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10CONTACTLESS ELECTROMAGNETIC METHOD FOR MONITORING MECHANICAL STRESSES
IN CYLINDRICAL PRODUCTS
Gorkunov B., Lvov S., Borysenko Y., Saliba Abdel Nour
National technical university "Kharkiv polytechnic institute "
Abstract
The paper considers methods for contactless mechanical stresses monitoring for cylindrical ferromagnetic products, absolute by amplitude and phase and compensatory by current and frequency. It is shown that compensatory monitoring methods using a parametric transducer significantly increase the sensitivity and can be used as the basis for constructing measuring instruments for load, forces, torsional shaft torque and other mechanical quantities.
Keywords: absolute and compensatory monitoring methods, magnetic permeability, electrical conductivity, mechanical stresses, parametric electromagnetic transducer.
To date, methods and means for measuring mechanical stress, force, pressure and other physical and mechanical variables have been studied. These are mainly mechanical, tensoresistive and magnetoelastic methods and transducers. [1-2]. The use of magnetoe-lastic method and means, implementing it has practical significance. While for magnetoelastic transducers with a laminated magnetic core (assembled from plates) there are equations for calculating its parameters in the low-frequency region of the electromagnetic field at various mechanical stresses in the plates, the theory of transducers with a solid magnetic core has not been sufficiently studied yet. Apparently, this is due to complex processes of decay of the magnetic field in a continuous ferromagnetic sample.
Incipient defects and zones of concentration of internal mechanical stresses are prevalent damage sources of the steel constructions, it hosts rapidly developing negative effects, such as continuity interruption, plastic strain and others. Conventional methods of non-destructive testing (NDT) do not allow at an early stage to diagnose and predict the behavior of a structure; hence, the reliability of the objects couldn't be guaranteed [3, 4].
For the early identification of areas of metal structures which are more likely to be damaged, as well as for a prompt response to the situations that have arisen, it is necessary to know their actual stress-strain state.
Electromagnetic methods with primary information in a form of electrical signals likely to be beneficial for noncontact monitoring of stresses. The mechanical and electromagnetic properties of materials are laid down at the level of the structure and are interrelated. All changes in the structure of the material in
the process of deformation, nucleation and development of microdamages are reflected in the corresponding changes in the electromagnetic parameters of the samples. [4].
The presence of internal stresses that arise both at the stage of formation of structure and during its operation is often due to the presence of external mechanical stresses, which lead to an increase in the internal energy of the crystal lattice. The internal energy of the material always tends to a minimum (stable equilibrium), an increase in the mechanical component of the internal energy of the crystal, in turn, leads to a change in the electromagnetic component [5].
Under the influence of an external magnetic field and mechanical stresses domain borders in the ferromagnetic material shifts and the domain structure changes its orientation. All these internal inconstancies affect the sample's behavior in the electromagnetic field, which in turn allows us to use electromagnetic transducers, which makes it possible to monitor the mechanical stresses in the sample by measuring its electromagnetic parameters [6].
This study proposes a model of the magnetic field, distributed in the cylindrical ferromagnetic sample, which is subjected to the mechanical stress (compression). Fig 1 shows cylindrical sample 1, placed in a magnetic field of strength H0. One side of the sample is firmly attached and the other is subjected to a force F. The sample is placed in an alternating electromagnetic field, which is created by the winding of a sliding electromagnetic transducer 2.
////////
J f ■ ;■
y j y
-t
-t
Figure 1 -A relation between the magnetoelastic effect and the electromagnetic transducer
Suppose in undisturbed state the relative magnetic permeability |r and specific electrical conductivity ct of a material are constant. Magnetic induction vector Bo is parallel to the lines of force of the excitating magnetic field Ho and is determined by the formula:
Bo = lolrHo , where ^o = 4n 10-7 H/m - magnetic constant.
(1)
Bo = H0H0 (Hr + AHr ) •
(2)
We use Maxwell's equations to obtain the electromagnetic transducer informative parameters functions of the mechanical stress CTm arising in the product under test [7]. If a strength of the probing electromagnetic field is low (Ho < 100 A/m), corresponding to the linear section of the material magnetization curve, provided we use sinusoidal varying magnetic and electric fields, we can write Maxwell's equations in the following form:
= CTE;
dH dr
dE E
—+— = -j®H oHrH ; or r
(3)
(4)
where ra - probing field cyclic frequency; r - actual radius of the product in a cylindrical coordinate
system; j = 4-1.
The formula for calculating the distribution of the magnetic field strength H along the radius r of the product (while r = a, H = Ho, a - product radius) is the solution of the system of equations taking into account boundary conditions
H (r )=H
lo( kr) I0 ( ka)
(5)
where Io - modified Bessel function of the first kind of zero order of imaginary arguments kr and ka , it = ^jai0|ar<j - complex parameter.
The magnetic flux ®2, penetrating the cross section S of the sample, is found by integrating expression (5), according to the formula
2naIl( kr)
Assume that when a force F ^ 0 is applied, the magnetic permeability in the longitudinal direction of the cylindrical sample us changed by A|r. In this case, in the longitudinal direction of the cylindrical sample, it becomes equal to the value |r = |r + A|r.
The magnitude of the magnetic induction vector in this case is determined by the formula:
®2 = H0HrH0
(6)
kI0 (ka)
where h - modified Bessel function of the first kind of the first order.
Thus, knowing the magnetic flux, for a specific electromagnetic transducer, it is possible to obtain functions of the output information parameters of the mechanical stress, arising in the tested product.
The results of [8] allows us to write expressions for the amplitude and phase of the complex parameter K, which characterizes the specific normalized magnetic flux per | r in a cylindrical product:
K = -
o9
2
1
ber[2 x + beij2 x
ber0 x + bei0 x
tg<?2 =
_ beijx(ber0 x + bei0 x) + berxx(ber0 x - bei0 x)
beijx(ber0x - bei0x) - berjx(ber0x + bei0x)
(7)
,(8)
where ®2 and ®o - magnetic fluxes inside the product and inside the transducer without the product respectively; ^ = a2/a^ - product fill factor, a and an - radiuses of the product and transducer winding; x = a^l|0|rCTra - generalized parameters; bero x,
beio x, berix and bei\x -zero and first order special Kelvin functions.
It should be noted that functions K = f1(x) and tg 92 = f2(x) are universal transformation functions that makes it possible to establish a certain sequence of measuring and calculation operations for the simultaneous determination of |r and ct of the stressed cylindrical product.
Figure 2 a, b has functions K = f1(x) and tg 92 = f2(x), obtained using the Bessel function [9].
K,
Sk
0,8
0,6
0,4
0,2
0
0,'
tg 92, S92 0,8
0,6
0.4
0,2
0
0,
Figure 2 a, b - Universal functions K = f1(x), tg 92 = f2(x) and sensitivity by amplitude SK and phase S9
tg 92
/ / ! / / / \ S92 \ \
! I 1 / / / / / / / -%- \ \ V
:>:> - i. . " I I, ::! .•: :; i X
6
A dotted line in the same figure shows the functions of the sensitivities by amplitude SK = AK/Ax
and phase S9 = A9/AX .
Thus, to determine the values of ct and |r at any constant mechanical stress in the product it is necessary to calculate tg 92 using informative signals of the transducer (®2, ®o and 9).
In the next place we can calculate x using function tg 92 = f2(x) then we can determine the amplitude value of K using function K = fi(x) and after that it is possible to calculate |r and ct using equations:
lr =
O0
ct = -
|0|r aa
2
(9) (10)
The authors utilized sliding parametric electromagnetic transducer (SPET) as a primary transducer (Fig. 1). Basic transducer parameters: length It = 40 mm; diameter dx = 3 mm; a coil was of
W = 350 turns using wire PEV-0,1 mm; coil active resistance Ro = 3.888 Ohm; coil inductance Lo = 7.45 uH.
The testing sample was steel wire of diameter d = 0.75 mm and initial (without a load) values | = 500 and ct = 8 MS/m.
Contactless monitoring of mechanical stresses in the cylindrical sample using SPET could be implemented in two ways. The first one is to use Wheatstone-bridge circuit to measure transducer's R and L. The second one is to measure amplitude U and phase 9 of the transducer's output signal.
Bridge circuits for connecting parametric transducers have high sensitivity, but industrial AC bridges operate at a fixed frequency. This limits their application for solving a wide range of tasks, since, as it can be seen from Figure 2, the maximum sensitivities of the Sk and S9 methods lie in the range 1.5 < x < 3. Changing the boundaries of this range is possible only by changing the frequency of the probing field, because other parameters present in the expression for determining x are parameters of the sample under study.
Table 1 contains results of calculations and measurements, carried out while performing electromagnetic monitoring of mechanical stresses Cтм = F/S inside
2
x
the cylindrical product. Wherein the parametric transducer was connected to the digital measuring bridge E7-8 with operating frequency of 1000 Hz. We define normalized flux ®h together with its phase shift according to [8] as follows:
tg9 = -
Ri - R
(12)
-(1 -"n)Lo ]'
where R1 and L\ - actual transducer's parameters; R0 and L0 - parameters of a transducer without a core.
O =
n
1
aLf
■V( R - Ro)2 +a2 [Li- (1 -rOLo ]2 ;(11)
Table 1
Results of calculations and measurements, carried out while performing electromagnetic monitoring of the me-
^ MPa L1, uH ^1, Ohm x lr ct, MS/m
0 170,6 4,377 2,0996 476 8,34
22 172,1 4,377 2,0869 478 8,20
44 176,2 4,381 2,0644 485 7,91
67 178,6 4,383 2,0509 490 7,74
89 182,6 4,389 2,0363 498 7,50
111 186,6 4,395 2,0223 507 7,27
133 190,8 4,403 2,0125 516 7,07
155 195,6 4,410 1,9967 526 6,82
133 195,6 4,412 2,0015 527 6,84
111 194,6 4,413 2,0107 527 6,91
89 193,4 4,411 2,0140 524 6,97
67 191,2 4,405 2,0146 518 7,06
44 189,1 4,400 2,0170 512 7,15
22 182,6 4,393 2,0466 500 7,54
0 179,4 4,389 2,0607 494 7,74
The application of an automated digital bridge makes it possible to measure the parameters up to 20 times, and in Table 1 are averaged values of the parameters, both for an increase in the load and for a decrease in the load on the sample.
As it can be seen from the Table 1 a variation of R1 affects CTm less, than a variation of L\ (>10%).
This makes it possible to effectively use resonance circuits with a parametric transducer for constructing devices for monitoring mechanical stresses.
Figure 3 a, b shows a diagram (a) of SPET connection and a vector diagram of voltages (b) when measuring the amplitude and phase of the output signal. In addition to simplicity, this scheme allows us to select
a rational mode of operation of the transducer according to the frequency of the probing field, regardless of the characteristics of the object under test.
The circuit contains a sinusoidal signal generator G with voltage and frequency adjustment, a phase shift meter ® for measuring the phase difference between the voltages Ue and Ux, resistor R for setting the current through the winding of the transducer, a primary parametric transducer with a test sample ZT, voltmeters V1, V2 and V3 for measuring voltages Ue, Ur and Ux, respectively. SPET and sample parameters are the same, as in the previous case. Excitation current of the transducer is adjusted in such a way, that the field strength corresponds to the linear part of the sample magnetization curve, i. e. |r is independent of H.
a)
Ur = IR0
Figure 3 - Parametric electromagnetic transducer connection diagram (a) and voltage vector diagram
(6)
The frequency is selected from the conditions when the value of the generalized parameter x is in the range 1.5 < x < 3.
Experiments to determine the relation of the amplitude and phase of the transducer signal and the value of the mechanical stress of the sample under study were carried out on a laboratory setup shown in figure 4.
Figure 4 - Laboratory setup for experimental study
Table 2 contains results of the experimental studies of a relation of amplitude and phase of SPET output signal and mechanical stress inside the cylindrical sample for a broad field frequency range.
Table 2
Results of the experimental studies of a relation of amplitude and phase of SPET output signal and mechanical
stress CTm.
MPa Ur, mV Ul, mV 92, deg. Sul, mV/MPa S92, deg / MPa
f = 500 Hz
0 150 68,7 6,54 0,01 0,005
22 150 68,6 6,56
44 150 68,7 6,68
67 150 68,8 6,81
89 150 69,0 6,99
111 150 69,1 7,16
89 150 69,5 7,43
67 150 69,4 7,28
44 150 69,2 7,06
22 150 68,9 6,83
0 150 68,7 6,49
f = 1000 Hz
0 150 71,4 10,96 0,02 0,01
22 150 71,6 11,04
44 150 71,7 11,19
67 150 72,9 11,35
89 150 73,4 11,69
111 151 74,7 11,88
89 150 75,3 12,15
67 150 75,0 12,01
44 150 74,6 11,78
22 150 74,1 11,47
0 150 73,5 11,15
f = 2500 Hz
0 150 95,5 17,30 0,06 0,003
22 150 95,6 17,37
44 150 95,8 17,45
67 150 96,4 17,52
89 151 99,0 17,55
111 151 103,3 17,64
89 149 104,0 17,67
67 151 103,6 17,65
44 151 102,4 17,59
22 151 101,3 17,50
0 151 100,3 17,39
From the Table. 2 it can be seen, that amplitude Su and phase S9 relative sensitivities of the parametric transducer for different frequencies of the probing field at a constant magnetization current is respectively equal to: forf = 500 Hz Su « 1.6%, S9 « 7,4%, forf = 1000 Hz Su « 2.7%, S9 « 8.1% and for f = 2500 Hz Su « 7%, S9 « 2.1%.
From this we can conclude that if we use simultaneously the change in the amplitude and phase of the output signal (two-parameter method), then the frequency of change in the magnetic field strength should be about 1000 Hz. This corresponds the range 1.5 < x < 3. At the same time relative errors of |r and
ct in this range [10] do not exceed 2% and 4%, respectively.
In the experiments carried out, the mechanical hysteresis during loading and unloading is approximately 2%. This means that this method operates on the limit of its capabilities.
In order to increase SPET resolution during con-tactless monitoring of a stress inside the cylindrical samples a compensatory method for monitoring by the magnetizing current or by the frequency of the excitat-ing field is proposed. The essence of the method is as follows. At first we measure transducer voltage phase shift for the specified magnetizing current (Ur = I0R, H < 100 A/m) and for set excitation field frequency f using condition 1.5 < x < 3 for unloaded (F = 0) sample. Then loading the sample stepwise by the quantity AF 4 0 we get new phase shift value. Then, smoothly changing generator G frequency (see figure 3 a) we make phase shift angle to be equal to 90. As a result, we obtain the dependence of the generator frequency on the mechanical stress CTm for the cylindrical product CTm f=T3(ctm). Table 3 contains experimental results.
Table 3
Experimental dependence of the generator frequency on the mechanical stress CTm
СТм? MPa Uro, mV Ul, mV Ue, mV ф2, deg. f, Hz
0 151 75,1 226 1000
22 151 75,0 226 990
44 152 75,0 227 979
67 152 75,0 227 962
89 152 75,0 227 942
111 152 75,0 227 919
133 152 75,0 222 897
155 151 75,6 227 11,14 867
133 152 75,0 222 876
111 152 75,0 227 908
89 152 75,0 227 932
67 152 75,0 227 954
44 152 75,0 227 970
22 151 75,0 226 980
0 151 75,1 226 990
To implement the compensatory method for the magnetizing current, we similarly set the initial values, as in the previous experiment. In this method, loading the tested product stepwise, the change in the
phase angle is compensated by a change in the magnetizing current of the transducer. The result is a function I = _/4(ctm). Table 4 contains experimental results.
Experimental Стм functions of the magnetizing current
Table 4
^ MPa Uro, mV Ul, mV Ue, mV ф2, deg
0 150 73,1 223
22 138 68,2 206
44 126 60,2 186
67 110 54,6 165
89 96 46,3 142
111 82 39,9 122
133 76 23,1 99
155 67 18,2 85 11,20
133 70 19,5 90
111 79 29,9 109
89 90 39,8 130
67 102 49,8 152
44 116 54,1 170
22 130 64,2 194
0 140 70,1 210
Using the data in the table. 3 and 4, one can calculate the relative sensitivity of the compensatory method in terms of frequency and current at 9o = const using formulas:
Af/f .
S/ =-
Sj =■
Mil
(13)
(14)
стм '
where Af AI and Астм - increments of frequency, current and mechanical stress at the transducer's operating point.
It should be noted that the average values of the relative sensitivity of the SPET to mechanical stress for compensation methods in frequency and current are
Sf « 15% and Si « 45% respectively. Thus, the relative sensitivity of the SPET to CTm when implementing the compensation method for current is approximately two times higher than the relative sensitivity of the SPET to the same values in case of using the compensation method by frequency.
It is interesting to determine the magnetoelastic sensitivity SM and tensoresistive Sct sensitivity of the sample material itself, which are found from the expressions:
s = (Mr -Mr0 V Mr M (стм "стмо)/CT (CT-Сто)/CTm
(стм -Стм0 VCT
(13)
where |r, ct and CTm - actual values of the relative magnetic permeability, specific electrical conductivity of the material at CTm 4 0; |r0, CT0 and CTm0 - these parameters values at CTm = 0 (F =0); |rmax, CTmax and CTm max - maximum values of these parameters.
using data from tables 3 and 4, and also formulas (9)-(12), we find out, that SM «10.8 % and Sct « 18 %. The latter shows that the magnetoelastic sensitivity of the sample material is higher than the tensoresistive sensitivity.
The absolute (by amplitude and phase of SPET) and compensatory (by frequency and current) methods considered in the work were used for contactless estimation of the ferromagnetic samples mechanical parameters. These methods can be used to predict the limiting mechanical values (strength limit, plasticity limit, etc.).
The developed methods and SPET one can use as the basis for developing measuring instruments for load, forces, torsional shaft torque and other mechanical quantities.
These methods and a parametric transducer can also be used for contactless monitoring of mechanical stresses in products made of the weak magnetic and non-magnetic materials, which have a more pronounced tensoresistive effect (for example, stainless steels, titanium alloys, alloys of the manganin type, constantan, etc.)
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1НФОРМАЦ1ЙНО - ВИМ1РЮВАЛЬНА СИСТЕМА ДЛЯ ПРЕЦИЗ1ЙНОГО ВИМ1РЮВАННЯ
КОЕФЩ1СНТУ ТЕПЛОПРОВ1ДНОСТ1
Воробйов Л.Й.
1нститут техн1чно'1 теплофизики НАН Украши, провгдний науковий ствробтник, с.н.с., доктор технгчних наук
Декуша Л.В.
1нститут техн1чно '1 теплофгзики НАН Украши, провгдний науковий ствробтник, с.н.с., доктор технгчних наук
Декуша О.Л.
1нститут технгчног теплофизики НАН Украши, старший науковий ствробтник, кандидат технгчних наук
INFORMATION - MEASURING SYSTEM FOR PRECISIONAL THERMAL CONDUCTIVITY
MEASUREMENT
Vorobiov L.,
Institute of Engineering Thermophysics NAS of Ukraine, Leading Researcher, S.R., D.Sc.
Dekusha L.,
Institute of Engineering Thermophysics NAS of Ukraine, Leading Researcher, S.R., D.Sc.
Dekusha O.
Institute of Engineering Thermophysics NAS of Ukraine,
Senior Researcher, Ph.D.
