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CHAOTIC　ANALYSIS　OF　subharmonic　resonant　waves　IN　UNDAMPED　AND　DAMPED　STRINGS

Journal of Hydrodynamics 1995年第1期7卷 92-104页

作者： Albert C. J. Luo, Ray P. S. Han (Department of Mechanical & Industrial Engineering,University of Manitoba,Winnipeg,Manitoba R3T 2N2, Canada)Xiang Yu-min(Department of Physics ,Sichuan Instutue of Industry and Chemical Technology,Zigong,Sichuan 643033, P. Univ of Manitoba Winnipeg Canada

The chaotic phenomena of subharmonic resonant waves in undamped and damped strings are investigated in this paper. The model consistS of a constant-tension, stretched string whose partial differential equation is derived by taking into account its exact configuration. Simplification via a Taylor series expansion of the curvature term and then employing the Galerkin method, an ordinary differential equation for the nonlinear dyamics is obtained. For the undamped case, we can formulate the Hamiltonian energy form of the conservative string, under the influence of an external periodic excitation. This permits the subharmonic resonant condition for this system to be derived. We truncate the resulting infinite number of subharmonic resonant waves to just two waves by renormalizing the Hamiltonian energy function near the subharmonic resonant orbit of the system. Adopting the renormalization group technique for the nit6raction of the two subharmonic resonant waves, an approximate chaotic condition associated with the subharmonic resonance of this system is determined. For the case fo the damped string, the minimum condition for the bifurcation of the subharmonic resonant wave is computed using the incremental energy balance method. For model verification, we carried out numerical simulations and they show good agreement with our analytical results.

关键词： chaos subharmonic resonant waves undamped and damped strings,incremental energy balance method renormalization group technique.

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