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{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 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "with(VariationalCalculus):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 15 "with(ODEtools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "with( plots ): with( plottools ): " }}{PARA 7 "" 1 "" {TEXT 
-1 50 "Warning, the name changecoords has been redefined\n" }}{PARA 7 
"" 1 "" {TEXT -1 59 "Warning, the names arrow and translate have been \+
redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "PDEtools[dec
lare]( (theta,phi,psi)(t), prime=t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#%dpderivatives~with~respect~to:~t~of~functions~of~one~variable~will~
now~be~displayed~with~'~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%J~theta(
t)~will~now~be~displayed~as~theta~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%F~phi(t)~will~now~be~displayed~as~phi~G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%F~psi(t)~will~now~be~displayed~as~psi~G" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{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 "" {TEXT 
-1 0 "" }{TEXT 257 51 "Here we define our position vectors and velocit
ies." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 257 "x[1]:=l[1]*sin(the
ta(t)):\ny[1]:=-l[1]*cos(theta(t)):\nx[2]:=-l[2]*sin(theta(t)):\ny[2]:
=l[2]*cos(theta(t)):\nx[4]:=x[1]-l[4]*sin(theta(t)+phi(t)):\ny[4]:=y[1
]+l[4]*cos(theta(t)+phi(t)):\nx[3]:=x[2]+l[3]*sin(theta(t)-psi(t)):\ny
[3]:=y[2]-l[3]*cos(theta(t)-psi(t)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 116 "R[1]:=vector(2,[x[1],y[1]]):\nR[2]:=vector(2,[x[2],y
[2]]):\nR[4]:=vector(2,[x[4],y[4]]):\nR[3]:=vector(2,[x[3],y[3]]):\n" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "`R[1]'` := map(diff,R[1]
,t):\n`R[2]'` := map(diff,R[2],t):\n`R[4]'` := map(diff,R[4],t):#the V
elocity vectors\n`R[3]'`:=map(diff,R[3],t):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }{TEXT 258 92 "Next, we calculate the kinetic and pote
ntial energies of the system and find the Lagrangian." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "T:=combine(simplify(collect(m[1]/2
*innerprod(`R[4]'`,`R[4]'`)+m[2]/2*innerprod(`R[3]'`,`R[3]'`),[l[1],l[
2],l[3],l[4],m[1],m[2],diff(theta(t),t),diff(phi(t),t),diff(psi(t),t),
cos(psi(t)),sin(psi(t))])),trig):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 159 "U:=collect(simplify(expand(m[1]*g*y[4]+m[2]*g*y[3]))
,[l[1],l[2],l[3],l[4],m[1],m[2],diff(theta(t),t),diff(phi(t),t),diff(p
si(t),t),cos(psi(t)),sin(psi(t)),g]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 129 "L:=simplify(collect(T-U,[l[1],l[2],l[3],l[4],m[1],m[
2],diff(theta(t),t),diff(phi(t),t),diff(psi(t),t),cos(psi(t)),sin(psi(
t))])):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 64 "Using \+
the EulerLagrange package to find our equations for motion" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Eul:=remove(has,EulerLagrange(L,t,[
theta(t),phi(t),psi(t)]),K[1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 44 "eq1:=op(select(has,Eul,(diff(psi(t),t)^2))):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 70 "eq2:=op(select(has,(remove(has,Eul,(diff(
psi(t),t)^2))),sin(psi(t)))):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 70 "eq3:=op(remove(has,(remove(has,Eul,(diff(psi(t),t)^2))),sin(psi(
t)))):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 49 "Setting
 up the initial conditions and Parameters." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 87 "INITS:=\{theta(0)=3*Pi/4,D(theta)(0)=0,phi(0)=Pi/4,
D(phi)(0)=0,psi(0)=Pi/4,D(psi)(0)=0\}:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "PARAM:=[g=9.8,l[1]=10,l[2]=100,l[3]=100,l[4]=21,m[1]=
1000,m[2]=1]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 74 "
Evaluating the general equation for motion with respect to our paramet
ers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sys:=eval([eq1,eq2,
eq3],PARAM):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 84 "W
e use a numeric ODE solver to approximate the change in theta and thet
a' over time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sol:=dsolv
e([op(sys),op(INITS)],numeric,output=listprocedure):" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 38 "Plotting time versus angular ve
locity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "odeplot(sol,[t,
diff(psi(t),t)],1..3,numpoints=1000,labels=[\"times(sec)\",\"Psi'(deg/
s)\"],labeldirections=[horizontal,vertical],font=[TIMES,ROMAN,20]):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "odeplot(sol,[t,diff(theta
(t),t)],t=1..3,numpoints=1000,labels=[\"time(sec)\",\"Theta'(deg/s)\"]
,labeldirections=[horizontal,vertical],font=[TIMES,ROMAN,20]):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "noffm:=23:" }}}{PARA 13 "" 
1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 264 51 "The animation and/or the strobe pl
ot for the model." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "divs:=1
0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 806 "for i from 0 to noff
m do\nx1:=eval(l[1]*sin(rhs(sol[6](i/divs))),PARAM):\ny1:=eval(-l[1]*c
os(rhs(sol[6](i/divs))),PARAM):\nx2:=eval(-l[2]*sin(rhs(sol[6](i/divs)
)),PARAM):\ny2:=eval(l[2]*cos(rhs(sol[6](i/divs))),PARAM):\nx3:=eval(x
2+l[3]*sin(rhs(sol[6](i/divs))-rhs(sol[4](i/divs))),PARAM):\ny3:=eval(
y2-l[3]*cos(rhs(sol[6](i/divs))-rhs(sol[4](i/divs))),PARAM):\nx4:=eval
(x1-l[4]*sin(rhs(sol[6](i/divs))+rhs(sol[2](i/divs))),PARAM):\ny4:=eva
l(y1+l[4]*cos(rhs(sol[6](i/divs))+rhs(sol[2](i/divs))),PARAM): rod1[i]
:=line([x2,y2],[x1,y1]):\nrod3[i]:=line([x2,y2],[x3,y3]):\nrod2[i]:=li
ne([x1,y1],[x4,y4]):\n\nstructure[i]:=polygon([[-20,-70],[0,0],[20,-70
],[-20,-70]]):\nma1[i]:=disk([x4,y4],2,colour=red):\nma2[i]:=disk([x3,
y3],1,colour=blue):\nanima[i]:=display(\{structure[i],rod1[i],rod2[i],
rod3[i],ma1[i],ma2[i]\}):\nend 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 558 469 469 
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