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Damped Harmonic Oscillation
In the previous chapter, we encountered a number of energy conserving physical systems which
exhibit simple harmonic oscillation about a stable equilibrium state. One of
the main features of such oscillation is that, once
excited, it never dies away. However, the majority of the oscillatory
systems that we encounter in everyday life suffer some sort of irreversible energy loss whilst they are in motion, which is due, for instance, to frictional or viscous heat generation. We would therefore expect oscillations excited in such systems
to eventually be damped away. Let us examine an example of a
damped oscillatory system.
Consider the massspring system investigated in Section 2.1.
Suppose that, as it slides over the horizontal surface, the mass is subject to
a frictional damping force which opposes its motion, and is directly
proportional to its instantaneous velocity. It follows that the net force acting
on the mass when its instantaneous displacement is takes the form

(55) 
where is the mass, the spring force constant, and a constant (with the dimensions of angular frequency) which parameterizes the strength of the damping. The time evolution equation of the
system thus becomes [cf., Equation (2)]

(56) 
where
is the undamped oscillation frequency [cf., Equation (6)]. We shall
refer to the above as the damped harmonic oscillator equation.
Let us search for a solution to Equation (56) of the form

(57) 
where , , , and are all constants. By analogy
with the discussion in Section 2.1, we can interpret the
above solution as a periodic oscillation, of fixed angular frequency
and phase angle , whose amplitude decays exponentially in
time as
. So, (57) certainly seems like a plausible solution for
a damped oscillatory system. It is easily demonstrated that
so Equation (56) becomes
Now, the only way in which the above equation can be satisfied at all times is if the
coefficients of
and
separately equate to zero, so that
These equations can be solved to give
Thus, the solution to the damped harmonic oscillator equation is written

(65) 
assuming that
(since
clearly cannot be negative). We conclude that the effect of a relatively small amount of damping, parameterized
by the damping constant ,
on a system which exhibits simple harmonic oscillation about a stable equilibrium state is to reduce the
angular frequency of the oscillation from its undamped value to
, and to cause the amplitude of the oscillation to decay exponentially
in time at the rate . This modified type of oscillation, which we shall refer to as damped harmonic oscillation, is illustrated in Figure 6. [Here,
,
, and . The solid line shows , whereas the dashed lines
show .]
Incidentally, if the damping is sufficiently large that
, which we shall assume is not the case, then
the system does not oscillate at all, and any motion simply decays away exponentially
in time (see Exercise 3).
Figure 6:
Damped harmonic oscillation.

Note that, although the angular frequency, , and decay rate, ,
of the damped harmonic oscillation specified in Equation (65) are determined by the constants appearing in the damped harmonic oscillator equation, (56), the initial amplitude, , and the phase angle, ,
of the oscillation
are determined by the initial conditions. In fact, if and
then it follows from Equation (65) that
giving
Note, further, that the damped harmonic oscillator equation is a linear differential equation: i.e., if is a
solution then so is , where is an arbitrary constant. It follows
that the solutions of this equation are superposable,
so that if and are two solutions corresponding to different initial
conditions then
is a third solution, where and
are arbitrary constants.
Multiplying the damped harmonic oscillator equation (56) by ,
we obtain

(70) 
which can be rearranged to give

(71) 
where

(72) 
is the total energy of the system: i.e., the sum of the kinetic and potential energies. Clearly, since the righthand side of (71)
cannot be positive, and is only zero when the system is stationary, the
total energy is not a conserved quantity, but instead decays monotonically in time due to the presence of damping. Now, the net rate at which the force (55) does work
on the mass is

(73) 
Note that the spring force (i.e., the first term on the righthand side) does negative work
on the mass (i.e., it reduces the system kinetic energy) when and are of the same sign, and does positive work when they are of the opposite sign. On average,
the spring force does no net work on the mass during an oscillation cycle. The
damping force, on the other hand, (i.e., the second term on the righthand side)
always does negative work on the mass, and, therefore, always acts to reduce the
system kinetic energy.
Next: Quality Factor
Up: Damped and Driven Harmonic
Previous: Damped and Driven Harmonic
Richard Fitzpatrick
20101011