A Study On Tooth Profile Modification Of Planetary Gears Based On Nonlinear Dynamic Response

Reductions of vibration and noise are important concerns in planetary gears design.Tooth Profile Modification(TPM)has been commonly employed to compensate for the elastic gear tooth deflection under different loads in order to minimize meshing excitation variation,as a result vibration and noise can be reduced.Gear tooth backlash is necessary in design while it has a side effect allowing meshing gear teeth to separate each other when they undergo large deflections near resonance,resulting strong nonlinear effects such as vibration jump phenomena.There are meshing phase differences between each gear meshing pairs of planetary gears,which leads to the mutual cancellation or amplification of each harmonic meshing forces.This unique structure of planetary gears makes the nonlinear dynamic behavior of the planetary gear system extremely complex.Different from the traditional TPM strategy based on minimizing static transmission error,the TPM strategy for planetary gears based on minimizing nonlinear dynamic response is proposed.It is of great theoretical significance and engineering value by improving the nonlinear dynamic model and excitation theory of planetary gears with TPM.The main work of this paper is summarized as follows:(1)An improved analytical method based on the potential energy theory for calculation of gear TVMS is proposed,and the TVMS of sun-planet and ring-planet under different load conditions are discussed.The effect of Extended Tooth Contact(ETC)is taken into account to improves the accuracy of the calculation of TVMS.The special FE/CM software Calyx is used to verify the correctness of the analytical model of TVMS.As the excitation casuse of the nonlinear dynamic model,the Time Varying Mesh Stiffness(TVMS)of the planetary gears are analyzed systematically.(2)Analytical nonlinear dynamic model of planetary gears with and without TPM are established based on the lumped parameter method respectively,and the nonlinearity caused by gear contact loss is characterized by TVMS multiplied by a meshing separation function.An improved nonlinear dynamic model was introduced to simulate the dynamics of planetary gears with TPM,in which the TVMS is taken as stiffness excitation factor and the Non-Loaded Static Transmission Error(NLSTE)is deemed as displacement excitation item.Modal analysis of the planetary gear system is carried out,and the torsional vibration mode,translation vibration mode and planetary gear vibration mode of planetary gears are discussed,the obvious modes and degenerate modes are analyzed.Base on the above analysis,it is pointed out that the dynamic response analysis of the planetary gear system should focus on the two obvious pure torsional modes,1940 Hz and 5332 Hz.The planet phasing is analyzed in detail,and the laws of vibration response/mode suppression of the planetary gears by phasing is revealed.Meanwhile,the result shows that the planetary gears are in-phase meshing,and the translation vibration response/mode is well suppressed.However,the composite torque formed by the meshing force is magnified and the torsional vibration response / mode is excited.(3)Harmonic Balance Method(HBM)and Method of Multi-Scale(MMS)are employed to solves the nonlinear dynamic equations of planetary gears with and without TPM.Compared with the traditional numerical integration method,HBM and MMS effectively capture the "jump" characteristics of the planetary gears near the natural frequencies and the nonlinear branches under the conditions of increasing and decreasing speeds,and a closed-form approximation of the analytical solution near the primary response is obtained directly by MMS.The calculation results of this analytical solution are in good agreement with the finite element and numerical integration results published in the literature.Combined with the advantages of HBM and MMS with the nonlinear dynamic response characteristics of planetary gears,HBM is used to solve the wide frequency span response,and MMS is advocated to solve the narrow frequency span near the primary response of concerned modes.(4)The total analytical planetary gears TPM calculation based on the nonlinear response is explored through combining the improved gear TVMS calculation model,the improved nonlinear dynamic model of planetary gears with TPM and MMS,consequently,the complex model,complex calculation and complex problems are studied synchronously.By using the method of parameter study,the influences of different modification length and amount on the nonlinear dynamic response of planetary gears under different working loads are studied.The study shows that the optimal amount of TPM increases correspondingly with the increase of load under the same length of TPM,based on minimizing nonlinear dynamic response.The optimal TPM are different due to different load conditions,so the TPM should be designed taking the actual working loads into account when developing planetary gears.Among the parameters of TPM for planetary gears,the length and the amount of modification are a pair of parameters that affect each other.On the contrary,the improper amount of modification magnifies the nonlinear dynamic response of the planetary gear system,which is caused by the additional NLSTE excitation due to TPM.The experimental results further show that the peak value of noise in the primary order near the obvious pure torsional mode(1940Hz)of the planetary gears is obviously suppressed in the selected TPM scheme,and the total sound pressure level at the corresponding speed(frequency)is also significantly reduced.In addition,the noise corresponding to other working speeds(frequencies)far away from the mode is also well controlled.