MS01.1: Reduced-Order Modeling and System Identification

Frequency-based substructuring is a common and easy-to-apply technique for coupling or decoupling linear systems. If the systems are nonlinear, working in the frequency domain cannot readily be done. Recent works suggest controlling the response amplitude of a subsystem to gain quasi-linear FRFs. With that approach, each FRF, for each response level, can be coupled to another linear substructure, and the response can be predicted. The works in this direction are in their first stages, and with this abstract, we explore if the quasi-linear FRFs can be obtained from impact testing to lessen the experimental burden.

Koopman operator theory has attracted a lot of attention in recent years, since it allows for linear representations of quite general (autonomous) nonlinear systems. Two questions that frequently arise in this context are the dimension of the linear Koopman system, and the question how a linear system may represent a nonlinear system with multiple attractors/repellors. We use a bistable duffing oscillator as a simple example to discuss these questions which are both intertwined with the concept of the so-called principal Koopman eigenfunctions. We numerically calculate the five principal eigenfunctions of the bistable Duffing oscillator and show how the nonlinearity, discontinuity and complicated range of the principal Koopman eigenfunctions conceal the effort that is required to obtain a seemingly simple linear system in the Koopman framework.

For smooth nonlinear systems, available nonlinear order reduction techniques generally require computing the eigenvalues of the underlying linear system. However, the presence of nonsmooth nonlinearities in the dynamics excludes the application of these methods since the system cannot be locally linearized. In this work, we study model order reduction of continuous piecewise linear systems (hereafter, CPWL). We start with a continuous matching method based on the min-max functions and show how singular perturbation reduction can be applied to soft-stiff mechanical systems with CPWL nonlinearity. Finally, we demonstrate how this results in a more general formulation of the invariant cone problem that takes external forcing into account.

The dynamic analysis for a Finite Element (FE) model requires careful consideration of all coupled modes, especially when strong dynamic coupling occurs, e.g. internal resonances; however, it is not trivial to identify these modes without a priori knowledge of their existence. In this work, we show a method to detect dynamic coupling in FE models based on indirect reduced-order modelling, which is suitable for commercial FE software. Specifically, we develop a technique which is applicable to FE models with a large number of Degrees-of-Freedom (DoFs), removing the need to solve the complete eigenproblem. To demonstrate the method, a clamped-pinned beam is analysed, in which the present method efficiently reveals the location of the dynamic coupling in the frequency domain.

We address in this communication the use of phase resonance testing, practically realized by phase lock loop experimental continuation of periodic solutions, to measure the conservative backbone curves of a geometrically nonlinear structure. We focus on highly flexible structures, such as cantilever elements, for which the transverse stiffness is related only to bending, without additional nonlinear stretching effects such as in plates, shells or axially restrained 1D structures. In this case, the geometrical nonlinearity is weak and noticeable only at very large amplitude. The communication will show that phase resonance testing is a way of measuring the nonlinear modes of a structure independently of damping effects and their nature (linear, nonlinear etc.), provided it is smooth. Then, by monitoring the input forcing, it is possible to estimate the damping law as a function of the amplitude of the motion.