UNIVERSITY OF ILLINOIS
Department of Civil and Environmental Engineering
CEE 573
B.F. Spencer, Jr.
Spring 2019
Due: April 4
Problem 1
Consider the simple 3-DOF system from Homework #2 and presented in the figure at the right, where k 1 = k 2 = 200 kN/m,
f
k 3 = 150 kN/m, m = 1200 kg. Assume that the damping is given
as 1% critical damping in each of the first two modes and 0.5% of
critical damping in the third mode. Simulate the response of the
structure due to an impulsive load applied at the top floor of the
structure and add 2% noise to the responses and the input. Assume
that the pulse width is 0.1 seconds and half-sine shape. Remember to
apply a force window for the excitation and an exponential window
for the response.
m, k3
m, k2
m, k1
x3
x2
x1
a. From the simulated data, calculate and plot the three transfer
functions of the building using 5 averages. Compare these
results to the exact results. What is the optimal value to use
for the exponent in the response window?
b. What is the affect of changing the width of the pulse to 0.2 seconds.
Problem 2
Consider again the simple 3-DOF system in problem #1.
c. Use the simulation program you developed for previous homework to “experimentally”
obtain the transfer function from an input ground excitation to the 3rd floor relative
displacement (i.e., relative to the ground), H x3 ꞏꞏx g    . Use the Matlab code given below to
estimate the pulse response function from the transfer function data and compare this
result with that given by the Matlab “impulse” command. Remember to account for the
factor of 1  t .
function [y]=irf(H)
% H(w_k) complex valued scalar FRF (column vector)
Htemp=H.'; % Htemp is a row vector
N=length(H);
if isinteger(N/2)
Htemp=[Htemp conj(fliplr(Htemp(:,2:N)))].';
else
Htemp=[Htemp conj(fliplr(Htemp(:,2:N-1)))].';
end
y=fliplr(real(ifft(Htemp)).');
d. Use the “experimental” pulse response function obtained in part (a) as input to the SISO
ERA algorithm distributed to you. From the ERA discrete time model, determine the
natural frequencies/damping factors and compare them to the exact results. Also calculate
the transfer function associated with the ERA model obtained and compare it with the
exact result, as well as the original transfer function estimated in part (a). (Remember that
comparing the model to the original data is an all important final step to any model
development!)