Medium-frequency (mid-frequency) vibration analysis of complex structures plays an important role in automotive, aerospace, mechanical, and civil engineering. Flexible beam structures modeled by the classical Euler–Bernoulli beam theory have been widely used in various engineering problems. A kinematic hypothesis made in the Euler–Bernoulli beam theory is that the plane sections of a beam normal to its neutral axis remain planes after the beam experiences bending deformation, which neglects shear deformation. However, previous investigations found out that the shear deformation of a beam (even with a large slenderness ratio) becomes noticeable in high-frequency vibrations. The Timoshenko beam theory, which describes both bending deformation and shear deformation, would naturally be more suitable for medium-frequency vibration analysis. Nevertheless, vibrations of Timoshenko beam structures in a medium frequency region have not been well studied in the literature. This paper presents a new method for mid-frequency vibration analysis of two-dimensional Timoshenko beam structures. The proposed method, which is called the augmented Distributed Transfer Function Method (DTFM), models a Timoshenko beam structure by a spatial state-space formulation in the [Formula: see text]-domain. The augmented DTFM determines the frequency response of a beam structure in an exact and analytical form, in any frequency region covering low, middle, or high frequencies. Meanwhile, the proposed method provides the local information of a beam structure, such as displacement, shear deformation, bending moment and shear force at any location, which otherwise would be very difficult with energy-based methods. The medium-frequency analysis by the augmented DTFM is validated in numerical examples, where the efficiency and accuracy of the proposed method is demonstrated. Also, the effects of shear deformation on the dynamic behaviors of a beam structure at medium frequencies are examined through comparison of the Timoshenko beam and Euler–Bernoulli beam theories.

中文翻译：

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Timoshenko梁结构的中频振动分析

复杂结构的中频（mid-frequency）振动分析在汽车、航空航天、机械和土木工程中发挥着重要作用。经典欧拉-伯努利梁理论建模的柔性梁结构已广泛应用于各种工程问题。欧拉-伯努利梁理论中的一个运动学假设是，在梁经历弯曲变形后，垂直于其中性轴的平面截面仍然是平面，忽略了剪切变形。然而，之前的研究发现，梁的剪切变形（即使长细比很大）在高频振动中变得很明显。同时描述弯曲变形和剪切变形的 Timoshenko 梁理论自然更适用于中频振动分析。然而，Timoshenko 梁结构在中频区域的振动在文献中还没有得到很好的研究。本文提出了一种新的二维 Timoshenko 梁结构中频振动分析方法。所提出的方法称为增强分布式传递函数方法 (DTFM)，通过 [公式：参见文本] 域中的空间状态空间公式对 Timoshenko 梁结构进行建模。增强的 DTFM 在涵盖低、中或高频率的任何频率区域中以精确和解析的形式确定梁结构的频率响应。同时，该方法提供了梁结构的局部信息，如任意位置的位移、剪切变形、弯矩和剪力，否则，使用基于能量的方法将非常困难。在数值示例中验证了增强 DTFM 的中频分析，证明了所提出方法的效率和准确性。此外，通过比较 Timoshenko 梁和 Euler-Bernoulli 梁理论，研究了剪切变形对中频梁结构动态行为的影响。