# Numerical Design Method for CVT Supported in Standard Variable Speed Rubber V-Belts
## 4 Example Analysis and Discussion
The tension due to the centrifugal force adds a value of 995 N (Equation (14)) with a belt density of 1.32 kg/dm\({}^{3}\) (Equation (16)). Belts with deep and short teeth have low bending stress [36], considering a bending stress of 15 N/mm\({}^{2}\) the bending tension is 2013 N (Equation (17)). The minimum diameter \(\phi 1_{min}\) for a V-belt with a thickness of 32 mm is 193 mm (Equation (8)). With the value of the maximum CVT speed ratio \(R_{cwt\_max}\), it is determined that the maximum driven pulley diameter is 384 mm. The pulleys diameter to develop a minimum ratio of 1:1 is 278 mm. The minimum pulley center distance \(C\) is 300 mm (Equation (18)) using the \(\tau_{cc\_in}\geq 160^{\circ}\) approximation. However, the pulleys are almost rubbing against each other, and to correct this the contact angle is increased to 166\({}^{\circ}\) obtaining a value of \(C\) of 400 mm which allows for an increase in the tensile capacity. Finally, the belt's length is 1674 mm. The lateral belt displacement in the driver pulley is 23.57 mm and 29.95 mm in the driven pulley (Equation (22)), and this causes the belt to be crossed by 0.69\({}^{\circ}\) (Equation (27)). To enhance alignment, the driven pulley angle is reduced to 28\({}^{\circ}\), ergo its lateral displacement is reduced to 26.42 mm and the misalignment to 0.4\({}^{\circ}\), which reduces the crossing belt by 57.97%. This modification generates a small distortion between the lateral forces on both pulleys when the belt passes from one to the other. The maximum normal lateral pressure reaches 9 kg/cm\({}^{2}\). The lateral force on the belt is 7600 N on average (Equations (13) and (23)) or 8600 N as a result of the integral (Equation (26)) and a 160\({}^{\circ}\) contact angle. The lateral force to prevent the belt slip is in the range from 8540-1415 Nm (Figure 19b). Figure 18: Electric powertrain. Figure 17 shows the CVT ratio change zone, starting at the point where the torque starts decreasing (1500 rpm) and ending at the sector where the motor's efficiency is the highest (2500 rpm). The torque range is 840 to 200 Nm. In this case, for the design of the centrifugal actuator, we have a ramp and weight system. Using the Equations (28)-(38) we can find the form of the ramp, the values and relationships between the centrifugal and clamping force, and the angle and mechanical advantage as a rotational speed function. The data obtained are initial ramp radius 40 mm and a final ramp radius of 120 mm, lateral displacement of 23.57 mm, and the mass of the weights is found by iterations. If the weight grows the lateral displacement grows, and vice versa. For the required displacement, the total weight is 288 g for the three weights. For low rpm the centrifugal force is low (285 N) but the required clamping force is high (8540 N) so the mechanical advantage must be high (30:1). This is achieved by using a near radial angle (1.9\({}^{\circ}\)) on the ramp. The opposite situation occurs at high speed, high centrifugal force (2096 N) and low clamping force (1415 N). In this case, the angle will turn to the right with a higher value regarding the radial axis (46.2\({}^{\circ}\)). For the driven pulley, a spring and a cylindrical cam are designed to regulate the clamping force according to the torque and the change in diameter for the pulley. The driven pulley spring is compressed at maximum torque applying a maximum lateral clamping force. As the torque changes the spring expands and at the same time, it responds to displacement due to the speed ratio change that compresses it by a maximum of 24 mm. With a minimum spring diameter of 70 mm, and using Equations (51) and (52), the spring constant is determined with a value of 140 N/mm. The displacement of the spring is 60 mm and the free length is 175 mm, the diameter of the spring wire is 14 mm made of piano wire rope that provides a torsional stress of 727 N/mm\({}^{2}\) at maximum compression, and a 10 mm pre-compression for minimum clamping force. The cam that compresses the spring is designed with Equations (52)-(62).