Short Answer:

Forced vibration is the type of vibration that occurs when a system is continuously acted upon by an external periodic force. In this case, the system vibrates with the frequency of the applied force rather than its natural frequency. The amplitude of vibration depends on the magnitude and frequency of the external force.

Forced vibration is very common in machines where external forces such as unbalanced loads, engine operation, or rotating parts cause continuous vibration. If the frequency of the applied force matches the system’s natural frequency, resonance occurs, leading to large and sometimes dangerous vibrations.

Detailed Explanation :

Forced Vibration

Forced vibration occurs when a mechanical system is subjected to a continuous external force that makes it vibrate. The vibration does not occur naturally but is maintained by the external force applied to the system. The frequency of vibration in this case is equal to the frequency of the external force, not the natural frequency of the system.

This type of vibration is commonly seen in mechanical systems like engines, compressors, and turbines where moving or rotating parts produce repetitive or periodic forces. The system vibrates as long as the external force acts upon it, and the amplitude of vibration depends on how the external frequency relates to the system’s natural frequency.

Nature of Forced Vibration

In forced vibration, the external force supplies energy continuously to keep the system in motion. The amplitude of vibration depends on three main factors:

The frequency of the external force.
The natural frequency of the system.
The amount of damping present in the system.

If there is damping, part of the supplied energy is lost due to resistance or friction, reducing the vibration amplitude. If damping is absent, vibrations may increase, especially near resonance conditions.

Examples of Forced Vibration

The vibration of an engine body due to continuous firing of cylinders.
A washing machine vibrating due to unbalanced rotating parts.
A building shaking due to ground motion during an earthquake.
The vibration of a bridge when vehicles pass over it.

In all these examples, vibration continues as long as the external force (engine firing, unbalanced load, or moving vehicle) acts on the system.

Mathematical Representation

The forced vibration of a single degree of freedom system can be represented by the equation:

Where:

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

The general solution for the vibration consists of two parts:

Transient Part: It represents the free vibration, which dies out after some time.
Steady-State Part: It represents the forced vibration, which continues as long as the external force is applied.

After some time, the transient vibration disappears, and only the steady-state vibration remains in the system.

Resonance in Forced Vibration

Resonance is an important concept related to forced vibration. It occurs when the frequency of the external force () becomes equal to the natural frequency () of the system. At this point, the amplitude of vibration becomes extremely large, which may lead to mechanical failure or damage.

Effects of Resonance:

Produces maximum amplitude of vibration.
Causes excessive noise and wear in machines.
Can lead to the failure of parts or structures due to fatigue.

To avoid resonance, engineers ensure that the operating frequency of a machine is well below or above its natural frequency.

Role of Damping in Forced Vibration

Damping plays a very important role in controlling forced vibrations. Damping reduces the amplitude of vibration and prevents resonance from becoming dangerous.

When damping is increased:

The peak amplitude at resonance decreases.
The vibration energy is converted into heat and dissipated.
The system becomes more stable and quieter.

For this reason, damping materials such as rubber mounts, shock absorbers, and springs are commonly used in machines and vehicles.

Practical Importance of Forced Vibration

Forced vibrations are very common in everyday mechanical systems. Engineers study and analyze them to ensure safety and reliability. Some important applications and considerations include:

Machine Design: Designing machines to avoid resonance and control vibration.
Vehicle Systems: Using dampers and suspension systems to reduce road-induced vibration.
Structural Design: Ensuring buildings and bridges can withstand vibrations from wind or earthquakes.
Vibration Isolation: Using mounts and pads to isolate vibrations from sensitive instruments or components.

Understanding forced vibration allows engineers to predict vibration behavior, improve comfort, reduce noise, and increase the lifespan of mechanical components.

Energy Behavior in Forced Vibration

In forced vibration, energy is continuously supplied by the external force. Some of this energy is converted into kinetic and potential energy within the system, while the rest is dissipated as heat due to damping. The system reaches a steady-state condition where the energy supplied per cycle equals the energy dissipated per cycle.

Conclusion

Forced vibration occurs when an external periodic force continuously acts on a mechanical system, causing it to vibrate at the frequency of that external force. The amplitude of the vibration depends on the strength of the force, the damping in the system, and the difference between the external and natural frequencies. If both frequencies match, resonance occurs, which can be harmful. Hence, understanding and controlling forced vibration is essential for safe and efficient machine and structural design.