基于声子晶体理论的梁板类周期结构振动带隙特性研究
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摘要
声子晶体是具有弹性波带隙的周期性复合材料,带隙频率范围内的弹性波不能传播。声子晶体是凝聚态物理领域在研究光子晶体中电磁波传播特性的基础上提出的一个新方向,具有重要的理论和应用价值而受到广泛关注。声子晶体在带隙理论和带隙算法方面均取得重要进展,但在减振降噪领域的应用探索才刚开始。
长期以来,结构振动控制一直是理论与工程界关注和着力解决的热点问题之一。将声子晶体理论应用到工程结构设计中,使之具有振动带隙,利用带隙特性抑制振动的传播,将会为结构振动控制提供了一种新的原理和技术途径。
本文以此为背景,在国家重大基础研究项目(973项目)的资助下,提出利用声子晶体理论对周期结构的振动带隙特性进行深入研究。论文是以声子晶体理论和算法为基础,应用理论分析、有限元仿真和实验验证相结合的研究方法,对工程中广泛应用的梁板类周期结构的振动带隙特性展开系统深入的研究。主要研究内容与成果包括:
1.针对不同梁板类结构,改进和完善了声子晶体的带隙计算方法,为梁板类周期结构的振动带隙计算提供了强有力的计算工具。
(1)利用传递矩阵法,可以有效计算梁结构的弯扭耦合振动带隙,同时也可以计算各种杆件结构的局域共振带隙。
(2)在二维周期结构板的带隙计算中,改进的平面波展开法可以计算周期栅格结构的振动带隙,而且通过修改傅立叶级数的展开形式,可以大大提高计算结果的收敛性。
2.深入系统地研究了梁板类周期结构的布拉格振动带隙特性,深化了对周期结构振动带隙的认识,丰富了周期结构的研究内容。
(1)证实了一维周期杆结构的纵向振动、轴结构的扭转振动、梁结构的弯曲振动以及二维周期薄板结构、栅格结构的纵向振动和弯曲振动均存在振动带隙。
(2)揭示了表面局域态是一维杆状周期结构传输特性中带隙内产生共振峰的根本原因。发现当一维杆状周期结构纵向振动时,自由表面材料中的纵波速度比另一种材料中的纵波速度小时,在带隙内可以产生表面局域态。
(3)深入研究了梁的弯扭耦合振动带隙特性,揭示了弯扭耦合振动对带隙特性的影响规律。特别地,在薄壁梁的弯扭耦合振动中,发现晶格常数是影响薄壁梁耦合振动带隙相对宽度的一个重要几何参数,两种材料的杨氏模量差异是决定带隙相对宽度的另一个关键因素。
(4)揭示了二维周期结构薄板的纵向振动带隙与二维声子晶体的XY模态带隙之间的差异。对于二维板结构弯曲振动,揭示了转动惯量和剪切变形对带隙的影响规律。
(5)发现了周期栅格结构的弯曲振动可以存在完全振动带隙,完全带隙内的衰减比方向带隙内的衰减强。
3.将声子晶体理论中的局域共振带隙思想引入到周期结构中,针对低频、宽带、小尺寸、强衰减等目标,优化设计了不同的局域共振结构,得到了相应的局域共振带隙,揭示了大量有价值的规律,为声子晶体的低频减振降噪应用提供了新的思路和研究方向,拓展了周期结构的研究领域。
(1)设计了一种环状局域共振结构,得到了扭转振动的低频局域共振带隙,分析了轴的材料属性对带隙内衰减的影响规律。
(2)设计了一种双自由度的局域共振结构,深入分析了局域共振结构本身的耦合模式对带隙的影响。理论和实验研究均表明,在梁上周期分布这种局域共振结构,可以获得低频、宽带、强衰减的局域共振带隙。这种思想对局域共振带隙的设计以及结构的减振降噪均具有重要意义。
(3)优化设计了二维局域共振结构板,理论和实验研究表明二维周期结构薄板存在纵向振动的局域共振带隙,但是低频、强衰减带隙特性不可兼得。
(4)研究表明低频局域共振带隙不仅存在于三组元周期结构薄板弯曲振动中,二维二组元薄板结构也可以产生弯曲振动的局域共振带隙。二组元薄板结构产生的局域共振带隙频率范围比较窄,而三组元薄板结构可以产生低频、宽带、强衰减的局域共振带隙。
(5)分析了阻尼因素对局域共振带隙宽度和带隙内衰减的影响。发现当阻尼增大时,局域共振带隙频率范围变宽,但是带隙范围内有效衰减减小。这对带隙宽度和带隙内衰减的优化设计具有重要指导意义。
总之,本文将声子晶体理论和算法引入到周期结构的振动带隙特性研究中,应用理论分析、有限元仿真和实验验证相结合的研究方法,对梁板类周期结构的布拉格带隙和局域共振带隙展开了系统深入的研究。进一步深化了周期结构的振动带隙理论;为周期结构振动带隙计算提供了准确、高效的计算方法,为揭示周期结构新的带隙特性提供了有力工具;拓展了周期结构的研究领域,特别是局域共振带隙思想的引入,为梁板类周期结构的低频减振降噪提供了新的思路和技术途径。这些研究对推动声子晶体理论在减振降噪领域中的工程应用具有重要的理论意义和工程参考价值。
Phononic crystalsare periodic composites with elastic waves band gaps within which the propagation of vibration and sound are forbidden. Phononic crystals are the counterparts of photonic crystals in elasticity. It becomes a hotspot topic in condensed matter physics recently. Phononic crystals exhibit rich new physics and promising potential applications, which have attracted much attention from various disciplines. Many important progresses in formation mechanism and calculation methods have been achieved in last decade. But its application is just to get under the way.
Controlling vibrations in structures has been one of the pop research topics in academic and engineering for a long time. If the philosophy of Phononic crystals is introduced into their design, engineering structures will exhibit frequency band gaps and attenuate their vibration, which provides a new approach for vibration control. Under this circumstance, funded by the State Key Development Program for Basic Research of China (973 program), this dissertation addresses the vibration band gaps in periodic structures using the Phononic crystals theory. With the calculation methods of Phononic crystals and their extensions, together with finite element method and experimental tests, the vibration band gaps in periodic beams and plates, which are applied widely in engineering, are investigated deeply and systemically. The main work and achievements are as follows:
1. The calculation methods of Phononic crystals are modified and improved to deal with different periodic beam or plate structures, which are validated by experiments to be powerful numerical tools for investigating vibration band gaps of periodic structures.
(1) The transfer matrix method is extended to deal with the coupling effects of flexural and torsional vibration in beams. It is illustrated that the extension can calculate locally resonant band structures of shafts or beams effectively.
(2) The plane wave expansion method is improved to calculate vibration band gaps in the periodic grid structure, as well as by changing the expansion form of Fourier series so that it can handle locally resonant gaps in two-dimensional plate structures with better convergence.
2. The vibration gaps with Bragg scattering mechanism in periodic beams and plates are studied deeply and systemically, which enrich the elastodynamics of periodic structures.
(1) Different vibration gaps are found theoretically and experimentally in the different periodic structures, such as longitudinal vibration gap in rods, torsional vibration gap in shafts, flexural vibration gap in beams, longitudinal and flexural vibration gaps in two-dimensional thin plates and grids.
(2) It is found that surface localized modes are the essential reason for the transmission peaks in one-dimensional periodic rods. It is found that when the longitudinal velocity of the material with free surface is less than that of the other material, there will exist surface localized modes in the free surface of periodic rods.
(3) The coupled flexural and torsional vibration gaps in periodic beams are studied deeply. Particularly, for the coupled vibration gaps in thin-walled beams, it is found that the lattice constant is an important factor that affects the normalized gap width in addition to the contrast of Young’s modulus.
(4) The differences between the longitudinal vibration gap in two-dimensional thin plates and the XY mode gaps in two-dimensional Phononic crystals are illustrated. The effects of rotary inertia and shear deformation on gaps in two-dimensional periodic plates are investigated and summarized.
(5) It is shown that complete flexural vibration gaps exist in periodic grids. The vibration attenuation in the complete gaps is stronger than that in the directional gaps.
3.The locally resonant gap mechanism of Phononic crystals is introduced in the design of periodic structures. Several small size periodic structures with different locally resonators are designed and tested. Wide gaps in low frequency with strong attenuation are observed in vibration experiments. Further theoretical and experimental studies opened out some valuable phenomena, which provide new idea for the applications of PCs in low-frequency vibration/noise control. All these related works expand the study field of periodic structures.
(1) It is illustrated that low frequency torsional gap exists in theshaft with periodic cylindrical locally resonators. It is found that torsional stiffness of the shaft has important influence on the propagating attenuation in the frequency ranges of gaps.
(2) The locally resonator with two-degree-of-freedom, i.e. the resonator with vertical and rotational vibration, is invented to obtain wide gaps in low frequency with strong attenuation in beams. Experimental results confirm the theoretical design reasonably. Such an idea is extremely promising for the design of locally resonant structures, as well as their applications in the vibration/noise control.
(3) Theoretical and experimental results show there are locally resonant gaps of longitudinal vibration in an optimized periodic plate. But it is impossible to obtain the gaps of longitudinal vibration in low frequency with strong attenuation in such plates.
(4) It is found that locally resonant gaps of flexural vibration not only exist in the two-dimensional ternary thin plates, but also in binary thin plates. But it is noteworthy that gaps in binary thin plate are much narrower than that in its ternary counterpart, and the attenuation in gaps are much less either.
(5) The effects of the damping on the width of the locally resonant gap and its attenuation are investigated. It is shown that damping can widen the frequency range while bate its attenuation.
In summary, the theories and calculation methods of PCs are introduced to the research of vibration gaps in periodic structures. The vibration gaps with Bragg scatter mechanism and locally resonant mechanism in periodic beams and plates are studied theoretically and experimentally. These systematic studies are valuable for periodic structure in deepening the theory, enriching the calculation methods, and expanding the study field. The research results in this dissertation are meaningful in the theory of the PCs as well as its application to vibration/noise control.
长期以来,结构振动控制一直是理论与工程界关注和着力解决的热点问题之一。将声子晶体理论应用到工程结构设计中,使之具有振动带隙,利用带隙特性抑制振动的传播,将会为结构振动控制提供了一种新的原理和技术途径。
本文以此为背景,在国家重大基础研究项目(973项目)的资助下,提出利用声子晶体理论对周期结构的振动带隙特性进行深入研究。论文是以声子晶体理论和算法为基础,应用理论分析、有限元仿真和实验验证相结合的研究方法,对工程中广泛应用的梁板类周期结构的振动带隙特性展开系统深入的研究。主要研究内容与成果包括:
1.针对不同梁板类结构,改进和完善了声子晶体的带隙计算方法,为梁板类周期结构的振动带隙计算提供了强有力的计算工具。
(1)利用传递矩阵法,可以有效计算梁结构的弯扭耦合振动带隙,同时也可以计算各种杆件结构的局域共振带隙。
(2)在二维周期结构板的带隙计算中,改进的平面波展开法可以计算周期栅格结构的振动带隙,而且通过修改傅立叶级数的展开形式,可以大大提高计算结果的收敛性。
2.深入系统地研究了梁板类周期结构的布拉格振动带隙特性,深化了对周期结构振动带隙的认识,丰富了周期结构的研究内容。
(1)证实了一维周期杆结构的纵向振动、轴结构的扭转振动、梁结构的弯曲振动以及二维周期薄板结构、栅格结构的纵向振动和弯曲振动均存在振动带隙。
(2)揭示了表面局域态是一维杆状周期结构传输特性中带隙内产生共振峰的根本原因。发现当一维杆状周期结构纵向振动时,自由表面材料中的纵波速度比另一种材料中的纵波速度小时,在带隙内可以产生表面局域态。
(3)深入研究了梁的弯扭耦合振动带隙特性,揭示了弯扭耦合振动对带隙特性的影响规律。特别地,在薄壁梁的弯扭耦合振动中,发现晶格常数是影响薄壁梁耦合振动带隙相对宽度的一个重要几何参数,两种材料的杨氏模量差异是决定带隙相对宽度的另一个关键因素。
(4)揭示了二维周期结构薄板的纵向振动带隙与二维声子晶体的XY模态带隙之间的差异。对于二维板结构弯曲振动,揭示了转动惯量和剪切变形对带隙的影响规律。
(5)发现了周期栅格结构的弯曲振动可以存在完全振动带隙,完全带隙内的衰减比方向带隙内的衰减强。
3.将声子晶体理论中的局域共振带隙思想引入到周期结构中,针对低频、宽带、小尺寸、强衰减等目标,优化设计了不同的局域共振结构,得到了相应的局域共振带隙,揭示了大量有价值的规律,为声子晶体的低频减振降噪应用提供了新的思路和研究方向,拓展了周期结构的研究领域。
(1)设计了一种环状局域共振结构,得到了扭转振动的低频局域共振带隙,分析了轴的材料属性对带隙内衰减的影响规律。
(2)设计了一种双自由度的局域共振结构,深入分析了局域共振结构本身的耦合模式对带隙的影响。理论和实验研究均表明,在梁上周期分布这种局域共振结构,可以获得低频、宽带、强衰减的局域共振带隙。这种思想对局域共振带隙的设计以及结构的减振降噪均具有重要意义。
(3)优化设计了二维局域共振结构板,理论和实验研究表明二维周期结构薄板存在纵向振动的局域共振带隙,但是低频、强衰减带隙特性不可兼得。
(4)研究表明低频局域共振带隙不仅存在于三组元周期结构薄板弯曲振动中,二维二组元薄板结构也可以产生弯曲振动的局域共振带隙。二组元薄板结构产生的局域共振带隙频率范围比较窄,而三组元薄板结构可以产生低频、宽带、强衰减的局域共振带隙。
(5)分析了阻尼因素对局域共振带隙宽度和带隙内衰减的影响。发现当阻尼增大时,局域共振带隙频率范围变宽,但是带隙范围内有效衰减减小。这对带隙宽度和带隙内衰减的优化设计具有重要指导意义。
总之,本文将声子晶体理论和算法引入到周期结构的振动带隙特性研究中,应用理论分析、有限元仿真和实验验证相结合的研究方法,对梁板类周期结构的布拉格带隙和局域共振带隙展开了系统深入的研究。进一步深化了周期结构的振动带隙理论;为周期结构振动带隙计算提供了准确、高效的计算方法,为揭示周期结构新的带隙特性提供了有力工具;拓展了周期结构的研究领域,特别是局域共振带隙思想的引入,为梁板类周期结构的低频减振降噪提供了新的思路和技术途径。这些研究对推动声子晶体理论在减振降噪领域中的工程应用具有重要的理论意义和工程参考价值。
Phononic crystalsare periodic composites with elastic waves band gaps within which the propagation of vibration and sound are forbidden. Phononic crystals are the counterparts of photonic crystals in elasticity. It becomes a hotspot topic in condensed matter physics recently. Phononic crystals exhibit rich new physics and promising potential applications, which have attracted much attention from various disciplines. Many important progresses in formation mechanism and calculation methods have been achieved in last decade. But its application is just to get under the way.
Controlling vibrations in structures has been one of the pop research topics in academic and engineering for a long time. If the philosophy of Phononic crystals is introduced into their design, engineering structures will exhibit frequency band gaps and attenuate their vibration, which provides a new approach for vibration control. Under this circumstance, funded by the State Key Development Program for Basic Research of China (973 program), this dissertation addresses the vibration band gaps in periodic structures using the Phononic crystals theory. With the calculation methods of Phononic crystals and their extensions, together with finite element method and experimental tests, the vibration band gaps in periodic beams and plates, which are applied widely in engineering, are investigated deeply and systemically. The main work and achievements are as follows:
1. The calculation methods of Phononic crystals are modified and improved to deal with different periodic beam or plate structures, which are validated by experiments to be powerful numerical tools for investigating vibration band gaps of periodic structures.
(1) The transfer matrix method is extended to deal with the coupling effects of flexural and torsional vibration in beams. It is illustrated that the extension can calculate locally resonant band structures of shafts or beams effectively.
(2) The plane wave expansion method is improved to calculate vibration band gaps in the periodic grid structure, as well as by changing the expansion form of Fourier series so that it can handle locally resonant gaps in two-dimensional plate structures with better convergence.
2. The vibration gaps with Bragg scattering mechanism in periodic beams and plates are studied deeply and systemically, which enrich the elastodynamics of periodic structures.
(1) Different vibration gaps are found theoretically and experimentally in the different periodic structures, such as longitudinal vibration gap in rods, torsional vibration gap in shafts, flexural vibration gap in beams, longitudinal and flexural vibration gaps in two-dimensional thin plates and grids.
(2) It is found that surface localized modes are the essential reason for the transmission peaks in one-dimensional periodic rods. It is found that when the longitudinal velocity of the material with free surface is less than that of the other material, there will exist surface localized modes in the free surface of periodic rods.
(3) The coupled flexural and torsional vibration gaps in periodic beams are studied deeply. Particularly, for the coupled vibration gaps in thin-walled beams, it is found that the lattice constant is an important factor that affects the normalized gap width in addition to the contrast of Young’s modulus.
(4) The differences between the longitudinal vibration gap in two-dimensional thin plates and the XY mode gaps in two-dimensional Phononic crystals are illustrated. The effects of rotary inertia and shear deformation on gaps in two-dimensional periodic plates are investigated and summarized.
(5) It is shown that complete flexural vibration gaps exist in periodic grids. The vibration attenuation in the complete gaps is stronger than that in the directional gaps.
3.The locally resonant gap mechanism of Phononic crystals is introduced in the design of periodic structures. Several small size periodic structures with different locally resonators are designed and tested. Wide gaps in low frequency with strong attenuation are observed in vibration experiments. Further theoretical and experimental studies opened out some valuable phenomena, which provide new idea for the applications of PCs in low-frequency vibration/noise control. All these related works expand the study field of periodic structures.
(1) It is illustrated that low frequency torsional gap exists in theshaft with periodic cylindrical locally resonators. It is found that torsional stiffness of the shaft has important influence on the propagating attenuation in the frequency ranges of gaps.
(2) The locally resonator with two-degree-of-freedom, i.e. the resonator with vertical and rotational vibration, is invented to obtain wide gaps in low frequency with strong attenuation in beams. Experimental results confirm the theoretical design reasonably. Such an idea is extremely promising for the design of locally resonant structures, as well as their applications in the vibration/noise control.
(3) Theoretical and experimental results show there are locally resonant gaps of longitudinal vibration in an optimized periodic plate. But it is impossible to obtain the gaps of longitudinal vibration in low frequency with strong attenuation in such plates.
(4) It is found that locally resonant gaps of flexural vibration not only exist in the two-dimensional ternary thin plates, but also in binary thin plates. But it is noteworthy that gaps in binary thin plate are much narrower than that in its ternary counterpart, and the attenuation in gaps are much less either.
(5) The effects of the damping on the width of the locally resonant gap and its attenuation are investigated. It is shown that damping can widen the frequency range while bate its attenuation.
In summary, the theories and calculation methods of PCs are introduced to the research of vibration gaps in periodic structures. The vibration gaps with Bragg scatter mechanism and locally resonant mechanism in periodic beams and plates are studied theoretically and experimentally. These systematic studies are valuable for periodic structure in deepening the theory, enriching the calculation methods, and expanding the study field. The research results in this dissertation are meaningful in the theory of the PCs as well as its application to vibration/noise control.
引文
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