# Math 2250
# Maple Lab 8: Earthquake project
# F2010
#
# Name _____________________________________ Class Time __________
#
# Project 8. Solve problems L8.1 to L8.5. The problem headers:
#
# _______ PROBLEM L8.1. EARHQUAKE MODEL FOR A BUILDING.
# _______ PROBLEM L8.2. TABLE OF NATURAL FREQUENCIES AND PERIODS.
# _______ PROBLEM L8.3. UNDETERMINED COEFFICIENTS STEADY-STATE SOL
# _______ PROBLEM L8.4. PRACTICAL RESONANCE.
# _______ PROBLEM L8.5. EARTHQUAKE DAMAGE.
#
# FIVE FLOOR Model.
# Refer to the textbook of Edwards-Penney, section 7.4, page 437.
# Consider a building with five floors each weighing 50 tons. 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.
#
#
# PROBLEM L8.1. BUILDING MODEL FOR AN EARTHQUAKE.
# Model the 5-floor problem in Maple.
# Define the 5 by 5 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.
#
# Sanity check: Mass m=3125, and the 5x5 matrix contains fraction 16/5.
#
# Then find the eigenvalues of the matrix A to six digits, using the
# Maple command "eigenvals(A)."
# Sanity check: All six eigenvalues should be negative.
#
# 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]]);
# with(linalg): evalf(eigenvals(A));
# Problem L8.1
# Define k, m and the 5x5 matrix A.
# with(linalg): evalf(eigenvals(A));
# PROBLEM L8.2. TABLE OF NATURAL FREQUENCIES AND PERIODS.
# Refer to figure 7.4.17, page 437.
#
# Find the natural angular frequencies omega=sqrt(-lambda) for the
# five story building and also the corresponding periods
# 2PI/omega, accurate to six digits. Display the answers in a table .
# Compare with answers in Figure 7.4.17, page 437, for the 7-story case.
#
# Sample code for a 4x3 table, 4-story building.
# 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 L8.2
# ev:=[fill this in]: 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 L8.3. 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 5-vector of excursions from equilibrium
# of the corresponding 5 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.
#
# Define b:=0.25*w*w*vector([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]):
# A:=matrix([ [-20,10,0,0], [10,-20,10,0],
[0,10,-20,10],[0,0,10,-10]]);
# Au:=evalm(A+u*diag(1,1,1,1));
# c:=linsolve(Au,-b):
# evalf(c[1],2);
# PROBLEM L8.3
# Define w, u, b, A, Au, c
# evalf(c[1],2);
# PROBLEM L8.4. PRACTICAL RESONANCE.
# Consider the forced equation x'=Ax+cos(wt)b of L8.3 above with
# b:=0.25*w*w*vector([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.
#
# 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=1 to T=5, ordinates
# Max=0 to Max=10, the vector c(w) being the answer produced in L8.3
# 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]):
# A:=matrix([ [-20,10,0,0], [10,-20,10,0], [0,10,-20,10],
[0,0,10,-10]]);
# Au:=evalm(A+u*diag(1,1,1,1));
# C:=ww -> subs(w=ww,linsolve(Au,-b)):
# plot(Max(C(2*Pi/r)),r=1..5,0..10,numpoints=150);
# PROBLEM L8.4. WARNING: Save your file often!!!
# w:='w': Max:= c -> norm(c,infinity): u:=w*w:
# Define b
# Define A
# Define Au
# Define C
# plot(Max(C(2*Pi/r)),r=1..5,0..10,numpoints=150);
# PROBLEM L8.5. EARTHQUAKE DAMAGE.
# The maximum amplitude plot of L8.4 can be used to detect the
# 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 L8.4.
#
# (a) Replot the amplitudes in L8.4 for periods 1.5 to 5.5 and
# amplitudes 5 to 10.
# There will be several spikes.
# (b) Create several zoom-in plots, one for each spike, choosing a
# T-interval that shows the full spike.
# (c) Determine from the several 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. This example for the 4-floor problem.
#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 L8.5. WARNING: Save your file often!!
#(a) Re-plot the spikes.
# plot(Max(C(2*Pi/r)),r=1.5..5.5,5..10,numpoints=150);
#(b) Plot zoom-in graphs.
# one:=1.79..1.83:plot(Max(C(2*Pi/r)),r=one,5..10,numpoints=150);
# two:=???:plot(Max(C(2*Pi/r)),r=two,5..10,numpoints=150);
# three:=???:plot(Max(C(2*Pi/r)),r=three,5..10,numpoints=150);
# four:=???:plot(Max(C(2*Pi/r)),r=four,5..10,numpoints=150);
# five:=???:plot(Max(C(2*Pi/r)),r=five,5..10,numpoints=150);
#(c) Print period ranges.
# PeriodRanges:=[one,two,three,four,five];