electro magnetic forming a russis based book landscape

8/13/2019 Electro Magnetic Forming a Russis Based Book Landscape

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CHAPTER 1

PHYSICAL PRINCIPLES OF ELECTROMAGNETIC METAL FORMING

1.1 BASIC EQUATIONS OF ELECTROMAGNETIC FORMING TECHNIQUE

The procedure of electromagnetic metal forming is described by a system of differential equations characterizing theelectromagnetic, mechanical and heat thermal phenomena.

Let us present the equations, describing the electromagnetic processes:

rot = ; (1.1)

rot ; (1.2)

(1.3)

Here

div =0, (1.4)

div (1.5)

where - is the electric field intensity; is magnetic permeability of metal; - is the magnetic field intensity; isthe material electrical conductivity; is the current density; is the velocity of the moving conductor.


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If we consider the variation of as a function of temperature, the last equation of the system transforms to:

= , (1.6)

where 0 - the electrical conductivity at zero temperature, t - is the temperature coefficient; c m - material specific heatcapacity and is the temperature of the conductor.

The thermal processes in electromagnetic forming are described by the equation:

(1.7)

where Q is a heat generated in the conductor,t - specific thermal conductivity of the conductor.

The boundary conditions on the workpiece surface - heat convection through the free surface should be considered:

, (1.8)

where n is the normal to the surface, 0is the ambient temperature; k 0 is the heat exchange coefficient.

The force per unit volume acting on the conductor placed in the magnetic field with field intensity H is equal to:

(1.9)

Considering (1.2) the equation (1.9) assumes the form:

(1.10)


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Finally, to obtain the complete picture of electromagnetic forming technique, the aforementioned equations must besupplemented by the elastic-plastic characteristics of the workpiece deformation.

Denoting the total force, acting on the workpiece elements as F c, we obtain the equation of the elastic deformation ofthe system:

D1 (1.11)

where is the workpiece displacement vector; D 1(S) is an operator that depends on the workpiece shape and theapplied theory of elastic deformation. For instance, in the general case of elastic body

(1.12)

where , is the Lame constant, G l .is the shear modulus.

The equation of plastic deformation can be written in the general form:

D2 (1.13)

where is the velocity vector of the workpiece displacement; D 2( ) is the operator describing the plastic deformationthat depends on the workpiece shape and the plastic deformation theory used. For instance, in the general case therigid-plastic body D 2( ), the resistance to the plastic deformation and the pressure in metal are taken into account:

D2( )= - grad p m + D 3( ) (1.14)

where D 3( ) includes the resistance to plastic deformation; p m is the pressure in metal.

Von Mises and Tresca conditions can be used as a yield criteria.


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The equations of elastic (1.11) and plastic (1.13) deformation can be solved along with boundary conditions.

1.2 CAPACITOR BANK DISCHARGE THROUGH THE SYSTEM OF TWO AND MORE INDUCTIVELYCONNECTED CIRCUITS.

Electrical schematic of the inductors of different type can be represented in the form of one, two and more inductivelyconnected circuits.

Fig.1

Lc- the self-inductance of the discharge circuit; Rc -self-resistance of the discharge circuit; L u3; R u3 - are the inductance and active resistance of the single circuitsystem.

The single circuit systems are inherent to the technological schemes, where the operations are performed bydischarging the main circuit through the workpiece. The two-circuit systems are typical to the wound spiral inductors


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and single turn solenoids; multi-circuit systems are characteristic of the induction systems with the magnetic fieldconcentrators and inductors with matching circuits.

Figures 1a, 2a and 3a show the technological operations, carried out by the three previously indicated methods (1-workpiece, 2 - induction coil, 3, 4, 5 - matching circuits made in the form of split turns). Figures 1b, 2b and 3brepresent the equivalent diagrams corresponding to the single circuit, two- and multi-circuit systems, respectively,where C is the capacitance, L c, R c are the self-inductance and resistance of the circuit; L i-w R i-w are the inductance andactive resistance of the single circuit system; L iR i are the inductance and resistance of the inductor; L wR w are theinductance and resistance of the workpiece; L 1R 1 are an inductance and resistance of the primary winding of thematching circuit; L 2, L 3, L n .....,L n+1 ,R 2, R 3....R n+1 - the inductances and resistances of the matching circuit; i d - is thecurrent in the discharge circuit; i w is the current in the workpiece; i 1, i 2 , in......i n+1are the currents in the matching circuit.

These schematics are designed for the compression. Electrical schematics for other operations (expansion, sheet metal

forming) will be the same.

From electrical point of view an inductor together with the workpiece connected to the discharge circuit of themagnetic pulse device can be interpreted as a chain of magnetically connected circuits with the capacitor in the primarycircuit.


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Fig.2 L1R1 are an inductance and resistance of the primary winding of the matching circuit; L2, L3, Ln .....,Ln+1,R2, R3....Rn+1 - the inductances and resistances of

the matching circuit; i1, i 2 , in......in+1 are the currents in the matching circuit , Lu, Ru - inductance and resistance of the inductor.


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Fig.3

Lc- the self-inductance of the discharge circuit; Rc -self-resistance of the discharge circuit; L 3, R 3 are the workpiece inductance and resistance, Lu, Ru -inductance and resistance of the inductor.


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Solving the equation for the single-circuit by the operator procedure (t=0, I=0, U c=U 0), and denoting:

introducing the dimensionless time 0 t = 0 and 1/ 0 = , we can find the relative value of the discharge current forthree cases:

1. 0 > 1 or 1 1:

. (1.17)


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Correspondingly, for the relative voltage on the inductor:

1. 1 1:

(1.20)


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Fig. 4

Fig.4 shows the family of curves =f ( 0) for different values of 1. As is obvious from the figure,increasing 1 results in the decrease of discharge current amplitude; for instance if 1=1 i d equals to 0.36 of the non-damped discharge current value. The primary goal is to reach minimum value of 1 .As a rule, 1 has a value of 0.1 to0.3. Consequently, for the further analysis of the electromechanical processes primary attention should be given to thefirst discharge process ( 1 < 1), since this process has the greatest practical significance.

In solving the equations describing the electromagnetic processes in the two-circuit system for the primary andsecondary currents during the discharge, we find the primary component of the discharge current:

(1.21)


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where i is the scattering coefficient,

; (1.22)

- is the angular frequency,

; (1.23)

-is the damping decrement,

, (1.24)

where

The current in the workpiece

(1.25)

If we consider that in the cases of practical importance >> 2, the expressions for the primary and secondary currentsin the workpiece will have the form

(1.26)


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(1.27)

respectively, where L e is the equivalent inductance of the inductor-workpiece system,

Le= iL i (1.28)

Under the given assumptions the damping coefficient is

(1.29)

Here R e is an equivalent resistance of the two-circuit system, which can be approximately determined from theexpression:

R e = R 1 + R 2 (1- i ). (1.30)

1.3 ELECTROMAGNETIC WAVES IN METAL

Usually in the electromagnetic forming equipment the frequency of the discharge current reaches several tens ofkilohertz. For such frequencies the magnetic field outside the metal can be considered quasistationary; on the contrary,the field in the metal has a wave nature.


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Neglecting the e.m.f. of the workpiece movement, from equations (1.1) to (1.3) we obtain the equation of the magneticfield diffusion in the metal for the uniaxial case:

(1.31)

The system of coordinates (fig. 5) is selected, where the origin is placed on the surface, through which the magneticfield enters the plate, the axis coincides with the positive direction of vectors , ; the y axis coincides with the

positive direction of vector ; z axis points inside the workpiece. It points the direction of motion of the forward waveof the field and positive direction of the vector Umov-Pointing:

(1.32)

It is of interest to solve an equation (1.31) with the following boundary conditions:

H(0,t) =0 if -


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Fig.5

Let us assume the current history in the inductor in the form of damping sine wave

i= . (1.35)

Then the magnetic field intensity on the surface of the inductor will be expressed by the formula:

H(0,t)= . (1.36)

Considering (1.36) the solution of the equation (1.31) is represented in the form:


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. (1.48)

From equation (1.48) for induction on the workpiece surface we find:

. (1.49)

The magnetic induction in the workpiece is analogous to also represents a traveling wave:

. (1.50)

The total current in the workpiece per unit length is:

. (1.51)

Integrating this equation and considering the equality (1.48), we have:

. (1.52)

From the equation it is clear, that the equivalent current could be generated by a layer of thickness with uniformly

distributed current density , where the equivalent current would have a phase shift of /4 with respect to the actualcurrent density on the surface at z=0. The thickness determined by (1.45) is called the equivalent depth of

penetration of the magnetic field into the metal (skin depth). As is obvious from (1.45), depends on the frequency ofthe induced current and the physical constants of the material.

The total current through the skin depth of the workpiece is


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, (1.53)

where l w is the workpiece length. By integrating we can obtain:

i=H mlw sin t. (1.54)

The magnetic flux through the entire half-space per unit length is

. (1.55)

Substituting the value of B s.s from (1.50) in this equation and integrating we finally find:

. (1.56)

Then for the original

. (1.57)

Consequently for the sinusoidal regime the magnitude of the equivalent layer of the magnetic flux in metal for thetime t= :

. (1.58)

In the most general situation the equivalent depth of penetration of the magnetic field can be found in terms of theinduction on the workpiece surface. Then, if we consider that the magnetic permeability remains constant, thefollowing expression is valid:


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. (1.59)

Considering the base value of , the value obtained for it during the sinusoidal process (1.58) is equal to:

. (1.60)

Let us substitute the value of H s(z,t) from (1.42) in (1.60). If the magnetic field intensity varies as the damping sinewave with time, considering (1.39), (1.40) the relative value transforms to:

. (1.61)

Finally, substituting the value of H (z,t) from (1.38) in (1.60) for the value of in the transient regime we obtain:

(1.62)


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The values of the relative depth of the magnetic flux in the stationary regime, calculated by the formula (1.61) for

different values of , are listed in Table 1, where the values of is less than one, which indicates the deceleration ofthe penetration of the magnetic field in the damping sine wave regime.

Lets determine the value of the relative depth of penetration of the magnetic flux for the mostly essential practicalcases during the electromagnetic metal forming: = /2, = . The values of calculated for different , are listedin Table 1.

Table 1.

0 0.1 0.2 0.3 0.4

, =1 1.42 1.425 1.49 1.56 1.75

, =1/2 1.56 1.46 1.45 1.5

Table 2.

0 0.1 0.2 0.3 0.4

, 1 0.95 0.91 0.89 0.86

Consequently, the equivalent value of e.t is appreciably greater than its value in the stationary sinusoidal regime.


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However, as shown in Table 2, its relative value varies insignificantly within a wide range of values of , .

For instance, for the practically important range = 0.1 - 0.3, = varies from 1.425 to 1.56; which can beconsidered in the practical calculations as the mean coefficient equal to 1.46.

Considering the expressions ; = t, d and are converted respectively:

; (1.63)

. (1.64)

Applying equation (1.38) we can plot the graphical dependence for =0.2, 0.3 and =1.57; 3.14.

The maximum value in the first half period is taken as the base value of the intensity H .


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Figures 6, 7 show the magnetic field intensity distribution with respect to the workpiece thickness for two valuesof and for the two most typical points in time respectively (curves 1 and 3). For the comparison the plots of thestationary process are shown on the same figures (curves 2 and 4).

1.4. PRESSURE ON THE WORKPIECE DURING ELECTROMAGNETIC METAL FORMING

It is well known that the lateral surface of the magnetic field tube experiences pressure numerically equal to the specificenergy density of the magnetic field per unit volume at the same space point:


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. (1.65)

The total pressure on the workpiece corresponds to the pressure difference on the surface tubes of the workpiece fieldlocated on both sides of it:

. (1.66)

where H 1, H 2 are the magnetic field intensities on the workpiece surface on the field input side and on the opposite side.

The formula (1.66) is universal. It expresses an instant pressure in terms of instant values of H and it corresponds toany law of magnetic field variation in time and space. However, it doesnt allow determination of the force distribution

per unit volume with respect to the workpiece thickness.

Lets find the force distribution per unit volume with respect to the workpiece thickness, assuming that the magneticwave damps completely in the metal. Lets isolate an elementary volume dv m inside the workpiece, as shown in Fig. 5.The base area of the elementary parallelepiped will be taken as equal to one. According to the equation (1.9) anelementary force acting on the current in the volume dv m, is:

dF = Bdv m. (1.67)

Since E = and B = H, we obtain

dF = EHdz. (1.68)

The specific force density

f = EH. (1.69)

Substituting the values of H c, Ec from (1.42) and (1.43) in equation (1.69), after all transformations we have:


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(1.70)

Integrating expression (1.70) from zero to z and considering t, = , we obtain the pressure distribution withrespect to the workpiece thickness:

(1.71)

The pressure distribution with respect to the workpiece thickness for various and time moment /4 is shown in Fig.8. As shown here, there is no pressure on the surface.

Curves 1, 2, 3 correspond to the values =0, 0.2, 0.3. The total force per unit workpiece surface or the pressure on theworkpiece is equal:

(1.72)

or

. (1.73)


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The law of variation of the specific force density in the transition regime can be obtained by substituting the values ofH, E from (1.38), (1.41) in equation (1.69). As it was shown numerically, the transient process of establishing theelectrodynamics forces practically lasts one period, however in the first half-period the difference between themagnitudes of the dynamic forces in the stationary and transient regimes are negligible and do not exceed 6 to 12%.

The incomplete damping of the magnetic field waves in the workpiece wall can happen during the metal forming ofthin-walled sheets and tubes, and also products made of metals with high resistivity. The minimum skin depth for the


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Fig. 9

The attenuation coefficient can be found from Fig. 10.


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Fig. 10


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CHAPTER 2

EQUIVALENT INDUCTANCES AND RESISTANCES OF THE TWO-CIRCUIT AND MULTI-CIRCUITSYSTEMS

2.1 GENERAL REMARKS AND BASIC EQUATIONS

The circuit diagram of the inductor-workpiece system represents a combination of magnetically bound circuits (in thesimplest case - two magnetically bound circuits). In the more complex systems there can be significantly more suchcircuits and they can be connected in series or in parallel or in series-parallel groups, which is shown in Fig. 11, a,b,respectively.


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Fig. 11

However, there are always two magnetically connected circuits in the inductor systems which are formed by theworkpiece and the inductor or by the other elements of the complex inductor systems, for instance, with magnetic field

concentrators or with the matching circuits. Therefore, any inductor with a large number of circuits can be representedin the form of two single-turn circuits, included in the discharge circuit through the air transformer or the transformerwith ferromagnetic core which has a single-turn winding with one or several parallel branches at the output.

The primary windings of the transformer can have any number of turns, any schematic and can be included directly in`the discharge circuit or be connected to it through the analogous devices with any number of turns both in the primaryand secondary windings as well.

Inasmuch, as was explained, two single-turn circuits (or one single-turn and another multi-turn circuit) are the common

element for any electric schematic of the inductor, the equivalent parameters of the inductor system can be determinedin two steps: first, determine the equivalent parameters of the simplest system, then they have to be generalized for amore complex electrical circuit.

The determination of the inductance of the simple or complex inductive system is related to the calculation of themagnetic fields of this system.

The exact calculation of the electric and magnetic fields of the inductor-workpiece system is a quite a complicated problem. The general nature of the current distribution in the inductor cross-section and in the workpiece can bedetermined as a result of studying the pattern of the resultant magnetic field of the system considering the skin-effect athigh frequencies characteristics of the oscillating capacitor discharge (tens of thousands Hertz).

At such high frequencies the electromagnetic field distribution is characterized by a strong skin effect. The depth of penetration of the magnetic field into the metal usually has a value on the order of fractions of millimeter, or simply isvery small compared to the diameters of the inductor and the workpiece; therefore the magnetic field is primarilyconcentrated in the gap between the inductor and the workpiece, and in the surface layers of the inductor and theworkpiece. The inductor and the workpiece currents leak in the surface layers. Therefore, the currents in the inductor


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and the workpiece are always concentrated in the surface layers faced to each other, independently of whether theworkpiece is inside or outside the inductor.

If the magnetic circuit doesnt contain ferromagnetic materials, for the flux coupling in the air gap (hereafter called the

flux coupling of the circuit):

=Li (2.1)

where L is the circuit inductance; I- is the instant current in the circuit, i=I msin t.

The overall flux coupling of the circuit is:

, (2.2)

where di is the elementary current tube of the circuit; - is the magnetic induction flux coupled to the elementarycurrent tube.

Usually if there is a phase shift between the current and the flux, the circuit inductance is a variable, the value of whichduring the half period varies within quite broad limits; therefore, to built the equivalent schematic, it is necessary to usethe mean inductance value.

Averaging expressions (2.1), (2.2) in the interval of the first half-period of harmonic current function, we obtain:

(2.3)

and

, (2.4)


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where L m is the mean value of the circuit inductance over the first current half-period.

From the last two equations it follows that:

. (2.5)

If the investigated schematic is reduced to the given circuit, the determination of its equivalent inductance is, accordingto (2.5) reduced to finding the self external and internal flux couplings of the circuit obtained. The equivalent circuitresistance is:

(2.6)

where i is the resistivity of the circuit element; i- is the equivalent depth of the penetration of the magnetic field waveinto the circuit element; V m is the integrated volume of the metal.

2.2 EQUIVALENT INDUCTANCE AND ACTIVE RESISTANCE OF THE TWO-CIRCUIT SYSTEMS

In the general case the two single-turn circuits formed by the forming workpiece and the operating surface of the

inductor adjacent to it (hereafter called the "inductor") can be represented by the system shown in Figure 12, where theconical workpiece 1 placed into the single-turn (Fig. 12,a), multi-turn (Fig. 12, b), coaxial (Fig.12,c), conicalinductor 2 is illustrated, through which the discharge current of the capacitor bank is leaking. The semi-angle at theapexes of the circuit is i. The diameters of the upper and lower bases are denoted by D and d , respectively. The lengthof the generatrix of the inductor is l .. The magnitude of the air gap is h a,a w is the workpiece wall thickness, a i is thethickness of the inductor.

The following assumptions are made here:


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1. The air gap between the inductor and the workpiece is small compared to the outer diameter of theworkpiece : hd.

2. The electromagnetic wave damps completely in the workpiece and inductor thickness: a w>3 w, a i>3 i.

3. The total flux coupling in the air gap is constant along the entire length of the workpiece for the single-turn inductor.

4. The workpiece is stationary with respect to the inductor.

5. The induction in the air gap of the coaxial inductor varies according to the law:

. (2.7)

Consequently, the equivalent inductance of the two-circuit system

L tc=L g+Lw +L i (2.8)

where L g is the inductance of the air gap; L w is the workpiece self-inductance and L i is the self-inductance of theinductor.


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; (2.9)

multi-turn

; (2.10)

coaxial:

; (2.11)

cylindrical inductor:

single-turn

; (2.12)

multi-turn

; (2.13)

coaxial


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(2.14)

disk inductor:

single-turn

; (2.15)

multi-turn

; (2.16)

coaxial

; (2.17)

Considering the equivalent active resistance of indicated systems and (2.6) we can obtain:

conical inductor:

single-turn


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(2.18)

multi-turn

; (2.19)

coaxial

(2.20)

cylindrical inductor:

single-turn

; (2.21)

multi-turn

; (2.22)

coaxial


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; (2.23)

disk inductor:

single-turn

; (2.24)

multi-turn

; (2.25)

coaxial

; (2.26)

For a small workpiece length of significantly developed toroidal zones which occurs in the massive single-turnsolenoids and magnetic field concentrators, the indicated factors can affect the magnitudes of the equivalent

parameters. For instance, for the standard schematic (Fig.13), the coefficient describing the decrease in the equivalentinductance has the form:


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, (2.27)

where l w - is the length of the operating zone; k is the slope of the toroidal generatrix to the inductor axis; D 0 =2R 0 - isthe large diameter of the toroidal zone.

The ratio of the toroidal currents to the total current in the inductor is represented by the function:

(2.28)


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Consequently, for the actual ratios =10 to 12, , h g=(1-1.5) x10 -3 m for exclusion of the significant effect of theend current on the magnitude of the working current, the following relationship should be satisfied:

(2.29)


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(2.30)

where k s is the coupling coefficient; ; p-is the number of turns of the concentrator; p=mn ( n - is the number ofsections of the winding connected in series; m- is the number of groups of the series included sections connected in parallel).

For the equivalent inductance of the inductor considering the number of grooves and the circuit diagram of the workingwinding we have:

(2.31)

To determine k s let us write the following system of equations:

(2.32)

(Le.hg

- is the equivalent inductance of one half of the groove for L ep

=0). Omitting the intermediate conversions for the coupling coefficient we find:

(2.33)


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In order to find the equivalent inductance of the groove with the surface of the working winding L e.gs going with it, letuse the equation (2.16) for a disk type multi-turn winding. Instead of D, d the outer and inner diameters of the grooveare substituted respectively, and the equivalent depth of penetration I, w are replaced by s, w - where g is the depth of

penetration of the magnetic field into the wall material of the concentrator grooves; w is the depth of magnetic field penetration into the material of

the working winding. The parameter h, just as in the initial formula, represents the magnitude of the air gap. Formula(2.16) is used due to the constant inductance in the air gap between the working winding and the wall of the groove.This is explained by the adopted system of current distribution, which indicates the existence of two identical currentlayers in the working winding located near the groove walls. Since the currents in the indicated layers leak in onedirection, even in the case of the multi-turn working winding the flux cannot leak between the turns, and the magneticcoupling between them is absent.

Considering the given remarks:

(2.34)

(where - is the number of turns in the groove).

The inductance of the current layer in the groove wall can be determined by the approximate expression for theinductance of the pancake (disk) coil [7], which is accurate enough in the range of variation of expressions D g dg:

. (2.35) Solving the system of equations (2.33)-(2.35) and considering the fact that g= n g, we finally find:

. (2.36)


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Now let us transform the equation (2.30):

. (2.37)

Substituting the inductance value from (2.9) instead of L e.p, and also L c.s., L d.s.s from (2.12), (2.15) considering (2.35), weobtain the expression for the equivalent inductance of the inductor with the magnetic field concentrator.

The conical shape of the working surface of the inductor :

(2.38)


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Cylindrical shape of the work surface:

(2.39)

Disk shape of the work surface:


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(2.40)

For the inductor that has a concentrator with a smooth surface, equation (2.37) will be reduced to the form:

(2.41)

where s - is the number of turns of the cylindrical working winding; L e.w - is the equivalent inductance of the workingzone defined, just as before, as a function of the form of the working surface by the formulas ( 2.9), (2.12), (2.15).Finally, in order to find L 2 we have to use the expression:

, (2.42)

where is the value determined by the curves in Fig. 15, 16 plotted in terms of functions l o /(d o- o ), (d o - o) /l o,respectively, then the expression for k acquires the form:


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, (2.43)

where d o - is the outer diameter of the smooth surface of the concentrator; l o - is the outer axial dimension of the

concentrator; h - is the insulating gap between the outer smooth surface of the concentrator and the surface of theworking winding adjacent to it.

CHAPTER 3

MECHANICAL PROCESSES DURING ELECTROMAGNETIC FORMING

3. 1 THE DETERMINATION OF THE WORK OF DEFORMATION OF THE WORKPIECE DURINGELECTROMAGNETIC FORMING

One of the important values characterizing the process of molding during electromagnetic forming is the work ofdeformation. Usually, the specific work of deformation during the tube and sheet metal forming is defined by theequation:

(3.1)


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where a s - is the intensity of the specific work of deformation; is the deformation intensity; B, m m are the mechanicalcharacteristics of the forming materials.

In the most general case the deformation intensity is equal:

(3.2)

( 1, 2, 3 are the components of the deformation in three mutually perpendicular directions).

Then the total deformation work:

A=a sVw (3.3)

(V w is the total volume of deformed material).

In practical sheet metal forming the mean intensity value is used which can be determined by the averaging of the truevalues of .

Figure 17 shows the features of the tube metal forming and the corresponding diagrams of strain distribution (the dottedline gives the mean value of strain), Fig. 17, a shows the cylinder forming, where the mean value of strain is:

(3.4)

Fig. 17, b shows the forming of the cone, sphere and rift, where a is equal:

(3.5)


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(3.6)

(3.7)

Fig. 17

During the sheet metal forming the elements formed can be shown in Fig. 18, where the forming of the sphere (fig. 18,a), the conical form (Fig. 18, b) and the flat stamping (Fig. 18, c) is illustrated. Let us give the values of the mean strainfor the given shapes of sheet metal forming:


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(sphere) (3.8)

(conical shapes) (3.9)

(flat stamping) (3.10)

The mean strain of the toroidal spherical shape (Fig. 19)

(3.11)

When determining the specific work of deformation of complex parts it is necessary to isolate the elementaryconfigurations for which we then determine the work of deformation.

The values of the mechanical constants B m for some materials are listed in Table 2a. The approximate values of theseconstants can be obtained by the method of direct tension [43], and m m = n, the B value can be calculated at the momentwhen necking takes place.

(3.12)


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Fig. 18

Here

(3.13)

If there is a diagram of the tensile testing in terms of P - load and l - total elongation, then this diagram must bereconstructed in the following coordinates: true stress- true strain.

By the magnitude of the total elongation l taken from the initial diagram, the true strain can be determined:

; (3.14)

and the true stress:


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. (3.15)

If only reference data are available (the yield point, UTS and non-uniform elongation at the necking n ) then:

. (3.16)

The constant B can be found from equation (3.12).

Consequently, having the set of formulas for determining the work of deformation of the simplest configurations, wecan find the work of deformation of more complex configurations, breaking them down into the simplest ones.

Fig. 19

Table 2a.

Material b x10

7 N/m

2 B x107 N/m 2

mm Material b x10

7 N/m

2 B x107 N/m 2

mm

Sheet steel 30 50 0.3 Lead 1.5 3.2 0.37


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10 kp

Steel 3 43 77 0.24 Tin 2.6 6.3 0.51

Steel 20 45 72 0.17 Zinc 11 13 0.05

1X18H9TSteel

62 118 0.29 Nickel 50 103 0.36

AnnealedAl

AnnealedCopper 23 48 0.38

Duralumin

AMn-Am

D16-Am

B-25M

12

21

21

22

33

31

0.2

0.15

0.12

Brass:

L-68

LMn-58

30

40

74

72

0.44

0.24

Ti alloy BT-1.D 47 90 0.12

3.2 EFFICIENCY OF ELECTROMAGNETIC FORMING

The efficiency of electromagnetic forming depends on numerous parameters (C, U, f) and in the first approximationcan be represented by the ratio of the sum of the maximum kinetic energy of the moving workpiece and the total workof deformation to the capacitor bank total discharge energy:


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, (3.17)

or in the more detailed form:

. (3.18)

Substituting the maximum velocity in equation (3.18) we find the efficiency for the deformation of the cylindricalworkpiece and the sheet metal workpiece:

(3.19)

,

where

; (3.20)

Let us write the expressions for the efficiency of the deformation of the cylindrical and flat workpieces, respectively, insomewhat simplified form:

(3.21)


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, (3.22)

where

Analogously (3.22) we can write for the launching of the disk by the single -turn inductor:

, (3.23)

where

.

It is interesting to note that in the disk launching or sheet metal forming the maximum efficiency occurs when

, (3.24)

or

. (3.25)

In both cases

, (3.26)


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means that efficiency depends only on the active losses in the circuit.

CHAPTER 4

TECHNOLOGICAL OPPORTUNITIES OF ELECTROMAGNETIC FORMING

4.1 GENERAL REMARKS As the experience in introducing this method has indicated, the electromagnetic metal forming has the followingadvantages compared to other metal forming techniques:

1. High efficiency of the technological process. The major factor limiting the efficiency increase is the verysignificant time spent on the process preparation (installation of the workpiece, adjustment and taking out


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the finished part). If necessary electromagnetic equipment can be designed with an output capacity of3600 operations per hour or even more.

2. The technological process can be easily automated and mechanized. It is possible to control the equipmentremotely. The tool (inductor), creating the magnetic field, is not connected to the workpiece mechanically.The forming energy can be dosed precisely up to 1% and with the remote control.

3. The great technological flexibility of the process. The same inductor can be used to form the workpiecesof different configurations.

4. Simplicity of the technological equipment. Only one die or plunger is used.

5. Absence of a transfer medium during forming process. This feature allows to form the metallicworkpieces through insulating coatings or the wall of a vacuum chamber.


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6. The possibility of obtaining high specific pressures. Nowadays the pressures of up to 10 8 H/m 2 can beobtained without destroying the inductor and pressures of up to 10 9 H/m 2 can be obtained with thedisposal inductors.

7. High culture of production and simplicity of equipment maintenance. The modern EMF equipmentoperates noiselessly. The tool and the assemblies of the electromagnetic equipment dont need lubrication.There is no aggressive environment. The equipment is completely automated. The monitoring and controlof the operation can be performed by single worker.

8. The improvement of the characteristics of the formed materials. The majority of aluminum alloys formedelectromagnetically show an increased ductility when compared to the static deformation. Themicrostructure of the alloys with the same amount of strain has fewer distortions while formed by EMF ifcompared to the static deformation.

9. It is possible to perform EMF in hard-to-reach areas. The tool (inductor) can be connected to the capacitor bank by a flexible bus bar, which allows to perform the technological operations far away from thecapacitor bank (expanding the long tubes in the central section, corrugation and punching holes in largearea metal sheets).

EMF disadvantages :


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1. It is difficult to obtain parts with deep drawing by using electromagnetic forming procedure. In order toobtain deep drawings it is necessary to form the workpiece by various inductors. Each subsequentoperation must be performed by the inductor, the shape of which repeats the shape of the formedworkpiece.

2. Not all metals and alloys can be formed using EMF. Low-conductive materials require high-conductive"drivers" to be formed.

3. Not any shape is suitable for forming electromagnetically. The forming forces are created as a result of theinteraction of the current induced in the workpiece with the magnetic field of the inductor. In order toobtain the induced current the defined conditions must be met.

4. Not all the geometries of the workpiece are suitable for EMF. There are some restrictions with respect tothickness and diameter of the tubular workpieces.

5. The low mechanical strength of the inductors in the case of deformation of steel workpieces. Themechanical and electrical characteristics of the modern inductors permit multiple repetition oftechnological operations without destruction of the inductor during metal forming of relatively light


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metals and their alloys (aluminum, copper and magnetic alloys). On metal forming of the steel workpiecesthe strength of the inductor decreases significantly.

The presented advantages and disadvantages must be considered when introducing EMF. Considering all thosefeatures, it is necessary to remember that the application of EMF is not always economically justified. The significantcost benefits can be obtained in the case where the workpiece is especially designed as applied to the new technique soall the possible advantages can be used.

4.2 REQUIREMENTS IMPOSED ON THE SHAPES OF THE WORKPIECES

Since EMF is performed due to interaction of the magnetic field with the induced current in the forming workpiece, itis necessary that the shape of the workpiece provide for continuity of the path of the induced current. Fig. 20 shows theadmissible and the inadmissible versions of the initial workpieces for all types of possible technological operations. Itshows that when a cylindrical workpiece is formed with cylindrical inductor, currents leak along its periphery. If thecylindrical workpiece has a through slit along the generatrix of the cylinder or a large number of through openings, theforces acting on the workpiece are weakened and forming is low-effective.

Due to low duration of the pressure pulse, it does not appear possible to use EMF for forming of solid metalworkpieces.

Forming procedure is suitable for sheet and tubular workpieces with the thickness a w up to 5 mm. The lower limit ofthe thickness is determined by the operating frequency of the discharge circuit

aw>=0.5 (4.1)

under the condition of forming of the workpiece in a non-conductive die or a mandrel (compression of metal tube onceramic mandrel, sheet metal forming in a glass-plastic die, etc.)


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aw>=2.0 (4.2)

under the condition of forming of the workpiece in a die or a mandrel (compression of metal tube on the metal mandrel,sheet metal forming in to metal die, etc.)

Note, that the failure to satisfy the condition (4.1) lowers the efficiency of the forming process, and the failure to satisfythe condition (4.2), in addition, leads to the formation of the so-called "magnetic cushion", which prevents filling of thedie or the mandrel. The upper limit of the thickness of the workpiece depends on the energy consumption of themagnetic pulse device, material specific density and the strength of the inductor. With the same loading conditions thestrength of the inductor decreases if massive workpieces are formed. The geometry of the workpiece (diameter, themachined area) are determined by the energy stored in the capacitor bank, the thickness of the machined material, thestructural execution of the inductor and the technological process equipment.

The maximum diameter of the workpiece can reach 2 m, and the area - up 1m 2.


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Efficient forming Low-efficient forming

Fig. 20

The workpieces for EMF can be welded. The quality of the weld must be high. As a result of the high formingvelocities an extra mass of the metal of the weld distorts the symmetry of the deformation.

Removing the weld metal it is necessary to preserve its integralness. The destruction of integrity of the weld prevents

the induced current from leaking in the workpiece, and hence, lowers the forming efficiency.

The dimensions and the shape of the workpiece are established the same way as for the conventional forming.

4.3 REQUIREMENTS IMPOSED ON THE DEFORMED MATERIALS


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The forming efficiency depends on the electrical conductivity of the material.

EMF with the discharge circuit frequency range of 10-20 kHz is the most efficient for materials which have the specificelectrical conductivity four times higher than the one of copper (silver, gold, copper, magnesium, aluminum and theiralloys). Table 3 lists the electrical and mechanical properties of the materials most suitable for forming. The groups ofmaterials that marked by an asterisk can be formed by means of "drivers" only.

To form materials of low conductivity it is necessary to use the EMF machines with high discharge frequency (60-100kHz) or drivers (highly conductive inserts) placed between the inductor and the workpiece. The thickness of the driveris selected from the condition:

(4.3)

The proper material for the driver is well-annealed M1 type copper. Its dimensions are limited by the deformation zone.It is desirable to extend the driver beyond the limits of the deformed workpiece.

The same requirements are imposed on the shape of the driver as on the deformed workpiece (it must provide allconditions for continuity of the current path in the driver). To form tubular workpieces a tape winding of thin coppercan be used as the driver. Here it is necessary to provide good contact between the turns of the tape.

The galvanic coatings can be used only if it is economically justifiable. The same requirements are imposed on thethickness of the galvanic coating as on the driver. Non-metallic materials can also be formed by means of drivers.

The significant ductility increase due to simultaneous induction heating of the workpiece has not been noticed.

Table 3


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Material Elasticstrengthx107N/m2

Yieldstrengthx107N/m2

Straintofailure,%

Specificelectricalresistance,x10-8* .m

Skindepth,,mm

1kHz

Skindepth,

,mm

5kHz

Skindepth,,mm

10kHz

Skindepth,,mm

20kHz

Skindepth,

,mm

50kHz

Gold

Silver

Copper

8-15

15-30

24

-

-

7

-

-

50

2.2

1.6

1.78

2.36

2.01

2.12

1.055

0.898

0.95

0.746

0.636

0.67

0.527

0.449

0.48

0.334

0.285

0.3

Brass

L96

L90

L80

L70

L68

L62

LMz 58-2

LC 59-1

24

26

32

32

32

33

40

40

6

6

12

12

10

10

11

16

14

50

45

50

55

55

50

40

45

4.3

4.0

6.0

6.9

7.2

7.2

10.8

6.8

3.3

3.18

3.9

4.18

4.25

3.86

6.23

4.15

1.48

1.42

1.74

1.87

1.91

1.73

2.34

1.86

1.04

1.01

1.23

1.32

1.35

1.22

1.65

1.31

0.82

0.71

0.87

0.93

0.95

0.86

1.17

0.93

0.467

0.45

0.552

0.592

0.602

0.546

0.881

0.587

Bronze 40 25 65 17.6 6.67 4.1 2.11 1.49 0.945


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BrOF65-04

BrOz4-3

BrA5

BrAMn9-2

BrKMn-1

BrB-2

35

38

40 40

50

65

16

30 20

30

40

65

25 50

30

8.7

9.95

11 15

6.8

4.69

5.02

5.28 6.16

4.15

2.1

2.25

2.36 2.75

1.86

1.48

1.59

1.67 1.95

1.31

1.05

1.12

1.18 1.38

0.93

0.664

0.71

0.748 0.872

0.587

Table 3 continued

Aluminumalloys highmelting

ADM

ADM

AMnM

AMnN

8 15

13

22

3 10

5

18

35 6

23

5

2.92 2.98

3.76

4.45

2.72 2.75

3.08

3.34

1.22 1.23

1.38

1.5

0.86 0.87

0.98

0.06

0.61 0.61

0.69

0.75

0.385 0.389

0.436

0.473


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AMgM

AMg3M

AMg6M

DIM

DIT

19

19

30

21

42

10

9

15

11

24

23

15

18

18

15

5.09

4.95

7.1

3.72

3.7

3.59

3.54

4.24

3.08

3.06

1.6

1.58

1.9

1.37

1.37

1.13

1.12

1.34

0.97

0.97

0.8

0.79

0.95

0.7

0.68

0.508

0.501

0.6

0.436

0.433

DI6M

DI6AT

DI6M

DI6T

B95M

B95T

18

42

21

46

22

55

10

28

11

30

10

46

18

18

18

11

15

10

4.35

5.87

4.15

5.95

4.15

4.2

3.32

3.85

3.25

3.88

3.25

3.82

1.48

1.72

1.45

1.74

1.43

1.46

1.05

1.22

1.02

1.23

1.02

1.03

0.74

0.86

0.72

0.87

0.74

0.73

0.47

0.545

0.46

0.548

0.46

0.54

Magnesiumalloys

MA8M-M

MA-IM

26

21

19

12

18

8

5.1

6.1

3.6

3.93

1.61

1.76

1.15

1.24

0.8

0.88

0.51

0.556

Low-carbonsteels

CT3

CT10

38

40

24

25

37

35

16.4

19.25

6.45

7.0

2.9

3.12

2.04

2.21

1.44

1.56

0.913

0.99


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CT20 48 30 33 24.2 6.0 2.7 1.9 1.34 0.85

Alloyedsteels

30XGSA

1X18H9T

70

99

45

30

22

50

72 13.5 6.04 4.2 3.02 1.91

Titaniumalloys

BT5-1

Bt14

90

100

80

90

14

10

138 18.7 8.36 5.91 4.18 2.65

In order to form magnesium, titanium and their alloy it is necessary, just as for other forming procedures, to provide the preliminary heating and also to consider an increase in the specific electrical resistance, and, hence, the depth of penetration of the magnetic field due to heating must be taken in to account.

4.4 REQUIREMENTS ON THE PROCESS EQUIPMENT

The dies used in the high-velocity forming operations must have the holes for exit of the air. In each closed cavity holesmust be made, the number of which is selected from the workpiece design and the volume of the closed cavities. The

spacing between the holes l p=(20-40)x10 -3 m. The holes in the die must be located where the contact of the workpiecewith the die in its deformation process takes place in the next turn.

The hole diameter is calculated so that d


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In order to improve the exit of the air (Fig. 21) the hole with large diameter: D=(2-3)d is made from the outside the die.The depth of the hole l 1=(6-10)x10 -3 m. The die geometry can be found in [13]. The die must have geometry and shapesuch as to insure free removal of the formed part. This can be done by placing a small cone along the knockout path.

The proper selection of the die material or an insert and their heat treatment is one of the basic factors that determinestheir productivity. To select the die material one has to consider the production scale (series or experimental), the typeof operation, the geometry and the shape of the workpiece, material properties and the depth of penetration of themagnetic field in the workpiece.

If magnetic field leaks through the deformed workpiece w>aw and the dies made of highly conductive materials areused, then the damping effect can occur because of counterpressure created by the magnetic field between the die andthe workpiece.

Sometimes this can result in the separation of the deformed material and the distortion of the workpiece shape; arcingcan occur in the contact parts of a segment die due to inductive current inside the die, and forming efficiency decreases.

Considering what has been discussed, and when it is possible with the production conditions and the workpiece design,dies made of insulating epoxy-based materials with different fillings or metal alloys with high electrical resistivity are

preferred.

To form the workpieces of aluminum and copper alloys with the thickness of a w3different epoxy-based composites can be used as the die and mandrel's materials [13].


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Fig. 21

For the forming of solid materials and also materials with relatively small radius of curvature


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The half-dies can be insuring that they will not come open on the discharge. Because of the short pulse load the numberof bolts pulling the die together and also the joining force are inverse proportional to the die mass. Sometimes underunit production conditions it is sufficient to use a heavy weight to avoid opening of the half-dies. If it is necessary toreduce the weight of the large dies and maintain their stiffness, dies can be made with stiffening ribs.

4.5 CLASSIFICATION OF THE TECHNOLOGICAL OPERATIONS

All of the workpieces formed by EMF technique can be divided into three schematics with respect to the inductor usedand technological equipment according to the proposal of the Kharkov Polytechnical Institute.

1. The helical solenoid with the coaxial tubular workpiece inside. These type operations are called"compression".

2. The helical solenoid with the coaxial workpiece outside. These type operations are called " bulging"operations.

3. The spiral flat inductor (pancake type) or the concentrator located above the flat workpiece ("sheet metalforming").

Depending on the type of technological operation the workpieces are divided in to groups and sub-groups.

The first group: workpieces obtained by assembling several parts (the assembly operations); the second group:workpieces obtained with shape changing operations (bending, drawing, dressing, relief stamping, bulging, rifling,


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etc.); third group: with separating operations. EMF can provide combined operations consisting of all three groups andschematics.

"COMPRESSION"-TYPE OPERATIONS (FIG. 22)

The workpiece is a cylindrical tube. The pressure is created by a helical solenoid or inductor with magnetic fieldconcentrator. Depending on the energy consumption, the diameter of the workpieces can be in the range of 3x10 -3 to 2m. The thickness of the workpiece can be up to 5x10 -3 m. For multi-action inductors with copper concentrators thespecific pressure must not exceed 10 8 N/m 2.

Note that during "compression"-type operations a short pressure pulse can heat the workpiece surface and subsequentlycool it down. As a result, additional stresses occur which promote improvement of the compression quality in theassembly operations. When the forming and separation-type operations with metal mandrels are performed, theindicated effect complicates the workpiece removal from the mandrel. The mandrels must be made sectioned or it isnecessary to provide a small cone insuring easy removal of the parts from the mandrel.

Due to complexity of the picking up of the finished part from the mandrel and loss of stability of the workpiece wall(buckling) the forming operations of "compression" type have not found broad application. Compression is the mostwidely accepted in assembly and welding operations.

The combined operations made up of any combinations of operations with respect to position 6 to 25 (Fig. 22) are also possible.

Assembly and welding operations. Joining of metal parts to ceramics, glass, plastics and other non-metallicmaterials (1) . The brittle materials such as glass, glazed pottery, porcelain, etc. if not specially protected, can bedestroyed during EMF. The surface of the metal workpiece and of the compressed brittle workpiece must have no


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scratches and unevenness. The pressed surface of the brittle workpiece must experience compressive stresses only, andthe gap between the workpiece and the pressed part must be minimal and uniform with respect to entire surface.Observation of the condition m w


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Fig. 22


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The electric contact resistance of the joint obtained by EMF is 1.5 to 2 times less than the electrical contact resistanceof the joint obtained by conventional hydraulic press. The filling coefficient of the pressed part is close to one.

The length of the pressed part of the cable l can be found by the ratio = 1.5 to 2.0, and the length of pressed part of thecable or line operating under tension can be found by the expression = 2.5 to 3.5.

Pressing of high-pressure hoses (3) . The hose with a union nipple is sealed by pressing of the tube fitted on the hose. Inorder to improve the quality of pressing, the surface of the connection is made rifled. The length of the pressed part ofthe connection = 2.5-3.0. Tin order to avoid destruction of the hose during pressing, the pressure of the magnetic fieldat the end of the tube on the hose side at a length of l 1=(4-5)x10 -3 is not created.

Pressing of the tube on metal tips. (4) . It is possible to obtain sealed joints that can stand the test pressures up to107 N/m 2. These joints can substitute the same joints made with nipples, rivets, threads and are typically stronger thanthose.

In order to provide solid and sealed joint on the metal tips, two or three grooves 2a w deep and (3-4) a w wide are made.

In order to avoid cutting the tube, the sharp edges of the groove must be blunted. If a w


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To obtain the maximum impact velocity an active (deformed) tube must have the smaller mass of the wall per unitsurface area.

Fig. 23 shows two possible inductor schematics: a - outer tube compression; b- outer tube compression andsimultaneous bulging of the inside tube.

For a sufficiently large diameter of the welded tubes (d 0>50 mm) the inductor 3 can be placed inside the tubes. Then aninternal tube 1 is accelerated and hits the outer one. As shown in Fig. 23, two inductors, connected in series can beused. With that schematic tubes 1, 2 are accelerated towards each other.

Considering what has been discussed , and when it is possible with the production conditions and the workpiece design,dies made of insulating epoxy-based materials with different fillings or metal alloys with high electrical resistivity are

preffered.

To form the workpieces of aluminum and copper alloys with the thickness of a w3different epoxy-based composites can be used as the die and mandrel's materials [13].

To prevent the deformation of the passive tube (Fig. 23) a metallic insert 4 is placed inside the tube for giving itmechanical strength. Prior to welding surfaces must be cleaned by using metal brush until they shine. Directly beforewelding the joined surfaces of the tube are degreased. The gap between tubes can be made by bulging of one of thetubes and further aligning them in the inductor.


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Fig. 23

Forming operations . Transverse undulating dressing (Fig.22, positions 6,13 ) d1/d0>0.8. Strain distribution withrespect to the workpiece thickness is

aw2=aw aw1>aw aw30.9. Strain distribution is

aw2>aw aw10.5


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Strain distribution through the thickness is a w2>aw aw10.9 a w10.8. To preserve the circular shape of the tube without loss in stability, it isnecessary that d 0/aw0.8. Flanging can be performed in two operations with the further sizing of theflange by inductor.

Dimpling, making of shreads, rifling, sizing (Fig. 22, positions 16-20 ).

The subgroup includes the technological operations that change the outer shape of the workpiece insignificantly: d 1-d0= (1-2)a w, but the curvature of the deformed surface is significant: r=(1-5) a w.

To perform those operations, the diameter of the workpiece must be taken (1-2)a w larger than the mandrel diameter.


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Fig. 24

The workpiece must be centered in the inductor relatively to the insert. The gap between the workpiece and insert must be uniform.

Separation operations. Hole punching and blanking (Fig. 22, positions 21-24 ).

Holes can be punched in the mandrels having a hole with a cutting edge. Long holes located along the generatrix of theworkpiece 23 are not recommended since they destroy the continuity of induced current. The corner of the rectangularhole must have a radius of r=3a w. To increase the process efficiency, a gap between a workpiece and a die must befixed.

To perform the separation operations, the cut-off edge on the cutting tool side has burrs and on the inductor side,rounding. During EMF procedure the burrs are less than with conventional press hole punching.


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Fig. 25

The precision of punched holes depends on the workpiece geometry and is in the range of precision class 2-4, and thefinish of the cut corresponds to the class of surface roughness 4-5. It is possible to punch the faceted holes. Theefficiency of the process depends on the ratio of the diameter of the punched hole and the ratio of the width of the

groove to the thickness of the formed workpiece. It is recommended that d 1/aw>8 a/a w>8.

Cutting of tubes (Fig. 22, position 25) is performed in mandrels by ring punching. Fig. 25 shows the workpieces made by "compression" technique.


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"BULGING" OPERATIONS (FIG. 26)

The forming workpiece is a cylindrical tube. The pressure is created by a helical solenoid, placed inside the workpiece.The diameter of the forming workpieces, depending on the energy capacity of the machine, can vary from 30x10 -3 to 2m. The wall thickness can be up to 5x10 -3m.

The bulging of the tubes with diameters less than 40 mm has some difficulties, since the inductor does not havesufficient mechanical strength. However, it is possible to form the tubular workpieces with d 0


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the workpiece is determined by the outer easily opened sectioned die. That's why if the workpiece has sufficiently largediameter d 0 > 0.1 m, the given shape can be obtained by bulging technique.

Transverse undulated dressing (Fig. 26, position 6, 13) d 1/d0


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Assembly and welding operations Forming operations Separation operartions

Fig. 26

d1/d0


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Strain distribution is a w2


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Fig. 28

Separation operations. Hole punching, blanking and trimming (Fig. 26, positions 21-25).

All the features of the process are described in section "compression".

Assembly and welding operations Forming operations Separation operartions


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Fig. 29

"SHEET METAL FORMING" (FIG. 29)


88/272

The forming workpiece is a sheet. The pressure is created by a flat spiral inductor or flat concentrator, placed above theworkpiece. The shape of the workpiece and the deformation process are determined by the die. The area of the formedworkpieces depends on the power consumption of the machine fluctuates from 10 -4 to 0.02 m 2. The thickness of theworkpiece can be up to 5x10 -3 m.

To built the technological procedure it is necessary to remember that it is quite difficult to create the uniform field pressure over the entire area of the workpiece. Practically during EMF (technique of induced currents in the formedworkpiece) the magnetic field pressure in the central part of the inductor (the normal component) is almost zero, andthe workpiece is not subject to pressure in the central part over the area limited to (0.5-6)x10 -4m2.

Fig. 29 shows all the technological operations with indicated EMF features.

With "sheet metal forming" schematic the combined operations made up of any variety of combinations of positions 6-25 (Fig. 29) can be performed.

Assembly and welding operations. Assembly of the disk with the shaft (Fig. 29, position 1). If the helical inductor hasthe larger diameter, than the workpiece, than magnetic field component arises, that tries to decrease the inside openingin the disk. This way it is possible to reduce the hole in the disk by compressing it on the axis and also insignificantlydiminish the holes in the sheets. Strain distribution : a w1>aw.

Disk flanging (Fig. 29, position 2) with subsequent compression on a cylindrical base a/d 1


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Forming operations . EMF can be applied to produce different shapes of the flat sheet with relatively small drawing ofthe material. EMF is especially efficient under the conditions of experimental and small-series production.

The process equipment is quite simple - usually either the female die or the plunger is used - that reduces theexpenditures on the equipment material by a factor of two and lowers the labor consumption of its manufacture bythree or four times. If sheet is formed in a solid die the hole for the air exit must be provided, sometimes - die must beevacuated.

Dishing operations (Fig. 29, position 6 ).Since pressure in the central part is zero, workpieces of "dish" type can beformed on the punch with flanging of the outline of the sheet. The depth of the drawing is

b/d 0aw .

Transverse and longitudinal bead forming (Fig. 29, positions 8,9).Performed in the die. The depth of dimple drawing is b


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The shallow flanging of the annular and rectangular grooves (Fig. 29, positions 13, 14 ). The flanging depth isdetermined by the dimensions of the grooves d 1, b, radius r 1, r 2 and also by the material thickness. Flanging is

performed in the die. It is recommended to take a3a w, r 2>5a w, a/d 1


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Fig. 30

Separation operations. Hole punching, blanking and trimming (Fig. 29, positions 21-25).

The characteristics and the requirements are described in the section "compression".

Fig. 30 shows the workpieces formed by electromagnetic sheet metal forming technique.


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CHAPTER 5

ELECTROMAGNETIC FORMING EQUIPMENT 5.1 GENERAL COMMENTS

Irrespective of the purpose and the accepted schematic electromagnetic forming equipment can be interpreted as acomplex consisting of technological processes and power equipment.

The power equipment (see the functional schematic of EMF Fig. 31) consists of storage capacity C; the charging unitincluding the step-up device (the high-voltage transformer) PY and the rectifying device BY; the commuting device P;the inductor I; the start regulator PPY; the igniting circuit BPY, the automation module BA and the controller Z; theshielding device made up of the short-circuiting device KS; the blocking and other elements; measuring equipment,including the voltage divider DH.


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Fig. 31

C- storage capacity; 3- a controller; BA- the automation module; B Y- the igniting circuit; DH - the voltage divider; PY- the start regulator; Y-the high-voltage transformer; BY- the rectifying device; K3- the short-circuiting device; U - the inductor.

Depending on the output capacity, consumed energy, operating conditions and the purpose of EMF equipment, theschematics and design of individual elements making it up and the device as a whole are defined.

5.2 ENERGY DISTRIBUTION IN THE EMF DISCHARGE CIRCUIT

To estimate the efficiency of the magnetic pulse device, the relation of the work required to deform the workpiece tothe discharge circuit parameters can be derived from the equation of the energy balance in EMF discharge circuit. Forthe time moment t we have:


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(5.1)

where is the magnetic energy, stored in the machine inductance (the capacitor, the busbar and the dischargedevice);

- is the kinetic energy, associated with the work of deformation of the elements of the magnetic pulse device and the movement of the arc in the switch;

- is the magnetic energy, stored in the inductor-workpiece system;

- are the active losses in the machine elements (the capacitor bank, the busbar and the switch);

- are the active losses in the inductor-workpiece system;

- is the kinetic energy required to deform the workpiece;

- is the energy of the capacitor bank at time t 1.

As follows from (5.1) to design EMF with the maximum work of deformation, the following conditions must be met:


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1. The self-parameters of the discharge circuit R c and L c determined by the active resistances andinductances of the capacitors, the switches and the busbar and also the equivalent resistance of inductor-workpiece system R ie must be minimized.

2. The parameters L i, L c can not vary during the discharge, that means that the busbar, the coil of theinductor, the clamping elements of the switch and the capacitor should not be deformed by theelectrodynamic forces.

3. The magnetic energy stored in the inductor-workpiece system must be much larger than the magneticenergy, stored in the machine self-inductances.

By neglecting the work required to deform the workpiece, the equation of the energy balance in the discharge circuitassumes the form:

, (5.2)

where the active resistance of the individual inductor elements in accordance to the given assumptions, is determined

by equation (2.6). For a concentrator with annular grooves the resistance of the working winding of the concentrator is determined by:

(5.3)

(b r - is the size of the wire of the working winding in the radial direction).


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The resistance of the groove surfaces adjacent to the working winding is:

; (5.4)

The resistance of the working surface of the concentrator is:

; (5.5)

The resistance of the workpiece is:

. (5.6)

The relations I 2=f (I 3) and I1 =f (I 3) can be represented by the expressions:

; (5.7)

. (5.8)

The inductance of the workpiece L 4 and the inductance of the working zone of the concentrator L 3 can be found by

formula (2.42):

; (5.9)

(5.10)


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The losses in the operating surface of the concentrator:

; (5.15)

The losses in the workpiece:

(5.16)

The presented expressions for the distribution of losses in the inductor-concentrator system with deep annular groovesare valid also for the concentrators with a smooth surface with the only difference that it is necessary to take n=1 and

2p=1, and calculate the resistance according to (2.6). If consider the self-resistance of the discharge circuit, the determination of the relative losses in the device are similarto (5.13):

, (5.17)

The relative losses of the remaining elements can be found according to (5.13)-(5.16), where R 1 should be substitutedwith R 1+R c.

Applying the expressions for the equivalent resistance of the operating surface-workpiece system, we can find theequivalent resistance:

. (5.18)


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The formula (5.18) is valid for the workpiece of cylindrical shape. For the workpieces of the disk-type and conicalshapes the corresponding values of the equivalent resistance can be determined by formulas in Section 2.2 and thensubstituted into (5.18) instead of R e.c,.

Let us present the expressions for the relative losses:

The losses in the discharge circuit

; (5.19)

The losses in the working winding

; (5.20)

The losses in the surfaces of the grooves adjacent to the working winding

; (5.21)

The losses in the working surface

; (5.22)

The losses in the workpiece


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; (5.23)

5.3 AUTOMATION OF THE DISCHARGE CIRCUIT PARAMETERS CONSIDERING THE INDUCTOR SYSTEMAND THE WORKPIECE

If neglect the movement of the workpiece, which in practice is always valid for the first current maximum, theefficiency of the conversion of the magnetic field energy to the work of deformation is:

(5.24)

where m is the coefficient characterizing the transfer of the magnetic energy to the inductor -workpiece system; a - isthe coefficient corresponding to the energy losses in the active resistance of the discharge circuit.

For the first current maximum the coefficientsm

,a

can be expressed in terms of the parameters of the dischargecircuit and inductor:

; (5.25)

. (5.26)


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Using the expressions for the equivalent inductance and equivalent resistance, and considering (1.45), we can find thevalue of the relative attenuation coefficient:

. (5.27)

Using the value of the self-frequency of the discharge circuit 0, formula (5.25) can be converted to the followingexpression:

(5.28)

( 0 is the working cycle frequency, equal to 2 f).

From (5.28)

. (5.29)

Solving equations (5.24), (5.26), (5.27), (5.29) simultaneously, for the pressure amplitude we have:

(5.30)

Investigating the function (5.30) for the extremum with respect to m and performing the numerical calculations, themaximum value of P m that corresponding to the m, can be found in the range of 0.8-0.9.

h d h h h ( f ) h f h b k d d l


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In the inductors with the matching circuit (concentrator, transformer) the energy of the capacitor bank is expended alsoon the creation of the scattering fluxes in the primary winding; therefore the energy efficiency coefficient can beexpressed analogously to (5.24):

(5.31)

( k is the transfer coefficient of the magnetic energy of the concentrator to the working zone). Since the value of k isalways less than one, the total energy use coefficient of capacitor bank+concentrator is slightly less than in the typicalinductor, therefore the value of the coefficient must be maximal. Applying equations (5.7), (5.8), (2.37), we obtain thecoefficient:

. (5.32)

Investigating the function (5.33) with respect to to the extremum, we find the relative value of the grooveinductance:

. (5.33)

Considering (2.36), (2.43), for concentrators with deep grooves and a smooth outer surface, we obtain:

; (5.34)

. (5.35)

S b i i h l f h i l i d f (5 33) i i (5 32) b i


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Substituting the value of the optimal inductance from (5.33) in equation (5.32), we obtain:

. (5.36)

If consider (2.36), (2.43) we can determine the value of the energy transfer coefficient from equation (5.36)

for the concentrator with deep grooves:

; (5.37)

for the concentrator with a smooth surface:

. (5.38)

Thus it becomes clear from equation (5.36), that the optimum energy transfer coefficient depends on the couplingcoefficient for the primary winding only, and according to equations (2.36), (2.43), it depends on D g and d g, and on the

gap between the winding and the wall or the outer cylindrical surface of the concentrator. All those circumstances must be taken into account in the design of inductors which include magnetic field concentrators.

Usually in actual constructions the value of k g is in the range of 0.95-0.98. The coefficient s is in the range 0.5-0.7.


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5.4 SELECTION OF OPERATING VOLTAGE FOR THE CAPACITOR BANK IN THE MAGNETIC PULSE UNIT

The value of U c at W c = const determines the discharge circuit frequency:

(5.39)

and the insulation distances, and, therefore, the design parameters of the elements in the discharge circuit of thecapacitor bank and of the inductor. So in order to design or purchase the machine this value must be determined first,whereas the C value can be found from W c, U c.

To select the rational value for U c the following aspects should be considered: 1) the cost of the unit; 2) pressure of themagnetic field developed in the inductor-workpiece system; 3) duration of the pressure pulse; 4) expendituresassociated with manufacture of inductors; 5) convenience of operation and safety conditions.

Let us examine the technique for selecting the nominal voltage of the capacitor bank for given energy W c in order to provide the maximum magnetic field pressure on the workpiece.

For the case of a "massive" workpiece, when [6], the maximum magnetic field pressure in " compression"-typeoperation is equal to:

(5.40)

(k is a coefficient that depends on ratio).

Th l i h h l f U W d d i h i i il i f


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The analysis shows that value of U c at W c=const and m=const determines the pressure maximum primarily in terms ofscattering coefficient i. At low voltages, i increases since the skin depth of the inductor and of the workpieceincreases (C goes up, goes down), at high voltages i increases since h I - the thickness of the inductor insulationincreases.

There exists an optimum voltage U c at which i is minimum, and P m1 is close to maximum. For given geometry of theinductor, the coefficient i depends mainly on the calculated gap:

(5.41)

(h i is the insulation thickness between the inductor and the workpiece).

Therefore, U c may be determined in accordance with the condition of the minimum calculated gap. From (5.41) we

have:

(5.42)

where

.

Applying the relationship between the cast insulation thickness and the nominal voltage we can obtain:

(5.43)

(k 3 - is a coefficient, accounting for the reserve strength of the insulation). For epoxy insulation we assume =2.38x10 -11, n=1.6.

Fi di g t f (U ) h


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Finding an extremum of (U c) we have:

(5.44)

If i=1.75x10 -8, m =0.8; k 3 =3; =2.38x10 -11, n=1.6 (epoxy insulation) formula (5.44) can be used to calculate thenominal voltages U opt =f( w , W c, , L self ). Fig. 32 shows the nomogram for determining U c . The dashed lines representthe nominal voltage of the capacitor bank.

In the case of a "semi-transparent" workpiece, when w >a w the value of U c affects the attenuation coefficient k at only,that drops from one to zero if U c decreases lower than a certain value.

Fig.32

The determination of the EMF nominal voltage for a series of technological operations with variable electrical


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The determination of the EMF nominal voltage for a series of technological operations with variable electricalconductivity of the workpieces is a highly complicated problem, that can not always be solved uniquely. The mostefficient range of voltages for the machines with 5-50 kJ energy and materials with a specific resistance of 1.75x10 -8;5x10 -8 ohms-m is: a) for the "massive" workpieces U c =4-12 kV; b) for the " semi-transparent" ones -- U c =4 -30 kV. Inthe last case the higher voltage is selected for the metal forming of the "semi-transparent" workpieces in the metal dies

or mandrels.

5.5 CAPACITOR BANK

Capacitor bank is the most expensive power unit of electromagnetic forming equipment.

It must satisfy the following requirements: 1) low self-inductance and ability to stand the high number of pulse

discharges; 2) minimum weight, dimensions and magnitude of the depreciation deductions for one process operation.

The parameters of the capacitor bank to a significant degree determine the technical and economic characteristics of theEMF equipment (lifetime, operation cost, overall dimensions and weight). Hence, in order to design and select theequipment right parameters of capacitor bank must be chosen.

The capacitor bank energy, W c is the basic parameter that determines the number of capacitors, the overall dimensionsof the capacitor bank and weight. It can be determined from the required work of deformation for the heaviest and themost complicated operation:

, (5.45)

where A - is the work of deformation. For the simplest shapes of the workpieces it can be found according toexpressions in section 3.1.

The calculations of the energy required for the forming process were performed on the machines of the Kharkov


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The calculations of the energy required for the forming process were performed on the machines of the KharkovPolytechnical Institute. They showed that in the case of optimal structural design of the inductor and the workpieces ofgood conductivity, the efficiency can assume different values depending on the type of the operation, the material ofthe workpiece, the deformation conditions (free forming, forming in a die), the die material, the size of the gap betweenthe inductor and the workpiece, the frequency of the discharge circuit and the diameter of the workpiece.

Fig. 33

The approximate values of the efficiency are listed in Table 4 for different operations; the characteristics of the low-


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The approximate values of the efficiency are listed in Table 4 for different operations; the characteristics of the low-inductive pulse capacitors produced in the USSR are shown in Table 5 [8]. Figures 33 and 34 show the dimensions ofthe low-inductive pulse capacitors used in electromagnetic forming equipment.

Fig. 34

To design the capacitor bank a number and type of capacitors can be determined in terms of basic parameters, W c, U c.The number of capacitors:


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(5.46)

where W ic - is the nominal energy of one capacitor. If (5.46) gives the fractional number, the number of capacitorsshould be rounded to the larger nu

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