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Numerical simulation of the magnetic pressure in tube electromagnetic bulging Li Chunfeng*, Zhao Zhiheng, Li Jianhui, Wang Yongzhi, Yang Yuying School of Material Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China Received 27 February 2001

Abstract Electromagnetic forming is a kind of high energy rate forming; in which magnetic pressure analysis is the foundation of theoretical analysis. Usually equivalent methods are used to resolve any problems, but because of the limitations of the calculation methods, only the average force which acts on the workpiece can be obtained. Especially, the tolerances at the end of the workpiece are higher, which results in great difficulties in the deformation analysis of the workpiece. This paper carries out a numerical simulation of the magnetic pressure acting on the workpiece during electromagnetic bulge forming by use of the FEA software ANSYS. The boundary conditions of the analysis model are established on the basis of theoretical analysis of the magnetic field properties. Through measuring the magnetic induction intensity between the coil and the workpiece, the simulative results are proven to be accurate. This is the first time that the magnetic pressure distribution in the thickness direction of the workpiece has been explored through FEM. # 2002 Published by Elsevier Science B.V. Keywords: Electromagnetic forming; Magnetic pressure; Numerical simulation; Magnetic field property

1. Introduction The theoretical analysis of electromagnetic forming includes two parts: magnetic pressure analysis and workpiece deformation analysis with the action of impact force. So far, usually equivalent methods have been used to resolve the magnetic pressure acting on the workpiece [1], but only the average magnetic pressure which acts on the workpiece can be obtained, and the stress of any location in the workpiece cannot be obtained by calculation; especially the tolerances at the end of the workpiece are greater [2]. Lately, there have been some reports on resolving the magnetic pressure by the finite element method [2,3]. This paper carries out the numerical simulation of magnetic pressure using ANSYS in electromagnetic bulging and obtains the dimensions and distribution of the magnetic pressure acting on the workpiece; whilst at the same time, the effects of technological parameters (electrical resistivity, workpiece length, discharge frequency) on the magnetic pressure are analyzed. Through measuring the magnetic induction intensity within the gap between the coil and the workpiece, the simulation results are confirmed. * Corresponding author. E-mail address: [email protected] (L. Chunfeng).

0924-0136/02/$ – see front matter # 2002 Published by Elsevier Science B.V. PII: S 0 9 2 4 - 0 1 3 6 ( 0 2 ) 0 0 0 6 3 - 8

2. Electromagnetic field properties of the solenoid coil The solenoid coil used in tube electromagnetic bulging is an axisymmetric coil. For the solenoid coil, according to its symmetric characteristic, only one-fourth of the field region is needed in analysis. When carrying out the ANSYS finite element simulation, the boundary conditions of the symmetric axis and the symmetric plane are needed. The magnetic theoretical analysis of the solenoid coil is the foundation of the finite element simulation of magnetic pressure, and it is also an important basis to ensure the finite element model’s boundary conditions during electromagnetic bulging. 2.1. Vector magnetic potential properties on the symmetric axis of the bulging coil For the axisymmetric coil, the helicity of every convolution can be ignored, and the axisymmetric coil regarded as the superposition of annular coils. The annular coil indicates the single-turn circular circuit, and the distribution is line current. The annular coil’s geometry is simple, so it is easy to analyze it and to calculate its magnetic field. Suppose the annular coil’s radius is a, the current is I, and both of these are shown in polar coordinates. The annular plane is

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perpendicular to the z-axis, and the center of the annular lies in the point ð0; 0; hÞ. At any point Pðr; j; zÞ, the vector magnetic potential of the magnetic field induced by the annular coil has only a circumferential component [4], and its value is Z p=2 m0 Ia 2 sin2 a 1 ~ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ q pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ da~ Aðr; j; zÞ ¼ ej 1 k2 sin2 a p ðr þ aÞ2 þ ðz hÞ2 0

the magnetic induction intensity on the symmetric plane only has a Bz component, and Br and Bj are zero, viz. the magnetic induction intensity on the symmetric plane is normal to the symmetric plane.

(1)

The structure of the electromagnetic bulging solenoid coil is shown in Fig. 1, the dimensions of the coil being shown in Table 1, and the workpiece parameters are in shown in Table 2.

7

in which m0 ¼ 4p 10 (constant), qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ . k ¼ 4ra=½ðr þ aÞ2 þ ðz hÞ2 When point P is located in the symmetric axis, r ¼ 0, and Eq. (1) can be written as ~ Að0; j; zÞ ¼ 0

(2)

The magnetic field meets the superposition principle, and the axisymmetric coil is equivalent to the superposition of many annular coils, so the vector magnetic potential ~ A ¼ 0, which is located in the symmetric axis of the bulging coil. 2.2. The magnetic induction intensity properties on the symmetric plane of the bulging coil The solenoid coil, of which the radius is a, the number of turns is n, and the current of every convolution is I, is shown in polar coordinates. The symmetric axis is the z-axis, and the axial coordinate of the coil’s upper end edge is Z2, whilst the axial coordinate of the lower end edge is Z1. According to the Biot–Savart law, the magnetic induction intensity at any point Pðr; j; zÞ has three components: Br, Bj, Bz, there into, Br, Bj are [4] rﬃﬃﬃ m nI a ½f ðk2 Þ f ðk1 Þ Br ðr; j; zÞ ¼ 0 (3) 2p r where sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4ra ; k2 ¼ 2 ðr þ aÞ þ ðz z2 Þ2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4ra ; k1 ¼ 2 ðr þ aÞ þ ðz z1 Þ2 2 2 k KðkÞ EðkÞ; f ðkÞ ¼ k k ( ) 1 X ð2n 1Þ! 2 p 2n 1þ KðkÞ ¼ k ; 2 ð2nÞ! n¼1 ( )

1 X p ð2n 1Þ! 2 k2n 1 EðkÞ ¼ ; 2 ð2nÞ! 2n 1 n¼1 Bj ðr; j; zÞ ¼ 0

3. The numerical simulation of the magnetic pressure of electromagnetic bulging

3.1. The model and boundary conditions of FEA in electromagnetic bulging ANSYS is used to resolve the magnetic field in electromagnetic bulging and to obtain the size and distribution of the magnetic pressure. The following limits in the analysis are assumed: (1) the coil current is distributed equally in its crosssection and is equal to setting value; (2) the magnetic conductivity and electric conductivity are invariable and isotropic; (3) the displacement current is ignored.

Fig. 1. Structure diagram of the solenoid coil used for electromagnetic bulging: 1, wire; 2, insulated support; 3, line-to-line insulation.

Table 1 Coil structure parameters Coil label d1 (mm) d2 (mm) h (mm)

p (mm)

b (mm)

Turns (N)

C1

6.3

4.6

19

67.4

57.2

120

Table 2 Workpiece parametersa

(4)

If the coordinate system is built in the center of the solenoid, namely Z1 ¼ Z2, then on the symmetric plane of the solenoid coil ðz ¼ 0Þ, Br ðr; j; zÞ ¼ 0, so that considering Eq. (4);

Material Al

d (mm) 75

a

t (mm) 2

l (mm) 120

r (O m)

m 8

2:7 10

1

d: inner diameter; t: wall thickness; l: length; r: electrical resistivity; m: relative permeability.

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Fig. 2. Sketch of the electromagnetic bulging system model.

The FEA model of bulging is shown in Fig. 2, where A1 is the coil, A2 the workpiece and A3 the air far-field model. The FEA mesh division of the magnetic field is shown in Fig. 3, where the eight-node quadrangle element is adopted, the total number of subdivision nodes is 1621, and the element number is 527. If the workpiece is treated as a single-turn solenoid coil, according to the principle of superposition, the magnetic field in electromagnetic bulging can be regarded as the superposition of magnetic field inducted by two solenoid coils (the discharge coil, and the workpiece). Because the coil and the workpiece are coaxial and symmetric, according to previous analysis of the solenoid coil’s magnetic field properties, the boundary conditions and excitation can be established as follows: (1) in the Cartesian coordinate system, B (the magnetic induction intensity) is normal to y ¼ 0; (2) in the Cartesian coordinate system, ~ A (the vector magnetic potential) is equal to zero at x ¼ 0; (3) in the polar coordinate system, an infinite mark is built at r ¼ 12h; (4) the coil current density is 93 kA, and the frequency is 10 kHz (the measured value).

Fig. 4. Stress diagram in electromagnetic bulging.

subjected to both radially outwards expansive force and axially downwards pressure; whilst the other parts are subjected to only radially outwards expansive force. This type of distribution of the magnetic pressure can be explained by the magnetic line’s distribution from the simulation. A diagram of the magnetic lines during bulging is shown in Fig. 5. The magnetic lines flowing from the inner parts of the coil mostly are forced into the narrow slit between the coil and the workpiece, except at the end of the workpiece, and include only the axial component, so the workpiece is subjected only to radially outwards expansive force; whilst at the ends of the workpiece, the magnetic lines are emanative, and comprise both the axial component and the radial component, which results in the ends of the workpiece being subjected to both the radially outwards expansive force and the axially downwards pressure. Because of the existence of this axially downwards pressure, the tube blank is in the constrained state at the two ends during electromagnetic bulging, which is different from the normal free-bulging technology: it is more useful to the workpiece deformation.

3.2. Analysis of the results of magnetic field simulation The stress diagram of the workpiece obtained from simulation is shown in Fig. 4, where the ends of the workpiece are

Fig. 3. Mesh subdivision diagram of the electromagnetic bulging system model.

Fig. 5. Magnetic lines diagram in electromagnetic bulging.

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magnetic field dynamic testing system was built. The dimensions and distribution of the magnetic induction intensity B were measured through a series of ‘‘spot’’ induction coils, and then the magnetic pressure p was obtained: in general, the test results accord well with the simulation results.

4. Conclusions

Fig. 6. The diagram of the radial expansive magnetic pressure acting on the workpiece in the thickness direction.

The magnetic pressure distribution in the center of the workpiece in the thickness direction is shown in Fig. 6. Its is obvious that the whole workpiece is subjected to radially outwards expansive magnetic pressure during electromagnetic bulging, and that this is induced by the permeation of the magnetic field. This kind of distribution of stress is also a characteristic in which electromagnetic bulging differs from other bulging technologies; in the common bulging technology, only the inner surface of the workpiece is subjected to the external force. It can be seen from Fig. 6 that all the four layers subdivided in the direction of workpiece thickness have the outwards expansive magnetic pressure, and that the closer to the coil, the greater the force. Further, this force weakens very quickly from the inside to the outside (the weakening is induced by the Kelvin effect). In order to validate the simulation results, according to the equation p ¼ B2 =2m0 and referring to Ref. [5], a digitized

Through the numerical simulation of electromagnetic bulging, the dimensions and distribution of the magnetic pressure acting on the workpiece are obtained. The end of the workpiece is subjected both to radially outwards expansive force and axially downwards pressure, which is useful to the workpiece bulging. During bulging, for the penetration of the magnetic field, the whole workpiece is subjected to the radially outwards expansive force, and is different from common bulging technology, where only the inner surface of the workpiece be subjected to external force. The magnetic pressure weakens very quickly from the inside to the outside in the direction of the workpiece thickness in accordance with the principle of Kelvin effect.

References [1] V. Belyy, S.M. Fertik, L.T. Khimenko, Electromagnetic Metal Forming Handbook, Department of Materials Science and Engineering, Ohio State University, 1996. [2] S.H. Lee, A finite element analysis of electromagnetic forming for tube expansion, J. Eng. Mater. Technol. 116 (4) (1994) 250–254. [3] S.H. Lee et al., Estimation of magnetic pressure in tube expansion by electromagnetic forming, J. Mater. Process. Technol. 57 (1996) 311–315. [4] L. Yinzhao, Calculation of Axisymmetric Solenoid Coil Magnetic Field, China Measurement Press, 1991. [5] H. Jansen, Some measurements of the expansion of a metallic cylinder with electromagnetic pulses, IEEE Trans. Ind. Gen. Appl. IGA-4 (1968) 428–440.