HW1soln

HW1soln - CE 573: Structural Dynamics Solution to HW#1...

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CE 573: Structural Dynamics Solution to HW#1 Problem #1 Given : A concrete slab (mass = 20,000 kg) forms the floor of a manufacturing facility. To prevent excessive vibrations from reaching the equipment placed on this floor, the slab is supported by six pairs of beams connected to the slab, the ground, and each other as shown. Each beam is made from structural steel (E = 200 GPa) and has a circular cross section with a diameter of 0.2 m. The vertical beams have a length of 2.5 m, while the horizontal beams have a length of 2.0 m. The vertical beams are rigidly fixed to the ground, the horizontal beams are rigidly fixed to the slab, and the blue circles represent ball and socket joints joining the horizontal and vertical beams. Required : (a) Determine the natural frequency for small amplitude vibrations in the x -direction. (b) Determine the natural frequency for small amplitude vibrations in the y -direction. (c) Determine the natural frequency for small amplitude vibrations in the z -direction. (d) Suppose that the system is redesigned to use a very soft material for the vertical beams – the Young’s modulus for the vertical beams is now smaller by a factor of 100. It is observed that the vibration frequencies in the x - and y -directions become much smaller and are nearly equal to each other. On the other hand, the vibration frequency in the z -direction is only slightly lowered. Explain qualitatively why each of these observations occurs. Hint : You may ignore the possibility that the slab rotates and simply consider pure translations in each of the three directions when calculating the frequencies.
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Solution : Before solving for the various natural frequencies, there are two observations to make that hold for all cases: 1) Suppose that k represents the stiffness of one of the vertical/horizontal beam pairs for a given type of motion. Since all six beam pairs must displace in identical manners, the six beam pairs act in parallel with each other, which implies that the total stiffness of the system is 6 total kk = . 2) For a given vertical/horizontal beam pair, if you displace the fixed vertical end a certain amount in one of the given directions, the corresponding ball and socket joint will not necessarily displace the same amount. However, the force needed to displace the fixed vertical end must be transmitted completely to the ball and socket joint and thus completely through to the ground. Since the force carried by both beams is the same, their stiffnesses act in parallel, which implies that for appropriate values of and . 11 1 vert horiz kk k −− − =+ vert k horiz k Thus, for each displacement case, the main task is to determine the appropriate values of and . Since the mass of the slab is given, the requested frequency can be computed as vert k horiz k / total k ω = m , where is found using the two results given above.
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HW1soln - CE 573: Structural Dynamics Solution to HW#1...

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