Short Answer:

Forced vibration is a type of vibration that occurs when an external force or periodic excitation acts continuously on a mechanical system. In this condition, the system vibrates with the frequency of the applied force instead of its natural frequency. Examples include the vibration of machinery caused by unbalanced rotating parts or engines mounted on vehicles.

In forced vibration, energy is continuously supplied by the external force to maintain the vibration. When the frequency of the external force matches the system’s natural frequency, a condition called resonance occurs, leading to large amplitude vibrations that can cause damage if not controlled.

Detailed Explanation :

Forced Vibration

Forced vibration occurs when an external periodic force acts on a system and continuously drives it to vibrate. In this case, the motion of the system depends on the magnitude, frequency, and direction of the applied external force. The system’s vibration is not due to its own stored energy but because of the continuous energy supplied from outside sources.

This type of vibration is commonly observed in mechanical systems like engines, motors, or any rotating machinery where unbalanced forces act repeatedly. Unlike free vibration, where the system oscillates naturally after an initial disturbance, forced vibration continues as long as the external excitation is applied.

Causes of Forced Vibration

1. Unbalanced Rotating Masses:
   When rotating machines like fans, turbines, or engines have unbalanced parts, they produce periodic forces that cause the whole structure to vibrate.
2. Reciprocating Forces:
   Engines with reciprocating pistons create periodic forces during operation, which act as an external excitation.
3. External Mechanical Inputs:
   Vibrations can occur due to mechanical impacts such as a hammer blow, continuous pressing, or gear meshing.
4. Environmental Factors:
   External vibrations can also be caused by surrounding sources like nearby machines, moving vehicles, or seismic activity.

Characteristics of Forced Vibration

1. Continuous Energy Input:
   In forced vibration, the system does not rely on its own potential or kinetic energy. The vibration continues as long as the external force acts.
2. Dependent on Excitation Frequency:
   The vibration frequency matches the frequency of the external force, not the system’s natural frequency.
3. Amplitude Variation:
   The amplitude of forced vibration depends on how close the excitation frequency is to the natural frequency and the damping present in the system.
4. Damping Effect:
   Damping reduces the vibration amplitude and prevents the system from reaching dangerously high oscillations.

Mathematical Expression

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

Where,

•  = mass of the system
•  = damping coefficient
•  = stiffness of spring
•  = amplitude of the external force
•  = angular frequency of external force
•  = displacement

The steady-state amplitude of vibration depends on the ratio of excitation frequency to natural frequency and damping present in the system.

Resonance in Forced Vibration

Resonance is a special condition of forced vibration when the frequency of the external force becomes equal to the system’s natural frequency. At this point, the amplitude of vibration increases sharply and may cause serious damage to machinery or structures.

To prevent resonance:

• Proper damping should be used.
• Operating speed should be kept away from natural frequency.
• System components should be balanced and designed to minimize vibration effects.

Examples of Forced Vibration

1. Vehicle Engine Vibration:
   The continuous firing of engine cylinders creates vibrations transmitted to the vehicle body.
2. Machine Tools:
   Unbalanced rotating components or periodic impacts cause vibrations during operation.
3. Buildings near Traffic or Railways:
   The passage of trains or heavy vehicles produces ground vibrations that act as external forces.
4. Loudspeakers:
   The diaphragm of a loudspeaker vibrates due to the alternating electric signal applied, producing sound waves.

Importance of Studying Forced Vibration

Understanding forced vibration is essential in mechanical engineering to design systems that can withstand external excitations safely. Improper management can lead to:

• Reduced efficiency,
• Noise generation,
• Component failure, and
• Structural damage.

Engineers use vibration isolators, dampers, and tuned absorbers to minimize these effects and maintain stable operation.

Conclusion

Forced vibration occurs when an external periodic force acts continuously on a system, causing it to vibrate at the frequency of that force. The amplitude of vibration depends on the relationship between the excitation frequency and the system’s natural frequency. If both frequencies coincide, resonance occurs, leading to very large vibrations. Hence, understanding and controlling forced vibrations is vital for the safety, stability, and performance of mechanical systems.