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Various frictional forces may act on a harmonic oscillator, tending to reduce its successive
amplitudes. Such a motion is called damped harmonic motion. Suppose a particle of mass m is
subject to a restoring force proportional to the distance from a fixed point on the x-axis and a
damping force proportional to the velocity. Then the equation of motion becomes
x being the displacement of the particle from the fixed point at any instant t, k and  are positive
constants. The damping force is – x& where  is the damping coefficient. Equation (3.1) can be
written as
where 2b = /m and = √ is the natural frequency of the oscillator. The relaxation time is defined by
An expression for the displacement of the damped harmonic
oscillator where the damping force is proportional to the velocity.
Discuss the effect of the damping on the displacement and frequency
of the oscillator.
The differential equation of the damped harmonic motion is given by
Eqn. (3.2):
where (–2mb𝑥̇ ) = -𝑥̇ & ) is the damping force acting on the particle of mass m and
 is the natural frequency of the oscillator.
Let x = exp (t) be the trial solution of above equation. Then, we have
The two roots of are
So, the general solution of Eqn. (3.2) is
where A1 and A2 are two constants whose values can be determined from the initial
Case I: b >  (the damping force is large)
The expression (3.8) for x represents a damped dead beat motion, the displacement
x decreasing exponentially to zero (Fig. 3.1).
Case II: b < (the damping is small)
In this case
and Eqn. (3.8) becomes
If we write B1 = R cos 
R sin ,then the above equation reduces to
Equation (3.10) gives a damped oscillatory motion (Fig. 3.1). Its amplitude R exp
(–bt) decreases exponentially with time. The time period of damped oscillation is
Whereas the undamped time period is
Thus the time period of damped oscillation is slightly greater than the undamped
natural time period when b  In other words, the frequency of the damped
’ is less than the undamped natural frequency 
Let us consider the simple case in which  = 0 in Eqn. (3.10). Then cos ’t =±1
when t=0,t1=
, etc. Suppose that the values of x in both directions
corresponding to these times are x0, x1, x2, x3 etc. so that
Considering the absolute values of the displacements, we get
The quantity
is called logarithmic decrement. Here ν is the frequency of the damped
oscillatory motion. The logarithmic decrement is the logarithm of the
ratio of two successive maxima in one direction = ln (xn/xn+2). Thus the
damping coefficient b can be found from an experimental measurement
of consecutive amplitudes. Since
We have from Eqn.
The energy equation of the damped harmonic oscillator:
We can regard equation (3.10) as a cosine function whose amplitude R
exp (– bt) gradually decreases with time. For an undamped oscillator of
amplitude R, the mechanical energy is constant and is given by E =
kR2. If the oscillator is damped, the mechanical energy is not constant
but decreases with time. For a damped oscillator the amplitude is\ R exp
(– bt) and the mechanical energy is
is the initial mechanical energy.
Like the amplitude the mechanical energy decreases exponentially with
Case III: b =  (critically damped motion)
When b = , we get only one root  = – b. One solution of Eqn. (3.2) is
and the other solution is
The motion is non-oscillatory and the particle approaches origin slowly.
Quality Factor of Damped Harmonic Oscillation:
The displacement x of the mass m at any time is given by
Total energy at any instant is given by
Loss of energy of the mass in time dt is
Assuming that e–2bt remains almost constant over a cycle, the average
energy over a complete cycle at that time is
since the average value of cos2(’t-) and sin2(’t-) are each ½ and
that of sin2(’t-) is zero