Research On Periodic Structures With Low-frequency And Tunable Bandgaps

A periodic structure or phononic crystal is a material or structural system that usually exhibits some form of periodicity,which can be in the constituent material phases,or the internal geometry,or even the boundary conditions.Wave propagation of elastic or acoustic waves in periodic structures is prohibited or attenuated in certain frequency ranges,which are termed as bandgaps.The dispersive properties of periodic structures implies a promising application in micro-vibration reduction of motional platform from ultraprecision manufacturing equipment.In comparison with frequencies of vibration needed to be reduced in the motional platform,the lowest frequency of bandgap(BG)of periodic structures is relatively higher.And the adaption of bandgaps is also needed to be enhanced.As a result,the biggest challenge of application of periodic structures lays in the decrease of BGs frequency and the enhancement of adaptation.Inspired by this unsolved key problem,this dissertation at one hand explores the tuning mechanism of shunting property to the properties of BGs resulting from the coupling between piezoelectric material and shunting circuit,by taking advantages of the integration of attenuating ability of locally resonant periodic structures and the excellent dissipative property of piezoelectric materials;on the other hand,based on research into the impact of nonlinearity on performance enhancement and transducing bandwidth extension of a nonlinear transducing structure,it investigates the influential mechanism of nonlinearity on dispersive properties of nonlinear periodic systems.The major contributions of the dissertation are as follows:(1)A model of piezoelectric bimorph circular metallic plate connecting to equal frequency shunting circuits is proposed,frequency BGs with large attenuation factors are obtained.The findings shows that there is no Bragg BG in the low-frequency range,but exist some vibration attenuation bands(VABs)with small attenuation factor.A equal frequency shunting circuit resulted from tuning the resonance frequency of each circuit into the same are designed to obtain an integrating locally resonant(LR)BG.The properties of the LR BG and VABs can be significantly affected by the geometric and circuit parameters.The frequency of LR BG decreases,while the width becomes narrower,with the increasing of the inductance.The transmission factor with the LR BG can be reduced dramatically,but the frequency keeps fixed when resistance decreases.For LR BG,there exist optimum resistances to obtain a small transmission factor corresponding to its central frequency,meanwhile that transmission factors on its both sides can be not large.Attenuation in VABs is strengthened and the frequency of these gaps decrease when the concentrated mass increases.Both frequencies of the LR BG and VABs decrease with increasing material ratio.(2)A periodic structure model of circular plate which is attached alternately by annular piezoelectric unimorphs connecting to resonant shunt circuits is proposed for vibration attenuation.Two kinds of equal frequency resonant shunt circuits are designed to achieve LR circuits.A semi-analytical method is utilized to obtain general solutions to motional equation governing portions of the structure attached piezoelectric patches.The findings shows that there is no Bragg BG in the low-frequency range,but there exist some vibration VABs with small attenuation factor.With the application of equal frequency circuit in LR unit cell,an integrated LR BG with strong attenuation is obtained.Influence of variation in geometric parameters,boundary conditions and electric parameters on BGs are revealed.The larger the base inductance of equal frequency circuit is,the lower the frequency of the LR BG is,the narrower the width is.The transmission factor of the LR BG reduces dramatically,but the frequency remains unchanged when resistance decreases.For the LR BG,there exist optimum resistances to obtain a small transmission factor on its central frequency,meanwhile that transmission factors on its both sides can be not large.Attenuation in VABs is strengthened and the frequency of these gaps decrease when the concentrated mass increases.Both frequencies of the LR BG and VABs decrease with increasing material ratio.(3)The performance of a pinned-pinned nonlinear power-transducing cell near the resonant frequency is analyzed.The basic nonlinear behavior of the system due to large deformation has been predicted.The dependence of the multi-value property and jump phenomena of the output power upon applied pressure,electric load and structural parameters was studied.Two methods to broaden the operation bandwidth are proposed: The first method is to expand the operation frequency of the left single-value region through optimal design including an increase in applied sound pressure,an increase in spanthickness ratio and an decrease in thickness ratio.The second method is to excite optimal initial conditions with a controlling voltage source to extend the operation bandwidth from the left single-value region to the whole multi-value region.And the required optimal voltage amplitude,region of frequency and corresponding phase are obtained by a deeply analysis of the system's basin of attraction.(4)Nonlinear LR unit cells are introduced to configure a nonlinear periodic structure.Finite element discretization based perturbation approach is utilized to predict the dispersion relationship of the nonlinear periodic system.The dependence of dispersive shifts on the magnitude of nonlinearity and amplitude has found.The resulting findings demonstrates that for a hardening system,the dispersive curves shift upward,while for a softening system,the curves shifts downward.The shifts of both acoustic branch curve and the second optical branch curve occur near the X,while the shift of the first optical curve occurs near the(38)point.As for a harden system,both the width and central frequency of BGs are increased with an increasing amplitude.In this dissertation periodic structures or phononic crystals containing linear/nonlinear LR unit cells are studied to investigate the impacting mechanisms of parameters,including structural parameters,material parameters,electrical parameters of piezoelectric shuntings and nonlinear parameters,on their dispersive properties such as BG width,frequency ranges and attenuation.A series of significant findings in the region of tunable low-frequency bandgaps has been discovered,which contribute a lot to the design and application of periodic structures in low-frequency situations.