Undamped oscillation occurs when there is no resistive force acting on an object that is vibrating. As the object oscillates, the energy is continuously transformed between kinetic and potential energy, but the total energy remains constant. In undamped vibration, the object will continue oscillating sinusoidally without losing amplitude. In reality, it is difficult to achieve true undamped vibration as even air provides some resistance.

1. Undamped oscillation

3. What is Undamped
oscillation
In undamped vibrations, no resistive force acts on the vibrating object.
As the object oscillates, the energy in the object is continuously transformed from kinetic
energy to potential energy and back again, and the sum of kinetic and potential energy
remains a constant value.
In practice, it’s extremely difficult to find undamped vibrations. For instance, even an object
vibrating in air would lose energy over time due to air resistance.

5. • Let us consider an object undergoing simple harmonic motion. Here,
the objet experiences a restoring force towards the equillibrium
point, and the size of this force is proportional to displacement.
• If the displacement of the object is given by x, then for an object with
mass m in simple harmonic motion, we can write:

6. Undamped oscillation
• This is a differential equation. A solution to this equation can be written in
the form:
•
•
• If vibration is undamped, the object continues to oscillate sinusoidally.

8. Undamped oscillation

14. Difference Between Damped and Undamped Vibration
Presence of Resistive Forces
• In undamped vibrations, the object oscillates freely without any
resistive force acting against its motion.
• In damped vibrations, the object experiences resistive forces.

15. Energy Loss
• In undamped vibrations, the sum of kinetic and potential energies
always gives the total energy of the oscillating object, and the value of
its total energy does not change.
• In damped vibrations, the total energy of the oscillating object
decreases over time. This energy is dissipated as the object does work
against the resistive forces.

16. Value of the Damping Coefficient
• For undamped vibrations,
• For damped vibrations,