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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 2250" }}{PARA 256 "" 0 "" 
{TEXT -1 18 "Earthquake project" }}{PARA 257 "" 0 "" {TEXT -1 13 "Nove
mber 2003" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
66 "Name _____________________________________   Class Time __________
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Proje
ct 3.  Solve problems 3.1 to 3.6. The problem 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 NATURAL FREQUENCIES AND PERIODS
." }}{PARA 0 "" 0 "" {TEXT -1 83 " _______     PROBLEM 3.3. UNDETERMIN
ED COEFFICIENTS STEADY-STATE PERIODIC SOLUTION." }}{PARA 0 "" 0 "" 
{TEXT -1 46 " _______     PROBLEM 3.4. PRACTICAL RESONANCE." }}{PARA 
0 "" 0 "" {TEXT -1 44 " _______     PROBLEM 3.5. EARTHQUAKE 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 13 "with(linalg):" }}}{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 57 "Answer check:  All s
even eigenvalues should be negative. " }}{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 ab
out evalf and eigenvals." }}{PARA 0 "" 0 "" {TEXT -1 74 " A := matrix(
[ [-20,10,0,0], [10,-20,10,0], [0,10,-20,10], [0,0,10,-10]]);" }}
{PARA 0 "" 0 "" {TEXT -1 24 " evalf(eigenvals(A),6); " }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{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 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 46 "3.2. TABLE OF NATURAL FREQUENCIES 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 96 "Find the natural angular frequencies  ome
ga=sqrt(-lambda) for the seven story building and also " }}{PARA 0 "" 
0 "" {TEXT -1 95 "the corresponding periods 2PI/omega, accurate to six
 digits.  Display the answers in a table . " }}{PARA 0 "" 0 "" {TEXT 
-1 46 "The answers appear in Figure 7.4.17, page 437." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "# Sample code for a  4
x3  table." }}{PARA 0 "" 0 "" {TEXT -1 48 "# Use maple help to learn a
bout nops and printf." }}{PARA 0 "" 0 "" {TEXT -1 63 " ev:=[-10,-1.206
147582,-35.32088886,-23.47296354]: n:=nops(ev):" }}{PARA 0 "" 0 "" 
{TEXT -1 32 " Omega:=lambda -> sqrt(-lambda):" }}{PARA 0 "" 0 "" 
{TEXT -1 37 " format:=\"%10.6f  %10.6f  %10.6f\\n\": " }}{PARA 0 "" 0 
"" {TEXT -1 62 " printf(\"%s    %s        %s\\n\",\"Eigenvalue\", \"Fr
eq\",\"Period\");" }}{PARA 0 "" 0 "" {TEXT -1 72 " seq(printf(format,e
v[i],Omega(ev[i]),2*evalf(Pi)/Omega(ev[i])),i=1..n);" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "# Problem \+
3.2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 62 "3.3. UNDETERMINED COEFFICIENTS STEADY-STA
TE PERIODIC SOLUTION." }}{PARA 0 "" 0 "" {TEXT -1 95 "Consider the for
ced equation  x''=Ax+cos(wt)b  where  b  is a constant vector. The ear
thquake's" }}{PARA 0 "" 0 "" {TEXT -1 89 "ground vibration is  account
ed 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 97 "Sought here is not the general solution, which ce
rtainly contains transient terms, but rather the" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "steady-state periodic solution, which is known from the t
heory to have the form x(t)=cos(wt)c" }}{PARA 0 "" 0 "" {TEXT -1 80 "f
or some vector  c  that depends only on  A  and  b. See the textbook, \+
page 433." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
12 "PROBLEM 3.3." }}{PARA 0 "" 0 "" {TEXT -1 98 "Define  b:=0.25*w*w*v
ector([1,1,1,1,1,1,1]):  in Maple and find the vector  c  in the undet
ermined" }}{PARA 0 "" 0 "" {TEXT -1 102 "coefficients solution  x(t)=c
os(wt)c. Vector  c  depends on w. As outlined in the textbook, vector \+
 c " }}{PARA 0 "" 0 "" {TEXT -1 98 "can be found by solving the linear
 algebra problem  -w^2 c = Ac + b; see page 433. Don't print  c," }}
{PARA 0 "" 0 "" {TEXT -1 71 "as it does not fit on one page; instead, \+
print c[1] as an illustration." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 79 "# Sample code for defining  b and A, then
 solving for  c  in the  4-floor case." }}{PARA 0 "" 0 "" {TEXT -1 52 
"# See maple help to learn about vector and linsolve." }}{PARA 0 "" 0 
"" {TEXT -1 46 "  w:='w': u:=w*w: b:=0.25*u*vector([1,1,1,1]):" }}
{PARA 0 "" 0 "" {TEXT -1 82 "  Au:=matrix([ [-20+u,10,0,0], [10,-20+u,
10,0], [0,10,-20+u,10], [0,0,10,-10+u]]);" }}{PARA 0 "" 0 "" {TEXT -1 
22 "  c:=linsolve(Au,-b): " }}{PARA 0 "" 0 "" {TEXT -1 23 "  'c[1]'=ev
alf(c[1],2);" }}{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 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "3.4 PRACT
ICAL RESONANCE." }}{PARA 0 "" 0 "" {TEXT -1 100 "Consider the forced e
quation  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 -1 279 "Practical resonance can occ
ur if a component of  x(t)  has large amplitude compared to the vector
\nnorm of  b. For example, an earthquake might cause a small 3-inch ex
cursion on level ground, but\nthe building's floors might have 50-inch
 excursions, enough to destroy the building." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "PROBLEM 3.4." }}{PARA 0 "" 0 "
" {TEXT -1 105 "Let Max(c) denote the maximum modulus of the component
s of vector c. Plot  g(T)=Max(c(w)) with w=(2*Pi)/T" }}{PARA 0 "" 0 "
" {TEXT -1 106 "for periods T=0 to T=6, ordinates Max=0 to Max=10, the
 vector c(w) being the answer produced in 3.3 above." }}{PARA 0 "" 0 "
" {TEXT -1 60 "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-floor building." 
}}{PARA 0 "" 0 "" {TEXT -1 64 "# Use maple help to learn about norm, v
ector, subs and linsolve." }}{PARA 0 "" 0 "" {TEXT -1 15 "  with(linal
g):" }}{PARA 0 "" 0 "" {TEXT -1 46 "  w:='w': Max:= c -> norm(c,infini
ty); u:=w*w:" }}{PARA 0 "" 0 "" {TEXT -1 32 "  b:=0.25*w*w*vector([1,1
,1,1]):" }}{PARA 0 "" 0 "" {TEXT -1 82 "  Au:=matrix([ [-20+u,10,0,0],
 [10,-20+u,10,0], [0,10,-20+u,10], [0,0,10,-10+u]]);" }}{PARA 0 "" 0 "
" {TEXT -1 38 "  C:=ww -> subs(w=ww,linsolve(Au,-b)):" }}{PARA 0 "" 0 
"" {TEXT -1 50 "  plot(Max(C(2*Pi/r)),r=0..6,0..10,numpoints=150);" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
47 "# PROBLEM 3.4. WARNING: Save your file often!!!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{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 23 "3.5. EARTHQUAKE DAMAGE." }}{PARA 0 "
" 0 "" {TEXT -1 103 "The maximum amplitude plot of 3.4 can be used to \+
detect the likelihood of earthquake damage for a given" }}{PARA 0 "" 
0 "" {TEXT -1 101 "ground vibration of period T.  A ground vibration (
1/4)cos(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 -1 106 "(a) Replot the amplitudes in 3.4 for periods 1.14
 to 4 and amplitudes 5 to 10. There will be four spikes. " }}{PARA 0 "
" 0 "" {TEXT -1 100 "(b) Create four zoom-in plots, one for each spike
, choosing a T-interval that shows the full spike. " }}{PARA 0 "" 0 "
" {TEXT -1 93 "(c) Determine from the four zoom-in plots approximate i
ntervals for the period  T  such that " }}{PARA 0 "" 0 "" {TEXT -1 88 
"some floor in the building will undergo excursions from equilibrium i
n excess of 5 feet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 85 "# Example: Zoom-in on a spike for amplitudes 5 feet to \+
10 feet, periods 1.96 to 2.01." }}{PARA 0 "" 0 "" {TEXT -1 58 "with(li
nalg): w:='w': Max:= c -> norm(c,infinity); u:=w*w:" }}{PARA 0 "" 0 "
" {TEXT -1 80 "Au:=matrix([ [-20+u,10,0,0], [10,-20+u,10,0], [0,10,-20
+u,10], [0,0,10,-10+u]]);" }}{PARA 0 "" 0 "" {TEXT -1 30 "b:=0.25*w*w*
vector([1,1,1,1]):" }}{PARA 0 "" 0 "" {TEXT -1 36 "C:=ww -> subs(w=ww,
linsolve(Au,-b)):" }}{PARA 0 "" 0 "" {TEXT -1 55 "plot(Max(C(2*Pi/r)),
r=1.96..2.01,5..10,numpoints=150);\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "# PROBLEM 3.5. WARNING: Save your file often!!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "#(a) Plot four spikes on sep
arate graphs" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "#(b) Plot f
our zoom-in graphs." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "#(c)
 Print period ranges." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " 
}}}{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 17 "3.6. FIVE FLOORS." }}{PARA 0 "" 0 "" {TEXT -1 104 "Consider a b
uilding with five floors each weighing 40 tons. Assume each floor corr
esponds to a restoring" }}{PARA 0 "" 0 "" {TEXT -1 107 "Hooke's force \+
with constant k=5 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 -1 212 "Model the 5-floor problem in Maple. Plot
 the maximum amplitudes against the period 1 to 15 and amplitude\n4 to
 10. Determine from the graphic the period ranges which cause the ampl
itude plot to spike above 4 feet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 110 "Sanity checks:  Recall that a ton equals
 2000 pounds, and that a pound of force equals mass (in slugs)  times \+
" }}{PARA 0 "" 0 "" {TEXT -1 111 "the acceleration of gravity, 32 ft/s
ec/sec.  From this you can work out how to convert tons to slugs.  You
 can " }}{PARA 0 "" 0 "" {TEXT -1 111 "check your reasoning on the ori
ginal model, see text before equation (1) page 437.  There are five sp
ikes, but " }}{PARA 0 "" 0 "" {TEXT -1 36 "you may have to zoom in to \+
see them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 46 "# PROBLEM 3.6. WARNING: Save your file often!!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# Define k, m and the 5x5 ma
trix A." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# Amplitude plot
 for T=0..15,C=4..10" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# P
lot five zoom-in graphs" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 
"# From the graphics, five T-ranges give amplitude\n# spikes above 4 f
eet. These are determined by left \n# mouse-clicks on the graph, so th
ey are approximate values only. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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