Boundary Element Method in Structural Mechanics for Periodic Structure Vibration Protection
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Abstract
The article presents the results of a study on the adaptation of the direct boundary element method (BEM) for solving urgent problems in structural mechanics, specifically the analysis of stationary vibrations of beams and plates. These elements are considered fundamental components of modern load-bearing structures subjected to intense dynamic loads. The primary focus is on the design of high-efficiency vibration protection systems for buildings with periodic structures, such as cross-bar systems, multi-span floors, and pile foundations. The scientific novelty of this work lies in the development and substantiation of a numerical scheme that allows for high-precision modeling of bending wave propagation processes in periodically fixed structural elements. This approach is critically important for calculating the parameters of foundations for heavy industrial equipment and ensuring the reliable stability of buildings against seismic impacts and man-made vibrations caused by transport or manufacturing processes. The authors provide a detailed description of the procedure for finding fundamental solutions for one-dimensional (beams) and two-dimensional (plates) systems using the Fourier transform and contour integration methods. The paper demonstrates the advantages of BEM over the traditional finite element method (FEM), particularly regarding problem dimensionality reduction and modeling accuracy in infinite domains, which is decisive when investigating the interaction between foundations and soil mass. The proposed methodology enables the identification of so-called "stop-bands" in periodic structures — frequency ranges within which vibrations do not propagate. The obtained results create a theoretical foundation for designing innovative construction metamaterials capable of completely blocking unwanted oscillations, thereby ensuring the necessary level of vibration safety for residential and industrial facilities.
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