# Numerical Design Method for CVT Supported in Standard Variable Speed Rubber V-Belts
## 2 Numerical Continuously Variable Transmission (CVT) Design Method
### Torque Sensing Cam Design  
The spring is required to provide enough lateral force so that the belt does not slip during the start of high torque. This requirement calls for a spring that does not decrease in force as the motor /engine speed increases, therefore, decreasing the amount of available torque, as shown in Figure 14. The effect of having lateral force that is higher than required (Figure 14, blue line) results in greater friction losses between the belt and the pulleys and, therefore, in a drop in system efficiency. To solve this problem, engineers added a cam on the driven pulley located on the output shaft. This cam responds to shaft torque, compresses at high torque values, and decompresses at low torque values. The result is equivalent to having multiple springs, shown as a green dotted line in Figure 14. The result is that when the torque is increased, both axes acquire an offset angle \(\varepsilon_{sp\_i}\) and an axial displacement \(\sigma_{axi\_j}\) this is due to the ramp as seen in Figure 15.  
The grey area of Figure 14 shows the ratio change zone. From there, the maximum and minimum values of torques; displacements; forces; reduction ratio; and angular speed are obtained and thus the ranges are established and divided into differentials \(n_{uv}\). Each lateral displacement differential for the cam follower is associated with its required lateral force, a torque on the driver and driven pulley, an angular speed, and a speed ratio, then the cam profile angle is calculated.  
The location of the starting point for the first ramp differential is an input parameter in the iterative method. For each iteration to generate continuity the points of the previous differential are connected to the starting points of the next one. The result is the actuator ramp as shown in the example in Section 4. The design of the torque sensor cam (Equations (53)-(63)) is divided into \(n_{uv}\) parts that consider the following:  
* The torque range to be affected (\(T_{i}\)). \(T_{ini}\) is the initial torque and \(T_{min}\) the minimum torque.
* The spring force range (\(F_{sp\_i}\)) according to the torque. \(F_{sp\_ini}\) is the initial spring force.
* The axial displacement of the cam \(\Delta\sigma_{axi\_j}\).  
\[\Delta T_{i}=\frac{T_{min}-T_{ini}}{n_{uv}}, \tag{53}\] \[T_{i}=T_{ini}+\Delta T_{i}i,,\] (54) \[F_{ta\_j}=\frac{T_{i}}{R_{cam\_sp}},\] (55) \[\Delta F_{sp\_i}=\frac{F_{axi\_max}-F_{axi\_min}}{n_{uv}},\] (56) \[F_{sp\_i}=F_{sp\_ini}+\Delta F_{sp\_i}i,\] (57) \[\sigma_{axi\_max}=\frac{\left|F_{axi\_max}-F_{axi\_min}\right|}{K _{sp}}, \tag{58}\]  
Figure 14: Force graph including the speed ratio change and the sensing torque cam effect.  
Figure 15: Torque sensing cam diagram.  
\[\Delta\sigma_{axi\_i}=\frac{\sigma_{axi\_max}}{n_{ave}}, \tag{59}\]  
\[\sigma_{axi\_i}=\sum_{1}^{i}\Delta\sigma_{lat\_i}, \tag{60}\]  
\[\psi_{num\_i}=\tan^{-1}\left(\frac{F_{sp\_i}}{F_{ta\_i}}\right), \tag{61}\]  
\[MA_{sen}=\frac{F_{ta\_i}}{F_{sp\_i}}, \tag{62}\]  
\[\sigma_{per\_i}=\sigma_{axi\_i}MA_{sen}, \tag{63}\]  
where \(F_{ta\_i}\) is the tangential force in the torque sensing cam; \(\mathrm{R_{cam\_SP}}\) is the torque sensing cam radius; \(F_{axi\_max}\) is the maximum axial force of the torque sensing cam; \(F_{axi\_min}\) is the minimum axial force of the torque sensing cam; and \(\psi_{num\_i}\) is the angle of the torque sensing cam.