Most structural beams are neither clamped- clamped nor simply supported but can be considered to have partial end fixity. At x = 0 and at x = L the uniform beam shown in Fig. P13.7 cannot displace lat- erally; that is, v(0, t) = v(L, t) = 0. However, at each end, rotation through (slope) angle is permitted, but rotation is resisted by a spring that induces a bending moment at the end of the beam that is proportional to the slope of the end of beam. Determine the character- istic equation for the beam shown in Fig. P13.7, where 0 B o is a dimensionless parameter controlling the amount of rotational restraint ( = 0 corresponds to a simply supported end; = corresponds to a fixed end). K = B (E) X v(x,t) L Figure P13.7 Ko