Page 304 - Anatomy of a Robot

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MECHANICS 289
where m is the mass, r is the radius of rotation, and u is the angular position of the rotat-
ing mass. This is the best equation for measuring the energy, but there’s an easier way.
If all the weight, W1, of the tire were at the edge (radius r), then each particle of the
tire would be moving at a speed of V. Each tire’s rotational energy would be
E 0.5 W2 V 2

As a practical matter, not all of the tire’s mass is at the rim. Some of the mass is within
the spokes. For the bicycle, the previous equation is a good conservative estimate, but
for wheels shaped like a hockey puck, significant weight would exist on the inside of the
wheel, closer to the axle. The rotational energy of the wheel would be lower than the pre-
vious number. It would take a bit of calculus to compute the proper number. However,
estimating the number can be done in an easier way. The energy of a rotating particle of
mass grows as r , but the number of such particles grows with the circumference of travel
2
3
as r increases. The calculus shows the energy increasing as r . If we want to estimate the
3
3
rolational energy in the wheel, we want to find r1 such that r1 0.5 r . This radius, r1,
turns out to be about 80 percent of r. Although the outside of the wheel might be mov-
ing at a speed of V, the average part of the wheel at a radius of r1 is moving at .8 V. So
a good first approximation for the rotational energy in a solid core wheel would be
2
E 0.5 W2 10.8 V2 0.32 W2 V 2
This would put the total energy within the bike between the following two energies:

High estimate This estimate assumes all the mass of the wheel is at the edge
near the rim:

2
2
E 0.5 1W1 W2 W22 V 2 10.5 W2 V 2
E 0.5 1W1 4 W22 V 2

Low estimate This estimate assumes all the mass of the wheel is evenly dis-
tributed throughout the wheel:

2
2
E 0.5 1W1 W2 W22 V 2 10.32 W2 V 2
E 0.5 1W1 2.64 W22 V 2

Do not forget that imparting energy to parts within the robot cannot be done effi-
ciently. These equations are only theoretical and are used to estimate only the energy