Identification of complex nonlinearities using cubic splines with automatic discretization

One of the major challenges in nonlinear system identification is the selection of appropriate mathematical functions to model the observed nonlinearities. In this context, piecewise polynomials, or splines, offer a simple and flexible representation basis requiring limited prior knowledge. The generally-adopted discretization for splines consists in an even distribution of their control points, termed knots. While this may prove successful for simple nonlinearities, a more advanced strategy is needed for nonlinear restoring forces with strong local variations. The present paper specifically introduces a two-step methodology to select automatically the location of the knots. It proposes to derive an initial model, using nonlinear subspace identification, and incorporating cubic spline basis functions with fixed and equally-spaced abscissas. In a second step, the location of the knots is optimized iteratively by minimizing a least-squares cost function. A single-degree-of-freedom system with a discontinuous stiffness characteristic is considered as a case study

Nonlinear Modal Analysis of the SmallSat Spacecraft

Non-linear elements are present in practically all spacecraft structures. When such non-linear effects are important linear modal analysis can no longer be applied. The development of a non-linear analogue of linear modal analysis is therefore an important endeavor. The objective of this paper is to show that nonlinear normal modes (NNMs) represent a useful and practical tool in this context. A full-scale spacecraft structure is considered and is modeled using the finite element method. Its NNMs are computed using advanced numerical algorithms, and the resulting dynamics is then carefully analyzed. Nonlinear phenomena with no linear counterpart including nonlinear modal interactions are also highlighted

Identification of mechanical systems with local nonlinearities through discrete-time Volterra series and Kautz functions

peer reviewedMathematical modeling of mechanical structures is an important research area in structural dynamics. One of the goals of this area is to obtain a model that accurately predicts the dynamics of the system. However, the nonlinear eff ects caused by large displacements and boundary conditions like gap, backlash or joint are not as well understood as the linear counterpart. This paper identifies a non-parametric discrete-time Volterra model of a benchmark nonlinear structure consisting of a cantilever beam connected to a thin beam at its free end. Time-domain data experimentally measured are used to identify the Volterra kernels, which are expanded with orthogonal Kautz functions to facilitate the identification process. The nonlinear parameters are then estimated through a model updating process involving optimization of the residue between the numerical and experimental kernels. The advantages and drawbacks of the Volterra series for modeling the behavior of nonlinear structures are finally indicated with suggestions to overcome the disadvantages found during the tests

Bayesian model updating of nonlinear systems using nonlinear normal modes

This paper presents a Bayesian model updating methodology for dynamical systems with geometric nonlinearities based on their nonlinear normal modes (NNMs) extracted from broadband vibration data. Model parameters are calibrated by minimizing selected metrics between identified and model-predicted NNMs. In the first approach, a deterministic formulation is adopted, and parameters are updated by minimizing a nonlinear least-squares objective function. A probabilistic approach based on Bayesian inference is next investigated, where a Transitional Markov Chain Monte Carlo is implemented to sample the joint posterior probability distribution of the nonlinear model parameters. Bayesian model calibration has the advantage to quantify parameter uncertainty and to provide an estimation of model evidence for model class selection. The two formulations are evaluated when applied to a numerical cantilever beam with geometrical nonlinearity. The NNMs of the beam are derived from simulated broadband data through nonlinear subspace identification and numerical continuation. Accuracy of model updating results is studied with respect to the level of measurement noise, the number of available datasets, and modeling errors

DMTs and Covid-19 severity in MS: a pooled analysis from Italy and France

We evaluated the effect of DMTs on Covid-19 severity in patients with MS, with a pooled-analysis of two large cohorts from Italy and France. The association of baseline characteristics and DMTs with Covid-19 severity was assessed by multivariate ordinal-logistic models and pooled by a fixed-effect meta-analysis. 1066 patients with MS from Italy and 721 from France were included. In the multivariate model, anti-CD20 therapies were significantly associated (OR = 2.05, 95%CI = 1.39–3.02, p < 0.001) with Covid-19 severity, whereas interferon indicated a decreased risk (OR = 0.42, 95%CI = 0.18–0.99, p = 0.047). This pooled-analysis confirms an increased risk of severe Covid-19 in patients on anti-CD20 therapies and supports the protective role of interferon

Low-order local modelling of structural nonlinearities

The present paper addresses the problem of characterising structural nonlinearities in view of system identification. A low-order local modelling strategy is proposed and encapsulated in a recently-introduced frequency-domain nonlinear subspace method for the estimation of model parameters. The complete methodology is first demonstrated using two academic examples, namely a Duffing oscillator and a five-degree-of-freedom system comprising two nonlinearities. The identification of an experimental beam involving nonlinear geometrical behaviour is finally addressed

10 years of advances in nonlinear system identification in structural dynamics: A review

Nonlinear system identification is a vast research field, today attracting a great deal of attention in the structural dynamics community. Ten years ago, an MSSP paper reviewing the progress achieved until then concluded that the identification of simple continuous structures with localised nonlinearities was within reach. The past decade witnessed a shift in emphasis, accommodating the growing industrial need for a first generation of tools capable of addressing complex nonlinearities in larger-scale structures. The objective of the present paper is to survey the key developments which arose in the field since 2006 towards developing these tools

Frequency-domain subspace identification of nonlinear mechanical systems

The objective of the present paper is to address the identification of a real-life strongly nonlinear space structure, the EADS-Astrium SmallSat spacecraft. To this end, a new nonlinear subspace identification method formulated in the frequency domain is exploited, referred to as the FNSI method. The frequency response functions of the underlying linear spacecraft and the amplitudes of the nonlinear internal forces are estimated based on a periodic-random data set corrupted by noise. This application is challenging for several reasons, including high modal density, highly non-proportional damping and the non-smooth nature of the nonlinearities

A new subspace-based approach to identify nonlinear mechanical structures in the frequency domain

peer reviewedThis paper introduces a new frequency-domain subspace-based method for the identification of nonlinear mechanical systems. The technique exploits frequency data and interprets nonlinearities as feedback forces exciting the underlying linear system. It is demonstrated using two academic examples, a Duffing oscillator and a five degree-of-freedom system comprising two nonlinearities

Grey-box nonlinear state-space modelling for mechanical vibrations identification

In the present paper, a flexible and parsimonious model of the vibrations of nonlinear mechanical systems is introduced in the form of state-space equations. It is shown that the nonlinear model terms can be formed using a limited number of output measurements. A two-step identification procedure is derived for this grey-box model, integrating nonlinear subspace initialisation and maximum likelihood optimisation. The complete procedure is demonstrated on the Silverbox benchmark, which is an electrical mimicry of a single-degree-of-freedom mechanical system with one displacement-dependent nonlinearity