264
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
{ p} =
⎧
⎨
⎩
43.872
120.155
167.00
⎫
⎬
⎭
and
p
2
=
⎧
⎨
⎩
1924.733
14437.20
27889.00
⎫
⎬
⎭
[] =
⎡
⎣
1.0000 1.0000 1.000
0.7594 −0.8047 −2.427
0.3361 −1.1572 2.512
⎤
⎦
[]
T
[M][] =
⎡
⎣
1.000 0.7594 0.3361
1.000 −0.8047 −1.1572
1.000 −2.427 2.512
⎤
⎦
× 20,000
⎡
⎣
1 0 0
0 1 0
0 0 1
⎤
⎦
⎡
⎣
1.000 1.000 1.000
0.7594 −0.8047 −2.427
0.3361 −1.1572 2.512
⎤
⎦
=
⎡
⎣
1.6896 0
0
0 2.9867 0
0
0 13.200
⎤
⎦ × 20,000
[]
T
{F(t)} =
⎡
⎣
1.000 0.7594 0.3361
1.000 −0.8047 −1.1572
1.000 −2.427 2.512
⎤
⎦
⎧
⎨
⎩
P cos ωt
0
0
⎫
⎬
⎭
=
⎧
⎨
⎩
P cos ωt
P cos ωt
P cos ωt
⎫
⎬
⎭
=
¯
F
cos ωt
The rth equation for this harmonic excitation is [from Eq. (7.5)]
¨
ξ ˙
r + p
2
r ξ r =
F r
¯
m r
cos ωt
(a)
Steady-state solution of Eq. (a) is
ξ r =
F r cos ωt
¯
m r
p 2
r − ω 2
(b)
The displacement at the top level is [Eq. (7.9)]
x 1 =
n
r =1
φ
(r )
1 ξ r
(c)
Let us first write the expression for x 1 (t) for all the modes, and then, we shall take
out the terms for the necessary truncation