On the Efficacy of Sparse Representation Approaches for Determining Nonlinear Structural System Equations of Motion

Abstract A sparsity-based optimization approach is presented for determining the equations of motion of stochastically excited nonlinear structural systems. This is done by utilizing measured excitation-response realizations in the formulation of the related optimization problem, and by considering a library of candidate functions for representing the system governing dynamics. Note that a novel aspect of the approach relates to treating, also, systems endowed with fractional derivative elements. Clearly, this is of significant importance to a multitude of diverse applications in engineering mechanics taking into account the enhanced modeling capabilities of fractional calculus. Further, the fundamental theoretical and computational aspects of various representative, state-of-the-art, numerical schemes for solving the derived sparsity-based optimization problem are reviewed and discussed. A Bayesian compressive sampling approach that exhibits the additional advantage of quantifying the uncertainty of the estimates is considered as well. Furthermore, comparisons and a critical assessment of the employed numerical schemes are provided with respect to their efficacy in determining the nonlinear structural system equations of motion. In this regard, two illustrative numerical examples are considered pertaining to a nonlinear tuned mass-damper–inerter vibration control system and to a nonlinear electromechanical energy harvester, both endowed with fractional derivative elements.