0
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,
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

Abstract

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).

<>

References

Chen, L.-Q., and Wang, B., 2009, “Stability of Axially Accelerating Viscoelastic Beams: Asymptotic Perturbation Analysis and Differential Quadrature Validation,” Eur. J. Mech.-A/Solids, 28(4), pp. 786–791.
Ding, H., and Chen, L.-Q., 2010, “Galerkin Methods for Natural Frequencies of High-Speed Axially Moving Beams,” J. Sound Vib., 329(17), pp. 3484–3494.
Kong, L., and Parker, R. G., 2004, “Approximate Eigensolutions of Axially Moving Beams With Small Flexural Stiffness,” J. Sound Vib., 276(1–2), pp. 459–469.
Stylianou, M., and Tabarrok, B., 1994, “Finite Element Analysis of an Axially Moving Beam, Part I: Time Integration,” J. Sound Vib., 178(4), pp. 433–453.
Pakdemirli, M., and Ulsoy, A. G., 1997, “Stability Analysis of an Axially Accelerating String,” J. Sound Vib., 203(5), pp. 815–832.
Pakdemirli, M., Ulsoy, A. G., and Ceranoglu, A., 1994, “Transverse Vibration of an Axially Accelerating String,” J. Sound Vib., 169(2), pp. 179–196.
Ding, H., Yan, Q.-Y., and Zu, J. W., 2014, “Chaotic Dynamics of an Axially Accelerating Viscoelastic Beam in the Supercritical Regime,” Int. J. Bifurcation Chaos Appl. Sci. Eng., 24(5), p. 1450062.
Yan, Q.-Y., Ding, H., and Chen, L.-Q., 2014, “Periodic Responses and Chaotic Behaviors of an Axially Accelerating Viscoelastic Timoshenko Beam,” Nonlinear Dyn., 78(2), pp. 1577–1591.
Ghayesh, M. H., Kafiabad, H. A., and Reid, T., 2012, “Sub- and Super-Critical Nonlinear Dynamics of a Harmonically Excited Axially Moving Beam,” Int. J. Solids Struct., 49(1), pp. 227–243.
Ghayesh, M. H., 2011, “Nonlinear Forced Dynamics of an Axially Moving Viscoelastic Beam With an Internal Resonance,” Int. J. Mech. Sci., 53(11), pp. 1022–1037.
Ghayesh, M. H., Yourdkhani, M., Balar, S., and Reid, T., 2010, “Vibrations and Stability of Axially Traveling Laminated Beams,” Appl. Math. Comput., 217(2), pp. 545–556.
Sze, K. Y., Chen, S. H., and Huang, J. L., 2005, “The Incremental Harmonic Balance Method for Nonlinear Vibration of Axially Moving Beams,” J. Sound Vib., 281(3–5), pp. 611–626.
Huang, J. L., Su, R. K. L., Li, W. H., and Chen, S. H., 2011, “Stability and Bifurcation of an Axially Moving Beam Tuned to Three-to-One Internal Resonances,” J. Sound Vib., 330(3), pp. 471–485.
Wickert, J. A., 1992, “Non-Linear Vibration of a Traveling Tensioned Beam,” Int. J. Non-Linear Mech., 27(3), pp. 503–517.
Marynowski, K., and Kapitaniak, T., 2007, “Zener Internal Damping in Modelling of Axially Moving Viscoelastic Beam With Time-Dependent Tension,” Int. J. Non-Linear Mech., 42(1), pp. 118–131.
Chen, L.-Q., and Yang, X.-D., 2006, “Vibration and Stability of an Axially Moving Viscoelastic Beam With Hybrid Supports,” Eur. J. Mech.-A/Solids, 25(6), pp. 996–1008.
Riedel, C. H., and Tan, C. A., 2002, “Coupled, Forced Response of an Axially Moving Strip With Internal Resonance,” Int. J. Non-Linear Mech., 37(1), pp. 101–116.
Ghayesh, M., 2012, “Stability and Bifurcations of an Axially Moving Beam With an Intermediate Spring Support,” Nonlinear Dyn., 69(1–2), pp. 193–210.
Suweken, G., and Van Horssen, W. T., 2003, “On the Transversal Vibrations of a Conveyor Belt With a Low and Time-Varying Velocity. Part II: The Beam-Like Case,” J. Sound Vib., 267(5), pp. 1007–1027.
Öz, H. R., Pakdemirli, M., and Özkaya, E., 1998, “Transition Behaviour From String to Beam for an Axially Accelerating Material,” J. Sound Vib., 215(3), pp. 571–576.
Özkaya, E., and Pakdemirli, M., 2000, “Vibrations of an Axially Accelerating Beam With Small Flexural Stiffness,” J. Sound Vib., 234(3), pp. 521–535.
Pakdemirli, M., and Öz, H. R., 2008, “Infinite Mode Analysis and Truncation to Resonant Modes of Axially Accelerated Beam Vibrations,” J. Sound Vib., 311(3–5), pp. 1052–1074.
Xu, G. Y., and Zhu, W. D., 2010, “Nonlinear and Time-Varying Dynamics of High-Dimensional Models of a Translating Beam With a Stationary Load Subsystem,” ASME J. Vib. Acoust., 132(6), p. 061012.
Ghayesh, M. H., 2012, “Subharmonic Dynamics of an Axially Accelerating Beam,” Arch. Appl. Mech., 82(9), pp. 1169–1181.
Ghayesh, M. H., 2012, “Coupled Longitudinal–Transverse Dynamics of an Axially Accelerating Beam,” J. Sound Vib., 331(23), pp. 5107–5124.
Öz, H. R., Pakdemirli, M., and Boyacı, H., 2001, “Non-Linear Vibrations and Stability of an Axially Moving Beam With Time-Dependent Velocity,” Int. J. Non-Linear Mech., 36(1), pp. 107–115.
Pakdemirli, M., 2001, “A General Solution Procedure for Coupled Systems With Arbitrary Internal Resonances,” Mech. Res. Commun., 28(6), pp. 617–622.
Pakdemirli, M., and Boyacı, H., 2003, “Non-Linear Vibrations of a Simple–Simple Beam With a Non-Ideal Support in Between,” J. Sound Vib., 268(2), pp. 331–341.
Pakdemirli, M., and Özkaya, E., 2003, “Three-to-One Internal Resonances in a General Cubic Non-Linear Continuous System,” J. Sound Vib., 268(3), pp. 543–553.
Ghayesh, M., 2014, “Nonlinear Size-Dependent Behaviour of Single-Walled Carbon Nanotubes,” Appl. Phys. A, 117(3), pp. 1393–1399.
Ghayesh, M. H., 2012, “Nonlinear Dynamic Response of a Simply-Supported Kelvin–Voigt Viscoelastic Beam, Additionally Supported by a Nonlinear Spring,” Nonlinear Anal.: Real World Appl., 13(3), pp. 1319–1333.
Chen, L.-Q., Ding, H., and Lim, C. W., 2012, “Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification,” Shock Vib., 19(4), pp. 527–543.
Chen, L.-Q., Tang, Y.-Q., and Lim, C. W., 2010, “Dynamic Stability in Parametric Resonance of Axially Accelerating Viscoelastic Timoshenko Beams,” J. Sound Vib., 329(5), pp. 547–565.
Chen, L.-Q., and Tang, Y.-Q., 2011, “Combination and Principal Parametric Resonances of Axially Accelerating Viscoelastic Beams: Recognition of Longitudinally Varying Tensions,” J. Sound Vib., 330(23), pp. 5598–5614.
Ghayesh, M. H., 2011, “On the Natural Frequencies, Complex Mode Functions, and Critical Speeds of Axially Traveling Laminated Beams: Parametric Study,” Acta Mech. Solida Sin., 24(4), pp. 373–382.
Ghayesh, M. H., Kazemirad, S., and Darabi, M. A., 2011, “A General Solution Procedure for Vibrations of Systems With Cubic Nonlinearities and Nonlinear/Time-Dependent Internal Boundary Conditions,” J. Sound Vib., 330(22), pp. 5382–5400.
Gholipour, A., Farokhi, H., and Ghayesh, M., 2014, “In-Plane and Out-of-Plane Nonlinear Size-Dependent Dynamics of Microplates,” Nonlinear Dyn., 79(3), pp. 1771–1785.
Ghayesh, M. H., and Farokhi, H., 2015, “Nonlinear Dynamics of Microplates,” Int. J. Eng. Sci., 86, pp. 60–73.
Farokhi, H., and Ghayesh, M. H., 2015, “Nonlinear Dynamical Behaviour of Geometrically Imperfect Microplates Based on Modified Couple Stress Theory,” Int. J. Mech. Sci., 90, pp. 133–144.

Figures

Fig. 1

Schematic representation of a 3D axially accelerating beam

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

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

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

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

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

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

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

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

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.

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

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

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

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.

Errata

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 Proceedings Articles
Related eBook Content
Topic Collections