Short Answer:

The natural frequency of a spring–mass system is the frequency at which the system vibrates freely after being displaced from its equilibrium position and released without any external force or damping. It depends only on the mass (m) of the body and the stiffness (k) of the spring.

Mathematically, the natural angular frequency (ωₙ) is given by

and the natural frequency (fₙ) in hertz (cycles per second) is

This means stiffer springs and lighter masses produce higher natural frequencies.

Detailed Explanation :

Natural Frequency of a Spring–Mass System

A spring–mass system is one of the simplest examples of a single-degree-of-freedom (SDOF) mechanical vibration system. It consists of a mass attached to a spring that can move freely in one direction, either vertically or horizontally. When the mass is displaced from its equilibrium position and released, it oscillates due to the spring’s restoring force. The frequency at which this system vibrates freely, without external force or damping, is called the natural frequency of the spring–mass system.

The natural frequency determines how fast the system vibrates naturally and is a key parameter in mechanical and structural vibration analysis. It depends on two main properties — the stiffness of the spring (k) and the mass of the body (m).

Derivation of Natural Frequency for a Spring–Mass System

Consider a simple system where a mass  is attached to a light spring of stiffness . The spring is fixed at one end, and the other end supports the mass. The following assumptions are made:

1. The spring follows Hooke’s Law, i.e., .
2. The motion of the mass is frictionless (no damping).
3. The displacement is small, so the motion is linear.

Now, let’s derive the expression for natural frequency step by step.

1. Forces Acting on the Mass

When the mass is displaced downward by a distance  from its equilibrium position and released, the spring exerts a restoring force proportional to the displacement:

where the negative sign indicates that the force acts opposite to the direction of displacement.

According to Newton’s second law,

where  is the acceleration of the mass.

Since the restoring force is the only force acting (for free vibration), we can equate the two:

This is the equation of motion for a spring–mass system under free vibration.

2. Simplifying the Equation

The above equation can be rewritten as:

Let,

Then the equation becomes:

This is the standard differential equation of simple harmonic motion (SHM), where  represents the natural angular frequency.

3. Solution of the Equation

The general solution to this second-order differential equation is:

where,

•  = displacement at time
•  = amplitude of vibration (maximum displacement)
•  = phase angle (depends on initial conditions)
•  = natural angular frequency (rad/s)

Thus, the motion of the mass is sinusoidal, and the body oscillates around the equilibrium position with angular frequency .

4. Expression for Natural Frequency

From the definition of angular frequency,

To express the natural frequency in cycles per second or hertz (Hz):

Hence, the natural frequency of a spring–mass system depends on the stiffness and mass as shown below:

• Larger stiffness (k) → higher natural frequency
• Larger mass (m) → lower natural frequency

5. Time Period of Vibration

The time period (T) is the time required for one complete vibration:

Thus, heavier masses or less stiff springs lead to longer time periods of vibration.

6. Physical Meaning of Natural Frequency

The natural frequency represents the inherent vibration behavior of the spring–mass system:

• It is the frequency at which the system tends to oscillate when no damping or external force is present.
• It shows how fast energy alternates between potential energy (stored in the spring) and kinetic energy (stored in the mass).
• During vibration:
  • At maximum displacement, potential energy is maximum, and kinetic energy is zero.
  • At the equilibrium position, kinetic energy is maximum, and potential energy is zero.

This continuous exchange of energy results in harmonic motion at the system’s natural frequency.

7. Example Calculation

Let a spring have a stiffness of , and a mass of  is attached to it.

Then,

and,

Hence, the system completes approximately 4.77 vibrations per second.

8. Factors Affecting Natural Frequency

1. Mass of the System (m):
   Increasing the mass decreases the natural frequency because the system becomes heavier and slower to respond.
2. Spring Stiffness (k):
   Increasing the spring stiffness increases the natural frequency because the system becomes harder to deform and oscillates faster.
3. System Configuration:
   The arrangement of springs (series or parallel) changes the effective stiffness and, therefore, the natural frequency.

9. Importance of Natural Frequency in Engineering

• Resonance Avoidance:
  Systems must be designed so that the natural frequency does not match the operating or excitation frequency, preventing excessive vibration.
• Machine Design:
  In machinery like engines and compressors, spring–mass models help predict vibration behavior for stable operation.
• Structural Engineering:
  Helps in designing buildings, bridges, and components to withstand dynamic forces like earthquakes or wind.
• Testing and Measurement:
  Natural frequency is used to identify stiffness, damping, and structural integrity through vibration tests.

Conclusion

The natural frequency of a spring–mass system is the rate at which it naturally oscillates when displaced and released without external force or damping. It is given by

This simple yet fundamental concept forms the foundation for understanding vibrations in mechanical systems. A stiffer spring or lighter mass increases the frequency, while a softer spring or heavier mass decreases it. Knowledge of natural frequency helps engineers design safe, stable, and vibration-free machines and structures.