# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 1 Introduction  
As technology advances, gear transmission mostly moves in the directions of greater bearing capacity, increased stability, reduced vibration and noise, and exquisite structure [1]. Theoretically, beveloid gears offer a type of precision transmission with adjustable back-lash [2]. Beveloid gears are advantageous for transmitting tiny tilt angles because they may adapt their design parameters to suit various gear transmission circumstances. The back-lash may be efficiently adjusted by shifting the axial position of the gear teeth, according to some studies, which enhances the transmission's precision [3]. In some circumstances, the gear system's vibration and noise can have a significant negative impact [4]. In order to provide a theoretical foundation for the ongoing development of gear system vibration reduction and noise reduction technology, it is crucial to study the vibration characteristics of beveloid gears and analyze their nonlinear vibration characteristics. [PERSON] studied the gear teeth's meshing conditions in 1956 and provided a technique for machining external beveloid gears using a basic machine tool [5]. [PERSON] et al. thoroughly examined and analyzed the gear hobbing technique of beveloid gears in order to advance beveloid gears from theory to practice [6, 7]. [PERSON] and others developed a method to process external beveloid gears in 2014 based on the gear processing principle of a gear hobbing machine [8]. [PERSON] and others proposed a method to process the internal beveloid gears using a gear shaper in 2000, which offered a more trustworthy approach for the actual processing research of the internal beveloid gear [9]. [PERSON] suggests processing internal beveloid gear using a gear shaper under multiple degrees of freedom [10]. A beveloid non\(-\)circular gear transmission scheme was developed by [PERSON] to improve stability and accuracy, reducing meshing impact and allowing backlash adjustment [11]. An adaptive design model is proposed by [PERSON] et al. for the continuous generation of beveloid gears in common gear grinding machines [12]. A gear dynamics model that takes into account mass and equivalent spring force was developed based on [PERSON]'s initial hypothesis that, under certain assumptions, a pair ofmeshing gears can be substituted by an elastic nonlinear model. This hypothesis was first put forth in 1950 [13; 14]. With the advancement of computers, numerical techniques can be utilized to examine the nonlinearity of gear teeth. The Runge-Kutta method is the most popular of them. [PERSON] analyzes the nonlinear properties of the transmission through Poincare sections, bifurcation diagrams, power spectra, etc. using the variable step size Gill technique [15]. A higher damping coefficient and the proper backlash can suppress the chaotic region, according to analysis by [PERSON] et al. using the Runge-Kutta numerical integration method [16]. The six degree of freedom gear's vibration analysis model was numerically studied by [PERSON] et al. The findings demonstrate that when the gear teeth's peeling area increases, the dynamic response's oscillation at its beginning and end becomes more important [17]. [PERSON] et al. created a gear tooth torsional vibration analysis model taking into account variables like multiple harmonic excitations [18]. [PERSON] examined the planetary gear's nonlinear dynamics using numerical and analytical techniques in 2010, and he evaluated the precision of the calculations using finite elements and numerical integrated simulation [19]. [PERSON] et al. examined the contact behavior between the meshing tooth surfaces of curved beveloid gear pairs and a fractal contact mechanics model for the rough surface of a beveloid gear with elliptical asperities was established [20]. A slice\(-\)based mesh stiffness modeling method is proposed by [PERSON] et al. for straight beveloid gear pair for the calculation of mesh stiffness of irregular tapered tooth profile [21; 22].