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Research Papers

Component-Centric Reduced Order Modeling for the Prediction of the Nonlinear Geometric Response of a Part of a Stiffened Structure

[+] Author and Article Information
Yuting Wang

SEMTE,
Faculties of Mechanical and
Aerospace Engineering,
Arizona State University,
501 E. Tyler Mall,
Tempe, AZ 85287-6106

X. Q. Wang

SEMTE,
Faculties of Mechanical and
Aerospace Engineering,
Arizona State University,
501 E. Tyler Mall,
Tempe, AZ 85287-6106

Marc P. Mignolet

Professor
Fellow ASME
SEMTE,
Faculties of Mechanical and
Aerospace Engineering,
Arizona State University,
501 E. Tyler Mall,
Tempe, AZ 85287-6106

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 19, 2018; final manuscript received September 9, 2018; published online October 29, 2018. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 13(12), 121006 (Oct 29, 2018) (12 pages) Paper No: CND-18-1112; doi: 10.1115/1.4041472 History: Received March 19, 2018; Revised September 09, 2018

Component-centric reduced order models (ROMs) have recently been developed in the context of linear structural dynamics. They lead to an accurate prediction of the response of a part of structure (referred to as the β component) while not requiring a similar accuracy in the rest of the structure (referred to as the α component). The advantage of these ROMs over standard modal models is a significantly reduced number of generalized coordinates for structures with groups of close natural frequencies. This reduction is a very desirable feature for nonlinear geometric ROMs, and thus, the focus of the present investigation is on the formulation and validation of component-centric ROMs in the nonlinear geometric setting. The reduction in the number of generalized coordinates is achieved by rotating close frequency modes to achieve unobservable modes in the β component. In the linear case, these modes then completely disappear from the formulation owing to their orthogonality with the rest of the basis. In the nonlinear case, however, the generalized coordinates of these modes are still present in the nonlinear stiffness terms of the observable modes. A closure-type algorithm is then proposed to finally eliminate the unobserved generalized coordinates. This approach, its accuracy and computational savings, is demonstrated first on a simple beam model and then more completely on the 9-bay panel model considered in the linear investigation.

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Figures

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Fig. 1

Clamped-clamped beam model supported by linear springs

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Fig. 2

Linear modes 1–4 of the beam model

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Fig. 3

Linear modes 3 and 4 of the beam in bay 1: (a) before rotation and (b) after rotation

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Fig. 4

Magnitude of the frequency responses at the center of the first bay of the beam using Eq. (1) the four linear mode ROM (dark line), the ROM without mode 4 (black), and the ROM lumping mode 4 onto mode 3 (dotted line)

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Fig. 5

Scatter plots obtained from a very short time dynamic simulation showing the modal responses of (a) rotated modes 3 and 4, (b) linear mode 1 and rotated mode 4, (c) linear mode 2, and rotated mode 4 (d), same as (a) but obtained from a set of nonlinear static solutions

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Fig. 6

Power spectral densities of the (a) transverse and (b) in-plane displacements at node 7 determined from the baseline ROM, the component-centric ROM (with lumping), and the baseline ROM with mode 4 ignored

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Fig. 7

Power spectral densities of the second Piola–Kirchhoff stress Sxx at node 7 for the (a) linear and (b) nonlinear dynamic cases and determined from the baseline ROM, the component-centric ROM (with lumping), and the baseline ROM with mode 4 ignored

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Fig. 8

Finite element model of the 9-bay fuselage sidewall panel: (a) isometric view and (b) top view from Ref. [29]

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Fig. 9

Normalized mean modal projection of a sample of snapshots

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Fig. 10

Power spectral densities of the transverse responses of bay 5 center obtained with nastran (lighter line), the 74-mode baseline ROM (black) for OASPL of 136 dB

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Fig. 11

Power spectral densities of the transverse responses of bay 5 center obtained with nastran (lighter line), the 74-mode baseline ROM (black) for OASPL of 144 dB

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Fig. 12

Estimated modal responses for bay 1 based on Eq. (4) for linear mode selection

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Fig. 13

Power spectral density of the transverse response of bay 1 center, overall sound pressure level of 136 dB, 74-mode baseline ROM (lighter line), 46-mode component-centric ROM with seven linear modes lumped (black), and 46-mode ROM with the same linear modes eliminated (dotted)

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Fig. 14

Power spectral density of the transverse response of bay 1 center, overall sound pressure level of 144 dB, 74-mode baseline ROM (lighter line), 46-mode component-centric ROM with seven linear modes lumped (black), and 46-mode ROM with the same linear modes eliminated (dotted)

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Fig. 15

Power spectral density of the transverse response of bay 4 center, overall sound pressure level of 136 dB, 74-mode baseline ROM (lighter line), 45-mode component-centric ROM with four linear modes lumped (black), and 45-mode ROM with the same linear modes eliminated (dotted)

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Fig. 16

Power spectral density of the transverse response of bay 4 center, overall sound pressure level of 144 dB, 74-mode baseline ROM (lighter), 45-mode component-centric ROM with four linear modes lumped (black), and 45-mode ROM with the same linear modes eliminated (dotted)

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