# Dynamic linear time invariant / variable model of gear transmission system

Before the s of the th century, gear dynamics was based on the theory of impact. People used meshing impact as the basis of describing and explaining the dynamic excitation and response of gear. The dynamic response of single degree of freedom system under impact was used to express the dynamic behavior of gear transmission system. In, the first spring mass model was put forward, in which the concept of equivalent meshing stiffness of gear teeth laid the theoretical foundation for the dynamics of gear transmission system. Since then, the dynamic model of gear transmission system has been put forward. The dynamic model of gear transmission system has experienced the development process from linear to nonlinear, from steady to variable. According to the different nonlinear factors, the dynamic models can be classified into the following types: linear time invariant model, linear time-varying model, nonlinear time invariant model, nonlinear time-varying model.

（1） Linear time invariant model

In the linear time invariant model of gear transmission system, all the nonlinear factors are ignored, and the time-varying meshing stiffness is replaced by the constant meshing stiffness. The linear time invariant model is still in the scope of linear vibration theory. This process does not consider the dynamic stability problem caused by time-varying meshing stiffness, and avoids the nonlinearity caused by clearance and other factors. The early research based on linear time invariant model is reviewed in detail. Recently, scholars at home and abroad have established a multi degree of freedom linear time-varying dynamic model for more complex gear transmission forms, including fixed shaft gear rotor system, planetary gear transmission system, herringbone wheel planetary gear transmission system, composite planetary gear transmission and bevel gear transmission system. In the model, the Timoshenko beam element is used to build the transmission shaft, and the influence of the gyroscopic moment caused by high speed is considered. Linear time invariant model is often used in two aspects: one is the natural characteristics of natural frequency, mode shape and critical speed, and the sensitivity of natural characteristics to system parameters; the other is to calculate and study the undamped free vibration response of the system and the forced vibration response under external excitation by numerical integration method.

（2） Linear time-varying model

In order to ensure the continuity of gear transmission, the coincidence degree of gears is generally greater than that of gears, resulting in the change of the number of teeth engaged in the process of gear transmission. In a very short time, the meshing stiffness changes sharply, resulting in meshing impact. The meshing stiffness is one of the main incentives of the gear transmission system, which is the main reason for the increase of the vibration of the gear transmission system and the gear transmission It is different from other vibration systems. In the linear time-varying model, the time-varying of meshing stiffness and the static transmission error caused by tooth profile error are considered, but the influence of nonlinear factors such as clearance is ignored. The dynamic characteristics of the gear transmission system are very sensitive to the amplitude changes of the frequency components in the meshing stiffness. At present, the time-varying meshing stiffness is usually calculated under the quasi-static condition by the finite element method, boundary element method, analytical method, empirical formula and test method. There are generally three ways to deal with the time-varying mesh stiffness in the gear dynamic model: directly replace the time-varying mesh stiffness model into the dynamic model; expand the time-varying mesh stiffness into the form of series; ignore the small change of mesh stiffness caused by the change of mesh point and adopt the mesh stiffness in the form of rectangular wave. Some scholars introduce the friction of tooth surface into the dynamic model at the same time. The study of linear time-varying model focuses on the stability of parametric vibration and the dynamic response under internal and external excitation. 