Forced oscillations under friction
77
The dissipative function itself has an important physical significance: it
gives the rate of dissipation of energy in the system. This is easily seen by
calculating the time derivative of the mechanical energy of the system. We
have
aL
=
Since F is a quadratic function of the velocities, Euler's theorem on homo-
geneous functions shows that the sum on the right-hand side is equal to 2F.
Thus
dE/dt==2-2F,
(25.13)
i.e. the rate of change of the energy of the system is twice the dissipative
function. Since dissipative processes lead to loss of energy, it follows that
F > 0, i.e. the quadratic form (25.11) is positive definite.
The equations of small oscillations under friction are obtained by adding
the forces (25.8) to the right-hand sides of equations (23.5):
=
(25.14)
Putting in these equations XK = Ak exp(rt), we obtain, on cancelling exp(rt),
a set of linear algebraic equations for the constants Ak:
(25.15)
Equating to zero their determinant, we find the characteristic equation, which
determines the possible values of r:
(25.16)
This is an equation in r of degree 2s. Since all the coefficients are real,
its roots are either real, or complex conjugate pairs. The real roots must be
negative, and the complex roots must have negative real parts, since other-
wise the co-ordinates, velocities and energy of the system would increase
exponentially with time, whereas dissipative forces must lead to a decrease
of the energy.
§26. Forced oscillations under friction
The theory of forced oscillations under friction is entirely analogous to
that given in §22 for oscillations without friction. Here we shall consider
in detail the case of a periodic external force, which is of considerable interest.
78
Small Oscillations
§26
Adding to the right-hand side of equation (25.1) an external force f cos st
and dividing by m, we obtain the equation of motion:
+2*+wox=(fm)cos = yt.
(26.1)
The solution of this equation is more conveniently found in complex form,
and so we replace cos st on the right by exp(iyt):
exp(iyt).
We seek a particular integral in the form x = B exp(iyt), obtaining for B
the value
(26.2)
Writing B = exp(i8), we have
b tan 8 = 2xy/(y2-wo2).
(26.3)
Finally, taking the real part of the expression B exp(iyt) = b exp[i(yt+8)],
we find the particular integral of equation (26.1); adding to this the general
solution of that equation with zero on the right-hand side (and taking for
definiteness the case wo > 1), we have
x = a exp( - At) cos(wtta)+bcos(yt+8)
(26.4)
The first term decreases exponentially with time, so that, after a sufficient
time, only the second term remains:
x = b cos(yt+8).
(26.5)
The expression (26.3) for the amplitude b of the forced oscillation increases
as y approaches wo, but does not become infinite as it does in resonance
without friction. For a given amplitude f of the force, the amplitude of the
oscillations is greatest when y = V(w02-2)2); for A < wo, this differs from
wo only by a quantity of the second order of smallness.
Let us consider the range near resonance, putting y = wote with E small,
and suppose also that A < wo. Then we can approximately put, in (26.2),
22 =(y+wo)(y-wo) 22 2woe, 2ixy 22 2ixwo, SO that
B = -f/2m(e-ii))wo
(26.6)
or
b f/2mw01/(22+12),
tan 8 = N/E.
(26.7)
A property of the phase difference 8 between the oscillation and the external
force is that it is always negative, i.e. the oscillation "lags behind" the force.
Far from resonance on the side < wo, 8 0; on the side y > wo, 8
-77.
The change of 8 from zero to - II takes place in a frequency range near wo
which is narrow (of the order of A in width); 8 passes through - 1/2 when
y = wo. In the absence of friction, the phase of the forced oscillation changes
discontinuously by TT at y = wo (the second term in (22.4) changes sign);
when friction is allowed for, this discontinuity is smoothed out.
§26
Forced oscillations under friction
79
In steady motion, when the system executes the forced oscillations given
by (26.5), its energy remains unchanged. Energy is continually absorbed by
the system from the source of the external force and dissipated by friction.
Let I(y) be the mean amount of energy absorbed per unit time, which depends
on the frequency of the external force. By (25.13) we have I(y) = 2F, where
F is the average value (over the period of oscillation) of the dissipative func-
tion. For motion in one dimension, the expression (25.11) for the dissipative
function becomes F = 1ax2 = Amx2. Substituting (26.5), we have
F = mb22 sin2(yt+8).
The time average of the squared sine is 1/2 so that
I(y) = Mmb2y2. =
(26.8)
Near resonance we have, on substituting the amplitude of the oscillation
from (26.7),
I(e) =
(26.9)
This is called a dispersion-type frequency dependence of the absorption.
The half-width of the resonance curve (Fig. 31) is the value of E for which
I(e) is half its maximum value (E = 0). It is evident from (26.9) that in the
present case the half-width is just the damping coefficient A. The height of
the maximum is I(0) = f2/4mx, and is inversely proportional to . Thus,
I/I(O)
/2
€
-1
a
FIG. 31
when the damping coefficient decreases, the resonance curve becomes more
peaked. The area under the curve, however, remains unchanged. This area
is given by the integral
[ ((7) dy = [ I(e) de.
Since I(e) diminishes rapidly with increasing E, the region where |el is
large is of no importance, and the lower limit may be replaced by - 80, and
I(e) taken to have the form given by (26.9). Then we have
"
(26.10)
80
Small Oscillations
