Nonlinear nonplanar dynamics of parametrically excited cantilever beams

被引:78
|
作者
Arafat, HN [1 ]
Nayfeh, AH [1 ]
Chin, CM [1 ]
机构
[1] Virginia Polytech Inst & State Univ, Dept Engn Sci & Mech, Blacksburg, VA 24061 USA
基金
美国国家科学基金会;
关键词
beams; internal resonance; parametric resonance; bifurcations; chaos;
D O I
10.1023/A:1008218009139
中图分类号
TH [机械、仪表工业];
学科分类号
0802 ;
摘要
The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.
引用
收藏
页码:31 / 61
页数:31
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