# Numerical Design Method for CVT Supported in Standard Variable Speed Rubber V-Belts
## 2 Numerical Continuously Variable Transmission (CVT) Design Method
### Rubber V-Belt Selection  
CVT belts are commercially available, and feature both metric and imperial systems. The belt tension capacity (\(F_{Tmax}\)) is a function of the section dimensions. For a wider belt there is a tendency to have a greater number of reinforcement members; and at higher heights, it tends to have a thicker friction surface and a higher minimum diameter. The belt reinforcement's quality indicates the specific tension value in (N/mm\({}^{2}\)) it can withstand. The maximum tension supported by the belt is the product of the sectional area multiplied by its maximum specific stress. Also, the belt must resist tensile forces such as working tension (\(F_{Twork}\)), centrifugal force (\(F_{Tce}\)) tension which is a function of the radius, the linear density of the belt, the angular speed (rpm), and bending tension (\(F_{Tdo}\)) when passing through the pulleys. This latter tension is a function of the belt thickness and the radius pulleys (Figure 2). The sum of the working tension, the centrifugal force tension, and the bending tension must not be greater than the maximum tension of the belt (Equation (9)). Therefore, an iterative process based on commercial references is presented to determine the minimum driver pulley diameter and the belt section selection (Figure 3). The minimum pulley diameter depends on CVT belts specifications, such as belt thickness, topology, whether it has teeth on the inside and outside face which decrease the bending stress, and the side section area of the belt. The process begins by selecting an initial belt section from where to obtain transversal area value (\(A_{belt}\)); section dimensions such as width (\(W_{belt}\)); thickness (\(h_{belt}\)); belt angle (\(\varphi_{belt}\)); maximum tensile stress; and belt density (\(\rho_{belt}\)). Then, these tensions and the minimum operational diameter are determined. The minimum driver pulley diameter (\(\phi_{1\_min}\)) is determined using Equation (8); this equation was developed by iterating the minimum diameter until the bending stress is significant. The diameter can also be checked in norms and standards [34]. It is verified that for the entire range of motor angular speeds (rpm) the sum of tensions is not greater than the tension that the belt withstand (Equation (9)). Finally, it is validated that the recommended lateral pressure is not exceeded either. \[\phi_{1\_min}=15.5+\big{(}2.5h_{belt}{}^{1.3}\big{)}, \tag{8}\]  
\[F_{Tmax}>F_{Twork}+F_{Tce}+F_{Td\sigma}, \tag{9}\]  
Figure 3: Belt selection flowchart. Figure 2: Working tensions in the rubber V-belt. The maximum stress that the belt can withstand is a function of the material's properties. In Equation (10) this stress is calculated, where \(\sigma_{belt}\) is the maximum tensile stress. \[F_{Tmax}=A_{belt}\sigma_{belt}, \tag{10}\]  
The working tension is given by Equation (11), where (\(\phi_{1\_min}\)) is the minimum driver pulley diameter. \[F_{Twork}=\frac{T\sigma_{mt}}{(0.5\phi_{1\_min})}, \tag{11}\]  
Assuming that the non-tensioned side of the belt is not significant when compared to the tensioned side, due to the high friction coefficient \(\mu\) between the pulley-belt and the belt angle (\(\phi_{bel}\)), the approximation in Equation (12) is proposed. By calculating the working force it is possible to solve the minimum pulley diameter (\(\phi_{1\_min}\)) using Equation (11). \[F_{Twork}\approx F_{1}, \tag{12}\]  
\[\frac{F_{1}}{F_{2}}=\frac{e^{\mu\tau_{cc}}}{\sin(0.5\sigma_{bel})}, \tag{13}\]  
The tension produced by the centrifugal force is mainly given to the linear speed of the belt (\(V_{Lbelt}\)). \[F_{Tcc}=A_{belt}\rho_{bel}(V_{Lbelt})^{2}, \tag{14}\]  
The linear speed belt can be determined from the following expression, where \(\omega_{mt}\) is the angular speed at maximum engine/motor torque. \[V_{Lbelt}=\frac{\pi}{60}\phi_{1\_min}\omega_{mt}, \tag{15}\]  
The belt density can be determined using Equation (16), where \(\rho l_{bel}\) is the belt's lineal density from manufacturers.