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Technical Briefs

On Veering of Eigenvalue Loci

[+] Author and Article Information
N. G. Stephen

Mechanical Engineering, School of Engineering Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UKngs@soton.ac.uk

J. Vib. Acoust 131(5), 054501 (Sep 10, 2009) (5 pages) doi:10.1115/1.3147130 History: Received September 26, 2008; Revised April 29, 2009; Published September 10, 2009

Eigenvalue veering is studied in the context of two simple oscillators coupled by a (presumed weak) spring, variants of which have been considered by several authors. The concept of a center of veering is introduced, leading to a coordinate translation; a subsequent coordinate rotation, dependent on the degree of asymmetry of the system, reduces the frequency equation to a standard north-south opening hyperbola. Thus veering occurs even when coupling is strong, and may be characterized by these coordinate transformations and geometric features of the hyperbola, rather than eigenvalue and eigenvector derivatives.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Two degree-of-freedom spring-mass system and its graph, denoted P2

Grahic Jump Location
Figure 2

The graph of a chain of n coupled oscillators, denoted Pn

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Figure 3

Eigenvalue loci for system in Fig. 1, with m1=m2=k2=1 and varying k1; the coupling spring has stiffness s=0.05. Also shown is the center of veering, and anticlockwise coordinate rotation.

Grahic Jump Location
Figure 4

Eigenvalue loci for system in Fig. 1, with m2=k1=k2=1 and varying m1; the coupling spring has stiffness s=0.05

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