A quasi-static non-linear modal analysis procedure extending Rayleigh quotient stationarity for non-conservative dynamical systems

Abstract

Non-linear Modal Analysis (NMA) refers to a class of analysis procedures that seek to characterize non-linear dynamical systems similar to how classical linear modal analysis characterizes the natural frequencies and mode shapes of linear systems. The current study proposes an extension to the stationarity of Rayleigh quotients, a classical technique for linear modal analysis, for non-linear, non-conservative, dynamical systems. The approach, termed Rayleigh Quotient-based Nonlinear Modal Analysis (RQNMA), formalizes each mode as a finite non-trivial perturbation about a static solution that is locally stationary in the work done. Apart from offering a theoretical basis for the concept of non-linear modes, this circumvents several limitations in previous methods (for example, Quasi-Static Modal Analysis (QSMA)), such as inconsistencies in handling static forces, assumptions on mode-shape change, etc. As with other NMA procedures, RQNMA is formulated for the characterization of the amplitude-dependent natural frequency (stiffness) and damping ratio (dissipation) near/at the resonances. The estimated stiffness and dissipation characteristics are compared with modal backbones generated from frequency-domain approaches, which typically are computationally more expensive than the presented approach. Comparisons are conducted using different benchmark models, placing special emphasis on structures with pre-stressed frictional contacts, in order to bring out the strengths and shortcomings of the presented approach to contextualize its applicability.