## 2 Nonlinear Dynamic Modeling of Spur Gear Systems
### Derivation of the Dynamic Model  
In order to better describe the deformation of gears, driveshafts, and bearings under a nonuniformly distributed meshing force, five DOF are introduced for each gear and two translational DOF are introduced for each bearing. Therefore, this section proposes an 18-DOF bending-torsion-swing-coupled nonlinear dynamic model based on [PERSON]'s equation. Let the generalized coordinates of the driving and driven gears be \(z_{1}=\left[x_{1},y_{1},x_{b1},y_{1},x_{b2},y_{b2},\theta_{x1},\theta_{y1}, \theta_{z1}\right]^{T}\) and \(z_{2}=\left[x_{2},y_{2},x_{b3},y_{b3},x_{b4},y_{b4},\theta_{x2},\theta_{y2}, \theta_{z2}\right]^{T}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Driving Gear** & **Driven Gear** \\ \hline Teeth number \(z\) & 30 & 36 \\ Mass (g) & 600 & 850 \\ Inertia torque (kg\(\cdot\)mm\({}^{2}\)) & 600 & 1200 \\ Modulus (mm) & 3 & \\ Pressure angle (\({}^{\circ}\)) & 20 & \\ Face width (mm) & 12 & \\ Average meshing stiffness \(k_{0}\) (N/mm) & \(2.39\times 10^{5}\) & \\ Tooth side clearance 2\(b\) (mm) & 0.1 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the gear system. Figure 8: Time-varying meshing stiffness curve. The kinetic energy of the gear system mainly consists of the translational and rotational kinetic energy of the gears and the rotational kinetic energy of the supporting bearings. Therefore, the kinetic energy function \(T_{1}\) of the driving gear and the kinetic energy function \(T_{2}\) of the driven gear can be expressed as  
\[T_{1} =\frac{1}{2}m_{1}(\dot{x_{1}}^{2}+\dot{y_{1}}^{2})+\frac{1}{2}I_{x1 }\dot{\theta}_{x1}^{2}+\frac{1}{2}I_{y1}\dot{\theta}_{y1}^{2}+\frac{1}{2}I_{z1 }\dot{\theta}_{z1}^{2}+\frac{1}{2}m_{b1}(\dot{x_{b1}}^{2}+\dot{y_{b1}}^{2})+ \frac{1}{2}m_{b2}(\dot{x_{b2}}^{2}+\dot{y_{b2}}^{2}) \tag{24a}\] \[T_{2} =\frac{1}{2}m_{2}(\dot{x_{2}}^{2}+\dot{y_{2}}^{2})+\frac{1}{2}I_{x2 }\dot{\theta}_{x2}^{2}+\frac{1}{2}I_{y2}\dot{\theta}_{y2}^{2}+\frac{1}{2}I_{z2 }\dot{\theta}_{z2}^{2}+\frac{1}{2}m_{b3}(\dot{x_{b3}}^{2}+\dot{y_{b3}}^{2})+ \frac{1}{2}m_{b4}(\dot{x_{b4}}^{2}+\dot{y_{b4}}^{2}) \tag{24b}\]  
where \(m_{i}(i=1,2)\) is the mass of the driving gear and the driven gear; \(m_{bj}(j=1,2,3,4)\) is the equivalent mass of the rotating body of the supporting bearings; \(I_{xi}\), \(I_{yi}\), \(I_{zi}(i=1,2)\) denotes the rotational inertia of the gears. In this study, the spring-damping model is used to calculate the potential energy and dissipation energy of the elastic components. The potential energy function and the dissipation energy function of the driving and driven gear are \(U_{1}\), \(U_{2}\), \(V_{1}\) and \(V_{2}\), respectively, i.e.,  
\[U_{1} =\frac{1}{2}k_{b1}\Delta_{1}x^{2}+\frac{1}{2}k_{b1}\Delta_{1}y^{2 }+\frac{1}{2}k_{bxy}({x_{b1}}^{2}+{y_{b1}}^{2}+{x_{b2}}^{2}+{y_{b2}}^{2}) \tag{25a}\] \[U_{2} =\frac{1}{2}k_{b1}\Delta_{2}x^{2}+\frac{1}{2}k_{b1}\Delta_{2}y^{2 }+\frac{1}{2}k_{bxy}({x_{b3}}^{2}+{x_{b4}}^{2}+{y_{b3}}^{2}+{x_{b4}}^{2})\] (25b) \[V_{1} =\frac{1}{2}c_{b1}\Delta_{1}\dot{x}^{2}+\frac{1}{2}c_{b1}\Delta_{1 }\dot{y}^{2}+\frac{1}{2}c_{bxy}(\dot{x_{b1}}^{2}+\dot{x_{b2}}^{2}+\dot{y_{b1}} ^{2}+\dot{y_{b2}}^{2})\] (25c) \[V_{2} =\frac{1}{2}c_{b1}\Delta_{2}\dot{x}^{2}+\frac{1}{2}c_{b1}\Delta_{2 }\dot{y}^{2}+\frac{1}{2}c_{bxy}(\dot{x_{b3}}^{2}+\dot{x_{b4}}^{2}+\dot{y_{b3}} ^{2}+\dot{y_{b4}}^{2}) \tag{25d}\]  
where \(k_{b1}\) and \(c_{b1}\) are the equivalent bending stiffness and damping of the two driveshafts. \(k_{bxy}\) and \(c_{bxy}\) are the stiffness and damping of the four supporting bearings. Let the input and output torques of the gear system be \(T_{s}\) and \(T_{c}\), respectively, and the base radii of the driving and driven gear be \(r_{b1}\) and \(r_{b2}\). According to Section 2.3, the generalized forces acting on the driving and driven gears can be expressed by the column vectors \(Q_{1}\) and \(Q_{2}\) as  
\[Q_{1}=\left[ \begin{array}{c}-F\sin\varphi\\ -F\cos\varphi\\ 0\\ 0\\ 0\\ -Fd\cos\varphi\\ -Fd\sin\varphi\\ T_{s}-Fr_{b1}\end{array} \right],\ Q_{2}=\left[