# Time varying transient center excitation of reducer variable speed integrated gear

The circular gear mainly includes two kinds of internal excitation stiffness excitation and error excitation. As the power transmission of the reducer variable speed integrated gear is carried out through the meshing between the teeth of the small cylindrical gear and the non-circular gear, it also has these two typical internal excitation. In addition, it also has a special internal excitation, the transient center excitation, which is caused by variable speed transmission.

Firstly, the mechanism of time-varying instantaneous center excitation is described, and the small cylindrical gear is simplified as a cylindrical rigid body with radius Rb1, where RB1 = r1cos α, α is the pressure angle of the indexing circle of the small cylindrical gear, and the non-circular gear is simplified into a cylindrical rigid body with radius of RB1 For the non cylindrical rigid body of rr22, the elastic contact between teeth is replaced by linear spring. For the convenience of analysis, the damping between teeth is temporarily ignored. The force direction of spring is consistent with the normal force between teeth, and Fig. 1a is obtained.

In Fig. 2-5A), point m is the coincidence point of pitch curve of spur gear and non-circular gear, where the linear velocity of the two gears is equal. Figure 2-5b) is the projection of the equivalent model along the direction of the end face of the small cylindrical gear, where  is the pressure angle of the small cylindrical gear, then the normal relative displacement X between the teeth of the gear is:

Where Q1 and Q2 are the rotation angles of the two gears caused by the elastic deformation of the teeth

Where km is the comprehensive meshing stiffness between teeth, then the torque of elastic meshing force between teeth on non-circular gear is:

In addition, the inertia torque of non-circular gear due to variable speed rotation is as follows:

It can be seen from the formula that even if the gear tooth stiffness is constant, the parameter R2 will also cause the change of the meshing force between teeth, and the parameter R2 is precisely determined by the time-varying instantaneous center, so the time-varying instantaneous center will cause the change of normal relative displacement between teeth, thus generating dynamic meshing force to dynamically excite the gear system. It can be seen from the formula that the torque of the non-circular gear changes with the force arm R2 of the circular force, so the time-varying instantaneous center can also generate dynamic torque by changing the arm of the circular force of the gear. In the formula, the inertia torque of non-circular gear changes periodically, and also generates dynamic excitation. Therefore, the time-varying instantaneous center excitation is to excite the gear system by changing the relative normal displacement between teeth, the circumferential force arm and generating dynamic inertia torque. The first two modes are similar to the meshing stiffness excitation of teeth, belonging to parametric excitation, and the latter one is similar to error excitation. 