(a) For the system in Problem 9.13, determine the natural vibration frequencies and modes. Express the frequencies in terms of m, , and L, and sketch the modes.
(b) The structure is pulled through a lateral displacement u1(0) and released. Determine the free vibration response.
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An umbrella structure has been idealized as an assemblage of three flexural elements with lumped masses at the nodes as shown in Fig. P9.13.
(a) Identify the DOFs to represent the elastic properties and determine the stiffness matrix. Neglect axial deformations in all members.
(b) Identify the DOFs to represent the inertial properties and determine the mass matrix.
(c) Formulate the equations of motion governing the DOFs in part (b) when the excitation is (i) horizontal ground motion, (ii) vertical ground motion, (iii) ground motion in direction b–d, (iv) ground motion in direction b–c, and (v) rocking ground motion in the plane of the structure.