monitoring one half of the vessel.
Mathematical calculations for the location
of an acoustic emission source in a
three-dimensional space become simpler
with this arrangement.
As shown in Fig. 12, four transducers
are located at (
r
0
, [±
π
÷ 2], 0) and
(
r
0
, 0, ±
h
) in cylindrical coordinates.
When values of
i
= 1 and 2, the distance
from an acoustic emission source at an
arbitrary position (
r
,
θ
,
z
) to each
transducer is:
(18)
where:
(19)
and:
(20)
For
i
= 3 and 4, the distance from an
acoustic emission source at an arbitrary
position (
r
,
θ
,
z
) to each transducer is:
(21)
where
a
3
2
=
a
4
2
and
a
3
2
=
r
0
2
+
r
2
– 2
r
0
r
cos
θ
.
When
j
= 1, 2, 3 and 4 and the subscripts
i
and
j
indicate transducer numbers, then:
(22)
The basic relations in Eqs. 18 to 22 lead
to a quadratic equation for
z
in terms of
r
and
θ
for each pair of transducers. The
solution of the quadratic equation for
transducer 1 and transducer 2 is:
(23)
For transducer 1 and transducer 3:
(24)
Equations 25 to 27 define the terms used
in Eq. 24:
(25)
(26)
(27)
To make Eq. 24 valid for transducer 1
and transducer 4,
h
and
Δ
t
13
in Eqs. 25 to
27 should be replaced by –
h
and
Δ
t
14
,
respectively.
The purpose for the development of
Eqs. 23 and 24 is to simplify computer
programming. Both equations can be used
to calculate values of
z
for each pair of
transducers from given values of
r
and
θ
.
Therefore, all possible solutions for
z
can
C
r r
v
t
v
t
r
r
r r
v
t
h v
t
h r r
h
=
−
(
)
+
−
+
(
)
+
+
(
)
−
+
+
(
)
+
4
4
4
2
2
2
0
2
2
2
4
13
4
2
13
2
0
2
2
0
2
13
2
2
2
13
2
2
0
4
cos
sin
cos
sin
sin
cos
θ
θ
θ
θ
θ
θ
Δ
Δ
Δ
Δ
B
hr r
hA
=
−
(
)
−
4
2
0
cos
sin
θ
θ
A
h
v
t
=
−
(
)
4
2
2
13
2
Δ
z
B
B
AC
A
=
−
±
−
2
4
2
z
r r
v
t
v
t
r
r
=
⎛
⎝
⎜
+
−
−
⎞
⎠
⎟
4
1
4
0
2 2
2
2
12
2
2
12
2
0
2
2
0 5
sin
.
θ
Δ
Δ
d
d
v
t
i
j
ij
+
=
Δ
d
h
z
a
i
i
=
±
(
)
+
2
2
a
r
r
r r
2
2
0
2
0
2
=
+
+
2
sin
θ
a
r
r
r r
1
2
0
2
0
2
=
+
+
2
sin
θ
d
z
a
i
i
=
+
2
2
129
Acoustic Emission Source Location
F
IGURE
12.
Source location in three dimensions for thick
walled cylinder.
Transducer 3
(
r
o
, 0,
h
)
Transducer 2
(
r
o
, [–
π
÷ 2], 0)
Transducer 1
(
r
o
, [
π
÷ 2], 0)
Transducer 4
(
r
oa
, 0, –
h
)
h
–
z
d
3
a
3
d
2
d
1
z
a
2
a
1
r
a
3
θ
Legend
a
1
,
a
2
,
a
3
=
horizontal distance (meter) from source to transducers 1, 2 and
3, respectively
d
1
,
d
2
,
d
3
=
direct distance (meter) from source to transducers 1, 2 and 3,
respectively
h
=
vertical length or height (meter) of cylinder in
z
direction
r
=
cylindrical coordinate for radius (meter)
z
=
cylindrical coordinate for vertical distance (meter)
θ
=
cylindrical coordinate for angular distance (radian)