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Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper
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CIRP-1679; No. of Pages 4 CIRP Annals - Manufacturing Technology xxx (2017) xxx–xxx
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CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper Brad Kinsey a,*, Ali Nassiri b,c a
Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USA Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA c Simulation Innovation and Modeling Center (SIMCenter), The Ohio State University, Columbus, OH 43210, USA b
Submitted by Scott Smith (1), Charlotte, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Forming Modelling Electromagnetic tube compression
In this study, a computationally cost effective, pure analytical model was developed for a multi-turn, axisymmetric coil with field shaper to predict the magnetic pressure and velocity during electromagnetic tube compression. This model is electro-magnetic-mechanically coupled with tube position affecting the magnetic field generated at each time increment. The mechanics-based analytical approach is different than past research and includes experimentally determined coupling coefficients between the coil, field shaper, and tube. To validate the analytical model, experimental tests with Photon Doppler Velocimetry (PDV) were conducted. The results show reasonably good agreement between the analytical and experimental results. © 2017 CIRP.
1. Introduction Electromagnetic Forming (EMF) is advantageous in forming, joining, and welding of the lightweight structures ranging from aerospace and automotive industry to medical devices and household appliances. Improved strain distribution and formability; short cycle time; and reduced wrinkling are among the benefits. EMF is suitable for tubular compression and expansion, sheet metal forming, and powder compaction [1]. The EMF process is based on Faraday’s law of electromagnetic induction. In this process, a capacitor bank is charged with a significant amount of electrical energy which is quickly dissipated into a specially designed coil. A magnetic field is generated that induces eddy currents in nearby conductive materials. These eddy currents produce a repulsive magnetic field, and Lorentz forces cause the workpiece to plastically deform away from the coil at a high velo`city. See schematic in Fig.1. One of the key process parameters in EMF is workpiece velocity. If the relative workpiece velocity is sufficient (>300 m/s [3]), this technique can be used for welding of dissimilar metals, i.e., Magnetic Pulse Welding (MPW), where a shear instability creates the characteristic wavy morphology at the weld interface [4,5]. While some multiphysics finite element packages exist that are capable of modelling the process and estimating the critical process parameters (e.g., velocity), there is a lack of simplified and accurate analytical modelling tools available. Such analytical models help to eliminate empirical investigations to determine the necessary process parameters and velocities to produce successful EMF and MPW parts.
* Corresponding author. E-mail address: [email protected] (B. Kinsey).
In this research, a purely analytical model for determining the pressure distribution applied to the workpiece and the subsequent workpiece velocity for a multi-turn, axi-symmetric coil with field shaper was investigated. In the proposed model, at each time increment, the magnetic field is automatically updated in response to the tube deformation. Compared to previous analytical modelling efforts [6–9], a rigid body or a rigid-plastic material assumption was eliminated. Alternatively, plastic deformation of the workpiece during the forming process was taken into account at each time increment. In addition, the coupling factors between the coil, field shaper, and workpiece were experimentally measured and incorporated into the analytical model. The analytical model was then experimentally verified using Photon Doppler Velocimetry (PDV) to measure the workpiece velocity. Reasonably accurate velocity and radial location results were obtained.
Fig. 1. Schematic of EMF/MPW process [2].
2. Analytical model The basic EMF process consists of three fundamental parts: a capacitor bank to store energy, a coil to create the magnetic field, and a workpiece to be formed. Based on the multiphysics nature of the
http://dx.doi.org/10.1016/j.cirp.2017.04.121 0007-8506/© 2017 CIRP.
Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121
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EMF process, the analytical model was divided into three stages. First, electrical theory was used to determine primary current and voltage out of the capacitor bank and passing through the coil based on the charge energy. Second, the magnetic field distribution and the effective magnetic pressure that was developed on the workpieces were calculated from electromagnetic analyses. Lastly, classical mechanics theory was used to determine the radial position and velocity of the workpiece caused by the effective magnetic pressure. Since the magnetic field distribution strongly depends on the gap distance between the field shaper and workpiece, the magnetic and mechanical processes were coupled. To calculate the workpiece displacement, a simplified constitutive law for the material was incorporated into the model. At each time increment, the magnetics field geometry was updated with an incremental displacement of the workpiece and hence the model includes the new gap between the field shaper and workpiece. In this study, a commercially purchased multi-turn, axisymmetric coil (Poynting, model: SMU-K100-4/65) was used. The workpiece was an Al6061-T6 tube with a length of 100 mm, wall thickness of 0.88 mm and outer diameter of 25.36 mm. In tube compression processes, a “field shaper” is often positioned between the coil and workpiece to concentrate the magnetic field generated (see Fig. 2a and b). A quick discharge of the capacitor bank causes a damped sinusoidal current flowing through the coil which induces a related electromagnetic field. In the field shaper, which acts as a short circuited second winding of a transformer, a secondary current is induced. Due to the skin effect and Lenz’ law, this induced current flows opposite to the coil current at the outer surface of the field shaper [10]. At the axial slot, the current is directed to the inner surface of the field shaper where the current direction is the same as in the coil (see Fig. 2c). Compared to the outer surface of the field shaper, the inner area is much smaller, resulting in a higher current density and field strength. For this analysis, a series of elements were generated along the surface of the field shaper. The elements were equally spaced, dz, in the axial direction and the width of each element, dr, (see Fig. 2d) was equal to the skin depth, d, defined as:
d¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rr
m0 mr v
ð1Þ
where v is the frequency of the current through a conductor of resistivity rr, mr is the relative magnetic permeability of the conductor, and m0 is the magnetic permeability of free space.
Fig. 3. Configuration of three circuits and two coupled systems.
primary and induced circuits are coupled with the secondary circuit through the mutual inductances (i.e., M1 and M2) which are a measure of induction between the two circuits. A “coupling factor” ðkÞ represents the loss of magnetic flux with a value between 0 k 1. In this study, the two coupling factors (i.e., k1 and k2 corresponding to M1 and M2 respectively) were experimentally measured by placing a Rogowski coil in two different locations in the experimental set-up and calculating the ratios of the secondary to primary current (k1 = 0.78) and induced to secondary current (k2 = 0.91). Applying Kirchhoff’s voltage law and summing the voltages around the primary circuit, a differential equation is obtained with respect to time, t [10] Z dip ðtÞ 1 ¼0 ð2Þ ip ðtÞdt þ ip ðtÞR þ L dt Cm where Cm is the capacitance of the machine, ip is the current in the primary circuit, R = Rm + Rc is the total resistance, and L = Lm + Lc is the total inductance. Note that the resistance and inductance of the machine, Rm and Lm, can be determined experimentally with a known capacitance by calculating the damped natural frequency and damping ratio of the RLC circuit. Resistance and inductance of the coil, Rc and Lc, are functions of the material properties (i.e., resistivity of the coil), the geometry of the coil, current condition (i.e., the angular frequency), and the cross-sectional area of the coil contained from the skin depth to the surface of the coil. For detailed information see Refs. [9,11]. 2.2. Magnetic theory The magnetic field produced from a given axi-symmetric coil can be determined with respect to the gap distance between the field shaper and tube along the axial direction. As is clear from Fig. 2d, because of the tapered geometry of the field shaper, the gap distance is varied along the axial length of the field shaper (i.e., gðzÞ). The magnetic flux density, B, produced by the coil induces eddy current in the workpiece with a current Jdensity. The current density, J, is related to the magnetic field, H, through a partial derivative in the radial direction. A Lorentz force based on His created which acts as a volume force, F , [10] F ¼ mm H
@H 1 @ðH2Þ ¼ mm 2 @r @r
ð3Þ
The body force, F , is integrated through the thickness of the tube to determine the magnetic pressure acting on the tube surface Zw Pm ¼
1 F dr ¼ mm H2gap 2
ð4Þ
0
Fig. 2. Schematic of multi-turn, axi-symmetric coil with field shaper and tube: (a) full view, (b) half view, (c) current directions, (d) cross section.
2.1. Electrical theory The electrical circuit consisting of the capacitor bank, coil, field shaper and workpiece can be represented by ideal electrical elements and circuits (see Fig. 3). As is clear from Fig. 3, both the
where the integration limit, w, is the tube thickness (see Fig. 2d) and Hgap is the gap magnetic field strength. In this study, the penetrated magnetic field was neglected due to the skin effect [10]. The magnetic field strength, Hgap, is the resultant field of a superposition of magnetic field strength from many current carrying differential elements, dHgap (see Fig. 2d). For the axi-symmetric coil investigated in this study, only the magnetic field strength along the coil’s axis is of interest (i.e., tangential to the workpiece) because this will create a force in the radial
Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121
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direction according to the cross product F ¼ J B[12], and can be calculated from " # i gðzÞ r gðzÞ þ r þ Hz ¼ ð5Þ 2p ðgðzÞ rÞ2 þ z2 ðgðzÞ þ rÞ2 þ z2 where gðzÞ is the varying gap between the coil element and the workpiece and i is the current in each element with height of dz and length of dr. As will be clear from Eq. (12), the magnetic field strength strongly depends on the radial gap between the field shaper and workpiece. 2.3. Mechanical theory Knowing that the workpiece in the forming process is plastically deformed, a simple plasticity model was employed to evaluate the exact deformation of the workpiece at each time increment. Due to the dynamic effects in the process, the radial stress (which acts as a resistance/inertia pressure) is calculated from
s R ¼ Pinertia ¼
ðmassÞa ð2pr Lz wÞ r a ¼ ¼ w ra 2pr Lz 2pr Lz
ð6Þ
strength at a given location in the gap: X Hgap ¼ Hz
3
ð12Þ
Substituting Eq. (12) in Eq. (4) yields a pressure distribution acting on the workpiece at each time increment. From the radial position data, the velocity of the workpiece was also determined. 3. Analytical Results Results from the analytical model include electrical, magnetic, and mechanical predictions. Electrical analysis determines the electrical parameters of the coil and the electrical response of the EMF machine. The predicted circuit response at 2.4 kJ energy discharge is shown in Fig. 4 while the spatial magnetic pressure distribution along the workpiece with respect to time is shown in Fig. 5. Clearly for such a coil based on Fig. 5, the pressure pulse from the first half cycle of the current pulse is more significant than the later oscillations. Thus, the forming event occurs during this timeframe. From the analytical model, the effective stress values for the 2.4 kJ and 3.6 kJ cases are 446 MPa and 457 MPa respectively.
where the denominator in the first equation represents the area that the dynamic force is applied over, r is the density of material, and a is the acceleration of the workpiece. Note that for the final calculation, the distance, Lz, is not required, and w and a (as well as P and r in Eq. (6)) are updated with each increment. Longitudinal and hoop stresses were calculated from the applied external pressure (including the inertia pressure resisting motion in the radial direction with respect to the hoop stress, but not in the axial direction) and radius of the workpiece, r, at a given axial location (see Fig. 2d):
sL ¼
Pm r 2w
&
sH ¼
ðPm s R Þr w
ð7Þ
These equations are derived with respect to quasistatic loading but have been validated for this case as well through numerical simulations. The effective stress, assuming von Mises yield criterion, was then calculated as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ðs H s L Þ2 þ ðs H s R Þ2 þ ðs L s R Þ2 s¼ ð8Þ 2
Fig. 4. Predicted current and voltage during discharge.
The effective stress, strain and strain-rate relation was defined as
s ¼ C en e_ m
ð9Þ
0
where C is the strength coefficient and n and m are the strain and strain-rate hardening exponents, respectively. For this study, these parameters were obtained from Kolsky Bar test at the strain rate of 5000 s1) (i.e., C0 = 447.2MPa, n = 0.067, and m = 0.006). The strain rate term in this equation was simply used to shift the power hardening curve based on the strain rate in the test. From Eq. (9),e was calculated. Neglecting the longitudinal strain (eL = 0) and considering deH + det = 0 (note that this assumption was confirmed again through numerical simulations and by experimentally measuring the axial elongation), the effective strain was evaluated, again assuming von Mises yield criterion, as 2 e ¼ pffiffiffieH ð10Þ 3 Finally, the hoop strain, eH was calculated from
eH ¼
X
r ln i1 ri
ð11Þ
to obtain the radial location change from time increment, i 1, to i, at each axial location, which were summed to obtain the total strain induced. An updated gap between the field shaper and workpiece was then calculated inside a loop by knowing the gap at the previous step and the current incremental displacement of the workpiece. Finally, the updated z-component of the magnetic field was calculated for each element (see Fig. 2d) using Eq. (5). Summation of these values provides the magnetic field
Fig. 5. Predicted induced pressure distribution (i.e., Pm–Pinertia) along the tube length with time for a 2.4 kJ discharge.
4. Experimental test To verify the analytical model, experimental tests with 2.4 kJ and 3.6 kJ and the same geometry and material used in the analytical model were conducted. The experimental setup is shown in Fig. 6a. The primary circuit current was measured with a Powertek CWT 3000B Rogowski coil. The comparison between the experimental test and the analytical model with a 2.4 kJ energy is shown in Fig. 4. Due to the shape of the commercially purchased coil and difficulty to measure the velocity during the process from the outside, a device was designed and fabricated to measure the velocity on the inside surface of the tube (see Fig. 6b). In this setup, an angled optical path was created by a laser mirror (see Fig. 6c),
Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121
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Fig. 8. Radial displacement from analytical model and experimental tests.
Fig. 6. (a) Experimental setup, (b) velocity measurement device, (c) schematic of the coil and laser mirror location.
and tube velocity was measured by Photon Doppler Velocimetry (PDV). This method is based on the Doppler shift between a reference laser beam and a reflected velocity-shifted beam. As shown in Fig. 4, the analytical model result for the current trace is close to the experimental measurement but with some discrepancies. These are due to complicated electromagnetic impulse process parameter interactions. The experimental results for workpiece velocity in two different energy levels along with the analytical predictions are shown in Fig. 7. To ensure the consistency and repeatability of the experimental measurement system, three tests were conducted at each energy level (variations shown with error bars). As is clear from Fig. 7, the predicted velocity from the analytical model is in reasonably good agreement with the experimental test, i.e., errors of 8% and 9.6% for the 2.4 kJ and 3.6 kJ cases respectively. The time of the peak velocity is predicted later in the process, but this parameter is less critical for the effectiveness of the process. A comparison between the predicted radial displacement from the analytical model and experimental test is shown in Fig. 8. The analytical model approximates the experimental forming behaviour well with final radial position errors of 3% and 5.6% for the 2.4 kJ and 3.6 kJ cases respectively. Note that the radial displacements from the analytical model converged to 3.3 mm and 5.3 mm for the 2.4 kJ and 3.6 kJ cases respectively, while for the experimental tests converged to 3.4 mm and 5.6 mm. Many factors can influence the experimental measurement results. For example, the system inductance and resistance as well as the material properties will change during the experiments because of resistance heating. These factors affect the accuracy of the input capacitor bank inductance and resistance values and material properties used in the analytical model. Also, current and velocity measurement devices have calibration errors. In addition, the analytical model includes various assumptions and simplifica-
Fig. 7. Velocity results from the analytical model and experimental tests.
tions. Finally, the incremental nature of the analytical model causes a time delay in the peak value. Considering these factors, the obtained comparisons are deemed reasonable. 5. Conclusions In this study, the electromagnetic impulse process was analytically modelled in detail. Compared to the previous analytical modelling efforts, the experimentally determined coupling coefficients in the system and plastic deformation of the workpiece during the process were incorporated. The analytical model is able to predict the magnetic pressure distribution on the workpiece and workpiece radial location and velocity during tube compression. The computational time of the analytical model is considerably less compared to FEA simulations. To validate the analytical model, experimental tests with a PDV system and a periscopic velocity measurement device were conducted. The results show reasonably good agreement between the analytical and experimental results. Acknowledgement Funding from the U.S. National Science Foundation (CMII0928319 and CMMI-1537471) is gratefully acknowledged.
References [1] Mamalis AG, Manolakos DE, Kladas AG, Koumoutsos AK (2004) Electromagnetic Forming and Powder Processing: Trends and Developments. Applied Mechanics Reviews 57(4):299–324. [2] Blakely M (2008) Filler Wire is for WIMPS (Welding) Fabricating and Metalworking, the Business of Metal Manufacturing, Hirotec America: 31–32. [3] Nassiri A, Chini G, Vivek A, Daehn G, Kinsey B (2015) Arbitrary Lagrangian– Eulerian Finite Element Simulation and Experimental Investigation of Wavy Interfacial Morphology During High Velocity Impact Welding. Materials & Design 88:345–358. [4] Nassiri A, Chini G, Kinsey B (2014) Spatial Stability Analysis of Emergent Wavy Interfacial Patterns in Magnetic Pulsed Welding. CIRP Annals—Manufacturing Technology 63(1):245–248. [5] Nassiri A, Kinsey B, Chini G (2016) Shear Instability of Plastically-Deforming Metals in High-Velocity Impact Welding. Journal of the Mechanics and Physics of Solids 95:351–373. [6] Weddeling C, Demir OK, Haupt P, Tekkaya AE (2015) Analytical Methodology for the Process Design of Electromagnetic Crimping. Journal of Materials Processing Technology 222:163–180. [7] Hahn M, Weddeling C, Lueg-Althoff J, Tekkaya AE (2016) Analytical Approach for Magnetic Pulse Welding of Sheet Connections. Journal of Materials Processing Technology 230:131–142. [8] Thibaudeau E, Kinsey BL (2015) Analytical Design and Experimental Validation of Uniform Pressure Actuator for Electromagnetic Forming and Welding. Journal of Materials Processing Technology 215:251–263. [9] Nassiri A, Campbell C, Chini G, Kinsey B (2015) Analytical Model and Experimental Validation of Single Turn: Axi-Symmetric Coil for Electromagnetic Forming and Welding. Procedia Manufacturing 1:814–827. [10] Psyk V, Risch D, Kinsey BL, Tekkaya AE, Kleiner M (2011) Electromagnetic Forming—A Review. Journal of Materials Processing Technology 211(5):787–829. [11] Zhang P, Kimchi M, Shao H, Gould JE, Daehn GS (2004) Analysis of the Electromagnetic Impulse Joining Process with a Field Concentrator. AIP Conference Proceeding 1253–1258. [12] Al-Hassani STS (1975) Magnetic Pressure Distributions in Sheet Metal Forming. Conference on Electrical Methods of Machining Forming and Coating 1–10.
Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121
CIRP-1679; No. of Pages 4 CIRP Annals - Manufacturing Technology xxx (2017) xxx–xxx
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper Brad Kinsey a,*, Ali Nassiri b,c a
Department of Mechanical Engineering, University of New Hampshire, Durham, NH 03824, USA Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA c Simulation Innovation and Modeling Center (SIMCenter), The Ohio State University, Columbus, OH 43210, USA b
Submitted by Scott Smith (1), Charlotte, USA
A R T I C L E I N F O
A B S T R A C T
Keywords: Forming Modelling Electromagnetic tube compression
In this study, a computationally cost effective, pure analytical model was developed for a multi-turn, axisymmetric coil with field shaper to predict the magnetic pressure and velocity during electromagnetic tube compression. This model is electro-magnetic-mechanically coupled with tube position affecting the magnetic field generated at each time increment. The mechanics-based analytical approach is different than past research and includes experimentally determined coupling coefficients between the coil, field shaper, and tube. To validate the analytical model, experimental tests with Photon Doppler Velocimetry (PDV) were conducted. The results show reasonably good agreement between the analytical and experimental results. © 2017 CIRP.
1. Introduction Electromagnetic Forming (EMF) is advantageous in forming, joining, and welding of the lightweight structures ranging from aerospace and automotive industry to medical devices and household appliances. Improved strain distribution and formability; short cycle time; and reduced wrinkling are among the benefits. EMF is suitable for tubular compression and expansion, sheet metal forming, and powder compaction [1]. The EMF process is based on Faraday’s law of electromagnetic induction. In this process, a capacitor bank is charged with a significant amount of electrical energy which is quickly dissipated into a specially designed coil. A magnetic field is generated that induces eddy currents in nearby conductive materials. These eddy currents produce a repulsive magnetic field, and Lorentz forces cause the workpiece to plastically deform away from the coil at a high velo`city. See schematic in Fig.1. One of the key process parameters in EMF is workpiece velocity. If the relative workpiece velocity is sufficient (>300 m/s [3]), this technique can be used for welding of dissimilar metals, i.e., Magnetic Pulse Welding (MPW), where a shear instability creates the characteristic wavy morphology at the weld interface [4,5]. While some multiphysics finite element packages exist that are capable of modelling the process and estimating the critical process parameters (e.g., velocity), there is a lack of simplified and accurate analytical modelling tools available. Such analytical models help to eliminate empirical investigations to determine the necessary process parameters and velocities to produce successful EMF and MPW parts.
* Corresponding author. E-mail address: [email protected] (B. Kinsey).
In this research, a purely analytical model for determining the pressure distribution applied to the workpiece and the subsequent workpiece velocity for a multi-turn, axi-symmetric coil with field shaper was investigated. In the proposed model, at each time increment, the magnetic field is automatically updated in response to the tube deformation. Compared to previous analytical modelling efforts [6–9], a rigid body or a rigid-plastic material assumption was eliminated. Alternatively, plastic deformation of the workpiece during the forming process was taken into account at each time increment. In addition, the coupling factors between the coil, field shaper, and workpiece were experimentally measured and incorporated into the analytical model. The analytical model was then experimentally verified using Photon Doppler Velocimetry (PDV) to measure the workpiece velocity. Reasonably accurate velocity and radial location results were obtained.
Fig. 1. Schematic of EMF/MPW process [2].
2. Analytical model The basic EMF process consists of three fundamental parts: a capacitor bank to store energy, a coil to create the magnetic field, and a workpiece to be formed. Based on the multiphysics nature of the
http://dx.doi.org/10.1016/j.cirp.2017.04.121 0007-8506/© 2017 CIRP.
Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121
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2
EMF process, the analytical model was divided into three stages. First, electrical theory was used to determine primary current and voltage out of the capacitor bank and passing through the coil based on the charge energy. Second, the magnetic field distribution and the effective magnetic pressure that was developed on the workpieces were calculated from electromagnetic analyses. Lastly, classical mechanics theory was used to determine the radial position and velocity of the workpiece caused by the effective magnetic pressure. Since the magnetic field distribution strongly depends on the gap distance between the field shaper and workpiece, the magnetic and mechanical processes were coupled. To calculate the workpiece displacement, a simplified constitutive law for the material was incorporated into the model. At each time increment, the magnetics field geometry was updated with an incremental displacement of the workpiece and hence the model includes the new gap between the field shaper and workpiece. In this study, a commercially purchased multi-turn, axisymmetric coil (Poynting, model: SMU-K100-4/65) was used. The workpiece was an Al6061-T6 tube with a length of 100 mm, wall thickness of 0.88 mm and outer diameter of 25.36 mm. In tube compression processes, a “field shaper” is often positioned between the coil and workpiece to concentrate the magnetic field generated (see Fig. 2a and b). A quick discharge of the capacitor bank causes a damped sinusoidal current flowing through the coil which induces a related electromagnetic field. In the field shaper, which acts as a short circuited second winding of a transformer, a secondary current is induced. Due to the skin effect and Lenz’ law, this induced current flows opposite to the coil current at the outer surface of the field shaper [10]. At the axial slot, the current is directed to the inner surface of the field shaper where the current direction is the same as in the coil (see Fig. 2c). Compared to the outer surface of the field shaper, the inner area is much smaller, resulting in a higher current density and field strength. For this analysis, a series of elements were generated along the surface of the field shaper. The elements were equally spaced, dz, in the axial direction and the width of each element, dr, (see Fig. 2d) was equal to the skin depth, d, defined as:
d¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 rr
m0 mr v
ð1Þ
where v is the frequency of the current through a conductor of resistivity rr, mr is the relative magnetic permeability of the conductor, and m0 is the magnetic permeability of free space.
Fig. 3. Configuration of three circuits and two coupled systems.
primary and induced circuits are coupled with the secondary circuit through the mutual inductances (i.e., M1 and M2) which are a measure of induction between the two circuits. A “coupling factor” ðkÞ represents the loss of magnetic flux with a value between 0 k 1. In this study, the two coupling factors (i.e., k1 and k2 corresponding to M1 and M2 respectively) were experimentally measured by placing a Rogowski coil in two different locations in the experimental set-up and calculating the ratios of the secondary to primary current (k1 = 0.78) and induced to secondary current (k2 = 0.91). Applying Kirchhoff’s voltage law and summing the voltages around the primary circuit, a differential equation is obtained with respect to time, t [10] Z dip ðtÞ 1 ¼0 ð2Þ ip ðtÞdt þ ip ðtÞR þ L dt Cm where Cm is the capacitance of the machine, ip is the current in the primary circuit, R = Rm + Rc is the total resistance, and L = Lm + Lc is the total inductance. Note that the resistance and inductance of the machine, Rm and Lm, can be determined experimentally with a known capacitance by calculating the damped natural frequency and damping ratio of the RLC circuit. Resistance and inductance of the coil, Rc and Lc, are functions of the material properties (i.e., resistivity of the coil), the geometry of the coil, current condition (i.e., the angular frequency), and the cross-sectional area of the coil contained from the skin depth to the surface of the coil. For detailed information see Refs. [9,11]. 2.2. Magnetic theory The magnetic field produced from a given axi-symmetric coil can be determined with respect to the gap distance between the field shaper and tube along the axial direction. As is clear from Fig. 2d, because of the tapered geometry of the field shaper, the gap distance is varied along the axial length of the field shaper (i.e., gðzÞ). The magnetic flux density, B, produced by the coil induces eddy current in the workpiece with a current Jdensity. The current density, J, is related to the magnetic field, H, through a partial derivative in the radial direction. A Lorentz force based on His created which acts as a volume force, F , [10] F ¼ mm H
@H 1 @ðH2Þ ¼ mm 2 @r @r
ð3Þ
The body force, F , is integrated through the thickness of the tube to determine the magnetic pressure acting on the tube surface Zw Pm ¼
1 F dr ¼ mm H2gap 2
ð4Þ
0
Fig. 2. Schematic of multi-turn, axi-symmetric coil with field shaper and tube: (a) full view, (b) half view, (c) current directions, (d) cross section.
2.1. Electrical theory The electrical circuit consisting of the capacitor bank, coil, field shaper and workpiece can be represented by ideal electrical elements and circuits (see Fig. 3). As is clear from Fig. 3, both the
where the integration limit, w, is the tube thickness (see Fig. 2d) and Hgap is the gap magnetic field strength. In this study, the penetrated magnetic field was neglected due to the skin effect [10]. The magnetic field strength, Hgap, is the resultant field of a superposition of magnetic field strength from many current carrying differential elements, dHgap (see Fig. 2d). For the axi-symmetric coil investigated in this study, only the magnetic field strength along the coil’s axis is of interest (i.e., tangential to the workpiece) because this will create a force in the radial
Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121
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direction according to the cross product F ¼ J B[12], and can be calculated from " # i gðzÞ r gðzÞ þ r þ Hz ¼ ð5Þ 2p ðgðzÞ rÞ2 þ z2 ðgðzÞ þ rÞ2 þ z2 where gðzÞ is the varying gap between the coil element and the workpiece and i is the current in each element with height of dz and length of dr. As will be clear from Eq. (12), the magnetic field strength strongly depends on the radial gap between the field shaper and workpiece. 2.3. Mechanical theory Knowing that the workpiece in the forming process is plastically deformed, a simple plasticity model was employed to evaluate the exact deformation of the workpiece at each time increment. Due to the dynamic effects in the process, the radial stress (which acts as a resistance/inertia pressure) is calculated from
s R ¼ Pinertia ¼
ðmassÞa ð2pr Lz wÞ r a ¼ ¼ w ra 2pr Lz 2pr Lz
ð6Þ
strength at a given location in the gap: X Hgap ¼ Hz
3
ð12Þ
Substituting Eq. (12) in Eq. (4) yields a pressure distribution acting on the workpiece at each time increment. From the radial position data, the velocity of the workpiece was also determined. 3. Analytical Results Results from the analytical model include electrical, magnetic, and mechanical predictions. Electrical analysis determines the electrical parameters of the coil and the electrical response of the EMF machine. The predicted circuit response at 2.4 kJ energy discharge is shown in Fig. 4 while the spatial magnetic pressure distribution along the workpiece with respect to time is shown in Fig. 5. Clearly for such a coil based on Fig. 5, the pressure pulse from the first half cycle of the current pulse is more significant than the later oscillations. Thus, the forming event occurs during this timeframe. From the analytical model, the effective stress values for the 2.4 kJ and 3.6 kJ cases are 446 MPa and 457 MPa respectively.
where the denominator in the first equation represents the area that the dynamic force is applied over, r is the density of material, and a is the acceleration of the workpiece. Note that for the final calculation, the distance, Lz, is not required, and w and a (as well as P and r in Eq. (6)) are updated with each increment. Longitudinal and hoop stresses were calculated from the applied external pressure (including the inertia pressure resisting motion in the radial direction with respect to the hoop stress, but not in the axial direction) and radius of the workpiece, r, at a given axial location (see Fig. 2d):
sL ¼
Pm r 2w
&
sH ¼
ðPm s R Þr w
ð7Þ
These equations are derived with respect to quasistatic loading but have been validated for this case as well through numerical simulations. The effective stress, assuming von Mises yield criterion, was then calculated as rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ðs H s L Þ2 þ ðs H s R Þ2 þ ðs L s R Þ2 s¼ ð8Þ 2
Fig. 4. Predicted current and voltage during discharge.
The effective stress, strain and strain-rate relation was defined as
s ¼ C en e_ m
ð9Þ
0
where C is the strength coefficient and n and m are the strain and strain-rate hardening exponents, respectively. For this study, these parameters were obtained from Kolsky Bar test at the strain rate of 5000 s1) (i.e., C0 = 447.2MPa, n = 0.067, and m = 0.006). The strain rate term in this equation was simply used to shift the power hardening curve based on the strain rate in the test. From Eq. (9),e was calculated. Neglecting the longitudinal strain (eL = 0) and considering deH + det = 0 (note that this assumption was confirmed again through numerical simulations and by experimentally measuring the axial elongation), the effective strain was evaluated, again assuming von Mises yield criterion, as 2 e ¼ pffiffiffieH ð10Þ 3 Finally, the hoop strain, eH was calculated from
eH ¼
X
r ln i1 ri
ð11Þ
to obtain the radial location change from time increment, i 1, to i, at each axial location, which were summed to obtain the total strain induced. An updated gap between the field shaper and workpiece was then calculated inside a loop by knowing the gap at the previous step and the current incremental displacement of the workpiece. Finally, the updated z-component of the magnetic field was calculated for each element (see Fig. 2d) using Eq. (5). Summation of these values provides the magnetic field
Fig. 5. Predicted induced pressure distribution (i.e., Pm–Pinertia) along the tube length with time for a 2.4 kJ discharge.
4. Experimental test To verify the analytical model, experimental tests with 2.4 kJ and 3.6 kJ and the same geometry and material used in the analytical model were conducted. The experimental setup is shown in Fig. 6a. The primary circuit current was measured with a Powertek CWT 3000B Rogowski coil. The comparison between the experimental test and the analytical model with a 2.4 kJ energy is shown in Fig. 4. Due to the shape of the commercially purchased coil and difficulty to measure the velocity during the process from the outside, a device was designed and fabricated to measure the velocity on the inside surface of the tube (see Fig. 6b). In this setup, an angled optical path was created by a laser mirror (see Fig. 6c),
Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121
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Fig. 8. Radial displacement from analytical model and experimental tests.
Fig. 6. (a) Experimental setup, (b) velocity measurement device, (c) schematic of the coil and laser mirror location.
and tube velocity was measured by Photon Doppler Velocimetry (PDV). This method is based on the Doppler shift between a reference laser beam and a reflected velocity-shifted beam. As shown in Fig. 4, the analytical model result for the current trace is close to the experimental measurement but with some discrepancies. These are due to complicated electromagnetic impulse process parameter interactions. The experimental results for workpiece velocity in two different energy levels along with the analytical predictions are shown in Fig. 7. To ensure the consistency and repeatability of the experimental measurement system, three tests were conducted at each energy level (variations shown with error bars). As is clear from Fig. 7, the predicted velocity from the analytical model is in reasonably good agreement with the experimental test, i.e., errors of 8% and 9.6% for the 2.4 kJ and 3.6 kJ cases respectively. The time of the peak velocity is predicted later in the process, but this parameter is less critical for the effectiveness of the process. A comparison between the predicted radial displacement from the analytical model and experimental test is shown in Fig. 8. The analytical model approximates the experimental forming behaviour well with final radial position errors of 3% and 5.6% for the 2.4 kJ and 3.6 kJ cases respectively. Note that the radial displacements from the analytical model converged to 3.3 mm and 5.3 mm for the 2.4 kJ and 3.6 kJ cases respectively, while for the experimental tests converged to 3.4 mm and 5.6 mm. Many factors can influence the experimental measurement results. For example, the system inductance and resistance as well as the material properties will change during the experiments because of resistance heating. These factors affect the accuracy of the input capacitor bank inductance and resistance values and material properties used in the analytical model. Also, current and velocity measurement devices have calibration errors. In addition, the analytical model includes various assumptions and simplifica-
Fig. 7. Velocity results from the analytical model and experimental tests.
tions. Finally, the incremental nature of the analytical model causes a time delay in the peak value. Considering these factors, the obtained comparisons are deemed reasonable. 5. Conclusions In this study, the electromagnetic impulse process was analytically modelled in detail. Compared to the previous analytical modelling efforts, the experimentally determined coupling coefficients in the system and plastic deformation of the workpiece during the process were incorporated. The analytical model is able to predict the magnetic pressure distribution on the workpiece and workpiece radial location and velocity during tube compression. The computational time of the analytical model is considerably less compared to FEA simulations. To validate the analytical model, experimental tests with a PDV system and a periscopic velocity measurement device were conducted. The results show reasonably good agreement between the analytical and experimental results. Acknowledgement Funding from the U.S. National Science Foundation (CMII0928319 and CMMI-1537471) is gratefully acknowledged.
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Please cite this article in press as: Kinsey B, Nassiri A. Analytical model and experimental investigation of electromagnetic tube compression with axi-symmetric coil and field shaper. CIRP Annals - Manufacturing Technology (2017), http://dx.doi.org/10.1016/j. cirp.2017.04.121