Modal Analysis of Non-Conservative Combined Dynamic Systems

The emergence of the use of mechanical metamaterials for vibration suppression and the creation of frequency gaps in structures require an understanding of the fundament underlying dynamics partial differential equations coupled to ordinary differential equations. Essentially periodic structures consist of a distributed parameter structure connected (embedded) to a series of spring-mass-dampers. Such systems in the past have been studied as combined dynamical systems. This work deals with the modal analysis of non-conservative combined dynamic systems formed by assembling distributed parameter structures and linear, viscously damped, lumped parameter oscillators. The mathematical model of the forced response of such dynamic systems is presented via differential operators. The related non-linear eigenproblem is formulated next and a proper solution is provided. Furthermore, the orthogonality of the eigenfunctions is studied and the completeness of the generated solution space is verified. Coupled modal coordinate differential equations result through modal analysis, thus revealing the non-proportional damping configuration, while the proportional damping conditions are also derived and discussed. The theory is applied to non-conservative Euler-Bernoulli beams subject to different types of boundary conditions and coupled to linear, viscously damped oscillators. A numerical example yields interesting conclusions about the non-proportionality and the applicability of the associated methods to solving the coupled differential equations.