Short Answer:

Forced vibration is the type of vibration that occurs when a body or mechanical system is subjected to a continuous external periodic force. In this case, the body vibrates with the frequency of the external force, not at its natural frequency.

In simple words, when an external force repeatedly acts on a body — such as an unbalanced engine part or rotating machinery — it keeps the body vibrating. Examples include the vibration of vehicles due to road irregularities, machine parts due to rotating imbalance, or buildings during earthquakes.

Detailed Explanation :

Forced Vibration

In the study of mechanical vibrations, forced vibration refers to the motion of a system that is continuously driven by an external time-dependent force. Unlike free vibration (which occurs due to an initial disturbance only), forced vibration continues as long as the external force is applied.

This kind of vibration is very common in machines, engines, and structures because they are constantly exposed to external forces such as unbalanced loads, wind pressure, or ground motion. The response of a system to such a force depends on the frequency of the applied force and the natural frequency of the system.

When both frequencies match, a phenomenon known as resonance occurs, causing extremely large vibrations that may damage or destroy the system.

Definition

Forced vibration can be defined as:

“The type of vibration in which a body or system vibrates under the continuous influence of an external periodic force is known as forced vibration.”

In this vibration, the system’s motion depends on both the magnitude and frequency of the applied force. The external force may be periodic (such as a rotating imbalance) or non-periodic (such as an earthquake).

Explanation of Forced Vibration

When a body is subjected to an external force that varies with time, it starts moving according to the nature of that force. The external force continuously supplies energy to the system, which maintains the vibration.

For example, consider a spring-mass system where a periodic external force acts on the mass. The system moves up and down following the applied force. If there is damping, part of the energy supplied by the force is used to overcome resistance, while the rest maintains motion.

If the frequency of the external force is low, the body follows the motion of the force easily. As the frequency increases, the amplitude of vibration also changes. When the forcing frequency becomes equal to the system’s natural frequency, resonance occurs — the amplitude increases sharply and may cause mechanical failure.

Mathematical Representation

Consider a mass-spring-damper system subjected to an external harmonic force:

Let,

= mass of the body
= damping coefficient
= stiffness of the spring
= displacement
= external periodic force

The equation of motion is given by:

where,

= angular frequency of the external force (in rad/s),
= amplitude of the applied force.

The solution of this equation has two parts:

Transient part – Represents the free vibration that dies out with time.
Steady-state part – Represents the forced vibration that remains as long as the external force acts.

The steady-state vibration occurs with the same frequency as the external force , not the natural frequency of the system.

Characteristics of Forced Vibration

External Force:
The vibration is maintained by a continuously acting periodic or non-periodic external force.
Vibration Frequency:
The system vibrates with the frequency of the applied force, not its natural frequency.
Amplitude Variation:
The amplitude of vibration depends on the relation between the forcing frequency and the natural frequency.
Resonance Phenomenon:
When the forcing frequency equals the natural frequency, the amplitude becomes maximum, leading to resonance.
Damping Effect:
Damping reduces the amplitude and prevents damage during resonance.

Resonance in Forced Vibration

Resonance occurs when the frequency of the external force () equals the natural frequency () of the system.

At this point:

The amplitude of vibration increases rapidly because the energy supplied by the external force adds up at the same rate as the system’s own vibration.

This can cause excessive vibrations and mechanical failure if not controlled. Examples include the collapse of bridges, vibration of aircraft wings, or breaking of machinery parts.

Examples of Forced Vibration

Vehicle Suspensions:
The road surface continuously applies varying forces on the wheels, causing forced vibrations in the vehicle body.
Engines and Rotating Machinery:
Unbalanced rotating components generate periodic forces, producing forced vibrations in the frame or foundation.
Buildings During Earthquakes:
Ground motion applies external forces on the structure, causing forced vibrations.
Bridge Structures:
Forces due to moving vehicles or wind loads produce forced vibrations.
Loudspeakers:
The diaphragm vibrates due to electrical signals acting as external periodic forces.

Effect of Damping in Forced Vibration

Damping plays a crucial role in controlling forced vibration.

Without damping: Amplitude becomes very large at resonance, which can cause damage.
With damping: Amplitude remains within safe limits, and resonance effects are reduced.

Thus, damping is intentionally introduced in machines (such as shock absorbers and vibration isolators) to reduce harmful vibrations.

Applications and Importance

Machine Design:
Helps engineers design systems that can withstand continuous external loads safely.
Automotive Engineering:
Used in suspension systems and vibration isolators for smooth rides.
Structural Engineering:
Important for designing earthquake-resistant buildings and bridges.
Rotating Equipment:
Used in balancing and tuning shafts and rotors to prevent excessive vibration.
Aerospace Engineering:
Used in reducing vibration in aircraft wings and engines to ensure safety and comfort.

Vibration Control Methods

Balancing Rotating Parts: To minimize unbalanced forces.
Damping: Using dampers or shock absorbers to reduce amplitude.
Isolation: Mounting machines on rubber or spring bases to isolate vibration.
Tuning: Adjusting stiffness or mass to shift natural frequency away from operating frequency.
Maintenance: Regular inspection to prevent loose or damaged components.

Conclusion

Forced vibration is the continuous oscillation of a body under the influence of an external periodic force. It differs from free vibration because it requires continuous excitation. The system vibrates at the frequency of the applied force, not its natural frequency. When the external frequency matches the natural frequency, resonance occurs, leading to large amplitude vibrations that can be dangerous. Understanding forced vibration is essential in mechanical engineering for designing safe, reliable, and efficient machines and structures, as well as for preventing failures caused by excessive vibration.