Page:On Governors.pdf/5

Rh of this pressure is taken off by a spring which acts on the centrifugal piece. The force acting on $$\scriptstyle\ B$$ to turn it round is therefore

and if we remember that the velocity varies within very narrow limits, we may write the expression

where $$\scriptstyle\ F$$ is a new constant, and $$\scriptstyle\ V_1$$ is the lowest limit of velocity within which the governor will act.

Since this force necessarily acts on $$\scriptstyle\ B$$ in the positive direction, and since it is necessary that the break should be taken off as well as put on, a weight $$\scriptstyle\ W$$ is applied to $$\scriptstyle\ B$$ tending to turn it in the negative direction; and, for a reason to be afterwards explained, this weight is made to hang in a viscous liquid, so as to bring it to rest quickly.

The equation of motion of $$\scriptstyle\ B$$ may then be written

where $$\scriptstyle\ Y$$ is a coefficient depending on the viscosity of the liquid and on other resistances varying with the velocity, and $$\scriptstyle\ W$$ is the constant weight. Integrating this equation with respect to $$\scriptstyle\ t$$, we find

If $$\scriptstyle\ B$$ has come to rest, we have

or the position of the machine is affected by that of the governor, but the final velocity is constant, and

where $$\scriptstyle\ V_1$$ is the normal velocity.

The equation of motion of the machine itself is

This must be combined with equation (7) to determine the motion of the whole apparatus. The solution is of the form

where $$\scriptstyle\ n_1,\ n_2,\ n_3$$ are the roots of the cubic equation

If $$\scriptstyle\ n$$ be a pair of roots of this equation of the form $$\scriptstyle\ a\pm \sqrt{-1}b$$, then the part of $$\scriptstyle\ x$$ corresponding to these roots will be of the form