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Research Papers

Quantification of Vibration Localization in Periodic Structures

[+] Author and Article Information
A. Chandrashaker, M. I. Friswell

College of Engineering,
Swansea University,
Swansea SA1 8EN, UK

S. Adhikari

College of Engineering,
Swansea University,
Swansea SA1 8EN, UK
e-mail: s.adhikari@swansea.ac.uk

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 7, 2015; final manuscript received November 12, 2015; published online January 18, 2016. Assoc. Editor: Jeffrey F. Rhoads.

J. Vib. Acoust 138(2), 021002 (Jan 18, 2016) (9 pages) Paper No: VIB-15-1115; doi: 10.1115/1.4032032 History: Received April 07, 2015; Revised November 12, 2015

The phenomenon of vibration mode localization in periodic and near periodic structures has been well documented over the past four decades. In spite of its long history, and presence in a wide range of engineering structures, the approach to detect mode localization remains rather rudimentary in nature. The primary way is via a visual inspection of the mode shapes. For systems with complex geometry, the judgment of mode localization can become subjective as it would depend on visual ability and interpretation of the analyst. This paper suggests a numerical approach using the modal data to quantify mode localization by utilizing the modal assurance criterion (MAC) across all the modes due to changes in some system parameters. The proposed MAC localization factor (MACLF) gives a value between 0 and 1 and therefore gives an explicit value for the degree of mode localization. First-order sensitivity based approaches are proposed to reduce the computational effort. A two-degree-of-freedom system is first used to demonstrate the applicability of the proposed approach. The finite element method (FEM) was used to study two progressively complex systems, namely, a coupled two-cantilever beam system and an idealized turbine blade. Modal data is corrupted by random noise to simulate robustness when applying the MACLF to experimental data to quantify the degree of localization. Extensive numerical results have been given to illustrate the applicability of the proposed approach.

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References

Figures

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Fig. 1

The 2DOF spring–mass system

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Fig. 2

Eigenvalue veering and mode localization in the 2DOF discrete spring–mass system for a variation in m2 = ϵm1 and α1 = 0.1: (a) Eigenvalue veering and (b) eigenvectors

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Fig. 3

Eigenvalue veering in a coupled beam system subjected to ±25% variation in mass density: (a) Modes 1 to 2, (b) Modes 3 to 8, and (c) Modes 9 to 15

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Fig. 4

Vibration modes of the coupled beam system subjected to variations in mass density of one beam: (a) Mode 1: 0%, (b) Mode 2: 0%, (c) Mode 3: 0%, (d) Mode 1: 2.5%, (e) Mode 2: 2.5%, (f) Mode 3: 2.5%, (g) Mode 1: 25%, (h) Mode 2: 25%, and (i) Mode 3: 25%

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Fig. 5

MACLF applied to the coupled beam system for a variation of density in one beam

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Fig. 6

Eigenvalues of the bladed disk system subjected to variations in the mass density of one blade: (a) Eigenvalues λ1:λ2; (b) Eigenvalues λ3:λ6; (c) Eigenvalues λ7:λ9; (d) Eigenvalues λ10:λ12; (e) Eigenvalues λ13:λ25; and (f) Eigenvalues λ26:λ36

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Fig. 7

MACLF for a bladed disk system for a variation in density

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Fig. 8

Mode 15 of the bladed disk system subjected to variations in density of one blade: (a) Mode 15–0%, (b) Mode 15–2.5%, and (c) Mode 15–20%

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Fig. 9

Mode 21 of the bladed disk system subjected to variations in density of one blade: (a) Mode 21–0%, (b) Mode 21–10%, and (c) Mode 21–17.5%

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Fig. 10

Mode 31 exhibiting extreme localization: (a) Mode 31–20% and (b) Mode 31–15%

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Fig. 11

The MACLF values for a bladed disk system for a variation in the stiffness of a single blade

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Fig. 12

Modes with a −10% variation in the Young's modulus of a single blade: (a) Mode 31–10%, (b) Mode 32–10%, and (c) Mode 33–10%

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Fig. 13

Modes with a −10% variation in the Young's modulus of a single blade: (a) MACLF with no noise, (b) MACLF with 10% noise, (c) MACLF with 20% noise, and (d) Change in MACLF for different strengths in noise

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