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.
§29
Resonance in non-linear oscillations
91
This equation, however, does not suffice to determine the resulting ampli-
tude of the oscillations. The attainment of a finite amplitude involves non-
linear effects, and to include these in the equation of motion we must retain
also the terms non-linear in x(2):
= cos(2wo+e)t. (29.11)
The problem can be considerably simplified by virtue of the following fact.
Putting on the right-hand side of (29.11) x(2) = b cos[(wo++)+8], where
b is the required amplitude of the resonance oscillations and 8 a constant
phase difference which is of no importance in what follows, and writing the
product of cosines as a sum, we obtain a term (afb/3mwo2)
of the ordinary resonance type (with respect to the eigenfrequency wo of the
system). The problem thus reduces to that considered at the beginning of
this section, namely ordinary resonance in a non-linear system, the only
differences being that the amplitude of the external force is here represented
by afb/3wo2, and E is replaced by 1/6. Making this change in equation (29.4),
we have
Solving for b, we find the possible values of the amplitude:
b=0,
(29.12)
(29.13)
1
(29.14)
Figure 33 shows the resulting dependence of b on € for K > 0; for K < 0
the curves are the reflections (in the b-axis) of those shown. The points B
and C correspond to the values E = To the left of
B, only the value b = 0 is possible, i.e. there is no resonance, and oscillations
of frequency near wo are not excited. Between B and C there are two roots,
b = 0(BC) and (29.13) (BE). Finally, to the right of C there are three roots
(29.12)-(29.14). Not all these, however, correspond to stable oscillations.
The value b = 0 is unstable on BC, and it can also be shown that the middle
root (29.14) always gives instability. The unstable values of b are shown in
Fig. 33 by dashed lines.
Let us examine, for example, the behaviour of a system initially "at rest"
as the frequency of the external force is gradually diminished. Until the point
t This segment corresponds to the region of parametric resonance (27.12), and a com-
parison of (29.10) and (27.8) gives 1h = 2af/3mwo4. The condition 12af/3mwo3 > 4X for
which the phenomenon can exist corresponds to h > hk.
+ It should be recalled that only resonance phenomena are under consideration. If these
phenomena are absent, the system is not literally at rest, but executes small forced oscillations
of frequency y.
92
Small Oscillations
§29
C is reached, b = 0, but at C the state of the system passes discontinuously
to the branch EB. As € decreases further, the amplitude of the oscillations
decreases to zero at B. When the frequency increases again, the amplitude
increases along BE.-
b
E
E
A
B
C D
FIG. 33
The cases of resonance discussed above are the principal ones which may
occur in a non-linear oscillating system. In higher approximations, resonances
appear at other frequencies also. Strictly speaking, a resonance must occur
at every frequency y for which ny + mwo = wo with n and m integers, i.e. for
every y = pwo/q with P and q integers. As the degree of approximation
increases, however, the strength of the resonances, and the widths of the
frequency ranges in which they occur, decrease so rapidly that in practice
only the resonances at frequencies y 2 pwo/q with small P and q can be ob-
served.
PROBLEM
Determine the function b(e) for resonance at frequencies y 22 3 wo.
SOLUTION. In the first approximation, x(1) = -(f/8mwo2) cos(3wo+t) For the second
approximation x(2) we have from (29.1) the equation
= -3,8x(1)x(2)2,
where only the term which gives the required resonance has been retained on the right-hand
side. Putting x(2) = b cos[(wo+)+8] and taking the resonance term out of the product
of three cosines, we obtain on the right-hand side the expression
(3,3b2f(32mwo2) cos[(wotle)t-28].
Hence it is evident that b(e) is obtained by replacing f by 3,8b2f/32wo², and E by JE, in
(29.4):
Ab4.
The roots of this equation are
b=0,
Fig. 34 shows a graph of the function b(e) for k>0. Only the value b=0 (the e-axis) and
the branch AB corresponds to stability. The point A corresponds to EK = 3(4x2)2-A3)/4kA,
t It must be noticed, however, that all the formulae derived here are valid only when the
amplitude b (and also E) is sufficiently small. In reality, the curves BE and CF meet, and at
their point of intersection the oscillation ceases; thereafter, b = 0.
