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4 Numerical Methods in Structural Dynamics …
Fig. 4.4 Variation of
acceleration
˙
x i + 1 = ˙
x i +
t
0
¨
x dt
or
˙
x i + 1 = ˙
x i +
t
0
¨
x i +
¨
x i + 1 − ¨
x i
t
(t − t i )
dt
˙
x i + 1 = ˙
x i +
1
2
¨
x i + 1 + ¨
x i
t
(4.16)
Similarly, the displacement at time t i + 1 can be derived [an expression similar to
Eq. (4.9)]
x i + 1 = x i + ˙
x i t +
t
2
6
¨
x i + 1 + 2 ¨
x i
(4.17)
Equations (4.16) and (4.17) indicate that the values of x i + 1 and ˙
x i + 1 are dependent on ¨
x i + 1 , which in turn is related to x i + 1 and ˙
x i + 1 through the equation of
motion. Thus, respective values at any time instant are to be evaluated in an iterative
manner.
Various schemes may be adopted for the iterative procedure. Some are given
below:
(1) By assuming ¨
x i + 1 = ¨
x i
(2) A linear variation of acceleration is assumed in two successive time intervals.