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Research Papers

On the Inclusion of Time Derivatives of State Variables for Parametric Model Order Reduction for a Beam on a Nonlinear Foundation

[+] Author and Article Information
David B. Segala

Naval Undersea Warfare Center,
1176 Howell Street,
Newport, RI 02841
e-mail: david.segala@navy.mil

Peiman Naseradinmousavi

Dynamic Systems and Control Laboratory,
Department of Mechanical Engineering,
San Diego State University,
San Diego, CA 92115
e-mails: pnaseradinmousavi@mail.sdsu.edu;
peiman.n.mousavi@gmail.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 6, 2016; final manuscript received January 7, 2017; published online May 24, 2017. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 139(8), 081009 (May 24, 2017) (7 pages) Paper No: DS-16-1487; doi: 10.1115/1.4035759 History: Received October 06, 2016; Revised January 07, 2017

The computational burden of parameter exploration of nonlinear dynamical systems can become a costly exercise. A computationally efficient lower dimensional representation of a higher dimensional dynamical system is achieved by developing a reduced order model (ROM). Proper orthogonal decomposition (POD) is usually the preferred method in projection-based nonlinear model reduction. POD seeks to find a set of projection modes that maximize the variance between the full-scale state variables and its reduced representation through a constrained optimization problem. Here, we investigate the benefits of an ROM, both qualitatively and quantitatively, by the inclusion of time derivatives of the state variables. In one formulation, time derivatives are introduced as a constraint in the optimization formulation—smooth orthogonal decomposition (SOD). In another formulation, time derivatives are concatenated with the state variables to increase the size of the state space in the optimization formulation—extended state proper orthogonal decomposition (ESPOD). The three methods (POD, SOD, and ESPOD) are compared using a periodically, periodically forced with measurement noise, and a randomly forced beam on a nonlinear foundation. For both the periodically and randomly forced cases, SOD yields a robust subspace for model reduction that is insensitive to changes in forcing amplitudes and input energy. In addition, SOD offers continual improvement as the size of the dimension of the subspace increases. In the periodically forced case where the ROM is developed with noisy data, ESPOD outperforms both SOD and POD and captures the dynamics of the desired system using a lower dimensional model.

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Figures

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Fig. 1

A simply supported beam on a nonlinear elastic foundation where external forcing is applied at each node and the nonlinear foundation is represented by nonlinear springs at each node

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Fig. 2

Subspace robustness for SOD (⋄), POD (∇), and ESPOD (◯) based ROMs for periodic forcing

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Fig. 3

Phase space portrait for the forcing amplitude f = 5.5 (top left) and the resulting SOD, POD, and ESPOD-based phase portrait

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Fig. 4

Energy captured in each subspace for SOD (⋯), POD (--), and ESPOD (—)

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Fig. 5

Subspace robustness for SOD (⋄), POD (∇), and ESPOD (◯) based ROMs for periodic forcing with normally distributed noise, N(0,0.5)

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Fig. 6

Phase space portrait for the forcing amplitude f = 5.5 (top left) and the resulting SOD, POD, and ESPOD based phase portrait

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Fig. 7

Energy captured in each subspace for SOD (⋯), POD (--), and ESPOD (—)

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Fig. 8

Subspace robustness for SOD (⋄), POD (∇), and ESPOD (◯) based ROMs for random forcing

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Fig. 9

Phase space portrait for the forcing amplitude f = 5.5 (top left) and the resulting SOD, POD, and ESPOD based phase portrait

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Fig. 10

Energy captured in each subspace for SOD (⋯), POD (- -), and ESPOD (—)

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