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Research Papers

On the Quantification of Eigenvalue Curve Veering: A Veering Index

[+] Author and Article Information
Jonathan L. du Bois1

Department of Aerospace Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, UKjon.dubois@bristol.ac.uk

Sondipon Adhikari

School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UKs.adhikari@swansea.ac.uk

Nick A. J. Lieven

 University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, UKnick.lieven@bristol.ac.uk

1

Corresponding author.

J. Appl. Mech 78(4), 041007 (Apr 13, 2011) (8 pages) doi:10.1115/1.4003189 History: Received March 03, 2009; Revised December 06, 2010; Posted December 08, 2010; Published April 13, 2011; Online April 13, 2011

Eigenvalue curve veering is a phenomenon that has found relevance and application in a variety of structural dynamic problems ranging from localization and stability studies to material property determination. Contemporary metrics for quantifying veering can be ambiguous and difficult to interpret. This manuscript derives three normalized indices in an effort to reconcile the deficit; two of these quantify the physical conditions which produce the behavior while the third provides a definitive measure of the overall intensity of the effect. Numerical examples are provided to illustrate the application of the methods, which are expected to form a basis for the development of advanced analytical tools.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

A geometric interpretation of the cross-sensitivity quotient and modal dependence factors described by CSQij=cos2(2β), MDFij=cos2(γi), and MDFji=cos2(γj). Depicted is a plane or subspace in the normal coordinate system containing two eigenvectors ϕi and ϕj. These vectors are separated from the veering datum vectors for that subspace, ϕi(0) and ϕj(0), by angle β. The corresponding eigenvector derivatives are pictured forming angles γi and γj with the subspace.

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Figure 2

Two degree of freedom spring-mass system with light spring coupling s between the masses

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Figure 3

Two DOF system plotted for k1=k2=3, m1=2, s=0.0625, and m2=1…3. Dotted lines indicate the half-CSQ parameter bandwidth.

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Figure 4

Four degree of freedom spring-mass system with light spring couplings s1–3 between the masses

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Figure 5

Four DOF system plotted for m1=m2=m3=m4=1, s1=s2=0.6, s3=0.05, k1=0.1+0.03δ, k2=0.75+0.03δ, k3=2.2, k4=3.2, and δ=1…150

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Figure 6

The four DOF system plotted for s1=s2=s3=0.6

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Figure 7

Natural frequencies for the cantilever plate

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Figure 8

Veering indices for the cantilever plate with respect to (a) the change in stiffness, and (b) the change in mass. Note the different scaling of the ordinate axes. Dashed lines correspond with the mode shapes in Fig. 9.

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Figure 9

Mode shapes for the tenth mode of the isotropic plate, with aspect ratios a/b of (a) 0.58, (b) 0.75, (c) 1.12, and (d) 1.475. The cantilever root is along the bottom right edge in these diagrams.

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Figure 10

Mode shapes for the uncoupled modes where they cross the tenth mode of the isotropic plate: (a) mode 11, (b) mode 9, (c) mode 8 (pre-veering) and (d) mode 8 (post-veering). The cantilever root is along the bottom right edge in these diagrams.

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