【作者】 王俊；

【导师】 周晓勤；

【作者基本信息】 吉林大学 ， 机械制造及其自动化， 2018， 博士

【摘要】 弹性超材料的概念自提出起就因其带隙内弹性波衰减特性在减振降噪领域受到许多关注,尤其是在低频减振方面。而局域共振型弹性超材料相比Bragg散射型声子晶体,其带隙中心频率不受晶格尺寸的制约,可以实现小尺寸低频率的弹性波带隙,在低频减振领域具有先天优势。局域共振型弹性超材料实现带隙的基本结构单元常见为单极共振单元、偶极共振单元,其中单极共振单元能产生负的等效弹性模量,偶极共振单元能产生负的等效质量,两者同时存在时可以产生双负的等效材料参数。其中偶极共振单元组成的负等效质量弹性超材料比较适合于对结构刚度有要求的减振应用,但其也存在带隙过窄的缺点,且带隙内衰减系数大的区域主要集中共振频率附近。故对超材料结构的优化设计是必要的,为此,本文将从理论建模、有限元数值仿真与振动实验测量三方面对局域共振型弹性超材料进行研究,主要内容包括:(1)利用扩展哈密顿原理推导了基于mass-in-mass模型的超材料杆、梁的理论模型,并用于求解铝杆附加环形振子结构的纵向与弯曲振动带隙特性,所得结果利用有限元模型与振动实验测量两方面进行了验证。结果表明,增加弹性超材料杆、梁的附加振子质量、采用密度更低或更细的支撑杆、降低单元长度均可以扩宽归一化带隙的频率范围。而增加附加振子质量,降低振子与支撑杆间连接弹簧的弹性系数均可以降低局域共振频率,从而使带隙频率降低。将相同截面梁模型扩展到变截面梁结构模型,并用来求解了方形格栅梁附加振子结构的能带结构。所得结果分别利用了二维有限元模型与三维有限元模型进行了验证。结果表明在格栅的面积与材料不变的情况下,格栅内孔的形状变化对局域共振带隙的影响很小。此外通过研究截面几何参数对带隙的影响发现,可以通过对附加振子的几何参数调整来实现带隙的平移,通过对格栅梁的几何调整来扩展带隙带宽。(2)利用分步均质化方法建立了复频mass-in-mass单元的等效质量模型,并与复频单元的能带结构进行对比,结果表明分步均质化方法计算的负的等效质量频率范围与能带结构显示的带隙频率范围吻合。附加振子数量增加之后,复频结构在离散链截至频率下的带隙数量与带隙宽度都增加了,这意味着增加附加振子数量是一个扩展局域共振带隙区域的有效方法。对双振子模型相关参数的研究发现,双振子模型的两个带隙中心频率在两附加振子质量差异较大时相距较近,较大的内外质量比与弹性系数比均有利于增加第一与第二带隙宽度。基于类晶格模型建立包含两个附加振子的复频结构的有限元模型,有限元模型的结果很好的验证了上述解析解。利用锤击法实验测量了铝杆附加双环形振子结构的振动传输特性,实验结果证实附加双振子结构可以生成两个相邻的带隙,用来扩展带隙范围。(3)研究了二维正方手性结构面内带隙特性,结果表明正方手性结构的第一带隙起始模态源于中间节点的刚性旋转模态,受韧带的倾斜角度影响明显。同时也发现节点边长增加有利于扩宽带隙频率范围、降低带隙频率,而增加韧带长度也有利于降低带隙频率;研究了正方手性晶格板中的Lamb波能带结构,结果表明正方手性晶格板中的第一带隙上下模态均源自与弯曲波主导的振动模态。同时发现弯曲波主导的能带曲线随板厚度的降低而增多,在板厚度低于10mm时就已经导致源于平面内旋转模态的完全带隙消失。板5×20单元阵列的振动传递特性显示板面内振动传递特性与板厚无关,振动衰减较大的区域与二维正方手性结构的相似;研究了中字型切缝超材料板中弯曲波能带结构,结果表明增加中字形中部与上下切缝长度均可以有效降低带隙频率。此外研究了尺寸梯度变化时中字型切缝超材料的振动带隙特征。(4)设计了多层弹性超材料结构,以多层悬臂振子杆状结构为例,推导了多层超材料的等效模型,并讨论了各组成层厚度与材料对带隙的影响。结果表明多层结构的振动带隙特征与组成基板的各层厚度与材料密切相关。当各组成层厚度相同时,多层结构的带隙中心频率将与厚度大小无关且位于各组成层的单层局域共振频率的最大值与最小值之间。当组成基板由刚度较小的阻尼层与刚度较大的无阻尼层组成时,多层结构可以在保持较高刚度的同时引入可控的阻尼特性,从而在不牺牲太多峰值衰减的情况下,有效利用阻尼特性降低带隙频率并小幅增加了带隙的宽度。本文通过对多组元与单组元弹性超材料振动带隙特性的研究,揭示了合理的设计参数,以实现低频与宽频带隙。研究了复频结构的多频带隙特征,从而达到扩宽带隙频率范围的目的;设计多层超材料结构从而为单组元超材料引入额外的带隙调控参数。文中结果对推进弹性超材料在减振降噪的应用具有重要意义。更多还原

【Abstract】 Acoustic/Elastic metamaterials have attracted much attention in the field of vibration and noise reduction due to the attenuation characteristics of elastic waves in the band gap,especially in the low frequency vibration reduction.Compared with Bragg scattering phononic crystals,locally resonant elastic metamaterials can build with small geometric size to achieve low frequency elastic wave band gaps,which have inherent advantages in the field of low frequency vibration reduction.The fundamental structural elements in locally resonant elastic metamaterials are unipolar resonance elements and dipole resonance elements,in which the unipolar resonance element can generate negative effective elastic modulus and the dipole resonance element can produce negative effective mass.When both of them exist,the double-negative elastic metamaterials would be achieved with simultaneously negative effective mass density and elastic modulus.Negative effective mass elastic metamaterials composed of dipole resonance elements are suitable for engineering applications which can be designed with a high stiffness base.However,the band gaps obtained from local resonances are usually in narrow frequency regions,and the attenuation performances become poor away from the resonance frequencies.Therefore,the optimizing process for the bandgaps of these elastic metamaterials is very necessary.In this paper,the local resonance elastic metamaterial are researched by the theoretical model,finite element numerical model and vibration experimental measurement,the main contents include:(1)The theoretical model of metamaterial bars and beams based on mass-in-mass model is modeled by the extended Hamiltonian principle,which is used to solve the longitudinal and bending vibration band gap characteristics of the aluminum bar with ring resonators.The analytic results are well verified by the finite element models and vibration experimental measurements.The results show that the frequency range of the normalized band gap can be widened by increasing the mass of the additional ring resonators,adopting lower or finer support rod,or reducing the length of the unit.The normalized reference frequency would be reduced by increasing the mass of the additional oscillator or decreasing the elastic coefficient of the connecting spring.The beam model with the same cross-section is extended to solve the variable cross-section beam model,and the band structure of the square grid beam with additional oscillator is solved.The results are verified by two-dimensional finite element model and three-dimensional finite element model respectively.The results show that the shape of the hole has little effect on the local resonant band gap when the area and material of the grid remain unchanged.In addition,it is found that the band gap can be shifted by adjusting the geometric parameters of the additional oscillator,and the band gap can be extended by adjusting the geometric parameters of the grid beam.(2)The effective mass of the multi-frequency mass-in-mass model is established by the two-step homogenization method.The results show that the negative effective mass frequency ranges calculated by the two-step homogenization method coincide with the frequency ranges of band gap.With the increase of the number of additional oscillators,the number and width of band gaps below the discrete frequency increase,which means that increasing number of the additional oscillators is an effective method to extend the local resonant band gap regions.It is found that the center frequencies of the two band gaps in the dual-oscillator model are close to each other when the lumped masses of the two additional oscillators are different,and the larger mass ratio and the elastic coefficient ratio are beneficial to broaden the width of the first and second band gaps.The finite element model of multi-frequency structure with two additional oscillators based on the mimicking lattice systems is modeled in Comsol.The finite element solutions have good agreement with the analytical results.Vibration transmission characteristics of double-ring oscillators with aluminum bar are measured by hammering method.The experimental results indicate that the double-ring oscillator can generate two adjacent band gaps which can be used to extend the frequency range of the band gap.(3)The band structure of two-dimensional square chiral structures are numerical studied in COMSOL.The results show that start modes of the band gap in square chiral structures originate from the rigid rotation modes of the intermediate nodes,which are significantly affected by the angle of the ligaments.Meanwhile,it is obviously that the increase length of the node edge is beneficial to lower and wider band gap,while the increase length of the ligament is also beneficial to reduce the band gap frequency.The Lamb wave band structure in the square chiral lattice plate is further studied.The results show that the upper and lower modes of the first band gap in the square chiral lattice plate both originate from the bending vibration mode.It is also found that the number of bending band curves increase with the decrease of plate thickness.When the plate thickness is less than 10 mm,the complete band gap originating from in-plane rotational modes will be disappeared by bending band curves.The vibration transmission of the 5×20 array show that the in-plane vibration transfer characteristics are independent of the plate thickness,and the vibration attenuation region is similar to that in the two-dimensional square chiral structure.In addition,the bending wave band structures in the kerf metamaterial plate are numerically studied.(4)The multi-layer elastic metamaterial is analytically and numerically studied.The bar-like multi-layer cantilever-in-mass structure is selected as an example,whose effective mass model is deduced based on the mass-in-mass model.By solve the effective mass model,the effects of the thickness and material of each component layer on the band gap are discussed.The results indicate the negative effective mass depend highly on the material parameters and the thickness of each layer.When the thickness of each component layer is the same,the resonance frequencies of layered structures will be independent of layer thickness,and the numeric value of the resonance frequencies are between the maximum and minimum local resonance frequency of their constituent layers.The dissipative multi-layer structures modeled by stacking a dissipative layer with the metal layers are numerically researched,the obtained results indicate that the damped structure can own both damping characteristics and high mechanical strength,and the total damping characteristics in the damped LCIMs can be affected by the specific gravity of the dissipative layer in the laminated structures.In summary,this paper present detailed studies on the vibration band gap of different elastic metamaterial structures,which reveal the reasonable parameters for generating lower and wider band gap.The multi-bands characteristics of multi-frequency structures are also studied which have benefit for widening the frequency range of band gap.And the layered structure is designed to bring in the additional band gap control parameters for the single-component metamaterials.The present studies have further benefit for the application of elastic metamaterials in vibration and noise reduction.更多还原