Instrumentation & Measurement Magazine 24-4 - 52

proposed SHM approach. The solution is to generate an enhanced
magnetic field over a steel plate using magnets and
investigate how this field is affected by the defects. The magnetic
field is caused by the interaction between two electrics
fields on top of each other (e.g., positive and negative charges).
The efficacy of the proposed method is verified through experiments
on a 297 mm × 210 mm steel plate with a diameter
of 0.8 mm. This steel plate is equipped with four Neodynium
magnets with a radius of 15 mm and a thickness of 3 mm at its
four corners. Gauss rating of a Neodynium disc magnet is a
magnetic field in the center of the disc. The discs are thin and
therefore, one can approximate their magnetic field with a current
loop. The magnetic field at the center of each disc (B0
) can
be calculated using the following equation:
Bk

2I
r
(1)
where I is the effective current, r is the radius of the loop (disc),
μ0
k 
 0
4
The magnetic dipole of the disc is denoted by m:
3
m Ir2
  
Br0
2k
(3)
Assuming the magnetic field of a dipole on a point of distance
R from the dipole is in a plane normal to the dipole, then
we will have:
m
Bk 3
 
R
(4)
The minus sign merely indicates that the direction of the
magnetic field is opposite to that of the magnetic dipole. Expressing
m in terms of B0
:
B  
B r
2
0 
R


3
(5)
The center of the plane is at equal distances from four identical
magnetic discs. Therefore, the magnetic field is four times
the above equation and can be represented by Bt
:
BB
t  2 0

R


r
3
(6)
Referring to Fig. 5, the magnetic field at height z from the
center of the plate (Bz
equations:
) can be calculated using the following
cos 22, sin
aa
Rr

,
22 5
a rr a ,
 a    
2
2
22 6
r RR a,
 a    
2
2
52
2
(9)
2
(8)
(7)
and
BB B

z 11  sin
2
cos 



(12)
We assume that all of the magnets tend to cross a closed
path between the N- and the S-pole. Therefore, most of the
magnetosphere is created in space. Fig. 4b compares the B−H
relations between the ferromagnets, paramagnets, diamagnets
and free space. The changes of the magnetic field can be
detected by the magnetometer Apps available in smartphones
or tablets, as shown in Fig. 4c [18]. The magnetometer sensor
in a smartphone uses modern solid-state technology to create
a miniature Hall-effect detector that measures the magnetic
field of the Earth in the X, Y and Z axes. The Hall-effect sensor
generates voltage that is commensurate with the strength and
polarity of the magnetic field along the axis of each sensor. The
voltage detected is converted to a digital signal reflecting the
frequency of the magnetic field. Other magnetometer technologies
may include magneto-resistive devices that change the
resistance measured based on magnetic field changes.
Experimental and Numerical Studies
A series of tests have been performed on steel plates to verify
the proposed SHM approach. Four circular Neodynium magnets
were attached to the corners of the plate to induce the
magnetic field. The changes of the magnetic field created by
the magnets were measured on an intact steel plate and plates
with 14 mm, 34 mm and 54 mm cracks. The abovementioned
smartphone magnet detection software was used to record the
IEEE Instrumentation & Measurement Magazine
June 2021
(2)
magnetic constant and k is a constant depending on the system
of units:
BB sin
z 
1
2
 
BB
tz

Fig. 5. Calculations of the magnetic field at altitude.
B  1 4r
 
0I
cos cos 



 



a
4 aa
22
00
5
2 2
6


II
a
30
(10)
a II I
 
 30 5 6aa a
2



4,
15
260I
a
 
6 30
00 0
6
6
5  
a
, (11)

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