290
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
on the spectral diagrams of Fig. 3.35. Also find the maximum displacement of each
mass.
The natural frequencies and mode shapes are to be calculated first. The values
obtained for the problem are indicated in the table below.
Mode
1
2
3
p r
4.92 rad/s
13.45 rad/s
18.7 rad/s
T r
1.277 s
0.467 s
0.336 s
φ
(r )
1
1.000
1.000
1.000
φ
(r )
2
0.759
−0.804
−2.462
φ
(r )
3
0.336
−1.157
2.580
The mode participation factors are then computed in the following table.
1 st mode
2nd mode
3rd mode
Mass
m ∗
j
m j φ
(1)
j
m j
φ
(1)
j
2
m j φ
(2)
j
m j
φ
(2)
j
2
m j φ
(3)
j
m j
φ
(3)
j
2
1
1
1.000
1.000
1.000
1.000
1.000
1.000
2
1
0.759
0.576
−0.804
0.646
−2.462
6.061
3
1
0.336
0.113
−1.157
1.339
2.580
6.656
2.095
1.689
−0.961
3.027
1.118
13.717
B 1 =
2.095
1.689 = 1.240 B 2 = −
0.961
3.027 = −0.317 B 3 =
1.118
13.717 = 0.082
*As all masses are same and there are same masses in both the numerator and the denominator,
which cancel out, they are taken as unity in this table for simplicity
The values of spectral ground acceleration and spectral ground displacement are
then determined from Fig. 3.38, considering 5% damping in each mode.
Mode
T r
S
(r )
a
mm/s 2
S
(r )
d
mm
1
1.227
1150
47.50
2
0.467
1650
9.16
3
0.336
1900
5.44
Applying Eq. (7.53), the relative displacement with respect to the ground for each
storey is determined.