Maple Assignment 3 - Earthquake Project

Math 2250-2 Summer 2007 Name: ??????

restart;

with(LinearAlgebra):

Problem 1

Define the mass m and spring constant k

m:= ????; k:= ????;

The mass matrix M is

M:= m*Matrix(6,6,shape=identity);

The stiffness matrix K is

K:= Matrix([[k,k,k,k,k],[-2*k,-2*k,-2*k,-2*k,-2*k,-k],[k,k,k,k,k]],shape=band[1,1],scan=band[1,1]);

Convert the system Mx'' = K x to x''=Ax where A is

Minv:= MatrixInverse(M);

A:= MatrixMatrixMultiply(Minv,K);

Problem 2

Find the eigenvalues of A

lambda := evalf(Eigenvalues(A));

Now find the natural frequencies of vibration

freq:= v -> sqrt(-v);

omega0:= evalf(map(freq,lambda));

and the periods are

period:= w0 -> ?????;

T0:= evalf(map(period,omega0));

Problem 3

Define the vector b that corresponds to the amplitude of the external forcing term for the system x'' = Ax + f(t) where f(t) = b cos(w*t)

beta:= ?????;

b:= Matrix([[beta],[beta],[beta],[beta],[beta],[beta]]);

The long term behavior of the system is given by the particular solution xp(t) = c cos(w*t) where c is found by solving (A+w^2*I) c = -b

P:= (A+omega^2*Matrix(6,6,shape=identity));

Pinv := MatrixInverse(P):

c:= MatrixMatrixMultiply(Pinv,omega^2*b):

The components of vector c give the amplitude/size ofthe earthquake motion experienced by each floor. The maximum amplitude of vibration is found by finding the biggest element of c

A:= w -> subs(omega=w,simplify(Norm(c))):

Plot of this maximal amplitude as a function of the forcing frequency w

plot(A(w),w=0..10,y=0..10,numpoints=10000);

Which will cause a greater damage, an earthquake of frequency 1.8/sec or that with frequency 0.8/sec?

What does each spike in your plot correspond to?

Problem 4

Replot the function A(w) to assess possible damage to an earthquake. Using the first spike of the previous plot, graph the function A(w) over a small enough interval and determine an approximate interval within which some floor of the building will undergo oscillations in excess of five feet.

plot(A(w),w=??????,y=0..10,numpoints=20000);

Estimate of forcing frequency interval that will cause damage

?????