Class Note for ECE 449 at UA
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Date Created: 02/06/15
A we 449549 ontinnons System whaling Modeling of Multibody Systems 0 In this lee 1Tl ew special problems M quot f7 s Ldeling of complex 7 p 0 We shall 3 i Z 39 iblems have 0 It shall be shown how matnX 39caiICullus makes it possible to keep the definitions ver compact September 24 2003 v start Presentation E AV cm 449549 Glontinnnns System mutualitth Table of Contents September 24 2003 Start Presentation i lll A l we 449549 toutinunux ystem mammal What is a Multibody System A multib odly TsiyteT 39Z39i39f s em s combination of mechanical pa each other to 50 If Hmm Maybe this is not yet the most luxurious j 5i 1 model but abstraction is everything after all Start Presentatjon September 24 2003 A we 449549 Glontinuous 93mm momma The Choice of State Variables Until now weihaxke al39a c39uledtthehdif er fritial equations that result from m 39 lr n 39 b w masses This iple had been a l nil defined for theind WA Therefore it m to i system library V In the MBS library the relative positions and relative velocities between bodies that are connected to each other are de ned as state variables September 24 2003 39I A I we 449549 untinnons System mutualitth u Structural Singularities I 6 no structural ody systems laps structural ures exhibit a freedom than s You may etimes also used ir security doors freedom y L x Kinema cally closed loops SitartPresentation September 24 2003 To avoid quotf 5 associated with these 39 v V nematically 0 Cut joints y tors thereby avoiding 1n r o uo lon bf structural singularities This is more efficient that to rely on the Pantelides algorithm September 24 2003 7 Start Presentation z I we 449549 ontinnons System mutualitth 39 A Algebraic Loops Closed kl I 35 39bad algebraic loops inth es l ma T se are usually ver large 39 v quot variables of the The autom 7 quot g variables motions Which I y39 find a set of suitable tearing variables constrain qd qdd September 24 2003 Start Pesenlaqu l l em 449549 Glontinnnns System mutualitth Choice of Potential and Flow Variables Die lVIBS lib r any 39 39 ton of a body 11 the body uld have been September 24 2003 A I we 449549 ontinnons System whaling Mechanical Connectors connector Frame Position r03 quotDistance of the frame from the inertial systemquot Real S3 3 quotTransformation matrix of the frame to the inertial systemquot Velocity v3 quotAbsolute velocity ofthe framequot AngularVelocity w3 quotAbsolute angular velocity of the framequot Acceleration a3 quotAbsolute acceleration of the framequot AngularAcceleration z3 quotAbsolute angular acceleration of the frame quot flow Force S quotForce acting on the framequot ow Torque 3 quotTorque acting on the framequot end Frame Because of the sign conventions an empty onqmt frame must always be connected to a full input frame I September 24 2003 Start Presentation 5 39l I we 449549 Glontinnnns System moteliml Mechanical Bodies I 0 Mechanical bodies define the D Alembert Principle fof t u 26f 39 H e e39s and torques model BorgBase quotInertia and mass properties of a rigid bodyquot BOdVBase extends Frame a a Mass m Position rCM3 quotDistance from frame to center of gravityquot Inertia I3 3 rCM Frame equation Center of l f ma crossz rCAJ crossw crossw rCJM Grav39ty t Iz crossw Iw crossrCM f end BodyBase The coordinates of the frames are first converted to the center of gravity The D Alembert Principle is then formulated for the center of gravity The resulting force f and torque t are finally transformed back to the frame by means of their relative movement under introduction of the accompanying centripetal and Coriolis forces Start Presentation 52 September 24 2003 it 449549 Glontinuoux System mowing Mechanical Bodies II Body model Body extends Frameia parameter Position rCM 3 0 0 0 quotVector from frameia to center of mass resolvedin frameia parameter Mass m0 quotMass ofbody kg quot parameter Inertia III 0 quot11 element of inertia tensor parameter Inertia 1220 quot22 element of inertia tensor parameter Inertia 1330 quot33 element of inertia tensor parameter Inertia I21 0 quot21 element of inertia tensor parameter Inertia I31 0 quot31 element of inertia tensor parameter Inertia 1320 quot32 element of inertia tensor BodyBase body equation connect frameia body from e711 bodym m b0dyrCM rCM body 1 11 1 121 131 121 122 132 131 132 133 end Body quotRigid body with one cut September 24 2003 Start Presentation lt1EIgt new 449549 Giuntinuuus system 1111111216th Mechanical Bodies III Body I i a nun component name uoss ModelicaAddi onsM ultIEodyf39a s Bu model Bod quotRigid body with one cut extends rame a parameter Pos1 39on rCM3 00 Vector from ame a to cen er 0 matsi r85 v d in frameia parameter assm quotMass 0 0 g quot parameter Inertia 1110 quot 11 e1emen 0 1nert1atensor Darameter Inertia 1220 quot r e ement o Vinertia tensor r quot yr UK r one body wt e e We Parameter o ereut Value Desulmiun rEMI3I ELELD Vector lmm lrameia to center o ass Iesolved In flameia m m E Mass ol body kg kg H1 u n 1 eIement or meme tensor ko m2 I22 I 221 eIement or meme tensor kg m2 I33 H 331 eIement or meme tensor ko m2 121 El 21 element of Inertia tensor kg m2 3 U 31 element of IneItIa tensor kg m2 132 D 32 element of InertIa tensor kg m2 4 Mourners I Information summer I extracted from tvne deolar a on September 24 2003 Start Presentation A I was 449549 ontinnons System mammal Mechanical Bodies IV Coordinate transformation frame a frame I r frameTra anon x r frame bodry lama rCM Body calculated relative to lrame a Bodies With more than two joints have to be constructed by the modeler using additional frame translations Such elements are not available in the MRS library as pre designed modulesl StartiPresentation September 24 2003 I was 449549 Glontinnnns System whaling Mechanical Bodies V BoxBody quotIgt Geometry for the animation frameTranslation Geometry for the computation of mass and inertia matrix not represented graphically since modeled by means of equations Start Presentation z September 24 2003 A I we 449549 ontinnons System mnaml Mechanical Bodies VI model BoxBody quotRigid body with box shape also used for animationquot extends MuliiBody Interfaces TonreeFramex parameter SIunits I osition r30100 quotVector from frameia to frameib resolved in frameiaquot parameter SIunits Position r030 0 0 quotVector from frameia to le box plane resolved in frameia parameter SIunits Posi onLengthDirection3r r0 quotVector in length direction resolved in frarneia parameter SIunits Position WidihDireciion301 0 quotVector in Width direction resolved in frameiaquot parameter SIunits LengthLengih xqrir r0 r r0 quotLength ofbox parameter SIunits Length Width0 1 quotWidth ofbox parameter SIunits Length Height 1 quotHeight of boxquot parameter SIunits Length InnerWidi 0 quotWidth of inner box surfacequot parameter SIunits Length InnerHeighi0 quotHeight of inner box surfacequot parameter Real rho7 7 quotDensity of box material gcrn 3 parameter Real Materid41 0 0 0 5 quotColor and specular coef cient SIunits Mass mo mi Real Sbox3 3 SIunits Length l w h wi hi FrameTran slationframeTranslaiion r r MultiBody Interfaces BodyBase body VisualShape box Shape box r0r0 LengihDireciionLengihDireciion WidihDireciion WidihDireciion LengthLengih WidthWidih HeighiHeighi Maierid Maierid September 24 2003 Start Presentation 2 I we 449549 Glontinnnns System muslingl Mechanical Bodies VII connect body ameia frameia connect frameia m eTranxlaiion w eia connect ameTramlaiioni ameib m e b boxS Sa boxr r0a boxthape Sbox l Length w Width 11 Heighi wi InnerWidih hi InnerHeighi Mass properties ofbox mo 1000rholwh mi 1000rholwihi bodym mo mi bodyrCM r0 l2boxnLengih bodyj Sboxdiagonalmoww hh miwiwi hihimoll hh mill hihimoll ww mill wiw112 transposeSbox end BoxBody September 24 2003 A I was 449549 hummus System muslingl Mechanical Joints I model Prismatic quotPrismatic joint 1 degreeof freedom usedin spanning tree airlifter extends Mul riBody Interfaces T reeJoim I Ip parameterReal n3100 quotAxis oftranslation resolved in frameia same as in frame bquot parameter SIunits Position q00 quotRelative distance offsetsee infoquot parameter Boolean xtartValueFixedfdxe quottrue if staIt values of q qd are xed SIunits Position and xedxtmtValueFixed Slunits Velocity qund xedxtmtValueFixed SIunits Acceleration 11111 SIunits Position qq Real rm3 SIunits Velocity vawc3 Modelica Mechanic s TranslationaLInterfaces Flangea axis Modelica Mechanic s Translational Interfaces Flangeb bearing P1ismatic Frame 1 Frame b September 24 2003 Start Presentation f2 l I was 449549 Gluntinnnns System mammal Mechanical Joints II WNW Transform the kinematic quantities from framea to frameb WISS 5 and the force and torque acting at frameh to framea bearingS 0 general equations of a quotTreeJointquot specialized to this class Sb Sa define states r0b r0a Sar rela qll derq 7 111111 de m vawc crosswa rirela vb va virela vawc normalize axis vector wb WE rm nxqrt 39l n ab aa airela crossza rirela crosswa vawc 2v7rela kinematic quantities V zb m Sire identity3 qqqq0 fa fb rirela nnqq ta rb Crossr7rela fa virela rm qd airela nnqdd ll d39Alemberts principle wirela zeros3 Maj nnfb zirela ZBIOS3 endPrixmaric September 24 2003 Start Presentation 52 lime 449549 Giantinnons System Whalingquot 39A l u Causalities I 439 lib dgp ends on the In the dir m 39 el EL V 39 problem the desired movements are predetermined Whereas the forces and torques that are needed to produce the desired movements are to be found September 24 2003 Causalities II o The ef 39c iencytof the generated code depends r 1 51quot ehuations Small changes 11 modify the efficiency 39 iof resulting equations nearly With the number of power the automatic t a f7 atid r off the equations by use of the Pantelides algorithm and the previously presented heuristics for finding small sets of tearing variables September 24 2003 10 Item 449549 ontinnons System whalingll 39A l C ausalities III 0 Matrix ca u s very elegant suited for notation 39 mm 1 automatic transfonna a f For this am 311 quot 39al are quotl i into scalar symbolica y e equations pjri or t the detenni nati on of the correct causalities September 24 2003 Start Pre entation 2 I we 449549 Glontinnnns System whaling 39 l An Example I mdauavL iv quot rodL ruvque ythEl agx i a 3 9 Eu l5 I 00025 ms i menial trans WWW V X lib n 39 SWIKM W 71 W mum September 24 2003 7 Start Presentation 2 ll A was 449549 ontinnons System whaling An Example II NdBo V h m mama2 v mque yWhEEl 1 1 gt eu wsmnqn v cY39H pm Wmquot September 24 2003 Star Prggentatjon I was 449549 Glontinnnns System whaling An Example III mdaaavL quue nythsI m i D Q tau 17 I L 00025 memal quotans 9 v X n WIND cyP H mu V39mquot September 24 2003 Start Presentation 2 l we 449mm minimums system mowing xpenrnerl 55m 7 txanslace odel r a mini ace Finiahe d gages w Translauion seazced mu nuni 2174 unknonm scalaza and 2474 scale equacinna lt 776 constants found 225 parameter bound vazlables found 1213 alias variables found 250 remaining time dependent variables lt quot Finished An Example IV hovalnw w in Mad u aucl elp quotModelicazlcldicions ulci ody Examples LoopaEngine1 Equations after expansion of the matrix expressions Elimination of trivial equations of the type a b V H Remaining equations after the symbolic tra nsformationi September 24 2003 39 Start P senlaticn 5m 449549 Gloutiuuous System whaling An Example V luvque yWhee f geZHI leu 1T file Ammalm iewp elp Tir na wounle V a m who amnion September 24gt 2003 13 A we 449549 Ginntinunus System mowing References N 23963r 1996 lathe Ob39ect September 24 2003 n PzesemgQiu i 14
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