Determine the free vibration response of the structures of Problem 10.11 and Problem 9.7) if they are displaced as shown in Fig. P10.15a, b, and c and released. Plot floor displacements versus t/T1 and comment on the relative contributions of the three vibration modes to the response that was produced by each of the three initial displacements. Neglect damping.
For the three-story shear buildings shown in Fig. P9.7:
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(a) Determine the natural vibration frequencies and modes; express the frequencies in terms of m, and h. Sketch the modes and identify the associated natural frequencies.
(b) Verify that the modes satisfy the orthogonality properties.
(c) Normalize each mode so that the modal mass Mn has unit value. Sketch these normalized modes. Compare these modes with those obtained in part (a) and comment on the differences.
Using the definition of stiffness and mass influence coefficients, formulate the equations of motion for the three-story shear frames with lumped masses shown in Figs. P9.7. The beams are rigid in flexure, and the flexural rigidity of the columns is as shown. Neglect axial deformations in all elements.