Short Answer:

Forced vibration is a type of vibration that occurs when an external periodic force acts on a body. In this case, the object vibrates due to the influence of an external source, such as an engine, unbalanced rotating machine, or other mechanical system. The amplitude of vibration depends on the frequency and magnitude of the external force.

When the frequency of the external force is close to the natural frequency of the system, resonance occurs, leading to large vibrations. Forced vibration is important in engineering to design systems that can safely handle such external forces without causing damage or failure.

Detailed Explanation:

Forced Vibration

Forced vibration occurs when an external time-varying or periodic force acts on a mechanical system and causes it to vibrate. Unlike free vibration, where the body vibrates due to its own natural frequency after being disturbed, forced vibration is continuously maintained by an external force. The external source could be a motor, unbalanced rotor, engine, or any machine that applies a repeating force on the system.

In forced vibration, the system vibrates with the frequency of the applied external force, not with its natural frequency. However, if the frequency of the external force becomes equal to or very close to the natural frequency of the system, the amplitude of vibration increases greatly. This condition is known as resonance, which can cause severe damage if not controlled properly.

Explanation of Working

When a mechanical system such as a spring–mass–damper arrangement is subjected to an external force that varies with time, the system starts to oscillate. The external force provides continuous energy to the system, making it vibrate in synchronization with that force.
Mathematically, the motion of forced vibration can be represented by the differential equation:

where:

= mass of the body,
= damping coefficient,
= stiffness of the spring,
= amplitude of the external force,
= frequency of the external force.

The total response of the system consists of two parts:

Transient (free vibration) part – which dies out over time due to damping.
Steady-state (forced vibration) part – which continues as long as the external force acts.

Thus, after a certain time, the vibration of the system becomes purely steady-state and vibrates only at the frequency of the external force.

Effect of Damping

Damping plays a very important role in controlling forced vibration. When damping is present, it reduces the amplitude of vibration and prevents the system from reaching excessively high values at resonance. In practical systems, some level of damping is always present, which helps in maintaining stability and protecting the structure.

If damping is small, the amplitude at resonance can become very high. However, with increased damping, the resonance peak becomes smaller and broader. Engineers often design systems with proper damping to ensure that resonance effects are minimized.

Resonance Condition

Resonance is a special case of forced vibration. It occurs when the frequency of the external force becomes equal to the natural frequency of the system. Under this condition, even a small external force can produce large amplitude vibrations, leading to failure of mechanical parts or structures.

To avoid resonance, engineers either:

Change the natural frequency of the system by modifying mass or stiffness, or
Introduce damping to limit the amplitude of vibration.

Examples of resonance can be seen in buildings, bridges, turbines, and engines when they are subjected to periodic forces at critical frequencies.

Practical Examples of Forced Vibration

The vibration of a car body due to an unbalanced engine or rough road.
The vibration of a washing machine during the spin cycle.
Vibrations in bridges due to continuous vehicle movement.
Vibrations in rotating machinery caused by unbalanced parts or external loads.

All these are examples where an external force continuously acts and causes the structure or body to vibrate with that frequency.

Importance in Engineering

Forced vibration analysis is essential in mechanical and structural engineering. By studying how systems respond to external forces, engineers can design components that minimize vibration and noise, improve performance, and prevent failure. This knowledge is especially important in rotating machines, engines, turbines, vehicles, and building structures.

Vibration isolators, dampers, and tuned mass absorbers are commonly used to control forced vibrations and ensure smooth operation.

Conclusion

In summary, forced vibration is the vibration of a system under the continuous action of an external force. The system vibrates with the frequency of that force, and if this frequency matches the natural frequency, resonance occurs, leading to large and dangerous amplitudes. Understanding forced vibration helps engineers design machines and structures that are safe, stable, and efficient under dynamic loading conditions.