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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "with(linalg):with(plots):with(DEtoo
ls): #tools for project" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the p
rotected names norm and trace have been redefined and unprotected\n" }
}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been \+
redefined\n" }}{PARA 7 "" 1 "" {TEXT -1 45 "Warning, the name adjoint \+
has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# m
in and max numbers in student id:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "mins:=1; maxs:=8;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%%minsG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%maxsG\"\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 60 "# mass matrix.  We have to make sure we get the uni
ts right." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "# I have decid
ed to use slugs, but other things are ok too." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 41 "# In this case, the mass per floor is :# " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fm:= maxs*2000/32;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#fmG\"$+&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "M:=evalm(fm*array(1..7, 1..7, identity));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"MG-%'matrixG6#7)7)\"$+&\"\"!F+F+F+F+F+7)
F+F*F+F+F+F+F+7)F+F+F*F+F+F+F+7)F+F+F+F*F+F+F+7)F+F+F+F+F*F+F+7)F+F+F+
F+F+F*F+7)F+F+F+F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# K matrix.  Making sur
e the unit are right again." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "# the units of the k values should be lbs/ft" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 15 "sk:= mins*2000;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#skG\"%+?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "K:=mat
rix([[-2*sk,sk,0,0,0,0,0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "     \+
     [sk,-2*sk,sk,0,0,0,0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "    \+
      [0,sk,-2*sk,sk,0,0,0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "   \+
       [0,0,sk,-2*sk,sk,0,0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "  \+
        [0,0,0,sk,-2*sk,sk,0]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " \+
         [0,0,0,0,sk,-2*sk, sk]," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 
"          [0,0,0,0,0,sk , -sk]]);# " }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"KG-%'matrixG6#7)7)!%+S\"%+?\"\"!F,F,F,F,7)F+F*F+F,F,F,F,7)F,F+F*
F+F,F,F,7)F,F,F+F*F+F,F,7)F,F,F,F+F*F+F,7)F,F,F,F,F+F*F+7)F,F,F,F,F,F+
!%+?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "A:=evalm(inverse(M)
&*K);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'matrixG6#7)7)!\")\"
\"%\"\"!F,F,F,F,7)F+F*F+F,F,F,F,7)F,F+F*F+F,F,F,7)F,F,F+F*F+F,F,7)F,F,
F,F+F*F+F,7)F,F,F,F,F+F*F+7)F,F,F,F,F,F+!\"%" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "evals:=array(1..7);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&evalsG-%&arrayG6$;\"\"\"\"\"(7\"" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 27 "evals:=evalf(eigenvals(A));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&evalsG6)$!\"%\"\"!$!+WS'y_\"!\"*$!+'f8s/\"!\")$
!+T>>[<!#5$!+#HsP;(F+$!+&[/`L\"F.$!+mj$3`\"F." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "evals:=sort([evals]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&evalsG7)$!+mj$3`\"!\")$!+&[/`L\"F($!+'f8s/\"F($!+#Hs
P;(!\"*$!\"%\"\"!$!+WS'y_\"F/$!+T>>[<!#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "fmt:=\"%10.5f   %10.5f   %10.5f   \\n\";" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$fmtGQ=%10.5f~~~%10.5f~~~%10.5f~~~|+6\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for i from 1 to 7 do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "  if (i=1) then printf(\"Eigenvalue
      Frequency     Period\\n\") end if;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "  printf(fmt,evals[i], sqrt(-evals[i]), evalf(2*Pi/sq
rt(-evals[i])));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 6 "
" 1 "" {TEXT -1 36 "Eigenvalue      Frequency     Period" }}{PARA 6 "
" 1 "" {TEXT -1 39 " -15.30836      3.91259      1.60589   " }}{PARA 
6 "" 1 "" {TEXT -1 39 " -13.35304      3.65418      1.71945   " }}
{PARA 6 "" 1 "" {TEXT -1 39 " -10.47214      3.23607      1.94161   " 
}}{PARA 6 "" 1 "" {TEXT -1 39 "  -7.16377      2.67652      2.34752   \+
" }}{PARA 6 "" 1 "" {TEXT -1 39 "  -4.00000      2.00000      3.14159 \+
  " }}{PARA 6 "" 1 "" {TEXT -1 39 "  -1.52786      1.23607      5.0832
0   " }}{PARA 6 "" 1 "" {TEXT -1 39 "   -.17482       .41811     15.02
745   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "B:=vector([1,1,1,1,1,1,1]);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"BG-%'vectorG6#7)\"\"\"F)F)F)F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# it may not be clear, but t
he amplitude of the forcing function does" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "# not matter" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "
# in the analysis of resonance.  The factor of E will just scale the" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "# amplitudes" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 68 "# of each floor in the same manner.  So, acc
ordingly, I ignore E and" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 
"# E*omega^2" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "# (Actually I gave \+
E an arbitrary value to get numbers easier to read)" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 7 "E:=500;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"EG\"
$+&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "F0:=evalm(inverse(M)
&*B*E);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F0G-%'matrixG6#7)7#\"\"
\"F)F)F)F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "    #The \+
F0 in the normalized equation (30), page 433" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "Iden:=array(1..7,1..7,identity);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%IdenG-%&arrayG6&%)identityG;\"\"\"\"\"(F)7\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Aleft:=omega->evalm(A + om
ega^2*Iden);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&AleftGf*6#%&omegaG6
\"6$%)operatorG%&arrowGF(-%&evalmG6#,&%\"AG\"\"\"*&)9$\"\"#F1%%IdenGF1
F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "    #the matrix
 function on the left side of (32)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "c:=omega->evalm(-inverse(Aleft(omega))&*F0*omega^2);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cGf*6#%&omegaG6\"6$%)operatorG
%&arrowGF(-%&evalmG6#,$*&-%#&*G6$-%(inverseG6#-%&AleftG6#9$%#F0G\"\"\"
)F:\"\"#F<!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  \+
   #the vector c(omega) in (32)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# c(omega) will \+
be a vector in R^7 and will be a bit ugly." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 70 "# I decided not to print it out in all its glory (s
ome with its norm)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "c(om
ega): #see equation (35) page 433" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot(no
rm(c(omega)),omega=0..6,y=0..(10)," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "       numpoints=200,color=`black`);" }}{PARA 13 "" 1 "" 
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