Short Answer:

Free vibrations are those vibrations that occur when a body or system, after being disturbed from its equilibrium position, is allowed to vibrate on its own without any external periodic force acting on it. The motion continues only due to the system’s internal energy.

In simple words, free vibration happens when an object is displaced and then released, such as a pendulum swinging after being pushed or a spring oscillating after being stretched. The vibration gradually decreases due to damping forces like air resistance or friction. The frequency of vibration in such motion is known as the natural frequency of the system.

Detailed Explanation :

Free Vibrations

Free vibration is one of the most fundamental types of vibration found in mechanical systems. It occurs when a body or system, after being given an initial disturbance, vibrates freely without any continuous external force acting on it. The system moves only due to the energy that was initially supplied to it. This energy is exchanged between potential energy (stored when displaced) and kinetic energy (due to motion).

When an external force is removed, the system’s internal restoring forces cause it to move back and forth around its equilibrium position. Over time, due to damping or resistance, the amplitude of vibration gradually reduces, and finally, the motion stops.

A system undergoing free vibration always vibrates with its natural frequency, which depends on its physical properties such as mass and stiffness.

Mathematical Representation

For a simple mass-spring system, free vibration can be described using Newton’s second law of motion.

Let:

m = mass of the system (kg)
k = stiffness of the spring (N/m)
x = displacement from the mean position (m)

The restoring force in the spring is proportional to the displacement and acts in the opposite direction:

According to Newton’s second law:

Equating both, we get:

This is the differential equation of free vibration.

The general solution of the equation is:

where,

A = amplitude of vibration,
ωₙ = √(k/m) = natural angular frequency (rad/s),
φ = phase angle,
t = time (s).

Hence, in free vibration, the system oscillates at its natural frequency, fₙ, given by:

This equation shows that the natural frequency depends only on the stiffness and mass of the system.

Characteristics of Free Vibrations

No External Force:
Once the system is set into motion, no continuous external force acts on it.
Natural Frequency:
The system vibrates at its own natural frequency, which depends on mass and stiffness.
Energy Exchange:
Energy continuously converts between potential and kinetic forms during vibration.
Gradual Energy Loss:
Due to damping (like air resistance or internal friction), the amplitude of vibration decreases with time.
Sinusoidal Motion:
The motion is periodic and follows a sine or cosine waveform.
Depends on System Properties:
The mass and stiffness determine the behavior and speed of free vibration.

Types of Free Vibrations

Undamped Free Vibrations:
These vibrations occur when no damping force (resistance) acts on the system. The body keeps vibrating indefinitely with a constant amplitude. This is an ideal condition that does not exist in real systems.
Example: A perfectly frictionless pendulum.
Damped Free Vibrations:
These vibrations occur when damping or resistance (such as air friction or internal material damping) is present. The amplitude gradually decreases over time until the motion stops.
Example: A car suspension system after hitting a bump.

Examples of Free Vibrations

Simple Pendulum:
When displaced from its equilibrium position and released, a pendulum swings back and forth freely due to gravitational restoring force.
Spring-Mass System:
When a spring is stretched or compressed and then released, the mass attached to it vibrates freely around the mean position.
Tuning Fork:
When struck against a surface, the prongs of a tuning fork vibrate freely at their natural frequency, producing a musical sound.
Vibrating Beam:
When a cantilever beam is bent and released, it vibrates freely due to its elasticity.
Vehicle Suspension:
After hitting a bump, a car’s suspension vibrates freely until the damping effect stops the motion.

Importance of Free Vibrations in Engineering

Determination of Natural Frequency:
Helps engineers calculate the natural frequency of a system, which is essential for design safety.
Avoiding Resonance:
By understanding free vibration behavior, machines can be designed so that their operating frequencies do not coincide with natural frequencies, preventing resonance and damage.
Structural Analysis:
In buildings and bridges, studying free vibrations helps in ensuring stability against wind and earthquake forces.
Machine Design:
Free vibration analysis is used to design shafts, rotors, and other parts that experience oscillations during operation.
Dynamic Testing:
Engineers use free vibration testing to find stiffness, damping, and mass properties of materials and components.

Energy in Free Vibrations

In free vibration, the total mechanical energy remains constant (if damping is ignored). The energy alternates between:

Potential Energy (PE): Stored when the system is displaced.
Kinetic Energy (KE): Present when the system is moving through the mean position.

At the extreme positions, all energy is potential, and at the mean position, all energy is kinetic.

When damping is present, part of this energy is lost as heat or sound, and the amplitude of vibration decreases over time.

Conclusion

Free vibrations are the natural vibrations of a system that occur when it is disturbed and then allowed to vibrate without any external force. The system vibrates at its own natural frequency, and the motion gradually decreases due to damping. Understanding free vibrations is essential in mechanical engineering for analyzing machine performance, avoiding resonance, and ensuring the stability and safety of structures and mechanical systems.