{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }2 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "V\344r\344htelev\344 jous isysteemi " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Jousen puristumista ja venymist\344 voidaan kuvata varsin yksin kertaisella matemaattisella mallilla" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "m*diff(x,`$`(t,2)) = -k*x;" "6#/*&% \"mG\"\"\"-%%diffG6$%\"xG-%\"$G6$%\"tG\"\"#F&,$*&%\"kGF&F*F&!\"\"" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "miss\344 " }{TEXT 272 1 "x" }{TEXT -1 42 " on jousen poikk eama tasapainoasemasta ja " }{TEXT 273 1 "k" }{TEXT -1 33 " jouselle t yypillinen jousivakio." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "T\344ss\344 esimerkiss\344 tarkastelun kohteena on \+ oheisen kuvion mukainen jousisysteemi, jossa kolme kappaletta massoilt aan " }{XPPEDIT 18 0 "m[1];" "6#&%\"mG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "m[2];" "6#&%\"mG6#\"\"#" }{TEXT -1 4 " ja " }{XPPEDIT 18 0 "m[3];" "6#&%\"mG6#\"\"$" }{TEXT -1 76 " on liitetty toisiinsa ja kiinnikkeeseen nelj\344ll\344 jousella jousivakioiltaan " }{XPPEDIT 18 0 "k[1];" "6#&%\"kG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k[2]; " "6#&%\"kG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "k[3];" "6#&%\"kG6 #\"\"$" }{TEXT -1 4 " ja " }{XPPEDIT 18 0 "k[4];" "6#&%\"kG6#\"\"%" } {TEXT -1 57 ". 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::::::::::::::::::::`vvPnUDP`uFW@:::::::::::::::::::::::::::::::GXGYed W`H=:::::::::::::::::::::::::::::::pUH`VaufH=::::::::::::::::::::::::: ::::::BjkrfW`H=:::::::::::::::::::::::::::::::j[INiLTacm:::::::::::::: ::::::::::::::::::lV@YNHhoGuVaoS::::::::::::::::::::::::::::::::HRYPlJ uVaoS::::::::::::::::::::::::::::::::tu\\`j>`::: ::::::::::::b:KR]R>]Rr?ir?iB@K ChWhmtF3:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 31 "Kappaleiden paikkakoordinaatit " }{XPPEDIT 18 0 "x [1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[2];" "6#&% \"xG6#\"\"#" }{TEXT -1 4 " ja " }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\" $" }{TEXT -1 434 " ilmoitetaan poikkeamina lepotilasta, yl\366sp\344in positiivisena ja alasp\344in negatiivisena. Newtonin lain mukaan kuhu nkin kappaleeseen vaikuttava voima on toisaalta kappaleen massa kerrot tuna sen kiihtyvyydell\344 eli paikkakoordinaatin toisella aikaderivaa talla, toisaalta kyseess\344 on jousien aiheuttama harmoninen voima. S ysteemin liikeyht\344l\366t ovat siten toisen kertaluvun differentiaal iyht\344l\366it\344, yksi yht\344l\366 jokaista kappaletta kohden. \n " }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "m[1]*diff(x[1],`$`(t,2)) = -k[1]* x[1]+k[2]*(x[2]-x[1])+k[3]*(x[3]-x[1]);" "6#/*&&%\"mG6#\"\"\"F(-%%diff G6$&%\"xG6#F(-%\"$G6$%\"tG\"\"#F(,(*&&%\"kG6#F(F(&F-6#F(F(!\"\"*&&F76# F3F(,&&F-6#F3F(&F-6#F(F;F(F(*&&F76#\"\"$F(,&&F-6#FGF(&F-6#F(F;F(F(" } {TEXT -1 1 "," }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "m[2]*diff(x[2],`$`(t ,2)) = -k[2]*(x[2]-x[1])-k[4]*(x[2]-x[3]);" "6#/*&&%\"mG6#\"\"#\"\"\"- %%diffG6$&%\"xG6#F(-%\"$G6$%\"tGF(F),&*&&%\"kG6#F(F),&&F.6#F(F)&F.6#F) !\"\"F)F>*&&F76#\"\"%F),&&F.6#F(F)&F.6#\"\"$F>F)F>" }{TEXT -1 1 "," }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "m[3]*diff(x[3],`$`(t,2)) = -k[3]*(x[3 ]-x[1])+k[4]*(x[2]-x[3]);" "6#/*&&%\"mG6#\"\"$\"\"\"-%%diffG6$&%\"xG6# F(-%\"$G6$%\"tG\"\"#F),&*&&%\"kG6#F(F),&&F.6#F(F)&F.6#F)!\"\"F)F?*&&F8 6#\"\"%F),&&F.6#F4F)&F.6#F(F?F)F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Tutkitaan jousisysteemin \+ k\344ytt\344ytymist\344 ratkaisemalla differentiaaliyht\344l\366ryhm \344\n" }}{PARA 0 "" 0 "" {TEXT -1 88 "Laskujen aluksi on syyt\344 h \344vitt\344\344 mahdollisista aiemmista laskuista j\344\344neet muutt ujat. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "As etetaan kappaleiden massoille ja jousivakioille numeeriset arvot:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "m[1]:= 2: m[2]:= 1: m[3]:= 1: k[1]:= 4: k[2]:= 2: k[3]:= 2: k[4] := 1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Muodostetaan systeemin differentiaaliyht\344l\366ryhm\344 . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "ryhma:= m[1]*diff(x[1](t), t$2)=-k[1]*x[1](t)+k[2]*( x[2](t)-x[1](t))+k[3]*(x[3](t)-x[1](t)), m[2]*diff(x[2](t), t$2)=-k[2] *(x[2](t)-x[1](t))-k[4]*(x[2](t)-x[3](t)), m[3]*diff(x[3](t), t$2)=-k[ 3]*(x[3](t)-x[1](t))+k[4]*(x[2](t)-x[3](t));" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Ker\344t\344\344n kappa leiden sijaintia osoittavat funktiot listaksi, samoin n\344iden deriva atat. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "funktiot:= x[1](t), x[2](t), x[3](t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "derivaatat:= [D(x[1])(t), D(x[2])(t ), D(x[3])(t)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "Systeemi\344 tarkastellaan siten, ett\344 se sys\344 t\344\344n hetkell\344 " }{TEXT 261 2 "t " }{TEXT -1 230 "= 0 liikkeel le lepotilasta antamalla kullekin kappaleelle jokin alkunopeus. Alkueh dossa annetaan siten paikkakoordinaateille arvo 0 ja nopeuksille eli p aikkakoordinaattien derivaatoille jotkin haluttua liiketilaa vastaavat arvot: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "alkusijainti:= x[1](0)=0, x[2](0)=0, x[3](0)=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "alkunopeus:= [D(x[1])(0)=-10 , D(x[2])(0)=10, D(x[3])(0)=30];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Saatu differentiaaliyht\344l\366ry hm\344 ratkaistaan " }{TEXT 262 5 "Maple" }{TEXT -1 2 "n " }{TEXT 263 6 "dsolve" }{TEXT -1 28 "-komennolla ja sievennet\344\344n." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "liike:= dsolve(\{ryhma, alku sijainti, alkunopeus[]\}, \{funktiot\}):\nsimplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 160 "Tulokses ta n\344hd\344\344n, ett\344 systeemin liiketila muodostuu kolmesta er ilaisesta v\344r\344htelyst\344. N\344iden taajuudet ilmenev\344t laus ekkeissa esiintyvist\344 sinifunktioista, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "sin(2*t);" " 6#-%$sinG6#*&\"\"#\"\"\"%\"tGF(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "sin (sqrt(2)*(sqrt(5)-1)*t/2);" "6#-%$sinG6#**-%%sqrtG6#\"\"#\"\"\",&-F(6# \"\"&F+F+!\"\"F+%\"tGF+F*F0" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "sin(sqrt (2)*(sqrt(5)+1)*t/2);" "6#-%$sinG6#**-%%sqrtG6#\"\"#\"\"\",&-F(6#\"\"& F+F+F+F+%\"tGF+F*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 81 "Seuraava koodi m\344\344rittelee kapp aleen liikkeess\344 tarvittavat ty\366kalut. Kysess\344 on " }{TEXT 271 5 "Maple" }{TEXT -1 29 "lla kirjoitettu ohjelmakoodi." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with (plots): with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1126 "jousi:= proc(hor, ala, yla, paa, jaksot)\n local ampl, jakso;\n ampl:= 3; \n jakso:= (yla-ala-2*paa)/jaksot;\n plot([[hor+ampl*sin (2*Pi*r/jakso),\n r+ala+paa, r=0..jaksot*jakso],\n [hor, ala+r, \+ r=0..paa], \n [hor, yla-r, r=0..paa]], color=black);\nend:\n\nkappa le:= proc(hor, ver, lev, kor, teksti)\n display(rectangle([hor-0.5*le v,\n ver+0.5*kor], [hor+0.5*lev,\n ver-0.5*kor], color=gray),\n \+ textplot([35, ver, teksti])\n ); \nend:\n\njousisyst:= proc(y0, t0 )\n local lev, kor, paa, lepo1, lepo2,\n lepo3, y1, y2, y3;\n lev:= 20; kor:= 10; paa:= 5; lepo1:= 50;\n lepo2:= 150; lepo3:= 100;\n y1 := -lepo1+subs(t=t0/(2*Pi), \n subs(y0, x[1](t)));\n y2:= -lepo2+s ubs(t=t0/(2*Pi),\n subs(y0,x[2](t)));\n y3:= -lepo3+subs(t=t0/(2*P i),\n subs(y0,x[3](t)));\n display(rectangle([-50,0], [50,5],\n \+ color=gray),\n kappale(0, y1, lev, kor, 'm1'), \n kappale(0, y2 , lev, kor, 'm2'), \n kappale(lev/2, y3, lev, kor, 'm3'), \n jou si(0, y1+kor/2, 0, 5, 5),\n jousi(-lev/2, y2+kor/2,\n y1-kor/2 , 5,10),\n jousi(lev/2, y3+kor/2,\n y1-kor/2, 5, 5),\n jous i(lev/2, y2+kor/2, \n y3-kor/2, 5, 5)\n );\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 189 "Animaati o on lista per\344kk\344isi\344 kuvia, jotka on ensin laskettava. Anim aatio k\344ynnistyy viem\344ll\344 hiiren osoitin kuvan p\344\344lle j a valitsemalla hiiren oikealla napilla esiin tulevasta valikosta " } {TEXT 264 1 "A" }{TEXT 256 1 "n" }{TEXT 265 7 "imation" }{TEXT -1 4 " \+ -> " }{TEXT 266 1 "P" }{TEXT 257 1 "l" }{TEXT 267 2 "ay" }{TEXT -1 55 ". Animaatiosta saa jatkuvan valitsemalla em. valikosta " }{TEXT 268 1 "A" }{TEXT 258 1 "n" }{TEXT 269 7 "imation" }{TEXT -1 4 " -> " } {TEXT 259 1 "C" }{TEXT 270 9 "ontinuous" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "displa y(seq(jousisyst(liike, u), u=0..29), insequence=true, axes=NONE, scali ng=CONSTRAINED);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 7 "Teht\344v\344 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 407 "Kiin nostava tilanne syntyy, jos systeemin liikkeess\344 esiintyy vain yksi edell\344 mainituista kolmesta taajuudesta. T\344llaiseen on mahdolli sta p\344\344st\344 valitsemalla alkunopeudet sopivasti. Seuraava lis ta sis\344lt\344\344 kutakin taajuutta vastaavat alkunopeuksien suhtee t. Seuraavat kolme komentoa asettavat alkunopeusasetukseksi kunkin n \344ist\344 vuorollaan. Lukija tutkikoon, millaisia systeemin liikkeit \344 t\344ll\366in syntyy! " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "moodit:=[0, -1, 1], [1/2*(-1 -sqrt(5)), 1, 1], [1/2*(-1+sqrt(5)), 1, 1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "alkunopeus:= zip((x, y)->x=y, subs(t=0, derivaatat ), 15*moodit[1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "alkuno peus:= zip((x, y)->x=y, subs(t=0, derivaatat), 15*moodit[2]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "alkunopeus:= zip((x, y)->x=y , subs(t=0, derivaatat), 15*moodit[3]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 260 24 "SKK & JP & MS 12.07.2001" }}}}{MARK "0 2 0" 94 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }