130
4 Numerical Methods in Structural Dynamics …
4.2.1 Finite Difference Method
The central difference scheme is adopted here. Time span under consideration is
divided into n equal time intervals t(=T/n). The solution is obtained in a progressive
manner from time t = 0 to t = T, at the time instants 0, t, 2t, 3t, …, it, …, T.
Let x i , ˙
x i and ¨
x i be the displacement, velocity and acceleration respectively, at
the time instant it.
These terms, expressed in finite difference form are
¨
x i =
1
t 2 (x i + 1 − 2x i + x i − 1 )
(4.1)
Also
¨
x i =
1
2t
( ˙
x i + 1 − ˙
x i − 1 )
(4.2)
and
˙
x i =
1
2t
(x i + 1 − x i − 1 ).
(4.3)
From Eqs. (4.1) and (4.3), it can be shown that
x i + 1 = x i + ˙
x i t +
1
2
¨
x i t
2
(4.4)
˙
x i + 1 = 2 ¨
x i t + ˙
x i − 1 .
(4.5)
Equations (4.1)–(4.5), together with the differential equation of motion, are used
for obtaining the necessary numerical solution. Initiation of the computation is
dependent on the initial acceleration. Two conditions are encountered.
4.2.1.1 When the Initial Acceleration is not Zero
We start with subscript 1 to indicate initial conditions. It is assumed that the initial
acceleration remains constant during the first interval. Based on this assumption, the
following relations can be written:
¨
x 2 = ¨
x 1
˙
x 2 = ¨
x 1 t
and x 2 =
1
2
¨
x 1 t
2
⎫
⎬
⎭
(4.6)