0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.

References

Hart, J. D. , Ford, G. W. , and Saure, R. , 1992, “ Mitigation of Wind Induced Vibration of Arctic Pipeline Systems,” ASME 11th International Conference on Offshore Mechanics and Arctic Engineering, Calgary, AB, Canada, June 7–12.
Castanier, M. P. , and Pierre, C. , 2006, “ Modeling and Analysis of Mistuned Bladed Disk Vibration: Status and Emerging Directions,” J. Propul. Power, 22(2), pp. 384–396. [CrossRef]
Nikolic, M. , 2006, “ New Insights into the Blade Mistuning Problem,” Ph.D. thesis, Imperial College, London.
Nikolic, M. , Petrov, E. P. , and Ewins, D. J. , 2008, “ Robust Strategies for Forced Response Reduction of Bladed Disks Based on Large Mistuning Concept,” ASME J. Eng. Gas Turbines Power, 130(2), pp. 285–295. [CrossRef]
Blair, A. J. , 1997, “ A Design Strategy for Preventing High Cycle Fatigue by Minimising Sensitivity of Bladed Disks to Mistuning,” Master's thesis, Wright State University, Dayton, OH.
Chen, Y. F. , and Shen, I. Y. , 2015, “ Mathematical Insights Into Linear Mode Localization in Nearly Cyclic Symmetric Rotors With Mistune,” ASME J. Vib. Acoust., 137(4), p. 041007. [CrossRef]
Pestel, E. C. , and Leckie, F. A. , 1963, Matrix Methods in Elastomechanics, McGraw-Hill, New York.
Soong, T. T. , and Bogdanoff, J. L. , 1963, “ On the Natural Frequencies of a Disordered Linear Chain of n Degrees of Freedom,” Int. J. Mech. Sci., 6(3), pp. 225–237. [CrossRef]
Lin, Y. K. , and Yang, J. N. , 1974, “ Free Vibration of a Disordered Periodic Beam,” ASME J. Appl. Mech., 41(2), pp. 383–391. [CrossRef]
Yang, J. N. , and Lin, Y. K. , 1975, “ Frequency Response Functions of a Disordered Periodic Beam,” J. Sound Vib., 38(3), pp. 317–340. [CrossRef]
Kissel, G. J. , 1988, “ Localization in Disordered Periodic Structures,” Ph.D. thesis, MIT, Boston.
Kissel, G. J. , 1992, “ Localization Factor for Multichannel Disordered Systems,” Phys. Rev. A, 44(2), pp. 1008–1014. [CrossRef]
Lin, Y. K. , and Cai, G. Q. , 1991, Disordered Periodic Structures, Springer, Dordrecht, The Netherlands.
Lin, Y. K. , and Cai, G. Q. , 1995, Probabilistic Structural Dynamics, McGraw-Hill, New York.
Xie, W. C. , and Ariaratnam, S. T. , 1994, “ Numerical Computation of Wave Localization in Large Disordered Beamlike Lattice Trusses,” AIAA J., 32(8), pp. 1724–1732. [CrossRef]
Xie, W. C. , and Ariaratnam, S. T. , 1996, “ Vibration Mode Localization in Disordered Cyclic Structures: Single Substructure Mode,” J. Sound Vib., 189(5), pp. 625–645. [CrossRef]
Xie, W. C. , and Ariaratnam, S. T. , 1996, “ Vibration Mode Localization in Disordered Cyclic Structures: Single Substructure Mode,” J. Sound Vib., 189(5), pp. 647–660. [CrossRef]
Ariaratnam, S. T. , and Xie, W. C. , 1995, “ Wave Localization in Randomly Disordered Nearly Periodic Long Continuous Beams,” J. Sound Vib., 181(1), pp. 7–22. [CrossRef]
Fang, Z. , 1995, “ Dynamic Analysis of Structures With Uncertain Parameters Using the Transfer Matrix Method,” Comput. Struct., 55(6), pp. 1037–1044. [CrossRef]
Mitchell, T. P. , and Moini, H. A. , 1992, “ An Algorithm for Finding the Natural Frequencies of a Randomly Supported String,” Probab. Eng. Mech., 7(1), pp. 23–26. [CrossRef]
Langley, R. S. , 1996, “ A Transfer Matrix Analysis of the Energetics of Structural Wave Motion and Harmonic Vibration,” Proc. R. Soc. Ser. A, 452(1950), pp. 1631–1648. [CrossRef]
du Bois, J. L. , Adhikari, S. , and Lieven, N. A. J. , 2009, “ Mode Veering in Stressed Framed Structures,” J. Sound Vib., 322(4–5), pp. 1117–1124. [CrossRef]
Liu, X. L. , 2002, “ Behaviour of Derivatives of Eigenvalues and Eigenvectors in Curve Veering and Mode Localization and Their Relation to Close Eigenvalues,” J. Sound Vib., 256(3), pp. 551–564. [CrossRef]
du Bois, J. L. , Adhikari, S. , and Lieven, N. A. J. , 2011, “ On the Quantification of Eigenvalue Curve Veering: A Veering Index,” ASME J. Appl. Mech., 78(4), p. 041007. [CrossRef]
Allemang, R. J. , 2003, “ The Modal Assurance Criterion - Twenty Years of Use and Abuse,” Sound Vib., 37(8), pp. 14–23.
Pierre, C. , 1988, “ Mode Localization and Eigenvalue Loci Veering Phenomena in Disordered Structures,” J. Sound Vib., 126(3), pp. 485–502. [CrossRef]
Fox, R. L. , and Kapoor, M. P. , 1968, “ Rates of Change of Eigenvalues and Eigenvectors,” AIAA J., 6(12), pp. 2426–2429. [CrossRef]
Adhikari, S. , 2000, “ Calculation of Derivative of Complex Modes Using Classical Normal Modes,” Comput. Struct., 77(6), pp. 625–633. [CrossRef]
Adhikari, S. , 2001, “ Eigenrelations for Non-Viscously Damped Systems,” AIAA J., 39(8), pp. 1624–1630. [CrossRef]
Rao, J. S. , 2006, “ Mistuning of Bladed Disk Assemblies to Mitigate Resonance,” Adv. Vib. Eng., 5(1), pp. 17–24.
Vijayan, K. , and Woodhouse, J. , 2014, “ Shock Transmission in a Coupled Beam System,” J. Sound Vib., 333(5), pp. 1379–1389. [CrossRef]
Vijayan, K. , and Woodhouse, J. , 2013, “ Shock Transmission in a Coupled Beam System,” J. Sound Vib., 332(16), pp. 3681–3695. [CrossRef]
Friswell, M. I. , and Mottershead, J. E. , 1999, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, UK.
Mills-Curran, W. C. , 1988, “ Calculation of Eigenvector Derivatives for Structures With Repeated Eigenvalues,” AIAA J., 26(7), pp. 867–871. [CrossRef]
Friswell, M. I. , 1996, “ The Derivatives of Repeated Eigenvalues and Their Associated Eigenvectors,” ASME J. Vib. Acoust., 118(3), pp. 390–397. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

The 2DOF spring–mass system

Grahic Jump Location
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

Grahic Jump Location
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

Grahic Jump Location
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%

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
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

Grahic Jump Location
Fig. 7

MACLF for a bladed disk system for a variation in density

Grahic Jump Location
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%

Grahic Jump Location
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%

Grahic Jump Location
Fig. 10

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

Grahic Jump Location
Fig. 11

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

Grahic Jump Location
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%

Grahic Jump Location
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

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In