# Study on the Influence of Gear Spoke Hole Numbers on Meshing Noise
## 2. Gear Meshing Model  
The gear meshing model can be regarded as a simple dynamic system by neglecting the deformation of the shaft system in the traditional rigid contact theory. That is, only the rotation of the two gears around each axis is considered. The traditional rigid contact theoretical model is shown in Figure 1[15]. The dynamic equations of the above model give [10]. \[I_{c}\phi_{c}{}^{r}+b_{c}C_{n}x^{\prime}+b_{c}K_{n}x=M_{C} \tag{1}\]  
\[I_{p}\phi_{p}{}^{r}-b_{p}C_{n}x^{\prime}-b_{p}K_{n}x=-M_{p} \tag{2}\]  
\[x=\int\limits_{l_{0}}^{t}b_{c}\theta_{c}^{\prime}dt-\int\limits_{l_{0}}^{t}b_{ p}\theta_{p}^{\prime}dt \tag{3}\]  
where \(x\) is the normal offset value; \(\theta_{p}\) is the base circle rotation angle of the driving gear; \(b_{p}\) is the base circle diameter of the driving gear; \(\theta_{c}\) is the base circle rotation angle of the  
Figure 1. Traditional gear meshing model. driven gear; \(b_{\rm c}\) is the base circle diameter of the driven gear; \(K_{n}\) is the meshing stiffness of the gears with the rigid spokes; \(C_{n}\) is the meshing damping; \(M_{\rm C}\) is the driven gear torque; and \(M_{p}\) is the driving gear torque. As is known, gear meshing stiffness is affected by various factors. In addition, the coupling relationship of the factors and their influence on meshing stiffness have not been studied by the contact model in previous pieces of research. This paper aims to determine the influence of gear spoke numbers on the meshing stiffness and noise. The first task of the research is to add the gear spoke stiffness to the meshing stiffness of the gears with the rigid spokes \(K_{n}\). Thus, the gear with the spoke can be decomposed into the flexible gear with the rigid no-hole spoke and the flexible spoke in the analysis of the meshing stiffness. The inner boundary force of the flexible gear with the rigid no-hole spoke can be regard as the boundary condition of the flexible spoke by considering the coupling relationship between the two structures. The model is shown in Figure 2.  
Therefore, the dynamic equations of the gears with spokes are [16]  
\[I\theta^{\prime\prime}+K_{T}\theta^{\prime}=M_{EX}-M_{NEX} \tag{4}\]  
\[I_{L}\theta^{\prime\prime}+K_{TL}\theta^{\prime}=M_{NEX}-M_{\rm C} \tag{5}\]  
where \(I\) and \(K_{T}\) are the moment of inertia and the torsional stiffness of the flexible gear with rigid no-hole spoke, respectively; \(I_{L}\) and \(K_{TL}\) are the moment of inertia of and the torsional stiffness of the flexible spoke, respectively; \(M_{NEX}\) is the torque of the rigid spokes given by the flexible gear with the rigid no-hole spoke; and \(M_{EX}\) is the torque of the driven gear given by the driving gear. Since the gear with the spoke had been decomposed into the flexible gear with the rigid no-hole spoke and the flexible spoke, the force and displacement of the gear meshing are basically annular. Thus, the gear model can be divided by the FE method along the annular shape. The torsional angle of the gear unit is distributed to each ring, as shown in Figure 3.  
Figure 2: Improved gear meshing model. The dynamic equation of the flexible gear with the rigid no-hole spoke can be obtained by taking the stiffness and moment of inertia parameters of each ring into the Equation (4), which gives:  
\[\left\{ \begin{array}{c}M_{EX}-M_{1}\\ M_{1}-M_{2}\\ \vdots\\ M_{n-1}-M_{n}\\ M_{n}-M_{NEX}\end{array} \right\}=\left\{ \begin{array}{c}I_{1}\\ I_{2}\\ \vdots\\ I_{n}\\ I_{n+1}\end{array} \right\}\left\{ \begin{array}{cccccccc}\theta^{\prime\prime} _{1}&\theta^{\prime\prime}_{2}&\cdots&\theta^{\prime\prime}_{n}&\theta^{\prime \prime}_{n+1}\\ \\ I_{n}\\ I_{n+1}\end{array} \right\}\] \[+\left\{ \begin{array}{c}K_{T1}\\ K_{T2}\\ \vdots\\ K_{Tn}\\ K_{T(n+1)}\end{array} \right\}\left\{ \begin{array}{cccccccc}\theta_{1}- \theta_{2}&\theta_{2}-\theta_{3}&\cdots&\theta_{n}-\theta_{n-1}&\theta_{n+1}- \theta_{n+1}\\ \\ \\ K_{T(n+1)}\end{array} \right\} \tag{6}\] \[-\left\{ \begin{array}{c}0\\ K_{T1}\\ \vdots\\ K_{T(n-1)}\end{array} \right\}\left\{