Flexural Vibration Band Gaps in a Parallel Double-Beam Periodic Structure With Local Resonators on Elastic Foundations

PurposeTo further investigate the band gaps within the low frequency ranges for periodic double beams structure, a novel parallel double-beam periodic structure with periodically local resonators on elastic foundations is proposed and studied in this paper.MethodsThe band structure of the infinite double-beam system is obtained by the plane wave expansion method. Then the existence of band gaps is verified by analysing the transmission characteristic obtained through finite element analysis on the double-beam system. Furthermore, the band gap formation mechanism is studied in terms of the eigenmodes and transverse deformation pattern. Simple calculation formulas of the starting and ending frequencies of the band gaps are derived based on the eigenmodes. Parametric studies are also conducted to investigate the structural and foundation properties on the flexible vibration band gap characteristics.ResultsIt indicates that alterations in foundation stiffness directly affect the vibration mode of the structure, consequently leading to modifications in the band gap. Modifying the frequencies of band gaps by tuning the stiffness of the springs connected with the two beams is also feasible. The bounding frequencies f1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{1}$$\end{document} to f5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{5}$$\end{document} can be determined from the band structure. The lattice constant a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a$$\end{document} is insensitive to f1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{1}$$\end{document}f2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{2}$$\end{document} and f3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{3}$$\end{document}. The bounding frequencies f4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{4}$$\end{document} and f5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f}_{5}$$\end{document} decrease as the lattice constant increases. The foundation stiffness kf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${k}_{f}$$\end{document} is equal to 80kN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$80 kN$$\end{document}, the band gap reaches its maximum of 93.8 Hz.ConclusionThe research provides a valuable reference for the vibration control of parallel double-beam periodic structures with local resonators, providing a foundation for future engineering applications.