# Time invariant / variable model of dynamic nonlinearity of gear transmission system

All the models are developed and established on the basis of the concept of equivalent tooth mesh stiffness, which are collectively referred to as lumped parameter models. With the continuous improvement of the theory and technology of finite element analysis, the finite element model of gear transmission system has developed rapidly. The finite element model of gear transmission is established based on the theory of finite element contact mechanics, and the dynamic finite element model of gear transmission system is established based on the finite element analysis software. The advantage of the finite element model is that it does not need to determine the time-varying stiffness, tooth profile error and tooth profile modification. These factors are naturally included in the dynamic model. The finite element model is used as the benchmark of the verification of the lumped parameter model, but the calculation efficiency of the finite element model is lower than that of the lumped parameter model.

Nonlinear time invariant model

In the process of gear transmission, in order to avoid tooth jamming, clearance should be left between the tooth sides. At the same time, the manufacturing error of the tooth and the wear of the tooth surface will cause clearance between the gears. In the nonlinear time invariant model, the influence of clearance is considered and the constant meshing stiffness is used instead of the time-varying meshing stiffness. Clearance function is a strong non-linear factor, which makes the gear transmission system have three different motion states: non impact state, unilateral impact state and bilateral impact state. With the change of parameters, there will be amplitude jump and many complex motion forms, such as subharmonic motion, quasi periodic motion and chaotic motion. Based on the research of nonlinear time-varying model, domestic and foreign scholars focus on the knock characteristics of gear, the dynamic characteristics under the mixed action of random parameters and determined parameters, and the nonlinear vibration form of gear transmission system under fitting clearance function. Because of the strong coupling between the clearance function and the time-varying meshing stiffness, the general clearance function and the time-varying meshing stiffness should be considered in the dynamic model of the gear transmission system.

Nonlinear time-varying model

The nonlinear time-varying model contains many nonlinear factors such as time-varying stiffness and clearance. Recently, with the development of nonlinear dynamics theory and the improvement of computer performance, scholars at home and abroad pay more attention to the nonlinear time-varying model of gear transmission and conduct extensive and in-depth research. At present, the research trend of nonlinear time-varying model of gear transmission system mainly focuses on two aspects: first, the degree of freedom of single mass point in the dynamic model increases gradually, from pure torsional dynamic model to transverse torsional coupling dynamic model to three-dimensional full freedom dynamic model; second, the nonlinear factors in the dynamic model are gradually diversified, not only including Time-varying meshing stiffness, backlash, tooth profile error and tooth surface friction, as well as geometric eccentricity and root crack. Nonlinear time-varying model is widely used in the study of dynamic characteristics of gear transmission system.

In the above nonlinear time-varying model, the modeling of gear engagement process is the core of the dynamic model. The test shows that the vibration response of the gear transmission system shows obvious nonlinear characteristics, and the time-varying stiffness and clearance become the most important nonlinear factors in the mesh model. At the same time, in order to establish a precise lumped parameter dynamic model, the lateral vibration caused by the deformation of the transmission shaft or bearing should be considered. In the previous meshing model of gear transmission system, the clearance was mostly fixed, and the time-varying meshing stiffness was obtained by quasi-static method and showed periodic time-varying characteristics. Many scholars have carried out in-depth research by using this meshing model, and the test results show that this meshing model can better predict the dynamic characteristics of the gear transmission system.

In fact, the meshing parameters, such as meshing angle, clearance and time-varying stiffness, are affected to some extent by the change of center distance. When the bending stiffness of the transmission shaft of the gear transmission system is large or the load is small, the influence of the change of the center distance on the meshing parameters may be very small and can be ignored. When the bending stiffness of the transmission 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 the change of gear center distance on the meshing parameters and dynamic characteristics of gears. The influence of center distance variation on meshing angle and coincidence degree is analyzed for the fixed shaft gear transmission system and the planetary gear transmission system respectively, but the clearance is ignored in the dynamic model. The influence of center distance on the clearance is analyzed, but the meshing stiffness is obtained by quasi-static method. In order to improve the calculation efficiency, the mesh stiffness is calculated under the quasi-static condition and substituted into the dynamic model. The model considers the influence of the change of the center distance and other factors on the mesh stiffness, but the clearance and mesh angle in the dynamic model are fixed. Considering the phase difference caused by the change of the meshing state of the tooth surface and the tooth back, the meshing stiffness model in the form of asymmetric rectangular wave is adopted to study the impact of the speed and the friction of the tooth surface on the knocking characteristics of the gear.

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