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Research Papers

# Veering and Strong Coupling Effects in Structural Dynamics

[+] Author and Article Information
Elisabetta Manconi

Department of Industrial Engineering,
University of Parma,
Parco Area delle Scienze 181/A,
Parma 43100, Italy
e-mail: elisabetta.manconi@unipr.it

Brian Mace

Department of Mechanical Engineering,
University of Auckland,
Private Bag 92109,
Auckland 1142, New Zealand
e-mail: b.mace@auckland.ac.nz

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 5, 2016; final manuscript received October 5, 2016; published online February 8, 2017. Assoc. Editor: Matthew Brake.

J. Vib. Acoust 139(2), 021009 (Feb 08, 2017) (10 pages) Paper No: VIB-16-1163; doi: 10.1115/1.4035109 History: Received April 05, 2016; Revised October 05, 2016

## Abstract

Mode veering is the phenomenon associated with the eigenvalue loci for a system with a variable parameter: two branches approach each other and then rapidly veer away and diverge instead of crossing. The veering is accompanied by rapid variations in the eigenvectors. In this paper, veering in structural dynamics is analyzed in general terms. First, a discrete conservative model with stiffness, mass, and/or gyroscopic coupling is considered. Rapid veering requires weak coupling: if there is instead strong coupling then there is a slow evolution of the eigenvalue loci rather than rapid veering. The uncoupled-blocked system is defined to be that where all degrees-of-freedom (DOFs) but one are blocked. The skeleton of the system is the loci of the eigenvalues of the uncoupled-blocked system as the variable parameter changes. These loci intersect at certain critical points in the parameter space. Following a perturbation analysis, veering is seen to comprise rapid changes of the eigenvalues in small regions of the parameter space around the critical points: for coupling terms of order $ε$ veering occurs in a region of order $ε$ around the critical points, with the rate of change of eigenvalues being of order $ε−1$. This is accompanied by rapid rotations in the eigenvectors. The choice of coordinates in the model and application to continuous systems is discussed. For nonconservative systems, it is seen that veering also occurs under certain circumstances. Examples of 2DOFs, multi-DOFs (MDOFs), and continuous systems are presented to illustrate the results.

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## Figures

Fig. 1

Veering: eigenvalues of uncoupled-blocked and coupled DOFs xj and xk. Inlays show directions of eigenvectors in (xj,xk) plane.

Fig. 2

2DOF weakly coupled system

Fig. 3

Eigenvalue loci for the 2DOF system of Fig. 2: and skeleton of the uncoupled-blocked system; and eigenvalues of the coupled system; - - - position of the critical point

Fig. 4

(a) Slope and (b) curvature of the eigenvalue loci in the 2DOF system of Fig. 2; position of the critical point

Fig. 5

Eigenvector angle in the 2DOF system of Fig. 2; - - - position of the critical point

Fig. 6

Strong coupling in the 2DOF system of Fig. 2. Note a gradual evolution of the eigenvalues of the coupled system as opposed to veering around the critical points.

Fig. 7

(a) Real and (b) imaginary parts of the natural frequency loci for the 2DOF system of Fig. 2 in the (P,ω) plane: and skeleton of the uncoupled-blocked system; and frequencies of the coupled system

Fig. 8

The ith mass in a mass-spring chain of n masses

Fig. 9

Veering in a 6DOF system: mi = 1; k0 = 1; ki = k0(1+i/6)/2; kn = k0/4; kn+1 is a variable stiffness. (a) Skeleton: eigenvalues of the uncoupled-blocked system, eigenvalues of fixed system; eigenvalue of system comprising mn, kn, and kn+1. Note intersections at the critical values of p. (b) Eigenvalues of coupled system and skeleton. Note rapid veering around the critical points.

Fig. 10

Rectangular simply supported plate with an attached oscillator

Fig. 11

Veering in a rectangular simply supported plate with an attached oscillator: Ly/Lx=1.5;ma=0.008;εa=20;(x¯,y¯)=(0.43,0.22); eigenvalues of the coupled system; eigenvalues of the uncoupled-blocked system

Fig. 12

Strong coupling: veering in the plate system of Sec. 5.3.2; eigenvalues of the coupled system; eigenvalues of the uncoupled-blocked system. Note that a gradual evolution of the eigenvalues of the coupled system as opposed to veering around the critical points occurs for increasing values of ε.

Fig. 13

Simply supported plate coupled to a simply supported beam by a resilient layer of stiffness per unit length ε

Fig. 14

Simply supported plate coupled to a simply supported beam by a resilient layer of stiffness ε: Ly/Lx=42/17;y0=Ly/6; eigenvalues of the uncoupled-blocked system. Note intersections at the critical values.

Fig. 15

Simply supported plate coupled to a simply supported beam by a resilient layer of stiffness ε: (a) weak coupling and (b) strong coupling; Ly/Lx=42/17;y0=Ly/6. eigenvalues of the coupled system; eigenvalues of the uncoupled-blocked system

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