Page:On Governors.pdf/4

but the motion itself is not liable to disturbances depending on the mutual action of the machine and the governor.

Distinction between Moderators and Governors.

In regulators of the first kind, let $\scriptstyle \ P$ be the driving-power and $\scriptstyle \ R$ the resistance, both estimated as if applied to a given axis of the machine. Let $\scriptstyle \ V$ be the normal velocity, estimated for the same axis, and $\scriptstyle \ dx/dt$ the actual velocity, and let $\scriptstyle \ M$ be the moment of inertia of the whole machine reduced to the given axis. Let the governor be so arranged as to increase the resistance or diminish the driving-power by a quantity $\scriptstyle \ F(dx/dt-V)$, then the equation of motion will be

(1)

${\frac {d}{dt}}\left(M{\frac {dx}{dt}}\right)=P-R-F\left({\frac {dx}{dt}}-V\right)$

When the machine has obtained its final rate the first term vanishes, and

(2)

${\frac {dx}{dt}}=V{\frac {P-R}{F}}$

Hence, if $\scriptstyle \ P$ is increased or $\scriptstyle \ R$ diminished, the velocity will be permanently increased. Regulators of this kind, as Mr Siemens^[1], has observed, should be called moderators rather than governors.

In the second kind of regulator, the force $\scriptstyle \ F(dx/dt-V)$, instead of being applied directly to the machine, is applied to an independent moving piece, $\scriptstyle \ B$, which continually increases the resistance, or diminishes the driving-power, by a quantity depending on the whole motion of $\scriptstyle \ B$.

If $\scriptstyle \ y$ represents the whole motion of $\scriptstyle \ B$, the equation of motion of $\scriptstyle \ B$ is

(3)

${\frac {d}{dt}}\left(B{\frac {dy}{dt}}\right)=F\left({\frac {dx}{dt}}-V\right)$

and that of $\scriptstyle \ M$

(4)

${\frac {d}{dt}}\left(M{\frac {dx}{dt}}\right)=P-R-F\left({\frac {dx}{dt}}-V\right)+G_{y}$

where $\scriptstyle \ G$ is the resistance applied by $\scriptstyle \ B$ when $\scriptstyle \ B$ moves through one unit of space. We can integrate the first of these equations at once, and we find

(5)

$B{\frac {dy}{dt}}=F\left(x-Vt\right)$

so that if the governor $\scriptstyle \ B$ has come to rest $\scriptstyle \ x=Vt$, and not only is the velocity of the machine equal to the normal velocity, but the position of the machine is the same as if no disturbance of the driving-power or resistance had taken place.

Jenkin’s Governor. In a governor of this kind, invented by Mr Fleeming Jenkin, and used in electrical experiments, a centrifugal piece revolves on the principal axis, and is kept always at a constant angle by an appendage which slides on the edge of a loose wheel, $\scriptstyle \ B$, which works on the same axis. The pressure on the edge of this wheel would be proportional to the square of the velocity; but a constant portion

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1. “On Uniform Rotation,” Phil. Trans. 1866, p. 657.