Question 16.P.1: A simply supported beam of span L, mass per unit length m, a...
A simply supported beam of span L, mass per unit length m, and flexural rigidity E I vibrates under the following support motions. Obtain expressions for the displacement response of the beam. (a) Both the left and the right hand ends undergo identical lateral motions given by u_{g}=A_{0} \sin \Omega t. (b) The left hand end of the beam translates in a lateral direction according to the equation u_{g}=A_{0} \sin \Omega t.
(a) u(x, t)=\frac{4 A_{0}}{\pi}\left(\Sigma \frac{\beta_{n}^{2}}{1-\beta_{n}^{2}} \frac{1}{n} \sin \frac{n \pi x}{L}\right) \sin \Omega t \quad n=1,3,5, \ldots
(b) u(x, t)=\frac{2 A_{0}}{\pi}\left(\Sigma \frac{\beta_{n}^{2}}{1-\beta_{n}^{2}} \frac{1}{n} \sin \frac{n \pi x}{L}\right) \sin \Omega t \quad n=1,2,3,4, \ldots
\omega_{n}=n^{2} \pi^{2} \sqrt{E I / m L^{4}}, \beta_{n}=\Omega / \omega_{n}, and u is the lateral displacement relative to the pseudo-static displacement imposed by support motion.