Consider the beam model shown in Fig. P11.14 . Each beam node has a vertical translational dof and a counter-clockwise rotational dof. No axial dofs are included. Take the beam length to be 60 mm, rectangular cross section of width 6 mm and depth 1 mm (thus, I = 0.5 mm 4 ), and obtain the first three natural frequencies for the different cases that follow. Plot the mode shapes. (After obtaining the output from eigensolver, which contains nodal displacements of the mode shapes, you can interpolate using Hermite cubic shape functions and then use MATLAB or other programs to plot the discretized curve.) Take material to be steel, with E = 200 GPa and r= 7850 kg/m 3 . (a) Left end is fixed. (b) Left end is fixed and a concentrated mass M is attached to the right end (node 5). Take M = 5% of the beam mass. (c) Beam is unconstrained, and a mass M is attached to the right end as in (b). For cases (a) and (b), you can use program INVITR, JACOBI, or GENEIGEN. For (c), use program JACOBI or GENEIGEN.

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