Problem 1.10 Consider the linear flexible structure in Fig. P1.10, with n masses
and equation of motion:
M€ x þ Kx ¼ F sinðXtÞ;
ð1:133Þ
where M ¼ diag m 1 m 2 Á Á Á m n
f
g ; F ¼ f 1 0 Á Á Á 0
f
g
T , and:
K ¼
k 1
Àk 1
0
0
Á Á Á
0
Àk 1 k 1 þ k 2
Àk 2
0
Á Á Á
0
0
Àk 2
k 2 þ k 3 Àk 3
Á Á Á
0
. .
.
. .
.
. .
.
. .
.
. .
.
. .
.
0
0
0
Àk nÀ2 k nÀ2 þ k nÀ1 Àk nÀ1
0
0
0
0
Àk nÀ1
k nÀ1
2
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
5
:
ð1:134Þ
a) Show that the stationary output at mass n is:
x n ¼ a n sinðXtÞ;
ð1:135Þ
where a n = {a} n is the amplitude of the n’th mass, and
a ¼ ðK À X
2
MÞ
À1 F:
ð1:136Þ
b) Let f 1 = 1, m i = k i = 1, i = 1, n, n = 10, and use a numerical tool to compute and
plot the natural frequencies, the mode shapes, and the frequency response
|H(X)| = |a n (X)|/f 1 . This corresponds to a uniform elastic structure, so |H(X)|
should show distributed resonance spikes.
Fig. P1.9
1.10 Problems
47