# Dynamic model of three ring reducer

These documents do not consider the problem of over constraint and elastic deformation of the mechanism when analyzing the forces on the three ring reducer, and the model of meshing force is too simple, so it is quite different from the actual situation.

Aiming at the problem of virtual constraint in the three ring reducer, the deformation coordination condition based on the contact deformation of the moving pair is proposed, and the multiple parallel crank force models with virtual constraint are established. But in the force analysis, the equilibrium equation is completely static.

Later, considering the constraints of the mechanism, the elasticity of the planetary bearing, the elasticity of the meshing gear pair, the rigid inertia force of the inner gear plate, etc., according to the force balance equation of the gear plate, the different positions of the mechanism are obtained

The meshing force of gear, the force of planetary shaft and planetary bearing.

During this period, it is considered that the main deformation of the three ring reducer includes the contact deformation of each component, the torsional bending deformation of the shaft, the tension and compression deformation of the internal gear plate, etc. It is considered that the main deformation modes are shaft bending and torsion deformation and internal tooth plate tension and compression deformation. In order to solve the problem simply, the inner tooth plate is regarded as a bar with constant cross section. On this basis, the balance equation of the internal gear plate is obtained, and the stress of each bearing and gear in different positions of the mechanism can be solved. However, when dealing with the deformation of the internal gear plate, it is simply equivalent to a bar with the same area, which has a large error with the actual situation. Later, the internal serrated plate is treated differently, as shown in, and the sectional equation of the section curve of tension and compression is given. Considering the practical problem, this paper uses the least square method to fit the actual curve into an approximate function composed of segmented oblique lines, so that it can be solved by a simple function.

The common feature of the above analysis is that the first and second derivative terms related to the elastic displacement are ignored, and the equations of motion established are algebraic equations rather than differential equations. The difference between each model is that different factors are recorded. Therefore, according to these equations, the natural frequency, mode shape, amplitude and other dynamic elements can not be calculated. 