Parametric Sensitivity Characteristics of Numerical Simulations on EMF Free Bulging of Circular Sheet Metal 1 2 3 3 4 Evandro Paese , Martin Geier , Roberto P. Homrich , Rodrigo Rossi , Pedro A. R. C. Rosa 1. Universidade de Caxias do Sul, Campus Universitário da Região dos Vinhedos, Bento Gonçalves, RS 95700-000, Brazil 2. Universidade do Vale do Rio dos Sinos, São Leopoldo, RS 93.022-750, Brazil, 3. Universidade Federal do Rio Grande do Sul, Porto Alegre, RS 90035-190, Brazil 4. Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal Mon-Mo-Po1.02-01 [13] I. INTRODUCTION An electromagnetic forming (EMF) can be considered essentially like primary and secondary electrical circuits coupled by mutual inductances, composed of an actuator coil and a conductive workpiece. The EMF system geometry and parameters of pulse unit are very important to reach a high, optimum rise time, and adequate distribution of electromagnetic pressure, aming a higher or adequate workpiece displacement. This study focuses on carrying out a parametric analysis by performing numerical simulations of free bulging of circular sheet metal by the EMF system using a spiral actuator coil with a single layer. II. OBJECTIVES Demonstrate the influence of EMF system parameters on the electromagnetic pressure, and consequently achieving higher or adequate workpiece displacement; ‚ Use a numerical method to solve the fully coupled electric-magnetic problems applying an in-house script developed in the Matlab. This aproach can predict the reflected inductances from secondary(workpiece) circuits to a primary one(machine); ƒ Solve the uncoupled mechanical problem using the ABAQUS/Explicit Finite Element Software; „ Show comparative numerically predicted deflections versus experimental measurements to verify the calculation method; … Outline desing principles for EMF systems of sheet metals. III. BASIC METHODOLOGY OF ANALYSIS The calculations of the electromagnetic problem use a method based on discretization of the actuator spiral coil into concentric (N) elements and the workpiece into multilayered concentric elements. The RLC primary circuit is fully coupled with several secondary RLs circuits and can be modeled using a set of ODEs applying Kirchhoff’s law. The inductances couple the electric-magnetic phenomena. This system of ODEs is solved by employing the explicit Runge–Kutta method inside Matlab 2015b (ODE45 function). The Biot-Savart’s law is applied to calculate the magnetic field B and B , which are used to calculate the self z r and mutual inductances, and the electromagnetic force in the axial direction, respectively. Both B , and B use numerical z r methods to solve the equations. The total electromagnetic force and consequent pressure along of the r-axis is calculated using The skin effect in the workpiece is an important electrical parameter to be evaluated, aiming the efficiency of the EMF process. In the present numerical method, the skin effect is implicitly considered in the performance of the EMF system, as it uses fully coupled electric-magnetic method, and multilayer discretization. [] ( ) [ ] ( )( ) [] ( )( ) [] ( ) ( ) 1 1 1 1 1 1 1 mn mn x mn mn x mn mn i M R d d i t - + + + + + + = - 0 c a dv i C dt = + DISCRETIZATION OF EMF SYSTEM SYSTEM PARAMETERS USED FOR EXPERIMENTAL RESULTS 2,5 8 5,5 67,5 110 t = t 0 L a R a i a C V 0 Primary Circuit (RLC) R i L i Discretized Workpiece (L to L ) 11 mn R ij L ij i qij r z i m j n Secundary Circuits (m x n RLs Circuits of the Discretized Workpiece) i a i a i a i a i a i a N - Windings of the Discretized Actuator Coil Parameter Value Single-layer copper spiral coil Number of windings ( N) 6 Outer diameter ( D 0 ) 67.5mm Inner diameter ( D i ) 7.5mm Pitch (P) 5.5mm Cross section ( A a ) 20mm 2 Self-inductance (L a ) 0.9m H (calculated) EMF machine Capacitance ( C) 60m F Maximum energy (U) 1.5kJ Workpiece Material AA1050 (annealed) Diameter 110 mm Thickness 1mm I. IV. EXPERIMENTAL RESULTS Radial Point of Workpiece (mm) Axial Deflection of W orkpiece (mm) 5 10 15 20 25 30 35 40 0 0 5 10 15 20 25 30 35 40 45 50 55 5.5 13 20.5 700J 50ms 125ms 210ms 0 50 100 150 200 250 300 350 0 10 20 Sensor at z = 4.5mm 0 50 100 150 200 250 300 350 Amplitude (V) 0 10 20 Sensor at z = 12mm t (ms) 0 50 100 150 200 250 300 350 0 10 20 Sensor at z = 19.5mm EXPERIMENTAL APPARATUS (a) (b) Holder Die Workpiece r z ( c ) Springs Die Electrical Connections Actuator Coil Holder Out R L V R I F OPV302 OP 906 Laser OPV302 Photodiode OP906 VI. CONCLUSIONS There exists an optimum time for the maximum electromagnetic pressure to happen, producing higher sheet displacements and this optimum time is related to the spiral coil geometry, and mainly with pulse unit parameters; ‚ The greatest electromagnetic pressure is obtained for the highest number of coil windings along with the lowest capacitance, but the best EMF system configuration (geometry/machine) is observed for the capacitance value of 210µF; ƒ The experimental results for N=6 and 60µF have good correlation with the calculated workpiece movements, demonstrating the effectiveness of this solution methodology; „ The EMF process is analyzed using a fully coupled electric-magnetic, and uncoupled mechanical solution method, as a result the method can predict the reflected impedances from secondary circuits to a primary one; … Finally, this methodology can outline optimum geometry and machine parameters aiding in the design of the EMF equipment. 360 330 300 270 240 210 180 150 120 90 60 30 3 4 5 6 7 0.45 0.4 0.3 0.35 0.25 0.55 0.5 8 Skin Effect Capacitance (mF) Number of Windings Skin Depth (mm) 360 330 300 270 240 210 180 150 120 90 60 30 3 4 5 6 7 0 20 40 60 80 100 8 Capacitance (mF) Number of Windings 2 Electromagnetic Pressure (MN/m ) Maximum Electromagnetic Pressure 700J r z N=3 Radial Point of Workpiece (mm) Axial Deflection of W orkpiece (mm) 5 10 15 20 25 30 35 40 0 0 5 10 15 20 25 30 35 40 45 50 55 N=4 N=5 N=6 N=7 N=8 N=8(210mF) 60mF 2 Electromagnetic Pressure (MN/m ) Distribution of Electromagnetic Pressure 5 10.5 16 21.5 27 32.5 38 43.5 49 54.5 0 10 20 30 40 50 60 70 80 N=3 N=4 N=5 N=6 N=7 N=8 60mF N=8(210mF) r (mm) V. PARAMETRIC RESULTS Calculated Self-Inductance as Function of the Number of Coil Windings Number of Windings 3 4 5 6 7 8 0.3 0.6 0.9 1.2 1.5 Inductance (mH) 360 330 270 210 150 90 30 3 4 5 6 7 65 60 50 45 55 40 8 Peak of Discharge Current (kA) Number of Windings Capacitance (mF) 360 330 270 210 150 90 30 3 4 5 6 7 0 5 10 15 20 8 Rise Time of the Discharge Current (ms) Capacitance (mF) Number of Windings Maximum Discharge Current as Function of the Number of Coil Windings and Capacitance The Peak of Electromagnetic Pressure Happens at the Same Moment of the Maximum Discharge Current -1/2 d=(pf km) f ( ) 2 2 1 zj zj j j f P r r p + = - 1 m zj rij ij ij i f B i l q = = å f is based on f Fourier analysis Martin Geier would like to acknowledge the financial support of the National Council for Scientific and Technological Development CNPq - Brazil, process n. 430725/2016-7