3.5.4 The Method of Multiple Scales
For linear systems the amplitudes and frequencies of oscillations are independent
quantities. For example, the oscillation frequency of a freely swinging pendulum is
independent of the oscillation amplitude, as long as the amplitudes are very small.
Nonlinear systems, by contrast, typically display dependency between amplitude
and frequency, For example, large-amplitude oscillations of a free pendulum occur
at a lower frequency than do small oscillations.
The straightforward expansion technique described above fails to correctly
represent a proper relation between amplitude and frequency. By the method of
multiple scales this deficiency is overcome by permitting the solution to be a
function of multiple independent time-variables, or -scales. For example, a fast
scale can be used for capturing motions at frequencies comparable to the linear
natural frequency of the system, while a slow scale accounts for slow modulations
of amplitudes and phases. This will allow for a proper amplitude-frequency relation
to be represented, which is free of secular terms.
By the method of multiple scales one generally assumes a uniformly valid
expansion of the form:
hðs; eÞ ¼ h 0 ðT 0 ; T 1 ; T 2 ; . . .Þ þ eh 1 ðT 0 ; T 1 ; T 2 ; . . .Þ þ e
2 h 2 ðT 0 ; T 1 ; T 2 ; . . .Þ þ Á Á Á
ð3:63Þ
where h j , j = 0, 1, … are functions to be determined, and T j , j = 0, 1, … are
independent time-scales:
T j ¼ e
j s; j ¼ 0; 1; . . .; e ( 1
ð3:64Þ
As many independent time-scales are needed as there are terms in the expansion
(3.63). That is, if the expansion is carried out to order O(e
n ) one needs the scales T j ,
j = 0, n. It will appear shortly, that the presence of several independent time-scales
allows one to impose conditions that will eliminate secular terms of the solution.
As with the straightforward expansion, the assumed expansion (now (3.63)) is
substituted into the equations of motion, and coefficients to like powers of e are
zeroed to yield a set of perturbation equations. Resonant terms will appear, causing
secular terms in the solution. Due to the independent time-scales, however, the
perturbation equations do not uniquely define the set of expansion functions h j . It is
like fitting a parabola through two data points only; there is no unique way to do
this, unless an additional restriction is imposed. For uniquely defining the expansion functions h j , one is thus free to impose the condition that the solution should
contain no secular terms. There is no cheating or magic about this; the condition
merely ensures the expansion to be uniformly valid, as assumed. That is, if a
solution of the form (3.63) exists, then it must be free of secular terms because
otherwise the solution would not be uniformly valid.
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3 Nonlinear Vibrations: Classical Local Theory
For linear systems the amplitudes and frequencies of oscillations are independent
quantities. For example, the oscillation frequency of a freely swinging pendulum is
independent of the oscillation amplitude, as long as the amplitudes are very small.
Nonlinear systems, by contrast, typically display dependency between amplitude
and frequency, For example, large-amplitude oscillations of a free pendulum occur
at a lower frequency than do small oscillations.
The straightforward expansion technique described above fails to correctly
represent a proper relation between amplitude and frequency. By the method of
multiple scales this deficiency is overcome by permitting the solution to be a
function of multiple independent time-variables, or -scales. For example, a fast
scale can be used for capturing motions at frequencies comparable to the linear
natural frequency of the system, while a slow scale accounts for slow modulations
of amplitudes and phases. This will allow for a proper amplitude-frequency relation
to be represented, which is free of secular terms.
By the method of multiple scales one generally assumes a uniformly valid
expansion of the form:
hðs; eÞ ¼ h 0 ðT 0 ; T 1 ; T 2 ; . . .Þ þ eh 1 ðT 0 ; T 1 ; T 2 ; . . .Þ þ e
2 h 2 ðT 0 ; T 1 ; T 2 ; . . .Þ þ Á Á Á
ð3:63Þ
where h j , j = 0, 1, … are functions to be determined, and T j , j = 0, 1, … are
independent time-scales:
T j ¼ e
j s; j ¼ 0; 1; . . .; e ( 1
ð3:64Þ
As many independent time-scales are needed as there are terms in the expansion
(3.63). That is, if the expansion is carried out to order O(e
n ) one needs the scales T j ,
j = 0, n. It will appear shortly, that the presence of several independent time-scales
allows one to impose conditions that will eliminate secular terms of the solution.
As with the straightforward expansion, the assumed expansion (now (3.63)) is
substituted into the equations of motion, and coefficients to like powers of e are
zeroed to yield a set of perturbation equations. Resonant terms will appear, causing
secular terms in the solution. Due to the independent time-scales, however, the
perturbation equations do not uniquely define the set of expansion functions h j . It is
like fitting a parabola through two data points only; there is no unique way to do
this, unless an additional restriction is imposed. For uniquely defining the expansion functions h j , one is thus free to impose the condition that the solution should
contain no secular terms. There is no cheating or magic about this; the condition
merely ensures the expansion to be uniformly valid, as assumed. That is, if a
solution of the form (3.63) exists, then it must be free of secular terms because
otherwise the solution would not be uniformly valid.
120
3 Nonlinear Vibrations: Classical Local Theory