Transcript
!"#$%&%'%()*+,-%.**/%01*.),$+%2&3%04+5% ! "#!$%&!'((')*'+,-!.,/'0)!&.!-1/2,!0-3)'45!+/4!0'..-6-34!7')*&)'45!.,&%!0&%3!2!)/6.2*-!'3*,'3-0!24!23!238,-! &.!9:!°;!&3-!&3!4&3-));!!?@#!!$=-!-1/24'&3!0-)*6'+'38!4=-! 7-,&*'45!<6&.',-!'3!-2*=!.,/'0!=2)!4=-!8-3-62,!.&6(!!
u(y) =
! gsin " 2 y + Ay + B ;! 2µ
%=-6-!"!230!#!26-!/3>3&%3!*&3)4234)!&.!'34-8624'&3#!!A')4!2,,!4=-!+&/30265!*&30'4'&3)!3--0-0!4&! 0-4-6('3-!4=-!*&3)4234)!&.!'34-8624'&3!.&6!-2*=!.,/'0!230!2!6-2)&3!.&6!-2*=!*&30'4'&3#!!B&!*6-0'4!%',,!+-! 8'7-3!.&6!2!*&30'4'&3!%'4=&/4!23!-C<,2324'&3#!DE<4)F! ! ! "F!! @! y = 0,!!!!!!!!!u1 = 0 ! ! 4=-6-!')!2!3&G),''3-4'*!-3-685F!2))&*'24-0!%'4=!4=24!.,&%!(25!+-!3-8,'8'+,-#! ! ! ! ! ! 9#!N=24!.&6*-D)F!26-!'(<&64234!.&6!,'.4P!N=24!.&6*-D)F!26-!'(<&64234!.&6!0628P!D@!<4)F! ! A'.4!S!<6'(26',5!<6-))/6-!.&6*-)! R628!S!<6-))/6-!230!)=-26!D.6'*4'&3F!.&6*-)
H#!R-4-6('3-!4=-!)46-2(!./3*4'&3!*&66-)<&30'38!4&!4=-!7-,&*'45!<&4-34'2,! " = x 3 ! 3xy 2 #!T>-4*=!4=-! )46-2(,'3-!!U!:!%='*=!<2))-)!4=6&/8=!4=-!&6'8'3#!!D":!<4)F! ! T46-2(!./3*4'&3V!E!<4)! T>-4*=V!H!<4)! !
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M#!!W,/'0!-34-6)!23!-C<23)'&3!'3!2!<'<'38!)5)4-(! %'4=!*&3)4234!7-,&*'45!$"!230!-C'4)!24!7-,&*'45!$@L! 4=-6-!')!2!<6-))/6-!06&#!DM! <4)F! ! X.!";!µ!230!#,!&3,5!4%&!&.!4=-!4=6--!26-!'30-<-30-34L!4=24!');!8'7-3!4%&!&.!4=-)-!%-!*23! 0-4-6('3-!4=-!4='60!/)'38!#=µ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!u !!u = X sin " t + µ 2 ! !t !y
98212!"!),!+82!:0'4)+672!(5!+82!'12,,612!<1:7)2*+@!:*7!8:,!6*)+,!2B6)=:42*+!+(!'12,,612D42*<+8#! ! E(*?7)02*,)(*:4)F2!+82!:3(=2!2B6:+)(*!6,)*(61!125212*/2!':1:02+21,#!!$G)*+H!':>! :++2*+)(*!+(!>(61!6*)+,I-! !
! 72+210)*:+)(*!(5!7)02*,)(*42,,!=:1):342,H!J!'+,! 76D7+H!K!'+,! 76D7>H!K!'+,! 7%6D7>%H!L!'+,! 5)*:4!2B6:+)(*H!L!'+,!$?%!)5!+82>!7(*M+!,)0'4)5>-!
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K#!!P:+21!54(9,!+81(6<8!+82!')')*,+20!,8(9*!324(9@!98)/8!),!0:72!6'!(5!&#NQ?)*/8!7):02+21! <:4=:*)F27!)1(*!')'2W!+82!+(+:4!42*<+8!(5!')'2!32+922*!'()*+!$"-!:*7!+82!2X)+!),!%"!)*/82,W!+82!243(9,!:12! 12<64:1@!Y&!°!:*7!+812:727#!!C82!Z2+!:+!+82!2X)+!12:/82,!:!82)<8+!K!)*/82,!:3(=2!+82!')'2!(6+42+W!+82!')'2! (6+42+!),!L!)*/82,!:3(=2!+82!12,+!(5!+82!')')*,+20#! ! S2+210)*2!+82!'12,,612!:+!'()*+!$"-!*22727!+(!'1(76/2!+8),!54(9#!$%Q!'+,-!
!! [)0'4)5>!2*21<>!2B6:+)(*H!J!'+,! [(4=2!5(1!UKH!Q!'+,! [(4=2!5(1!T2H!%!'+,! R)*7!2DSH!"!'+! S25)*2!82:7!4(,,H! R)*7!51)/+)(*!5:/+(1H!%!'+,! R)*7!0)*(1!4(,,2,H!%!'+,! R)*7!V"H!Q!'+,! ! ! !
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%Q!'+,
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vr = 2r + 3r 2 sin ! #! S2+210)*2!+82!+:*<2*+):4!=24(/)+>!/(0'(*2*+! v! #!$"\!'+,-!! ! !
! ;*/(0'12,,)342!54(9!2B6:+)(*H!$%!'+,-! [63,+)+6+2!5(1!1U1H!$L!'+,-! [(4=2!5(1!721)=:+)=2!(5!1U1H!$L!'+,-! [2':1:+2!:*7!)*+2<1:+2H!$Q!'+,-! [(4=2!5(1!U]H!K!'+,!$?"!)5!+82>!7(*M+!'6+!)*!5$1-! ! $"\!'+,-!
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