Hence we see that the co-ordinate Qa corresponds to a normal vibration antisymmetrical
about the y-axis (x1 = x3, y1 = -y3; Fig. 29a) with frequency
The co-ordinates qs1, qs2 together correspond to two vibrations symmetrical about the
y-axis (x1 = -X3, y1 y3; Fig. 29b, c), whose frequencies Ws1, W82 are given by the roots
of the quadratic (in w2) characteristic equation
1
When 2 x = 75, all three frequencies become equal to those derived in Problem 1.
PROBLEM 3. The same as Problem 1, but for an unsymmetrical linear molecule ABC
(Fig. 30).
A
FIG. 30
SOLUTION. The longitudinal (x) and transverse (y) displacements of the atoms are related
by
mAX1+mBX2+mcx3 = 0, mAy1tmBy2+mcy3= 0,
MAhy1 = mcl2y3.
The potential energy of stretching and bending can be written
where 2l = li+l2. Calculations similar to those in Problem 1 give
for the transverse vibrations and the quadratic (in w2) equation
=
for the frequencies wil, W12 of the longitudinal vibrations.
$25. Damped oscillations
So far we have implied that all motion takes place in a vacuum, or else that
the effect of the surrounding medium on the motion may be neglected. In
reality, when a body moves in a medium, the latter exerts a resistance which
tends to retard the motion. The energy of the moving body is finally dissipated
by being converted into heat.
Motion under these conditions is no longer a purely mechanical process,
and allowance must be made for the motion of the medium itself and for the
internal thermal state of both the medium and the body. In particular, we
cannot in general assert that the acceleration of a moving body is a function
only of its co-ordinates and velocity at the instant considered; that is, there
are no equations of motion in the mechanical sense. Thus the problem of the
motion of a body in a medium is not one of mechanics.
There exists, however, a class of cases where motion in a medium can be
approximately described by including certain additional terms in the
§25
Damped oscillations
75
mechanical equations of motion. Such cases include oscillations with fre-
quencies small compared with those of the dissipative processes in the
medium. When this condition is fulfilled we may regard the body as being
acted on by a force of friction which depends (for a given homogeneous
medium) only on its velocity.
If, in addition, this velocity is sufficiently small, then the frictional force
can be expanded in powers of the velocity. The zero-order term in the expan-
sion is zero, since no friction acts on a body at rest, and so the first non-
vanishing term is proportional to the velocity. Thus the generalised frictional
force fir acting on a system executing small oscillations in one dimension
(co-ordinate x) may be written fir = - ax, where a is a positive coefficient
and the minus sign indicates that the force acts in the direction opposite to
that of the velocity. Adding this force on the right-hand side of the equation
of motion, we obtain (see (21.4))
mx = -kx-ax.
(25.1)
We divide this by m and put
k/m= wo2, a/m=2x; =
(25.2)
wo is the frequency of free oscillations of the system in the absence of friction,
and A is called the damping coefficient or damping decrement.
Thus the equation is
(25.3)
We again seek a solution x = exp(rt) and obtain r for the characteristic
equation r2+2xr + wo2 = 0, whence ¥1,2 = The general
solution of equation (25.3) is
c1exp(rit)+c2 exp(r2t).
Two cases must be distinguished. If   wo, we have two complex con-
jugate values of r. The general solution of the equation of motion can then
be written as
where A is an arbitrary complex constant, or as
= aexp(-Xt)cos(wta),
(25.4)
with w = V(w02-2) and a and a real constants. The motion described by
these formulae consists of damped oscillations. It may be regarded as being
harmonic oscillations of exponentially decreasing amplitude. The rate of
decrease of the amplitude is given by the exponent X, and the "frequency"
w is less than that of free oscillations in the absence of friction. For 1 wo,
the difference between w and wo is of the second order of smallness. The
decrease in frequency as a result of friction is to be expected, since friction
retards motion.
t The dimensionless product XT (where T = 2n/w is the period) is called the logarithmic
damping decrement.
76
Small Oscillations
§25
If A < wo, the amplitude of the damped oscillation is almost unchanged
during the period 2n/w. It is then meaningful to consider the mean values
(over the period) of the squared co-ordinates and velocities, neglecting the
change in exp( - At) when taking the mean. These mean squares are evidently
proportional to exp(-2xt). Hence the mean energy of the system decreases
as
(25.5)
where E0 is the initial value of the energy.
Next, let A > wo. Then the values of r are both real and negative. The
general form of the solution is
-
(25.6)
We see that in this case, which occurs when the friction is sufficiently strong,
the motion consists of a decrease in /x/, i.e. an asymptotic approach (as t ->
00)
to the equilibrium position. This type of motion is called aperiodic damping.
Finally, in the special case where A = wo, the characteristic equation has
the double root r = - 1. The general solution of the differential equation is
then
(25.7)
This is a special case of aperiodic damping.
For a system with more than one degree of freedom, the generalised
frictional forces corresponding to the co-ordinates Xi are linear functions of
the velocities, of the form
=
(25.8)
From purely mechanical arguments we can draw no conclusions concerning
the symmetry properties of the coefficients aik as regards the suffixes i and
k, but the methods of statistical physics make it possible to demonstrate
that in all cases
aki.
(25.9)
Hence the expressions (25.8) can be written as the derivatives
=
(25.10)
of the quadratic form
(25.11)
which is called the dissipative function.
The forces (25.10) must be added to the right-hand side of Lagrange's
equations:
(25.12)
t See Statistical Physics, $123, Pergamon Press, Oxford 1969.
