132
4 Numerical Methods in Structural Dynamics …
x 2 =
t
0
t
0
¨
x 1 +
¨
x 2 − ¨
x 1
t
t
dt dt
or
x 2 =
1
6
(2 ¨
x 1 + ¨
x 2 )
t
2
(4.9)
Similarly
˙
x 2 =
1
2
( ¨
x 1 + ¨
x 2 ) t
(4.10)
With ¨
x 1 = 0, Eqs. (4.8), (4.9) and (4.10) are to be solved, to evaluate the
necessary quantities to start with. Then the procedure indicated in Art. 4.2.1.1 is to
be followed. Both the above cases have been dealt with in the examples given below.
Example 4.1 Solve numerically the differential equation for a SDF system starting
from rest with the force applied on it, as shown in Fig. 4.2. The equation of motion
for the SDF system of Fig. 4.2a is given by
2 ¨
x + 2000x = F(t)
Fig. 4.2 Example 4.1