Civil Engineering Reference
In-Depth Information
∞
2
π
F
(w)
=
f
(r)rdr
exp[
j(rw
cos
(ψ
−
ϕ))
]d
ϕ
(8.B.6)
0
0
Using (Abramovitz and Stegun 1972)
2
π
exp[
ju
cos
(ψ
−
ϕ)
]d
ψ
=
2
π
J
0
(u)
(8.B.7)
0
Equations (8B.5) and (8B.6) can be rewritten
∞
1
2
π
f
(r)
=
F
(w)w
J
0
(wr)
d
w
(8.B.8)
0
∞
F
(w)
=
2
π
f
(r)r
J
0
(wr)
d
r
(8.B.9)
0
The functions F and F are related by F
=
2
π
F.
Response of a linear system
If the response to an excitation exp[
vy)
] is a Z vector component like
u
s
z
or a
scalar which only depends on
r
, the total response
U
z
of the system for the excitation
f(
r)
is given by
−
j(ux
+
∞
2
π
1
4
π
2
U
z
(r)
=
F
(w)u
z
(w)w
d
w
exp[
j(rw
cos
(ψ
−
ϕ))
]d
ψ
(8.B.10)
0
0
which can be rewritten
∞
1
2
π
U
z
(r)
=
F
(w)u
z
(w)
J
0
(wr)w
d
w
(8.B.11)
0
where F/2
π
can be replaced by F.
If the response is a radial component which only depends on
r
and not on
θ
, the total
response in the direction
x
is obtained by projecting in the direction
x
the responses for
the different orientations of the excitation in the
XY
plane. Using
2
π
j
2
π
J
1
(u)
=
exp
(
−
ju
cos
θ)
cos
θ
d
θ
0
(Abramovitz and Stegun 1972) leads to
∞
=
−
j
2
π
U
x
(r)
F
(w)u
r
(w)
J
1
(wr)w
d
w
(8.B.12)
0
