Short Answer:

The controlling force is the force that acts on the governor balls to balance the centrifugal force during engine operation. It is directed towards the axis of rotation and helps to maintain the stable position of the balls at a particular speed.

In simple words, the controlling force ensures that the governor balls neither move outward nor inward unnecessarily. It maintains the equilibrium of the governor mechanism, allowing it to function properly and control the engine speed effectively under different load conditions.

Detailed Explanation :

Controlling Force

The controlling force in a governor is one of the most important factors that determine its stability and performance. It is defined as the radial inward force which acts on the governor balls to keep them in equilibrium for a particular speed of rotation. In other words, when the governor balls rotate along with the spindle, centrifugal force acts outward, while the controlling force acts inward toward the axis of rotation. The balance between these two forces decides the working position of the governor balls and the engine’s mean speed.

The controlling force is provided either by the gravity of the balls (in simple governors like Watt’s governor) or by springs (in spring-controlled governors like Hartnell or Porter governors). The correct balance between the controlling force and the centrifugal force ensures smooth and accurate control of engine speed.

1. Meaning and Role of Controlling Force:
   When a governor operates, the engine shaft rotates along with the governor spindle, and the attached balls move in a circular path. The centrifugal force acting on the balls tends to move them outward. To keep the balls in a stable position, an inward force known as the controlling force is required. This controlling force resists the outward movement of the balls and maintains the equilibrium of the governor mechanism.

The role of the controlling force is to balance the centrifugal force at a particular radius of rotation, ensuring that the governor works accurately and smoothly. If the controlling force becomes too high or too low, the governor becomes unstable, resulting in irregular speed control.

2. Mathematical Expression for Controlling Force:
   Let the mass of each ball be m, the radius of rotation be r, and the angular velocity of the balls be ω(in radians per second).
   The centrifugal force acting outward on each ball is given by:

For the governor to be in equilibrium, the controlling force (F) acting inward must be equal to the centrifugal force (F_c).

Hence,

This relation shows that the controlling force depends directly on the speed of rotation and the radius of the governor. As speed increases, the controlling force also increases to maintain equilibrium.

3. Sources of Controlling Force:
   The controlling force in different governors is provided by various mechanisms:

• Gravity-controlled governors:
  In this type, such as the Watt governor, the controlling force is provided by the weight of the balls and links acting under gravity. The downward force helps to balance the centrifugal force when the governor balls rise.
• Spring-controlled governors:
  In governors like the Hartnell and Wilson-Hartnell types, the controlling force is provided by one or more helical springs. These springs compress or extend depending on the movement of the balls. The spring tension provides the necessary controlling force, which can be easily adjusted to obtain the required governor performance.

4. Characteristics of Controlling Force:
   The behavior of a governor can be studied using a controlling force diagram, which shows how the controlling force varies with the radius of rotation. The characteristics can be explained as follows:

• Stable Governor:
  When the controlling force increases with an increase in radius, the governor is stable. It means that for a higher radius, the speed must also increase to balance the higher controlling force.
• Unstable Governor:
  When the controlling force decreases with an increase in radius, the governor becomes unstable. In this case, the governor cannot maintain equilibrium and results in irregular speed control.
• Isochronous Governor:
  When the controlling force remains constant for all radii, the governor is said to be isochronous, meaning it maintains the same speed for all positions of the balls. However, this condition is ideal and not practical for most mechanical systems.

5. Importance of Controlling Force:
   The controlling force plays a vital role in ensuring the proper functioning of a governor. Its importance can be summarized as follows:

• It maintains the equilibrium of the governor mechanism.
• It determines the stability and sensitivity of the governor.
• It controls the range of speed variation for the engine.
• It ensures smooth adjustment of fuel supply under different load conditions.
• It affects the overall performance and efficiency of the governor system.

Without a proper controlling force, the governor would not be able to maintain engine speed accurately, leading to fluctuations and inefficient operation.

6. Factors Affecting Controlling Force:
   Several factors influence the magnitude and behavior of the controlling force:

• Mass of the governor balls
• Radius of rotation
• Type and stiffness of the spring (in spring-controlled governors)
• Linkage geometry
• Friction between moving parts
  These factors must be designed carefully to achieve the desired governor performance and maintain engine stability.

7. Relation Between Controlling Force and Speed:
   The controlling force determines the equilibrium speed of the governor. As speed increases, the centrifugal force increases, and the controlling force must rise proportionally to maintain stability. The relation between these forces helps engineers design governors that provide stable speed control under various operating conditions.

Conclusion:

The controlling force is the inward-acting force that balances the centrifugal force in a governor. It ensures that the governor balls stay in a stable position and the engine speed remains nearly constant. It can be produced by gravity or springs, depending on the type of governor. A proper controlling force is essential for achieving smooth, accurate, and efficient speed regulation in engines, making it a key element in the design and performance of all governors.