Problem 1: The solution for spatial dependence beam vibration is Z(x) = C sin Bx + C2 cos Bx + C; cosh fx + C, sinh Bx where EI For a beam with the same boundary conditions on both ends, it is possible to explore free vibrations that are symmetric and antisymmetric about the mid-point of the beam by considering separate frequency equations. Consider a beam of length & with the origin of the coordinate system taken at the mid- point of the beam, so that the beam occupies the region x = +L/2. Derive the two frequency equations for the symmetric and antisymmetric natural vibrations. - X E, I. A, p. L