom included in the vibration model, the dynamic analysis model can also be divided into single degree of freedom and multi degree of freedom analysis models.

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