{VERSION 4 0 "SUN SPARC SOLARIS" "4.0" } {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 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 9 "Math 2250" }}{PARA 257 "" 0 "" {TEXT -1 18 "Earthquake project" }}{PARA 258 "" 0 "" {TEXT -1 10 "Apri l 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 -1 106 "Project 3. Please fill in all areas below marked ????? and solve problems 3.1 to 3.6. The pr oblem headers:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 " _______ PROBLEM 3.1. BUILDING MODEL FOR AN EARTHQUAK E." }}{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 COEFFICIENTS STEADY-STATE PERIODIC S OLUTION." }}{PARA 0 "" 0 "" {TEXT -1 46 " _______ PROBLEM 3.4. PRA CTICAL RESONANCE." }}{PARA 0 "" 0 "" {TEXT -1 44 " _______ PROBLEM 3.5. EARTHQUAKE DAMAGE." }}{PARA 0 "" 0 "" {TEXT -1 39 " _______ \+ PROBLEM 3.6. THREE FLOORS." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "with(DEtools):with(plots):with(lina lg):" }}}{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. Consi der 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 equa tion (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 the ei genvalues of the matrix A to six digits, using the Maple command \"eig envals(A).\"" }}{PARA 0 "" 0 "" {TEXT -1 85 "Justify in particular tha t 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 77 "# Use \+ maple help to learn about augment, stackmatrix, diag, transpose, evalm ." }}{PARA 0 "" 0 "" {TEXT -1 27 " A1:=diag(-20,-20,-20,-10):" }} {PARA 0 "" 0 "" {TEXT -1 75 " A2:=augment(vector([0,0,0,0]),stackmatri x(diag(1,1,1),matrix([[0,0,0]]))):" }}{PARA 0 "" 0 "" {TEXT -1 19 " A3 :=transpose(A2):" }}{PARA 0 "" 0 "" {TEXT -1 26 " A:=evalm(A1+10*A2+10 *A3);" }}{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" }}}{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 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 "# Sample code for a 4x3 table." }} {PARA 0 "" 0 "" {TEXT -1 39 "# Use maple help to learn about printf." }}{PARA 0 "" 0 "" {TEXT -1 38 " ev:=[-38.3,-33.4,-26.2,-17.9]: n:=4: " }}{PARA 0 "" 0 "" {TEXT -1 32 " Omega:=lambda -> sqrt(-lambda):" }} {PARA 0 "" 0 "" {TEXT -1 36 " 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" }}}{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 "Consider the forced equation x'=Ax+cos(wt)b where b is a co nstant vector. The earthquake's" }}{PARA 0 "" 0 "" {TEXT -1 89 "ground vibration is accounted for by the extra term cos(wt)b, which has per iod 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 fl oors." }}{PARA 0 "" 0 "" {TEXT -1 97 "Sought here is not the general s olution, which certainly contains transient terms, but rather 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 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 10 " w:='w': " }}{PARA 0 "" 0 "" {TEXT -1 32 " b:=0.25*w*w*vector([1,1,1,1]):" }}{PARA 0 "" 0 "" {TEXT -1 29 " A1:=diag(-20,-20,-20,-10): " }}{PARA 0 "" 0 "" {TEXT -1 76 " A2:=augment(vector([0,0,0,0]),stackmatrix(diag(1,1,1),m atrix([[0,0,0]]))):" }}{PARA 0 "" 0 "" {TEXT -1 38 " A:=evalm(A1+10*A 2+10*transpose(A2));" }}{PARA 0 "" 0 "" {TEXT -1 46 " c:=linsolve(eva lm(A+w*w*diag(1,1,1,1)),-b): " }}{PARA 0 "" 0 "" {TEXT -1 14 " evalf( c[1]);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "# PROBLEM 3.3" }}}{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 component of x(t) h as large amplitude compared to the vector\nnorm of b. For example, an earthquake might cause a small 3-inch excursion on level ground, but \nthe building's floors might have 50-inch excursions, enough to destr oy 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)=M ax(c(w)) with w=T/(2*Pi)" }}{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 answe r 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 de fine the function Max(c), 4-floor building." }}{PARA 0 "" 0 "" {TEXT -1 39 "# Use maple help to learn about \"norm.\"" }}{PARA 0 "" 0 "" {TEXT -1 15 " with(linalg):" }}{PARA 0 "" 0 "" {TEXT -1 31 " 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 13 "# PROBLEM 3. 4" }}}{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 detec t 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 81 "Replot the amplitudes in 3.4 for periods 0 to 6 an d amplitudes 5 to 10. Determine" }}{PARA 0 "" 0 "" {TEXT -1 79 "approx imate ranges for the period T such that some floor in the building w ill" }}{PARA 0 "" 0 "" {TEXT -1 56 "undergo excursions from equilibriu m in excess of 5 feet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "# PROBLEM 3.5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# Plot it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "# Print period ranges ???." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "3.6. THREE FLOORS." }}{PARA 0 "" 0 "" {TEXT -1 110 "Consider a buildi ng with only three floors each weighing 20 tons. Assume each floor cor responds to a restoring" }}{PARA 0 "" 0 "" {TEXT -1 107 "Hooke's force with constant k=4 tons/foot. Assume that ground vibrations from the e arthquake 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 3-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 13 "# PROBLEM 3.6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "# Define k=??? and m=???, then matr ix A=???." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# Amplitude pl ot for T=0..6,C=4..10" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "# From the graphics, T-ranges are ???? \n# which give amplitude spikes \+ above 4 feet. These are\n# determined by mouse-clicks on the graph, so they\n# are approximate values only. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "15 0 0" 39 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }