HW 4 Soln - ’ mm i Let R“ R2 and R3 lwe fie restoring...

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Unformatted text preview: ’ mm i Let R“ R2 and R3 lwe fie restoring forCéS‘ in springs. Ezuai-l‘ons of mafion of mass m in x and y. Jirecfu‘ons axe “'3‘ 3" Z‘ R; “’5 ‘YL ' (£1) i=1 .. 3 . my 2 Z: R; St" ‘3; (£1) 41:1 where R£= -- «£( 1: cos Og- + 3- Sin 0(3) (53) Figs-(E1) Jan (as) am ' _ a II . 1 I ‘ ' - *- mx+ £§1«‘(xmsog+y S‘lfld‘wofl) ,_ o (54) no 3 m y + 2 k; (2: sh og- cos“ at; + y 15:34:) = o (5:) i=1 For ql=r+5°, 0Q: I35°, «3: 270° cum! ku=1<z= *2 = 4‘: 589454) am; (8;) reduce to m32+kx =0 (56) “mgr-kit?- =0 (E7) TheSe Gallo-Hon? m uncouplecl‘ For harmonic mofn‘on . 'XUT) :2 X (.08 (wt +95) , aft) -_- Y (95‘ (wt + , auan {lance {:31 = E for mot-"on in it. clt‘rech'én ‘ (.31: P“ for mofi‘on in 4;- AirecHOn Tn Natural made? we 9M" 53 At) = X wd‘g' t + $61) 34+): Y ms t + 952) w'wre X, 951. Y cmcl ¢z co,“ in Agégrmhed from inifa‘al candftu‘onf- prflfiglflq all Eauadfions of moh‘on {n tar-n13 of ’X. and 9‘. m§£+ kl(z-£,e)-+ k2 (x+ £2 a; 5.- 491: (z— 1.9)“? “2.32 (1* 22.6) For our-ea vibration, we): >< manta) 9(f):® CasQJt-g-Fi) 9) 2'0 (El) =0 (52.) (53). k‘fii C54) and EESIE.) W (E?) bamme - m caL+ 19+ k2 "(kl’ql‘ 1:121) Freguencg eauatron is —ma3 4.» “km- in; .x a: -(kl9l "" 1‘2 22) '1. z z { —3;&9 + 1:,2, +k121] ‘——(1c££,-'4<1£1) Jag—4 =0 1 2. ‘(kt’tt‘kzfifl «35031+4<,£. + “Na I'ven) 2. figs + 5000 too o 100 mod 632+ 2030 1"8') 6 O-3C34- 3530 (492' +- (044 xao == 0 ['ea 1 3 w = 6735. 3313 J «493:- 31‘?3 '5‘ 03:3"- 70-5735 rad/sec; J Q91: 82.3732 rant/53¢ mode Shapes: (-looo 69,24- 5x106) X + o-I K106 ® =o or X! I "Od X10 6 a 59. ‘looo GS.Z+ 5x :06 and 6 x -O-£ Xlo fl} = . 6 = 0.056'OI Q (‘32. ‘looo a9:+ 5x“) Ila 41. .I 1' W ‘ ’9 ' llllliillllillll'lllllllillullllllll lliillilllllliiiiiilllIlllli 31. . V». E uivalenh s stem is shown i ' are: 4< kg 5 z 5’ r " H ‘ m : 2 “E ‘2) 3 L-l.2. L. t) L—prb) L? x.(. 2. E1 «I: «2 = “k = 48 3 "tr-=11“, m1=m «‘1' ha» fl 4‘3 K2 (za- it!) EM“ ‘5’ mam“ “#5:: + (“PF 4‘01!" 1‘1"; =0 “Lag- k2“. + k, X, :: o . . _ I ._ t For {warmer-“c moteon xzfiy) .. xi cos (mt+ , 4... l.2, we 3e (“balm-1- kl+ k1) - ‘kz x1 ___ o [ '“2 (—w‘mw ""(E') FreZuEnc‘j game—tron is k (us-{BLTHE + ky‘f‘ K1) - kg. = O 1 . OT m’mz .— {.9 (mzk'+m2 «1+ “H1123 + ksz :0 w“ = (“1249+ mzkz+m.f<2_) j; Jénzk, +m1kz+mkgz _ 4mlm1k|kz (E) 1m|m1 )- For given data, 2 1. g, 1, mt+mk+zmk in, mt+m4<+1rnk -8rn ‘K * 4c + di_ (.9 =( ) C- ) “3(i“fi) “at: 05"“: if; = 3-1435 EL 3 031: #3066 F; = 9-0524 [EST 3 mi, . X“) flwfmi-f‘kgdr #1 “1m 0.1+.2-k -2{b.1fizqk)+2k .3..— m M...— : ___________________ = r1: 0) ‘ R: k Mk XI = t"H42 1 - 2k (7-: 1g: __ -¢3:ml+1"+k?— :ame.” 2—2([1o:1<)+ <11 " kt" “" Mk ————-————---—"‘" X1 = “WW” 1-4142 , ' —-I-+M~2 Mao‘s shapes cure: . ten—*— 10 n “m i ‘ W 1.1 X = (-4142 4(1) ha (1:) X = Xi . "'4’4'2‘ Farsi: moJe a L°Cahon °Tf m‘le’ (“0 “0J9? (one na 6) 3» 1- 3r 1.444; 1 I’ 3: was“ warm '5' (on) Ezua’cions of I‘noh‘on: Assmme: 9., 92 are Small. MDmen’c eauifflrfium ‘égumh‘ong 053 HM Two masses aJaou'l: Fame. 5).: mil '6}!- mgflet + “K 42(91'”92.>=° 2“ 1 U) ml 62+m3161—Ikal (a..-ez):o ' (a) (1:) Natural fret Mamie? and mode Ska-E69 Assume: Harmonic mofl‘on wtth «(1(69-"9‘3 95k) = (,9; cos (wt—15),. ;: Lg) ‘ I where e, ml @2 m K ' ‘ me! “Fumes “'9‘ 9* “‘4 92a :mxe' are; respectively, as Is the. ” 6' L l, haiurai freyuencj , and! 5.6 m} m? is “'13 Phage “’“BLE' Free 13on diagram Using {gaff}; 535-0) and. (1) (am he exPresseal in mairix form cu! -(gmrf firm My” ,;;;‘:kd«]{:;}={:}m 0 @a Fre- no 2.8,;41} : W1 7?- m‘ c. 2. —wm£ +mjfl+k¢1 flkd .2. :0 “RA?- ~wzmfll+mgfl+ 1‘01 or a. wq—caz(fl+ “in +1;+ Egg—>20 (5) 19 Q fl m L Solu’cion of» E .(5‘) 3N3; 1 ' LBIST , 2 fl mg?— By SubSHEu'h'ng for 09.1 and 692’ Info (4), we obtain (E); ‘0)" {flu }{')_ i ‘0 ' (a?) “1 ‘" can * MG” 92.)“..— 51 or {3}”) ='- {,1 Jz‘a'm @I Tim the matron a; the masses '«'n “u”— fwo m4“ "‘ 3”“ b5 ’ (0‘ (a) ‘ ’9" )(t) = {9' W}: (6). } “‘(‘aat + #5:) (7) az‘"(t) 91”) (t) (c) Free vara/h‘on res onse: . Using “near SUPETPOSI‘h‘on of natural “modes, fle free vibraifon response of H18 fiyfitem is given by 5%): c. 3")(twc23mm (9) By choosing c,:.— 0;: hmfin no loss of generality , E59. (1) to (9) teapot to c . 910:) =-' @Enws (wit *3“) + @laJ “’5 (wit + (’53) (to) 916:) = @f') cos (cad: 4:33,) .— (3?) cos (c.9215 + 561) C“) where (Bil), $3” In) and $32 axe wnStantS‘ ta be aeterminEOI 39mm Hue inifiaal condffl'ons. when elm);— ofl 92(9):. o, ($.00): 0 «ma é1(o)=o: 5135.00) mm! Cu) yield 04 = (33?) 0’5 ibl + @952) cos it 0 - (‘1’?) C03 #| "“ 0:2) cos ¢2p ('2) 0 = .- a" Q}? Sin ¢l -- {192 Sl‘n $2. {535. (12.) can be So‘vea' for (95') n 55,, @El) and $52 "('0 abf'ou'n __. {32*53 £49 (£91 Q‘Uz) .- 0U cos mug—J 4: .ws 2: t 2 ‘ ('3) 928:); 00 Sin wzwzfi' Sin “92:”! f I P0 '2 If... (4)11 rflfi ’ {A} {I . {Ag-E ,1 5 {A} ‘5‘, gfl/ nggéw 6 QM "56%) (0') Canditfons for beat”! : 42 gm Q ‘ . when 1&7: << % or 'k << :12. ’ '0‘") the fwo frewuency components ha 53:2 (I3) , namely, wz—zw' ml “1+ ‘3’ , can be Mfrflgmteg as I . .51.: wi””' 95 .15.. Ci 4 05) 9- 2m W r and 2 _, “92+”! w 1‘ 61 ('6) “2"”? =- \/€[ +3.: W TIM'S implfes that the mo‘ta'ons a; the. Peanulums axe given by r 9105).": a.) gas .n-‘t. cos ail-1t 2 _ ('7) 81(17):: a: sin .11.,t- sen .5th f This motion. Egg-(I7), Ashotes bemh‘na Phenomenon. Pro/55am 6’ Ezuaition of mafia?) of— mass m: 'm Ii = «- kg (1:— r9) ---(E,) SW 0’6— o‘B Cylinder of mass m, anal mus mow-emf: of ,1; = __ k‘ra'e .. «2 (r9- x)!’ "‘59 inertia. J; = limorz 1 ‘ I For «(Hzxmasmm M4 900:9 moat-+4). 239.050 anal (Ea) give the :Freauencj caution a: O 2. 2. 2 2. “"' k1!“ #Jimar ((9 4' 7- 1‘! a. 1: 1+ k .k Le a? —-G9 ‘2": + S ' 1})+ 3— |“: :0 "‘9 m m 79mg 5/ m— WM 5%. Wt; 7:1: mi“, {9; Tn; Xizn—k (xiv—10 (“figs “'2. = ##(zz- 1:) i-e. mlil+ {<1} - k 111:: O .—-—-— (El) 3" .. Assuming 1mg): X; 05051”. 55) 5 ;=.,z ’ E8345.) MA (5;) (M Gums +«)(-—c.9m2- (-9 g1. +49 .. kl :0 or (3+(m.mz+ “'33) __ 592(m,4<+1m14< + 4:33) =0 Ezuaifons of moh‘on: , ty(£’)= .. Y cast-3t mx, = -‘kg(xu";‘)-kz(x.*"1) «. ° “115;: 2': #kzflczv x.) ' M: 1 .. 290%) or mI x, + (1c,+ k109i. - kg}; ‘7: kfl’o C05 63“: ~~- (5.) 4‘: mzxz+ Kg 7&2 -- k1 X, 2 O “"0293 m2. "'1, X} C.th ; J1: 1,2 I __._ (5'3) E85.(EI) CURE! 71m G (—- (327":4- m- m) x, .. 1c; X1 = leo l «10%) = W“ (‘3 "'3 + «1) k' Y" _ ads of (-m‘ 091+ «.4- kz) G- m; w"+ 4‘s) -- it: «.49. Yo GM|C3L+ “5+ kz)(-mzw2'+ k1,) -' K: moat xztt) : ...
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This note was uploaded on 04/07/2008 for the course ME Mechanical taught by Professor Nersersov during the Spring '08 term at Villanova.

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HW 4 Soln - ’ mm i Let R“ R2 and R3 lwe fie restoring...

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