ADS_HW02_2009 - h b h Column section p (t) 3 p (t) 2 p (t)...

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Unformatted text preview: h b h Column section p (t) 3 p (t) 2 p (t) 1 Mass m/2 Mass m Mass m h h v (t) g c c ADVANCED DYNAMICS OF STRUCTURES / Home Work 2 / November 25, 2009 H.Boduro ğ lu / Z. Celep ADVANCED DYNAMICS OF STRUCTURES / Home Work 2/ November 25, 2009 H.Boduro ğ lu / Z. Celep Consider the column which can be represented as a system of three degrees-of-freedom shown: a. Write down the equations of motion of the system shown by including the ground motion ) ( t v g and the external load ) ( t p i and evaluate the mass matrix m , the rigidity matrix k, and the flexibility matrix k = d-1 , b. Determine the three circular frequencies and the periods of the free vibration i ω and i T and the corresponding mode shapes φ i . Give their graphical representation ( i =1, 2, 3 ), c. Check the orthogonality of the modes with respect to the mass matrix and the stiffness matrix φ 1 T m φ 2 , φ 1 T m φ 3 and φ 1 T k φ 2 , φ 1 T k φ 3 , d. Evaluate the generalized masses and stiffness M i = φ i T m φ...
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