7.9 Direct Integration for Determining Response of MDF Systems
293
a o =
1
ααt 2 ; a 1 =
δ
ααt
; a 2 =
1
ααt
; a 3 =
1
2α
− 1;
a 4 =
δ
α
− 1; a 5 =
t
2
δ
α
− 2
; a 6 = t(1 − δ); a 7 = δδt
(4) The effective stiffness matrix may be formed. It is given by
[K ] = [K ] + a o [M] + a 1 [C]
(7.66)
(5) [K ] matrix is inverted.
For a given problem, steps 1 through 5 are to be carried out once only. The
following steps are to be performed for each time interval.
(1) The effective load at (i+1)th time instant is given by
{F} i+1 = {F} i+1 + [M](a o {x} i + a 2 { ˙
x} i + a 3 { ¨
x} i )
+ [C](a 1 {x} i + a 4 { ˙
x} i + a 5 { ¨
x} i )
(7.67)
(2) Solve for displacements at (i +1)th time instant
{x} i+1 = [K ]
−1
{F} i+1
(7.68)
(3) Calculate accelerations and velocities at (i +1)th time instant
{ ¨
x} i+1 = a o
{x} i+1 − {x}
i
− a 2 { ˙
x} i − a 3 { ¨
x} i
(7.69)
{ ˙
x} i+1 = { ˙
x} i + a 6 { ¨
x} i + a 7 { ¨
x} i+1
(7.70)
The method has been demonstrated in the following example
Example 7.8 Calculate the displacement response of two degrees of freedom system
having the following data:
[K ] =
72 −72
−72 144
; [M] =
8 0
0 8