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Recovery of Differential Equations from Impulse Response Time Series Data for Model Identification and Feature Extraction

1
Department of Mechanical Engineering, Hamburg University of Technology, 21073 Hamburg, Germany
2
Centre for Audio Acoustics and Vibration, University of Technology Sydney, 2007 Broadway, Australia
3
Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK
*
Author to whom correspondence should be addressed.
Vibration 2019, 2(1), 25-46; https://doi.org/10.3390/vibration2010002
Received: 6 December 2018 / Revised: 4 January 2019 / Accepted: 7 January 2019 / Published: 10 January 2019
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Abstract

Time recordings of impulse-type oscillation responses are short and highly transient. These characteristics may complicate the usage of classical spectral signal processing techniques for (a) describing the dynamics and (b) deriving discriminative features from the data. However, common model identification and validation techniques mostly rely on steady-state recordings, characteristic spectral properties and non-transient behavior. In this work, a recent method, which allows reconstructing differential equations from time series data, is extended for higher degrees of automation. With special focus on short and strongly damped oscillations, an optimization procedure is proposed that fine-tunes the reconstructed dynamical models with respect to model simplicity and error reduction. This framework is analyzed with particular focus on the amount of information available to the reconstruction, noise contamination and nonlinearities contained in the time series input. Using the example of a mechanical oscillator, we illustrate how the optimized reconstruction method can be used to identify a suitable model and how to extract features from uni-variate and multivariate time series recordings in an engineering-compliant environment. Moreover, the determined minimal models allow for identifying the qualitative nature of the underlying dynamical systems as well as testing for the degree and strength of nonlinearity. The reconstructed differential equations would then be potentially available for classical numerical studies, such as bifurcation analysis. These results represent a physically interpretable enhancement of data-driven modeling approaches in structural dynamics. View Full-Text
Keywords: signal processing; sparse regression; system identification; impulse response; optimization; feature generation; structural dynamics; time series classification signal processing; sparse regression; system identification; impulse response; optimization; feature generation; structural dynamics; time series classification
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Stender, M.; Oberst, S.; Hoffmann, N. Recovery of Differential Equations from Impulse Response Time Series Data for Model Identification and Feature Extraction. Vibration 2019, 2, 25-46.

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