## 2 Proposed Meshing Stiffness Calculation Model with Tooth Surface Crack
As displayed in Figure 3a, the parameters \(A_{x}\) and \(I_{x}\) remain unchanged when the contact line is located between \(G\) and \(S\). Thus, the mesh stiffness of gear tooth pair in this area is the same as that of healthy gear teeth. As illustrated in Figure 3b, the crack part can still bear shear, axial compressive, and bending force when the contact line is located between \(S\) and \(P\), and the mesh stiffness of gear teeth in this area is also the same as that of normal gear. When the contact line passes through the end point \(P\) of the crack, the crack turns to an open state due to the presence of meshing force, and the parameters \(A_{x}\) and \(I_{x}\) of the cracked tooth zone will decrease, which is presented in Figure 3c. Through the above analysis, the cross section area \(A_{x}\)' and the moment of inertia \(I_{x}\)' of the gear with tooth surface crack are calculated as follows,  
\[A_{x}^{\prime}=\left\{ \begin{array}{ll}2h_{x}W-L_{2}D^{\prime}&x_{1}\leq x \leq x_{1}+x_{2}\,\text{and}\,\,d>x_{1}+L_{1}\\ 2h_{x}W&\text{else}\end{array} \right. \tag{10}\]  
Figure 3: Meshing positions of gear tooth with surface crack fault: (**a**) left of the fault zone, (**b**) above the fault zone, (**c**) right of the fault zone. \[I_{x}^{t}=\left\{ \begin{array}{ll}\frac{2}{3}h_{x}^{3}W-\left[\frac{1}{12}D^{ \prime 3}L_{2}+D^{\prime}L_{2}\Big{(}h_{x}-\frac{D^{\prime}}{2}\Big{)}^{2} \right]&x_{1}\leq x\leq x_{1}+x_{2}\,\text{and}\ d>x_{1}+L_{1}\\ \frac{2}{3}h_{x}W&\text{else}\end{array} \right. \tag{11}\]  
where \(D^{\prime}\) represents the reduced height of the cross section due to the tooth surface crack, it can be denoted as,  
\[D^{\prime}=h_{x}-(h_{x_{1}}-D) \tag{12}\]  
where, \(h_{x1}\) is the cross-sectional height with \(x_{1}\) distance from the dedendum. According to the research conclusions of [PERSON] and [PERSON], the magnitude of the Hertz contact stiffness \(k_{h}\) of healthy meshing tooth pairs is a constant throughout the meshing line. For the meshing gear teeth with surface crack, the crack will not affect the effective contact width of the gear teeth during the whole meshing process. Therefore, the Hertz contact stiffness of surface cracked teeth is consistent with the healthy case. Its expression is [34],  
\[k_{h}=\frac{\pi EW}{4(1-v^{2})} \tag{13}\]  
where \(W\) and \(v\) represent tooth width and Poisson's ratio, respectively. Under the action of meshing force, deformations of the tooth fillet-foundation also affect the gear meshing stiffness. The formula widely adopted to compute the fillet-foundation deformation of gear tooth was developed by [PERSON] et al. in Ref. [18]. It is indicated as,  
\[\frac{1}{k_{f}}=\frac{\cos^{2}\alpha_{1}}{WE}\left\{L\left(\frac{u_{f}}{S_{f}} \right)^{2}+M\left(\frac{u_{f}}{S_{f}}\right)+P\Big{(}1+Q\tan^{2}\alpha_{1} \Big{)}\right\} \tag{14}\]  
where \(k_{f}\) denotes the tooth fillet-foundation stiffness. The parameters \(u_{f}\) and \(S_{f}\) are displayed in Figure 2 in [18], and the parameters \(L\), \(Q\), \(M\), and \(P\) are functions of \(\theta_{f}\) and \(h_{f}\), which can be presented as [18],  
\[X_{i}\Big{(}\theta_{f},h_{f}\Big{)}=A_{i}/\theta_{f}^{2}+B_{i}h_{f}^{2}+C_{i}h _{f}/\theta_{f}+D_{i}/\theta_{f}+E_{i}h_{f}+F_{i} \tag{15}\]  
where the meaning of symbols \(h_{f}\), \(\theta_{f}\), \(A_{i}\), \(B_{i}\), \(C_{i}\), \(D_{i}\), \(E_{i}\), and \(F_{i}\) can be observed in [18]. Finally, the comprehensive meshing stiffness of one tooth pair can be calculated as,  
\[k_{t}=\frac{1}{\frac{1}{k_{h}}+\frac{1}{k_{h}}+\frac{1}{k_{h1}}+\frac{1}{k_{h1 }}+\frac{1}{k_{h1}}+\frac{1}{k_{h1}}+\frac{1}{k_{h2}}+\frac{1}{k_{h2}}+\frac{1 }{k_{h2}}+\frac{1}{k_{h2}}} \tag{16}\]  
where the subscripts \(1\) and \(2\) mean the pinion (gray) and gear (driven gear), respectively.