Damped Simple Harmonic Motion:
We know that a simple pendulum does not continue to oscillate endlessly. The
amplitude of oscillation keeps decreasing and after some time the pendulum
stops oscillating. If the pendulum is placed under water, it will swing only
briefly. This happens because, in the first case, air, and in the second,
water, exert a drag force on the pendulum and transfer energy from the motion
of the pendulum.
When the motion of an oscillator is reduced due to some external force, we
say the motion is 'damped'.

Consider a block of
mass m attached to a spring with spring constant k and oscillating
vertically. Let a vane connected by a rod to the block (both assumed to be massless) be immersed in a liquid as shown. As the mass
oscillates, the liquid will exert a drag force on the vane and thus on the
entire system. The mechanical energy of the blockvane system is gradually
transformed to thermal energy of the liquid and the vane.
The damping force F_{d}
is proportional to the velocity of the vane.
Thus F_{d} = b_{v} where b is a constant. The
negative sign indicates that F_{d}
opposes the motion.
The force F_{s} on the block due to the spring is given by
F_{s} = k_{x}
Then, ma = b_{v}  k_{x} (Newton's 2nd law)
i.e., m.

d^{2}x

= b

dx

 k_{x}



dt^{2}

dt

or m .

d^{2}x

+ b

dx

+ k_{x} = 0



dt^{2}

dt

The solution of this equation gives
x = A e^{}^{bt}^{/2 }^{m} cos
(w't + f), [where A is the
amplitude and w' is the angular frequency of the damped oscillator.]
w' =
If there is no damping, b = 0 and w = ,
which gives the angular frequency of an undamped
oscillator.
