## 2 Nonlinear Dynamic Modeling of Spur Gear Systems
### Dynamic Center Distance  
When subjected to a large load, the elastic deformation of the driveshafts and supporting bearings can cause time-varying center distances and result in the difference between the actual and theoretical line of action (LOA). As a result, the direction of the meshing force between the gears is no longer along the theoretical LOA.  
As shown in Figure 6, the dashed line indicates the initial positions of the driving gear and the driven gear, while the solid line indicates their positions at an arbitrary time. The straight line PQ represents the actual dynamic LOA at an arbitrary time. \(L\) and \(L_{0}\) denote the actual dynamic center distance and the theoretical center distance, respectively. Meanwhile, \(\alpha\) indicates the dynamic pressure angle in the working process, and \(\beta\) indicates the angle between the dynamic LOA and the \(X\)-axis.  
Let the position vectors of the two gears' centroids be \(\mathbf{R_{1}}\) and \(\mathbf{R_{2}}\); then, the center distance \(L\) of the gear pair at an arbitrary given time is  
\[L=\left|\mathbf{R_{2}}-\mathbf{R_{1}}\right|=\sqrt{\left(\mathrm{L_{0}+\Delta x }\right)^{2}+\Delta y^{2}} \tag{9}\]  
where \(\mathbf{R_{1}}=x_{1}\mathbf{i}+y_{1}\mathbf{j}\), \(\mathbf{R_{2}}=(x_{2}+\mathrm{L_{0}})\mathbf{i}+y_{2}\mathbf{j}\), \(\Delta x=x_{2}-x_{1}\), \(\Delta y=y_{2}-y_{1}\). \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors for the \(X\) and \(Y\) axes, respectively.  
The pressure angle of the driving and driven gears can be obtained as  
\[\alpha=\arccos\left(\frac{r_{b1}+r_{b2}}{L}\right) \tag{10}\]  
where \(r_{b1}\) and \(r_{b2}\) are the base radii of the driving gear and the driven gear, respectively.  
According to the geometric relationship of the global coordinate system, the angle between the two gears' center line and the \(X\)-axis at an arbitrary time is  
\[\beta=\arctan\left(\frac{\Delta y}{L+\Delta x}\right) \tag{11}\]  
Therefore, taking into account the influence of elastic deformation and gear transmission errors on the gear center distance, the meshing deformation between two gears is obtained as  
\[\lambda=(x_{1}-x_{2})\sin(\alpha-\beta)-(y_{1}-y_{2})\cos(\alpha-\beta)+r_{b1 }\theta_{z1}-r_{2}\theta_{z2}-e(t) \tag{12}\]  
where \(e(t)\) is the gear transmission error, which causes the actual gear meshing position to deviate from the theoretical position and thus influences the meshing accuracy of the gear system.  
Generally, the simple harmonic function is used to represent the gear transmission error [1], i.e.,  
\[e(t)=e_{0}+e_{1}\sin(\omega t+\varphi) \tag{13}\]  
Figure 6: Schematic diagram of dynamic LOA.  
where \(e_{0}\) and \(e_{1}\) denote the constant value and amplitude of the transmission error, respectively; \(\omega\) is the meshing frequency of the gear pair; \(\varphi\) is the initial phase angle, generally taking the value of 0.  
Actually, the swing of the gear mainly influences the position and the deformation of the driveshaft, and its influence on the meshing force is weak. The dynamic meshing force between the gears can be described by the Hertz contact model, as shown in Figure 7. In Figure 7, \(k(t)\) is the time-varying stiffness introduced in Section 2.4, and \(c(t)\) is the meshing damping, i.e.,  
\[c(t) =2\xi\sqrt{\frac{I_{1}I_{2}}{{r_{\text{10}}}^{2}I_{1}+{r_{\text{1 0}}}^{2}I_{2}}k(t)} \tag{14}\]  
where \(\xi\) is the mesh damping ratio of the gear and generally takes a value between 0.03-0.17; \(I_{1}\) and \(I_{2}\) are the moments of inertia, and \(r_{\text{10}}\) and \(r_{\text{0}}\) are the base circle radii of the driving and driven gear, respectively.  
Besides, the backlash function \(f(\lambda)\) is defined as a piecewise function [6], i.e.,  
\[f(\lambda)=\left\{\begin{array}{ll}\lambda-b&\lambda>b\\ 0&-b\leq\lambda\leq b\\ \lambda+b&\lambda<-b\end{array}\right. \tag{15}\]  
Therefore, the dynamic meshing force is finally expressed as  
\[F=k(t)f(\lambda)+c(t)\dot{f}(\lambda) \tag{16}\]