Homework 8 - Duke University

CEE 201L. Uncertainty, Design, and Optimization
Department of Civil and Environmental Engineering
Duke University
Homework 8, due: Wednesday April 15, 2015
1. (20 points) Wile E. Coyote wants to spring off the end of a diving board onto the
Road-Runner. Of course, the Road-Runner has painted the end of the diving board
with tar. When Wile E. Coyote lands on the tar he sticks to the board and he and the
diving board oscillate with a natural frequency ωn . Neglecting the mass of the diving
board, derive an expression for ωn in terms of EI of the diving board, L, and the mass
M of Wile E. Coyote.
h
D
C
B
3L/2
A
L
2. (20 points) Continuing from problem 1, as Wile E. Coyote’s leaps onto the end of
the board, his feet reach a maximum height of h above the diving board, as shown.
Neglecting damping, derive an expression for the maximum flexural deflection of the
end of the diving board, D, in terms of M , g, h, EI, and L.
3. (30 points) An under-damped simple oscillator is driven by sinusoidal base motion,
z(t) = Z¯ cos ωt. At time = 0 the position and velocity of the mass relative to the
ground are do and vo . Derive an expression for the response displacement x(t) that
includes both the transient response and the steady-state response, as a function of
time, t, forcing frequency ω, undamped natural frequency ωn , damping ratio ζ, and
initial conditions, do , and vo .
4. (15 points) The magnitude of the resonant peak of the frequency response of the relative
displacement of an under-damped single-degree-of-freedom oscillator to the base-motion
displacement (X/Z) is exactly equal to 2. What is the damping ratio? (Use three
significant figures.)