126 lines
5.4 KiB
Text
126 lines
5.4 KiB
Text
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
|