Parametric Sensitivity Characteristics of Numerical Simulations on EMF Free Bulging of Circular Sheet Metal

1 2 3 3 4Evandro Paese , Martin Geier , Roberto P. Homrich , Rodrigo Rossi , Pedro A. R. C. Rosa1. Universidade de Caxias do Sul, Campus Universitário da Região dos Vinhedos, Bento Gonçalves, RS 95700-000, Brazil2. 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, Brazil4. 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 1mn mn x mn mn x mn mni M R

d

di

t

-

+ + + + + += -

0ca

dvi C

dt=+

DISCRETIZATION OF EMF SYSTEM

SYSTEM PARAMETERS USED FOR EXPERIMENTAL RESULTS

2,5

8

5,5

67,5

110

t = t0

La Raia

C

V0 Primary Circuit (RLC)

RiLi

DiscretizedWorkpiece(L to L )11 mn

Rij

Lij iqij

r

z

i

mj

n

Secundary Circuits(m x n RLs Circuitsof the Discretized

Workpiece)

iaia

iaiaiaia

N - Windingsof the Discretized

Actuator Coil

Parameter

Value

Single-layer copper spiral coil

Number of windings (N)

6

Outer diameter (D0)

67.5mm

Inner diameter (Di)

7.5mm Pitch (P)

5.5mm

Cross section (Aa) 20mm2

Self-inductance (La) 0.9mH (calculated)

EMF machine Capacitance (C)

60mF

Maximum energy (U)

1.5kJ

Workpiece

Material

AA1050

(annealed)

Diameter

110 mm

Thickness

1mm

I. IV. EXPERIMENTAL RESULTS

Radial Point of Workpiece (mm)

Axi

al D

efle

ctio

n of

Wor

kpie

ce (

mm

)

5

10

15

20

25

30

35

40

00 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 3500

10

20Sensor at z = 4.5mm

0 50 100 150 200 250 300 350Am

plit

ude

(V)

0

10

20Sensor at z = 12mm

t (ms)0 50 100 150 200 250 300 350

0

10

20Sensor at z = 19.5mm

EXPERIMENTAL APPARATUS

(a)

(b)

Holder

Die

Workpiece

rz

( c )

Springs

Die

Electrical Connections

Actuator CoilHolder

Out

RL

VRIF

OPV302 OP 906

Laser OPV302Photodiode 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.

36033030027024021018015012090603034

56

7

0.45

0.4

0.3

0.35

0.25

0.55

0.5

8

Skin Effect

Capacitance (mF)

Number of Windings

Ski

n D

epth

(m

m)

36033030027024021018015012090603034

56

7

0

20

40

60

80

100

8

Capacitance (mF)

Number of Windings

2E

lect

rom

agne

tic

Pre

ssur

e (M

N/m

)

Maximum Electromagnetic Pressure

700Jr

z

N=3

Radial Point of Workpiece (mm)

Axi

al D

efle

ctio

n of

Wor

kpie

ce (

mm

)

5

10

15

20

25

30

35

40

00 5 10 15 20 25 30 35 40 45 50 55

N=4N=5N=6N=7

N=8

N=8(210mF)

60mF

2E

lect

rom

agne

tic

Pre

ssure

(M

N/m

)

Distribution of Electromagnetic Pressure

5 10.5 16 21.5 27 32.5 38 43.5 49 54.50

10

20

30

40

50

60

70

80

N=3N=4N=5N=6N=7N=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

0.30.6

0.91.2

1.5

Indu

ctan

ce (m

H)

360330

270210

15090

3034

56

7

65

60

50

45

55

408

Pea

k of

Dis

char

ge C

urre

nt (

kA)

Number of Windings Capacitance (mF)

360330

270210

15090

3034

56

7

0

5

10

15

20

8

Ris

e T

ime

of t

he D

isch

arge

Cur

rent

(m

s)

Capacitance (mF)Number of Windings

Maximum Discharge Currentas Function of the Number of

Coil Windings and Capacitance

The Peak of Electromagnetic PressureHappens at the Same Moment of the

Maximum Discharge Current

-1/2d=(pf �km)f

( )2 21

zj

zj

j j

fP

r rp +

=-

1

m

zj rij ij ijif B i lq=

=å

f ��is based onf

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