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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 2002" }}{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 37 " _______     PROBLEM 3.6. SIX 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 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 "" }}{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 -1 121 "Find the natural angular frequencies  omega=sqrt(
-lambda) for the seven story building and also the corresponding perio
ds" }}{PARA 0 "" 0 "" {TEXT -1 115 "2PI/omega, accurate to six digits.
  Display the answers in a table . 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  4x3  table." }}{PARA 0 "" 0 "" {TEXT -1 
48 "# Use maple help to learn about nops and printf." }}{PARA 0 "" 0 "
" {TEXT -1 63 " ev:=[-10,-1.206147582,-35.32088886,-23.47296354]: n:=n
ops(ev):" }}{PARA 0 "" 0 "" {TEXT -1 32 " Omega:=lambda -> sqrt(-lambd
a):" }}{PARA 0 "" 0 "" {TEXT -1 37 " format:=\"%10.6f  %10.6f  %10.6f
\\n\": " }}{PARA 0 "" 0 "" {TEXT -1 72 " seq(printf(format,ev[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" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "3.3. UNDETERMINED COEFFICIENTS \+
STEADY-STATE PERIODIC SOLUTION." }}{PARA 0 "" 0 "" {TEXT -1 94 "Consid
er the forced equation  x'=Ax+cos(wt)b  where  b  is a constant vector
. The earthquake's" }}{PARA 0 "" 0 "" {TEXT -1 89 "ground vibration is
  accounted 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 certainly contains transient terms, but rather the" }}{PARA 0 "
" 0 "" {TEXT -1 93 "steady-state periodic solution, which is known fro
m the theory to have the form x(t)=cos(wt)c" }}{PARA 0 "" 0 "" {TEXT 
-1 52 "for some vector  c  that depends only on  A  and  b." }}{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*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. 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 61 "as it is too complex; instead, print c[1] as an illustrat
ion." }}{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-flo
or case." }}{PARA 0 "" 0 "" {TEXT -1 52 "# See maple help to learn abo
ut 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,-1
0+u]]);" }}{PARA 0 "" 0 "" {TEXT -1 22 "  c:=linsolve(Au,-b): " }}
{PARA 0 "" 0 "" {TEXT -1 16 "  evalf(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 "" }}}{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 -1 279 "Practical resonance can occur if a compo
nent of  x(t)  has large amplitude compared to the vector\nnorm of  b.
 For example, an earthquake might cause a small 3-inch excursion on le
vel 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 components 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, vector, s
ubs and linsolve." }}{PARA 0 "" 0 "" {TEXT -1 15 "  with(linalg):" }}
{PARA 0 "" 0 "" {TEXT -1 46 "  w:='w': Max:= c -> norm(c,infinity); 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,-2
0+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 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "3.5. EART
HQUAKE 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 perio
d T.  A ground vibration (1/4)cos(wt), T=2*Pi/w, will be assumed, as i
n 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 amplitud
es 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 zo
om-in plots approximate intervals for the period  T  such that " }}
{PARA 0 "" 0 "" {TEXT -1 88 "some floor in the building will undergo e
xcursions from equilibrium in excess of 5 feet." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "# Example: Zoom-in on a s
pike for amplitudes 5 feet to 10 feet, periods 1.97 to 2.01." }}{PARA 
0 "" 0 "" {TEXT -1 58 "with(linalg): w:='w': Max:= c -> norm(c,infinit
y); 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=1.97..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 separate graphs" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "#(b) Plot four zoom-in graphs." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "#(c) Print period ranges." }}}{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 16 "3.6. SIX FLOORS.
" }}{PARA 0 "" 0 "" {TEXT -1 103 "Consider a building with six floors \+
each weighing 50 tons. Assume each floor corresponds to a restoring" }
}{PARA 0 "" 0 "" {TEXT -1 107 "Hooke's force with constant k=5 tons/fo
ot. Assume that ground vibrations from the earthquake are modeled by" 
}}{PARA 0 "" 0 "" {TEXT -1 68 "(1/4)cos(wt) with 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 6-floor problem in Maple. Plot the maximum amplitudes a
gainst the period 0 to 6 and amplitude\n4 to 10. Determine from the gr
aphic the period ranges which cause the amplitude plot to spike above \+
4 feet." }}{PARA 0 "" 0 "" {TEXT -1 88 "Sanity check:  m=3125, and the
 6x6 matrix contains fraction 16/5. There are five spikes." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "# PR
OBLEM 3.6. WARING: Save your file often!!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 35 "# Define k, m and the 6x6 matrix A." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# Amplitude plot for T=0..6,C=4..10
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# Plot five zoom-in gra
phs" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "# From the graphics
, five T-ranges give amplitude\n# spikes above 4 feet. These are deter
mined by left \n# mouse-clicks on the graph, so they are approximate v
alues only. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK 
"4 0" 66 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 
}