## 3 Establishment of the Dynamic Model for Spiral Bevel Gears
### Loaded Tooth Contact Analysis (LTCA)
It is required that they coincide in a certain position in terms of normal direction and position vector, as shown in Figure 5. The contact Equations (22) and (23) of the gear pair can be expressed as the following:  
\[r_{m}^{(1)}(u_{1},\theta_{1},\varphi_{1}) =r_{m}^{(2)}(u_{2},\theta_{2},\varphi_{2}) \tag{22}\] \[\mathbf{n}_{m}^{(1)}(u_{1},\theta_{1},\varphi_{1}) =\mathbf{n}_{m}^{(2)}(u_{2},\theta_{2},\varphi_{2}) \tag{23}\]Combining the tooth surface Equation (12) for spiral bevel gears, the solution is obtained using Equations (21) and (22). Taking the rotation angle \(\varphi_{1}\) of the pinion 1 as the independent variable, the meshing points on the tooth surface can be determined. Substituting the rotation angle of gear 2 into Equation (5) yields the transmission error for spiral bevel gears [34]. The spiral bevel gear transmission system can be equivalent to an eight-degrees-of-freedom dynamic model, where the system's degrees of freedom are represented as a column vector:  
\[Q=\left\{X_{1},Y_{1},Z_{1},\theta_{1x},X_{2},Y_{2},Z_{2},\theta_{2y}\right\} \tag{24}\]  
where \(X_{i}\), \(Y_{i}\), and \(Z_{i}\) (\(i=1,2\)) represent the pinion and gear's movements along the x-axis, y-axis, and transverse z-axis, respectively. \(\theta_{1x}\) and \(\theta_{2y}\) represent the torsional vibrations of the pinion around the x-axis and the passive gear around the y-axis. The relative displacement along the normal direction of the meshing points on the tooth surfaces due to vibration and errors is given by the following equation:  
\[\mathbf{F_{k}p_{k}+w_{k}=Z+s_{k}\;(k=1,2)} \tag{17}\]0  
The parameters \(a_{1}\), \(a_{2}\), and \(a_{3}\) are related to the pressure angle and spiral angle. \(e_{n}(t)\) represents the normal static error of the gear pair; \(r_{1}\) and \(r_{2}\) are the meshing point radius. The tooth profile error is expressed using a sine harmonic function:  
\[\mathbf{F_{k}p_{k}+w_{k}=Z+s_{k}\;(k=1,2)} \tag{17}\]1  
where \(e_{0}\) and \(e_{r}\) are the constant and amplitude of the meshing error, with \(e_{0}\) typically being 0; \(\Omega_{h}\) is the meshing frequency; and \(\varphi\) is the phase angle, typically set to \(\varphi=0\). The normal dynamic load and force components along the coordinate directions during the meshing of the gear pair are as follows:  
\[\left\{ \begin{array}{l}F_{n}=k_{n}(t)\lambda_{n}+c_{h}\dot{\lambda}_{n}\\ F_{x}=-F_{n}a_{4}\\ F_{y}=F_{n}a_{5}\\ F_{z}=F_{n}a_{6}\end{array} \right. \tag{27}\]  
The time-varying meshing stiffness is given by the following equation:  
\[k_{h}(t)=k_{m}+\sum_{l=1}^{n}A_{kl}\cos(l\Omega_{h}t+\Phi_{kl}) \tag{28}\]  
Figure 5: Dynamic model of spiral bevel gear. where \(c_{h}\) represents the meshing damping; \(k_{m}\) is the average meshing stiffness; \(A_{k1}\) is the amplitude of the \(l\)-th harmonic of the stiffness; \(\Phi_{kl}\) is the initial phase of the \(l\)-th harmonic of the stiffness. The meshing stiffness of the spiral bevel gear can be obtained and fitted according to the principles of LTCA. In this study, it is obtained and fitted through finite element analysis (FEA). According to empirical formulas, the meshing damping of the gear is given by the following equation:  
\[c_{h}=2\xi_{g}\sqrt{\frac{k_{h}I_{1}I_{2}R_{1}^{2}R_{2}^{2}}{R_{1}^{2}I_{1}+R_ {2}^{2}I_{2}}} \tag{29}\]  
where, \(J_{1}\) and \(J_{2}\) are the moments of inertia for the pinion and wheel; \(R_{1}\) and \(R_{2}\) are the base circle radii for the pinion and wheel; \(\xi\) is the meshing damping ratio; and \(\xi\) is taken as 0.1 in this study. \(c_{h}\) represents damping in tooth surface contact, and it is applied to the damping results in three directions in the general equation of dynamic Equation (36). Considering the influence of friction on the dynamic process, the relationship between the friction coefficient and friction force is first examined.