局域共振型周期结构振动带隙形成机理
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摘要
为了揭示局域共振型周期结构振动带隙的形成机理,以二维局域共振型周期结构为对象,依据动力学理论,提出了振动带隙形成的基础理论假设:振子和基体的振动模式,由其主模态主导,主模态依据模态参与因子通过模态切换形成不同的振动模式,并依此建立了振动带隙形成机理模型,研究发现:局域共振型周期结构中存在6种广义振动模式,各模式由基体的广义主模态主导,而广义主模态又由广义模态依据模态叠加原理形成,当被抑制时,广义子带隙形成。研究结果表明:振子广义主模态依据模态参与因子,通过抑制或释放基体的广义主模态实现振动带隙的打开与闭合;振动带隙特性主要由振子广义主模态的等效刚度和等效质量决定,通过设计振子可主动设计与调控结构的带隙特征。该研究不仅为工程结构的振动控制提供了新方法,为带隙的主动设计奠定了理论基础,也完善了周期结构的基本理论。
Detailed formation mechanisms of vibration band gaps in locally resonant periodic structures are investigated based on a two-dimensional locally resonant periodic structure. Firstly, based on the dynamic theory a basic theoretical hypothesis of the formation mechanism for the band gap of locally resonant periodic structure is proposed following the modal superposition principle. Secondly, the detailed formation mechanism of vibration band gaps is further clarified according to the formation mechanism model of band gaps. Finally, the formation mechanism of vibration band gaps is verified by the interaction between the elastic wave and the structure during the formation of vibration band gaps in a typical two-dimensional locally resonant periodic structure. The result shows that there exist six types of generalized propagation modes of waves in the locally resonant periodic structure, while these generalized propagation modes are formed by the mutual transformation of the main modes and these main modes are generated by corresponding 12 modes based on the superposition principle. The vibration mode dominated by the main mode of the vibrator determines the formation of a vibration band gap by suppressing or releasing the generalized propagation mode. When the oscillator's main mode suppresses the generalized propagation mode of the wave, a generalized sub-band gap that can only suppress the corresponding propagation mode is formed. The influencing mechanism of the band gap is further studied and an active design method of the band gap is proposed. This research may perfect the basic theory of periodic structure, lay a theoretical foundation for the study of band gap theory of periodic structure and the active design of band gap characteristics, and provide a new method for vibration reduction of engineering structures.
Detailed formation mechanisms of vibration band gaps in locally resonant periodic structures are investigated based on a two-dimensional locally resonant periodic structure. Firstly, based on the dynamic theory a basic theoretical hypothesis of the formation mechanism for the band gap of locally resonant periodic structure is proposed following the modal superposition principle. Secondly, the detailed formation mechanism of vibration band gaps is further clarified according to the formation mechanism model of band gaps. Finally, the formation mechanism of vibration band gaps is verified by the interaction between the elastic wave and the structure during the formation of vibration band gaps in a typical two-dimensional locally resonant periodic structure. The result shows that there exist six types of generalized propagation modes of waves in the locally resonant periodic structure, while these generalized propagation modes are formed by the mutual transformation of the main modes and these main modes are generated by corresponding 12 modes based on the superposition principle. The vibration mode dominated by the main mode of the vibrator determines the formation of a vibration band gap by suppressing or releasing the generalized propagation mode. When the oscillator's main mode suppresses the generalized propagation mode of the wave, a generalized sub-band gap that can only suppress the corresponding propagation mode is formed. The influencing mechanism of the band gap is further studied and an active design method of the band gap is proposed. This research may perfect the basic theory of periodic structure, lay a theoretical foundation for the study of band gap theory of periodic structure and the active design of band gap characteristics, and provide a new method for vibration reduction of engineering structures.
引文
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[19] 温激鸿,郁殿龙,赵宏刚,等.人工周期结构中弹性波的传播:振动与声学特性 [M].北京:科学出版社,2018:6.
[20] 李锁斌,窦益华,陈天宁,等.声子晶体板中低频宽禁带形成机理 [J].西安交通大学学报,2018,52(12):159-166.LI Suobin,DOU Yihua,CHEN Tianning,et al.Forming mechanisms of low frequency broad band gaps in locally resonant phononic crystal plates [J].Journal of Xi’an Jiaotong University,2018,52(12):159-166.
[2] 陈荣,吴天行.非对称周期结构中耦合波的传播特性 [J].振动与冲击,2015,32(1):68-73.CHEN Rong,WU Tianxing.Coupled wave propagation in asymmetric periodic structures [J].Journal of Vibration and Shock,2015,32(1):68-73.
[3] MAASCH M.Wave propagation in periodic structures [M].Berlin,Germany:Springer International Publishing,2016:26.
[4] 吴九汇,马富银,张思文,等.声学超材料在低频减振降噪中的应用评述 [J].机械工程学报,2016,52(13):69-78.WU Jiuhui,MA Fuyin,ZHANG Siwen,et al.Application of acoustic metamaterials in low-frequency vibration and noise reduction [J].Journal of Mechanical Engineering,2016,52(13):69-78.
[5] SIGALAS M,ECONOMOU E N.Band structure of elastic waves in two dimensional systems [J].Solid State Communications,1993,86(3):141-143.
[6] LIU Z Y,ZHANG X X,MAO Y W,et al.Locally resonant sonic materials [J].Science,2000,289:201-205.
[7] HO K M,CHENG C K,YANG Z,et al.Broadband locally resonant sonic shields [J].Applied Physics Letters,2003,83(26):5566-5568.
[8] WU T T,HUANH Z G,TSAI T C,et al.Evidence of complete band gap and resonances in a plate with periodic stubbed surface [J].Applied Physics Letters,2008,93(11):111902.
[9] HSU J.Local resonances-induced low-frequency band gaps in two-dimensional phononic crystals slabs with periodic stepped resonators [J].Journal of Physics:D Applied Physics,2011,44(5):055401.
[10] ASSOUAR M B,OUDICH M.Enlargement of a locally resonant sonic band gap by using double-sides stubbed plate [J].Applied Physics Letters,2012,100(12):100123506.
[11] LIF M,WANGY Z.Elastic wave propagation and localization in band gap materials:a review [J].Science China,2012,55(10):1734-1746.
[12] CHEN Jiujiu,HAN Xu,LI Guangyao.Asymmetric lamb wave propagation in phononic crystal slabs with graded grating [J].Journal of Applied Physics,2013,113(18):184506.
[13] 高明,吴志.一维三振子周期结构带隙设计 [J].物理学报,2013,62(14):140507.GAO Ming,WU Zhiqiang.Band gap design for one-dimensional periodic structure with three oscillators [J].Acta Physica Sinica,2013,62(14):140507.
[14] LI S B,CHEN T N,WANG X P,et al.Expansion of lower-frequency locally resonant band gaps using a double-sided stubbed composite phononic crystals plate with composite stubs [J].Physics Letters:A,2016,380(25/26):2167-2172.
[15] 李锁斌,陈天宁,奚延辉,等.声子晶体板中低频完全禁带形成机理研究 [J].西安交通大学学报,2016,50(12):51-57.LI Suobin,CHEN Tianning,XI Yanhui,et al.Forming mechanisms of low frequency complete band gaps in phononic crystal plate [J].Journal of Xi’an Jiaotong University,2016,50(12):51-57.
[16] LI S B,XI Y H,CHEN T N,et al.Modulating Lamb wave band gaps using an elastic metamaterial plate [J].Acoustic Journal,2017,63(5):508-516.
[17] 徐青,张俊杰.周期复合板的带隙特性和辐射噪声衰减特性分析 [J].振动与冲击,2017,36(11):188-191.XU Qing,ZHANG Junjie.Research on the band gap and attenuation characteristic of sound radiation for periodic compound plate [J].Journal of Vibration and Shock,2017,36(11):188-191.
[18] 石志飞,程志宝,向宏军.周期结构理论及其在隔震减振中的应用 [M].北京:科学出版社,2017:21.
[19] 温激鸿,郁殿龙,赵宏刚,等.人工周期结构中弹性波的传播:振动与声学特性 [M].北京:科学出版社,2018:6.
[20] 李锁斌,窦益华,陈天宁,等.声子晶体板中低频宽禁带形成机理 [J].西安交通大学学报,2018,52(12):159-166.LI Suobin,DOU Yihua,CHEN Tianning,et al.Forming mechanisms of low frequency broad band gaps in locally resonant phononic crystal plates [J].Journal of Xi’an Jiaotong University,2018,52(12):159-166.