**Free Vibrations: **If a given body is once set into vibrations and then let free to vibrate with its own natural frequency, the vibrations are said to be free vibrations. The natural frequency of free vibrations depends on the nature and structure of the body and in ideal situation, the amplitude, frequency and the energy of the vibrating body remain constant.

**Forced Vibrations: **The vibrations in which a body oscillates under the effect of an external periodic force, whose frequency is different from the natural frequency of oscillating body, are called forced vibrations. In forced vibrations, the oscillating body vibrates with the frequency of external force and amplitude of oscillations is generally small.

If an external driving force is represented by F(t) = Fₒ cos ω_{d}t

The motion of particle is under combined action of

i) Restoring force (-kx)

ii) Damping force (-bv), and

iii) Driving force F(t)

Now, ma = – kx – bv + Fₒ cos ω_{d}t

Or \(\frac{{{d}^{2}}x}{d{{x}^{2}}}=-\frac{kx}{m}-\frac{b}{m}\frac{dx}{dt}+\frac{{{F}_{0}}\text{ }cos\text{ }{{\omega }_{d}}t}{m}\).

The solution of this equation gives x = xₒ sin (ω_{d}t + φ) with amplitude

\({{x}_{0}}=\frac{\frac{{{F}_{0}}}{m}}{\sqrt{\left( \omega _{0}^{2}-\omega _{d}^{2} \right)+{{\left( \frac{b\omega }{m} \right)}^{2}}}}\).

\(\operatorname{Tan}\theta =\frac{\omega _{0}^{2}-\omega _{d}^{2}}{\frac{b\omega }{m}}\).

\({{\omega }_{0}}=\sqrt{\frac{k}{m}}\).

= natural frequency

**Damped Vibrations: **When a body is set in free vibrations, generally there is a dissipation of energy due to dissipative causes like viscous drag of a fluid, frictional force, hysteresis, electromagnetic damping force, etc.., and as a result amplitude of vibration regularly decreases with time. Such vibrations of continuously falling amplitudes are called damped vibrations.

In these oscillations, the amplitude of oscillation decreases exponentially and hence, energy also decreases exponentially.

If the velocity of an oscillator is v, the damping force

F_{d }= – bv

Where, b = damping constant.

Resultant force on a damped oscillator is given by F = F_{R} + F_{D} = -k x – b v** **

Or \(\frac{m{{d}^{2}}x}{d{{t}^{2}}}+\frac{bdx}{dt}+kx=0\).

Displacement of a damped oscillator is given by \(X={{x}_{m}}{{e}^{\frac{-bt}{2m}}}\sin (\omega ‘t+\phi )\).

\(\omega ‘=\sqrt{\omega _{0}^{2}-{{\left( \frac{b}{2m} \right)}^{2}}}\).

Where, ω’ = angular frequency of the damped oscillator for a damped oscillator, if the damping is small then the mechanical energy decreases exponentially with time as \(E=\frac{1}{2}kx_{m}^{2}{{e}^{-bt/m}}\).

**Resonant Vibrations: **It is a special case of forced vibrations in which frequency of external force is exactly same as the natural frequency of oscillator. As a result the oscillating body begins to vibrate with large amplitude leading to the resonance phenomenon to occur. Resonant vibrations play a very important role in music and tuning of station/channel in a radio/TV.