Short Answer:
Free vibration is the type of vibration that occurs when a body or system is displaced from its equilibrium position and then allowed to vibrate without any external periodic force acting on it. The body continues to oscillate on its own due to the balance between its inertia and elastic restoring force.
In simple terms, when a mechanical system like a spring-mass setup or a tuning fork is disturbed once and then left free, it vibrates at its natural frequency. This vibration gradually decreases due to damping or resistance from the surroundings.
Detailed Explanation :
Free Vibration
Free vibration is a fundamental concept in the study of mechanical vibrations and dynamics. It describes the natural motion of a system when it is disturbed from its equilibrium position and allowed to oscillate freely under the influence of its own internal forces, without any external periodic excitation.
Every mechanical system, such as a mass-spring, beam, or rotating shaft, has a natural frequency at which it tends to vibrate. In free vibration, the system vibrates at this natural frequency depending on its mass, stiffness, and damping characteristics.
For example, when you pull a pendulum to one side and release it, it swings back and forth freely — this is an example of free vibration. Similarly, when you strike a tuning fork, it vibrates freely and produces a sound at its natural frequency until it gradually stops due to air resistance.
Definition
Free vibration can be defined as:
“The vibration of a system that occurs when it is displaced from its equilibrium position and allowed to vibrate on its own without any external periodic force acting on it.”
This means that the system vibrates only due to the restoring forces developed within it, and the motion continues until the energy is dissipated through damping or resistance.
Conditions for Free Vibration
- Initial Disturbance:
The system must first be displaced from its equilibrium position by applying an initial force. - No Continuous External Force:
After releasing the body, no external force should act on it to maintain or alter its motion. - Internal Restoring Force:
The motion is maintained by the restoring force (like spring elasticity or gravitational pull). - Energy Exchange:
The system continuously converts potential energy into kinetic energy and vice versa during vibration.
Explanation of Free Vibration
When a system (like a spring and mass) is displaced from its equilibrium position, it stores potential energy in the spring due to deformation. Once released, this energy is converted into kinetic energy as the mass moves. As it passes the equilibrium position, the restoring force acts in the opposite direction, causing it to oscillate.
The system continues this back-and-forth motion until damping (friction, air resistance, or internal material resistance) dissipates all the energy, and the motion stops.
Mathematical Representation
Consider a mass-spring system where:
- = mass of the body
- = stiffness (spring constant)
- = displacement from mean position
According to Hooke’s law, the restoring force is:
From Newton’s second law:
Rearranging, we get the equation of motion:
Let , which is the natural circular frequency of the system.
Thus, the solution for displacement becomes:
Where:
- and depend on initial conditions,
- is the natural angular frequency, and
- is the natural frequency (in Hz).
This equation shows that the system vibrates freely at its natural frequency without external input.
Characteristics of Free Vibration
- Occurs at Natural Frequency:
The vibration always takes place at the system’s own natural frequency. - No Continuous External Force:
Once initiated, no periodic external force acts on the body. - Energy Interchange:
Energy alternates between potential and kinetic forms. - Amplitude Decreases with Time:
In real systems, due to damping, the amplitude gradually reduces until motion stops. - Depends on System Parameters:
The natural frequency depends on mass (m) and stiffness (k) of the system.
Types of Free Vibration
- Undamped Free Vibration:
- Ideal condition where there is no energy loss.
- The body oscillates indefinitely with constant amplitude.
- Example: Theoretical spring-mass system without friction.
- Damped Free Vibration:
- Occurs when energy is gradually lost due to damping.
- Amplitude decreases with time until motion stops.
- Example: A pendulum swinging in air.
Examples of Free Vibration
- Tuning Fork:
When struck, it vibrates freely at its natural frequency producing a musical note. - Pendulum:
When displaced and released, it swings freely due to gravity. - Spring-Mass System:
When a weight attached to a spring is pulled and released, it vibrates freely. - Bridge or Structure:
When disturbed by wind or a vehicle and left to settle, it experiences free vibration. - Machine Components:
Shafts or blades vibrate freely after external excitation stops.
Advantages of Free Vibration Study
- Helps determine the natural frequency of systems.
- Useful in predicting and preventing resonance in machinery.
- Aids in designing components for vibration isolation.
- Essential for analyzing dynamic behavior of machines and structures.
- Helps improve the stability and lifespan of engineering systems.
Practical Applications
- Design of Machine Tools:
To avoid resonance by keeping the operating speed away from the natural frequency. - Vehicle Suspension Systems:
To provide comfort by controlling free vibrations of the vehicle body. - Bridge and Building Design:
To ensure safety under wind or seismic loads. - Rotating Machinery:
To balance rotating parts and reduce vibration. - Musical Instruments:
To generate desired frequencies of sound using controlled free vibrations.
Conclusion
Free vibration is the oscillatory motion of a system that occurs when it is disturbed once and allowed to move without any continuous external force. The system vibrates at its natural frequency, determined by its mass and stiffness. In ideal conditions, vibration continues indefinitely, but in real systems, damping gradually reduces amplitude until motion stops. Understanding free vibration is essential in mechanical engineering to design stable, efficient, and vibration-resistant machines and structures, preventing unwanted noise, fatigue, or failure.