The Effects of Wheel and Tire Rotational Energy

[aardvarcus]

There have been some threads on changing tire and wheel sizes and weights, and one issue that comes up is how adding weight to the wheels and tires affects the car differently than just putting weight on the vehicle. I have been working on quantifying how the weight of wheels and tires affects the amount of energy it takes to get the car up to speed. If we model the car as just a mass and we model the wheels and tires as hollow cylinders of a set weight we can use the below equations. Note that this is a simplification because the mass of the sidewalls do not actually occur at the set radius of the rim and the mass of the sidewalls of the tire do not occur at the outside radius of the tire tread, but should get close enough for a general idea.

Starting with the basic formulas:
Total Kinetic Energy (ft-lb) = Translational Kinetic Energy (ft-lb) + Rotational Kinetic Energy (ft-lb)
Translational Kinetic Energy (ft-lb) = ½ Mass (slugs = lb/32.1740) * Velocity (ft/s) ^2
Rotational Kinetic Energy (ft-lb) = ½ * Moment of Inertia (slug-ft^2) * Angular Velocity (rad/sec) ^2
Moment of Inertia (slugs-ft^2) = Mass (slugs = lb/32.1740) * Radius (ft) ^2

And simplifying them using conversion factors into a more “car friendly” fashion:
Translational Kinetic Energy of the Car (ft-lb) = 0.03342 * Car Weight (lbs) * Car Velocity (miles/hour) ^2
Angular Velocity (rad/sec) = 35.2 * Car Velocity (miles/hour) / Tire Outer Diameter (inches)
Rotational Kinetic Energy of Tire(s) (ft-lb) = 0.00002698 * Weight of Tire(s) (lbs) * Tire Outer Diameter (inches) ^2 * Angular Velocity (rad/sec) ^2
Rotational Kinetic Energy of Wheel(s) (ft-lb) = 0.00002698 * Weight of Wheel(s) (lbs) * Wheel Outer Diameter (inches) ^2 * Angular Velocity (rad/sec) ^2

I encourage anyone who has some time on their hands to check my work here, there were a lot of conversions to get them into this form and it would have been very (very) easy for me to have made a mistake.

Here are the results I got:

Example of a large truck with big tires.
Car Weight 7000 lbs
Car Velocity 60 mph
Tire Outer Diameter 32.5 inches
Each Tire Weight 53 lbs
Wheel Outer Diameter 17 inches
Each Wheel Weight 30 lbs
Angular Velocity 64.98 rad/sec Percent Total
Translational Kinetic Energy 842184 ft-lb 96.62%
Tires Rotational Kinetic Energy 25513 ft-lb 2.93%
Wheels Rotational Kinetic Energy 3951 ft-lb 0.45%
Total Kinetic Energy 871649 ft-lb
Net Weels Tires 8.12%

Example of a car with medium tires:
Car Weight 2500 lbs
Car Velocity 60 mph
Tire Outer Diameter 26 inches
Each Tire Weight 22 lbs
Wheel Outer Diameter 16 inches
Each Wheel Weight 22 lbs
Angular Velocity 81.23 rad/sec Percent Total
Translational Kinetic Energy 300780 ft-lb 95.37%
Tires Rotational Kinetic Energy 10590 ft-lb 3.36%
Wheels Rotational Kinetic Energy 4011 ft-lb 1.27%
Total Kinetic Energy 315381 ft-lb
Net Weels Tires 11.67%

From these results, it looks like one pound on the wheels and tires is equivalent to about 1.75 +/- pounds on the car from an energy to get up to speed standpoint.

Anyone have any thoughts, ideas, or corrections?