Short Answer:

Free vibration is the type of vibration in which a body or system vibrates freely after being given an initial disturbance without any continuous external force acting on it. The vibration continues only due to the system’s internal restoring forces, such as the elasticity of materials or spring tension.

In free vibration, the system vibrates with its natural frequency, which depends on its mass and stiffness. Examples include a tuning fork vibrating after being struck or a pendulum swinging after being displaced and released. Energy loss due to air resistance or friction causes the vibration to gradually die out.

Detailed Explanation:

Free Vibration

Free vibration is a basic concept in the study of mechanical vibrations. It occurs when a system is displaced from its equilibrium position and allowed to move freely under the influence of its internal restoring forces. There is no external force acting on the system after it starts vibrating. The motion continues due to the conversion of potential energy into kinetic energy and vice versa, until the energy is lost due to damping effects like air resistance or friction.

Principle of Free Vibration

When an elastic system is disturbed from its mean position, it tends to return to that position due to the restoring force. This force is proportional to the displacement and acts in the opposite direction, following Hooke’s law, which states:

where,

•  = restoring force,
•  = stiffness or spring constant,
•  = displacement from equilibrium.

As the system returns to its original position, its inertia causes it to overshoot, leading to oscillatory motion around the equilibrium point. This back-and-forth movement is called free vibration.

Conditions for Free Vibration

Free vibration occurs under the following conditions:

1. The body is displaced from its equilibrium position.
2. No external periodic force acts on it after release.
3. The motion is maintained by internal elastic and inertial forces.

For example, when a spring-mass system is stretched and released, it vibrates freely about its mean position until frictional forces stop it.

Natural Frequency

Every system has its own natural frequency, which depends on its stiffness and mass. During free vibration, the system vibrates at this frequency. The natural frequency of a simple spring-mass system is given by:

where,

•  = natural frequency (Hz),
•  = stiffness of spring (N/m),
•  = mass of body (kg).

A higher stiffness increases the frequency, while a larger mass decreases it. This concept is essential in machine design because operating a machine near its natural frequency can cause resonance and lead to mechanical failure.

Examples of Free Vibration

1. A pendulum swinging after being displaced.
2. A tuning fork vibrating after being struck.
3. A mass-spring system oscillating up and down after being stretched and released.
4. Vibrations in machine components when suddenly released from stress.

These examples show that in all cases, the vibration occurs due to an initial disturbance and internal forces, without continuous external influence.

Mathematical Expression for Free Vibration

The motion of free vibration can be expressed as:

where,

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

This equation shows that the motion is periodic, with the amplitude depending on the initial disturbance and the frequency being constant.

Energy in Free Vibration

In free vibration, energy alternates between kinetic and potential forms:

• When displacement is maximum, potential energy is maximum, and kinetic energy is zero.
• When the mass passes through the equilibrium position, kinetic energy is maximum, and potential energy is zero.

In an ideal undamped system, the total energy remains constant. However, in real systems, energy gradually decreases due to damping forces like friction and air resistance.

Types of Free Vibration

1. Undamped Free Vibration:
   • No energy loss occurs.
   • Amplitude remains constant.
   • Theoretical condition (ideal case).
2. Damped Free Vibration:
   • Some energy loss due to damping.
   • Amplitude gradually decreases with time.
   • Motion eventually stops.

Importance in Engineering

Free vibration analysis is important in mechanical engineering for several reasons:

• To determine the natural frequencies of structures and machines.
• To avoid resonance in mechanical design.
• To study the behavior of materials and components under oscillatory motion.
• To design vibration isolators and dampers for smoother operation.

For example, in automobiles, engineers study free vibrations of suspension systems to ensure ride comfort and stability. In bridges or tall buildings, natural frequencies are analyzed to prevent resonance during wind or earthquake conditions.

Conclusion:

Free vibration occurs when a system vibrates due to an initial disturbance and no external force acts afterward. The motion continues at the system’s natural frequency and gradually dies out due to damping. Understanding free vibration is essential for analyzing machine parts, designing stable structures, and preventing harmful resonant conditions in mechanical systems.