and
v 1ss þ
X N
k¼1
c k v 0 sinðm k s þ w k Þ À p 1 ðtÞ sinðms þ wÞ
þ X
À2
v 1tt þ cv 1t þ x
2
v 1
Â
Ã
þ X
À1 2v 1ts þ cv 1s
½
þ
X N
k¼1
c k v 1 sinðm k s þ w k Þ À v 1 sinðm k s þ w k Þ
h
i
ð
Þ
#
¼ 0;
ð7:117Þ
For the case of interest, where resonant effects are significant but not dominating,
the term X
−2
x
2
v 1 in (7.117) is comparable in magnitude to the dominating first
terms, so the equation of fast motion can be written:
v 1ss þ X
À2
x
2
v 1 ¼ À
X N
k¼1
c k v 0 sinðm k s þ w k Þ
þ p 1 ðtÞ sinðms þ wÞ þ OðX
À1
Þ;
ð7:118Þ
with stationary solution (valid when X is away from sharp resonance):
v 1 ¼
X N
k¼1
m
2
k I À X
À2
x
2
À
Á À1 c k v 0 sinðm k s þ w k Þ
À m
2
I À X
À2
x
2
À
Á À1 p 1 ðtÞ sinðms þ wÞ þ OðX
À1
Þ:
ð7:119Þ
Inserting into (7.116) and evaluating the average, the equation for the slow
motions becomes:
v 0tt þ cv 0t þ x
2
þ Dx
2
À
Á v 0 ¼ p 0 ðtÞ þ Dp 0 ðtÞ;
ð7:120Þ
where Dx
2 denotes the apparent change in stiffness:
Dx
2
1
2
X N
k;j¼1
c k m
2
j I À X
À2
x
2
À1 c j d m k m j cosðw k À w j Þ;
ð7:121Þ
and Dp 0 (t) denotes an apparent change in static load:
Dp 0 ðtÞ
1
2
X N
k¼1
c k m
2
I À X
À2
x
2
À
Á À1 p 1 ðtÞd m k m cosðw k À wÞ:
ð7:122Þ
Expressions (7.120)–(7.122) provide straight-forward predictions of apparent
changes in stiffness and static loading, and facilitate predictions of related quantities
7.6 A General Class of Linear Continuous Systems
431