On New Analytic Free Vibration Solutions of Doubly Curved Shallow Shells by the Symplectic Superposition Method Within the Hamiltonian-System Framework
On New Analytic Free Vibration Solutions of Doubly Curved Shallow Shells by the Symplectic Superposition Method Within the Hamiltonian-System Framework
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DOI：
10.1115/1.4047701
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发表时间：
2021-02
期刊：
影响因子：
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通讯作者：
Rui Li;Chao Zhou;Xinran Zheng
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文献类型：
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作者：
Rui Li;Chao Zhou;Xinran Zheng
本文首次尝试用辛叠加法探索双弯曲浅壳的新的解析自由振动解，重点研究了由于高阶偏微分方程边值问题难以处理而难以用经典解析方法求解的非l<s:1>型壳。与传统的在欧几里德空间中求解的基于拉格朗日系统的表达式相比，本文的问题描述是在哈密顿系统中，求解过程是在辛空间中实现的，结合了辛特征值问题的表述和辛特征展开。具体地说，首先将一个原问题转化为两个子问题，用上述策略求解得到辛解。根据原问题与子问题叠加的等价性要求，得到解析频率解和模态振型解。对具有代表性的非l型壳的综合计算结果进行了制表或作图，结果与数值有限元法的结果吻合较好。由于数学推导的严格性和解的准确性，该方法为寻求更多的解析解提供了坚实的途径。
This study presents a first attempt to explore new analytic free vibration solutions of doubly curved shallow shells by the symplectic superposition method, with focus on non-Lévy-type shells that are hard to tackle by classical analytic methods due to the intractable boundary-value problems of high-order partial differential equations. Compared with the conventional Lagrangian-system-based expression to be solved in the Euclidean space, the present description of the problems is within the Hamiltonian system, with the solution procedure implemented in the symplectic space, incorporating formulation of a symplectic eigenvalue problem and symplectic eigen expansion. Specifically, an original problem is first converted into two subproblems, which are solved by the above strategy to yield the symplectic solutions. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition of subproblems. Comprehensive results for representative non-Lévy-type shells are tabulated or plotted, all of which are well validated by satisfactory agreement with the numerical finite element method. Due to the strictness of mathematical derivation and accuracy of solution, the developed method provides a solid approach for seeking more analytic solutions.