{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "_cstyle1" -1 204 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle1" -1 200 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_psty le2" -1 201 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "_pstyle3" -1 202 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 200 "" 0 "" {TEXT 204 9 "Math 2250" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 31 "Maple Lab 8: Earthquake project" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 5 "F2008" }}{PARA 200 " " 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 "Name _________ ____________________________ Class Time __________" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 61 "P roject 8. Solve problems L8.1 to L8.5. The problem headers:" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 57 "_______ PROBLEM L8.1. EARHQUAKE MODEL FOR A BUILDING." } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 67 "_______ PROBLEM L 8.2. TABLE OF NATURAL FREQUENCIES AND PERIODS." }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 68 "_______ PROBLEM L8.3. UNDETERMINED COEFFICIENTS STEADY-STATE SOL" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 46 "_______ PROBLEM L8.4. PRACTICAL RESONANCE." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 44 "_______ PROBLEM L8.5. E ARTHQUAKE DAMAGE." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 16 "SIX FLOOR Model." }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 63 "Refer to the textbook of Edwards-Penne y, section 7.4, page 437." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 63 "Consider a building with six floors each weighing 50 tons. Eac h" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 64 "floor corresponds to a restoring Hooke's force with constant k=5" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 64 "tons/foot. Assume that ground vibratio ns from the earthquake are" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 45 "modeled by (1/4)cos(wt) with period T=2*Pi/w." }{TEXT 204 0 " " }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 47 "PROBLEM L8 .1. BUILDING MODEL FOR AN EARTHQUAKE." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 36 "Model the 6-floor problem in Maple. " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 "Define the 6 by 6 mass matrix M a nd Hooke's matrix K for this system" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 64 "and convert Mx''=Kx into the system x''=Ax where A is defined by" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 32 "textboo k equation (1), page 437." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "Sanity check: Mass m=3125, and the 6x6 matrix contains fraction 16/5." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 67 "Then find the eigenvalues of the matrix A to six digits, using the " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 29 "Maple command \"eigenvals(A ).\"" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 53 "Sanity check: \+ All six eigenvalues should be negative." }{TEXT 204 0 "" }}{PARA 200 " " 0 "" {TEXT 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 46 "# Sample \+ Maple code for a model with 4 floors." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 52 "# Use maple help to learn about e valf and eigenvals." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 72 "# A:=matrix([ [-20,10,0,0], [10,-20,10,0], [0,10,-20, 10],[0,0,10,-10]]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 36 "# with(linalg): evalf(eigenvals(A));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 14 "# Problem L8.1" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 35 "# Define k, m and the 6x6 matrix A." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 36 "# wit h(linalg): evalf(eigenvals(A));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 55 "PROBLEM L8.2. TABLE OF NATURAL FREQUENCIES AND PERIODS." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 33 "Refer to figure 7.4.17, pag e 437." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 "Find the natural angular frequencies omega= sqrt(-lambda) for the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 53 "six story building and also the corresponding periods" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 "2PI/omega, accurate to six digi ts. Display the answers in a table ." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "Compare with answers in Figure 7.4.17, page 437, fo r the 7-story case." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 " " }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 48 "# Sample code for a 4x3 table , 4-story building." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 48 "# Use maple help to learn about nops and printf." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 64 "# ev: =[-10,-1.206147582,-35.32088886,-23.47296354]: n:=nops(ev):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 33 "# Ome ga:=lambda -> sqrt(-lambda):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 " > " 0 "" {MPLTEXT 1 0 38 "# format:=\"%10.6f %10.6f %10.6f\\n\": " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 74 "# seq (printf(format,ev[i],Omega(ev[i]),2*evalf(Pi)/Omega(ev[i])), i=1..n); " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> \+ " 0 "" {MPLTEXT 1 0 14 "# Problem L8.2" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 34 "# ev:=[fill this in]: n:=nops(ev) :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 69 "# Omega:=lambda -> sqrt(-lambda): format:=\"%10.6f %10.6f %10.6f\\n\" :" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 73 "# seq(printf(format,ev[i],Omega(ev[i]),2*evalf(Pi)/Omega(ev[i])),i=1..n );" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 41 "PROBLEM L8.3. UNDETERMINE D COEFFICIENTS " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 45 " \+ STEADY-STATE PERIODIC SOLUTION." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 "Consider the forced equation x'=Ax+cos(wt)b where b is a constant" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 "vector. The earthquake's ground vibration is accounted for by \+ the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 47 "extra term cos( wt)b, which has period T=2Pi/w." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 "The solution x(t) is the 6-vector of excursions from equilibrium" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 30 "of the corresponding 6 floors." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 "Sought here is not the general solution, which certainly conta ins" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "transient terms , but rather the steady-state periodic solution, which " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 64 "is known from the theory to have \+ the form x(t)=cos(wt)c for some" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 43 "vector c that depends only on A and b." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 65 "Define b:=0.25*w*w*vector([1,1,1,1,1,1]): in Maple and find the " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 69 "vector c in the \+ undetermined coefficients solution x(t)=cos(wt)c. " }{TEXT 204 0 "" } }{PARA 200 "" 0 "" {TEXT 204 64 "Vector c depends on w. As outlined \+ in the textbook, vector c " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 "can be found by solving the linear algebra problem -w^2 c = Ac + b;" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 "see pa ge 433. Don't print c, as it is too complex; instead, print" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 24 "c[1] as an illustration." } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 201 "> " 0 " " {MPLTEXT 1 0 56 "#Sample code for defining b and A, then solving fo r c " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 21 "#in the 4-floor case." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> \+ " 0 "" {MPLTEXT 1 0 52 "# See maple help to learn about vector and lin solve." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 47 "# w:='w': u:=w*w: b:=0.25*u*vector([1,1,1,1]):" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 73 "# A:=matrix([ [-20,1 0,0,0], [10,-20,10,0], [0,10,-20,10],[0,0,10,-10]]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 31 "# Au:=evalm(A+u*diag (1,1,1,1));" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 23 "# c:=linsolve(Au,-b): " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 17 "# evalf(c[1],2);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 14 "# PROBLEM L8.3" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 28 "# Define w, u, b, A, Au, c " } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 16 "# eva lf(c[1],2);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 200 "" 0 "" {TEXT 204 34 "PROBLEM L8 .4. PRACTICAL RESONANCE." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 64 "Consider the forced equation x'=Ax+cos(wt)b of L8.3 above wi th" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 34 "b:=0.25*w*w*vect or([1,1,1,1,1,1])." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 64 " Practical resonance can occur if a component of x(t) has large" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "amplitude compared to the vector norm of b. For example, an earthquake" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 61 "might cause a small 3-inch excursion o n level ground, but the" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 "building's floors might have 50-inch excursions, enough to des troy" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 13 "the building. " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 68 "Let Max(c) denote the maximum modulus of the componen ts of vector c." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 70 "Plo t g(T)=Max(c(w)) with w=(2*Pi)/T for periods T=0 to T=6, ordinates" } {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 66 "Max=0 to Max=10, the \+ vector c(w) being the answer produced in L8.3" }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 68 "above. Compare your figure to the tex tbook Figure 7.4.18, page 438." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 68 "# Sample maple c ode to define the function Max(c), 4-floor building." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 64 "# Use maple help to \+ learn about norm, vector, subs and linsolve." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 16 "# with(linalg):" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 47 "# w: ='w': Max:= c -> norm(c,infinity); u:=w*w:" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 33 "# b:=0.25*w*w*vector([1,1 ,1,1]):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 73 "# A:=matrix([ [-20,10,0,0], [10,-20,10,0], [0,10,-20,10], [0,0,1 0,-10]]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 32 "# Au:=evalm(A+u*diag(1,1,1,1));" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 39 "# C:=ww -> subs(w=ww,lins olve(Au,-b)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 51 "# plot(Max(C(2*Pi/r)),r=0..6,0..10,numpoints=150);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 48 "# PROBLEM L8.4. WARNING: S ave your file often!!!" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 49 "# w:='w': Max:= c -> norm(c,infinity): u:=w*w: \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 10 "# \+ Define b" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 10 "# Define A" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 11 "# Define Au" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 " > " 0 "" {MPLTEXT 1 0 12 "# Define C " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 50 "# plot(Max(C(2*Pi/r)),r=0..6,0..1 0,numpoints=150);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 32 "PROBLEM L8.5. EARTHQUAKE DAMAGE." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 61 "The maximum amplitude plot of L8.4 can be used to d etect the " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 32 "of earth quake damage for a given" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 63 "ground vibration of period T. A ground vibration (1/4)cos(wt) ," }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 38 "T=2*Pi/w, will be assumed, as in L8.4." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 81 "(a) Replot the amplitudes in L8 .4 for periods 1.5 to 5.5 and amplitudes 5 to 10. " }{TEXT 204 0 "" }} {PARA 200 "" 0 "" {TEXT 204 27 "There will be five spikes. " }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 61 "(b) Create five zoom-in plo ts, one for each spike, choosing a" }{TEXT 204 0 "" }}{PARA 200 "" 0 " " {TEXT 204 38 "T-interval that shows the full spike. " }{TEXT 204 0 " " }}{PARA 200 "" 0 "" {TEXT 204 67 "(c) Determine from the five zoom-i n plots approximate intervals for" }{TEXT 204 0 "" }}{PARA 200 "" 0 " " {TEXT 204 64 "the period T such that some floor in the building wi ll undergo" }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 48 "excursio ns from equilibrium in excess of 5 feet." }{TEXT 204 0 "" }}{PARA 200 "" 0 "" {TEXT 204 0 "" }}{PARA 201 "> " 0 "" {MPLTEXT 1 0 63 "# Exampl e: Zoom-in on a spike for amplitudes 5 feet to 10 feet," }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 22 "#periods 1.97 to 2.01." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 59 "#with(linalg): w:='w': Max:= c -> norm(c,infinity); u:=w*w:" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 80 "#Au:= matrix([ [-20+u,10,0,0], [10,-20+u,10,0], [0,10,-20+u,10],[0,0,10,-10+ u]]);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 31 "#b:=0.25*w*w*vector([1,1,1,1]):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 37 "#C:=ww -> subs(w=ww,linsolve(Au,- b)):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 55 "#plot(Max(C(2*Pi/r)),r=1.97..2,01,5..10,numpoints=150);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 47 "# PROBLEM L8.5. WARNING: S ave your file often!!" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 " " {MPLTEXT 1 0 29 "#(a) Re-plot the five spikes." }{MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 54 "# plot(Max(C(2*Pi/r)),r= 1.5..5.5,5..10,numpoints=150);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 30 "#(b) Plot five zoom-in graphs." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 65 "# one :=1.79..1.83:plot(Max(C(2*Pi/r)),r=one,5..10,numpoints=150);" } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 58 "# two :=???:plot(Max(C(2*Pi/r)),r=two,5..10,numpoints=150);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 62 "# three:=???:plot( Max(C(2*Pi/r)),r=three,5..10,numpoints=150);" }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 60 "# four:=???:plot(Max(C(2*P i/r)),r=four,5..10,numpoints=150);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 60 "# five:=???:plot(Max(C(2*Pi/r)),r =five,5..10,numpoints=150);" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 25 "#(c) Print period ranges." }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 201 "> " 0 "" {MPLTEXT 1 0 42 "# PeriodRanges:=[one, two,three,four,five];" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 201 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 202 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 2 0 " 1 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }