which includes terms of the first two orders. To determine the limits of instability, we again
treat the coefficients as constants, obtaining
cos(wote)t-
+[-2woebo+thwo861] sin(wo+e)t.
+[-30002a1+thanoPao] cos 2(wote)t+
sin 2(wote)t+[c1wo+thwo2ao] 0.
Hence a1 = hao/6, b1 = hbo/6, C1 = -thao, and the limits aret € = -5h2wo/24, € = h2wo/24.
PROBLEM 3. Find the conditions for parametric resonance in small oscillations of a simple
pendulum whose point of support oscillates vertically.
SOLUTION. The Lagrangian derived in §5, Problem 3(c), gives for small oscillations
( < 1) the equation of motion + wo2[1+(4a/1) cos(2wo+t)) = 0, where wo2 = g/l.
Hence we see that the parameter h is here represented by 4all. The condition (27.11), for
example, becomes |
§28. Anharmonic oscillations
The whole of the theory of small oscillations discussed above is based on
the expansion of the potential and kinetic energies of the system in terms of
the co-ordinates and velocities, retaining only the second-order terms. The
equations of motion are then linear, and in this approximation we speak of
linear oscillations. Although such an expansion is entirely legitimate when
the amplitude of the oscillations is sufficiently small, in higher approxima-
tions (called anharmonic or non-linear oscillations) some minor but qualitatively
different properties of the motion appear.
Let us consider the expansion of the Lagrangian as far as the third-order
terms. In the potential energy there appear terms of degree three in the co-
ordinates Xi, and in the kinetic energy terms containing products of velocities
and co-ordinates, of the form XEXKXI. This difference from the previous
expression (23.3) is due to the retention of terms linear in x in the expansion
of the functions aik(q). Thus the Lagrangian is of the form
(28.1)
where Nikl, liki are further constant coefficients.
If we change from arbitrary co-ordinates Xi to the normal co-ordinates Qx
of the linear approximation, then, because this transformation is linear, the
third and fourth sums in (28.1) become similar sums with Qx and Qa in place
t
Generally, the width AE of the region of instability in resonance near the frequency
2wo/n is given by
AE =
a result due to M. BELL (Proceedings of the Glasgow Mathematical Association 3, 132, 1957).
§28
Anharmonic oscillations
85
of the co-ordinates Xi and the velocities Xr. Denoting the coefficients in these
new sums by dapy and Hapy's we have the Lagrangian in the form
(28.2)
a
a,B,Y
We shall not pause to write out in their entirety the equations of motion
derived from this Lagrangian. The important feature of these equations is
that they are of the form
(28.3)
where fa are homogeneous functions, of degree two, of the co-ordinates Q
and their time derivatives.
Using the method of successive approximations, we seek a solution of
these equations in the form
(28.4)
where Qa2, and the Qx(1) satisfy the "unperturbed" equations
i.e. they are ordinary harmonic oscillations:
(28.5)
Retaining only the second-order terms on the right-hand side of (28.3) in
the next approximation, we have for the Qx(2) the equations
(28.6)
where (28.5) is to be substituted on the right. This gives a set of inhomo-
geneous linear differential equations, in which the right-hand sides can be
represented as sums of simple periodic functions. For example,
cos(wpt + ag)
Thus the right-hand sides of equations (28.6) contain terms corresponding
to oscillations whose frequencies are the sums and differences of the eigen-
frequencies of the system. The solution of these equations must be sought
in a form involving similar periodic factors, and so we conclude that, in the
second approximation, additional oscillations with frequencies
wa+w
(28.7)
including the double frequencies 2wa and the frequency zero (corresponding
to a constant displacement), are superposed on the normal oscillations of the
system. These are called combination frequencies. The corresponding ampli-
tudes are proportional to the products Axap (or the squares aa2) of the cor-
responding normal amplitudes.
In higher approximations, when further terms are included in the expan-
sion of the Lagrangian, combination frequencies occur which are the sums
and differences of more than two Wa; and a further phenomenon also appears.
86
Small Oscillations
§28
In the third approximation, the combination frequencies include some which
coincide with the original frequencies W Wa+wp-wp). When the method
described above is used, the right-hand sides of the equations of motion there-
fore include resonance terms, which lead to terms in the solution whose
amplitude increases with time. It is physically evident, however, that the
magnitude of the oscillations cannot increase of itself in a closed system
with no external source of energy.
In reality, the fundamental frequencies Wa in higher approximations are
not equal to their "unperturbed" values wa(0) which appear in the quadratic
expression for the potential energy. The increasing terms in the solution
arise from an expansion of the type
which is obviously not legitimate when t is sufficiently large.
In going to higher approximations, therefore, the method of successive
approximations must be modified so that the periodic factors in the solution
shall contain the exact and not approximate values of the frequencies. The
necessary changes in the frequencies are found by solving the equations and
requiring that resonance terms should not in fact appear.
We may illustrate this method by taking the example of anharmonic oscil-
lations in one dimension, and writing the Lagrangian in the form
L =
(28.8)
The corresponding equation of motion is
(28.9)
We shall seek the solution as a series of successive approximations:
where
x(1) = a cos wt,
(28.10)
with the exact value of w, which in turn we express as w=wotw1)+w(2)+....
(The initial phase in x(1) can always be made zero by a suitable choice of the
origin of time.) The form (28.9) of the equation of motion is not the most
convenient, since, when (28.10) is substituted in (28.9), the left-hand side is
not exactly zero. We therefore rewrite it as
(28.11)
Putting x(1)+x(2), w wotwi and omitting terms of above the
second order of smallness, we obtain for x(2) the equation
= aa2 cos2wt+2wowlda cos wt
= 1xa2-1xa2 cos 2wt + 2wow1)a cos wt.
The condition for the resonance term to be absent from the right-hand side
is simply w(1) = 0, in agreement with the second approximation discussed
