massive metal billets working.

This chapter will be devoted to calculations of the fundamental characteristics of the tools in the traditional magnetic pulse metalworking. The final aims of the calculations are the recommendations permitting to choice the optimum parameters of the inductor systems and to make sure the maximum force influence on the metal billets to be worked.

Before we pass to the calculations, we would like do some digression and show, what characteristics are fundamental and how they have influence upon the inductor system efficiency.

The traditional magnetic pulse metalworking permits to realize the following main groups of the technology operations:

- the compression and expansion of the tube workpieces;

- the flat stamping of the articles from the sheet metals;

- the welding operations and others.

The field influence efficiency in the time of this operations execution is being made sure by using of the massive well conducting metal workpieces, in the main, as the raw materials. In the whole, the geometrical and electrodynamical parameters of the articles to be worked must ensure the minimum depth of the electromagnetic field penetration into metal during the pulse action time. Execution of this condition means the work in the abrupt surface phenomena regime - the regime of the maximum efficiency of the magnetic pulse method in its traditional version.

The practical realization of this regime is being determined by the inductor field parameters too. For the better understanding of their role in this process we will consider the magnetic pulse metal working scheme in the whole (Fig.2.1).

The principle of scheme action consists in the following: the capacitor bank of energy (1) is discharging on the inductor winding (3) after the commutator (2) closure. During the discharge time the inductor excites the power magnetic field. It induces the whirlwind currents into the workpiece (4). The currents and the external inductor field interaction conducts to appearance of the magnetic pressure forces on the workpiece. Ultimately, the last one will be deformed in accordance with the matrix profile.

(1) is the capacitor bank; (2) is the commutator; (3) is the magnetic field source-inductor;

The discharge current and, consequently, the inductor system magnetic field form depend on the scheme parameters. Practically, the metal articles working occurs in the oscillation regime, when . The according current time form is the exponentially attenuating sine-function with the angular frequency  and the relative attenuation factor  (really,  ¸ 100) kHz and = (0.1¸ 0.3)).

The efficiency of the capacitor bank energy transformation in the inductor system magnetic energy is being determined by the inductance of the last one and the scheme active resistance. These facts define the demands to the inductor system parameters and the scheme in the whole.

If the inductor system inductance is high enough in comparison with the summary inductance of the capacitor bank and the current-carrying conductors, the effective transformation takes place.

From other hand, increase of this parameter leads to the acting field frequency lowering. In this case the diffusion processes becomes the more intensive in the workpiece metal. The abrupt surface effect condition is being broken.

Thus, some optimum has to exist for the inductor system inductance. It must make sure the efficiency of the whole article forming process.

The calculations and optimization of the inductor system properly demand the separate consideration in each concrete case.

There are some different approaches in the calculations of the integral characteristics of the tools for the magnetic pulse metalworking. The first one is being based on the circuits theory methods. This approach suggests introducing of the replacing equivalent electrical schemes. The second approach is the more rigorous. It is being based on the electromagnetic field theory methods. In dependence on the view of the boundary problem the Maxwell equations is integrated with help of the analytical or numerical methods. The second approach owns by the obvious advantage. It consists in the following: the found solution of the according field problem gives the possibility to choose the inductor system form making sure the production operation fulfillment not only, but the achievement of the optimum values of the electrodynamic characteristics in the chosen construction.

Having discussed the main component of the magnetic pulse metalworking scheme we will pass to another its elements.

The capacitor bank and current-carrying conductors inductance can be varied at the expense of the special capacitors application and the constructive solutions ensuring the minimum summary value of this parameter.

The resistance value is closely concerned with the inductance optimum of the all discharge circuit. Their correlation determines the discharge character (oscillated or aperiodical). Besides, the small resistance means the negligible dissipation of the electromagnetic energy and the higher efficiency of the working process in the whole. The scheme resistance lowering can be achieved at the expense of the well conducting metal use for the current-carrying conductors and the inductor windings manufacture.

Ultimately, the maximum effective regime of the magnetic pulse metalworking in its traditional personification is being made sure by the correct choice of the scheme inductance and resistance. The inductance properly inductor system has to take in account the conducting workpiece presence without fail. It is necessary to mark that the working regime efficiency depends on the space-time shape of the magnetic intensity distribution in the working zone of the inductor system essentially too.

The last two factors, connected with the magnetic pulse method tool, are basic. Ultimately, the inductor system inductance is the efficiency value and the field distribution is the magnetic pressure forces distribution, what means the production task execution.

These two most important parameters of the tool of the magnetic pulse method we will further determine as the fundamental characteristics of the inductor system.

The problem formulation for their calculations and based choice are being determined by the stringent technical conditions. In accordance with them the inductor system construction must ensure:
 

1) the ordered frequency of the discharge current in the circuit;

2) the effective transformation of the capacitors electrical energy in the inductor system magnetic energy;

3) the pressure forces distribution which is necessary for the assigned technology operation execution;

4) the high mechanical strength.
 

Solving of the according field problem can be realized with help of the two different analytical approaches.

The first of them supposes to replace the inductor system real configuration in the working zone by some adequate geometrical model permitting to get the rigorous mathematical solution of the according boundary problem.

The second approach supposes the preliminary assumptions about the character of the investigated physical process. On their base the simple mathematical correlations can be received for the fundamental characteristics of the considered inductor systems. The final results of such approach need the experimental check essentially more than the conclusions received by rigorous solving in the limits of the approximate geometrical model.

The application of the considered approaches for the theoretical investigations in the traditional magnetic pulse metal working will be illustrated by the calculations of the inductor systems with magnetic field concentrators for the massive metal billets compression.

The chosen illustrative objects are not accidental. Here is a practical sense.

The concentrators usage improves the working characteristics of the inductor systems in the whole essentially. This fact explains the primary application of the system like that for any technology operation, where it is possible only.

We will explain the action principle of such construction with help of scheme on the Fig.2.2.

The magnetic field of the current in the inductor winding (1) induces the whirlwind current in the concentrator body (2). This current is being closed along the circuit (L), which is shown by the dotted line upon the cross-section (b).

There is some factor defining the efficiency of the construction like that. But at first we will introduce several definitions for the best understanding of the touched upon problem.

From Fig. 2.2 a we may see that the all inner surface of the concentrator consists of the central cylindrical and two lateral conical surfaces.

According to the special technical terminology, the cylindrical surface is defined as "the working surface" or "the working zone" of the concentrator. The current flowing there is named by "the working current".

The lateral surfaces are defined as "the end surfaces". The according currents are named by "the end currents".

Now we will return to the concentrator efficiency problem. Quantitatively, this parameter is being established by the transformation degree of the inductor winding current in the working current.

The business consists in the following. The induced current is being divided and flows along the parallel circuits: on the two end surfaces and on the working surface of the concentrator. The end currents presence lowers the working current value. Ultimately, the inductor system efficiency falls down too.

Thus, the choice of the construction with the minimum ratio of the end and working currents is the main problem in the time of the magnetic field concentrator elaboration.

Discussing the conditions of the inductor systems efficiency it is necessary to pay attention to the serious disadvantage in the investigations of these problems. From our viewpoint, this fact was a reason of the magnetic fields unsuccessful application for solving of the enough numerous technology problems. We keep in mind the synthesis of the fields with the assigned space-time configuration in the inductor systems intended for the execution of the concrete technology operation only. The information about similar works is absent in the science publications.

We will explain this statement.

As it follows from the scientific articles devoted to the traditional magnetic pulse metal working, the main advantage of the method consists in the equipment universality for the execution of the different technology operation. It means there is possibility to work up the different articles (but similar certainly!) by one and the same inductor.

From our viewpoint, it is the error. It can be advertisement but no more.

The tool intended for some concrete operation can not be successfully applied for solving of other technology problems.

The unlike (just a little) article working demands of other inductor system taking into account the all peculiarities of the odder operation. Why?

Let us return to the inductor system intended for the metal workpieces compression (Fig.2.2). It seems this system is universal, if it is necessary to work up any cylindrical billets with according sizes. But it seems only!

The workpiece compression will not be azimuthly homogeneous in consequence of the circle symmetry distortion of the acting field picture. The reason of this fact is the slit presence in the concentrator body.

This inductor system is universal and it can be applied for the execution of the rough operation, when the rigorous demands to the azimuth homogeneity are absent. If the technology operation conditions demand the uniform compression, the cross-form of the working zone in the concentrator will have to be another. It has to eliminate the distortion brought by the slit and synthesize the homogeneous field on the billet surface to be worked. In this case the results of the separate magnetic pulse influences will be the same. The quality repetition of the odder technology operation can be ensured for the manufacturing industry on a large scale.

From our view point the individual approach to each concrete production problem will give the possibility to except the failures of the magnetic pulse metal working in the broad application.

2.1. The magnetic field in the inductor system with concentrator and short workpiece.

The fundamental characteristics of the inductor system will be calculated below. We want to illustrate practically one of the methodical approaches based upon rigorous solving of the boundary problem in the limits of the adopted model for the calculation.

Let we have the inductor with magnetic field concentrator and the short workpiece. The workpiece length is limited by the working zone length.

In practice the considered case can be occurred under pressing of cable tips, for instance.

The calculation model of the concentrator with the short workpiece is presented on Fig.2.3.

The diameter of hole in the working zone is . The air gap between the working surface and the workpiece is h. The working zone length is . The slope angle of the end surfaces to the system axis is . The outer diameter of concentrator is 

The choice of the geometry for the calculation model is conditioned by the following statements:
 

- the boundary surfaces in the adopted sphere-conical coordinates system permit to divide the variables both in the differential equation and in the boundary conditions;

- there is the physically acceptable adequacy to the investigation object.

In accordance with adopted model for the calculation we will introduce the following assumption:

- the field in the inner volume of the concentrator owns by the axis symmetry (the slit influence is negligible);

- the outer radius is great enough, so 

- the workpiece edges are limited by the sphere surfaces with the radius ;

- the field penetration depth is less of all characteristic sizes of the system, it permits to adopt the condition of the abrupt surface effect in the time of the calculation.

It is necessary to mark, that the last assumption seems wrong. And that is why. It can not be adopted previous to the knowledge of the inductor system fundamental characteristics and the according field time parameters, which are the final aims of the calculation.

Discrepancy? Not at all.

Really, the problem will be solved in the neglect of the diffusion processes. But the found results permit to determine the justice of the adopted model and assumptions always, this is to establish the limits of the abrupt surface effect existence in the practice.

Now we will pass to solving of the formulated problem.

In accordance the made assumptions the excited field will be described by the only non-trivial component of the vector potential . Besides, the electromagnetic process is stationary.

If we take into account these factors, the Maxwell equations (1.1) can be simplified. Ultimately, they is being conducted to the view:

We will define the integral of the equation (2.1) in the area bounded by the conical surface  and by the sphere  as the functions sum:

where  is the addendum satisfying to the boundary conditions:

It is necessary to mark the boundary conditions (2.5) and (2.6) were received in the assumption that the field is plane-parallel and uniform in the air gap. The justice of such assumption follows from the conducted experiments.

The function of the view satisfies to the equation (2.1) and the boundary conditions (2.3) and (2.6):

where  are the constants of the integration;  is the conical function of the first order, .

The function can be represented by the integral:

where  are the factors which are being determined by the reverse Fourier transform (they are not adduced, the formulas for them are cumbersome enough).

Substituting  in the (2.10) and comprising the found result with the correlation (2.11) we determine the unknown constants of the integration - :

If to take into account the values of  and , the formula (2.10) can be conducted to the form:

where

The general solution of the equation (2.1) has the form satisfying to the boundary condition (2.7) and the limit condition under :

where  is the unknown constants of the integration;  is the Legendreís function; ;  is the discrete series of numbers, which can be found from the equality .

If  we have, that

With the help of the boundary conditions (2.8) and (2.9) the unknown constants D_k can be determined. For we have,

where

Taking into account the dependences (2.2),(2.12)¸ (2.14) we may write the formula for the azimuth component of the vector potential in the present area:

The current density on the end surface  can be found from one of the Maxwell equations (1.1). Using the abrupt surface effect we are receiving

Integrating the function (2.17) we are finding the total current flowing on the end of the concentrator with the short workpiece:

The working current can be defined by the dependence (without the edge effect):

Further, the total current  and the system inductance can be found with the help of the dependences (2.18) and (2.19).

We are receiving after the simple, but cumbersome enough, transforming, that

The coefficients  and  are the function of the geometrical parameters of the concentrator: of the slope angle of the end surfaces - b and of the ratio of the workpiece radius to the air gap 

The  and  values were calculated for the practical important cases of the concentrator elaboration. They are presented below for and .

b	p / 6	p / 5	p / 4	p / 3	p / 2,5	p / 2
	0,842	1,021	1,301	1,814	2,282	3,14
	0,057	0,086	0,120	0,204	0,262	0,34

These results were used for the calculations of the fundamental characteristics values.

The inductance dependence on the slope angle of the end surfaces is given on Fig.2.4.

The dependence of the inductance of the magnetic field concentrator on the slope angle of the end surfaces L(b ), 

The practically important conclusion may be done with the help of the found dependence. It consists in the necessarity of the slope angle increase for the growth of the concentrator efficiency.

This conclusion can be easy based, if to write the relation of the working current  to the total current 

We may find with the help of the formulas (2.19),(2.20) and (2.21), that

As it follows from the correlation (2.22), the inductance growth takes place under the slope angle increase. It leads to the proportional growth of the relative value of the working current. This fact means increasing of the energy part which can be spent for deforming of the short metal workpiece.

The experimental verification of the found correlations has been conducted at the special installation carried out for the practical investigations of the inductor system in the magnetic pulse metalworking.

The scheme of this installation is shown on Fig. 2.5.

The elements of the charging unit (1) are chosen thus that the charge of the capacitor bank (2) occurs during the first half-period of the entrance voltage of the industry frequency. During the second half-period the commutator (3) is being opened. The capacity discharges at all on the winding of the inductor with the magnetic field concentrator (4) and the workpiece (5).

The scheme returns to the initial statement to beginning of the next period by the entrance charging voltage. After this, the "charge - discharge" cycle is repeated all over again.

The frequency of passing of the discharging pulses is enough for getting of the steady oscillogram of the current in the discharging circuit. It permits to realize the stationary investigation of the field distribution in the inductor system, to determine their equivalent parameters and others.

The coil sondes (6) have been applicated for the measurement of the magnetic induction. They have been situated in the different points of the end surfaces of the magnetic field concentrator. The axes of the coil sondes have coincided with the induction vector direction in zone of the measurement.

The electrical signal from the coil sonde has come to the oscillograph (8) entrance through the integrating circuit (7). The values of the end currents have been determined with the oscillograms help.

The relative values of the end and working currents have been found (to the total current) for comparison with the calculations results. In the experiments the concentrators have been used with the slope angles  and . The differences between the experimental and calculated dates have not exceed 13% what was acceptable fully.

Finishing the consideration of the suggested instance we have to mark the present analytical approach does not give the according accuracy of the calculations in spite of the rigor of the mathematical operations. This fact can be explained by the introduction of the geometrical model. Though it permits to get the analytical solution, but the model corresponds to the real construction of the inductor system approximately.

2.2. The magnetic field of the concentrator with the long workpiece.

The other analytical approach (from the methodical viewpoint) for the calculations of the fundamental characteristics of the inductor system can be realized, if to build the physical (not geometrical!) model of the investigated process from some qualitative preliminary consideration. The approach like that has been named as phenomenological in the methodical literature.

Undoubtedly, this problem solution may be the more approximate than the result received with help of the geometrical model of the considered construction. But this approach owns by the undoubted advantage: the analytical formulas in a form easy to grasp can be received at the expense of the simplified physical assumption about the process character very often.

Let us consider the inductor system with the concentrator and the enough long workpiece. The system axis coincides with the workpiece axis.

The model for the calculation is presented on Fig.2.6.

We can do some assumption before solving of the formulated problem:
 

- the system geometry is that

where  is the air gap in the working zone of the magnetic field concentrator;

is the radius of the cylindrical workpiece;

- is the length of the working zone;

- is the outer radius of the concentrator;

- is the slope angle of the conical end surfaces;

- the received geometry permits to suppose, that the influence of the edge effects are negligible enough;

- the abrupt surface effect takes place in the inductor system which is being investigated;

- the influence of the slit in the concentrator body is negligible enough, and the magnetic field has the axial symmetry;

- there are some conditional surfaces both the working and end zones; they are such, that the magnetic field intensities are the same for all points on each of them; the intensity vectors coincide with the outer normal vectors of these surfaces; we will name them by the surfaces of the equal intensities of the magnetic field.

The last assumption about the character of the physical process is the basic, because its giving concrete (this is the indication of the surfaces sharpís) permits to get the formulas for the magnetic field vectors.

It is obvious from the phenomenological consideration that the surfaces of the equal intensities have the plane circular shape with the inner radius  and the outer  in the working zone.

The magnetic field intensity vector has the only -component;

where F is the magnetic flux;

is the magnetic permeability of vacuum.

From the same phenomenological consideration, we may suppose that the surfaces of the equal intensities can have the toroidal shape in the end zones of the concentrator, for example. Further, we will show that the choice of this or other shape does not exert the essential influence on the value of the magnetic field intensity, if the present choice is being done in the limits of the physical intelligence.

In accordance with Fig. 2.6, these toroids are formed by the rotation of the circumstances arcs around the system longitudinal axis. These arcs are limited by the conical end surface of the concentrator and by the cylindrical surface of the workpiece. The circumstances centers are situated on crossing of the conical end surfaces and the surface of the cylinder 

In the arbitrary point (p,B), on the end surface the magnetic field intensity will be proportional to ratio of the magnetic flux and the area  of the according toroidal surface of the equal intensity which is formed by the rotation of the arc  around axis .

The calculation of this surface area is illustrated by Fig. 2.7.

The calculation of the area of the surface of the equal intensity in the concentrator

end zones: 

With the help of the known formula from the mathematical analysis we is finding, that

where - is the equation of the arc , 

is the radius of the circumstance, where the arc BC is;

is the limits of the integration, 

If we take into account formula (2.24), we may write the correlation for the value of the magnetic field intensity on the end surface of the concentrator as a function of radius r (it is to take into account that on the conical surface ),

With the help of the correlation (2.23) and the condition of the abrupt skin-effect we may find the working current:

By the analogy way we will determine the end currents with the help of the skin-effect condition and the formula (2.25). Integrating the (2.25) on and supposing that , we is getting

Now the total current may be found as the sum of the correlations (2.26) and (2.27):

The inductance of the concentrator with the long workpiece will be defined by the relation:

The last thing which will be necessary for analysis is the ratio of the working current  to the total current of the concentrator .

With help of the formulas (2.26), (2.28) and (2.29) we may find, that

It is necessary to mark, that correlation (2.30) conforms fully with the analogue relationship for the short workpiece (2.22), if . Before passing to the analysis of the found results, we must be convinced in their authenticity.

As it was indicated before, the essence of the present phenomenological approach consists in the preliminary assumptions about the process character with the obligatory experimental verification of the justice of the made assumptions.

By the same way as it was done in the case of the concentrator with the short workpiece the measurements of the end and working currents were conducted in the inductor system with the long billet. These works were realized on the experimental installation described before.

The concentrators with the slope angles  were used. For them the differences between the experimental and calculated dates do not exceed 11%. This fact confirms the justice of the adopted physical model for the calculations of the fundamental characteristics of the inductor systems with the concentrator and the long workpiece.

But the question appears. Why are the differences between the experimental and calculated dates here less than in the case of the rigorous mathematical approach?

Do not be surprised! The explanation is simple enough and consists in the following: it is obvious the phenomenological model is more faithful to the reality than the adopted geometrical model in the case of the concentrator with the short workpiece.

It is necessary to add else the important remark: if the surfaces of the equal intensities exist really, then the small variation of their shape does not exert the essential influence on the results of calculations. And if these surfaces are conical, for example, this fact will not reflect on the calculated dates for the slope angles . A small distinction will be defined by the difference between the areas of the bodies of rotation which are formed by the rotation of the circumstances arcs and tightening chord (it is shown graphically on Fig. 2.8 a).

However, if , as it follows from the physical considerations, the toroidal shape of the surfaces of the equal intensities can be adopted only. It becomes obvious, if to pay attention to the illustration on Fig. 2.8 b.

Let us pass to the analysis of the received theoretical results. It is clear from the physical consideration, that the concentrator efficiency is being defined by the relative value of its working current. If to remove the current branches to the end, the all energy of the magnetic field will be concentrated in the working zone of the concentrator.

As it follows from formula (2.30), under  (accordingly, , the correlation (2.27). In this case, the inductance of the concentrator will equal to the inductance of its working zone practically.

The every thing presented above means that the slope angle of the end surfaces must be chosen from the relation ship:

The instance of the concentrator construction with the optimum shape of the working and end surfaces is shown on Fig. 2.9.

The recommendations for the choice of the slope angles have been received for the concentrator with the long workpiece. These conclusions remain just for the inductor system with the short detail.

These conclusions do not follow evidently from the results of the previous consideration. But the identity of the formulas for the relative working currents (2.22) and (2.30) permits to speak about the identity of the electromagnetic processes in the compared systems.

The instance of the construction of the magnetic field concentrator with the minimum branches of the currents to the end surfaces.

Finishing the summary of the analysis of the electromagnetic processes in the inductor system with help of the phenomenological approach it is necessary to mark its simplicity and possibility to be easy-to-interpret. The dependences and formulas received by the method like that are simple and easy-to-interpret naturally. And what is more the present approach have the enough broad possibilities of the application. Probably, the preliminary assumptions can be done about the processes character in the concentrators with the short workpieces, if to adopt the geometrical model of this system shown on Fig.2.10. The toroids can be by the surfaces of the equal intensities in this case too.

The calculation model of the bimetallic "element" of the concentrator working zone.

The main criterion of the justice of the assumptions must be the experiment. Its results either will confirm or refute the adequacy the adopted model for the calculations. In the second case, it is necessary to reconsider the previous assumptions and to formulate the new statements.

Finally, the last remark on the adequacy of the calculation models to the real processes may be done.

The received results were found under the condition of the abrupt surface effect. To take into account the electromagnetic field penetration into the concentrator metal and the workpiece material it is necessary to conduct the known substitutions. Instead of the geometrical values  and  the equivalent parameters must be replaced:

; ;

where  and - are the field penetration depth into the real metal of the concentrator and workpiece accordingly.

2.3. The heat and mechanical processes in the metal of the magnetic field concentrator.

The instance of the optimum construction of the magnetic field concentrator is the main conclusion of the previous consideration. This result is interesting from the practical viewpoint, because this technical solution owns by the high efficiency of the electromagnetic energy usage. For the practical realization of this suggestion, it is necessary to fulfill the analysis of the questions which become for this construction important enough.

As it is seen from Fig. 2.9, the horizontal part of the radial section in the working zone of the concentrator must be manufactured enough thin-walled (should remember the requirement ). But lowering of its thickness leads to the significance growth of the heat and mechanical processes in the concentrator metal. Therefore, the new suggestions for the heat lowering and the mechanical strength increase of this construction are necessary.

We will not describe all possible methods of the formulated problem solutions. Try to stop and to throw light upon one of them which is the most effective and interesting from our view point.

In an article by the Japanese scientists M. Date and A. Yamagishi the attractive construction was described. In the application to the proposed construction of the magnetic field concentrator their suggestion can be reduced to what the horizontal part in the working zone must be fulfilled in a form of a bimetallic tube. If to look at the longitudinal section, one of the layers faced to the workpiece has to be thin enough and to own by the high conductivity (the main part of the current must flow in it). The second layer has to be relatively thick and to own by the low conductivity. Its purpose is to ensure the mechanical strength of the construction and the heat abstraction from the first layer.

This recommendation needs the ground. It must be fulfilled both mathematically and physically. The analysis of the passing processes may be conducted with help of the calculation model shown on Fig. 2.11.

Let the inner layer of the concentrator working zone (it is faced to the workpiece) is the coating with the conductivity  and thickness . The outer layer is fulfilled from the hard metal with the conductivity  and thickness . The working zone length is .

The possible phenomenological model of the magnetic field concentrator with the short workpiece.

The heat and mechanical processes of interest to us are attributed by the electromagnetic field penetration. The calculation of the intensities may be fulfilled in the plane approximation. The "element" of the profile of the concentrator working zone on Fig.2.11 is being supposed by the bimetallic plate. We will conduct the consideration in the Decart rectangular system of the coordinates.

Naturally, the received results will characterize the indicated processes qualitative and quantitative, if  only. In the opposite case the qualitative description of the diffusion, heat and mechanical phenomena stays just only.

The formulation of the problem is being defined by the following assumptions:

- the dependence upon the only space coordinate  takes place;

-the reason of the appearance of the magnetic field with the only Y-component is the current in the bimetal of the working zone ;

- the spectrum of the exciting current has the components with the frequencies w ; these frequencies satisfy to the thin-walled conductor condition (1.9), this is 

The L-transformed Maxwell equations can be written for the non-trivial components of the electromagnetic field vectors in the adopted system of the coordinates:

The integrals of the system (2.32) is being presented by the following dependences:

1) ,

where ;

is the tangent component of the magnetic field intensity on the inner surface of the working zone;
is the wave factor in the substance with conductivity - _,

,

is the unknown tangent component of the magnetic field intensity on the bound dividing the metal layers, where ;

where ;

is the unknown tangent component of the magnetic field intensity on the outer bound of the bimetal, where 

where  is the wave factor in vacuum,  ;

- is the velocity of the light in vacuum;  - is the wave resistance of vacuum.

The unknown  and  can be found from the equality of the tangent components of the electric field intensities on the boundaries which divide the different substances with the help of the according formulas (2.33), (2.34) and (2.35).

The adopted assumption about the limited spectrum of the exciting current permits to suppose that . Fulfilling the limit transforming in the intermediate formulas for  and , after the reverse Laplace transform we is getting the correlations for the before unknown intensities of the magnetic fields diffused through the layers of the bimetal:

By the same way we can find the space-time distributions of the magnetic fields intensities inside the bimetal of the considered concentrator.

Fulfilling the limit transforming in the formulas (2.33) and (2.34) under  with the received result (2.36) we are getting, that

It is necessary to note that the magnetic field intensity of the inductor winding is presented as ratio  in formula (2.37). It can be done in accordance with the law of the total current.

The space distribution of the current density may be determined by differentiating of the functions in the (2.37).

As it follows from (2.38), the ratio of the current densities in the different layers of the bimetal of the concentrator working zone will equal:

The local heating value is being defined by the heat quantity in the volume unit. With the help of the Joule-Lenz law and the formula (2.38) we can find, that

The ratio of the heat quantities in the volume unit of the different layers of the bimetal may be determined from the correlations (2.40):

The ratio of the heating temperatures of the bimetal layers may be found with the help of the formula (2.41):

where c₁ and c₂ are the thermal capacitances of the different layers accordingly.

For the concrete analysis let us suppose that the current in the concentrator can be assigned by the sine function of the time:, where  is the amplitude and  is the angular frequency.

The limit of the current can be calculated with the help of the formulas (2.40).

Let the permissible temperature of heating of the first layer is  within the quarter period of the acting current , then

where  is the material density of the first layer.

As it is seen from the (2.43), the second layer existence () leads to the increase of the limit of current  times as much. It means the increase of the permissible temperature of heating until the value .

Thus, the efficiency of the heat abstraction can be saluted quantitatively by the date equaled to .

The amplitude of the limit of the current density in the first layer is being found by the substitution of the (2.43) in the correlation (2.38):

It is possible to evaluate the main characteristics of the diffusion and heat processes in the considered construction of the magnetic field concentrator with the help of the received results.

We will illustrate the practical application of the found results by the following example: let the current with the frequency  flows in the working zone by

the length ; the first layer is manufactured from the copper with conductivity , the second one (as the all concentrator) from the steel with 

We may get some evaluations from the thin-walled conductor condition connecting the allowable spectrum of the current and the thickness of the bimetal layers in the working zone of the concentrator: , .

Further we can calculate with the help of the correlations (2.41) and (2.44) that  and, if  and . Graphically, the dependences illustrating the conducted analysis are given on Fig.2.12 where the distributions are shown through the working zone thickness of the magnetic field relative intensity (Fig. 2.12 a), the current relative density (Fig. 2.12 b) and of the relative temperatures (2.12 c).

We have considered the problem of the heating. The problem of the mechanical strength of the magnetic field concentrator will be the next object of our consideration.

The plane model for the mechanical analysis supposes the appearance of the normal stresses only. They can be determined from the static condition of their equality to the magnetic pressure forces (naturally with the reverse sign). But these stresses do not fix the mechanical strength of the concentrator, because their values are a little relatively.

A new quality appears under the consideration of the cylindrical systems. It is connected with the excitation of not the normal stresses only but of the tangent components of the inner electrodynamics forces attributed by the interaction of the field with the concentrator metal.

Let us return to the calculation model on Fig.2.11 for the analysis of the present process. We will consider this model like as the "element" of the cylindrical geometry.

The appearance of the inner mechanical forces is being characterized by the radial  and tangent stresses in the cylindrical constructions. As it follows from the practice of the magnetic pulse metal working the tangent stresses reach the essential values. They can give rise to the destruction of the working zones of the concentrators.

The method of lowering of the tangent stresses  in the especially mechanical constructions (for instance, the gun barrel) was based by the Russian academician A. V. Gadolin in the middle of the last century.

According to this suggestion the cylindrical tubes are being made with a preload interference fit in the composed constructions. Under the interference fit the pressure forces appear in the places of the connections. Their action leads to lowering of the tangent stresses in the material of the inner cylinder.

The magnetic field penetration through the bimetal of the concentrator can lead to the appearance of the pondermotor forces between the layers under the some conditions. Their action will be analogous to the pressure at the expense of the interference fit in the A. V. Gadolin method.

We will apply the superposition principle for the explanation of the physical mechanism of the tangent stresses lowering in the bimetal construction at the expense of the choice of the rigorous determined thickness of the inner layer.

At the beginning we will solve separately the problem of the appearance of the mechanical forces in the material under the action of the pressure on the inner surface of bimetal from side of the magnetic field with the intensity 

Then we will find the inner mechanical forces at the expense of the action of the magnetic field with the intensity  (the penetrated field). And at last we will sum the received solutions.

Now we will write the space-time distributions of the relative total stresses in the present construction of the concentrator. For this we will use the well known mechanical insights of the stresses as the forces of the counteraction and the formula for the magnetic pressure on the conductor (1.3).

We will get,

where

(2.46)

For easy-to-interpret we will pass to the space variable  connected immediately with the bimetal in the Decart rectangular system of the coordinates in the received correlations.

The cylindrical layers are thin-walled enough. It permits to keep the infinitely small values of the first order relatively to  in the formulas (2.45) and (2.46). Then we may use the correlation (2.36) connecting the acting field intensities and 

We will get after the simple but awkward transformations, that

Further we will need the formulas for the stresses under the action of the magnetic pressure forces on the inner bimetal surface only, where .

As it follows from the correlation (2.36),  under , if  Fulfilling the limit transforming in the formulas (2.47) and (2.48) we are finding for the first layer of the bimetal:

The formulas (2.49) can be interpreted:

- if the field penetration is absent, the greatest tangent and radial stresses appear at the points of the inner surface of the concentrator working zone, where ;

- the ratio of the radial and tangent stresses is the value of order  and consequently at the adopted approximation the tangent stresses will establish the mechanical strength of the magnetic field concentrator, because the maximum tangent stresses will be the  times as much as the radial stresses. With the help of the formulas (2.48) the some relation between the bimetal layers can be found. It makes sure the least tangent stresses at the point of the inner surface of the concentrator working zone.

We are getting from the approximate equality  that

In the example considered before we had that the bimetal had been presented by the copper and the steel layers (). With the help of the equation (2.50) we may find, that  if 

The graphic illustration of the tangent stresses distribution through the bimetal thickness, when the inner layer is unloaded completely almost, is represented on Fig. 2.13.

The space distribution of the relative tangent stresses in the bimetal of the concentrator working zone.

a) under the action of the field with the intensity H₁(t);

b) under the action of the field with the intensity H₂(t);

c) the final distribution of the relative tangent stress.

The calculated results confirm the authenticity the statement formulated before: the magnetic field diffusion in bimetal leads (under the according conditions) to the same result as in the case of the preload interference fit for the composed cylindrical constructions according to the A. V. Gadolin method.

In conclusion of the present consideration it is necessary to mark the very important circumstance.

The conducted analysis of the heat and mechanical processes were executed for the instance of the rigorously fixed construction of the magnetic field concentrator. But the found results keep the justice in the qualitative relation for the concentrators of any form.

The recommendation about the bimetal application and the conclusions about the result of the action like that (we keep in mind the strength increase and the heat lowering) are staying true accordingly.
 

2.4. The example of the optimum construction of the magnetic field concentrator for the compression of the massive metal workpieces.
 

Before received results of the investigations of the electromagnetic, heat and mechanical processes permit to formulate the example of the optimum construction of the inductor system with the concentrator in the traditional magnetic pulse working.

We want to mark that the present construction is neither universal nor typical. Should not forget that any technology operation needs the separate analysis. This analysis will give the recommendations for the construction of the concrete tool. Should consider the presented example of the construction as the theoretical and experimental results of the investigation in the inductor systems elaboration only.

We will begin the optimum construction description from the mention of the some common statements. They will help us further to set off its main advantages in the comparison with the known analogous things.

The offered inductor system is intended for the compression of the cylindrical workipieces from metals with the high conductivity by the pressure of the pulse magnetic field.

The concentrators in the known inductor systems have the enough great end surfaces. The essential part of the total current branches out on them. For instance, the value of the end currents is 50 -70% of the total current of the concentrator, if the ratio of the working zone length to its diameter is one less. Therefore, for the execution of the required operation it is necessary the current increase in the winding. But it means the growth of the capacitor bank energy. The last circumstance gives rise the essential lowering of the magnetic pulse metal working efficiency in the whole, the increase of the inner electrodynamics forces in the material of the inductor and the essential lowering of the usage term.

The radial section of the offered concentrator has the form of the cross section of the beam with two horizontal flanges. One of them faces to the workpiece (the working zone).

The constructive solution like that permits increasing of the efficiency of the electromagnetic energy usage and growing of the mechanical strength essentially in the suggested inductor system.

The concentrator is being manufactured from the solid metal. Its outer surface (where the winding is situated), the surfaces of the radial slit and the inner cavity are being covered by the layer of metal with the high value of the conductivity.

The possible concrete materials and some character sizes were named before under the previous consideration of the mechanical stresses and the thermal states, so, the concentrator may be done from the steel, but the covering with the thickness ~ m is being made from the copper.

The total thickness of the "inner" horizontal flange (faced to workpiece) of the radial section of the concentrator will equal  ~m.

The adopted geometry ensures unloading of the working zone from the stretching destroying stresses. As far as the resistance of the working zone to the radial stresses are concerned the construction needs the additional correction: the thickness of the present horizontal flange must be agreed with the condition of the mechanical balance under the action of the magnetic field.

From these consideration we may write the inequality:

where  is the absolute magnetic permeability of vacuum;

is the amplitude of the magnetic field intensity in the center of the air gap between the concentrator and the workpiece;

is the ultimate strength;

is the diameter of the working zone of the concentrator.
 

The suggested inductor system is represented on Fig. 2.14.

This construction consists of the magnetic field concentrator (1) having the stiffening ribs (2). Before indicated surfaces are covered by the layer of the well conducting metal (3). The slots (4) are in the concentrator body from the side of its outer surface. The slots receive the winding (5). The hole (6) of the working zone is connected by the radial slit (7) with the slots (4). The workpiece (8) is situated into the hole of the working zone. The workpiece is separated from the concentrator by the dielectric grommet (9).

The discharging circuit with the capacitor bank (10) is being closed by the commutator (11).

The example of the optimum construction of the inductor system with the magnetic field concentrator for the compression of the massive metal billets:

a) the radial section of the system;

b) the view in accordance with the arrowed line A.

The suggested inductor system for the magnetic field metalworking acts by the following way.

After the switching on of the commutator (11), the power current pulse appears in the closed circuit, consisting of the capacitor bank (10) and the inductor winding (5).

The magnetic field of the current in the winding excites in the walls of the slots (4) the whirlwind currents. They flow on the slit surfaces (7) and on the surface of the working zone (6). The magnetic field is being excited in the air gap between the concentrator and workpiece. The interaction of this field with the whirlwind current induced into the workpiece leads to the appearance of the pressure forces that deform the workpiece as it needs.

The all current concentrates in the working zone practically, because the radial section has the form of the beam with two horizontal flanges. It is the reason of the efficiency increase of the magnetic field action on the billet to be worked.

The copper coating and the stiffening ribs increase the mechanical strength of the concentrator. The operation term of the inductor system grows in the whole accordingly. The thermal regime of its work becomes normal too.

In the whole, the suggested construction of the concentrator is optimum. But we should not forget that the problem of the magnetic field intensity distribution in the cross section of the working zone remains (we keep in mind the distortion of the field picture at the expense of the radial slit existence) and demands of the additional consideration, if it is necessary in accordance with the conditions of the technology operation.