a) b) 3) 4) 5) 6) 1. [50 points] (Hand Calculation) Find u(x, t) for the string of length 1 when the mitial velocity 1s zero and the initial deflection 15 0.01 * x(1 x). Sketch the u(x, t). du 3 d%u _ = ot dx2 S | The small free vertical vibrations of a uniform elastic beam are modeled by the fourth order PDE 0%u 3 d*u _ e at? dx?t z _ EI Re fp!l E = Young's modulus of elasticity, I = moment of inertia of the cross section, A = Cross-sectional area Find solutions u,, = F,(x)G, (x) corresponding to zero imtal velocity and satisfying the boundary conditions for a simply supported beam. Find the solution that satisfies the conditions in section (bl) as well as the initial condition. u(x,0) = f(x) = x(L x). Compare the results of section (a) and (b2). What 1s the basic difference between the frequencies of the normal modes of the vibrating string and the vibrating beam? Show that F in section (bl) satisfies clamped beam at both ends boundary conditions if fL 1s a solution of the equation. cosh fL cos L = 1. Determine approximate solutions. If the beam 1s clamped at the one end and free at the other end show that F in section (bl) satisfies these conditions if fL 1s a solution of the equation cosh fL cos L = 1. Determine approximate solutions