Short Answer:

Free vibration is the type of vibration that occurs when a mechanical system is displaced from its equilibrium position and then allowed to vibrate freely without any external force acting on it. The system continues to oscillate at its natural frequency, which depends on its mass and stiffness.

In free vibration, the motion starts due to an initial disturbance, such as a pull, push, or twist, and continues until energy is lost due to damping or friction. Examples include a swinging pendulum or a stretched spring returning to its position after release.

Detailed Explanation :

Free Vibration

Free vibration is a natural motion that occurs in mechanical systems when they are disturbed and then allowed to vibrate without continuous external influence. It happens because of the restoring force developed in the system that tries to bring the body back to its original position after being displaced. However, due to inertia, the body overshoots the equilibrium position, and the motion repeats in the opposite direction, leading to oscillations.

The frequency at which the system vibrates in free vibration is called its natural frequency. It depends only on the system’s physical properties, such as stiffness, mass, and boundary conditions. Free vibrations are fundamental to the study of mechanical vibrations, as they help engineers analyze how systems behave naturally and how they respond to disturbances.

Characteristics of Free Vibration

1. The system vibrates freely after the initial disturbance without any continuous external force.
2. The vibration occurs at the system’s natural frequency.
3. The amplitude of vibration depends on the initial disturbance.
4. Energy changes continuously between potential energy and kinetic energy.
5. The motion continues until it stops due to damping or resistance.

In ideal conditions, without any resistance (air friction, material damping, etc.), free vibration would continue indefinitely. However, in real systems, some energy loss always occurs, causing the vibration to gradually die out.

Examples of Free Vibration

• A tuning fork vibrating after being struck.
• A pendulum swinging freely after being displaced.
• A mass attached to a spring that is stretched and released.
• The vibration of a plucked guitar string.

These examples show that free vibration is present in both simple systems like springs and complex systems such as engines or machines.

Mathematical Concept of Free Vibration

For a simple spring–mass system, the vibration can be expressed mathematically.
Let:

•  = mass of the system
•  = stiffness of the spring
•  = displacement from equilibrium position

According to Newton’s law of motion,

Solving this equation gives the displacement as:

where:

•  = amplitude of vibration
•  = natural angular frequency
•  = phase angle

The frequency of free vibration (in Hertz) is given by:

This shows that the natural frequency depends only on the stiffness and mass of the system. A stiffer system vibrates faster, while a heavier system vibrates slower.

Energy Transfer in Free Vibration

In free vibration, energy constantly changes between potential and kinetic forms:

• When the system is at its extreme position, energy is completely potential.
• When the system passes through the mean position, energy is completely kinetic.
• The total energy remains constant if there is no damping.

This continuous exchange of energy keeps the system vibrating until external forces or resistance reduce the motion over time.

Damped and Undamped Free Vibration

Free vibration can be further classified based on the presence of damping:

1. Undamped Free Vibration:
   • No resistance or energy loss.
   • The system vibrates with constant amplitude.
   • This is an ideal condition and rarely exists in real life.
2. Damped Free Vibration:
   • Includes resistance or damping forces (like friction).
   • The amplitude gradually decreases with time.
   • The system finally comes to rest after a few cycles.

Damping is very important in practical systems because it prevents excessive vibration and ensures stability.

Importance of Free Vibration in Engineering

Free vibration analysis is essential for designing and maintaining mechanical systems. Engineers study it to:

• Determine the natural frequency of machines and structures.
• Avoid resonance, which can cause damage or failure.
• Design systems with proper stiffness and mass to achieve desired performance.
• Develop effective damping mechanisms to control vibration levels.

Examples include the design of vehicles, turbines, engines, and building structures, where understanding free vibration ensures smooth operation and longer life.

Conclusion

Free vibration is the natural oscillation of a system after being disturbed and released without any continuous external force. It occurs at the system’s natural frequency and depends on its mass and stiffness. This concept is very important in mechanical engineering because it helps in designing stable, efficient, and safe systems. By studying free vibration, engineers can predict system behavior, prevent resonance, and control unwanted oscillations for better performance.