Parametrically Excited Nonlinear Two-Degree-of -Freedom Systems with Repeated Natural Frequencies

The method of normal forms is used to study the nonlinear response of two-degree-of-freedom systems with repeated natural frequencies and cubic nonlinearity to a principal parametric excitation. The linear part of the system has a nonsemisimple one-to-one resonance. The character of the stability and various types of bifurcation including the formation of a homoclinic orbit are analyzed. The results are applied to the flutter of a simply supported panel in a supersonic airstream.

[1]  A. H. Nayfeh,et al.  The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance , 1986 .

[2]  Raouf A. Ibrahim,et al.  AUTOPARAMETRIC RESONANCE IN A STRUCTURE CONTAINING A LIQUID, PART I: TWO MODE INTERACTION , 1975 .

[3]  M. Nagata.  Nonlinear Faraday resonance in a box with a square base , 1989, Journal of Fluid Mechanics.

[4]  A. Nayfeh,et al.  Nonlinear Analysis of the Lateral Response of Columns to Periodic Loads , 1978 .

[5]  P. Holmes.  Chaotic motions in a weakly nonlinear model for surface waves , 1986, Journal of Fluid Mechanics.

[6]  Ali H. Nayfeh,et al.  Surface waves in closed basins under parametric and internal resonances , 1987 .

[7]  E. Dowell,et al.  Aeroelasticity of Plates and Shells , 1974 .

[8]  S. Nemat-Nasser,et al.  On the stability of steady-state response of certain nonlinear dynamic systems subjected to harmonic excitations , 1972 .

[9]  Earl H. Dowell,et al.  Parametric Random Vibration , 1985 .

[10]  A. H. Nayfeh,et al.  The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance , 1983 .

[11]  Ali H. Nayfeh,et al.  The Method of Normal Forms , 2011 .

[12]  P. R. Sethna,et al.  Resonant surface waves and chaotic phenomena , 1987, Journal of Fluid Mechanics.

[13]  A. H. Nayfeh,et al.  Parametrically excited non-linear multidegree-of-freedom systems with repeated natural frequencies , 1982 .

[14]  Jerry P. Gollub,et al.  Surface wave mode interactions: effects of symmetry and degeneracy , 1989, Journal of Fluid Mechanics.

[15]  K. Asmis,et al.  Combination and Internal Resonance in a Nonlinear Two-Degrees-of-Freedom System , 1972 .

[16]  Procaccia,et al.  Low-dimensional chaos in surface waves: Theoretical analysis of an experiment. , 1986, Physical review. A, General physics.

[17]  S. Ciliberto,et al.  Chaotic mode competition in parametrically forced surface waves , 1985, Journal of Fluid Mechanics.

[18]  K. Asmis,et al.  Multiple parametric resonance in a non-linear two degree of freedom system , 1974 .

[19]  L. P. Šil'nikov,et al.  A CONTRIBUTION TO THE PROBLEM OF THE STRUCTURE OF AN EXTENDED NEIGHBORHOOD OF A ROUGH EQUILIBRIUM STATE OF SADDLE-FOCUS TYPE , 1970 .

[20]  Ali H. Nayfeh.  Parametrically excited multidegree-of-freedom systems with repeated frequencies , 1983 .

[21]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

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

[23]  Z. C. Feng,et al.  Symmetry-breaking bifurcations in resonant surface waves , 1989, Journal of Fluid Mechanics.

[24]  Sia Nemat-Nasser,et al.  Stability of solution of systems of linear differential equations with harmonic coefficients. , 1972 .

[25]  Charles M. Krousgrill,et al.  Combination parametric resonance leading to periodic and chaotic response in two-degree-of-freedom systems with quadratic non-linearities , 1988 .

[26]  Ali H. Nayfeh,et al.  Modal Interactions in Dynamical and Structural Systems , 1989 .

[27]  Ali H. Nayfeh,et al.  Parametric excitation of two internally resonant oscillators , 1987 .

[28]  Two-degree-of-freedom systems with quadratic non-linearities subjected to parametric and self excitation , 1991 .

[29]  A. Nayfeh,et al.  Surface waves in closed basins under principal and autoparametric resonances , 1990 .

[30]  H. Barlow.  Surface Waves , 1958, Proceedings of the IRE.

[31]  A. H. Nayfeh,et al.  The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation , 1983 .