CEE 502 Problem Set 6 Winter 2008
(Due 4:00 PM, Monday, Feb 25, in 233 More)
(YOU MAY WORK IN GROUPS OF TWO ON THIS ASSIGNMENT)
Reading: Chopra Sections 5.1-5.5 (skip 5.2)
Machinery with rotating parts imparts a sinusoidal force, F(t), to the tower shown below.
The tower will be modeled as a single-degree-of-freedom system with mass, m, and
stiffness, k and damping ratio, ζ. The sinusoidal force has a period Tb and maximum
intensity ±Fmax
Assume that the weight of the tower is 20 kips, the natural period of the tower, Tn, is 0.50
seconds, and the damping ratio is 5%. The maximum intensity of the driving force, Fmax,
is 2 kips, and the period of the sinusoidal forcing function, Tb, is 0.25 seconds.
1. Determine the total displacement response of the system, u(t), by superimposing the
particular (steady-state) and characteristic (transient) solutions. You are free to use
solutions that are derived in the textbook.
2. Using the Constant Acceleration Method, calculate the tower's displacement response
for the first 3 seconds. Use a time step, h, of:
a) h = Tn/50
b) h = Tn/20
c) h = Tn/10
d) h = Tn/3
3. a) On a single plot, show the analytical solution and the numerical solutions for h
= Tn/50 and h= Tn/100.
b) On a single plot, show the analytical solution and the numerical solutions for h
= Tn/20 and h= Tn/10.
c) On a single plot, show the analytical solution and the numerical solutions for
h= Tn/3.
3. For each time step considered in Problem 1 (a through d), calculate the maximum
normalized error as: () ()
()
i
numerical i exact
exact
Maximum u t u t
Normalized Error Maximum u t
−
=
Summarize your results by plotting the normalized error vs h/Tn on a log-log plot.
Determine the slope of the line.
F(t)
t
Fmax
Tb F(t)
