## 2 Proposed Meshing Stiffness Calculation Model with Tooth Surface Crack  
As demonstrated in Figure 1, the tooth surface crack fault is usually located at the middle of the tooth surface and is symmetrical along the center plane of gear tooth width. The boundary of tooth surface cracks is assumed as consisting of straight lines for the convenience of modeling and calculation. \(L_{1}\) represents the crack length, \(L_{2}\) stands for the crack width, and \(D\) refers to the crack depth. The potential energy method is diffusely adopted in the gear mesh stiffness calculation due to its fast calculation and simple modeling [16; 17], which is also applied in this paper. The gear tooth is assumed to be a cantilever beam with a variable cross section when establishing the meshing stiffness calculation model of a gear tooth with surface cracking, which is displayed in Figure 2. The bending energy \(U_{b}\), shear energy \(U_{s}\), and axial compressive energy \(U_{a}\) deposited in a tooth can be expressed as [16; 17],  
\[U_{b}=\frac{F^{2}}{2k_{b}}=\int_{0}^{d}\frac{\left[F_{b}(d-x)-F_{a}h\right]^{2 }}{2 EI_{x}}dx,U_{s}=\frac{F^{2}}{2k_{s}}=\int_{0}^{d}\frac{1.2F_{b}^{2}}{2 GA_{ x}}dx,U_{a}=\frac{F^{2}}{2k_{a}}=\int_{0}^{d}\frac{F_{a}^{2}}{2 EA_{x}}dx \tag{1}\]  
where \(k_{b}\), \(k_{s}\), and \(k_{a}\) signify the bending stiffness, shear stiffness, and axial compressive stiffness, respectively. \(F\) refers to the meshing force acting on the gear tooth. \(E\) and \(G\) are the [PERSON]'s modulus and shear modulus, respectively, and it is assumed that the material of the gear is uniform and the [PERSON]'s modulus and shear modulus of its surface and internal parts are the same in this paper. \(d\) denotes the distance from the meshing point to the dedendum circle, and \(h\) is the distance from the meshing point to the center line of the gear tooth. \(F_{a}\) and \(F_{b}\) represent the horizontal and vertical components of the meshing force \(F\), respectively, which are expressed as,  
\[F_{a}=F\sin\alpha_{1},F_{b}=F\cos\alpha_{1} \tag{2}\]  
According to Equations (1) and (2), the stiffness \(K_{b}\), \(K_{s}\), and \(K_{a}\) can be deduced as,  
\[\frac{1}{k_{b}}=\int_{0}^{d}\frac{\left[(d-x)\cos\alpha_{1}-h\sin\alpha_{1} \right]^{2}}{EI_{x}}dx,\,\frac{1}{k_{s}}=\int_{0}^{d}\frac{1.2\cos\alpha_{1} ^{2}}{GA_{x}}dx,\,\frac{1}{k_{a}}=\int_{0}^{d}\frac{\sin\alpha_{1}^{2}}{EA_{x }}dx \tag{3}\]  
According to the characteristic of involute tooth profile of healthy gear tooth, the expressions of parameters \(d\), \(x\), \(h\), \(h_{x}\), \(A_{x}\), and \(I_{x}\) are as follows,  
\[d=R_{b}[(\alpha_{1}+\alpha_{2})\sin\alpha_{1}+\cos\alpha_{1}]-R_{f}\cos\alpha _{3} \tag{4}\]  
Figure 1: Geometric parameters of tooth surface crack. Figure 2: The cantilever beam model of gear with tooth surface crack fault. \[x=R_{b}[(\alpha+\alpha_{2})\sin\alpha+\cos\alpha]-R_{f}\cos\alpha_{3} \tag{5}\]  
\[h=R_{b}[(\alpha_{1}+\alpha_{2})\cos\alpha_{1}-\sin\alpha_{1}] \tag{6}\]  
\[h_{x}=\left\{ \begin{array}{ll}R_{b}\sin\alpha_{2}&0<x<x_{0}\\ R_{b}[(\alpha+\alpha_{2})\cos\alpha-\sin\alpha]&x>x_{0}\end{array} \right. \tag{7}\]  
\[A_{x}=2h_{x}W \tag{8}\]  
\[I_{x}=\frac{2}{3}h_{x}^{3}W \tag{9}\]  
where \(h_{x}\) is the cross-sectional height with \(x\) distance from the dedendum. \(R_{b}\) and \(R_{f}\) are the base circle radius and dedendum circle radius, respectively. \(\alpha\) represents the gear rotation angle, \(\alpha_{2}\) and \(\alpha_{3}\) refer to half of tooth angle of the base circle and dedendum circle, respectively. \(A_{x}\) is the area of the effective section and \(I_{x}\) represents the area moment of inertia. The meshing positions of the gear with tooth surface cracking are presented in Figure 3. Point \(G\) represents the position of dedendum, point \(P\) denotes the position at the beginning of the crack, point \(S\) refers to the position at the end of the crack, and point \(T\) stands for the position of the addendum.