Page:On Governors.pdf/8

8 entering into the calculation of the general equations of motion of these pieces, we may confine ourselves to the case of small disturbances, and write the equations

where $$\scriptstyle\ \theta$$, $$\scriptstyle\ \phi$$, $$\scriptstyle\ \chi$$ are the angles of disturbance of the main shaft, the centrifugal arm, and the moveable wheel respectively, $$\scriptstyle\ A$$, $$\scriptstyle\ B$$, $$\scriptstyle\ C$$ their moments of inertia, $$\scriptstyle\ X$$, $$\scriptstyle\ Y$$, $$\scriptstyle\ Z$$ the viscosity of their connexions, $$\scriptstyle\ K$$ is what was formerly denoted by $$dA/d\phi$$, and $$\scriptstyle\ T$$ and $$\scriptstyle\ J$$ are the powers of Thomson's and Jenkin's breaks respectively.

The resulting equation in $$\scriptstyle\ n$$ is of the form

or

I have not succeeded in determining completely the conditions of stability of the motion from this equation; but I have found two necessary conditions, which are in fact the conditions of stability of the two governors taken separately. If we write the equation

then, in order that the possible parts of all the roots shall be negative, it is necessary that

I am not able to show that these conditions are sufficient. This compound governor has been constructed and used.

Mr C. W. Siemens’s Liquid Governor. Let $$\scriptstyle\ \rho$$ be the density of the fluid, $$\scriptstyle\ k$$ the section of the tube at a point whose distance from the origin measured along the tube is $$\scriptstyle\ s$$, $$\scriptstyle\ r$$, $$\scriptstyle\ \theta$$, $$\scriptstyle\ z$$ the coordinates of this point referred to axes fixed with respect to the tube, $$\scriptstyle\ Q$$ the volume of liquid which passes through any section in unit of time. Also let the following integrals, taken over the whole tube, be

the lower end of the tube being in the axis of motion.