Orthogonality and Modal Matrices

# Orthogonality and Modal Matrices - CE 573 Structural...

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CE 573: Structural Dynamics Example on Orthogonality and Modal Matrices 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. 4 600 in I = Required: Determine the natural frequencies and natural vibration modes for this building model. Check orthogonality of the mode shapes with respect to the mass and stiffness matrices. Determine the modal mass and stiffness matrices for this frame, and confirm the formula for natural frequencies that uses the modal mass and stiffness values. Solution: Begin by finding the property matrices – start with the mass matrix: () ( ) [] 2 2 2 2 2 2 kip kip kip ft ft ft kip-sec kip-sec kip-sec 12 3 in in in in in in sec sec sec 0.8 20 ft 0.75 20 ft 0.5 20 ft 0.04141 , 0.03882 , 0.02588 , 386.4 386.4 386.4 0.04141 0 mm m M == = = ⇒= 2 kip-sec in .03882 .

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Orthogonality and Modal Matrices - CE 573 Structural...

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