## 3 Establishment of the Dynamic Model for Spiral Bevel Gears  
### Dynamic Transmission Error  
Dynamic transmission error mainly occurs when the gear does not operate at the speed dictated by the gear ratio. Dynamic transmission error is a primary dynamic parameter of gears and can be expressed in terms of angles. It can be represented by the following formula:  
\[\delta(\varphi_{1})=\left(\varphi_{2}-\varphi_{2}{}^{0}\right)-i_{21}\left( \varphi_{1}-\varphi_{1}{}^{0}\right) \tag{5}\]  
where \(\varphi_{1}{}^{0}\) and \(\varphi_{2}{}^{0}\) represent the initial angular positions of the small and large gears, respectively, while \(\varphi_{1}\) and \(\varphi_{2}\) represent their actual angular positions. \(Z_{1}\) and \(Z_{2}\) denote the number of teeth on the small and large gears, respectively [30].  
Gear transmissions are inherently conjugate, ensuring a constant transmission ratio. Therefore, under normal circumstances, the transmission error is zero. However, in actual gear transmissions, the gears are subjected to forces, and the rotational motion between two gears deviates from the ideal transmission ratio, leading to the generation of dynamic transmission error.  
The linearity of transmission error can be derived by taking the following:  
\[\left(\varphi_{2}-\varphi_{2}{}^{0}\right)=\mathrm{F}\left(\varphi_{1}-\varphi _{1}{}^{0}\right) \tag{6}\]  
Performing a Taylor series expansion, we can obtain the following:  
\[\mathrm{F}\left(\varphi_{1}-\varphi_{1}{}^{0}\right)=\mathrm{m}_{21}\left( \varphi_{1}-\varphi_{1}{}^{0}\right)+\frac{1}{2}\dot{\mathrm{m}}_{21}\left( \varphi_{1}-\varphi_{1}{}^{0}\right)^{2}+\frac{1}{6}\dot{\mathrm{m}}_{21} \left(\varphi_{1}-\varphi_{1}{}^{0}\right)^{3}\dots \tag{7}\]  
Omitting higher-order terms in the formula, the expression for the transmission error is as follows:  
\[\delta(\varphi_{1})=\frac{1}{2}\dot{m}_{21}\left(\varphi_{1}-\varphi_{1}{}^{0 }\right)^{2} \tag{8}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Postprocessing Operation** & **Roughness/\(\upmu\)m** \\ \hline Unground machining & 0.8 \\ Grinding machining & 0.4 \\ Polishing & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Manufacturing methods for spiral bevel gears.  
Figure 2: Surfaces under different fractal dimensions.  
Equation (8) indicates a quadratic parabolic equation for calculating the transmission error. A negative value assigned to the first derivative of \(m_{21}\) results in a downward-opening parabolic curve for the transmission error. Dynamic transmission error curves for spiral bevel gears are derived by shifting each graph following the engagement cycle. The impact of the transmission error curve is twofold: Firstly, the acceleration curve of the quadratic parabolic transmission error affects the dynamic characteristics of the gear pair at the engagement transition point. Secondly, the angle between the two tangents at the meshing transition point influences the gear pair's dynamic performance, with an increase in the angle reducing the impact on the gear teeth. Figure 3 illustrates the process of toothcontact analysis.  
The TCA (tooth contact analysis) method is conducted by applying the principles of contact mechanics to calculate parameters such as contact stress, contact area, and contact pressure distribution on gear tooth surfaces. The contact stress, contact area, and pressure distribution mentioned in the text are reflected in subsequent sections through finite element analysis. This typically involves numerical simulation methods and finite element analysis, with subsequent analysis primarily developed using finite element methods.