Page:On Governors.pdf/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

(14)	$\left({\frac {F}{M}}+{\frac {Y}{B}}\right){\frac {Y}{B}}-{\frac {G}{B}}={\mbox{a positive quantity}}$

.

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 $\scriptstyle \ P'$,

(15)	$F={\frac {P-P'}{V-V'}}$

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

(16)	$G={\frac {P-P'}{y-y'}}$

General Theory of Chronometric Centrifugal Pieces.

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

(17)	${\frac {d\theta }{dt}}\left(A{\frac {d\theta }{dt}}=L\right)$

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

${\tfrac {1}{2}}A\left[{\frac {d\theta }{dt}}\right]^{2}+{\tfrac {1}{2}}B\left[{\frac {d\phi }{dt}}\right]^{2}$

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

(18)	$E={\tfrac {1}{2}}A\left[{\frac {d\theta }{dt}}\right]^{2}+{\tfrac {1}{2}}B\left[{\frac {d\phi }{dt}}\right]^{2}+P=\int Ld\theta$

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

(19)

${\begin{array}{lcr}{\frac {d\phi }{dt}}\left({\tfrac {1}{2}}{\frac {A}{d\phi }}\left[{\frac {d\theta }{dt}}\right]^{2}+{\tfrac {1}{2}}{\frac {dB}{d\phi }}\left[{\frac {d\phi }{dt}}\right]^{2}+{\frac {dP}{d\phi }}\right)+A{\frac {d\theta }{dt}}{\frac {d^{2}\theta }{dt^{2}}}+B{\frac {d\phi }{dt}}{\frac {d^{2}\phi }{dt^{2}}}\\=L{\frac {d\theta }{dt}}={\frac {d\theta }{dt}}\left({\frac {dA}{d\phi }}{\frac {d\theta }{dt}}{\frac {d\phi }{dt}}+A{\frac {d^{2}\theta }{dt^{2}}}\right)\end{array}}$