Wheels Spinning

What does it mean for a wheel to spin? Repeated, endless gyroscopic motion? But some of the most useful wheels do not endlessly spin. They start, stop and resist turning and spinning, just like the wheels on your Grand Prix car.

Perhaps you have noticed that a wheel that is not perfectly round or is weighted wobbles when it spins. Or that wheels of different size and weight are not all easy to start spinning. These observations demonstrate a physical quality of spinning objects called moment of inertia.

Below a procedure to find the moment of inertia of Grand Prix wheels is given. It uses the wheel geometry model already developed in this manual.

Moment of inertia tells us how much kinetic energy is stored in a rotating object just like mass tells us how much kinetic energy is stored in a moving object. This relationship to mass is no accident. Some theories suggest that mass itself is a whirlpool in the fabric of space, taking on the properties of matter because of spinning. A physicist named D'Broilly even estimated how fast some atomic particles must spin.

It is then no surprise that the moment of inertia, I, acts in a similar way to mass, m, in equations of motion. Newton's Second Law becomes

T = I@ like F = ma

The "T" is for torque, a measure of force, F, acting through a distance around an axis.

"@" is angular acceleration instead of linear acceleration, a.

In the same way that linear equations of motion can be formulated using F = ma, angular equations of motion can be stated using T = I@.

But even after such equations are established, one is often left wondering what is I and how can it be measured. This is certainly a necessary question to answer for modeling a Grand Prix race.

How To Estimate The Moment Of Inertia For A Wheel

There are general ways to solve for the value of the moment of inertia for an object of arbitrary shape spinning about an arbitrary axis. They invlove the methods of calculus and can be found in any college level mechanics book. Reference books are also available that have performed the analysies already for many shapes and axies.

A simple wheel is like a cylinder. For a cylinder of mass, m, with outer radius, R, and inner radius, r, its moment of inertia about its axis is I = m(R2+r2)/2. Generally, we don't measure mass directly, it's not a force. Instead weight is measured. To get the mass, the weight, W, must be divided by gravitational acceleration, g = 386.088 in/s2. The fact that the width of the cylinder is not an explicit factor in the equation is a bit deceiving. It is accounted for by the mass. Wider cylinders, have more mass. A denser cylinder will spin like one less dense but wider.

Let R = 0.591 in, r = 0.049 in, W = 0.0833 oz.

Then m = W/g = 0.0002157539 ozs2/in.

I = 0.0002157539((0.591)2+(0.049)2)/2 = 0.0002157539x0.351682/2 I = 0.00003793822 ozins2 (ie., ounce-inch-seconds squared, only the seconds are squared)

More complex wheels can generally be thought of as a series of concentric cylinders. In particular, the following example uses the measurements of an AWANA Grand Prix kit wheel.

Begin with the cylinders from the geometry model of the AWANA kit wheel. The kit wheel's plastic has a uniform composition and elasticity. Because of uniform composition, use the calculated volume of each cylinder, Vi = wpi(Ri2-ri2), to get their masses, mi.

The total volume of the wheel is V = V1 + V2 + V3 + V4 + V5. The proportion of the wheel's volume occupied by a cylinder, i, is the ratio of the cylinder's volume, Vi, to the total wheel volume, V, which is Vi/V. So it's mass, mi, is the wheel weight multiplied by its portion of volume, WVi/V. This means Vi/V can be interpretted as the proportion of total wheel mass contained in each cylinder, %mi. So %mi = Vi/V.

Moment of inertia about a common axis is additive, so it was computed for each cylinder without its mass as (Ri2+ri2)/2 which is Ii/mi.

Since the mass of each cylinder is its percent/100 times the total wheel mass (mi = m%mi/100), each percent was multiplied by the Ii/mi quantity to get each cylinder's contribution to the total moment of inertia.

Ii/m = %miIi/100mi

After adding each Ii/m together, the total wheel mass, m, was multiplied by it to get the total moment of inertia for the wheel.

I = 0.000048637 ozins2.

Weight(_0.08333_ oz)/386.088 = Total Mass (0.0002157539 ozs2/in).

I/m (0.2253489) x Total Mass (0.0002157539) = Total Wheel Moment of Inertia, I = (0.000048637 ozins2).

Notice, the roughly estimated moment of inertia for the wheel in example 1 is less by about 20% than the more accurate model in this example. The difference is the distribution of the weight. The weight of the "bulky" wheel in example 1 is, on the whole, closer to the hub. If it were divided into cylinders with the same radii, its chart would look like this:

I/m (0.17587145) x Total Mass (0.0002157539) = Total Wheel Moment of Inertia, I = (0.00003794479 ozins2).

Note that over 50% of the wheel's mass is in the fourth ring, whereas over 50% of the mass of the AWANA kit wheel is in the fifth ring, further away from the hub.

Here is the same kind of chart for a Boy Scout kit wheel