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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 49 "Mass Spring Dashpot (dam
ped harmonic oscillator) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 68 "Finding the general solution of  mx\"+bx'+kx =0,  \+
x(0)=x0, x'(0)=v0, " }}{PARA 0 "" 0 "" {TEXT -1 27 "via the matrix exp
onential." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
62 "Let's convert the second order ODE into a 2x2 linear system by" }}
{PARA 0 "" 0 "" {TEXT -1 49 "the standard trick: x1=x(t), x2=x'(t),  y
ielding:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
24 "vec(x)'(t) =A vec(x)(t)," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 43 "where vec(x) =[x1,x2], (as a column vector)" }}
{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 26 "A = [ \+
 0          1    ]  " }}{PARA 0 "" 0 "" {TEXT -1 23 "       [  -k/m  -
b/m  ]" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 
73 "Setting the ratio k/m = 1  ( k =1, m =1), this leaves one free par
ameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
77 "We need the eigenvalues of A, which are the roots of det( A - lamb
da I ) = 0," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 58 "These end up being    lambda_1 = ( -b + sqrt( b^2 -4) ) /2" }}
{PARA 0 "" 0 "" {TEXT -1 69 "                                 lambda_2
 = ( -b - sqrt(b^2 -4) ) / 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 94 "If   b < 2,  then we have a negative value insi
de the sqrt, and hence complex conjugate roots." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "For notational simplicity
,  set  lambda_1 = -b/2 + i* sqrt( 1 - (b/2)^2 )  := -b/2 + i*w;" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 66 "Here, w \+
will play the role of a \"damped\" or \"retarded\"  frequency." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 86 "The eigen
value lambda_1  has eigenvector  x = [  -b/2 - i*w,  1 ] (as a column \+
vector)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
52 "We then can set up the matrices  P, Rot, and P^(-1):" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The columns of P a
re the Re(x) and Im(x) respectively." }}{PARA 0 "" 0 "" {TEXT -1 64 "T
he matrix Rot is  [  exp(-b/2) cos(w t)   exp(-b/2) sin(w t)  ]" }}
{PARA 0 "" 0 "" {TEXT -1 80 "                                   [ -exp
(b/2) sin(w t)     exp(-b/2) cos(w t) ]" }}{PARA 0 "" 0 "" {TEXT -1 
31 "and P^(-1) is the inverse of P." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 82 "The matrix exponential   is  exp(A t) = \+
P * Rot * P^(-1), which is a 2x2  matrix, " }}{PARA 0 "" 0 "" {TEXT 
-1 60 "and the general solution is    [ x1(t) ]  =  exp(A t) [ x0 ]" }
}{PARA 0 "" 0 "" {TEXT -1 78 "                                        \+
  [ x2(t) ]                     [ v0 ]" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 17 "Let's try it out:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}{PARA 7 "" 1 "" {TEXT -1 
45 "Warning, the name adjoint has been redefined\n" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 
45 "Warning, the name adjoint has been redefined\n" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 16 "b:='b'; w :='w';" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"bGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wGF$" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "P:=matrix([[-b/2, -w],[1,0
]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PG-%'matrixG6#7$7$,$%\"bG#
!\"\"\"\"#,$%\"wGF-7$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "Pinv:=inverse(P);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%PinvG-%'matrixG6#7$7$\"\"!\"\"\"7$,$*&F+F+%\"wG!\"\"F0,$*&%\"bGF+F/
F0#F0\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "Rot:=matrix(
 [[exp(-b/2 *t)*cos(w*t), exp(-b/2 *t)*sin(w*t)],[-exp(-b/2 *t)*sin(w*
t), exp(-b/2*t)*cos(w*t)]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Rot
G-%'matrixG6#7$7$*&-%$expG6#,$*&%\"bG\"\"\"%\"tGF1#!\"\"\"\"#F1-%$cosG
6#*&%\"wGF1F2F1F1*&F+F1-%$sinGF8F17$,$F;F4F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 29 "expAt :=multiply(P,Rot,Pinv);" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#>%&expAtG-%'matrixG6#7$7$,$*&,&*(%\"bG\"\"\"-%$expG6#
,$*&F.F/%\"tGF/#!\"\"\"\"#F/-%$sinG6#*&%\"wGF/F5F/F/F6*(F=F/F0F/-%$cos
GF;F/F7F/F=F7F7,(*(F.F/F0F/F?F/F6*(F=F/F0F/F9F/F/*&#F/F8F/*&*&F,F/F.F/
F/F=F7F/F77$,$*&*&F0F/F9F/F/F=F7F7,&*&F0F/F?F/F/*&#F/F8F/*&*(F0F/F9F/F
.F/F/F=F7F/F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "x0:='x0'; \+
v0:='v0'; xinit:=vector([x0,v0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#x0GF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0GF$" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%&xinitG-%'vectorG6#7$%#x0G%#v0G" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 28 "xsol:=multiply(expAt,xinit);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#>%%xsolG-%'vectorG6#7$,&*&*&,&*(%\"bG\"\"\"-%$ex
pG6#,$*&F.F/%\"tGF/#!\"\"\"\"#F/-%$sinG6#*&%\"wGF/F5F/F/F6*(F=F/F0F/-%
$cosGF;F/F7F/%#x0GF/F/F=F7F7*&,(*(F.F/F0F/F?F/F6*(F=F/F0F/F9F/F/*&#F/F
8F/*&*&F,F/F.F/F/F=F7F/F7F/%#v0GF/F/,&*&*(F0F/F9F/FAF/F/F=F7F7*&,&*&F0
F/F?F/F/*&#F/F8F/*&*(F0F/F9F/F.F/F/F=F7F/F7F/FJF/F/" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 28 "There, now wasn't that nice?" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Let's try some specific
 values of b (and hence w), and some specific initial conditions:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "First, th
e undamped case (b = 0), w = sqrt( 1 - (b/2)^2) = 1;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 40 "b:=0; w :=sqrt( 1-(b/2)^2); x0:=1;v0:=0;
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{XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G
\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#v0G\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "xsol;" }}{PARA 11 "" 1 "" {XPPMATH 
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cp$\"3nH<5J[ic!)F07$Fgep$\"3$*o5HY?5>!)F07$F[gp$\"3P@'y$3uGzzF07$F_hp$
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0 2 9 1 2 2 1.000000 51.000000 24.000000 0 0 "Curve 1" "Curve 2" "Curv
e 3" "Curve 4" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "As you can see
, the oscillations are damped, and dying off inside the envelopes of e
xp(b/2 t) and -exp(b/2 t)!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 32 "Now increase the damping: b = .1" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "b:=.1; w:=sqrt(1-(b/2)^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"\"\"!\"\"" }}{PARA 11 "" 1 "
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{XPPMATH 20 "6#>%#xtG,&*&-%$expG6#,$%\"tG$!+++++]!#6\"\"\"-%$sinG6#,$F
+$\"+y@\\()**!#5F/$\"+X<E1]F.*($\"+++++5!\"*F/F'F/-%$cosGF2F/F/" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vtG,$*&-%$expG6#,$%\"tG$!+++++]!#6
\"\"\"-%$sinG6#,$F+$\"+y@\\()**!#5F/$!+\\BD,5!\"*" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 51 "plot(\{xt,vt,exp(-b/2 *t),-exp(-b/2*t)\},t=0
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