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ADVANCED DYNAMICS OF STRUCTURES / Midterm Exam / December 16, 2009 HBoduro ğ lu/ZCelep Problem # 1 Consider the system of two degrees-of-freedom shown: a. Evaluate the flexibility d matrix, the mass matrix m and the rigidity matrix k = d -1 and the load vector p . b. Determine the circular frequencies and the periods of the free vibration i ω and i T in terms of EI , m and h . Obtain the corresponding two mode shapes φ i and give their graphical representation ( i =1, 2 ), c. Check the orthogonality of the modes with respect to the mass matrix and the stiffness matrix φ 1 T m φ 2 , and φ 1 T k φ 2. 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 the relationship i 2 = K i / M i . ( i =1, 2 ), h v (t) 2 v (t) 1 m 3m 2h p (t) 2 p (t) 1 EI EI Pa 3EI EI a P 3 Pa 2EI 2 Ma 2EI EI a M 2 Ma EI v(x, t) x V(x, t) M(x, t) m, EI k l M k v t M Problem # 1 Problem # 2 h d 12 2h 1 EI d 11 h d 22 2h 1 EI d 21 Problem # 2: Consider the distributed parameter system shown where m is the mass per unit length and EI is the bending rigidity of the cross section. The beam has two lumped masses of M at the two ends. Write down the boundary conditions for the free vibration of the
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