Determine the natural vibration frequencies and modes of the system defined in Problem 9.6. 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, identifying all DOFs.
Figure P9.6 shows a two-story frame with lumped masses subjected to lateral forces, together with some of its properties; in addition, the flexural rigidity is E I for all columns and beams.
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(a) Identify the DOFs to represent the elastic properties and determine the stiffness matrix. Neglect axial deformations in all elements.
(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).