Study On Low Frequency Band Gap Characteristics And Control Mechanism Of Two-dimensional Local Resonant Phononic Crystals

Phononic crystals are composed of materials with periodically distributed elastic constants and densities.This periodic structure can generate elastic wave band gaps,which suppress the propagation of elastic waves at certain frequencies.Due to the significant application value of this feature,this structure has received extensive attention and research.The mechanism of band gap formation was first proposed as the Bragg scattering mechanism.Previous studies have shown that the Bragg scattering type phononic crystals are difficult to obtain lower band gaps in small sizes.In order to overcome this shortcoming,researchers have explored and obtained local resonant type phononic crystals,which can achieve "small size control and large wavelength".In order to better describe the vibration modes at the lower and upper boundaries of the first band gap of the local resonant phononic crystal,an equivalent mass spring system and corresponding formulas have been proposed,from which it can be seen that reducing the equivalent stiffness and increasing the equivalent mass can obtain a low-frequency band gap.For local resonant phononic crystals,the following researches have been done in this paper:Firstly,this paper considers reducing the equivalent stiffness of the equivalent mass spring system to obtain a low frequency band gap,and designs an ideal two-dimensional connected plate type phononic crystal.The structure is to wrap a layer of annular silicone rubber around the lead cylindrical scattering body,with 8 evenly distributed through-holes opened in the silicone rubber,and embed the scattering body and silicone rubber into four epoxy resin connecting plates.The dispersion curve and corresponding vibration modes of the structure were calculated using the finite element method.The results show that the structure has a band gap at 29.37～354.07 Hz.Compared with phononic crystal structures without connecting plates,they have lower initial frequencies and wider band gaps,indicating that connecting plate structures are easier to obtain low-frequency band gaps.By analyzing the vibration modes of the displacement field and combining the mass spring model,the reason for the formation of the band gap is explained.Based on this,the effects of the radius of the lead cylindrical scatterer,the radius of the silicone rubber coating layer,the radius of the silicone rubber through-hole,and the width of the connecting plate on the band gap are discussed.Secondly,this paper considers increasing the equivalent mass in the equivalent mass spring system to obtain a low frequency band gap,and designs a new multivibrator phononic crystal structure with four tungsten oscillators wrapped in silicone rubber.The dispersion curve,vibration mode,and transmission loss spectrum of the structure are calculated by finite element method.The results show that the band gap range of the structure is 18.85～225.28 Hz,which coincides with the frequency attenuation range of the transmission loss spectrum.Compared with traditional structural analysis of multivibrator phononic crystals,the effects of the longitudinal and transverse spacing between oscillators and the gap angle of the phononic crystal plate on the band gap are mainly discussed.The research shows that when the transverse or longitudinal distance between the oscillators increases,the lower and upper boundaries of the band gap both rise,and the rise amplitude of the upper boundary is larger than the lower boundary,resulting in an increase in the width of the band gap,thereby optimizing the band gap of the phononic crystal structure;When the gap angle of the phononic crystal plate decreases,the lower boundary of the band gap remains almost unchanged,while the increase in the upper boundary of the band gap increases the width of the band gap.In addition,the notch design of the phononic crystal plate can also save material and reduce the weight of the structure.Finally,piezoelectric materials are introduced into phononic crystals to actively regulate the band gap using the coupling effect of acoustic and electromagnetic fields in piezoelectric materials.In this paper,based on the multivibrator phononic crystal plate,the scattering body and coating material are respectively changed into piezoelectric material PIN-PMN-PT and plexiglass to form a piezoelectric phononic crystal plate.By applying short circuit and open circuit electrical boundary conditions to a piezoelectric phononic crystal plate,the effects of electrical boundary conditions and piezoelectric constants of piezoelectric materials on the band gap are discussed.Based on a piezoelectric phononic crystal plate with open circuit electrical boundary conditions,the 1× 5 supercell is constructed in the x and y directions.Subsequently,the scattering body at the middle position of the supercell was changed to a short circuit boundary condition to introduce point defects,and the influence of different point defect locations on the band gap was further discussed.