Free-vibrations and sensitivity analysis of a defective 2-degree-of-freedom system

Free oscillations of a two degree-of-freedom system with nonproportional damping are analyzed. By a suitable choice of parameters, a family of defective systems having a noncomplete set of eigenvectors is selected. Free motions of underdamped and overdamped defective systems are studied in the four-dimensional state space, and their main characteristics are discussed. In particular, the rate at which the trajectories are attracted by the eigenvectors is determined. Small perturbations of order epsilon of the parameters are then considered, and asymptotic expressions for the modified system eigensolutions are obtained. These allow qualitative discussion of the effects of modifications on the mechanical behavior of nearly defective systems. Marked sensitivities of order epsilon(1/2) or epsilon(1/4) are found. These depend strongly on the damping magnitude. An extensive numerical analysis is performed.