## 3 Establishment of the Dynamic Model for Spiral Bevel Gears
### Loaded Tooth Contact Analysis (LTCA)
According to empirical formulas, the friction coefficient can be expressed as the following:  
\[\mu=\mu_{0}+\mu_{r}\cdot R \tag{30}\]  
where \(\mu_{0}\) is the static friction coefficient between surfaces. In the case of metal-metal friction, it typically ranges from 0.6 to 1.0. For this study, considering good lubrication conditions, we use \(\mu_{0}=0.6\). \(\mu_{r}\) is an additional friction coefficient related to roughness, which can vary with changes in roughness. In this study, we use \(\mu_{r}=0.01\). \(R\) is the effective roughness between the contacted surfaces [35]. The relationship between the roughness \(R\) and fractal dimension \(D\) is commonly approximated as the following:  
\[R=CL^{(1-D)} \tag{31}\]  
where \(R\) represents roughness, typically the root mean square roughness of the surface. \(D\) stands for the fractal dimension, describing the geometric complexity of the surface. \(L\) represents the characteristic scale on the surface, also known as the measurement length. \(C\) is a constant related to the specific system and measurement units. In this study, \(C=1\) is adopted, and Table 2 gives the result. The friction force is expressed as the following:  
\[F_{f}=\mu F_{n} \tag{32}\]  
\[D=1.515/R_{a}^{0.088} \tag{33}\]  
\[G=10^{-5.26/R_{a}^{0.042}} \tag{34}\]  
The integrated relationship between the friction coefficient and the fractal dimension is obtained as follows:  
\[\mu=\mu_{0}+\mu_{r}(CL^{(1-D)}) \tag{35}\]  
Combining the above analysis, this paper establishes an eight-degrees-of-freedom dynamic model for spiral bevel gears, considering the fractal dimension. The model \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Postprocessing** & **Surface** & & **Characteristic Scale** \\
**Operation** & **Roughness/\(\mu\)m** & **Fractal Dimension** & **Coefficient** \\ \hline Unground machining & 0.8 & 1.55 & \(5\times 10^{-6}\) \\ Grinding machining & 0.4 & 1.60 & \(3\times 10^{-6}\) \\ Polishing & 0.2 & 1.65 & \(2\times 10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The relationship between roughness and fractal dimension. decomposes friction forces in various directions. Incorporating [PERSON]'s second law, the dynamic equation for spiral level gears considering the fractal dimension are derived as follows:  
\[\left\{ \begin{array}{l}m_{1}\overset{\cdot\cdot}{X}_{1}+c_{x1}\dot{X}_{1}+k_{x1 }X_{1}=F_{x}-F_{f1x}\\ m_{1}\dot{Y}_{1}+c_{y1}\dot{Y}_{1}+k_{y1}Y_{1}=F_{y}-F_{f1y}\\ m_{1}\ddot{Z}_{1}+c_{z1}\dot{Z}_{1}+k_{z1}Z_{1}=F_{z}-F_{f1z}\\ J_{1}\ddot{\theta}_{1x}=T_{1}-F_{z}r_{1}+F_{f}rf_{1x}\\ m_{2}\ddot{X}_{2}+c_{x2}\dot{X}_{2}+k_{x2}X_{2}=F_{x}-F_{f2x}\\ m_{2}\ddot{Y}_{2}+c_{y2}\dot{Y}_{2}+k_{y2}Y_{2}=F_{y}-F_{f2y}\\ m_{2}\ddot{Z}_{2}+c_{z2}Z_{2}+k_{z2}Z_{2}=F_{z}-F_{f2z}\\ \ddot{\theta}_{2x}=T_{2}-F_{z}r_{2}+F_{f}rf_{2x}\end{array} \right. \tag{36}\]  
where, \(m_{1}\) and \(m_{2}\) are the masses of the pinion and wheel; \(c_{ij}\) and \(k_{ij}\) (\(i=x,y,z;j=1,2\)) are the damping and stiffness coefficients of the pinion and wheel along the \(x,y,z\) directions; \(T_{1}\) is the input torque; and \(T_{2}\) is the load torque. \(F_{fij}\) represents the component of frictional force along the j direction on gear \(i\); \(r_{i}\) is the lever arm of the normal force component on tooth \(i\); and \(r_{fix}\) is the lever arm of the frictional force on tooth \(i\).