1995/01/01 by A. Luongo, Angelo Luongo · 4 citations
Engineering · #Dynamics and Control of Mechanical Systems #Vibration and Dynamic Analysis #Vehicle Dynamics and Control Systems
paper · doi:10.2514/3.12341
Free oscillations of a two degree-of-freedom system with honproportional 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 e 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 £2 or e* are found. These depend strongly on the damping magnitude. An extensive numerical analysis is performed. N structural dynamics it is by no means rare to encounter systems having multiple eigenvalues. This does not pose any substantial difficulties when the system is conservative, but it involves specific problems when the system is nonconservative. Very often indeed, the geometric multiplicity of the eigenvalue is less than the algebraic multiplicity, and so the system has an incomplete set of eigenvectors, insufficient to form a base for the state space. Systems of this type are called defective. The free evolution of defective systems is well known to persons working in the automatic control or system theory field; moreover, basic notions can be found in any good book on linear algebra. However, in the author's opinion, the problem is not sufficiently known to the structural analyst. Indeed, it is common practice to assume damping of proportional type, so that the eigenvectors coincide with those of the corresponding undamped system (always forming a complete set), and defective systems cannot occur.