# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 2 Dynamic Model of Internal Beveloid Gear System  
Table 1 displays the geometric specifications of the internal beveloid gear transmission system. From the large end of the beveloid gear, the dividing circle pressure angle on the end face and dividing circle helix angle can be expressed as:  
\[\alpha_{\text{tL}(\text{R})}=\text{arc}\tan\biggl{(}\frac{\tan{a_{\text{n}}} \cos{\delta}}{\cos{\beta^{\prime}}}\pm\sin{\delta}\tan{\beta^{\prime}}\biggr{)} \tag{1}\]  
\[\beta_{\text{R}(\text{L})}=\text{arc}\tan\biggl{(}\cos{\delta}\tan{\beta^{ \prime}}\mp\frac{\tan{a_{\text{n}}}\sin{\delta}}{\cos{\beta^{\prime}}}\biggr{)} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Gear Parameters** & **External Gear** & **Internal Gear** \\ \hline Number of teeth & \(Z_{1}=45\) & \(Z_{2}=60\) \\ Module & \(m=2.5\) & \(m=2.5\) \\ Pressure angle of rack cutter/\({}^{\circ}\) & \(\alpha=20\) & \(\alpha=20\) \\ Machining pitch angle/\({}^{\circ}\) & \(\delta=5\) & \(\delta=5\) \\ Coefficient of addendum \(h_{a}^{*}\) & 0.55 & 0.55 \\ Coefficient of bottom clearance \(h_{i}^{*}\) & 0.25 & 0.25 \\ Face width/mm & \(b_{1}=22.5\) & \(b_{2}=22.5\) \\ Mass/kg & 1.345 & 1.312 \\ Moment of inertia/kg\(\cdot\)m\({}^{2}\) & 0.002550 & 0.009016 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of internal beveloid gear transmission system. where \(\delta\) is the machining pitch angle of beveloid gear (rad), \(\beta^{\prime}\) is the cutting tool helix angle (rad), and \(\alpha_{\rm n}\) is the dividing circle normal pressure angle (rad). For parallel\(-\)axis beveloid gears, both gears are manufactured using straight\(-\)tooth racks. The dividing circle pressure angle and the dividing circle helix angle on the end face can be expressed as:  
\[\alpha_{\rm tL1}=\alpha_{\rm tR2}=\arctan\tan(\tan\alpha_{\rm n}\cos\delta) \tag{3}\]  
\[\beta_{\rm L1}=\beta_{\rm R2}=\arctan\tan(\tan\alpha_{\rm n}\sin\delta) \tag{4}\]  
In order to assist the study, the dynamic model is constructed with a number of fundamental assumptions. It is assumed that both gears are ideal rigid bodies and that the meshing force of the gears is equal to the spring\(-\)damper model in the direction of the path of contact; it is assumed that the installation position of the two gears is ideal and unchanged, and there are no influencing factors such as tooth eccentricity, regardless of the impact between the tooth meshing and the influence of the gear meshing and bearing\(-\)related friction factors. Therefore, the internal beveloid gear's bend\(-\)torsion\(-\)bend 3\(-\)DOF vibration model is established. The model is represented by Figure 1.  
The internal beveloid gear transmission's variable coordinate \(X\) is  
\[X=\left[y_{p},z_{p},\theta_{p},y_{g},z_{g},\theta_{g}\right] \tag{5}\]  
where \(y_{i},z_{i},\theta_{i}(i=p,g)\) are the relative vibration displacement and angular vibration angular displacement of the center of mass of the pinion and gear in the \(y\) and \(z\) axes, respectively. At the point where two gears mesh, the connection between their radial and axial vibration displacements may be represented as:  
\[\overline{z}=\overline{y}\tan\beta_{p} \tag{6}\]  
Figure 1: Dynamic model of internal beveloid gear. The expression for the radial and axial vibrational displacement offset at the meshing point of the external gear is:  
\[\overline{y}_{1}=\overline{y}_{p}+\overline{\theta}_{p}R_{p} \tag{7}\]  
\[\overline{z}_{1}=\overline{z}_{p}-\left(\overline{y}_{p}+\overline{\theta}_{p} R_{p}\right)\tan\beta_{p} \tag{8}\]  
The expression for the radial and axial vibrational displacement offset at the meshing point of the ring gear is:  
\[\overline{y}_{2}=\overline{y}_{g}-\overline{\theta}_{g}R_{g} \tag{9}\]  
\[\alpha_{\rm tL1}=\alpha_{\rm tR2}=\arctan\tan(\tan\alpha_{\rm n}\cos\delta) \tag{3}\]0  
The gear transmission system's properties can be described as meshing stiffness \(k_{m}\), meshing damping \(c_{m}\), transmission error \(e\), and normal tooth backlash 2 \(b\) on the \(y-\) and \(z-\)axes, respectively:  
\[