Band-gap Characteristics Of Elastic Metamaterial Plate Based On The Finite Element And Spectral Element Hybrid Method

With the development of science and technology,various equipment and engineering structures are gradually developing towards light quality,large flexibility,high precision,and good reliability.By the influence of strong vibration or vibration combined with other environments,like temperature,humidity,and centrifugation,the product or the structure is difficult to play its effectiveness.The application of vibration control and absorption is essential.Vibration control is divided into active control and passive control.Active control needs external energy input and has poor vibration control effect on high frequency,general reliability,and environmental adaptability.Passive control needs no additional energy,and has simple structure,economic,convenience,good reliability and stability,and good vibration attenuation effect on full frequency.It is widely used in the field of vibration attenuation.The elastic metamaterial structure composed of periodically distributed artificial substructure has the vibration band-gap characteristics.The elastic wave cannot propagate in the band-gap and then the passive control of the vibration is realized.By adjusting the material and structure parameters,the number,position,and width of band-gaps can also be changed,so that it has a good application prospect in vibration attenuation.In this paper,three kinds of locally resonant elastic metamaterial plates are proposed and designed,with many equal-section homogeneous straight rods,composite periodic rods or axially deformed beams periodically distributed in the middle of the two thin plates as the local resonance.For this structure,finite element and spectral element hybrid method(FE-SEHM)is proposed,and the dynamic model is established.Combined with frequency response curves of numerical simulation and experiment,the influence of band-gap characteristics and the change of different parameters on the vibration attenuation performance is analyzed to improve the vibration suppression effect.The main works of this article are as follows:A metamaterial plate consisting of two thin plates and straight rods distributed periodically between the plates is designed.The dynamic stiffness matrices of plate element and rod element are derived by finite element method(FEM)and spectral element method(SEM)respectively based on bending theory of the plate and wave equation of the rod.According to the node position relationship between two elements in the unit-cell of the metamaterial plate,the integration method of finite element stiffness matrix is adopted to obtain the dynamic stiffness matrix of the unit-cell and the FE-SEHM model of the whole structure is constructed.It lays a theoretical foundation for the verification of analysis method,the analysis of band-gap characteristics and the design of metamaterial plates.Comparing the similarities and differences of the numerical simulation results of two metamaterial plates with rods and mass-springs as the local resonance,the designed structure has band-gap characteristics.The dynamical analysis of metamaterials plates with rod core are performed by FE-SEHM and FEM.The results of FEM converge to the results of FE-SEHM with the number of FEM elements increasing.It means that the FE-SEHM model has a higher accuracy.The specimen and the matching fixture are designed and processed.The band-gap characteristics of the metamaterial plate are verified by modal experiment results.The causes of the band-gap quantity and location generated in metamaterials plate are analyzed.The band-gap positions are found to be correlated with natural frequencies and mode shapes of the local resonance,with distinct band-gaps near the odd order natural frequency of the rod.The lower-order band-gaps are wide and the higher-order band-gaps are narrow.The influence of material and structural parameters and periodic distribution,like the loss factor,mass ratio,quantity and natural frequency of the rod,changes on the band structure are analyzed.The specific application of band-gap characteristic on vibration attenuation is also explored.The elastic sandwich metamaterial plate with composite periodic rod core is presented,and the model of the composite periodic rod and whole structure are obtained by using FE-SEHM.Compared with the metamaterial plate containing rod core,the band-gaps of this metamaterial plate are produced near natural frequencies of the composite periodic rod.The number of band-gaps is increased,and the positions of band-gaps are adjustable.Vibration attenuation by different multiple frequency band-gaps is realized.The influence of the period number of the composite periodic rod and the stiffness of the sub-rod on the band-gap characteristics are also analyzed.The metamaterial plate structure with axially deformed beam cores is designed.The local resonances are Timoshenko beam which can bear the vertical pull-press and horizontal bending and shearing.The models of thin plate element with in-plane displacement,axially deformed beam element,the unit-cell,and the whole metamaterial plate are obtained by using FE-SEHM.The metamaterial plate has the band-gap characteristic of reducing vibration on the vertical and horizontal load.Compared with the above two metamaterial plates,its application range is more extensive,and it achieves a good vibration control effect.