Multipulse Shilnikov orbits and Chaotic Dynamics for Nonlinear Nonplanar Motion of a Cantilever Beam

The multipulse Shilnikov orbits and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam are studied in this paper for the first time. The cantilever beam studied here is subjected to a harmonic axial excitation and two transverse excitations at the free end. The nonlinear governing equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system under combined parametric and forcing excitations. The resonant case considered here is principal parametric resonance-1/2 subharmonic resonance for the first mode and fundamental parametric resonance-primary resonance for the second mode. The parametrically and externally excited system is transformed to the averaged equation by using the method of multiple scales. From the averaged equation, the theory of normal form is used to find their explicit formulas. Based on normal form obtained above, the dissipative version of the energy-phase method is utilized to analyze the multipulse global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam. The energy-phase method is further improved to ensure the equivalence of topological structure for the phase portraits. The analysis of global dynamics indicates that there exist the multipulse jumping orbits in the perturbed phase space of the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the multipulse Shilnikov type chaotic motions can occur for the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations are given to verify the analytical predictions. It is also found from the results of numerical simulation in three-dimensional phase space that the multipulse Shilnikov type orbits exist for the nonlinear nonplanar oscillations of the cantilever beam.

[1]  George Haller,et al.  N-pulse homoclinic orbits in perturbations of resonant hamiltonian systems , 1995 .

[2]  M. Yao,et al.  Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam , 2005 .

[3]  Gregor Kovačič,et al.  A Melnikov Method for Homoclinic Orbits with Many Pulses , 1998 .

[4]  Santosha K. Dwivedy,et al.  Simultaneous combination and 1:3:5 internal resonances in a parametrically excited beam–mass system , 2003 .

[5]  A. Nayfeh,et al.  On the Transfer of Energy between Widely Spaced Modes in Structures , 2003 .

[6]  Stephen Wiggins,et al.  Global Bifurcations and Chaos , 1988 .

[7]  Z. C. Feng,et al.  Global Bifurcations in Parametrically Excited Systems with Zero-to-One Internal Resonance , 2000 .

[8]  N. Sri Namachchivaya,et al.  Chaotic Motion of Shallow Arch Structures under 1:1 Internal Resonance , 1997 .

[9]  Ali H. Nayfeh,et al.  Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams , 1998 .

[10]  T. H. Young,et al.  Dynamic stability and response of fluttered beams subjected to random follower forces , 2003 .

[11]  Wei Zhang.  Global and chaotic dynamics for a parametrically excited thin plate , 2001 .

[12]  Kazuyuki Yagasaki,et al.  Periodic and Homoclinic Motions in Forced, Coupled Oscillators , 1999 .

[13]  Wei Zhang,et al.  Vibration analysis on a thin plate with the aid of computation of normal forms , 2001 .

[14]  M. R. Silva,et al.  Out-of-plane vibrations of a beam including non-linear inertia and non-linear curvature effects , 1978 .

[15]  G. Kovačič,et al.  Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation , 1992 .

[16]  N. Sri Namachchivaya,et al.  Multipulse Orbits in the Motion of Flexible Spinning Discs , 2002, J. Nonlinear Sci..

[17]  Stephen Wiggins,et al.  On the existence of chaos in a class of two-degree-of-freedom, damped, strongly parametrically forced mechanical systems with brokenO(2) symmetry , 1993 .

[18]  Ebrahim Esmailzadeh,et al.  Periodic behavior of a cantilever beam with end mass subjected to harmonic base excitation , 1998 .

[19]  M. R. Silva,et al.  Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion , 1978 .

[20]  A. Nayfeh,et al.  COUPLING BETWEEN HIGH-FREQUENCY MODES AND A LOW-FREQUENCY MODE: THEORY AND EXPERIMENT , 1993 .

[21]  Wei Zhang,et al.  Global Analysis for a nonlinear Vibration Absorber with Fast and Slow Modes , 2001, Int. J. Bifurc. Chaos.

[22]  Thomas A. Wettergren,et al.  Homoclinic orbits in the dynamics of resonantly driven coupled pendula , 1996 .

[23]  N. Sri Namachchivaya,et al.  Chaotic Dynamics of Shallow Arch Structures under 1:2 Resonance , 1997 .

[24]  Joseph P. Cusumano,et al.  Chaotic non-planar vibrations of the thin elastica: Part II: Derivation and analysis of a low-dimensional model , 1995 .

[25]  N. Sri Namachchivaya,et al.  Global dynamics of parametrically excited nonlinear reversible systems with nonsemisimple 1:1 resonance , 1995 .

[26]  J. Zu,et al.  Computation of Normal Forms for High Dimensional Nonlinear Systems and Application to Nonplanar Motions of a Cantilever Beam , 2003 .

[27]  M. R. Silva,et al.  Non-linear non-planar resonant oscillations in fixed-free beams with support asymmetry , 1979 .

[28]  C. L. Zaretzky,et al.  Experimental Investigation of Non-Linear Modal Coupling in the Response of Cantilever Beams , 1994 .

[29]  Joseph P. Cusumano,et al.  Chaotic non-planar vibrations of the thin elastica: Part I: Experimental observation of planar instability , 1995 .

[30]  M. R. Silva,et al.  Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. II. Forced Motions , 1978 .

[31]  Tasso J. Kaper,et al.  MULTI-BUMP ORBITS HOMOCLINIC TO RESONANCE BANDS , 1996 .

[32]  S. Wiggins,et al.  Orbits homoclinic to resonances: the Hamiltonian case , 1993 .

[33]  M. F. Golnaraghi,et al.  Large free vibrations of a beam carrying a moving mass , 2003 .

[34]  Ali H. Nayfeh,et al.  Non-linear non-planar oscillations of a cantilever beam under lateral base excitations , 1990 .

[35]  B. O. Al-Bedoor,et al.  Comparison of analytical techniques for nonlinear vibrations of a parametrically excited cantilever , 2001 .

[36]  P. Yu,et al.  Global Dynamics of a Parametrically and Externally Excited Thin Plate , 2001 .

[37]  Wei Zhang,et al.  Global dynamics of the cable under combined parametrical and external excitations , 2002 .

[38]  J. Zu,et al.  Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillations of a cantilever beam , 2004 .

[39]  Ali H. Nayfeh,et al.  Non-linear non-planar parametric responses of an inextensional beam☆ , 1989 .

[40]  George Haller,et al.  Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schro¨dinger equation , 1995 .

[41]  Z. C. Feng,et al.  Global bifurcations in the motion of parametrically excited thin plates , 1993 .