Controlling Force Equation and Governor Behavior

In a spring-controlled governor, the controlling force (F) is the net radial force acting on the governor balls, which balances the centrifugal force at the equilibrium speed. This force determines the position of the governor balls for a given speed. The behavior of a spring-controlled governor, specifically its stability, can be analyzed by examining the relationship between this controlling force and the radius of rotation (r) of the governor balls.

Governor Stability Defined

The stability of a governor refers to its ability to maintain a definite speed with a definite radius of rotation for the governor balls.

• Stable Governor: A governor is considered stable if, for every speed within its operating range, there is a specific, definite radius of rotation for the governor balls. This means that as the equilibrium speed increases, the radius of rotation of the balls also increases. For a stable governor, the controlling force (F) must increase with the radius of rotation (r).
• Unstable Governor: A governor is said to be unstable if, for a given speed, there is no definite radius of rotation for the governor balls. In an unstable governor, as the speed increases, the balls tend to fly out to their maximum radius, or if the speed decreases, they tend to fall to their minimum radius in an uncontrolled manner. This occurs when the controlling force (F) decreases as the radius of rotation (r) increases.
• Isochronous Governor: An isochronous governor is a special type of stable governor where the equilibrium speed is constant for all radii of rotation within the working range. This means the governor maintains a constant speed irrespective of the load on the engine (within its operational limits).
• Insensitive Governor: An insensitive governor refers to a governor that requires a relatively large change in speed to produce a noticeable change in the sleeve lift. This is often due to friction in the governor mechanism.

Analyzing the Given Controlling Force Equation

The question provides the controlling force equation for a spring-controlled governor as:

\($ F = p \cdot r + q $\)

where \(F\) is the controlling force, \(r\) is the radius of rotation of the governor balls, and \(p\) and \(q\) are constants.

To determine the stability of the governor from this equation, we need to examine how the controlling force \(F\) changes with the radius of rotation \(r\). This is determined by the derivative of \(F\) with respect to \(r\):

\($ \frac{dF}{dr} = \frac{d}{dr}(p \cdot r + q) $\)

\($ \frac{dF}{dr} = p $\)

The value of the constant \(p\) determines the slope of the controlling force curve when plotted against the radius \(r\).

• If \(p > 0\), then \($ \frac{dF}{dr} $\) is positive. This means that as the radius of rotation \(r\) increases, the controlling force \(F\) also increases. This condition corresponds to a stable governor.
• If \(p < 0\), then \($ \frac{dF}{dr} $\) is negative. This means that as the radius of rotation \(r\) increases, the controlling force \(F\) decreases. This condition leads to an unstable governor. If the controlling force decreases as the balls move outwards, there is less opposing force, causing them to move further out uncontrollably.
• If \(p = 0\), then \($ \frac{dF}{dr} = 0 $\), meaning \(F = q\) (a constant force). This scenario is generally not practical for governor operation as it would not provide the necessary variable resistance for speed control.
• For an isochronous governor, the controlling force \(F\) must be directly proportional to the radius \(r\) such that the equilibrium speed remains constant for all radii. This typically means \(q=0\) and \(p\) has a specific positive value related to the mass of the balls and the desired constant speed.

Conclusion on Governor Type

Given the controlling force equation \($ F = p \cdot r + q $\), the stability of the governor depends entirely on the sign of the constant \(p\).

If the governor is stated to be unstable, it implies that the constant \(p\) in the controlling force equation must be negative (\(p < 0\)). When \(p\) is negative, an increase in the radius of rotation \(r\) results in a decrease in the controlling force \(F\), leading to an uncontrolled outward movement of the governor balls, which is characteristic of an unstable governor.

Therefore, based on the provided correct answer, the governor is unstable, implying that the coefficient \(p\) in the given controlling force equation \($ F = p \cdot r + q $\) is negative.