Purpose To 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. Methods The 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. Results It 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${f}_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${f}_{5}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation> can be determined from the band structure. The lattice constant <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq3.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$a$</EquationSource> <EquationSource Format="MATHML"><math> <mi>a</mi> </math></EquationSource> </InlineEquation> is insensitive to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${f}_{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${f}_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${f}_{3}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>. The bounding frequencies <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${f}_{4}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq8.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">${f}_{5}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>f</mi> <mn>5</mn> </msub> </math></EquationSource> </InlineEquation> decrease as the lattice constant increases. The foundation stiffness <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">${k}_{f}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>k</mi> <mi>f</mi> </msub> </math></EquationSource> </InlineEquation> is equal to <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42417_2024_1603_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$80 kN$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>80</mn> <mi>k</mi> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation>, the band gap reaches its maximum of 93.8 Hz. Conclusion The 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.

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

期刊论文
发表日期：

2025年1月15日

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Hongping Zhu,
Zehui Shen,
Shun Weng,
Ying Zhang,
Hao Luo

摘要

Purpose

To 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.

Methods

The 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.

Results

It 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 ${f}_{1}$ $f_{1}$ to ${f}_{5}$ $f_{5}$ can be determined from the band structure. The lattice constant $a$ $a$ is insensitive to ${f}_{1}$ $f_{1}$ ${f}_{2}$ $f_{2}$ and ${f}_{3}$ $f_{3}$ . The bounding frequencies ${f}_{4}$ $f_{4}$ and ${f}_{5}$ $f_{5}$ decrease as the lattice constant increases. The foundation stiffness ${k}_{f}$ $k_{f}$ is equal to $80 kN$ $80 k N$ , the band gap reaches its maximum of 93.8 Hz.

Conclusion

The 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.