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Research Papers

Nonlinear Dynamical Behavior of Axially Accelerating Beams: Three-Dimensional Analysis

[+] Author and Article Information
Mergen H. Ghayesh

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

Hamed Farokhi

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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 15, 2014; final manuscript received February 14, 2015; published online June 30, 2015. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 11(1), 011010 (Jan 01, 2016) (16 pages) Paper No: CND-14-1316; doi: 10.1115/1.4029905 History: Received December 15, 2014; Revised February 14, 2015; Online June 30, 2015

The three-dimensional (3D) nonlinear dynamics of an axially accelerating beam is examined numerically taking into account all of the longitudinal, transverse, and lateral displacements and inertia. Hamilton’s principle is employed in order to derive the nonlinear partial differential equations governing the longitudinal, transverse, and lateral motions. These equations are transformed into a set of nonlinear ordinary differential equations by means of the Galerkin discretization technique. The nonlinear global dynamics of the system is then examined by time-integrating the discretized equations of motion. The results are presented in the form of bifurcation diagrams of Poincaré maps, time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).

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References

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Figures

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Fig. 1

Schematic representation of a 3D axially accelerating beam

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Fig. 2

Bifurcation diagrams of Poincaré points for increasing amplitude of the axial speed variations on the system with c0 = 1.00; ((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

Period-2 oscillations for the system of Fig. 2 at c1 = 0.2480: ((a) and (b)) time traces of the q1 and q2 motions, respectively; ((c) and (d)) phase-plane diagrams of the q1 and q2 motions, respectively; and ((e) and (f)) Poincaré sections of the q1 and q2 motions, respectively

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Fig. 4

Bifurcation diagrams of Poincaré points for increasing amplitude of the axial speed variations on the system with c0 = 1.15; ((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. 5

Chaotic oscillations for the system of Fig. 4 at c1 = 0.1536: ((a) and (b)) time traces of the q1 and q2 motions, respectively; ((c) and (d)) phase-plane diagrams of the q1 and q2 motions, respectively; and ((e) and (f)) Poincaré sections of the q1 and q2 motions, respectively

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Fig. 6

Bifurcation diagrams of Poincaré points for increasing amplitude of the axial speed variations on the system with c0 = 1.40; ((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. 7

Bifurcation diagrams of Poincaré points for increasing mean axial speed on the system with c1 = 0.12; ((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. 8

Periodic oscillations for the system of Fig. 7 at c0 = 1.856: ((a) and (b)) time traces of the q1 and q2 motions, respectively; and ((c) and (d)) phase-plane diagrams of the q1 and q2 motions, respectively

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Fig. 9

Bifurcation diagrams of Poincaré points for increasing mean axial speed on the system with c1 = 0.16; ((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. 10

Period-2 oscillations for the system of Fig. 9 at c0 = 1.104: ((a) and (b)) time traces of the q1 and q2 motions, respectively; ((c) and (d)) phase-plane diagrams of the q1 and q2 motions, respectively; and ((e) and (f)) FFTs of the q1 and q2 motions, respectively. ωd is the dimensionless frequency.

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Fig. 11

Bifurcation diagrams of Poincaré points for increasing mean axial speed on the system with c1 = 0.20; ((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. 12

Quasi-periodic oscillations for the system of Fig. 11 at c0 = 1.768: ((a) and (b)) time traces of the q1 and q2 motions, respectively; ((c) and (d)) phase-plane diagrams of the q1 and q2 motions, respectively; and ((e) and (f)) Poincaré sections of the q1 and q2 motions, respectively

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Fig. 13

Bifurcation diagrams of Poincaré points for increasing mean axial speed on the system with c1 = 0.40; ((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. 14

Chaotic oscillations for the system of Fig. 13 at c0 = 1.848: ((a) and (b)) phase-plane diagrams of the q1 and q2 motions, respectively; and ((c) and (d)) FFTs of the q1 and q2 motions, respectively. ωd is the dimensionless frequency.

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