Exact analytical solutions for the free vibrations of Thin-Walled shallow shells are presented in this work. The shells considered have rectangular planform and two principal constant curvatures. Past treatment of this problem included only one possible case of boundary conditions, and was limited to symmetric modes only. The new solutions address six different cases of in-plane and out-of-plane boundary conditions, and provide an interesting insight into the dynamics of such shells. The formulation may be applied to cylindrical, spherical, elliptical paraboloidal, and hyperbolic paraboloidal shallow shells. Previous works in this field have not addressed the present boundary conditions in exact analytical form.

这项工作提出了薄壁浅壳自由振动的精确解析解。所考虑的壳体具有矩形平面形状和两个主要的恒定曲率。过去对该问题的处理仅包括边界条件的一种可能情况，并且仅限于对称模式。新的解决方案解决了平面内和平面外边界条件的六种不同情况，并为此类壳的动力学提供了有趣的见解。该制剂可以应用于圆柱形，球形，椭圆形抛物面和双曲抛物面浅壳。该领域的先前工作尚未以精确的分析形式解决当前的边界条件。