0
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

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 ε1. 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.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Webster, J. J. , 1968, “ Free Vibrations of Rectangular Curved Panels,” Int. J. Mech. Sci., 10(7), pp. 571–582. [CrossRef]
Petyt, M. , and Fleisxer, C. C. , 1971, “ Free Vibration of a Curved Beam,” J. Sound Vib., 18(1), pp. 17–30. [CrossRef]
Nair, P. S. , and Durvasul, S. , 1973, “ On Quasi-Degeneracies in Plate Vibration Problems,” Int. J. Mech. Sci., 15(12), pp. 975–986. [CrossRef]
Leissa, A. W. , 1974, “ On a Curve Veering Aberration,” J. Appl. Math. Phys. (ZAMP), 25(1), pp. 99–111. [CrossRef]
Perkins, N. C. , and Mote, C. D., Jr. , 1986, “ Comments on Curve Veering in Eigenvalue Problems,” J. Sound Vib., 106(3), pp. 451–463. [CrossRef]
Pierre, C. , 1987, “ Localisation of Vibration and Vibrations by Structural Irregularity,” J. Sound Vib., 114(3), pp. 549–564. [CrossRef]
MacKay, R. S. , 1986, “ Stability of Equilibria of Hamiltonian Systems,” Nonlinear Phenomena and Chaos, S. Sarkar , ed., Adam Hilger, Bristol, UK, pp. 54–70.
Balmes, E. , 1993, “ High Modal Density, Curve Veering, Localization: A Different Perspective on the Structural Response,” J. Sound Vib., 161(2), pp. 358–363. [CrossRef]
Triantafyllou, M. S. , and Triantafyllou, G. S. , 1991, “ Frequency Coalescence and Mode Localization Phenomena: A Geometric Theory,” J. Sound Vib., 150(3), pp. 485–500. [CrossRef]
Arnold, V. I. , 1989, Mathematical Method of Classical Mechanics, 2nd ed., Springer-Verlag, New York.
Vidoli, S. , and Vestroni, F. , 2005, “ Veering Phenomena in Systems With Gyroscopic Coupling,” ASME J. Appl. Mech., 72(5), pp. 641–647. [CrossRef]
Doll, R. W. , and Mote, C. D. , 1976, “ On the Dynamic Analysis of Curved and Twisted Cylinders Transporting Fluids,” ASME J. Pressure Vessel Technol., 98(2), pp. 143–150. [CrossRef]
Triantafyllou, M. S. , 1984, “ The Dynamics of Taut Inclined Cables,” Q. J. Mech. Appl. Math., 37(3), pp. 421–440. [CrossRef]
Du Bois, J. L. , Adhikari, S. , and Lieven, N. A. J. , 2009, “ Eigenvalue Curve Veering in Stressed Structures: An Experimental Study,” J. Sound Vib., 322(4–5), pp. 1117–1124. [CrossRef]
Giannini, O. , and Sestieri, A. , 2016, “ Experimental Characterization of Veering Crossing and Lock-In in Simple Mechanical Systems,” Mech. Syst. Signal Process., 72–73, pp. 846–864. [CrossRef]
Cooley, C. G. , and Parker, R. G. , 2014, “ Vibration of Spinning Cantilever Beams With an Attached Rigid Body Undergoing Bending-Torsional-Axial Motion,” ASME J. Appl. Mech., 81(5), p. 0510021.
Shi, C. , and Parker, R. G. , 2013, “ Vibration Models and Natural Frequency Veering in Three Dimensional, Cyclically Symmetric Centrifugal Pendulum Vibration Absorber Systems,” ASME J. Vib. Acoust., 136(1), p. 0110141. [CrossRef]
Klauche, T. , Strehlau, U. , and Kuhhorn, A. , 2013, “ Integer Frequency Veering of Mistuned Integrated Disk,” ASME J. Turbomach., 135(6), p. 0610041.
Han, Y. , and Mignolet, M. P. , 2015, “ A Novel Perturbation-Based Approach for the Prediction of the Forced Response of Damped Mistuned Bladed Disks,” ASME J. Vib. Acoust., 137(4), p. 041008. [CrossRef]
Timoshenko, S. , Young, D. H. , and Weaver, W., Jr. , 1967, Vibration Problems in Engineering, Wiley, New York.
Bishop, D. , and Price, W. G. , 1977, “ Coupled Bending and Twisting of a Timoshenko Beam,” J. Sound Vib, 50(4), pp. 469–477. [CrossRef]
Lacarbonara, W. , Arafat, H. N. , and Nayfeh, A. H. , 2005, “ Non-Linear Interactions in Imperfect Beams at Veering,” Int. J. Non-Linear Mech., 40(7), pp. 987–1003. [CrossRef]
Chiba, M. , and Sugimoto, T. , 2003, “ Vibration Characteristics of a Cantilever Plate With Attached Spring-Mass System,” J. Sound Vib., 260(2), pp. 237–263. [CrossRef]
Hodges, C. H. , 1982, “ Confinement of Vibration by Structural Irregularity,” J. Sound Vib., 82(3), pp. 411–424. [CrossRef]
Pierre, C. , 1988, “ Mode Localization and Eigenvalue Loci Veering Phenomena in Disordered Structures,” J. Sound Vib., 126(3), pp. 485–502. [CrossRef]
Chen, P. T. , and Ginsberg, J. H. , 1992, “ On the Relationship Between Veering of Eigenvalue Loci and Parameter Sensitivity of Eigenfunction,” ASME J. Vib. Acoust., 114(2), pp. 141–148. [CrossRef]
Lee, S. Y. , and Mote, C. D., Jr. , 1998, “ Traveling Wave Dynamics in a Translating String Coupled to Stationary Constraints: Energy Transfer and Mode Localization,” J. Sound Vib., 212(1), pp. 1–22. [CrossRef]
Chandrashaker, A. , Adhikari, S. , and Friswell, M. I. , 2016, “ Quantification of Vibration Localization in Periodic Structures,” ASME J. Vib. Acoust., 138(2), p. 021002. [CrossRef]
Vakakis, A. F. , and Cetinkaya, C. , 1993, “ Mode Localization in a Class of Multidegree-of-Freedom Nonlinear Systems With Cyclic Symmetry,” SIAM J. Appl. Math., 53(1), pp. 265–282. [CrossRef]
Gil-Santos, E. , Ramos, D. , Pini, V. , Calleja, M. , and Tamayo, J. , 2011, “ Exponential Tuning of the Coupling Constant of Coupled Microcantilevers by Modifying Their Separation,” Appl. Phys. Lett., 98(12), p. 123108. [CrossRef]
Manav, M. , Reynen, G. , Sharma, M. , Cretu, E. , and Phani, A. S. , 2014, “ Ultrasensitive Resonant MEMS Transducers With Tuneable Coupling,” J. Micromech. Microeng., 24(5), p. 055005. [CrossRef]
Gallina, A. , Pichler, L. , and Uhl, T. , 2011, “ Enhanced Meta-Modelling Technique for Analysis of Mode Crossing, Mode Veering and More Coalescence in Structural Dynamics,” Mech. Syst. Signal Process., 25(7), pp. 2297–2312. [CrossRef]
Mace, B. R. , and Manconi, E. , 2012, “ Wave Motion and Dispersion Phenomena: Veering, Locking and Strong Coupling Effects,” J. Acoust. Soc. Am., 131(2), pp. 1015–1028. [CrossRef] [PubMed]
Kreyszig, E. , 1962, Advanced Engineering Mathematics, Wiley International Edition, 4th edition, Wiley, New York, pp. 399–409.

Figures

Grahic Jump Location
Fig. 1

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

Grahic Jump Location
Fig. 2

2DOF weakly coupled system

Grahic Jump Location
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

Grahic Jump Location
Fig. 4

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

Grahic Jump Location
Fig. 5

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

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
Fig. 8

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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 10

Rectangular simply supported plate with an attached oscillator

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
Fig. 13

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

Grahic Jump Location
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.

Grahic Jump Location
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

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