CC BY
6
UDC 624.318
doi: 10.20998/2074-272X.2019.5.07
Yu.V. Batygin, E.A. Chaplygin, S.A. Shinderuk, V.A. Strelnikova
NUMERICAL ESTIMATES OF CURRENTS AND FORCES IN LINEAR TOOLS OF THE MAGNETIC-PULSE ATTRACTION OF METALS. PART 1: LOW ELECTRICAL CONDUCTANCE METALS
Purpose. The electrodynamics processes study in the linear tools of magnetic-pulsed attraction, the final result of which should be the physics-mathematical dependencies for the characteristics of the flowing processes, under the conditions of intensive penetration in metal of acting electromagnetic fields, as well as numerical estimates of these processes main characteristics. Methodology. To carry out research, we used the fundamental statements of the electromagnetic field theory and the mathematical simulation with help of the standard codes from the Wolfram Mathematica package. Results. The functional dependencies for the space-temporal distributions of the currents and forces excited in the linear tools of magnetic-pulsed attraction under intensive penetration of the acting electromagnetic fields through the tool's conducting construction elements are used for the numerical estimates. From the calculation results it follows that from a physical point of view, the cause of the increase in attractive forces while decreasing the operating frequencies can be considered the intensification of penetration processes of the excited fields, which leads to increasing the magnetic pressure from the outside. Thus, the transition to low operating frequencies of the exciting currents can significantly increase the efficiency of the magnetic-pulsed attraction of the sheet metals with the linear tools. Originality. It was firstly determined that from the physical point of view, the reason for the increase in attractive forces with a decrease in operating frequencies can be considered the intensification of penetration processes of the excited fields, which leads to increasing the forces of magnetic pressure on the conductors from side of their external surfaces. Practical value. A usage of the obtained results will allow creating new, more efficient linear tools of magnetic-pulsed attraction of the low-electrical conduction sheet metals operating under conditions of intensive penetration of the electromagnetic fields being excited. References 10, figures 9.
Key worlds: linear tool for eliminating dents in sheet conductors, intensive penetration of the field into metals with low electrical conductance, magnetic-pulse attraction of conductors with unidirectional currents.
Метою cmammi e проведення чисельних ощнок основних характеристик emKmpodunaMiHHUx процеав в Мншних тструментах магттно-шпульсного притягання прoвiднuкiв з односпрямованими струмами в умовах ттенсивного проникнення дючих електромагттних пoлiв в метали з низькою питомою електрoпрoвiднiстю. Методика. Для проведення обчислень використовувалися положения теори електромагнтного поля, що виникчють, з рiвнянь Максвелла, i стaидaртиi математичт програми з пакета Wolfram Mathematica Результати. За допомогою аналтичних вираз'ш розраховаш aмплiтуднo-чaсoвi зaлежнoстi струмiв i електрoдuнaмiчнuх зусиль в лтшних магштно-шпульсшх жструментах, дieвiсть яких заснована на збудженн сил взаемного притягання прoвiднuкiв з односпрямованими струмами. Вперше встановлено, що збтьшення сил притягання, в першу чергу, можливо за рахунок ттенсифжацп процеав проникнення полв, що забезпечуеться переходом до досить низьких робочих частот збуджуючих струмiв. Практична знaчuмiсть. Використання отриманих результaтiв дозволить створювати нoвi бтьш ефективж лiнiйнi тструменти мaгнiтиo-iмпульснoгo притягання заданих дтянок листових металв, що працюють в умовах ттенсивного проникнення збуджених електрoмaгнiтиuх пoлiв. Бiбл. 10, рис. 9. Ключовi слова: лшшний шструмент усунення вм'ятин в листових проввдниках, штенсивне проникнення поля в метали з низькою питомою електропроввдшстю, магштночмпульсне притягання проввдниюв з односпрямованим струмами.
Целью статьи является проведение численных оценок основных характеристик электродинамических процессов в линейных инструментах магнитно-импульсного притяжения проводников с однонаправленными токами в условиях интенсивного проникновения действующих электромагнитных полей в металлы с низкой удельной электропроводностью. Методика. Для проведения вычислений использовались положения теории электромагнитного поля, вытекающие из уравнений Максвелла, и стандартные математические программы из пакета «Wolfram Mathematica». Результаты. С помощью аналитических выражений рассчитаны амплитудно-временные зависимости токов и электродинамических усилий в линейных магнитно-импульсных инструментах, действенность которых основана на возбуждении сил взаимного притяжения проводников с однонаправленными токами. Впервые установлено, что увеличение сил притяжения, в первую очередь, возможно за счет интенсификации процессов проникновения полей, что обеспечивается переходом к достаточно низким рабочим частотам возбуждающих токов. Практическая значимость. Использование полученных результатов позволит создавать новые более эффективные линейные инструменты магнитно-импульсного притяжения заданных участков листовых металлов, работающие в условиях интенсивного проникновения возбуждаемых электромагнитных полей. Библ. 10, рис. 9.
Ключевые слова: линейный инструмент устранения вмятин в листовых проводниках, интенсивное проникновение поля в металлы с низкой удельной электропроводностью, магнитно-импульсное притяжение проводников с однонаправленными токами.
Introduction. The methods of attraction of specified development of technologies for removing dents and
sections of sheet metals using the energy of pulsed leveling sheet metal surfaces during the restoration of
magnetic fields for various kinds of processing aircraft and care bodies [4].
technologies are becoming increasingly relevant in _
various industries [1-3]. Particular attention is paid to the © Yu.V. Batygin, E.A. Chaplygin, S.A. Shinderuk, V.A. Strelnikova
As practice shows, more than 50 % of damages are dents in areas with difficult or completely closed reverse access [4]. In this regard, a very attractive method of magnetic-pulse attraction of thin-walled sheet metals, which can be the basis for creating an effective tool for external straightening of car bodies, is the attraction by linear tools. The physical essence of this method consists in the force interaction of unidirectional parallel currents (Ampere law [10]). The first is the current in the main current lead of a linear instrument (an analog of an inductor, in the generally accepted terminology [1]). The second is the current in the metal being processed. Both currents are excited by direct serial or parallel point connection to a power source. The main conductor is the linear tool firmly fixed. The metal being processed is deformed by attraction. An obvious obstacle to the effective excitation of electrodynamic forces can be induction effects, respectively to which the current in the main current lead induces a countercurrent in the sheet workpiece, and the current in its metal induces a countercurrent in the main current lead of a linear tool.
In general, the attractiveness of this method is due to the simplicity of technical implementation and fairly high energy performance [5, 6]. An analysis of electrodynamic processes in such a scheme will establish the requirements, the fulfillment of which will provide not only ability, but also the effectiveness of the tool of the proposed principle of action.
In fairness, it should be noted that a similar problem has already been considered previously by the authors [7, 8]. However, the direct use of the obtained formulas and relations obtained in the cited works for practical calculations is not possible, due to the limited results found by the conditions for the formulation and solution of this problem.
The goal of the work is an analysis of the temporal characteristics of electrodynamic processes in linear instruments of magnetic-pulse attraction of sheet metals, the final result of which should be physical and mathematical dependencies for the excited currents and forces under conditions of intensive penetration of electromagnetic fields into the conducting elements of tools.
Physico-mathematical model, assumptions and problem definition. For the analysis of electromagnetic processes in the considered metal processing scheme, we adopt a calculation model (Fig. 1) corresponding to the cross section of the studied electromagnetic system with currents J1 and J2 [9].
Az
Main current lead *
jfc
3
Sheet metal
1 Y
X>
Fig 1. Calculation models (J12 - flowing currents)
The workpiece being processed is a rather thin sheet metal of thickness d with rather large transverse dimensions and electrical conductance of the material y.
The main current lead is also «transparent» to the existing fields, so that its metal does not affect the flowing electromagnetic processes. There is a geometric symmetry of the system relative to the ZOX coordinate plane. Along the abscissa axis OX, the system has a sufficiently large extent, so that the differentiation operator d/dx = 0. In the main current lead in the direction of the OX axis, an exciting current uniformly distributed along OY with linear density j(t)=jmjt(t) flows, where jm'= IJ2a is the amplitude of the current density (Im is the maximum of the current), and jt(t) is the time dependence of the current.
The frequency characteristics of the exciting current are such that the condition of quasistationarity according
to Landau - —• b <<1 [10], where a> is the cyclic
c
frequency of the process; c is the speed of light in vacuum; b is the arbitrary characteristic size of the considered electromagnetic system. Let an electromagnetic field with nonzero strength components Ex 4 0; Hyz 4 0 be excited in the system. It is required to estimate the currents and forces in the electromagnetic system under consideration.
Calculation relations. For further calculations, we use the analytical expressions obtained by the authors of [9]. According to the cited publication, on the assumption that the main current lead and the given section of the deformed metal are parallel, the excited currents taking into account induction effects can be described by the following relationship. Total current in the main lead:
^(t) = Im
j (')-B )i
h
-a— ^
f . f a^\ sin| a — d
a
a — d
X e d ¿SkGP4Í ^.epkt\ada
k=0
k ^k^II dt
(1)
where
G{Pk ,a) =
(1 - cos pk )+|P
•sin Pk;
0(pk,a) = cospk \a2 + 2a-pi]+ ipk sinpk •[a +1];
4 is the Kronecker symbol; pk are the roots of the equation:
ctgPk = 0,5[a-a\ ; Pk •(Pk +a2),
la pk ) (^d2)
k = 0, + 1, 2...;
a Pk
djt (t).
dt
,Pkt
is the convolution of
functions.
The integral force of attraction, excited by the interaction of identical unidirectional parallel currents for the same conductors - the main current lead and the given section of sheet metal, is written in the form [5]:
X
,-Bl .( r (s)
^(t) (t) I -
2
(2)
2n V J h where ^0 is the magnetic constant; l is the conductor length; h is the distance between the main current lead and sheet metal.
We carry out numerical estimates for the following initial data. Sheet metal - d = 0.001 m, y = 0.4-107 (Q-m)-1 (steel). The main current lead is made of the same metal and the same thickness as the object being processed (the electrodynamic system «steel -steel» is considered). The geometry of the system is l = 0.06 mm, 2a = 0.01 ... 0.06 m, h = 0.002 m, the working area of the tool is the region lx(2a). The main current lead and the given section of the metal being processed are parallel and identical, so that the currents flowing in the circuit of each of them are the same.
The parameters of current pulses are determined by the characteristics of the power source - magnetic pulse installation MHyC-2 [2], developed and created at the Laboratory of Electromagnetic Technologies of Kharkiv National Automobile and Highway University.
The maximum voltage at the capacitive storage is U = 2000 V. The maximum operating frequency when connected directly to the MHyC-2 electrical output is fmax = 7000 Hz (relative attenuation coefficient is dim = = 0.3). The operating frequency in the discharge circuit of the installation is f = 1000 ... 1500 Hz (relative attenuation coefficient is dim = 0.3) when it is connected through a matching device with a coefficient of energy transfer to the instrument working area K ~ 4. Magnetic-pulse installation MHyC-2 allows to operate in 2 modes. The first mode is the generation of exponentially decaying (oscillating) current pulses. The second mode is the generation of current pulses of an aperiodic (unipolar) time shape. It should be noted that the latter mode is more preferable for practice, since the operation of thyristor switches of this installation with unipolar signals is characterized by a longer term of their operation.
The calculations were carried out using the standard Wolfram Mathematica software package. The approximate interval of variation of the integration variable a e [0; amax] is determined by the functional form of the Fourier image of the transverse distribution of the exciting current. For values a e [0; amax] the modulus of the distribution function must be nonzero. As shown by numerical estimates for the maximum a e [0; 1]. In the general case, the calculated dependence fik = fik(a) can be approximated by the totality of the sequence of linear functions. Numerical estimates showed that for the adopted geometry of the main current lead (inductor), the approximation of fik(a) by the analytical dependence
t-\f2a + k-n .
k = 0, found the integrand
±1, ±2... seems to be quite dependence pk = fik(a) expression
of
is the
A
satisfactory. The substituted into dependence (1).
The improper integrals and the sums of the series in them are calculated using the standard NIntegrate and NSum programs. Summation in rows is performed for different numbers of higher harmonics. Their number taken into account is considered sufficient if an increase in the
limiting value of «k» per unit does not lead to a result different from the previous one by more than 5 ... 10 %. Correction of the integration interval a e [0; amax] is made for the condition a < amax accepted a priori by comparing the final results of the calculations. Their discrepancy is allowed ~ 5 ... 10 %.
The graphs of the characteristic calculated dependencies are grouped below by the type of current pulses generated by the magnetic pulse installation.
Graphic illustrations. In accordance with the capabilities of the power source in terms of generating current signals of various time shapes, numerical estimates of the parameters under consideration were performed initially for oscillating current pulses (Fig. 2-5).
Option No. 1. Operating frequency isf = 7000 Hz, voltage is U = 2000 V, working area: l*(2a) = 0.06 m x 0.06 m.
From the results of calculations for high operating frequencies (f = 7000 Hz) it follows that in the mode of oscillating pulses of the exciting current, the maximum potential of the power source corresponds to a maximum of the attractive force of ~ 1000 N. The transition to the unipolar pulse of the exciting current leads to a decrease in the amplitude of the attractive force by almost ~ 2 times, i.e. up to ~ 500 N.
I
30000
20000
10000
-10000
/ 11
/ \
1 \
\ / !
\
,rad
10
12
Fig. 2. Exciting current
4(<p), A
10000
-10000
-20000
/
/ \
\ /
\J
.rad
10
12
Fig. 3. Induced current
Fig. 4. Total current in the circuit of each of the interacting conductors
soo
¿100
400
200
/ \
/ \
\
\
J
0 2 4 8 10 12
(p.rad
Fig. 5. Excited attractive force
Fig. 6. Exciting current
Fig. 7. Induced current
Fig. 8. Total current in the circuit of each of the interacting conductors
1750 1500 1250 1000 750 500 250 0
/
/ \
/ \
j \
1 \
j \
J - -
çuad
Fig. 9. Excited attractive force
From a physical point of view, as the reason for a decrease in the attractive force with an aperiodic shape of the exciting current, a decrease in its temporal duration in comparison with an oscillating and exponentially decaying harmonic signal can be considered (see Fig. 2, where y e [0, (4-n)]). This, obviously, leads to a reduction in the development time of electrodynamic processes in the processed object. All further calculations were performed for the unipolar shape of the exciting current, as more preferable for practice (increase in the operating life of the thyristor switches of the installation) (Fig. 6-9).
Option No. 2. Working frequency is f = 1500 Hz, voltage is U = 2000 V, working area: lx(2a) = 0.06 m x x 0.06 m.
From the results of calculations for low operating frequencies (f = 1500 Hz) it follows that:
• the maximum potential of the power source corresponds to the maximum of the attractive force ~ 1800 N, which is approximately 2 times higher than the similar maximum for high frequencies (f = 7000 Hz) of the exciting current;
• from a physical point of view, as the reason for the increase in attractive forces with a decrease in working frequencies, the intensification of the processes of penetration of excited fields into the workpiece metal and tool can be considered, which leads to an increase in magnetic field strengths from the outside of the system conductors and, accordingly, to an increase in magnetic pressure forces them from the outside.
Conclusions.
1. The amplitude-temporal dependencies of the characteristics of electromagnetic processes in linear instruments of magnetic-pulse attraction of sheet metals with low electrical conductance at the intense penetration of the excited magnetic fields into the conductive elements of the tools and the processed objects are calculated.
2. It has been established that the transition to low operating frequencies of exciting currents can significantly increase the amplitudes of the attractive forces of metal (for example, at f = 1.5 kHz, the force indicators for the steel being processed increase by about 2 times compared with the case when f = 7 kHz) .
3. It is shown that as the physical reason for the increase in attractive forces with a decrease in working frequencies, the intensification of the processes of penetration of the excited fields through the metal of the tool and the workpiece can be considered, which leads to an increase in the magnetic pressure forces on the conductors from the outside.
Acknowledgment. The work was carried out by the Department of Physics of Kharkiv National Automobile and Highway University in the framework of the Research Project «Energy-saving low-cost technologies of power supply and repair of vehicles» No. 08-53-19, funded by the Ministry of Education and Science of Ukraine.
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Received 16.04.2019
Yu.V. Batygin', Doctor of Technical Science, Professor, E.A. Chaplygin', Candidate of Technical Science, Associate Professor,
S.A. Shinderuk1, Candidate of Technical Science, Associate Professor,
V.A. Strelnikova1, Postgraduate Student, 1 Kharkiv National Automobile and Highway University, 25, Yaroslava Mudrogo Str., Kharkov, 61002, Ukraine, phone +380 57 7003852,
e-mail: [email protected]; [email protected]; [email protected]; v.strelnikova91 @gmail.com
How to cite this article:
Batygin Yu.V., Chaplygin E.A., Shinderuk S.A., Strelnikova V.A. Numerical estimates of currents and forces in linear tools of the magnetic-pulse attraction of metals. Part 1: Low electrical conductance metals. Electrical engineering & electromechanics, 2019, no.5, pp. 40-44. doi: 10.20998/2074-272X.2019.5.07.
