The expansion is substituted into (3.107) by noting that:
_
h ¼ D 0 þ eD 1
ð
Þ h;
€ h ¼ D
2
0 þ 2eD 0 D 1 þ Oðe
2
Þ
À
Á
h;
ð3:110Þ
where D i
j
∂
j /∂T i
j . Performing the substitution, and equating to zero the coefficients to like powers of e one finds that, to order e
0 :
D
2
0 h 0 þ h 0 ¼ 0;
ð3:111Þ
and that, to order e
1 :
D
2
0 h 1 þ h 1 ¼ À2D 0 D 1 h 0 À 2bD 0 h 0 À ch
3
0 þ qx
2 cosðxT 0 Þh 0 :
ð3:112Þ
The general solution of the zero order problem (3.111) can be written:
h 0 ¼ AðT 1 Þe
iT 0 þ AðT 1 Þe
ÀiT 0 ;
ð3:113Þ
where A is an arbitrary function of the slow scale T 1 , which is determined so as to
make the solution of the first-order problem (3.112) free of secular terms.
Substituting the zero order solution (3.113) into the first-order problem (3.112)
we obtain:
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
iðx þ 1ÞT 0 þ Ae
iðxÀ1ÞT 0
þ cc;
ð3:114Þ
where A′ dA/dT 1 , the term cos(x T 0 ) has been expressed in complex exponential
form, and cc denotes complex conjugates of the preceding terms.
The function A(T 1 ) is determined by the requirement that the solution for h 1 in
(3.114) should be free of secular terms, that is, free of terms containing T 0 e
iT 0 as a
factor. Since secular solution terms are caused by resonant excitation terms, we
examine the right side of (3.114) for the presence of resonant terms. Terms oscillating at a frequency equal to the natural frequency of the homogeneous system are
resonant, and their sum should be equated to zero to get rid of secular solution
terms. The natural frequency of the homogeneous system here equals unity (square
root of the coefficient to h 1 ), and so we should equate to zero the sum of all terms
containing e
iT 0 as a factor, since this term oscillates at unit frequency (a term e
irt
cos(rt) + isin(rt) oscillates at frequency r). The first parenthesis of (3.114) is readily
seen to make up a resonant term. However, the term
1
2 qx
2 e
iðxÀ1ÞT 0 will also be
3.6 The Forced Response – Multiple Scales Analysis
131
_
h ¼ D 0 þ eD 1
ð
Þ h;
€ h ¼ D
2
0 þ 2eD 0 D 1 þ Oðe
2
Þ
À
Á
h;
ð3:110Þ
where D i
j
∂
j /∂T i
j . Performing the substitution, and equating to zero the coefficients to like powers of e one finds that, to order e
0 :
D
2
0 h 0 þ h 0 ¼ 0;
ð3:111Þ
and that, to order e
1 :
D
2
0 h 1 þ h 1 ¼ À2D 0 D 1 h 0 À 2bD 0 h 0 À ch
3
0 þ qx
2 cosðxT 0 Þh 0 :
ð3:112Þ
The general solution of the zero order problem (3.111) can be written:
h 0 ¼ AðT 1 Þe
iT 0 þ AðT 1 Þe
ÀiT 0 ;
ð3:113Þ
where A is an arbitrary function of the slow scale T 1 , which is determined so as to
make the solution of the first-order problem (3.112) free of secular terms.
Substituting the zero order solution (3.113) into the first-order problem (3.112)
we obtain:
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
iðx þ 1ÞT 0 þ Ae
iðxÀ1ÞT 0
þ cc;
ð3:114Þ
where A′ dA/dT 1 , the term cos(x T 0 ) has been expressed in complex exponential
form, and cc denotes complex conjugates of the preceding terms.
The function A(T 1 ) is determined by the requirement that the solution for h 1 in
(3.114) should be free of secular terms, that is, free of terms containing T 0 e
iT 0 as a
factor. Since secular solution terms are caused by resonant excitation terms, we
examine the right side of (3.114) for the presence of resonant terms. Terms oscillating at a frequency equal to the natural frequency of the homogeneous system are
resonant, and their sum should be equated to zero to get rid of secular solution
terms. The natural frequency of the homogeneous system here equals unity (square
root of the coefficient to h 1 ), and so we should equate to zero the sum of all terms
containing e
iT 0 as a factor, since this term oscillates at unit frequency (a term e
irt
cos(rt) + isin(rt) oscillates at frequency r). The first parenthesis of (3.114) is readily
seen to make up a resonant term. However, the term
1
2 qx
2 e
iðxÀ1ÞT 0 will also be
3.6 The Forced Response – Multiple Scales Analysis
131