# Transmission Performance Analysis of RV Reducers Influenced by Profile Modification and Load
## 2 Analytical Model of RV Reducer with Tooth Profile Modifications
### Contact Force Calculation
#### 2.1.1 Contact Force Model
To calculate the contact force more accurately, the Gaussian quadrature method is used to discretize the contact areas in the present study [22]. The compression region is divided into many small differential units, as indicated in the enlarged view of Figure 3. Thus, each differential element can be replaced with a spring-damped model. Therefore, the normal contact force of the contact area can be expressed as follows:  
\[F_{Ni}\approx\sum_{j=1}^{n}H_{j}F(\delta_{ij})\,, \tag{13}\]  
where \(n\) represents the number of differential elements, and \(H_{j}\) is the weight coefficient. As shown in Figure 3, the penetration of the _jth_ small element can be calculated based on the difference between \(r_{rp}\) and \(O_{pi}B\) as follows:  
\[\delta_{ij}=r_{rp}-O_{pi}B \tag{14}\]  
The area of the differential element may be calculated in the following manner:  
\[dA_{ij}=\frac{1}{2}r_{rp}Ld\tau \tag{15}\]  
Because the Lankarani-Nikravesh contact force model exhibits a higher accuracy based on the Hertz contact force model, it is adopted to calculate the contact force of a single differential element. The contact force model is as follows [23]:  
\[dF_{Nij}=dK_{ij}\delta_{ij}^{n}+dD_{ij}\dot{\delta}_{ij}, \tag{16}\]  
where \(K_{ij}\) is the Hertz contact stiffness of the differential element, which is calculated as follows [19]:  
Figure 3:0  
Here, \(W_{ij}\) is the thickness of the elastic layer, and \(D_{ij}\) is the damping coefficient, the modified formula of which is [24]:  
Figure 3:1  
where \(\dot{\delta}_{ij}^{(-)}\) is the initial penetration velocity upon impact of the differential element of the _ith_ contact area, and \(c_{c}\) is the restitution coefficient. The exponent \(n\) is set to 1.5 for metallically circular or elliptical surfaces. Thus, the contact force of the entire contact area of the _ith_ pin is as follows:  
Figure 3:2  
The contact force \(F_{Ni}\) can be divided into the radial and tangential components in the direction and vertical direction of the center line \(\overline{OO_{1}}\). The contact force can then be expressed as follows:  
Figure 3:3  
\[\left\{ \begin{array}{l}F_{Ni}^{t}=F_{Ni}sin\psi_{i}\\ F_{Ni}^{r}=F_{Ni}cos\psi_{i}\end{array} \right. \tag{21}\]  
In this study, the Ambrosio friction force model is adopted to calculate the friction force between meshing surfaces. Compared with the traditional Coulomb friction model, the dynamic correction coefficient \(c_{d}\) is added to the Ambrosio friction model [25], effectively preventing the friction from changing direction when the tangential velocity is zero. Therefore, the improved friction model creates numerical stability and is more suitable for a computer simulation. The calculation formula is as follows:  
\[F_{fi}=-c_{f}c_{d}F_{Ni}\frac{V_{f}}{\|V_{f}\|}=F_{fi}^{x}i+F_{fi}^{y}j, \tag{22}\]  
\[\left\{ \begin{array}{l}F_{fi}^{x}=-F_{fi}sin(\varphi-\psi_{i})\\ F_{fi}^{y}=F_{fi}cos(\varphi-\psi_{i})\end{array} \right., \tag{23}\]  
where \(F_{fi}^{x}\) and \(F_{fi}^{y}\) represent the horizontal and vertical components of friction, respectively; \(c_{f}\) is the friction coefficient; V\({}_{f}\) is the relative tangential velocity; and \(c_{d}\) is the dynamic correction coefficient, which can be expressed in the following manner:  
\[c_{d}= \begin{cases}0&if\ \|V_{f}\|\leq v_{d}\\ \frac{\|V_{f}\|-v_{d}}{v_{u}-v_{d}}&if\ v_{d}\leq\ \|V_{f}\|\leq v_{u}\\ 1&if\ \|V_{f}\|\geq v_{u}\end{cases} , \tag{24}\]  
where \(v_{d}\) and \(v_{u}\) are the given tolerances for the relative tangential velocity of the contact surfaces. Then, the torque balance equation of cycloid transmission may be expressed as follows:  
\[T_{o}-J\ddot{\partial}_{pl}=2\sum_{i=m_{1}}^{m_{2}}\left[F_{Ni}^{t}(r_{c}^{ \prime}+a)-F_{fi}^{x}\cdot y_{pi}^{\prime}+F_{fi}^{y}\cdot x_{pi}^{\prime}\right] \tag{25}\]  
where \(m_{1}\) is the serial number of pins that begin to mesh, and \(m_{2}\) is the serial number of pins that exit the meshing, which are determined from Equation (10). The simulation flow of all calculation processes is shown in Figure 4.