This is a design study to see what factors affect the performance of an electric vehicle, miniature or full-size. We’ll look at the power required and the range, too, for a spread of conditions. We’ll consider this range of variables– we will need to calculate the tractive force, which we will find from a free-body diagram of the drive wheel.
As we saw last week, the motor generates a driving force, a tractive force at the drive wheels. This is the force that overcomes the resistance to motion, but what determines the resistance to motion? We will consider three types of resistance to motion– rolling resistance, aerodynamic resistance, and hill-climbing. Rolling resistance is conventionally represented as weight times a coefficient of rolling friction. We will assume that the coefficient of rolling friction does not depend upon speed. Here’s an example for you to try. Pause the video and give it a go.
We’ve got a value of 206 newtons.
Aerodynamic drag - aerodynamic resistance is conventionally determined from this beautiful equation. We’ve seen it before. Rho is the air density, A is the cross-sectional area, CD is the coefficient of drag, and v is the speed. C is determined from wind tunnel tests. Here’s an example for you to try. Pause the video and find aerodynamic resistance for a vehicle with the following dimensions. Note that 1 metre per second is 3.6 kilometres per hour, and take the area as the box defined by the three dimensions above.
We got FA– that is, the aerodynamic resistance– equals 198 newtons. Hill-climbing resistance– we can get drag from hill-climbing directly from the free-body diagram. It is the component of the weight that acts down the hill and is calculated as weight time sine of the angle of the incline. Notice that the tractive effort in this case is shown at the front wheels. It is a front-wheel drive car. Small-angle approximation– for small angles, sine is approximately equal to 10, and 10 is the ratio of height gained to distance travelled. Hills are often expressed as height gained to distance travelled– 1 in 10, for instance. Here’s an example for you to try. Pause the video and give it a go.
Find the gravity drag for a 2,000-kilogramme car climbing a 1 in 10 hill.
The answer is easy. The gravitational drag force is 2,000 times 9.8 divided by 10 newtons, which is 1,960 newtons. This gradient is what you might find on a steep section of motorway. Notice that it is much larger than the rolling drag and the aerodynamic drag we calculated earlier. Now we’ll put it all together. We can show these forces on a free-body diagram of the car. In this case, we’ve shown a point force for the aerodynamic drag, but actually it is distributed all over the car. We’ve shown rolling resistance at the rear wheels for convenience. It actually happens at all the wheels.
Again, we have assumed a front-wheel drive car. Assuming no acceleration, we can write some of the forces in the x-direction equals 0. Notice that the x-direction is aligned with the slope. Summing the forces in the x-direction this way gives us FT minus Da minus Dr minus Dg equals 0. So the tractive force is FT equals Da plus Dr plus Dg. To get the power required at the wheels, you need to know that power in watts is tractive force in newtons time speed in metres per second. And to find the power needed from the motor, we need to know how much of the motor power makes it to the wheels. There are losses. We account for losses by using motor efficiency.
Motor efficiency equals mechanical power out of the motor divided by the electrical power supplied.
We also need to know transmission efficiency. Transmission efficiency equals power at the wheels divided by mechanical power out of the motor. If you know the required power at the wheels, you divide that power by the transmission efficiency to get the power required at the output of the motor. Then you divide that power by the motor efficiency to get the power required from the battery. Now you can find the power needed from the motor at any given speed. Now we can estimate the range. Step 1 - calculate the power required at the wheels for a given speed. Step 2 - divide by transmission efficiency to get the power required at the output of the motor.
Step 3 - divide by motor efficiency to get the power required at the output of the battery. Step 4 - find the battery capacity in kilowatt-hours when it’s supplying power at the rate that we determined from step 3. Step 5 - divide the hours found in step 4 by the power found in step 3. Step 6 - multiply those hours by speed to get the range.
Now, for your design exploration - 1, download the PDF file of vehicle specifications. There are several vehicles to choose from. 2 - choose operating conditions that interest you. You can vary payload, speed, rolling resistance, drag coefficient, and gradient. Find the required power at the motor. You should compare with the motor power given in the PDF to make sure that your vehicle can actually supply that amount of power. 4 - find the power required from the battery. 5 - find the range. And 6 - report your findings in the discussion. And 7 - have fun.