Domagalski Łukasz
Faculty of Civil Engineering, Architecture and Environmental Engineering, Łódź University of Technology, 90-924 Lodz, Poland.
Materials (Basel). 2021 Dec 11;14(24):7628. doi: 10.3390/ma14247628.
This paper deals with the linear natural vibrations analysis of beams where the geometric and material properties vary periodically along the beam axis. In contrast with homogeneous prismatic beams, the frequency spectra of such beams are irregular as there exist enlarged intervals between some adjacent frequencies. Presented here are two averaged models of beams based on the tolerance modelling approach. The assumptions of classical Euler-Bernoulli and Timoshenko-Ehrenfest beam theories are adopted as the foundations. The resulting mathematical models are systems of differential equations with constant, weight-averaged coefficients. This makes it possible to apply any classical method of solution suitable for homogeneous beams, such as Galerkin orthogonalization. Here, emphasis is placed on the comparison of natural frequencies neighbouring the frequency band-gaps that are obtained from these two theories. Two basic cases of material and geometric property distribution in a periodicity cell are studied, and the natural frequencies and mode shapes are obtained for a simply supported beam. The results are supported by a comparison with the finite element method and partially exact solutions.
本文研究几何和材料特性沿梁轴呈周期性变化的梁的线性固有振动分析。与均质等截面梁不同，此类梁的频谱不规则，因为某些相邻频率之间存在较大间隔。本文基于公差建模方法提出了两种梁的平均模型。采用经典的欧拉 - 伯努利梁理论和铁木辛柯 - 埃伦费斯特梁理论的假设作为基础。所得数学模型是具有恒定加权平均系数的微分方程组。这使得可以应用任何适用于均质梁的经典求解方法，如伽辽金正交化方法。在此，重点是比较从这两种理论获得的与频带间隙相邻的固有频率。研究了周期性单元中材料和几何特性分布的两种基本情况，并获得了简支梁的固有频率和振型。通过与有限元方法及部分精确解的比较对结果进行了验证。
