Determine the natural vibration frequencies and modes of the systems defined in Problem 9.9. Express the frequencies in terms of m, , and h and the joint rotations in terms of h. Normalize each mode to unit displacement at the roof and sketch it, including all DOFs.
Figures P9.9 show three-story frames with lumped masses subjected to lateral forces, together with the flexural rigidity of columns and beams.
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(a) Identify the DOFs to represent the elastic properties and determine the stiffness matrix. Neglect the axial deformation of the members.
(b) Identify the DOFs to represent the inertial properties and determine the mass matrix. Assume the members to be massless and neglect their rotational inertia.
(c) Formulate the equations governing the motion of the frame in the DOFs in part (b).