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Title Page

Abstract

Contents

Ⅰ. Introduction 9

1.1. Background of flexural metamaterial 9

1.1.1. Wave, Vibration and Metamaterial 9

1.1.2. Flexural wave and Flexural metamaterial 9

1.1.3. Previous research 10

1.2. Research objective and organization 10

Ⅱ. Theoretical basis for bandgap by flexural metamaterials 11

2.1. 1D flexural metamaterial 11

2.1.1. Mass-spring system for beam 11

2.1.2. Analysis of dispersion curve 13

Ⅲ. Band gap formation in two dimensions through theoretical development 14

3.1. 2D flexural metamaterial 14

3.1.1. Mass-spring system for plate 14

3.1.2. Analysis of dispersion curve 17

3.1.3. Unit cell design to meet conditions for band gap formation 19

3.2. Experimental validation 25

3.3. Experimental validation 27

Conclusion 29

Ⅴ. Appendix 30

A. Unit cell with connectors arranged horizontally 30

B. Unit cell with asymmetric structure between x and y directions 31

References 33

List of Figures

Figure 1. Mass-spring system for continuum beam. 11

Figure 2. Dispersion curve in 1D flexural metamaterial 13

Figure 3. Mass-spring system for continuum plate. 14

Figure 4. Unit cell consisting of a cubic mass and one connector. 17

Figure 5. Dispersion curve without bandgap in diagonal direction. 18

Figure 6. I/m change depending on mass shape. 19

Figure 7. Derivation of (a) shear stiffness, (b) bending stiffness, and (c)torsional stiffness. 20

Figure 8. Thickness and length of connector. 20

Figure 9. Parameter study regarding to (a) shear stiffness, (b) bending stiffness, (c) and torsional stiffness depending on thickness of connector. 20

Figure 10. Parameter study regarding to (a) shear stiffness, (b) bending stiffness, (c) and torsional stiffness depending on length of connector. 21

Figure 11. Design at (a) d＝0 mm, (b) d＝30 mm. 21

Figure 12. (a) Shear stiffness, (b) bending stiffness, (c) torsional stiffness corresponding to distance between two connectors. 22

Figure 13. Unit cell design with spherical mass and two connectors. 22

Figure 14. Dispersion curve with bandgap through theoretical analysis. 23

Figure 15. Dispersion curve with bandgap through numerical analysis. 24

Figure 16. Mode shape corresponding to (a) shear, (b) bending, (c) torsion. 24

Figure 17. Simulation setting in (a) x-direction and (b) diagonal direction. 25

Figure 18. Simulation setting in 2D. 25

Figure 19. Simulation result in x-direction at (a) first pass band, (b) bandgap, (c) second pass band and diagonal direction at (d) first pass band, (e) bandgap, (f) second pass band. 26

Figure 20. Simulation result in x-y plane at (a) second pass band, (b) bandgap, (c) first pass band. 26

Figure 21. Experimental setting. 27

Figure 22. Transmission coefficient at (a) 3rd, (b) 5th, (c) 5'th unit cell.[이미지참조] 28

Figure 23. Unit cell with connectors arranged horizontally. 30

Figure 24. Dispersion curve of the designed unit cell. 30

Figure 25. Asymmetric unit cell. 31

Figure 26. Dispersion curve of the designed unit cell. 32

초록보기

 Vibration-related issues are crucial consideration in various industries. Most vibrational phenomena are closely associated with flexural wave that vibrates along a transverse axis and propagates along structures such as plates and beams. Accordingly, studies on flexural metamaterials aimed at controlling flexural waves for the purpose of vibration isolation has been consistently conducted recently. Metamaterial is structures in which artificial shapes smaller than the wavelength of propagating waves are periodically arranged. Through such structures, it's possible to control the propagation of the wave in a new way different from the general way. There are various phenomena caused by these metamaterials, such as negative refraction, non-reciprocity, band gap, wave localization, and more. In this thesis, bandgap, the most representative phenomenon among those cause by metamaterials, was utilized to block flexural waves in a specific frequency band on x-y plane. Previous studies on flexural metamaterials have been mainly conducted in one dimension, and the Timoshenko beam theory, which considers the shear deformation of the structure, converts a continuous beam into a mass-spring system with two degrees of freedom of shear and bending motions. Thus, broadband vibration isolation is implemented at low frequencies. However, in these studies, only waves in the x-direction can be controlled. Therefore, in this thesis, we extend the existing one-dimensional flexural metamaterial to two dimensions and convert the plate into a mass-spring system with three degrees of freedom by additionally considering the torsional motion rotating along the y-axis in addition to the existing shear and bending motions. As the degree of freedom is added, the frequency components that make up the band gap in the dispersion curve change, and conditions for forming a two-dimensional band gap are added accordingly. In this thesis, two-dimensional bandgap formation was achieved through the design of a unit cell with shear stiffness, bending stiffness, and torsional stiffness that satisfies the conditions, and it was verified through numerical analysis and experiments that waves do not propagate in all direction in the bandgap. Through the research conducted in this thesis, it became possible to block flexural waves in all directions in a two-dimensional structure, and it is expected that flexural metamaterials will be able to effectively block vibrations in plates.

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