# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 5 Conclusions  
This paper primarily explores the nonlinear vibration characteristics of parallel\(-\)axis internal beveloid gears under external factors such as input torque, rotational fluctuation, and input speed, as well as internal factors such as bearing backlash and tooth backlash. By investigating the impact and role of parameter excitation on the nonlinear vibration characteristics of the transmission system under the combined influence of internal and excitations, it aims to provide references for the structural design optimization and reduction in vibration noise in internal beveloid gear transmission systems. The main achievements and research findings obtained through analysis are as follows:  
(1) By utilizing the Runge-Kutta method, the gear dynamic differential equations can be solved to investigate the effects of various parameters on the vibration characteristics of the system. Research shows that: when the input torque changes with a tiny amplitude, it mostly influences the peak value of the system's steady vibration response and has a significant effect on the axial vibration amplitude of the gear; torque fluctuation will increase the instability of the system vibration, and its influence degree is closely related to the input torque amplitude; the input rotational speed of the transmission system directly determines its meshing angular frequency; the increase in its rotational speed under the non\(-\)resonant low rotational speed will increase the vibration amplitude rapidly, and when it is close to the natural frequency, there will be a sudden vibration change, which will affect the stability of the transmission system; the bearing backlash is one of the nonlinear parameters, which will affect the vibration stability of the system and the offset of the gear vibration balance position, which will make the gear vibration balance point shift greatly; and the existence of tooth backlash makes the vibration amplitude and stability of the system change, and the beveloid gear can adjust the tooth backlash to a certain extent. (2) The harmonic balance method is used to calculate and analyze the vibration differential equation of the single\(-\)stage internal gear, and the approximate analytical solution of the gear transmission system and the corresponding vibration speed and acceleration time domain response curve are obtained. It is compared with the time domain response curve of vibration velocity obtained by numerical method, and the two are consistent. The sweep curve is drawn to analyze the main and harmonic resonance phenomena of the system, the difference between the linear and nonlinear sweep curves is compared, and the influence of damping on the resonance characteristics is analyzed. It is found that the meshing damping has a strong inhibitory effect on the amplitude of harmonic resonance. This article has undertaken a certain level of investigation into the vibration characteristics of internal beveloid gear transmission systems. Nevertheless, there are aspects that can still be further enhanced. The distinction between radial and axial backlash of the bearing was not considered in this study, and the influences of factors such as gear eccentricity and lubrication were not taken into account in the dynamic model. Furthermore, in the investigation of the relationship between changes in gear parameters and the improvement of system vibration characteristics, the methods of gear modification to enhance gear vibration characteristics were not considered. [PERSON] conceived and designed the study, and wrote the article. [PERSON] conducted data extraction and performed the analyses. [PERSON] applied for the funding which financially supported the study. [PERSON] and [PERSON] provided their comments and suggestions during discussions and by writing, especially on the data extraction method and paper formatting. All authors have read and agreed to the published version of the manuscript. **Funding:** This work is supported by the National Natural Science Foundation of China under Grant Nos.