§27
Parametric resonance
81
constants: x1(t+T) = 1x1(t), x2(t+T) = u2x2(t). The most general functions
having this property are
(t) = 111t/TII1(t), x2(t) = M2t/T112(t),
(27.3)
where II1(t), II2(t) are purely periodic functions of time with period T.
The constants 1 and 2 in these functions must be related in a certain way.
Multiplying the equations +2(t)x1 = 0, 2+w2(t)x2 = 0 by X2 and X1
respectively and subtracting, we = = 0, or
X1X2-XIX2 = constant.
(27.4)
For any functions x1(t), x2(t) of the form (27.3), the expression on the left-
hand side of (27.4) is multiplied by H1U2 when t is replaced by t + T. Hence
it is clear that, if equation (27.4) is to hold, we must have
M1M2=1.
(27.5)
Further information about the constants M1, 2 can be obtained from the
fact that the coefficients in equation (27.2) are real. If x(t) is any integral of
such an equation, then the complex conjugate function x* (t) must also be
an integral. Hence it follows that U1, 2 must be the same as M1*, M2*, i.e.
either 1 = M2* or 1 and 2 are both real. In the former case, (27.5) gives
M1 = 1/1*, i.e. /1112 = 1/22/2 = 1: the constants M1 and 2 are of modulus
unity.
In the other case, two independent integrals of equation (27.2) are
x2(t) = -/I2(t),
(27.6)
with a positive or negative real value of u (Iu/ # 1). One of these functions
(x1 or X2 according as /x/ > 1 or /u/ <1) increases exponentially with time.
This means that the system at rest in equilibrium (x = 0) is unstable: any
deviation from this state, however small, is sufficient to lead to a rapidly
increasing displacement X. This is called parametric resonance.
It should be noticed that, when the initial values of x and x are exactly
zero, they remain zero, unlike what happens in ordinary resonance (§22),
in which the displacement increases with time (proportionally to t) even from
initial values of zero.
Let us determine the conditions for parametric resonance to occur in the
important case where the function w(t) differs only slightly from a constant
value wo and is a simple periodic function:
w2(1) = con2(1+h cosyt)
(27.7)
where
the
constant h 1; we shall suppose h positive, as may always be
done by suitably choosing the origin of time. As we shall see below, para-
metric resonance is strongest if the frequency of the function w(t) is nearly
twice wo. Hence we put y = 2wo+e, where E < wo.
82
Small Oscillations
§27
The solution of equation of motion+
+wo2[1+hcos(2wot)t]x
(27.8)
may be sought in the form
(27.9)
where a(t) and b(t) are functions of time which vary slowly in comparison
with the trigonometrical factors. This form of solution is, of course, not
exact. In reality, the function x(t) also involves terms with frequencies which
differ from wother by integral multiples of 2wo+e; these terms are, how-
ever, of a higher order of smallness with respect to h, and may be neglected
in a first approximation (see Problem 1).
We substitute (27.9) in (27.8) and retain only terms of the first order in
€, assuming that à ea, b ~ eb; the correctness of this assumption under
resonance conditions is confirmed by the result. The products of trigono-
metrical functions may be replaced by sums:
cos(wot1e)t.cos(2wote)t =
etc., and in accordance with what was said above we omit terms with fre-
quency 3(wo+1e). The result is
= 0.
If this equation is to be justified, the coefficients of the sine and cosine must
both be zero. This gives two linear differential equations for the functions
a(t) and b(t). As usual, we seek solutions proportional to exp(st). Then
= 0, 1(e-thwo)a- - sb = 0, and the compatibility condition
for these two algebraic equations gives
(27.10)
The condition for parametric resonance is that S is real, i.e. s2 > 0.1 Thus
parametric resonance occurs in the range
(27.11)
on either side of the frequency 2wo.ll The width of this range is proportional
to h, and the values of the amplification coefficient S of the oscillations in the
range are of the order of h also.
Parametric resonance also occurs when the frequency y with which the
parameter varies is close to any value 2wo/n with n integral. The width of the
t An equation of this form (with arbitrary y and h) is called in mathematical physics
Mathieu's equation.
+ The constant u in (27.6) is related to s by u = - exp(sn/wo); when t is replaced by
t+2n/2wo, the sine and cosine in (27.9) change sign.
II If we are interested only in the range of resonance, and not in the values of S in that
range, the calculations may be simplified by noting that S = 0 at the ends of the range, i.e.
the coefficients a and b in (27.9) are constants. This gives immediately € = thwo as in
(27.11).
§27
Parametric resonance
83
resonance range (region of instability) decreases rapidly with increasing N,
however, namely as hn (see Problem 2, footnote). The amplification co-
efficient of the oscillations also decreases.
The phenomenon of parametric resonance is maintained in the presence
of slight friction, but the region of instability becomes somewhat narrower.
As we have seen in §25, friction results in a damping of the amplitude of
oscillations as exp(- - At). Hence the amplification of the oscillations in para-
metric resonance is as exp[(s-1)t] with the positive S given by the solution
for the frictionless case, and the limit of the region of instability is given by
the equation - X = 0. Thus, with S given by (27.10), we have for the resonance
range, instead of (27.11),
(27.12)
It should be noticed that resonance is now possible not for arbitrarily
small amplitudes h, but only when h exceeds a "threshold" value hk. When
(27.12) holds, hk = 4X/wo. It can be shown that, for resonance near the fre-
quency 2wo/n, the threshold hk is proportional to X1/n, i.e. it increases with n.
PROBLEMS
PROBLEM 1. Obtain an expression correct as far as the term in h2 for the limits of the region
of instability for resonance near 2 = 2wo.
SOLUTION. We seek the solution of equation (27.8) in the form
x = ao cos(wo+1e)t +bo (wo+le)t +a1 cos 3( (wo+le)t +b1 sin 3(wo+le)t,
which includes terms of one higher order in h than (27.9). Since only the limits of the region
of instability are required, we treat the coefficients ao, bo, a1, b1 as constants in accordance
with the last footnote. Substituting in (27.8), we convert the products of trigonometrical
functions into sums and omit the terms of frequency 5(wo+1) in this approximation. The
result is
[
- cos(wo+l)
cos 3(wo+1e)tt
sin 3(wo+1e)t = 0.
In the terms of frequency wothe we retain terms of the second order of smallness, but in
those of frequency 3( (wo+1) only the first-order terms. Each of the expressions in brackets
must separately vanish. The last two give a1 = hao/16, b1 = hbo/16, and then the first two
give woe +thwo2+1e2-h2wo2/32 = 0.
Solving this as far as terms of order h2, we obtain the required limits of E:
= theo-h20003.
PROBLEM 2. Determine the limits of the region of instability in resonance near y = wo.
SOLUTION. Putting y = wote, we obtain the equation of motion
0.
Since the required limiting values of ~~h2, we seek a solution in the form
ao cos(wote)t sin(wote)t cos 2(wo+e)t +b1 sin 2(wo+e)t- +C1,
84
Small Oscillations
§28
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
§29
Resonance in non-linear oscillations
87
at the beginning of this section. Solving the inhomogeneous linear equation
in the usual way, we have
(28.12)
Putting in (28.11) X wo+w(2), we obtain the equa-
tion for x(3)
= -
or, substituting on the right-hand side (28.10) and (28.12) and effecting
simple transformation,
wt.
Equating to zero the coefficient of the resonance term cos wt, we find the
correction to the fundamental frequency, which is proportional to the squared
amplitude of the oscillations:
(28.13)
The combination oscillation of the third order is
(28.14)
$29. Resonance in non-linear oscillations
When the anharmonic terms in forced oscillations of a system are taken
into account, the phenomena of resonance acquire new properties.
Adding to the right-hand side of equation (28.9) an external periodic force
of frequency y, we have
+2x+wo2x=(fm)cos = yt - ax2-Bx3;
(29.1)
here the frictional force, with damping coefficient A (assumed small) has also
been included. Strictly speaking, when non-linear terms are included in the
equation of free oscillations, the terms of higher order in the amplitude of
the external force (such as occur if it depends on the displacement x) should
also be included. We shall omit these terms merely to simplify the formulae;
they do not affect the qualitative results.
Let y = wote with E small, i.e. y be near the resonance value. To ascertain
the resulting type of motion, it is not necessary to consider equation (29.1)
if we argue as follows. In the linear approximation, the amplitude b is given
88
Small Oscillations
§29
near resonance, as a function of the amplitude f and frequency r of the
external force, by formula (26.7), which we write as
(29.2)
The non-linearity of the oscillations results in the appearance of an ampli-
tude dependence of the eigenfrequency, which we write as
wo+kb2,
(29.3)
the constant K being a definite function of the anharmonic coefficients (see
(28.13)). Accordingly, we replace wo by wo + kb2 in formula (29.2) (or, more
precisely, in the small difference y-wo). With y-wo=e, the resulting
equation is
=
(29.4)
or
Equation (29.4) is a cubic equation in b2, and its real roots give the ampli-
tude of the forced oscillations. Let us consider how this amplitude depends
on the frequency of the external force for a given amplitude f of that force.
When f is sufficiently small, the amplitude b is also small, so that powers
of b above the second may be neglected in (29.4), and we return to the form
of b(e) given by (29.2), represented by a symmetrical curve with a maximum
at the point E = 0 (Fig. 32a). As f increases, the curve changes its shape,
though at first it retains its single maximum, which moves to positive E if
K > 0 (Fig. 32b). At this stage only one of the three roots of equation (29.4)
is real.
When f reaches a certain value f k (to be determined below), however, the
nature of the curve changes. For all f > fk there is a range of frequencies in
which equation (29.4) has three real roots, corresponding to the portion
BCDE in Fig. 32c.
The limits of this range are determined by the condition db/de = 8 which
holds at the points D and C. Differentiating equation (29.4) with respect to
€, we have
db/de =
Hence the points D and C are determined by the simultaneous solution of
the equations
2-4kb2e+3k264+2 0
(29.5)
and (29.4). The corresponding values of E are both positive. The greatest
amplitude is reached where db/de = 0. This gives E = kb2, and from (29.4)
we have
bmax = f/2mwod;
(29.6)
this is the same as the maximum value given by (29.2).
§29
Resonance in non-linear oscillations
89
It may be shown (though we shall not pause to do so heret) that, of the
three real roots of equation (29.4), the middle one (represented by the dotted
part CD of the curve in Fig. 32c) corresponds to unstable oscillations of the
system: any action, no matter how slight, on a system in such a state causes
it to oscillate in a manner corresponding to the largest or smallest root (BC
or DE). Thus only the branches ABC and DEF correspond to actual oscil-
lations of the system. A remarkable feature here is the existence of a range of
frequencies in which two different amplitudes of oscillation are possible. For
example, as the frequency of the external force gradually increases, the ampli-
tude of the forced oscillations increases along ABC. At C there is a dis-
continuity of the amplitude, which falls abruptly to the value corresponding
to E, afterwards decreasing along the curve EF as the frequency increases
further. If the frequency is now diminished, the amplitude of the forced
oscillations varies along FD, afterwards increasing discontinuously from D
to B and then decreasing along BA.
b
(a)
to
b
(b)
f<f
b
(c)
f>tp
B
C
Di
A
E
F
€
FIG. 32
To calculate the value of fk, we notice that it is the value of f for which
the two roots of the quadratic equation in b2 (29.5) coincide; for f = f16, the
section CD reduces to a point of inflection. Equating to zero the discriminant
t The proof is given by, for example, N.N. BOGOLIUBOV and Y.A. MITROPOLSKY, Asymp-
totic Methods in the Theory of Non-Linear Oscillations, Hindustan Publishing Corporation,
Delhi 1961.
4
90
Small Oscillations
§29
of (29.5), we find E2 = 3X², and the corresponding double root is kb2 = 2e/3.
Substitution of these values of b and E in (29.4) gives
32m2wo2x3/31/3k.
(29.7)
Besides the change in the nature of the phenomena of resonance at fre-
quencies y 22 wo, the non-linearity of the oscillations leads also to new
resonances in which oscillations of frequency close to wo are excited by an
external force of frequency considerably different from wo.
Let the frequency of the external force y 22 two, i.e. y = two+e. In the
first (linear) approximation, it causes oscillations of the system with the same
frequency and with amplitude proportional to that of the force:
x(1)= (4f/3mwo2) cos(two+e)t
(see (22.4)). When the non-linear terms are included (second approximation),
these oscillations give rise to terms of frequency 2y 22 wo on the right-hand
side of the equation of motion (29.1). Substituting x(1) in the equation
= -
using the cosine of the double angle and retaining only the resonance term
on the right-hand side, we have
= - (8xf2/9m2w04) cos(wo+2e)t.
(29.8)
This equation differs from (29.1) only in that the amplitude f of the force is
replaced by an expression proportional to f2. This means that the resulting
resonance is of the same type as that considered above for frequencies
y 22 wo, but is less strong. The function b(e) is obtained by replacing f by
- 8xf2/9mwo4, and E by 2e, in (29.4):
62[(2e-kb2)2+12] = 16x2f4/81m4w010.
(29.9)
Next, let the frequency of the external force be 2= 2wote In the first
approximation, we have x(1) = - (f/3mwo2) cos(2wo+e)t. On substituting
in equation (29.1), we do not obtain terms representing an
external force in resonance such as occurred in the previous case. There is,
however, a parametric resonance resulting from the third-order term pro-
portional to the product x(1)x(2). If only this is retained out of the non-linear
terms, the equation for x(2) is
=
or
(29.10)
i.e. an equation of the type (27.8) (including friction), which leads, as we
have seen, to an instability of the oscillations in a certain range of frequencies.