The rotary inertia defined by Timoshenko to account for the angular velocity effect in flexural vibration of beams has been questioned by some researchers in recent years, and it caused some confusions. This paper discusses the appropriate rotary inertia in Timoshenko beam theory (TBT) and evaluates the influence of the two forms of the rotary inertia on the prediction of the higher-mode frequencies of transversely vibrating beams. Based on the theory of elasticity and variational principle, this work shows that the rotary inertia in the original TBT, defined in terms of the rotation of beam cross-section induced by bending deformation, is variational consistent and is capable of yielding good results of the phase velocities of transversely vibrating beams even in the case where the wavelength of vibrating beams approaches the beam height. On the other hand, the so-called corrected TBT, in which the rotary inertia is defined in terms of the slope of beam deflection, is neither variational consistent nor accurate when the wavelength of vibrating beams approaches the beam height. Therefore, the rotary inertia in TBT defined by Timoshenko is correct and should be used in the dynamic analysis of beams.

中文翻译：

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论铁木辛科梁理论中的适当转动惯量

铁木辛科定义的用于解释梁弯曲振动中角速度效应的转动惯量近年来受到了一些研究人员的质疑，并引起了一些混淆。本文讨论了 Timoshenko 梁理论 (TBT) 中的适当转动惯量，并评估了两种形式的转动惯量对横向振动梁高模频率预测的影响。基于弹性理论和变分原理，本工作表明，原TBT中的转动惯量，定义为弯曲变形引起的梁截面转动，是变分一致的，能够产生良好的结果。即使在振动光束的波长接近光束高度的情况下，横向振动光束的相速度。另一方面，所谓的校正 TBT，其中转动惯量是根据光束偏转的斜率来定义的，当振动光束的波长接近光束高度时，它既不是变分一致的，也不是准确的。因此，Timoshenko 定义的 TBT 中的转动惯量是正确的，应该用于梁的动态分析。