Flight vehicles such as airplanes and rockets are unconstrained, and therefore they can undergo rigid- body motion. The free-free uniform beam gives some insight into the behavior of such free-free structures. Consider transverse vibration of the uniform beam (EI = constant, p A = constant) shown in Fig. P13.6. (a) Show that the beam has two zero-frequency (i.e., rigid-body) modes, one mode in translation and one in rotation. (b) Determine the characteristic equation governing the nonzero frequencies. Your answer should contain the parameters and L, where is defined by Eq. 13.13. (Note that this is the same equation as obtained in Problem 13.5 for a uniform clamped-clamped beam.) (c) Determine an expression for the fundamental nonzero frequency of the beam (3 rad/s). (Note: For the rigid- body modes, w = = 0.) (d) Determine an expression for the fundamental flexible transverse-vibration mode shape, 3(x), and use MATLAB, or another computer program, to plot the fundamental flexible mode. Scale the mode so that its maximum value is 1.0. X v(x, t) L Figure P13.6