INTERNAL RESONANCES IN WHIRLING STRINGS INVOLVING LONGITUDINAL DYNAMICS AND MATERIAL NON-LINEARITIES

Abstract Internal resonance mechanisms between near-commensurate longitudinal and transverse modes of a taut spatial string are identified and studied using an asymptotic method, and the influence of material non-linearities on the resulting solutions is considered. Geometrical non-linearities couple longitudinal motions to in-plane and out-of-plane transverse motions, resulting in resonant and non-resonant interactions between linearly orthogonal string modes. Past studies have included only transverse modes in the description of string motions and have predicted periodic, quasi-periodic, and chaotic whirling motions arising from the geometrical non-linearities. This study considers further the inclusion of longitudinal motions and a non-linear material law, which are both appropriate for the study of rubber-like strings. An asymptotic analysis captures the aforementioned whirling motions, as well as a new class of whirling motions with significant longitudinal content. Periodic, quasi-periodic, and aperiodic (likely chaotic) responses are included among these motions. Their existence, hardening–softening characterization, and stability are found to be highly dependent on the magnitude of the material non-linearities.

[1]  O. Gottlieb,et al.  Nonlinear Dynamics of a Taut String with Material Nonlinearities , 2001 .

[2]  Anil K. Bajaj,et al.  Amplitude modulated and chaotic dynamics in resonant motion of strings , 1989 .

[3]  Angelo Luongo,et al.  On the Reconstitution Problem in the Multiple Time-Scale Method , 1999 .

[4]  John W. Miles,et al.  Resonant, nonplanar motion of a stretched string , 1984 .

[5]  J. Oden.  Finite Elements of Nonlinear Continua , 1971 .

[6]  Ali H. Nayfeh,et al.  Nonlinear Response of a Taut String to Longitudinal and Transverse End Excitation , 1995 .

[7]  Noel C. Perkins,et al.  Nonlinear oscillations of suspended cables containing a two-to-one internal resonance , 1992, Nonlinear Dynamics.

[8]  A. K. Bajaj,et al.  On the amplitude dynamics and crisis in resonant motion of stretched strings , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[9]  Nicholas B. Tufillaro,et al.  Torus doubling and chaotic string vibrations: Experimental results , 1990 .

[10]  L. Meirovitch.  Principles and techniques of vibrations , 1996 .

[11]  M. B. Rubin,et al.  NUMERICAL SOLUTIONS OF FORCED VIBRATION AND WHIRLING OF A NON-LINEAR STRING USING THE THEORY OF A COSSERAT POINT , 1996 .

[12]  Leon O. Chua,et al.  Practical Numerical Algorithms for Chaotic Systems , 1989 .

[13]  Roddam Narasimha,et al.  Non-Linear vibration of an elastic string , 1968 .

[14]  Oliver M. O’Reilly,et al.  Global bifurcations in the forced vibration of a damped string , 1993 .

[15]  Philip Holmes,et al.  Non-linear, non-planar and non-periodic vibrations of a string , 1992 .