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Research Papers

Three-Dimensional Nonlinear Global Dynamics of Axially Moving Viscoelastic Beams

[+] Author and Article Information
Hamed Farokhi

Department of Mechanical Engineering,
McGill University,
Montreal, QC H3A 0C3, Canada
e-mail: hamed.farokhi@mail.mcgill.ca

Mergen H. Ghayesh

School of Mechanical,
Materials and Mechatronic Engineering,
University of Wollongong,
Wollongong, New South Wales 2522, Australia e-mail: mergen@uow.edu.au

Shahid Hussain

School of Mechanical,
Materials and Mechatronic Engineering,
University of Wollongong,
Wollongong, New South Wales 2522, Australia
e-mail: shussain@uow.edu.au

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received February 25, 2015; final manuscript received September 11, 2015; published online October 26, 2015. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 138(1), 011007 (Oct 26, 2015) (11 pages) Paper No: VIB-15-1064; doi: 10.1115/1.4031600 History: Received February 25, 2015; Revised September 11, 2015

The three-dimensional nonlinear global dynamics of an axially moving viscoelastic beam is investigated numerically, retaining longitudinal, transverse, and lateral displacements and inertia. The nonlinear continuous model governing the motion of the system is obtained by means of Hamilton's principle. The Galerkin scheme along with suitable eigenfunctions is employed for model reduction. Direct time-integration is conducted upon the reduced-order model yielding the time-varying generalized coordinates. From the time histories of the generalized coordinates, the bifurcation diagrams of Poincaré sections are constructed by varying either the forcing amplitude or the axial speed as the bifurcation parameter. The results for the three-dimensional viscoelastic model are compared to those of a three-dimensional elastic model in order to better understand the effect of the internal energy dissipation mechanism on the dynamical behavior of the system. The results are also presented by means of time histories, phase-plane diagrams, and fast Fourier transforms (FFT).

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Figures

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Fig. 1

Schematic representation of a three-dimensional axially moving viscoelastic beam

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Fig. 2

Bifurcation diagrams of the Poincaré sections for increasing forcing amplitude on the system with c = 1.35: (a) and (b) the first two generalized coordinates of the transverse motion; (c) and (d) the first two generalized coordinates of the lateral motion; and (e) and (f) the first two generalized coordinates of the longitudinal motion

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Fig. 3

Bifurcation diagrams of the Poincaré sections for increasing forcing amplitude on the system with c = 1.50: (a) and (b) the first generalized coordinate of the transverse and lateral motions, respectively

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Fig. 4

Chaotic motion of the system of Fig. 2 at f1 = 0.1680: (a) and (b) time trace and phase-plane diagram of the q1 motion, respectively

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Fig. 5

Bifurcation diagrams of the Poincaré sections for increasing axial speed of the system with f1 = 0.15: (a) and (b) the first generalized coordinate of the transverse and lateral motions, respectively

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Fig. 6

Bifurcation diagrams of the Poincaré sections for increasing axial speed of the system with f1 = 0.20: (a) and (b) the first generalized coordinate of the transverse and lateral motions, respectively

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Fig. 7

Quasi-periodic motion of the system of Fig. 5 at c = 1.760: (a)–(d) time trace, phase-plane diagram, FFT, and Poincaré map of the q1 motion, respectively

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Fig. 8

Bifurcation diagrams of the Poincaré sections for increasing forcing amplitude on the system with c = 1.50: (a) and (b) correspond to the first generalized coordinate of the transverse and lateral motions of the viscoelastic system and (c) and (d) are the counterparts of (a) and (b) for the elastic system; f2 = 0.1 f1

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Fig. 9

Bifurcation diagrams of the Poincaré sections for increasing axial speed for the system with f1 = 0.15: (a) and (b) correspond to the first generalized coordinate of the transverse and lateral motions of the viscoelastic system and (c) and (d) are the counterparts of (a) and (b) for the elastic system; f2 = 0.1 f1

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