294
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
{F (t)} =
0
480t
; {x} o =
0
0
i
; { ˙
x} o =
0
0
α =
1
4
and δ =
1
2
t = 0.2s
Choose
The constants of integration are calculated first. They are:
a o = 100.0, a 1 = 10.0, a 2 = 20.0, a 3 = 1.00
a 4 = 1.00, a 5 = 0.00, a 6 = 0.1, a 7 = 0.1
The effective stiffness matrix is
[K ] =
72 −72
−72 144
+ 100
8 0
0 8
=
872 −72
−72 944
= 8
109 −9
−9 118
[K ]
−1
=
1
8 × 12781
118 −9
−9 109
For every time interval, the following calculation is to be performed.
F
i+1
=
0
480t
i+1
+
8 0
0 8
(100{x} i + 20{ ˙
x} i + { ¨
x} i )
From Eq. (7.65), it is seen that as {x} 0 = { ˙
x} 0 = {F} 0 = {0}
Therefore, { ¨
x} 0 = {0}
F
1
=
0
96
+
0
0
=
0
96
From Eq. (7.68), we get
{x} 1 = [K ]
−1
(F} 1
or
{x} 1 =
1
8 × 12781
118 −9
−9 109
0
96
=
−0.00845
0.10234
From Eqs. (7.69) and (7.70), we get
{ ¨
x} 1 = 100({x} 1 − {x} 0 ) − 20{ ˙
x} 0 − 1.0{ ¨
x} 0