{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 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 "Warning" -1 7 1 {CSTYLE "" -1 -1 " Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 63 "First we i nitialize all of the packages we'll use in this study" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "restart:\nwith(VariationalCalculus ):\nwith(ODEtools):\nwith(DEtools):\nwith(plots):\nwith(plottools):\nw ith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changec oords has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 59 "Warning, th e names arrow and translate have been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint has been redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace hav e been redefined and unprotected\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "PDEtools[declare]((theta)(t),prime=t):" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%J~theta(t)~will~now~be~displayed~as~theta~G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%dpderivatives~with~respect~to:~t~of~f unctions~of~one~variable~will~now~be~displayed~with~'~G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 51 "Here we define our positi on vectors and velocities." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "x[1]:=l[1]*sin(theta(t)):\ny[1]:=-l[1]*cos(theta(t)):\nx[2]:=-l[2 ]*sin(theta(t)):\ny[2]:=l[2]*cos(theta(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "R[1]:=vector(2,[x[1],y[1]]):\nR[2]:=vector(2,[x[2] ,y[2]]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "`R[1]'`:=map(di ff,R[1],t):\n`R[2]'`:=map(diff,R[2],t):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 92 "Next, we calculate the kinetic and potent ial energies of the system and find the Lagrangian." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "T:=factor(collect(simplify(m[1]/2*innerp rod(`R[1]'`,`R[1]'`)+m[2]/2*innerprod(`R[2]'`,`R[2]'`)),[m[1],m[2],l[1 ],l[2],diff(theta(t),t)])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "U:=simplify(m[1]*g*y[1]+m[2]*g*y[2]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "L:=T-U:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 260 64 "Using the EulerLagrange package to find our equations fo r motion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "EuL:=EulerLagra nge(L,t,theta(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eq:=o p(remove(has,EuL,K[1])):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 259 49 "Setting up the initial conditions and Parameters." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "INITS:=\{theta(0)=3*Pi/4,D(t heta)(0)=0\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "PARAM:=[g= 9.8,l[1]=10,l[2]=100,m[1]=1000,m[2]=1]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 74 "Evaluating the general equation for motio n with respect to our parameters." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "eq2:=eval(eq,PARAM):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 84 "We use a numeric ODE solver to approximat e the change in theta and theta' over time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "sol:=dsolve(\{eq2,op(INITS)\},numeric,output=listp rocedure):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 38 "Plo tting time versus angular velocity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "odeplot(sol,[t,180/Pi*diff(theta(t),t)],0..5,numpoin ts=1000,labels=[\"time(sec)\",\"Angular Speed(Deg/Sec)\"]):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 51 "The animation and/ or the strobe plot for the model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "noffm:=70:\ndivs:=10:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 533 "for i from 0 to noffm do\nx1:=eval(l[1]*sin(rhs(sol[ 2](i/divs))),PARAM):\ny1:=eval(-l[1]*cos(rhs(sol[2](i/divs))),PARAM): \nx2:=eval(-l[2]*sin(rhs(sol[2](i/divs))),PARAM):\ny2:=eval(l[2]*cos(r hs(sol[2](i/divs))),PARAM):\nrod[i]:=line([x1,y1],[x2,y2]):\nstructure [i]:=polygon([[-20,-70],[0,0],[20,-70],[-20,-70]],colour=green):\nma1[ i]:=disk([x1,y1],2,colour=red):\nma2[i]:=disk([x2,y2],1,colour=blue): \n#grnd[i]:=polygon([[-100,-40],[100,-40],[100,-70],[-100,-70]],colour =brown):\nanima[i]:=display(\{structure[i],rod[i],ma1[i],ma2[i]\}):\ne nd do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "display(seq(anima [i],i=0..noffm),scaling=constrained,axes=none,insequence=true);" }} {PARA 13 "" 1 "" {GLPLOT2D 535 490 490 {PLOTDATA 2 "6%-%(ANIMATEG6co7& -%)POLYGONSG6$7U7$$\"+8y1r!*!\"*$\"+6y1rqF.7$$\"+;sHb!*F.$\"+yUt@tF.7$ $\"+NTB3!*F.$\"+'eZ%ovF.7$$\"+&y?1$*)F.$\"+;pJ2yF.7$$\"+t6oB))F.$\"+f^ dM!)F.7$$\"+-=5*o)F.$\"+;$QmC)F.7$$\"+o]+H&)F.$\"+B?;S%)F.7$$\"+#z:fM) F.$\"+(H%47')F.7$$\"+.9sU\")F.$\"+iOsf()F.7$$\"+(REE#zF.$\"+;>s!)))F.7 $$\"+,=5*o(F.$\"+W3=t*)F.7$$\"+T/$eW(F.$\"+8BkN!*F.7$$\"+_)[m>(F.$\"+o 77n!*F.7$$\"+un[XpF.Feo7$$\"+%=0jp'F.$\"+7BkN!*F.7$$\"+DQ.`kF.F[o7$$\" 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