The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview
TL;DR: In this article, a different approach is adopted, and proper orthogonal decomposition is considered, and modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data.
Abstract: Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study.
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TL;DR: In this article, a review of the past and recent developments in system identification of nonlinear dynamical structures is presented, highlighting their assets and limitations and identifying future directions in this research area.
1,000 citations
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01 Aug 2014
TL;DR: In this article, a comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells is presented. But the authors do not consider the effect of boundary conditions on the large-amplitude vibrations of circular cylinders.
Abstract: Introduction. 1. Nonlinear theories of elasticity of plates and shells 2. Nonlinear theories of doubly curved shells for conventional and advanced materials 3. Introduction to nonlinear dynamics 4. Vibrations of rectangular plates 5. Vibrations of empty and fluid-filled circular cylindrical 6. Reduced order models: proper orthogonal decomposition and nonlinear normal modes 7. Comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells 8. Effect of boundary conditions on a large-amplitude vibrations of circular cylindrical shells 9. Vibrations of circular cylindrical panels with different boundary conditions 10. Nonlinear vibrations and stability of doubly-curved shallow-shells: isotropic and laminated materials 11. Meshless discretization of plates and shells of complex shapes by using the R-functions 12. Vibrations of circular plates and rotating disks 13. Nonlinear stability of circular cylindrical shells under static and dynamic axial loads 14. Nonlinear stability and vibrations of circular shells conveying flow 15. Nonlinear supersonic flutter of circular cylindrical shells with imperfections.
862 citations
TL;DR: In this article, the authors acknowledge the partial support of the National Science Foundation Graduate Fellowship and the National Defense Science and Engineering Graduate Fellowship for a research grant from King Abdullah University of Science and Technology (KAUST) and Stanford University.
Abstract: The first author acknowledges the partial support by a National Science Foundation Graduate Fellowship and the partial support by a National Defense Science and Engineering Graduate Fellowship. The second and third authors acknowledge the partial support by the Motor Sports Division of the Toyota Motor Corporation under Agreement Number 48737, and the partial support by a research grant from the Academic Excellence Alliance program between King Abdullah University of Science and Technology (KAUST) and Stanford University. All authors also acknowledge the constructive comments received during the review process.
591 citations
Cites methods from "The Method of Proper Orthogonal Dec..."
...Within these contexts, the projection-based model reduction technique has been successfully applied to problems in aerodynamics [8, 9, 10, 11, 12], aeroelasticity [13, 14, 15, 16, 17], control [18], structural dynamics [19, 20, 21, 22], and non destructive evaluation and parameter estimation [23], to name only a few....
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28 Feb 2019TL;DR: In this paper, the authors bring together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science, and highlight many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy.
Abstract: Data-driven discovery is revolutionizing the modeling, prediction, and control of complex systems. This textbook brings together machine learning, engineering mathematics, and mathematical physics to integrate modeling and control of dynamical systems with modern methods in data science. It highlights many of the recent advances in scientific computing that enable data-driven methods to be applied to a diverse range of complex systems, such as turbulence, the brain, climate, epidemiology, finance, robotics, and autonomy. Aimed at advanced undergraduate and beginning graduate students in the engineering and physical sciences, the text presents a range of topics and methods from introductory to state of the art.
563 citations
TL;DR: In this paper, a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks is introduced. But the LSTM neural networks perform inference of highdimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor.
Abstract: We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.
356 citations
References
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8,333 citations