We present the first full analytical solution to the mode-shape equation of the gravity-loaded Rayleigh–Timoshenko beam, thus describing in closed form the vibration modes of beam–columns, bending cables, and stiff marine risers under self-weight. The modes are given in terms of a family of complex “Hankel–Airy” functions, along with their asymptotic expansions, orthogonality relations and series representations. It is shown that the new solutions generalize multiple approximate and asymptotic solutions in the literature and that their numerical approximations closely match past numerical and experimental results as well as models built using the ANSYS software package.

我们提出了重力加载瑞利-铁木辛哥梁的振型方程的第一个完整解析解，从而以封闭形式描述了梁-柱、弯曲电缆和自重下刚性海洋立管的振动模式。这些模式是根据一系列复杂的“Hankel-Airy”函数以及它们的渐近展开、正交关系和级数表示给出的。结果表明，新的解决方案概括了文献中的多个近似和渐近解决方案，并且它们的数值近似与过去的数值和实验结果以及使用 ANSYS 软件包构建的模型密切匹配。