INCREASING AIR DEFENSE CAPABILITY BY OPTIMIZING BURST DISTANCE

Transcription

1 INCREASING AIR DEFENSE CAPABILITY BY OPTIMIZING BURST DISTANCE A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF MIDDLE EAST TECHNICAL UNIERSITY BY MEHMET TÜRKUZAN IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN ELECTRICAL AND ELECTRONICS ENGINEERING DECEMBER 010

2 Approval of he hesis: INCREASING AIR DEFENSE CAPABILITY BY OPTIMIZING BURST DISTANCE submied by MEHMET TÜRKUZAN in parial fulfillmen of he requiremens for he degree of Maser of Science in Elecrical and Elecronics Engineering Deparmen, Middle Eas Technical Universiy by, Prof. Dr. Canan Özgen Dean, Graduae School of Naural and Applied Sciences Prof. Dr. İsme Erkmen Head of Deparmen, Elecrical and Elecronics Engineering Prof. Dr. Erol Kocaoğlan Supervisor, Elecrical and Elecronics Engineering Dep., METU Examining Commiee Members: Prof. Dr. Mübeccel Demirekler Elecrical and Elecronics Engineering Dep., METU Prof. Dr. Erol Kocaoğlan Elecrical and Elecronics Engineering Dep., METU Prof. Dr. Kemal Leblebicioğlu Elecrical and Elecronics Engineering Dep., METU Assis. Prof. Dr. Afşar Saranlı Elecrical and Elecronics Engineering Dep., METU Dr. Hüseyin Yavuz ASELSAN Inc. Dae:

3 I hereby declare ha all informaion in his documen has been obained and presened in accordance wih academic rules and ehical conduc. I also declare ha, as required by hese rules and conduc, I have fully cied and referenced all maerial and resuls ha are no original o his work. Name, Las name : MEHMET TÜRKUZAN Signaure : iii

4 ABSTRACT INCREASING AIR DEFENSE CAPABILITY BY OPTIMIZING BURST DISTANCE Türkuzan, Mehme M. Sc., Deparmen of Elecrical and Elecronics Engineering Supervisor: Prof. Dr. Erol Kocaoğlan December 010, 56 pages In his hesis, burs disance is opimized o increase air defense capabiliy for sysems uilizing airburs muniions. A simulaor program is creaed o use during he sudy by aking advanage of he MATLAB environmen. While creaing he simulaor program, a muniion pah model is derived by using fourh order Runge- Kua mehod. Then, simulaions are conduced a differen burs disances and relaed informaion are recorded. By using leas square opimizaion mehod and gahered daa, opimum burs disance is found. Moreover, he effecs of several facors on opimum burs disances are analyzed, including: he weighs of he objecives in he opimizaion, arge dimensions, arge range, wind, arge posiion ambiguiy, firing angle, and velociy ambiguiy afer burs. Furhermore, a firing mehod is proposed. The resul of he proposed firing mehod and he opimum soluion are compared and success is presened. To sum up, his sudy presens a way o find opimum burs disance, analyzes he facors ha may affec opimum burs disance, and suggess a firing mehod for effecive shos. Keywords: Airburs muniions, burs disance opimizaion, fire conrol. iv

5 ÖZ HAA SAUNMA KABİLİYETİNİN ARTTIRILMASI AMACIYLA PARALANMA MESAFESİ OPTİMİZASYONU Türkuzan, Mehme Yüksek Lisans, Elekrik Elekronik Mühendisliği Bölümü Tez Yöneicisi: Prof. Dr. Erol Kocaoğlan Aralık 010, 56 sayfa Bu çalışmada, havada paralanan mühimmala hava savunması yapan sisemlerde ekinliğin arırılması amacıyla paralanma mesafesi opimizasyonu yapılmışır. Çalışmada kullanılmak üzere MATLAB oramından faydalanılarak bir simülaör yazılmışır. Simulaor gelişirme aşamasında mühimma yolu modeli dördüncü dereceden runge-kua meodu kullanılarak çıkarılmışır. Daha sonra farklı mesafelerde paralanma mesafesi simülasyonu yapılarak gerekli veriler oplanmışır. Bu veriler kullanılarak en küçük kareler meoduyla opimum paralanma mesafesi bulunmuşur. Ayrıca; opimizasyonda kullanılan paramerelerin, hedef boyuunun, hedef uzaklığının, rüzgârın, hedef belirsizliğinin, aış açısının, paralanmadan sonra parçacıkların hızlarındaki değişimin paralanma mesafesine ekileri incelenmişir. İlaveen, bir de aış şekli önerisinde bulunulmuşur. Bu aış şeklinin vermiş olduğu paralanma mesafesiyle opimum paralanma mesafesi karşılaşırılmışır. Özele, bu çalışma opimum paralanma mesafesi bulma yolu gösermişir. Opimum paralanma mesafesini ekileyen fakörleri incelemişir. Ayrıca, ekin bir aış yapabilmek için bir aış şekli önerisinde bulunmuşur. Anahar Kelimeler: Havada paralanan mühimmalar, paralanma mesafesi opimizasyonu, aış konrol. v

6 To My Parens and Love vi

7 ACKNOWLEDGEMENTS I would like o express my sincere hanks and graiude o my supervisor Prof. Dr. Erol Kocaoğlan for his guidance, valuable advices, and suppor. I owe my deepes graiude o Burak Kekeç and Muharrem Tümçakır for heir suppor and encouragemens. I cerainly would like o hank Emin İlker Çeinbaş for his reviews and feedback during he wriing process. I wan o express my graiude o my colleagues a Aselsan Inc. I wan o hank Aselsan Inc. for faciliies provided. I would also like o hank TÜBİTAK for providing me financial suppor for my sudy. Finally, my hanks go o my parens İdris and Türkan, o my wife Ayşe, and o my broher Melih for heir endless love, encouragemens, and suppor. vii

8 TABLE OF CONTENTS ABSTRACT...I ÖZ... TABLE OF CONTENTS... III LIST OF TABLES... X LIST OF FIGURES...XI ABBREIATION LIST... XIII CHAPTERS 1 INTRODUCTION BACKGROUND AND SCOPE OF THE THESIS OUTLINE OF THE THESIS... LITERATURE SUREY FIRE CONTROL AIRBURST MUNITIONS OPTIMIZATION PROBLEM LEAST SQUARES OPTIMIZATION METHOD RUNGE-KUTTA METHOD LITERATURE SUREY ABOUT AIRBURST MUNITIONS DERIING THE SYSTEM MODEL ASSUMPTIONS FLIGHT PATH MODEL COST FUNCTION A FIRING METHOD IN THE CASE OF PERFECT TARGET LOCALIZATION A FIRING METHOD IN THE PRESENCE OF TARGET LOCALIZATION ERRORS SIMULATOR PROGRAM ANALYSIS AND SIMULATIONS viii

9 4.1 THE EFFECT OF FIRING ANGLE THE EFFECT OF RANGE THE EFFECT OF THE TARGET DIMENSIONS THE EFFECTS OF THE WEIGHTS IN THE COST FUNCTION THE EFFECTS OF AMBIGUITY IN TARGET LOCATION THE EFFECT OF WIND THE EFFECT OF ELOCITY AMBIGUITY AFTER BURST SIMULATION WITHOUT AMBIGUITY SIMULATION WITH COMPLETE MODEL CONCLUSION AND FUTURE WORK CONCLUSION FUTURE WORK REFERENCES ix

10 LIST OF TABLES TABLES Table 4-1: Deviaion weighs of he objecive parameers in oal deviaion Table 4-: Objecive parameers of he opimum soluion and he soluions of he firing mehods Table 4-3: Objecive parameers of he opimum soluion and he "firing mehod" soluion x

11 LIST OF FIGURES FIGURES Figure -1: Illusraion of general fire conrol problem [1]... 5 Figure -: 35mm ime programmable airburs muniion [10]... 7 Figure -3: Descripion of he pars [10]... 8 Figure -4: General airburs sysem configuraion... 8 Figure -5: End par of cannon... 9 Figure 3-1: Coordinae sysem for six-degrees-of-freedom rajecories Figure 3-: Ejecion of subprojeciles, ploed by simulaor program in MATLAB Figure 3-3: An illusraion of subprojecile dispersion Figure 3-4: 3D subprojecile componen illusraion Figure 3-5: The aim of he firing mehod is illusraed... 5 Figure 3-6: The area where arge cenre posiion exiss mosly... 7 Figure 3-7: The radius of he aimed circle... 7 Figure 3-8: The screen sho of he simulaor... 8 Figure 3-9: The illusraion of he arge cross-secion seen by he muniion a Y-Z axis Figure 3-10: Targe area which is seen on he pah of he muniion Figure 3-11: Area covered by muniion... 3 Figure 3-1: The inersecion of he arge and area covered by muniion... 3 Figure 3-13: The descripion of he covered area Figure 4-1: A represenaive figure of coordinae sysem Figure 4-: A represenaive figure of firing angle Figure 4-3: An illusraion of firing angle simulaion Figure 4-4: The effec of firing angle on opimum burs disance Figure 4-5: The effec of range on opimum burs disance Figure 4-6: The effec of arge dimensions on opimum burs disance Figure 4-7: The graph of equal weigh cos funcion wih respec o burs disance 40 xi

12 Figure 4-8: The change in burs disance wih increasing velociy consan Figure 4-9: The change in burs disance wih increasing coverage consan... 4 Figure 4-10: The change in burs disance wih increasing paricle consan... 4 Figure 4-11: The change in burs disance wih increasing variance Figure 4-1: The change in burs disance wih increasing wind velociy agains he movemen on Z axis Figure 4-13: The change in burs disance wih increasing wind velociy in he same direcion wih he movemen on Z axis Figure 4-14: The change in burs disance wih increasing wind velociy on X axis45 Figure 4-15: Change in he burs disance wih respec o differen velociy incremens due o burs Figure 4-16: Change in he cos value wih respec o differen velociy incremens due o burs Figure 4-16: Burs disance cos value graph. Opimum burs disance and he soluion of he proposed mehod are also indicaed Figure 4-17: Burs disance cos value graph. Opimum burs disance and firing mehod soluion are also indicaed xii

13 ABBREIATION LIST α : Aack Angle : ecor elociy of he Projecile W S : ecor elociy of he Wind : Projecile Reference Area : Air Densiy C D : Drag Force Coefficien x : Uni ecor Along he Projecile's Roaional Axis of Symmery C La : Lif Force Coefficien d : Projecile Reference Diameer C Npa : Magnus Force Coefficien I y : Projecile Transverse Momen of Ineria I x : Projecile Axial Momen of Ineria h Ineria : ecor Angular Momenum Divided by he Transverse Momen of C Nq C Na : Pich Damping Coefficien g : ecor Acceleraion Due o Graviy : ecor Coriolis Acceleraion T m : Rocke Thrus Force : Mass of he Projecile xiii

14 r e : Disance from Projecile Cener of Mass o he Rocke Nozzle Exi xiv

15 CHAPTER 1 INTRODUCTION 1.1 BACKGROUND AND SCOPE OF THE THESIS Missile hurling was a skilled craf housands of years before wriing was developed [1] Throughou he hisory, human beings needed o figh for food or defense. A Sone ages, hey used hurled sones and hen hey creaed spears and javelins. Afer he invenion of gunpowder, firearms were invened, hus weapons have been changed. Lehaliy of weapons and effecive ranges of he weapons have been improved wih ime. However, as he effecive ranges of hese weapons increased, i became more difficul o accuraely aim hese weapons. This problem is named as Fire Conrol Problem and defined as he firing of a projecile from a weapon in order o hi a seleced arge [1]. Parallel o he developmen of guns; muniions are improved and diversified. In his sudy, model is derived for medium caliber ammuniion. 0 millimeers (mm) hrough 60mm ammuniion is grouped as medium caliber ammuniion which was firs used in he World War I. Previously he main purpose of his size ammuniion involved an ani aircraf role. Is early use in ground applicaions was agains lighly armored vehicles [3]. Convenional medium caliber ammuniion is grouped ino wo; high explosive ammuniions, which are used in poin deonaing or poin deonaing delay mode, and airburs ammuniions. In World War II, airburs muniions of he ime were used as ani aircraf. Manually aimed guns, which fire fragmenal (airburs) muniions, had provided an effecive air defense agains bomber aircrafs of ha 1

16 ime. The mos known ones were he Germans flak (fliegerabwehrkanone) guns which fired grooved projeciles [9, 18]. Medium caliber ammuniion has improved over ime. Incremen of he ground ani armor peneraion requiremens during he 1990 s gave rise o he high performance armor piercing ammuniion. The echnological advances had improved airburs muniions [3]. Once more hey would be used agains aircrafs like in he World War II. Today, he mos effecive air defense sysems are acceped as he ones deploying air bursing muniions, wih heir greaer area of engagemen [9]. In he lieraure, here are some sudies abou airburs muniions, such as; he burs ime opimizaion [8, 13, 19], various ways of seing he fuse imer [6, 7, 8]. Abou he burs ime opimizaion, general endency is on keeping he opimum burs disance consan. There are wo ways of seing fuse imer which are; seing imer a he muzzle of he gun [6, 8] and seing imer as lae as possible a somewhere on is fligh pah [7]. Boh of hese mehods require he opimum burs disance. In his hesis, opimum burs disance is found. Moreover, he parameers ha may affec opimum burs disance is analyzed which are: firing angle, range of he arge, dimensions of he arge, presence of wind, imporance of he objecives, ambiguiy in he arge posiion, and variaion of he paricles velociy afer burs. Addiionally, a firing mehod is proposed. However; arge posiion esimaion, firing angle calculaion, arge and muniion pah calculaions are no in he scope of his hesis. They are implemened o provide inpus for he sudy. 1. OUTLINE OF THE THESIS In his hesis, opimum burs disance of airburs muniions is sudied. Simulaions are carried ou and daa is gahered for opimizaion. In he simulaions, arge movemen is no racked and i is no considered because he sudies like racking he arge and esimaing he arge posiion are well known sudies which had been

17 already sudied a lo. Thus, during he simulaions, inersecion poin of he muniion pah and arge pah is assumed as given. In Chaper, fire conrol problem and opimizaion problem is explained. Airburs muniions are described and oher ypes of muniions are menioned briefly as background informaion. Leas square opimizaion mehod and fourh order Runge Kua mehod is presened. Lieraure survey in he field of airburs muniions is presened by menioning several ousanding sudies briefly. Chaper 3 derives he sysem model, describes he cos funcion, shows he proposed firing mehod, and presens he simulaor program wrien in MATLAB. Therefore, assumpions abou sysem model derivaion are given in his par. In Chaper 4, opimizaion problem is solved. Resuls are presened. The effecs of he parameers on opimum burs disance are analyzed. Proposed firing mehod's soluion is compared o he opimum soluion. Chaper 5 concludes he sudy by giving a summary of he work done. I also menions possible fuure work o guide he researchers in his area. 3

18 CHAPTER LITERATURE SUREY This chaper gives general informaion abou he field of he sudy and background on he subjec. In Secion.1, general informaion abou fire conrol is presened. In Secion., airburs muniions are described. In Secion.3, general informaion abou opimizaion problem is presened. In Secion.4, Runge-Kua mehod is presened. In he las secion, general paened mehods abou increasing effeciveness of he sysems ha use airburs muniions are presened..1 FIRE CONTROL Launching a projecile from a gun sysem o hi a arge is called he fire conrol problem. Fire conrol mainly means offseing he gun direcion from line of sigh in order o solve he fire conrol problem which is hiing he seleced arge as illusraed in Figure -1 [1]. In fire conrol problem, boh arge and gun sysem may be moving. 4

19 Figure -1: Illusraion of general fire conrol problem [1] Offse angle is called predicion angle ha is he angle beween line of sigh and gun direcion, called weapon line. Predicion angle is he soluion provided by a fire conrol sysem by available informaion. This soluion, predicion angle, is achieved as he resul of offse componens in elevaion and azimuh. Soluion daa are applied up o he ime of firing for guns and rocke launchers, whereas for guided missiles, soluion daa are also applied a some inervals or coninuously afer firing [1]. Fire conrol has mainly hree funcions. These funcions are; acquiring appropriae inpu daa, calculaing he elevaion and azimuh angles required for he projecile o inersec he arge, and applying hese angles o he fire conrol mechanism o posiion he gun correcly. These hree funcions are associaed wih acquisiion and racking sysems, compuing sysems and gun poining sysems. In some siuaions fire conrol includes soluion of wo addiional problems. The firs problem is mainaining awareness of he gun-arge siuaion which means ha gun urre is following he arge. Gun is always ready o fire. This problem is more 5

20 significan for fas arges. In his case, gun aims o he arge all he ime, since aiming jus before firing is someimes physically impossible due o he need of very fas movemens of gun urre. The second problem is conrolling he ime and volume of fire o achieve maximum effeciveness of fire and minimize wase of ammuniion, which involves making projeciles explode when hey reach he viciniy of he arge by means of ime fuzes prese by a fuze-ime compuer. Thus, fire conrol may be broadly defined as quaniaive conrol over one or more of he following iems o deliver effecive weapon fire on a seleced arge: 1. The direcion of launch. The ime and volume of fire 3. The deonaion of he missile [1] The firs iem, namely direcion of launch, and he second iem namely ime and volume of fire are subjecs of oher sudies. The deonaion of missile conrol is he problem of he sysems which uilize airburs muniions. In his hesis, airburs muniions are considered and deonaion of missile conrol is he subjec of his sudy.. AIRBURST MUNITIONS Muniions are diversified according o heir sizes; small caliber, medium caliber and large caliber. Small caliber muniions diameer are smaller han 0mm. Medium caliber muniions are considered as 0mm hrough 60mm and large caliber muniions are bigger han 60mm. Muniions are also classified by he ypes of heir fuzes, namely; impac, ime, command, inferenial and proximiy. Impac fuzes operae as muniions hi he arge, ime fuzes operae afer a predefined ime passes, command fuzes operae by a signal from a remoe conroller, inferenial fuzes funcion if precondiions are me, and proximiy fuzes funcion if muniions are a defined disance [0]. 6

21 Airburs muniions have wo ypes of fuzes, menioned above, which are ime fuzes and command fuzes [7, 8]. Since his hesis concenraes on he opimum burs disance, he fuze ype of he muniion is no imporan, as long as ime of burs is conrolled. The concep of airburs muniions is o burs muniions in he air above he arge or in fron of he arge. The aim is o pu maximum fragmens on he arge area. Since airburs muniions have los of fragmens, heir probabiliy of hi is greaer han a single piece muniion. Alhough, he main aim of airburs muniions is increasing he probabiliy of hi, hey may be used o engage more arges which are close o each oher, if i is necessary []. In Figure -, 35mm base fused ime-programmable airburs muniion is shown. As seen from he figure, here are a lo of sub projeciles inside. Figure -: 35mm ime programmable airburs muniion [10] In Figure -3, he pars of he muniion are shown. Differen muniions include differen number of sub projeciles inside. The one in he figures includes 15 ungsen sub projeciles. 7

22 Figure -3: Descripion of he pars [10] In Figure -4, he configuraion of a sysem uilizing airburs muniions is shown. The working principle of he sysem can be summarized as follows: Afer deecion of arge, posiion and velociy informaion are sen o gun compuer. Then, gun makes required calculaions including arge pah esimaion, ime of fligh and burs ime. Finally, muniion is fired. Figure -4: General airburs sysem configuraion [10] 8

23 In Figure -5, he end par of he cannon of he above configuraion is shown. This device is used for calculaing he acual muzzle velociy of he projecile in several sysems. The device has hree coils and he working principle of he device is he following. When muniion is sensed by firs coil, gun compuer sars a imer. When muniion is sensed by second coil, gun compuer sops he imer. Wih his informaion, he compuer calculaes acual muzzle velociy. Then gun compuer calculaes burs ime and se his ime on he fuze using he hird coil. Figure -5: End par of cannon [10] There is a differen sysem ha also fires airburs muniions. Their muniions are of differen kinds. One of hem is rigged by a specific radio signal, in which he sysem deecs racks of rajecories and gives he fire command accordingly. Afer firing, boh he arge and he muniion are racked wih radars. A he bes poin fire signal is sen o burs he muniion..3 OPTIMIZATION PROBLEM Opimizaion problem is defined wih he following quadruple (S, m, v, C). Where S is a se of soluions (burs disances), m(x) is he se of objecive parameers, given an insance v(x) is he se of cos values, given an insance x S, x S, 9

24 C is he cos funcion. The aim is o find an opimal soluion, X o S, in he se of soluions S: v x C m x x v( X o ) min v( x) S x S (-1) In his hesis, opimizaion is used o find burs disance. Simulaions are carried on, objecive parameers are recorded. Then, hese records are processed by he cos funcion o find he opimum soluion. The cos funcion of he opimizaion is creaed by using leas squares opimizaion mehod..3.1 LEAST SQUARES OPTIMIZATION METHOD The cos funcion of leas square opimizaion problems is expressed as a sum of squares [9]. The bes fiing curve, according o leas squares, has he minimal sum of he deviaions squared (leas square error) from a given se of daa [31]. There are a se of daa poins x, ), x, ),..., x, y ) where x is he ( 1 y1 ( y ( n n independen variable and y is he dependen variable. The deviaions (error) d of he fiing curve f (x) from each daa poin are d y f ), d y f ),..., 1 1 ( x 1 ( x d n y n f ( x n ). According o he mehod of leas squares, he bes fiing curve is he curve which saisfies he minimum squared error as in Equaion(-) [31]. d n n 1 d dn di [ yi f ( xi )] i 1 i 1... (-) In his hesis, leas squares mehod is used o derive cos funcion of he opimizaion. Acual values of he objecive parameers are subraced from he ideal objecive parameers. Each deviaion is squared and hey are summed up as formulaed in Equaion (3-15). Minimum value means minimum deviaion from he ideal in he leas squares sense. 10

25 .4 RUNGE-KUTTA METHOD In numerical analysis, Runge-Kua mehods are an imporan family of implici and explici ieraive mehods. Runge-Kua mehods are used for he approximaion of he soluions of ordinary differenial equaions [31]. An ordinary differenial equaion of he form of Equaion (-3) can be ieraively solved wih he 4h order Runge-Kua mehod whose formula is given by Equaion (-4). dy dx f ( x, y) y ( 0) y (-3) 0 y h k k k k i x i y 1 i f ( x, y ) i f ( x i f ( x i f ( x i 1 ( k 6 x i i 1 1 h, yi 1 h, yi h, y i k k 1 k1h) 1 kh) k h) 3 3 k ) h 4 (-4) Hence, y i 1 is calculaed by he presen value yi plus he produc of he inerval h and an esimaed slope. This esimaed slope is a weighed average of slopes: k 1 is he slope a he beginning of he inerval, k is he slope a he midpoin of he inerval, using slope k 1 o deermine he value of y a he poin n h using Euler's mehod, k 3 is again he slope a he midpoin, bu now using he slope k o deermine he y-value, k 4 is he slope a he end of he inerval, wih is y-value deermined by using k 3 in previous sep [31]. 11

26 In his hesis, fourh order Runge Kua mehod is used o solve he ieraive differenial equaions of he fligh pah model..5 LITERATURE SUREY ABOUT AIRBURST MUNITIONS In lieraure, here are some paens awarded for increasing effeciveness of airburs muniions. All hese paened mehods concenrae on he calculaion of burs ime. In his secion, he differences and similariies of hese mehods are presened. One paened mehod [8] aims a deermining burs ime of airburs muniions. I is possible ha hi probabiliy of airburs muniions can be improved by using his mehod. In order o score beer hi probabiliies, mehod suggess keeping opimum disance beween he burs poin and he hi poin consan. The mehod calculaes a ime correcion value for keeping burs disance consan. Calculaion of ime correcion value is basically found by muliplying he velociy difference beween esimaed muzzle velociy and acual muzzle velociy by a consan, as shown in Equaion(-5)[8, 1, 13, 19]. esimaed is he average muzzle velociy of he previous shos. calculaed is he burs ime calculaed wih a esimaed muzzle velociy. burs is he burs ime correced by measuring acual muzzle velociy. ) (-5) burs calculaed ( esimaed acual The acual muzzle velociy is measured by a device locaed a he muzzle of he gun as shown in Figure -5. Burs ime is correced and he success of he projecile is improved [8, 1, 13, 19]. There is a paened mehod [6] ha measures muzzle velociy by couning revoluions of he projecile in he barrel. Couning he revoluions of he projecile in he barrel is he difference of he presen mehod from he above menioned mehod. This mehod also keeps he opimum burs disance consan. The mehod uses a device which measures he revoluion of he projecile. By his revoluion couning device acual muzzle velociy is measured. The mehod says ha defined number of revoluion is normally compleed in ime. The revoluions couning 1

27 device couns defined revoluions in ime m. Then, Equaion (-6) gives he correced burs ime calculaion as: Tburs Tesimaed ( m ) (-6) is he consan. T burs is he correced ime. T esimaed is he esimaed burs ime depending on he previous experimens [6]. A hird paened mehod [1] differs from he ohers by is device for ransferring informaion o projeciles. This device is placed in he barrel of he gun. Burs ime depends on he posiion of his device in he barrel. Thus, performance of he sho can be uned by changing he posiion of he device. Furhermore, his invenion uses many compuing unis and filer blocks o calculae burs ime beer in order o mainain opimal burs disance. Anoher paened invenion [7] differs from he ohers by waching projecile and arge acively. This mehod deermines burs ime by keeping opimum burs disance consan like ohers. However, projecile is remoely fragmenable. Burs ime is no downloaded o he projecile. Radar and gun compuer acively wach he projecile and arge. When disance beween hem is equal o he opimum burs disance, an RF signal is sen and projecile is bursed. The above menioned mehods all assume ha an opimum burs disance is available. They sugges differen mehods o keep opimum burs disance as i is. However, here is no publicly available sudy abou finding opimum burs disance. Hence, his hesis concenraes on opimum burs disance. Therefore, he resul of his sudy can be used by all mehods menioned. 13

28 CHAPTER 3 DERIING THE SYSTEM MODEL In his chaper, derivaions of he models which are used during he sudy are presened. Firsly, muniion pah model is derived. Iniial model includes all forces ha ac on he projecile. However, he model used in he simulaions is a simplified version of he iniial model. The simplificaion is done by using assumpions given in Secion 3.1. Nex, he cos funcion o be opimized is presened. Leas-square error minimizaion mehod is used o find he opimum burs disance. Afer cos funcion presenaion, wo differen firing mehods are presened. Firs firing mehod is for he case when arge locaion is known wih zero error. Second firing mehod is for he case when arge locaion is no known perfecly. Finally, he program wrien in MATLAB o solve equaions ha are derived in his chaper is presened. 3.1 ASSUMPTIONS Simplificaion is very imporan for modeling [8]. Targe is assumed as an ellipsoid in his hesis. As mahemaical definiion of an ellipsoid is simple and easy o express, arge is assumed o be of ellipsoid shape. Air densiy and graviaional acceleraion are assumed as consan. In Figure 3-1, here is an illusraion of a projecile wih rajecory pah. The angle beween he axial direcion of a projecile and he angen o he rajecory is he aack angle. Aack angle is assumed as zero during he simulaions. Besides, projecile is 14

29 assumed as non-roaional. Since, he sudy concenraes on very shor range air defense, hose las wo assumpions have minor effecs on he projecile pah [6]. There are some assumpions o define projecile and is behavior. Projecile includes 181 paricles. Afer burs, paricles fly in a cone shape wih 10 degrees apical angle [8]. Furhermore, paricles gain 150 m/s velociy in average due o he burs effec. To define coordinae sar poin, muzzle of he gun is assumed as he origin of he coordinae sysem. Figure 3-1: Coordinae sysem for six-degrees-of-freedom rajecories [6] 3. FLIGHT PATH MODEL The aim of his par is o derive he fligh pah model of an airburs muniion. Firsly, a model ha includes all forces, which is known as six degrees of freedom rajecory model, is presened as given in Equaion (3-1). Then, he equaion is simplified by inroducing some assumpions. The six-degrees-of-freedom vecor differenial equaions of moion, for a rigid, roaionally symmeric projecile aced on by all significan aerodynamic forces are summarized in Equaion(3-1) [6]. 15

30 d d vsd( C vsc m Nq m D C v Na ) SC m La [ v x h x g ( v x) v] gt m x I SdC mr m y Npa I I y x mre h m h x x x v (3-1) In his equaion, is he velociy vecor of he projecile wih respec o he Earh fixed coordinae sysem. W is he velociy vecor of he wind wih respec o he Earh fixed coordinae sysem. v is he velociy vecor of he projecile wih vsc respec o he air ( v W ). The firs erm ( D v ) in Equaion (3-1) is m relaed o drag force, and v indicaes he norm of he vecor v. S is he projecile reference area, is he air densiy, m is he mass and CD is he drag force coefficien [6]. SC La The second erm ( [ v x ( v x) v] ) is relaed o he lif force. x is he uni m vecor along he projecile's roaional axis of symmery. coefficien. is he do produc operaor [6]. C La is he lif force The hird erm ( SdC m Npa I I y x h x x v ) in Equaion (3-1) is relaed o he magnus force. d is he projecile reference diameer. CNpa is he magnus force coefficien. I y is he projecile ransverse momen of ineria. I x is he projecile axial momen of ineria. h is he vecor angular momenum divided by he ransverse momen of ineria and is he cross produc operaor [6]. The fourh erm ( vsd( C Nq m C Na ) h x ) in Equaion (3-1) is relaed o he pich damping force. C C is he pich damping coefficien [6]. Nq Na 16

31 g is he acceleraion vecor due o graviy. is he coriolis acceleraion vecor. gt m x I y mr m re m h x are he rocke relaed forces. T is he rocke hrus force. m is he projecile mass. re is he disance from he cener of mass of he projecile o he rocke nozzle exi [6]. Equaion (3-1) is a general expression. We will explain five simplificaions of Equaion (3-1) for our own problem. The simplificaions are done by he following hree assumpions: 1. h is zero,. angle of aack which is presened in Figure 3-1 by is zero, 3. is zero. This sudy concenraes on airburs muniions. Airburs muniion mass is consan during he fligh. They are no rocke like muniions. So, gt m x is zero, as T is zero. I y mr mre m h x is zero, since used projecile is non-roaional which means h is zero. SC La The second erm ( [ v x ( v x) v] ) in he Equaion(3-1) diminishes o zero. m The reason can be simply explained as: The angle of aack is aken as zero in his sudy, so x and v will have he same direcion. As a resul v x v, norm of v. Hence, v x ( v x) v can be wrien as v x v v which is equal o v x v x, and hence he desired resul. Anoher consequence of zero angle of aack is ha 17

32 x v 0, since x and v have he same orienaion. Thus, he hird erm SdCNpa I y ( h x x v m I x ) in he Equaion (3-1) diminishes o zero. Since he projecile used in he sudy is non-roaional, h is zero, hence he fourh erm ( vsd( C Nq m C Na ) h x ) in he Equaion(3-1) becomes zero. The force due o coriolis is ignorable compared o graviaional and air drag forces. Therefore, coriolis force is aken as zero during he sudy. Finally, six degrees-offreedom equaions converge o he equaions of Poin-Mass rajecory given in Equaion (3-). d d vsc m D v g (3-) Trajecory formulaion is derived. Then, he rajecory model of he muniion which is used during he sudy should be defined. Firsly, he sae vecor of he model is presened in Equaion (3-3). x x y y z (3-3) z The sae space is six dimensional Caresian space, 6 S. The sae vecor S consiss he posiion p 3, and he velociy v 3 [8]. The general formulaion of he saes are presened in Equaion (3-4) and Equaion (3-5). p v (3-4) v g v v 18

33 (3-5) Above menioned ieraive fligh pah differenial equaions are solved by fourh order Runge-Kua inegraion mehod. X, Y and Z componens of he muzzle velociy is given by Equaion (3-6) where 0 is he muzzle velociy and θ is he firing angle in he Y-Z plane. Z cos 0 Y 0 sin (3-6) X 0 The iniial condiion vecor is hen given as: 0 X 0 ( 0) (3-7) 0 Y Z Whereas Equaion (3-8) given below indicaes he acceleraions: dz d dy d dx d g Z X Y (3-8) The value of sae vecor a he end of one ime sep will hen be given as: 1 ) (0) ( k1 k k3 k ) 6 ( 4 19

34 0 In he above equaion, he i k values (i=1,, 3, 4) are found by using Equaion (3-9). ), (0) ( ), (0) ( ), (0) ( (0),0) ( k f k k f k k f k f k (3-9) Using funcions f and, he explici form of i k values are obained as: 1 ) (0),0 ( Z Z Y Y X X g f k (3-10)

35 1 ) (1 ) (1 ) )(1 ( ) (1 ) (1 ) (1 ), (0) ( 1 1 g k g k f k Z Z Y Y X X Z Z Y Y X X (3-11) )) (1 (1 )) (1 (1 )) (1 )(1 ( )) (1 (1 )) (1 (1 )) (1 (1 ), (0) ( 3 g k g k f k Z Z Y Y X X Z Z Y Y X X (3-1) ))) (1 (1 (1 ))) (1 (1 (1 ))) (1 (1 )(1 ( ))) (1 (1 (1 ))) (1 (1 (1 ))) (1 (1 (1 ), (0) ( g k g k f k Z Z Y Y X X Z Z Y Y X X (3-13)

36 These calculaions are performed for each ime sep o ierae saes forward unil he burs of he muniion. Then, a burs insan he effec of he burs is added o he saes and calculaions sar again unil he paricles hi o he arge. Figure 3- illusraes he ejecion of subprojeciles a burs insan. The ejecion of subprojeciles can be assumed o sar wih firing he muniion from he gun. Muniion flies unil i reaches o he burs poin. Afer burs, 181 paricles sar o fly in a cone shape wih 10 degrees apical angle. Each paricle has a fligh pah and hese pahs are raced by he formulaion which is previously given. Firsly, iniial saes of he paricles are defined. Posiions of he paricles are all he same, namely he burs poin. However, velociies of he paricles are differen, and are given as shown by Equaion (3-14). From his poin, each paricle is raced as menioned in he equaions from 3-8 o Figure 3-: Ejecion of subprojeciles, ploed by simulaor program in MATLAB Figure 3-3 is he illusraive picure of he paricles posiion a lile afer he burs. I is creaed by using MATLAB.

37 Figure 3-3: An illusraion of subprojecile dispersion Afer burs, paricles are scaered in an order. There are 36 paricles on each circle. Their models are given in Equaion (3-14). Figure 3-4: 3D subprojecile componen illusraion In Figure 3-4, AD is he pah of he subprojeciles. In his figure, i is shown ha pah has hree componens, namely X, Y and Z. The disance AB is he disance from burs poin, A, o he cener of he covered area, B. The direcion from poin A 3

38 o B is he direcion of he Z componen of he sub-projeciles. The direcion from poin B o poin C is he direcion of he X componen of he sub-projeciles. And he direcion from C o D is he Y componen direcion. In Equaion (3-14), is he elevaion angle, is he azimuh angle, B is he elevaion angle from -5 o 5 degrees and is used for creaing he cone shape. 150 m/s is he velociy incremen gained by paricles due o burs. Toal X ( burs ) Y ( burs ) Z ( burs ) ( ) ( 150 )cos( B Xp Toal )sin( ) ( ) ( 150)sin( g (3-14) Yp Toal B ) ( ) ( 150)cos( B Zp Toal )cos( ) In Equaion (3-14), is aken as zero a he ime of burs. The sae vecor is refreshed a he burs, and calculaions performed for muniion are repeaed for paricles o find heir pah afer burs. 3.3 COST FUNCTION During he sudy, leas square mehod is used for opimizaion. The variables of he opimizaion process are hi velociy, number of paricles ha hi he arge and coverage. The weighs of he objecives changes wih respec o he arge. Because, some arges are sronger, peneraion of he sub-projecile will be saisfacory if velociy is higher, and for some weak arges, peneraion is saisfacory for low speeds so he imporan componens are disribuion and coverage. Increasing number of sub-projeciles which hi he arge is a common need for all ypes of arges. Equaion (3-15) is a general leas squares cos funcion formulaion. The aim is o find he burs disance which minimizes he cos funcion value. The weighs of he objecive parameers are a, b, and c in Equaion (3-15). Maximum velociy is deermined separaely for each simulaion, he only excepion being he simulaion where he change of burs disance wih respec o firing angle is invesigaed. In he above menioned excepional case, he maximum velociy is 4

39 chosen as he same for each calculaion. Maximum coverage is %100, and maximum number of paricles is 181. C a(max_ hielociy acual_ hielociy ) b(100 Coverage) (3-15) c( oalparicles paricleswhichhi) During he sudy, he weighs are seleced as equal wih, a=b=c=1. This choice indicaes ha he effecs of he differen deviaions have equal imporance. To invesigae he effecs of cos funcion weighs on burs disance change, a separae sudy is conduced in Chaper A FIRING METHOD IN THE CASE OF PERFECT TARGET LOCALIZATION In his par, a firing mehod is presened. The success of he mehod is shown in Chaper 4. Figure 3-5: The aim of he firing mehod is illusraed Figure 3-5, illusraes he aim of he firing mehod. Since arge posiion is perfecly known, hi probabiliy is 100%. Thus, kill probabiliy should be increased. To increase he kill probabiliy, number of paricles ha hi he arge should be maximized. Moreover, coverage should be maximized o give damage more pars of he arge, since he chance of survival for arges decreases wih increasing 5

40 affeced area. On he oher hand, if hi velociy is no enough o give he desired damage, burs disance is decreased. So, coverage is decreased. A Secion 3., he models of he sub-projeciles were derived. Targe surface equaion is given o be as: ( X X ) arge x R ( Y Y ) arge y R ( Z Z ) arge z R 1 (3-16) The rajecory of he muniion is found ieraively and posiions of he paricles are known a each ieraion. In Equaion (3-16): X, Y, and Z are replaced wih he posiions of paricles. When he resul is equal o or smaller han 1, paricle his he arge. In oher words, paricles are checked for his a each ieraion. The cos funcion is calculaed for each assumed burs disance, and as a resul, he burs disance giving he minimum cos is obained. 3.5 A FIRING METHOD IN THE PRESENCE OF TARGET LOCALIZATION ERRORS In his par, he "firing mehod", menioned in Secion 3.4, is changed by handling localizaion errors. Tha means ha he posiion of he arge is no known exacly. The cener of he arge is assumed o have an uncerainy of Gaussian ype. This uncerainy is shown in Figure

41 Figure 3-6: The area where arge cenre posiion exiss mosly In Figure 3-6, covered area represens he area where arge cener posiion exiss mosly. The covered area is illusraed as a circle, however i is an ellipse. The aim of his firing mehod can be shown by a circle whose radius is presened in Figure 3-7 wih a red line. Figure 3-7: The radius of he aimed circle 7

42 In Figure 3-7, circular area indicaes he area where arge cener posiion mosly exiss, i is he circle shown in Figure 3-6. The ellipse represens he arge. The red line shows he radius of he wors case circle ha is he aim of he firing mehod. The sar poin of he line is he cener of he circular area and he end poin of he red line is on he ellipse as shown in he Figure SIMULATOR PROGRAM The simulaor sofware is programmed by using MATLAB. I has a graphical user inerface. Screensho of he program is shown in Figure 3-8. Figure 3-8: The screen sho of he simulaor 8

43 The simulaor program calculaes he pah of he muniion, simulaes burs, and races he pahs of he paricles. The program calculaes he number of paricles ha hi he arge, i calculaes he area ha he muniion covers on he arge surface, and i gives he hi velociy of he paricles o he arge. Targe's posiion ambiguiy is achieved by Mone Carlo simulaions. The model of he muniion derived in Chaper 3. is direcly used in he simulaor. The posiion of he arge is deermined by Mone Carlo simulaions as explained in he following pars. Random number generaion algorihm for Mone Carlo simulaion is creaed using Malab funcion, namely randn(). This funcion (randn) generaes random numbers whose mean is zero, variance is one. In his sudy, random number generaion is done wih he following formula: "mean + variance*randn(1)". Hence, random numbers generaed by Mone Carlo simulaions (errors on he arge posiion) have Gaussian probabiliy densiy funcion. By using Equaion(3-14) and Figure 3-4, he relaion beween azimuh and elevaion angles can be found in order o model he cone shaped disribuion of he sub-projeciles. This condiion is saisfied by he Equaion (3-17). AB AD cos( )cos( ) (3-17) Since cone shaped disribuion exiss, AD is consan for he same cosine muliplicaion angles ( cos( )cos( ) ). AB is he burs disance, hus i is consan. According o 3-17, he relaion beween elevaion angle and azimuh angle is derived, which indicaes ha muliplicaion of cosines of he angles is consan. The arge cross-secion seen by he muniion is illusraed in Figure 3-9 and is deermined by Equaion (3-18). 9

44 Figure 3-9: The illusraion of he arge cross-secion seen by he muniion a Y-Z axis z r y Z r Y 1 z y r r Y ' Y ' sin cos ( ry ' sin r Z ) ( r Y ' cos r Y ) 1 r Y ' r Z cos r r Z Y r Y sin (3-18) In Equaion (3-18), z and y are he coordinaes of he poin ha red line inersecs he upper half of he ellipse. r Z is he radius of he arge in he Z axis. r Y is he radius of he arge in he Y axis. r Y ' is he radius of he cross-secion seen by he muniion in he Y axis wih respec o he recangular coordinae sysem wih he origin of coordinaes locaed a he inersecion poin of he muniion pah and he arge. Coverage calculaion is done by using an image processing funcion of MATLAB. To calculae coverage, an area is defined whose geomeric sizes are 30

45 deermined by using probable errors and dimensions of arge as in he Equaion(3-19). X ' Y' ( r ( r X r r X Y ' Y ' ) ) (3-19) X ' and Y ' are he magniudes of he sides of he area. r X is he radius of he arge in he X axis and r X is he magniude of he max error in he X axis. Then wo zero marices are defined wih he sizes of he area found in Equaion(3-19). The random middle poin which is found by Mone Carlo simulaion is pu on one of he marix. This poin is he cener of he ellipse wih radiuses rx and r Y '. The oher marix is for saving muniion's covered area. Boh of he marices represen he same area perpendicular o he pah of he muniion. Then paricle posiions are combined by roipoly funcion of he Image Processing Toolbox of he MATLAB o define he area ha muniion covers. Then, biwise AND operaion is performed for hese marices. Thus, covered area by he muniion on arge surface is found. Figure 3-10, 3-11, and 3-1 show he images in sequence o describe coverage calculaion beer. Figure 3-10: Targe area which is seen on he pah of he muniion In Figure 3-10, arge is placed o he marix. The ellipic area is one, grey par is zero in he marix. This ellipic area represens he arge area seen by he muniion. 31

46 Figure 3-11: Area covered by muniion In Figure 3-11, covered area by he muniion is se o one and he res is zero. Figure 3-1: The inersecion of he arge and area covered by muniion In Figure 3-1, he inersecion area of Figure 3-10 and Figure 3-11 is shown. This area is achieved by performing biwise AND operaion beween arge area and covered area of he muniion as described in Figure

47 Figure 3-13: The descripion of he covered area In Figure 3-13, A is he inersecion of arge area (Figure 3-10) and area covered by muniion (Figure 3-11). B is he arge area no covered by muniion. No damage is done o ha par of he arge. C is he covered area ha does no cover any arge area. The aim hroughou he sudy can be summarized by making use of he Figure 3-13 as increasing A, decreasing C and B as much as possible. 33

48 CHAPTER 4 ANALYSIS AND SIMULATIONS In his par, resuls of he simulaions o find opimum burs disance are presened. Furhermore, he parameers ha affec opimum burs disance are analyzed. Throughou he simulaions, burs disances are swep by changing burs ime. The incremen beween burs disances is abou 1 m. Hence, he accuracy of burs disance is 1 m. However, calculaed burs disances, which belong o ha burs ime, naurally have fracions which is no meaningful when accuracy is 1 m. To eliminae his siuaion, found burs disances from he simulaions are rounded o inegers a his chaper. As known from Chaper 3, he coordinae sysem used during he simulaions is Caresian coordinae sysem which is given in Figure 4-1. Figure 4-1: A represenaive figure of coordinae sysem In Secion 4.1, he effec of firing angle on burs disance is simulaed and resuls are presened by a graph. In Secion 4., arge disance from he gun is changed o see he effec of ha on burs disance. In Secion 4.3, dimension of he arge is 34

49 changed and change in he burs disance wih respec o arge dimension is presened. In he nex secion, effecs of he weighs of he opimizaion funcion on burs disance are analyzed separaely and resuls are presened wih graphs. In Secion 4.5, effec of he ambiguiy in arge locaion on burs disance is analyzed and resuls are presened. In he nex par, he effec of he wind on burs disance is simulaed in hree differen ways; wind agains movemen, wind suppors movemen, and side wind. The change in he burs disance wih respec o wind velociy is graphed and presened. In Secion 4.7, he effec of he velociy difference afer burs on opimum burs disance is analyzed. In Secion 4.8, a scenario is buil wihou ambiguiies and burs disance is found. Moreover, he resul of he firing mehod is compared o he opimum burs disance. In he las par, a scenario is simulaed such ha here are ambiguiies in arge posiion burs velociy. Burs disance is hen calculaed in he presence of such ambiguiies. Furhermore, he resul of he firing mehod is compared o he opimum burs disance. 4.1 THE EFFECT OF FIRING ANGLE An imporan par of he simulaions is he effec of firing angle on burs disance. Firing angle, also known as elevaion angle, is he angle beween gun urre and ground as shown in Figure 4-. Figure 4-: A represenaive figure of firing angle 35

50 Five differen firing angles; 15, 30, 45, 60, and 75 degrees are simulaed respecively wih he following cos funcion defined in Chaper 3: C (max_ hielociy (181 acual _ hielociy ) acual _ numberofparicles ) (100 acual _ Coverage) The inersecion poin of he muniion pah and he arge pah is assumed as 1000 meers away from he gun which is represened by 'r' in Figure 4-3. Figure 4-3: An illusraion of firing angle simulaion Dimensions of he arge are aken as follows: X radius is 10 m, Y radius is 5 m, and Z radius is 10 m. The resuls of his simulaion are presened in Figure 4-4 which shows he change in burs disance wih respec o firing angle. 36

51 Figure 4-4: The effec of firing angle on opimum burs disance As shown in Figure 4-3, firing angles are changed from 15 degrees o 75 degrees. A 15 degrees burs disance is 10 m and a 75 degrees burs disance is 109 m. From hese simulaions, i is seen ha burs disance increases wih firing angle. This is an expeced resul, since cross-secional area of he arge increases wih angle which is clear by Equaion (3-18) and burs disance increases wih crosssecional area as given in Secion 4.3. Table 4-1: Deviaion weighs of he objecive parameers in oal deviaion Firing angle (degree) Opimum burs disance (m) Deviaion square in velociy Deviaion square in coverage Deviaion square in # of paricles Toal cos value

52 To observe he conribuion of each error erms o he oal cos value, a able (Table 4-1 given on he previous page) is consruced. The able shows he firing angle, he corresponding opimum burs disances, he individual error squares, and finally he oal opimum cos value. I can be easily observed ha he oal cos value, and he cos due o number of paricles hiing he arge decreases dramaically as he firing angle increases. This is an expeced resul, because as firing angle increases he cross secional area of he arge facing he paricles increases. 4. THE EFFECT OF RANGE The effec of he range of he arge on he burs disance is an imporan par of he simulaions. The dimensions of he arge used during he simulaions are he same as in Secion 4.1. Range in hese simulaions sands for he disance beween he gun and he inersecion poin of he muniion pah and he arge pah. Five differen ranges are simulaed in his par o see he change in he burs disance. These ranges correspond o very shor range air defense in real life, as in he case of a demonsraion of 35 mm airburs muniion handled by Army Research Laboraory []. These simulaed ranges are: 500 m, 1000 m, 1500 m, 000 m, and 500 m a 45 degrees of firing angle. Figure 4-5 shows he change in burs disance wih respec o range. Figure 4-5: The effec of range on opimum burs disance 38

53 As shown in Figure 4-5, ranges are changed from 500 m o 500 m. When range is 500 m burs disance is 110 m and when range is 500 m burs disance is 100 m. Thus, i is inferred from hese simulaions ha burs disance decreases wih ranges which is expeced. Angle of he muniion wih respec o ground decreases on he rajecory wih range. Hence, opposie of Secion 4.1, burs disance decreases wih decreasing angle. 4.3 THE EFFECT OF THE TARGET DIMENSIONS The effec of he arge dimensions on burs disance is anoher imporan par of he simulaions. Since arge is modeled as an ellipsoid, dimensions are he radii in X, Y, and Z axes. Simulaions a his par are handled for five differen arge dimensions. The radii are saed in he (x, y, z) forma; (5, 3, 5), (10, 5, 10), (15, 7, 15), (15, 15, 15), and (0, 10, 0). Firing angle is 45 degrees during he simulaions. Moreover, hi poin, inersecion of he muniion pah and he arge pah, is chosen as 1000m from he gun. Figure 4-6 shows he change in burs disance wih respec o arge volume. Figure 4-6: The effec of arge dimensions on opimum burs disance 39

54 If arge area increased wih he same burs disance, coverage would decrease. To increase coverage objecive, burs disance should be increased. According o Figure 4-6, burs disance increases wih arge dimensions which is an expeced resul. If arge dimension increases, burs disance will increase. 4.4 THE EFFECTS OF THE WEIGHTS IN THE COST FUNCTION In his par, he effecs of he weighs of he cos funcion on opimum burs disance are analyzed. Iniially, he weighs of he objecive parameers are equal o 1. Effec of an objecive parameer is simulaed, while weighs of he oher objecive parameers are kep consan. Firing angle is 45 degrees during he simulaions. Hi poin, inersecion poin of he muniion pah and he arge pah, is aken as 1000 m from he gun. Dimensions of he arge are he same as in Secion 4.1. Figure 4-7 is he graph of equal weigh cos funcion wih respec o burs disance. Opimum burs disance is 107 m. Figure 4-7: The graph of equal weigh cos funcion wih respec o burs disance Targes wih sronger skin are hard o damage. Paricles should hi wih higher velociy o hese kinds of arges. Hence, his makes he imporance of he hi 40