Aerospace Engineering - Strutture Aerospaziali

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Course of Space Structures February 15th , 2023Course of Space Structures Written test, February 15th , 2023 Question 1  True/False (put a T (true) or F (false) at the end of the sentence) 1.The solution obtained using the Ritz method guarantees that the equilibrium conditions are satised at anypoint of the domain: F 2.A three-node truss element allows linear axial strains to be represented: T Question 2  Multiple Choice (circle the correct answer) 1.The Mindlin and Kirchho plate models: (a)share the same boundary conditions (b)are characterized by same essential but dierent natural conditions(c)are characterized by dierent essential and natural conditions2.A beam is characterized by non-negligible shear deformability. (a)The solution has to be evaluated with a force-based approach (b)The solution has to be evaluated with a displacement-based approach (c)The Timoshenko model has to be preferred over the Euler-Bernoulli one Course of Space Structures February 15th , 2023Exercise 1 Consider the three-beam system in the gure. The beams are connected each other at 90 degrees and have length equal tol, bending stinessE Jand torsional stinessGJ. The eect of shear deformability is assumed to be negligible. A uniformly distributed load of intensityqis applied as shown in the gure. Determine the reaction forceVby solving the problem with a force-based approach. (Unit for result:N)Data l= 1000mm E J= 1010 N mm2 GJ= 2E J q= 1N=mm 2 Course of Space Structures February 15th , 2023Exercise 2 Model the structure in the gure by using Euler-Bernoulli assumptions. Solve the static problem with the Ritz method with 1 and 2 degrees of freedom. Evaluate the displacement at the elastically restrained end and plot the deected shapes obtained with the two approximations. Discuss how the 1-dof solution is expected to behave when!+1. (Unit for result:mm)Data l= 1000mm E J= 108 N mm2 k=E J=l3 = 100 q= 1N=mm 7 Course of Space Structures February 15th , 2023Exercise 3 Let's consider the pin-jointed plane frame in the Figure, consisting in a set of trusses of axial stinessE A iand mass per unit length mi. The frame is xed at points A and F and is connected to a viscous dampercat point C. The structure is sub jected at point D to a forcef(t)alongy-direction, assumed to be an ergodic random process of zero mean value and PSD function Sf f( !) =! 2(1+ 2 1! 2 )(1+2 2! 2 )(  1;  2> 0).1.Derive the equations of motion (EOM) governing the dynamics of the structure according to a suitable nite element discretization of the frame, where the mass of each truss is lumped at the nodes(clearly dene al l the quantities involved in the model and specify the dimensions of the vectors and matrices). 2.(Starting from the EOM at point 1.)Derive the reduced-order modalstate-space model (state and output equations) of the structure when the system's output is the vertical displacement of point E, i.e.,y(t) =v E( t). 3.(Starting from the state-space model at point 2.)Derive the equations to compute the root-mean-square value ofv Eaccording to the Lyapunov method. 15 202315FebmarySpaceStruchuresExercise 3- solution yx Flt )^*3V6s4xxus7 ^74 6}6X i I Pî - - D?8DequVj î IJ î sun akzsk5 ^ W Z5U53x * sll T Nodedisplacements wilt ), Vilt ) ( i-1...,.6) Displocementvectoroftrun O ilt ) ( i-1....8) Cexouple : th -TU,V, uz V :JT ts =[ uz V 2U5UsJT uz -[uz v 246Vo JT ) Displocementvector êLt )-Tu,Y, u ,Y. ...4 o 005' ( sise 12 x1)^ O Stiffnenmetsisoftum O : KI -TITKITI Ki -EAi i [? where } = ^ litl( i-1, 2,3,5,6,7)lø -lo-Fl andTi - cisioO [ oociSi } ci- cosxi ,si- Sniai OO MenmatisoftunO : Mi -mif'ie. } oo1ooo Assembly Ê -ŻLikiLi ( rize 12 x12)i=18 â .ylîmili ( size 12 x12)I where ii are ladisationmoticesofsise4712relotingdisplecemntsHioftrussDwiththeelementsoftheglobaldisplocementvector ê@ Cexomples : L 1-[IxQ JLa -[ Q 4r IxQ 4xo] ) Dempingmaetix ? - clite ( fine 12+12)where t-t00000 0 0oooo o ) Snicemodes1 and4ere graunded Tu ,v,usV, usV , 4Vy4 g V H 6VoJ' Ht -llllill 1oooo H MCt ) - Tu2 V 2 4 s V 3 45 V 546 V 6J+ xx 1)siseC M 5I obtainedbyremoingzovskolums1,2,7,8 =a K - E h~zn- C=>C n " Ifise 858) F- 3 O Theequationofmotion are MÜLTEYLE )+EQC-IFLE) ()where 6- P000000 0, j+ Undorgedeigenveluepoblem ( ult)-yeicty E -WMl J 4=Q-sui, wi( i- 1,2)Modalexponien : Ilt )-QfLt kbere Q = [ u. uz ] ( rire 872) qC 1-191 92 J+ ( size 2 x1) Modalequation : DiogLmilälHtYT ę GfLtDieg /miwisqlh-YEF.2 ()where mi - III (i-7,2) Üçu -uîçuii=1 4 State -space form : ILt )- As = 55 H3+ bsFLt ) 3 e () Ewhore Ic -/f). A "Epinglai") - DigSSQi). biI BingsmibQi ?) Yl -V1-To3 0 01 0 0oo JaLH - tyVfLD -[LyY qJEsHJ - CsXs(+) (3 b) TheLyapmovmethodcannotbeapplieddirectly on Eqs . Ce )-bmie flts isnotaahitenoise . We can introduce a shaping filter .i yCH - IIlFHs ( Aia.' fumutatia )-s ( filter)5 O Todelerinethe filtes ,wecanusethefellowingreletirer : SfflwJ -HIjwJHIjoJSuiwIw)-HfL-jwJHfljw) vlex Sfflus . atzia 0 Jceziw ?) Itfollowsthat HEIS ) - HISJCAE 2SJ -11 Helssfss w Wi ' 12Merz 4 : Fs , 1 { 1 ss: Inbicas 'teat2jst1 ls 2( s) 6 O Iethetimedomain ZZI (HT Z 1+22 JECH +ZC=WHS { FLHJ -zCt State -space formoftheshaging filter FILH -SEICHY, TE -AATEtbIw FLH -CfXf where Af -I1^ I as2-24+% ne . Jist .) ef -lo^ } Theoveralldyaniss isgovernedbythefollawingsystem : l Xs - AsYtbyf - AsIstbyCfXf * f -AfFftbEWY - CSTS ZO Definingtheongmentedshetsvector I ILH . IH s 3, vebeve Ilt )-AICH + b (y /t) - Ye L t ) - EIL where Q -TA S BHE J .b-TEJ.E-TGe3 TheRMSvolueofy conbecompuitedfromyansi y -f*Ç" whereJhx isthesolutionofthe hyjonoreqnation Áfktjk |tbbT-E 8