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Research Papers

Robust and Dynamically Consistent Model Order Reduction for Nonlinear Dynamic Systems

[+] Author and Article Information
David B. Segala

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

David Chelidze

Department of Mechanical Engineering,
University of Rhode Island,
Kingston, RI 02881
e-mail: chelidze@egr.uri.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 30, 2014; final manuscript received August 24, 2014; published online September 24, 2014. Assoc. Editor: Prashant Mehta.

J. Dyn. Sys., Meas., Control 137(2), 021011 (Sep 24, 2014) (8 pages) Paper No: DS-14-1049; doi: 10.1115/1.4028470 History: Received January 30, 2014; Revised August 24, 2014

There is a great importance for faithful reduced order models (ROMs) that are valid over a range of system parameters and initial conditions. In this paper, we demonstrate through two nonlinear dynamic models (pinned–pinned beam and thin plate) that are both randomly and periodically forced that smooth orthogonal decomposition (SOD)-based ROMs are valid over a wide operating range of system parameters and initial conditions when compared to proper orthogonal decomposition (POD)-based ROMs. Two new concepts of subspace robustness—the ROM is valid over a range of initial conditions, forcing functions, and system parameters—and dynamical consistency—the ROM embeds the nonlinear manifold—are used to show that SOD, as opposed to POD, can capture the low order dynamics of a particular system even if the system parameters or initial conditions are perturbed from the design case.

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Figures

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Fig. 1

A simply supported beam with two nonlinear springs positioned left and right of center

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Fig. 2

A simply supported pate with a nonlinear spring positioned at the center node

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Fig. 3

(Left) Subspace robustness and (Right) dynamical consistency for both periodically and randomly forced POD-based (∇ and , respectively) and SOD-based (◇ and ◯, respectively) ROMs for the beam

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Fig. 4

(Left) Subspace robustness and (Right) dynamical consistency for both periodically and randomly forced POD-based (∇ and , respectively) and SOD-based (◇ and ◯, respectively) ROMs for the plate

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Fig. 5

(Left) Phase portrait for the eight DOF's full scale trajectory with periodic forcing with amplitude f = 1.0 for the beam that we wish to reconstruct. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the periodically forced beam.

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Fig. 6

(Left) Phase portrait for the 21st DOF's full scale trajectory with periodic forcing with amplitude f = 100.0 for the plate that we wish to reconstruct. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the periodically forced plate.

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Fig. 7

(Left) Phase portrait for the eight DOF's full scale trajectory with random forcing with amplitude f = 0.6 for the beam that we wish to reconstruct. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the randomly forced beam.

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Fig. 8

(Left) Phase portrait for the 21st DOF's full scale trajectory with random forcing with amplitude f = 100.0 for the plate that we wish to reconstruct. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the randomly forced plate.

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Fig. 9

(Left) Phase portrait for the eight DOF's full scale trajectory with periodic forcing with amplitude f = 2.0 and random initial condition for the beam that we wish to reconstruct. Subspaces were constructed with system parameters f = 1.3 and zeroes as the initial condition. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the periodically forced beam off-design configuration.

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Fig. 10

(Left) Phase portrait for the 21st DOF's full scale trajectory with periodic forcing frequency of ω = 4π/5 and a random initial condition for the plate that we wish to reconstruct. Subspaces were constructed with system parameters ω = 3π/5 and zeroes as the initial condition. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the periodically forced plate.

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Fig. 11

(Left) Phase portrait for the eight DOF's full scale trajectory with zeroes as the initial condition and nonlinear spring coefficient β = 10 for the beam that we wish to reconstruct. Subspaces were constructed with system parameters zeroes as the initial condition and nonlinear spring coefficient β = 5. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the randomly forced beam off-design configuration.

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Fig. 12

(Left) Phase portrait for the 21st DOF's full scale trajectory with random initial condition and nonlinear spring coefficient β = 16 for the plate that we wish to reconstruct. Subspaces were constructed with system parameters zeroes as the initial condition and nonlinear spring coefficient β = 8. (Right) Corresponding L2-norm between the full scale trajectory and the POD-based ROM (–) and the SOD-based ROM (- -) for the randomly forced plate.

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