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Abstract

In this paper, a phononic crystal is designed using a Helmholtz resonator with a membrane wall, in which the coupled vibration of air and membrane is utilized. The structure of the Helmholtz resonator is a two-dimensional structure. On the basis of the square Helmholtz resonator, a " W”-type outlet is used as a cavity outlet to increase the air quality involved in resonance, and the cavity wall is replaced with a membrane with distribution mass to increase the number of resonance units. The finite element method is used to calculate the band gaps and transmission loss of sound below 1700 Hz. The results show that the starting frequency of the first band gap of the structure is further reduced. At the same time, it is lower than the starting frequency of ordinary Helmholtz structure and the natural frequency of membrane under the same conditions. Then, a new peak of transmission loss is obtained, and its value is greater than the original structure’s. And although the width of the first band gap is reduced, some new band gaps appear in the low-frequency range, so that the total band gap width is improved. By analyzing the vibration mode of the membrane and sound pressure distribution, it is found that the sum of the sound pressures of the outer cavity is zero at the starting frequencies of the band gaps, and the sound pressure of the inner and outer cavity are respectively positive and negative at the cut-off frequency. With the increase of frequency, the vibration mode of the membrane gradually turns from low-order to high-order, but no anti-symmetric-type mode participation is found at the starting and cut-off frequency. The components of the structure can be made equivalent to corresponding ones, respectively, i.e. air in the outlet is equivalent to uniform flexible rod, and the air in the inner and outer cavity are equivalent to a spring. So that the structure can be equivalent to a series system consisting of a rod, a spring and a membrane at starting frequency of the band gap, and a loop system consisting of a rod, two springs and a membrane at cut-off frequency. Thus, by the transfer matrix method and the Rayleigh-Ritz method considering the influence of tension and elastic modulus, it is possible to calculate the range of band gap which is extremely close to the result from the finite element method. Through the analysis of the formulas, it can be found that the new band gap is caused by the new vibration mode produced by the membrane or the air in the cavity outlet, and the lower starting frequency of the first band gap is due to the reduction of the equivalent extent of the system by the membrane. By adjusting the relevant parameters of the membrane and the cavity outlet respectively, it can be found that the band gaps of the structure correspond to the modes of different orders of the air in the cavity outlet and the membrane. In other words, the change of the natural frequency of a certain mode of air in the outlet or membrane only has a greater influence on the corresponding band gap but has less influence on other band gaps, also, the trends of change are the same, and the change values are very close to each other. But, changing the volume of the inner cavity and the outer cavity has a great influence on all the band gaps. Therefore, it is possible to adjust some band gaps through this method.

Keywords:

Authors and contacts

Corresponding author:Zhao Jing-Bo,chjzjb@163.com

Funds:Project supported by the National Natural Science Foundation of China (Grant No. 11504429)

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References

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l_s/10^–3m	4	6	8	10	12	14
1阶固有频率	252.4	237.9	226.8	218.2	211.6	206.6
2阶固有频率	751.3	782.8	794.0	814.8	843.4	879.1

DownLoad: CSV

l_s/10^–3m	第一带隙下限		第一带隙上限		第二带隙下限		第二带隙上限		第三带隙下限		第三带隙上限
l_s/10^–3m	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%
4	89.2	3.7	121.4	4.4	314.2	2.8	557.7	0.6	790.7	2.2	900.3	1.6
4	92.5	3.7	126.7	4.4	323.0	2.8	561.2	0.6	808.3	2.2	914.7	1.6
6	88.9	3.6	120.1	4.2	297.7	1.8	526.8	–0.8	789.1	0.4	905.2	0.8
6	92.2	3.6	125.1	4.2	303.1	1.8	522.8	–0.8	792.4	0.4	912.2	0.8
8	88.7	3.5	118.9	3.9	285.1	0.9	508.3	–2.1	795.6	–1.1	916.1	–0.5
8	91.9	3.5	123.5	3.9	287.6	0.9	497.5	–2.1	786.8	–1.1	912.0	–0.5
10	88.6	3.4	117.9	3.5	275.6	–0.1	497.9	–3.5	815.9	–3.7	932.5	–2.3
10	91.6	3.4	122.0	3.5	275.4	–0.1	480.5	–3.5	786.0	–3.7	910.8	–2.3
12	88.5	3.1	117.2	3.0	268.4	–1.0	492.6	–4.8	843.8	–6.9	953.9	–5.2
12	91.3	3.1	120.7	3.0	265.6	–1.0	469.1	–4.8	785.3	–6.9	904.4	–5.2
14	88.5	2.8	116.7	2.3	262.9	–2.0	490.5	–5.9	878.5	–11	966.3	–8.2
14	91.0	2.8	119.4	2.3	257.6	–2.0	461.5	–5.9	779.9	–11	887.3	–8.2

DownLoad: CSV

T/10⁶N·m^–1	第一带隙下限		第一带隙上限		第二带隙下限		第二带隙上限		第三带隙下限		第三带隙上限
T/10⁶N·m^–1	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%
0.5	74.3	3.4	89.3	2.8	259.2	0.7	440.3	–1.2	574.1	0.1	770.1	0.5
0.5	76.8	3.4	91.8	2.8	261.0	0.7	434.9	–1.2	574.4	0.1	774.3	0.5
1.5	96.3	3.4	143.1	4.6	345.5	3.2	589.6	0.2	952.1	2.2	1035.1	1.7
1.5	99.6	3.4	149.7	4.6	356.7	3.2	591.0	0.2	973.0	2.2	1053.1	1.7
2.5	103.3	2.9	174.6	4.6	415.0	4.2	648.0	0.7	1217.4	2.8	1274.1	2.3
2.5	106.4	2.9	182.7	4.6	432.6	4.2	652.4	0.7	1251.0	2.8	1303.2	2.3
3.5	106.8	2.7	196.4	4.4	474.9	4.8	691.4	1.1	1434.1	3.0	1478.6	2.5
3.5	109.6	2.7	205.0	4.4	497.6	4.8	698.8	1.1	1477.6	3.0	1516.1	2.5
4.5	108.8	2.5	212.5	4.1	528.2	5.1	729.6	1.4	1621.6	3.2	1642.5	3.3
4.5	111.5	2.5	221.3	4.1	555.2	5.1	740.1	1.4	1673.4	3.2	1696.5	3.3
10	113.0	2.1	257.8	3.0	757.3	5.9	907.9	2.7	1645.9	5.8	1741.8	3.1
10	115.3	2.1	265.6	3.0	801.7	5.9	932.8	2.7	1740.6	5.8	1796.2	3.1
100	116.2	1.8	311.0	1.2	1654.1	5.2	1737.3	3.1	2270.8	9.0	2375.3	6.1
100	118.3	1.8	314.9	1.2	1740.4	5.2	1791.6	3.1	2475.7	9.0	2520.3	6.1

DownLoad: CSV

l₁/mm	第一带隙下限		第一带隙上限		第二带隙下限		第二带隙上限		第三带隙下限		第三带隙上限
l₁/mm	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%
99	88.4	4.4	119.1	5.7	302.1	3.4	533.0	1.3	772.3	3.4	891.4	2.4
99	92.3	4.4	125.9	5.7	312.5	3.4	540.0	1.3	798.6	3.4	912.9	2.4
148	74.5	3.4	101.5	5.1	301.2	3.5	513.3	1.1	772.6	3.4	873.7	2.4
148	77.1	3.4	106.7	5.1	311.7	3.5	519.1	1.1	798.9	3.4	894.4	2.4
197	66.0	2.9	89.9	4.7	301.9	3.5	500.0	1.1	772.4	3.3	836.0	1.9
197	67.9	2.9	94.1	4.7	312.5	3.5	505.3	1.1	798.2	3.3	851.8	1.9
246	60.1	2.6	81.4	4.5	303.2	3.5	488.5	1.0	697.6	1.2	734.1	0.9
246	61.7	2.6	85.1	4.5	313.7	3.5	493.3	1.0	705.9	1.2	740.6	0.9
295	55.8	2.5	75.0	4.3	304.7	3.4	475.7	0.9	587.8	0.7	637.5	0.7
295	57.2	2.5	78.2	4.3	315.2	3.4	479.9	0.9	591.9	0.7	642.3	0.7
344	52.5	2.4	69.8	4.2	306.3	3.4	458.4	0.7	507.5	0.6	558.9	3.1
344	53.7	2.4	72.7	4.2	316.6	3.4	461.5	0.7	510.8	0.6	576.4	3.1

DownLoad: CSV

V₂/10^–4m³	第一带隙下限		第一带隙上限		第二带隙下限		第二带隙上限		第三带隙下限		第三带隙上限
V₂/10^–4m³	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%	FEM TMM	误差/%
7.07	108.5	6.3	120.5	5.8	406.9	1.0	566.2	1.0	809.9	1.9	953.1	1.2
7.07	115.3	6.3	127.6	5.8	411.0	1.0	571.8	1.0	825.3	1.9	964.2	1.2
10.57	103.3	4.8	121.1	4.8	361.4	1.6	558.3	–0.1	796.5	1.9	925.4	1.4
10.57	108.3	4.8	126.8	4.8	367.2	1.6	557.8	–0.1	811.4	1.9	938.7	1.4
14.07	98.0	4.2	121.0	4.4	335.0	1.9	550.0	–0.1	790.9	1.8	913.1	1.4
14.07	102.2	4.2	126.4	4.4	341.3	1.9	549.4	–0.1	804.9	1.8	925.8	1.4
17.57	93.3	3.9	120.9	4.3	317.7	2.1	544.4	–0.1	788.3	1.6	906.5	1.3
17.57	96.9	3.9	126.1	4.3	324.4	2.1	543.9	–0.1	801.1	1.6	918.1	1.3
19.32	91.1	3.8	120.8	4.3	311.1	2.2	542.3	–0.1	787.4	1.6	904.1	1.2
19.32	94.5	3.8	126.0	4.3	317.9	2.2	541.8	–0.1	799.7	1.6	915.3	1.2

DownLoad: CSV

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Vol. 68, Issue 21 - 2019-11-05

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