TankMunitionsandVectorCalculusRichardJardineIntroductionSuccess in a tank battle is often determinedbythe superiority of the weaponryinvolved.The increased lethalityofmoderntanksisduetotheeffectivenessofarmorpiercingprojectiles,suchasthatshown inFigure1.Armyresearchlaboratoriescontinuetoimproveperformanceofarmorpenetrators.Atthefoundationof theresearcharemathematicalmodels involvingvectorcalculusFigure1.Tankroundtravelling inexcessof1500meterspersecondphotocourtesyofArmyResearchLaboratories(ARL)Inthischapter,wedescribetheuseofvectorcalculustomodel armorpiercingtankrounds.Alongtheway,wereviewmathematicaloperationswithvectors,asfamiliaritywiththemathematicsis prerequisiteto understanding theapplication.Mathematical coveragebegins withthebasic operations ofvectorarithmetic andcontinues withexercises applyingthe dotproduct,cross product, andgradient.Vectorcalculus isspecificallyappliedtothemodelingofsabot rounds,atypeoftankammunitiondescribed inthefollowingparagraphs.Inordertodiscovertheconnection between the mathematics and the application,it is important that youdo the Exercise Sets.You may havetorefertoyour multivariable calculus text todothe exercises.Forthis reason,keepingyourtextbooksforfuturereferencesisprobablyagoodidea.Topenetratethemassivearmorplatingprotectingthecrewofmoderntanks,aprojectilemustbefast-moving,aerodynamic,and havemassivemomentum.The sabot round pictured in Figure 2 is such a projectile. Do not let the slim53
53 Tank Munitions and Vector Calculus Richard Jardine Introduction Success in a tank battle is often determined by the superiority of the weaponry involved. The increased lethality of modern tanks is due to the effectiveness of armor piercing projectiles, such as that shown in Figure 1. Army research laboratories continue to improve performance of armor penetrators. At the foundation of the research are mathematical models involving vector calculus. Figure 1. Tank round travelling in excess of 1500 meters per second, photo courtesy of Army Research Laboratories (ARL) In this chapter, we describe the use of vector calculus to model armor piercing tank rounds. Along the way, we review mathematical operations with vectors, as familiarity with the mathematics is prerequisite to understanding the application. Mathematical coverage begins with the basic operations of vector arithmetic and continues with exercises applying the dot product, cross product, and gradient. Vector calculus is specifically applied to the modeling of sabot rounds, a type of tank ammunition described in the following paragraphs. In order to discover the connection between the mathematics and the application, it is important that you do the Exercise Sets. You may have to refer to your multivariable calculus text to do the exercises. For this reason, keeping your textbooks for future references is probably a good idea. To penetrate the massive armor plating protecting the crew of modern tanks, a projectile must be fast-moving, aerodynamic, and have massive momentum. The sabot round pictured in Figure 2 is such a projectile. Do not let the slim
profilemislead.Theprojectile ismade of extremelydensematerials andpropelledtoachieveincrediblespeeds,givingthepenetratorthemomentumnecessarytopunctureprotectivearmortensofcentimetersthick.Figure2.Computer model of penetrator with sabot, courtesyof ARL.The slenderpenetratormust beguideddownthegreaterdiametertankmaingunbarrel.Theguiding sleeve,or sabot, consists of petalswhichbreak apartuponexitfromtheguntube,asinFigure3.Optimaldesignofthesabotroundmustensurethesleeveseparatesfromthepenetratorwithoutinterferingwiththeflightdynamics of the penetrator,which will speed to a target.Testing the design byrepeatedfiringoftheveryexpensiveroundsiscostineffective.Computersimulations are an efficient alternativetothe destructive and costlytrial-and-errormethods.Programmedintothesimulationsarealgorithms implementingapplicationsofthevectorcalculus.Figure3.Model of sabot petal separation,courtesy of ARL54
54 profile mislead. The projectile is made of extremely dense materials and propelled to achieve incredible speeds, giving the penetrator the momentum necessary to puncture protective armor tens of centimeters thick. Figure 2. Computer model of penetrator with sabot, courtesy of ARL. The slender penetrator must be guided down the greater diameter tank main gun barrel. The guiding sleeve, or sabot, consists of petals which break apart upon exit from the gun tube, as in Figure 3. Optimal design of the sabot round must ensure the sleeve separates from the penetrator without interfering with the flight dynamics of the penetrator, which will speed to a target. Testing the design by repeated firing of the very expensive rounds is cost ineffective. Computer simulations are an efficient alternative to the destructive and costly trial-anderror methods. Programmed into the simulations are algorithms implementing applications of the vector calculus. Figure 3. Model of sabot petal separation, courtesy of ARL
Thecomputermodelsareguitesophisticatedandwell-developedFiniteelement(meshing)methodsareusedtodiscretizeandmodelthesabotround,as shown inFigure 4.Each of theapparent rectanglesare in factquadrilaterals.Eachquadrilateralistheexternalfaceofafiniteelementbrick.Thebricksarethebuilding blocks of the model of the sabot round. Depending on the detailsrequired in the simulations,thesabot roundmodel canbe divided intothousandsofthesebricks(meshes)bycomputer.The ModelInthemillisecondsaftertheexplosionwhichpropelsthesabotroundoutthetubeof thetank cannon,theroundissubjectto intenseheatandpressure.Irnordertomodeltheinfluenceoftheheatandfluidpressureontheflightdynamicsoftheprojectile,millionsofcomputationsareperformedintheprocessofthecomputersimulation.Theeffectsofheatandpressureontheblocksaresimulatedusingvectors.Figure4.Discretizedmodel of sabot round,graphic courtesyofARLThepressurefieldaffectingthe sabot round ismodeledwithavector-valuedfunction.As anexample,thefunctionp(x,y,z)=pi(x,y,=)i+ p2(x,y,=)j+pi(x,y,=)krepresentsthepressurefieldthatexists intheguntube.Thecomponentofthepressureinthexyandzdirectionsarepi,P2andp3,respectively.Tobespecificwith an example, suppose p(x,y,z)= 2xyi+xz j+ 3yk is the pressure fieldfunction.Then thepressure inthe tubeatthepoint A(1,2,3)would bep(x,y,z)=4i+3j+5k.Obviously,thispressurefieldhasan influenceonthemotionoftheprojectile.Withmathematicalmodelsof thepressureinfluence,engineers candeterminetheflightcharacteristics ofthepenetrator.Onemathematical tool usedtomodel the influenceofthepressurefield onthesabotroundistoapplythedotproductoperation.Bycalculatingthe dotproductofthepressurefieldwithaunitvectornormaltothesurfaceoftheprojectile,wecanpinpointthepressureat thatpoint onthesabotround.Figure5isavectorrepresentationofthedotproductcomputation.Priortodoingthatcalculationwereviewtheprocess offindinga unitvectornormal to asurfaceandtheprocess of evaluating the dot product.55
55 The computer models are quite sophisticated and well-developed. Finite element (meshing) methods are used to discretize and model the sabot round, as shown in Figure 4. Each of the apparent rectangles are in fact quadrilaterals. Each quadrilateral is the external face of a finite element brick. The bricks are the building blocks of the model of the sabot round. Depending on the details required in the simulations, the sabot round model can be divided into thousands of these bricks (meshes) by computer. The Model In the milliseconds after the explosion which propels the sabot round out the tube of the tank cannon, the round is subject to intense heat and pressure. In order to model the influence of the heat and fluid pressure on the flight dynamics of the projectile, millions of computations are performed in the process of the computer simulation. The effects of heat and pressure on the blocks are simulated using vectors. Figure 4. Discretized model of sabot round, graphic courtesy of ARL. The pressure field affecting the sabot round is modeled with a vector-valued function. As an example, the function p( , , ) ( , , )i ( , , ) j ( , , ) k 1 2 1 x y z = p x y z + p x y z + p x y z represents the pressure field that exists in the gun tube. The component of the pressure in the x, y and z directions are p1, p2 and p3, respectively. To be specific with an example, suppose p(x, y,z) = 2xy i + xz j + 3y k is the pressure field function. Then the pressure in the tube at the point A(1,2,3) would be p(x, y,z) = 4 i + 3 j + 5k . Obviously, this pressure field has an influence on the motion of the projectile. With mathematical models of the pressure influence, engineers can determine the flight characteristics of the penetrator. One mathematical tool used to model the influence of the pressure field on the sabot round is to apply the dot product operation. By calculating the dot product of the pressure field with a unit vector normal to the surface of the projectile, we can pinpoint the pressure at that point on the sabot round. Figure 5 is a vector representation of the dot product computation. Prior to doing that calculation, we review the process of finding a unit vector normal to a surface and the process of evaluating the dot product
Figure5.Dotproductofpandnistheinfluenceofthevectorpontheplanarsurface.Let's begin by working towardfinding the vector normal to the surface of theprojectile.Beforewedothethree-dimensional problem,westartwiththesimplesttwo-dimensionalcase:findingthenormalvectortoaline.Onerepresentation ofa line l istheequationax+by+c=O.Recall the slope--a.From this form we seeinterceptformoftheequationoftheline,y:bb.A second line, h, perpendicular to li wouldthattheslopemofljis mi=bbhaveslopem2, the negativereciprocal of maEXERCISESET1:Giventhelinel:3x+2y+4=0,a.Plot the lineb.Findtheslopeofthe linelhc. Find the slope of a line l perpendicular to l.d. Plot the line l on the same graph as li.Avectorvcoincidentwith(orparallelto)thelinel,hascomponentsv=bi-a j. Of course, this is not the only vector parallel to l,as any scalarmultipleofvwillbeparalleltovandl.Unitvectorsareusedtostandardize56
56 Figure 5. Dot product of p and n is the influence of the vector p on the planar surface. Let's begin by working toward finding the vector normal to the surface of the projectile. Before we do the three-dimensional problem, we start with the simplest two-dimensional case: finding the normal vector to a line. One representation of a line 1 l is the equation ax + by + c = 0 . Recall the slopeintercept form of the equation of the line, b c x b a y - - = . From this form we see that the slope m1of 1 l is b a m - 1 = . A second line, 2 l , perpendicular to 1 l would have slope a b m2 = , the negative reciprocal of m1 . EXERCISE SET 1: Given the line 1 l : 3x + 2y + 4 = 0, a. Plot the line b. Find the slope of the line 1 l . c. Find the slope of a line 2 l perpendicular to 1 l . d. Plot the line 2 l on the same graph as 1 l . A vector v coincident with (or parallel to) the line 1 l has components v = b i - a j . Of course, this is not the only vector parallel to 1 l , as any scalar multiple of v will be parallel to v and 1 l . Unit vectors are used to standardize
operations with vectors. A unit vector has length one. For example, the unitb-avectorforvisthevectoru:iNa?+b2Va?+b2EXERCISESET2:Given the linel:3x+2y+4=0,a. Use v=bi-aj to finda vector parallelto li and sketch thatvectoronlib.Findaunit vectoruparallel toljC. Find a unit vector n normal (perpendicular) to li and sketch thatvectoronl.Fromthelast exercise,notethata vectornnormaltovhas componentsn =ai+b j.A general way to obtain that result is to use the gradient operator,%i+ % +% k. You may reall hat role of the vector from theVf=SaxOyOzmultivariable calculus.By evaluating thegradient of the linearfunctionf(x,y)=ax+by+c,weobtain thenormal vectortothe line lj.This result canbegeneralized not onlytofunctions otherthan linearfunctions,but to surfaces inthree dimensions. In doing the following exercises you will rediscover thatapplicationofthe gradient.EXERCISESET3a.Use the gradient operatortofind a vector perpendicular to theline3x+2y+4=0.b.Findavector normaltotheline7x-2y+6=0,then sketchtheline and the normal vector to the line at the point P(o,3),c. Plot the parabola x2 + 5x -7y =0 and the unit normal vector atthe point P(2,2).d. Plot the outward unit normal to the circle of radius 3 at the3/23/2point22e.Sketchtheplane 3x+2y+5z=10 andthe unit normal to theplane at the point P(1,1,I). Note that just as n= ai+b j was normal57
57 operations with vectors. A unit vector has length one. For example, the unit vector for v is the vector u i j 2 2 2 2 a b a a b b + - + + = . EXERCISE SET 2: Given the line 1 l : 3x + 2y + 4 = 0 , a. Use v = b i - a j to find a vector parallel to 1 l and sketch that vector on 1 l . b. Find a unit vector u parallel to 1 l . c. Find a unit vector n normal (perpendicular) to 1 l and sketch that vector on 1 l . From the last exercise, note that a vector n normal to v has components n = a i + b j . A general way to obtain that result is to use the gradient operator , i j k z f y f x f f ¶ ¶ + ¶ ¶ + ¶ ¶ Ñ = . You may recall that role of the vector from the multivariable calculus. By evaluating the gradient of the linear function f (x, y) = ax + by + c , we obtain the normal vector to the line 1 l . This result can be generalized not only to functions other than linear functions, but to surfaces in three dimensions. In doing the following exercises you will rediscover that application of the gradient. EXERCISE SET 3: a. Use the gradient operator to find a vector perpendicular to the line 3x + 2y + 4 = 0 . b. Find a vector normal to the line 7x - 2y + 6 = 0 , then sketch the line and the normal vector to the line at the point P(0,3). c. Plot the parabola 5 7 0 2 x + x - y = and the unit normal vector at the point P(2,2). d. Plot the outward unit normal to the circle of radius 3 at the point ÷ ÷ ø ö ç ç è æ 2 3 2 , 2 3 2 P . e. Sketch the plane 3x + 2y + 5z = 10 and the unit normal to the plane at the point P(1,1,1). Note that just as n = a i + b j was normal
to the line ax+by+c=,n=ai+b j+ckisnormal totheplaneax+by+ cz+d =0.f. Sketch the cylinder ? + x? = 9 and find the outward unit normal(3/223/2)at the point P2Asignificant portionof thesabot canbemodeledwitha cylinder,suchasthat inthelastexercise.Tofindthe influenceofthepressurefield onthecylinder,weusethedotproductvectoroperation.Bycalculatingthedotproductofthevectorpressurefieldwiththenormal vectortothesurface,weobtainthescalarprojection of the vector field on the unit normal vector.That projectionrepresentsthescalarpressureatthatpointonthesurfaceofthecylinderKnowingthepressureonthesurfaceoftheprojectile,engineerscanmakepredictions aboutthe flight dynamics of the sabot round.EXERCISESET4:a. Find the dot product of the vectors u=3i+2j+3k andv=6i-5j+4k.b.Find aunit vector of u=2i+2j+k,then evaluate the dotproductofthe unitvectorwiththevectorfunctionf(x,y,z)= 2xyi+3zj+2yk.c. A circular ring of diameter 6 exists in a field modeled by thevector function f(x,y)=2xyi-3xj.Find the effect of the field at(3/2 3/2)thepoint P2”2d.The fin of a sabot round is modeled by a portion of the plane2x-3z=0.GivenapressurefieldmodeledbyP(x,y=)= 号i-6xyj -2yzk, find the pressure at the point2A(3,1,2).e.Thenose of thepenetrator can be modeledby a cone.Developthe model of a cone with cross section shown in the sketch below.22on the nose of theFindthepressureatthepoint B(3"3)penetrator in the field P(x y,z)= 号i-6xj-2yzk.258
58 to the line ax + by + c = 0 , n = a i + b j + c k is normal to the plane ax + by + cz + d = 0 . f. Sketch the cylinder z x 2 2 + = 9 and find the outward unit normal at the point P 3 2 2 2 3 2 2 , , æ è ç ö ø ÷ . A significant portion of the sabot can be modeled with a cylinder, such as that in the last exercise. To find the influence of the pressure field on the cylinder, we use the dot product vector operation. By calculating the dot product of the vector pressure field with the normal vector to the surface, we obtain the scalar projection of the vector field on the unit normal vector. That projection represents the scalar pressure at that point on the surface of the cylinder. Knowing the pressure on the surface of the projectile, engineers can make predictions about the flight dynamics of the sabot round. EXERCISE SET 4: a. Find the dot product of the vectors u = 3i + 2 j + 3k and v = 6i - 5 j + 4k . b. Find a unit vector of u = 2i + 2 j + k , then evaluate the dot product of the unit vector with the vector function f (x, y, z) = 2xy i + 3z j + 2y k . c. A circular ring of diameter 6 exists in a field modeled by the vector function f ( x, y ) = 2xyi - 3xj . Find the effect of the field at the point ÷ ÷ ø ö ç ç è æ 2 3 2 , 2 3 2 P . d. The fin of a sabot round is modeled by a portion of the plane 2x - 3z = 0 . Given a pressure field modeled by P( x, y,z) i j k xy 2 = - 6xy - 2yz , find the pressure at the point A(3,1,2). e. The nose of the penetrator can be modeled by a cone. Develop the model of a cone with cross section shown in the sketch below. Find the pressure at the point B 2 3 1 2 3 , , æ è ç ö ø ÷ on the nose of the penetrator in the field P( x, y,z) i j k xy 2 = - 6xy - 2yz
f.Acomputerprogramdiscretizesthepenetratorintoquadrilateral finiteelementblocks.Theoutput ofthediscretizationprogram(Table1)identifiesthecoordinatesofthe four corners of the external faceof one block.NodeCoordinates1234(2,1,1)(3,4,1)(1,5,2)(0,1,2)Table 1(1)Findthepressureofeachcornerof theface inthepressure field P(x,y,=)=号i-6xyj-2yzk. Use the2cross product to find a vector normal to the planepassingthrough3of thepoints,orfindtheequation oftheplanepassingthrough3ofthepointsandusethegradient to find the normal.Then evaluatethe dotproduct ofthepressurevector withthenormal totheplane at each point of interest. Howmight the pressureon that portion oftheprojectile(onthe entirefiniteelementblock)berepresented?(2) Is there a bettermethodto representthepressureon theelement?Forexample,considerevaluatingthepressureatthe centroid of the element face.g. A fin of the penetrator is shown in the sketch below. ThetemperatureaffectingthefinismodeledwiththefunctionT(x,y,2)= 3x2 +4y? + 7z -9. The resistance to heat flowdeterminesthedistributionofthecompositematerialsthatcomprisethefin.Sketchthe isotherms(curves of constanttemperature)on thefin.Thenfindthedirectionof maximumtemperature change from the point C(1,l). Plot a unit vector in that59
59 f. A computer program discretizes the penetrator into quadrilateral finite element blocks. The output of the discretization program (Table 1) identifies the coordinates of the four corners of the external face of one block. Node Coordinates 1 (2,1,1) 2 (3,4,1) 3 (1,5,2) 4 (0,1,2) Table 1 (1) Find the pressure of each corner of the face in the pressure field P( x, y,z) i j k xy 2 = - 6xy - 2yz . Use the cross product to find a vector normal to the plane passing through 3 of the points, or find the equation of the plane passing through 3 of the points and use the gradient to find the normal. Then evaluate the dot product of the pressure vector with the normal to the plane at each point of interest. How might the pressure on that portion of the projectile (on the entire finite element block) be represented? (2) Is there a better method to represent the pressure on the element? For example, consider evaluating the pressure at the centroid of the element face. g. A fin of the penetrator is shown in the sketch below. The temperature affecting the fin is modeled with the function ( , , ) 3 4 7 9 2 2 T x y z = x + y + z - . The resistance to heat flow determines the distribution of the composite materials that comprise the fin. Sketch the isotherms (curves of constant temperature) on the fin. Then find the direction of maximum temperature change from the point C(1,1) . Plot a unit vector in that
direction.Whatportionofthefinshouldbecomposedof themostheatresistantmaterials?y3.4ConclusionThe last exerciseand this chapterhavedemonstratedtheapplicabilityofelementaryvectorcalculusoperationsinunderstandingmoderntankmunitionsfromanengineer'sperspective.EngineersattheWeaponsTechnologyDirectorateoftheArmyResearchLaboratorydesignandtest prototypeprojectiles using sophisticated supercomputing facilities.Vector calculuscomputationsarecodedintothose computersimulations;programsusedtodesignandevaluatepresentandproposedpenetratorsCadetsandfacultyfromtheUnited StatesMilitaryAcademyhaveparticipated intheresearcheffort inconjunctionwiththeARLscientists.Thesuccessoffutureweapons development is contingent on theability of civilian and militaryresearcherstoeffectivelyusemathematicaltoolsReferencesTheeffectofsabotfrontboreriderstiffnessonthelaunchdynamicsoffin-stabilized kinetic energy ammunition, P. Plostins, 1. Celmins, and J. Bornstein,AlAA28thAerospaceSciencesMeeting,January8-11,1990,Reno,Nevada.DYNA-3D,Anon-linear,explicit,three-dimensionalfiniteelementcodeforsolidandstructuralmechanics-UserManual,R.G.Whirley,UCRL-MA-107254,LawrenceLivermoreNational Laboratory,1991Calculus,2ndEd.,R.FinneyandG.Thomas,Addison-Wesley,NewYork,1994.AcknowledgementsSpecial thankstoDr.PeterPlostinsandDr.StevenWilkersonfortheirsupportofUSMAparticipation inthisactivity.60
60 direction. What portion of the fin should be composed of the most heat resistant materials? Conclusion The last exercise and this chapter have demonstrated the applicability of elementary vector calculus operations in understanding modern tank munitions from an engineer’s perspective. Engineers at the Weapons Technology Directorate of the Army Research Laboratory design and test prototype projectiles using sophisticated supercomputing facilities. Vector calculus computations are coded into those computer simulations; programs used to design and evaluate present and proposed penetrators. Cadets and faculty from the United States Military Academy have participated in the research effort in conjunction with the ARL scientists. The success of future weapons development is contingent on the ability of civilian and military researchers to effectively use mathematical tools. References The effect of sabot front borerider stiffness on the launch dynamics of finstabilized kinetic energy ammunition, P. Plostins, I. Celmins, and J. Bornstein, AIAA 28th Aerospace Sciences Meeting, January 8-11, 1990, Reno, Nevada. DYNA-3D, A non-linear, explicit, three-dimensional finite element code for solid and structural mechanics – User Manual, R.G. Whirley, UCRL-MA-107254, Lawrence Livermore National Laboratory, 1991. Calculus, 2nd Ed., R. Finney and G.Thomas, Addison-Wesley, New York, 1994. Acknowledgements Special thanks to Dr. Peter Plostins and Dr. Steven Wilkerson for their support of USMA participation in this activity