Determination of maximum force on static sheet of aluminum subjected to magnetic field of hammer coi

Ball, B. J.

Hill, L. J.

Hyde, G. S.

Wier, D. D.

English

NASA Technical Reports Server

. 
A STUDY OF INTENSE MAGNETIC FIELDS FOR 
HIGH ENERGY FORMING AYD STR'U'CT't'RAL ASSEMaLY 
INTEREM REPORT 
D. D. Wier, B. J. Ball 
L. J. Hill and G. S. Hyde 
PRINCIPAL INVESTIGATOR: 
D. D. Wier 
Associate Professor 
Department of Electrical Engineering 
Mississippi State University 
Box 326, State College, Mississippi 39762 
Telephone: 601+323-4321, Ext. 396 
https://ntrs.nasa.gov/search.jsp?R=19670010628 2020-03-16T18:05:47+00:00Z
TABLE OF CONTENTS 
INTRODUCTION 
INVESTIGATION OF HALLEN'S EQUATION 
EDDY CURRENT METHOD 
EQUIVALENT CIRCUIT MEtHOD 
CONCLUSION 
PAGE 
1 
-
3 
4 
5 
13 
INTRODUCTION 
. 
The purpose of t h i s  s tudy is  t o  determine t h e  maximum f o r c e  on a 
s t a t i c  shee t  of aluminum subjected t o  t h e  magnetic f i e l d  of t h e  hammer 
c o i l  now being used a t  t h e  Marshall  Space F l i g h t  Center a t  Hun t sv i l l e ,  
Alabama. 
r e s u l t  of t h i s  fo rce ,  t h e  present  u se  of t h e  hammer c o i l  i s  i n  smoothing 
metal su r faces .  This  a l lows only a d i f f e r e n t i a l  movement which can be 
c l o s e l y  approximated by a s t a t i c  shee t .  I f  t r u e  forming is  at tempted,  
Although movement of the metal shee t  does t ake  p l ace  as a 
t h e  movement of t h e  metal  s h e e t  would create vo l t ages  due t o  motion of 
a conducting f ia ter ia l  i n  a magnetic f i e l d .  
i n  t h i s  s tudy are those  c rea t ed  i n  a s t a t i o n a r y  conducting material 
sub jec t ed  t o  a t i m e  varying magnetic f i e l d .  I f  t h e  f o r c e  c rea t ed  i n  a t r u e  
forming ope ra t ion  is  t o  be determined, t h e  cu r ren t  due t o  both induced 
vo l t ages  would have t o  be considered. 
The only  vo l t ages  considered 
This  s tudy  i s  d i r e c t l y  concerned with t h e  f o r c e s  produced on four  
th icknesses  of s i x  d i f f e r e n t  aluminum a l l o y s .  The th icknesses  under 
cons ide ra t ion  are 0.060, 0.375, 0.500 and 0.750 inches .  The a l l o y s  and 
some of t h e i r  c h a r a c t e r i s t i c s  are l i s t e d  i n  Table I. The va lues  of t h e  
e lec t r ica l  and magnetic p r o p e r t i e s  were furn ished  through t h e  cour tesy  of 
t h e  Aluminum Company of America. 
Two methods of determining the f o r c e  exer ted  on t h e  aluminum shee t  
are  suggested i n  t h i s  r e p o r t .  One method r e q u i r e s  a de te rmina t ion  of t h e  
magnitude of t h e  c u r r e n t s  induced i n  t h e  aluminum as a func t ion  of t h e  
cu r ren t  i n  t h e  c o i l .  The i n t e r a c t i o n  of t h e  f i e l d s  produced by these  two 
c u r r e n t s  w i l l  then  be used t o  determine t h e  f o r c e  exer ted  on t h e  s h e e t .  
The o t h e r  method is  t o  determine the  s e l f  and mutual inductances of t h e  
2. 
. 
Relative Magnetic 
Alloy Temper P erne ab i 1 i t y Conductivity@20c Resistivity@20c 
% International (Microhm- 
Annealed - Copper cent ime t er ) 
Standard (Equal Vol) 
2219 T87 1.0000196 
7075 T6 1.0000159 
7075 T651 1.0000159 
6061 T6 1.0000199 
5456 H343 *l. 00002 
5456 H321 *l. 00002 
32 
33 
33 
43 
29 
29 
5.4 
5.2 
5.2 
4.0 
5.9 
5.9 
*Estimated Value 
Characteristics of Aluminum Alloys 
Table I 
3 .  
c o i l  and t h e  aluminum shee t .  
inductances as c o e f f i c i e n t s ,  an equat ion f o r  t h e  f o r c e  exer ted  as a 
func t ion  of cu r ren t  w i l l  be  developed. 
i n d i c a t i o n  which, i f  e i t h e r ,  of these  two methods are b e t t e r  o r  more 
accu ra t e .  
From a system of equat ions  having t h e s e  
A t  t h e  p re sen t  t i m e  t h e r e  i s  no 
I n v e s t i g a t i o n  of Ha l l en ' s  Equation 
An i n v e s t i g a t i o n  was made of t h e  equat ion 
where T i s  t h e  t o t a l  f o r c e  per  u n i t  area a c t i n g  on t h e  elemental  s u r f a c e ,  
n i s  t h e  u n i t  vec to r  i n  t h e  d i r e c t i o n  of t h e  outward normal t o  t h e  
elemental  s u r f a c e ,  and B and a r e  t h e  magnetic f i e l d  q u a n t i t i e s  immediate- 
l y  o u t s i d e  t h e  elemental  s u r f a c e  of a magnetized body immersed i n  a f l u i d  
- 
- 
such as air.' 
ferromagnet ic  macer ia i  and w a s  not a p p l i c a b l e  t o  this i n v e s t i g a t i o n .  
It w a s  determined t h a t  t h i s  equat ion  only appl ied  t o  a 
The s tudy  of t h i s  equat ion  was performed by using a balance t o  measure 
t h e  f o r c e  exer ted  on a shee t  of metal  when sub jec t ed  t o  a t i m e  varying 
magnetic f i e l d .  A photograph of t h i s  ba lance  i s  shown i n  Figure I. 
The s h e e t  of m e t a l  w a s  balanced without any magnetic f i e l d  being app l i ed .  
A magnetic f i e l d  w a s  then app l i ed  energizing a c o i l  d i r e c t l y  beneath t h e  
shee t .  The shee t  w a s  balanced and a reading w a s  made of t h e  d i f f e r e n c e  i n  weight 
requi red  t o  ba lance  t h e  mater ia l  with and without  t h e  presence of t h e  f i e l d .  
Readings were a l s o  made of t h e  vo l t age ,  c u r r e n t ,  and power inpu t  t o  t h e  c o i l .  
These readings  are shown i n  Table 11. 
The curves of Figures  2 and 3 show t h e  v a r i a t i o n  of power d i s s i p a t i o n  
and r e f l e c t e d  r e s i s t a n c e  as a func t ion  of t h e  th ickness  of t h e  s h e e t .  
4 .  
Figure  4 shows t h e  f o r c e  exer ted  o n - d i f f e r e n t  t h i cknesses  of aluminum 
as a func t ion  of th ickness .  
c u r r e n t  of f o u r  amperes. 
A l l  po in t s  i n  F igure  4 were made wi th  a c o i l  
Eddy Current Method 
An i n s p e c t i o n  of Figure 4 i n d i c a t e s  t h a t  a l i m i t  i s  approached 
i n  t h e  f o r c e  exer ted  as t h e  th ickness  of t h e  m e t a l  i nc reases .  This  is  
due t o  t h e  so c a l l e d  p e n e t r a t i o n  depth of t h e  magnetic f i e l d  i n  t h e  
metal. 
which creates a c u r r e n t .  
t h e  change i n  t h e  app l i ed  magnetic f i e l d .  It ,  t h e r e f o r e ,  tends t o  cance l  
ou t  t h e  magnetic f i e l d ,  and, as a r e s u l t ,  t h e  magnetic f i e l d  reaches  ze ro  
magnitude a t  some po in t  w i t h i n  t h e  metal. 
dependent on frequency as given by t h e  equat ion  
The t i m e  vary ing  magnetic f i e l d  induces a vo l t age  i n  t h e  metal 
This  cu r ren t  is  i n  such a d i r e c t i o n  as t o  oppose 
The p e n e t r a t i o n  depth is  
1 6 =  
where f is  t h e  frequency, l~ is t h e  permeabi l i ty  of t h e  m a t e r i a 1 , a  i s  
t h e  r e s i s t i v i t y  of t h e  material, and 6 is  t h e  depth a t  which t h e  induced 
L v o l t a g e s  i s  1/, t i m e s  t h e  va lue  of induced v o l t a g e  a t  t h e  su r face .  
From photographs taken of t h e  cu r ren t  wave of t h e  hammer c o i l ,  t h e  
n a t u r a l  frequency is  approximately 3 . 3  KHz. This  frequency i s  being used 
i n  t h e  pre l iminary  c a l c u l a t i o n s  t o  determine t h e  p e n e t r a t i o n  depth.  This  
g ives  a depth of 0.100 inches which i n d i c a t e s  t h a t  only one of t h e  
proposed th i cknesses  need inves t iga t ion .  A l l  s h e e t s  t h i c k e r  than  0.100 
inches  should have t h e  same induced c u r r e n t .  
5. 
It is proposed t o  d i v i d e  the s h e e t  i n t o  concen t r i c  c i r c u l a r  segments 
about  t h e  c e n t e r  l i n e  of t h e  c o i l .  From-the-cross-sec t ion  area, l eng th ,  and 
p e n e t r a t i o n  depth,  t h e  r e s i s t a n c e  of t h e  segments w i l l  be ca l cu la t ed .  The 
c u r r e n t  f low i n  t h e  segment due t o  t h e  induced v o l t a g e  w i l l  then  be  determined. 
From t h e  va lue  of t h e  induced cu r ren t  and t h e  va lue  of t h e  c o i l  c u r r e n t ,  f o r c e  
w i l l  be  determined i n  an  equat ion of t h e  form 
F - K 1 2  (3 )  
where I is t h e  cu r ren t  i n  t h e  hammer c o i l ,  K i s  a cons tan t  which inc ludes  
t h e  rise t i m e  of t h i s  c u r r e n t ,  and F i s  t h e  f o r c e  exer ted  on t h e  aluminum 
shee t .  An at tempt  w i l l  a l s o  be made t o  express  t h i s  f o r c e  i n  terms of t h e  
app l i ed  vol tage .  
Equivalent C i r c u i t  Method 
This  method a l s o  necessitates d iv id ing  t h e  s h e e t  i n t o  concen t r i c  
c i r c u l a r  segments and determining t h e  c u r r e n t s  i n  each segment 8s a f u n c t i m  
of t i m e .  The equi l ibr ium equat ion  f o r  t h e  hammer c o i l  then  becomes 
3 di2. + , . . + L di16 
d t  2 -  l6 dt 
e - E IL1 9 dt + L  1 
6. 
Equation 4 is based on approximating the hammer coil by sixteen 
circular conducting paths each carrying the hammer coil current and on 
dividing the aluminum sheet into N concentric circular segments. 
self inductance of each segment is indicated by L and the segment identified 
by a single subscript. The mutual inductances between two segments i s  indicated 
by M and the circuits involved are identified by a double subscript. 
term in equation 4 is the resistance drop in the hammer coil. 
The 
The Ri 
For the individual segments in the metal sheets, a family of equations 
may be set up as shown in equations 5 .  
k=16 n=N 
0 = E ' Mak _dik 
k=1 C 
. . . . . . . . . . . . . . . . . . . . .  
where it is taken in each respective equation that Ma, = La, 
and MW = LN. 
Mbb = Lby . . . 
Equation 5 is a family of differential equations using the inductances 
as constant coefficients. It is believed that the mutual inductances will 
be a function of separation only. Once this relationship is definitely 
determined, the solution of these equations should be obtainable by use of 
the digital computer. 
7. 
W 
Z 
(3 a 
z 
W 
Z 
2 
E 
W 
I- 
W 
0 
I- 
I 
n 
n 
W Z  
W O  
1 .  
8. 
9. 
' S W t i O  
10. 
. 
11. 
l -  
. 
Thickness Force Frequency Current Voltage Power Res i s t ance  
( inches)  (ounces) (Hz) (amperes) ( v o l t s )  (wat t s )  (ohms) 
0.054 0.72 60 4 116 56.0 3.50 
0.080 1.50 60 4 115 59.5 3.72 
0.125 2.45 60 4 114 62.0 3.87 
0.175 3.40 60 4 113 63.0 3.94 
0.375 4.20 60 4 110 56.0 3.50 
0.500 5.00 60 4 108 55.0 3.44 
0.54 0.36 60 3 86 28 
0.375 2.25 60 3 82 30 
0.375 1.00 60 2 54 14 
0.375 0.25 400 .87 202 
0.6375' 0.25 400 .87 202 
Data Obtained as Function of Thickness of Material 
Table I1 
12.  
The va lue  of each inductance term may be determined by 
In equat ion  6 ,  p is the  pe rmeab i l i t y -o f . t he -  surronndlng medium, r 
is  t h e  r a d i u s . o f - t h e  conductor,  A,., max is t h e  v e c t o r  magnetic p o t e n t i a l  a t  
t h e  center '  o f .  t h e  conductor  "a" due t o  t h e  cu r ren t -  in- the: conductor ,Ia, 
and La is  t h e  self- inductance.  
Mab = 2npr *ab max 
I b  
( 7 )  
I n  equat ion  7 ,  t h e  symbols have t h e  same meaning except t h a t  Amax i s  
t h e  v e c t o r  magnetic p o t e n t i a l  i n  conductor ''a'' due t o  t h e  cu r ren t  i n  conductor 
"b", I b  ,and Mab is t h e  mutual inductance between conductor 
The f o r c e  on conductor segment "a" i n  t h e  metal shee t  
found by 
II 11 a and "b". 
may then  be  
dab dac 'an 
where P is  t h e  vec to r  f o r c e  on segment "art, il, i s  t h e  cu r ren t  i n  t h e  
hammer c o i l ,  ia is t h e  cu r ren t  i n  segment "a", and dab i s  t h e  s e p a r a t i o n  
between segments "a" and "b" . 
Equation 8 could then  be used t o  f i n d  t h e  f o r c e  on each segment i n  
t h e  aluminum shee t .  
CONCLUSION 
A s  prev ious ly  stated, it is no t  known a t  t h e  present  time which of 
t h e  two methods w i l l  g ive b e t t e r  o r  more accu ra t e  r e s u l t s .  
t i m e ,  no equipment is a v a i l a b l e  f o r  checking experimental ly  the  t h e o r e t i c a l  
r e s u l t s  obtained.  A much b e t t e r  eva lua t ion  of  t he  t h e o r e t i c a l  r e s u l t s  
would be poss ib l e  if a hammer c o i l  and power supply were a v a i l a b l e .  
A t  t he  p re sen t  
The use of  t he  magnetic vec to r  po ten t ia l ,A,  t o  o b t a i n  va lues  of  
inductance is  the  r e s u l t  of a previous s tudy made a t  Miss i s s ipp i  S t a t e  
Univers i ty  under the  sponsorship of  N.A.S.A., Marshall  Space F l i g h t  
Center .  
l i c a t i o n  on t h i s  sub jec t .  
t h e  Electrical Engineering s t a f f  a t  Miss i s s ipp i  S t a t e  Univers i ty  and the  
t e c h n i c a l  s t a f f  a t  Marshall  Space F l i g h t  Center .  
A t  t he  present  t ime, a paper i s  being prepared f o r  poss ib l e  pub- 
This paper  w i l l  be a j o i n t  e f f o r t  of some of  
The present  p lans  are to cont inue i n v e s t i g a t i o n  o f  both methods 
o u t l i n e d .  I f  both methods should prove success fu l ,  an eva lua t ion  w i l l  
then be made as t o  which is  t h e  more p r a c t i c a l  approach. 
14, 
REFERENCES 
1. Hallen, Eric, Electromagnetic Theory. New York: John Wiley & Sons, 
Inc., 1962. 
2. Kraus, John D., Electromavnetics. New York: McGraw-Hill, 1953. 
3 .  Wier, D. D., Ball, B. J., Catledge, C. G., and Hill, L. J., 
Development of a Valid Mathematical Formula or Group of Formulas 
t o  Establish Within an Accuracy of 5% the Inductance Audio Range 
Resulting in Beryllium Coil Assemblies, Final Report, unpublished, 
Mississippi State University, 1966. 


A study of intense magnetic fields for high energy forming and structural assembly Interim report

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