The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance. 1. Introduction Band saws, fibre textiles, magnetic tapes, paper sheets, aerial tramways, pipes transporting fluids, thread lines, and belts are some technological examples classified as axially moving continua. Analytical models for axially moving systems have been extensively used in the last few decades. The vast literature on axially moving continua vibration has been reviewed by Wickert and Mote Jr. [1] up to 1988. While a linear analysis provides natural frequencies, mode shapes, and critical speeds, its validity regarding the response of the system diminishes as the vibration amplitude becomes sufficiently large or as the critical speed is approached [2]. In these cases one must resort to a nonlinear analysis. Wickert and Mote Jr. [3, 4] studied the transverse vibration of axially moving strings and beams using an eigenfunction method. They also studied the dynamic response of an axially moving string loaded suspended mass. Wickert [5] presented a detailed study of the nonlinear vibrations and bifurcations of moving beams using the Krylov-Bogoliubov-Mitropolsky asymptotic method. Chakraborty et al. [6, 7] investigated both free and forced vibration of the nonlinear traveling beam using complex normal modes. There are papers devoted to the analysis of the dynamic behavior of traveling systems with time-dependent axial velocity or with time-dependent axial tension force. ？z and Pakdemirli [8] investigated principal parametric resonances and combination resonances of sum and difference types for any two modes for an axially
References
[1]
J. A. Wickert and C. D. Mote Jr., “Current research on the vibration and stability of axially moving materials,” Shock and Vibration Digest, vol. 20, pp. 3–13, 1988.
[2]
C. D. Mote Jr., “On the nonlinear oscillation of an axially moving string,” Journal of Applied Mechanics, vol. 33, pp. 463–464, 1966.
[3]
J. A. Wickert and C. D. Mote Jr., “Classical vibration analysis of axially moving continua,” Journal of Applied Mechanics, vol. 57, no. 3, pp. 738–744, 1990.
[4]
J. A. Wickert and C. D. Mote Jr., “Travelling load response of an axially moving string,” Journal of Sound and Vibration, vol. 149, no. 2, pp. 267–284, 1991.
[5]
J. A. Wickert, “Non-linear vibration of a traveling tensioned beam,” International Journal of Non-Linear Mechanics, vol. 27, no. 3, pp. 503–517, 1992.
[6]
G. Chakraborty, A. K. Mallik, and H. Hatwal, “Non-linear vibration of a travelling beam,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 655–670, 1999.
[7]
G. Chakraborty and A. K. Mallik, “Non-linear vibration of a travelling beam having an intermediate guide,” Nonlinear Dynamics, vol. 20, no. 3, pp. 247–265, 1999.
[8]
H. R. ？z and M. Pakdemirli, “Vibrations of an axially moving beam with time-dependent velocity,” Journal of Sound and Vibration, vol. 227, no. 2, pp. 239–257, 1999.
[9]
H. R. ？z, M. Pakdemirli, and H. Boyaci, “Non-linear vibrations and stability of an axially moving beam with time-dependent velocity,” International Journal of Non-Linear Mechanics, vol. 36, no. 1, pp. 107–115, 2001.
[10]
A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, New York, NY, USA, 1979.
[11]
A. H. Nayfeh and B. Balachandran, Applied Nonlinear Dynamics-Analytical, Computational and Experimental Methods, John Wiley & Sons, New York, NY, USA, 1994.
[12]
A. H. Nayfeh and B. Balachandran, “Modal interactions in dynamical and structural systems,” Applied Mechanics Reviews, vol. 42, pp. 175–201, 1989.
[13]
C. H. Riedel and C. A. Tan, “Coupled, forced response of an axially moving strip with internal resonance,” International Journal of Non-Linear Mechanics, vol. 37, no. 1, pp. 101–116, 2002.
[14]
E. ？zkaya, S. M. Ba？datli, and H. R. ？z, “Nonlinear transverse vibrations and 3？:？1 internal resonances of a beam with multiple supports,” Journal of Vibration and Acoustics, vol. 130, no. 2, Article ID 021013, 11 pages, 2008.
[15]
S. M. Ba？datli, H. R. ？z, and E. ？zkaya, “Non-linear transverse vibrations and 3？:？1 internal resonances of a tensioned beam on multiple supports,” Mathematical and Computational Applications, vol. 16, no. 1, pp. 203–215, 2011.
[16]
C. Chin and A. H. Nayfeh, “Three-to-one internal resonances in parametrically excited hinged-clamped beams,” Nonlinear Dynamics, vol. 20, no. 2, pp. 131–158, 1999.
[17]
L. N. Panda and R. C. Kar, “Nonlinear dynamics of a pipe conveying pulsating fluid with parametric and internal resonances,” Nonlinear Dynamics, vol. 49, no. 1-2, pp. 9–30, 2007.
[18]
L. N. Panda and R. C. Kar, “Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances,” Journal of Sound and Vibration, vol. 309, no. 3–5, pp. 375–406, 2008.
[19]
K. Y. Sze, S. H. Chen, and J. L. Huang, “The incremental harmonic balance method for nonlinear vibration of axially moving beams,” Journal of Sound and Vibration, vol. 281, no. 3–5, pp. 611–626, 2005.
[20]
J. L. Huang, R. K. L. Su, W. H. Li, and S. H. Chen, “Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances,” Journal of Sound and Vibration, vol. 330, no. 3, pp. 471–485, 2011.
[21]
L. Chen, Y. Tang, and C. W. Lim, “Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams,” Journal of Sound and Vibration, vol. 329, no. 5, pp. 547–565, 2010.
[22]
H. Ding and L. Chen, “Galerkin methods for natural frequencies of high-speed axially moving beams,” Journal of Sound and Vibration, vol. 329, no. 17, pp. 3484–3494, 2010.
[23]
H. Ding, G. C. Zhang, and L. Q. Chen, “Supercritical equilibrium solutions of axially moving beams with hybrid boundary conditions,” Mechanics Research Communications, vol. 38, no. 1, pp. 52–56, 2011.
[24]
K. Marynowski and T. Kapitaniak, “Kelvin-Voigt versus Bürgers internal damping in modeling of axially moving viscoelastic web,” International Journal of Non-Linear Mechanics, vol. 37, no. 7, pp. 1147–1161, 2002.
[25]
K. Marynowski, “Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension,” Chaos, Solitons and Fractals, vol. 21, no. 2, pp. 481–490, 2004.
[26]
K. Marynowski and T. Kapitaniak, “Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension,” International Journal of Non-Linear Mechanics, vol. 42, no. 1, pp. 118–131, 2007.
[27]
M. Pakdemirli and H. R. ？z, “Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations,” Journal of Sound and Vibration, vol. 311, no. 3–5, pp. 1052–1074, 2008.
[28]
S. V. Ponomareva and W. T. van Horssen, “On the transversal vibrations of an axially moving continuum with a time-varying velocity: transient from string to beam behavior,” Journal of Sound and Vibration, vol. 325, no. 4-5, pp. 959–973, 2009.
[29]
M. H. Ghayesh, “Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance,” International Journal of Mechanical Sciences, vol. 53, no. 11, pp. 1022–1037, 2011.
[30]
M. H. Ghayesh, H. A. Kafiabad, and T. Reid, “Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam,” International Journal of Solids and Structures, vol. 49, no. 1, pp. 227–243, 2012.
[31]
M. H. Ghayesh, “Coupled longitudinal-transverse dynamics of an axially accelerating beam,” Journal of Sound and Vibration, vol. 331, pp. 5107–5124, 2012.
[32]
M. H. Ghayesh, “Subharmonic dynamics of an axially accelerating beam,” Archive of Applied Mechanics, vol. 82, pp. 1169–1181, 2012.
[33]
M. H. Ghayesh and M. Amabili, “Steady-state transverse response of an axially moving beam with time-dependent axial speed,” International Journal of Non-Linear Mechanics, vol. 49, pp. 40–49, 2013.
[34]
G. Chakraborty and A. K. Mallik, “Stability of an accelerating beam,” Journal of Sound and Vibration, vol. 27, no. 2, pp. 309–320, 1999.
[35]
M. P. Paidoussis, “Flutter of conservative systems of pipe conveying incompressible fluid,” Journal of Mechanical Engineering and Science, vol. 17, no. 1, pp. 19–25, 1975.
[36]
M. Pakdemirli and H. Boyaci, “Comparision of direct perturbation methods with discretization-perturbation methods for nonlinear vibrations,” Journal of Sound and Vibration, vol. 186, pp. 837–845, 1985.
[37]
A. H. Nayfeh and P. F. Pai, Linear and Nonlinear Structural Mechanics, Wiley-Interscience, New York, NY, USA, 2004.
