# Numerical Design Method for CVT Supported in Standard Variable Speed Rubber V-Belts
## 4 Example Analysis and Discussion  
A powertrain design composed of a CVT and a fixed gear transmission for a common city vehicle transmission is presented. Typically mountainous towns have small buses (22-28 passengers), delivery cars, and fire and police trucks, among others, with cargo capacities of approximately 3000 or 4000 kg. These vehicles do not exceed speeds of 80 km/h in their operation and their average operating speed is 30 km/h. Chosen routes have between 4 and 15 km of maximum travel, with inclination angles between 4\({}^{\circ}\) and 16\({}^{\circ}\). One of the reasons for these vehicles to be small is that they travel on city slopes and these roads are usually narrow and with low curvature radius. The characteristics used for the analysis of the vehicle powertrain are: gross vehicle mass of 5500 kg, aerodynamic coefficient of 0.45, front area 3.5 m\({}^{2}\), air density 0.98 kg/m\({}^{3}\), the rolling coefficient is 0.03, wheel diameter approximately 0.75 m (R17.5). The aim is for the loaded vehicle to achieve maximum speed on slopes within 5 s. This results in a climb acceleration of 1.7 m/s\({}^{2}\) and a maximum speed of 30 km/h on slopes and 80 km/h on flat routes. Using equations (1)-(3) it is determined that the peak power for the vehicle is 130 KW on a 16\({}^{\circ}\) slope and 45 KW continuous power on flat routes. Recent recurring problems in vehicle design are the reduction of polluting emissions, increasing overall efficiency, and improving energy performance. A solution to this problem is the use of EV, therefore, this type of vehicle is the one chosen. New electric motors tend to operate at high rpm (>6000) to decrease the weight per power unit. However, for these angular speeds the belts would exceed the allowable tangential speed (approximately 30 m/s) and a fatigue of the V-belt could occur, as a result of the available tensile capacity being reduced by the effect of the centrifugal force. Therefore, a motor with a maximum rpm in the order of 4000 rpm should be selected with the requirements of having a continuous torque value of 500 Nm at 1500 rpm and a peak value of 840 Nm. After analyzing the possibilities in the market, the motor described in Figure 17 was chosen. The ratio of the fixed transmission system is obtained using Equations (4) and (5), the \(R_{gear}\) has a value of 6.2. Configurations may vary, for instance, it is possible to have a single gearbox and a differential or a transaxle. According to Equations (6) and (7) the maximum ratio of the CVT is \(R_{CVT\_max}\) = 1.94. Finally, the maximum speed ratio of the system is 12. In this case, the system is composed of an electric motor, a CVT, and a differential as show in Figure 18.  
Figure 17: Electric motor curves: (**a**) continuous graph; (**b**) peak graph. The belt section is determined using the iterative process in Figure 3, for which the commercial belt section that most closely approximates the parameters is the W100 reference. It has a width \(W_{bel}\) of 100 mm, a thickness \(h_{bel}\) of 32 mm, a belt angle \(\varphi_{bel}\) of 30\({}^{\circ}\), and a cross-sectional area of 3050 mm\({}^{2}\). The maximum tension present in the belt is 11,858 N determined by the sum of the three tensions. Assuming that the belt tensile strength (\(\sigma_{belt}\)) is 4 N/mm\({}^{2}\), the minimum cross-sectional area of the section is obtained using the Equation (10) of 2965 mm\({}^{2}\), afterwards the aforementioned belt is selected as it is the closest to this value. The tensions that interact on the belt are the working tension, the tension due to the centrifugal force, and the bending tension. Using Equation (11) it is determined that the value of the working tension is 8840 N with a starting torque of 840 Nm and a diameter \(\phi 1_{min}\) of 193 mm. A more precise value of this tension is obtained using Equation (12) with a result of 8750 Nm, considering a friction coefficient \(\mu\) between the belt and the pulley of 0.5.