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Research Papers

Hybrid Optimization and Anti-Optimization of a Stochastically Excited Beam

[+] Author and Article Information
Isaac Elishakoff

Fellow ASME
Department of Ocean and
Mechanical Engineering,
Florida Atlantic University,
Boca Raton, FL 33431

Maurice Lemaire

Institut Français de Mécanique Avancée,
Aubière, Cedex 63175, France

Guy Gadiot

Nederland Instituut voor
Vliegtuigontwikkeling en Ruimtevaart,
NIVR,
Delft 2600 AA, Netherlands

Manuscript received February 21, 2013; final manuscript received August 26, 2013; accepted manuscript posted September 12, 2013; published online October 16, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 81(2), 021017 (Oct 16, 2013) (12 pages) Paper No: JAM-13-1082; doi: 10.1115/1.4025402 History: Received February 21, 2013; Revised August 26, 2013; Accepted September 12, 2013

Random vibrations of the damped Bernoulli–Euler beam with two supports and subjected to a stationary random excitation are studied. The supports are symmetrically placed with respect to the middle cross-section of the beam. We investigate the mean square displacement of the beam with the goal of determining the optimum location of supports in order to minimize the maximum probabilistic response. This study falls in the category of hybrid optimization and anti-optimization, since we are looking for the worst maximum response, constituting the anti-optimization process; subsequently, we are looking for optimization of the structure to make the maximum response minimal by properly the spacing supports.

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Figures

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Fig. 2

Half beam that ought to be considered for either the symmetric or antisymmetric cases

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Fig. 1

Beam supported by two point supports

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Fig. 4

The ten first mode shapes for  = 0.4: (a) symmetric mode shapes, and (b) antisymmetric mode shapes support locations are denoted by bold vertical lines

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Fig. 3

The ten first mode shapes for  = 0.2: (a) symmetric mode shapes, and (b) antisymmetric mode shapes support locations are denoted by bold vertical lines

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Fig. 5

Limiting cases: (a) corresponds to two C-F beams of length L/2, and (b) corresponds to one beam S-S at both ends

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Fig. 6

The Dw2(x) for various values of ε (symmetric case)

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Fig. 7

Variation of the maximum value of Dw2(x) with regard to ε (symmetric case)

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Fig. 9

Variation of the maximum value of Dw2(x) with regard to ε (antisymmetric case)

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Fig. 10

Variation of the maximum value of Dw2(x) with regard to ε (general case)

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Fig. 8

The Dw2(x) for various values of ε (antisymmetric case)

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Fig. 12

Variation of the maximum value of Dw2(x) with regard to ε (point load case)

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Fig. 11

Location of the point loading on the beam

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