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Abstract

Local-resonance bandgap and Bragg bandgap can coexist in a metamaterial beam, and their coupling effect can be used to realize ultra-wide bandgap, which has great potential applications in the field of wide-band vibration reduction. Previous studies usually considered the single-order coupling between the local-resonance bandgap and Bragg bandgap in metamaterial beams with a single array of local resonators, which can only achieve the single-order ultra-wide coupling bandgap and cannot meet the wide-band vibration reduction requirements of double/multiple target frequency bands. In this paper, metamaterial beams with double arrays of local resonators are considered, and the regulation design and analysis of double-order coupling of local-resonance and Bragg bandgaps are carried out based on an analytical model of bending wave dispersion relation. Moreover, the vibration reduction characteristics of the double-frequency-resonator metamaterial beams with double-order coupling bandgaps are studied by using spectral element method and the finite element method. The main conclusions are as follows. 1) A design method is proposed for realizing double-order coupling wide bandgap in a metamaterial beam with double arrays of local resonators. By using this method, the resonance frequencies of the local resonators can be quickly designed on conditions that host beam parameters, lattice constant and added mass ratio of the local resonators are given. 2) The double-order coupling bandgaps in a metamaterial beam carrying double arrays of local resonators are compared with the single-order coupling bandgaps in metamaterial beams with a single array of local resonators. It is found that through proper design, the total normalized width of the double-order coupling bandgap can be much broader than that of the single-order coupling bandgap, so the double-order coupling bandgap is more beneficial to wide-band vibration reduction. 3) It is found that for a given total added mass ratio of the double arrays of local resonators, it is necessary to optimize the mass distribution ratio of the double resonators to maximize the total normalized width of double-order coupling bandgap. An approximate formula for designing the optimal mass distribution ratio of the double resonators is further established. 4) The spectral element method is used to study the vibration reduction characteristics of the metamaterial beams carrying double arrays of local resonators designed based on double-order bandgap coupling. The accuracy of the spectral element method is verified by comparing with the finite element method. The results show that significant vibration reduction can be achieved in two wide frequency bands corresponding to the double-order coupling bandgaps. The influences of number of unit cells and resonator damping on the vibration reduction characteristics of the metamaterial beam are further analyzed. It is shown that the increase of number of unit cells can enhance the vibration reduction performance in the bandgap, and the increase of resonator damping can effectively broaden the vibration reduction frequency band.

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Authors and contacts

Corresponding author:Xiao Yong,xiaoy@vip.sina.com

Funds:Project supported by the Major Program of the National Natural Science Foundation of China (Grant No. 11991032) and the National Natural Science Foundation of China (Grant No. 51875569).

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References

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密度 ρ/(kg·m^–3)	杨氏模量 E/Pa	宽 b/mm	厚 h/mm	晶格常数a/mm
2700	7×10¹⁰	20	4	200

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耦合带隙设计情形		单阶耦合带隙(n= 1)	单阶耦合带隙(n=2 )	双阶耦合带隙
第1带隙边界频率	f_1,1/Hz	108.300	130.190	120.970
第1带隙边界频率	f_1,2/Hz	226.531	230.890	229.121
第2带隙边界频率	f_2,1/Hz	230.881	542.630	231.051
第2带隙边界频率	f_2,2/Hz	355.171	923.548	265.621
第3带隙边界频率	f_3,1/Hz	922.821	924.158	595.631
第3带隙边界频率	f_3,2/Hz	955.951	1341.580	923.168
第4带隙边界频率	f_4,1/Hz	—	—	923.848
第4带隙边界频率	f_4,2/Hz	—	—	1288.140
带隙总宽度	$\sum G$	1.1656	1.4458	1.5177

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	h/mm	a/m	γ₁	γ₂	f_r,1/Hz	f_r,2/Hz
原始基体梁	4	0.2	0.625	0.375	196.48	726.48
新基体梁	36	0.6	0.625	0.375	196.48	726.48

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γ₁	γ₂	f_r,1/Hz	f_r,2/Hz
0.625	0.375	196.48	726.48

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	附加质量比γ	共振频率f_r/Hz
单振子1	1	169.46
单振子2	1	576.66

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	密度ρ/(kg·m^–3)	杨氏模量E/Pa	泊松比	损耗因子
橡胶	1300	2×10⁶	0.49	0.05
铜	8960	110×10⁹	0.35	0.05

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	b/mm	w₁/mm	h₁/mm	w₂/mm	h₂/mm	f_r/Hz
振子1	20	4	5.5	10	15.1	196.08
振子2	20	7.2	2.5	10	9	726.89

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Vol. 73, Issue 15 - 2024-08-05

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