# Math 2250 # Maple Lab 6: Earthquake project # Spring 2006 # # Name _____________________________________ Class Time __________ # # Project 3. Solve problems L6.1 to L6.6. The problem headers: # # _______ PROBLEM L6.1. BUILDING MODEL FOR AN EARTHQUAKE. # _______ PROBLEM L6.2. TABLE OF NATURAL FREQUENCIES AND PERIODS. # _______ PROBLEM L6.3. UNDETERMINED COEFFICIENTS STEADY-STATE # PERIODIC SOLUTION. # _______ PROBLEM L6.4. PRACTICAL RESONANCE. # _______ PROBLEM L6.5. EARTHQUAKE DAMAGE. # _______ PROBLEM L6.6. SIX FLOORS. # > with(linalg): # # 1. BUILDING MODEL FOR AN EARTHQUAKE. # Refer to the textbook of Edwards-Penney, section 7.4, page 437. # Consider a building with 7 floors. # # Let the mass in slugs of each story be m=1000.0 and let the spring # constant be k=10000.0 (lbs/foot). # Define the 7 by 7 mass matrix M and Hooke's matrix K for this system # and convert Mx''=Kx into # the system x''=Ax where A is defined by textbook equation (1) , page # 437. # # PROBLEM L6.1 # Find the eigenvalues of the matrix A to six digits, using the Maple # command "eigenvals(A)." # Justify in particular that all seven eigenvalues are negative by # direct computation. # # # Sample Maple code for a model with 4 floors. # # Use maple help to learn about evalf and eigenvals. # A:=matrix([ [-20,10,0,0], [10,-20,10,0], [0,10,-20,10], # [0,0,10,-10]]); # evalf(eigenvals(A)); # > # Problem L6.1 > # # 2. TABLE OF NATURAL FREQUENCIES AND PERIODS. # Refer to figure 7.4.17, page 437. # # PROBLEM L6.2. # Find the natural angular frequencies omega=sqrt(-lambda) for the # seven story building and also the corresponding periods # 2PI/omega, accurate to six digits. Display the answers in a table . # The answers appear in Figure 7.4.17, page 437. # # # Sample code for a 4x3 table. # # Use maple help to learn about nops and printf. # ev:=[-10,-1.206147582,-35.32088886,-23.47296354]: n:=nops(ev): # Omega:=lambda -> sqrt(-lambda): # format:="%10.6f %10.6f %10.6f\n": # seq(printf(format,ev[i],Omega(ev[i]),2*evalf(Pi)/Omega(ev[i])),i=1..n # ); # > # Problem L6.2 > # # 3. UNDETERMINED COEFFICIENTS STEADY-STATE PERIODIC SOLUTION. # Consider the forced equation x'=Ax+cos(wt)b where b is a constant # vector. The earthquake's # ground vibration is accounted for by the extra term cos(wt)b, which # has period T=2Pi/w. # The solution x(t) is the 7-vector of excursions from equilibrium # of the corresponding 7 floors. # Sought here is not the general solution, which certainly contains # transient terms, but rather the # steady-state periodic solution, which is known from the theory to have # the form x(t)=cos(wt)c # for some vector c that depends only on A and b. # # PROBLEM L6.3. # Define b:=0.25*w*w*vector([1,1,1,1,1,1,1]): in Maple and find the # vector c in the undetermined # coefficients solution x(t)=cos(wt)c. Vector c depends on w. As # outlined in the textbook, vector c # can be found by solving the linear algebra problem -w^2 c = Ac + b; # see page 433. Don't print c, # as it is too complex; instead, print c[1] as an illustration. # # # Sample code for defining b and A, then solving for c in the # 4-floor case. # # See maple help to learn about vector and linsolve. # w:='w': u:=w*w: b:=0.25*u*vector([1,1,1,1]): # Au:=matrix([ [-20+u,10,0,0], [10,-20+u,10,0], [0,10,-20+u,10], # [0,0,10,-10+u]]); # c:=linsolve(Au,-b): # evalf(c[1],2); # > # PROBLEM L6.3 > # # 4 PRACTICAL RESONANCE. # 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]). # Practical resonance can occur if a component of x(t) has large # amplitude compared to the vector # norm of b. For example, an earthquake might cause a small 3-inch # excursion on level ground, but # the building's floors might have 50-inch excursions, enough to destroy # the building. # # PROBLEM L6.4. # Let Max(c) denote the maximum modulus of the components of vector c. # Plot g(T)=Max(c(w)) with w=(2*Pi)/T # 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. # Compare your figure to the textbook Figure 7.4.18, page 438. # # # Sample maple code to define the function Max(c), 4-floor building. # # Use maple help to learn about norm, vector, subs and linsolve. # with(linalg): # w:='w': Max:= c -> norm(c,infinity); u:=w*w: # b:=0.25*w*w*vector([1,1,1,1]): # Au:=matrix([ [-20+u,10,0,0], [10,-20+u,10,0], [0,10,-20+u,10], # [0,0,10,-10+u]]); # C:=ww -> subs(w=ww,linsolve(Au,-b)): # plot(Max(C(2*Pi/r)),r=0..6,0..10,numpoints=150); # > # PROBLEM L6.4. WARNING: Save your file often!!! > # # # 5. EARTHQUAKE DAMAGE. # The maximum amplitude plot of 3.4 can be used to detect the likelihood # of earthquake damage for a given # ground vibration of period T. A ground vibration (1/4)cos(wt), # T=2*Pi/w, will be assumed, as in 3.4. # # PROBLEM L6.5. # (a) Replot the amplitudes in 3.4 for periods 1.14 to 4 and amplitudes # 5 to 10. There will be four spikes. # (b) Create four zoom-in plots, one for each spike, choosing a # T-interval that shows the full spike. # (c) Determine from the four zoom-in plots approximate intervals for # the period T such that # some floor in the building will undergo excursions from equilibrium in # excess of 5 feet. # # # Example: Zoom-in on a spike for amplitudes 5 feet to 10 feet, # periods 1.97 to 2.01. # with(linalg): w:='w': Max:= c -> norm(c,infinity); u:=w*w: # Au:=matrix([ [-20+u,10,0,0], [10,-20+u,10,0], [0,10,-20+u,10], # [0,0,10,-10+u]]); # b:=0.25*w*w*vector([1,1,1,1]): # C:=ww -> subs(w=ww,linsolve(Au,-b)): # plot(Max(C(2*Pi/r)),r=1.97..2,01,5..10,numpoints=150); # > # PROBLEM L6.5. WARNING: Save your file often!! > #(a) Plot four spikes on separate graphs > #(b) Plot four zoom-in graphs. > #(c) Print period ranges. > # # # 6. SIX FLOORS. # Consider a building with six floors each weighing 50 tons. Assume each # floor corresponds to a restoring # Hooke's force with constant k=5 tons/foot. Assume that ground # vibrations from the earthquake are modeled by # (1/4)cos(wt) with period T=2*Pi/w (same as the 7-floor model above). # # PROBLEM L6.6. # Model the 6-floor problem in Maple. Plot the maximum amplitudes # against the period 0 to 6 and amplitude # 4 to 10. Determine from the graphic the period ranges which cause the # amplitude plot to spike above 4 feet. # Sanity check: m=3125, and the 6x6 matrix contains fraction 16/5. # There are five spikes. # > # PROBLEM L6.6. WARING: Save your file often!! > # Define k, m and the 6x6 matrix A. > # Amplitude plot for T=0..6,C=4..10 > # Plot five zoom-in graphs > # From the graphics, five T-ranges give amplitude > # spikes above 4 feet. These are determined by left > # mouse-clicks on the graph, so they are approximate values only. >