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 引用本文: 宋子文, 孙直, 朱一超. 基于渐近分析法的复杂截面梁振动分析. 力学学报, 2024, 56(2): 494-505.
Song Ziwen, Sun Zhi, Zhu Yichao. Vibration analysis of beam with complex cross-section based on asymptotic analysis method. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(2): 494-505.
 Citation: Song Ziwen, Sun Zhi, Zhu Yichao. Vibration analysis of beam with complex cross-section based on asymptotic analysis method. Chinese Journal of Theoretical and Applied Mechanics, 2024, 56(2): 494-505.

## VIBRATION ANALYSIS OF BEAM WITH COMPLEX CROSS-SECTION BASED ON ASYMPTOTIC ANALYSIS METHOD

• 摘要: 梁结构在工程工业中发挥着重要的作用, 梁的自振问题能够体现其本征动力学特性, 是梁动力学分析中的基本问题. 经典梁理论的提出往往基于不同的假设, 文章采用渐近分析的方法, 以梁的截面尺寸与长度之比为特征参数, 严格地推导了三维梁结构的一维等效振动分析模型. 基于梁结构几何特征的参数的引入, 有助于确定梁结构应力场和位移场等物理量之间的真实量级关系. 渐近分析结果表明, 纯弯曲变形只是细长梁振动分析中的首阶项, 对于复杂截面梁结构仍然可以使用与欧拉梁模型复杂度相似的一维等效梁模型进行振动分析. 以精细三维有限元的计算结果为基准, 验证了所提出的一维等效梁模型用于振动分析的准确性和有效性. 一维等效梁模型可以在商业有限元软件中通过自定义截面等内嵌模块便捷地实现. 对比有限元软件中多种梁单元的计算结果发现, 在相似的模型复杂度下, 等效模型对细长梁自振频率的分析精度与稳定性显著优于在商业有限元软件中直接选择复杂梁截面使用梁单元计算的结果.

Abstract: The beam structures play a very important role in the engineering industry. The free vibration problem of the beam structures reflects their basic dynamic characteristics, which is also a vital problem in the dynamics analysis. The classical theories of the beam structures are often based on different assumptions. The present paper uses the asymptotic analysis method to strictly derive a one-dimensional equivalent beam model for the vibration analysis of the three-dimensional beam structures with the ratio of the section size to its length as the characteristic parameter. The parameter which is based on the geometric characteristic of the beam structures, helps to determine the true order of the magnitude relationship between physical quantities of the beam structures, such as the stress field and the displacement field. The asymptotic analysis results show that the pure bending deformation is only the first order term in the vibration analysis process of the beam structures. For the slender beams with complex cross-sections, the one-dimensional equivalent beam model with similar complexity to the Euler beam model can still be used for the vibration analysis. Based on the results of the precise three-dimensional finite element method, the accuracy and effectiveness of the proposed one-dimensional equivalent beam model are verified when it is applied in the vibration analysis. The proposed one-dimensional equivalent beam model can be conveniently implemented in the commercial finite element software through the user-defined section and other embedded modules. Comparing the natural frequency calculated by various beam elements in the finite element software, it is found that with the similar computational complexity, the proposed equivalent beam model is significantly superior. The results of the slender beam structures calculated by the proposed method are more accurate and stable than that calculated by selecting the complex cross-sections and using the beam elements directly in the commercial finite element software in the vibration analysis.

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