Abstract This article concerns with the analysis of the frequency range within which Timoshenko’s model can be applied for the study of vibrating beams, possibly without incurring in large engineering approximations. Timoshenko's equations are solved in exact way for all classical boundary conditions; frequencies and modes are compared with exact or asymptotically exact solutions of the elasticity theory based on the plane stress assumptions. The analytical procedures, the comparisons and the related findings show that a threshold previously conjectured as cut-off frequency (could be a useful indication, to a certain extent, but it does not seem generalizable; incidentally, the analytical procedure, naturally provides frequencies falling in what could be defined as a single spectrum of Timoshenko's beam theory.

中文翻译：

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论铁木辛柯梁理论的频率范围

摘要 本文涉及对 Timoshenko 模型可用于研究振动梁的频率范围的分析，可能不会引起大型工程近似。Timoshenko 方程在所有经典边界条件下都以精确的方式求解；频率和模式与基于平面应力假设的弹性理论的精确或渐近精确解进行比较。分析程序、比较和相关发现表明，先前推测为截止频率的阈值（在某种程度上可能是一个有用的指示，但它似乎不可推广；顺便说一句，分析程序自然提供了下降频率在什么可以定义为铁木辛哥的梁理论的单一频谱。