Research On The Multi-frequency Local Resonance Elastic Metamaterial Designs And Vibration Bandgaps

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.