Flight vehicles such as airplanes and rockets are unconstrained, and therefore they can undergo rigid- body...

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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 λ 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, w₁ = ₂ = 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 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 λ 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, w₁ = ₂ = 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

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a The two rigidbody modes are 1 Translation mode The entire beam translates without deformation This ... View the full answer