## 2 The Analytical Model of Mesh Stiffness of Helical Gear
### The Relationship between Neighboring Sliced Gears  
According to slice theory, one helical gear can be divided into \(n\) independent sliced gears along the width direction. The sliced gears are connected by slice coupling stiffness. \(k_{\mathrm{t}}^{\mathrm{ij}+1}\) and \(k_{\mathrm{f}}^{\mathrm{i}\mathrm{j}+1}\) denote the coupling stiffness between teeth and the coupling stiffness between the fillet-foundation, as shown in Figure 1.  
Based on the relationship between neighboring sliced gears, for each sliced gear, the following equation can be deduced:  
\[\left\{ \begin{array}{l}k_{1}^{1}(\delta_{\mathrm{t,1}}-\delta_{\mathrm{t,1}})+k _{\mathrm{t}}^{1,2}(\eta_{1}\delta_{\mathrm{t,1}}-\eta_{2}\delta_{\mathrm{t,2}} )=F_{1}\\ k_{1}^{1}\delta_{\mathrm{t,1}}+k_{\mathrm{t}}^{1}(\delta_{\mathrm{t,1}}-\delta_{ \mathrm{t,1}})+k_{\mathrm{f}}^{1,2}(\delta_{\mathrm{t,1}}-\delta_{\mathrm{t,2 }})=0\\ k_{\mathrm{t}}^{2}(\delta_{\mathrm{t,2}}-\delta_{\mathrm{t,2}})+k_{\mathrm{t}}^{1,2}(\eta_{2}\delta_{\mathrm{t,2}}-\eta_{1}\delta_{\mathrm{t,1}})+k_{\mathrm{t} }^{2,3}(\eta_{2}\delta_{\mathrm{t,2}}-\eta_{3}\delta_{\mathrm{t,3}})=F_{2}\\ k_{\mathrm{t}}^{2}\delta_{\mathrm{t,2}}+k_{\mathrm{t}}^{2}(\delta_{\mathrm{t,2 }}-\delta_{\mathrm{t,2}})+k_{\mathrm{f}}^{1,2}(\delta_{\mathrm{t,2}}-\delta_{ \mathrm{t,1}})+k_{\mathrm{f}}^{2,3}(\delta_{\mathrm{t,2}}-\delta_{\mathrm{t,3} })=0\\ \dots\dots\\ k_{\mathrm{t}}^{n}(\delta_{\mathrm{t,n}}-\delta_{\mathrm{t,n}})+k_{\mathrm{t}}^{ n-1,n}(\eta_{n}\delta_{\mathrm{t,n}}-\eta_{n-1}\delta_{\mathrm{t,n-1}})=F_{n}\\ k_{\mathrm{f}}^{n}\delta_{\mathrm{t,n}}+k_{\mathrm{t}}^{n}(\delta_{\mathrm{t,n}} -\delta_{\mathrm{t,n}})+k_{\mathrm{f}}^{n-1,n}(\delta_{\mathrm{t,n}}-\delta_{ \mathrm{t,n-1}})=0\end{array} \right. \tag{1}\]  
where \(\delta_{\mathrm{f,i}}\), \(\delta_{\mathrm{t,i}}\) represent the deformation of the fillet-foundation and tooth, respectively. \(k_{\mathrm{f}}^{\mathrm{i}}\), \(k_{\mathrm{t}}^{\mathrm{i}}\) represent the stiffness of the fillet-foundation and tooth, respectively. \(F_{\mathrm{i}}\) represents the external force. \(\eta_{\mathrm{i}}\) donates the value of the deflection of the loading point divided by the deflection of the tooth center. It should be noted that the above symbols refer only to a sliced gear. The above equations can be rewritten as follows:  
\[\left\{\begin{array}{l}k_{\mathrm{t}}^{1}+\eta_{1}k_{\mathrm{t}}^{1,2}-k_{ \mathrm{t}}^{1}-\eta_{2}k_{\mathrm{t}}^{1,2}&0&0&0\\ -k_{\mathrm{t}}^{1}&k_{\mathrm{t}}^{1}+k_{\mathrm{t}}^{1}+k_{\mathrm{t}}^{1,2} &0&-k_{\mathrm{t}}^{1,2}&0&0\\ -\  
\[\delta_{\rm p}=[\delta_{\rm t,L},\ \ \delta_{\rm f,L},\ \ \delta_{\rm t,L},\ \ \delta_{\rm t,2},\ \ \delta_{\rm f,2},\ \ \ldots,\ \ \delta_{\rm t,n},\ \ \delta_{\rm f,n}]^{T}\]  
\[F_{\rm p}=[F_{1},\ \ \ 0,\ \ \ F_{1},\ \ 0,\ \ \ \ldots\ \ F_{\rm n},\ \ 0]^{T}\]  
Obviously, \(K_{\rm p}\) is a function of \(k_{\rm t}^{1},k_{\rm t}^{2},\ldots,k_{\rm t}^{n},k_{\rm t}^{1},k_{\rm t}^{2}, \ldots,k_{\rm f}^{n},k_{\rm t}^{1,2},k_{\rm t}^{2,3},\ldots,k_{\rm t}^{n-1,n}\) and \(k_{\rm f}^{1,2},k_{\rm f}^{2,3},\ldots,k_{\rm f}^{n-1,n}\). Thus, the displacement of the sliced driving gear considering the slice coupling stiffness can be deduced:  
\[\delta_{\rm p}=K_{\rm p}^{-1}F_{\rm p} \tag{4}\]  
Similarly, the displacement of the sliced driven gear also can be obtained:  
\[\delta_{\rm w}=K_{\rm w}^{-1}F_{\rm w} \tag{5}\]  
The total deformation of each sliced gear pair considering the slice coupling stiffness can be obtained as follows:  
\[\delta=\delta_{\rm p}+\delta_{\rm w}+\delta_{\rm h}=K_{\rm p}^{-1}F+K_{\rm w}^ {-1}F+K_{\rm h}^{-1}F=\Big{[}K_{\rm p}^{-1}+K_{\rm w}^{-1}+K_{h}^{-1}\Big{]}F=AF \tag{6}\]  
As a result, the mesh stiffness of each sliced gear pair can be expressed as follows:  
\[k_{\rm i}=\left(\sum_{\rm j=1}^{n}A_{2{\rm i}-1,2{\rm j}-1}+\sum_{\rm j=1}^{n} A_{2{\rm i},2{\rm j}-1}\right)^{-1} \tag{7}\]  
The total mesh stiffness of a helical gear is the sum of each sliced gear pair considering slice coupling.