What is the natural frequency of an undamped SDOF system?

Short Answer:

The natural frequency of an undamped single-degree-of-freedom (SDOF) system is the frequency at which the system freely oscillates when disturbed from its equilibrium position without any damping or external force. It depends only on the stiffness (k) and mass (m) of the system.

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

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

This frequency is a fundamental property of the system and is independent of the amplitude of vibration.

Detailed Explanation :

Natural Frequency of an Undamped SDOF System

The natural frequency of a mechanical system represents the rate at which it naturally vibrates when displaced and released, provided no external force or damping acts on it. In an undamped single-degree-of-freedom (SDOF) system, the oscillations continue indefinitely with constant amplitude because there is no energy loss.

An SDOF system consists mainly of a mass (m), spring (stiffness k), and sometimes a damper (damping coefficient c). In an undamped condition, the damping is neglected (i.e., ), which means the system can vibrate freely forever once disturbed.

The natural frequency depends on two physical quantities — the stiffness (k), which provides the restoring force, and the mass (m), which resists motion. The ratio of these two parameters determines how fast the system oscillates.

Derivation of Natural Frequency of an Undamped SDOF System

Consider a simple spring–mass system as shown conceptually: a mass  attached to a spring of stiffness , fixed at one end. When the mass is displaced by a small distance  and released, the spring exerts a restoring force that tries to bring the mass back to the equilibrium position.

Step 1: Forces Acting on the Mass

According to Hooke’s Law,

where,

  •  = restoring force of the spring (N)
  •  = stiffness of the spring (N/m)
  •  = displacement from equilibrium (m)

According to Newton’s second law,

where,

  •  = mass of the body (kg)
  •  = acceleration of the mass (m/s²)

Since the restoring force opposes motion, we can write:

This is the differential equation of free vibration for an undamped SDOF system.

Step 2: General Solution of the Equation

The differential equation is:

Let,

Hence, the equation becomes:

This is a standard equation of simple harmonic motion (SHM), whose general solution is:

where,

  •  = amplitude of vibration (m)
  •  = natural angular frequency (rad/s)
  •  = phase angle (radians)

Step 3: Expression for Natural Frequency

From the above, the natural angular frequency is:

The natural frequency (fₙ) in hertz (cycles per second) is:

Thus, the natural frequency depends only on the system’s stiffness and mass.

Physical Meaning of Natural Frequency

The natural frequency represents how fast a system vibrates when disturbed and allowed to move freely without any energy loss.

  • A stiffer system (high k) vibrates faster (high natural frequency).
  • A heavier system (high m) vibrates slower (low natural frequency).

Therefore:

  • Increasing stiffness (k) → increases
  • Increasing mass (m) → decreases

This explains why small, rigid objects (like a tuning fork) vibrate rapidly, while large, heavy structures (like a bridge) vibrate slowly.

Graphical Representation

If we plot the displacement  versus time , the motion appears sinusoidal:

  • At , the body starts from rest.
  • It reaches maximum displacement (amplitude) after one-quarter of a cycle.
  • It completes one full vibration in a period , given by

The time period  is the time required for one complete oscillation.

Example Calculation

Let a mass  be attached to a spring with stiffness .

Then,

The natural frequency in hertz is:

Hence, the system completes about 3.18 cycles per second.

Significance of Natural Frequency

  1. Resonance Prediction:
    When an external periodic force acts at the natural frequency, large amplitude vibrations (resonance) occur, which may damage the system.
  2. System Design:
    Engineers design mechanical and structural systems so that their natural frequencies do not coincide with operating frequencies.
  3. Vibration Control:
    By adjusting mass and stiffness, designers can tune systems to achieve desired vibration behavior.
  4. Diagnostic Analysis:
    Measuring a system’s natural frequency helps detect faults like cracks or looseness in machinery.
  5. Material Selection:
    Proper materials are chosen to achieve required stiffness and natural frequency characteristics.

Key Observations

  • The natural frequency is independent of amplitude for linear systems.
  • It depends only on mass (m) and stiffness (k).
  • Damping does not affect the natural frequency in the undamped condition.
  • The motion is sinusoidal and continuous.
Conclusion

The natural frequency of an undamped single-degree-of-freedom system is the rate at which it vibrates freely when disturbed, without any damping or external force. It is determined by the system’s stiffness and mass and is given by . A higher stiffness increases the frequency, while a larger mass decreases it. Understanding natural frequency is essential in mechanical design to avoid resonance and ensure safe and stable operation of machines, vehicles, and structures.