| The lattice constant of ternary locally resonant phononic crystal(PC) canbe 1～2 orders of magnitude lower than wavelength in its spectral gaps whichinduced by Bragg scattering. This characteristic can be used to control longclassical elastic wave by composite material of small dimensions. Hence it possesspotential for manufacturing of new wave guide, filters and switches. This thesiscompute the e?ect on resonance frequency, position of bandgap and transmissionspectrum of 2D and 3D locally resonant phononic crystal by changing the shapeof resonant unit. The change of resonant unit mainly includes the shape, fillingratio of three components and the cell structure for lower symmetry unit. Bystudying the units'shape and structure dependence of resonance frequency andtransmission characteristic, we obtain the way to tune the resonance frequencyand transmission spectrum of PC slab.In the first chapter, acoustics and brief deduction of sound velocity weregiven, then were generalized to the wave equation in ?uid. The concept ofelastic wave field was proposed to usher in the description of PC. The origin,progress and the current research of PC were introduced in detail. The tradi-tional noise, noise reducer and two kind of classical resonance phenomena werereviewed brie?y. Then the formalism of band gap in locally resonant PC and theprospect in noise reduce field was discussed. The orthogonal curvilinear coordi-nates transformation and the elastic wave equation in rectangular coordinates wasintroduced in Chapter two, from which the equations in cylindrical and sphericalcoordinates can be obtained. After a brief description of several kinds of bound-ary conditions in solid-?uid coupled problems, a spherical resonator example wasdiscussed to illustrate the elastic wave field computation.In chapter 3, both the least square collocation method and the Lagrange mul- tiplier method were ushered in to the computation of resonance unit with irreg-ular boundaries. Combined with the single-pole multiple expansion of displace-ment potential, the semi-analytical method was formulated. The semi-analyticalway possessed merits of both analytical method and numerical method: shortercomputation time and more general application scope, and hence is suit for prac-tical problems computation. The dynamic e?ective mass density was proposedto figure out the position and width of band gap, which can save time. Both theresonance frequency and normalized band gap width of resonance unit can beoptimized by adjusting the shape of the unit.In chapter 4, the resonance modes and the transmission spectrum of 2Dternary PC were studied. The unit cell consists of two conjugate placed asym-metric elliptic cylinders coated with silicon rubber and embedded in a rigid ma-trix. The modes are obtained by the semi-analytic method in the least squarecollocation scheme and confirmed by the finite element method simulations. Tworesonance modes, corresponding to the vibration of the cylinder along the longand short axes, give rise to resonance re?ections of elastic waves. One mode inbetween the two modes, related to the opposite vibration of the two cylindersin the unit cell in the direction along the layer, results in the total transmissionof elastic waves due to zero e?ective mass density at the frequency. The reso-nance frequency of this mode, which has not yet been identified before, changescontinuously with the orientation angle of the elliptic resonator.The multiple expansion with single pole was extended to multiple multipolemethod, so as to obtain the resonance frequency of PC unit with more generalresonator shape. Although the wave fields are expanded into a set of nonorthogo-nal basis functions, the convergence was enhanced obviously, and the sti? matrixattribute to the irregular shape can be solved more accurately. The second partof this chapter was dedicated to the mathematical basis of the finite elementmethod, and we tried to generate the two dimensional mesh for irregular shapedresonator.
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