Short Answer:

Undamped vibrations are the type of vibrations in which a system continues to vibrate with constant amplitude and energy because there is no resistance or energy loss due to damping forces like friction or air resistance. The amplitude remains constant, and the vibration continues indefinitely.

In simple words, undamped vibration occurs when a body or system oscillates freely without any energy being lost to the surroundings. This is an ideal condition that cannot exist in real life because all practical systems have some damping. However, it is useful for understanding the basic vibration principles and for mathematical analysis.

Detailed Explanation :

Undamped Vibrations

Undamped vibrations are the simplest form of vibration where the amplitude of motion remains constant with time because there are no resistive or damping forces acting on the vibrating system. When a body vibrates, it alternately stores energy in two forms — potential energy (due to deformation) and kinetic energy (due to motion). In undamped vibration, this energy is not lost but continuously exchanged between the two forms.

In reality, every material and system experiences some damping due to air resistance, internal material friction, or surface contact. Therefore, undamped vibration is a theoretical concept used for understanding vibration fundamentals and deriving mathematical relationships for free vibration and natural frequency.

Mathematical Expression of Undamped Vibrations

Consider a simple mass-spring system, where:

• m = mass of the vibrating body (kg),
• k = stiffness or spring constant (N/m),
• x = displacement from the mean position (m).

When the system is displaced from its equilibrium position, a restoring force acts on it according to Hooke’s law:

According to Newton’s second law,

Equating both expressions gives:

Dividing by m:

Let,

where ωₙ is the natural angular frequency (in radians per second).

Thus, the equation of motion becomes:

The general solution is:

where,

• A = amplitude of vibration,
• ωₙ = natural angular frequency,
• t = time,
• φ = phase angle.

This equation shows that the displacement changes sinusoidally with time, and since there is no damping, the amplitude A remains constant for all time.

Characteristics of Undamped Vibrations

1. Constant Amplitude:
   The amplitude of vibration does not decrease with time because there is no energy loss.
2. Constant Energy:
   The total mechanical energy remains constant, continuously transforming between potential and kinetic energy.
3. No Damping Force:
   No resistive force like air resistance, friction, or material damping acts on the system.
4. Natural Frequency:
   The system vibrates only at its natural frequency, which depends on the system’s stiffness and mass.
5. Theoretical Condition:
   Perfectly undamped vibration cannot exist in real systems because every material has some internal resistance.

Energy Transfer in Undamped Vibrations

In undamped vibrations, energy continuously exchanges between potential and kinetic forms:

• At extreme positions, the displacement is maximum and the energy is fully potential.
• At the mean position, the velocity is maximum and the energy is fully kinetic.

The total energy remains constant throughout the vibration:

This perfect energy conversion without loss makes undamped vibration an ideal case for studying free vibration systems.

Natural Frequency in Undamped Vibrations

The natural frequency (fₙ) is the frequency at which a system vibrates freely when disturbed and left without any external force or damping. It is expressed as:

where,

• k = stiffness of the system (N/m),
• m = mass of the system (kg).

Thus, the natural frequency increases with stiffness and decreases with mass.

In undamped vibrations, this frequency remains constant since no energy loss occurs. It is the most important property for analyzing vibration behavior in engineering systems.

Examples of Undamped Vibrations

Although truly undamped systems do not exist, some real systems can approximate undamped vibrations for short durations, especially when damping is very small. Examples include:

1. Tuning Fork:
   When struck lightly, the vibration amplitude remains almost constant for a few seconds before damping becomes noticeable.
2. Pendulum in Vacuum:
   A pendulum oscillating in a vacuum, where air resistance is absent, approximates undamped motion.
3. Ideal Spring-Mass System:
   A frictionless, perfectly elastic spring system in theoretical analysis represents an undamped vibration.
4. Molecular Vibrations:
   At the microscopic level, atoms in a molecule vibrate nearly without damping.
5. Electromagnetic Oscillations:
   In a lossless LC electrical circuit, energy transfers between electric and magnetic fields similar to undamped vibration.

Importance of Studying Undamped Vibrations

1. Foundation for Vibration Theory:
   Undamped vibration provides the basic understanding of oscillatory motion, natural frequency, and energy transfer.
2. Simplified Analysis:
   It allows engineers to develop simple mathematical models for free vibration before including damping and external forces.
3. Design of Machines:
   Knowing the natural frequency helps avoid resonance in mechanical systems, reducing the risk of failure.
4. Comparison with Real Systems:
   Engineers compare undamped and damped responses to estimate damping effects and improve performance.
5. Educational Use:
   It forms the base concept for advanced topics such as forced vibration, resonance, and vibration isolation.

Limitations of Undamped Vibrations

• In reality, no system is completely free from damping.
• Over time, even small friction or air resistance will cause energy loss and amplitude reduction.
• Hence, undamped vibration models are mainly used for idealized or theoretical analysis rather than real applications.

Despite these limitations, they are very useful for understanding the fundamental behavior of vibrating systems.

Conclusion

Undamped vibrations are ideal vibrations where the amplitude remains constant and energy is not lost because no damping or resistance is present. The system vibrates indefinitely at its natural frequency. Although it is an ideal condition that does not occur in real systems, the concept of undamped vibration is very important in engineering because it forms the foundation for understanding free, forced, and damped vibrations. It helps engineers design stable, efficient, and vibration-resistant mechanical systems.