A vibrating cantilever beam is embedded at its left end (x = 0) andfree at its right

Chapter 13, Problem 10

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A vibrating cantilever beam is embedded at its left end (x = 0) andfree at its right end(x = 1). See FIGURE 13.7.2. The 1113.8 Fourier Series in Two Variables transverse displacement u(x, t) of the beam is determined from a4u a2u - + - = 0 0 < x < 1, t > 0 ax4 at2 ' u(O, t) = 0, - = O a2u I ax2 x=I ' au I = o t > ax o x=O ' a3u ----:3 I = 0, t > 0 ax- x= I u(x, 0) = f(x), au I = g(x), 0 < x < 1. at t=O This boundary-value problem could serve as a model for the displacements of a vibrating airplane wing. (a) Show that the eigenvalues of the problem are determined from the equation cos a cosh a = -1. (b) Use a CAS to find approximations to the first two positive eigenvalues of the problem. [Hint: See in Exercises 13.4.]