Research Papers

Component-Wise Method Applied to Vibration of Wing Structures

[+] Author and Article Information
E. Carrera

Professor of Aerospace Structures and Aeroelasticity
e-mail: erasmo.carrera@polito.it

A. Pagani

Ph.D. student
e-mail: alfonso.pagani@polito.it

M. Petrolo

Research Assistant
e-mail: marco.petrolo@polito.it
Department of Mechanical and Aerospace Engineering,
Politecnico di Torino,
Corso Duca degli Abruzzi 24,
10129 Torino, Italy

1Corresponding author.

Manuscript received May 18, 2012; final manuscript received October 15, 2012; accepted manuscript posted October 22, 2012; published online May 16, 2013. Assoc. Editor: Marc Geers.

J. Appl. Mech 80(4), 041012 (May 16, 2013) (15 pages) Paper No: JAM-12-1198; doi: 10.1115/1.4007849 History: Received May 18, 2012; Revised October 15, 2012; Accepted October 22, 2012

This paper proposes advanced approaches to the free vibration analysis of reinforced-shell wing structures. These approaches exploit a hierarchical, one-dimensional (1D) formulation, which leads to accurate and computationally efficient finite element (FE) models. This formulation is based on the unified formulation (UF), which has been recently proposed by the first author and his coworkers. In the study presented in this paper, UF was used to model the displacement field above the cross-section of reinforced-shell wing structures. Taylor-like (TE) and Lagrange-like (LE) polynomial expansions were adopted above the cross-section. A classical 1D FE formulation along the wing's span was used to develop numerical applications. Particular attention was given to the component-wise (CW) models obtained by means of the LE formulation. According to the CW approach, each wing's component (i.e. spar caps, panels, webs, etc.) can be modeled by means of the same 1D formulation. It was shown that Msc/Patran® can be used as pre- and postprocessor for the CW models, whereas Msc/Nastran® DMAP alters can be used to solve the eigenvalue problems. A number of typical aeronautical structures were analyzed and CW results were compared to classical beam theories (Euler-Bernoulli and Timoshenko), refined models (TE) and classical solid/shell FE solutions from the commercial code Msc/Nastran®. The results highlight the enhanced capabilities of the proposed formulation. In fact, the CW approach is clearly the natural tool to analyze wing structures, since it leads to results that can only be obtained through 3D elasticity (solid) elements whose computational costs are at least one-order of magnitude higher than CW models.

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Fig. 1

Coordinate frame of the beam model

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Fig. 2

Cross-section L-elements in natural geometry: (a) four-point element, L4; (b) nine-point element, L9

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Fig. 3

Two assembled L9 elements in actual geometry

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Fig. 4

From a 3D problem to 1D CUF finite elements, TE and LE approaches

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Fig. 7

Three-stringer spar modeled with L9 elements

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Fig. 6

Component-wise approach to simultaneously model panels, stringers and ribs

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Fig. 5

Differences between the TE and LE models

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Fig. 17

Rectangular wing boxes

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Fig. 11

Shell-like ninth mode of the two-stringer spar: (a) 3 L9 model (14.97 Hz), (b) 4 L9 model (12.70 Hz), (c) SOLID model (11.58 Hz)

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Fig. 9

L9 distributions above the cross-section, LE models of the two-stringer spar: (a) 3 L9, (b) 4 L9

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Fig. 10

Frequencies and vibration modes versus beam models in the case of the two-stringer spar

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Fig. 12

MAC values between the CW models and the SOLID model, two-stringer spar: (a) 3 L9 CW model, (b) 4 L9 CW model

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Fig. 13

Three-stringer spar

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Fig. 14

MAC values, three-stringer spar

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Fig. 15

Local modes, 5 L9 (LE) model of the three-stringer spar: (a) mode 12, f12 = 31.21 Hz, (b) mode 21, f21 = 84.66 Hz, (c) mode 29, f29 = 104.99 Hz

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Fig. 16

Local modes, Msc/Nastran® solid models of the three-stringer spar: (a) mode 12, f12 = 29.69 Hz, (b) mode 19, f19 = 58.24 Hz

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Fig. 18

Natural frequencies values versus structural model. Comparison between the ribbed and the rib-free configuration.

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Fig. 19

A bending and a differential–bending modal shape. Seventh order (N = 7) TE model of the wing box without the rib: (a) mode 2, f2 = 60.39 Hz, (b) mode 3, f3 = 63.49 Hz.

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Fig. 20

Shell-like modes, LE models of the rectangular wing boxes: (a) mode 7, f7 = 13.38 Hz, (b) mode 14, f14 = 30.28 Hz, (c) mode 51, f51 = 135.74 Hz

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Fig. 21

Local modes, Msc/Nastran® solid models of the rectangular wing boxes: (a) mode 7, f7 = 12.58 Hz, (b) mode 22, f22 = 28.75 Hz, (c) mode 62, f62 = 66.16 Hz

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Fig. 22

Cross-section of the wing, dimensions in millimeters

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Fig. 23

Shell-like modes of the wing (configuration A) evaluated with the CW model: (a) mode 10 (89.35 Hz), (b) mode 26 (142.91 Hz)

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Fig. 24

Shell-like modes of the wing (configuration A) evaluated with the CW model. Midspan cross-section: (a) mode 8 (86.36 Hz), (b) mode 9 (88.94 Hz), (c) mode 54 (246.70 Hz).

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Fig. 25

Shell-like modes of the wing (configuration A) evaluated with the SOLID model. Midspan cross-section: (a) mode 10 (75.13 Hz), (b) mode 8 (73.85 Hz), (c) mode 51 (218.63 Hz).

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Fig. 26

MAC values between the CW models and the SOLID model, complete aircraft wing (configuration A)



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