3.6.4 The Near-Resonant Case
When x % 2 we are faced with the case of primary parametric resonance. Some
measure of the nearness of x to the resonant value 2 will be in need, so we let
x ¼ 2 þ er
ð3:120Þ
where r is the so-called detuning parameter. This definition expresses the
assumption that x differs from the value 2 only by a small quantity er. Substituting
(3.120) into the x-dependent oscillatory terms of (3.114), near-resonant terms will
convert into resonant terms. We do not substitute into x-dependent terms that are
non-oscillatory (such as
1
2 qx
2 ), since this is unnecessary for the purpose. By noting
that
ðx À 1ÞT 0 ¼ ð2 þ erÞ À 1
ð
Þ T 0 ¼ T 0 þ erT 0 ¼ T 0 þ rT 1 ;
ðx þ 1ÞT 0 ¼ ð2 þ erÞ þ 1
ð
Þ T 0 ¼ 3T 0 þ erT 0 ¼ 3T 0 þ rT 1 ;
ð3:121Þ
the substitution of (3.120) into (3.114) yields:
D
2
0 h 1 þ h 1 ¼ Ài2A
0
À i2bA À 3cA
2 A
À
Á
e
iT 0 À cA
3 e
i3T 0
þ
1
2
qx
2 Ae
i3T 0 þ Ae
iT 0
À
Á
e
irT 1 þ cc:
ð3:122Þ
Resonant terms (proportional to e
iT0 ) appear here, that will feed off secular terms
in the solution for h 1 . To eliminate secular terms we require the sum of all resonant
terms to vanish, that is, the solvability condition becomes:
Ài2A
0
À i2bA À 3cA
2 A þ
1
2
qx
2 Ae
irT 1 ¼ 0:
ð3:123Þ
This condition differs from that of the non-resonant case (3.115) only by an
added excitation term (last term of (3.123)). With the condition fulfilled, a particular
solution to (3.122) becomes
h 1 ¼
1
8
cA
3 e
i3T 0 À
1
16
qx
2 Ae
ið3T 0 þ rT 1 Þ
þ cc:
ð3:124Þ
For determining A(T 1 ) we let A =
1
2 ae
iu , insert into (3.123), separate real and
imaginary parts, and obtain the modulation equations for the resonant case:
a
0
¼ Àba þ
1
4
qx
2 a sin rT 1 À 2u
ð
Þ ;
au
0
¼
3
8
ca
3
À
1
4
qx
2 a cos rT 1 À 2u
ð
Þ :
ð3:125Þ
134
3 Nonlinear Vibrations: Classical Local Theory