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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 2250" }}{PARA 257 "" 0 "" 
{TEXT -1 18 "Earthquake project" }}{PARA 258 "" 0 "" {TEXT -1 13 "Nove
mber 2001" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
51 "Enter your name here: ?????????????????????????????" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Project 3. Please fill i
n all areas below marked ????? and solve problems 3.1 to 3.6. The prob
lem headers:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 59 " _______     PROBLEM 3.1. BUILDING MODEL FOR AN EARTHQUAKE." }}
{PARA 0 "" 0 "" {TEXT -1 67 " _______     PROBLEM 3.2. TABLE OF NATURA
L FREQUENCIES AND PERIODS." }}{PARA 0 "" 0 "" {TEXT -1 65 " _______   \+
  PROBLEM 3.3. UNDETERMINED COEFFICIENTS STEADY-STATE " }}{PARA 0 "" 
0 "" {TEXT -1 45 "                           PERIODIC SOLUTION." }}
{PARA 0 "" 0 "" {TEXT -1 46 " _______     PROBLEM 3.4. PRACTICAL RESON
ANCE." }}{PARA 0 "" 0 "" {TEXT -1 44 " _______     PROBLEM 3.5. EARTHQ
UAKE DAMAGE." }}{PARA 0 "" 0 "" {TEXT -1 38 " _______     PROBLEM 3.6.
 FIVE FLOORS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "restart:with(plots):with(linalg):" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 38 "3.1. BUILDING MODEL FOR AN EARTHQUAKE." }}{PARA 0 "" 0 
"" {TEXT -1 98 "Refer to the textbook of Edwards-Penney, section 7.4, \+
page 437. Consider a building with 7 floors." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "Let the mass in slugs of each \+
story be m=1000.0 and let the spring constant be k=10000.0  (lbs/foot)
." }}{PARA 0 "" 0 "" {TEXT -1 94 "Define the 7 by 7 mass matrix M and \+
Hooke's matrix K for this system and convert Mx''=Kx into " }}{PARA 0 
"" 0 "" {TEXT -1 74 "the system  x''=Ax where A is defined by textbook
 equation (1) , page 437." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 11 "PROBLEM 3.1" }}{PARA 0 "" 0 "" {TEXT -1 91 "Find t
he eigenvalues of the matrix A to six digits, using the Maple command \+
\"eigenvals(A).\"" }}{PARA 0 "" 0 "" {TEXT -1 85 "Justify in particula
r that all seven eigenvalues are negative by direct computation. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "# Sample \+
Maple code for a model with 4 floors." }}{PARA 0 "" 0 "" {TEXT -1 52 "
# Use maple help to learn about evalf and eigenvals." }}{PARA 0 "" 0 "
" {TEXT -1 72 " A:=matrix([ [-20,10,0,0], [10,-20,10,0], [0,10,-20,10]
, [0,0,10,-10]]);" }}{PARA 0 "" 0 "" {TEXT -1 21 " evalf(eigenvals(A))
;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "# Problem 3.1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PAGEBK }{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "3.2. TABLE OF NATURAL FRE
QUENCIES AND PERIODS." }}{PARA 0 "" 0 "" {TEXT -1 33 "Refer to figure \+
7.4.17, page 437." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 12 "PROBLEM 3.2." }}{PARA 0 "" 0 "" {TEXT -1 121 "Find the na
tural angular frequencies  omega=sqrt(-lambda) for the seven story bui
lding and also the corresponding periods" }}{PARA 0 "" 0 "" {TEXT -1 
68 "2PI/omega, accurate to six digits.  Display the answers in a table
 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "# S
ample code for a  4x3  table." }}{PARA 0 "" 0 "" {TEXT -1 45 "# Use ma
ple help to learn about nops, printf." }}{PARA 0 "" 0 "" {TEXT -1 32 "
 ev:=[-38.3,-33.4,-26.2,-17.9]: " }}{PARA 0 "" 0 "" {TEXT -1 32 " Omeg
a:=lambda -> sqrt(-lambda):" }}{PARA 0 "" 0 "" {TEXT -1 36 " format:=
\"%10.6f  %10.6f  %10.6f\\n\":" }}{PARA 0 "" 0 "" {TEXT -1 79 " seq(pr
intf(format,ev[i],Omega(ev[i]),2*evalf(Pi)/Omega(ev[i])),i=1..nops(ev)
);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "# Problem 3.2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "3.3. UNDETERMINED COEFFICIENTS STE
ADY-STATE PERIODIC SOLUTION." }}{PARA 0 "" 0 "" {TEXT -1 94 "Consider \+
the forced equation  x'=Ax+cos(wt)b  where  b  is a constant vector. T
he earthquake's" }}{PARA 0 "" 0 "" {TEXT -1 89 "ground vibration is  a
ccounted for by the extra term cos(wt)b, which has period  T=2Pi/w." }
}{PARA 0 "" 0 "" {TEXT -1 99 "The solution  x(t)  is  the  7-vector of
 excursions from equilibrium of the corresponding 7 floors." }}{PARA 
0 "" 0 "" {TEXT -1 78 "Sought here is not the general solution which c
ontains the general solution to" }}{PARA 0 "" 0 "" {TEXT -1 94 "to hom
ogeneous problem,  but rather the particular periodic solution with an
gular freqency w, " }}{PARA 0 "" 0 "" {TEXT -1 87 "which is known from
 the theory to have the form x(t)=cos(wt)c for some vector  c  that " 
}}{PARA 0 "" 0 "" {TEXT -1 95 "depends only on  A  and  b.  The idea i
s that in any real problem there would be a small amount" }}{PARA 0 "
" 0 "" {TEXT -1 101 "of damping, which would lead to transient homogen
eous solution, and steady periodic solution close to" }}{PARA 0 "" 0 "
" {TEXT -1 52 "the one you are finding for this undamped problem.  " }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "PROBLEM \+
3.3." }}{PARA 0 "" 0 "" {TEXT -1 99 "Define  b:=(1/4)*w*w*vector([1,1,
1,1,1,1,1]):  in Maple and find the vector  c  in the undetermined" }}
{PARA 0 "" 0 "" {TEXT -1 102 "coefficients solution  x(t)=cos(wt)c. Ve
ctor  c  depends on w. As outlined in the textbook, vector  c " }}
{PARA 0 "" 0 "" {TEXT -1 98 "can be found by solving the linear algebr
a problem  -w^2 c = Ac + b; see page 433. Don't print  c," }}{PARA 0 "
" 0 "" {TEXT -1 61 "as it is too complex; instead, print c[1] as an il
lustration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 79 "# Sample code for defining  b and A, then solving for  c  in th
e  4-floor case." }}{PARA 0 "" 0 "" {TEXT -1 52 "# See maple help to l
earn about evalm, diag, vector." }}{PARA 0 "" 0 "" {TEXT -1 10 "  w:='
w': " }}{PARA 0 "" 0 "" {TEXT -1 32 "  b:=0.25*w*w*vector([1,1,1,1]):
" }}{PARA 0 "" 0 "" {TEXT -1 73 "  A:=matrix([ [-20,10,0,0], [10,-20,1
0,0], [0,10,-20,10], [0,0,10,-10]]);" }}{PARA 0 "" 0 "" {TEXT -1 46 " \+
 c:=linsolve(evalm(A+w*w*diag(1,1,1,1)),-b): " }}{PARA 0 "" 0 "" 
{TEXT -1 16 "  evalf(c[1],6);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "# PROBLEM 3.3" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "3.4 PRACTICAL RESONANCE." }}
{PARA 0 "" 0 "" {TEXT -1 100 "Consider the forced equation  x'=Ax+cos(
wt)b  of 3.3 above with b:=0.25*w*w*vector([1,1,1,1,1,1,1])." }}{PARA 
0 "" 0 "" {TEXT -1 201 "Practical resonance occurs when any component \+
of  x(t)  has large amplitude compared to the vector\nnorm of  the for
cing amplitudes in b. For example, an earthquake might cause a small 3
-inch excursion " }}{PARA 0 "" 0 "" {TEXT -1 105 "on level ground, but
 the building's floors might have 50-inch excursions, enough to destro
y the building." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 12 "PROBLEM 3.4." }}{PARA 0 "" 0 "" {TEXT -1 190 "Let Max(c) \+
denote the maximum modulus of the components of vector c. Plot  g(T)=M
ax(c(w)) with w=(2*Pi)/T for periods T=0 to T=6, ordinates Max=0 to Ma
x=10, the vector c(w) being the answer " }}{PARA 0 "" 0 "" {TEXT -1 
84 "produced in 3.3 above.  Compare your figure to the textbook Figure
 7.4.18, page 438." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 68 "# Sample maple code to define the function Max(c), 4-floo
r building." }}{PARA 0 "" 0 "" {TEXT -1 77 "# Use maple help to learn \+
about norm, diag, vector, subs, evalm and linsolve." }}{PARA 0 "" 0 "
" {TEXT -1 15 "  with(linalg):" }}{PARA 0 "" 0 "" {TEXT -1 39 "  w:='w
': Max:= c -> norm(c,infinity); " }}{PARA 0 "" 0 "" {TEXT -1 54 "  B:=
w*w*diag(1,1,1,1): b:=0.25*w*w*vector([1,1,1,1]):" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "  C:=ww -> subs(w=ww,linsolve(evalm(A+B),-b)):" }}{PARA 
0 "" 0 "" {TEXT -1 36 "  plot(Max(C(2*Pi/r)),r=0..6,0..10);" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "re
start:with(plots):with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 47 "# PROBLEM 3.4. WARNING: Save your file often!!!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 23 "3.5. EARTHQUAKE DAMAGE." }}{PARA 0 "" 0 
"" {TEXT -1 103 "The maximum amplitude plot of 3.4 can be used to dete
ct the likelihood of earthquake damage for a given" }}{PARA 0 "" 0 "" 
{TEXT -1 101 "ground vibration of period T.  A ground vibration (1/4)c
os(wt), T=2*Pi/w, will be assumed, as in 3.4." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "PROBLEM 3.5." }}{PARA 0 "
" 0 "" {TEXT -1 91 "Replot the amplitudes in 3.4 for periods 0 to 6 an
d amplitudes 5 to 10. There will be three" }}{PARA 0 "" 0 "" {TEXT -1 
89 "spikes. Zoom-in on each spike, choosing a T-interval that shows th
e full spike. Determine" }}{PARA 0 "" 0 "" {TEXT -1 94 "from the three
 zoom-in plots approximate intervals for the period  T  such that some
 floor in " }}{PARA 0 "" 0 "" {TEXT -1 75 "the building will undergo e
xcursions from equilibrium in excess of 5 feet. " }}{PARA 0 "" 0 "" 
{TEXT -1 83 "# Example: Zoom-in on a spike for amplitudes 5 feet to 10
 feet, periods 1.4 to 1.5." }}{PARA 0 "" 0 "" {TEXT -1 39 "plot(Max(C(
2*Pi/r)),r=1.4..1.5,5..10);\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "# PROBLEM 3.5. WARNING: Save your file often!!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 33 "# Plot three spikes on one graph." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# Plot three zoom-in graphs.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "# Print period ranges.
" }}}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 17 "3.6. FIVE FLOORS." }}{PARA 0 "" 0 "" {TEXT -1 104 "Consider a b
uilding with five floors each weighing 20 tons. Assume each floor corr
esponds to a restoring" }}{PARA 0 "" 0 "" {TEXT -1 107 "Hooke's force \+
with constant k=4 tons/foot. Assume that ground vibrations from the ea
rthquake are modeled by" }}{PARA 0 "" 0 "" {TEXT -1 68 "(1/4)cos(wt) w
ith period T=2*Pi/w (same as the 7-floor model above)." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "PROBLEM 3.6." }}
{PARA 0 "" 0 "" {TEXT -1 211 "Model the 5-floor problem in Maple. Plot
 the maximum amplitudes against the period 0 to 6 and amplitude\n4 to \+
10. Determine from the graphic the period ranges which cause the ampli
tude plot to spike above 4 feet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(plots):with(lin
alg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# PROBLEM 3.6. WAR
NING: Save your file often!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 45 "# Define k=??? and m=???, then matrix A=???.\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 35 "# Amplitude plot for T=0..6,C=4..10" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# Plot 4 zoom-in graphs" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "# From the graphics, T=??..
??, ??..??, ??..??, ??..??\n# give amplitude spikes above 4 feet. Thes
e are\n# determined by left mouse-clicks on the graph, so they\n# are \+
approximate values only. \n" }}}}{MARK "59 0" 0 }{VIEWOPTS 1 1 0 1 1 
1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }