Calculation Method And Characteristic Studies Of Band Gaps In Locally Resonant Phononic Crystal Double Panel Structures

With the constant deepening and development of theory of vibration and noise reduction,phononic crystal(PC)plates have attracted extensive attention of scholars at home and abroad.Particularly,locally resonant(LR)PC plates with low frequency bandgap properties bring the possibility for the vibration and noise control in low frequency region.Comparing with single plate structures,double panel structures with excellent advantages are increasingly applied to the field of engineering.Therefore,it's necessary to research the calculation methods and characteristics of band gaps(BGs)in LRPC double panel structures.Based on the studies on traditional PCs and PC plates,the design idea of traditional LRPC is introduced to double panel structures in this paper.Efficient calculation methods are proposed for different PC plate structure forms.In addition,the formation mechanisms(FMs)and regulation rules(RRs)of BGs are investigated in detail.The main work and results are shown as follows:(1).The coupled plane wave expansion and finite element(PWE/FE)method is proposed to study the semi-infinite PCs.Firstly,model and formulize according to the structural features of semi-infinite beam-like PC and semi-infinite plate-like PC,respectively.Then,the accuracy and applicability of calculated results are verified by comparing the results with those calculated by finite element(FE)method.All the results show that the strong applicability and high accuracy are occurred to not only the semi-infinite beam-like PCs but the semi-infinite plate-like PCs.In addition,the BG properties of threecomponent semi-infinite plate-like LRPC are investigated.Numerical results and further analysis demonstrate that a band gap is opened by the coupling between the dominant vibrations of the rubber layer and the matrix modes.Moreover,the BG can be regulated and controled effectively by changing the Pb layer radius,rubber layer radius,lattice constant and the relevant viscidity parameters of rubber layer.(2).Flexural vibration BG properties in simplified model of LRPC single plates are investigated.At first,explicit matrix formulations of plane wave expansion(PWE)method are developed for the calculation of band structure,and the accuracy of calculated result is verified.Then,the flexural vibration BG properties of “spring-mass” simplified model of stubbed-on LRPC single plate are studied.Results show that a BG in the low frequency region is opened and the BG can be controlled effectively by adjusting the parameters of spring oscillator.Besides,by comparing the band structures of the “spring-mass” simplified model of stubbed-on LRPC single plate with and without the torsional spring added,the added torsional spring has obvious influences on the band structure.With the increase of torsional spring stiffness,the BG width increases and the starting frequency decreases.But the rotational inertia has nothing to do with the band structure.In the end,for the simplified model of composite LRPC single plate,the BG can be controlled effectively by adjusting the radius and relevant viscidity parameters of soft material.(3).Flexural vibration BG properties in simplified model of LRPC double panels are investigated.Firstly,PWE method is developed to calculate the band structure of simplified model,and the accuracy is confirmed by comparing the result with the one calculated by FE method and the transmission spectrum.Based on the observation of steady-state wave profiles of the finite structure under some specific frequencies and imposing additional constraints on the computing matrix,the band structures are divided into two types: the band structures vibrating in the symmetric mode and the antisymmetric mode.Hence,the formation of BG can be attributed to that two bands corresponding to the antisymmetric vibration mode open a wide band gap but is cut narrower by a band corresponding to the symmetric mode under 800 Hz.The BG can be controlled effectively by adjusting the parameters of spring oscillator.Then,the effect of additional springs added on BG is researched.Results show that the bands corresponding to symmetric vibration mode can be tuned to high frequency region easily,which makes the BG wider.Adding the number of additional springs and increasing the distance between the additional springs and the original resonator can lower the total stiffness of the additional springs.Besides,by comparing the band structures of the “spring-mass” simplified model of stubbed-on LRPC double panel with and without the torsional spring added,the added torsional spring has obvious influences on the band structure.With the increase of torsional spring stiffness,the BG width increases and the starting frequency decreases.But the rotational inertia has nothing to do with the band structure.In the end,the effects of the radius and viscidity of soft material on the band structure of the simplified model of composite LRPC double panel are investigated.Results show that the influence rules are similar to those of torsional spring.Increasing the radius of soft material can widen the BG and lower the starting frequency.And increasing the initial-final value ratio or decreasing the relax time of soft material plays a role in widening the BG and lowering the starting frequency.(4).BG properties in LRPC double panel structures are studied by applying FE method.Results show that the coupling between structure and acoustic in the cavity can be ignored.For the stubbed-on LRPC double panel structure,a BG with low starting frequency and wide band width is opened by the coupling between the dominant vibrations of the pillars and the plate modes of the upper and lower plates when the vibration source and the receiver are considered on different sides of the structure.By analyzing the displacement fields of eigenmodes in the edge bands,both symmetric and antisymmetric vibrating modes are occurred in not only the flexural vibration but also the longitudinal vibration.By comparing the band structure of the double panel and that of the single plate,the bands of the double panel can be treated as the division of the corresponding bands of the single plate by the effect of the upper and lower plates.With the help of “spring-mass” simplified model,the effects of rubber layer height,Pb layer height,pillar radius,plate thickness and lattice constant on band structure are studied and explained.Then,the BG properties of composite LRPC double panel structures with periodic holes are studied.Results show that the holes can not only widen the BG but also keep the starting frequency constant.The filled-in rubber in the plate makes it possible to control the vibration and noise in low frequency region,but reduces the width of BG.In addition,for the stubbed-on LRPC double panel structure surrounded by additional soft material shell,the additional soft material shell only affects the band in symmetric flexural vibrating mode and shifts it upward,which widens the BG.Results revel that the density of soft material shell has nothing to do with the band structure except for some flat bands.The Poisson's ratio of soft material shell has no effect on band structure,which illustrates that the additional soft material shell only affects the bands in flexural vibrating mode but not the bands in longitudinal vibrating mode.The Young's modulus of soft material shell affects the bands in flexural vibrating mode directly.With the increase of Young's modulus,the starting frequency keeps constant and the BG width increases first and then remains unchanged.