# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 2 Dynamic Model of Internal Beveloid Gear System
\begin{cases}k_{my}=k_{m}\cos\beta_{p}&k_{mz}=k_{m}\sin\beta_{p}\\ c_{my}=c_{m}\cos\beta_{p}&c_{mz}=c_{m}\sin\beta_{p}\\ e_{y}=e\cos\beta_{p}&e_{z}=e\sin\beta_{p}\\ b_{y}=b\cos\beta_{p}&b_{z}=b\sin\beta_{p}\end{cases}  \tag{11}\]  
The radial meshing component of the gear is  
\[ \begin{split} F_{y}&=k_{my}(\overline{t})\overline{f}_{y} \big{(}\overline{y}_{1}-\overline{y}_{2}-\overline{z}_{y}\big{)}+c_{my}\big{(} \dot{\overline{y}}_{1}-\dot{\overline{y}}_{2}-\dot{\overline{e}}_{y}\big{)}\\ &=\cos\beta_{p}\left[k_{m}\overline{f}_{y}\big{(}\overline{y}_{p}+ \overline{\theta}_{p}R_{p}-\overline{y}_{g}+\overline{\theta}_{g}R_{g}- \overline{e}_{y}\big{)}\right.\\ &\left.+c_{m}\Big{(}\dot{\overline{y}}_{p}+\dot{\overline{\theta}}_{p}R_{ p}-\dot{\overline{y}}_{g}+\dot{\overline{\theta}}_{g}R_{g}-\dot{\overline{e}}_{y} \Big{)}\right]\end{split}  \tag{12}\]  
The change in the modification coefficient of the beveloid gear creates a slight bevel angle along the axis, generating an axial force along the small end of the gear toward the large end. To the pinion:  
\[F_{\text{a}}=\frac{2\ \overline{T}_{pm}}{\text{d}_{p}}\frac{\tan\alpha_{\text{n }}}{\cos\beta}\sin\delta \tag{13}\]  
where d\({}_{p}\) is the dividing circle diameter of pinion (m) and \(\beta\) is helical angle of the dividing circle (rad). Considering the bevel angular component, the meshing component of the axial is:  
\[ \begin{split} F_{z}^{\prime}&=F_{\text{z}}\pm F_{ \text{a}}=k_{mz}(\overline{t})\overline{f}_{z}(\overline{z}_{1}-\overline{z}_{2 }-\overline{e}_{z})+c_{mz}\Big{(}\dot{\overline{z}}_{1}-\dot{\overline{z}}_{2 }-\dot{\overline{e}}_{z}\Big{)}\pm F_{\text{a}}\\ &=\sin\beta_{p}\left\{k_{m}\overline{f}_{z}\left[\overline{z}_{p} -\left(\overline{y}_{p}+\overline{\theta}_{p}R_{p}\right)\tan\beta_{p}- \overline{z}_{g}+\left(\overline{y}_{g}-\overline{\theta}_{g}R_{g}\right) \tan\beta_{p}-\overline{e}_{z}\right]\right.\\ &\left.+c_{m}\left[\dot{\overline{z}}_{p}-\left(\dot{\overline{y} }_{p}+\dot{\overline{\theta}}_{p}R_{p}\right)\tan\beta_{p}-\dot{\overline{z}}_ {g}+\left(\dot{\overline{y}}_{g}-\dot{\overline{\theta}}_{g}R_{g}\right) \tan\beta_{p}-\dot{\overline{e}}_{z}\right]\right\}\pm F_{\text{a}}\end{split}  \tag{14}\]  
On the basis of [PERSON]'s law, a dynamic equivalent model of the internal beveloid gear is developed:  
\[m_{p}\ddot{\overline{y}}_{p}+c_{py}\dot{\overline{y}}_{p}+k_{py}\overline{f} _{b}\big{(}\overline{y}_{p}\big{)}=-F_{y} \tag{15}\]  
\[m_{p}\ddot{\overline{z}}_{p}+c_{pz}\dot{\overline{z}}_{p}+k_{pz}\overline{f}_{ b}\big{(}\overline{z}_{p}\big{)}=F_{z}+F_{a} \tag{16}\]  
\[I_{p}\ddot{\theta}_{p}=\overline{T}_{p}-F_{y}R_{p} \tag{17}\]  
\[m_{g}\ddot{\overline{y}}_{g}+c_{gy}\dot{\overline{y}}_{g}+k_{gy}\overline{f} _{b}\big{(}\overline{y}_{g}\big{)}=-F_{y} \tag{18}\]\[m_{g}\ddot{\overline{z}}_{g}+c_{g}\dot{\overline{z}}_{g}+k_{gz}\overline{z}_{b} \left(\overline{z}_{g}\right)=-F_{z}-F_{a} \tag{19}\]  
\[I_{g}\ddot{\overline{g}}_{g}=-\overline{T}_{g}+F_{y}R_{g} \tag{20}\]  
According to Formulas (15)-(20), the gear transmission system's differential equations may be derived. Each gear has three generalized degrees of freedom and each degree of freedom has a vibration differential equation. \[ \begin{array}{l}m_{p}\ddot{\overline{y}}_{p}+c_{m}\left(\ddot{\overline{y}}_ {p}-\ddot{\overline{y}}_{g}+R_{p}\ddot{\overline{\theta}}_{p}+R_{g}\dot{ \overline{\theta}}_{g}-\dot{\overline{e}}_{y}\right)\cos\beta_{p}+k_{py} \overline{f}_{b}\left(\overline{y}_{p}\right)\\ +c_{py}\ddot{\overline{y}}_{p}+k_{m}(\overline{t})f_{y}\left(\overline{y}_{p}- \overline{y}_{g}+R_{p}\overline{\theta}_{p}+R_{g}\overline{\theta}_{g}- \overline{e}_{y}\right)\cos\beta_{p}=0\end{array}  \tag{21}\]  
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