142
4 Numerical Methods in Structural Dynamics …
From equation of motion [Eq. (4.21)]
¨
x 2 =
F 2 (t)
m
− 2ζ p ˙
x 2 −
k
m
x 2
¨
x 2 =
96
8
− 0.6 ˙
x 2 − 9 x 2
(4.30)
Substituting x 2 and ˙
x 2 from Eqs. (4.28) and (4.29) into Eq. (4.30) gives
¨
x 2 = 10.71, ˙
x 2 = 1.071 and x 2 = 0.072
Solution of equations as presented above is not general, which would have been
the case had the solution been obtained by successive approximation.
At t = 2 × 0.2.
Let ¨
x 3 = ¨
x 2 = 10.71.
Then, ˙
x 3 = 1.071 +
1
2
[10.71 + 10.71] × 0.2 = 3.213x 3 = 0.072 +
1.071 × 0.2 +
1
3
× 10.71 × 0.2
2
+
1
6
× 10.71 × 0.2
2
= 0.5 and
¨
x 3 = 24 − 0.6 × 3.213 − 9 × 0.5 = 17.572.
Let ¨
x 3 = 17.572
˙
x 3 = 1.071 +
1
2
[10.71 + 17.572] × 0.2 = 3.899
x 3 = 0.072 + 1.071 × 0.2 +
1
3
× 10.71 × 0.2
2
+
1
6
× 17.572 × 0.2
2
= 0.546
and ¨
x 3 = 24 − 0.6 × 3.899 − 9 × 0.546 = 16.746.
A new iteration is to be performed with ¨
x 3 = 16.746 and steps are to be
continued.
4.3 Numerical Evaluation of Duhamel’s Integral
The response of a SDF system subjected to a general type of forcing function, as
given by Duhamel’s integral is as follows [Eq. (4.31)].
x (t) =
t
0
F (τ )
mp
sin p (t − τ ) dτ
(4.31)
For practical cases, this integral is to be evaluated numerically, as F(τ) is such that
analytical solutions become difficult to obtain. The form of Eq. (4.31) is not suited
4 Numerical Methods in Structural Dynamics …
From equation of motion [Eq. (4.21)]
¨
x 2 =
F 2 (t)
m
− 2ζ p ˙
x 2 −
k
m
x 2
¨
x 2 =
96
8
− 0.6 ˙
x 2 − 9 x 2
(4.30)
Substituting x 2 and ˙
x 2 from Eqs. (4.28) and (4.29) into Eq. (4.30) gives
¨
x 2 = 10.71, ˙
x 2 = 1.071 and x 2 = 0.072
Solution of equations as presented above is not general, which would have been
the case had the solution been obtained by successive approximation.
At t = 2 × 0.2.
Let ¨
x 3 = ¨
x 2 = 10.71.
Then, ˙
x 3 = 1.071 +
1
2
[10.71 + 10.71] × 0.2 = 3.213x 3 = 0.072 +
1.071 × 0.2 +
1
3
× 10.71 × 0.2
2
+
1
6
× 10.71 × 0.2
2
= 0.5 and
¨
x 3 = 24 − 0.6 × 3.213 − 9 × 0.5 = 17.572.
Let ¨
x 3 = 17.572
˙
x 3 = 1.071 +
1
2
[10.71 + 17.572] × 0.2 = 3.899
x 3 = 0.072 + 1.071 × 0.2 +
1
3
× 10.71 × 0.2
2
+
1
6
× 17.572 × 0.2
2
= 0.546
and ¨
x 3 = 24 − 0.6 × 3.899 − 9 × 0.546 = 16.746.
A new iteration is to be performed with ¨
x 3 = 16.746 and steps are to be
continued.
4.3 Numerical Evaluation of Duhamel’s Integral
The response of a SDF system subjected to a general type of forcing function, as
given by Duhamel’s integral is as follows [Eq. (4.31)].
x (t) =
t
0
F (τ )
mp
sin p (t − τ ) dτ
(4.31)
For practical cases, this integral is to be evaluated numerically, as F(τ) is such that
analytical solutions become difficult to obtain. The form of Eq. (4.31) is not suited