## 3 Load Distribution Model
### Tooth Compliance Model  
With the central points of potential contact regions defined in the unloaded TCA, the next step is to determine the tooth compliances of both contacting members. Due to the larger tooth thickness of the dedendum and the shorter tooth height of the cycloid gear, contact deformation makes a much greater contribution to the tooth deformation than bending and shear deformations. Therefore, only the contact deformation is considered, which is sufficient for engineering applications [4]. The Boussinesq solution [29; 30] for the contact deformation of tooth pairs in an elastic half-space is introduced in this study. Figure 7 shows the contact state of a cycloid gear pair. The two coordinate systems \(S_{1}(x_{1},z_{1})\) and \(S_{2}(x_{2},z_{2})\) are defined with an identical origin at the initial contact point (region) from the side view,  
Figure 6: Coordinate systems for unloaded tooth contact analysis. as shown in Figure 7a. The tooth surface is defined in the \(S(x,y)\) coordinate system where \(x\) and \(y\) represent the gear profile and width directions, respectively. The potential contact region is discretized into a contact grid, with each grid cell represented by a contact point, as shown in Figure 7c. In the profile direction, the contact region of a tooth pair is divided into \(p\) grid cells of identical arc length. In the gear width direction, the same potential contact region is divided into \(q\) grid cells of equal length. The area of each contact grid cell is \(\Omega^{\prime}=2\xi\times 2\eta\). The contact deformation \(\delta_{\varepsilon}\) at a point \((x_{i},y_{j})\) can be represented as  
\[\delta_{c}=(\frac{1-\
u_{1}^{2}}{\pi E_{1}}+\frac{1-\
u_{2}^{2}}{\pi E_{2}}) \iint\limits_{\Omega}\frac{P(x^{\prime},y^{\prime})}{\left[(x-x^{\prime})^{2} +(y-y^{\prime})^{2}\right]^{1/2}}dx^{\prime}dy^{\prime}, \tag{14}\]  
where \(\
u\) and \(E\) are the [PERSON]'s ratio and [PERSON]'s modulus, and subscripts 1 and 2 refer to the roller and the cycloid gear, respectively. \(P(x^{\prime},y^{\prime})\) is the distributed pressure in the contact area \(\Omega\). The initial separations of the two surfaces are defined as \(\omega_{1 ij}\) and \(\omega_{1 ij}\) along the \(z\) axis, respectively, which can be expressed as follows:  
\[\omega_{1 ij}=\left\{\rho_{1}^{2}-[2\xi i-2\xi(p/2+1)]^{2}\right\}^{1/2}-\left| \rho_{1}\right|,i=1,2,\ldots,p;j=1,2,\ldots,q, \tag{15}\]  
\[\omega_{2 ij}=\left\{\rho_{2}^{2}-[2\xi i-2\xi(p/2+1)]^{2}\right\}^{1/2}-\left| \rho_{2}\right|,i=1,2,\ldots,p;j=1,2,\ldots,q, \tag{16}\]  
where \(\rho_{1}\) is the curvature radius of the roller, and \(\rho_{2}\) is the curvature radius of the cycloid gear teeth at the contact point [5; 15]. For the case of FCM, according to Equation (5), the initial separations of the cycloid gear tooth surface can be rewritten as  
\[\omega_{2 ij}^{FCM}=\omega_{2 ij}+\left\{R_{f}^{2}-[2\eta j-2\eta(q/2+1)]^{2} \right\}^{1/2}-R_{f},i=1,2,\ldots,p;j=1,2,\ldots,q. \tag{17}\]  
Also, for the case of PCM, according to Equation (6), the initial separations of the cycloid gear tooth surface can be rewritten as  
\[\omega_{2 ij}^{PCM}=\omega_{2 ij}+\left\{\left\{R_{p}^{2}-[2\eta j-2\eta(q/2+1) ]^{2}\right\}^{1/2}-R_{p}\right\}H(t),i=1,2,\ldots,p;j=1,2,\ldots,q, \tag{18}\]  
where \(H(t)=\left\{ \begin{array}{l}1,t>0\\ 0,t\leq 0\end{array} \right.\) and \(t=(2j-1)/(q+1)-1-L_{0}/L\). Figure 7: (**a**) Side view of the initial contact region, (**b**) contact of a tooth pair, and (**c**) discretization of the contact region into a grid.