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ADVANCED DYNAMICS OF STRUCTURES / Home Work 3 / December 16, 2009 (to be collected at the Final Exam) HBoduro ğ lu/ZCelep Problem # 1 : Consider the system of three degrees-of-freedom shown: a. Write down the equations of motion of the system by including the ground motion ) ( t v g 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 in terms of EI , M and l . Obtain the corresponding mode shapes φ i and give their graphical representations ( 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 , φ 2 T m φ 3 and φ 1 T k φ 2 , φ 1 T k φ 3 , φ 2 T k φ 3 , d. Evaluate the generalized masses and stiffness M i = φ i T m φ i , and K i = φ i T k φ i , and assess ω i 2 = K i / M i . ( i =1, 2, 3 ), e. Obtain the first free vibration mode shapes φ 1 and the corresponding the first circular frequencies ω i of the system by using
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