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{SECT 0 {EXCHG {PARA 207 "" 0 "" {TEXT 220 33 "Two Pendulums Coupled b
y a Spring" }}{PARA 256 "" 0 "" {TEXT -1 29 "by Jason Schattman, Maple
soft" }}{PARA 256 "" 0 "" {TEXT -1 26 "applications@maplesoft.com" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 221 
172 "We model the motion of two identical pendulums swinging in parall
el planes, attached by a spring.  We describe the motion by the pendul
ums' angles of deflection over time, " }{XPPEDIT 2 0 "theta(t)" "6#-%&
thetaG6#%\"tG" }{TEXT 221 5 " and " }{XPPEDIT 2 0 "alpha(t)" "6#-%&alp
haG6#%\"tG" }{TEXT 221 3 ".  " }}{PARA 208 "" 0 "" {TEXT -1 0 "" }}
{PARA 208 "" 0 "" {TEXT 221 42 "We assume that the pendulums swing in \+
the " }{TEXT 222 4 "x-z " }{TEXT 221 24 "plane, their hinges are " }
{TEXT 222 1 "d" }{TEXT 221 188 " units apart on the y-axis, they have \+
unit length and unit mass, and their mass is concentrated at the ends.
  Pendulum 1 swings about the origin, and pendulum 2 swings about the \+
point (0, " }{TEXT 222 1 "d" }{TEXT 221 47 ", 0).  We assume the sprin
g has natural length " }{TEXT 222 1 "d" }{TEXT 221 3 ".  " }}{PARA 
208 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Initi
alizations" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 223 8 "restart;" 
}}{PARA 208 "> " 0 "" {MPLTEXT 1 223 95 "with(plots):with(plottools):w
ith(LinearAlgebra):with(VectorCalculus): interface(showassumed=0):" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "
Assume all values are real, and positive, if appropriate.  This will h
elp Maple simplify the formulas." }}}{EXCHG {PARA 208 "> " 0 "" 
{MPLTEXT 1 223 51 "assume( theta(t)::real, alpha(t)::real, t>0, d>0 );
" }}}{EXCHG {PARA 208 "" 0 "" {TEXT -1 0 "" }}{PARA 208 "" 0 "" {TEXT 
221 44 "The radial vectors of the pendulums at time " }{TEXT 222 1 "t
" }{TEXT 221 1 "." }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 223 113 "r
adius1 := Vector([ sin(theta(t)), 0, -cos(theta(t)) ]);\nradius2 := Ve
ctor([ sin(alpha(t)), d, -cos(alpha(t)) ]);" }}}{EXCHG {PARA 208 "" 0 
"" {TEXT -1 0 "" }}{PARA 208 "" 0 "" {TEXT 221 11 "The vector " }
{TEXT 224 1 "r" }{TEXT 221 26 " between the ends at time " }{TEXT 222 
3 "t, " }{TEXT 221 103 "and the distance.  Note that the distance form
ula can be distilled down to just one trigonometric term." }}}{EXCHG 
{PARA 208 "> " 0 "" {MPLTEXT 1 223 107 "r := radius2 - radius1;\ndist0
 := Norm( r, 2 );\ndist1 := simplify( dist0 );\ndist  := combine( dist
1, trig );" }}}{EXCHG {PARA 208 "" 0 "" {TEXT -1 0 "" }}{PARA 208 "" 
0 "" {TEXT 221 41 "The force of the spring on pendulum 1 is " }
{XPPEDIT 18 0 "k*(dist - d)" "6#*&%\"kG\"\"\",&%%distGF%%\"dG!\"\"F%" 
}{TEXT 221 24 ", by Hooke's Law, where " }{TEXT 258 1 "k" }{TEXT 221 
73 " is the spring stiffness.  The direction of the force is the unit \+
vector " }{XPPEDIT 18 0 "r/dist" "6#*&%\"rG\"\"\"%%distG!\"\"" }{TEXT 
221 2 ". " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 223 40 "springForc
e1 := k * (dist-d) * (r/dist);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT -1 
0 "" }}{PARA 208 "" 0 "" {TEXT 221 108 "The force of the spring on pen
dulum 2 is equal and opposite to the force on pendulum 1, by Newton's \+
3rd Law." }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 223 30 "springForce
2 := -springForce1;" }}}{EXCHG {PARA 208 "" 0 "" {TEXT -1 0 "" }}
{PARA 208 "" 0 "" {TEXT 221 40 "Gravity acts downwards in the z-plane.
  " }{TEXT 256 1 "m" }{TEXT 221 60 " is the mass of the balls at the e
nds of the pendulums, and " }{TEXT 257 1 "g" }{TEXT 221 32 " is the ac
celeration of gravity." }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 223 
82 "gravityForce1 := Vector([ 0, 0, -m*g ]); \ngravityForce2 := Vector
([ 0, 0, -m*g ]);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT -1 0 "" }}{PARA 
208 "" 0 "" {TEXT 221 356 "The vector sum of these forces induces a to
rque about the hinges.  The general definition of torque is r x F, but
 in this case, the pendulums have unit length, so we can greatly simpl
ify the formulas by avoiding the cross product altogether.  But to do \+
this, we need to work with the vectors tangent to the pendulums' paths
 instead of their radial vectors." }}{PARA 208 "" 0 "" {TEXT -1 0 "" }
}{PARA 208 "" 0 "" {TEXT 221 76 "Because the motion is circular, the t
angential vectors are perpendicular to " }{TEXT 222 7 "radius1" }
{TEXT 221 5 " and " }{TEXT 222 7 "radius2" }{TEXT 221 37 ".  There are
 two such vectors in the " }{TEXT 261 3 "x-z" }{TEXT 221 15 " plane, n
amely " }{XPPEDIT 18 0 "`<,>`(cos(theta(t)),0, sin(theta(t)));" "6#-%$
<,>G6%-%$cosG6#-%&thetaG6#%\"tG\"\"!-%$sinG6#-F*6#F," }{TEXT 221 5 " a
nd " }{XPPEDIT 18 0 "<-cos(theta(t)), 0, -sin(theta(t))>" "6#-%$<,>G6$
,$-%$cosG6#-%&thetaG6#%\"tG!\"\"6$\"\"!,$-%$sinG6#-F+6#F-F." }{TEXT 
221 105 ", which are negatives of each other.  We choose the one that \+
points in the direction of increasing angle." }}}{EXCHG {PARA 208 "> \+
" 0 "" {MPLTEXT 1 223 132 "tangentialVector1 := Vector([ cos(theta(t))
, 0, sin(theta(t)) ]); \ntangentialVector2 := Vector([ cos(alpha(t)), \+
0, sin(alpha(t)) ]);" }}}{EXCHG {PARA 208 "" 0 "" {TEXT -1 0 "" }}
{PARA 208 "" 0 "" {TEXT 221 163 "The torque on each pendulum about its
 hinge is the component of the net force in the tangential direction, \+
that is, the dot product of the force and the direction." }}}{EXCHG 
{PARA 208 "> " 0 "" {MPLTEXT 1 223 125 "torque1 := (gravityForce1 + sp
ringForce1) . tangentialVector1;\ntorque2 := (gravityForce2 + springFo
rce2) . tangentialVector2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 243 "To compute the motion of the system, we \+
solve the following system of differential equations, which is the ana
logue of Newton's 2nd law for rotations.  We have unit mass and length
, so we do not include the moments of inertia in the equations." }}}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 223 169 "DESystem := \{diff(thet
a(t),t,t)=torque1, \n             diff(alpha(t),t,t)=torque2, \n      \+
       theta(0)=theta0, alpha(0)=0,\n             D(theta)(0)=0, D(alp
ha)(0)=0\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 30 "Specify unit mass and gravity." }}}{EXCHG {PARA 208 "> \+
" 0 "" {MPLTEXT 1 223 11 "m:=1; g:=1;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "A procedure to animate th
e motion.  Enter the system DEsys, the initial deflection T0 of pendul
um 1" }{TEXT -1 59 ", the distance between the hinges dd, the spring s
tiffness " }{TEXT 260 1 "K" }{TEXT -1 27 ", and the number of swings \+
" }{TEXT 259 1 "n" }{TEXT -1 17 " to animate over." }}}{EXCHG {PARA 
208 "> " 0 "" {MPLTEXT 1 223 1464 "DrawPendulum := proc( DEsys, T0, dd
, K, n )\n\nlocal t,finalSys,sol,T,A,framesPerCycle,increment,\n      \+
x1,x2,z1,z2,post1,post2,L0,L1,L2,L3,C,i,frame;\n\nfinalSys  :=  subs( \+
\{d=dd, k=K, theta0=T0\}, DEsys );\n\nsol := dsolve( finalSys, numeric
, range=0..2*n*Pi, output=listprocedure) ;\n\nT := rhs(sol[4]); #the n
umerical solution theta(t) returned by dsolve\nA := rhs(sol[2]); #the \+
numerical solution alpha(t) returned by dsolve\n\n#The static backgrou
nd\n\nL0 := line( [0,-1,0], [0,dd+1,0], thickness=4, color=black ):\np
ost1 := line( [0,-1,0], [0,-1,-2], thickness=4, color=black ):\npost2 \+
:= line( [0,dd+1,0], [0,dd+1,-2], thickness=4, color=black ):\nground \+
:= plot3d( -2, x=-1..1,y=-1..dd+1, grid=[15,15], color=grey ):\nC0 := \+
pointplot3d( \{[0,0,0], [0,dd,0]\}, symbolsize=20, symbol=circle, colo
r=red ):\n\nframesPerCycle := 30:\nincrement := 2*Pi/framesPerCycle;\n
\nfor i from 1 to framesPerCycle*n do\n  t := increment*i;\n  x1 := si
n(T(t)); \n  z1 := -cos(T(t));\n  x2 := sin(A(t)); \n  z2 := -cos(A(t)
);\n  L1 := line( [0,0,0], [x1, 0, z1], thickness=2, color=blue ):\n  \+
L2 := line( [0,dd,0], [x2, dd, z2], thickness=2, color=blue ):\n  L3 :
= line( [x1, 0, z1], [x2, dd, z2], thickness=2, color=green ):\n  C :=
 pointplot3d( \{[x1,0,z1], [x2,dd,z2]\}, symbolsize=20, symbol=circle,
 color=red ):\n  frame[i] := display( L0,L1,L2,L3,C,C0,ground,post1,po
st2 ):\nend do:\n\ndisplay( seq( frame[i], i=1..framesPerCycle*n ), \n
         insequence=true, axes=none, scaling=unconstrained );\nend pro
c:\n" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 223 43 "DrawPendulum( D
ESystem, Pi/2, 2, 1.5, 6 ); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{PARA 209 "" 0 "" {TEXT -1 0 "" }}{PARA 210 "" 0 "" {TEXT 
-1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }