0
TECHNICAL PAPERS

Nonlinear Dynamics and Bifurcations of an Axially Moving Beam

[+] Author and Article Information
F. Pellicano

Dip. Scienze dell’lngegneria, Università di Modena e Reggio Emilia, 41100 Modena, Italy e-mail: frank@unimo.it

F. Vestroni

Dip. di Ingegneria Strutturale e Geotecnica, Università di Roma “La Sapienza,” 00184 Roma, Italy e-mail: vestroni@scilla.ing.uniroma1.it

J. Vib. Acoust 122(1), 21-30 (Aug 01, 1999) (10 pages) doi:10.1115/1.568433 History: Received August 01, 1999
Copyright © 2000 by ASME
Your Session has timed out. Please sign back in to continue.

References

Ashley,  H., and Haviland,  G., 1950, “Bending Vibrations of a Pipe Line Containing Flowing Fluid,” ASME J. Appl. Mech., 17, pp. 229–232.
Swope,  R. D., and Ames,  W. F., 1963, “Vibrations of a Moving Threadline,” J. Franklin Inst., 275, pp. 36–55.
Mote,  C. D., 1966, “On the Non-Linear Oscillation of an Axially Moving String,” ASME J. Appl. Mech., 33, pp. 463–464.
Naguleswaran,  S., and Williams,  J. H., 1968, “Lateral Vibration of Band-Saw Blades, Pulley Belts and the Like,” Int. J. Mech. Sci., 10, pp. 239–250.
Thurman,  A. L., and Mote,  C. D., 1969, “Free Periodic Nonlinear Oscillation of an Axially Moving Strip,” J. Appl. Mech., 36, pp. 83–91.
Shih,  L. Y., 1971, “Three-Dimensional Non-Linear Vibration of a Traveling String,” Int. J. Non-Linear Mech., 6, pp. 427–434.
Ames,  W. F., Lee,  S. Y., and Zaiser,  J. N., 1968, “Non-Linear Vibration of a Travelling Threadline,” Int. J. Non-Linear Mech., 3, pp. 449–469.
Simpson,  A., 1973, “Transverse Modes and Frequencies of Beams Translating Between Fixed end Supports,” J. Mech. Eng. Sci., 15, pp. 159–164.
Holmes,  P. J., 1978, “Pipes Supported at Both Ends Cannot Flutter,” ASME J. Appl. Mech., 45, pp. 619–622.
Paidoussis,  M. P., and Moon,  F. C., 1988, “Nonlinear and Chaotic Fluidelastic Vibrations of a Flexible Pipe Conveying Fluid,” J. Fluids Struct., 2, pp. 567–591.
Wickert,  J. A., and Mote,  C. D., 1990, “Classical Vibration Analysis of Axially Moving Continua,” ASME J. Appl. Mech., 57, pp. 738–744.
Meirovitch,  L., 1974, “A New Method of Solution of the Eigenvalue Problem for Gyroscopic Systems,” AIAA J., 12, No. 10, pp. 1337–1342.
Meirovitch,  L., 1975, “A Modal Analysis for the Response of Linear Gyroscopic Systems,” ASME J. Appl. Mech., 42, pp. 446–450.
D’Eleuterio,  G. M. T., and Huges,  P. C., 1984, “Dynamics of Gyroelastic Continua,” ASME J. Appl. Mech., 51, pp. 415–422.
Wickert,  J. A., and Mote,  C. D., 1991, “Traveling Load Response of an Axially Moving String,” J. Sound Vib., 149, No. 2, pp. 267–284.
Wickert,  J. A., 1992, “Non-Linear Vibration of a Travelling Tensioned Beam,” Int. J. Non-Linear Mech., 27, pp. 503–517.
Hwang,  S.-J., and Perkins,  N. C., 1992, “Supercritical Stability of an Axially Moving Beam Part I: Model and Equilibrium Analysis,” J. Sound Vib., 154, No. 3, pp. 381–396.
Hwang,  S.-J., and Perkins,  N. C., 1992, “Supercritical Stability of an Axially Moving Beam Part II: Vibration and Stability Analyses,” J. Sound Vib., 154, No. 3, pp. 397–409.
Al-jawi,  A. A. N., Pierre,  C., and Ulsoy,  A. G., 1995, “Vibration Localization in Dual-Span Axially Moving Beams, Part I: Formulation and Results,” J. Sound Vib., 179, No. 2, pp. 243–266.
Al-jawi,  A. A. N., Pierre,  C., and Ulsoy,  A. G., 1995, “Vibration Localization in Dual-Span Axially Moving Beams, Part II: Perturbation Analysis,” J. Sound Vib., 179, No. 2, pp. 267–287.
Al-jawi,  A. A. N., Ulsoy,  A. G., and Pierre,  C., 1995, “Vibration Localization in Band-Wheel Systems: Theory and Experiment,” J. Sound Vib., 179, No. 2, pp. 289–312.
Wickert,  J. A., 1993, “Free Linear Vibration of Self-Pressurized Foil Bearings,” ASME J. Vibr. Acoust., 115, pp. 145–151.
Lakshmikumaran,  A. V., and Wickert,  J. A., 1996, “On the Vibration of Coupled Travelling String and Air Bearing System,” ASME J. Vibr. Acoust., 118, pp. 398–405.
Perkins,  N. C., and Mote,  C. D., 1987, “Three-Dimensional Vibration of Travelling Elastic Cables,” J. Sound Vib., 114, No. 2, pp. 325–340.
Beikmann,  R. S., Perkins,  N. C., and Ulsoy,  A. G., 1996, “Free Vibration of Serpentine Belt Drive System,” ASME J. Appl. Mech., 118, pp. 406–413.
Moon,  J., and Wickert,  J. A., 1997, “Non-Linear Vibration of Power Transmission Belts,” J. Sound Vib., 200, No. 4, pp. 419–431.
Lin,  C. C., 1997, “Stability and Vibration Characteristics of Axially Moving Plates,” Int. J. Solids Struct., 34, No. 24, pp. 3179–3190.
Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag, New York.
Carrier,  G. F., 1945, “On the Non-Linear Vibration Problem of the Elastic String,” Q. Appl. Math., 3, pp. 157–165.
Luongo,  A., Rega,  G., and Vestroni,  F., 1984, “Planar Nonlinear Free Vibrations of an Elastic Cable,” Int. J. Non-Linear Mech., 19, pp. 39–45.
Szemplinska-Stupnicka,  W., 1983, “Non-Linear Normal Modes and the Generalized Ritz Method in the Problems of Vibrations of Non-Linear Elastic Continuous Systems,” Int. J. Non-Linear Mech., 18, No. 2, pp. 149–165.
Semler,  C., and Paı̈doussis,  M. P., 1996, “Nonlinear Analysis of the Parametric Resonances of a Planar Fluid-Conveying Cantilevered Pipe,” J. Fluids Struct., 10, pp. 787–825.
Pakdemirli,  M., Nayfeh,  S. A., and Nayfeh,  A. H., 1995, “Analysis of One-to-One Autoparametric Resonances in Cables-Discretization vs. Direct Treatment,” Nonlinear Dyn., 8, pp. 65–83.
Pellicano,  F., and Zirilli,  F., 1998, “Boundary Layers and Nonlinear Vibrations of an Axially Moving Beam,” Int. J. Non-Linear Mech., 33, No. 4, pp. 691–711.

Figures

Grahic Jump Location
Representation of the first four linear modes in terms of the Lagrangian coordinates qi(i=1,[[ellipsis]],8) for two speed values: v=0.3 (solid line) and v=1.0 (dotted line)
Grahic Jump Location
Spatial form of the complex eigenfunctions for two speed values: v=0.3 (solid line) and v=1.0 (dotted line)
Grahic Jump Location
Bifurcation diagram, qi vs v
Grahic Jump Location
Real and imaginary part of the second derivative of the first mode (solid line) and approximation through a sine series expansion (dotted lines) using four (upper figures) and eight terms (lower figures)
Grahic Jump Location
Phase portraits, time histories and spectra for a perturbation of the trivial fixed point above the first critical velocity (v=1.1): (a, c, e), infinitesimal perturbation, i.e., homoclinic orbit; (b, d, f ), strong perturbation
Grahic Jump Location
Stable and unstable eigenmodes (a) and snapshots of the homoclinic orbit (b), v=1.1
Grahic Jump Location
Representation of the homoclinic motion in modal coordinates
Grahic Jump Location
Time histories and spectra for a perturbation of the trivial fixed point. Case v=1.25>v(1): (a) time history, (b) spectrum; case v=2>v(2): (c) and (e) time histories, (d) and (f ) spectra for slightly different initial conditions.
Grahic Jump Location
Energy versus q1,q2 for a two-degrees-of-freedom model with vf=0.104 and (a) v=1.1>v(1), (b) v=1.25>v(2)
Grahic Jump Location
Stability map for the straight equilibrium position: dotted region means instability
Grahic Jump Location
Snapshots of the first two modes in half a period of oscillation for v=1.0

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