{VERSION 4 0 "IBM INTEL NT" "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 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Math 2250" }}{PARA 0 "" 0 " " {TEXT -1 31 "Maple Lab 8: Earthquake project" }}{PARA 0 "" 0 "" {TEXT -1 9 "Fall 2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Name _____________________________________ Class Tim e __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Project 8. Solve problems L8.1 to L8.5. The problem headers:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "_______ PROBLEM L8.1. EARHQUAKE MODEL FOR A BUILDING." }}{PARA 0 "" 0 "" {TEXT -1 67 "_______ PROBLEM L8.2. TABLE OF NATURAL FREQUENCIES AN D PERIODS." }}{PARA 0 "" 0 "" {TEXT -1 68 "_______ PROBLEM L8.3. U NDETERMINED COEFFICIENTS STEADY-STATE SOL" }}{PARA 0 "" 0 "" {TEXT -1 46 "_______ PROBLEM L8.4. PRACTICAL RESONANCE." }}{PARA 0 "" 0 "" {TEXT -1 44 "_______ PROBLEM L8.5. EARTHQUAKE DAMAGE." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "SIX FLOOR Model." }}{PARA 0 "" 0 "" {TEXT -1 63 "Refer to the textbook of Edwards-Penney , section 7.4, page 437." }}{PARA 0 "" 0 "" {TEXT -1 63 "Consider a bu ilding with six floors each weighing 50 tons. Each" }}{PARA 0 "" 0 "" {TEXT -1 64 "floor corresponds to a restoring Hooke's force with const ant k=5" }}{PARA 0 "" 0 "" {TEXT -1 64 "tons/foot. Assume that ground \+ vibrations from the earthquake are" }}{PARA 0 "" 0 "" {TEXT -1 45 "mod eled by (1/4)cos(wt) with period T=2*Pi/w." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "PROBLEM L8.1. BUILDING MODEL FOR AN EARTHQUAKE. " }}{PARA 0 "" 0 "" {TEXT -1 36 "Model the 6-floor problem in Maple. \+ " }}{PARA 0 "" 0 "" {TEXT -1 68 "Define the 6 by 6 mass matrix M and H ooke's matrix K for this system" }}{PARA 0 "" 0 "" {TEXT -1 64 "and co nvert Mx''=Kx into the system x''=Ax where A is defined by" }}{PARA 0 "" 0 "" {TEXT -1 32 "textbook equation (1), page 437." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Sanity check: Mass m= 3125, and the 6x6 matrix contains fraction 16/5." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Then find the eigenvalues of the matrix A to six digits, using the " }}{PARA 0 "" 0 "" {TEXT -1 29 "Maple command \"eigenvals(A).\"" }}{PARA 0 "" 0 "" {TEXT -1 53 "Sanity check: All six eigenvalues should be negative." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# Sample Maple code for a model with 4 floors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# Use maple help to learn about evalf and eigenvals. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "# A:=matrix([ [-20,10,0 ,0], [10,-20,10,0], [0,10,-20,10],[0,0,10,-10]]);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 36 "# with(linalg): evalf(eigenvals(A));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# Pro blem L8.1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "# Define k, m \+ and the 6x6 matrix A." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# \+ with(linalg): evalf(eigenvals(A));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "PROBLEM L8.2. 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 65 "Find the natural angular frequenci es omega=sqrt(-lambda) for the" }}{PARA 0 "" 0 "" {TEXT -1 53 "six st ory building and also the corresponding periods" }}{PARA 0 "" 0 "" {TEXT -1 68 "2PI/omega, accurate to six digits. Display the answers i n a table ." }}{PARA 0 "" 0 "" {TEXT -1 70 "Compare with answers in Fi gure 7.4.17, page 437, for the 7-story case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "# Sample code for a 4x3 t able, 4-story building." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " # Use maple help to learn about nops and printf." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# ev:=[-10,-1.206147582,-35.32088886,-23.4729 6354]: n:=nops(ev):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "# Om ega:=lambda -> sqrt(-lambda):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "# format:=\"%10.6f %10.6f %10.6f\\n\": " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "# seq(printf(format,ev[i],Omega(ev[i]),2*evalf (Pi)/Omega(ev[i])), i=1..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# Problem L8.2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "# ev:=[fill this in]: n:=nops(ev):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "# Omega:=lambda -> sqrt(-lambda): f ormat:=\"%10.6f %10.6f %10.6f\\n\":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# seq(printf(format,ev[i],Omega(ev[i]),2*evalf(Pi)/Om ega(ev[i])),i=1..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "PROBLEM L8.3. UNDETERMINED COEFF ICIENTS " }}{PARA 0 "" 0 "" {TEXT -1 45 " STEADY-STATE PE RIODIC SOLUTION." }}{PARA 0 "" 0 "" {TEXT -1 69 "Consider the forced e quation x'=Ax+cos(wt)b where b is a constant" }}{PARA 0 "" 0 "" {TEXT -1 65 "vector. The earthquake's ground vibration is accounted fo r by the" }}{PARA 0 "" 0 "" {TEXT -1 47 "extra term cos(wt)b, which ha s period T=2Pi/w." }}{PARA 0 "" 0 "" {TEXT -1 68 "The solution x(t) \+ is the 6-vector of excursions from equilibrium" }}{PARA 0 "" 0 "" {TEXT -1 30 "of the corresponding 6 floors." }}{PARA 0 "" 0 "" {TEXT -1 65 "Sought here is not the general solution, which certainly contai ns" }}{PARA 0 "" 0 "" {TEXT -1 70 "transient terms, but rather the ste ady-state periodic solution, which " }}{PARA 0 "" 0 "" {TEXT -1 64 "is known from the theory to have the form x(t)=cos(wt)c for some" }} {PARA 0 "" 0 "" {TEXT -1 43 "vector c that depends only on A and \+ b." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Def ine b:=0.25*w*w*vector([1,1,1,1,1,1]): in Maple and find the" }} {PARA 0 "" 0 "" {TEXT -1 69 "vector c in the undetermined coefficien ts solution x(t)=cos(wt)c. " }}{PARA 0 "" 0 "" {TEXT -1 64 "Vector c depends on w. As outlined in the textbook, vector c " }}{PARA 0 "" 0 "" {TEXT -1 68 "can be found by solving the linear algebra problem \+ -w^2 c = Ac + b;" }}{PARA 0 "" 0 "" {TEXT -1 66 "see page 433. Don't p rint c, as it is too complex; instead, print" }}{PARA 0 "" 0 "" {TEXT -1 24 "c[1] as an illustration." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "#Sample code for defining b an d A, then solving for c " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "#in the 4-floor case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "# See maple help to learn about vector and linsolve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "# w:='w': u:=w*w: b:=0.25*u*vector ([1,1,1,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# A:=matri x([ [-20,10,0,0], [10,-20,10,0], [0,10,-20,10],[0,0,10,-10]]);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "# Au:=evalm(A+u*diag(1,1,1,1 ));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "# c:=linsolve(Au,-b ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "# evalf(c[1],2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "# PR OBLEM L8.3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "# Define w, u , b, A, Au, c " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# evalf( c[1],2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "PROBLEM L8.4. PRACTICAL RESONANCE." }} {PARA 0 "" 0 "" {TEXT -1 64 "Consider the forced equation x'=Ax+cos(w t)b of L8.3 above with" }}{PARA 0 "" 0 "" {TEXT -1 34 "b:=0.25*w*w*ve ctor([1,1,1,1,1,1])." }}{PARA 0 "" 0 "" {TEXT -1 64 "Practical resonan ce can occur if a component of x(t) has large" }}{PARA 0 "" 0 "" {TEXT -1 70 "amplitude compared to the vector norm of b. For example, \+ an earthquake" }}{PARA 0 "" 0 "" {TEXT -1 61 "might cause a small 3-in ch excursion on level ground, but the" }}{PARA 0 "" 0 "" {TEXT -1 66 " building's floors might have 50-inch excursions, enough to destroy" }} {PARA 0 "" 0 "" {TEXT -1 13 "the building." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "Let Max(c) denote the maximum m odulus of the components of vector c." }}{PARA 0 "" 0 "" {TEXT -1 70 " Plot g(T)=Max(c(w)) with w=(2*Pi)/T for periods T=0 to T=6, ordinates " }}{PARA 0 "" 0 "" {TEXT -1 66 "Max=0 to Max=10, the vector c(w) bein g the answer produced in L8.3" }}{PARA 0 "" 0 "" {TEXT -1 68 "above. \+ Compare your figure to the textbook Figure 7.4.18, page 438." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "# Sample \+ maple code to define the function Max(c), 4-floor building." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "# Use maple help to learn ab out norm, vector, subs and linsolve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "# with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "# w:='w': Max:= c -> norm(c,infinity); u:=w*w:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "# b:=0.25*w*w*vector([1,1,1 ,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "# A:=matrix([ [-20 ,10,0,0], [10,-20,10,0], [0,10,-20,10], [0,0,10,-10]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# Au:=evalm(A+u*diag(1,1,1,1));" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "# C:=ww -> subs(w=ww,lins olve(Au,-b)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "# plot(Ma x(C(2*Pi/r)),r=0..6,0..10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "# PRO BLEM L8.4. WARNING: Save your file often!!!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "# w:='w': Max:= c -> norm(c,infinity): u:=w*w: \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# Define b" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# Define A" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Define Au" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "# Define C " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# plot(Max(C(2*Pi/r)),r=0..6,0..10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 32 "PROBLEM L8.5. EARTHQUAKE DAMAGE." }}{PARA 0 "" 0 " " {TEXT -1 61 "The maximum amplitude plot of L8.4 can be used to detec t the " }}{PARA 0 "" 0 "" {TEXT -1 32 "of earthquake damage for a give n" }}{PARA 0 "" 0 "" {TEXT -1 63 "ground vibration of period T. A gro und vibration (1/4)cos(wt)," }}{PARA 0 "" 0 "" {TEXT -1 38 "T=2*Pi/w, \+ will be assumed, as in L8.4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "(a) Replot the amplitudes in L8.4 for periods 1 .5 to 5.5 and amplitudes 5 to 10. " }}{PARA 0 "" 0 "" {TEXT -1 27 "The re will be five spikes. " }}{PARA 0 "" 0 "" {TEXT -1 61 "(b) Create fi ve zoom-in plots, one for each spike, choosing a" }}{PARA 0 "" 0 "" {TEXT -1 38 "T-interval that shows the full spike. " }}{PARA 0 "" 0 " " {TEXT -1 67 "(c) Determine from the five zoom-in plots approximate i ntervals for" }}{PARA 0 "" 0 "" {TEXT -1 64 "the period T such that \+ some floor in the building will undergo" }}{PARA 0 "" 0 "" {TEXT -1 48 "excursions from equilibrium in excess of 5 feet." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "# Example: Zoom-i n on a spike for amplitudes 5 feet to 10 feet," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "#periods 1.97 to 2.01." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "#with(linalg): w:='w': Max:= c -> norm(c,infin ity); u:=w*w:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "#Au:=matri x([ [-20+u,10,0,0], [10,-20+u,10,0], [0,10,-20+u,10],[0,0,10,-10+u]]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "#b:=0.25*w*w*vector([1, 1,1,1]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "#C:=ww -> subs( w=ww,linsolve(Au,-b)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "# plot(Max(C(2*Pi/r)),r=1.97..2,01,5..10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "# PROBLEM L8.5. WARNING: Save your file often!!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "#(a) Re-plot the five spikes ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "# plot(Max(C(2*Pi/r)), r=1.5..5.5,5..10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "#(b) Plot five zoom-in graphs." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 65 "# one:=1.79..1.83:plot(Max(C(2*Pi/r)),r=one,5. .10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "# tw o:=???:plot(Max(C(2*Pi/r)),r=two,5..10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "# three:=???:plot(Max(C(2*Pi/r)),r= three,5..10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# four:=???:plot(Max(C(2*Pi/r)),r=four,5..10,numpoints=150);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "# five:=???:plot(Max(C(2*Pi/ r)),r=five,5..10,numpoints=150);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "#(c) Print period ranges." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 42 "# PeriodRanges:=[one,two,three,four,five];" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "12 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }