Determine the free vibration response of the system of Problem 9.2 (and Problem 10.2) due to each…

Determine the free vibration response of the system of Problem 9.2 (and Problem 10.2) due to each of the three sets of initial displacements: (a) u1(0) = 1, u2(0) = 0; (b) u1(0) = 1, u2(0) = 1; (c) u1(0) = 1, u2(0) = −1. Comment on the relative contribution of the modes to the response in the three cases. Neglect damping in the system.

Problem 10.2

For the system defined in Problem 9.2:

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(a) Determine the natural vibration frequencies and modes; express the frequencies in terms of m, , and L. 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.

Problem 9.2

A uniform simply supported beam of length L, flexural rigidity, and mass m per unit length has been idealized as the lumped-mass system shown in Fig. P9.2. The applied forces are also shown.

(a) Identify the DOFs to represent the elastic properties and determine the stiffness matrix. Neglect the axial deformations of the beam.

(b) Identify the DOFs to represent the inertial properties and determine the mass matrix.

(c) Formulate the equations governing the translational motion of the beam.




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