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7 Forced Vibration Analysis of Multiple Degrees of Freedom System
Fig. 7.8 Example 7.6
From Eqs. (3.153) and (3.157), it is known that the spectral displacement of the
rth mode is
z
(r )
max = S
(r )
d =
1
p dr
t
o
¨
x s (τ ) exp(− p r ζ r (t − τ )) sin p dr (t − τ )dτ
max
= −
¨
x so
p 2
r
(DLF) a,r
max
(7.51)
where ¨
x s0 is the maximum amplitude of the ground motion. Thus, Eq. (7.50) is
written as
|z
(r )
i | max = φ
(r )
i B r S
(r )
d
(7.51a)
Example 7.6 A two-storeyed bent is shown in Fig. 7.8. Compute the displacements
at the floor levels due to the motion of the support as shown. The natural frequencies
for this bent are 9.00 rad/s and 23.5 rad/s, respectively. The mode shape matrix is
[] =
1.00 1.00
1.57 −1.06
Let us next compute the mode participation factors. They are
B 1 =
n
j=1 m j φ
(1)
j
n
j=1 m j
φ
(1)
j
2 =
(5250) × 1.00 + (3160) × 1.57
5250 × (1.00)
2
+ (3160) × (1.57)
2
= 0.783
Similarly
B 2 =
(5250)(1.00) + (3160)(−1.06)
(5250)(1.00)
2
+ (3160)(−1.06)
2
= 0.216