Short Answer:

Dynamic equilibrium is a condition in which all the forces and moments acting on a moving body are balanced, even though the body is in motion. It means the resultant force and resultant torque on the body are zero, so the body moves with constant velocity or uniform rotation.

In simple terms, dynamic equilibrium occurs when the acceleration of a moving object becomes zero because all external forces, including inertia forces, are equal and opposite. This principle is widely used in the dynamic analysis of machines, where both real and inertia forces are considered to maintain equilibrium during motion.

Detailed Explanation :

Dynamic Equilibrium

Dynamic equilibrium is an important concept in mechanics and dynamics, describing the state of a moving body when the sum of all the forces and moments acting on it equals zero. Unlike static equilibrium, where the body remains at rest, in dynamic equilibrium, the body continues to move with constant velocity or angular velocity. This occurs when the applied forces are exactly balanced by the inertia forces generated due to the mass and motion of the body.

According to Newton’s second law, the net force acting on a body is equal to the product of its mass and acceleration:

If we consider the inertia force, which is an imaginary force equal in magnitude but opposite in direction to the product , the equation can be rewritten as:

This shows that when the real external forces and inertia forces are in balance, the body is said to be in dynamic equilibrium.

In other words, dynamic equilibrium is a state of balance between the applied forces and the inertial effects of motion. It allows engineers to analyze moving systems as if they were in a condition of rest by including the inertia forces in the analysis. This principle is very useful in the dynamic analysis of mechanisms, such as engines, linkages, and rotating machines.

Principle of Dynamic Equilibrium

The principle of dynamic equilibrium states that “a body is said to be in dynamic equilibrium when the sum of all external forces and the inertia force acting on it is zero.”
Mathematically,

where  is the inertia force.

Similarly, for rotational motion,

where  is the inertia torque,  is the mass moment of inertia, and  is the angular acceleration.

This principle converts a dynamic problem into a static problem, making it easier to analyze. Engineers use this concept to calculate forces in moving parts like pistons, crankshafts, and connecting rods of engines, where acceleration and deceleration take place continuously.

Examples of Dynamic Equilibrium

1. Vehicle Moving at Constant Speed:
   When a car moves on a straight road at uniform velocity, the driving force produced by the engine equals the resistive forces such as air resistance and friction. The net force becomes zero, and the car moves in dynamic equilibrium.
2. Aircraft Flying at Constant Altitude:
   During level flight, the lift force generated by the wings balances the weight of the aircraft, and the thrust produced by the engines equals the drag. Hence, the aircraft remains in dynamic equilibrium.
3. Rotating Flywheel at Constant Speed:
   When a flywheel rotates at a constant angular speed, the torque produced by the engine balances the resisting torque due to friction and load. There is no angular acceleration, which means the flywheel is in dynamic equilibrium.
4. Crank and Connecting Rod Mechanism:
   In a reciprocating engine, when all the forces acting on the piston, connecting rod, and crank are balanced by the inertia forces, the moving system is said to be in dynamic equilibrium. This helps maintain smooth motion without vibration.

Difference Between Static and Dynamic Equilibrium

• Static Equilibrium: The body remains at rest, and all forces and moments are balanced. Example: A book resting on a table.
• Dynamic Equilibrium: The body is in motion with constant velocity, and the sum of forces and moments is still zero. Example: A car moving at constant speed on a straight road.

The key difference is that in static equilibrium there is no motion, whereas in dynamic equilibrium motion exists but without acceleration.

Importance in Engineering

Dynamic equilibrium plays a major role in mechanical and structural design. Some of its main applications are:

1. Machine Dynamics: Used to calculate the forces in moving parts of mechanisms like engines and turbines.
2. Vehicle Stability: Helps in designing systems that maintain steady motion and balance forces during acceleration or braking.
3. Vibration Control: Helps predict and balance forces that cause oscillations in machinery.
4. Balancing of Rotating Masses: Used to ensure smooth operation of rotating machinery without excessive vibration.
5. Dynamic Analysis of Structures: Helps engineers study how structures respond to moving loads, such as bridges under traffic.

Mathematical Expression

For linear motion,

For rotational motion,

Here,  and  are the sums of all external forces and torques respectively,  is the inertia force, and  is the inertia torque.

When these conditions are satisfied, the system is in dynamic equilibrium.

Conclusion:

Dynamic equilibrium is a condition of balanced motion where the sum of all forces and torques, including inertia effects, is zero. Even though the body is moving, there is no acceleration. This principle is widely used in the study of machine dynamics to simplify the analysis of moving systems. By considering inertia forces, engineers can treat dynamic problems as static ones, ensuring smooth and stable operation of machines and mechanical structures.