Modal approaches for linear and nonlinear dynamical systems with non-classical damping

Abstract

Dynamical systems with non-classical damping appear frequently in engineering practice, be that in flexible structures with localized dampers, acoustic waveguides with radiating boundaries or any other system with unevenly distributed dissipation mechanisms. In this work, the use of different modal approaches to study dynamical systems with non-classical damping is explored. Focus is given on how different modal basis can be used to treat non-classically damped problems that also contain nonlinear terms. The aim of this paper is twofold: (1) provide a comprehensive review of the main modal approaches enabling the decoupling of non-classically damped linear problems and (2) illustrate how these methods can be used to treat nonlinear problems via tensor formulations stemming from modal projection. Three different methods are investigated, namely: the well-established approach using the modes of the associated conservative system (real normal modes); the complex modal approach which decouples damped linear systems in state-space; and a more recent approach named phase synchronization, which uses a real, time-varying transformation. Firstly, a review of the three methods is presented for both discrete and continuous systems, in linearized contexts. Subsequently, the treatment of nonlinear terms and the calculation of the associated tensors is detailed for each case. The benefits and limitations of each approach are discussed with respect to particular target applications. Finally, to clarify the associated procedures , an illustrative example is presented involving a nonlinear waveguide with a resistive boundary. Additionally, to give the interested reader a basis for implementation, Matlab codes for solving both discrete and continuous systems using all three approaches are provided.