This paper proposed an energy method, based on the Donnell thin shell theory, to derive the vibration characteristic equations of underwater ribbed and ring-stiffened functionally graded (FG) cylindrical shells, and the characteristic equations are solved by MATLAB to obtain their natural frequencies. Through comparative analysis with the existing literature, the correctness and effectiveness of the calculation method are verified. By using numerical examples, the effects of the shell size, material composition, rib section length to width ratio, and rib arrangement on the natural frequency are analyzed, and it has been proved that the aforementioned parameters are the key parameters influencing natural frequency. It is found that the natural frequency of the FG cylindrical shell increases with the decrease in the length–diameter ratio of the shell, increases with the increase in the shell thickness–diameter ratio, and increases with the decrease in the rib spacing. When the cylindrical shell is short and thin, the addition of longitudinal ribs can significantly change the natural frequency of the cylindrical shell.

中文翻译：

---

带肋和环加筋的FG圆柱壳在静水压力下的振动分析

本文基于Donnell薄壳理论，提出了一种能量方法，推导了水下肋肋和环形加筋功能梯度圆柱壳的振动特性方程，并通过MATLAB求解了该特性方程，得到了其固有频率。通过与现有文献的对比分析，验证了该计算方法的正确性和有效性。通过数值例子，分析了壳体尺寸，材料组成，肋截面长宽比和肋的排列方式对固有频率的影响，并证明上述参数是影响固有频率的关键参数。结果发现，FG圆柱壳的固有频率随着壳长直径比的减小而增加，随着壳厚直径比的增加而增加，并且随着肋间距的减小而增加。当圆柱壳短而薄时，增加纵向肋可以显着改变圆柱壳的固有频率。