浅球薄壳轴对称极凹陷的非轴对称的动力不稳定性<sup>*</sup>

浅球薄壳轴对称极凹陷的非轴对称的动力不稳定性^*

基金项目: * 国家自然科学基金资助项目

摘要: 当壳厚-升高比ε²<<1时,由二次型载荷产生的轴对称极凹陷的浅球薄壳是动力不稳定的.微小的扰动足以使它变成非轴对称凹陷.在两种情形下,问题都可归结为特征值问题Tw_n=c_nw_n.其中T当ε²<<1时近似为一Sfrum-Liouville算子.本文用谱理论和Hilbert定理均可证明T至少存在一实的特征值;它意味着轴对称极凹陷是动力不稳定的.再进一步,本文还找出代表不稳定凹陷壳体的非轴对称变形的、属于一特征值的T的特征函数.

Abstract: If the parameter ε², which measures the thickness-to-rise of the sliell, is small, the axismnnetrie polar dimpling of shallow spherical shell due to quadratic pressure distribution is dynamic instability, i.e., a small perturbation can change it to an asymmetric polar dimple mode.In two cases, the problem can be reduced to an eigenvalue problem Tw_n=c_n+w_n, where T can approximately be reduced to a Sturm-Liouvi/le operator if ε²<<1. The existence of at least one real eigenvalue of T, which means that the axisyntmetric polar dimpling is dynamically unstable, is proved by spectral theorem or Hilbert theorem.Furthermore, an eigenfunction, which represents one of the asymmetric modes of the unstable dimple shell, belonging to an eigenvalue of T, is found.

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