局域共振型结构的带隙调控与减振降噪特性研究
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摘要
近年来，声物理发展前沿关于“局域共振型声子晶体”和“声学超材料”等新概念的提出为工程结构的减振降噪设计提供了新的思路。类比局域共振型声子晶体和声学超材料的设计思想，通过在杆、梁、板等传统工程结构上周期性地附加局域共振单元，可以得到一类局域共振型人工周期结构（本文简称为“局域共振型结构”）。目前已有的初步探索研究表明：局域共振型结构具有显著的弹性波带隙特性，在其带隙频率范围内可以高效地抑制结构振动。这些研究为结构减振降噪技术发展开辟了一条新的途径。然而，要实现局域共振型结构的减振降噪应用，在其性能预报、行为调控、机理揭示以及轻质宽带设计等方面还面临大量的基础问题有待解决。
本文紧紧围绕局域共振型结构在减振降噪应用方面目前面临的关键理论和技术问题展开系统深入研究，研究的主要内容包括：(1)局域共振型结构的带隙及振动与声学特性计算方法；(2)局域共振型结构的带隙调控与形成机理；(3)局域共振型结构的带隙展宽方法及宽带减振降噪特性。论文研究取得的主要创新成果包括：
1、提出了计算局域共振杆/梁结构中振动传递特性的波动/谱元法，并将平面波展开法推广用于计算局域共振板结构的带隙特性及声透射特性，为局域共振型结构的带隙与减振降噪特性研究提供了高效工具。
2、系统分析了局域共振型结构的设计参数对其带隙特性的调控规律，并发现了有价值的新带隙现象及效应，深化了对于局域共振型结构带隙行为的认识。首次发现局域共振杆中两种带隙（局域共振带隙和Bragg带隙）可以精确地耦合成为一个超宽的完全带隙；首次发现当两种带隙被调节至相互靠近时，不仅局域共振单元的局域共振作用对Bragg带隙可以产生展宽效应，而且周期结构的Bragg散射作用对局域共振带隙也可以产生展宽效应；首次发现局域共振型结构中局域共振单元的谐振频率可以落在Bragg带隙范围内，甚至是通带范围内。
3、深入揭示了局域共振型结构的带隙形成机理，推导得到了封闭的解析设计公式，可以精确地或近似地预报局域共振型结构中设计参数对带隙边界频率的调控规律，以及确定结构中两种带隙发生相互耦合和相互转换的条件，为带隙设计提供依据。
4、提出了基于两种带隙相互耦合机制、结构参数失谐设计以及局域共振单元多频谐振作用的三种带隙展宽方法，为实现局域共振型结构在轻质条件下的宽带减振降噪设计提供了技术支持。
5、率先研究了局域共振板的声透射降噪特性，首次发现在亚波长晶格条件下，若将局域共振单元的谐振频率调谐至基体板声透射的质量密度定律区，局域共振板可以实现远高于同质量均质板的传声损失（隔声量）；若将谐振频率调谐至基体板的声透射吻合效应区，局域共振板可以抑制吻合效应，大幅提高整个吻合效应区的传声损失。这些特性对于局域共振板的降噪应用具有重要指导意义。
总之，本文主要针对局域共振型结构在减振降噪应用方面需要解决的关键理论与技术问题展开系统深入研究：进一步完善了局域共振型杆、梁、板结构的带隙及振动与声学特性计算方法，深入揭示了这类局域共振型结构的带隙特性调控规律与形成机理，并提出了实现其带隙及减振降噪频带展宽的三种方法，从相关研究中提炼出了系列设计公式、准则及建议。这些研究成果为局域共振型结构的减振降噪应用奠定了若干重要的理论基础和技术支撑。
In the past few years, the new concept of “locally resonant phononic crystals” and“acoustic metamaterials” developed in the frontiers of acoustical physics providedsome new ideas for the control of vibration and noise of engineering structures. Byanalogy with locally resonant phononic crystals and acoustic metamaterials, locallyresonant periodic engineering structures (called “locally resonant structures” in thisdissertation) can be constructed by attaching periodic arrays of local resonators tostructural waveguides such as rods, beams, plates, etc. Some initial studies have shownthat locally resonant structures can exhibit notable elastic wave band gaps, andstructural vibrations can be significantly attenuated within the frequency ranges ofband gaps. These studies provide a novel approach to control vibration and noise ofengineering structures. However, to achieve practical applications of locally resonantstructures in engineering, there exist still many important problems required to beexamined, such as the performance prediction, behavior tuning, mechanismsexplanation, as well as the lightweight and broadband design.
The purpose of this dissertation is to investigate some crucial theoretical andtechnical problems systematically to promote the applications of locally resonantstructures in the field of vibration and noise control. The following three aspects havebeen considered:(1) the methods for calculating band gaps and the methods forpredicting vibration/noise properties of locally resonant structures;(2) the tuningbehavior of band gaps and the underlying band-gap formation mechanisms;(3) themethods for broadening band gaps and the study of resulting performance ofbroadband vibration/noise reduction. The main findings of this dissertation are listed asfollows:
1. A wave/spectral element method is proposed to predict vibration transmittanceof general locally resonant rods/beams, and a plane wave expansion method isdeveloped for the calculation of band gaps and sound transmission properties ofgeneral locally resonant thin plates. These methods are provided as efficient tools forthe study of band gap and vibration/noise reduction properties of locally resonantstructures.
2. The tuning band gap behavior of locally resonant structures by changing designing parameters is analyzed in a systematic manner, and some valuable new bandgap phenomena and effects have been discovered. It is shown that two types of bandgaps (locally resonant and Bragg band gaps) can exactly couple together to form acombined super-wide band gap. It is also found that when the location of two types ofband gaps is tuned to be very close to each other, not only that the locally resonanteffect can broaden Bragg band gap, but also that the Bragg scattering effect can widenthe locally resonant band gap. In addition, it is found that the resonance frequency oflocal resonators can appear in a Bragg band gap, and even more surprisingly, in a passband.
3. The band gap formation mechanisms of locally resonant structures have beenclarified in depth. A number of explicit analytical design formulas are derived to enablean exact prediction of the dependence of band edge frequencies on the designingparameters of locally resonant structures. Furthermore, explicit formulas are alsoprovided to enable a direct determination of the conditions of coupling or transitionbetween two types of band gaps. These results provide guidelines for the band gapdesign in locally resonant structures.
4. Three band gap broadening methods are proposed, respectively based on theband gap coupling mechanism, the design of disordered structural parameters, and theutilization of multi-resonant local resonators. These methods provide technical supportto realize lightweight locally resonant structures with broadband vibration/noisereduction performance.
5. For the first time, the sound transmission loss (STL) behavior of locallyresonant thin plates is investigated. It is found that, under the condition ofsubwavelength lattice constant, if the resonance frequency of local resonators is tunedto the mass-law region of sound transmission for the host plate, the locally resonantplate can achieve much higher STL than a homogeneous plate with the same surfacemass. While, if the resonance frequency is tuned to the coincidence region of soundtransmission, the locally resonant plate can break the coincidence effect, resulting in asignificantly increased STL over the whole coincidence region. These properties are ofgreat interest for the application of locally resonant plates in the noise control field.
In summary, this dissertation is concerned with some crucial theoretical andtechnical problems involved in the investigations and applications of locally resonantstructures for the purpose of reducing structural vibration/noise. The maincontributions of this dissertation include:1) some efficient methods are developed for the calculation of band gaps and for the prediction of vibration/noise properties ofgeneral locally resonant structures;2) the dependence of band gap behavior on somekey designing parameters is explored, and the underlying band gap formationmechanisms are clarified;3) several band gap broadening methods are proposed, and anumber of design formulas, principles and suggestions are provided. The results of thisdissertation present significant theoretical foundations and technical guidelines tofacilitate the application of locally resonant structures in reducing structural vibrationand noise.
本文紧紧围绕局域共振型结构在减振降噪应用方面目前面临的关键理论和技术问题展开系统深入研究，研究的主要内容包括：(1)局域共振型结构的带隙及振动与声学特性计算方法；(2)局域共振型结构的带隙调控与形成机理；(3)局域共振型结构的带隙展宽方法及宽带减振降噪特性。论文研究取得的主要创新成果包括：
1、提出了计算局域共振杆/梁结构中振动传递特性的波动/谱元法，并将平面波展开法推广用于计算局域共振板结构的带隙特性及声透射特性，为局域共振型结构的带隙与减振降噪特性研究提供了高效工具。
2、系统分析了局域共振型结构的设计参数对其带隙特性的调控规律，并发现了有价值的新带隙现象及效应，深化了对于局域共振型结构带隙行为的认识。首次发现局域共振杆中两种带隙（局域共振带隙和Bragg带隙）可以精确地耦合成为一个超宽的完全带隙；首次发现当两种带隙被调节至相互靠近时，不仅局域共振单元的局域共振作用对Bragg带隙可以产生展宽效应，而且周期结构的Bragg散射作用对局域共振带隙也可以产生展宽效应；首次发现局域共振型结构中局域共振单元的谐振频率可以落在Bragg带隙范围内，甚至是通带范围内。
3、深入揭示了局域共振型结构的带隙形成机理，推导得到了封闭的解析设计公式，可以精确地或近似地预报局域共振型结构中设计参数对带隙边界频率的调控规律，以及确定结构中两种带隙发生相互耦合和相互转换的条件，为带隙设计提供依据。
4、提出了基于两种带隙相互耦合机制、结构参数失谐设计以及局域共振单元多频谐振作用的三种带隙展宽方法，为实现局域共振型结构在轻质条件下的宽带减振降噪设计提供了技术支持。
5、率先研究了局域共振板的声透射降噪特性，首次发现在亚波长晶格条件下，若将局域共振单元的谐振频率调谐至基体板声透射的质量密度定律区，局域共振板可以实现远高于同质量均质板的传声损失（隔声量）；若将谐振频率调谐至基体板的声透射吻合效应区，局域共振板可以抑制吻合效应，大幅提高整个吻合效应区的传声损失。这些特性对于局域共振板的降噪应用具有重要指导意义。
总之，本文主要针对局域共振型结构在减振降噪应用方面需要解决的关键理论与技术问题展开系统深入研究：进一步完善了局域共振型杆、梁、板结构的带隙及振动与声学特性计算方法，深入揭示了这类局域共振型结构的带隙特性调控规律与形成机理，并提出了实现其带隙及减振降噪频带展宽的三种方法，从相关研究中提炼出了系列设计公式、准则及建议。这些研究成果为局域共振型结构的减振降噪应用奠定了若干重要的理论基础和技术支撑。
In the past few years, the new concept of “locally resonant phononic crystals” and“acoustic metamaterials” developed in the frontiers of acoustical physics providedsome new ideas for the control of vibration and noise of engineering structures. Byanalogy with locally resonant phononic crystals and acoustic metamaterials, locallyresonant periodic engineering structures (called “locally resonant structures” in thisdissertation) can be constructed by attaching periodic arrays of local resonators tostructural waveguides such as rods, beams, plates, etc. Some initial studies have shownthat locally resonant structures can exhibit notable elastic wave band gaps, andstructural vibrations can be significantly attenuated within the frequency ranges ofband gaps. These studies provide a novel approach to control vibration and noise ofengineering structures. However, to achieve practical applications of locally resonantstructures in engineering, there exist still many important problems required to beexamined, such as the performance prediction, behavior tuning, mechanismsexplanation, as well as the lightweight and broadband design.
The purpose of this dissertation is to investigate some crucial theoretical andtechnical problems systematically to promote the applications of locally resonantstructures in the field of vibration and noise control. The following three aspects havebeen considered:(1) the methods for calculating band gaps and the methods forpredicting vibration/noise properties of locally resonant structures;(2) the tuningbehavior of band gaps and the underlying band-gap formation mechanisms;(3) themethods for broadening band gaps and the study of resulting performance ofbroadband vibration/noise reduction. The main findings of this dissertation are listed asfollows:
1. A wave/spectral element method is proposed to predict vibration transmittanceof general locally resonant rods/beams, and a plane wave expansion method isdeveloped for the calculation of band gaps and sound transmission properties ofgeneral locally resonant thin plates. These methods are provided as efficient tools forthe study of band gap and vibration/noise reduction properties of locally resonantstructures.
2. The tuning band gap behavior of locally resonant structures by changing designing parameters is analyzed in a systematic manner, and some valuable new bandgap phenomena and effects have been discovered. It is shown that two types of bandgaps (locally resonant and Bragg band gaps) can exactly couple together to form acombined super-wide band gap. It is also found that when the location of two types ofband gaps is tuned to be very close to each other, not only that the locally resonanteffect can broaden Bragg band gap, but also that the Bragg scattering effect can widenthe locally resonant band gap. In addition, it is found that the resonance frequency oflocal resonators can appear in a Bragg band gap, and even more surprisingly, in a passband.
3. The band gap formation mechanisms of locally resonant structures have beenclarified in depth. A number of explicit analytical design formulas are derived to enablean exact prediction of the dependence of band edge frequencies on the designingparameters of locally resonant structures. Furthermore, explicit formulas are alsoprovided to enable a direct determination of the conditions of coupling or transitionbetween two types of band gaps. These results provide guidelines for the band gapdesign in locally resonant structures.
4. Three band gap broadening methods are proposed, respectively based on theband gap coupling mechanism, the design of disordered structural parameters, and theutilization of multi-resonant local resonators. These methods provide technical supportto realize lightweight locally resonant structures with broadband vibration/noisereduction performance.
5. For the first time, the sound transmission loss (STL) behavior of locallyresonant thin plates is investigated. It is found that, under the condition ofsubwavelength lattice constant, if the resonance frequency of local resonators is tunedto the mass-law region of sound transmission for the host plate, the locally resonantplate can achieve much higher STL than a homogeneous plate with the same surfacemass. While, if the resonance frequency is tuned to the coincidence region of soundtransmission, the locally resonant plate can break the coincidence effect, resulting in asignificantly increased STL over the whole coincidence region. These properties are ofgreat interest for the application of locally resonant plates in the noise control field.
In summary, this dissertation is concerned with some crucial theoretical andtechnical problems involved in the investigations and applications of locally resonantstructures for the purpose of reducing structural vibration/noise. The maincontributions of this dissertation include:1) some efficient methods are developed for the calculation of band gaps and for the prediction of vibration/noise properties ofgeneral locally resonant structures;2) the dependence of band gap behavior on somekey designing parameters is explored, and the underlying band gap formationmechanisms are clarified;3) several band gap broadening methods are proposed, and anumber of design formulas, principles and suggestions are provided. The results of thisdissertation present significant theoretical foundations and technical guidelines tofacilitate the application of locally resonant structures in reducing structural vibrationand noise.
引文
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