Original Article Current frequency effect on electromagnetic tube expansion Proc IMechE Part C: J Mechanical Engineering Science 2020, Vol. 234(23) 4636–4644 ! IMechE 2020 Article reuse guidelines: sagepub.com/journals-permissions DOI: 10.1177/0954406220927059 journals.sagepub.com/home/pic SK Dond1 , Hitesh Choudhary2, Biswaranjan Dikshit2, Tanmay Kolge2 and Archana Sharma2 Abstract The frequency of capacitor discharge current is an important parameter in the electromagnetic forming process. In the present work, simulation and experimental study of electromagnetic expansion is carried out on aluminium tubes (Al 5052). The aim is to obtain the optimum frequency of discharge current that gives maximum tube expansion at constant discharge energy. A two-dimensional axisymmetric numerical simulation model is developed that couples electromagnetic and structural phenomenon sequentially. After validating experimental tube expansion results with numerical results, the simulation model is used to analyse the effect of variation in current frequency on the tube expansion. It is observed that maximum tube expansion and higher process efficiency occurred at 5 kHz, which is considered as the optimum frequency for the used experimental set-up. In contrast to sheet forming case observed in the literature, results of tube expansion show that maximum forming is obtained at a frequency where the ratio of skin depth to tube thickness is less than 1. It is also noticed that the optimum frequency depends on the inductance and resistance of the system. Keywords Electromagnetic tube expansion, optimum frequency, sequentially coupled simulation, plastic deformation Date received: 23 December 2018; accepted: 21 April 2020 Introduction Electromagnetic forming (EMF) is an impulse forming process and ﬁnds the applications in automobile and aerospace industries. The process has advantages like forming, crimping and welding of lightweight materials such as aluminium.1 In EMF process, a damped sinusoidal current in the range of kiloamperes is passed through the electromagnetic coil and coil current generates an eddy current in the magnetically coupled workpiece (WP). Lorentz force responsible for forming is generated by the interaction of current density induced in WP and the magnetic ﬂux density present between the coil and WP. A comprehensive review and assessment of the EMF process is presented.2 Several analytical and numerical studies have been done for EMF circuit analysis of tube and sheet forming. To investigate the EMF process, numerical simulation is the best choice as it helps to get a realistic insight of the procedure which is diﬃcult in practice. The numerical investigation is performed by various software using loosely coupled and sequentially coupled techniques.3–5 The advantage of the sequential approach is that the eﬀect of WP deformation on electromagnetic force distribution is considered and hence it has better accuracy over a loosely coupled approach.4,6 Electrical and mechanical parameters involved in the EMF process decide the resultant forming of WP. The eﬀect of coil–tube relative position and inertia on the tube forming shapes is reported by the author.7 Reducing the coil length reduces the inductance and resistance of the coil and that causes an increase in peak current and frequency. It is observed that magnetic pressure acting on tubular WP varies with length and resistivity of tubular material.8 Forming behaviour also strongly depends on the characteristic of the current passing through the coil. Not only peak current, but the duration of the current pulse also aﬀects the resultant sheet forming.9 Optimum selection of capacitance value and discharging energy play a key role in controlled forming of WP. Variation in capacitance varies the frequency of the current ﬂowing through the 1 Homi Bhabha National Institute, Mumbai, India Bhabha Atomic Research Centre, Mumbai, India 2 Corresponding author: SK Dond, Homi Bhabha National Institute, Mumbai 400094, Maharashtra, India. Email: email@example.com Dond et al. 4637 electromagnetic coil. The EMF process eﬃciency is less2 and hence, it is required that the system parameters be set so as to get maximum eﬃciency. For given discharge energy, there exists a frequency of the current in the coil that gives maximum deformation. Very few documents are available that have studied the eﬀect of frequency on WP forming. The eﬀect of frequency on tube compression is studied10 and relates the optimum frequency to high plastic strain energy. In the frequency domain approach,11 electromagnetic ﬁelds obtained from FEM modelling are used to estimate the optimum frequency but WP deformation is not considered during the analysis. For sheet forming case,12,13 the ratio of skin depth to tube thickness is found to be near or more than 1 when optimum frequency achieved. In the present work, the electromagnetic expansion study is performed on tubular WP made up of aluminium. A 2D axis-symmetric, sequentially coupled simulation is carried out using COMSOL FEM software and obtained results are compared with experimental ones. Numerical simulation model is further used to ﬁnd the optimum frequency of discharge current and to study the eﬀect of process parameters like the tube thickness, inductance and resistance of the system on optimum frequency. This study can be useful in deciding the system and coil design parameters to get more eﬃciency. Figure 1. Schematic of the system. EMF system EMF uses a high energy magnetic ﬁeld to deform materials. This ﬁeld is produced by the transient under-damped current that ﬂows through the coil. The time varying, pulsed magnetic ﬁeld is generated by the coil and is linked with surrounding metallic WP. Eddy current gets induced in a WP with direction opposite to that of electromagnet current direction. Figure 1 shows the schematic of the EMF process for tubular WP. Here, dot and cross show the direction of the current in the coil and WP. Initially, energy is stored in the electrostatic form in the capacitor bank by charging the capacitors. The energy stored in the capacitor is given by equation (1). Here W is the energy stored in the capacitor bank, V is the charging voltage of the capacitor bank and C is the capacitance value. High power rating switch is used to discharge the capacitive energy into the coil 1 W ¼ CV2 2 ð1Þ The electrical equivalent circuit of EMF is presented in Figure 2. Coil and WP are mutually coupled. RC and LC are coil resistance and inductance whereas RW and LW are tubular WP resistance and inductance. System resistance RS and inductance LS can be obtained from the current waveform of the system short circuit test. R and L are total resistance Figure 2. Electrical equivalent circuit of the EMF system. and inductance of the system, respectively. As the WP moves away from the coil, the magnetic coupling between the coil and WP decreases and resultant R and L also vary. However, in EMF, the most important part of the electric current that causes the displacement is very short14 and hence R and L can be assumed constant during forming. The damped sinusoidal current ﬂowing through the coil is given by equation (2) and it is determined by the resonant circuit formed by equivalent resistance, inductance and capacitance of the system. Here x is the angular frequency of the current and b is the damping coeﬃcient. x and b depends on the system parameters as given by equations (3) to (5) V t e sin !t !L pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ! ¼ !0 2 I¼ ð2Þ ð3Þ R 2L ð4Þ 1 !0 ¼ pﬃﬃﬃﬃﬃﬃﬃ LC ð5Þ ¼ 4638 Proc IMechE Part C: J Mechanical Engineering Science 234(23) 1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f0 r r ¼ h ð6Þ ð7Þ placed over the coil. The supporting structure is used to maintain the same axial position of the coil and tube. Figure 3 shows the coil and WP dimensions. The material properties are given in Table 1. The magnetic ﬁeld diﬀusion in the WP depends on the skin depth. The skin depth d is given by equation (6) where f is the coil damping current frequency, is the WP electrical conductivity and l0 and lr are permeability of free space and the relative permeability, respectively. dr is the relative skin depth as given by equation (7). It is the ratio of skin depth to WP thickness h. Experiment detail EMF set-up as shown in Figure 4 consists of the power supply, capacitor bank and coil–tube arrangement. The power supply has an energy capacity of 40 kJ. The capacitors are rated for 20 kV charging voltage and 224 mF capacitance. A solenoid coil (Figure 5) is used to generate the magnetic ﬁeld. The tube is Figure 5. The coil with reinforcement and used Al 5052 tube. Table 1. Material properties of the EMF system. Material Parameter Value Coil (copper) Number of turns Electrical conductivity Yield stress Electrical conductivity Density Poisson’s ratio 7 58 MS/m 89.6 MPa 20 MS/m 2680 kg/m3 0.33 Workpiece (Al 5052) Figure 3. Dimensions of the experimental system. Figure 4. Experimental setup and arrangement of coil and tube. Dond et al. 4639 Tube forming experiments are carried out at diﬀerent energies by varying the operating voltage and at a constant capacitance of 112 mF. Resistance R and inductance L of the EMF system are 18 mX and 1.7 mH, respectively. Discharging current through the coil is measured by Rogowski coil and recorded on an oscilloscope. Numerical simulation An axisymmetric, 2D sequentially coupled simulation is performed using COMSOL. Following assumptions are made in electromagnetic ﬁeld analysis: 1. The cylindrical coordinate system is used for the developed 2D model that consists of axial and radial components. 2. The electrical conductivity of material and permeability is constant and isotropic. 3. The eﬀect of temperature on material properties is not considered. In the electromagnetic ﬁeld model, magnetic vector potential method is used as given below to solve the Maxwell equations r B ¼ 0 r E ¼ ð8Þ @B @t r B ¼ J þ " ð9Þ @E @t ð10Þ where E is the electrical intensity (V/m), J is the current density in the coil (A/m2), B represents magnetic ﬂux density (T) and 2 is permittivity of the medium (F/m). Representing magnetic and electric ﬁeld using magnetic vector potential A such that B ¼ r A ð11Þ @A E ¼ @t ð12Þ The obtained force is used as an input in the mechanical model. In the structural mechanics model, tube displacement is calculated using the following equation @2 u r s ¼ fm @t2 Here, q is the mass density of WP, u is displacement vector, s is stress tensor and fm is magnetic force density. Figure 6 shows the ﬂowchart of the sequential coupled model used for simulation. The coil current obtained by the Rogowski coil is used as input to the simulation model. Time step is kept at 0.5 ms and at each time step tube geometry is updated based on Lorentz force. Numerical simulation model consists of a coil, tube and air. The coil and tube regions are meshed through mapped meshing. Figure 7 shows the meshing used in the numerical model. The electromagnetic and structural models are run till the tube gets ﬁnal displacement. Tubular WP ﬂow stress approximated by Johnson–Cook material model is as given by equation (16). The temperature eﬀect is ignored here as any small rise in temperature has little inﬂuence on electrical and mechanical properties of WP material15–17 T T0 m ¼ ½Aw þ Bw ðp Þn ½1 þ Cw lnð_p Þ 1 Tm T0 ð16Þ Here, 2p is an eﬀective plastic strain; _p is strain rate; and Aw, Bw, Cw, n and m are material Substitution of equations (11) and (12) in equations (8) to (10) can give r 1 @A r A ¼ J @t ð13Þ where is the conductance of medium (S/m) and @@tA is the induced current density in the tube (A/m2). According to the Lorentz formula, electromagnetic force acting on WP is given as 1 B F ¼ J B ¼ ðr BÞ ð15Þ ð14Þ Figure 6. Flowchart of sequential coupled simulation. 4640 Proc IMechE Part C: J Mechanical Engineering Science 234(23) Figure 8. Expanded tubes (a) 26.8% and (b) 40.2% change in diameter. Figure 7. The 2D axis symmetric arrangement and meshing. constants.18 First bracket represents stress as a function of strain. Second and third bracket represents stress as a function of strain rate and temperature, respectively. Results and discussion Comparison of experimental and simulation results Experimentally deformed tubes are shown in Figure 8. It is observed that the increase in discharge energy increases the tube expansion as expected. For the operating voltage of 16 kV and capacitance of 112 mF, 26.8% increase in tube central diameter is obtained whereas an increase in voltage to 18 kV showed a 40% increase in tube diameter. Maximum deformation of the tube occurred at the central region and less towards the tube ends. This happened because of the tube length being higher than the coil length. Maximum magnetic ﬂux concentration occurred at the tube surface facing coil central turn and that causes higher magnetic pressure at the middle part of the tube. Comparison of experimental and simulation results is shown in Figure 9. Numerical results of tube ﬁnal displacement and experimental results have a maximum error of 8% thereby validating our simulation model. The reason for error can be attributed to the following reasons. (a) The diﬀerence in experimental and literature values of material properties and plasticity model constants. (b) Assumptions taken in the numerical model. Figures 10 to 12 indicate the simulation results obtained at 16 kV, 112 mF condition. Figure 9. Radial displacement of the tube by experiment and by simulation. The displacement and velocity variation with time at the maximum displacement point of the tube is shown in Figure 10. Peak velocity of the tube is found to be delayed by 15 ms from peak coil current which might be due to tube inertia. Figure 11 shows the surface plot of the tube deformation. The formed shape of the tube is similar to the experimental one. Displacement of the tube along the surface is indicated by diﬀerent colours. Forming eﬃciency is obtained from the ratio of plastic strain energy of the tube to energy stored in the capacitor. Figure 12 shows these energy plots where 10.5% forming eﬃciency is obtained. This low eﬃciency can be understood in the following way: As observed in Figure 10, tube deformation completes well before the total current dies out. Only ﬁrst and second current pulses are utilized for WP deformation while the consequent pulses result in just heating. This aﬀects the eﬃciency of the process. Dond et al. 4641 Figure 10. Coil current, tube velocity and displacement at 16 kV, 112 mF. Figure 13. Radial displacement of the tube with a thickness of 2.4 mm. Figure 11. Radial displacement of tube obtained at 16 kV, 112 mF. Figure 12. Energy plots at 16 kV, 112 mF. Optimum frequency estimation by the numerical model The simulation model is further used to observe the eﬀect of frequency on the forming of the tube at constant discharge energy condition. The operating voltage and capacitance are varied in such a way to keep total discharge energy given by equation (1) as a constant at 16 kJ. Change in capacitance causes changes in damping frequency of coil current. Displacement of the tube is measured at the maximum forming point. Figures 13 and 14 show displacement obtained at diﬀerent frequencies for diﬀerent tube thickness. The maximum displacement of the tube is found to occur at 5 kHz and we called it optimum frequency. The relative skin depth dr at the optimum frequency in Figures 13 and 14 is less than 1. This observation is in contrast to the sheet forming case13,14 where dr is equal to and more than 1 when maximum forming occurs in the sheet. The optimum frequency phenomenon can be explained with the magnetic pressure proﬁle acting on tube and skin depth. Magnetic pressure is proportional to the square of magnetic ﬂux density and the magnetic ﬁeld is proportional to the coil current. Higher the current, higher the magnetic ﬁeld. Skin depth d increases with the decrease in frequency value (equation (6)). When d is more than the thickness of the tube, in other words, dr is more than 1, the magnetic ﬁeld penetrates through tube thickness. The eﬀective magnetic pressure acting on the tube gets reduced. That is why the tube forming reduces when the dr is more than 1. Increase in frequency increases the current peak value with a decrease in current pulse duration as shown in Figure 15. At high frequency, though the magnetic pressure is high because of higher peak current, the duration of this pressure pulse is very short due to rapid damping of the current pulse. As mentioned in Cui et al.,9 resultant WP forming is not only aﬀected by peak current but also by the duration of the current waveform. That is why the tube forming slowly reduces when the frequency is increased beyond an optimum value. It means that not only the peak current but the optimum combination of peak current and the duration of the current 4642 Figure 14. Radial displacement of the tube with thickness 3 mm. Figure 15. Coil current waveform at various frequencies. pulse resulted in maximum tube forming. For the used experimental set-up, the optimum combination occurred at 5 kHz. Further, the analysis is carried out to study the eﬀect of diﬀerent parameters on the optimum frequency and corresponding relative skin depth. Effect of tube thickness Figures 12 and 13 show the eﬀect of tube thickness variation at constant discharge energy. It is observed that tube forming is reduced with an increase in tube thickness. With an increase in tube thickness, the optimum frequency is unaﬀected but the relative skin depth dr at the optimum frequency is decreased. This is because the optimum combination of the current peak and duration of the current pulse (current frequency) have resulted in a maximum push to the material. The tube thickness has no eﬀect on the peak current or on the current frequency. The process eﬃciency of EMF is shown in Figure 16 for diﬀerent thickness of tubes. Maximum eﬃciency Proc IMechE Part C: J Mechanical Engineering Science 234(23) Figure 16. Process efficiency variation with frequency. Figure 17. Effect of frequency variation with higher inductance on tube expansion. occurred at same optimum frequency and it is more for thin tubes as these are more susceptible to deformation. Effect of system inductance and resistance Variation in WP resistivity, WP length, the coupling between the coil and the WP and coil material aﬀect the magnetic pressure acting on the WP. From the electrical circuit point of view, all these variations aﬀect the resultant inductance and resistance of the EMF system. To take into account this parametric variation, the eﬀect of resistance and inductance variation on optimum frequency is studied. The simulation model is used to study the eﬀect of inductance and resistance value on optimum frequency. Figures 17 and 18 depict the tube expansion performed at reduced damping coeﬃcient b value for 2.4 mm thickness tube. b depends on resistance and inductance of the system as given by equation (4). In the ﬁrst case, b is reduced from 5294 to 3500 S1 by increasing the inductance and in the second case b is Dond et al. 4643 Funding The author(s) received no ﬁnancial support for the research, authorship, and/or publication of this article. ORCID iD SK Dond https://orcid.org/0000-0002-1793-792X References Figure 18. Effect of frequency variation with lower resistance on tube expansion. reduced to 3500 S1 by reducing the resistance value. Compared to previous results, it is observed that optimum frequency increased with an increase in inductance, whereas it decreased with a decrease in resistance. Thus, the optimum frequency does not remain constant. It depends on system parameters and this fact should be taken into consideration during the design of EMF system conﬁguration. System parameters can be set in such a way that the coil current oscillates with optimum frequency. This gives higher process eﬃciency along with increase coil life. In this paper, a 2D axisymmetric numerical model is developed for the analysis. However, a close observation of formed tube shows that the tube forming is not exactly axisymmetric due to the helical nature of the coil. The future aim is to develop a 3D numerical model to improve the numerical simulation accuracy. Conclusions Electromagnetic expansion of aluminium tubes is carried out experimentally and the tube displacement results are compared with results of the 2D numerical simulation model. Eﬀects of system parameters on the optimum frequency and relative skin depth are further analysed using a simulation model. From the observed results, the following conclusions are drawn: (1) Maximum expansion of tubular WP occurred at a frequency where the relative skin depth is less than 1 in contrast to sheet forming case observed in the literature. (2) The frequency of current and the relative skin depth at which maximum forming occurs are not constant. The optimum frequency changes with resistance and inductance of the system. Declaration of Conflicting Interests The author(s) declared no potential conﬂicts of interest with respect to the research, authorship, and/or publication of this article. 1. Golovashchenko S. Electromagnetic forming and joining for automotive applications. 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