## 2 Nonlinear Dynamic Modeling of Spur Gear Systems
### Nonuniformly Distributed Meshing Force  
The meshing force is usually treated as an ideal uniform load to facilitate an analysis of the gear system. In practical applications, however, the external load of the gear system is more or less unbalanced due to machining errors, assembly errors, misalignment errors, etc., especially for the wind turbine gear system used in complicated working conditions, which will induce the swing effect to the gears and cause complicated nonlinear dynamic responses. In this subsection, the swing effect caused by the deflection torque around the \(X\) and \(Y\) axes is taken into consideration, which can better reflect the actual dynamic behavior of a gear system. As shown in Figure 1, plane P1P2-Q1Q2 denotes the driving and driven gear's plane of action. According to gear meshing theory, the meshing force between the gears is in the plane of action and acts on the straight line (P1P2) along the face width. However, due to misalignment errors and the elastic deformation of low-rigidity components, the actual meshing force is no longer uniformly distributed along the face width, as shown in Figure 2.  
Figure 1: Schematic diagram of the plane of action. Figure 2: Nonuniformly distributed load. According to the derivation of misalignment errors, the load of the unit face width can be approximately expressed, as it is in [9]. \[F(z)=F_{\max}\left[1-\left(\frac{z}{B^{\prime}}\right)^{3}\right] \tag{1}\]  
where \(F_{\max}\) denotes the maximum load, and \(B^{\prime}\) is the length of the actual contact line. Then, the meshing force \(F\) can be defined as follows [9]:  
\[F=\int_{0}^{B^{\prime}}F(z)dz=\frac{3}{4}B^{\prime}F_{\max} \tag{2}\]  
The maximum load can be expressed as [9]:  
\[F_{\max}=C_{\gamma}e_{m}\frac{B^{\prime}}{B} \tag{3}\]  
where \(C_{\gamma}\) denotes the spring constant, and \(e_{m}\) is the error on the plane of action. Then the length of the contact line can be deduced according to Equations (1)-(3), i.e.,  
\[B^{\prime}=\sqrt{\frac{4 FB}{3C_{\gamma}e_{m}}} \tag{4}\]  
As shown in Figure 3, the meshing force can be equivalent to the resultant force \(F\) and the deflection distance \(d\). Thus, the deflection torque \(T\) perpendicular to the plane P\({}_{1}\)P\({}_{2}\)-Q\({}_{1}\)Q\({}_{2}\) is obtained as:  
\[T=Fd \tag{5}\]  
where the meshing force \(F\) is derived from the Hertz contact theory: \(d=\frac{B}{2}-\frac{\int_{0}^{B^{\prime}}zF(z)dz}{\int_{0}^{B^{\prime}}F(z)dz}\). Based on the above analysis, a single-stage involute spur gear system, considering the elastic deformation of the supporting bearings and driveshafts, was researched, as shown in Figure 4. Due to the existence of the deflection moment, \(T\), the bearing deformations are different. Therefore, it is necessary to consider two translational DOF along the \(X\)-axis and \(Y\)-axis for each supporting bearing. Meanwhile, the translational DOF along the \(X\)-axis and \(Y\)-axis and three rotational DOF around the centroid coordinate system are considered for each gear. Therefore, an 18-DOF bending-torsion-swing-coupled dynamic model is proposed in this study to depict the actual dynamic behavior of the gear system subjected to a nonuniformly distributed meshing force. The inertial coordinate systems of the gears and the supporting bearings are O\({}_{i}\)X\({}_{i}\)Y\({}_{i}\)Z\({}_{i}\)\((i\)=1,2) and B\({}_{i}\)X\({}_{bi}\)Y\({}_{bi}\)Z\({}_{bi}\)\((i\)\(=1,2,3,4)\), respectively. The global coordinate system OXYZ coincides with O\({}_{1}\)X\({}_{1}\)Y\({}_{1}\)Z\({}_{1}\). The origin of each coordinate system is at the centroid of the gears. Besides, the axis X\({}_{1}\) coincides with the axis X\({}_{2}\); the axis Z\({}_{bi}\)\((i\)\(=1,2)\) coincides with the axis Z\({}_{1}\); the axis Z\({}_{bi}\)\((i\)\(=3,4)\) coincides with the axis Z\({}_{2}\). Meanwhile, \(x_{i},y_{i}\)\((i\)\(=1,2)\) denotes the displacements of the driving and driven gears with respect to the origin of their  
Figure 3: Schematic diagram of deflection torque. own coordinate system, respectively.