## 2 The Original Dynamic Model of Gear System
### Internal Excitation Calculation Based on Actual Meshing Performance
#### 2.2.2 Meshing-In Impact  
Under the load condition shown in Table 1, the gear pair had full tooth surface contact, so the initial engagement occurred at the tip of the gear and the tooth surface of the pinion. However, due to the concave form error and slope error shown by the yellow line in Figure 4 defined in this paper, the meshing-in end of the pinion protruded. On the contrary, the convex form error and the slope error shown by the blue line in Figure 4 made the meshing-out end of the pinion protrude. The time and position of meshing-in interference outside the theoretical line of action were different in these two cases. Therefore, two methods for calculating the meshing-in impact were proposed for the two cases above. Figure 9 shows the actual initial meshing positions in the two cases above. Point \(D\) was the theoretical meshing-in point on the theoretical line of action, \(N_{1}N_{2}\), without considering the manufacturing error and the deformation of the tooth surface. Point \(D_{0}{}^{\prime}\) was the position of meshing-in interference on the standard surface of the pinion, and point \(A\) was the tip of the gear. However, when the meshing-in end of the pinion protruded, as shown in Figure 9a, the meshing-in interference would occur much sooner, which was the actual meshing-in point \(D^{\prime}\) on the instantaneous line of action, \(N_{1}{}^{\prime}N_{2}{}^{\prime}\). When the meshing-out end of the pinion protruded, as shown in Figure 9b, there was still a certain gap at the position \(D_{0}{}^{\prime}\) between the tooth surface and the tip of the gear. As the gear pair continued to rotate, the meshing-in interference finally occurred at the actual meshing-in point \(D^{\prime}\). Figure 8: Time-varying mesh stiffness considering the effect of the manufacturing error. Figure 9: Sketch map of the actual contact position in the section under the situations of (**a**) the meshing-in end of the pinion protruding and (**b**) the meshing-out end of the pinion protruding. For the case of the meshing-in end protruding, the accurate position of interference point \(D^{\prime}\) was determined according to the meshing principle and geometric relation. In Figure 10, point \(D\) is the theoretical meshing-in point, and point \(D^{\prime}\) is the position of the meshing-in interference. Point \(E\) is the reverse rotation of point \(D\). In \(\Delta O_{1}D^{\prime}O_{2}\) and \(\Delta PD^{\prime}O_{2}\):  
\[r_{O_{1}D^{\prime}}=\sqrt{a^{2}+r_{a2}^{2}-2 ar_{a2}\cos(\angle PO_{2}D^{ \prime})} \tag{9}\]  
where  
\[\angle PO_{2}D^{\prime}=\gamma_{2}+\theta_{LTE}+\varphi_{2} \tag{10}\]  
\[\gamma_{2}=\pi/2-\alpha_{t}-\angle PDO_{2} \tag{11}\]  
\[\angle PDO_{2}=\arcsin\frac{r_{2}\sin\left(\frac{\pi}{2}+\alpha\right)}{r_{a2}} \tag{12}\]  
\[r_{O_{1}D}=\sqrt{a^{2}+r_{a2}^{2}-2 ar_{a2}\cos\gamma_{2}} \tag{13}\]  
\[ \begin{array}{c}\arcsin\left(\frac{r_{a2}\sin\left(\angle PO_{2}D^{\prime} \right)}{r_{O_{1}D^{\prime}}}\right)-\Delta\varphi_{2}\cdot i-\arcsin\left( \frac{r_{a2}\sin\left(\gamma_{2}\right)}{r_{O_{1}D}}\right)=\\ inv\left(\arccos\left(\frac{r_{b_{1}}}{r_{O_{1}D^{\prime}}}\right)\right)- inv\left(\arccos\left(\frac{r_{b_{1}}}{r_{O_{1}E}}\right)\right)\end{array}  \tag{14}\]  
In addition, the basic properties of the involute equation can be calculated by  
\[inv(\lambda)=\tan\lambda-\lambda \tag{15}\]  
By combining Equation (9) with Equation (15), the position of the actual meshing-in point \(D^{\prime}\) could be obtained. In the preceding formula and Figure 10, \(r_{1}\) and \(r_{2}\) are the pitch radii of the pinion and gear, \(r_{b1}\) and \(r_{b2}\) are the base radii of the pinion and gear, \(r_{a2}\) is the tip radius of the gear, \(\varphi_{1}\) and \(\varphi_{2}\) are the reverse rotation angles of the pinion and gear, \(\alpha_{t}\) is the transverse pressure angle, \(\theta_{LTE}\) is the loaded transmission error of point \(D^{\prime}\), and \(i\) is the transmission ratio. For the case of the meshing-out end protruding, the meshing position was obtained by tooth contact analysis (TCA).