{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 260 "Courier" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 " " 0 1 0 0 0 0 0 1 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 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } 0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 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 } {PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 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 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 " " 0 1 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 } {PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 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 }{PSTYLE "" 13 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 13 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 39 "Dynamics Class 2:\nxxxx xxxxxxxxxxxxxxxxx" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 18 "xXXXXXXXXX XXXXXXXX" }}}{EXCHG {PARA 257 "" 0 "" {TEXT 257 34 "(Version 2.0 DRAFT , February 2000)" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 16 "Setting up Ma ple" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Please run the following c ommand to load in the \"plots\" package (output suppressed):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "with(plots):" }{XPPEDIT 18 0 "" "6#%#%?G" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 134 "To save typing out all the code for this class, we recommend that you download an electronic version of this worksheet from the WWW at" } {TEXT -1 2 " " }{TEXT 260 64 "http://www.maths.surrey.ac.uk/personal/ st/M.Bartuccelli/dynamics" }{TEXT 259 0 "" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 46 "1. The pendulum with vert ically-moving support" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 261 46 "SETUP / LAGRANGIAN / EQUATIONS OF MOTION ....." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "We set up the equations in Maple as follo ws" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "vpendEqns:= diff(the ta(t),t)=thetadot(t),\n diff(thetadot(t),t)=\n -(1/l)*(g+a*omega^2*c os(omega*t))*sin(theta(t));" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*vpendEqnsG6$/-%%diffG6$-%&thetaG6#%\"tGF--%)thetadot GF,/-F(6$F.F-,$*&*&,&%\"gG\"\"\"*(%\"aGF8)%&omegaG\"\"#\"\"\"-%$cosG6# *&F%\"lG!\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "and then set the par ameters:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "g:=1.0: l:=10: \+ a:=2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "and some initial conditions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "pendInits:= theta(0)=Pi +0.1: thetadot(0):=0.0:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "an d a value for " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 1 " " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "omega:= 3.85:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "We run ds olve to generate a solution" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sol1:= dsolve(\{vpendEqns,pendInits\},\n \{theta(t),thetadot(t)\}, numeric);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1GR6 #%(rkf45_xG6'%\"iG%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoCopyright~(c)~19 93~by~the~University~of~Waterloo.~All~rights~reserved.G6\"C&>8&-%&eval fG6#9$@$52-%$absG6#,$F3!\"\"-F<6#,&&%,loc_controlG6#\"\"#\"\"\"F3F?4-% 'memberG6$&FD6#\"\"'<*FF!\"#FGF?$F?\"\"!$FPFR$FFFR$FGFRC%>FD-%%copyG6# =F06#;FG\"#LE\\[lB\"#KFR\"#9FR\"#8FR\"#7FR\"#?$\"+aEfTK!\"*\"#=FR\"#:F R\"#CFRFFFR\"#JFR\"#EFR\"#>FR\"\"*\"%+5\"#HFR\"#BFR\"#FFR\"\")\"&++$FG FFFhnFR\"#IFR\"\"$FR\"\"($FGFao\"\"%$FG!\")\"#DFR\"#@FR\"\"&Fdp\"#;FR \"#5FR\"#%'loc_y0G-FY6#=F06#;FGFFE\\[l#FFFRF GF_o>%'loc_y1G-FY6#=F0FdqE\\[l!@$0F;FRC$>&FD6#F`pF3@%1%'DigitsG-%'eval hfG6#FerC$>8%-%*traperrorG6#-Fgr6#-%=dsolve/numeric_solnall_rkf45G6,%& loc_FG-%$varG6#FD-Ffs6#F`q-Ffs6#Fhq-Ffs6#%'loc_F1G-Ffs6#%'loc_F2G-Ffs6 #%'loc_F3G-Ffs6#%'loc_F4G-Ffs6#%'loc_F5G-Ffs6#%)loc_workG@$/F[s%*laste rrorGC%>8'-%+searchtextG6$.Fgr-%(convertG6$-%#opG6$FG7#F[s%%nameG>8(-F eu6$.%)hardwareGFhu@%50FcuFR0FavFR-Fbs6,FdsFDF`qFhqF^tFatFdtFgtFjtF]u- %&ERRORG6#F[sFjv7$/%\"tGF7-%$seqG6$/&%$ordG6#,&8$FGFGFG&F`q6#Fjw/FjwFe qF06%FDF`qFhqF0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "And then use odeplot to plot this orbit in the phase plane" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "odeplot(sol1,[the ta(t),thetadot(t)],0..100,numpoints=400);" }{TEXT -1 0 "" }}{PARA 13 " " 1 "" {INLPLOT "6#-%'CURVESG6$7]dl7$$\"+aEfTK!\"*\"\"!7$$\"1^N!=)Qe]K 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A movie is especially useful for this. The following (not part icularly optimised) code will produce one for us. First, another packa ge is required:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(geo metry):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "We define a line for the vertical axis along which the pe ndulum support will oscillate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "line(vline,[point(A,0,-1.2*l),point(B,0,1.2*l)]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The follo wing procedure will create a \"frame\" for the movie at a particular v alue of " }{TEXT 264 1 "t" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 322 "movieframe:=proc(t)\n zval:=a*cos(omega*t); \n \+ thetaval:=rhs(select(has,sol1(t),theta)[1]);\n point(support,0,zval); \n point(mass,l*sin(thetaval),zval-l*cos(thetaval));\n segment(rod,[ support,mass]);\n draw([vline,support,mass,rod],\n view=[-1.2*l..1 .2*l,-1.2*(l+a)..1.2*(l+a)],\n axes=none,scaling=constrained);\nend :" }}{PARA 7 "" 1 "" {TEXT -1 44 "Warning, `zval` is implicitly declar ed local" }}{PARA 7 "" 1 "" {TEXT -1 48 "Warning, `thetaval` is implic itly declared local" }}}{EXCHG {PARA 260 "" 0 "" {TEXT -1 0 "" }} {PARA 259 "" 1 "" {TEXT -1 34 "Let's test \"movieframe\" for a few " } {TEXT 265 1 "t" }{TEXT -1 8 "-values:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "movieframe(0);" }}{PARA 13 "" 1 "" {INLPLOT "6)-%'POI NTSG6%7$\"\"!$\"\"#F'-%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBG$\"*++++\"!\" )F'F'-F$6%7$$!+0\"F4F*F.-%'CURVESG6%7$F&F7F*F.-F>6%7 U7$F'$\"$W\"!\"\"7$F'$\"+++S#Q\"F47$F'$\"+++![K\"F47$F'$\"+++?n7F47$F' $\"+++g47F47$F'$\"++++_6F47$F'$\"+++S%4\"F47$F'$\"+++!o.\"F47$F'$\"+++ +#z*!\"*7$F'$\"++++;#*Fjn7$F'$\"++++S')Fjn7$F'$\"++++k!)Fjn7$F'$\"++++ )[(Fjn7$F'$\"++++7pFjn7$F'$\"++++OjFjn7$F'$\"++++gdFjn7$F'$\"++++%=&Fj n7$F'$\"++++3YFjn7$F'$\"*++?.%F47$F'$\"*++gX$F47$F'$\"*+++)GF47$F'$\"* ++SI#F47$F'$\"*++!G " 0 "" {MPLTEXT 1 0 14 "movi eframe(3);" }{TEXT -1 0 "" }}{PARA 13 "" 1 "" {INLPLOT "6)-%'POINTSG6% 7$\"\"!$\"+_-\"H0\"!\"*-%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBG$\"*++++\"! \")F'F'-F$6%7$$!+JPF=G!#5$\"+8Q*[5\"F5F+F/-%'CURVESG6%7$F&F8F+F/-F?6%7 U7$F'$\"$W\"!\"\"7$F'$\"+++S#Q\"F57$F'$\"+++![K\"F57$F'$\"+++?n7F57$F' $\"+++g47F57$F'$\"++++_6F57$F'$\"+++S%4\"F57$F'$\"+++!o.\"F57$F'$\"+++ +#z*F*7$F'$\"++++;#*F*7$F'$\"++++S')F*7$F'$\"++++k!)F*7$F'$\"++++)[(F* 7$F'$\"++++7pF*7$F'$\"++++OjF*7$F'$\"++++gdF*7$F'$\"++++%=&F*7$F'$\"++ ++3YF*7$F'$\"*++?.%F57$F'$\"*++gX$F57$F'$\"*+++)GF57$F'$\"*++SI#F57$F' $\"*++!G " 0 "" {MPLTEXT 1 0 14 "movieframe(6);" }} {PARA 13 "" 1 "" {INLPLOT "6)-%'POINTSG6%7$\"\"!$!+3+!Q\"*)!#5-%&STYLE G6#%%LINEG-%'COLOURG6&%$RGBG$\"*++++\"!\")F'F'-F$6%7$$\"+!fe!>7!\"*$\" +tm.M!*F;F+F/-%'CURVESG6%7$F&F8F+F/-F?6%7U7$F'$\"$W\"!\"\"7$F'$\"+++S# Q\"F57$F'$\"+++![K\"F57$F'$\"+++?n7F57$F'$\"+++g47F57$F'$\"++++_6F57$F '$\"+++S%4\"F57$F'$\"+++!o.\"F57$F'$\"++++#z*F;7$F'$\"++++;#*F;7$F'$\" ++++S')F;7$F'$\"++++k!)F;7$F'$\"++++)[(F;7$F'$\"++++7pF;7$F'$\"++++OjF ;7$F'$\"++++gdF;7$F'$\"++++%=&F;7$F'$\"++++3YF;7$F'$\"*++?.%F57$F'$\"* ++gX$F57$F'$\"*+++)GF57$F'$\"*++SI#F57$F'$\"*++!G " 0 "" {MPLTEXT 1 0 81 "display([seq(movieframe(n/4) ,n=0..1000)],\n insequence=true, scaling=constrained);" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 13 "" 1 "" {INLPLOT "6$-%(ANIMATEG6ein7)-%'POINTSG6%7$\"\"!$\"\" #F+-%&STYLEG6#%%LINEG-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-F(6%7$$!+0\"F8F.F2-%'CURVESG6%7$F*F;F.F2-FB6%7U7$F+$\"$W\"!\"\"7 $F+$\"+++S#Q\"F87$F+$\"+++![K\"F87$F+$\"+++?n7F87$F+$\"+++g47F87$F+$\" ++++_6F87$F+$\"+++S%4\"F87$F+$\"+++!o.\"F87$F+$\"++++#z*!\"*7$F+$\"+++ +;#*F^o7$F+$\"++++S')F^o7$F+$\"++++k!)F^o7$F+$\"++++)[(F^o7$F+$\"++++7 pF^o7$F+$\"++++OjF^o7$F+$\"++++gdF^o7$F+$\"++++%=&F^o7$F+$\"++++3YF^o7 $F+$\"*++?.%F87$F+$\"*++gX$F87$F+$\"*+++)GF87$F+$\"*++SI#F87$F+$\"*++! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Since seq only goes up i n integer steps, which does not give enough time resolution, we rescal e time so as to get 100 frames for " }{TEXT 266 1 "t" }{TEXT -1 11 " = 0 up to " }{TEXT 267 2 "t " }{TEXT -1 5 "= 25." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 8 "Problems" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "1. Plot \+ a few orbits in phase space with initial conditions close to the origi n; give the parameter " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 102 " a small value (between 1.0 and 2.0), so as the motion is stab le about the downward vertical (i.e. in " }{TEXT 263 2 "xy" }{TEXT -1 22 "-space). Now increase " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" } {TEXT -1 68 " by steps of about 0.25 and find the smallest value of o mega which " }{TEXT 262 12 "destabilises" }{TEXT -1 28 " the motion of the pendulum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "2. Now examine the stability of the UPWARD vertical \+ equilibrium of the pendulum (theta=" }{XPPEDIT 18 0 "pi;" "6#%#piG" } {TEXT -1 58 ", thetadot=0); more precisely, find the smallest value of " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 10 " (call it " } {XPPEDIT 18 0 "omega[stable];" "6#&%&omegaG6#%'stableG" }{TEXT -1 83 " ) such that the motion about the equilibrium point is stablilised for \+ any value of " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" }{TEXT -1 13 " lar ger than " }{XPPEDIT 18 0 "omega[stable];" "6#&%&omegaG6#%'stableG" } {TEXT -1 35 ". Hint: start with high values of " }{XPPEDIT 18 0 "omeg a;" "6#%&omegaG" }{TEXT -1 91 " (say 20) for which the upward motion o f the pendulum is already stable, and then decrease " }{XPPEDIT 18 0 " omega;" "6#%&omegaG" }{TEXT -1 13 " from there. " }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 66 "2. The double pend ulum -- with vertically-moving support\n?????????" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} }{MARK "36 3 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }