Division of Mchanics Lund Univrsity MULTIBODY DYNMICS Examination 7033 Nam (writ in block lttrs):. Id.-numbr: Writtn xamination with fiv tasks. Plas chck that all tasks ar includd. clan copy of th solutions should b writtn in th spac availabl undr ach task. Th back sid of th papr may also b usd if ncssary. Clar motivations to your answrs should b supplid. Prmissabl aids: Summary (of boxs from th Lctur Nots), Som formulas in vctor algbra, Momnts of inrtia and mathmatical handbooks such as TEFYM may b usd. Summary of tst rsults: Task Commnts/judgmnt Points(0-3) 3 4 Sum Nam (signatur).. Lg:
. Th motion of a rigid body is govrnd by th translation u = u() t V and th rotation R= R() t SO( V ) whr is a fixd rduction point. Lt P b an arbitrary point in and lt r P and p P dnot th rlativ position vctors in th rfrnc and prsnt placmnts, rspctivly. Which of th following statmnts ar tru, in gnral, and which ar fals? Giv a short motivation for your answr! a) R() t = R () t T b) up () t = u() t + R() t r P c) pp () t = R() t r P d) up () t = u () t + W() t pp () t whr W () t is th spin tnsor of th body. ) u () t = u () t + W () t p () t P P nswr and motivation:
. homognous body with mass m is in th shap of a solid sphr with radius R. Its cntr of mass is dnotd C. Th point, on th boundary surfac of th body, has position vctor pc = x( R) whr = ( x y ) is a RON-basis. Lt I C and I dnot th inrtia tnsors of th body with rspct to C and, rspctivly. Which of th following statmnts ar tru and which ar fals? short motivation for your answrs is rquird. a) Th axis ( C, x ) is a fr axis. b) ll axs through C ar fr axs. c) IC = mr. d) ll axis (, n ), whr n is an arbitrary unit vctor, ar principal axs of inrtia at. ) [ I ] mr 0 0 7 7 0 0 mr = 0 mr 0 y C x nswr and motivation: 3
3. homognous body consists of an isotropic, linar lastic matrial. Th scond Piola-Kirchhoff strss tnsor S is thn givn by S= λ tre+ µ E () whr E is th Grn-St.Vnant strain tnsor and λ and µ ar constants. Lt B 0 dnot th body in its rfrnc placmnt and assum that th body is subjctd to a systm of xtrnal forcs with th xpndd powr P on. Th total lastic nrgy of th body is dnotd U. Which of th following statmnts ar tru and which ar fals? short motivation for your answr is rquird. a) U = µ E B 0 dv( X ) b) U = (tr ) dv( X ) λ E B0 c) U = S E dv( X ) B0 d) U = S E dv( X ) B ) 0 d U + dm = P dt x nswr and motivation: 4
4. Th configuration spac of a multibody is dscribd by a st of gnralid n coordinats q= q, q,..., q. Two rigid parts and of ar connctd by a sphrical joint. Lt acting from on f and M dnot th forc and momnt sums of th forcs and lt I k dnot th mchanical intraction btwn th parts.,,sys Lt x dnot th rlativ systm vlocity of th parts at point and lt dnot thir rlativ angular vlocity., ω Which of th following statmnts ar tru and which ar fals? short motivation for your answrs is rquird., a) ω = 0. b) Thr is a point O in spac such that x = 0.,,sys O,, sys, c) I k = x k ; f + ω ; k M, k =,..., n, whr is an arbitrary point. d) If th q - coordinat systm is compatibl with th constraint conditions, rprsnting th joint thn Ik = ω; k M, k =,..., n whr is an arbitrary point. ) If th q - coordinat systm is compatibl with th constraint conditions rprsnting th joint and if th constraint mchanism of th joint is idal thn I = 0, k =,..., n. k nswr and motivation:
. rigid homognous, right-circular con of mass m, bas radius r and hight h is connctd to a rigid fork-shapd structur by way of an idal rvolut joint with th horiontal axis ( O, ). Th con symmtry axis ( O, 3) is assumd to b prpndicular to ( O, ). is coupld to ground via an idal rvolut joint with th fixd vrtical axis ( O, ). Th momnt of inrtia for with rspct to ( O, ) is I. Th fork axis is subjctd to a driving momnt Md = M, M () d d = Md t. Introduc th angls ψθ, as configuration coordinats for th multibody consisting of and. S figur blow! a) Formulat th angular vlocitis for th con and th fork, rspctivly. b) Formulat th Lagrangian L= L( ψθψθ,,, ) = T V for th multibody consisting of and. c) Formulat th gnralisd xtrnal forc Q du to th driving momnt d) Formulat Lagrang quations of motion for th multibody. M d. 3 g = ( g) θ O ψ M =, d M d Solution: 6
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