Page:On Governors.pdf/4

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

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

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

Hence, if $$\scriptstyle\ P$$ is increased or $$\scriptstyle\ R$$ diminished, the velocity will be permanently increased. Regulators of this kind, as Mr Siemens, 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

and that of $$\scriptstyle\ M$$

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

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