where T 0 = t and T 1 = et are the fast and the slow time-scale, respectively, and u 0
and u 1 are the unknown functions to be determined. On substituting the expansion
into (3.178) one obtains:
D
2
0 þ 2eD 0 D 1 þ Oðe
2
Þ
À
Á
u 0 þ eu 1 þ Oðe
2
Þ
À
Á þ x
2
0 u 0 þ eu 1 þ Oðe
2
Þ
À
Á
¼ e q cos XT 0 À 2bx 0 D 0 þ eD 1
ð
Þu 0 þ eu 1 þ Oðe
2
Þ
À
Á
Â
Àc u 0 þ eu 1 þ Oðe
2
Þ
À
Á 3
i
;
ð3:180Þ
where D i
j
∂
j /∂T i
j . Equating to zero the coefficients to like powers of e yields, to
order e
0 :
D
2
0 u 0 þ x
2
0 u 0 ¼ 0;
ð3:181Þ
and to order e
1 :
D
2
0 u 1 þ x
2
0 u 1 ¼ q cos XT 0 À 2D 0 D 1 u 0 À 2bx 0 D 0 u 0 À cu
3
0 :
ð3:182Þ
The general solution to the zero order problem (3.181) is:
u 0 ¼ AðT 1 Þe
ix 0 T 0 þ AðT 1 Þe
Àix 0 T 0 ;
ð3:183Þ
where A(T 1 ) 2 C is the unknown function to be determined. Substituting the
solution for u 0 into the first-order problem (3.182) we obtain that:
D
2
0 u 1 þ x
2
0 u 1 ¼
1
2
qe
iXT 0 À i2x 0 A
0 e
ix 0 T 0
À i2bx
2
0 Ae
ix 0 T 0 À c A
3 e
i3x 0 T 0 þ 3A
2 Ae
ix 0 T 0
À
Á þ cc;
ð3:184Þ
where the term cos(XT 0 ) has been expressed in exponential form.
Terms proportional to e
ix 0 T 0 are resonant to the left-hand side of the equation.
These will cause secular terms (proportional to T 0 e
ix 0 T 0 ) to appear in the particular
solution for u 1 . Further, since by assumption X % x 0 , the term
1
2 qe
iXT 0 will be
near-resonant, causing small divisor terms to appear in the particular solution for u 1 .
To convert near-resonant terms to resonant terms a detuning parameter r is introduced, measuring the nearness to resonance by:
X ¼ x 0 þ er:
ð3:185Þ
Substituting this into (3.182) we obtain, upon noting that XT 0 = x 0 T 0 +
erT 0 = x 0 T 0 + rT 1 and rearranging terms:
154
3 Nonlinear Vibrations: Classical Local Theory