Almost Classically Damped Continuous Linear Systems

The dynamic response of a general class of continuous linear vibrating systems is analyzed which possess damping properties close to those resulting in classical (uncoupled) normal modes. First, conditions are given for the existence of classical modes of vibration in a continuous linear system, with special attention being paid to the boundary conditions. Regular perturbation expansions in terms of undamped modeshapes are then utilized for analyzing the eigenproblem as well as the vibration response of almost classically damped systems. The analysis is based on a proper splitting of the damping operators in both the field equations and the boundary conditions. The main advantage of this approach is that it allows application of standard modal analysis methodologies so that the problem is reduced to that of finding the frequencies and mode shapes of the corresponding undamped system. The approach is illustrated by two simple examples involving rod and beam vibrations.