AD-A268 975

PL-TR-92-2168Instrumentation Papers, No. 344

EFFECTS OF MAGNUS MOMENTS ONMISSILE AERODYNAMIC PERFORMANCE

George Y. Jumper, JrC. J. Frushon, Capt, USAFR. K. Longstreth, Capt, USAFJ. Smith, Capt, USAF DTiCJ. Willett ' LID.ICD. Curtis EL. E

10 December 1991

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110 December 1991 Scientific, Final4. TITLE AND SUBTITIeE S. FUNDING NUMBERS

Effects of Magnus Moments on Missile Aerodynamic PE 62101FPerformance PR 66W . TA 12 WU 12

6. AUTHOr(S)George Y. Jumper, Jr*; C1 J. Frushon, Capt, USAF;R. K. Longstreth, Capt, USAF; J. Smith, Capt, USAF;J. Willett; D. Curtis

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Phillips Laboratory (SXAI) PL-TR-92-2168Hanscom AFB, MA 01731-5000. IP, No. 344

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*AFOSR Summer Research Fellow, 1990

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13. ABSTRACT (Maximum 200 words)

A 3 degree of freedom computer simulation was used to study a newconfiguration of the 2.75" Folding Fin Aircraft Rocket (FFAR) designed tocarry the payload for the REFS (Rocket Electric Field Sounding) program.Since the shell of the payload section of the new configuration spins at adifferent rate than the motor casing, relatively large moments from the Mag-nus forces have some effect on the aerodynamic characteristics of the missile.Aerodynamic coefficients were based on existing flight data of the FFAR andpredictions of the USAF Missile DATCOM program.

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Magnus effect, Sounding rocket, 3 degree of freedom, 36Euler angles 16. PRICE CODE

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Contents

1 INTRODUCTION 1

2 THEORY 22.1 Rotational M otion: ..................... 22.2 The M oments ......................... 9

2.2.1 Roll M oments: .................... 92.2.2 Pitch and Yaw Moments: .............. 92.2.3 Magnus Moments .................. 12

2.3 Solution of Equations of Rotation ............ 17

3 RESULTS 18

4 CONCLUSIONS 28

References 29

A0esaion For

I D'I'IS R&TIC1 TABUrnna 1'mai('od (-J

pmTfl QUALITY i2LCTED 3 _

tit .ud/or-

Dlat spornial

List of Figures

1 The REFS Missile .......................... 32 The Yaw-Pitch Euler Angle Coordinate System ............ 53 The Angular Velocities in the Yaw-Pitch Euler Angle Coordi-

nate System ................................... 64 Predicted Roll Output Compared to the Data of Winn . ... 105 Magnus Lift and Drag Figure from Reference 9, Originally

From Reference 10, With the Curve Fits Through the Exper-imental Data Used for This Study ...................... 13

6 Elevation View of the Pitch Plane and Normal View in thePitch Plane of the REFS Missile Showing Magnus Lift andDrag Forces ....... ............................ 16

7 Pitch Plane Moments on the REFS Missile With Positive For-ward Section Rotation as a Function of Time ............. 19

8 Pitch Plane Moments on the REFS Missile With NegativeForward Section Rotation as a Function of Time ........ .. 20

9 Total Angle of Attack Versus Time With Positive and NegativeForward Section Rotation ....... .................... 21

10 Total Angle of Attack for Two Values of Static Margin forBoth Positive and Negative Angle Forward Section Rotation 23

11 Euler Angle 0 vs Euler Angle tk for Positive and NegativeForward Section Rotation ....... .................... 24

12 The Bank-Pitch Euler Angle Coordinate System ........ .. 25

List of Tables

1 Parameters for REFS System Analysis ................. 18

iv

Acknowledgements

Research sponsored by the Air Force Office of Scientific Research, AirForce Systems Command, United States Air Force, under contract F49620-88-C-0053 to UES, Inc., Dayton, Ohio. The United States Government isauthorized to reproduce and distribute reprints for governmental purposesnotwithstanding any copyright notation hereon. The authors want acknowl-edge the support of G. Kirpa, R. Steves and C. N. Stark. Also, we areindebted to Prof. Raymond Hagglund for insight into the world of Eulerangles, and M.J. McCrossan for the art work.

V

Effects of Magnus Moments onMissile Aerodynamic Performance

1 INTRODUCTION

The goal of the atmospheric research program, Rocket Electric Field Sound-ing (REFS), is to determine the min'mum electrical field strength that wouldpose a lightning strike hazard to ascending rockets such as the Space Shuttle.The 2.75-in. Folding Fin Aircraft Rocket (FFAR) was selected to carry thescientific payload. The FFAR missile is aerodynamically unstable at launch,until the fins spring out. To minimize the effects of the instability, severalversions of the missil:. have nozzle designs that provide a roll moment duringthrust. When a spinning cylindrical body experiences any pitch or yaw tothe local windstream, a side force is produced which is proportional to boththla angle of yaw and the angular velocity of the spin. This is the well knownMagnus effect. If the center of pressure for this force is too far from the centerof mass, the resulting moments can cause the projectile to be dynamicallyunstable.

1

The FFAR configuratior,, which will be used for the REFS experiment,has the potential for higher than normal moments from the Magnus forcesdue to the fact that the outer shell of the payload section of the missile isdesigned to spin at a rate that is different from the rate of the rearwardportion of the missile, as shown in Figure 1. The net effect is that any pitchor yaw of the missile produces magnus forces of different magnitudes for eachportion of tbe body, resulting in a moment on the missile.

The purpose of this study is to perform an approximate analysis of the ro-tatioral motlion of the REFS missile to determine the potential for instability

RWceived for publication i July 1992

due to the Magnus moments.

2 THEORY

The motion of a vehicle in flight is governed by Newton's Laws of motion,which strictly apply to each point mass that makes up the object. Whenthese laws are properly applied to each infinitesimal portion of the body, thetotal effect on a rigid body can be represented by three scalar equations oflinear motion of the center of mass of the vehicle and three scalar equationsof rotational motion of the body about the center of mass. An analysis thatconsiders both linear and rotational motion is known as a six degree of free-dom analysis (6DOF). There are several computer models that perform thiscomplete analysis (for example, MASS); 2 howevr, these complete simula-tions are often complex and difficult to use to isolate specific cause and effectrelationships.

For this work, only the rotational motion will be analyzed. The magni-tude of the velocity of the missile is considered an independent variable whichis specified to vary with time in a way similar to that observed in test flights.The velocity vector is assumed to always point in the same direction andmissile rotations are analyzed relative to this fixed direction. This simplifica-tion differs from reality in two important ways. First, the change in velocitydirection as a vehicle moves along the trajectory provides a continual forcingfunction to change the angular position of the vehicle. In this regard, theanalysis can be considered incomplete and possibly non-conservative since aperturbation is usually only introduced at the start of the analysis and, if itdamps out, the vehicle is assumed stable. Second, this analysis ignores theeffect that angular position relative to the velocity vector has on the velocityitself. For a reasonable missile vehicle which does not experience very largedivergence angles from the velocity vector, this is not a problem except forthe thrusting phase of flight. While thrusting, the vehicle, in effect, reducesangular divergences by causing changes to velocity to be in the direction ofthe flight vehicle if the rocket nozzles are aligned with the missile. Conse-quently, the angular motions predicted during thrusting are larger than thosethat would be act.ally experienced in flight.

2.1 Rotational Motion:

The equations governing the rotational motion about the center of mass ofan object are known as Euler's Equations.3 The three scalar components arerepresented by the single vector equation:

M ,(1)

2

~2.75-

P +AP

Figure 1: The REFS Missile

3

whereM, is the moment about the center of mass,

andHe,, is the time derivative of the angular momentum vector about the centerof mass.

The change in the H vector must be with respect to either an inertialcoordinate system or a non-rotating coordinate system through the center ofmass of the body. The latter is most often used for analysis of flight systems.The forces and moments experienced by a flight vehicle are usually expressedin terms of a coordinate system that is fixed to and rotating with the vehicle.Consequently, it is not convenient to express these forces and moments interms of a noin-rotating system. The equation can remain expressed in terrmsof the convenient system provided the following identity is used to correctlydetermine the derivative relative to a non-rotating system.

d A . y . ( A + (O x A (2 )

where A is a general vector, 1 is the angular velocity of the rotating xyzsystem relative to the non-rotating XYZ system, and (A-,) is the timederivative of the vector as inferred by an observer located on the rotatingxyz system. With that identity, Euler's equation becomes:

M = (-- 9 + fl x H, (3)

The task at hand is to define the relationship of the convenient coordinatesystem to the non-rotating coordinate system. This is often accomplished bydefining a sequential set of rotations of the convenient system about one axisat a time, beginning with the convea'ient system aligned with the non-rotatingsystem. These rotations are referrei to as Eu!er Angles.

The most convenient system for +b-js problem is a simplification of thestandard way that body rotations are described for aircraft.4 The axis ofsymmetry of the aircraft is first aligned with the X axis of a system thatmaintains its origin at the center of mass of the aircraft, but does not rotate.The X axis is horizontal pointing forward, the Y axis points to the right ofthe aircraft, and the Z axis points down. This is shown in Figure 2. Forpurposes of this study, the velocity stays aligned with the X axis. The firstrotation is a positive rotation about the Z axis an angle 4, which resultsin a new X and Y axis position, next is a rotation about the new Y ,axisan angle 0. A full, body-fixed axis system would now incorporate any rollabout the new X axis in a roll angle 4. Since the REFS missile spins ata substantial rate about the new X axis, and since the missile is assumed

4

xY', Y.

Figure 2: The Yaw-Pitch Euler Angle Coordinate System

to experience forces and moments that are independent of the roll position,the coordinate system for this study will not go through this final rotation.The convenient xyz system has the x axis aligned with the axis of symmetryof the missile pointing forward, the y axis always remains in the horizontalplane toward the right when looking at the missile from the rear, and the xzplane is always a vertical plane, with the z axis orthogonal to the x and yaxes. Due to symmetry, the zyz system is a principal axis system with nocross moments of inertia and with I, equal to I,,, which will be referred toas I'. I., will be referred to as simply I.

The components of the angular velocity vector w, expressed in the con-venient system, are expressed as follows: the angular velocity p about the xaxis, q about the y axis, and r about the z axis. The angular momentum, in

Y x

* z

Figure 3: The Angular Velocities in the Yaw-Pitch Euler Angle CoordinateSystem

the body system coordinates, is simply:

H,,,, = Ip i +'q j + I'r k (4)

where , j, and k aie unit vectors in the xyz system. Note that the spinningouter shell of the payload section could affect the overall angular momentumof the venicle, but for this missile it has a negligible contribution and hasbeen ignored. The angular velocity must also be expressed in terms of theangular velocities 3f the Euler angles. By inspection of Figure 3, it can beseen that the body- fixed rates are related to the Euler angle rates by theequations:

w•= p = q•-•sin8

w,= q -' 0 (5)

z "r = •Cos

6

The angular momentum of the missile, expressed in terms of the Eulerangles, is: ,H = I ( - iksin0) i + I' j + P' t,cos0 k (6)

Since the z axis always points to the nose, the moments ot' inertia in thissystem are independent of the rotations T'he tinte der-ivative of A as inferredby an observer on the zyz system is then:

di , (7)+I'i j + I, (€cos0- ?0sin0) k

The rotation of the xyz coordinate system is the same as the rotation of themissile except for the final ý about the x axis:

0 = -ý,sin 0 i + 0 j + €cos0 k (8)

When the cross product and the time derivative are substituted back intoEuler's Equation, the following scalar equations result:

•.z= If- IisinO- I~i/cos0

FTy = 1'G+(I'-I)•]2 sinOcosO+I4iPqcos9 (9)

ET, = l'cosO-(21'- I)?OsinO - Iq4

The above procedures yield equations of ro'ational motion with all thekinematic terms as functions of the Euler angles and with moments computedrelative to the convenient body pointing, but not fi,'lly body fixed, coordi-nates. The aerodynamic forces and moments are defined and most easilyexpressed in terms of these body pointing coordinates. To have a completeset of equations for solution, equations relating position relative to the bodypointing system to the Euler angle system must be developed. These areobtained by determining the direction of the velocity vector in both systems.Since, for this work, the velocity vector is always aligned with X, the com-ponents of velk'city relative to the body system are simple functions of theEuler angles. The velocity vector in terms of the fixed system is:

V = V I+0 J+0 K (10)

To obtain the components in terms of the new basis vectors of the bodysystem, the components in the original system are left multiplied by the tworotation matrixes that describe first a positive rotation ?k about the Z axisfollowed by a positive rotation 0 about the new Y axis3 as follows:

u cos60 0 -sin 0][ cos 0 sin tk 0][V= 1 0 -sin i cos4' 0 0 (11)

sin 0 cosO 0 0 1

7

This results in the three components:

U = VcosOcostv = -Vsino (12)w = Vsin0cosb

The position of the body relative to the velocity vector is expressed in termsof the angle of attack, a, and the angle of sideslip, P. A useful angle is the"total angle of attack", 0, a single angle incorporating the effect of bothangle of attack .- nd angle of sideslip. These angles can be expressed iD termsof the Euler angles by applying the definitions:

tan a = w/Usin/5 = v/V (13)cosO = u/V

when the component;s from Eq. 12 are substituted into the above, the follow-ing identities result:

0 = -0 (14)

0 = cos-' (cos 0 cos p)

For the sake of completeness, it is useful here to develop the angle of attackand sideslip relationshi ' b elative to a complete body fixed coordinate system.The output of 6DOF iimulations is usually in terms of these angles, whichare subscripted with BF in this work to distinguish them from the bodypointing system. In or-cit, to express the velocity vector in a full body fixedsystem, one more rote ,ion must be accomplished: a rotation of a positiveangle 0 about the new X axis. This is accomplished by the matrix operation:[UBF 1 I 1 0 0 iVCos 0Cos,1

VBF = cos sin 1 -Vsin1' (15)WBF.ý 0 -sine coslJ Vsin0cosol

This results in the fH Ilowing equations for the velocity components:

UpPF = VcosOcosOVBF = -Vcos4,sink + Vsin4,sin0cos4' (16)WBF = Vsin4,sint+Vcos4,sin0costk

Now apply the definitions of the angles in Eq. 13 to obtain:

aBF = tan 1 (sin4tanbsecO+cos4tan0) (17)/3RF = sin- (-cos4sine + sin4,sin0cos ¢)

These are usually simplified by making the small angle assumptions for theangles ¢, a, aBF, and flBF. The total angle of attack is the same in eithercoordinate system.

8

2.2 The Moments

2.2.1 Roll Moments:

The FFAR configuration that was used for this project has four exhaustnozzles. Each nozzle is "scarfed," that is, truncated by a plane that is notnorrn!, tz t ý •z.,lc ccatcrliw. TLs results -n a tangnijtie.l complienit tothe thrust. The nozzles are scarfed in a direction that results in the sidethrust being perpendicular to the centerline of the missile. The magnitudeof the moment of these canted nozzle sets is not contained in any of theknown documentation. The value of the roll moment and the roll momentcoefficients used in this analysis were deduced from rotation data acquiredby William P. Winn.' The roll moments considered for the simulation arethe moment due to thrust, LT, and roll damping, Lp. When substituted intoa simple, one dimensional roll rotation equation, the result is:

/• = (LT + Lp)/I (18)

The roll damping moment, expressed in terms of the non-dimensional rolldamping coefficient, is:

L, = (2-)(p--) C1 , p (19)

2 2

where:

C4, = The roll damping coefficient (n.d.)S = Reference Area (missile cross sectional area) (ft2 )A = Reference Length (missile diameter) (ft)p = Air density (slugs/ft3 )

Winn's data showed a roll velocity increase from about 1 rev/s at the exitof the launch tube to about 24 revs/s at burnout, then what appeared to bean exponential decay. A Cip value of -0.25 produced an initial roll decay ratethat matched Winn's data after burnout. Then with this roll damping, therocket thrust moment in the roll direction was determined that produced theobserved spinup. The resulting moment was 0.349 ft-lbs at the full thrust of734 lb. This represents a thrust offset of less than one-half degree with themomentum arm of each nozzle approximately 0.75 in. from the center line.A comparison of the roll simulation to Winn's data is shown in Figure 4.

2.2.2 Pitch and Yaw Moments:

Excep.` for the moments resulting from the Magnus Effect, the moments usedin this analysis are standard for aircraft and missile analysis. Therefore, thesewill be treated only briefly. Since the vehicle is symmetrical, the coefficients

9

25

,>,,20 -Model

"-1,5 - Winn's Data

"o 10,

0II0 1 2 3 4 5 6 7 8 9 0

Time (seconds)

Figure 4: Predicted Roll Output Compared to the Data of Winn

that apply to an angle or Pngfdar velocity in the pitch axis are the same asthos._ that apply for the yaw axis. The following disicussion will be limitedto the aerodynamic forces and moments for the pitch axis. These must beapplied in a similar manner to the yaw axis as well. Most of the values forthe coefficients were obtained from the USAF Missile DATCOM computerprogram." This program is a special adaptation of the USAF Stability andControl DATCOM methodology" for missile shapes.

The primary pitching moment is the result of the sum of all of the inplaneforces that results when a vehicle is at an angle of attack to the velocityvector. Using the "linear aerodynamics" assumption, the normal force, N,,is computed as follows:

N, = CN2,QSa (20)

where:

CN -= The normal force coefficient (n.d.)Q -2pV 2, the dynamic pressure (lb/ft2)

Care must be taken since the coefficient is often expressed either per de-gree or per radian of pitch angle. This coefficient is the result of the contri-butions from all the aerodynamic surfaces of the vehicle, which are computedseparately. The full vehicle coefficient is not a straight sum of each of the

10

individual effects since the presence of other parts of the body will mod-ify the contribution of the individual surface. In this case, the parts of thevehicle are the body and the fins. There does not seem to be much interfer-ence since the total effect is approximately equal to the sum of CN,, firt, andCNa body- The fin contribution is between 50 and 60 percent of the total. TheDATCOM CNi,, values were entered into the analysis as a tabular functionof mach number, and CN,, fi,,, was estimated to be a constant 60 percent ofCN,. Required values are determined by interpolation of the data.

The pitching moment on the vehicle, M, is computed in terms of thenon-dimensional pitching moment coefficient as follows:

M.y = C.QSAa (21)

The moment about the center of mass of the normal forces on the vehicle canbe collectively expressed as a single total normal force acting at a point calledthe center of pressure, XC,. This is a function of Mach number and is one ofthe outputs of the DATCOM program. The DATCOM data is also enteredas a table in the 3DOF program. The position of the center of mass, Xcm,has been determined from the actual flight hardware for the time before themissile is fired and the time after the propellent has been expended. It hasbeen assumed that X,, varies linearly with time from launch to burnout,then stays constant for the remainder of the flight. For this analysis, theaerodynamic pitching moment coefficient is computed as a function of threetabulated functicits:

C.. = CN.(XCp - X.)/IA (22)

This coefficient normally is the basis of the static stability of the vehicleabout the pitch axis. If the coefficient is negative, the vehicle is staticallystable; that is, if there is a positive angle of attack, the resulting moment willbe negative, causing a restoration to zero angle of attack. The sign of thecoefficient is negative if the center of pressure is located behind the center ofmass (positive x is in the direction of the front of the vehicle). The differencebetween the two lengths is a measure of vehicle stability and is known as thestatic margin, often expressed in multiples of the missile diameter.

There is a moment that results from the rate of change of pitch, q. As-suming linear aerodynamics, this moment, Mq is expressed in terms of anon-dimensional coefficient as follows:

Mq = C,,,A,5Q • (23)

The '-pitch damping derivative" C,,q can be computed from the CN0 for eachcomponent and the distance between XCP for the component and Xt,,, for thevehicle. For a missile, the primary contributor to the coefficient is the tail

11

surface, so the following approximate expressions was used for this analysis:

Cmq = -2CN.,t .o[(Xp fins - Xe ,,,vicle)/A]2 (24)

A second damping moment is jet damping, which can be significant in thevery early phase of the flight. Jet damping is the term used to describe thereduction of angular momentum of a body that is losing mass. The formulafor jet damping is:

Mi = Wr(l1, - ) (25)

where:

MJvi = Moment about the ith axisWi = Angular velocity about the ith axisrh = Jet exhaust mass flow rateI.Li = the distance from the center of mass to the nozzle exit, in a

direction perpendicular to ith direction.= the radius of gyration about the i axis = I/•rn

2.2.3 Magnus Moments

The classical analysis of the Magnus effect considers the aerodynamic effectsof a spinning cylinder in a cross flow. A potential flow theory analysis, whichrepresents the spinning cylinder by a source/sink couple with circulation,resu!ts in the prediction of a normal force (lift) and no drag. As White'points out, experiments show that the lift is, at best, only about one-halfthat predicted, and spinning causes the drag to increase above the valueexperienced by a non-rotating cylinder. The results are nicely summarizedin Figure 5, from White. The theoretical (potential theory) coefficient of lift,using White's nomenclature is:

L 27raw (26)QC,,QS U•

where:

a = cylinder radius (ft)b = cylinder length (ft)L = Normal force (Lift) (lb)

•2 0PU (lb/ft2)

S = 2 b a, the cross sectional area of the cylinder (ft2)U.0 -= Cross flow velocity (ft/s)w = angular velocity of cylinder (1/s)p = fluid density (slugs/ft3 )

12

in

c.-Theory Fit Through Experimental CL Data.

2aw,Lx In ri men I eL : ..

C1 . C; •-

4 Fit Through Worst Slope of the CD Data.

aw

U- ., .Theory C

CO) - a

4 6 8

Velocity ratio. !

U-

Figure 5: Magnus Lift and Drag Figure from Reference 9, Originally FromReference 5, With the Curve Fits Through the Experimental Data Used forThis Study

13

The experimental values for both drag and lift coefficients tend to ap-proach asymptotic val:ies at high velocity ratios, which means at higl, ro-tation rates, the lift and drag are independent of the rotational velocity.Similarly, the experimental coefficient of drag at low rotation rates is inde-pendent of rotation rate. In any flat region, the Magnus force for the twoparts of the REFS mfissile would be identical to the result if the entire missilewere spinning at the same rate, and the Magnus moments would be no largerthan for a typical spinning missile. It is in those regions where the coefficientshave the greatest slope that one would experience the greatest difference inforce for the two spinning sections, and, hence, the greatest Magnus mo-ments. Since this analysis is only an r -proximate attempt to determine theworst case Magnus effect, the coefficients will be assumed to follow a linearrelationship to velocity ratio, using the value through the regions of greatestslope. Two additional lines are shown on the curve. The equations of theselines are:

CL = (27)U.0aw

CD = . - 1 (28)

The approximate Magnus results are now applied to the dual spinningcylinders of the REFS missile. Two views of the missile are shown in Figure 6.The view on the lower left is an elevation view that is normal to the pitchplane and shows the missile at a total angle of attack a to the free streamvelocity vector V. It is assumed that U0, for the Magnus calculation is thecross wind component of V, or V sin a, which is customary for projectileanalysis. It is further assumed that the Magnus force for each section of thecylinder is not influenced by the spinning of the other section. One couldimagine that the swirl induced by the front might modify the flow over theback, but that has been ignored here. The projected view on the upperright is a view from a location in the pitch plane looking in the negative zdirection. From this location, the velocity vector is below the missile, andthe cross flow velocity is pointing out of the page. Assuming positive angularvelocity p of the rocket motor (the rear section) and a positive increment ofangular velocity Ap of the front cylinder, then both Magnus lift vectors pointtoward the left, or the negative y direction, and the lift on the front is largerthan that of the rear. On this side of the missile, the tangential velocitydue to rotation is in the same direction as the cross flow velocity. Using thesubscript 1 to denote the front cylinder of the missile and 2 to denote therear cylinder, the magnitude of the lift on the second cylinder is:

L2 = (CL)2Q., 2 b2 a2 (29)

14

r1L2 = (2ap) (P112)2 a= (30)

L2 = p Vsina 2 a2 b2 p (31)

Similarly, the lift on the front of the cylinder is:

L=p Vsina 2 a2 b, (p+Ap) (32)

The moment caused by these parallel forces has a magnitude about the cmequal to the sum of the product of each force times its moment arm, Ii. Thedirection of the moment is out of the page, which is the negative z direction.Therefore, the vector expression for the moment is:

Tm•gL = -(L1t- L 2 12 ) k (33)

In the spirit of the approximate nature of this analysis, the followingsimplifying assumptions are made: the force center (center of pressure) ofeach Magnus force is the center of each cylinder, the cylinders have thesame radius, a, and the cylinders are each the same length, b. Further, itis assumed that the cm is approximately at the joint of the two cylinders,which makes each moment arm b/2. When applied, the resulting expressionfor the moment is:

T,,,JL = -pVa 2 b2 (Ap) sin a k (34)

Note that the moment is no longer a function of the rocket motor angularvelocity, only the difference between the rotational speeds.

The Magnus drag forces are also shown in Figure 6 in the elevation view.Note that if both p and Ap are positive, and if we are in a region of increasingCD, the drag in the front will be larger than that in the back, causing thereculting moment to be into the page, which is in the positive y direction. Ifthe above assumptions are applied to the "worst case" linear portion of thedrag curve, the result i3:

TMEa9D = lpVa'bP(Ap) sin a j (35)

The moment from the drag has a magnitude of one-half of that from thelift, and is potentially destabilizing since an increase in a results in a highervalue of the moment that is trying to increase a. This is resisted by theinherent aerodynamic static stability of the missile provided by the fins, sothe magnitudes of each must be examined to determine the stability.

15

y

D~2

?igTC ~ ~eV~i~f Viw o th pich lane and N4ornmal View in the Pitch

pla~ne of the REFS Missile ShowinS 1a68LftadDagFre

2.3 Solution of Equations of Rotation

The aerodynamic moments discussed above are explicit functions of the angleof attack and sideslip and their angular velocities. These can be convertedinto the euler angles and the euler angular velocities through Eq. 5 andthe angle of attack and sideslip conversions mentioned in the descriptionsof the coordinate systems. The result is a system of three coupled, non-linear, non-homogeneous, second order ordinary differential equations, withnon-constant coefficients. While the equations yield analytical solutions foronly some special simplifications, they can be solved easily with numericalmethods. Observe that j and •b are the only second derivatives in the y andz component equations. The value of -k can then be substituted into the xcomponent equation to arrive at an x component equation with ý as the onlysecond derivative.

]" (LT + Iýsin0+ 10 cos0)/I

=(T, - (I'- I)sin0cosO- l0 . cosO)/r (36)0= (ET, + (21'- 1)00 sin0 + IOq)/I'cos0

Notice that the third equation is undefined if 0 goes to 90 degrees. Thisshould not be a problem for a well behaved missile, since this angle is theangle of attack. In practice, these equations are treated as three first orderequations in the angular velocities, and three additional first order equationsfor the angles complete the equation set:

= fjT4 dt + 0of = fore dt+ o

Tk = 'foT dt+7 (37)

0 fý 0 dt +40

0 = frO dt+0o

Vk = fore dt+ Oo

The initial positions of the angles and angular velocities are all that is re-quired to start the solution. The strategy for this analysis was to provide aslight perturbation at the beginning and observe the progress of the solution.The perturbation was to set the initial angle of attack, Oo, equal to 5 degrees,the roll velocity, 7o is set to 1 revolution per second, and all of the otherangles and angular velocities equal to 0. As the solution progressed, if theangle of attack tended to increase, the missile was deemed unstable. If theangle of attack tended towards 0, the missile was deemed stable.

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Parameter Valuea 0.1163 ftb 3.0 ftCIF -0.25CN. 7.99 to 16.87 per radian1. 0.00496 to 0.00363 slug-ft2

/, and 1, 2.146 to 1.514 slug-ft2

LT 0.349 ft-lbsMass 0.7736 to 0.5774 slugs,h 0.1266 slugs/sS 0.04246 ft2

V 0 to 2400 ft/sX1. 41.9 to 35.9 in from noseXIP 52.8 to 69.1 in from noseAP -15 rev/s and +15 rev/sA 0.2325 ft

Table 1: Parameters for REFS System Analysis

3 RESULTSThe principal result of this analysis is that the REFS missile configurationis very stable. The conditions for the analysis are listed in Table 1. Theinherent static stability from the fins is well in excess of that required toovercome the moments from Magnus lift and drag. Figures 7 and 8 showthe moments about the pitch axis with positive and negative forward sectionspin respectively. In both cases, the Aero Restoring moment (from Cm,,)gets larger than 30 ft-lbs while the Magnus moments are on the order of lessthan 0.5 ft.lb. Note that the Aero Damping moment (from C,,q) and thejet damping moments are the second and third largest moments. The pitchplane moment from Magvus lift is lower than from Magnus drag in apparentcontradiction to what is expected from the coefficients. Pitch plane momentfrom Magnus lift is the result of angle of sideslip. Since the initial conditionis an angle of attack only, sideslip angle is never very large compared to theangle of attack, so the resulting moments are much lower.

The effect on total angle of attack is shown in Figure 9. In these plots,angle of attack is shown for the DATCOM predicted C.,, for one-half thatvalue, and for one-tenth of that value. In every case the missile is stable,although the positive spin case with the lowest coefficient is noticeably slowerin damping the disturbance.

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40!30-- Ap = +15

00-% 20 [

-J 10

4--

• -10-

C*o-

F -20-•

3- 0 - Aero Restoring Torque- - Aero Damping Torque

-40 4

1.0 - -- "

Ap =+15

,- 0.5-,

L.0

-° .. •'=

I - • "n

L... " :/, Torq ue0.0 0.5.. 1. 1 2.0

=e~~im (s,:ec"):

Figure.7: Pitc Pln.oet nteR F isl ihPstv owr

Set- Magnsif Tioe

- .I 1

Section RotaionMagnusFuLiftn Torque

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40 IIH

30- iAp = -15301 ±

21)-

-J.10"

-20--

-30 - Aero Restoring Torque-0 Aero Damping Torque

-40 +, I

1.0 IAp -15

0.5--

0.0 : " %'

0

O"-,,Magnus Drag Torque

I-I• • .:-

0.0 0.5 1.0 1.5 2.0

Time (sec)

Figure 8: Pitch Plane Moments on the REFS Missile With Negative ForwardSection Rotation as a Function of Time

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Ap = -15"Static Margin = 1.0

4- ..... Static Margin = 2.0Ap 15

Static Margin = 1.0,........... ......Static Margin = 2.00 3 \--

0'V.00

C 2 \"6•0

00.

I-I

0.0 0.5 1.0 1.5 2.0

Time (sec)

Figure 9: Total Angle of Attack Versus Time With Positive and NegativeForward Section Rotation

21

As mentioned above, the REFS missile static margin is high. The staticmargin is in excess of 5 missile diameters, while 1 to 2 diameters is moretypical. Figure 10 shows the total angle of attack versus time with the moretypical static margins of 1 and 2 diameters. These margins appear sufficientto overcome any Magnus moments.

An interesting result of this analysis is that the effects of the Magnusmoments are slightly less severe when the front section is spun in the negativedirection. This is especially evident in Figure 10, which shows both forwardspin directions on the same plot.

Finally, the one clear effect of front section spin is in the direction ofprecession taken by the missile as the nose converges to the center. Notice inEquation 34 that, given a positive angle of attack and no sideslip, the sign ofthe moment about the z axis is the negative of the sign of the forward spin.Therefore, a positive Ap results in an initial tendency for negative z momentresulting in an initially negative tendency in 0, which is a positive sideslipangle #. So, when viewed from the rear, the nose of the REFS experimentwould precess in a counterclockwise direction with positive Ap and wouldprecess in a clockwise direction with negative Ap. This is demonstrated inthe plots in which the Euler angle 0 is shown as a function of the Euler angle4' (Figure 11).

Stability Considerations:The results of the above analysis invite a more fundamental look at the

problem to determine when the Magnus effects could cause a disturbance tomissile flight.

The Bank - Pitch Coordinate System: The Yaw-Pitch Coordinatesystem was used to analyze the angular motion of the REFS missile. Thereis another coordinate system that is convenient for stability arguments andfor visualization of the moments on the missile. That is the "Bank - Pitch"system, which is often used to analyze the motion of a top or gyroscope.3 Asshown in Figure 12, this system begins with a rotation an angle ?k about theX axis, followed by a rotation of 0 about the new Y axis. Again, the missileis allowed to rotate an angle 0 about the new X axis, but the coordinatesystem does not perform this final rotation. In top or gyroscopic motion,the angle 0 is referred to as the nutation angle and 0 is called the precessionangle.

If the new angular definitions are applied to the Ealer equations in amanner similar to the initial part of this report, the following scalar equationsof rotational motion result:

ET, = I#+I4cosO-1 0sinOET, = l'i- (I,- I)4 2 sinOcos0 + I4'4sinO (38)

T, = I4sin + (21'2- 1)46cos6 - Iý

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- - Cma/2

4. ..... Cmaz b

04) 2

0 . .... ...

05

Ap= +15

-- Cma = base..... Cma/2 .

4. Cma/10

0

0 .... ...

0-_____

1.52.

0.0 0.5 1.0

Time (sec)

Figure 10: Total Angle of Attack for Two Values of static, Margin for Both

Positive and Negative Angle Forward Section Rotation

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7.5 I",.Delta p +l5rev/a

5.0 0.5" -1 T•TO0L 2.5

, 0.0)a,,-2.5 -- 1.0

-5.0

-7.5 '

7.5.Delta p = -I5rev/s

5.0 T=o0

U,ý

V 2.5 0.5.

IV 0.0~--oo

1.0.-2 -2.5 10

-5.0

-7.5 1 1 i

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5

Psi (degrees)

Figure 11: Euler Angle 0 vs Euler .Angle ik for Positive and Negative Forward

Section Rotation24

x

X,x'

Y

y', y I , z

z

Figure 12: The Bank-Pitch Euler Angle Coordinate System

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The advantage of this system is that the angle 0 is the total angle ofattack referred to as E above. In addition, there is no sideslip angle. Thisis a convenient system for analysis if the motion is similar to a well behavedtop, where the nutation angle stays greater than zero and does not changerapidly. In the Bank - Pitch system, the coordinate x always points to thenose, the coordinate y is always normal to the angle of pitch and points to theright when viewed from the back of the missile; the coordinate z is always inthe pitch plane and started out pointing downward before any rolling takesplace. This system lends itself the simplest visualization of the aerodynamicforces and moments, and, since the entire cross wind component is containedin the angle 0, it is a good system to use for stability analysis.

When the equations of rotation are solved for the highest derivatives, thefollowing equations result:

4)= (2.T,-Ii cosO+I 60sinO)/I0 = (Tv, + (I'- I)b 2 sin0cosO- I¢4)sinO)/I' (39)

= (ET. + (2r- I)6Ocos0- I)/rI'sin0

The first equation is for the spin of the missile. The right side of the thirdequation can be substituted for the ' term, which reduces the equation to asingle second order derivative. The second equation contains the moments forthe pitch plane and determines the total angle of attack, or "nutation" angle.The third equation, for bank, or "precession" angle, is undefined for a totalangle of attack, 0, equal to zero. This is similar to the gimbal lock problem ofa real gyroscope. The disadvantage of this system for long term prediction ofrotational motion of a missile is that the nutation angle (pitch) of a missiledoes change rapidly and sometimes gets very close to zero, or even goesthrough zero. The former causes very high rates of angular accelerations ofthe angle ¢, which slows down the computation; the latter causes the solutionto have a divide by zero error. Consequently, the set is useful for the initialportion of the simulation, but cannot be used as 0 goes to zero.

The typical static stability analysis involves setting all angular rates tozero (except for spin), setting the angle of attack (0) equal to some smallangle, and determining if the angular acceleration in pitch (0) is positive(unstable), negative (stable), or zero (neutrally stable). With all angularrates equal to zero, the pitch angular acceleration equation reduces to:

i = ETl,/I (40)

The moments in pitch are the following: aero restoring, aero damping, jetdamping, and Magnus drag. Magnus lift does not affect the static stabil-ity. With zero angular rates, the two damping terms are zero and equationbecomes:

(C,,aQSAa, + 2pVa2b2(Ap)sina)/IF (41)

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With small angles, the sine of the angle approaches the value of the angle inradians. Since a is the radius, it is one half the reference length A, and Q ispV 2 /2. With these substitutions, the equation becomes:

A2

= (C,,,,QSA a + Q-.b 2 (Ap) a)/I' (42)

For a rocket such as REFS, with fins in the rear, Ci, is negative, and,with a positive a, the contribution of the term is negative. If the forwardspin is negative, then the Magnus drag provides anther negative term thatworks with the aero restoring moment to provide additional static stability.If the forward spin is positive, the Magnus drag works against the restoringforce and could be destablizing. This is in agreement with the simulations.Observe the more rapid decrease in angle of attack ii Figure 11 when theAP is negative.

The value of C,,.a necessary for neutral stability condition is obtained bysetting 0 to zero:

b 2 A(Ap) (43)S4V

Note that Cin is dimensionless except that it is per radian. So long as C,,is more negative than the expression on the right, the missile is staticallystable. Remember that the Magnus drag moment was based on the worstslope of the drag curve, so the analysis is conservative and overpredicts thedestablizing effects for a good part of the flight regime.

Equation (41) indicates that the critical flight condition occurs at min-imum velocity. The typical velocity profile for the REFS mission is for themissile to leave the launch tube at about 100 ft/s and rapidly accelerate to2400 ft/s at burnout. There is a local minimum at apogee of about 140 ft/s.When this minimum and the missile parameters from Table 1 are substitutedinto the above equation, the minimum magnitude of the coefficient is about8.3. Recall that the C., can be expressed as the product of CNC,, times thestatic margin. At low speeds, this missile has a CNv,, of about 9.3. Thisindicates that the static margin for neutral stability is about 0.88.

The equation for precession (the ¢ equation) reinforces the previous ob-servation that the Magnus lift term controls the direction of precession. Forthe initial condition when the only angular velocity is the spin and the onlyangle is a small angle of attack, the equation reduces to,

ý = ET,/I' sine (44)

With no q or r angular velocity, there is no z damping, so the only momentis the Magnus lift moment:

= -p V a2 b2 (Ap) sin c/Isin 0 (45)

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With this coordinate system, the Magnus Lift moment remains along the zax!,. and forces the precession in the diroction opposite to the sense of theforward relative rotation rate, Ap. A stable missile will begin to develop anegative 6, adding a positive kinetic term to the ' equation. Eventually theprecession acceleration is balanced by damping terms.

A final stability question that might be addressed is one usually analyzedfor tops and gyroscopes: are there any solutions with constant angle of attack(nutation angle) and constant precession rate. Since the gravitational termfor a top is always a pure Ty, the typical top has only very small T, termswhich are associated with damping of the usually low precession rates. Withno i terms, and essentially no moments, then the third equation of rotationshows that there will be no change to the precession rate. Then, assumingthat there is little or no change to the spin rate, the second equation providesthe precession rate required to cancel out the gravitational moment.

For the REFS niissile, the static stability is so high that the missile neverstays at an appreciable angle of attack for very long. If the static maigin werereduced to near neutral magnitudes, then the Magnus lift moment provides amoment in the z direction, which is not present in top motion. The Magnuslift moment accelerates the precession until balanced by kinetic and dampingterms at an equilibrium precession rate. This rate would probably not be thecorrect precession rate to balance any aerodynamic moments in the nutationequation. Even a top does not automatically find the correct precessionrate to eliminate nutation, but must be coaxed into it, which takes a littlepractice. To find a constant nutation angle solution for missile flight withMagnus lift moments would demand a very contrived set of conditions.

4 CONCLUSIONS

The REFS missile will not suffer any problems from Magnus moments. Themissile is more aerodynamically stable than typical missiles. If the staticmargin were reduced to typical levels of 1 to 2, the missile should show noill effects. The static stability analysis indicated that if the static marginwere below 1, some problems could develop at apogee if the AP is positive.If direction of rotation of the front section relative to the rear is an openchoice, AP should be in the negative sense (counterclockwise looking fromthe rear). This will provide additional aerodynamic stability.

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References[1] Sturek, W.B., Dwyer, H.A., Kayser, L.D., Nietubicz, Reklis, R.P., and

Opalka, K.O., (1978) "Computations of Magnus Effects for a Yawed,Spinning Body of Revolution", AIAA Journal, Vol. 16, No. 7, p. 687.

[2] AVCO Systems Division (1974) MASS Program User's Manual, AVCOSystems Division, Maryland Operations, Seabrook, Maryland, preparedfor NASA Goddard Space Flight Center, Greenbelt Maryland, ContractNAS5-23231

[3] Shames, I.H., (1980) Engineering Mechanics, Statics and Dynamics, 3ed., Prentice-Hall, Inc., Englewood Cliffs, NJ

[4] Nelson, R. C., (1989) Flight Stability and Automatic Control, McGraw-Hill Book Company, Inc., New York

[5] Winn, W.P. (1969) National Center for Atmospheric Research, Boulder,Colorado, personal communication.

[6] Vukelich, S.R., Stoy, S.L., Burns, K.A., Castillo, J.A., and Moore, M.E.(1986) Missile DATCOM, Volume I - Final Report, AFWAL-TR*-86-3091, Flight Dynamics Laboratory, Air Force Wright Aeronautical Lab-oratories, Air Force Systems Command, Wright-Patterson Air ForceBase, Ohio 45433-6553.

[7] Hoak, D.E., and Finck, R.D., (1978) USAF Stability and Control Dat-com, AFWAL TR-83-3048, October 1960, Revised 1978.

[8] Nielsen, J.N., (1960) Missile Aerodynamics, McGraw-Hill Book Com-pany, Inc., New York

[9] White, F.M., (1979) Fluid Mechanics, McGraw-Hill Book Company,NY, p. 476ff.

[101 Rouse, H. (1946) Elementary Mechanics of Fluids, Wiley, NY

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