(3.193). Letting a′ = w′ = 0, we first note that a = 0 is not a solution, by contrast to
the case of the parametrically excited pendulum (cf. Section 3.6.4). Squaring and
adding the two equations (with a′ = w′ = 0) we find that a is given by the implicit
solution to the following frequency response equation:
2bx
2
0
À
Á 2 þ 2rx 0 À
3
4
ca
2
2
!
a
2
¼ q
2
:
ð3:194Þ
The corresponding phase w is then given by
tan w ¼
Àbx 0
r À
3c
8x 0
a 2
:
ð3:195Þ
Having determined a from (3.194) and w from (3.195), the function A(T 1 ) is
given by (3.190) and (3.192) as:
AðT 1 Þ ¼
1
2
ae
iu
¼
1
2
ae
iðrT 1 ÀwÞ
;
ð3:196Þ
and so the approximate solution (3.189) becomes
uðt; eÞ ¼
1
2
ae
iðrT 1 ÀwÞ e
ix 0 T 0 þ e
c
64x 2
0
a
3 e
i3ðrT 1 ÀwÞ e
i3x 0 T 0 þ Oðe
2
Þ þ cc
¼ a cos rT 1 þ x 0 T 0 À w
ð
Þ
þ e
c
32x 2
0
a
3 cos 3 rT 1 þ x 0 T 0 À w
ð
Þ
ð
Þ þ Oðe
2
Þ:
ð3:197Þ
Inserting (3.185), T 0 = t and T 1 = et, we find that the stationary periodic solutions u(t) for the primary resonant Duffing equation are given by:
uðtÞ ¼ a cos Xt À w
ð
Þ
þ e
c
32x 2
0
a
3 cos 3 Xt À w
ð
Þ
ð
ÞþOðe
2
Þ;
ð3:198Þ
where the constants a and w are given by (3.194) and (3.195) with r = X – x 0 , and
where e now merely serves the purpose of indicating the level of approximation.
Note that the oscillations occur at the frequency of excitation X, and a higher
harmonic 3X of this.
Nonstationary (transient) solutions are also given by (3.198), though, with time
varying amplitudes and phases a(t) and w(t) as determined by the solutions to the
modulation equations (3.193) with relevant initial conditions.
Plotting Frequency Responses To see how the stationary amplitude a varies
with the excitation frequency X we plot the frequency response. Equation (3.194) is
not readily solved for a, since one has to locate the roots of a third order polynomial
156
3 Nonlinear Vibrations: Classical Local Theory