In this paper, the parametric vibration without backlash and the nonlinear vibration with backlash are studied respectively. The transmission system of the integrated gear is designed and manufactured. The actual vibration law of the gear is tested

(1) The time-varying transient center excitation of reducer variable speed integrated gear is a composite form of parametric excitation and displacement excitation. It excites the gear system by changing the relative displacement between teeth, the arm of dynamic torque and the inertia moment of driven gear.

(2) The pure torsional parameter vibration model of the reducer variable speed integrated gear is constructed, and the approximate analytical solution of the gear response is derived by using the multi-order harmonic balance method. Through quantitative analysis, it is found that with the increase of the speed, eccentricity, error amplitude, meshing stiffness amplitude ratio and torque, the vibration of the gear will increase; at high speed, the dynamic performance of the gear will increase At this time, the dynamic performance of the gear can be improved by tooth modification. The sensitivity of each parameter to vibration response varies with its own value. The increase of speed, error amplitude and meshing stiffness amplitude will increase the corresponding sensitivity, while the sensitivity of eccentricity will increase first and then decrease with the increase of eccentricity.

(3) Taking the backlash into account, the nonlinear vibration model of the reducer variable speed integrated gear is constructed. The approximate analytical solution of the dynamic response of the gear is derived by using harmonic balance method and discrete Fourier transform. Through quantitative analysis, it is found that with the increase of eccentricity, input speed and error amplitude, the gear undergoes no rattle, one-sided slap and two-sided slap, and with the increase of eccentricity, input speed and error amplitude, the gear undergoes the state of no rattle, single-sided slap and double-sided slap in turn When the moment increases, the gear will experience the opposite slap state; in the no slap and single side state, the dynamic response peak to peak value of the gear increases gradually, while when the bilateral impact occurs, the response peak to peak value curve jumps, and the gear vibration suddenly increases.

(4) The vibration data of the gear mechanism under the conditions of constant load and variable speed and constant speed and variable load are obtained. The test results show that the frequency components of the reducer variable speed integrated gear measured by the test are in good agreement with the assumed frequency components in the theoretical analysis. The influence law of the speed change and load change on the gear vibration response is basically consistent with the theoretical analysis results, which verifies the correctness of the theoretical analysis.