# Nonlinear dynamic model of lateral-torsional-pendulum coupling for involute spur gear drive

The three-dimensional solid model of involute spur gear drive is shown in the figure, and the following assumptions are used for dynamic modeling:

1.The gear wheel body is simplified as a rigid body and the gear pairs are coupled together by forces along the meshing surface.

2.Ignore the influence of the degree of freedom of gear swing on the contact line of gear teeth along the tooth width direction, and the meshing surface is always tangential to the base circle of both gears;

3.Theoretically, the involute spur gear drive has no axial force component. Therefore, each gear has two degrees of freedom, including two degrees of freedom for lateral translation, one degree of freedom for torsion and two degrees of freedom for swing.

The three-dimensional dynamic model of involute spur fixed-shaft gear drive is shown in the figure under the engagement state of tooth surface.There are two coordinate systems in the figure: the first is the local coordinate system of each gear, which is the theoretical center position of the drive shaft; the second is the fixed coordinate system, which coincides with the local coordinate system of the gear.It is the meshing surface between the gear pairs.Each gear consists of two degrees of freedom for lateral translation, one degree of freedom for torsion and two degrees of freedom for swing.

When using Lagrange method to model, the force between gear pairs and the direction of action as well as the centroid position vector of each gear should be determined first.In this section, firstly, the uneven load distribution in the direction of tooth width is considered, and the force between gear pairs is analyzed. Then, the centroid position vector of each gear is determined, and the influence of center distance change, tooth surface change and meshing state change on the direction of meshing force is analyzed. Finally, the yaw-torsion coupling non-linear dynamic model is established by Lagrange method. 