# Non-linear time-varying model of spur gear

In fact, the engagement parameters such as engagement angle, clearance and time-varying rigidity of gears are affected to some extent by the variation of center distance.

When the bending rigidity of the drive shaft of the gear drive system is large or the load is small, the influence of the change of center distance during gear movement on the engagement parameters may be small and negligible.When the bending rigidity of the drive shaft of the gear transmission system is small or the load is large, the change of the center distance of the gear will increase obviously, which further increases the influence of the change of the center distance on the meshing parameters.

Recently, some scholars have studied the influence of gear center distance change on meshing parameters and gear dynamic characteristics.For fixed-shaft gear drive system and planetary gear drive system respectively, the influence of center distance change on engagement angle and overlap degree is analyzed, but clearance is neglected in dynamic model.The influence of center distance change on clearance is analyzed, but the meshing stiffness is obtained by quasi-static method.

In order to improve the calculation efficiency, a model is used to calculate the engagement stiffness under quasi-static conditions and substituted into the dynamic model. The model takes into account the influence of such factors as the change of center distance on the engagement stiffness, but the clearance and engagement angle in the dynamic model are fixed values.Considering the phase difference caused by the change of meshing state of tooth surface and back of tooth, the impact characteristics of gear driven by rotating speed and friction of tooth surface were studied by employing asymmetric rectangular waveform meshing stiffness model.

These models are developed and established on the basis of the concept of equivalent meshing stiffness of teeth. They are collectively called centralized parameter models.With the continuous improvement of finite element analysis theory and technology, the finite element model of gear transmission system has been developed rapidly.Finite element model of gear drive is established based on contact mechanics theory of finite element, and dynamic finite element model of gear drive system is established based on finite element analysis software.

The advantage of finite element models is that no additional factors such as time-varying stiffness, tooth profile error and tooth profile modification need to be determined. These factors are naturally included in the dynamic model and the finite element model is used as the basis for verification of the centralized parameter model, but the calculation efficiency of the finite element model is lower than that of the centralized parameter model.

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