MDOF Free Vibration

MDOF Free Vibration - CE 573: Structural Dynamics Example:...

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CE 573: Structural Dynamics Example: Solving the Free Vibration Problem Given: A three-story building is modeled as shown. The dead load per unit length for each horizontal girder varies with each floor; you may assume that the weight of each column is negligible in comparison. Use E = 29,000 ksi and for all columns, and assume that the horizontal girders are rigid. The building is given the following initial displacement and initial velocity: 4 600 in I = {} {} 1.5 ft 0 ( 0) 2.7 ft ; ( 0) 0 . 4.4 ft 10 ft/sec ut ⎫⎧ ⎪⎪ == ⎬⎨ ⎩⎭ ± Required: Determine the free vibration response of this building. Solution: 1. Property matrices, natural frequencies, mode shapes, and modal matrices – from the previous example: [] 2 kip-sec kip in in 0.04141 0 0 299.29 209.78 0 0 0.03882 0 , 209.78 258.97 49.189 0 0 0.02588 0 49.189 49.189 MK ⎡⎤ ⎢⎥ ⎣⎦ {} { } rad 1 sec 1 rad 2 sec 2 rad 3 sec 3 25.08 ; 0.3927 0.5115 0.7643 . 53.58 ; -0.4673 -0.4019 0.7875 . 110.80 ; -0.7008 0.7017 -0.1283 . T T T a a a ω
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Putting in the known information: >> mmat = diag([0.04141 0.03882 0.02588]) mmat = 0.0414 0 0 0 0.0388 0 0 0 0.0259 >> kmat = [299.29 -209.78 0; -209.78 258.97 -49.189; 0 -49.189 49.189]
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This note was uploaded on 05/09/2008 for the course CE 573 taught by Professor Whalen during the Fall '05 term at Purdue.

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MDOF Free Vibration - CE 573: Structural Dynamics Example:...

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