The vibration of a cantilever beam is given by the partial differential equation: J²u Ju + = 0 at2 where u(x,t) is the displacement. The beam has length e; the clamped (fixed) end is at x = 0 and the free end is at x = l. Since the equation is fourth-order in r, it must have 4 boundary conditions. These are given as: u(0,t) == 0 Ju (0,t) = 0 მე: J²u მე-2 (l,t) == 0 J³u (lt) = 0 The first two of these state that the displacement and the slope of the beam at the clamped end must be zero. The latter two state that there is no force and no moment at the beam end (you may recognize this if you have taken MAE 213). (a) Using a separation constant of -14, apply separation of variables and find the spatial and temporal (in time) ODE's. (b) Find the general form of solution for spatial and temporal functions. (i.e., without satisfying boundary or initial conditions). Note that for any linear ODE with constant coefficients, you can assume a solution of the form er and then find the roots, r. Be sure to find all four roots. (Recall that 44 = 1.) (c) Show that the equation needed to find the eigenvalues (the A's) is cos Alcosh Al-1 In order to do this, you may want to recall that ae + be = A cosh z + B sinh z and ce¹³ + de-i C cos z + D sin z. Also, you may want to know that cosh² u-sinh² u = 1. Find the first two eigenvalues for 1, and plot the beam mode shapes corresponding to these eigenvalues. Tu(x,t) l