Short Answer:

Forced vibrations are the vibrations that occur when a system is subjected to a continuous external periodic force. In this type of vibration, the system does not vibrate at its natural frequency but at the frequency of the external force applied to it.

In simple words, when an external force such as an engine, motor, or unbalanced mass acts on a body, it causes the body to vibrate continuously. These vibrations remain as long as the external force is acting. Examples include vibrations of machines, engines, and vehicles during operation. If the forcing frequency equals the natural frequency, resonance occurs, leading to large vibrations.

Detailed Explanation :

Forced Vibrations

Forced vibrations occur when a system is made to vibrate by an external time-dependent force. Unlike free vibrations, where a system vibrates on its own after being disturbed, in forced vibrations the motion is maintained by an external agency. The system responds to this external force and vibrates at the same frequency as the applied force, not its natural frequency.

These vibrations are very common in mechanical and structural systems. For example, the vibration of a vehicle due to an uneven road, the oscillations of a washing machine during spinning, or the shaking of a building during an earthquake are all examples of forced vibrations. The amplitude and behavior of these vibrations depend on the relationship between the external frequency and the system’s natural frequency.

Mathematical Representation

Let:

• m = mass of the system (kg)
• c = damping coefficient
• k = stiffness of the system (N/m)
• x = displacement at any time t (m)
• F = amplitude of the external periodic force (N)
• ω = angular frequency of external force (rad/s)

The equation of motion for a damped forced vibration is given by:

Here,

• The left-hand side represents the resisting forces due to inertia, damping, and stiffness.
• The right-hand side represents the applied periodic external force.

The steady-state solution of this equation gives the amplitude of forced vibration as:

where,

• X = amplitude of forced vibration,
• r = ω / ωₙ = frequency ratio (ratio of external to natural frequency),
• ζ = c / c_c = damping ratio (ratio of actual damping to critical damping).

This equation shows that the amplitude of forced vibration depends on the forcing frequency, damping, and stiffness of the system.

Nature of Forced Vibrations

1. Dependence on External Force:
   The motion continues only as long as the external periodic force is applied. Once the force is removed, the vibration gradually stops.
2. Vibration Frequency:
   The vibration frequency is equal to the frequency of the external force, not the natural frequency of the system.
3. Amplitude Variation:
   The amplitude of forced vibration changes depending on how close the forcing frequency is to the natural frequency.
4. Effect of Damping:
   The presence of damping reduces the vibration amplitude and prevents resonance.

Resonance in Forced Vibrations

A special and important condition in forced vibration is resonance. It occurs when the frequency of the external periodic force becomes equal to the system’s natural frequency (ω = ωₙ).

At resonance:

• The amplitude of vibration becomes extremely large.
• The system can experience severe vibration and even mechanical failure if damping is not sufficient.

Example: When a vehicle engine runs at a speed matching the natural frequency of its frame, strong vibrations and noise are produced. Similarly, bridges and buildings can experience dangerous vibrations during earthquakes if resonance occurs.

Examples of Forced Vibrations

1. Vehicle on Rough Road:
   The irregular road surface applies periodic forces on the vehicle suspension, causing continuous vibrations.
2. Machine on Foundation:
   A motor or compressor produces unbalanced forces during rotation, which cause the foundation or floor to vibrate at the forcing frequency.
3. Bridge Vibrations:
   Continuous traffic or wind pressure applies periodic forces, leading to forced vibration of bridge structures.
4. Aircraft and Ships:
   Engines, propellers, and aerodynamic forces cause forced vibrations in the structure during operation.
5. Electrical Machines:
   Alternating magnetic forces in electric motors and transformers can cause forced mechanical vibrations in their components.

Importance of Forced Vibration Study

1. Avoiding Resonance:
   Understanding forced vibration helps engineers design systems so that the natural frequency does not coincide with the external forcing frequency, preventing excessive vibration.
2. Machine Design:
   Helps determine suitable damping and stiffness values to minimize vibration in rotating and reciprocating machines.
3. Structural Safety:
   Ensures that buildings, bridges, and towers can safely withstand wind, traffic, or seismic forces without resonating.
4. Comfort and Performance:
   In vehicles and aircraft, forced vibration analysis improves passenger comfort and reduces noise and fatigue.
5. Maintenance and Diagnosis:
   Engineers use vibration measurements to identify unbalanced or defective components in machinery during operation.

Effect of Damping in Forced Vibrations

Damping plays a vital role in controlling forced vibrations.

• In an undamped system, resonance causes very large amplitudes that may lead to failure.
• In a damped system, the amplitude at resonance is limited because damping absorbs part of the vibration energy.

Thus, properly designed damping devices such as shock absorbers, dampers, and vibration isolators are used in machines, vehicles, and structures to reduce forced vibration effects.

Energy Transfer in Forced Vibrations

During forced vibration, energy is continuously supplied to the system by the external source. Part of this energy is stored as potential energy, part as kinetic energy, and the rest is lost due to damping. At steady-state vibration, the energy supplied by the external force per cycle is equal to the energy dissipated due to damping per cycle, resulting in constant vibration amplitude.

This energy balance explains why forced vibrations continue as long as the external excitation remains active.

Conclusion

Forced vibrations occur when an external periodic force acts continuously on a system, causing it to vibrate at the same frequency as the applied force. The amplitude of vibration depends on the forcing frequency, natural frequency, and damping present in the system. When the forcing frequency equals the natural frequency, resonance occurs, leading to large vibrations that can cause damage. Hence, understanding and controlling forced vibrations are essential for designing stable, safe, and efficient mechanical systems.