~Ae! TheEngineering~ Resource For~ Advancing Mobility 400COMMONWEALTH DRIVEWARRENDALE. PA15096 r 850416 Mathematical Simulation oftheDynamics ofa MilitaryTank R.M.Krupka Northrop Corp. Electro-Mechanical Division International Congress &Exposition Detroit,Michigan February 25-March1,1985Downloaded from SAE International by North Carolina State Univ, Sunday, September 16, 2018Theappearance ofthecodeatthebottomofthefirstpageof thispaperindicates SAE'sconsentthatcopiesofthepapermaybemadeforpersona! orinternrluse,or forthepersonal orinternaluseofspecificclients.Thisconsent isgivenonthecon­ dition,however, thatthecopierpaythestatedperarticlecopyfeethrough the Copyright Clearance Center,Inc.,Operations Center,P.O.Box765,Schenectady, N.Y.12301,for.copying beyondthatpermitted bySections 107or108oftheU,S. Copyright Law.Thisconsent doesnotextendtootherkindsofcopying suchas copying forgeneral distribution, foradvertising orpromotional purposes, for creating newcollective works,orforresale, Paperspublished priorto1978mayalsobecopiedalaperpaperfeeof$2.50 undertheabovestatedconditions. SAEroutinely stocksprintedpapersforaperiodofthreeyearsfollowing dateof publication. DirectyourorderstoSAEOrdp.rDepartment. Toobtainquantity reprintrates,permission toreprintatechnical paperorper­ mission tousecopyrighted SAEpublications inotherworks,contact theSAE Publications Division. ISSN0148-7191 Copyright ©1985SocietyofAutomotive Engineers, Inc. Thispape-rissubjcrttorevision. Statements andopinions ad· vancedinpapersordiscussion arcthe-author's andarchis responsibility, notSAE's;however, thepaperhasbeenedited bySALforuniform stylingandformat. Discussion willbe printedwiththepaperifitispublished inSAlTransactions, I-'orpermission topublishthispaperinfullorinpart,conlill't theSAl::Publications Division.Persons wishing tosubmitpaperstoberonsidcred forpre-­ sentation orpublication through SAl::shouldsendthemanu­ scriptora300wordabstrart ofaproposed manuscript to: Secretary, Engineering Activity Board,SAL PrintedinU.S.A.Downloaded from SAE International by North Carolina State Univ, Sunday, September 16, 2018850416 Mathematical Simulation oftheDynamics ofa MilitaryTank R.M.Krupka Northrop Corp. Electro-Mechanical Division Theinertia forcesandcouples acting onthe hullortheturretaredefined byLagrange equa­ tionsofmotionforthebodyfixedaxis,which appearbelow:Theangular momentum vectorisrelated to theangular velocity vector bythefollowing relationship:ABSTRACT Theequations ofmotionassociated witha military tankarederived usingLagrange equa­ tions. Thetankistreated asthreedistinct bodies, eachhavingsixdegrees offreedom (threedimensional motion), representing the hull,theturretandthemaingun.Therigid bodiesaresubject toasetofkinematic con­ straint relationships, whichprovide thephysi­ calinterface ofthesystem. Eulerian coordi­ natesareusedtodescribe therigidbodies. Lagrangian coordinates areusedtodescribe the bending ofthemaingun.Thegunisassumed to bendintheelevation andazimuthal directions.[::1'[::xx-:xy -:xzJ[WXJA xyyyyzwy Ixz-IyzIzzWz(2) F Az,Id (~~xrA(£1_J dt-W (3)zavy WA(£LYA + Fav x,I y z d (~~yYA (~~J at-Wx +WA(£L)A FA zav y,Ix d(£LY_A(£LtWdtaV yavz x wA~£L)A+xav y(1) +!(w~lA+w~A+w~A/)2x x y y z z 0148·7191/85/0225·0416$02.50 Copyright 1985SocietyofAutomotive Engineers,lnc.2 !~r(v A+2 xLetAbeageneric termrepresenting the hull,H,ortheturret, T.Thekinetic energy oftranslation androtation ofthecenterof gravity ofthehuilortheturret, inthebody fixed,Eulerian coordinate systemisgivenby theexpression:THETANKEQUATIONS OFMOTION weredeveloped con­ sidering thetanktoconsist ofthreeseparate rigidbodies: thehull,theturretandthemain gun,whichareconnected byasetofkinematic constraints. Gunbending effects arealso included. Thispremise allowsthedevelopment ofasetofequations inwhichtheredundant coordinates areeliminated through useof Lagrange multipliers. Thespatial orientation ofthethreebodiesisdetermined bysolving a setofdirection cosineequations. INERTIA FORCES ANDCOUPLES AC