扁球面网壳的混沌运动研究

THE STUDY ON CHAOTIC MOTION OF THE SHALLOW RETICULATED SPHERICAL SHELLS

  • 摘要: 在圆形三向网架非线性动力学基本方程的基础上,用拟壳法给出了圆底扁球面三向网壳的非线性动力学基本方程. 在固定边界条件下,引入了异于等厚度壳的无量纲量,对基本方程和边界条件进行无量纲化,通过Galerkin作用得到了一个含二次、三次的非线性动力学方程. 为求Melnikov函数,对一类非线性动力系统的自由振动方程进行了求解,得到了此类问题的准确解. 在无激励情况下,讨论了稳定性问题. 在外激励情况下,通过求Melnikov函数,给出了可能发生混沌运动的条件. 通过数字仿真绘出了平面相图,证实了混沌运动的存在.

     

    Abstract: By using the method of quasi-shells,the nonlinear dynamical equations are established for three-dimensional shallow spherical shellswith circular bottom based on the nonlinear dynamical equationsfor circular reticulated structures with three-dimensional grids.The foundational equations and the boundary conditions are simplified byintroducing dimensionless quantities, and a nonlineardifferential equation of the third order is derivedunder the boundary conditions of fixed edges by using Galerkin method. In orderto obtain the Melnikov function, the free oscillation equation of a kind ofnonlinear dynamics system is solved, and then the exact solution to the problem isobtained. The stability is discussedon the condition of no external excitation. The conditions for chaoticmotion are given by solving for the Melnikov function underexternal excitations. Existence of the chaotic motion is proved bynumerical simulation and the phase planes are plotted.

     

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