Homework6_Su_2010_Solution

Homework6_Su_2010_Solution - ME 562 Advanced Dynamics...

This preview shows pages 1–8. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ME 562 Advanced Dynamics Summer 2010 HOMEWORK # 6 Due: July 23, 2010 Q1. Two wheels, each of mass m, are connected by a massless axle of length 1. Each wheel is considered to have its mass concentrated as a particle at its hub. The wheels roll without slipping on a horizontal plane. The hub of wheel A is attached by a spring of stiffness k and unstressed length l to a ﬁxed point 0. Use r, 0, and (1) as generalized coordinates. Then, ﬁnd: (a) the constraints satisﬁed by these three variables; (b) the relationships between the virtual displacements in the three variables; (0) if the constraints are holonomic or non-holonomic. (see Problem 6-25 in the text for a ﬁgure). Q2. (see Problem 6-7 in the text for a ﬁgure). A double pendulum consists of two massless rods of length l and two particles of mass m which can move in the vertical plane. Assume frictionless joints and deﬁne the conﬁguration of the system using the coordinates 9 and at). Recall that the system is in the vertical plane. (i) Derive the generalized forces for the generalized coordinates 9 and (1) corresponding to the weights forces of the two particles. (ii) Then, use Lagrange’s equations for holonomic systems and derive the differential equations of motion for the system. Q3. (see Problem 6-13 in the text for a ﬁgure). A smooth tube in the form of a circle of radius r is pinned at O and rotates in its vertical plane with a constant angular velocity (1). The position of a particle of mass m that slides inside the tube is given by the relative coordinate ti). (1) is the angle that the line joining the center of the ring/tube (0’) to the particle makes with 00’. Use Lagrange’s equations for holonomic systems to derive the differential equation for (1), the only generalized coordinate. Note that 19 = a) is constant and is speciﬁed, thus it is not a generalized coordinate. Q4. (see Problem 6-22 in the text for a ﬁgure). A dumbbell is composed of two particles, each of mass m, connected by a massless rod of length 1. One particle of the dumbbell is connected by a pin to the edge of a disk of radius r, which is massless except for a particle of mass m at its center. The disc can roll without slipping on a horizontal surface. Assume frictionless joints and deﬁne the conﬁguration of the system using the coordinates 8 and (1) which are absolute rotation angles. The system is in the vertical plane. Then, use Lagrange’s equations for holonomic systems and derive the differential equations of motion for the system. i743 : “ﬂ[¢"5)3i”[¢‘5 (grjéz’iVErCéQa gifgv/Sxfeez'éér) J[¢'9)§9 Jammie); 1% = 4&5 “9)5infﬂﬁ'6)§p +£9cosfgé—GMQ 42(425'64? to If“: T' gr _ IIIOA: Fgﬂ- rg'r IDA 2 i v E (n : TAB _: Brosw'eﬂﬂ'ﬂanw 9) 9 :[05[¢’9)£r+\$mw_6) £9 *' mgr ffcoﬂgﬁraﬂsmzw-w] foA - ’n : (1‘94 régeHmw—emrwinw-m 99]:— l‘casfcé'ﬂ) 4L ré Sin M ’9 I) 5: O (all : £05[¢‘6) v) an :FSM[¢'9)) a13 : “17.16 1: O ) 005(05-6) 5N rsinmre) 59 = O 5) Con51lram+i5 kinemaJicl Mu; non’Ao/anomic (D 161i! __.__—7 HM {6WD} T ﬂwsﬂi * “"9; ) Hirwhawl {DH -2: n‘ m3 :iqlr‘fC039+(05{9+¢)-}£ +{gm0+5m [6+¢)} g + {atom {mm zamgg] ’U: “:5 T2;£[fﬁégin9-[é+d)5fﬁ[g+¢?} 1" Ts—Lmyi'»l+-2mmlf2.1{2 Z —'—m1226'2+—§’-m122[é25ihq0+Ké+¢ﬁgingﬁ¥¢ﬂ 29[g+¢)5magm{am 4- ézcogze +[é+¢)2(052{64' + 29[Q+¢)6056605/9+ (75)] —’—m£1[é2+(é+¢3)2+2é(é+¢5)céacﬁ] @ ...__L 2.2+ - 2M9 2 5w: Q5 58+ 6195M: mgué-ér, + m9; -2 #: 'fgggma;,;gégmze+a} 99.? 729.2%???) ______ M. «Lions me Then’ Lagrange‘s (aqua 9: _ Fa]; : as z 5”“ 66 36 %:m2?é+m21[é+lé+d)+[Q+é)(05¢+éms¢] : 3m21é'+m£1¢'+ m£’[{5+ {5) E05 45 ' {é+d)¢55in¢ + 9.}qu - 955m : 3mﬂ1€5+ mfg-D} Zmézé'cosa‘ fqué.[05¢ wgzﬁém ¢+ (325:?) 5:0 a6 Mfr,“ .____ m£2[[3+2.cas(15) 1‘ [Nays (:5) 0‘5.— + mgﬂpsine Him/WW] '—' 0 EM] £15 ' d [31)“ 3%: 63¢ _m__ WW a' at? ad 3—: : mﬂ‘fémtéhmfzéfosfj 3&0 E3mﬂ2[é+é)+mﬂzécm¢ w “‘31 é 1" " £115 # Wéﬁmgﬁ F(a¢),m£(9+¢)+m £05¢m ﬂzdmﬂzésinﬂ’wvtd) a¢ # WM” * mﬁzé(l+cos¢)+mfzé.+m 2523M¢+mgﬂm ? ----—~— My"... _ _ Mu ..,._———_- ______' L ' A \$9 ezwqmlmL jw 05 Sq, F A IOA:{pgina+r5in{¢’6))Z_§+ rst-r(05[¢—6)}j °':' {.71 : rircoseéw cosw—eww— 5951i -p 3in659*5i"[¢—6)[S¢-J9)}5i :{ms[¢—a)_e+mn(¢~e)gW +{we «mm—am —[mme+rsm{¢—e))g}ga f2 =m9g' 5b! : 5-ng :[m3m;n{¢,e)]5¢—mgp[sme+smz¢way§e T 61¢ : m3r51n[¢_9) 25M = [Fém504' r(¢:é)m{¢.e)]_§ + [arésanQ-rrﬁvékinM’aﬂg = [WU €069+rf (1'4de ﬁ—e ﬂi ~ [rwme— r666 ~ux)m{¢»9)jg ram ' rm = FZwZCosZGJrZrZw [rivet/7 €056 cos/6L9) 4 r’ﬂ-wfcos 205,9) 4 riot/13m? .. 2r‘wM-w75in am (pf—9 )wzw-wa-e) . whammy—WaneJae/m)? = F20)?” 2’20” [dwmwekoa’fﬁmsaf-Jin ¢sin 9) (05¢ : r2w2+2r2w [éwu/Jca‘yﬂﬁrZKﬂ—wﬂ . .6 2 TPZLWI-m'foa)=ZLmr2[W2+2WW“W) Wm M w) J +ZWKd—w)(o:¢'+ ¢IZ’Z¢'W+WE7 =--Z}—mr2’[w2 :ELmr"[2w1+2w [d—who‘s’ﬁé MAME] %5%mrz[m605¢+2¢»2w] -: mﬂgucasﬁl-L "60) '-'- mr : mnzf-w dsin¢f+ {3] (it by: i: = ELmrEPZwM'waW: “WPZWM'Wbin/ﬁ M {cu/WWW] Maggi”512M2522'; 0 / 5’22. Ad'lfa'ny Indiu't't/uul kinG‘h'C Ener-gr'es’ . . L - 1 . I. T: gmle’ﬁm r-e) HM) fl<r5)w5€] r‘} MUN} "9.4.1::‘9 Hécos {h}; )- «réa'fne {-Ztésfn¢)z] 0f . T: .5. v: _ '3 ‘ 1- h ‘miro (-Sfjcnss) } \$5 _ Wu“ :9 Her [canary-rm: 15]} . 31' V: mé;(2r¢956-I¢osjﬁ)) .é.(§)ng-t+a—V1*zo a‘r . - - ‘ . ‘ . 3 3.; z: wxgmquose) mr2¢[60,-(9r¢)+mgj airmzmymze 6;: = "WIS-off? 41:03 9} ray-Ié‘tioﬁa'r‘ﬁwa‘]-1mr‘a 51.36 nmrféké fé)§rh(0tiﬁ)ké\$iué] ! ‘ r ' i a I ‘ 1 ~3mv16‘5m9.*mr35?’5’”(5"¢) 52 pfgaffonj mrléYfr-‘Ic859JPhr.);[cob(6f}ﬂ+ C1159!) “meiélmhé -mrf§5L[5i'-"t {6%) f' 5’3?) "‘22 m r 5:3{6 2 O ir— 2. 1111‘; +mﬁéfcas(é+¢}+cgsﬁ it : mfg???” T 1" n : - i ' ‘ ‘F r "i; ff.(§;.)= m1 ¢frnrlé caswwmwj Wlﬂfwl-ﬂwiﬁrﬁrﬁ g .. ﬁx ~mrza1£ltsﬁd9+fﬂﬁfn¢1i «:5 e’quuh'oq: ME}; +mr?§1¢cs(9r¢)'rwf ¢]-mr!6&fm{6 +56) 1‘ hid 1?: U M g . I I. ...
View Full Document

{[ snackBarMessage ]}

Page1 / 8

Homework6_Su_2010_Solution - ME 562 Advanced Dynamics...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online