Short Answer:

Free vibration is the type of vibration that occurs when a body or system is disturbed from its equilibrium position and then allowed to vibrate freely without any external periodic force acting on it. The motion continues due to the system’s own elastic and inertial properties.

In free vibration, the system oscillates at its natural frequency, which depends only on its mass and stiffness. Examples include a pendulum swinging after being displaced, or a spring-mass system moving up and down after being pulled and released.

Detailed Explanation :

Free Vibration

Free vibration is one of the most basic and important types of vibration in mechanical systems. It occurs when an object or mechanical system is displaced from its equilibrium position and then allowed to vibrate without the influence of any continuous external force. The movement happens because of the internal restoring forces (like spring elasticity or gravitational force) that try to bring the body back to its original position.

For example, when a spring is stretched and released, it oscillates freely around its mean position. The body vibrates because of the energy stored in the system during displacement, which gets converted between kinetic and potential energy during the motion.

1. Basic Concept of Free Vibration

In free vibration, once the initial disturbance is given, no further energy is added to the system. The system continues to move under the influence of its own natural forces until the motion dies out due to damping or resistance.

The frequency at which a system naturally vibrates when disturbed is called the natural frequency. The natural frequency depends on two factors — the stiffness of the system and its mass. Mathematically, for a simple spring-mass system:

Where,
= natural frequency,
= stiffness of the spring,
= mass of the body.

If there is no damping present, the motion continues indefinitely, but in practical systems, energy losses due to air resistance or friction cause the amplitude to gradually decrease.

2. Characteristics of Free Vibration

The key features of free vibration are as follows:

• No external force: After the initial disturbance, no continuous external force acts on the body.
• Natural frequency: The body vibrates at its own natural frequency, which is unique for every system.
• Energy exchange: Energy continuously shifts between kinetic and potential forms.
• Amplitude decay: In real-life systems, amplitude gradually decreases due to damping.
• Sinusoidal motion: The motion is usually simple harmonic in nature, following a sinusoidal pattern.

These characteristics help engineers to analyze the natural behavior of a system before external forces or damping are applied.

3. Examples of Free Vibration

Some common examples include:

• A pendulum swinging freely after being pushed once.
• A tuning fork vibrating after being struck.
• A spring-mass system oscillating after the mass is displaced and released.
• The beam of a bridge vibrating slightly after a vehicle passes over it.
• A machine component vibrating after being hit or displaced.

All these examples show free vibration where the system vibrates due to its own restoring force without external influence.

4. Importance of Free Vibration in Engineering

Free vibration study is very important in mechanical engineering for several reasons:

• Determining natural frequency: Helps in identifying the system’s natural frequency, which must be avoided during operation to prevent resonance.
• Machine design: Engineers design machines and structures so that their natural frequency does not match with the frequency of any external force.
• Failure prevention: Helps in avoiding excessive vibration that can cause fatigue or structural damage.
• Noise reduction: Proper vibration analysis reduces unwanted noise and increases machine life.
• Dynamic behavior analysis: Provides a basis for understanding the vibration response of mechanical systems.

Thus, the study of free vibration forms the foundation of vibration analysis in engineering.

5. Free Vibration with and without Damping

• Undamped Free Vibration:
  In an ideal case where no resistance or damping exists, the amplitude remains constant, and the motion continues indefinitely. However, this is only theoretical since some energy loss always occurs in practice.
• Damped Free Vibration:
  In real systems, damping forces such as air resistance, friction, or material internal friction cause the vibration amplitude to decrease gradually with time. Eventually, the motion stops as the energy is dissipated.

Even though damping reduces vibration, it helps in maintaining stability and preventing excessive oscillations.

6. Mathematical Representation of Free Vibration

For a simple spring-mass system, the equation of motion is:

The solution of this equation gives:

Where,
= displacement at time  ,
= amplitude,
= natural angular frequency,
= phase angle.

This represents a simple harmonic motion, which is the basis of free vibration behavior.

Conclusion

Free vibration is the oscillatory motion of a system that occurs after it has been displaced from its equilibrium position and released, without any external periodic force acting on it. It depends solely on the system’s mass and stiffness, and occurs at its natural frequency. Understanding free vibration helps engineers design machines and structures that avoid resonance, reduce wear, and operate safely. Therefore, free vibration study is a fundamental aspect of mechanical vibration analysis and system design.