Expanded Equations for Torque and Force on a Cylindrical Permanent Magnet Core in a Large- Gap Magnetic Suspension System

NASA Technical Paper 3638 Expanded Equations for Torque and Force on a Cylindrical Permanent Magnet Core in a Large- Gap Magnetic Suspension System Nelson J. Groom Langley Research Center Hampton, Virginia National Aeronautics and Space Administration Langley Research Center Hampton, Virginia 23681-0001 February 1997

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Abstract The expanded equations for torque and force on a cylindrical permanent magnet core in a large-gap magnetic suspension system are presented. The core is assumed to be uniformly magnetized, and equations are deeloped for two orientations of the magnetization ector. One orientation is parallel to the axis of symmetry, and the other is perpendicular to this axis. Fields and gradients produced by suspension system electromagnets are assumed to be calculated at a point in inertial space which coincides with the origin of the core axis system in its initial alignment. Fields at a gien point in the core are defined by expanding the fields produced at the origin as a Taylor series. The assumption is made that the fields can be adequately defined by expansion up to second-order terms. Examination of the expanded equations for the case where the magnetization ector is perpendicular to the axis of symmetry reeals that some of the second-order gradient terms proide a method of generating torque about the axis of magnetization and therefore proide the ability to produce sixdegree-of-freedom control. Introduction This paper deelops the expanded equations for torque and force on a cylindrical permanent magnet core in a large-gap magnetic suspension system. The core is assumed to be uniformly magnetized, and equations are deeloped for two orientations of the magnetization ector. One orientation is parallel to the axis of symmetry, and the other is perpendicular to this axis. Fields and gradients produced by suspension system electromagnets are assumed to be calculated at a point in inertial space which coincides with the origin of the core axis system in its initial alignment with a reference inertial axis system. Fields at a gien point in the core are defined by expanding the fields produced at the origin as a Taylor series. The assumption is made that the fields can be adequately described by expansion up to second-order terms. The expansion of the fields and gradients is presented in appendix A. The equations for torques and forces on a magnetic core that are produced by a large-gap magnetic suspension system hae been presented and discussed in a number of papers. For example, see references 1 through 6. The torques on the core are usually approximated as a function of the external or applied fields at the centroid of the core, and the forces on the core are usually approximated as a function of the gradients of the applied fields at the centroid. It is generally assumed that terms that are a function of second-order or higher gradients of the applied fields at the centroid can be neglected. In practical applications that inole large-gap magnetic suspension systems, these assumptions hae proen to be alid. For an axisymmetric core, such as a cylinder, it can be shown that if the direction of magnetization is along the axis of symmetry, then the torque about that axis, produced by the applied fields and gradients of the applied fields, is always zero (ref. 3). Various methods of oercoming this constraint, which include shaping the core and using nonuniform three-dimensional magnetization, are discussed in references 4 and 7. Howeer, examination of the expanded equations for a cylindrical core reeals that for the case of uniform magnetization perpendicular to the axis of symmetry, some of the secondorder gradient terms proide a method of generating torque about the axis of magnetization and therefore the ability to produce six-degree-of-freedom control (ref. 8). For completeness, all gradient terms for expansion of the fields up to second order are presented. Finally, instead of deeloping the torque and force equations from a set of goerning equations that are a function of core olume, core magnetization ector, and suspension system fields and gradients, appendix B presents a deelopment that begins at a more fundamental leel in an attempt to proide better insight into the origin of these equations than is commonly aailable in the literature. Symbols A area, m 2 a B B radius of core, m magnetic flux density ector, T expanded magnetic flux density ector, T [ B] matrix of field gradients, T/m [ B ] matrix of expanded field gradients, T/m F δf I l total force ector on core, N force ector on incremental olume of core, N coil current ector, A length of core, m m magnetic moment ector, A-m 2

δm M Q m r T δt δt T m magnetic moment ector of dipole with incremental olume, A-m 2 magnetization ector, A/m pole strength, A-m position ector, m total torque ector on core, N-m torque ector on incremental olume of core, N-m torque on incremental olume of core about core origin, N-m inertial coordinate to suspended-element coordinate ector-transformation matrix U potential energy core olume, m 3 W δ work incremental olume, m 3 x, y, z coordinates in orthogonal axis system, m θ Euler orientation for 3, 2, 1 rotation sequence, rad gradient operator Subscripts: x, y, z components along x-, y-, z-axes, respectiely ij (ij)k Matrix notations: partial deriatie of i component in j-direction partial deriatie of ij partial deriatie in k-direction [ ] T transpose of matrix row ector A bar oer a symbol indicates that it is referenced to suspended-element coordinates. Magnetic Torques and Forces The torques and forces on a cylindrical permanent magnet core are deeloped in this section by integrating the equations for torques and forces on an incremental olume of the core with magnetic moment Mδ oer the core olume. These equations are deeloped in appendix B. Figure 1 shows the cylindrical core and the core coordinate system. The core coordinate system consists of a set of orthogonal x, y, z body-fixed axes that are initially aligned with a set of orthogonal x-, y-, z-axes fixed in inertial space. In order to define the fields and gradients z, z l/2 l/2 y, y x, x a Figure 1. Core coordinate system. 2

at any point in the core, the fields and gradients at the origin of the core axis system are expanded as a Taylor series. It is assumed that the fields can be adequately described by expansion up to second-order terms. The expanded fields and gradients in both inertial and core coordinates are presented in appendix A. For simplicity in deeloping the equations in this section, relatie motion between the core and the reference inertial coordinate system is assumed to be zero. This assumption remoes the requirement to transform between the inertial and core coordinate systems and eliminates a significant number of components which are small relatie to the fundamental terms in the equations when small-angle assumptions are used. In particular, the transformation of second-order gradient terms from inertial to core coordinates is ery complicated, as illustrated by equation (A14). The torque on an incremental olume of the core, about the core origin, can be written as δt δt + ( r δf) (1) where δt and δf are the torque and force on the incremental olume due to the field at that location, and r x y z is the position ector of the incremental olume (fig. 2). The total torque on the core can be written as T δt [ δt + ( r δf )] (2) where the integration is oer the core. Substituting equations (B20) and (B21) results in T {( M B ) + [ r ( M ) B ]} d The total force on the core can be written as F ( M )B d The term ( M )B can be written as (ref. 5) ( M )B [ B ]M (3) (4) (5) z δ r y x Figure 2. Incremental core olume. 3

where T y M x B z d + M x ( B xxz B xzx ) d (12) and B B xx B xy B xz B yx B yy B yz B zx B zy B zz (6) T z M x B y d + M x ( B xyx B xxy ) d Ealuating B xzy d first (13) M M x M y M z In equation (6) the notation f ij f i j is used. Magnetization Along Axis of Symmetry (7) For orientation of the magnetization ector along the axis of symmetry (x-axis) of the permanent magnet core, the only nonzero term in M is M x. Expanding equation (5) results in Substituting equation (8) into the second part of equation (3) and expanding results in Expanding the first part of equation (3) results in B xx [ B ]M M x B xy B xz ( B xyz + B xzy) r ( M ) B M x ( B xxz B xzx) 0 M B M x B z ( B xxy + B xyx) B y (8) (9) (10) B xzy d B xz y d + B ( xx)z xy d + B ( xy)z y 2 d + B ( xz)z zy d (14) where B xz has been expanded by using a Taylor series, as detailed in appendix A, and the notation f ( ij)k ( f i / j )/ k has been used. All the integrals inoling first-order terms in equation (14) are zero. Ealuating y 2 d yields l/ y 2 d y 2 ( z) a y 2 dy dz a y 1 ( z) l/2 (15) where y 1 ( z) y 2 ( z) a 2 z 2, a is the radius of the permanent magnet core, and l is the length (fig. 1). Since the area of the face of the permanent magnet core is A πa 2 and the olume is Al, equation (15) reduces to Substituting in equation (14) results in ( a 4 4)πl ( a 4 4)πl ( a 2 4) B xzy d ( a 2 4)B ( xy)z Ealuating B xyz d in a similar manner results in dx (16) (17) The components of equation (3) become 4 T x 0 + M x ( B xzy B xyz) d (11) B xyz d ( a 2 4)B ( xy)z (18) which is equal to equation (17). Therefore the torques about the x-axis due to second-order gradients cancel

out and T x 0 as expected. Going next to equation (12), Finally, noting that B (ij)k B (ik)j B (jk)i... and collecting terms, the components of torque become B z d B z + ( l 2 24)B ( zx)x + ( a 2 8)B ( zy)y + ( a 2 8)B ( zz)z (19) T y T x 0 M x B z M x [( a 2 4) ( l 2 8 ) ]B ( xx)z M x ( a 2 8) ( B ( yy)z + B ) ( zz)z (25) (26) and ( B xxz B xzx ) d [ ( a 2 4) ( l 2 12) ]B ( xx)z Substituting into equation (12) results in (20) T z M x B y M x l 2 2 + [( 8) ( a 4 ) ]B( xx)y M x ( a 2 8) ( B ( yy)y + B ) ( yz)z (27) The force on the core, equation (4), can be ealuated in a similar manner. The components of equation (4) are T y Mx B z + ( l 2 24)B ( zx)x + ( a 2 8)B ( zy)y F x M x B xx d (28) + ( a 2 8)B ( zz)z + M x ( a 2 4) ( l 2 12) B ( xx)z Continuing to equation (13), (21) F y M x B xy d F z M x B xz d (29) (30) B y d B y + ( l 2 24)B ( yx)x + ( a 2 8)B ( yy)y and + ( a 2 8)B ( yz)z ( B xyx B xxy ) d [ ( l 2 12) ( a 2 4) ]B ( xx)y Substituting into equation (13) results in (22) (23) Expanding the integral of equation (28) results in B xx d ( B xx + B ( xx)y y + B ( xx)z z + B ( xx)x x d Since the integrals containing first-order terms in and z are zero, equation (31) reduces to F x M x B xx d M x B xx (31) x, y, (32) Ealuating equations (29) and (30) results in T z M x B y + ( l 2 24)B ( yx)x + ( a 2 8)B ( yy)y F y M x B xy (33) + ( a 2 8)B ( yz)z + M x [( l 2 12) and ( a 2 4) ]B ( xx)y (24) F z M x B xz (34) 5

Magnetization Perpendicular to Axis of Symmetry For orientation of the magnetization ector perpendicular to the axis of symmetry (x-axis), the only nonzero term in M is M z. Equation (5) becomes B xz [ B ]M M z B yz B zz and the second part of equation (3) becomes ( B yzz + B zzy) r ( M ) B M z ( B xzz B zzx) ( B xzy + B yzx) The first part of equation (3) becomes M B M z B y B x 0 Substituting (36) and (37) into (3) results in T x M z B y d + M z ( B zzy B yzz ) d T y M z B x d + M z ( B xzz B zzx ) d T z 0 + Mz ( B yzx B xzy ) d (35) (36) (37) (38) (39) (40) Ealuating the integrals as before and collecting terms results in T x M z B y M z ( l 2 24)B ( xx)y M z ( a 2 8) ( B ( yy)y + B ( yz)z ) T y M z B x + M z [( 3a 2 8) ( l 2 12) ]B ( xz)z + M z ( l 2 24)B ( xx)x + M z ( a 2 8)B ( xy)y T z M z [( l 2 12) ( a 2 4) ]B ( xy)z (41) (42) (43) The components of the force (eq. (4)) using equation (35) becomes Discussion of Results (44) (45) (46) Examination of equations (25) to (27), (32) to (34), (41) to (43), and (44) to (46) reeals that, for expansion of the applied fields up to second-order terms, no coupling exists between force and torque components. As stated earlier, it is generally assumed that the higher order torque terms, which are functions of second order gradients, can be neglected. Howeer, equation (43) indicates that for magnetization perpendicular to the axis of symmetry, torque about the axis of magnetization can be generated by controlling a higher order term directly, thus allowing the core to be controlled in six degrees of freedom. For a cylindrical permanent magnet core magnetized along the axis of symmetry, from equation (25), only fie-degree-of-freedom control is possible. Concluding Remarks This paper has deeloped the expanded equations for torque and force on a cylindrical permanent magnet core in a large-gap magnetic suspension system. The core was assumed to be uniformly magnetized, and equations were deeloped for two orientations of the magnetization ector. One orientation was parallel to the axis of symmetry of the core and the other was perpendicular to this axis. It is generally assumed that terms that are a function of second-order or higher gradients of the applied fields can be neglected. In practical applications inoling large-gap magnetic suspension systems, these assumptions hae proen to be alid. Howeer, in the case where the magnetization ector is perpendicular to the axis of symmetry of the core, the expanded equations indicate that torque about the magnetization ector can be produced by controlling a second-order gradient directly. This case allows the core to be controlled in six degrees of freedom whereas a cylindrical permanent magnet core magnetized along its axis of symmetry can be controlled only in fie degrees of freedom. NASA Langley Research Center Hampton, VA 23681-0001 October 24, 1996 F x M z B xz d F y M z B yz d M z B xz M z B yz F z M z B zz d M z B zz 6

Appendix A Expansion of Fields and Gradients About the Nominal Operating Point of a Cylindrical Permanent Magnet Core In appendix A the fields and gradients produced by the suspension system electromagnets are expanded by using a Taylor series about the initial suspension point of the permanent magnet core. The assumption is made that the fields can be adequately described by expansion up to second-order terms. Figure 1 shows the cylindrical core and core coordinate system. The core coordinate system consists of a set of orthogonal x, y, z body-fixed axes that define the motion of the core with respect to an orthogonal x, y, z system fixed in inertial space. The core coordinate system is initially aligned with the x, y, z system. The transformation from inertial coordinates to core coordinates is gien by x y z [ T m ] (A1) where [ T m ] is the orthogonal transformation matrix for a 3, 2, 1 (z, y, x) Euler rotation sequence and is defined as x y z x where r y and is the gradient operator. Using z compact notation, each element of B in equation (A3) can be written as where and 2 B ---------- i r 2 B i B i B i --------r ( 1 2)r T 2 B i + + ----------r r r 2 B i -------- r ( B i x) ---------------------------- x ( B i y) ---------------------------- x ( B i z) --------------------------- x B i -------- x B i -------- y B i -------- z ( B i x) ---------------------------- y ( B i y) ---------------------------- y ( B i z) --------------------------- y ( B i x) ---------------------------- z ( B i y) ---------------------------- z ( B i z) --------------------------- z (A4) (A5) (A6) Using the notation f ij f i / j and f ( ij)k ( f i / j)/ k, equations (A5) and (A6) can be written as [ T m ] cθ z cθ y sθ z cθ y sθ y ( cθ z sθ y sθ x sθ z cθ x ) ( sθ z sθ y sθ x + cθ z cθ x ) cθ y sθ x ( cθ z sθ y cθ x + sθ z sθ x ) ( sθ z sθ y cθ x cθ z sθ x ) cθ y cθ x and B i -------- B r ix B iy B iz (A7) (A2) where sin has been shortened to s, cos has been shortened to c, and θ z, θ y, and θ x are angles of rotation about the z-, y-, and x-axes, respectiely. The field B and gradients of B produced by the suspension system electromagnets, which are fixed in the inertial frame, are calculated at the origin of the x, y, z system. Expanding B about the origin of the x, y, z system as a Taylor series, up to second order, results in The first-order gradients of B can be written as where 2 B i ---------- r 2 B ( ix)x B ( ix)y B ( ix)z B ( iy)x B ( iy)y B ( iy)z B ( iz)x B ( iz)y B ( iz)z ( B i / j) B ij B ij + ------------------------- r r (A8) (A9) B B + ( r )B + ( 1 2) ( r ) 2 B (A3) ( B i / j) ------------------------- B r ( ij)x B ( ij)y B ( ij)z (A10) 7

The expanded fields can be expressed in core coordinates as Transforming back into core coordinates, B B + ( r )B + ( 1 2) ( r ) 2 B (A11) B x B x -------- [ T r m ] T r ( 1 /2 )r T [ T m ] 2 B x + + ----------- r 2 [ T m ] T r x where r y is the displacement in core coordinates, z B [ T m ]B, and [ T m ]. Since, from equation (A1), r [ T m ] T r (A12) each element of B can be expanded in inertial coordinates by substituting equation (A12) into equation (A4), B i B i B i -------- [ T r m ] T r ( 1 2)r T [ T m ] 2 B i + + ---------- r 2 [ T m ] T r (A13) B [ T m ] B y B y -------- [ T r m ] T r ( 1 /2 )r T [ T m ] 2 B y + + ----------- r 2 [ T m ] T r B z B z -------- [ T r m ] T r ( 1 /2 )r T [ T m ] 2 B z + + ----------- r 2 [ T m ] T r (A14) The expansion of equation (A14) can be simplified by using small-angle assumptions (ref. 6). Under smallangle assumptions, cosθ 1, sinθ θ, and products of angles are neglected. The transformation matrix [ T m ] then becomes [ T m ] 1 θ z θ y θ z 1 θ x θ y θ x 1 (A15) 8

Appendix B Torques and Forces on a Magnetic Dipole With Incremental Volume The torques and forces on a magnetic dipole in a steady magnetic field are identical to those on an infinitesimal current loop with the same magnetic moment (ref. 9). Therefore, the equations for torque and force on an infinitesimal current loop will be deeloped first by using the fundamental relationship for the force on a current-carrying-conductor element in a uniform, steady magnetic field. For a discussion of magnetic dipoles and infinitesimal current loops, see references 9 and 10. Infinitesimal Current Loop Consider a plane loop of conductor with steady current I located in the external, uniform, steady magnetic field B (fig. B1). In this region B B 0. The force on an element dl of the conductor is gien by the fundamental relationship (obtained from the Lorentz force law) df I dl B (B1) where df is a ector indicating magnitude and direction of force on the conductor element; I is the scalar magnitude of the current in the conductor element; dl is a ector whose magnitude equals the length of the conductor element and whose direction is in the positie direction of the current; and B is a ector indicating magnitude and direction of the flux density of the external field component. The torque on the loop can be written as [ T I r ( dl B ) ] (B2) where r is the position ector of dl and the integration is around the loop. By using the identity r ( dl B ) dl( r B) B( r dl) equation (B2) can be written as T I ( r B)dl B r dl ] [ (B3) (B4) Using Stokes s theorem and a related result (ref. 9, p. 289), the line integrals in equation (B4) can be transformed into surface integrals resulting in T I [ da ( r B) ] B ( r ) da s s (B5) where da is a ector whose magnitude is a differential area and whose direction is normal to the plane of the current loop in the sense of the right-hand rule relatie to the direction of current flow, is the gradient operator, and the integrals are oer the surface that is defined by the conductor loop. Since r is zero and ( r B ) B for constant B, equation (B5) simplifies to Taking the integral results in (B6) (B7) An infinitesimal current loop can be defined by letting A go toward zero and I go toward infinity, keeping the product IA finite. For an infinitesimal current loop, the requirement that B be uniform no longer exists. The product IA is called the magnetic moment of the loop and is designated by the letter m. Therefore equation (B7) becomes (B8) The torque T acts on the infinitesimal current loop in a direction to align the magnetic moment m with the external field B. If m and B are misaligned by the angle θ, the magnitude of the torque is (B9) To increase θ by the amount d θ, work dw must be done against the torque T resulting in an increase in potential energy du: (B10) The potential energy of an infinitesimal current loop in an external magnetic field can then be obtained by integrating equation (B10): (B11) where the constant of integration is chosen to be zero when m is perpendicular to B. The force on the infinitesimal current loop can be obtained from equation (B11). If an external force F displaces the infinitesimal current loop by the infinitesimal distance dr, then the work done dw will be equal to a decrease in potential energy, du: Therefore T I ( da B) s T IA B T m B T mbsinθ du dw T dθ mbsinθdθ U mbcosθ m B dw F dr du U dr F U ( m B) (B12) (B13) 9

The right-hand side of equation (B13) can be expanded as ( m B) m ( B) + ( m )B + B ( m) + ( B )m (B14) Since ( B), ( B), and ( m) are zero, equation (B13) can be written in the form F ( m )B (B15) This form is generally used in the deelopment of the equations for large-gap magnetic suspension systems (refs. 1 through 6). Magnetic Dipole With Incremental Volume The magnetic moment of a permanent magnet dipole with north and south poles separated by length l and with pole strength Q m is defined as m Q m l (B16) The magnetic moment m is a ector pointing from the south pole to the north pole. In the case of an actual magnet, Q m and l may be indefinite but m can be determined and is sufficient to specify the fields of the magnet at a large distance from it. At large distances, a magnetic dipole with magnetic moment Q m l can be treated the same as an infinitesimal current loop with magnetic moment IA and is identical in effect if Q m l IA. Therefore, in a steady magnetic field B the equations for torques and forces on a magnetic dipole with magnetic moment m are the same as equations (B8) and (B15). In theory, it can be assumed that a permanent magnet of a gien olume consists of a large number of uniformly distributed permanent magnet dipoles with incremental olumes δ which are oriented in the same direction. The magnetic moment δm of a gien dipole with incremental olume δ can be coneniently described by a quantity called the magnetization M, which is defined as the magnetic moment per unit olume. That is, (B17) The total magnetic moment m for a gien permanent magnet can then be written as (B18) where the integration is oer the olume of the permanent magnet. Magnetization is also a ector and has the same direction as m. If the permanent magnet is uniformly magnetized, that is, M is constant oer the olume of the permanent magnet, then (B19) For a discussion of magnetic dipoles and magnetization, see reference 10. The torques and forces on an incremental olume of permanent magnet material, in terms of the magnetization M, can then be written as and M m m δm δ from equations (B8), (B15), and (B17). M d M δt ( M B)δ δf ( M )Bδ (B20) (B21) 10

z I dl r B y x Figure B1. Plane loop of conductor with steady current I in uniform steady magnetic field B. 11

References 1. Basmajian, V. V.; Copeland, A. B.; and Stephens, T.: Studies Related to the Design of a Magnetic Suspension and Balance System. NASA CR-66233, 1966. 2. Stephens, Timothy: Design, Construction, and Ealuation of a Magnetic Suspension and Balance System for Wind Tunnels. NASA CR-66903, 1969. 3. Coert, Eugene E.; Finston, Morton; Vlajinac, Milan; and Stephens, Timothy: Magnetic Balance and Suspension Systems for Use in Wind Tunnels. Progress in Aerospace Sciences, Volume 14, D. Küchemann, P. Carriére, B. Etkin, W. Fiszdon, N. Rott, J. Smolderen, I. Tani, and W. Wuest, eds., Pergamon Press, 1973, pp. 27 107. 4. Britcher, C. P.: Some Aspects of Wind Tunnel Magnetic Suspension Systems With Special Applications at Large Physical Scales. NASA CR-172154, 1983. 5. Groom, Nelson J.: Analytical Model of a Fie Degree of Freedom Magnetic Suspension and Positioning System. NASA TM-100671, 1989. 6. Groom, Nelson J.; and Britcher, Colin P.: Open-Loop Characteristics of Magnetic Suspension Systems Using Electromagnets Mounted in a Planar Array. NASA TP-3229, 1992. 7. Goodyer, M. J.: Roll Control Techniques on Magnetic Suspension Systems. Aeronaut. Q., ol. xiii, Feb. 1967, pt. 1, pp. 22 42. 8. Groom, Nelson J.: Description of the Large Gap Magnetic Suspension System (LGMSS) Ground-Based Experiment. Technology 2000, NASA CP-3109, ol. 2, 1991, pp. 365 377. 9. Clemmow, P. C.: An Introduction to Electromagnetic Theory. Cambridge Uni. Press, 1973. 10. Kraus, John D.: Electromagnetics. McGraw-Hill Book Co., Inc., 1953. 12

Form Approed OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to aerage 1 hour per response, including the time for reiewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reiewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Serices, Directorate for Information Operations and Reports, 1215 Jefferson Dais Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503. 1. AGENCY USE ONLY (Leae blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS 6. AUTHOR(S) REPORT DOCUMENTATION PAGE February 1997 Technical Paper Expanded Equations for Torque and Force on a Cylindrical Permanent Magnet Core in a Large-Gap Magnetic Suspension System Nelson J. Groom WU 505-64-70-03 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) NASA Langley Research Center Hampton, VA 23681-0001 8. PERFORMING ORGANIZATION REPORT NUMBER L-17495 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) National Aeronautics and Space Administration Washington, DC 20546-0001 10. SPONSORING/MONITORING AGENCY REPORT NUMBER NASA TP-3638 11. SUPPLEMENTARY NOTES 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unclassified Unlimited Subject Category 31 Aailability: NASA CASI (301) 621-0390 13. ABSTRACT (Maximum 200 words) The expanded equations for torque and force on a cylindrical permanent magnet core in a large-gap magnetic suspension system are presented. The core is assumed to be uniformly magnetized, and equations are deeloped for two orientations of the magnetization ector. One orientation is parallel to the axis of symmetry, and the other is perpendicular to this axis. Fields and gradients produced by suspension system electromagnets are assumed to be calculated at a point in inertial space which coincides with the origin of the core axis system in its initial alignment. Fields at a gien point in the core are defined by expanding the fields produced at the origin as a Taylor series. The assumption is made that the fields can be adequately defined by expansion up to second-order terms. Examination of the expanded equations for the case where the magnetization ector is perpendicular to the axis of symmetry reeals that some of the second-order gradient terms proide a method of generating torque about the axis of magnetization and therefore proide the ability to produce six-degree-of-freedom control. 14. SUBJECT TERMS Magnetic suspension; Large-gap magnetic suspension; Magnetic leitation; Magnetic suspension model 17. SECURITY CLASSIFICATION OF REPORT 18. SECURITY CLASSIFICATION OF THIS PAGE 19. SECURITY CLASSIFICATION OF ABSTRACT Unclassified Unclassified Unclassified 15. NUMBER OF PAGES 16. PRICE CODE 20. LIMITATION OF ABSTRACT NSN 7540-01-280-5500 Standard Form 298 (Re. 2-89) Prescribed by ANSI Std. Z39-18 298-102 13 A03