Short Answer:

Forced vibrations are the vibrations that occur in a system when an external periodic force acts on it. These vibrations continue as long as the external force is applied. The amplitude and frequency of vibration depend on the magnitude and frequency of the applied force.

In simple words, forced vibration happens when a machine or structure is made to vibrate by an external source like an engine, motor, or unbalanced rotating part. Unlike free vibrations, forced vibrations depend on the external excitation and stop when the external force is removed.

Detailed Explanation :

Forced Vibrations

Forced vibrations are those vibrations that occur when a system is subjected to an external time-varying force. In this condition, the system does not vibrate at its natural frequency but at the frequency of the external force. The external force supplies energy to the system continuously, keeping it in motion.

These vibrations are very common in real-world mechanical and structural systems such as engines, turbines, vehicles, and machines. For example, when an unbalanced motor runs, it applies a periodic force on the structure it is mounted on, causing forced vibrations.

If the frequency of the external force approaches the natural frequency of the system, the amplitude of vibration increases sharply, leading to a condition known as resonance. Therefore, understanding forced vibration is essential to prevent excessive vibration and damage in engineering systems.

Principle of Forced Vibrations

When a mechanical system such as a spring–mass system is subjected to a periodic external force, the motion of the system depends on both the applied force and the damping present in the system. The equation of motion for a damped forced vibration system is given by:

Where:

= mass of the system
= damping coefficient
= stiffness of the system
= external periodic force
= frequency of external excitation

The solution of this equation shows that the total motion of the system consists of two parts:

Transient part (natural vibration): It occurs initially and dies out with time due to damping.
Steady-state part (forced vibration): It continues as long as the external force is applied and occurs at the frequency of the applied force.

Thus, the system finally vibrates at the frequency of the external force, not its own natural frequency.

Amplitude of Forced Vibrations

The amplitude of steady-state forced vibration is given by the expression:

Where:

= amplitude of vibration
= amplitude of external force
= stiffness of the system
= damping ratio
= frequency ratio (ratio of forcing frequency to natural frequency)

From this relation:

When is small (forcing frequency much less than natural frequency), the amplitude is small.
When is near 1 (forcing frequency ≈ natural frequency), the amplitude increases sharply — this is resonance.
When is large (forcing frequency much greater than natural frequency), the amplitude again becomes small.

Thus, the system’s response depends greatly on the ratio between external and natural frequencies and the damping present.

Effect of Damping in Forced Vibrations

Damping has a significant influence on forced vibration behavior:

At resonance: Damping reduces the maximum amplitude, preventing damage to the system.
Beyond resonance: Damping has little effect; amplitude remains small.
Low damping: The resonance peak is sharp and high.
High damping: The resonance peak becomes flatter and broader.

Hence, damping is necessary to control vibration amplitude during resonance and to maintain system stability.

Examples of Forced Vibrations

Engines and Motors:
Rotating unbalanced masses in engines or motors apply periodic forces, causing vibrations in machine frames or structures.
Vehicles:
The vibrations in automobiles caused by irregularities on the road surface are examples of forced vibrations.
Machinery Foundations:
Machines like compressors or pumps produce dynamic forces that create forced vibrations in their supporting structures.
Bridges and Buildings:
Wind loads, moving vehicles, or earthquakes apply external periodic forces that induce forced vibrations in structures.
Household Appliances:
Washing machines or fans with unbalanced loads cause forced vibration due to periodic external motion.

Advantages of Controlled Forced Vibrations

Although excessive forced vibrations are undesirable, controlled forced vibrations can be useful in several applications:

Vibration testing of materials and components: Used to determine natural frequencies and damping characteristics.
Industrial vibration equipment: Vibrating screens and compactors work based on forced vibrations.
Sound and musical instruments: Many instruments produce sound due to forced vibrations.

Effects of Excessive Forced Vibrations

If forced vibrations are not properly controlled, they can lead to serious problems:

Structural Damage: High amplitudes can cause fatigue and cracks in components.
Noise Generation: Continuous forced vibrations produce unwanted noise.
Reduced Machine Life: Repeated vibrations shorten component lifespan.
Loss of Efficiency: Vibration causes energy loss and unstable operation.
Resonance Failure: At resonance, the amplitude may become so large that it leads to catastrophic failure.

Hence, in mechanical design, damping devices and isolators are used to minimize the harmful effects of forced vibrations.

Control of Forced Vibrations

To prevent damage from forced vibrations, the following measures are commonly used:

Use of damping materials: Rubber mounts, viscous dampers, and oil dashpots absorb vibration energy.
Balancing rotating parts: Reduces unbalanced forces that cause vibrations.
Vibration isolators: Springs and shock absorbers prevent transmission of vibrations.
Proper design of natural frequency: Ensures that the natural frequency of the system is far from the operating frequency to avoid resonance.

Conclusion

In conclusion, forced vibrations are vibrations that occur when a mechanical system is subjected to an external periodic force. The system vibrates at the frequency of the applied force, not its natural frequency. The amplitude of these vibrations depends on the magnitude of the external force, the difference between the forcing and natural frequencies, and the amount of damping present. Forced vibrations are common in all mechanical systems and must be controlled using proper design and damping to prevent resonance and ensure smooth, safe operation.