Page:On Governors.pdf/7

whence we have, by eliminating $\scriptstyle \ L$,

(20)	${\frac {d}{dt}}\left({\frac {d\phi }{dt}}\right)={\tfrac {1}{2}}\left({\frac {dA}{dt}}\right)\left[{\frac {d\theta }{dt}}\right]^{2}+{\tfrac {1}{2}}B\left[{\frac {d\phi }{dt}}\right]^{2}-{\frac {dP}{d\phi }}$

The first two terms of the right-hand side indicate a force tending to increase $\scriptstyle \ \phi$ depending on the squares of the velocities of the main shaft and of the centrifugal piece. The force indicated by these terms may be called the centrifugal force.

If the apparatus is so arranged that

(21)	$P={\tfrac {1}{2}}A\omega ^{2}+{\mbox{const}}$

where $\scriptstyle \ \omega$ is a constant velocity, the equation becomes

(22)	${\frac {d}{dt}}\left(B{\frac {d\phi }{dt}}\right)={\tfrac {1}{2}}{\frac {dA}{d\phi }}\left(\left[{\frac {d\theta }{dt}}\right]^{2}-\omega ^{2}\right)+{\tfrac {1}{2}}B{\frac {dB}{d\phi }}\left[{\frac {d\phi }{dt}}\right]^{2}$

In this case the value of $\scriptstyle \ \phi$ cannot remain constant unless the angular velocity is equal to $\scriptstyle \ \omega$.

A shaft with a centrifugal piece arranged on this principle has only one velocity of rotation without disturbance. If there be a small disturbance, the equations for the disturbance $\scriptstyle \ \theta$ and $\scriptstyle \ \phi$ may be written

(23)	$A{\frac {d\theta ^{2}}{dt^{2}}}+{\frac {dA}{d\phi }}\omega {\frac {d\phi }{dt}}=L,$

(24)	$B{\frac {d^{2}\phi }{dt^{2}}}+{\frac {dA}{d\phi }}\omega {\frac {d\theta }{dt}}=0.$

The period of such small disturbances is $\scriptstyle \ (dA/d\phi )(AB)^{-1/2}$ revolutions of the shaft.

They will neither increase nor diminish if there are no other terms in the equations.

To convert this apparatus into a governor, let us assume viscosities $\scriptstyle \ X$ and $\scriptstyle \ Y$ in the motions of the main shaft and the centrifugal piece, and a resistance $\scriptstyle \ G\phi$ applied to the main shaft. Putting $\scriptstyle \ (dA/d\phi )\omega =K$, the equations become

(25)	${\begin{aligned}A{\frac {d^{2}\theta }{dt^{2}}}+X{\frac {d\theta }{dt}}+K{\frac {d\phi }{dt}}+G\phi &=&L,\\B{\frac {d\phi ^{2}}{dt^{2}}}+Y{\frac {d\phi }{dt}}+K{\frac {d\theta }{dt}}&=&0.\end{aligned}}$
(26)

The condition of stability of the motion indicated by these equations is that all the possible roots, or parts of roots, of the cubic equation

(27)	$ABn^{3}+(AY+BY)n^{2}+(XY+K^{2})n+GK=o.$

shall be negative; and this condition is

(28)	$\left({\frac {X}{A}}+{\frac {Y}{B}}\right)\left(XY+K^{2}\right)>GK.$

Combination of Governors. If the break of Thomson’s governor is applied to a moveable wheel, as in Jenkin’s governor, and if this wheel works a steam-valve, or a more powerful break, we have to consider the motion of three pieces. Without