AIAA-1970-533-510

bY JOHN D. NICOLAIDES University of Notre Dame Notre Dame, Indiana

T SS Y

A HISTORY OF ORDNANCE FLIGHT DYNAMICS

John D. Nicolaides" University of Notre Dame

Abstract

Ear ly ordnance flight dynamicists were concerned with the Particle Trajectory. Trajectory computation and drag reduction were their pr imary interests, However inaccuracies in ordnance due to projectile wobbling motion and yaw of repose lead to the development of a linear theory for the angular motion of a missile. This theory was investigated and perfected; first, by yaw card fir - ings and, later, by the development of the Aero- ballistic Range. The importance of the static moment, the dampirg moment, the magnus moment and the asymmetry moments were emons t r a t ed by eve r improving experiments and by ever improving linear theory .

The introduction of more sophisticated ordnance weapons revealed serious nonlinearities in the aerodynamic force and moment system which could not be ignored. Nonlinear Magnus moment lead t h e way, followed by a nonlinear roll moment and side moment which produced Catastrophic Yaw. Nonlinearities in the dynamic damping moment also appeared.

Various nonlinear theories and analysis techniques have been proposed and utilized in con- junction with new testing techniques in the range, in t h e wind tunnel, and in actual weapon flight tests.

At this point in history, where very sophisticated theories for the flight dynamics of ordnance exist and where excellent dynamic testing techniques are available, we find that serious flight dynamic stability problems and inaccuracies plague al l ordnance ; projectiles, bombs ,rockets, magnus rotors , re-entry missiles,parachutes. . . .et a l . . . .

It is esser.tia1 that the professional ordnance flight dynamicist redouble his efforts to imprcve the performance and accuracy of ordnance weapons.

Introduction

The origill of Exterior Ballistics, which is concerned with the flight performance of ordnance type missi les , may be traced to earliest rnan.Each of t h e ancient civilizations had their ballisticians who were often also t h e i r astronomers and their mathematicians, and sometimes even their philosophers; Aristotle, Philon, Uzziah ,Empedocles, Phi l ipnos ,et al . '4

Gal i led is generally regarded as the first

*Professor of Aero-Space

1

modern exterior ballistician . However many famous men, have followed in his steps; Newton, Torricelli , Robbins,Bernoulli.Euler .Tavlor-

I a ---,

D' Alembert , LaFange , Lapla&, Legendre, Mach, Zahm,et aLG4' A'

The problem of original concern was the determination of the trajectory of a missile, . . . its path thru the sky,with special emphasis on its impact point and on its accuracy.This is the classical problem of the Particle Trajectory, Fig. 1, which has occupied so much attention over the years and has greatly predominated exterior ballistics. An excellent history of both the analytical and the ape r imen ta l work carr ied out by the various principal investigators over the yea r s on the parti- cal trajectory problem i s g i g n in the last chapter of the book by McShane et al. No attempt will be made here to duplicate that account to which the reader is referred. In summary the primary research areas of the particle trajectory are gravity, ae rodynam ic drag, and rn athem at ical methods. The modern high speed computer was originally developed to compute the particle trajectory. se~ss.s

# #

DRAG c

/ 1

/ c

/ /

I

t-* t a

GRAVITY f \ \ \ \ \ \

Figure 1. Particie Trajectory

The advent of the elongated rifled projectile lcad to missile wobbling motions which have been neglected by the particle dynamicists and only re- centlv have been recognized to be of primary importance in t h e performance of projectiles. This wobbling motion produces large drag and, thus, it is not possible to compute a proper projectile trajectory unless this wobbling motion is fully con- s idere d .

The performance of fin stabilized missiles is also quite adversely affected by their wobbling motion. As a resul t ,our primary concern in t h i s present review will be with the history of the under standing of the wobbling motion of miss i l e s ,b th projectiles and finned missiles.

Rigid Sody Motion

Linear Flight Dynamics When fired.the elongated rifled projectile

could actually de seen to wobble in flight, even at supersonic speeds. In order to eliminate this wobbling motion extensive test firings were carried out on the various proving grounds.Even when the wobbling motion was eliminated by trial and error , the projectiles were observed to fall considerably to the right of the expected impact point. When the same projectile was given an opposite spin (counterclockwise) it would fall to the left. Clearly a non-particle problem was evolved. The understanding of this problem of "drift"3ccupied some early investigators. McShane et a1 suggeat that the drift was first explained by Fowlc&gt al; however the e x c e l k ~ t texts by Charbonnier and by Lancaster , particularly the latter treat the phenomena in a correct and excellent manner.

in fbllowing the curved trajectory has a pitching velocity which produces an aerodynamic damping moment.Such a moment acting on a clockwise spinning shell causes a right yaw or angle of sideslip.The horizontal lift resultiizg from this sideslip causes the observed d r i f t to the right. A magnus force also exists which tends to increase the range of the projectile. A greater contribution to range increase is the action of the static yaw moment which produces an angle of attack which i n t u r n produces a lift up in the vertical plane.

43 Larxaster p i n t s out that a spinning projectile

Due to recent excellent measurements of the wobbling motioqzc&actual projectiles in flight by Hayden,Hazeltine, and Kline, we now know that projectiles generally wobble to a considerable ex - tent throughout their entire flight .This wobbling motion can be serious at any point in the trajectory, however, the transonic flight regimes and subsonic flight regimes seem quite critical. A large wobbling motion creates large drag which causes the projectile to fall "short", thus , missing the target and often injuring our own close support troops. Shorts are a "no no", and must be elirninated.Thus , it is vita€ to fully understand the wobbling motion of missiles, be they projectiles, bombs,rockets, or re-entry missiles.

Originally, the wobbling motiori of t h e spin stabilized shell was compared with the motion of a simple "top" and three types of motion were identified (1) unstable flight (2) superstahle flight and (3) stable flight, Too little spin resulted in unstable flight, an too much spin produced super- stable flight * and the right spin was alleged to pro- duce stable flight. While t h e first two statements are true, the latter is not since even with the right spin the projectile can be dynamically unstable and fhus wobble violently and fall short.

In tk. case of a projectile with under spin, the 8

static overturning moment is to large for the small gyroscopic term and thus the projectile wobbles just like a top with to little spin.We can see this phenomena each weekend by watching a football when passed with too little spin.(%orts are also serious in football since they are easily intercepted).

In the case of a projectile with over spin, the gyroscopic term overwhelms the static moment and thus the projectile is rigid in space. It will not wobble, or change direction, or trail (i. e. follow the trajectory). A football punted with too much spin wil l keep its nose pointing towards the sky and actually bit tail first.Even today some operational p-ojectiles hit the ground tail first.

I

In brder to avoid these two f&$t failures, the early exterior ballisticians Cranz, Greenhil1,et a1 used thq Stability Factor:

( 1 ) pa 1; '41: rl, MdZ cpf&u'Sd = .I-

If S is greater than 1,under spin is avoided. If S is less than 2.5, serious overspin may be avoided. !n order to obtain stable flight of a projectile it is necessary (but not sufficient) to have 6 near 1.6 (generally; 1.2 C < 2 .0) .

If the projectile and the top have a proper stability factor, the wobbling motion will be com- posed of two components, a nutational motion (a fast motion) and a precessional motion (a slow motion) as pointed out by the early flight dynam icists.

Figure 2. Epicyclic Motion

In order to achieve dynamically stable flight it is essential that both of these motions damp to small size. If one or both do not damp, then the projectile will continue to wobble. Our primary concern, therefore, is to insure that both the

2

nutation arm and the precession arm damp:

Dynamic Stability

In the case of the "top" where just the overturning moment due to gravity is acting, the nutational and precessional motions will neither grow or damp since,hNtp=O and thus the wobbling motion persists.

In the case of the projectile,there are other aerodynamic forces and moments which act in addition to the static normal force and its overturning moment. It is the proper combination of these other aerodynamic forces and moments that produces projectile flight dynamic stability.

The various investigators over the years have had difficulties in identifying the important aerodynamic forces and moments, and have had difficulties in determining their proper balance tor dynamically stable flight. W e are now able to identify the important aerodynain ic forces and moments as the normal fvrce and its static moment, the damping force and its damping moment*, and the magnus force and its niagnus moment. Of these only t h e damping moment, the magnus moment, the normal force and the static moment are vital to projwti le dynamic j i oil i i y ; swci fically

Dynamic Stability Condition

(4)

projectile in flight real ly Has. A l s o , they were seeking an experimental determination of the t rue aerodynamic force and moment system acting on a projectile in flight. Their aerodynamic system contained the drag, the normal force and the static moment. "By analogy wi& treatment of the motion of a n aeroplane", t e'y included the damping moment ("But the values we obtain are too rough to enable us to study the variations. . .with any argument". ) Without reference to previous investigators, they introduce spin dependent aei s- dynamics as the roll damping moment, the magnus force, and the magnus moment ("no certain evidence that they exist is given by our experiments") .

h

They state that the magnus moment effect on damping of the nutational and precessional motions "is a priori unlikely to he comparable with the damping moment". However they state that t h e magnus effect" is uiuch larger than its expected value and of the opposite sign". They further state "an interesting feature of the (projectile) damping is that a velocity of about 900 f.s. ,the yaw (i.e. complex angle of attack) has a distinct tendency to increase (instead of decreasing with time;this happends with all four types of shel ls . Whether this is a real phenomenon or is caused by the impact on the card> is not yet clear."

-

'The main object of the jump card experiment, d e s c r i k d in this paper,is to determine the two periods of t h e initial angular oscillations of a shell, fired horizontally from a gun". They were quite successful in their objective in obtainin& and

U P , (5) W e are able to write t h i s important expression

because of the excell$n$ mathematicalbyork of

However, an ea r l i e r understanding of projegiile flight dynarnngs wz5,eZidenced by Ma ewski , Cranz','Lz Charbonnier , Sparre, and Lanchester .The latter has an appendix on projectiles in h i s excellent text on aircraft which correctly treats "drift", and also clearly identifies the importance of the damping moment and the magnus moment to projectile dynamic stability.

Fowler et a1~Kent~'"Nielsen and Synge , Kelley et aL"j 61

Their report is a excellent one,even if they, and most reviewers,were lead to believe that the YJ magnus moment might not be important. **

Tn carefully reviewing their paper, this re- viewer believes that Fowler et a1 actually ftiuli; the riagnus moment i n their exper imexs and that it was the same size and importance as the damping moment in &ternlining projectile dynamic flight stability.

57 Durinp World War I Fowler et a' fired 3 inch

shell "through a series of cards over a distance of 600 feet. From t h e diape of t h e holcs in the ca rds the actual motion of the axis of the shell can be re- constructed. " "From the periods of the oscillations of the axis of the shell,we can deduce the (static) moment. In the same way the nature of the decay of the oscillations c m be used to determine the damping forces". They also set forth a theory for the angular motion of a projectile.

The i r work is important because they under-

While overseas during World War I Kent mer with ,R?yler alld was greatly impressed by his work. he longed to repeat it and improve it in the U.S. In 1920 Kent developed h is toy "battery"with which he persisted in demonstratin& the funda- meiitals of projectile flight dynamics to all who would watch down through the yea r s until h i s pss- ing.

took to determine what the angular motion of a During World War I1 Kent was able to pursua&,

3

&ti Ik Synge, and Kelley et a1 to $?consider the mathematical work of Fowler et al.Their papers added

and excellence, arid encour- they closely follow Fowler

et a1 and add no new information on the magnus moment or OAI projectile dynamic Ttg+bility. Both Sterne in pl3jectiles and Zaroodng In mortars early pointed out the pr imary importance of the magnus moment but with little response.

One of the major contributions of Kent was to obtain from his fkiend von Karman the services of his student, Charters. Following World W a r I1 it was Charters who made the major contribution to exterior ballistics by designing, developing, and perfecting the Aeroballistic Range which is able to obtain spark-photographs of the complete motion of a projectile in flight. The excellent flight data obtained by Charters allowed the f i r s t accurate determination of projectile performance and thus the f j rs t accurate determination o,f the damping and magnus momF$Lf by Charters ,%alperin, Karpov , Twetsky , and Kopal, 'o'~'ab

This fine work on projectiles clearly estab- lished t h e equal importance of both the damping momesi and the magnus moment in determining the dynamic stability of projectiles. The first measurements of the magnus moment and the damping moment o n finned missiles is given in Ref. 7z .

69 -77 The Aeroballistic Range ,enabled Nicolaides to study the motion of finned missiles* in free flight. Special attention was given to the pure rolling motio#'%nd to pi5;ygw-roll coupling (ResoYBnce) in finned missiles.' A general Unified Theory was given which is applicable to both projectiles and finned missiles havin configurational, m a s s , and inertial asymmetries. 13*'x*''r %,Ill I

Tricyclic Motion"

i p f + NT e

Thus, we see that t h e wobbling motion of a projectile is Epicyclic (Eq. 2 ) and Lie motion of a finned missile is Tricyclic,Eq. (7). Actually the motion of both i s Quadricyclic since t h e projectiles

may have mass or inertial asymmetries and both projectiles and finned miss i les have "yaw of repose'' which produces drift.

Qua& ic yc lic Mot ion 145

This linear theory has been the basis for aeroballistic flight performance for many years now, and i t continues to provide the best first approach to any new ordnance missi le design.

Nonlinear Flight Dynamics * * , By the mid-fifties it became clcar that new

aeroballistic flight performance problems were raising their hairy heads. T h e Navy's Weapon A "-* heralded the new age of "Nonlinear Ordnance Flight Dynamics". This subsonic fin stabilized rocket when fired to the port from a ship was quite stable; however,the same weapon when fired to the star- bound became unstable and wobbled violently, often breaking up when it h i t the water. Th i fgxxu l i a r behavior was found by Jonc';Highberg,Hayes, Fredette and Mi1lerp"to be due to a magnus moment which was quite nonlinear with complex angle of attack and to be due to initial tip-off.

Outstanding wind tunnel measurements of this nonlinear magnus moment were carried out by Nestingin et a1 at NOL. Actually, ear l ier wind tunnel tests by Zaroodny oqnqrportars had revealed nonlinear magnus moment but had passed unnoticed due to a search for spin instability. Also, Turetsky'" had analysed aeroballistic range data on cone cylinder projectiles which revealed nonlinear magnus moment and static moment.

The Weapon A experience followed by s imi la r problems in the Weapon B , Slim Jim , ,y.d 20MM projectiles, all lead to a Quazi-Linear method for evaluating the dynamic stability of missiles acting upon by nonlinear aerodynamic forces and moments. Whi le t h e method was confined to single a rm motions (either pure nutation o r pure precession), numerical integrations of the e%act equations of motion on the NORC confirmed'this very simple and useful engineering method .''*4'oS

*The work on finned missi les in t h e Aeroballistics Range came about because there were no supersonic wind tunnels available at the end of the war to study guided missile models at Mach numbers greater than 2.0 and, also, because of flight stability problems in special weapons. 69

Ma( I'al)= M cro+Mcr 2

Mq < @I)= Mq, +Mq2 la 1 2 -i(M + W=M,(lZJ) Q + M ~ ( la() &

S l r G+

+* + M f i i l 4 ) k + M p a ( ( z ( ) P (1 W n r Y

\a()= ~ I L : ~ + ?bt ;u2 i 8 i 3 %ifa( ISl)=Mpao+ MP& l a I 2

4

10 I An excellent nonlinear method for v g l i n g

otions w' 3 set forth by Leitm nn and by Murphy he Pattei was concerned with projectile data obtained i n the Aeroballistic Range on the A N configuration and is largely responsible €or the present methods for handling nonlinearities in the static moment and the magnus moment. The mathematical development also included a nonlinear damping moment.

W,@d tunnel measurements on the AN projectile. revealed a nonlinear magnus moment and static moment which worsened as the length of the projectile was increased o r as the center of gravity was moved forward. Wind tunnel and Aeroballistic Range research programs on the A N projectile continue in most aeroballistic facilities throughout the world.

qs. rl+,ln

In recent years it has been possible to obtain the angular motion of an actual full scale projectile over i t s ent i re flight on the proving grounds. Special telemetry designed and developed by Hayden in England is now being used by Haseltine and by Kline. These flight data reveal that a typical projectile wobbles over its entire flight. In some parts of t h e flight path the wobbling damps, in other parts the wobbling grows. The yaw of repose is apparent At high angles of fire wobbling is critical.Transonic flight and subsonic flight are also serious. Various investigators are continuing to analyse this excellent flight data on actual projectiles.

Free flight wind tunncl tests on projectiles are being car r ied out by various activities and three degree of angular freedom tests in a vertical wind tunnel are v..nderwav by the Army, Navy and A i r Force. IfJb,'Y

In summary, it now appears that the flight stability problems of projectiles are still with us and that they may be more critical now than eve r before in history.

Nonlinear flight dynamics problems i n finned stabilized missi les appeared in bombs which were unable to achieve %sir design steady state rolling x *L

velocity due to cant: These bombs locked-in kt the nutation rate and their resulting violent wobbling motions were catastrophic. A study of the free rolling motion of a Basic Finner model in the wind tunnel revealed that a s the angle of attack increased the steady state rolling motion would stop.* It was found that a nonlinear induced roll moment due to angle of attack and roll orientation explained both the wind tunnel and the bomb drop roll phenornend'i'k was also found that a nonlinear side moment existed which was also due to angle of attack and rol l orientatiod? This side moment when

(

introduced into the equations of motion produced an amplified resonance called C a t a r o p h i c .Yaw :'wwfp '' Numerical integration of the differential equation of motion on the NORC confirmed the phenomena and the catastrophic yaw. IPr* '*

Unique three degreen$f freedom wind tunnel tests on the Basic Finner and other finned missiles in both horizontal and vertical wind tunnels have also revealed the ent i re phenomena and permitted its detailed study.

Catastrophic Yaw has "5% &qp+$s?;ved in the flights of numerous hombs, soun mg rockets:'' mortars , and other fin stabilized ordnance, It is a very serious problem in todays ordnance.

Special Ordnance

During W o r l d War I1 small parachutes were used to spread smal l munitions over a wide area. The actual area coverage achieved was relatively small since the parachute is only a drag device and thus follows a highly retarded particle trajectory. In recent times in order to achieve better dispersion the magnas rotor was introduced, primarily by Flatau, The magnus rotor flies like a flipped calling card ; it autorotates and it glides due to lift provided by the magnus force. In the old days some cannon balls picked up spin about a horizontal axis by rolling down the barrel during launching, When f i red at high angles of elevation they sometimes actually fell behind the gun due to this effect. Golf balls have underspin and, also, exhibit a large magnus lift which gives them their range.

The magnus rotor, as a munition,produces lift to drag ratios ranging from 1/4 to 1 and, thus, provides greatly improved dispersion and area coverage. Research on the flight stability and performance of magnus rotors has been carried out by Flatau, Nicolaicks, Feredette, Stilley, Bronk, et al. rc6

The recent development of the gliding parachute, particularly the Parafoil, with lift to drag ratios ranging from 4 to 6, promises a return of the non rigid systems as a method of munition area coverage and delivery.

Performance Analysis

'The six degree of freedom motion of a missile may be computed from numerical integration of the differential equations of motion on a high speed computer , provided that the complete aerodynamic force and moment system is known. One of the f i rs t and finest computer programs for handling

B a s i n g t!: angle of attack beyond 400 usually resulted in a speed up in rolling velocity to very large values. This@ the rolling or spin instability originally sought by ZarmdnyOin explaining fli@ tailures i n mortars . The continuing short in mor t a r s are now known to be d w to both magnus instability and catasrropn ic yaw.

nonlinear magnus and also catastrophic yaw was developed by a h e n at NWL. ''~'7

The inverse approach of determining the aere dynamic force and moment system from the motion of the missilg has long e n used in aeroballistics. Fowler et al used yaw cards to get the missi le motion and used the rates of the Epicyclic Theory to determine the static moment. From the damping of the arms an effort was made to obtain the damping and the magnus moments. Charters Karpov and Turetsky developed elegant methods for "fitting" the Epicyclic Theory to aeroballistic range data gicg the method of differential corrections. Nicolaides employed the same methods in fitting \$e Tricyclic Theory to finned missi le rage data. Murphy employed the same m e t h d s in analyzing projectile data containing nodinear magnus and static moments'? A potential problem with respect to the aeroballistic range is the relatively small number of cycles of motion and the small number of data points per cycle .This limitation can become serious when large nonlinearities a r e present.

In order to obtain more cycles of motion and more data per cycle, wind tunnel dynamic testing; techniques have been developed. The Basic Finne~"' w a s H g d in ear ly dynamic wind tunnel tests at NCL hnd, more recently, at Noue Darne.The large amount of motion data enables the fitting of many sine waves to the motion. T h u s , t k concept of sectional fitting was developed, where a sine wave is fit to the first couple of cycles,then a second sine wave is fit by dropping a few data points at the start and picking up a few data points at the end. In this way many cycles are f i t to all of the data. From each single cyclic f i t , a complete set of the aerodynamic forces and moments are determined. In initially reviewing the resu l t s of this sectional fitting technique it was noted that the aerodynamic stability coefficients were not constant Quite different values of the aerodynamic stability coefficients were obtained at the beginning of the motion as compared to those at the end. A s a matter of fact there was a systematic variation of the &fficients during the entire motion of the model. Thus, it was found that major nonlinearities were present. The methods developed for analyzing free motions and taking into account the various nonlinearities producing the motior. are contained in the WOBBLE program. '''*'*(

The nonlinearitie's mcovered in the aerodynamic coefficients were so large as to explain the differences in the values of the coefficients for the Basic Finner as obtained in the various testing facilities around,g,e ,$9yntry by using a linear method of analysis: "m the origirra! Basic Finner program these nonlinearities were unsus- pected. Thus, only linear techniques were uti'iized and even there, the data per cycle and number of cycles obtained were small. Althgtgh the resul ts in each case appeared to be good, M e n data from different facilities were compared large disagree -

ments in the aerodynamic coefficients were ob= served. It is now realized that it is absol.aely essent ial to utilize tRe nonlinear methods of data analysis in order to determine the true values for the aerodynamic coefficients.

'The Sasic Finner wind tunnel program includes combined pitching and yawing motion, and combined pitching, yawing and rolling motion.

frp.lrr

The Sandia Laboratories obtained an actual full scale free flight motion of a Tomahawk sounding rocket as it exited the ear th 's atmosphere and, also, as it re -entered the earth's atmosphere. * The data were obtained from a Whittaker-gyro platform which gave the two position coordinates, pitch and yaw, as well as the rolling velocity of the sounding rocket. The WOBBLE program which had been developed for the three degree of freedom wind tunnel case was then modified to handle this full scale sounding rocket angular motion. The motion of the sounding rocket was Quadricyclic in that it had both the nutation and precession a r m s and also the rolling t r im arm and the yaw of repose. An additional problem in the case of the sounding rocket was that its velocity changed during flight, the air density changed during flight, and also the aerodynamic coefficients changed due to the change in Mach number and Reynolds number. The concept of sectional fitting developed previously for the case of the simple pitching motion in a wind tunnel was able to be applied with good results to the Tomahawk sounding rocket dataivonce suitable corrections were made for the effects of frequency change. The success i n t i e analysis of the Tomahawk sounding rocket led to the receipt and reduction of flight data from the Apache sounding rocket. 146, I*

Actual d$ut6pn the free flight motion of re- entry missi les was obtained. This data was in t e rms of pitching and yawing rates rather than angular position coordinates. However, a method was developed for handling the rate data. As in the previous eee flight cases large non-linearities in the aerodynamic coefficients were observed."

Two particularly interesting motions from free flight missiles were obtained from the Weapons Research Establishment of Australia?wo bombs had been dropped, and the flight data on their motion determined. h once case, the bomb had cruciform fins and experienced Catastrophic Yaw In the second case, the bomb as stabilized by a split-skirt, and during its flight a large rolling velocity developed which led to magnus instability. Once again the WOBBLE technique was applied, with good success in revealing the non-linear nature of aerodynamic coefficients.'"

Free flight motion of f in stabilized missi les were obtained in the Cal Tech wind tunnel and also in the Army BRL wind tunnel. These motions were measured and also successfuIiy reduced.

6

A very outstanding three-degree-of-freedom eir bearing support system for wind tunnel dynamic work was developed by Ward et a1 at ARO for studying the stability of re-entry missiles. Excellent data was obtained by this system which was reduced by the WOBBLE program. The program has also been applied to extensive additional data obtained in the wind tunnel at Notre Dame on the three degree of freedom motion of various missiles, such as the Low Drag Bomb, the 2.57" rocketyre-entry bodies, etc. Not only has the wbbl ing motion been analyzed but also the nonlinear rolling motion at constant values of the angle of attack?"*

I l l u While the various applications of the WOBBLd

method proved successfd across a wide spectrum of missile motions, it should be emphasized that in developing the technique a number of very special precautions were taken. First, and fore- most, waa the evaluation of each method developed by utilizing the high speed computer in generating motion data. In other words, the non-linear differential equations were numerically integrated on the high speed computer to obtain generated data. Th i s data was then fitted by the particular WOBBLE technique for the determination of the ability trf the technique to represent the motion and for the determination of the associated nonlinear aerodynamic coefficients. 'In t h i s way WOBBLE determined nonlinear coefficients could be compared with the known nonlinear values of these coefficients as originally utilized in the differential equations of motion. Thus , by utilizing computer generated data, the accuracy of the technique could be evaluated. The effects of changing air density , changing missile velocity and changing coefficients due to Mach number and Reyncllds number were all studied.

Another promising method of analysing missile motion data is-to utilize the ,$ feren t ia1 equations of mozon directly; Goodman, C%apman and Kirk, White, Kline, and Eikenberry et al. . . .

The main advantage of working with the actual motion of the missi le is that you must directly cope with the true situation and with the actual nonlinear static and dynamic forces and moments that are acting on the missile during its flight. Taking the normal approach and working with the differential equations where you put in what is believed to be existing may not truly represent reality. Very minor differences between the computed motion and the actual motion can in fact be d w to large nonlinearities. This characteristic was observed over and over again in working wit1 generated data from the computer .*

***

In general, the value and importance of working directly with the actual motion of the missile has been validated over the years and has led to excellent full scale testing techniqws, as well as EO see r i r p t rests in wmc tunnels mid three ciegree

of freedom tests in the wind tunnels across the country

Current S t m s

When one reviews the excellent progress which has been ma& and appreciates the modern experimental and analytical methods which are now available, it is difficult to believe that any per - brmance problems could exist in current ordnance missiles. Unfirtunately, the opposite situation actually exists. Many bombs wobble in flight and are inaccurate. We know of no subsonic or transonic projectile which is dynamically stable. Mortars continue to fall short. Rockets are plagued with inaccuracy: some experience magnus instability and some experience catastrophic yaw Re-entry missi les have abnormal roll behavior and experience trajectory deviations .

The truth seems to be that as we l earn more about the flight characteristics of ordnance missiles, the more opportunities emerge for flight performance improvement.

=S,flJ- Non-Linear Motion , FIB 3 .

where

26.

27.

Re&rences Mayewski (General). Traite de Balistique Exterieure, Paris, 187~,Cawhier=ViUars, in-@ M ~ & l o n e l ) . de@

~ ~-

Philon of Byzantium, g&how O L C utrc 3 cen. , B. C.

, 1. aroiectiles ob1 o m s clam l'airt Revue d'artillerie,t.XII, 1878,et t. XIII, 1879. Tartaglia,Nuava Scienzia,Text, 1537.

Samba&, , hllbtics , German Text, 1561. Galileo, G X w u e s Concerning Two New Sciences. Text. 1436. ~~

2. 3, 4.

28. Poisson. Memoires sur lemnouvements des projectiles dans l'air. J. de 1'Ecole polytechni- que, 1839. Ronca (Capitaine de fregate) et Bassani Rofesseur A .. ). Balistica esterna, 1901, Livorno. Giusti, in-80.

5. 1. # alis Rincipia Mathematica

6. 7. 30.

31 . -' (Capitaine de fregate). Manuale di Balistica Esterna, in-8O , 1901 , Livorno ,Giusti . Sparre (Comte M. I%), Movement des proiectiles oblongs dans le tir de plein fouet,Paris, 1875, Gauthier Jillars. in-120.

-' (Capitaine de fregate). Manuale di Balistica Esterna, in-8O , 1901 , Livorno ,Giusti . Sparre (Comte M. I%), Movement des proiectiles oblongs dans le tir de plein fouet,Paris, 1875, Gauthier Jillars. in-120.

_ - - - Gre&hil/,G. ,Notes on Djfnamics. uplace, P. S. , Mechanique Celesti. Astier Colonel\.De I'influence de la rotation

8. 9. 10.

32 . -' Sur le mouvement des projectiles oblongs amour de leur centre de gravite. Memorial de 1'Artillerie de la Marine, t. XXII, 1894,

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nt

Method bf Determining the Aerodynami? Properties of Symmetric Bodies in Free Flight. NavalShip Research and Developrne Cm. ,Report 2872, March 1969. Chadwick, W. R. , Flight Dynamics of a Bomb with Cruciform Tail. J. SpacecraftJol. 4, No. 6.

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1 1

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,Ingram,C.W.,Martin,J.M.,

185.

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Innram . C . W .-. and Clare. Thomas . Y

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Freedom. Ph. D. DissertationJune 1970. N n s . Tohn D. Free Flight Dvnamics . Text, Univk;sity of Notre Dame, d1970.

Appendix A

Letter Symbols for Aeronautical Sciences, Appendix 11, 1959.

. I .

W369-199,Vol. 3, paper 31).

Documents

AIAA-1970-533-510