Everything has its own oscillating frequency, called natural frequency. It is the kind of frequency that an object shows when it oscillates without any kind of external force.
When a body is left to oscillate itself after displacing, the body oscillates in its own natural frequency. Let that natural frequency be denoted by \(\omega _n\). But the amplitude of the oscillation decreases continuously and the oscillation stops after some time.
The decrease in amplitude is caused by the non-conservative forces such as friction in the system of the oscillation. The non-conservative forces are also called dissipative forces.
In real life situations the dissipative forces do work on the oscillating systems and the oscillations die out after some time and such oscillations are called damped oscillations.
In Figure 1 from simple harmonic motion you can see a mass-spring system in which a box oscillates about its equilibrium position.
If the friction between the surface of the box and the floor is not neglected, the oscillations die out after some time. The force that causes damping of the oscillation is called damping force. The restoring force provided by the spring is
\[F_\text{spring} = - kx\]
Here we consider the simpler case of velocity dependent damping force. The expression for the damping force is,
\[{F_{dx}} = -b{v_x} \tag{1} \label{1}\]
The negative sign in the above equation shows that the damping force opposes the oscillation and \(b\) is the proportionality constant called damping constant. We consider that such a damping force is along x-axis as indicated by the subscript \(x\).
Therefore, the net force on the harmonic oscillator including the damping force is,
\[\begin{align*} {F_{{\rm{net}}}} &= - kx - b{v_x}\\ {\rm{or,}}\quad m{a_x} &= - kx - b{v_x}\\ {\rm{or,}}\quad m\frac{{{d^2}x}}{{d{t^2}}} &= - kx - b\frac{{dx}}{{dt}} \tag{2} \label{2} \end{align*}\]
We do not go into solving Eq. \eqref{2} (but you can go ahead and solve it), the solution is,
\[x = A{e^{ - bt/2m}}\cos (\omega t + \phi ) \tag{3} \label{3}\]
In Eq. \eqref{3} the amplitude of the damped oscillation is \(A' = A{e^{ - bt/2m}}\) which decreases exponentially with time (see Figure 1).
In Eq. \eqref{3} the angular frequency of the damped oscillation is \(\omega = \sqrt {\frac{k}{m} - \frac{{{b^2}}}{{4{m^2}}}} \). We apply a condition that when \(\omega = 0\), you'll get,
\[\begin{align*} \frac{k}{m} - \frac{{{b^2}}}{{4{m^2}}} &= 0\\ {\rm{or,}}{\kern 1pt} \quad b &= 2\sqrt {km} \tag{4} \label{4} \end{align*}\]
When the value of the damping constant is equal to \(2\sqrt {km} \) that is, \(b = 2\sqrt {km} \), the damping is called critical damping and the system is said to be critically damped. In critical damping an oscillator comes to its equilibrium position without oscillation.
And when \(b < 2\sqrt {km}\) the system is said to be under-damped and the damping is called under-damping. In under-damped oscillating system the oscillator oscillates but the amplitude of the oscillation decreases continuously and finally the oscillations stop.
In another case when \(b > 2\sqrt {km} \) the oscillating system is over-damped and the damping is called over-damping. In over-damped case the oscillator comes more slowly to its equilibrium position without oscillating.
