Internal resonance and nonlinear response of an axially moving beam: two numerical techniques

The nonlinear resonant response of an axially moving beam is investigated in this paper via two different numerical techniques: the pseudo-arclength continuation technique and direct time integration. In particular, the response is examined for the system in the neighborhood of a three-to-one internal resonance between the first two modes as well as for the case where it is not. The equation of motion is reduced into a set of nonlinear ordinary differential equation via the Galerkin technique. This set is solved using the pseudo-arclength continuation technique and the results are confirmed through use of direct time integration. Vibration characteristics of the system are presented in the form of frequency-response curves, time histories, phase-plane diagrams, and fast Fourier transforms (FFTs).

[1]  B. Y. Ballal.  Vibrations of rectangular plates , 1966 .

[2]  S. Naguleswaran,et al.  Lateral vibration of band-saw blades, pulley belts and the like , 1968 .

[3]  C. D. Mote,et al.  Free, Periodic, Nonlinear Oscillation of an Axially Moving Strip , 1969 .

[4]  L. Y. Shih.  Three-dimensional non-linear vibration of a traveling string , 1971 .

[5]  A. Simpson.  Transverse Modes and Frequencies of Beams Translating between Fixed end Supports , 1973 .

[6]  P. J. Holmes.  Pipes Supported at Both Ends Cannot Flutter , 1978 .

[7]  C. D. Mote,et al.  Current Research on the Vibration and Stability of Axially-Moving Materials , 1988 .

[8]  A. Galip Ulsoy,et al.  Transverse Vibration of an Axially Accelerating String , 1994 .

[9]  B. Tabarrok,et al.  Finite Element Analysis Of An Axially Moving Beam, Part I: Time Integration , 1994 .

[10]  A. Galip Ulsoy,et al.  STABILITY ANALYSIS OF AN AXIALLY ACCELERATING STRING , 1997 .

[11]  Thomas F. Fairgrieve,et al.  AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont) , 1997 .

[12]  E. Özkaya,et al.  Approximate Boundary Layer Solution of a Moving Beam Problem , 1998 .

[13]  H. R. Öz,et al.  Non-linear vibrations and stability of an axially moving beam with time-dependent velocity , 2001 .

[14]  T. Kapitaniak,et al.  Kelvin–Voigt versus Bürgers internal damping in modeling of axially moving viscoelastic web , 2002 .

[15]  G. Suweken,et al.  On the Weakly Nonlinear, Transversal Vibrations of a Conveyor Belt with a Low and Time-Varying Velocity , 2002 .

[16]  F. Vestroni,et al.  COMPLEX DYNAMICS OF HIGH-SPEED AXIALLY MOVING SYSTEMS , 2002 .

[17]  M. Pakdemirli,et al.  Non-linear vibrations of a simple–simple beam with a non-ideal support in between , 2003 .

[18]  Liqun Chen.  Analysis and Control of Transverse Vibrations of Axially Moving Strings , 2005 .

[19]  M. Amabili,et al.  Effect of concentrated masses with rotary inertia on vibrations of rectangular plates , 2006 .

[20]  T. Kapitaniak,et al.  Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension , 2007 .

[21]  S. E. Khadem,et al.  Non-linear vibration and stability analysis of a partially supported conveyor belt by a distributed viscoelastic foundation , 2007 .

[22]  M. Ghayesh.  Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide , 2008 .

[23]  Liqun Chen,et al.  Parametric resonance of axially moving Timoshenko beams with time-dependent speed , 2009 .

[24]  M. Ghayesh.  Stability characteristics of an axially accelerating string supported by an elastic foundation , 2009 .

[25]  M. Ghayesh.  Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation , 2010 .

[26]  Liqun Chen,et al.  Response in Coupled Planar Vibration of Axially Moving Viscoelastic Beams , 2010 .

[27]  Liqun Chen,et al.  Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model , 2010 .

[28]  M. Pakdemirli,et al.  A general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitation , 2010 .

[29]  M. Amabili.  Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass , 2010 .

[30]  Mergen H. Ghayesh,et al.  Vibrations and stability of axially traveling laminated beams , 2010, Appl. Math. Comput..

[31]  M. Ghayesh,et al.  Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams , 2010 .

[32]  Mergen H. Ghayesh,et al.  On the Natural Frequencies, Complex Mode Functions, and Critical Speeds of Axially Traveling Laminated Beams: Parametric Study , 2011 .

[33]  M. Ghayesh.  Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance , 2011 .

[34]  M. Ghayesh,et al.  Nonlinear dynamic response of axially moving, stretched viscoelastic strings , 2011 .

[35]  M. Ghayesh.  Parametrically excited viscoelastic beam-spring systems: nonlinear dynamics and stability , 2011 .

[36]  F. Alijani,et al.  An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system , 2011 .

[37]  S. H. A. Chen,et al.  Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances , 2011 .

[38]  M. Ghayesh,et al.  A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions , 2011 .

[39]  K. Hong,et al.  Stabilization of an axially moving web via regulation of axial velocity , 2011 .

[40]  M. Ghayesh.  Nonlinear dynamic response of a simply-supported Kelvin–Voigt viscoelastic beam, additionally supported by a nonlinear spring , 2012 .

[41]  M. Païdoussis,et al.  Subcritical parametric response of an axially accelerating beam , 2012 .

[42]  M. Ghayesh.  Coupled longitudinal–transverse dynamics of an axially accelerating beam , 2012 .

[43]  M. Ghayesh.  Stability and bifurcations of an axially moving beam with an intermediate spring support , 2012 .

[44]  M. Ghayesh,et al.  Free vibrations of beam-mass-spring systems: analytical analysis with numerical confirmation , 2012 .

[45]  Liqun Chen,et al.  Dynamical analysis of axially moving plate by finite difference method , 2012 .

[46]  M. Ghayesh,et al.  Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam , 2012 .

[47]  M. Ghayesh,et al.  Nonlinear vibrations and stability of parametrically exited systems with cubic nonlinearities and internal boundary conditions: A general solution procedure , 2012 .

[48]  M. Ghayesh,et al.  Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system , 2012 .

[49]  M. Païdoussis,et al.  Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis , 2012 .

[50]  M. Amabili,et al.  Experiments and simulations for large-amplitude vibrations of rectangular plates carrying concentrated masses , 2012 .

[51]  M. Ghayesh.  Subharmonic dynamics of an axially accelerating beam , 2012 .

[52]  M. Ghayesh,et al.  Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance , 2012 .

[53]  M. Ghayesh,et al.  Thermo-mechanical nonlinear dynamics of a buckled axially moving beam , 2013 .

[54]  M. Ghayesh,et al.  Steady-state transverse response of an axially moving beam with time-dependent axial speed , 2013 .