Low-frequency vibrations of a thin elastic annulus are considered. The dynamic equations of plane strain are subjected to asymptotic treatment beyond the leading-order approximation. The main peculiarity of the considered problem is a specific degeneration associated with the effect of the almost inextensible midline of the annulus, resulting in a few unexpected features of the mechanical behaviour. In particular, it is discovered that the leading-order even component of the circumferential stress is not uniform across the thickness, as is usually assumed, and can be determined only at the next order. The derived refined equations also govern vibrations of a cylindrical shell at the lowest cut-off frequencies. The two-term asymptotic formula obtained for the latter fully agrees with the expansion of the transcendental dispersion relation for plane strain but does not coincide in the second term with the prediction of the Kirchhoff–Love theory for thin shells.

中文翻译：

---

薄弹性环的精细动力学方程的渐近推导

考虑了薄弹性环的低频振动。平面应变的动力学方程经过超前阶近似的渐近处理。所考虑问题的主要特点是与环几乎不可伸展的中线的影响相关的特定退化，导致机械行为的一些意想不到的特征。特别是，发现周向应力的前序偶数分量在厚度上并不均匀，正如通常假设的那样，并且只能在下一个阶次确定。导出的精确方程还控制了最低截止频率下圆柱壳的振动。