§ 121. Flywheels.—A flywheel is a rotating piece in a machine, generally shaped like a wheel (that is to say, consisting of a rim with spokes), and suited to store and restore energy by the periodical variations in its angular velocity.
The principles according to which variations of angular velocity store and restore energy are the same as those of § 117, only substituting moment of inertia for mass, and angular for linear velocity.
Let W be the weight of a flywheel, R its radius of gyration, a2 its maximum, a1 its minimum, and A = 1⁄2 (α2 + α1) its mean angular velocity. Let
I/S = (α2 − α2) / A
denote the unsteadiness of the motion of the flywheel; the denominator S of this fraction is called the steadiness. Let e denote the quantity by which the energy exerted in each cycle of the working of the machine alternately exceeds and falls short of the work performed, and which has consequently to be alternately stored by acceleration and restored by retardation of the flywheel. The value of this periodical excess is—
e = R2W (α22 − α12), 2g,
(77)
from which, dividing both sides by A2, we obtain the following equations:—
e / A2 = R2W / gS
R2WA2 / 2g = Se / 2.
(78)