When a body rotates in a circular path, there is an inward radial force or centripetal force acting on it. In case of a governor running at a steady speed, the inward force acting on the rotating balls is known as controlling force. It is equal and opposite of the centrifugal reaction.

Controlling force, $F_c = m.w^2.r$

The controlling force is provided by the weight of the sleeve and balls as in porter governor and by the spring and weight as in Hartnell governor (or spring controlled governor)

When the graph between the controlling force (FC) as ordinate and radius of rotation of the balls ( r ) as abscissa is drawn, then the graph obtained is known as controlling force diagram. This diagram enables the stability and sensitiveness of the governor to be examined and also shows clearly the effect of friction.

Controlling force diagram for porter governor.

The controlling force diagram for a porter governor is a curve as shown in figure (a) we know that controlling force,

$F_c = m.w^2.r = m(\frac{ 2 \pi N}{60})^2 r$

$N^2 = \frac{1}{m} (\frac{60}{2 \pi})^2 (\frac{F_c}{r}) = \frac{1}{m} (\frac{60}{2 \pi})^2 (tan \ \phi)$

$N = \frac{60}{2 \pi} (\frac{tan \ \phi}{m})^{1/2}$ ...... ....... [$\because \frac{F_c}{r} = tan \ \phi]$

Where $\phi$ is the angle between the axis of radius of rotation and a line joining a given point (say A) on the curve to the origin O.

Controlling force diagram for spring – controlled governor.

The Controlling force diagram for spring controlled governor is a straight line, as shown in figure. We know that controlling force,

$F_C = m.w^2.r$ or $F_c/r= m.w^2$

The following points, for the stability of spring controlled governors, may be noted:

1. For the governor to be stable, the controlling force (FC) must increase as the radius of rotation (r) increases, i.e. FC/r must increase as r increases. Hence the controlling force line AB when produced must intersect the controlling force axis below the origin, as shown in figure (b).

The relation between the controlling force ($F_c$) and the radius of rotation (r) for the stability of spring controlled governors is given by the following equation:

$F_c = a.r - b$ ……………………(1)

Where a and b are constants.

2. The value of b in equation (1) may be made either zero or positive by increasing the initial version of the spring. If b is zero, the controlling force line CD passed through the origin and the governor becomes isochronous because FC/r will remain constant for all radii of rotation.

The relation between the controlling force and the radius of rotation, for an isochronous governor is, therefore,

$F_c = a.r$ ……….(2)

3. If b is greater than zero or positive, then FC/r decreases as r increases, so that the equilibrium speed of the governor decreases with an increase of the radius of rotation of balls, which is impracticable, such a governor is said to be unstable and the relation between the controlling force and the radius of rotation is, therefore

$F_c = a.r + b$