Page:On Governors.pdf/6

6 If $$\scriptstyle\ a$$ is a negative quantity, this will indicate an oscillation the amplitude of which continually decreases. If $$\scriptstyle\ a$$ is zero, the amplitude will remain constant, and if $$\scriptstyle\ a$$ is positive, the amplitude will continually increase.

One root of the equation (12) is evidently a real negative quantity. The condition that the real part of the other roots should be negative is

.

This is the condition of stability of the motion. If it is not fulfilled there will be a dancing motion of the governor, which will increase till it is as great as the limits of motion of the governor. To ensure this stability, the value of $$\scriptstyle\ Y$$ must be made sufficiently great, as compared with $$\scriptstyle\ G$$, by placing the weight $$\scriptstyle\ W$$ in a viscous liquid if the viscosity of the lubricating materials at the axle is not sufficient.

To determine the value of $$\scriptstyle\ F$$, put the break out of gear, and fix the moveable wheel; then, if $$\scriptstyle\ V$$ and $$\scriptstyle\ V'$$ be the velocities when the driving-power is $$\scriptstyle\ P$$ and $$\scriptstyle\ P'$$,

To determine $$\scriptstyle\ G$$, let the governor act, and let $$\scriptstyle\ y$$ and $$\scriptstyle\ y'$$ be the positions of the break when the driving-power is $$\scriptstyle\ P$$ and $$\scriptstyle\ P'$$, then

Sir W. Thomson’s and M. Foucault’s Governors. Let $$\scriptstyle\ A$$ be the moment of Inertia of a revolving apparatus, and $$\scriptstyle\ \theta$$ the angle of revolution. The equation of motion is

where $$\scriptstyle\ L$$ is the moment of the applied force round the axis. Now, let $$\scriptstyle\ A$$ be a function of another variable $$\scriptstyle\ \phi$$ (the divergence of the centrifugal piece), and let the kinetic energy of the whole be

where $$\scriptstyle\ B$$ may also be a function of $$\scriptstyle\ \phi$$, if the centrifugal piece is complex.

If we also assume that $$\scriptstyle\ P$$, the potential energy of the apparatus is a function of $$\scriptstyle\ \phi$$ then the force tending to diminish ?, arising from the action of gravity, springs, etc., will be $$\scriptstyle\ dP/d\phi$$.

The whole energy, kinetic and potential, is

Differentiating with respect to $$\scriptstyle\ t$$, we find