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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 2250" }}{PARA 256 "" 0 ""
{TEXT -1 31 "Maple Lab 6: Earthquake project" }}{PARA 257 "" 0 ""
{TEXT -1 11 "Spring 2006" }}{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 61 "Project 3. Solve problems L6.1 to L6.6. The problem head
ers:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 " \+
_______ PROBLEM L6.1. BUILDING MODEL FOR AN EARTHQUAKE." }}{PARA
0 "" 0 "" {TEXT -1 68 " _______ PROBLEM L6.2. TABLE OF NATURAL FRE
QUENCIES AND PERIODS." }}{PARA 0 "" 0 "" {TEXT -1 84 " _______ PRO
BLEM L6.3. UNDETERMINED COEFFICIENTS STEADY-STATE PERIODIC SOLUTION."
}}{PARA 0 "" 0 "" {TEXT -1 47 " _______ PROBLEM L6.4. PRACTICAL RE
SONANCE." }}{PARA 0 "" 0 "" {TEXT -1 45 " _______ PROBLEM L6.5. EA
RTHQUAKE DAMAGE." }}{PARA 0 "" 0 "" {TEXT -1 38 " _______ PROBLEM \+
L6.6. 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 36 "1. BUILDING MODEL FOR AN EARTHQUAKE.
" }}{PARA 0 "" 0 "" {TEXT -1 98 "Refer to the textbook of Edwards-Penn
ey, 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 b
y 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 12 "PROBLEM L6.1" }}{PARA 0 "
" 0 "" {TEXT -1 91 "Find the eigenvalues of the matrix A to six digits
, using the Maple command \"eigenvals(A).\"" }}{PARA 0 "" 0 "" {TEXT
-1 85 "Justify in particular that all seven eigenvalues are negative b
y 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 14 "# Problem L6.1" }}}{EXCHG
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 44 "2. TABLE OF NATURAL FREQUENCIES AND PERI
ODS." }}{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 13 "PROBL
EM L6.2." }}{PARA 0 "" 0 "" {TEXT -1 121 "Find the natural angular fre
quencies omega=sqrt(-lambda) for the seven story building and also th
e corresponding periods" }}{PARA 0 "" 0 "" {TEXT -1 115 "2PI/omega, ac
curate to six digits. Display the answers in a table . The answers ap
pear 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,-35.320
88886,-23.47296354]: n:=nops(ev):" }}{PARA 0 "" 0 "" {TEXT -1 32 " Ome
ga:=lambda -> sqrt(-lambda):" }}{PARA 0 "" 0 "" {TEXT -1 37 " format:=
\"%10.6f %10.6f %10.6f\\n\": " }}{PARA 0 "" 0 "" {TEXT -1 72 " seq(p
rintf(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
14 "# Problem L6.2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "3. UNDETE
RMINED COEFFICIENTS STEADY-STATE PERIODIC SOLUTION." }}{PARA 0 "" 0 "
" {TEXT -1 94 "Consider 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, whi
ch 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 correspo
nding 7 floors." }}{PARA 0 "" 0 "" {TEXT -1 97 "Sought here is not the
general solution, which certainly contains transient terms, but rathe
r the" }}{PARA 0 "" 0 "" {TEXT -1 93 "steady-state periodic solution, \+
which is known from 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 14 "PROBLEM L6.3." }}{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, ve
ctor c " }}{PARA 0 "" 0 "" {TEXT -1 98 "can be found by solving the l
inear 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 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 ma
ple 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:=lins
olve(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
14 "# PROBLEM L6.3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "4 PRACTIC
AL RESONANCE." }}{PARA 0 "" 0 "" {TEXT -1 100 "Consider the forced equ
ation 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 component of x(t) has large amplitude compared to the vector\n
norm of b. For example, an earthquake might cause a small 3-inch excu
rsion on level ground, but\nthe building's floors might have 50-inch e
xcursions, enough to destroy the building." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "PROBLEM L6.4." }}{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
48 "# PROBLEM L6.4. 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 21 "5. EARTHQ
UAKE DAMAGE." }}{PARA 0 "" 0 "" {TEXT -1 103 "The maximum amplitude pl
ot of 3.4 can be used to detect the likelihood of earthquake damage fo
r 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 13 "P
ROBLEM L6.5." }}{PARA 0 "" 0 "" {TEXT -1 106 "(a) Replot the amplitude
s in 3.4 for periods 1.14 to 4 and amplitudes 5 to 10. There will be f
our 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 s
pike. " }}{PARA 0 "" 0 "" {TEXT -1 93 "(c) Determine from the four zoo
m-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 47 "# PROBLEM L6.5. 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 14 "6. SIX FLOORS." }
}{PARA 0 "" 0 "" {TEXT -1 103 "Consider a building with six floors eac
h weighing 50 tons. Assume each floor corresponds to a restoring" }}
{PARA 0 "" 0 "" {TEXT -1 107 "Hooke's force with constant k=5 tons/foo
t. 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 13 "PROBLEM L6.6." }}{PARA 0 "" 0 "" {TEXT -1 211 "
Model the 6-floor problem in Maple. Plot the maximum amplitudes agains
t the period 0 to 6 and amplitude\n4 to 10. Determine from the graphic
the period ranges which cause the amplitude plot to spike above 4 fee
t." }}{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 46 "# PROBLEM
L6.6. 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 graphs"
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "# From the graphics, fiv
e T-ranges give amplitude\n# spikes above 4 feet. These are determined
by left \n# mouse-clicks on the graph, so they are approximate values
only. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "131 \+
0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }