Almost classically damped continuous linear systems

The dynamic response of a general class of continuous linear vibrating syst ems is analyzed which possess damping properties close to those resulting i n classical (uncoupled) normal modes. First, conditions are given for the e xistence of classical modes of vibration in a continuous linens system, wit h special attention being paid to the boundary conditions. Regular perturba tion expansions in terms of undamped modeshapes are then utilized for analy zing the eigenproblem as well as the vibration response of almost classical ly damped systems. The analysis is based on a proper splitting of the dampi ng operators in both the field equations and the boundary conditions. The m ain advantage of this approach is that it allows application of standard mo dal analysis methodologies so that the problem is reduced to that of findin g the frequencies and mode shapes of the corresponding undamped system. The approach is illustrated by two simple examples involving rod and beam vibr ations.