4.3 Numerical Evaluation of Duhamel’s Integral
147
Fig. 4.6 Variation of F (τ )
I 1 =
t i
i − 1
e ζ pτ cos p d τ dτ =
e ζ p τ
(ζ p) 2 + ( p d ) 2 (ζ p cos p d τ + p d sin p d τ )|
t i
t i − 1
(4.41)
I 2 =
t i
t i − 1
e
ζ pτ sin p d τ dτ =
e
ζ p τ
(ζ p)
2
+ ( p d )
2
(ζ p sin p d τ − p d cos p d τ )|
t i
t i − 1
(4.42)
I 3 =
t i
t i − 1
τ e
ζ p τ sin p d τ dτ = τ −
ζ p
(ζ p)
2
+ p
2
d
I
2 + 1
p d
(ζ p)
2
+ p
2
d
I
t i
t i − 1
(4.43)
I 4 =
t i
t i − 1
τ e
ζ p τ cos p d τ dτ =
ζ p
(ζ p)
2
+ p
2
d
I
1 +
p d
(ζ p)
2
+ p
2
d
I
2
t i
t i − 1
(4.44)
where I
1 and I
2 are the integrals of Eqs. (4.42) and (4.43) before the evaluation of
limits. A d (t i ) and B d (t i ) are evaluated as follows.
A d (t i ) = A d (t i − 1 ) +
F (ti − 1) − t i − 1
F i
t i
I 1 +
F i
t i
I 4
(4.45)
B d (t i ) = B d (t i − 1 ) +
F (ti − 1) − t i − 1
F i
t i
I 2 +
F i
t i
I 3
(4.46)