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Vibration
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Data-Driven Modelling of Nonlinear Dynamic Systems
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Data-Driven Modelling of Nonlinear Dynamic Systems

A special issue of Vibration (ISSN 2571-631X).

Deadline for manuscript submissions: closed (30 November 2020) | Viewed by 32898

Special Issue Editor

Dr. Jean-Philippe Noël

E-Mail Website
Guest Editor
Control Systems Technology Group, Department of Mechanical Engineering, Eindhoven University of Technology，5612 AZ, Eindhoven, The Netherlands
Interests: nonlinear data-driven modelling; nonlinear system identification; nonlinear dynamics; machine learning; nonlinear model-based control; mechanical engineering; mechanical vibrations

Special Issue Information

Dear Colleagues,

The data-driven modelling of nonlinear dynamic systems, also known as nonlinear system identification, is a science and engineering field that is progressing incredibly quickly. The reasons for this flourishing state are certainly multifaceted, but it might be argued that two facts have recently acted as major catalysts. First, over the past few years there has been a growing and convergent need across many high-tech engineering sectors for a new generation of modelling tools that can deal with nonlinear dynamic behaviours. Second, a synergy of means has emerged amongst the vast array of scientific communities interested in solving the fundamental issues associated with nonlinear data-driven modelling, further contributing to consolidating research efforts in the field.

This Special Issue of Vibration intends to provide an up-to-date snapshot of the most exciting and popular research trends in the field. A non-exhaustive list of subjects of interest could be formulated as follows:

  • Input design for nonlinear data-driven modelling.
  • Nonparametric data analysis towards model structure selection.
  • Machine learning mappings in nonlinear data-driven modelling.
  • Uncertainty quantification in nonlinear data-driven modelling.
  • Analysis, reduction and interpretation of nonlinear data-driven models.
  • Nonlinear model-based control.
  • Nonlinear model-based design.
  • Complex real-life applications.

Dr. Jean-Philippe Noël
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Vibration is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • data-driven modelling
  • system identification
  • experimental data
  • nonlinearity
  • dynamic systems
  • nonlinear vibrations

Published Papers (10 papers)

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Research

18 pages, 1642 KiB  
Article
Benchmarking Optimisation Methods for Model Selection and Parameter Estimation of Nonlinear Systems
by Sina Safari and Julián Londoño Monsalve
Vibration 2021, 4(3), 648-665; https://doi.org/10.3390/vibration4030036 - 05 Aug 2021
Cited by 2 | Viewed by 3155
Abstract
Characterisation and quantification of nonlinearities in the engineering structures include selecting and fitting a good mathematical model to a set of experimental vibration data with significant nonlinear features. These tasks involve solving an optimisation problem where it is difficult to choose a priori [...] Read more.
Characterisation and quantification of nonlinearities in the engineering structures include selecting and fitting a good mathematical model to a set of experimental vibration data with significant nonlinear features. These tasks involve solving an optimisation problem where it is difficult to choose a priori the best optimisation technique. This paper presents a systematic comparison of ten optimisation methods used to select the best nonlinear model and estimate its parameters through nonlinear system identification. The model selection framework fits the structure’s equation of motions using time-domain dynamic response data and takes into account couplings due to the presence of the nonlinearities. Three benchmark problems are used to evaluate the performance of two families of optimisation methods: (i) deterministic local searches and (ii) global optimisation metaheuristics. Furthermore, hybrid local–global optimisation methods are examined. All benchmark problems include a free play nonlinearity commonly found in mechanical structures. Multiple performance criteria are considered based on computational efficiency and robustness, that is, finding the best nonlinear model. Results show that hybrid methods, that is, the multi-start strategy with local gradient-based Levenberg–Marquardt method and the particle swarm with Levenberg–Marquardt method, lead to a successful selection of nonlinear models and an accurate estimation of their parameters within acceptable computational times. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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16 pages, 1753 KiB  
Article
Simultaneous Regression and Selection in Nonlinear Modal Model Identification
by Christopher Van Damme, Alecio Madrid, Matthew Allen and Joseph Hollkamp
Vibration 2021, 4(1), 232-247; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration4010016 - 13 Mar 2021
Cited by 2 | Viewed by 2104
Abstract
High fidelity finite element (FE) models are widely used to simulate the dynamic responses of geometrically nonlinear structures. The high computational cost of running long time duration analyses, however, has made nonlinear reduced order models (ROMs) attractive alternatives. While there are a variety [...] Read more.
High fidelity finite element (FE) models are widely used to simulate the dynamic responses of geometrically nonlinear structures. The high computational cost of running long time duration analyses, however, has made nonlinear reduced order models (ROMs) attractive alternatives. While there are a variety of reduced order modeling techniques, in general, their shared goal is to project the nonlinear response of the system onto a smaller number of degrees of freedom. Implicit Condensation (IC), a popular and non-intrusive technique, identifies the ROM parameters by fitting a polynomial model to static force-displacement data from FE model simulations. A notable drawback of these models, however, is that the number of polynomial coefficients increases cubically with the number of modes included within the basis set of the ROM. As a result, model correlation, updating and validation become increasingly more expensive as the size of the ROM increases. This work presents simultaneous regression and selection as a method for filtering the polynomial coefficients of a ROM based on their contributions to the nonlinear response. In particular, this work utilizes the method of least absolute shrinkage and selection (LASSO) to identify a sparse set of ROM coefficients during the IC regression step. Cross-validation is used to demonstrate accuracy of the sparse models over a range of loading conditions. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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24 pages, 7977 KiB  
Article
Investigating the Influence of Fluid-Structure Interactions on Nonlinear System Identification
by Skriptyan N. H. Syuhri, Hossein Zare-Behtash and Andrea Cammarano
Vibration 2020, 3(4), 521-544; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3040032 - 09 Dec 2020
Cited by 3 | Viewed by 3103
Abstract
A complex fluid-structure interaction can often create nonlinear dynamic behaviour in the structure. This can be better estimated using nonlinear modal analysis, capable of identifying and quantifying the nonlinearity in the structure. In this study, the case of a vibrating beam submerged in [...] Read more.
A complex fluid-structure interaction can often create nonlinear dynamic behaviour in the structure. This can be better estimated using nonlinear modal analysis, capable of identifying and quantifying the nonlinearity in the structure. In this study, the case of a vibrating beam submerged in liquid using a nonlinear parameter identification method is presented. This system is considered as an alternative propulsion mechanism, hence understanding the interaction between the fluid and the structure is necessary for its control. Here, impulse signals are used to characterise the numerical and experimental dynamics response of the system. Since the transient responses contain of a multi-component vibratory signals, a vibration decomposition method is used to separate the time response signals based on the dominant amplitude in the frequency response function. The separated time-series signals are then fitted to the nonlinear identification method to construct the backbone and damping curves. The modal parameters obtained from experimental data are then used as a base for the development of the analytical models. The analytical approaches are based on the Euler-Bernoulli beam theory with additional mass and quadratic damping functions to account for the presence of the fluid. Validations are carried out by comparing the dynamic responses of the analytical and experimental measurements demonstrating the accuracy of the model and hence, its suitability for control purposes. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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23 pages, 12244 KiB  
Article
Numerical Assessment of Polynomial Nonlinear State-Space and Nonlinear-Mode Models for Near-Resonant Vibrations
by Nidish Narayanaa Balaji, Shuqing Lian, Maren Scheel, Matthew R. W. Brake, Paolo Tiso, Jean-Philippe Noël and Malte Krack
Vibration 2020, 3(3), 320-342; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3030022 - 18 Sep 2020
Cited by 2 | Viewed by 3223
Abstract
In the present article, we follow up our recent work on the experimental assessment of two data-driven nonlinear system identification methodologies. The first methodology constructs a single nonlinear-mode model from periodic vibration data obtained under phase-controlled harmonic excitation. The second methodology constructs a [...] Read more.
In the present article, we follow up our recent work on the experimental assessment of two data-driven nonlinear system identification methodologies. The first methodology constructs a single nonlinear-mode model from periodic vibration data obtained under phase-controlled harmonic excitation. The second methodology constructs a state-space model with polynomial nonlinear terms from vibration data obtained under uncontrolled broadband random excitation. The conclusions drawn from our previous work (experimental) were limited by uncertainties inherent to the specimen, instrumentation, and signal processing. To avoid these uncertainties in the present work, we pursued a completely numerical approach based on synthetic measurement data obtained from simulated experiments. Three benchmarks are considered, which feature geometric, unilateral contact, and dry friction nonlinearity, respectively. As in our previous work, we assessed the prediction accuracy of the identified models with a focus on the regime near a particular resonance. This way, we confirmed our findings on the strengths and weaknesses of the two methodologies and derive several new findings: First, the state-space method struggles even for polynomial nonlinearities if the training data is chaotic. Second, the polynomial state-space models can reach high accuracy only in a rather limited range of vibration levels for systems with non-polynomial nonlinearities. Such cases demonstrate the sensitivity to training data inherent in the method, as model errors are inevitable here. Third, although the excitation does not perfectly isolate the nonlinear mode (exciter-structure interaction, uncontrolled higher harmonics, local instead of distributed excitation), the modal properties are identified with high accuracy. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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23 pages, 2627 KiB  
Article
Bayesian Joint Input-State Estimation for Nonlinear Systems
by Timothy J. Rogers, Keith Worden and Elizabeth J. Cross
Vibration 2020, 3(3), 281-303; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3030020 - 07 Sep 2020
Cited by 9 | Viewed by 2949
Abstract
This work suggests a solution for joint input-state estimation for nonlinear systems. The task is to recover the internal states of a nonlinear oscillator, the displacement and velocity of the system, and the unmeasured external forces applied. To do this, a Gaussian process [...] Read more.
This work suggests a solution for joint input-state estimation for nonlinear systems. The task is to recover the internal states of a nonlinear oscillator, the displacement and velocity of the system, and the unmeasured external forces applied. To do this, a Gaussian process latent force model is developed for nonlinear systems. The model places a Gaussian process prior over the unknown input forces for the system, converts this into a state-space form and then augments the nonlinear system with these additional hidden states. To perform inference over this nonlinear state-space model a particle Gibbs approach is used combining a “Particle Gibbs with Ancestor Sampling” Markov kernel for the states and a Metropolis-Hastings update for the hyperparameters of the Gaussian process. This approach is shown to be effective in a numerical case study on a Duffing oscillator where the internal states and the unknown forcing are recovered, each with a normalised mean-squared error less than 0.5%. It is also shown how this Bayesian approach allows uncertainty quantification of the estimates of the states and inputs which can be invaluable in further engineering analyses. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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15 pages, 8207 KiB  
Article
Experimental Identification of Backbone Curves of Strongly Nonlinear Systems by Using Response-Controlled Stepped-Sine Testing (RCT)
by Taylan Karaağaçlı and H. Nevzat Özgüven
Vibration 2020, 3(3), 266-280; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3030019 - 07 Sep 2020
Cited by 9 | Viewed by 3833
Abstract
In stepped-sine testing of strongly nonlinear structures with the classical force-control strategy, corrective force perturbations of a standard controller used to capture the reference signal in the proximity of turning points of frequency response curves may often lead to a premature jump before [...] Read more.
In stepped-sine testing of strongly nonlinear structures with the classical force-control strategy, corrective force perturbations of a standard controller used to capture the reference signal in the proximity of turning points of frequency response curves may often lead to a premature jump before reaching the actual resonance peak. Accordingly, a classical force-control approach is not suitable to identify backbone curves of strongly nonlinear structures. This paper shows that currently available commercial modal test equipment can accurately identify backbone curves of strongly nonlinear structures by using Response-Controlled stepped-sine Testing (RCT) and the Harmonic Force Surface (HFS) concept, both recently proposed by the authors. These methods can be applied to systems where there are many nonlinearities at several different (and even unknown) locations. However, these techniques are not applicable to systems where internal resonances occur. In RCT, the displacement amplitude of the driving point, rather than the amplitude of the applied force, is kept constant during the stepped-sine testing. Spectra of the harmonic excitation force measured at several different displacement amplitude levels are used to build up a smooth HFS. Isocurves of constant amplitude forcing on the HFS lead to constant-force frequency response curves with accurately measured turning points and unstable branches (if there are any), which makes it possible to identify backbone curves of strongly nonlinear structures experimentally. The validation of the proposed approach is demonstrated with numerical and experimental case studies. A five degree-of-freedom (DOF) lumped system with five cubic stiffness elements, which create strong conservative nonlinearity, is used in the numerical example. Experimental case studies consist of a cantilever beam and a control fin actuation mechanism of a real missile structure. The cantilever beam is supported at its free-end by two metal strips constrained at both ends to create strong stiffening nonlinearity. The control fin actuation mechanism exhibits very complex and strong nonlinearity due to backlash and friction. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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31 pages, 9920 KiB  
Article
Towards the Development of an Operational Digital Twin
by Paul Gardner, Mattia Dal Borgo, Valentina Ruffini, Aidan J. Hughes, Yichen Zhu and David J. Wagg
Vibration 2020, 3(3), 235-265; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3030018 - 04 Sep 2020
Cited by 26 | Viewed by 4919
Abstract
A digital twin is a powerful new concept in computational modelling that aims to produce a one-to-one mapping of a physical structure, operating in a specific context, into the digital domain. The development of a digital twin provides clear benefits in improved predictive [...] Read more.
A digital twin is a powerful new concept in computational modelling that aims to produce a one-to-one mapping of a physical structure, operating in a specific context, into the digital domain. The development of a digital twin provides clear benefits in improved predictive performance and in aiding robust decision making for operators and asset managers. One key feature of a digital twin is the ability to improve the predictive performance over time, via improvements of the digital twin. An important secondary function is the ability to inform the user when predictive performance will be poor. If regions of poor performance are identified, the digital twin must offer a course of action for improving its predictive capabilities. In this paper three sources of improvement are investigated; (i) better estimates of the model parameters, (ii) adding/updating a data-based component to model unknown physics, and (iii) the addition of more physics-based modelling into the digital twin. These three courses of actions (along with taking no further action) are investigated through a probabilistic modelling approach, where the confidence of the current digital twin is used to inform when an action is required. In addition to addressing how a digital twin targets improvement in predictive performance, this paper also considers the implications of utilising a digital twin in a control context, particularly when the digital twin identifies poor performance of the underlying modelling assumptions. The framework is applied to a three-storey shear structure, where the objective is to construct a digital twin that predicts the acceleration response at each of the three floors given an unknown (and hence, unmodelled) structural state, caused by a contact nonlinearity between the upper two floors. This is intended to represent a realistic challenge for a digital twin, the case where the physical twin will degrade with age and the digital twin will have to make predictions in the presence of unforeseen physics at the time of the original model development phase. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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18 pages, 6357 KiB  
Article
Multiharmonic Resonance Control Testing of an Internally Resonant Structure
by Alexander D. Shaw, Thomas L. Hill, Simon A. Neild and Michael I. Friswell
Vibration 2020, 3(3), 217-234; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3030017 - 03 Sep 2020
Cited by 1 | Viewed by 2506
Abstract
The experimental characterisation of a nonlinear structure is a challenging process, particularly for multiple degree of freedom and continuous structures. Despite attracting much attention from academia, there is much work needed to create processes that can achieve characterisation in timescales suitable for industry, [...] Read more.
The experimental characterisation of a nonlinear structure is a challenging process, particularly for multiple degree of freedom and continuous structures. Despite attracting much attention from academia, there is much work needed to create processes that can achieve characterisation in timescales suitable for industry, and a key to this is the design of the testing procedure itself. This work proposes a passive testing method that seeks a desired degree of resonance between forcing and response. In this manner, the process automatically seeks data that reveals greater detail of the underlying nonlinear normal modes than a traditional stepped sine method. Furthermore, the method can target multiple harmonics of the fundamental forcing frequency, and is therefore suitable for structures with complex modal interactions. The method is presented with some experimental examples, using a structure with a 3:1 internal resonance. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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17 pages, 9401 KiB  
Article
Experimental Characterization of Friction in a Negative Stiffness Nonlinear Oscillator
by Dario Anastasio and Stefano Marchesiello
Vibration 2020, 3(2), 132-148; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3020011 - 05 Jun 2020
Cited by 10 | Viewed by 2629
Abstract
Nonlinear dissipative phenomena are common features of many dynamical systems and engineering applications, and their experimental characterization has always been a challenge among the research community. Within the wide range of nonlinear damping mechanisms, friction is surely one of the most common, and [...] Read more.
Nonlinear dissipative phenomena are common features of many dynamical systems and engineering applications, and their experimental characterization has always been a challenge among the research community. Within the wide range of nonlinear damping mechanisms, friction is surely one of the most common, and with a high impact on the dynamical behavior of structures. In this paper, the nonlinear identification of friction in a negative stiffness oscillator is pursued. The structure exhibits a strong nonlinear behavior, mainly due to its polynomial elastic restoring force with a negative stiffness region. This leads to an asymmetric double-well potential with two stable equilibrium positions, and the possibility of switching between them in a chaotic way. Friction plays a crucial role in this context, as it derives from the continuous sliding between the central guide and the moving mass. The system is driven through harmonic tests with several input amplitudes, in order to estimate the variations in the energy dissipated per cycle. The identification of the frictional behavior is then pursed by minimizing the errors between the experimental measurements and the model predictions, using the harmonic balance method in conjunction with a continuation technique on the forcing amplitudes. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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15 pages, 8601 KiB  
Article
Simplified Analysis for Multiple Input Systems: A Toolbox Study Illustrated on F-16 Measurements
by Péter Zoltán Csurcsia, Bart Peeters, Johan Schoukens and Tim De Troyer
Vibration 2020, 3(2), 70-84; https://0-doi-org.brum.beds.ac.uk/10.3390/vibration3020007 - 17 Apr 2020
Cited by 13 | Viewed by 2874
Abstract
This paper introduces a nonparametric, nonlinear system identification toolbox called SAMI (simplified analysis for multiple input systems) developed for industrial measurements of vibro-acoustic systems with multiple inputs. It addresses the questions related to the user-friendly (semi-)automatic processing of multiple-input, multiple-output measurements with respect [...] Read more.
This paper introduces a nonparametric, nonlinear system identification toolbox called SAMI (simplified analysis for multiple input systems) developed for industrial measurements of vibro-acoustic systems with multiple inputs. It addresses the questions related to the user-friendly (semi-)automatic processing of multiple-input, multiple-output measurements with respect to the design of an experiment and the analysis of the measured data. When the proposed toolbox is used, with minimal user interaction, it is easily possible (a) to decide whether the underlying system is linear or not, (b) to decide whether the linear framework is still adequate to be used, and (c) to tell an inexperienced user how much can be gained using an advanced nonlinear framework. The toolbox is illustrated on openly accessible F-16 ground vibration testing measurements. Full article
(This article belongs to the Special Issue Data-Driven Modelling of Nonlinear Dynamic Systems)
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