University Duke Department of Civil and Environmental Engineering CE 130L. Uncertainty, Design, and Optimization Homework 6, Due Friday April 18, 2009
1. The structural system shown below is practically massless, except for the concentrated mass M shown at point B. The slender beam has exural rigidity EI and the cable has axial rigidity EA. The mass can vibrate freely in the vertical direction.
1111 0000 1111 0000
D
M
H
111 000 111 000 111 000
A
B
C
2L/3
L/3
(a) Find the natural frequency n of this system. Hint: start by nding the relationship between a static force displacement collocated with the concentrated mass (at point B). (You may neglect the eects of shear deformation in the beam.) (b) Express the natural frequency in terms of the vertical displacement at point B due to the weight of the mass, M g.
1
2. The vertical slender cantilever beam shown below is shaken with base accelerations z (t).
x(t) M
11 00 11 00 11 00 11 00 11 00 11 00 11 00
EI z(t)
H
111111 000000 111111 000000 111111 000000 111111111111 000000000000 111111111111 000000000000 111111111111 000000000000
(a) Find the stiness of the cantilever relating horizontal forces point at the end of the cantilever to the horizontal displacements at the end of the cantilever. (b) Neglecting the P eect discussed in class on Friday April 10, express the natural frequency n in terms of M , EI, and H. (c) Now, considering the P eect discussed in class on Friday April 10, at what height H will the lateral stiness drop to zero? (d) For EI = 1000 kN.m2 , H = 1 m, M = 1 ton (1000 kg), and assuming the damping ratio is 0.05 (ve percent), calculate the damped natural frequency d including the P eect. (e) If z(t) = 0 and at time t = 0 the initial displacement do is 0.1 m and the initial velocity vo is 0.3 m/s, create a plot of x(t) vs. t for 0 < t <...