{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 2250" }}{PARA 256 "" 0 "" {TEXT -1 18 "Earthquake project" }}{PARA 257 "" 0 "" {TEXT -1 10 "Apri l 2004" }}{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 "Project \+ 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. UNDETERMINED CO EFFICIENTS 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 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "3.1. BUILDING M ODEL FOR AN EARTHQUAKE." }}{PARA 0 "" 0 "" {TEXT -1 98 "Refer to the t extbook 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 "P ROBLEM 3.1" }}{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 57 "Answer check: All seven eigenvalues shou ld 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 about evalf and eigen vals." }}{PARA 0 "" 0 "" {TEXT -1 14 " with(linalg):" }}{PARA 0 "" 0 " " {TEXT -1 74 " A := matrix([ [-20,10,0,0], [10,-20,10,0], [0,10,-20,1 0], [0,0,10,-10]]);" }}{PARA 0 "" 0 "" {TEXT -1 24 " evalf(eigenvals(A ),6); " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 " \n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "# Probl em 3.1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(linalg): \n " }}}{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 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 96 "Find the nat ural angular frequencies omega=sqrt(-lambda) for the seven story buil ding and also " }}{PARA 0 "" 0 "" {TEXT -1 106 "the corresponding peri ods 2PI/omega, accurate to six digits. Display the answers in a simpl e handwritten " }}{PARA 0 "" 0 "" {TEXT -1 107 "table or a computer-ge nerated table as in the example below. The answers appear in Figure 7. 4.17, page 437," }}{PARA 0 "" 0 "" {TEXT -1 76 "although in a slightly different order than what would be computed in MAPLE." }}{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 51 " ev:=[-10,-1.2 06147582,-35.32088886,-23.47296354]: " }}{PARA 0 "" 0 "" {TEXT -1 13 " n:=nops(ev):" }}{PARA 0 "" 0 "" {TEXT -1 32 " Omega:=lambda -> sqrt(- lambda):" }}{PARA 0 "" 0 "" {TEXT -1 37 " format:=\"%10.6f %10.6f %1 0.6f\\n\": " }}{PARA 0 "" 0 "" {TEXT -1 62 " printf(\"%s %s \+ %s\\n\",\"Eigenvalue\", \"Freq\",\"Period\");" }}{PARA 0 "" 0 "" {TEXT -1 72 " seq(printf(format,ev[i],Omega(ev[i]),2*evalf(Pi)/Omega(e v[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 14 "with(linalg): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 62 "3.3. UNDETERMINED COEFFICIENTS STEADY-STATE PERIODIC SOLUTION. " }}{PARA 0 "" 0 "" {TEXT -1 95 "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 t erm 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 equilibri um of the corresponding 7 floors." }}{PARA 0 "" 0 "" {TEXT -1 97 "Soug ht here is not the general solution, which certainly contains transien t terms, but rather the" }}{PARA 0 "" 0 "" {TEXT -1 93 "steady-state p eriodic solution, which is known from the theory to have the form x(t) =cos(wt)c" }}{PARA 0 "" 0 "" {TEXT -1 80 "for some vector c that dep ends 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*vector([1,1,1,1,1,1,1]): i n Maple and find the vector c in the undetermined" }}{PARA 0 "" 0 " " {TEXT -1 102 "coefficients solution x(t)=cos(wt)c. Vector c depen ds 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 -1 101 "as it does not fit on one page; instead, print c[1] as an illustr ation. You should get -0.09304656278" }}{PARA 0 "" 0 "" {TEXT -1 30 "w hen c[1] is evaluated at w=1." }}{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 26 " subs(w=1,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 14 "with(linalg): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# subs(w=1,evalf(c[1],2));" }}}{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 fo rced 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 c an occur 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-i nch excursion on level ground, but\nthe building's floors might have 5 0-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 comp onents 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=4, ordinates Max=0 to Max=10 , the vector c(w) being the answer produced in 3.3 above." }}{PARA 0 " " 0 "" {TEXT -1 102 "Compare your figure to the textbook Figure 7.4.18 , page 438. Your figure is expected to show 6 spikes." }}{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, subs and li nsolve." }}{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,-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..4,0..10,numpoints=400);" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "# PRO BLEM 3.4. WARNING: Save your file often!!!" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "with(linalg): \n# plot(Max(C(2*Pi/r)),r=0..4,0..10, numpoints=400);\n" }}}{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)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 -1 105 "(a) Replot the amplitudes in 3.4 for graph window x=0.95 to 3.5 and y=5 to 10. There will be six spikes. " }}{PARA 0 " " 0 "" {TEXT -1 87 "(b) Create one zoom-in plot near T=3, choosing a T -interval that shows the full spike. " }}{PARA 0 "" 0 "" {TEXT -1 88 " (c) Determine from the zoom-in plot (near T=3) one approximate T-inter val such that some" }}{PARA 0 "" 0 "" {TEXT -1 88 " floor in the b uilding will undergo excursions from equilibrium in excess of 5 feet. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "# Exa mple: Zoom-in on a spike for amplitudes 5 feet to 10 feet, periods 1.9 6 to 2.01." }}{PARA 0 "" 0 "" {TEXT -1 58 "with(linalg): w:='w': Max:= c -> norm(c,infinity); u:=w*w:" }}{PARA 0 "" 0 "" {TEXT -1 80 "Au:=ma trix([ [-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 92 "plot(Max(C(2*Pi/r)),r=1.96..2.01,5..10,nu mpoints=400);\nprintf(\"Period T from 1.96 to 2.01\");" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# PROBL EM 3.5. WARNING: Save your file often!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "#(a) Plot six spikes on one graph" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "#(b) Plot one zoom-in graph near T=3." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "#(c) \+ Report one approximate T-interval near T=3." }}}{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 -1 108 "Hooke's force with constant k=10 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 -1 104 "Model the 6-floor problem in Maple. Plot the maximum amplitudes i n graph window x=1 to 4 and y=4 to 10. " }}{PARA 0 "" 0 "" {TEXT -1 109 "Determine from the graphic one period range near T=3.5 which caus es the amplitude 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 slug s) times " }}{PARA 0 "" 0 "" {TEXT -1 115 "the acceleration of gravit y, 32 ft/sec/sec. From this you can work out how to convert tons to s lugs. Use (5) and " }}{PARA 0 "" 0 "" {TEXT -1 110 "(6) page 425 to f ind the matrices M and K (on paper) and then write down A as the inver se of M times K. Check " }}{PARA 0 "" 0 "" {TEXT -1 106 "your reasonin g on the original model: your logic should reproduce the text before e quation (1) page 437. " }}{PARA 0 "" 0 "" {TEXT -1 108 "There are fiv e spikes. To see them, follow the examples above, especially, use the \+ plot option numpoints=400" }}{PARA 0 "" 0 "" {TEXT -1 10 "or larger." }}{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 6x6 matrix A." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# Amplitude plot for T=1..4, C=4..10" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# Plot one zoom- in graphic near T=3.5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# Print the T-range near T=3.5 for the zoom-in plot above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "155 0 0" 38 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }