Wind-tunnel simulation of store jettison with the aid of magnetic artificial gravity

TABLE OF CONTENTS Section Title Page No.

1 . 0 INTRODUCTION 1 2 . Q SUMMARY OF PRESENT STUDY 2

3 . 0. SUMMARY OF SCALING LAMS FOR STORE JETTISON

WITH ARTIFICIAL GRAVITY 5 4 . a MAGNETIC FORCES AND ARTIFICIAL GRAVITY 6 4.1 FORCE FIELD UNIFORMITY 12 5.a IRON-CORE MAGNET SYSTEMS 1 6 6. 0. SINGLE-AXIS, CONSTANT GRADIENT AIR CORE COIL 17 7 . 0 COMBINED VERTICAL AND HORIZONTAL FORCES 22 8 . 0 FORCE FIELD NON-UNIFORMITIES 2 4 9 . Q STABILITY OF SATURATION MAGNETIZATION 2 6 1Q. 0 FIELD AND FORCE ANALYSIS OF ARBITRARY AIR-CORE COIL CONFIGURATIONS 27 ll.Q PRACTICAL COIL CONFIGURATIONS 28 11.1 ANALYSIS AND OPTIMIZATION OF COIL SYSTEMS 30 11.2 GEOMETRIC PARAMETERS AND CONSTRAINTS 31 11.3 EFFECTS OF NON-ZERO WINDING CROSS- SECTIONAL AREA 34 11.4 OPTIMIZATION OF 4-PARAMJLZTERS FOR UNIFORM FORCE 34 PERFORMANCE PARAMETERS OF MAGNET SYSTEM 36 11.5 11.6 COST RELATED PARAMETERS 4 9 V

TABLE OF CONTENTS (Concluded) Section Title Page No.

12.0. GENERAL POWER SUPPLY REQUIREMENTS 12.1 POWER REQUIREMENTS FOR STEADY OPERATION 52 ENERGY REQUIREMENTS FOR STARTUP OR 12.2 INTERMITTENT OPERATION 53 12.3 RESPONSE OF MAGNET SYSTEM TO POWER INPUT VARIATIONS 5 4 13.0 MULTIPLE SIMULTANEOUS STORE JETTISON TESTS 5 7 AJpendices A MAGNETIC FIELDS DUE TO AN ARRAY OF STRAIGHT LINE CURRENT ELEMENTS AND CORRESPONDING FORCES ON A FERROMAGNETIC SPHERE B COMPUTER PROGRAM FOR CALCULATING AND TABULATING MAGNETIC F.IELD AND FIELD AN ARRAY OF GRADIENT COMPONENTS DUE TO STRAIGHT LINE CURRENT ELEMENTS, AND THE CORRESPONDING FORCES ON A FERROMAGNETIC SPHERE 71 C COMPUTER PROGRAM FOR PLOTTING THE MAGNITUDE AND DIRECTION OF THE MAGNETIC FORCE ON A FERROMAGNETIC SPHERE DUE TO AN ARRAY OF STRAIGHT LINE CURRENT ELEMENTS 85 D COMPUTATION OF CURRENT ELEMENT END POINTS 103 E STORE DROP IN A COMPUTER SIMULATIONOF MAGNETIC ARTIFICIAL GRAVITY FACILITY 110 REFERENCES vi

I

LIST OF ILLUSTRATIONS Figure Title Page NO.

la Prototype Parent Aircraft and Store in Earth- Fixed Reference 7 lh Model Parent Aircraft and Store in Wind- Tunnel Reference Frame 7 2 Distribution of Vertical Field Strength for Uniform Vertical Force Along the Vertical Axis, for an Unsaturated Ferromagnetic Sphere of igh 13 3 Distribution of Vertical Field Strength for Uniform Vertical Force Along the Vertical Axis, for a Saturated Ferromagnetic Sphere 15 4 Arrangement of Circular Coils to Provide a Vertical Gradient of the Vertical Field and an Ambient Vertical Field Along the Vertical Axis Arrangement of Coils to Provide Combined Axial and Vertical Forces v - .

Practical Arrangement for a Working Two- Component.Magnetic Artificial Gravity Facility Generalized Dimensions of Practical Air Core Coil Configuration. 3 2 Approximation of the Axial Gradient and the Vertical Gradient Coil Pairs for Estimation of the Mutual Inductance Between Each Pair of Coils 9 Coil Pair Geometry for Calculation of 4 4 Mutual Inductance 10 Critical Properties of a Typical Super- conducting Coil Material 4 7 11 Region of Maxinum Field within Coil 4 8 Conductor vii LIST OF ILLUSTRATIONS (Concluded) Figure No.

12 Two Iron Spheres Immersed in a Saturating Ambient Magnetic Field 5 8 13 Variation of Strength and Direction of Magnetic Force on a Volume Element dv Due to a Saturated Sphere and a Saturating Ambient Field, At a Given Radius. 60 14 Angle Dependent Factor X ( 0 ) and Direc-

1-1

tion J, of Total Magnetic Force on a PFl Magnetized Volume Element, Due to a Sphere Magnetized Parallel to the Element 61 A- 1 Definition of Current Element and Field Point Posi.tions 6 7 D- 1 Generalized Dimensions of Practical Air Core Coil Configuration 1 0 7 D- 2 Straight-Line Approximation to Rounded Corner D-3 Illustration of Nomenclature Used in Subdivision of Windings into Multiple Loops E - 1 General Flow Chart for Store 118 E- 2 Aerodynamic Coefficients in Store Axis E-3 Coordinate Systems and Euler Angles 120 viii LIST OF SYMBOLS A Cross-sectional area of coil winding (Eq. 11.10-13) Principal axes of magnetization of a body Magnetic field strength Coil winding buildup (Fig. 7) B Demagnetizing factors of a body (Eqs. 4.7-9) Electrical energy (Eq. 12.8) Magnetic force (Eq. 4.5) Coil winding packing factor (Eq. 11.35) F P Gravitational acceleration Acceleration due to combined gravitational and magnetic forces acting on store model (Eq. 4.1) Acceleration of store model due to magnetic force gmag (Eqs. 4.2,3) I Electrical current J Electrical current density (amps/unit area) K Radii of gyration of store about x,y,z axes (Eq. 3.2) XIYIZ Conversion factor in magnetic force and moment kt relations (Eq. 4.19, 23) Coefficient of inductive coupling between axial kxx/ z z gradient and vertical gradient coil systems (Eqs.

12.16,17) L Characteristic length of store (Eq. 3.2) Self inductance of coil or coils (Eqs. 11.26-29) L Length of coil mean turn (Eqs. 11.7-9) R Mean length of one side of a square coil Eq. 11.49) Mach number (Eq,. 3.1) M ix LIST OF SYMBOLS (Continued) Magnetic moment of a magnetized body (Eq. 4.5) axial gradient and Mutual inductancebetween 11.30) vertical gradient coil system (Eq.

+ m Magnetization (magnetic moment/unit volume) (Eq. 4.6) Saturation magnetization of ferromagnetic material Msat 4.15) (Eq.

Coil winding mass (Eq. 11.74) m Number of turns of conductor in a coil system N Number of turns of conductor on a single coil n Electrical power (Eq. 12.1) P Magnetic performance parameter (Eqs. 11.18-21.)

Q Dynamic pressure (Eq. 3.3) Electrical resistance (Eqs. 11.22-25) Radius of iron sphere in store model (Eq. le.1) R Reynolds number (Eqs. 3.5,6) Re One half 0.f clear in,side dimension of coil RO assembly (Fig. 71 coil corner radii (Fig. 7 ) r Radial measure in spherical coordinates (Eqs. 13.1,2) r Coil resistance parameters (Eqs. 11.22-25) S Outside length of a square coil (Eq. 11.41) S Laplace transform variable (Eqs. 12.14-25) S T Absolute temperature Self-inductance parameters (Eqs. 11.26-29) T Magnetic torque on a ferromagnetic body (Eq. 4.13) Volume of magnetic material (Eq. 4.6) Volume of coil windings (Eq. 11.72) V X LIST OF SYMBOLS (Continued) V Coil or coilsystemterminalvoltage(Eqs. 11.26-29) Mutual inductance parameter-axial and vertical wxx/zz gradient coil system (Eq. 11.30) W Weight XfYlZ Principal axes of store xtytz Tunnel-fixedcoordinates(Fig.lb) X' ,Y' ,Z' Earth-fixed coordinates (Fig. la) a( 1 Coil winding buildup ratios (Eqs. 11.1,2) 16 ( 1 Coil winding buildup ratios .(Eqs. 11.3-6) Y Geometricparameterusedinformulaforself- inductance of a square coil CEq. 11.41) 8 Ratioofmeanaxialsp.acingofsquarecoilsto near length of one side (Eq. 11.49)

rl /F (Eq. 11.49)

0 Dive angle of parent aircraft (Eqs. 4.2,3) 0 Elevation component in spherical coordinates (Eq. 13.1) Permeability of free space P O Mass density of store (Eqs. 3.3,6,7) P S P Electrical resistivity (Eq. 11.35) 'I Coilorcoilsystemtimeconstant (L/R) (Eqs. 12.14-17) 9 Angleparametersrelatedtocoilsystemgeometry(Fig. 7) 9 Azimuthal component in spherical coordinates (Eq. 13.3) X Magnetic susceptibility (Eqs. 4.7-9) X (Eq. 11.49) VJ Directionofmagneticforceonaferromagneticsphere due to adjacent spheres, relative to direction of applied saturating field (Fig. 13, Eq. 13.13) $ Vector gradient operator (Eq. 4.5) xi L I S T OF SYMBOLS (Concluded) SUBSCRIPTS Principal magnetic axes of ferromagnetic body Magnetic Conditions for model Conditions for prototype Conditions for store Measured in the x,yIz direction Quantities related to axial ambient field coil sys tem (e. g.

IX) Quantities related to vertical ambient field coil system Quantities related to axial gradient field coil sys tem Quantities related to vertical gradient field coil system xii WIND TUNNEL SIMULATION OF STORE JETTISON WITH THE AID OF MAGNETIC ARTIFICIAL GRAVITY by Timothy Stephens and Ronald Adams Massachusetts Institute of Technology Aerophysics Laboratory 1.0 1,NTRODUCTION An important component of the problem of simulation of store jettison by means of small scale drop tests in a wind tunnel arises from the appearance of gravity in the scaling relationships. Generally, for accurate reproduction of the full scale trajectory, the ratio of gravity force to aerody- namic force must be the same for the model as for the full scale store (Reference 1). When other necessary conditions for simulation are imposed, it is generally found that a greater than normal gravity is required for small scale models.

A basic method of providing the required increment of body force corresponding to the "gravity" needed for such tests has been proposed by Covert (Reference 2 ) . This method involves the use of magnet coils surrounding the wind tunnel test sec- tion which interact with ferromagnetic material imbedded in the model of the jettisonable store. In this manner, a magnetic "artificial gravity" field is provided which is approxi- mately uniform throughout the test section. It is feasible to extend this method to provide control of the angulation of the resultant "gravity" field, thereby allowing simulation of diving or climbing attitudes.

Since a "magnetic artificial gravity" facility appears to offer a solution to the difficulties encountered in conventional store jettison test methods, a study was undertaken for the design of such a facility.

2.0 SUMMARY OF PRESENT STUDY I n c l u d e d i n t h i s s t u d y were t h e f o l l o w i n g items: 1. Review of t h , e s c a l i n g laws a p p l i c a b l e t o t h e wind t u n n e l s i m u l a t i o n o f , s t o r e j e t t i s o n , i n terms of a c o n t r o l l a b l e a r t i f i c i a l g r a v i t y field.

2. D e f i n i t i o n of t h e d e s i g n c o n s t r a i n t s i n v o l v e d i n t h e i n t e g r a t i o n of t h e f a c i l i t y w i t h a windtunnel.

3 . Development of a d e t a i l e d p e r f o r m a n c e a n a l y s i s p r o c e - dure. The p e r f o r m a n c ea n a l y s i s i s a p p l i c a b l e t o a i r - core magnetsystemsandprovides a d e t a i l e d d i s t r i b u - t i o n of t h e s t r e n g t h and u n i f o r m i t yo ft h e a r t i f i c i a l g r a v i t yf i e l d .

4 . Establishmentof a p r a c t i c a l magnetconfigurationand analysisofthemagneticperformance of t h ec o n f i g u r a - t i o n .

5 . Extensionoftheperformanceanalysisprocedure t o i n c l u d e the e f f e c ’ t s o f r e s i d u a l n o n u n i f o r m i t i e s i n t h e a r t i f i c i a l g r a v i t y f i e l d o nt y p i c a l store trajectories.

T h i s t h e r e f o r e p r o v i d e s a n e v a l u a t i o n o f t h e f a c i l i t y i n terms oftheenduse.

6. E x p l o r a t i o n of t h e r e l a t i v e merits ofiron-coreand air-core magnetconfigurations.

7. D e t e r m i n a t i o no ff a c t o r si n v o l v e di n the choiceofthe mode of o p e r a t i o no ft h ef a c i l i t y . The a l t e r n a t i v e s c o n s i d e r e d a r e : Continuousoperationof a normal(non- superconducting) c o i l system, i n t e r m i t t e n t o p e r a t i o n o f a normal c o i l system, and c o n t i n u o u so p e r a t i o n of a superconducting c o i l s y s t e m .

f o l l o w i n ga r e some of t h ef a c t o r si n v o l v e d : S i n c e t h e store-droppingprocedure is i n t e r m i t t e n t , i n t e r m i t t e n t o p e r a t i o n o f t h e m a g n e t s may be f e a s i b l e .

(b) I n t e r m i t t e n t o p e r a t i o n of the magnetsreducesthe average power r e q u i r e d t o operate"normal"magnets.

This is b e n e f i c i a l f o r t w o reasons: i) Reduces the c o i l coolingsystemrequirements, and ii) reduces the cost of electrical power.

(c) I n c o n t r a s t t o i n t e r m i t t e n t o p e r a t i o n , it has been d e t e r m i n e d t h a t c o n t i n u o u s o p e r a t i o n o f a normal s y s t e m would t y p i c a l l y r e q u i r e a p p r o x i m a t e l y 1 0 t o 1 0 megawatts f o r a w i n dt u n n e lf a c i l i t yi n t h e 3 ' t o 4 ' s i z er a n g e . T h i s is t y p i c a l l ya t least oneorderofmagnitudehigherthan t h e power r e q u i r e d t o r u n the windtunnel i t s e l f .

(d) I n t e r m i t t a n t o p e r a t i o n r e q u i r e s r e l a t i v e l y s o p h i s t i - c a t e d i n t e r m i t t e n t power supplysystems, which i n c o r p o r a t e means o f s t o r i n g and c o n t r o l l i n g t h e r e l e a s e o f l a r g e q u a n t i t i e s of e l e c t r i c a l e n e r g y .

(e) Continuousoperationofthemagnetsystem is feas- i b l e w i t h t h e use of s u p e r c o n d u c t i n gc o i l s .I n t h i s c a s e , power c o s t s a r e r e l a t i v e l y small and t h e power s u p p l i e s need be ofonlymodestcapacity.

( f ) U s e of superconducting c o i l s involves t h e u s e of more e l a b o r a t e and expensivematerialsand con- s t r u c t i o nt e c h n i q u e s . The engineering of such a system is more complicated,becauseadditional f a c t o r sa r ei n v o l v e d :

i) Thermal design - The c o i l s are immersed i n a

t r i p l e - w a l l e d c o n t a i n e r of complex shapedesigned t o c o n t a i n l i q u i d h e l i u m , l i q u i d n i t r o g e n and t h e r m a li n s u l a t i o n .

ii) S t r u c t u r a ld e s i g n - The magnet system must

s u p p o r t the self-inducedmagnetic stresses and g r a v i t y forces, i n a manner which is compatible w i t h t h e thermaldesign.

iii) Material s e l e c t i o n - The s u p e r c o n d u c t i n gm a t e r i a l

t h a t is s e l e c t e dm u s tb ec a p a b l eo fs t a b l ea n d reliable s u p e r c o n d u c t i n go p e r a t i o n a t t h e maximum d e s i g n m a g n e t i c f i e l d l e v e l s .

(9) A s u p e r c o n d u c t i n g f a c i l i t y w i l l r e q u i r e a supply of l i q u i dh e l i u ma n dl i q u i dn i t r o g e n .T h i s w i l l e i t h e r b ep r o v i d e di nb a t c h e sa n dt h e" b o i l o f f "d i s c a r d e d , o r by a c l o s e d c y c l e u s i n g a r e f r i g e r a t i o n s y s t e m t o conservetheheliumandnitrogen.

8 . It h a sb e e nd e t e r m i n e dt h a tu n d e rc e r t a i nc o n d i t i o n s , it i s f e a s i b l et os i m u l a t em u l t i p l es i m u l t a n e o u ss t o r e l a u n c h e si n a f a c i l i t yo ft h i sk i n d . The main l i m i t a - t i o n s stem from e r r o r s i n t r o d u c e d by t h e m u t u a l i n t e r a c t i o no ft h e stores. Since these magnetic i n t e r a c t i o n f o r c e s v a r y i n v e r s e l y w i t h t h e f o u r t h power o ft h es e p a r a t i o nb e t w e e n t h e centers o f g r a v i t y , it may beassumed t h a t n e g l i g i b l e p e r t u r b a t i o n s t o t h e t r a j e c - t o r i e s are i n c u r r e d i f t h e s e p a r a t i o n is largeenough.

( T y p i c a l l yo nt h eo r d e ro f two s t o r e d i a m e t e r s . ) Under t h ec u r r e n tw o r k ,g e n e r a ls p e c i f i c a t i o n sf o rm a g n e t i c a r t i f i c i a 1 , g r a v i t y f a c i l i t i e s are b e i n gc o n s i d e r e df o rb o t h i n t e r m i t t e n t and continuous operation. I t i s n e c e s s a r yt o a c c u m u l a t ea d d i t i o n a lt e c h n i c a li n f o r m a t i o n ,p a r t i c u l a r l yi n t h e a r e a o f c u r r e n t d e s i g n p r a c t i c e i n v o l v i n g l a r g e r n u l t i - component superconductingmagnetsystems,before it i s p o s s i b l e t o make t h es e l e c t i o nb e t w e e nt h ei n t e r m i t t e n to p e r a t i o n( n o r m a l conductor) case and t h ec o n t i n u o u so p e r a t i o n( s u p e r c o n d u c t o r ) c a s e . I n view of t h ep r e s e n tu n c e r t a i n t ya s t o t h e mode of o p e r a t i o n , it appearspremature a t t h i s t i m e t o a t t e m p t t o p r e p a r e d e t a i l e d cost and time estimates f o r t h e d e s i g n of a f a c i l i t y f o r a medium-sizedwindtunnel.

S i n c et h ee m p h a s i so ft h ep r e s e n t work hasbeen on t h e of t h e b a s i c d e s i g n o f t h e m a g n e t i c a r t i f i c i a l development g r a v i t y f a c i l i t y , t h e d e t a i l e d s t u d y of additionalequipment requirementshasbeendeferred t o a f u t u r e time. Included i n t h i s c a t e g o r ya r es u c h items ascameras,timingunits,strobo- s c o p i c f l a s h u n i t s , etc., n e c e s s a r yt op e r f o r m store j e t t i s o n tests i n a transonic/supersonicwindtunnel. I t is considered t h a t s p e c i f i c a t i o n of such items a t t h i s p o i n t is premature; however, it i s e x p e c t e dt h a tc o n v e n t i o n a l wind t u n n e l s t o r e j e t t i s o n test techniquesusingsuch items may beemployedwith t h e a r t i f i c i a l g r a v i t y f a c i l i t y , a n dn oi m p o r t a n tr e s t r i c t i o n s on t h e i r u s e w i l l be i n c u r r e d b e c a u s e o f any p a r t i c u l a r c h a r a c - teristics o f t h e f a c i l i t y i t s e l f .

3.0 SUMMARY OF S C A L I N G LAWS FOR STORE JETTISON WITH A R T I F I C I A L GRAVITY The f o l l o w i n ga r er e l a t i o n s h i p s among t h e test c o n d i t i o n s o c c u r r i n g i n t h e s i m u l a t i o n o f s t o r e . j e t t i s o n and s i m i l a r p r o b - lems i n t h e wind t u n n e l w i t h t h e "free drop"method(References 1 - 4 ) . The valueofgravity,"gfl",under t h e t e s t c o n d i t i o n s is c o n s i d e r e dt o be a v a r i a b l e t o a l l o w f o r a magneticcomponent of body f o r c e .

1. Model a n dp r o t o t y p ea r eg e o m e t r i c a l l ys i m i l a r .

2 . Mach number i s t h e same f o r modeland prototype.

i . e . , M M = M P 3 . Mass d i s t r i b u t i o n is t h e same f o r model and p r o t o t y p e .

4 . Ratio of a e r o d y n a m i ca c c e l e r a t i o nt o" g r a v i t a t i o n a l " a c c e l e r a t i o n is t h e same f o r model a n dp r o t o t y p e ,a t g e o m e t r i c a l l y s i m i l a r p o i n t s i n t h e t r a j e c t o r y .

5. Induced a n g l e s of a t t a c k are t h e same f o r model and prototype.

i . e . , - = ' M TM - ( 3 . 4 ) g LM T13 (Assuming 2 and 4 h o l i ) 6 . Reynolds Number r a t i o ( f o r a i r ) (Reference 5) : o r , w i t h c o n d i t i o n s 3.1 - 3 . 5 s a t i s f i e d , ( P s ) "- ReM - LM M [ L 1 0 . 2 6 R e L P P ( Ps) TM P 7. Store d e n s i t yr a t i o (from 3 . 4 , 3 . 5 ) L i m i t s onReynolds Number S c a l i n g 3 . 6 i l l u s t r a t e s a f u n d a m e n t a ll i m i t a t i o nt oR e y n o l d s Equation Number s c a l i n gd u e t o t h e p r a c t i c a l limits a v a i l a b l e f o r t h e store d e n s i t y r a t i o ( P , ) ~ / ( P ) and t h e t e m p e r a t u r e r a t i o Tp/TM.

s P' S i n c e t h e Reynolds Number r a t i o is onlyweaklydependent on T /TM, P t h e s t r o n g e s tc o m p e n s a t i o nf o r small scale factor is providedby t h e store d e n s i t y r a t i o . However, it is n o ta l w a y sf e a s i b l e t o i n c r e a s e t h e d e n s i t y of t h e model store t o suchanextent as t o fullycompensate f o r t h e scale factor (L /L ) , andproduce f u l l M P s c a l e Reynolds N u m b e r . I ng e n e r a l therefore, the Reynolds Number r a t i o w i l l be related moststrongly t o t h e scale f a c t o r .

4 .O MAGNETIC FORCES AND A R T I F I C I A L GRAVITY The following is a summary o ft h er e l a t i o n s h i p sg o v e r n i n g t h e scaled g r a v i t y o b t a i n e d by magneticforcesactingon a s t o r e model c o n t a i n i n gf e r r o m a g n e t i cm a t e r i a l .

The t o t a l body force a c t i n g on t h e s t o r e model is t h e sum o ft h eg r a v i t y andmagneticforces.In terms o ft h es c a l e d + g r a v i t y gM, t h i s is: F i g u r e l a . P r o t o t y p eP a r e n t Aircraft and Store i nE a r t h - FixedReferenceFrame.

F i g u r e lb. Model P a r e n tA i r c r a f ta n dS t o r ei n Wind-Tunnel Reference Frame.

I n terms of t h ed i v ea n g l e , 0 , relative t o t h eh o r i z o n t a l ,t h e magnetic gravity components are: (See Figs. l a , b ) cos 0 - g (gmag 1 z = g M (grnag'x = SM s i n 0 -+ The "magneticgravity"component, gmag, is given by -f is: The magneticforcecomponent F mag + where M is t h e t o t a l m a g n e t i c moment ofthemagnetized material imbedded i nt h e store model and i s t h e m a g n e t i c f i e l d g r a d i e n t t e n s o r .

Magnetizationof Model Core(Reference 6 ) The magnetic moment fi i s given by -+

The averagemagnetization m, f o rt ' S o f t "m a g n e t i cm a t e r i a l s

+ s u c h a s i r o n , c a n b e r e l a t e d t o t h e a p p l i e d m a g n e t i c f i e l d B as follows: Where x i s t h em a g n e t i cs u s c e p t i b i l i t yo f t h e m a t e r i a l , t h e f a c t o r s D a , D b , D are t h ed e m a g n e t i z i n gf a c t o r sa s s o c i a t e dw i t h C t h et h r e ep r i n c i p a lm a g n e t i ca x e sa ,b ,a n d c of t h ef e r r o - magneticbody,anddepend upon t h e e x t e r n a l s h a p e of thebody, + and Ba, Bb, Bc are t h e a, b , c components of B.

The d e m a g n e t i z i n gf a c t o r s are r e l a t e d as follows:

Da + Db + Dc = 1 ( 4 . 1 0 )

I f t h e material is m a g n e t i c a l l y s a t u r a t e d , t h e r e l a t i o n s h i p b e t w e e n t h e a p p l i e d f i e l d a n d t h e r e s u l t a n t m a g n e t i z a t i o n is more complicatedandthecomponents are no longeruncoupled.

For t h e s p e c i a l case of e q u a l demagnetizing factors, however, themagnetizationcomponentsreduce to: ( 4 . 1 1 ) where ( 4 . 1 2 ) 3 3 The averagemagnetization m , (andthemagnetic moment M ) a r e t h u s p a r a l l e l t o t h e a p p l i e d f i e l d 6.

Torque-FreeCondition I n t h e p a r t i c u l a r case o f i n t e r e s t , it is a requirement t h a t n oe x t r a n e o u st o r q u e sb ei n t r o d u c e d by t h e m a g n e t i c f i e l d .

(4.13)

Thus f o r T' t o be z e r o , it is n e c e s s a r yt h a t M b ep a r a l l e lt o

3 mag B. F o ru n s a t u r a t e d o r s a t u r a t e d material, t h i sc o n d i t i o n i s s a t i s f i e di ft h et h r e ed e m a g n e t i z i n g factors, D a , Db, and Dc are e q u a l ,a n dt h e material p o s s e s s e s l o w r o t a t i o n a l h y s t e r e s i s (Reference 7 ) .

i . e . , T = 0 i f Da = Db = Dc = 1 / 3 mag T h i sc o n d i t i o n i s s a t i s f i e d by a s p h e r e , o r othershapessuch as a cube or a s h o r tc y l i n d e r . For t h i s case (Da = Db = D c ) , t h e a v e r a g em a g n e t i z a t i o n component i n t h e t u n n e l - f i x e d frame

- - -

m m m are related d i r e c t l y t o t h e f i e l d components B x t x ' y' 2 Y 1 and BZ, w i t h o u t t h e n e c e s s i t y of a t r a n s f o r m a t i o n i n v o l v i n g t h e a t t i d u d e o f t h e i r o n core relative t o t h e t u n n e l - - = 1/3)

i.e. , f o r (Da = Db - D c

i) U n s a t u r a t e d core 3 I B I < Msat

-

- i

mx = 3Bx; m = 3B - m = 3 B Z

Y Y, z

ii) S a t u r a t e d core, 3 1 B I - > Msat

MagneticForce Components The m a g n e t i cf o r c ec o m p o n e n t si nt h er e c t a n g u l a rc o o r d i - n a t e s (x,y,z) , f o re q u a ld e m a g n e t i z i n gf a c t o r s ( D a = Db E Dc) ,

*

are a sf o l l o w s : (4.16) F B B Z B l

= K[&Bxy + & Byy + z y

'mag B B Z F Z - BX " (4.18)

'mag BZz = & * y z + (Br B z z l

The c o e f f i c i e n t K h a s . t h e f o l l o w i n g values depending upon t h el e v e lo fm a g n e t i z a t i o n :( f r o m Eqs. 4 . 1 4 , 1 5 ) m s a t )

i) Unsaturated ( < 7

K = kt 3lg1 1 4 . 1 9 )

*

The f o l l o w i n gn o t a t i o n i s used t o r e p r e s e n t t h e g r a d i e n t com- ponents : B aBX - " " - a etc.

BxY ay Bxx ax ii) S a t u r a t e d

-

(4.20) K = kt .msat Magnetic - .. - F i e l d - .. - -Gradient - I n t e r r e l a t i o n s The g r a d i e n t components of t h e s t e a d y f i e l d B i n free s p a c e (or a i r ) are r e l a t e dt h r o u g h Maxwell's Equations as follows: + B + B Z Z = O (4.21) Bxx YY and - - BZx; B = B B ' = B (4.22) X y y x ; Bxz Y Z ZY ForceUnits I n o r d e r t o use common u n i t s . o f measurement, a conversion f a c t o r "kt1' is r e q u i r e d i n t h e force e q u a t i o n .

Fmag = k t M - V B i . e . , (4.23) 'mag f o r F = pounds M = k i l o g a u s s B = k i l o g a u s s VB = k i l o g a u s s / i n .

V = ( i n ) mag kt = 1 . 1 4 ( i n - l b ) ( i n ) - 3 ( K g a u s s ) - * Example (a) Consider an i r o n s p h e r e of diameter d = 1" having mag a s a t u r a t i o nm a g n e t i z a t i o n msat = 21 k i l o g a u s s ( t y p i c a l f o r i r o n ) , immersed i n a m a g n e t i cf i e l d B = 1 0 k i l o g a u s s ,w i t h a . g r a d i e n t Z B Z z = 0 . 1 k i l o g a u s s / i n . The d e n s i t y P of t h es p h e r e i s mag

0.280 l b / i n . C a l c u l a t et h e t o t a l magnetic force, and t h e force

p e ru n i tw e i g h t ,o nt h es p h e r e .

S i n c e3 [ B [ > Msatf t h es p h e r e is s a t u r a t e d ,a n d FZ = V mag kt Msat Bzz = (T/6) ( 1 ) 3 ( 1 . 4 ) (21) (0.1) = 1 . 2 5 l b Total magneticforce = 1 . 2 5 l b .

- - V Weight of s p h e r e , w Pmag mag mag = ( 0 . 2 8 0 ) ( ( ~ / 6 ) 13) = 0.148 l b .

i.e. ,w mag

- - 1.259 -

- 8.459 Magnetic force/unit weight = Fz/wmag o.148 T h u s ,t h et o t a lf o r c e( m a g n e t i cp l u sg r a v i t y )a c t i n g on t h ei r o n s p h e r e is 9.45 times t h ew e i g h to ft h es p h e r e ,a n dw i t h no addi- t i o n a l mass, t h e s p h e r e w o u l db ea c c e l e r a t e di nt h ez - d i r e c t i o n (downwards) w i t h9 . 4 5g ' s .

For a s a t u r a t e d i r o n s p h e r e , t h e a c c e l e r a t i o n d u e t o mag- n e t i c f o r c e i s mode 1.

gm

o r VB = 0.019(- - 1) (w (4.25)

g mag 4 . 1 FORCE FIELD UNIFORMITY I t i s n o t p o s s i b l e t o s o l v e t h e e q u a t i o n s r e l a t i n g t h e f o r c e s a n ds t e a d ym a g n e t i cf i e l d s t o f i n d a m a g n e t i c f i e l d c o n f i g u r a t i o n whichproduces a uniform force f i e l do v e ra ne x t e n d e dt h r e e dimen- s i o n a lr e g i o n of spaceand also s a t i s f i e s t h e e q u a t i o n s r e l a t i n g t o oneanother.

t h e f i e l d g r a d i e n t s I t i s possible,however, t o produce a magneticforcewhich is uniformalong a l i n e . Two cases are discussedbelow: Uniform Force onanUnsaturatedIronSphere ConsidertheFZcomponentsalongthe z-axis, and assume t h a t 1 2 B are zero.

a t x = 0 , y = 0 , thegradientcomponents B xy' BxZr. yz From E q s . ( 4 . 1 8 , 1 9 ) ( u n s a t u r a t e d case) ( 4 . 2 6 ) as s ume i . e . BZ = kl z 1 / 2 ( 4 . 2 7 ) It is f e a s i b l e t o p r o d u c e a m a g n e t i c f i e l d d i s t r i b u t i o n approximatelyaccording t o E q . ( 4 . 2 7 ) over a l i m i t e dd i s t a n c e , by means ofanaxisymmetriccoilarrangement.

T h i s f i e l dd i s t r i b u t i o n is shown i nF i g u r e 2 . The maximum f i e l d s t r e n g t h i s governed by t h e s a t u r a t i o n o f t h e s p h e r e and t h eu s a b l er a n g e is l i m i t e d by t h e p r a c t i c a l p r o b l e m of producing t h e i n c r e a s i n g l y l a r g e g r a d i e n t i n t h e n e g a t i v ez - d i r e c t i o n .

I I I I I 2 min max Figure 2. D i s t r i b u t i o no fV e r t i c a lF i . e l dS t r e n g t hf o r Uniform yertical Force'Along t h e V e r t i c a l Axis, f o r a n UnsaturatedFerromagneticSphere of High Permeability.

I f it i s assumed t h a t the maximum p r a c t i c a l g r a d i e n t i s twice t h a tc o r r e s p o n d i n g t o t h e p e a k( s a t u r a t i o nl i m i t e d )f i e l d , B-(max),then it can be shown t h a t t h e u s a b l er a n g e i s given z by m 1 s a t - " Z - z max min 8 B z z ( z max' ExamDle The usablerangeof u n s a t u r a t e d o p e r a t i o n f o r a t y p i c a l s i t u a t i o n is c a l c u l a t e dh e r e . If msat = 2 1 k i l o g a u s s ,g r a v i t y scale f a c t o r gM/g = 2 0 , l e n g t h scale f a c t o r L ~ ~ / L ~ = 1 / 2 0 , mass

r a t i o (ws/w ) = 2 , p r o t o t y p e store l e n g t h L = l o o " , then

mag P from Eq. (4.25) = (0.0119) (20-1) (2) BZz = 0.452kilogauss/in = 5.8inches b u t , model l e n g t h Lm = 5.0 inches.

T h e r e f o r e ,i n t h i s example,the maximum usefullengthofuniform- f o r c er e g i o n i s o n l y s l i g h t l y g r e a t e r t h a n the l e n g t h of the store model. Note t h a t i f a l l f a c t o r s r e m a i n t h e same o t h e r t h a nt h eg r a v i t ya n dl e n g t h scale factors, t h e r a t i o of t h e u s e f u l r a n g et os t o r el e n g t h is approximatelyconstant.

Due t o the s e v e r e l i m i t a t i o n s on t h eu s e f u lr a n g eo fu n i f o r m f o r c e i n h e r e n t i n t h i s method ( u n s a t u r a t e d core), t h i sa p p r o a c h was n o tp u r s u e df u r t h e r . The followingapproach,which is based upon a s a t u r a t e d core, was found t o be more s u i t a b l e .

UniformForceon a S a t u r a t e d I r o n S p h e r e ConsidertheFZcomponent a t l o c a t i o n sa l o n gt h ez - a x i sa n d

assume t h a t a t x = 0 , y - - 0 t h e g r a d i e n t components B

xy' BXZ, B are z e r o , and B BZ are a l s o zero.

Y Z Y ' From Eqs. ( 4 . 1 8 , 2 0 ) ( s a t u r a t e d case) W (4.28)

0 BZ = a + alz

( 4 . 2 9 ) m s a t

and BZ > 7

T h u s , t h e v e r t i c a l m a g n e t i c field s t r e n g t h v a r i e s l i n e a r l y w i t h v e r t i c a l d i s t a n c e , and i n t h e r e g i o n of uniformforcemust be greater than t h a t r e q u i r e dt os a t u r a t e the sphere. T h i s f i e l d d i s t r i b u t i o n is shown i n F i g u r e 3 .

TUNNEL Q

*=p+*

B Z /

-

&

F CE I LING RANGE OF UNIFORMFORCE FLOOR W,ND O f WIND TUNNEL

*(Ai I + = TUNNEL HEIGHT 4 TUNNEL

F i g u r e 3 . D i s t r i b u t i o n of Vertical Fi.eld S t r e n g t h f o r Uniform Vertical ForceAlong t h e Vertical Axis, f o r a SaturatedFerromagneticSphere.

Example The f i e l d r e q u i r e m e n t s f o r s a t u r a t e d o p e r a t i o n in a t y p i c a l s i t u a t i o n are c a l c u l a t e d h e r e .

m = 2 1 k i l o g a u s s ; - LM - - 1/20

s a t LP Tunnel test s e c t i o nh e i g h t (Az) = 4 8 " t Usefulrangeof z , (Az), = .75 (Az), = 3 6 " From E q . (4.25) = 0 . 4 5 2 k i l o g a u s s / i n .

B Z Z B~ = 7 . 0 k i l o g a u s s @ z = -12"

= 7 . 0 + ( 3 6 ) ( 0 . 4 5 2 ) = 2 3 . 3 k i l o g a u s s @ z = + 2 4 "

BZ ( f l o o r o f t u n n e l ) 5 .O IRON-CORE MAGNET SYSTEMS I nt h ec o u r s e of t h ep r e l i m i n a r yl a y o u ta n da n a l y s i s of p o s s i b l e magnet c o n f i g u r a t i o n s , a d e c i s i o n was made t o e x c l u d e from f u r t h e rc o n s i d e r a t i o ns y s t e m se m p l o y i n gi r o n o r o t h e r ferromagnetic material i nt h em a g n e t i cc i r c u i t .T h i sd e c i s i o n was based upon t h e f o l l o w i n g f a c t o r s : a ) " M a t e r i a le f f e c t s , " , namely v a r i a t i o n si nm a g n e t i c p r o p e r t i e so ft h ef e r r o m a g n e t i cm a t e r i a l , would make p r e d i c t i o na n dc o n t r o lo ft h em a g n e t i cf o r c ef i e l d e x t r e m e l y d i f f i c u l t , s i n c e t h e material wouldbe p a r t l y o r w h o l l y s a t u r a t e d .

b ) S i n c et h er e q u i r e dm a g n e t i cf i e l d s are w e l l abovethe s a t u r a t i o n l e v e l o f t h e best m a g n e t i cm a t e r i a l s , and t h e e f f e c t i v e a i r gapswouldbelarge,theuseofiron offers o n l ym a r g i n a lr e d u c t i o ni n magnet power ( o r amp- t u r n ) r e q u i r e m e n t s .

c ) Geometrical c o n s i d e r a t i o n sf o rt h i sp a r t i c u l a ra p p l i c a t i o n ( f o r example,requirements of model v i s i b i l i t y andspace test s e c t i o n )p r e v e n tt h ei r o nf r o m f o r t h e windtunnel beingused t o advantage.

The performanceofiron-coremagnetsystemswithlarge a i r gaps is d i f f i c u l t t o analyzewithaccuracy; it i s u s u a l l y n e c e s s a r y t o b u i l d p r e l i m i n a r y small-scale modelsandmeasure i n d e t a i l t h e m a g n e t i c f i e l d c o n f i g u r a t i o n s f o r a range of magnet c u r r e n t s .

S i n c e it i s p o s s i b l e ,o nt h eo t h e rh a n d , t o analyze air-core magnetsystemsin a s t r a i g h t f o r w a r d mannerby u s i n g methodsof l i n e a r s u p e r p o s i t i o n t o o b t a i n v e r y a c c u r a t e estimates of mag- n e t i cp e r f o r m a n c eo fa r b i t r a r yc o i lc o n f i g u r a t i o n s ,t h e r e a p p e a r e dt ob e n o a d v a n t a g e i n b u i l d i n g small scale working modelsof c o i ls y s t e m si nt h ep r e l i m i n a r yd e s i g ne v a l u a t i o np h a s e .

I n f a c t , t h e p r o c e s so fd e s i g no p t i m i z a t i o n may beperformed q u i t e r e a d i l y f o r a i r - c o r es y s t e m su s i n gp u r e l ya n a l y t i c a l methods .

For t h er e a s o n so u t l i n e da b o v e ,n o small scale working c o i l w e r e c o n s t r u c t e d .

systemmodels 6 . 0 SINGLE-AXIS, CONSTANT G R A D I E N T AIR-CORE C O I L SYSTEM The f i e l d c o n f i g u r a t i o n shown i n F i g u r e 3 c a n beproduced approximately by s u p e r p o s i t i o n o f t h e f i e l d c o n t r i b u t i o n s f r o m f o u rc o a x i a lc o i l s . (See Fig. 4 . ) The basicapproach i s a s f o l l o w s : (i) Two i d e n t i c a lc i r c u l a rc o i l s , c o a x i a lw i t ht h ez - a x i s , are arrangedsymmetricallyaboveandbelowthe x-y planeand spaced a d i s t a n c e 22, a p a r t . Each c o i l h a s NZ t u r n s ,a n d both coils are i n series e l e c t r i c a l l y andconnectedsuch t h a t t h e c u r r e n t Iz p a s s e st h r o u g ht h e c o i l s i n t h e same s e n s e so as t o produce a n e t v e r t i c a l f i e l d B Z ( O , O , O ) a t t h ec e n t e ro f symmetry. I f 22, = R Z , suchanarrangement is known as a "Helmholtzpair,"and by v i r t u e of t h ep a r - t i c u l a r c h o i c e o f s p a c i n g , w i l l produce a uniform BZ f i e l d o v e r a large volume o fs p a c es u r r o u n d i n gt h ec e n t e ro f symmetry.

VERTICAL-GRADIENT -

OF-VERT1 CAL-FI EL0 ( B z z 1 COILS VERTICAL-AMBIENT7

- FIELD (6,) COILS

" WINDTUNNEL Figure 4. Arrangement of Circular Coils to Provide a Vertical and an Ambient Gradient of the VerticalField Along the Vertical Field Vertical Axis.

(ii) Added to the Helmholtz pair is a pair of "gradient coils," of radius RZZ, coaxial with the z-axis, and arranged symmetrically above and below the x-y plane.

These coils are spaced a distance 22,, apart. Each c o i l has N turns and both coils are in series electrically, and are connected such that the upper coil produces a negative BZ, ar,d the lower coil produces a positive BZ. The net effect of these two coils is a magnetic field which is z e r o a t t h e c e n t e r ofsymmetry,andwhich has a g r a d i e n t B Z z a l o n g t h e z - a x i s . I f z * * = JZR t h i s g r a d i e n t is c o n s t a n to v e r 2 22' an a p p r e c i a b l ed i s t a n c e .

Thus, t h e Helmholtzcoilsproduce a uniformambientverti- c a l f i e l d , and t h eg r a d i e n tc o i l sp r o d u c e a u n i f o r mg r a d i e n t of t h ev e r t i c a l field. T h i s arrangement i s analyzedquanti- t a t i v e l y below.

Analysis

-

1. C i r c u l a r C o i l s The v e r t i c a l f i e l d BZ on the z-axisdue t o t h e c o i l arrangement shown i n F i g u r e 4 is: where i s t h e p e r m e a b i l i t y of free space.

"'0 For the Helmholtzpair: For t h e g r a d i e n t c o i l s , 2 . $quare C o i l s If the c i r c u l a r coils are r e p l a c e d by s q u a r e coils of dimension 2RZ and 2 R Z z on a side, t h e f i e l d e q u a t i o n s are: I/z No1 1 -1/2

c (-) . [ (l+- u?) ( l + u ? ) - 5

BZ = - R 1 3 I T 2 1

i , j=1

( i = z , z f z z f z z ) z -z z - 2 z - 2 z * * +

*

- * * -

u =

; u2 = *- I u3 - ; u4 - ( 6 . 7 )

RZ R Z RZz RZZ a 2BZ z X

For - - - 0 @ X f Y f Z = 0; - = 0 . 5 5

( 6 . 1 0 ) az RZ

where NZIz = - t o t a l ampturns i n z - c o i l s

a 3 B

Z Z k *

and f o r - - - 0 @ x , y , z = 0; - - - 0 . 9 4

( 6 . 1 2 ) az Rxx BzzRzz PO and (N I ) = 0.81 - (6.13) zz zz/RZz I T where NzzIzz = t o t a la m p t u r n si n zz-coils.

Example : As an example of the magnitudes involved, consider the following case, which is an extension of the example on page 17.

Z , = 30" RZ = 54.6" Square coils: Z , , =. 4 2 ' ' = 44.71' RZz BZ = 13.42 kilogauss (@x3yrz=O) B Z z ( 0 , O t O ) = 0.452 k.gauss/in.

From 6.11.

(12.42)(54.6) NZIZ= (39.4in/m) ( ~ I I x ~ O - ~ ) = 4.31~10' (total ampturnsin Helmholtz coils.

And from Equation 6.13, Bzz~O,O,O)Rzz i.e. NZzIZz= 0 . 8 1 ' o / I I

- - (0.452) (54.6)*

(0.81~4nxlO-~/n) (39.4) = 1 0 . 5 ~ 1 0 ~ (total ampturns in gradient coils.

7.0 COMBINED VERTICAL AND HORIZONTAL FORCES I n o r d e r t o s i m u l a t e s t o r e j e t t i s o n f r o m a d i v i n g climbing aircraft, it i s n e c e s s a r y t o p r o v i d e a magnetic f o r c e component a l o n g t h e t u n n e l a x i s i n a d d i t i o n t o t h e v e r t i c a l component, as d e f i n e d by Equations 2 4 , 26. The a x i a l f o r c e component Fx canbeprovided by a set o f f o u r c o i l s c o a x i a l w i t h t h e x - a x i s and spacedsymmetrical.lyaboutthe c e n t e r of symmetry o ft h ez - c o i l s .T h u s ,a tt h ec e n t e r of symmetry, for a s a t u r a t e di r o ns p h e r e , BZ ktmsat Bzz Bxx(O,O,O) = K x x I x x - ~ 1 K I b u t , rom 4 . 2 1 z zz z and - - K I ) (Kxxlxx 2 z z 2 2 (7.7) = k m Kzlz - - K 1 I ) 2 1 / 2 (KZzxZz 2 xx xx I ) 2 + ( K I ) x x z z From Equations (5.6, 5.71, it i s s e e n t h a t t h e maximum combined m a g n e t i cf o r c e , i s obtained i f t h e d i r e c t i o n o f the Fmag 1 2 2 " .

AXIAL-GRADIENT-OF- A X I AL-eFI ELD COI LS T VERT1 CAL-GRAD1 ENT- OF-VERTICAL-FIELD COI LS

\ I

-AM61 ENT- LS

\\

AX I AL- AMBIENT- FIELD COILS Figure 5. Arrangement of Coils to 'Provide Combined Axial and Vertical Forces.

x- c u r r e n t s i s o p p o s i t e i n s i g n t o t h e d i r e c t i o n o f t h e z- c u r r e n t s .

i . e . , (7.9,lO) T h i s s i t u a t i o n i s shown s c h e m a t i c a l l y i n F i g u r e 5, i n which the x-currentshavebeenchosen t o be n e g a t i v e ( i . e . , t h e x - f i e l d and x - f i e l d g r a d i e n t c o i l i n t h e p o s i t i v e x half-space producenegative Bx a l o n g t h e x - a x i s ) .

I n a d d i t i o n t o e n h a n c i n g t h e s t r e n g t h o f t h e r e s u l t a n t f o r c e , it can be shown t h a t t h eg r a d i e n t component B i s YY reduced, as a r e the c o r r e s p o n d i n gy - d i r e c t e df o r c e sa tl o c a t i o n s o u t s i d e the y=O plane.

8.0 FORCE FIELD N O N U N I F O R M I T I E S Cons3der t h e f o r c e f i e l d d u e t o a u n i f o r m v e r t i c a l g r a d i e n t o f the v e r t i c a l f i e l d superimposedon a uniform ambient vertical f i e l d , a c t i n g on a s a t u r a t e d i r o n s p h e r e .

B A s s u m e t h a t B are n e g l i g i b l e .

Bxz XY Y Z ~ Note t h a t : - 1 1 -

- B = c o n s t ; Bx - - "B * X

Bxx - - 2 z z 2 z z 1 1

B = - - B = c o n s t ; B = -

"B * y 2 z z YY 2 z z Y BZ = B + B Z Z . z .

zo Then or I zo zo The z-force is uniform along the z-axis. The x and y force components are approximately proportianal to the product of the z-force and the x or y displacements. These forces thus appear to be repulsive, away from the z-axis, and increasing in strength with distance from the z-axis. Also, for a given displacement from the z-axis, the x and y forces vary approxi- mately inversely with z displacement, and likewise, they vary approximately with the ambient field level, BZ . Thus, specifications of the desired degree of uniformity 8f the force field will be directly reflected in the required level of the ambient field.

TABLE - Example of Off-Axis Forces, for BZ = 12.42 kilogauss BZz = 0.452 kilogauss/in. Normaliged with Respect to the Force at the Center of Symmetry, FZ 0 0 0 0% 0% 0% 12" 0 0 +lo. 65% 0% +2.4% 12" 12" -12" +18.5% +18.5% +12.2% 12 12" 0 +lo. 17% +4.7% +10.17% 12 12" +12" +7.4% +2.3% +7.4% 12" 12" +24" +5.8% +5.8% +1.4% 9.0 STABILITY OF SATURATION MAGNETIZATION The magnetic force on a saturated body is proportional to the saturation magnetization of the body. Thus, it is of interest to examine the accuracy with which this quantity can be estimated, and also the variation of this quantity under the conditions experienced in the wind tunnel environment.

The most important effect on m is due to changesin sat temperature. If msat(Tref) is the saturation magnetization measured at a reference temperatureT the saturation ref magnetization m (T) for iron at another temperature T is sat related to the Curie temperature Tc approximately as follows: (Reference 7 ) where T, Tc = absolute temperature Tref, k = 0.11 (empirical, for iron) = 1870OR for iron TC The temperature coefficient of magnetization can be written T 1/2 k (7)

a (msat)

" -m 8T sat (Tref) I C am sat / Ta= 48ppm/OF @ T=Tref=53O0R(7OoF.) for iron.

e.g.

msat 10.0 FIELD AND FORCE ANALYSIS OF ARBITRARY AIR-CORE COIL CONFIGURATIONS, In the cases described above, each magnet coil has been represented by a concentrated current element. This leads to great simplification in the initial design and allows promising configurationsto be readily conceived and evaluated approximately. However, in order to evaluate the force field due toa magnet system composed of coils having large winding buildup, over a large region of space, greater detail is required in theanalysis. In this case, it is most convenient to usea digital computer, since the number of computation steps in the evaluation procedure' becomes very large.

In the analysis that has been developed for this applica- tion, the magnet coils have been approximated by a series of straight-line current elements. This choice was made since it appeared at the outset that the most probable configuration, to be compatible with a typical wind tunnel test section, would involve square or rectangular coils.

The total magnetic field and field gradient components at a point in space due to an array of current carrying elements surrounded by a medium of uniform magnetic susceptibility can be evaluated by linear superposition of the effects of each individual current element.

The magnetic force field on a ferromagnetic sphere is calculated from the total field and field gradient components using Equations (4.16-18).

The relations used in the computation of the magnetic field, field gradient components, and the magnetic forces, are outlined in Appendix A.

A computer program has been developed to provide a tabu- lation of the magnetic field components, component derivatives, and magnetic force components for a set of field positions.

"TABLE" and is described in detail in This program is called B.

Appendix has b.een developed as an extension A computer program of of TABLE which provides a qualitative graphical display magnetic force field. This program the distribution of the described in detail in Appendix C.

is called "PLOT" and is 11.0 PRACTICAL COIL CONFIGURATIONS A family of practical coil configurations has been de- veloped from the basic arrangement of Figure 5. An example showing the basic elements and the proportions that are expected to be typical of 'a magnetic artificial gravity facil- ity is outlined in Figure 6.

In general, the wind tunnel test section structure will most probably be of closed jet design. In the case of a porous- wall transonic test.section, designed to operate over a range of Mach Number, the clearance between the inner and outer walls must be larger than required for purely structural reasons, since plenum volume must be provided, and provision must be made for controlled mass removal from the plenum in order to control the test section Mach Number.

AXIAL-AMBIENT- FIELD COILS ENT- OF- AXIAL-GRAD1 AXIAL- FIELD co I LS ~ I WIND TUNNEL VERTICAL-GRADIENT-OF-- VERTICAL-AMBIENT- TEST SECTION VERTICAL- FIELD COILS FIELD COILS ,

R

MODEL 'PORT I I Fi.gure 6. Practical Arrangement for a Working Two-Component Magnetic Artificial Gravity Facility, The acquisition of store jettison data will most probably be by photographic means. This requires at least one large viewing window in the test section wall.

Access to the test section can be provided a short dis- tance from the center of the test section.

The parent model support structure is supported itself by the test section, and means must be provided for remote control of the release of the store models, as in conventional facil- ities.

Operations may call for sharing of the tunnel circuit with other types of facilities, for example conventional foxce and moment balance. This may dictate that the magnetic system be removable without major disassembly. This would in turn require that the test section be removable along with the magnet system, as indicated in Figure 6.

11.1 ANALYSIS AND OPTIMIZATION OF COIL SYSTEMS In order to analyze the coil geometry shown in FLgure 6 in a systematic way, the configuratlon is defined in terns of parameters and constraints. Such parameters and constraints can be categorized as follows: (a) Geometric parameters These apply to a particular configuration, and are dimensionless "shape factors" or length ratios sufficient to define the shape of the configuration.

(b) Geometric constraints These applyto a particular configuration, and define limits to the geometric parameters imposed by mechanical' interference.

(c) Performance parameters These apply to a particular configuration, and define the relevant performance characteristics of the configu- ration in terms of the geometric parameters, material .

parameters, and a characteristic linear dimension of the configuration.

(d) Material parameters These are related to the material and operating temperature chosen for the coil conductor, and may be contained in the performance parameters. .

( 3 ) Cost-related parameters Approximate cost factors can be estimated from the geometric, performance, and material parameters, for some parts of the system.

11.2 GEOMETRIC PARAMETERS AND CONSTRAINTS The coil geometry outlined in Figure 6 is shown in greater detail in Figure 7 in which all relevant dimensions are identified by symbols. The geometric parameters relating these dimensions are defined below.

Geometric Parameters: (Square coils) (a) Angles to winding centroids (in vertical plane through x-axis) @x'@xx'%'@zz B , B , I L

-

(b) Buildup factors ; a 2 - - - (11.1.2)

% -

RO

- wX wXX

- " (11.3,4)

; B , - -

' x B, B, I I (11.5'6) (c) Internal radius ratios

t'

Figure 7. Generalized Dimensions of Practical Air Core Coil Configuration.

(d) Mean-turn l e n g t h p a r a m e t e r s Define as t h e l e n g t h of t h e mean t u r n ( l o c a t e d j o a t t h ew i n d i n gc e n t r o i d ) of a s i n g l e j-coil.

Then I R r z z a o r 6 2 II zzl

" - 8 [ (l+a1eT)

zz)l

- (l- (%- + Ro

(11.9) (e) Winding a r e a p a r a m e t e r s Define A as t h e area of a s i n g l ej - c o i l .

j o Then ( 1 1 . 1 0 ) A xxO -& = a 26 '(11.11) 1 xx RO (11.12) (11.13)

Geometric Constraints (Square - Co$ls)

(Bx + B , ) I tam$xx - tan$x

c1 2 (1++ (Interference of x-coil and xx coil) (11.13a) r z1

crl(BZ+BZZ) + 42 (&

< tan$z - tan$zz

2 (l++ (Interference of z-coil and xx-coil) (11.13b) 11.3 EFFECTS OF NON-ZERO WINDING CROSS-SECTIONAL AREA The magnetic field distribution from a system of coils of non-zero cross sectional area is different from that due to a similar system of concentrated current elements located at the winding cross section centroids of the former system.

The analysis of the fields due to a system of coils of non- zero cross-sectional area may be carried out by dividing each coil cross-section.into zones and representing the current flowing through each zone by a concentrated current element located at the centroid of the zone. With the coil geometry specified in terms of the'parameters shown in Figure 7, it is most convenient to determine the end-points of the current elements using a computer program, since the number of elements can become quite large.A program to accomplish this is out- lined in Appendix D. In this program, the rounded coil corners are approximated in the field computations by forty- five degree bevels.

11.4 OPTIMIZATION OF $ - P m T E R S FOR UNIFORM FORCE FIELD A force-field-uniformity maximization procedure based upon the analysis outlined above proceeds as follows: First, a set of buildup factors ( a ' s and B t s ) are selected compatible with a set of initial values of the @ t s chosen to provide approximately uniform fields and gradients (i.e., $x = 2 9 O ,

@xx = 4 3 " f @ z = 61°, @,, = 47O) based upon single-current-

element square loops. The field property Bx is computed at the center of symmetry of the coil system, and at two points on the x-axis (+Ax) and (-Ax) from the center of symmetry, due to currents flowing only in the x-coils. The angle, @x, is adjusted as follows: @x('ll, B ) x optlmum -f B x ( + A x , O , O , I x ) + B x ( - A ~ , D , O , I x ) = 2 B x ( O , O , O , I x ~ (11.14) Similarly, the other angles $ x x f @ z f $ z z are optimized as follows: @xx(cll, $ 1 xx opt+ Bxx (+Ax,O,O,1: xx ) + B x x ~ - A x f O f O , I ~ ~ = 2 B ~ ~ O f O f O f I ~ (11.15) (11.17) So far, this adjustment procedure has been performed by trial and error (not directly by the computer program) : how- ever, it is evidently a simple matter to implement this step automatically by iteration. The variationsin the values of the optimum @Is are typically small, of the order of two or three degrees, to achieve the maximum force field uniformity.

3 5 11.5 PERFORMANCE F,7WWETEqS OF MAGNET SXSTEM The performance of the magnet system can be def;ined in terms of parameters relatfng the magnetic and electrical properties of the system. Of interest are the following items: (a) MagnetiC performanc'e paranlete'rs ( Q . ) 3- These parameters relate the magnetic field properties at the center of symmetry to the ampere-turns in each coil subsystem and are computed from the geometric parameters, as defined below: define : Bx(O,O,O) Bxx(O,O,O)Ro Qx= I ( N ~ I ~ / R ~ 1 1 (11.18) Qxx= ( N ~ ~ I ~ ~ / R ~ ) 1 (11.19) (b) Electric.al performance parameters.

of These parameters relate the electrical properties the coil.system to the geometric parameters, and the mater-ial parameters. Included in this category are the electrical resistance and self-inductance of each coil subsystem, and the mutual inductance of coil subsystems which are magnetically coupled.

ci) Resistance parameters . .

(-5

RORX s = - (11.23) X N : Sz=- RoRz (11.25) N ; where Rx, Rxx * etc. are the resistances of the x, xx, etc.

coil subsystem respectively, as described below.

"I " " l "."" 1

(ii) Self-inductance parameters. (Txi) Lx T = . - , X Tz=- LZ T Z Z " = Z Z (11,281 (11.29) RON: RON;Z where Lx, Lxxr etc. are the self-inductances of the x, x x , etc. coil subsystems respectively, as described below.

(iii) Mutual-inductance parameter (wxx/zz 1 Mxx/zz (11.30) Wxx/zz=R N . N 0 xx z z is the mutual inductance between the xx coil where Mxx/zz subsystem and the zz coil subsystem, as described below.

Due to the summetry of this particular coil configuration, the xx and zz subsystems are the only pair with non-zero net magnetic coupling.

Magnetic Performance Parameters (i) Approximate Values These may be estimated from the mean-turn geometry, with acceptable accuracy for the purposes of preliminary analysis, from Equations 6.11,13, as follows: (11.31) Bxx (O,O,O)Ro 0. 81p0 I I (11.32) Qxx

"-

Nxxlxx/Ro (11.33) (11.34) (ii) Accurate Values These may be calculated using the field analysis computer program described in detail in Appendix B, in conjunction with the current element location program detailed in Appendix D.

Resistance Parameters The resistance parameter S of a single coil " 'I of jo constant current density, of the configuration defined in Figure 7 , can be estimated as follows: (11.35) where R = resistance of jo coil $0 = average resistivity of j-coil conductor material

-

X = length of mean turn of single coil (or controid jo filament) fi; = total cross sectional area of single j-coil jo

( % I j o = average packing area factor of conductor (ratio of

conductor cross section to total winding cross section) n = number of turns in singlej coil jo The resistivity p depends upon the conductor material j and the operating temperature, the packing factor F depends P 3 8 upon the construction used, and the number of tuxns depends upon the .desired hpedance level. The remaznfng factors are seen to be the reciprocal of the windlhg area parameter, and t b mean length parameter. (See Eqs. 11.7-13).

For the particular configuration, N = 2njo, and the .total j resistance parameters are: (11.36) (11.37)

-

1 p x x R X X O ) ( Ro ) - RoRxx ) 11.38)

( - -

sxx - 2 = 2 'r Ro

P = Axxo Nxx - RoRz 1 ! z R Z RO2 - (11.39) s Z - - = 2 2 (Fp)zz (-1 Ro ( " 1 A, N Z (11.40) Self-Inductance Parameters The self-inductance of a symmetric pair of coils com- prising one of the coil subsystems is found from the self- inductance of an isolated coil and the mutual inductance between the two coils.

(i) Self-inductance of individual coils. (Reference 8 , The self-inductance L of an isolated square coil is . jo given by the formula: GnZcrohenrles/inch turnL) where: s = outside length of one side of j-coil (inches) j S - - j t .fV ~j t = radial thickness of j-coil j V = axial thickness of j-coil j n = number of turns in j-coil j For the particular coil configuration: (11.42) (11.43)

"

sZ 2 " 2 8 2 2 " (11.44)

- 2 [ ( l + ~ r ~ + ~ ) tan$ + 1

RO z z C11.451 (11.46) (11.47) (11.48) Ciil Mutual inductance between two identical parallel, square, coaxial coils. (Reference 8) The mutual inductance M between the two identical j/j

(11 . 49)

hicrohenries/inch turn ) where: R = mean length of one side of j-coil j n = number o f turns in j-coil j t i i = ratio of mean axial spacing of coils to J mean length I t R " j 'I; = For the particular coil configuration: R

' %

-

(11.50)

= - 2 c 1 +

R o - " RO

2 5

- = 2(1 + tan+. (11.51)

RO (11.52) (iii) Total self-inductance parameters The total self-inductance L of each coil pair is given by: j (11. 531 L = 21Ljo"j,j] j N for the particular coil confiigurat$.on, n - -8.

j " 2 (11.54) (11.55) (11.56) - Lzz 1 LZZO MZz/zz (11.57) = Z I (

TZz - 2 1 3

1"

RoNzz Ro (Nzz/2) Ro (NZz/2) Mutual Inductance Parameter (Gradient Coils) (Ref. 8 ) The axial gradient coil pair is magnetically coupled with the vertical gradient coil pair due to the way in which the terminals of these coils are necessarily connected. As a result, changes in current through one pair of coils will result in net induced voltages in the other pair. This effect must be considered since it influences the specifications of the power supplies required to regulate the currents in the two systems.

The particular gradient coil configuration can be approx- imated by the arrangement shown in Figure 8.below.

Vertical Gradient CoilPair Axial Gradient CoilPair Figure 8 . Approximation of the Axial Gradient and the Yertical Gradient Coil Pairs for Estimation of the 14utual Inductance Between Each Pair of Coils.

From the symmetry of the figure, further simplification of the analysis can be made by observing that the mutual inductance, MAC' between coils A and C is equal to the 4 3

mutual inductances M m , MBCr

MBD Therefore, it i s necessary only to f b d an expressi’an f o r M 9s. shown Ln FZguxe 9.

AC Figure 9. Coil Pair Geometry for Calculation of Mutual Inductance.

MAC = M 1 5 + M37 - M17 - M35

(11.58) R 2 Z4=0,00508n n !?, [21nA+(1+-)LnB+C-E] (11.59) 1 2 1 (Ref. 8 ) where :

R1 R 2 &2 D

(11.60) A = 1 (l+r)+J(l+-) 2+ (+ 23 1 &1 1 I I I -

B = -

(11.61) 4 4 "1 "1 (11.63) where :

a2 = -1 , J D 2 = (d-R1) L + (c+RZ) '

a3 = -1 , J D 3 = (d+Rl) L + (c-2,)

a4 = +l , JD4= (d+R,) L + (c+R2) L'

"

R 2 = R (l+a +-)tan$zz 0 1 2 c = R (l+T) "1 tan+xx a d = R (l+" +-) 0 1 2 a 4 z 2 1 R 2 Mxx/ - )=0.00508(1+2-) c (2?nA.+(1+ )!?nB.+p - E . ) wxx/zz " ( R N N

3 5 3 1 1

0 xx 22 j=l (11.64) (microhenries/inch turn ) Current Density The current density is of interest, particularly with superconductors.

The Overall current density J in each d n d i n g jo 4 5 (11.66) where Bj=Bx; ROBxx; BZ; RoBzz (11.68) Peak Magnetic Field Strength Inside Coil Conductor (Ref.9) In the case of superconducting coil material, the current density is constrained by the properties of the particular material and the local magnetic field strength. The properties of typical superconducting materials are shown in Figure 10 below.

4 6 U c C I

0 .5 I .o 1.5 2.0

C r i t icol Field Figure 10. Critical Properti.es of Typical Superconducting Coil Material.

For discussion purposes, the properties of a typical superconducting material can be approximated by the relation:

J<Jcrit - + dJcrit

(11.69) - JrefdBcri,tCBref-B) If superconducting materialis used, it is necessary to find the maximum value of B within the coil winding (Bmax)cond and the corresponding current density J in order ( B ' B m a x ) cond. ' 4 7 t o assume that the c r i t i c a l current d e n s i t y i s n o t exceeded.

The c o n s t r a i n t c a n he s t a t e d a l t e r n a t e l y as fallow-: . L w .

c r i t J c r i t Jref+aBcrit ‘Bmax) cond For t h e c o i l geometryunderconsideration, it hasbeen found that t h e l o c a t i o n of the maximum f i e l d p o i n t is i n t h e p o s i t i v e - z ,n e g a t i v e - xq u a d r a n t of t h e s y s t e m , i n t h e y = 0 plane. T h i s c o n d i t i o n i s shown i nF i g u r e 11.

I Region o f Maximum

F i e l d I

Figure 11. Region of M a x i m u m F i e l d Wi.thin C o i l Conductor.

4 8 ° his f i . e l d s t r e n g t h c a n be e s t i m a t e d w i t h s a t i s 4 a c t o r y accuracy C 5%1. by assunring th3t the windsngs are. e q u i 3 a l e n t t o i n f s n i t e f l ' l a m e n t s l o c a t e d a t t E e w h d i x g c e n t r o i d s , a n d c a r r y r n gt h ec o r r e s p o n d h gc u r r e n t s . The effect of other q u a d r a n t s is ignored.

T I E n e tm a g n e t i cf i e l d [BmaxIcond a t the p o i n t of i n t e r e s t is:

uo J2 a +b +2ab cosOzz

- " (11.71) CBmax)cond 2II .

where r r r X xx z J J z z j a 2 f 3 2 z z R 2 0 b = ( 1 r Z Z

r = Ro-2 9%

i X a r = R~ ( 1 + 2 ) 1 ( t a n @ z - t a n @ x x ) + T "lBXX Z 11.6 COST-RELATED PARAMETERS Several parameters which may be e x p e c t e d t o be c l o s e l y r e l a t e d t o the c o s t of the f a c i l i t y c a n b e e v a l u a t e d f r o m t h e geometric,material,andperformanceparametersof the c o i l system. Included among these are the following: Car Winding volume psxahe'ters: (Y ./R. 1 J 0..

T h e tots1 volume "/w. ox c ~ i 1 windbgs i s

c11.72) j-1 Ro 3 (11.73) (j=x,xx,z,zz) where (F) = density of conductor in j-coils.

(F ) .= winding packing factor - j-coils.

P J (c) Current-length product The cost of superconducting material is often quoted in terms of (price)/ (unit current x unit length) (eg., dollars/kiloamp. foot). Thus, it is desirable to derive a parameter in terms of this quantity.

The total length R of conductor in the j-coil J tot pair is (11.75) The current-length product 8 I . for the j-coils h o t j ' can he ralqtedto' tke pexfQ;rrPqnce pqrameter Q j as follows: C11.761 where 12.0 GENERAL POKER SUPPLY FU3QUIREMENTS Electrical power supplies are required to provide control- led currents through each of the four coil pairs.

The fields CBxo, B Z o ) may be controlled independently of the gradients CBxxo, BZzo) by use of four separate power sup- plies, one for each o f , B x , Bxx, B Z , and BZZ. Considerations that are involved are as follows: (a) Control of force amplitude while maintaining saturation.

The operati.on of the system requires that the spher- ical iron core of the store model be saturated at all points within the useful volume of the tunnel test section. This in turn places a lower 1imit.onthe net magnetic field strength at any point. With the combined axial and ver- tical ambient field strength held at a level adequate to saturate the sphere, the magnetic force may then be varied by adjustment of the axial and vertical field gra- dient coil currents.

(b) Minimization of force field nonuniformities,.

It has been demonstrated that the nonuniformities in the force field due to the gradient fields can be re- duced hy increqsea $ n the uniform ambient fi\eld leyel [Sect. 8l. T h u s , .the.xe i s an advantqge Ih opexat%g the -2ent f2eld at the maxlSrmm leyel, Independent of the force required. Alternately. stated, t F i e maxihum instal- led power available for the ambient field CBx, BzI coi.1 should be used at all times (in a ratio consistent with the requiked force field inclination, of course).

The alternative would be to have only two independent powe supplies, wLth the Bxx coils connected in series with the Bx coils to one power supply, and the BZz and BZ connected in series to the other power supply. Adjustment in the force level can be accomplished to some degree by adjustment of the currents, variation of the size of the spherical core relative to the store model, or choice of magnetic material having dif- ferent saturation magnetization level.

12.1 POWER REQUIREMENTS FOR STEADY OPERATION The d.c. power required forsteady'operation of the system can be found from the performance parameters as follows: Total d.c. power P = CP'. (dc) (12.1) total 3 P = I.>R j (d.c. 1 1 1 sx sxx - or P =R B ( - ) ; R )'(-I x(d.c.) o xo Pxx(d.c.)-RoCBxxo o Qx QXX 2 (12.3) (12.4) Discussion Several assumptions may be justified concerning the probable values of the parametersin.the expression 12.7 above.

(i 1 CBx 'kax = const.

0 (depend upon saturation magneti-

) zation of spherical core in

= const.storemodelj 'Bz 'max Cii) Ro(B=o m a = const* (gradients (and forces) inversely (Bxo)max proportional to scale factor, due to aerodynamic scaling re- quirements) Ro CBzzo'max = const.

i B Z 0 ) S .

Note that the other. factors (2) are related to shape, material, and constructi.on Qj method.

On the basis of the assumptions and observations above, note that the maximum d.c. power required for the coil system scales as the first power of the linear dimension.

12.2 ENERGY REQUIREMENTS FOR STARTUP OR INTERMITTENT OPERATION The magnet system not only dissipates energy due to ohmic losses, but also stores energy by way of the magnetic fields to which are produced. Thus, the electrical energy required produce change in the field level will be a function of the time taken tQ effect the desired change, in addition ts the difference 2n stored enexgy between the two 1 e Y e . l . s .

Cdi.ss?pationl In particular, I (12.10) Note that the resistance parameters (S.) are included under the integral sign. Recall that S is proportional to the resis- j tivity p of the conductor'material which in general may be j affected by changes in temperature of the conductor, due in turn to the dissipation itself. The exception to this of course is the case of superconduzting material, in which the dissipati term may be negligible.

12.3 RESPCNSE OF KAGNET SYSTEM TO POWER INPUT'VAXIATIONS If it is assumed that the resistance parameters are con- stant, the four coil systems respond to voltage inputs as follows below: 5 4 (a) Four independent power supplies Vx, Vu, V , , V,, (i) Non-superconducting coils (12.13) (12.14) (12.15) (12.16) where : - - Mxx/zz - wxx/zz - kxx/zz /EXXLZ z JTxxTz z s = Laplace transform operator.

(ii) Superconducting coils (R = 0) j (12.17) (12.18) (12.19) (12.20)

(b) Two independent power supplies (Vx + Vxx), (Vz + Vzz)

with x and xx coils in series (I - ) and z and zz coils in X - IXX series (Iz = IZZ) (i) Non-superconducting coils.

(12.21) -k 1 xx/ z z

+ -

. ’ I 4 (Rx +Rxx) (RZ+RZz 1 D* SI (VX+VXX) ( s ) (12.22)

Lx+Lxx . Lz+Lzz

- - where ‘ I = x* Rx+Rxx ’ ‘Iz* RZ+RZz (ii) Superconducting coils (R = 0) j 13.0 MULTIPLE SIMULTANEOUS STORE JETTISON TESTS When two or more stores are jettisoned simultaneously in the magnetic artificial gravity facility, perturbations to the trajectories will be experienced due to mutual magnetic attraction or repulsion between the store models. Consequently, it is of interest to estimate the magnitude of these interaction forces and their variation with the spacing and relative orien- tation of the store models in the ambient magnetic fields.

Analvsis a) "Two-body problem. It The problem can be illustrated by analyzing the forces between two spherical iron bodies immersed in a saturating magnetic field. The situation is shown in Figure 12.

Ambient Field Figure 12. Two Iron Spheres Immersed in a Saturating Ambient Magnetic Field.

Sphere 1 is magnetically saturated, and the magnetization is assumed to be parallel to the ambient field Bx. (i.e. The perturbation to the magnetization of sphere 1, due to sphere 2, is neglected.)

The external fi.eld due to sphere 1, in spherical coordi- nates, is (13.1) m R 3 sat(1) 1 sing = I 3 3 - (13.2)

B e ) (1)

r3 (13.3) (13.4) (13.7) or, for z = 0, - = dFx -kt (msat)( 1) (m R 3 dV sat)(a) '[cos9 (2-5sin29)] (13.8) r" The magnitude of the total force1 SI on an element is: (13.10) i.e.

I r 4 (13.11) The direction $ of the total elemental force is

lml

(13.13) The angle-dependent terms K ldFl ( 0 ) and $ are plotted in ldFl Figures 13 and 14.

Figure 13. Variation of Strength and Direction of Magnetic Force on a Volume Element dV Due to a Saturated Sphere and a Saturating Ambient Field, at a Given RadiuS'.

6Q

I I I I I I I

I Saturatinq I I 180" , 150" 120" 90" 60"

30" -

LL U 0 "

- 30"

- 60"

-30"

- 120"

- 150"

- I 80"

0 10" 20" 30" 40" 50" 60" 70" 80" 90" Figure 14. Angle Dependent Factor K ( 8 ) and Di'rection I )

P I

1 - 1 Of

Magnetic Force on a Magnetized Volume Element, Due to a Sphere Magnetized Parallel to the Element.

The force components on a volume element( d V , ) thus vary inversely with the fourth power of the distance from the cen of sphere 1, are functions of the angle to the volume element from the center of sphere 1, relative to the ambient field, and are proportional to the product of the saturation magnetization of sphere 1 and that of the volume element.

The total forces B Fe exerted on sphere 2 due to the r ' gradients of the field from sphere 1 are found by integrating the elemental forces over the volume of sphere 2.

i.e., F (13.14) r ( 2 ) (13.15) F = o (13.16) @ (2) The integrations indicated in 13.14,15, for the general case, are not evaluated here in their exact form. Instead, a con- servative approximate. form is presented which illustrates the situation in a relatively simple way. The following assumptions are made: (i) The direction of the total force corresponds to that of the elemental force calculated for 8 = e12.

(ii) The magnitude of the total force is estimated by integration of a volume force which varies with (l/s4) over the volume of sphere 2, where s is a distance in a rectangular coordinate system.

The results are as follows: and, angle of F12 = I ) , , = tan -1 Itane121 4-5sin2ol2 1 I-n (13.18) $12 2-5sin2 0 12 The total force is thus identical to the elemental force at the center of sphere 2 multiplied by the volume of sphere 2, and weighted by a function of (R2/rli) which approaches unity as (R2/rl2) approaches zero, as is expected.

If it isassuned'thatthe two spheres. are identical, of radius R, and saturation magnetization rn and their centers sat' separated by a distance r, i.e. , R1 = R2 = R m = m = m sat2 sat satl r = r then : (13.19) R

Observe that (r) = 0.5 (contact)

r n a x R Values of the weighting function of in Eq. 13.19 are tabulated below.

4.33 x 2.52 x 1.58 x 1.01 x The acceleration (as)& of the store model due to the perturbing force estimated above is as follows: (13.20) (13.21) (13.22) Example : Let (3) = 0.5 wS m = 21 kilogauss sat = 0.283 lb/in3 Pmag 6 4 I r LetR = 0.5" ; (E) = 6 ; e = 0 , then = 2.9 g i.e., For two store models, each containing 1" diameter iron spheres, with a separation 3" between centers of gravity, the mutual perturbing acceleration is of the order of 1.5 to 3 g's, depending upon the orientation of the line between the centers relative to the applied field.

For greater separation, the perturbation is less. For example, by increasing the separation from (r/R) = 6 to (r/R) = 8 , the perturbing acceleration diminishes from 2.9 g's to 0.88 g.

For this example, the probable gravity scale factor would be of the order of 20. Thus, the perturbation acceleration is a significant fraction of the "normal" acceleration at short separation distances, but diminishes rapidly with separation.

Note the effect of scale: the ratio of perturbation ac- celeration to "normal" acceleration is independent of scale to a first approximation, due to the reduced normal acceleration required with increasing model size.

(b) Multi-body problem.

The total force on a single sphere is the vector Sum of the forces produced by all surrounding spheres as calculated for the two-body problem. That is, the forces add linearly.

APPENDIX A MAGNETIC FIELDSDUE TO AN ARRAY OF. STRAIGHT LINE CURRENT ELEMENTS AND CORRESPONDING FORCES ON A FERROMAGNETIC SPHERE The following is an outline of the relations required to compute the magnetic field strength and gradient components and the corresponding forces on a ferromagnetic sphere, produced by an array of straight-line current elements, and by extension, by an assembly of square or rectangular magnet coils. These relations apply for regions of constant permeability, and there- fore do not allow for ferromagnetic material in the vicinity, as would be the case if iron cores were to be used in associat with the coil assembly.

The relationships are based upon the principle of linear superposition. That is, the field strength at a particular point in space is found by adding the effects of individual current elements.

Field and Gradient Components from a Single Current Element The field strength at a point in space due to a straight- line current element is found by application of the Biot-Savart In terms of rectangular coordinates and the guhntities illustrated in Figure A - 1 , the field components are as follows: Y X Figure A-1. D e f i n i t i o n of C u r r e n t Element a n dF i e l d Point P o s i t i o n s .

n

- + - + H n = l + P 1 P 2 let (A- 81 PlP2 (A-9) ( P 1+P,2 1 10-21 if let (A-10) Gn Hn P1P2 (A-11) (A-12) (A-14) by letting Y1 = U Y2 = V Y 3 = W I. (A-15) Equations (A-2,4,6) can be reduced by index notation to the form The gradient components are thus, (.A- 1 7 1 6 8 (A-18)

let p1 + p 2 = P

1p 2 = Q expanding ( A - 1 8 ) , (A- 19 ) (A- 20) Expanding (A- 2 0 ) ( A - 2 1 ) The total field properties at a point (xofyofzo) due to N cur- rent elements are thus given by the following: N (A-22)

(Bxi) - - 1 (IGYi), i=1,2,3

n=l Magnetic Forces The magnetic force components can be written in index notation as follows: j=1,2,3 (A-24) where K = 3 I B I if 3 1 B l < msat

APPENDIX B COMPUTER PROGRAM FOR CALCULATING AND TABULATING MAGNETIC FIELD AND FIELD GRADIENT COMPONENTS DUE TO AN ARRAY OF STRAIGHT LINE CURRENT ELEMENTS, AND THE CORRESPONDING FORCES ON A FERROMAGNETIC SPHERE The following is a brief description of a computer program "TABLE" written in.Fortran IV language which is used to compute and tabulate the magnetic field properties and the magnetic force components on a ferromagnetic sphere; 'using the relations outlined in Appendix A.

Tabulating Program ("TABLE") TABLE was developed to provide a tabulation of the magnetic field components, component derivatives, and magnetic force components for a set of field positions.

The input data for this program includes the end points of the current elements as defined in Figure A-1, the current flowing in each element, and the saturation magnetization of the sphere material. Variable names associated with the input data can be found in the "Input Variable List" of the Program Listing on page 72.

Sample input and output data are shown on pages 81 and 82.

G

t ARTIFICIAL GFUlVIlY - 'I'&&E

C A C C O E T L C A L C U L A T E T t - E M A G h E T I C f O H C E S ON A E O D Y I N A F I E L D PRODUCED C B Y C O I L S CdNSISTiNC; DF S T R A I G h T L I N E C U R R E N T E L E H E K T S . 0 T H E F O R C E S C AND F I E L D C H A R A C T E R I S T I C S A R E T A B U L A T E 0 A T C O R R E S P O N O I N G P L I I N T S I N C T H R E E O I H E A S I G N A L SPACE.

C C U R R E N T E L E M E N T S A R E C O U N T E D C O U h T E K - C L C C K W I S E A @ C L T T H E C C R R E S P G N D I N G C C O U R D I h A T E C I K E C T I U N S . A L L C U R R E N T S C W € P O S I T I V E C O U N T E R - C L U C K h I S E o C C C I N P U T V A R I P E L E L f S J C C V A R I A P L E h A M E O € f 1 f i I T I C R C C A D E M A G N E T i Z I N G C t N S T b k T F O R T H E P G C E L o C A M 5 S A T U K A T I C h PAGNET I Z A T I 8 N F O R ThE MGDEL.

C X K T M A G N E T I C F C R C E C C h S T A h T .

M A G N E T I C P E R M E A B I L I I Y G F F R E E SPACE.

4 ' XMU w c I h M THE hUMBEW C f C A T A SETS.

C I NPOPT I N P U T C P T I C h a C C O N F G OESCRIPTIdN OF A R T I F I C I A L G R A V I l Y C C N F I G U R A T I C N m C I M NUMBER OF I N C R E M E h T S I N T b E X - D I R E C T X J N o C JC NUMBER OF I h C R E M E h ' T S I N T H € Y - O I R E C T I C N m C K M NUMBER UF AIhCKEMEi'4TS I N T t - E Z - D I R i i C T I O N .

C LP T O T A L h U M t 3 t P OF C U R R E N T E L E M E N T S C c x ' D E L T A ' X.

C O Y ' D t L T A ' Y .

C DZ ' I I E L T A ' Z.

C X ( 1 ) X C G d R D I N A T E O F S T b R T I K GP L I h T FCR I N C R E H E N T I N G

c Y ( l 1 Y C G G R D I N A T E OF S T A R T I K G P O I N T FOR I N C R E M E N T I N G O

C Z ( 1 ) 2 CCJGRDINATE O f S T 4 R T I h G P C I N T FOR I N C R E M E N T I N G O C X i r Y 1.21 C O O R D I N A T E S OF T H E ENC P O I h T S OF T H E S T R A I G H T L I N E C X Z t Y 2 . 2 2 ( 1 U K R f l L T f L € M E h T S M A K I k G UP T H E C O I L S .

C CUR M A G h I T L D E GF T H E C U R R E h T I h AHFERES,+ F H C M 1 T U 2 .

C C U R T C U R R € N T F L O W I N G IN A LOCP OF FdUH C U R R k N T E L E M E N T S .

C C b K X 1 THE C L R H E N T I P T H E + X T G T A L F I t l C C C I L o C C U R X 2 I h T H E + X T H E T d T A L C b R R E N T G R A C I E h TC O I L .

C C U R X 3 T H E T O T A L C L H R E k T I h T H E - X F I E L C C C I L o c C U R X 4 T H E T O T A L C U R R E N T I h T H E - X G R A D I E h T C U I L .

C C U R 2 1 THE T t J T A L C L R K E h T I h T P E + Z F I k L C C O I L .

c C U R 2 2 T h E T G T A L C C R h E N T I h THE +Z G K A C I E h l C C I L .

C CURZ3 T H E T U T A L C . U i ( K t N T I N T H E -Z F I E L D C O I L .

C CUR24 T H E T G T A L C L R H E N T I h T h E - 2 G R A C I E h T C O I L .

C 1 5 3 F U H M A T ( 2 4 X , 1 8 A 4 ) l t . 5 f C P M A T ( F 1 O . O ) 106 F C E M A T ( 6 f 8 . 4 ) C I N P U T T H E M A G N E T I Z A T I C N C C ~ S T A N T F J K THE G E C P E T K Y C f T H E BCOYI THE C M A G N I T U C E J f T H E S A T U R A T I G N M A G N E T I Z A T I O N F U R T H E P A T E R I A L , T H E C M A G N t T I C F O R C E C C N S T A h T p T H E P E R M E A B ' I - L I T Y G F T H k MECIUM9 THE NUMHER C D A T A S E T S , A N G THE I h P U T CPTION.

C I N P U P T = I C O R K E S P Z N O S T 3 I b i P U T I N G THE CURRENT I N EACH E L E M E h T o I N P O P T C - 2 C O R H t S F d N D S T O I N P U T I I \ ( G 1b.E C U R R E N T I N EACH L C C P O f F O U R ELEMENTS.

H E A C ( S r l l 2 ) C A I A M S , X K T , X C ~ , I ~ C , i h Q C P T XMF=XHL/(4.*3.1416) I F ( I N P L ) P T ~ E O Y ~ ) GG TO 1 7 1 C I N P O P T = 2 C INPUT THE O E S C R I P T I U N ' O f T h € C C N % i G % g A T l O N (LE. ' 7 2 C H A R A C T E R S ) .

R E A C ( 5 9 1 5 5 ) ( C O N f G ( I C ) r I C = 1 , 1 8 ) C I N P U T T h E M A X I M U M N U M e E R OF X INCREMEkTSp T H E M A X I M C H N U M B E R Of Y 2 C I N C R E ' H E k T S , THE MAXIMUM NUMBER C F 2 I k C R E H E N T S , T k E NUMBER OF C U R R E N T C E L E M E N T S 9 D E L T A X 9 V 9 AND 2 1 P N D T H E C C R k E R P O I N T O F T k € P L O T ( F A X I H U M C V A L U E O F X , H I h l I M U M V A L U E OF Z I AhrD Y F C S I T I C N ) .

C N i l T E T h A T C K IS N E G A T I V E AND CZ IS. P C S I T I V E .

REAO(5,lGC) I M ~ J M , K M ~ C M ~ D X ~ O Y ~ O Z I X o , Y ( 1 ) , Z ( I ) C I N P U T T H E E N D P C l I N T 5 CF T H E C U R R E N T ELEMENTS.

R E A C ( - 5 9 1 6 6 ) ( X ~ ( ~ I ) T Y ~ ( ~ I ) , Z ~ ( N I ) ~ X Z ( N I ) T Y ~ ( N ~ ) T ~ ~ ( N I ) ~ N ~ = ~ , L M ) N T M = L W 4 C I N P O P T = l 1 7 1 DU 2OC I N f = l r I N M I F ( I N P D P T o E Q . 2 ) GD TO A74 C I N P U T THE C E S C R I P T I C h GF T k E C C N F I G U R A T I O N (LE. 7 i C k A K A C T E R S ) .

R E A D ( 5 , 1 5 5 ) ( C O N F G ( I C ) , I C = 1 , 1 8 ) * C I N P U T T h E M A X I M U M NUMBEH OF X I h C R E M E h T S , THE M A X I Y U M IJUKBEA C f Y C I N C H E H E N T S , THE M A X I M U M NUMZER C F Z I h C H E M E h T S , T k E N U M B E R OF C U R R E N l C E L E M E N T S 9 D E L T A X 9 Y 9 AND 2 9 Atdo THE CCHPiER F C I N T CF T H E P L U T ( M A X I M U M C V A L U E UF X t N I N I M U M V A L U E OF 2, AND V P O S I T I C N ) .

C N O T E T H A T C X IS N E G A T I V E ANC C Z IS P C S I T I V E .

R E A O ( 5 , l C i ) I M , J H T K C T L ~ , D ~ , D Y , O Z , X ( ~ ~ ~ Y ( ~ ) T ~ ( ~ ) C I N P U T T H E EN0 P C I I N T S 3F T H E C U R R E N T E L E M k N J S P I W THE C U R R E h T FLOPrIhiG C C A L C U L A T E 1 ; AND I T S D E R I V I T X V E S I N T H E X t Y t Z DIRECTIUNS.

DC 23C M1=1,3 O G A = H ~ * ( K M + K D R ) * D S ( M l ) ~ G B = H S ~ ( S M ~ ( C P ( M l ) * O D o J + O Q ( M l J * ( R M + R O R J j DG(Ml)-(CGA-DGB)/(RM+(RM+ROR))*~2 230 CCrUTIhUE GO TC 3 2 G = ( ( R S I I ( R M - R O R ) J / ( R M * ~ ~ R ~ ~ X ~ ) 00 2 4 0 P2=1,3 O G A = ( R S ~ ( D P ( M 2 ) - D D ( ~ 2 ) ) + D S ( M 2 ) * ( R t " H C P ) ) * R M * H # R * * 2 D G B = H S * ( R M - K D P ) * ( R M ~ 2 . r H X H * O C ( M 2 ) + O P o ~ ~ X R ~ * 2 ) D G ( M 2 ) = ( D G A - U G B ) / ( H M * ~ X ~ * * 2 ) * . * 2 240 C C h T INUE C C A L C U A L T E T H E F I E L D C C h T H I B O T I C N S O f E A C H C U R R € N T E L E M E N T .

3 GGX=OG( 1 ) D G V = C G ( 2 1 U D G Z = C G ( 3 00 CURP=XMP~CUR(L)~iOeG0./3S.37 CURP=Xh!P*CUR(L)*G*lOUCUo a X l = C L R M * U BY1=CURH*V BZl=C(lRM*ck C CALCULATE THE G R A C I E N T CChTRIBUTICNS OF EACH CURRENT ELEMENT.

B X X l = C C H P * U * D G X BXY l = C U R P * ( G * ( E-F ) +U*DGY 1 B X Z l = C U R Q * ( G ~ ~ ( D - C ) + U * C G Z ~ 6 Y Y l=GURP*V*OGY BYZl=CUHP*(G*(A-B)+V4CGZI B Z L l = C U H P * D G Z * W C SUM T H E INOIVUAL C U N T R I B U T I C h S T O ' T H E F I E L D AND G R A D I E h T TC GET THE C TOTAL F I E L C AloU GRAULENTS.

B X = B X t C 3 X 1 B Y = B Y + B Y l B Z = B Z + B Z 1 B r x = e x x + B x x l B X Y = B X Y + B X Y l B % z = E ! x L + e x z 1 B Y Y - e w t m Y l

BY z= @ Y L+ BY 2 1

e z z = E z m x l

21r3 COhTINLiE C C A L C U L A T E AhD TEST T H E M A G N i T J Z A l I C h C F THE B O D Y FOR S A T U R A T I O N .

XDK=XK T /UA R 8 = ( e X * ~ 2 + ~ Y ~ ~ 2 t ~ Z r ~ ~ ) ~ : * C o ~ AM=( L / C A I * K B I F ( A I U . - P M S ) l i l 9 i 0 t l l C C A L C U L A T E THE FORCES PRODUCEC CN T H k EOCY.

IO F X = X O K * ( H X = B X X + B Y * B X Y + B Z - B X Z 1 F z = x D ~ x t ~ x * ~ x z + B Y ~ ~ Y z + ~ z * ~ z z ~ F Y = X D K * ( U X * B X Y + B Y * B Y Y + B Z * B Y Z ) GC T O 12 C C A L C U L A T E THE C C M P d N E h T S C F ThE M A G N E T I Z A T I C N AT S A T U R A T I C k .

11 BIJ,Y=( l ? Y / R B ) * A M S U 19 B F X = ( @ X / R B ) * A M S E W = ( E L / K @ ) * A M S C C A L C U L A T E THE F i l R C E S PKODUCEC CN TH€ B C D Y o F X = X K T * ( H M X * B X X + B M Y * B X Y + ~ H Z T B X Z ) FZ=XKT*( B N X * B X Z + @ ~ Y ~ @ Y Z + e C L r B Z Z ) F Y = X K T * ( a H X * ~ X Y + B M Y * B Y Y + B M Z ~ B Y Z ) 12 C O N T I N U E

io I P = I P t l

C IF AT ThE BOTTOM OF ThE PAGE9 S K I PT C NEXT PAGE AhC WRITE C T H E kEADING FOR THE N U M E R I C A L TABULATIGh.

I F ( 1P-rti.l) 50951951 5 1 ClRi f E (6 9 1 1 8 ) W R I T E ( 6 9 1 5 9 ) ( C G N F G ( I O C ) r I O C = l r l 8 ) W P I T E ( 6 , 1 1 5 ) M R I T E ( b 9 1 5 1 ) C U R X 1 9 C U R X 2 W R i T E ( S r l 5 2 ) C U R X 3 9 C U R X Q W R I T t ( b r 1 5 3 ) C U R Z l v C U R Z 2 W A I T E ( 6 9 i 5 4 ) C U R Z 3 , C U R Z 4 W R I T E ( 6 r l l 4 1 I

TABLE Sample I n p u t - Current Element End P o i n t s and

Currents.

X Y L R X TNCHFS ZAUSS/l’J.

43.9 -18.9 21.973 -2.3RQ 1207.1 -?.9Ar!

-45.1 -25.R 2 1 . 2 1 2 1 2 5 1 - 4 -37.5 - 3 . 3 6 7 - 1 2R. 6 29.155 1192.5 -1.71 0 lPQ6.S -195.4 - 3 9 . I 1 9 . 6 7 9

- 9.904

1 Ct-3.9 -242.1 -45.7 17.227 912.2 - 2 6 9 . 7 -51.? 1 5 . Q 3 3 - 1 . O P l 1 4 . 4 6 h 976.5 -2A1.2 -56.7 -3.769 13.247 -3. 5 3 0 749.4 -25n.3 “ 5 1 . 9 - 7 7 n . 7 -66.5 12.1 5 0 -1.214 6 8 1 - 7 -253.6 -71.9 11.211 -2.845 623.7

574.6 -237.8 - 7 4 . 5 1 ” . ? 6 6 - 2.402

‘3.541 533.7 -299.2 -79.“ -2.11R

5rr.7 - I e4.1 - 9 1 . ’ ’ 3 . - 2 2 - 1.582

479.1 -159.3 - 8 3 . 5 9.491 -1.140 9 . “ 4 ? -?. 6 9 7 4 5 1 . h - 1 3 2 . 3 -R5.7 - 9 7 . 4 -1.254 4 3 5 . 2 -106.4 7 . 5 5 0 9.194 423.1 - R 7 . 9 -9E.9 7 . 3 4 ?

-55.9 - A 9 . 9 7 . ? 8 2 415. cI r . 6 3 ~ - 9 1 . 4 6 . 0 7 2 1.571 4 1 r . 4 -31.5 400,” -7.7 - 0 2 . 9 4 . 7 1 9 1.515

-y. 0

1 5 . h 6 . 5 0 1 1 . 9 6 2 4 1 9 . 9 415.6 3A.4 - 9 9 . 5 5.514 7 . 4 1 4 423.3 6“.7 - 7 7 . 7 6 . 4 7 9 2.916 434.” R2.6 -89. 5 5.49r. 3.309 44A.‘ 1 0 4 . 7 -8e.6 5 . 5 3 7 3.936 5.536 465.4 125.“ -87.5 4.301 4A0.7 4 . I R 7 4. 968 1 4 5 . 4 -A6.1 0 . 907 5.41 n 517.7 lS5.1 -94.9 -A?.? 7 . 7 7 5 5.995 542.6 183.7 p n r . . q - A l . 6 7 . 6 3 4 6 . h n l 5 1 % 5 6 7 0 . q 215.9 - 7 9 . 9 R . I R S 7.233 7 . !In9 h F . 1 2 2 7 . 7 - 7 7 . 9 9 . 5 k 7 777.? 234.7 -7h.‘ 9.337 9.561 - 7 9 . 0 797.5 235.1 1 7 . 1 7 4 9 - 2 3 ] 9.977 Q6A. 1 226.1 - 7 1 . 9 1 1 . 1 7 4 l “ . 4 5 R 0 4 6 . Q 234.R - 5 7 . 5 1 2 . 3 4 5 - 6 7 . G 19.69‘ 1q. 975 1 p 3 2 . 2 167.6 1 1 1 7 . 2 -65.7 1 5 . 1 3 Q 1 1 . 2 1 4 111.R - 6 3 . 11.267 36.5 TABLE Sample Output.

APPENDIX C COMPUTER PROGRAM FOR PLOTTING THE MAGNITUDE AND DIRECTION OF THE MAGNETIC FORCE ON A FERROMAGNETIC SPHERE DUE TO AN ARRAY OF STRAIGHT LINE CURRENT ELEMENTS The following is a brief description of a computer program "PLOT" written in Fortran IV language which is used to plot the magnitude and direction of magnetic force on a ferromagnetic sphere using the relations outlined in Appendix A .

Plotting Program ("PLOT") PLOT was developed to provide a qualitative graphical display of the distribution of the magnetic force field on a ferromagnetic sphere due to an array of straight-line current elements.

Each display is in two parts, consisting of: a) A plot of the magnitude of the magnetic force, and b) A plot of the angle of the magnetic forces for an array of field points in a y = const. plane.

The magnetic forces to be plotted are calculated as described in Appendix B, for TABLE. A symbol representing the force range in which the magnetic force lies is then matched to the force. This symbol is placed in the Jth and Kth spacial position (corresponding to Jx increments and Kz increments from a starting point) of the force display, and the entire display array is then printed. In this way, the magnetic force at discrete points in the xz plane is represented by a symbol in the output field. Detailed examples of the use of this technique can be found in Ref. 10.

Since the range of force magnitude is generally large, and a multitude of symbols would be required to represent a 8 3 typical rangeof constant force increments, a logarithmic increment has been used instead. Thus, each force range represents a constant percentage of the value of the upper (or lower) limit of the range.

By this method, the force magnitude is first reduced to an exponent of a base number, i.e., F = An logF n = logA Then, the exponent is reduced to a positive integer n + N N is an integer greater than zero F (K,J) = FSymbol (N) ( C - 3 ) where FSymbol (N) is the Nth symbol in the array r- - -.--- ?: ng force magnitudes.

Reducing n to a positive integer can be accomplished in several ways. In PLOT, n was reduced by adding 1 and truncating in the case of positive exponents or adding 1 plus the magnitude of the largest negative exponent to be considered in the case of negative exponents.

for n > 0, N = truncated (n+l) ((2-4 1 for n < 0, N = truncated (n + 1 + la] 1 ( C - 5 ) a = magnitude of largest negative exponent (i.e., F = 0 when n < -a) Thus, the force represented by FSymbol(N) has a magnitude n+ E N + ~ - E

between A and A , where E is small. (See the sample

output for PLOT, page 102for an example.)

The array representing angle magnitudes is similarly 8 4 constructed. In this case, a constant angle increment is used so that the symbol subscript is calculated by dividing the angle by the increment, adding one, and truncating, i.e., F -1 x

8 = tan -

FZ

e

M = truncated (m + 1)

Again, the 8 array is printed so that a symbol is associated with the angle at each point inthe xz plane.

C A H T I F ICAL GRAV I T Y - P L O T C C C A CODE TU C A L C U L A T E T h E F O R C E GN A M A G N € T I C @ O D Y IN A F I E L D PRODUCED C BY CclILS C O N S I S T I N G OF S T R A I G H T L I N E CURRENT E L E H E h T S o T h E M A G N E T I C C F O R C E ANC AhiGLE ARE F I E L O P L C T T t i D I N 7H€ X i ! PLANE.

C CURREhT ELEMENTS AKE C C U N T E O C C U N T E K - C L C C K W I S E A B C U T T h E C O K R E S P J N D I N G C C O O R D I N A T E D L R E C T I U N S . A L L C U R R k N T S A R E P U S I T I V E C G U N T E R - C L O C K W I S E .

C C C I N P U T V A R I A e L E L I S T C C V A R I A B L E N A M E D € F I N I T I L N C C F M T 1 FORMAT STATEMENTS FCR IIvPUT AND CUTPUT OPERATIONS C FMT2 R E Q U I R I N G V A R I A B L E FORMAT.

I 8 I 1 8 I 8 8 8 8 @ I 8 8

s f C 1 3

I 8 8 1 II I 8 8 8 *I 8 1 C FM 14 C P h L G T k E B A S E F G R L d G A R I l H M I C FORCE PLOTS.

C ANMX A B S C L U T E VALUE CF T t - E L A R G ' E S T N E G A T I V E PCWER C OF ANLG T O B E L S E D I N FORCE PLO'ISv P L L S 1 .

c L B h M X

THE NUHBER O F P L O T T I N G S Y M B O L S R E P R E S E N T I N G FORCES C OF P A G N I T U O E G R E A T E R T H A N 10 C S Y H R l P L d T T I N G S Y M B O L S FCR FORCE PL3TS.

nn II I1 n II I1 II I1 n nn C SYMO2 r O I O I ( 8 n* n 40 Hn - m I# C SYMt33 I1 w n #I N I 8 r #I n #I nn C SY M B 4 u #I # 8 W m n 91 N t l n 81 C O C T v A S T E R C C T h E T A A N G L E I h C R t M E N T FOR bNGLE F L G T e C hAhG NUMeER OF SYHBClLS F C R PNGLE PLOT.

C A S Y M B 1 P L O T T I N G S Y M B C L S F C P A k G L E FCOTC,.

?IN n IO II *I 88 II H H nn C ASY M 82 C O A D € M A G N E T I Z I h G C C k S T P N T FOR THE POCEL.

S A ~ U R A T I U N MAGNE'FI-ZATIO~V F c R T ~ E MCCEL.

C AMS

c X K T MAGNET IC F O R C EC O h S T A h T

C XPU M A G N E T I C P E P H E A B I L I T Y CF FREE S F A C E .

C I N M NUHt3fR CF CbTA SETS F O R T P I S RUho l N P O P T lNPCiT O P T I C R o C CONF G A U E S C K I P T I G N i l F T b E MAGNET CONFIGURATZONo C IH NUHBER OF INCREHEhTS I R T k € X - D I R E C T I C h .

C NUMBER OF INCREMENTS i N T F E Y - O I R E C T i C h o C JM K t NUYeEH OF IhCREMEIUTS IN T k E Z - O I K E C f I C N o C TOTAL NUMBEP O f CURREhT ELEFENTZ C L H C X X.

C ' D E L T A ' C C Y ' D E L T A ' Y o

C c z ' D E L T A ' Z o

X ( 1 ) X CGUHDINATE OF S T b R T I t v G PCIEiT FOR INCHEMENTING C Y ( L J Y C J J R O I N A T E OF S T A R T I N G P O I N T FUR INCKEMENTIhG.

C C Z ( 1 J L COORDINATE d FS T A R T I N GP O I N T FOR INCREHENTING.

C X 1 ? Y l ? Z l COORDINATES OF THE ENC PUIhTS O F THE STRAIGHT L I N E CURRENT ELEMEEJTS MAKING UP THE COILS.

C X 2 , Y 2 r Z 2 MAGNITUCE G F THE CtiRRENT I N AMPERES,+ FiiCM 1 TO 20 C CUR CURRENT F L C k I h G I N CI LOOP GF FGbR CURRENT ELEHEhTS.

CURT TkE TOTAL CLRRENT I N T h E + X F I E L D C C I L .

C C U R X l THE T t i T A L CURRENT I h THE +X G R A C i E k T C O I L .

C C U R X 2 THE T O T A L CURRENT I h THE - X F I E L C C C I L o C CURX3 THE TOTAL CURRENT Ih; T H E - X G R A t I E l u T C O I L .

C CURX4 THE T O T A L CURRENT I h THE +Z F I E l C CCIL.

C C U R Z l THE TOTAL CURRENT I h THE +Z G R A C i E h f C O I L .

C CUR 2 2 CUR23 THE TOTAL CURRENT I h THE -2 F I E L C COIL.

C THE TOTAL CURRENT I h THE - 2 G R A C I E h T COIL.

C C U R 2 4 C 1 1 5 F a R M A T ( / / / ) 116 F U ~ M B T ( ~ X , ~ H I N C H € S ~ ~ ~ X ~ ~ H G A ~ S S ~ ~ ~ ~ ~ ~ ~ G A L ~ ~ / I N ~ ~ ~ 1 1 8 F C R M A l ( l H 1 ) 119 F L ~ R M A T ( ~ A ~ T L A ~ ~ ~ A ~ ~ ~ A ~ ) 1 2 7 F O F M A T ( 2 4 X t l d h l N P U T L A T A ) 1 2 8 F O R M A T ( Z X t b I i X l ( I N ) t 2 X t b H Y l ( I ~ ) , 2 X t 6 ~ Z l ( I ~ ) , ~ X ~ ~ H X ~ ( I N ) T ~ X T ~ H Y ~ ( I N ) 1 , 2 X , 6 H Z Z ( I N ) , l X T l O H C U R H E ~ T ( A ) ) 1 2 9 F C R M A T ( i X t 6 F 8 . 4 T F l D . 0 ) 1 3 1 F O R M A T ( i F l i l o d ) 132 F U R M A T ( 3 X p 6 I i S Y M B O L ~ S X t 2 4 H F J R C E R A h G E ( L B F / C U o I N 0 1 1 133 F 3 R M A T ( S X t 1 A Z , 7 X t F 9 o 3 r 2 X , 2 H T C l l X 1 F 9 . 3 ) 13s F O R M A T ( 3 X t 2 9 H P L C T OF FX I N X - 2 PLLNE A T Y = , f k o l , l X , 3 H I N o ) 1 3 6 F C R M 4 T ( 3 X , 2 7 H h 3 K I Z U N T A L C G G K D A N A T E I S X , F S ~ l , l X t 3 H I N o t 4 H T O t F 5 . 1 l t l X t 3 H I h o ) 137 F ~ ~ M A T ( 3 X t 2 5 H V E R T I C A L C L j G R O I k A T E IS 2 t F S o l r l X t 3 H I h o t 4 l + T O e F 5 . 1 ~ 1 l X t 3 H I N o ) O3 03 138 F Q R M A T ( 3 X t 2 9 H P L C T CF FY I h X - Z F L P h t AT Y = t F 6 0 l t l X , 3 l - I h o ) 1 4 2 F O R M A T ( l X t 1 2 3 H o 0 0 0 0 e 0 a 0 0 0 0 0 0 0 e 0 0 0 0 0 0 1 ~ ' . . . . Q . . . . 0 . . . . . . . 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 20 1 143 F O P M A T ( Z F B a 4 t I 5 ) 144 f C P M b T ( Z A 2 ) 1 4 5 F O R M A T ( 3 X t 2 5 H I O E N O T E S k 6 T l i R A T I C h L I N f i ) 150 F G P M P T ( 4 F l O o O ) 1 5 1 F d K M A T ( 3 X t 3 2 H T U T A L C U R K E h T I N + X F . I E L ( 3 C G I L = , F l C o O t 6 h A M P S o t 3 X . 3 5 l H T O T A L C U R R E N T I N + X G H A C I E h T C O I L = , F L U o O t 6 H A M F S o ) 1 5 2 F O P " A T ( 3 X t 3 L H T O T A L C U R K E h l I N - X F I E L D C O I L = t F l i r e C , b H A M P S e t 3 X t 3 5 1 H T C T A L C U R R E N T I N - X G R A C I E t V T C O I L = t F L G o 0 , 6 H A M P S , ) 153 F 3 R M A T ( 3 X t 3 L H T O T A L C U R R E h T I N + Z F I E L D C C I L = t F i 0 . 6 , 6 H A M P S O r 3 X t 3 5 . l H T O T P L C U R R E N T I N +Z G N A C I E N T C O I L = r F 1 0 0 0 t 6 H P H P S o ) 1 5 4 F O R M A T ( 3 X t 3 2 H T O T A L C U R R E h T I N -2 F I E L D C C I L = t F l O o G t 6 F PMPSor3X.35 l H T C T A L C U R R E N T I N -2 G K A C I E N T C O I L = , F l O o O t 6 H A M P Z o ) 1 5 5 F G H M A T ( 1 8 6 4 ) 159 F C f i N A T ( 2 4 X t l S A 4 ) 160 F U H M A T ( F b o 3 t 1 5 ) IC2 F d R M A T ( 3 X t 6 H Z Y M B C L t 9 X , l € ! H b h G L E K A h G E ( D E G o ) 1 163 F C G M A T ( j X , 4 7 H P L O T OF T O T A L M A G N E T I C F C R C E I N X-2 P L A N E A T Y = , F b . i , l l X ~ 3 h I N . I 1 6 4 F O E M b T ( 3 X , 4 7 h P L C T OF P A G h E T I C F G H C € ANGLE IN 8-Z PLANE A T Y=,F6ol, 1 1 x 9 3 H I N . 1 165 f C P M E T ( f l O m 0 ) 166 F O F M A T ( 6 F B . 4 ) C INPUT T H E F d R M A T C d D E S FOP V A R I A U L E F C R F A T T E D S T A T E M E N T S .

C F M T l S P E C I F I E S T k E S I Z E dF Tt'E F C R C E F I E L D PLOT. F O R E X A M P L E , C F M T l = ( i H o , 3 l A 2 1 CORRESPi2hO.S T O A P L O T T I N G R E G I C h Xz-30. TO X=+30.

C h I T V P N ilYCREMENT ( I r E o D X ) C F 2s I N C h E S o C F C T 2 IS T H E I N P U T C D D E FCR P L O T T I h G S Y M B C L S R E P K E E E N T I N G F O R C E S L E S S C ThAIU 1 , AND F M T 3 I S F G R S Y M B C L S R E P R E S E h T I N G F O R C E S G R E A T E R THAN 10 i EXAMPLES A P E f " T 2 = ( L Z A Z ) AIUU F M T 3 = ( 3 t A 2 / 2 2 A 2 ) FOR 12 AND 5.f S Y M B O L S C R E S P E C T I V I t L Y e FMT4 I S F O P P k G L E S Y C E C L S , F P T 4 = ( h A h G M A 2 1 .

R E A 0 ( 5 , 1 1 9 1 F M T i , F M T L , F H l , 3 , F M T 4 oo C I N P U T T H E E A S E FOR F'dRCE P L C l T i h GA N T ' A B S O L 6 1 €V A L L E OF THE L A R G E S T 9 C N E G A T I V E POWER 3F T F € B A S E , P L U S 1, P N C THE NUMBER O F SYMBGLS FCR C GREATER THAN CNE.

R E A L ( 5, i4.3) AFILG, bNMX,LEh!!X L A h M X = A h M X - l r C I N P U T L E T T E R V A L U E S FOR T F E F C R C E ANC ANGLE M A G N I T U D E S FUR P L O T T I N G .

C S Y N B l AND S Y M 8 2 A R E S Y M B O L S FCR P O S I Y I V E FOFcCE M A G N I T U D E S LESS THAN C bkE G R E A T E R T H A N 1 R E S P E C T I V E L Y .

R€AD(S,FMT2) ( S Y M B l ( N l ) , N l = l , L A ~ C X ) R E A G ( 5 r F A T 3 ) ( S Y M B 2 ( N 2 ) , N ~ = l , L B ~ P ~ ) C SYMt33 ANC S Y Y B 4 A R E S Y F B C L S FCR h E G b T I V E FORCE M A G N I T U D E S L E S S T H A N C ANL; G R E A T E R T H A N ONk R E S P E C T I V E L Y .

REPC(5,FHTL) ( S Y M B 3 ( N 3 ) , N 3 = 1 , L A N W X ) R E A C ( S , F M T 3 ) ( S Y M 8 4 ( h 4 ) , ~ 4 = 1 , L E h ~ X ) H E A C ( 5 , 1 4 4 1 D O T , A S T E R

c INPUT T ~ E MAGNITUCE ' d ~ T ~ E A ~ G L E INC~KEMENT ~ N ' G T ~ E NUMBER UF ANGLE

C I N C R E M E N T S e R k b C ( 5 , l O O ) U T H E T A p N A h G P C P S Y Y t 3 1 A N C A S Y M B 2 A K E S Y M B O L S F C R P O S I T I V E ANC N E C A T I U E A N G L E C M A G N I T U D E S R E S P E C T I V E L Y .

K € A O ( S , F M T 4 ) ( A ~ Y W B l ( N T H l ) , N T H l = l , N A h G M ) R E b 0 ( 5 r F M T Q J ( A Z Y M 8 2 ( N T H ~ ) r N T H 2 = 1 t ~ A ~ G M ) C I N P U T T H E M A G N E T i Z A T I C h C C N S T A h T FUR THE G E C M E T R Y CF TFE BODY, THE C MAGNITUDE O F THE S A T U R A T I O N M A G N E T I Z A T I G N FOR T H E P A T E R I A L 9 T H E C Y A G N I T I C FCRCE C C h S T A h T t T k t P E R M E A B I L I T Y OF FREE SPACE9 THE NUMB€R Of C DATA SETS, AND THE I h P L T C P T I C N o C I N P O P T = l CORkESPCNOS Ti) I N P U T I N G THE CURRENT I N EACH ELEMEAT. I N P O P T C =2 CQRKESPUNOS T O I l I P U T I h G T H E CURREFiT I N EACH L C L P C F F C U R ELEkENTS.

R E A C l ( 5 ~ 1 1 2 ) D A , A M S , X ~ T , X ~ U , I N M , " ' ~ ' ~ P C ~ T XHP=XHU/(4.*3.14161 C CALCULATE THE L i l G A R I T H M U f THE B P S € FCR FORCE Plc31S.

X L G = A L C G ( A h L G ) A M l = l o U S * A M S C CALCULATE THE MAGNITUCES C F F O R C E RAACES FOR FORCE PLCTSo Di3 43C J i ) = l r L A N M X J L = J O - ( L A h M X + l ) J2= J l + 1 F k I N l ( J d ) = A N L G * * J l F M A X l ( J O ) = A N L G * * J Z 4 C U C C A T I A L E DO 900 K D = l , l B N F % K I = K C l - l K 2 = K C F M I N Z ( K O I = A N L G * * K l FPAXL (KO)=ANLG**K2 S O 0 ' C C N T I N U E f)r I N 1 (1 )=00 F C I N 2 ( 1 ) = 1 o d C CALCULATE THE ANGLE R A N G E S F C R ANGLE F L C T I N G o AhGl(1 )=do A N G 2 ( 1) =DTHETA hAFiGh=NANGM-l OC 504 hANG=1 ,NANC-k A N G l ( N A N G + l ) =ANG1( NAhG) +GTtETA A N G Z ( N A i q G + i ) = A N G 2 ( N A h G ) + O T k E T A 504 C C l v T I h U E I f ( I N P U P T o i i Q o 1 ) G C T C 1 7 1

c INPOPT=Z

C I N P U T T H E C E S C H I P T I C k CF T h E C C N F I G U P h T I d N (LE. 7 2 CHARACTERS)* R E A C ( 5 ~ 1 5 5 1 ( C L l N f G ( I C J r i C = 1 , 1 8 J C INPUT T k E WAXIMUH NUMBER GF % INCREMENTS, TI..€ HAXIRUM hUMBfiR CF Y C INCREMENTS, THE M A X I M U M NUWafR LF Z IhCHEMENTS, TkE NUMBER OF CURRENT C ELEMkNTS, DELTA X , Y , ANLI 2, AhU THE CGRNER F C I N T OF THE FLOT (MAXIMUM C VALUE O f X,MINiMUM VALUE OF 2 , Ah0 Y P G S I T I O N J .

C NOTE THAT C X IS NEGATIVE A N D CZ I S P C S I T I V E o R E A D ( 5 , 1 C l G ) ~ M , J M , K M ~ L M , O X ~ U Y , D Z , X ( l ) ~ Y ( l ) , Z ( 1 ) C I N P U T THE EhC P O I N T S CF TP‘E CUHHENT ELENEhTSo R E A O ( S , l b 6 J ( X L ( N I ) T Y ~ ( N I J , Z ~ ( N I ) ~ X ~ ( N I ) , Y Z ( N I ) , Z ~ ( N I J , ~ I = ~ ~ L ~ ~ h T P = L P / 4 C I N P G P T = l 1 7 1 U C 200 I N P = l , i N M i F ( I k F O P T o E U e 2 ) GO TC 1 7 4 C INPUT THE OESCRIPTZON CF THE CCNFIGUPATIOh (LE. 7 2 C H P R I C T E R S ) .

w H E A O ( 5 9 1 5 5 ) ( C O N F G ( I C ) , I C = l r l 8 J r C INPUT T k E MAXIMUM NUMBER CF X INCCEIMcE’NTS, TFE HAXIMUM NUMBER OF. Y C INCREHkfUTSw THE MAXIMUM NUPlBkF G F Z IhCREMENTS, TkE hUM6ER OF CURRENT C ELEMENTS, D E L T A X, Y , ANU Z, AND THE CORNER P C I N T CF THE PLOT ( Y A X I H U M C VALUE OF X,NIkiIMUW VALUk C f Z , L A D Y F O S I T I G N I e C NUT€ T H A T C X I S NEGATIVE ANU C Z IS P C S I T I l r E o R E A D ( S r 1 0 O ) I H T J M , K C , L ~ , C X ~ D Y , C Z ~ X ( ~ ) ~ L ( ~ ) C INPUT TH1-5 EbiO P O I N T S CF THE ClrRRENT ELECEhTS AND THE CURRENT FLCWING C I N EACH E L E P E h T I REAO(5,llOJ ( X L ( N ) ~ Y ~ ( N ) T Z ~ ( N ) , X ~ ( ~ ~ , Y ~ ( ~ ) ~ Z ~ ( N J , C U R ( h l r N = l , L M ) GO TO 1 7 3 C INPCPT=2 C I N P U T THE CURRENT FLOWING I N EACH LCCF CF FCUR CUFRENT ELEMENTS.

174 R E A C ( 5 r A 6 5 ) ( C U K T ( h T J v N T = l , N T M I KL=-:!

K L l = O C ASSIGN CURRENT M A G N I T U C E S T C EACH CURRENT ELEHENT.

DO 17C JT=l,NTM K L = K L + 4 K L I = K L 1 + 4 D G Y = D G ( 2 ) CGZ=CG(3) CUPP=XCP*CUR(L1*10030./3~.37 CUR#=XMP*CUR(LlrG~lOuOO.

bXl=CCIRM+U B Y l = C L ' R M * V B Z l = C U R M * k C C A L C U L A T E THE GRACIEAT C O N T R I E U T I C N S OF EACH C t i R R E h T ELtMENT.

B X X l = C t K P * U * D G X B X Y l = C U K P * ( G*( E-F J +U*DGY 1 R X Z l . = C U R P ~ ( G ~ ( O - C ) + U ~ G ~ ~ ) B Y Y l = C U R P * V * D G Y B Y Z l = C U K P * ( G * ( A - B ) + V * D G Z J E ! Z Z l = C l I R P * D G Z * n c SUM THE I N D I V U A L C O h T H I B U T I O R S T O T I - € F I E L D AND GPADIEhT TO GET THE C TOTAL F I E L C AkD GRADIENTS.

B X = B X + B X l t D B Y = f l Y + e Y l VI e z = E z + e z l R n L = m x + B x x 1 t 3 X Y = B X Y t B X Y l e x z = e x z + B x z 1 B V Y = 8 L Y + B Y Y l e Y z = E Y z + w z l

e z z = e z z + e z z l

210 C C N T I N U E C CALCULATE AlvD T E S T THE HAGNET I Z A T I O N CF T H E B O D Y FOR SATURATION.

X D k = X K T / C A R E = ( @ X * ~ 2 + B Y ~ * 2 + 8 2 * * 2 ) * * G o ~ A P = ( I / C A ) * K B I F ( A M - A l r l S ) 1 0 v l O g i 1 C CALCULATE T h E F O R C E S P R G D U C E C CN THE EOOY.

10 F X = X D K * ( B X * B X X + B Y * B X Y + B Z * E X Z J f Z = X C K ~ ( a X ~ B X Z + B Y ~ ~ Y Z * B Z , E Z Z ) F Y = X D K U ( B X * B X Y + B Y S @ Y Y + @ Z * @ Y Z ) GO Ti: 12 C C A L C U L A T E THE CDMPONEhTS OF THE M A G N E T I Z A T I C N A T SPTURATiCN.

11 i3MY=( e Y / R B ) * A H S B H X = ( @ X / K B ) * A M S E M Z = ( E Z / R B J + A M S C C A L C U L A T E T H E F O R C E S PROGUCEC ' C N T H E ' BCDY.

F X = X K T * ( B M X * B X X + B ~ Y * E X Y + B ~ Z * B X Z ) F Z = X U T ~ ( B M X * B X Z + B M Y 3 ~ Y Z + B ~ Z * ~ Z Z ) F Y = X K T * ( B M X ~ B X Y + B M Y + @ Y Y + e ~ Z ~ B Y Z ) C OEFINE THE S A T U R A T I O N L I N E BY A S S I G N I N G ASTER (3) T C TH€ FGRCES AND C A h G L E S P L C N G THE S b T U R A T I C h L I N E .

I f ( A " A l r 1 ) 71G9710912 710 F O R C E ( K , I ) = A S T E K FCHCY ( K v I ) = A S T E R I ANGL( K , I ) = A S T E R GC T 3 3U0 12 C L h T I hllE v C D E F I N E THE X AND 2 A X E S @ Y A S S l G N I N G COT ( 0 1 Ti3 Tkii FCRCES AND AkGLES m C ALChG ' X = O o P N C 2=00 I F ( Z ( K ) o E Q o i i . o ~ R o ) ( ( i ) . t C . O o ) GG T C 14 G O T O 15 1 4 F G H C E ( K t I )=DLiT F O R C Y ( K I I )=DOT . A N G L ( K , I ) = i ) D T G C TO 300 C CALCULATE THE T O T A L M A B N E T I C F C K C E AhC ANGLE I NT k E X-Z PLANE.

15 FT=(fX3FX+FZ*FZI**3.5 E T A = f X / F Z T H E T A = A T A k ( E T A ) r l 8 0 . / 3 . 1 4 1 6 C MATCH A S Y M B O L H l T H T H E C O K R E S P C h D I N G F O R C E A N C A h G L E M A G N I T U D E S .

C FOHCE M A T C H I N G I S A C C C M P L I S h k C EY F I R S T C A L C U L A T I h G T k E € X P C N E N T OF C TkE B A S E NUMBER, THEN T h E E X F C N k k T I S T R U h C A T E D TC A N I k T E G E R C C O R R E S P L N C I h G T 3 A P O S I T I C N I h SYMBOL E R R P Y . ANGL€ M A T C H I N G I S C A C C O M P L I S H E U d Y T R U N C A T I N G T k E C I V I O E A D OF THE ANGLE AhC T k E P H G L E C I N C R E M E N T T U G E T E R M I N E T h E A R R A Y P O S I T I O N .

I F ( T H E T A ) 5ijl953.L 95CZ S O 1 N G A 2 = l s - ( T H f T A / O T F E T A ) 5 c 3 7 2 7 3 7 s 7 7 7 b 7 9 7 0 \o 8 2

a o

8 3 5 c 5 5 66 5C8 5 i3 5 1 3 SV MR @I- A P A 0 r. R S n E T F II P. r.

V Y w X Y z PLOT’Symbols - Force Magnitude Ranges.

~ a . . . . . . . . . . . . . . . . . . * * . * * * * * ~ * * * * * * *

. -9-9 9 9 A A 7 7 6 6 . 9- 6 -2-2

.) - 9 9 9 R 7 '7 6 . 6 6 7 8-9-7 - 3 - 3 - 3 - 3

. - 9 - 9 9 R R 7 7 7 - 9 - 5

. - A -0-9 9 9 A A 7 7 7 7 7 R -9-8-7 -6

- - A -9 9 9 R A 7 . a 9 - 9 -8 -7

.. - 9 - 9 9 9 R R A . R R 9-9-9 -8 - 8

. - R - R - 9 9 9 8 8 8 A . 8 8 R 9 -9 - 8 - 8 a - A - A - 9 9 9 R A R 8 R 8 m R R A 9 9-9-9-9-9-9-9-9 9

. - 9 - 9 - 9 9 9 R A 8 A 8 R o R A A . 9 9 9 9 9 9 9

c . - R - 9 9 9 R P 8 R A A . R A A R 9 9 9 8 8

. - P - 9 9 9 R R 9 A B . 9 8 B A A R U

" - 9 - 9 - 9 9 9 A 8 8 R R . B R R R 8 S S 8 8 R 8 8 7

. - 3 - 9 9 Q 8 9 R R R P . A R 8 4 8 8 8 8 R 8 8 7 7

. -9-G 0 9 A R R A 8 A . R A R R R R O B 7 7

. - 9 - 9 9 9 9 9 A R R R R . R R 7 7 .-9 c . r) r) R R R R A A A . 7 7 7

. 9 9 9 R R R R R A B . 7 7 7 7 6

. R P R R A R R A . 7 7 7 7 7 h

R R R A R 8 R R R . 7 7 7 7 7 7 6 6

. R A J A R P F ! R R . 7 7 7 7 7 7 6 6 h

. . * . . . . . . . . . . . . . . . . . . . . . . . . . * . * *

a 7 . 7 7 7 7 7 6 6 6

,. 7 7 7 7 . 7 7 7 7 6 6 6

. 7 7 7 7 7 7 . 7 7 7 6 6 6 6 5

a h 7 7 7 7 7 7 7 7 7 . 7 7 6 6 6 6 5 5 . h 7 7 7 7 7 7 7 7 7 7 7 . 7 6 6 6 5 5 5 . A 7 7 7 7 7 7 7 7 7 7 . 7 6 6 6 5 5 5 5

. h h 7 7 7 7 7 7 7 7 7 7 . 6 6 6 5 5 5

. h 7 7 7 7 7 7 7 7 7 . 6 6 5 5 4 4

. 5 6 6 7 7 7 7 7 7 7 7 7 . 6 6 5 5 4 4 4 4 - 5 6 6 7 7 7 7 7 7 7 7 . h h 5 5 ' 4 4 . 5 h h 7 7 7 7 7 7 7 7 . 4 6 5 5 4 4 3 3 a 5 A h 7 7 7 7 7 7 7 7 . 6 6 5 4 4 3 3 3 - 5 h h 7 7 7 7 7 7 7 . 7 6 6 5 4 3 2 2 - 5 h h 7 7 7 - 7 6 4 3 2 2 - 5 h h 7 7 0 7 7 6 4 3 2 1 1 1 1

a h 7 7 7 8 ' 8 8 . 7 1

. i- h 7 7 8 A R R R . R 6 3 -1-1-1-1-1

. h h 7 7 R A A . 8 -1-2-2-2-2-2-2

. h 7 7 8 9 . ' 9 9 . ? 9 9 A - 2 - 3 - 3 - 3

7 1 R R 9 9 9 - 9 . -R-8-8"7-4 -5 -4

PLOT Sample Output, Force Angle.

10 1 ~ ~ ~ ~ ~ * e . o . . . e 0 0 . . 0 . . * 0 0 0 * 0 ~ 0 * * * ~ ~ ~ * 0 0 v v v !J I J T c * a Y M w I ) 1 1 T U V 4 v v U I! '1 T F e o O N P (1 v 0 v v I I I J I J T < R . 3 P ' 7 0 0 \I 0 !I IJ T T . S R e 0 P P P P O R S !I v e I J I J I J T T S F R e 0 8 0 R S T I !

I J I1 T T T 5 5 0 i 3 O C i ) G i R S T I !

e T T T T s s s e R P R S T T T T T T T s s e R R R R S T 0 s s s r 9 R R R R R F ! H S T T

. s s s s s s s , R R P H R R P s s

0 s s s s s s s s s s a ? P . R R R R s s s * 9 s s s s s s s P R R R s s s s s o R R s s s 0 R R 5 s s s R P 0 S 0 R . R e e 8 R Q e 0 R R R R O r ) R P P e I< I 4 e o R P R e R R . a R R . P 5 - e R Y R P R R e e ~ a . e . o e e . . o o o e e a e e e o o e o e e e ~ e * * 0 0 R R R R e R R P R H R R R R R R K R R H " a R P R R R o S R R R R R R H R H R H R H R a 0 R P R R R o P R R R R R R Q R R R R P P P 0 0 0 R 9 R R R R . S R R R 9 R R R R R R R R R R m o R H R R R , S R R R R R R K H R H R R R R 0 P R Q R R . Q R R R P . R R R R R P . R R R Q R R R H e R R R R R R R R 9 R P f ?

o R R R R . . R R R R. R R R R R P R 0 s s R R e Q 9 9 9 R e s s s 5 s s s s s R P m R R R S

a s s s s s s s s s s

R R I R R s s e s s s R R o Q Q 0 0 0 R R S

e T T T T s s s

R R o 9 0 0 0 0 R S

e T T T T s s R R o 0 0 O R S T

0 I1 T T T s s 9 R e O P P P P P P S T

e I J U T T S S R - 0 P 0 0 0 P Q R . T I J e I J I 1 T T S S R e P 0 N N n O R 5 a v I J U I J T S R R e P O N M 0 R S T U V e v I J I J T , S S R e P O L J J R e v v v U t J T T S R o 0 0 K E T U V PLOT SampleOutput, Force Magnitude.

APPENDIXD COMPUTATION O F CURRENT ELEMENT END P O I N T S C C C C ' C C C C C C C C C C . c C C C C C C C C C C C C C C C C C C C C.

c C C C C C c C 10.4 znn 1 C c C C C C C C C c c

""-

L!zfwzz -I

+ Figure D-1.Generalized Dimensions of Practical Air Core Coil Configuration.

-+-

I "

i

Straight-Line Approximation Figure D-2. t o Rounded Corner.

Figure D - 3 . Illustration of Nomenclature Used in Subdivision of Windings into Multiple Loops.

APPENDIX E COMPUTER SIMULATION OF STORE DROP IN A MAGNETIC ARTIFICIAL GRAVITY FACILITY A computer code has been developed for use in the evalua- tion of coil configurations considered in the artificial gravity program. This code, designated STORE, provides a measure of the correlation of non-uniformities in the artificial gravity field with trajectory errors by calculating ideal or constant gravity trajectories and comparing with the trajectories calculated for a store released in the artificial gravity field.

All aerodynamic .forces and moments are considered. The code itself consists of a main controlling program (MAIN), and the following four subroutines: INPl (for input), ARGMV (calculates the artificial gravity components), TRAJ (provides the trajectory coordinates), and OUTPUT (controls the output). A general flow chart depicting the interrelation of these four routines and MAIN appears in FigureE-1. Input for STORE includes character- istics of the artificial gravity coil system as well as dynamic and aerodynamic characteristics of the store model used in the evaluation. A complete listing of input variables can be found in the "Input Variable List" of the input subroutine (INP1) listed on page 125. A sample output sheet is on page 1 4 1 .

LIST OF SYMBOLS D Store diameter Moments of i n e r t i a a b o u t p r i n c i p l e axes of store IxIIy?Iz L S t o r el e n g t h m Mass of store Dynamic p r e s s u r e r R e l a t i v e d i s t a n c e b e t w e e n s t o r e and a i r c r a f t S Reference area o f s t o r e t T i m e T i m e increment A t a Angle of a t t a c k Angle of s i d e s l i p B O x I Of Coordinatecomponents z R e f e r r e dt os t o r ec o o r d i n a t es y s t e m 0 s Referred t o earthcoordinatesystem ()E Theory The store trajectories are calculated by determining the linear and angular accelerations through application of Newton's Second Law for a rotating frame (the store coordinate system of Figure E-31, then integrating twice by Simpson's Rule. From Ref. 11, the vector equations of motion are,

+ - E A +

"-

as m w S x 3s + I C P X g

+

where as and kS are the linear and angular accelerations in a

frame fixed to the principle axes of the store and [CIT is the transpose of the rotation matrix defined below. (See A , B, C).

Appendices In this convenient vector form, the aerodynamic force coefficients are: " The aerodynamic moment coefficients, as defined in Fig. E-2, are: C D P

Em =fc Ll

CrL The vector 2 arising from the rotating frame is: The aerodynamic force coefficients are calculated by first calculating lift, drag, and side force coefficients by the conventional definitions Ci.e., perpendicular and parallel to the wind vector), then the angles of attack and side slip are used to determine the resulting forces in the store coordinate system Csee Fig. E-21. Thus, = (C sina-C cosa)cosg+C sing Cx (E- 7 1 L D 6

c = ( C cosasin8+Cscos8) (E- 8 1

Y D where, (E- 10) cL = Lo a (E-12)

cs = cso+cs B

The moment coefficients are calculated from the static and dynamic derivatives as, (E-13)

c =o

P (E-14) (E-15) At this point, the static and dynamic derivatives for moment coefficients and the derivatives for force coefficients ,etc.) form part of the input for the calculation and (cLa'cs@ must be either estimated or determined from experimental data.

The 'subzero' factors (CL ,C ?Cr ,C )are generally a result 0 90 0 so of aerodynamic interference from the releasing body (i.e., aircraft model) since the stores are otherwise symmetric. For simplicity, these are approximated by a power series in l/r, i.e. , (E-16) where, again,a, b, c,and d can be Zetermined from a 'curve fit' of experimental data, or estimated. The series can be extended to higher order terms with minor modifications to the input and trajectory subroutines. It is noted that the goal of the calculation is a reasonable evaluation of the coil system so that a set of coefficients which produces a representative trajectory is sufficient.

The first integration of the dynamical equations is per- formed in the store coordinate system. Using Simpson's Rule based on half the time increment, the store velocity is

where as = as(tn + At/2). In order to determine both the

n+- acceleration and velocity (for the next integration) at the

half interval point, ti + At/2, the velocity is approximated

as a quadratic in time between tn and tn+l, = c t 2 . + c2tn +c3 (E-18)

% I n

n + dys +

= “&l= 2c t + c2 (E-19)

I n where the coefficients are -t

c = (as - 2 )/2At (E-20)

n+l ‘ n + - (E-21) c2 - asn - . 2cltn (E-22)

c3 = 3 S - (Cltk + C2tJ

n so that -k (E-23)

as = 2cl (tn + At/2) + c2

n+Z

3 s 1 = C1 (tn + At/2) + c2 (tn + At/2) +c3

n+T (E-24) The angular acceleration is integrated in the same manner. Next, the linear and angular velocities are transferred to the non- rotating earth axis (see Figure E-3) by application of the proper rotation matrices, i.e., -k + (E-25)

= I a n vs

n+l vEn+l .

+ + (E-26) ‘n+l = D l n us n+l where [Cl and [Dl are as follows: posencos$n sin$ n sine,cos$, -cos$nsin$n cosOnsin$, sin$nsin8nsin$n

I C ] , =

tcos$ncos$n -sine in$ncosen n

-

(E-27) '

-1 sin$,tanen cos$ntanen 1

-sin$n cos$nsecen sin@,secen

la 1

(E-28) and 9 , 8 , and $ are the Euler angles defined in Figure E-3.

NOW, the linear velocities and Euler rates are integrated, again by Simpson's Rule, to determine the next coordinates of the store c.g. and the Euler angles locating the store principal axes, ire, + -t

+ LL ("v +4GE +G 1

= XE (E-29) ' E n + 1 n En n+- En+l (E-30) where (E- 3 1) + (E-32) The computer code developed for the trajectory calculation has been tested for thecases of a non-rotating sphere with constant acceleration and for a purely rotating sphere. Under these conditions, the dynamical equations reduce to (33-33) and (E-34) respectively, where ..

-b -b

6 = [Dl b t , , en = [Dl; b = const. (E-35)

n

- 1 ARGRAV

- T R A J

I OUTPUT

A

Figure E-1. General Flow C h a r tf o rS t o r e .

Figure E-2. Aerodynamic Coefficients in Store Axis.

I Figure E-3. CoordinateSystems and Euler Angles.

C C C C C C C

c

C C P h, P* C C C C C I C C ~ H A T B E T A ~ C ~ A G ~ V S ~ V E T T I ~ E T A F ~ C E L A , C E S ~ ~ C D O ~ C ~ A Z ~ C M A ~ C ~ A D ~ C M T D ~ C N B ~ B D , C N P S I D T ~ T R T I T R A T I T R B T B E T S T C , A L P S T L T B S T ~ , A S T ~ T B ~ T V M A G ) C A L P H A I ( I T R ) = A L P H A ( I T R ) r 5 7 o 3 T I M E ( I T R ) = T I M E ( ITR-1 )+DELT C C CALCULATE THE LJCAL G ANGLE A N D MAGNITUDE.

GESl(ITRI=GESl(lJ L . E I A l ( I T R J = Z E T A l ( l ) C C IS THE S T O K E U U T S I O E OF THE L I M I T S OF THE REGION OF. INTEREST.

I F ( X T K ( I T R , l ) . G E o X M A X o U K ~ X T R ( I T R , ~ ~ p L E o X M I N s O R . X T R ( I T R ~ 3 ) . G E o Z M A X o l J R ~ ; X T R ( I T R , 3 ) . L E r , Z M I N ) GO TU 940 G O T U 3 2 1 C 940 I M A X = i T R C C CALCULATE THE ACTUAL TRAJECTORYo !

C DEFINE STARTING CdNDITIONSo DU 950 NM=1p 3 T H E T A ( ~ ~ M ) = T H E f A 1 ( N M ) XAC(NM)=XAC1(NM) 950 X E ( N M l = X E l ( N M ) ITRA=O I T R B = 8 I T R = 1 CALL ROTA(THETAtC1 C 9 2 6 CALL A R G R A V ( D A T A M S , X K T T X H U , X R , X l T Y l T Z l , X Z T Y ~ T Z Z t C U R t L ~ t R H O I , A X T A Y ~ 1 A Z p I T R ) A G ( I T I ) = A X ( I T R ) A G ( l t Z ) = A Y ( i T R ) A G ( l r 3 J = A Z ( i T R ) + G C C CALCULATE THE LuCAL G ANGLE A N D MAGNITUDEo G E S 2 ( I T R ) = ( ( A X ( I T R ) 3 ~ Z + ( A Z ( I T R ) + G ) ~ ~ 2 ) * ~ ~ ~ ) ~ G .. .

C C INPUT SUBROUTINE C C I N P U T V A R I A B L E L I S T C C V A R I A B L E NAME OEF I N I T I O N C C CONFS D E S C R I P T I O N UF STOKE CONFIGURATION.

C X E l I N I T I A L S T C R E P O S I T I O N I N THE EARTH SYSTEM ( X T Y , Z ) O C THETA1 I N I T I A L EULER ANGLES ( P H I T T H E T A , P S I ) ~ C X A C l P 3 S I T I D N O F A I R C R A F T ~ X ~ V T Z ) ~ I N I T I A L P C V A C VELOCITY CUMPONENTS OF TUNNEL W I N O ( V X I V Y , V Z ) ~ I XM STCJRE M A S S AND MOMENTS O F I N E R T I A 0 X M ( ~ T ~ ) = X M ( ~ , ~ ) =

K c

C X M ( ~ T ~ ) = M A S S T X M ( 2 , l ) = I X p X M ( 2 , 2 ) = I Y , AND X M ( 2 * 3 ) = I Z C AF CONSTANTS FOR D E T E R H I N I G AERODYNAMIC INTERFERENCE C FIELD0 A F ( l r l ) , A f f l r Z ) ~ A F ( l 0 3 ) AND A F ( l t 4 ) ARE FOR C S I D E FORCE9 A F ( ~ T ~ ) T A F ( ~ T ~ ) ~ A ~ ( ~ T ~ ) , AND A F ( 2 9 4 ) ARE C FOR NUKMAL fORCE9 A F ( 3 r I ) FOR P I T C H I N G MUMENTT AND C AF(49 I ) F9R Y A X I N G MOMENT0 C CURVE CLOPE (DCC/OALPHA) CECA L I F T C CESB S I D E F O R C E D I R I V I T I V E ( D C S / O B E T A ) U C BASE DRAG.

coo

C C 3 A 2 DCD/DALPHA**2 DCM/DALPHA (PITCHING MOMENT)

c CMA

CMAD DCM/DALPHADOT @ ' ' @

C DCM/DWYDOT ' @ C C!4 TO OCN/OBETA (YAWING MONENT 1 C CNB DCN/DBETADOT " " C CNBD C 2 0 0 r 2 , 3 ) ~ V E ( l 0 0 0 r 2 ~ 3 ) , A F ( 4 , 4 ~ , c u ~ ~ ( l ~ ~ ) C C

CNPS I D DCN/DWZDUT ' *

C XN s DISTANCE FROM S T O A € COG. TO NOSEo C OfLT T I M E INCREMENT FOR TRAJECTORY CALCULATIONSO C RHOZE DENSITY OF TUNNEL ATMOSPHERE.

C BD S l O R E D I A M E T E R o C B L STORE LENGTH.

C X M I N T X M A X X L I M I T S OF R E G I O N OF I N T E R E S T o C Z M I N T Z M A X 2 L I M I T S O f R E G I O N OF I N T E H E S T r C BSTL CONSTANT FOR S I D E F O R C E C O E F F I C I E N T C A L C U L A T I O N P A S 7 C S T A L L o C ASTL CONSTANT FOR L I F T C O E F F I C I E N T C A L C U L A T I O N P A S T S T A L L C B E T S T L S T A L L I N G ANGLE OF S I D E S L I P .

C A L P S T L S T A L L I N G ANGLE OF ATTACKo C G ACCELERATION DUE T O GRAVITY.

P C vs I N I T I A L V E L O C I T Y COMPONENTS. AND R O T A T I O N HATES O r

h, m c STORE R E L A T I V E 10 STORE C c Se WS( l p I ) CORRESPONDS C T 3V E L O C I T I E S AND V S ( 2 , f ) TO R O T A T I O N RATES.

C VE CURRESPONDING VELOCITIES AND R O T A T I O N R A T E S I N C E A R T H A X 1 Sa C F O R M A T ( 1 8 A 4 ) 1 6 4 C C i)A DEMAGNETIZING CONSTANT FOR THE MODEL.

C AMS S A T U R A T I O N M A G N E T I Z A T I O N F O R T H E MODELc C MAGNET I C FORCE COhSTANTo XK T C XM U M A G N E T I C P E R H E A B I L I T Y irF FREE SPACE.

RHO I D E N S I T Y OF MAGNETIC MATERIAL OF SPHEREo C C LM TCITAL NUMBER OF CURRENT ELEMENTS C INPCIPT I N P U T O P T I O N .

C CONFG A D E S C R I P T I O N OF THE MAGNET CGNFIGURATIONo C x 1 , Y l T 21 COORDINATES OF THE END P O I N T S OF T H E S T R A I G H T L I N E C X 2 r Y Z T Z 2 CURRENT ELEMENTS MAKING UP THE COILS.

C CURT CURRENT FLOWING I N A LOOP OF FdUR CURRENT ElEMENTSo C CUR MAGNITUDE OF THE CURRENT I N AMPERES*+ f R 3 M 1 TO 2 0 C CURX 1 T H E TOTAL CURRENT I N THE +X F I E L D C O I L e C C U K X 2 THE TOTAL CURRENT I N THE + X GRAOIENT C O I L .

C CURXS THE TOTAL CURRENT IN THE - x FIELD c o n o

C CURX4 THE TOTAL CURRENT I N THE - X G R A D I E N T C O I L .

C C U R 2 1 THE TC)TAL CURRENT I N THE +Z F I E L D C O I L e C C U R 2 2 THE TOTAL CURRENT I N THE + Z GRADIENT C O I L S C C U R 2 3 THf T O T A L CURRENT I N THE -2 F I E L D COIL.

C C U R 2 4 THE TOTAL CURRENT I N THE - Z GRADIENT C O I L 0 C C NOTE THAT D D U B L E S U B S C R I P T E D V A R I A B L E S SUCH AS X M ( i , J ) A R E READ I N THE C ORDER X M ( ~ , ~ ) , X M ( ~ , ~ ) , X M ( ~ , ~ ) , ~ ~ ~ , X M ( ~ ~ ~ ) , X M ( ~ , ~ ~ T O . O ~ ETC. UNLESS C S P E C I F I E D A S I V S T A T E M E N T R f A D ( 5 , 1 4 2 ) .

C R E A D ( 5 9 164) CONFS R E A D ( 5 , 1 6 0 ) X ~ l , T H E T A l ~ X A C L r V A C ~ X M , A F 169 F O R M A T ~ 3 F 1 4 ~ 8 / 3 F 1 4 ~ 8 / 3 F l 4 0 8 / 3 F l 4 ~ 8 / 6 F l Z ~ 6 / 8 F 9 ~ 5 / 8 F 9 ~ 5 ~ P h, READ(5,161) CELAtCESB,CDO,CDAZ,CHA 4 R € A D ( 5 ~ l b l ) : C M A D ~ C M T D q C N B ~ C N B D , C N P S I O R E A D ( 5 ~ 1 6 1 1 XNStD€CT,RHOZE,BD,BL R E A D ( 5 , 1 6 1 ) X M ~ N T X M A X T Z M I N ~ Z M A X T ~ S T L 161 F O R M A T ( 5 F 1 4 a 8 1 R E A D ( S r l 6 3 ) A S T L ~ B E T S T L ~ A L P S T L , G 163 F O R M A T ( 4 F 1 4 0 8 ) READ(59162) ( ( V S ( ~ T I N , I M ) , I M = ~ T ~ ) , ~ N = ~ , ~ ) R E A D ( 5 ~ 1 6 2 1 ( ( V E ( ~ T J N ~ K M ) ~ K M = L T ~ ) ~ J N = ~ ~ ~ ~ 162 FDRMAT(6Fl236) C REA0(5,112) D A , A M S T X K T T X M U T R H O I , L M T I N P O P T R E A 0 ( 5 , 1 6 4 J CONFG C i N P U P T = l CORRESPUNOS TO I N P U T I N G THE CURRENT I N EACH ELEMENTo XNPOPT C =2 CORRESPONDS T O I N P U T I N G T H E CURRENT I N EACH COOP OF FOUR ELEMENTS.

If ( I N P O P T a E Q a 1 ) G O TO 1 7 1 R E A 0 ( 5 , 1 6 6 l ( X L ( ~ I J , Y l ( N I J , Z Z ( N I J , X Z ( ~ r J , ~ Z ( ~ ~ J T Z Z ( ~ f J , ~ ~ = ~ T ~ M ~ 166 F O R M A T ( 6 F 8 0 4 J N T M = L M / 4 .. .

SGK(3)=EGR T G K ( 1 )=BGR TGR (2 )=DGR TGR ( 3 1 =FGR C C A L C U A L T E U t V, AND W.

UGR=C GKzcF GR-DGR*EGH VGR=EGRdRGK-FGR*AGR WGR=AGK*SLIGR-tjGR*CGR C C A L C U L A T E H H I J l A N D R H 0 2 0 RG1=( AGR*AGR+CGRvCGR+EGR*EGR 1 **e 5 R G 2 = ( B G K ~ B G R + O G R ~ D G R + F G R ~ F G K I**" 5 C C A L C U L A T E T H E SUM, PRODUCT, DOT PRODUCT, AND CROSS PRODUCT OF R H O l

c AND RH029

S S = R G I + R G 2 K M = K G l * K G 2 RDR=AGK~OGR+CGR*DGR+EGR*FGR P G, KXK=UGK+VGK+WGK 0 C C A L C U L A T E T H E D E H I V I T I V E S O F THE SUM9 ETC. 13F R H O l A N D R H 0 2 0 DO 220 M=1,3 D P ( M ) = - ( S G R ( M ) ~ K G 2 / R G l + T G R ( M ) ~ R G l / ~ G 2 ) DS(M)=-(SGR(M)/RGl+TGR(M)/RG2) D D ( M ) = - ( S G R ( M ) + T G R ( M ) ) 2 2 3 C O N T I N U E D C ( l ) - F G K - E G K + C G R - D G R DC12)=EGR-f6R+BGR-AGR DC ( 3 1 =DGR-CGR+AGR-BGR C C A L C U L A T E A N D T E S T H TO D E T E R M I N E f i Q U A T I D N FOR G TO B E USEDo H = ( R M + K D K ) / R M I F ( H - O o 0 1 ) 2,191 C C A L C U L A T E G AND I T S D E R I V I T I V E S i N THE X , Y , Z D I R E C T I U N S c 1 GGR=KS/(RM*(RM+RDR) 1 DO 230 M 1 = 1 9 3 D G A = R M s ( R M + K D R ) - D S ( M l ) D G B = K S ~ ~ R M ~ ( O P ( M l ~ + D D ( M l ~ ) + D P ( M l J * ( R M + R D R ) f D G ( M L ) = ( D G A - O G B ) / ( R ~ P ( K M + R O R ) )**2 230 C O N T I N U E GO TO 3 2 G G K = ( ( K S ) ~ ( R M - R O K ) ) / ( R ~ ~ R X ~ ~ : ~ X R ) DO 240 M 2 = 1 9 3 D G A = ( R S ~ ( D P ( M Z ) - U D ( M 2 ) )+DS(M2)*(RM-RDR) J*RM*RXR*’*Z D G B = R S ~ ( R M - R O R ) ~ ( R M * Z o ~ K X R ~ ~ C ( M 2 ) + D P ( M Z ) ~ R X R ~ ~ 2 ) D G ( M Z ) = ( D G A - D G B ) / ( R M * R X R ~ * Z ) ~ * Z 243 CONTINUE C CALCUALTE THC F I k L D C O N T R I B U T I O N S O F EACH CURRENT ELEMENT.

3 UGX=DG( 1 1 .DGY=DG( 2 D G Z = D G ( 3 I C U R P = X M P ~ C U R ( L ) * l O O U 0 . / 3 9 . 3 7 CURM=XMP*CUR(L)*GGR*lODl)O.

BXl=CCIRM*UGR BZl=CURM*HGR B Y l = C l R M * V G R p C CALCULATE THE G R A D I E N T C O N T R I B U T I O N S OF EACH CURRENT ELEHENTo W B X X l = C U K P W G R * D G X P BXYl=CURP~~IGGR~(EGR-FGR)+UGR)+UGR.~DGY) DXZl=CUKP-*(GCR*(DGR-CGR)+UGR*DGZ) R Y Y l = C U H P ~ V G R * D G Y B Y Z 1 = C U H P ~ ~ ( G G R s ( A G R = B G R ) + V G R p D G Z ) BZtl=CURP*DGZ*WGR C SUM THE I N D I V U A C C O N T R I B U T I O N S TO THE F I E L D AN0 G R A O I E N T TO GET THE C TOTAL F I E L D AND G R A D I E N T S * B X = B X + B X l B Y = O Y + B Y l BZ=BZ +RZ 1 B X X = B X X + B X X l R X Y = B X Y + B X Y l R X Z = 8 X Z + B X f l B Y Y = B Y Y + B Y Y l B Y Z = B Y Z + B Y Z 1 B Z Z = B Z Z + B Z Z l f 210 C O N T I N U E C CALCULATE AND TEST THE MAGNETIZATION Of THE BODY FOR S A T U R A T I O N o K R = ( B X ~ ~ 2 + B Y ~ ~ 2 + 6 2 * 4 2 ) ~ ~ ~ . 5 X D K = X K T / O A A M = ( l / D A ) * R B I F ( A M - A M S ) 1 U ~ l t ) t l l C CALCULATE THE FORCES PRODUCEO ON THE BODY.

I !

I ! ! F X = X D K * ( B X ~ B X X + B Y * B X Y + B Z V B X Z 1 FY=XDK*:( B X A B X Y + B Y * R Y Y + B Z ~ B V Z ) F Z = X D K 4 ( B X ~ B X Z + B Y * B Y Z ~ B Z B Z Z ) I GO TO 12 C CALCULATE THE COMPONENTS OF THE M A G N E T I Z A T I O Q AT SArURATIONo 1 1 B M Y = ( B Y / R B ) @ A M s I j M X = ( B X / R B ) * A M S 0 M Z = ( BZ/RB)sAMS C CALCULATE THE F O R C E S P K O O U C E O ON THk 8 O D V e F X = X K T a ( H M X ~ B X X + B M Y ~ B X Y + B M Z * B X Z 1 F Y = X K T V ( B M X ~ B X Y + R M Y ~ B Y Y + B M Z * B Y Z ) P F Z = X K T * ( B M X ~ B X Z + B M Y ~ B Y Z + B M Z : ~ B Z Z ) LC, N 12 C O N T I N U E C A X ( I T R ) = F X / R H O I A Y ( I T R ) = F Y / R H O I A Z ( I T R ) = F Z / K H O I C R E T U R N END P w

j

G, C E S t C F O ( 1 ) + B E T A ( I T H I J i - C E S B CEL=CF0(2)+CELA*ALPHA(ITR) C C TEST FOR S T A L L A N D C A L C U L A T E T H E L I F T AND S I D E FORCE AT STALL.

I F ( B E T A ( I T R ) o G E . B E T S T L ) GO TO 200 GO TO 201 200 I T R B = I T R 6 + 1 IF( I T R B o E Q . 1 )C E S S T L z C E S C E S = C E S S T L - B S T L * ( B E T A ( 1 T R ) - B E T S T L J*:*2 201 I F ( A L P H A 4 I T R ) m G E o A L P S T L ) GO TO 2 0 2 GO T O 203 202 I T R A = I T R A + l 3 I F ( I T R A . E Q o 1 ) C E L S T L = C E L CEL=C€LSTl.-ASTL*(ALPHA(ITR)-ALPSTL)**2 C C C A L C U L A T E THE FORCE C O f F F I C I E N T S R E L A T i V E T O THE STORE C e S o

I

p 203 C F ( l ~ l ~ = ( C E L * S I N ( A L P H A ( I T R ) ~ ~ C ~ E * C O S ( A L P H A ~ I T ~ ~ ~ ~ * C O S ~ ~ E T A ~ I T R ~ ~ + C G, l b 1 E S * S I N ( B E T A ( I T R ) 1

I

C F ( l , Z ) = C D E ~ C O S ( A L P H A ( I T R ) ) * S X N ( B E T A ( I T R J ) + C E S * C U S ( B E T A ( I T R ) 1 i C f ( l ~ 3 ) = - ( C O E * S I N ~ 4 L P H A ( I T R ) ) + C E L * C O S ~ A L P H A ~ I T R ~ ~ ~ C C C A L C U L A T E THE MOMENT C O E F F I C I E N T S o C F ( Z t l ) = O o C F ~ ~ ~ ~ ~ ~ ~ C ~ ~ ~ ~ ~ + C N B * ~ E ~ A ~ I T R ~ + C ~ ~ O ~ B E T A D + C ~ P S I O ~ V S )*BL CF(2,2)=(CFD(3)+CMA*ALPHA( I T R ) + C M A D * A L P H A D + C M T U ~ V S ( I T R t 2 t 2 ) ) * B L !

C C

i

1 C C A L C U L A T E T H E A C C E L E R A T I O N S D U E TO T H E R d T A T I N G STORE C o S c I ! W X V ( ~ ~ ~ ) = V S ~ I T R ~ ~ ~ ~ I ~ V S ~ ~ ~ R ~ ~ ~ ~ ~ ~ V S ( I T R I ~ ~ ~ W X V ( ~ ~ ~ ) = V S ( I T R ~ ~ ~ ~ ) ~ V S ~ ~ ~ ~ ~ ~ ~ ~ J - V S ( I T ~ T Z ~ ~ ~ ~ V ~ J X V ( ~ ~ ~ ) = V S ~ I T R ~ ~ ~ ~ ) ~ V S ( : I T R . ~ ~ ~ ~ ~ - V S ( X T R T ~ ~ ~ ) * V S ( I T R ~ ~ ~ ~ C W X V ( Z t l J = V S ( ~ T R ~ ~ ~ ~ J ~ V S ( I T R ~ ~ ~ ~ ) ~ ~ ~ X M ~ Z ~ ~ ) - X M ( ~ ~ ~ ) ) / X ~ W X V ( ~ ~ Z ) = V S ~ I T R ~ ~ ~ ~ ) ~ V S ( I T R ~ ~ ~ ~ ) ~ ( X M ( ~ ~ ~ ~ - X M ( ~ ~ ~ ) ) / X M ~ Z I ~ W X V ( 2 t 3 ) = V S ( I T R t 2 r 2 ) a V S ( I T R , 2 ~ l ) ~ ( X M ( Z , 2 ) - X ~ ( 2 ~ l ) ) / X M ( 2 t 3 ~ C T I M H = T I M E ( I T R ) + D E L T / Z m C DO 120 N F = l r 2 DO 140 M F = 1 ~3 GS (NF ,MF 1 =9.> D O 138 L F = 1 , 3 C TKAMSFEK THE MAGNETIC AND GRAVITY FORCES TO THE STORE CmSm 130 GS(NF,MF)=GS(NF,MF)+C(NF1LF,MF)FAG(NflLF C C A L C U L A T E THE A C C E L E R A T I O N OF THE STORE I N THE STORE C o s .

A C S ( I T K + L , N f , M F ) = C F ( N F 1 M F ) C Q S / X M ( N F I M F ) - ~ X V ( ~ ~ T M F ) + G S ( N F T M f ) I F ( I T R s E Q o l ) A C S ( l r N F 9 M F I=ACS(2,NF,Mf 1 C CALCULATE THE CONSTANTS FOR THE POWER S E R I E S EXPANSiON I N T I M E F O R C V E L O C I T Y , ( I o E o V=Cl*T**L+CZ*JT+C3 lo C L ( N F T ~ F ) = ( A C S ( I T R + ~ , N F T M F ) - A C S ( I ~ ~ T ~ F T M ~ ~ ) / ( O E L ~ * Z ~ ) C 2 ( N f , M F ) = A C S ( I T R , ~ f , M ~ ) ~ 2 . ~ : C l ( N F , ~ ~ ) * T I M E ( I T R ~ C ~ ( N F , M F ) = V S ( I T R , N F T M F ) - ( C ~ ( ~ F T M F ) ~ T I M E ( I T R ) ~ * Z + C ~ ( ~ F T M F ) * T I M E ( I T R 1 ) 1 P I w C C A L C U L A T E THE A C C E L E R A T I J N A N D V E L O C I T Y AT THE CENTER OF T H E I N T E R V A L VI V S H ( N F , M F ) = C ~ ( N ~ I M F ) ~ T X M ~ * * ~ + C ~ ( N F , M F ) ~ T I M H + C ~ ( N F T M F ) ACSH(NF,MF)=Z.UCl(NF,MF )*TIMH+C2(NF,MF) C USE S I M P S O N ' S R U L E T O C A L C U L A T E THE S T O R E V E L O C I T Y A N D R O T A T I O N RATES.

140 V S ( I T R + ~ , N F I M F ) = V S ( I T R , N F T M F ) + ( A C S ( I T R ~ N F , M F ) + ~ ~ * A C S H ( N F , M ~ ) ~ A C S ( ~ l T R + l , N F , M F 1 )*DELT6 DO 120 NE3113 V f ( I T R + ~ , N F T N E ) = D * VEHlNF,NE)=da DO 150 M E = 1 , 3 C T R A N S F E R V E L O C I T I E S AND ROTATICIN RATES TO THE EARTH COORDINATE SYSTEM V E ( I ~ R + ~ T N F ~ N E ) = V E ~ I T R + ~ ~ N F ~ N E ) + C ( N F T N E T M E ~ ~ V S ~ I T R + ~ ~ ~ F ~ M € ~ 150 V E H ( N F , N E ) = V E H ( N F , N E ) + C ( N F T N E , M E ) ~ V S H ( N f , M € ) C USE S I M P S O N ' S R U L E T O C A L C U L A T E STORE P O S I T I O N ( S P A C I A L COORDINATES C AND EULER ANGLES).

I F ( N F . E Q o 1 ) X E ( N E ) = X E ( N E ~ + ( V E ( I T K ~ ~ ~ N E ) + ~ ~ ~ V E H ( ~ T N E ~ + V E ( I T R + ~ ~ ~ T N E 1 1 )*DELT6 IF(NFOEQu2) T H E T A ( N E ) = T H E T A ( N E ) + ( V E ( I T ~ , ~ T N E ) + ~ ~ ~ V E H ( ~ T N ~ ) + V E ~ I T R + 1 1 , 2 , N E ) ) * D f L T 6 123 CONT I NU€ C CALCULATE THE ROTATION M A T K T X FOR A X I S ROTATION.

CALL ROTA(THETA,C) DO 180 LE=1,3 C CALCULATE THE RELATIVE POSITION OF S T O R € COG. AND N O S E I N THE EARTH C COORDINATE SYSTEM.

X A C ( L E ) = X A C ( L E ) + V A C ( L E I * D E L T XREL( I T R + l r L E ) = - 1 2 , ~ ( X A C ( L E ) - X E ( L E ) ) 180 X N E ( I T R + 1 ,LE J=XREL( I T R + l t L E ) + 1 2 u *C( I r L E , 1 )*XNS C I T R = I T K + l C C CALCULATE THE MAGNITUDE OF THE STORE VELOCITY, STOKE ANGLE OF ATTACK, C AND A N G t E OF S I O f S L I P 0 V M A G = ( V S ( I T R , 1 ~ 1 ) * * 2 + V S ( I T ~ ~ l , 2 ) ~ ~ Z + V S ( I T ~ ~ ~ ~ 3 ~ * ~ 2 ) ~ ~ . 5 I F ( V M A G e E Q o O a 1 GO TO 2 2 0 P B E T A ( I T H ) = A R S I N ( V S ( I T R , l r 2 ) / V M A G ) LJ ALPHA(ITR)=ARSIN(VS(ITK,l,3)/VMAG) c n GO TO 2 2 1 2 2 0 B E T A ( I T R ) = i ) D A L P H A ( I T R ) = Q s 2 2 1 RETURN END i I C R E T U R N I END P SUBROUTINE O U P U T ( C O N F G , X ~ T Y ~ T Z ~ ~ X ~ , ' I r rp r I CONSTANT GRAVITY AN0 ACTUAL TRAJECIORIES FOR MINO TUNNEL STORE DROP Y I T H A R T I F I C I A L G R A V I T Y ARTIFICIAL GRAVITY CONFIGURATION: EIGHT ORTHOGJNAL HELHHOLTZ COILS, SUPERCONO. (ORTH8-5.41 STORE CONFIGURATIONS TEST THREE 01 DEBUG--MOOEL OF 15 FT. ST3RE TOTAL CURRENT I N * X F I E L O C O I L .

0 . AMPS. TOTAL 0 . AMPS. CURRENT I N +X GRADIENT COIL- TOTAL CURRENT I N - X F I E L OC O I L - 0. AMPS. 0 . AMPS. TOTAL CURREYT I N - X GRADIENT COIL- T O T A L CURHEWT IN + z FIELO c o n . = 4 5 5 0 ~ . ~ 0 . AMPS. TOTAL CURRENT I N +z GRADIENT COIL= 4165000. AMPS.

TOTAL CURRLNT I N - 2 F I E L O C O I L . 4550000. AMPS. TOTAL CURREYT I N -2 GRADIENT COIL- -4165000. AHPS.

TUANEL WINO VELOCITY. 500.6350 FPS Z ERROR IS NPRHALILEO TO STORE LENGTH.

CONST AN1 r ACCELERATION ACTUAL Z TR :AJECTORY E R I IOR T I M E ( S 1 X ( I N 1 Y l I N 1 Z I I N I G ' S I X Z I G ANGLE X ( I N 1 Y ( I N 1 L ( I N l G ' S ( X 2 l G ANGLE ZERRORX ZERRORY ZERRORZ 3.0 0.L 0 . 3 1.5033 10.4805 0.0 0.0 b.0 1.5OCO 10.4805 0.0 0 . 0 0.0 0.0 3.001 -2,3052 3.30co 1e532J 10.4805 0. L, -0.u002 o.oot.1 1.5020 10.4805 -0.0000 0 . 0 6.0000 n . o o.ou2 -0:cc07 3. G3hJ 1.5J81 10.4805 0. 0 -0.0007 o.oooa 1.5081 10.4805 -0.0001 -0.0000 0.0000 0.0 5.003 -%PO15 0.503 1 1.5181 10.4835 0. 0 -0.0015 0.0001 1.5181 10.4806 -6.0002 0.0 0.0000 -o.ooon

0.304 - i ~ . c'C26 0.3033 1.5322 13.4805 0.0 -3.0026 0.0002

1.5322 10.4807 -0.0003 0 . 0 n.0000 -a. 5030 0.005 -3.CC41 3.30C4 1.5503 10.4805 0.0 -3.0041 0.0004 1.5503 10.4808 -6.OOC5 0.0 0.3000 -O.OO@O 0.G66 -3.C058 0.0606 1.5724 li~. 4805 0.0 -000058 0.0006, 1.5724 lC.4808 -0.0006 0.0 0.0000 -0.0000 0 . 007 -1).1~080 6.0039 1.5985 13.4805 0.0 -3.0080 0.0009 1.5985 10.4809 -0.0009 0 . 0 0.0000 -0.0000 0.018 -C.0104 0.9011 1.6286 10.4805 3 . 0 -0.0104 0. b o l l 1.6286 10.4811 -0.0011 0.0 0.0000 -0.0000 0.009-0.0132 0.3015 1.6627 10.4865 0.0 -0.0132 0.0015 1.6627 10.4613 -0.0014 0.0 o.nooo -0.0300 3.010 -0.0163 (r.0016 1.70@8 10.4805 5.0 - 3 . 0 16 3 0.0018 1.7008 16.4814 -0.0018 0.0 0.0000 -0.0000 0.011 -0.t197 0.0022 1.7433 10.4805 3.0 -0.0197 0.0022 1.7430 10.4816 -0.0021 0.0 0.0000 -0.0000 3.012 -0ob234 0.0026 1.7891 10.4805 0.0 -3.0234 0.0025 1.7891 10.4818 -b.)G25 o.n 0.0000 -0.0000 1.013 -0.0275 0.003 1 1.8392 10.48C5 0.0 -3.0275 6.0031 1.8392 10.4823 -0.0030 0.0 o.ooon -0.0001 0.014 -0.0319 0.3036 1.8933 10.4805 5.0 -0.031 9 0.0036 1.8933 16.4823 -0.0035 0.0 0.0039 -0.0001 0.015 -0.0366 o.no41 5.0 1.3514 10.48r15 -3.0366 0.0041 1.9514 1'2.4825 -0.004C 0.0 0.0060 -0.0001 0.016 -IJ.C417 0.0047 2.0135 10.4805 0. 0 4 . 0 4 1 7 3.0047 2.6135 10.4828 -0.0045 0 . 0 0.0000 -0.0001 0.01 7 -0.0470

3.3352 2.0796 12.4805 3.0 -0.0470 11.0052 2.0796 lb.4831 -L,. J051

6.0 0.0000 -o.onr)l 3.018 -3.0527 0.0058 2.1497 15.4805 0 . 0 -0.0527 0.0058 2.1497 10.4835 -0.0057 0.0 0.JOOO -0.0002 0.019 -0. 0588 0.0065 252237 10.4805 0.0 -0.3588 0.0065 2.2238 10.4838 -0.0064 0.0 0.0000 -0.0002 0.020 -0.C651 0.0071 2.3018 1d.48.35 0.0 -.~.n651 0.9071 2.3018 10.4841 -0.0070 0.0 0.0000 -0.0003 0.021 -Ge0718 0.3378 203838 111.48G5 0.0 - ~ . 0 7 1 8 (r.0078 2.3838 10.4845 -0.0078 0.0 0.0000 -0.0003 0.022 -0.0786 U.3C84 2.4698 10.4805 0.0 -0.0788 0.508Q 2.4698 10.4849 -0.0085 0 . n 0.0000 -3. on04 3.023 -&LC861 3. 0091 2.5597 10.4805 0.0 ' -@.OB61 0.0991 2.5598 10.4853 -C.0093 0.0 0.0000 -0.0004 3.024 -0.0937 O.OC98 2.6537 10.4805 0. 0 -0.0937 0.0098 2.6538 10.4857 -0.01Cl C.@ 0.0000 -3.0035 3.025 -0.1017 0.31C5 2.7516 1q.4805 3.0 -n.1017 n. 0 105 2.7517 10.4862 -D.0110 0.3301 0.0000 -0.000 6 0.026 -0.1lLt C.0113 2.8535 10.4805 0.0 - @ . l l O O 0.0113 2.8536 10.4866 -0.t119 0.0001 0.0900 -0.0007 5.027 -0.1166 0.3120 2.5593 13.4805 3. hi -0.1186 0.0120 2.9595 10.4870 -0.0126 C.COC1 n . oooq -0.0008 3.028 -0.1275 0.5127 3.0691 15.4805 0.0 -0.1276 0.0127 3.0693 10.4875 -0.0138 0.0031 0.0000 -0.0009 0.029 -0.1368 0.3135 3.1829 10.48P5 0.0 -0.1368 0.0 134 3.1831 10.4880 -0.CL48 o.oon1 0 . 0 0 3 0 -0.0011 3.330 -0.1464 3.3142 3.3050 13.4805 5.0 -1'.l464 0.0142 3.3009 10.4885 -0.0159 0.0001 3.3000 -0.0012 0.031 -0.1563 0.0149 3.4223 19.4805 3.0 -3.1563 0.0149 3.4226 10.4893 -0.0169 c.0002 0.0000 -0.0014 3.032 -0.1666 J n i l l S b 3.5483 1U.4805 0.0 -J. 1666 0.0156 3.5482 10.4894 -0.C181 c.0002 0.0033 -0.0016 0.033 -n.1772 0 . J164 3.6775 10.4835 e. 0 3 . 1 7 7 2 C.Cl64 3.6779 10.4930 -C.0192 0 . 0 0.0000 -3.0018 0.034 -6.1880 0.0171 3.8111 10.4805 3.0 -3.1880 O.t'l71 3.8114 16.4905 -0.02@4 c.. 0 0.3co3 -3.0020 3.335 -0.1992 0.0178 1 ; . 4805 3.3486 3.5 -0.1992 t.3179 3.9490 10.4910 -0.C1216 0.3 -0.0000 -3.0022 0.036 -0. 2106 0.01 84 4.0903 10.48C5 0.0 -3.213b U.Cl84 0.0010 -0.3090 -0.0025 9.037 +e2223 0.0191 4.2353 1 ; . 4805 0. c -0.2225 0.619i 0.3010 -0.3039 -0.0028 9.038 -J.234* 0. b198 4.3846 15.4105 5.0 -0.2346 0. c 198 0.0010 -0.1000 -o.ot-31

b.-e?.?-=0..2468 .." I J . 020c. 4.5373.10n.4805 0.0 - yp.247c

0.0204 C..O310 "0. PO??

:3. 00?4.

STORE Sample Output-Trajectories REFERENCES 1. Sandahl, CarlA. and Faget, Maxime A . : Similitude Relations for Free-Model Wind-Tunnel Studies of Store-Dropping Problems, NACA TN-3907, 1957.

2. Covert, E. E.: Wind Tunnel Simulation of Store Jettison with the Aid of Artificial Gravity Generated by Magnetic Fields, J. Aircraft, Vol. 4, No. 1, pp. 48-51 (Jan.-Feb.

1967).

3 . Scherberg, Max and Rhode, R. V.: Mass Distribution and Performance of Free Flight Models, NACA TN-268 (1927).

4. Neihouse, Anshal I. and Pepoon, Philip W.: Dynamic Similitude Betweena Model and a Full-scale Body for Model Investigation at Full-scale Mach Number, NACA TN-2062 (1950).

5. Ames Research Staff: Equations, Tables, and Charts for Compressible Flow, NACA Rept. 1135 (1953) (p. 19; Eq. Al) 6. Slater, John C., Frank, Nathaniel H . : Electromagnetism, McGraw-Hill.Book Company, Inc., 1947, pp. 69-74.

7. Bozorth, Richard M . : Ferromagnetism, Bell Laboratories Series, D. Van Nostrand Company, Inc. (1951).

8 . Terman, Frederick Emmons: Radio Engineers Handbook, McGraw-Hill Book Company, Inc. (1943).

9. Montgomery, D. Bruce: Solenoid Magnet Design, Wiley- Interscience (1969).

lo. McCracken, Daniel D.: A Guide to Fortran IV Programming, John Wiley and Sons, Inc. (1965).

11. Etkin, Bernard: Dynamics of Flight - Stability and Control, John Wiley and Sons, Inc. (1959).

* U . S . GOVERNMENT P R I M I N G OFFICE : 1972 735-029llo67 CR-1955 - 1