We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.

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边界条件和转动惯量对弹性环振动模式的影响

我们展示了具有周期性和夹紧边界条件的圆形弹性环的小振幅振动。我们将杆建模为不可伸长的、各向同性的、自然直的基尔霍夫弹性杆，并通过分析获得环的振动模式，用于周期性边界条件，并通过数值获得夹紧边界条件。特别令人感兴趣的是振动模式对环中扭转应力的依赖性以及杆的转动惯量对模式频率和振幅的影响。在重新缩放 Kirchhoff 方程时，我们引入了一个与杆的纵横比成反比的参数。该参数可以捕捉杆的转动惯量的影响。我们发现转动惯量对振动模式的影响很小，但与环中高频扭曲变形相对应的特定类别的模式除外。此外，根据环中施加的扭转应力，一些振动模式在弹性杆上方或下方扭曲。