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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "cuploscc.mws " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 26 "coupled oscillator
s 7/8/96" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
63 "This 'c' version will  do two coupled oscillators between walls" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "It also \+
teases out the normal modes (x2/x1 at each mode freq)" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "L:= m1* v1^2/2  +m2
*v2^2/2- k1* x1^2 / 2 -kc*(x2-x1)^2/2-k1*x2^2/2 ;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "d1:=diff(L,v1 );  d2:=diff(L,v2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "p1:=subs(v1=v1(t),d1); p2:=s
ubs(v2=v2(t),d2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "First do the
 derivatives, then tell it you have a function of t." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "d1:=diff(p1,t)-di
ff(L,x1)=0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 77 "Same thing for derivative with respect to x1. First the
 derivative wrt to x1," }}{PARA 0 "" 0 "" {TEXT -1 40 "then tell it th
at x1 is a function of t." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 28 "d2:=diff(p2,t)-diff(L,x2)=0;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f1:=subs(x1=x1(t),x2=x2(t),v1(t)=di
ff(x1(t),t),d1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f2:=sub
s(x1=x1(t),x2=x2(t),v2(t)=diff(x2(t),t),d2);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "h1:=subs(v1(t)=diff(x1(t),t),f1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "h2:=subs(v1(t)=diff(x1(t),t),f2);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "z:=lhs(h1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 50 "Solve for the simple harmonic oscillator \+
equation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "sol:=dsolve(\{h1,x1(0)
=0,D(x1)(0)=1,h2,x2(0)=0,D(x2)(0)=0\},\{x1(t),x2(t)\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sb:=\{m1=1,m2=1,k1=1,kc=1/10\};" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sol1:=subs(sb,sol);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q1:=simplify(evalf(rhs(sol1[
1])));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "q2:=simplify(eval
f(rhs(sol1[2])));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot(
\{q1,q2\},t=0..50,title=`coupled oscillators`);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "hx:=subs(diff(x1(t),t,t)=-omega^2*x1(t),h1);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x21:=solve(hx,x2(t));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "hy:=subs(diff(x2(t),t,t)=-o
mega^2*x2(t),h2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "x22:=s
olve(hy,x2(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "eqn2:= r
*x1(t) = x22;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ratio1:=so
lve(eqn2,r);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "This puts us in p
osition to substitute frequencies and find the mode configurations!" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "so
lfreq:=solve(x22=x21,omega);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 17 "frq1:=solfreq[1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "
frq2:=solfreq[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "omega1
:=evalf(subs(sb,frq1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "
omega2:=evalf(subs(sb,frq2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "mode2x2_over_x1:=subs(sb,omega=omega2,ratio1);" }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 0 21 116 "In mode 2, the masses move together, and \+
the center spring is not compressed or expanded. The amplitude ratio i
s 1:1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "mode1x2_over_x1:=s
ubs(sb,omega=omega1,ratio1);" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 
128 "In mode 1, the masses move oppositely, compressing the spring, an
d operating at a higher frequency. The amplitude ratio is -1:1." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "The 'tran
sfer time' from one oscillator to the other should be argued from " }}
{PARA 0 "" 0 "" {TEXT -1 109 "'beats', where the plot looks just the s
ame.  Want 1/4 of the 'period' of the difference frequency, (f1-f2)/2
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Mario
n problem 12-5, different masses, same k" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "solm:=solve((-m*x+2*
k)*(-M*x+2*k) = k^2,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
A_B_ratio:=(-m*x+2*k)/k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 
"rat1:=subs(x=solm[1],A_B_ratio);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "rs:=simplify(rat1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "subs(M=10*m,rs);" }}}{EXCHG {PARA 0 "> " 0 "" 
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