The free-vibration solution of a two-degree-of-freedom system can be determined by solving the equations with x = [x (1) x2 (1), using the initial conditions x(t = 0) = xo [m]x + [k]x = 0 = Lx10 [x0) and x(t = 0) = x = = 0 Jx10l Lizel If w and I w2 are the natural frequencies and and 2 are the mode shapes of the system obtained from the solution of the characteristic equation [[m]s + [k]]u (E.2) with s = = w, w (characteristic roots), the solution of Eq. (E.1), X(t), can be found as a linear combination of different solutions as: X(t): Cue it + Cue+ e+iwit + C3ue-iwt + Cauze +int (E.1) (E.3)