In the linear time-invariant model of gear transmission system, all non-linear factors are neglected and the time-varying meshing stiffness is replaced by the constant meshing stiffness.Linear time-invariant model is still in the scope of linear vibration theory. This process does not consider the dynamic stability caused by time-varying meshing stiffness and avoids the nonlinearity caused by clearance and other factors.

Early research on linear time-invariant model is summarized in detail.Recently, domestic and foreign scholars have established multi-degree-of-freedom linear time-varying dynamic models for more complex gear transmission forms, including fixed-shaft gear-rotor system, planetary gear transmission system, herringbone planetary gear transmission system, compound planetary gear transmission and bevel gear transmission system.

In the model, the drive shaft is mostly built with Timoshenko beam unit and the influence of gyroscopic moment caused by high speed is considered.Linear time-invariant models are often used in two aspects:

First, the inherent characteristics such as natural frequency, vibration mode and critical speed as well as the sensitivity of the inherent characteristics to the system parameters.

Second, the numerical integration method is used to calculate the undamped free vibration response of the system and the forced vibration response under external excitation.