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Unformatted text preview: ﬂehwmwwa CE 573: Structural Dynamics
Exam #1 Problem #1 (25 pts.) 5 ft 5 ft
Given: A compressor unit weighing 3000 lb is supported by two parallel, simply supported steel beams (E = 30, 000 ksi for each beam). The motor in the unit runs at 300 revolutions per minute and has an unbalance W'e = 180 lbin. You may assume that the mass of the beams is negligible, the supports share
all loads equally, and that the damping ratio 5 for the system as a whole is negligible. Reguired: (a) You wish to choose a value for] such that the force that the beams exert on the supports in
the steady state is always downward. There are two possible ranges of I that will accomplish this.
Detetmine each range. (Hint: there is more than one contribution to the force exerted on a support!) (b) There is a nonzero value for 6 Such that any choice of I will lead to an always downward force on the supports in the steady state. What is this value of 6 ? (You may assume that this value is small.) _ PL’
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Rigid base Given: You are asked to analyze a landing platform used to cushion supplies being drOpped from
airplanes into remote regions. The platform consists of a two rigid bases of negligible mass, two springs
having identical stiffnesses kl 2 = 80 kN/m, and a viscous damper having a damping coefﬁcient c = 6.4
kNsec/m. The distance between the top and bottom base is initially h = 026 m, which you may assume
is the unstretched length of each spring. The platform is designed to carry a mass m = 400 kg. When it
lands, you may assume that the downward speed of the combined system is 4.5 m/s and that the platform
does not bounce (tie, the bottom base remains in contact with the surface). Reguired: At what time aﬂer impact does the top base come closest to the bottom base, and what is the
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spring of stiffness k and a viscous damper with a damping coefﬁcient 0. Each mass is subjected to an
external force, respectivelyF, (t) and FZU) . The displacements of each mass are ua (t) and 112(1) , and the
system is in static equilibrium when 141 = u2 = 0. Deﬁne u,(t) =u2 (t) —uI (t) as the relative . _ 1 l 1
displacement between the two masses, and deﬁne meq (the “equwalent mass”) Via — = — + — . m mm eq 1 2 Reguired: (a) Show that the equation of motion for the relative displacement ur (t) is 2
m
mm, d Md”r maniac).
dt dt m1 m2
(b) Let m1 = 40 kg, m2 = 50 kg, k = 200 N/m, (2 = 4Ns/m. F10) = 0 and F2(t) = 135 cos 2t N. Find the steadystate solution for u, (t). meg +ku, =— (0) Suppose that both mass #1 and mass #2 are initially at rest when the force F20) in part (b) is applied. Would you expect the M solution for ur(t) (i. e., transient plus steadystate solution) to be the same as the steadystate solution found in part (b)? Explain why or Why not. (Note: you do n_ot need to solve for
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.7 A: uJMM‘oH‘M: —_§7¢(M{).‘14?M3=—.338:¢O WWM /MMMM\WM%W~ Problem #4 (20 pts.) Given: A free vibration test is conducted on an elevated water tank as shown. A cable attached to the tank is pulled (by a very strong technician!) such that it applies a force of 20.02 kips at an angle 9 =35°. At
this time, you measure an initial horizontal displacement for the tank of 0.20 in. The cable is then released
and the subsequent free vibration of the tank is recorded. It is noted that it takes 2.0 seconds to complete
four full cycles of vibration and that the measured amplitude of the vibration at that time is 0.10 in. Reguired: Estimate the following quantities:
(a) the damping factor 4‘ , (b) the undamped natural frequency a) , (c) the mass, stiffness, and viscous damping coefﬁcient for the tank, (d) the number of m cycles it would take for the vibration amplitude to go below 0.02 inches. (6) The test is now repeated, but this time the initial horizontal displacement of the tank is 1% larger.
The initial displacement and amplitude of vibration after four full cycles are again measured, and it is
found that it still takes 2.0 seconds to complete four cycles. Which test (small initial displacement or large
initial displacement) would you expect to give the best estimates for the quantities in parts (a)(c)? Give at
least one reason to explain your choice. (Assume that you can make your measurements with the same
level of precision in both tests.) Scum: (a3 Amawmwiﬂewwtm/
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 Fall '05
 Whalen
 initial displacement, steadystate solution, initial horizontal displacement, relative displacement ur

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