NPROCEEDINGSNATIONAL CONFERENCE ON ADVANCES IN ARMAMENT TECHNOLOGY 21-22 Nov 2008

Armament Research & Development Establishment Pashan, Pune - 411 021

INDEXPaper Author Title

S1S1.1 S1.2 S1.3 S1.4 S1.5 S1.6 S1.7

Aerodynamics & Structures and Inbore, External & Terminal BallisticsPeyada NK, Sen A, Dutta GG, Singhal A, Rajan KM, Joshi DK, Ghosh AK Sonawane PD, Dr. Singh Amarjit Misra A, Ghosh AK, Ghosh K Chand KK Giri SN, Chand KK Raj Venkatnarayan, Varadrajan S Gite LK, Padmanabham M, Joshi DK Effect of Mach Number on Supersonic Wrap Around Fin Aerodynamics Slender Aerodynamic Bodies for High Speed Armaments Stability & Control of Aircraft Bombs at High Angle of Attack using Cascade Fins Understanding Trajectory Modeling via Error & Uncertainty Analysis Role of Applied Physics & Ballistics in Armaments : A Review Weapon Locating Radar Parameter Estimation Techniques : Application to Prediction Algorithm for Weapon Locating Radar Ballistic Needs for Weapon Locating Radar

S1.8

Padmanabham M, Joshi DK, Gite LK

S2S2.1 S2.2 S2.3 S2.4 S2.5

Warheads and Non Lethal WeaponsRajeev Prakash Anguswamy V, Sahu AC Brig. HSN Sastry (Retd) Dhote KD, KPS Murthy Harikrishnan S, KPS Muthy, Surendra Kumar Mohammadali Partanian, Pralhad RN Padhan PK, Biswas PP, Dayanad, Bhattacherjee RN Kamlesh Kumar, Satwinder Singh, KPS Murthy Non Lethal Weapons : Naval Perspective Dual Type Fragmentation Warhead for Hard/Soft Targets Non Lethal Weapons : Concepts, Technologies & developments Velocities of Pre-made Fragments in Multi-Layers Design & Evaluation of a Light Weight Blast Barrier for Tandem Warhead of a Third Generation Anti Tank Missile Analysis of Jet Penetration of Hollow Charge from Concept of Hydrodynamic Theory & Fracture Mechanics An Overview on End Game for Surface to Air Missile (SAM) Thermobaric Warhead for Under-Ground Targets I

S2.6

S2.7 S2.8

NCAAT 2008

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Author

Title

S3

Weapons, Launch Technology, SAM, Fuze & Armament Electronics

S3.1

Sen A, Subrahmanyam S, Wahi P, Ghosh AK Rao MV, Mahalingam VS Salutagikar CS, Mukane SB, Pravker RC Pawale SO, Rojatkat PT, Rajan KM Mahmoud Zarrini & Pralhad RN Pandey RP, Verma Gaurav, Dixit Aishwarya, Gade SV, Datar AM Pandey RP, Verma Gaurav, Dixit Aishwarya, Gade SV, Datar AM Khilare TB

Five-Degree of Freedom Dynamic Wind Tunnel Test Rig for Estimation of Dynamic Derivatives of Scale Down Models Pinaka Weapon System Integration With ACCCS : Challenges & Solutions Enhancing Accuracy of Free Flight Rocket By Minimizing Dynamic Unbalance : A Case Study Estimation of Shock Velocity & Pressure for the Bullets Impacting on Human Body Lethality Assessment of Small Arms Ammunition Effect of Small Arms Bullet Construction on Penetration of Hard Targets Enhancement of Lethality of Rocket Assisted Bomb Over Standard HE Mortar Bomb Integrated G-hardened Telemetry System for Cannon Launched Munition Flight Testing Application of Wireless Sensor Network in the Selective & Intelligent Actuation of Scattered Munition Remote Time Data Setting for Munitions Study of Frequency Selection for Communication with Underground Munitions Hybrid Composite Technology For Gun Barrel Development Testing, System Integration & Evaluation of Indigenous Surface to Air Weapon System

S3.2 S3.3

S3.4 S3.5 S3.6 S3.7

S3.8

Saini Vaibhav, Kumar Vinay, Saini DS, Rana SC, Kapil Deo Saha PK, Sahu BK, Dhande P, Ms Mahajan CP Dhande P, PK Saha, BK Sahu Vaishnavi Sharma, Ms Mahajan CP Ms Savita, Ms Patel Prachi, Dhande Pramod, Saha PK, Ms Mahajan CP Deodhar KD, Kharat DK Sachin Sood, Narasimha Rao G, Chandramouli G, Panyam RR

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S3.10 S3.11

S3.12 S3.13

S4S4.1 S4.2

Test & Evaluation and Guidance & ControlKolhe JP, Shaik JB, Bokil GA, Talole SE Bommu Mallikarjuna, Venkata Rama Subbaiah B, B Srinivas Kalyan, C Arjun Rao, Talole SE Robust Roll Autopilot Design Missile Autopilot Design Using Dynamic Inversion and Fuzzy Logic Control

NCAAT 2008

II

Paper

Author

Title

S4.3 S4.4

Singh VK, Sharma IS, Dayal Arun and Rajiv Gupta Nayak Navin, Bisval TK

Inferencing from Limited Trials in Weapons Evaluation High Speed Imaging & Radar Based Measurements for Performance Evaluation of Armaments Dynamic Test & Evaluation Methodologies for Measurement of Critical Test Parameters of Armaments : Case Studies Next Generation of Environmental Testing Analytical Method for Predicting Sabot Discard Phenomena Study of Gun Dynamics Using Data Mining Advanced Miniature Precision Effectors Guidance for Perpendicular Hit for Shaped Charge Warhead Testing & Revalidation of Imported Cartridge PDO for Su-30 Fighter Aircraft Testing & Evaluation of Canopy Severance System for Tejas Trainer Aircraft Impact Studies on Spaced Armour

S4.5

Bisval TK, Nayak Navin

S4.6 S4.7 S4.8 S4.9 S4.10 S4.11 S4.12 S4.13

Saha Subhas, Kolhatkar DR Bhange Niteen P, Ghosh AK Lakra S, PK Das Gupta, V Anguswamy Dr. Fleming R, Dr Gorge McConnell Jha PK, Kumar Krishna Shaikh AD, Salkar YB, Parate DA, Dr. Virendra Kumar, Dr. Kharat DK Katare RV, Ravidas Rakesh Mathew T, Chandel P, Rajeev Kumar, Athwal M, Sewak KB

S5S5.1 S5.2 S5.3

Modeling & Simulation and Safety, Quality & ReliabilitySharma IS, Singh VK, Dayal Arun, Rajiv Gupta Singh SK, Kumar Manoj, Sharma VK, Srinivasa Rao HV Kamble AD, Guru Prasad, Gandhi Vandana Modeling Weapon Effectiveness in Integrated Combat Scenario Mathematical Modeling for Earth Penetrating Bomb Design of MIL - STD - 1553B Communication Interface Protocol & Computer Models for a RTOS Based Distributed Mission Critical Electro - Hydraulic Control System Swarm Intelligence Based Algorithms for Design Optimization of Stiffened Cylindrical Shells for Underwater Weapons Evaluation of Probability Grid Using Simulation of Fragmentation Pattern on Ground Development of Fuzes, Safe/Arm System & Seekers III

S5.4

KKK Sanyasi Raju, D Chakraborti

S5.5

Deodhar RS, Anupam Anand

S5.6

Philppe Ecolivet

NCAAT 2008

Paper

Author

Title

S5.8

Sunil Kumar K. Surya Kiran CH, Jaya Ramiah M, Vara Prasad Rao S, Sekharan VG Dr. Rajagopal Chitra Mishra GK Wadhawa KC Aguru NK

Modeling & Simulation on Explosive Dynamics Phenomena during Stage Separation Mechanism Ammunition Storage and Safety Regime in India. Comparison with Other Countries. Disposal of Life Expired and Regiime in India : Comparison with other Countries Fire Safety Aspects of Armaments & Ammunition Enhancement of Explosive safety in Indian Army & Reduce Land Requirement by Converting Ammunition of un Hazard Division 1.1 To 1.2 Numerical Simulation on Explosively Formed Projectile

S5.9 S5.11 S5.12 S5.13

S5.14

Bhupinder Sewak, Sharma AC, Saroha DR, Singh M

S6S6.1 S6.2

Explosives, Propellants, Pyrotechnics and Propulsion TechnologyProf. Barpande Girish, Dr. Singh Amarjit Sanghavi RR, Shelar SD, Tope BG, Chakraborthy TK, Singh Amarjit, A Subhananda Rao A Sudheer Babu CFD Simulation and Analysis of Flow Through a Supersonic Combustor Studies on Effect of Picrite on Insensitive Nitramine Gun Propellants Prediction of Impact Sensitivity of CaHbNcOdExplosives Using Artificial Neural Networks Development of Aluminised Explosive Composition for Thermobaric Warhead Applications Development of Laser Initiated Pyrodevices for Futuristic Defence Applications Experimental Studies on Accuracy Improvement of Pyrotechnic Delays High Intensity Illuminating Flares Based on Energetic Binder Studies on Thermal and Structural Analysis of Pyrotechnic Thermite System Designed for Fast Melting of Metals Estimation of Flow Parameters of Explosion at Shock Front for Different Explosives IV

S6.3

S6.4

Saji J, Vishwakarma AK, Roy S, Apparao A

S6.5

Daniel R, Danali SM, Kohadkar MJ, Raha KC, A Subhananda Rao Danali SM, Swarge NG, Gaikwad VD, Kohadkar MJ, Raha KC Palaiah RS, Soman RR, Thakur BR, Kohadkar MJ, Raha KC, A Subhananda Rao Debnath S, Rohit, VP Ambekar, PM Deshmukh, Danali SM, V Ramaswamy, Raha KC Swami Umesh, Pralhad RN

S6.6 S6.7

S6.8

S6.9

NCAAT 2008

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Author

Title

S7

Advanced Materials and Manufacturing Techniques

S7.1

Ramakrishna B, Mishra B, Madhu VT, Balakrishna Bhat I Nithiya Priya, HH Kumar, DK Kharat Premkumar S, B Praveen Kumar, HH Kumar, DK Kharat Kolhe SM, HH Kumar, Koparkar PV, Lonkar CM, DK Kharat Chakradeo NM, Dighe NA

Perforated Armour : Improving The Ballistic Efficiency of HH Armour Steel Rheological study of Aqueous PZT slurry for Tape Casting Characteristics of 3-3 porous PZT Composites Evaluated by Finite Element Analysis Piezoelectric Polymer 0-3 Composite for Power Harvesting Application Use of CAD/CAM Technology for Manufacture of Critical Armament Components Manufacturing of Barrel for "Under Barrel Grenade Launcher" (UBGL) Advances in Spring Materials for Armament Application Effect of Fracture Toughness & Charpy Impact Value on Inbore Structural Integrity of KE Projectiles Design of Cartridge Signal Flare 26mm by using Aluminum Case Ti-based Gate Dielectrics on Si-passivated n-GaAs for MOSFET Applications

S7.2 S7.3

S7.4 S7.5

S7.6 S7.7 S7.8

Chakradeo NM, Dighe NA Dharia CJ B Murlidhar, SM Nirgude, M Ashok Kumar Jadhav SC, Parate BA, Moon LC, Dr. Virendra Kumar Das T, Das PS, Chakraborty A, Majhi B, Hota MK, Mallick S, Verma S, Maiti CK

S7.9 S7.10

NCAAT 2008

V

S1Aerodynamics & Structures Inbore, External & Terminal Ballistics

NEFFECT OF MACH NUMBER ON SUPERSONIC WRAP AROUND FIN AERODYNAMICSPeyada NK, Sen A, Dutta GG, Singhal A, Ghosh AK IIT Kanpur 208 016 Rajan KM, Anand Raj Armament Research & Development Establishment, Pune 411 021ABSTRACT Wrap around fins (WAF) have been used on tube launched missiles and dispenser launched projectiles. Modern advances in stealth technology have made the use of missiles equipped with WAFs desirable because they can be stowed to reduce the radar cross section of the aircraft. Recent studies have identified several roll and pitch moment instabilities. The rolling moment of the WASF is positive at subsonic velocities (defined here to means that the missile rolls towards the concave side of fin). A roll reversal occurring at M = 1.0 indicates that the magnitude of the rolling moment decreases with increasing Mach number and that a second rolling moment reversal may occur at high supersonic speed. Tilman et al and Bowersox examined experimentally and numerically the flow structure in the vicinity of a single fin mounted onto blended cylindrical body at Mach 3.0 and 5.0 Those studies indicated that the flow field was highly asymmetric about WAF, with a stronger bow shock structure on the concave side of the fin. This resulted in creating a very high-pressure region between the fin and its center of curvature, which resulted in relatively high surface pressures near the mid span of the fin. In contrast, on the convex side of the fin, surface pressure was relatively independent of location along the span. This resulted in asymmetric pressure loading on the fin, which caused generation of out-of-plane moment at angle of attack, it is suspected that this side moment is symptomatic of WAF configurations. When testing such configurations, the test engineer should ensure that this side moment is obtained and stability boundaries are computed. This side moment can have a dramatic effect on trajectory computation based on the conventional aerodynamic coefficients and derivatives. Experiments have been conducted to study the effect of fin shapes on generation induced rolling moment and outof-plane moment coefficients. Four sets of WAF configuration were tested. The results were then compared with the flat fin configuration. It was observed that addition of back sweep helps in reducing rolling moment as well as out of plane moment substantially at supersonic Mach numbers. 1. INTRODUCTION Wrap around fins (WAF) have been used on tube launched missiles and dispenser-launched projectiles. Modern Advances in stealth technology have made the use of missiles equipped with WAF's desirable because they can be stowed to reduce the radar cross section of he aircraft1. Recent studies have identified several roll and pitch moment instabilities2. The roll reversal at transonic conditions is the most recognized instability. The rolling moment of the WAF is positive at subsonic velocities (defined here to means that the missile rolls towards the concave side of fin)1. A roll reversal occurs at M = 1.0 have indicated that the magnitude of the rolling moment decreases with increasing Mach number and that a second rolling moment reversal may occur at high supersonic speed. Tilman et al3 and Tilman and Bowersox4 examined experimentally and numerically the flow structure in the vicinity of a single fin mounted onto a blended cylindrical body at Mach 3.0 and 5.0. Those studies indicated that the flow field was highly asymmetric about WAF, with a stronger bow shock structure on the concave side of the fin. This resulted in creating a very high-pressure region between the fin and its center of curvature, which resulted in relatively high surface pressures near the mid span of the fin. In contrast, on the convex side of the fin, surface pressure is relatively independent of location along the span. This resulted in asymmetric pressure loading on the fin. This asymmetric pressure loading caused generation of out-of-plane moment at angle of attack, It is suspected that this side moment is this side moment is symptomatic of WAF configurations, when testing such configurations, the test engineer should ensure that this side moment is obtained and stability boundaries are computed. Designers of such configurations should also consider the possibility of this side moment because it can have a dramatic effect on trajectory computation based on the conventional aerodynamic coefficients and derivatives5.

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2. RESULT AND DISCUSSION Experiments have been conducted to study the effect of fin shapes on generation induced rolling moment and outof-plane moment coefficients. Four sets of WAF configurations were tested. The results were then compared with the flat fin configuration. Configurations of the fin tested are shown in Fig. 1.

Configuration - I Fig. 1 : Fin shapes tested for lateral characteristics

Configuration -II

It is observed that addition of trailing edge sweep helps in reducing induced rolling moment as well as out of plane moment substantially at supersonic Mach numbers. Figure 1 and 2 present comparison of induced rolling moment and out of plane moment coefficient for these configurations. It is clearly seen that the fin with a larger back sweep resulted in lesser at play Cl , Cy and Cnasupersonic Mach number. These values of Cna important role in establishing dynamic stability. A preliminary study shows that the design of fin must ensure lower value of C na 6 ) (

Fig. 2 : The comparative variation of Cl , Cy and Cn as function of Mach Number

a

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N3. RESTRICTED VALIDATION BY FLIGHT TEST Three sets of rockets having 20 degrees trailing edge sweep were fired. The trajectory coordinates (x, y, z, V) of these rockets in flight were acquired by Doppler radar. The trajectory variables were used to estimate drag coefficients using method proposed in Ref. 5. The trajectory variables used for estimation of CD is presented in Fig 3.

Fig. 3 : Trajectory variables used for estimation of CD

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Estimated values of CD are graphically presented in Fig. 4

Fig. 4 : Estimated value of CD as function of Mach Number

It could be appreciated that the estimated CD is almost 8 12% higher tan the wind tunnel estimation. This information may help in tuning the trajectory model for field Application.

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N4. CONCLUSION The introduction of trailing edge sweep helps in reducing induce roll and out-of-plane moment coefficient. Such an approach may help in avoiding any possibilities of dynamic instability associated with WAF rocket at supersonic speeds. The estimation procedure presented in Ref. 6 may be used to estimate overall drag coefficient of the rocket using measured flight data. 5. ACKNOWLEDGEMENT The numerous suggestions provided by Shri D.K. Joshi, Joint Director, ARDE, Pune is highly acknowledged. REFERENCES 1.Mc Intyre T.C., Bowersox, R.D., and Larry, P.G., Effects of Mach number on Supersonic Wrap Ariound Fin Aerodynamics, Journal of Space craft and Rocket, Vol. 35, No. 6, Nov Dec 1998. 2.Abate, G.L. and Winchenbach, G.L., Aerodynamic of Missiles with Slotted in Configurations, AIAA paper 910676 January 1991. 3.Tilmann, C.P., Huffman, R.E., Jr Buter, T.A. and Bowersox, R.D.W., Experimental Investigation of the flow structure Near a Single Wrap Around Fin, Journal of Spacecraft and Rocket, vol. 34, No. 6, 1997. 4.Tilman, C.P. and Bowersox, R., Characterisation of the flow field near a wrap around fin at Mach number 4.9, AIAA paper 98-0684, January 1998. 5.Winchenbach, G.L. and Randy, S.B., Whyte, T.A. and Hathway, W.H., Subsonic and Transonic Aerodynamics of a Wrap Around Fin configuration, Journal of Guidance. Vol. 9, No. 6, Nov Dec 1986. 6.G.G. Dutta, Ankur Singhal and A.K. Ghosh, "Estimation of Drag Coefficient using Real Radar Tracked Data of an Artillery Shell," Proceedings AIAA, Paper No. 2006-6149, AIAA Atmospheric Flight Mechanics Conference and Exhibit06, Keystone, Colorado21-24 Aug, 2006.

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NSLENDER AERODYNAMIC BODIES FOR HIGH SPEED ARMAMENTSPushkaraj D. Sonawane Vishwakarma Institute of Information Technology, Pune Dr. Amarjit Singh Defence Institute of Advanced Technology, Pune

ABSTRACT The requirement of defense against ballistic missiles has led to a renewed interest in the aerodynamics of slender, blunt-nosed, conical bodies for high speed armaments. Different slender aerodynamic bodies which can be used for the high speed armaments are as follows: Slender Bodies: Jorgensen has shown that there are distinct aerodynamic advantages in using elliptic cones rather than circular cones. Reggiori has demonstrated that a cone with anhedral wings can provide good high lift characteristics at high angles of attack in hypersonic speeds. Singh, Kumar, and Tiwari conducted a parametric study to determine the effects of nose bluntness on the entire flowfield over slender bodies (blunted cones and ogives) under different hypersonic free stream conditions. Specific results obtained for spherically-blunted cones and ogives demonstrate that there are significant differences in flow-field and surface quantities between sharp and blunted bodies. Jorgensen investigated the aerodynamic advantages of elliptic cones over circular cones experimentally to determine the force and moment characteristics for elliptic cones at high Mach numbers. Power Law Bodies Power-law bodies can be an alternative to the blunted cone configurations because a power-law body has a greater internal volume than a blunted cone of same fineness ratio and secondly, various theories predict that a power-law body represents the minimum drag at hypersonic speeds. Spencer et al found from experiments on a series of 2/3-power law-wave-drag bodies at high Mach numbers that increase in the lift, drag, and pitching moment coefficients and lift-drag ratios at positive angles of attack and at all test Mach numbers. Jorgensen et al performed an experimental investigation to determine the effects of side strakes on the aerodynamic characteristics of a body of revolution, at High Mach Numbers and incidences from 0 to 58. The data demonstrated that the aerodynamic characteristics for a body of revolution with side strakes can be significantly affected by changes in nose fineness ratio, nose bluntness, Reynolds number, Mach number and angle of attack. Amarjit Singh performed the force measurements for a one-half-power-law body and found that the lift, drag and pitching moment coefficients increased smoothly with incidence. The strakes were found to improve the L/D ratio of the configuration without any significant change in the pitching moment characteristics. Conclusions: Bodies with elliptic cross-section exhibit higher lift to drag ratios than those of circular cross-section of the same fineness ratio and volume. Power-law bodies can be an alternative to the blunted cone configurations because a power-law body has a greater internal volume than a blunted cone of same fineness ratio and represents the minimum drag case at hypersonic speeds. The additions of strakes to the slender bodies are found to improve the L/D ratio of the slender bodies configuration without any significant change in the pitching moment characteristics. NOMENCLATURE M8 Re8 L D Cl Cd free-stream Mach number free-stream Reynolds number lift drag lift coefficient based on body planform area drag coefficient based on body planform area

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NCm Y Rb d X l n pitching moment coefficient based on body length and planform area the local radius of the body body base radius diameter axial distance from the nose body length the power exponent. incidence angle

1. INTRODUCTION The requirement of defence against ballistic missiles has led to a renewed interest in the aerodynamics of slender, blunt-nosed, conical bodies flying at hypersonic speeds. In hypersonic flows, nose bluntness for leading edges of the aerofoil is extremely useful to reduce stagnation heating. Many aerospace vehicles use blunt noses and leading edges not just because it will be very difficult to keep them sharp in hypersonic flow but also because of the added advantage of reduced aerodynamic heating and increased internal space. Theoretical and experimental investigations(1,2) have shown that an elliptical cone may have significantly higher lift to drag ratios than a circular cone of the same cross sectional area per unit length. 2. SLENDER BODIES Jorgensen(2) has shown that there are distinct aerodynamic advantages in using elliptic cones rather than circular cones. Reggiori(3) has demonstrated that a cone with anhedral wings can provide good lift characteristics at high angles of attack in hypersonic speeds. Singh, Kumar, and Tiwari(4) conducted a parametric study to determine the effects of nose bluntness on the entire flow-field over slender bodies (blunted cones and ogives) under different hypersonic free stream conditions. Specific results obtained for spherically-blunted cones and ogives demonstrate that there are significant differences in flow-field and surface quantities between sharp and blunted bodies. Depending upon the flow conditions and geometry, the differences are found to persist as far as about 300 nose radii downstream. For different angle of attacks and nose bluntness the post shock flow field is studied(5) in detail from the contour plots of Mach number, density, pressure, and temperature. The effect of nose bluntness for slender cones persists as far as 200 nose radii downstream. Jorgensen(6) investigated the aerodynamic advantages of elliptic cones over circular cones experimentally to determine the force and moment characteristics for elliptic cones at high Mach numbers. Results showed that bodies of elliptic cross-section exhibited higher lift to drag ratio than those of circular cross-section. 3. POWER-LAW BODIES Power-law bodies can be an alternative to the blunted cone configurations because a power-law body has a greater internal volume than a blunted cone of same fineness ratio and secondly, various theories predict (7-8) that a power-law body represents the minimum drag at hypersonic speeds. Spencer et al(9) found from experiments on a series of 2/3-power-law wave-drag bodies at high Mach numbers that increase in the lift, drag, and pitching moment coefficients and lift to drag ratios at positive angles of attack and at all test Mach numbers. The successive increases in major-to-minor axis ratio, with the major axis horizontal, resulted in gains in lift-curve slope for bodies of the same fineness ratio. The surface co-ordinates of a power-law body (Figure 1) are given by Y/Rb = (X/l)n, The n = 1 value represents a sharp cone and a reduction in the value of n from 1 to 0 continuously increases the bluntness of the body. In general nose bluntness is useful to reduce the stagnation heating as well as to increase the volume for a given fineness ratio (l/d) body.

Fig. 1 : Surface contour of power-law bodies

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NSpencer and Fox(10) compared the aerodynamic characteristics of power-law bodies, of circular and elliptic crosssection with that of a body of minimum-wave drag shape determined under the constraint of prescribed body length and volume. They found that the increase in the ellipticity ratio for a given power-law or the theoretical minimum-wave-drag body results in an almost constant incremental increase in maximum L/D ratio, independent of body longitudinal contour. Ashby(11) obtained experimental data for power-law bodies found that at the highest Reynolds number, the powerlaw body for minimum drag is blunter (exponent n lower) than predicted by inviscid theory (n approximately 0.6 instead of n = 0.667). Westby and Regan(12) conducted an experimental investigation of the longitudinal aerodynamic characteristics of power-law bodies of revolution in a gun-tunnel also found the same reults. 4. SLENDER BODIES WITH STRAKES Jorgensen et al(6) performed an experimental investigation to determine the effects of side strakes on the aerodynamic characteristics of a body of revolution, at High Mach Numbers and incidences from 0 to 58. The data demonstrated that the aerodynamic characteristics for a body of revolution with side strakes can be significantly affected by changes in nose fineness ratio, nose bluntness, Reynolds number, Mach number and angle of attack. Removing the strakes from the cylindrical after section greatly decreased the lift and moved the center of pressure forward. Levinsky et al(13) showed theoretically and experimentally that relatively large increases in lift with even the smallest stakes sizes, i.e. 10% of the body radius, are possible. Amarjit Singh(15) performed the force measurements for a one-half-power-law body is found to be in good agreement with the other existing results. The lift, drag and pitching moment coefficients are found to increase smoothly with incidence. The strakes are found to improve the lift to drag ratio of the configuration without any significant change in the pitching moment characteristics. Kontis et al(17) performed experiments and collected data for two pairs of elliptic cones with and without strakes at Mach number 8.2 and unit Reynolds number 9.3 x 106 /m. The strakes produced a significant increase in the lift and drag coefficients, in the incidence range of 0 to 20. The addition of strakes moved the centre of pressure slight forward for the right elliptic model and backwards for the power-law body at all incidences tested. Both experimental and computational studies(18) showed that the addition of strakes caused a reduction of pressure on the leeward side and an increase of pressure on the windward side. The paper concludes with suggesting that, in the design stage, CFD can be used to identify important areas and important flow conditions before the model is cut, which can save unnecessary waste in the experimental tests. After data collection from the experiments, detailed numerical simulation can be conducted for the experimental conditions. 5. EXPERIMENT Experiments(15) were conducted to obtain lift, drag and pitching-moment coefficients for a one-half-power-law body at the College of Aeronautics gun tunnel by one of the authors. The axisymmetric nozzle provided a useful jet of 15 cm diameter at M8 of 8.2 and Re8 equal to 9.35 x 106/m. A three-component strain gauge balance was used to measure axial, and normal forces and pitching moment. The balance was calibrated using a locally made test rig. The calibration of the balance was repeated many times and showed insignificant difference.

Fig. 2 : Power-law elliptic body model with and without strakes

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NFigure 2 shows the basic dimensions of the wooden models. These models were187.5 mm long. The strakes are approximately 1.67 mm square at the base. Dead weights were used at the base of the models to get the C. G. position at about 26 mm from the base. The mass of the models was 56 grams and 33 grams with and without strakes respectively. The models had an elliptical cross section and a one half-power-law contour along the length. The models were rigidly screwed on to the balance sting and fixed at zero roll angle. 6. NAVIER STOKES COMPUTATIONS A commercial solver has been used to calculate the flow field around the half-power-law elliptic body shown in 0 figure 2. Flow field results for incidence angle = 0 , 30, 60, 90, 120,150 are obtained assuming a perfect gas and a laminar 6 boundary layer over the full length of the body. The free-stream conditions used are M8 = 8.2 and Re8 = 9.35x10 / m. Approximately 800000 polyhedral cells are used for discretization of the flow field. The Navier-Stokes solution converged to the experimental shock position after about 16,000 iterations on a Quadra core Linux workstation. Figure 3 shows the Mach number contours which are reasonably good in agreement with the experimental results(15) . Model without strakes has been run and model with strakes will be run soon to optimize the shape size and location of strakes to obtain maximum lift to drag ratio without affecting the pitching moment characteristics.

Fig. 3(a) : Mach number contours of a half-power-law elliptic body

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Fig. 3(b) : Mach number contours of a half-power-law elliptic body

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7. RESULTS AND DISCUSSIONS Lift Coefficient Figure 4 (a) shows the variation of lift coefficient with incidence. The computational results compare reasonably well with the experimental values. The experimental values are slightly above the computational values. The gap widens above = 5, the angle around which the leeward flow is expected to separate forming counter rotating leeward vortices.

Fig. 4(a,b,c) : Lift, Drag, Moment Coefficient vs Incidence angle

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Drag Coefficient Figure 4 (b) compares the drag values. The computational values are lower than the computational values. As the incidence angle is increasing the variations in the results is more. Pitching Moment Coefficient The variation in pitching moment coefficient, figure 4 (c), follows the experimental results at lower incidences. At higher incidences the difference in experimental results and the computational results is significant due to effect of base flow and effect of balance string has not been modeled in the computational results. Lift to Drag Ratio The computational L/D ratio shows improvement over the experimental above 7 incidence due to increased suction over the leeward side, figure 4 (d).

Fig. 4(d,e) : L/D ratio and Xcp/L vs Incidence angle

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N8. CONCLUSION 1. Bodies with elliptic cross-section exhibit higher lift to drag ratios than those of circular cross-section of the same fineness ratio and volume. 2. Power-law bodies can be an alternative to the blunted cone configurations because a power-law body has a greater internal volume than a blunted cone of same fineness ratio and represents the minimum drag case at hypersonic speeds. 3. The additions of strakes are found to improve the L/D ratio of the configuration without any significant change in the pitching moment characteristics. 4. Mach number, lift and the pitching moment characteristics of for the computational and experimental results of a halfpower-law body gave a reasonably good agreement .However, the agreement is not so good for the drag and pitching moment as effect of base flow and effect of balance string has not been modeled in the computational results. REFERENCES 1. Fraenkel, L. E. , Supersonic flow past slender bodies of elliptic cross-section, ARC A&M 2954, 1955 2. Jorgensen, L. , Elliptic cone alone and with wings at supersonic speeds, NACA TN-4045, 1957 3. Reggiori, A., Lift and drag of wing-cone configuration in hypersonic flow, AIAA Journal, vol. 9, No. 4 , April 1971, pp 744-745. 4. Tiwari, S. N., Singh, D. J., Sehgal, A. K., Combined effect of nose bluntness and angle of attack on slender bodies in viscous hypersonic flow, AIAA Paper 92-0755, January 1992. 5. Ashby, G. C., and Cary, A. M., A parametric study of the aerodynamic characteristics of nose-cylinder-flare bodies at Mach number of 6.0, NASA TN D-2854, June 1965. 6. Jorgensen, L. H. , Nelson, E. R., Experimental aerodynamics characteristics for a cylindrical body of revolution with side strakes and various noses at angles of attack from 0 to 58 and Mach numbers from 0.6 to 2.0, NASA TM X3130,1975 7. Eggers, A. J., Resnikoff, M. M., and Dennis, D. H., Bodies of revolution having minimum drag at high supersonic speeds, NACA Report 1306, 1957. 8. Cole, J. D., Newtonian flow theory for slender bodies, Journal of Aeronautical sciences, vol. 24, no. 6, pp 448-455, June 1957. 9. Spencer, B., Phillips, W. P. , Supersonic aerodynamic characteristics of a series of bodies having variations in fineness ratio and cross-section ellipticity, NASA TN D-2389, 1964. 10. Spencer, B., and Fox, C. H., Hypersonic aerodynamic performance of minimum wave-drag bodies, NASA TR R-250, November 1966. 11. Ashby, G. C., Jr., Longitudinal aerodynamic performance of a series of power-law and minimum wave drag bodies at Mach 6 and several Reynolds numbers, NASA-TM-X-2713, August 1974. 12. Westby, M. F., and Regan, J. D., The aerodynamic characteristics of power-law bodies in continuum and transitional hypersonic flow, RAE Technical Memorandum Aero 2164, August 1989. 13. Levinsky, E. S., Wei, M. H. Y., Non-linear lift and pressure distribution of slender conical bodies with strakes at low speeds, NASA CR-1202, 1968. 14. Singh, Amarjit, Force measurement on a one-half-power-law elliptic cone with and without strakes, Cranfield University, College of Aeronautics Report No. 9505, August, 1995. 15. Singh, Amarjit, Experimental study of slender vehicles at hypersonic speeds, PhD thesis, Cranfield University, 1996. 16. Singh, Amarjit, Chauhan, Y. S., A study of the effect of strakes on the aerodynamic characteristics of a power-law body of elliptical cross-section at hypersonic speeds, Proceedings of 51st AGM and Seminar on Advances in Aerospace Technologies, Jan 21-22, 2000. 17. Kontis, K., Stollery, J. L., Edwards, J. A., Hypersonic effectiveness of slender lifting elliptic cones with and without strakes, AIAA 97-521, pp1-11,1997. 18. Kontis, K., Qin, N., Stollery, J. L., Computational and experimental investigation of hypersonic performance of a lifting elliptic cone with and without strakes, AIAA 97-2252, pp1-11, 1997. 19. Kontis, K., Flow control effectiveness of jets, strakes, and flares at hypersonic speeds Proc. IMechE Vol. 222 Part G: J. Aerospace Engineering, pp 585-603, 2008. 20. Sonawane, P. D., Singh, Amarjit, Bhatkar, V. W. and Oak, S. M., CFD Analysis of Hypersonic Flow around a Power Law Body Engineering Design 2007 Conference, Indian Institute of Science, Bangalore, Aug 09-11, 2007. 21. Sonawane, P. D., Mittal, Nitika, Barpande, G. S., Jaware, V. B., Singh, Amarjit, Slender Bodies at Hypersonic Speeds, IDST-MSAT 2007, Missile Complex, DRDL, Hyderabad, Oct 26-27, 2007. 22. Bertin J. J., Hypersonic aerothermodynamics. AIAA education series, 1994. 23. Cox, R. N. and Crabtree, L. F., Elements of hypersonic aerodynamics, Academic Press, New York, 1965. 24. Rasmussen, M., Hypersonic Flow, John Wiley & Sons, New York, 1994.

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NSTABILITY AND CONTROL OF AIRCRAFT BOMBS AT HIGH ANGLE OF ATTACK, USING CASCADE FINSMisra A, Ghosh AK, Ghosh K IIT, Kanpur 208016

ABSTRACT Most guided bombs use planar fins that are mounted parallel to the body axes. These fins rotate back and forth to generate forces in the horizontal and / or vertical planes that cause the bomb to yaw right and left, pitch up and down, or roll as the bomb maneuvers towards the target. Lattice or grid fins are a relatively new Aero-mechanical technology for tail controlled ordnances. One of the advantages of grid fins is their capability to produce effective aerodynamic control forces at high angle of attack. Unlike conventional planar fins, grid fins do not experience classical stall at high angle of attack. These characteristics lead to more effective stability and control at intermediate and large angles of attack. In addition, the hinge moments produced by grid fins are very small due to small chord. This leads to requirement of very small actuators for deflecting these fins. This advantage becomes more and more significant as the bomb size increases. The major disadvantage of grid fins is high drag leading to lower aerodynamic efficiency compared to conventional planar fins. In order to address the issue of low aerodynamic efficiency of grid fins, a new category of fins called cascade fins is proposed. Cascade fins consist of planar members placed parallel to each other at optimized gap to chord ratio. Unlike grid fins, no cross members are present, only a base plate and an optional end plate (EP) is present. Planar members, in cascade fins, are of symmetric airfoil cross section. Due to cascade effect the lift increases with increase in number of planar members for a given angle of attack. These could also be flat plates. These fins produce lesser drag as compared to grid fins (better CL/CD). Absence of cross members in a cascade fin does not alter its stall prevention characteristics. Cascade fins preserve both the basic advantages of grid fins, namely delayed stall, and very low hinge moments. Flat plate cascade fins provide more stability and better restoring moments, whereas Aerofoil cascade fins provide superior aerodynamic efficiency. Some of the parameters affecting the aerodynamics of cascade fins are gap to chord ratio, number of planar members, presence of end plate and the cross section shape of the planar members. 1. INTRODUCTION Most guided missiles use planar fins that are mounted parallel to the body axes. These fins rotate back and forth to generate forces in the horizontal and / or vertical planes that cause the vehicle to yaw right and left, pitch up and down, or roll as the missile maneuvers towards the target. These moving control fins are located aft of the weapon's center of gravity (C.G.) in a tail control layout. At the CG in a wing control format, or forward of the CG in a canard control configuration. Missile concepts with forward control fins or canard, have been used for many years. However, previous studies have shown that concepts with canards can suffer from adverse induced rolling moments1-5. Lattice or grid fins are a relatively new aero-mechanical technology for tail controlled missiles. The design of these grid fins allows an effective aerodynamic control device stowed along with the body of a missile without increasing the overall dimensions. Therefore, the fin configuration promises good stow ability for potential tube-launched and internal carriage dispenser-launched applications. The internal grid structure, which forms webbing for a tail fin, provides a high strength to weight ratio compared to planar fins that can be quite large. This in-turn, allows for the fins to have advantage of a low hinge moment due to smaller chord and hinge control effectiveness. Therefore, small and light actuators can be used for missiles with Grid / Lattice fin controls. The use of grid fins or lattice controls, for the tail control surfaces instead of conventional planar fins was proposed by the U.S. Army Aviation and Missile Command (AMCOM) personnel as a possible remedy for the roll control problem5. A grid fin is an unconventional lifting and control surfaces of small chord6. One of the advantages of grid fins is their capability to produce effective aerodynamic control forces at high angle of attack over wide Mach number ranges. Unlike conventional planar fins, grid fins do not experience classical stall at higher angle of attack. These characteristics lead to more effective stability and control characteristics at intermediate and large angles of atatck7. Grid fins are normally mounted transverse to the longitudinal axis of the missile so that the flow passes through the grid numbers. The angle of attack of the fin is achieved by simple rotation of the fin about its horizontal axis in much the same way a conventional fin control surface is deflected. Although there is evidence8,9 that the lattice control concept has been around for some time, there use as missile

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Nstability and control surfaces is a fairly recent innovation. Russian federation armed forces already use grid fins on the AA-12 medium range air-to-air missile. This missile is understood to be equivalent to the US AMRAAM. Grid fins are also used on various ballistic missiles of the former Soviet Union and have even been used as emergency drag brakes on the SOYUZ TM22 spacecraft. United states uses grid fins on Massive Ordnance Air Blast (MOAB) bomb10,11. Thus, grid fine have found a number of applications within the sphere of missile aerodynamics The Research Development and Engineering Center of U.S. Army Aviation and Missile Command, Huntsville, Alabama has investigated the aerodynamics of the grid fins since 1985. A total nine wind tunnel tests have been conducted in order to gain greater insight into the aerodynamic characteristics of grid fins for Mach numbers ranging from 0.3 to 3.5. The grid fin is a unique device that can be used as either an aerodynamic stabilizer or a control surface. It's unique design and aerodynamic characteristics separate it from conventional planar fins. Planar fins can generally be described with information about the root chord, tip chord, span, and thickness. Grid fins require additional geometric parameters namely height, cell spacing element thickness12. The team continued their research and tested the leading edge profiles (flat, convex and concave) with respect to the free stream directions and found that effects of the curvature on the grid fin aerodynamics were negligible12. Next, the effect of frame cross-sectional shape as well as web-thickness on the drag and other forces were studied12. Their tests results yielded significant changes in drag characteristics at all Mach numbers tested, but the normal force (Lift) and hinge moment characteristics showed small changes. In early 1989, wind tunnel tests by Rockwell International Missile System Division, Georgia demonstrated several interesting trends12. The results showed that horizontal slat configuration tend to provide higher normal-force characteristics than open geometries13. Complex grid-like inner fin arrangements provided increase in normal force over open geometries and comparatively larger axial force. The cross fin configurations produced roughly the same normal force as the single slat fin. It was realized that the same fins because, of their low hinge actuators requirements, might be efficient as lifting surfaces when flown in the normal transverse direction but could also be used as drag brakes if turned to a horizontal position13. Interest in grid fins is primarily based on its potential use on highly maneuverable munitions due to their advantages over conventional planar controls at high angle of attack and high Mach number. The fin design yields favorable lift characteristics at high angle of attack and near zero hinge moments, which allow the use of small and light weight actuators1,2. Aerodynamics of grid fins has been investigated since 1985 by the U.S. Army Missile Research and Development Center (MRDEC). These investigations indicated that one of the advantages of grid fins is their ability to maintain lift at high angle of attack since they do not have the same stall characteristics as those of conventional planar fins. The main disadvantage was indicated to be higher drag compared to that of planar fins, although some techniques for minimizing drag by altering the grid fin frame cross sectional shape were demonstrated. Any modification of grid fin configuration must necessarily provide adequate lift at high angle of attack. This is possible if the stall is delayed and / or is of mildness nature compared to planar fin. The drag reduction of a grid fin configuration could be achieved, to the some extent, by altering the grid fin cross sectional shape. For performance enhancement such a configuration needs to generate higher values of aerodynamic efficiency. Undamentally the delay in stall could be linked to delayed separation which could be further attributed to the cascade effect14. A typical grid fin with criss cross members can be approximated as consisting of orthogonal slats4 (Fig. 1). One of the important parameter effecting stall could be the gap to chord ratio of the lifting members of the grid fin14. A typical cruciform configuration of grid fins as shown in Fig. 1 consists of two pairs.

Fig. 1: Schematic of the model used for design evaluatin and testing fo fgrid fins

One in elevator position and other in rudder position. In elevator position the lift producing members are horizontal and cross members are vertical. In rudder position the lifting members are vertical and cross members are horizontal. The cross members a part from providing structural rigidity to some extent, contribute to drag. In addition, elevator fins' cross members contribute to directional stability and rudder fins' cross members contribute to pitch stability. The cross members do not provide control in any case. In order to maximize we propose a new category of grid fins called cascade fins. The present work is a study of such cascade fins which do not have cross members as is the case with grid fins. A cascade fin has fins parallel to each other placed at optimized gap to chord ratio (g/c), forming a cascade of flow. Most of the grid fins configurations available in open literature addresses issues related to grids fins with high aspect ratio. The low aspect ratio fins, although induce large drag, are better suited for high angle of attack maneuvers. The selection of test configurations was also influenced by the immediate requirement of a lattice fin tail unit for an existing aircraft bomb.

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Accordingly the aspect ratio chosen for the planar member of cascade fin was 2.0. Further, to investigate the effect of cross sectional shape of planar members, two cross sectional shapes, namely Flat Plate and Symmetrical aerofoil (NACA 0012) were chosen. In order to eliminate any body fin interference, these cascade fins were tested in fin alone configuration with out body. In order to study the effect of number of planar members of a cascade fin on its aerodynamics the number of such planar members selected was 1, 3, 4 & 5. Here 1 planar member configuration refers to a single conventional planar fin. Another important parameters to be studied was gap to chord ratio of a cascade fin. Five different gap to chord ratios were tested (g/c = 0.5, 0.6, 0.7, 0.8, 0.9). A full scale model was tested at Re = 308,000. A sting (to mount the model in wind tunnel) available in the tunnel was unsuitable to simulate high angle of attack in vertical plane. This was overcome by mounting the cascade model on the floor of the wind tunnel as shown in Fig. 2.

Fig. 2: Cascade fin without and with end plate (Rear View at0 0

= 0)

The angle of attack range (0 to 50 ) was simulated in the horizontal plane by rotating the base turn table. A shroud was placed over the balance front and rear end adapter to isolate the balance from direct wind loads. A six component balance, mounted vertically on the floor, was used to measure the aerodynamic forces and moments (CN, Cy, Cm, Cl, Cn). Both dry and wind runs were made in order to asses the value of the offset to be removed from the measured data. The data obtained through these tests were processed to get numerical values of force and moment coefficients. The reference area of 0.02 m2 and chord length of 0.1 m were used for nondimensiolizing the force and moment coefficients. As stated earlier, in order to address the issue of low aerodynamic efficiency of grid fins, a new category of fins called cascade fins is proposed14. Cascade fins consist of planar members placed parallel to each other at optimized gap to chord ratio. Unlike grid fins, no cross members are present, only a base plate and an optional end plate (EP) are present. Planar members, in cascade fins, are of symmetric airfoil cross section. These could also be flat plates. These fins produce lesser drag as compared to grid fins (better CL/CD). Absence of cross members in a cascade fin does not alter its stall prevention characteristics. Cascade fins preserve both the basic advantages of grid fins namely delayed stall and very low hinge moments. Some of the parameters affecting the aerodynamics of cascade fins are gap to chord ratio, number of planar members, presence of end plate and the cross section shape of the planar members. Exhaustive wind tunnel tests were conducted to quantify the effect of these parameters on cascade aerodynamics. A brief discussion on the wind tunnel tests is described next. The effect of gap to chord ratio and end plate effect has been documented in Ref 14. This paper briefly presents the effect of number of planar member and cross sectional shape of planar member or cascade aerodynamics. 2. RESULT AND DISCUSSION Figure 3 shows the effect of number of planer members in a cascade fin. As expected the lift increases with increase in number of planar members for a given angle of attack. However, this increase in lift is not linear. This is due to cascade Effect. It can be seen referring Fig. 3 that there is significant increase in the stall angle of the cascade fin. The stall angle for single flat plate is found to be around 12 degrees. Whereas for cascade fins it lie near 22 degrees. The stall angle could easily be increased by reducing the gap to chord ratio14-. The cascade fins could be used for high angle of attack operation without facing conventional problems of stall.

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Fig. 3 CL vs. Alpha for different n (n=no. of planar members); g/c = 0.5 (WT Data)

Next issue was to increase CL/CD of the cascade fins. One approach followed was to replace flat plate cascade by aerofoil cascade. Figure 4 compares the aerodynamic characteristics of Flat Plate Cascade (FPC) and Aerofoil Cascade (AFC). The lift curves show that upto 20 degrees angle of attack, FPC gives marginally higher lift. From 20 degrees to 30 degrees angle of attack, AFC gives marginally more lift. Around 32 degree angle of attack, AFC's lift drops abruptly. The drag curves show that AFC is superior than FPC throughout the angle of attack range of 0 to 53 degrees. The aerodynamic efficiency (CL/CD) of AFC is significantly higher than FPC from 4 to 32 degrees angle of attack. For bombs whose operable range of angle of attack lie within 4 to 32 degrees, an AFC tail unit is a betterchoice from aerodynamic efficiency point of view. The pitching moment characteristics of the two cascades show that the pitch stiffness provided by AFC is nearly constant upto 22 degrees angle of attack. Beyond 22 degrees there is a sudden drop in pitch stiffness. FPC, on the other hand, provides higher stiffness than AFC upto 20 degrees. Beyond 20 degrees angle of attack, upto 40 degrees, AFC gives marginal stability. The reduction in pitch stiffness of cascade fins at higher angle of attack can advantageously be utilized to design the suitable controller for highly maneuverable missiles / aircraft bombs.

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Fig 4: Comparison of Flat Plate and Aerofoil Cascades' Aerodynamics

REFERENCES 1. Allen, J.M., Blair, A.B., Jr., Comparison of Analytical and Experimental Supersonic Aerodynamics Characteristics of a Forward Control Missile, Journal of Spacecraft and Rockets, 1982, 19(2), pp 155-159. 2. Blair, A.B., Jr., Dillon, J.L., Jr., Watson, C.B., Experimental Study of Tail-Span Effects on a Canard-Controlled Missile, Journal of Spacecraft and Rockets, 1993, 30(5), pp 635-640. 3. Blair, A.B., Jr., Supersonic Aerodynamics Characteristics of Maneuvering Canard Controlled Missile with Fixed and Free-Rolling Tail Fins, SAE paper 90-1993, Society of Automotive Engineers: Warrendale, PA, October 1990. 4. Burt, J.R., Jr., The Effectiveness of Canards for Roll Control, RD-77-8, U.S. Army Missile Command, Red Stone Arsenel, AL, 1976. 5. James De Spirito, Milton E., Vaughn, Jr., and W. David Washington, Numerical Investigation of Aerodynamics of Canard-Controlled Missile using Planar and Grid Fins, Part II: Subsonic and Transonic Flow, ARL TR 3162, March 2004. 6. Washington, W.D., Miller, M.S., Grid Fins A New Concept for Missile Stability and Control , AIAA paper 93-0035, American Institute of Aeronautics and Astronautics, Reston, VA, January 1993. 7. Washington, W.D., Miller, M.S., Experimental Investigation of Grid Fin Aerodynamics: A Synopsis of Nine Wind Tunnel and Three Flight Tests, Symposium on Missile Aerodynamics, Sorrento, Italy, 11-14 May, 1998. 8. Belotserkovskiy, S.M., etal, Wings with Internal Frame Work, Machine Translation, FTD-ID (RS)-1289-86, Foreign Technology Division, February 1987.

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9. 10. 11. 12. 13. 14.

Simpson, G.M. and Sadler, A.J., Lattice Controls: A Comparison with Conventional Planar Fins, Symposium on Missile Aerodynamics, RTO-MP-5, Italy, 11-14 May 1998. Drab, J., and Patterson, B.C., Massive Ordinance Air Blast Weapon Development, AFRLS, Munition Directorate, Elgin, AFBFL, 2004. Fulghum, David A., It is a Big One (MOAB Actually Fits in a B-2, Aviation Week and Space Technology, March 16, 2003. Washington, William D. and Miller, Mark S., Curvature and Leading Edge Sweep Back Effects on Grid Fin Aerodynamics Characteristics, AIAA paper 93-3480, Applied Aerodynamics Conference, Monterey, CA, August 1993. Miller, Mark S. and Washington, William D., An Experimental Investigation of Grid Fin Configurations, AIAA paper 94-1914-CP, WL Technical Library. Misra A., Ghosh A.K. and Ghosh K., Cascade Fins - An Alternate Tail Stabilization Unit, AIAA conference, Hawaai, Aug 18-20,2008.

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NUNDERSTANDING TRAJECTORY MODELLING VIA ERROR AND UNCERTAINTY ANALYSISKK Chand Proof & Experimental Establishment, Chandipur, Balasore, Orissa- 756 025

ABSTRACT Trajectory modeling and simulation has been addressed in many contexts by researches and scientists in many disciplines in its broad applications and usefulness. It is a computational tool to calculate the flight path of projectiles and its elements. There is a wide span of trajectory modeling and computations differing with respect to their performance and fidelity characteristics, which are used throughout the life cycle of a weapon system to improve the understanding of various aspects for a variety of tasks. Trajectory modeling of a projectile system is a process and its important characteristics lie in its mathematical representations. The applicability of a model is limited by the assumptions and the uncertainties in the evaluation data, understanding the judgments associated with the modeling process. Its analysis under uncertainty environments is an engineering discipline, and a powerful tool in any weapon systems developmental programmes. A fundamental understanding is necessary to comprehend the factors that influence accuracy and how to account for them in the determination of firing experimental data. Growing international competition in the technological and industrial applications has necessitated that armament designers, managers and practitioners ensure a level of performance of their products before releasing them for service use. The rational treatment of uncertainties in trajectory modeling has received increasing attention in recent years. Loading and boundary conditions, material properties, geometry and various other parameters show in some cases considerable variations at macro-scale or at micro-scale in a multi-levels physical context. Observations and measurements of physical processes as well as parameters at different scales clearly indicate their random characteristics. Uncertainty analysis has been an established engineering methodology, but is relatively new to epidemiology for the quantitative assessment of biases, hence the ballistician's interested to apply it to the armament field too. A systematic uncertainty analysis also provides insight into the degree of confidence of a measurement or experimental results in the existing data and models to assess how various possible model estimates should be weighed. In view of above, this paper will consist of two sections. Section one will discuss various types and sources of errors and uncertainties in projectile trajectory modeling from physical, mathematical, experimental and computational aspects. Section two will discuss a systematic approach to quantifying various parameters from a case study. 1. INTRODUCTION In our physical world, motion is ubiquitous and controlled by the application of two principal ideas, force and motion. The study of these ideas is called dynamics. Through the study of dynamics, we are able to calculate and predict the trajectory of a projectile system. Trajectory modelling is the process of identifying the principal physical dynamic effects by analyzing a system, writing the differential and algebraic equations from the fundamental laws. Modelling projectile motion and its simulation is nothing but as a computational tool to calculate the flight path and its characteristics. There is a wide span of trajectory modelling and simulation differing widely with respect to their performance and fidelity characteristics, such as specifying ammunition performance requirements, designing ammunitions, optimizing the design parameters, assessing ammunition performance, teaching users the correct use of weapon and fire control system. The study of projectile motion is called ballistics, that is, it is, in essence, the study of projectile motion and the conditions that influence that motion. As an applied and complex fields of science, ballistics has a wide meaning and a wide scope of subjects, ranging from the behaviors of projectile inside a gun barrel (Internal Ballistics), the flight of projectile through the air (External Ballistics) and the interaction between projectile and targets (Terminal Ballistics), which shown in Figure 1. Its development is very much related to the evolution of military technology and particularly in artillery projectile trajectory modeling system. Similarly, the science of investigating projectile trajectory modeling and computation in space is known as external ballistics or the science of flight dynamics. It existed for centuries as an art before its first beginnings as a science. For understanding the inflight behaviour of the projectile system is to develop the mathematical models that can predict all elements of the trajectory from launch to target. External ballistics variations are the main concern for artillery projectile

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system in the field. Figure 2 depicts an exterior ballistics environment of a projectile system. Figure 3 depicts the global motion of a projectile system in space. Recent involvement in designing and developing a trajectory and simulation model to allow interaction between physical, mathematical and computational processes has also led to interest in error and uncertainty propagation. The effectiveness of a trajectory modelling is closely linked to the accuracy to which it may be delivered to the intended target area. The trajectory of a projectile depends upon many factors, such as the components of initial/launch velocity, launch

Figure 3: Global Motion of a Projectile System

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Nangle of the projectile, its mass/weight, shape, size, rate of spin, drag characteristics, the density and temperature of the air, the direction and velocity of the wind at various altitudes, rotation-of-earth effects, forces and moments induced by projectile spin, and the latitude of the gun and the azimuth of fire etc. Various models may be used, with varying complexities, to incorporate any of the above effects. The main classes of trajectory models are known as: in-vacuo, point-mass, modified point-mass and six-degree-offreedom, universal and artificial neural networks. In development of models, we often combine many parameters together to produce models with hundreds and even thousands of parameters. The uncertainty in these models arises not only from the uncertainty in the model inputs but also due to other more fundamental problems associated with our ability to model complex systems. Often in presenting model outputs we fail to adequately represent these uncertainties. In this paper we intend to explore some of the issues associated with uncertainty in projectile trajectory modelling. Random firing vibration resulting projectile's initial disturbing having random error, which is got at the moment when they fly out of the muzzle. Thereby, projectile deviate the predicted trajectory, which resulted in dispersion of a series of, fired projectiles. So, dispersion of projectiles is decided by a few random elements. The following equation for muzzle velocity variation (MVV) is valid for our purposes: MVV (m/s) = Shooting Strength of Weapon + Ammunition Efficiency +Round to Round Variation; Random nature of various parameters in ballistic test and evaluation demands probabilistic approach, which provides the mathematical basis for evaluation of projectile's trajectory system performance via prediction of overall projectile's dispersion accuracy or consistency, which are random variable and assumed a normal multidimensional distribution. It is essential parameters for a projectile-weapon-ammunition system, which affect performance to simulate the firing process and predicting dispersion accuracy. In this paper, a general methodology for uncertainty analysis is presented that combines modeling, simulation and experimental techniques for an understanding of projectile's trajectory modeling system performance and is in accordance to the principles of the GUM. In view of the above, first, this paper discusses various types of errors and uncertainties, and its classifications in general. In second, a case study has been discussed for quantifying various parameters from a test firing system.

Projectile Motion As A Dynamical System A projectile is defined as an object, projected by an applied exterior force or impulse and continuing in motion by virtue of its own inertia over a very short period of time. A projectile with its weapon and ammunition form a projectile system. It is term that emerges from many disciplines and domains and has many interpretations, implications and problems associated with it. Generally, modern artillery projectile systems have many features in common, which include a fuze, ogive, bourrelet, and rotating band. Figure 4 shows the concept of a simple projectile system. Similarly, linked with projectile motion, a trajectory can be defined as an imagined trace of positions followed by an object moving through space. A projectile system, when subjected to propelled, fired, shot, launched, or thrown through space by the exertions of a force by a suitable means, behaves dynamically. This is a dynamic system, because the level of the elements changes in time. Mathematically the term trajectory refers to the ordered set of states (as a function of the time), which are assumed by a

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Ndynamical system over time. Mathematically a trajectory may be described either by the geometry of the path (i.e. the set of all positions taken by the object), or as the position of the object over time. To describe a trajectory completely it is sufficient to specify the x, y and z coordinates of the center of gravity of the projectile at any time t after the release by the projecting mechanism; in other words, a trajectory is regarded as defined by: x = x(t); y = y(t); z = z(t); where x(t), y(t) and z(t) are functions of the time t which are equal to zero when t = 0. The trajectory is determined by: (a) the position of the origin; (b) the conditions of projections;(c) the ballistic characteristics of the projectile; (d) the characteristics of the air through which it passes; Projectile Trajectory Modeling And Simulation Trajectory modeling and simulation, in the present context, means computing the flight path and other parameters, such as range, deviation, orientation, and angular rates of a projectile system from the start to the end of its motion. Trajectory modelling deals with mathematical models of the behavior of projectile and its subsystems (if any) during its test & evaluation and operations. The equations of motion determine the acceleration, velocity and position (range & deviation) of the projectile resulting from forces and moments due to gravity, thrust and aerodynamics. If all characteristics of a projectile, together with atmospheric conditions are exactly equal to a set of predicted values, the projectile will fly on a known trajectory. This trajectory is called nominal trajectory. In practice, there are always some differences between the real and predicted values. These differences make the projectile not to fly exactly on its nominal trajectory. Therefore, there are always some errors between the positions of a desired and real impact point. Estimation of these errors is very important from the operational point of view. Also, investigation of the error sources and their effects can help a projectile designer to optimize design parameters for the lowest impact point error. The guidance and control models account for subsystems such as the control system in case of a missile system. Particularly trajectory simulation requires a customized approach. The level of sophistication of simulations varies greatly depending on the applications ranging from unsophisticated two-dimensional models to very detailed six-seven degree of freedom models as per shapes and sizes of projectile systems. The requirements of a given trajectory simulation are derived from the objectives of the intended users requirements like, analysis, development, procurement and operation. The modeling phase is a decisive stage because the analysis of the model depends strongly on the model. The choice of the model depends on several criteria, the most important ones are; (i) data of the modeled system, (ii) system's complexity, (iii) availability of a computer tool implementing the model, (iv) compatibility of the model with the analysis leading to expected result. Mathematical Modeling Let us first consider 2-D (3-DOF) case, the projectile as a particle and the motion of the projectile of mass m influenced by gravity g and air resistance. Therefore the trajectory modelling of the projectile motion is described by:

d2y d 2x a x = - ( F sin + (1 / m m g a x = - FD q 2 = ) D q ) (1) = ) cos (1 / m dt dt 2Where

u tan ( v v =v v + = / v x )D =C D F 0.5 v 2 A r2 x 2 y 1 y

,

and

by incorporating following basic limitations/assumptions: (a) Ignoring of the earth's curvature or earth's flatness; (b) Planar Motion of the projectile; (c) No wind speed; (d) Constancy of drag index and projectile's mass; (e) Proportionality of drag force to squared velocity of the projectile; (f) Constancy of drag coefficient and air density; (g) Constancy of drag force or altitude independency of drag force; (h) Neglecting of centrifugal, coriolis, magnus forces and its cross-effects; The above equations are solved subject to the following state or initial conditions:

u = =q / dt = 0 v ==y =0; at t = 0 (2) dx / dt v0 cos 0 ; v = v0 sin q0 ; x dy v

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Where the drag force FD acting on the projectile depends on air density , velocity v, cross-section area A= r2, radius of the projectile r, and drag coefficient CD. If the drag force is neglected, the calculation of the projectile trajectory becomes trivial. If however the drag force is taken into account, the analytical solution is not soluble due to the drag dependency on the square of the velocity and presence of angle as argument of trigonometric functions. Target Density Parametersn

(i) Elevating probability error, (ii) Azimuth probability error,

(D Y) i

2

Ba = 0.6745

1

n1n

(D Z) i

2

Bb = 0.6745

1

n1

Where, Yi elevation coordination deviation from the center of clustering (meter), Yi=Yi - YcP ;, Zi azimuth coordination deviation from the center of clustering (meter), Zi=Zi - ZcP ; YcP , ZcP clustering coordinate (meter) n n Y 1 Z i, n - the number of the projectile; Yi and Zi elevation, azimuth coordinate of each projectile. 1 iYcP = Z cP = n n

2. GOAL OF UNCERTAINTY ANALYSIS/PROPAGATION Uncertainty is an integral part of almost every real-world problem due to its complex nature of the physical process. The objective of uncertainty analysis or propagation is to characterize the uncertainties in the system output given some knowledge of the uncertainties associated with the system parameters together with one or more computational models or experimental data. For realistic systems one must include uncertainties of various types in the mathematical model of the system. Therefore, researchers and scientists attempting to have a better understanding of the process have devoted tremendous efforts. The appropriate incorporation of uncertainty into the analyses of complex systems is a topic of importance and widespread interest. The particular context under consideration is shown in Figure 5. Projectile trajectory modelling system analyses and designs uncertainties can arise from the various sources including environmental uncertainties, model uncertainties, parameter uncertainties, data uncertainties, and operational uncertainties. Environmental uncertainty is associated with the inherent randomness of environmental processes such as the occurrence of wind speeds and humidity. A model is only an abstraction of the reality, which generally involves certain degrees of simplifications and idealizations. Model uncertainty reflects the inability of a model or design technique to represent precisely the system's true physical behavior. Parameter uncertainties resulting from the inability to quantify

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accurately the model inputs and parameters. Parameter uncertainty could also be caused by change in operational conditions of system structures, inherent variability of inputs and parameters in time and in space, and lack of sufficient amounts of data. Data uncertainties include (1) measurement errors, (2) inconsistency and non-homogeneity of data, (3) data handling and transcription errors, and (4) inadequate representation of data sample due to time and space limitations. Operational uncertainties include those associated with construction, manufacture, deterioration, maintenance, and human. Uncertainty analysis of measurement results is not a deterministic, but rather holistic and probabilistic in nature. Its complexity and ambiguity contributes that the uncertainty analysis is often misunderstood, misrepresented or even avoided. The process of developing uncertainty budgets requires a manufacturer to first find the causes of measurement uncertainty and quantify them. A manufacturer or designer or evaluator must understand the measuring process being evaluated, and a certain amount of experience, to identify and quantify uncertainty contributors. There are a couple of techniques that can help in systematically searching for uncertainty contributors. When collecting uncertainty contributors, it is important to know the areas where they may be lurking. These 10 areas are the most common a designer should use in looking for uncertainty contributors. The "circle of contributors" (Figure 6) technique involves using a list of areas where uncertainty contributors may be lurking as an aid in searching for them.

The process of generating and propagating the simulation values is shown in Figure 7. For each actual measured coordinate point, (Xi, Yi, Zi), the associated errors, (Xi, Yi, Zi), are generated by sampling from the appropriate estimated distribution. This process creates a set of new measurement points, (, which are then evaluated by the trajectory computation software according to the feature being measured. Figure 7 shows an example of a trajectory path being measured; is the final calculated trajectory of the projectile, and the function f represents the algorithm used to calculate the trajectory path from the set of n data points. This process will be repeated a number of times according to how the experiment is designed, and the statistics for will be used to estimate the combined standard uncertainty for the measurement of the trajectory path. Definition Of Error And Uncertainty The performance of a trajectory modelling system is adversely affected by error and uncertainty. According to the scope of the Guide to the Expression of Uncertainty in Measurement (GUM), the error and uncertainty can be defined as: (a) Error: A recognizable deficiency that is not due to lack of knowledge. (b) Uncertainty: A potential deficiency that is due to lack of knowledge. These two definitions imply that error is deterministic in nature and uncertainty is stochastic or non-deterministic in nature.

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Types of Errors Errors can be classified as either "acknowledged" or "unacknowledged": (A) Acknowledged errors are those, which have been identified in a simulation, but the analyst decides not to remove them because the results are adequate for the job requirements and budget (examples include round-off error and discretization error, and convergence error in an iterative numerical scheme). (B) Unacknowledged errors in a simulation are those the analyst is unaware of but they are recognizable. They have no foolproof procedures for finding them but they may be detected by redundant procedures and double-checking (examples are computer programming errors, or usage errors, including mistakes and blunders). Our knowledge of how the physical system works is limited by at least five kinds of uncertainty: (1) inadequate knowledge of all the factors that may influence something, (2) inadequate number of observations of those factors, (3) lack of precision in the observations, (4) lack of appropriate models to combine all the information meaningfully, and (5) inadequate ability to compute from the models. It is the incompleteness of the knowledge of the value of the measurand associated with the given result of a measurement. It is an interval within which a true value of a measurand lies with given probability. The reasons responsible to introduce uncertainty in a model may be: (a) The model structure, i.e., how accurately does a mathematical model describe the true system for a real-life situation; (b) The numerical approximation, i.e., how appropriately a numerical method is used in approximating the operation of the system; (c) The initial / boundary conditions, i.e., how precise are the data / information for initial and / or boundary conditions; (d) The data for input and/or model parameters; Types of Uncertainties The following three types of uncertainties can be identified: Type-I: Uncertainty due to variability of input and/or model parameters when the characterization of the variability is available (e.g., with probability density functions, (pdf )) Type-II: Uncertainty due to variability of input and/or model parameters when the corresponding variability characterization is not available; Type-III: Uncertainty due to an unknown process or mechanism; Type I uncertainty may be referred to as aleatory or quantitative (i.e., dependent on chance) or random or stochastic or "inherent" or "irreducible" uncertainty. It can be expressed using numerical values. It is not due to a lack of knowledge and cannot be reduced. Aleatory uncertainty, however, generally characterizes due to random processes or effects,

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Nsuch as the wind speed and direction at an arbitrary point in space as a flying projectile passes. Type II and III are referred to as epistemic or qualitative uncertainties. It can be expressed using qualitative terms and used when the sources of uncertainty cannot be estimated precisely using numerical values. Epistemic uncertainty deals with our state-of knowledge about portions of our model. This uncertainty has been called subjective, or "parameter," or state-ofknowledge or "reducible" uncertainty. Examples are such things as inaccuracies and measurement error in the physical burn-rate parameters characterizing a propellant. Epistemic uncertainty can be reduced or eliminated by an increase in knowledge about the system being analyzed or measurement of the parameters. Quantification of Uncertainty It often happens in real life applications that all three types of uncertainties are present in the systems under study. The goal of uncertainty quantification is to assign an appropriate mathematical model to real-world situation with respect to objective and subjective uncertainty. The choice of an appropriate uncertainty model primarily depends on the characteristics of the available information. That is, the underlying reality with the sources of the uncertainty dictates the model. In each particular case this information must be analyzed and classified to be eligible for quantification. Quantification or evaluation of uncertainties is very judgmental process, based on full scientific understanding, knowledge and experience of measured quantities, properties and characteristics of instrumentation used, measurement processes and procedures, and the data reduction procedures used to obtain the final measurement result. Uncertainty quantification intends to work toward reducing type II and III uncertainties to type I. The quantification for the type I uncertainty is relatively straightforward to perform. Techniques such as Monte Carlo are frequently used. Pdf can be represented by its moments (in the Gaussian case, the mean and covariance suffice), or more recent techniques. To evaluate type II and III uncertainties, the efforts are made to gain better knowledge of the system, process or mechanism. Methods such as fuzzy logic or evidence theory are used. In the planning phase of an experimental program, one focuses on the general or overall uncertainties. Consider a general case in which an experimental result, f, is a function of n measured variables Xi: f = X ,... X ) (3) f (X ,1 2 n

Equation (3) is the data reduction equation used for determining f from the measured values of the variables Xi. The overall uncertainty in the result is then given by 2 f f (4) f ... + U 2 U 2 = U 2 +U 2 + f x x 1 X1 2 X2 x n Xn

where are the uncertainties in the measured variables Xi. It is assumed that the relationship given by Eq.(3) is continuous and has continuous derivatives in the domain of interest, that the measured variables Xi are independent of one another, and that the uncertainties in the measured variables are also independent of one another. Uncertainties In Projectile's Trajectory Modeling Process Uncertainty in projectile trajectory modelling system has become of interest in recent years and emerged from a combination of various computational aspects in interior or exterior or terminal ballistics models. Decision-makers have also expressed a desire to have probabilities assigned to each scenario so that there is a better sense of whether certain scenarios are more likely than others. Difficulty in assigning probabilities to scenarios can stem from "epistemic" or "stochastic" sources of uncertainty. Epistemic sources of uncertainty are those that can be reduced by further study of the system, improving various states of knowledge, etc. Stochastic sources of uncertainty are those that are considered "unknowable"items such as variability in the system, the chaotic nature of the projectile system, and the indeterminacy of human systems. Generally, uncertainties in projectile trajectory modelling are considered are mainly attributed to ambiguity and vagueness in defining the variables and parameters of the systems and their relations. The ambiguity component is generally due to noncognitive sources, which include (a) physical randomness; (b) statistical uncertainty due to the use of limited information to estimate the characteristics of these parameters; and (c) model uncertainties that are due to simplifying assumptions in analytical and prediction models, simplified methods, and idealized representations of real performances. Similarly, the vagueness-related uncertainty is due to cognitive sources that include (a) the definition of certain variables or parameters, e.g., structural performance (failure or survival), quality, deterioration, skill and experience of construction workers and engineers, environmental impact of projects, conditions of exiting structures; (b) other human factors; and (c) defining the interrelationships among the parameters of the problems, especially for complex weapon systems. The accuracy of a projectile's trajectory modelling system firing such as most tank or artillery systems relies on understanding numerous sources of error and correcting for them. Random errors are addressed through projectile and its ammunition design, quality production, effective maintenance, and firing processes that minimize the magnitude of these error sources. Bias errors are different because their magnitude and direction can often be estimated and then compensated for while aiming the weapon.

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NCharacterizing Uncertainties Parameters A common feature of engineering models is the need to provide values for model parameters that define properties, or parameters, which appear in boundary or initial conditions. For example, according to drag law the combined effects of air resistance can be reduced to a drag co-efficient for a projectile. It states that the drag force is proportional to the crosssectional or reference area of the projectile, air density in which the projectile moves and its velocity squared. The proportionality constant is known as a drag coefficient (CD(a )), which is widely used to represent a projectile's ) or CD(M aerodynamic efficiency. Its value depends on a function of the angle of attack, a number, M. Mathematically, it can or Mach be written as Fdrag = CD(a )A( r ) or CD(M where CD(a )= the coefficient of drag,; cross-sectional or reference V2), ) or CD(M A = area of the projectile; the density of air (~1.2 Kg/m3);and V = the velocity of the projectile relative to the air. The value for r = this model parameter i.e. drag coefficient is estimated using experimental techniques utilizing samples of the projectiles. In practical terms the drag co-efficient is a function of velocity, usually represented as the Mach number (M) that varies with air temperature, which in turn reduces with altitude. The drag co-efficient is estimated during ammunition design, or since computer simulations became possible, it can be modelled. These estimates can then be confirmed or adjusted from 'range & accuracy' firing. Another widely used data is the ballistic co-efficient (CB), which reflects the rate at which velocity is lost as the projectile penetrates air. The ballistic co-efficient is sometimes called the 'carrying power' of a shell, meaning how far a given muzzle velocity will 'carry' it. It is expressed as CB=m/nd2, where CB = carrying power, m = projectile mass, d = projectile diameter and n = , where (kappa) is the co-efficient of shape, (sigma) the co-efficient of steadiness and (tau) the co-efficient of air density. This means that mass is the main determinant of carrying power and is the reason that artillery shells go further than rifle bullets even when the latter are fired at higher velocities. For a particular muzzle velocity and firing elevation angle the heavy shell will always go further (unless its shape and steadiness are significantly inferior). Generally, the parameters and co-efficients in models are not constant. The drag co-efficient varies with projectile velocity. Particularly, important parameter, the density of the atmosphere varies with altitude. In flight, a projectile's velocity decreases with gravity and air resistance until vertex is reached, when air resistance continues but gravity accelerates the projectile as it descends (hence the elliptic trajectory). The rate of spin (revolutions per second) also decreases with velocity throughout the shell's flight, which affects its stability. Gravity is not constant either; it decreases with distance from the earth's surface (which is uneven). Unfortunately, the methods used to characterize drag coefficient will always lead to errors in the measured coefficient due to experimental technique and equipment, due to uncertainty associated with uncontrolled environmental conditions, and due to uncertainty associated with variability from projectile to projectile because of the manufacturing process. A key question is how do we quantitatively characterize the sources and nature of this uncertainty. Looking at the parameters affecting a trajectory modelling, simulation and its analysis, six main categories can be differentiated as: (a) Parameters directly affecting the initial conditions or Fire Control: gun position, gun height, muzzle velocity, azimuth and elevation or launch angle; (b) Parameters indirectly affecting the initial conditions: propellant temperature and barrel abrasion or wear; (c) Parameters directly affecting the projectile: in flight containing all atmospheric properties depending on height (temperature, air density, air pressure, wind, speed of sound) and all projectile depending coefficients (drag, lift, Magnus and Coriolis forces, spin and yaw); (d) Parameters indirectly affecting the projectile: case thickness, case strength and explosive weight; (e) Parameters directly affecting the target: impact angle, velocity, location, and angle-ofattack; (f) Parameters indirectly affecting the target: thickness and strength; From a measurement point of view, all these parameters are error afflicted and therefore, every parameter can be quantified by its mean and standard deviation. When we develop mathematical or simulations models we sometimes deal with systematic uncertainties, which can originate from biases in measurements or biases in solutions methods. Examples of systematic uncertainties include: (a) Instrumentation system error when measuring gun tube elevation and azimuth angles; (b) GPS error in target location; (c) Measurement error in depth of penetration, range and deviation data; (d) Prediction error because a physical phenomena is not accounted for in the model equations; The gun itself also affects the projectile's flight due to 'droop' and 'jump'. When the axis of the bore at the breech differs slightly from the axis of the bore at the muzzle the 'droop' generates. Similarly, the movement of the projectile's 'whipaction', that means when the projectile leaves the axis of the bore at the muzzle (in the vertical plane) differs to its axis before

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Nfiring, the 'jump' generates. Similarly, a difference in height between a gun and its target also affects the trajectory. Therefore, the angle of sight has to be added to or subtracted from the gun's firing elevation angle. However the slant range to the target is greater than the horizontal range and applying the angle of sight does not compensate for this, which is called non-rigidity of the trajectory is shown in Figure 8. The size of the correction for this depends on the difference in height between the target and the gun, and the shell's angle of descent. The gun's muzzle velocity and the range to the target in turn determine this angle. Furthermore, altering the elevation angle also puts the trajectory through a different 'slice' of the atmosph