Short Answer:

A free response is the behavior or motion of a system when it is allowed to vibrate or move freely without any external force acting on it after an initial disturbance. It depends only on the system’s own properties such as mass, stiffness, and damping.

In mechanical systems, the free response represents the natural vibration that occurs when a system is displaced from its equilibrium position and released. It shows how the system returns to rest and helps determine important characteristics such as natural frequency and damping ratio.

Detailed Explanation :

Free Response

The free response of a system refers to its motion or vibration after an initial disturbance, when no external or continuous force acts on it. In this condition, the system vibrates purely under the influence of its own inertia, elasticity, and damping properties. The motion gradually reduces in amplitude due to damping and eventually stops at equilibrium.

Free response is one of the most fundamental concepts in mechanical vibrations because it reveals how a system behaves naturally, without outside interference. It helps engineers determine the natural frequency, damping behavior, and stability of machines and structures.

Basic Concept of Free Response

When a mechanical system such as a mass-spring-damper arrangement is displaced from its rest position and released, it starts vibrating freely due to the restoring force of the spring. If there is damping, the vibration amplitude decreases with time until the system stops moving.

The free response occurs after the removal of any external excitation or force. It depends only on the initial conditions such as initial displacement or initial velocity and on the system’s mass (m), stiffness (k), and damping coefficient (c).

The governing differential equation for the free response of a single-degree-of-freedom (SDOF) system is:

Here,

= mass of the system
= damping coefficient
= stiffness of the spring
= displacement

This is a homogeneous equation because there is no external force term. The solution of this equation gives the free response of the system.

Types of Free Response

The nature of the free response depends on the amount of damping present in the system. Based on the damping ratio (ζ), three types of free response occur:

Underdamped System (ζ < 1):
The system oscillates about its equilibrium position with gradually decreasing amplitude. This is the most common type in mechanical systems.
The displacement is given by:

where is the damped natural frequency.

Critically Damped System (ζ = 1):
The system returns to its equilibrium position as quickly as possible without oscillation.
The response is:

Overdamped System (ζ > 1):
The system returns to equilibrium slowly without oscillation. The motion decays exponentially and takes a longer time to settle.

Thus, the damping ratio plays a key role in determining whether the free response is oscillatory or non-oscillatory.

Free Response of an Undamped System

If there is no damping in the system (), the free vibration continues indefinitely with a constant amplitude and natural frequency.

The equation of motion becomes:

The solution is:

where,

= initial amplitude
= natural frequency
= phase angle

This represents simple harmonic motion, where the system vibrates continuously because there is no energy loss.

In real systems, however, some damping is always present, so the amplitude decreases gradually with time.

Free Response of a Damped System

When damping is included (), energy is gradually lost due to friction or resistance, and the amplitude of vibration decreases exponentially.

The governing equation is:

The general solution for the underdamped case () is:

Here,

: represents exponential decay of amplitude due to damping,
: damped natural frequency,
: damping ratio.

The exponential term shows that as time increases, the vibration amplitude decreases, and the system gradually stops moving.

Physical Meaning of Free Response

The free response shows the natural behavior of the system after it is disturbed and left alone.
Physically, it helps engineers understand:

How quickly a machine or structure will stop vibrating after being disturbed.
How damping influences the decay of motion.
What the natural frequency of the system is.

For example:

A tuning fork vibrating after being struck is a free vibration.
A bridge oscillating slightly after a vehicle passes is a free response.
A machine rotor slowing down after a power cut also exhibits free response.

Importance of Free Response in Engineering

Determining Natural Frequency:
The free response directly shows the system’s natural frequency, which is crucial for avoiding resonance.
Understanding Damping Behavior:
It helps determine how damping affects system stability and motion decay.
Predicting System Stability:
A system that continues to oscillate for a long time with large amplitude may be unstable, while one that settles quickly is stable.
Design of Machine Elements:
Helps in designing springs, shafts, and supports to withstand vibrations.
Vibration Testing:
Free vibration tests are used to experimentally determine system parameters like damping ratio and stiffness.

Examples of Free Response

A car suspension bouncing after hitting a bump and gradually coming to rest.
A pendulum swinging freely after being pushed.
A metal plate vibrating after being struck by a hammer.
A rotor spinning freely after the motor is switched off.

In all these cases, the system is excited initially and then left to vibrate on its own without any continuous external force.

Conclusion:

The free response is the motion or vibration of a system after an initial disturbance when no external force acts on it. It is determined entirely by the system’s mass, stiffness, damping, and initial conditions. The response can be oscillatory or non-oscillatory depending on the damping ratio. Studying the free response helps engineers understand natural frequency, damping characteristics, and system stability, which are vital for designing safe and reliable mechanical and structural systems.