Bandgap Properties And Design Of Phononic Crystals With Resonant Units

Phononic crystals (PCs) are a kind of acoustic functional composite materials/structures which have some form of spatial periodicity and exhibit elastic/acoustic bandgaps. Generally, there are two mechanims of the bandgap generation:Bragg scattering and local resonance. The wavelength of the wave in the bandgap is in the same order as the lattice constant. A relatively big lattice constant is needed to generate a Bragg bandgap at a low frequency, which is much diffcult for the pratical applications. While for a locally resonant PC, a small lattice constant can generate a bandgap corresponding to a large wavelength. So they have essential application prospects in reduction of vibration and noise at low frequencies. In this thesis, with repect to the scientific problems related to the mechanism of the generation of the locally resonant bandgap and the reduction of low-frequency vibration and noise, bandgap properties of the PCs with resonant units and its design are studied deeply and systemically via the combination of numerical simulation, theoretical model analysis and experimental investigation. The main contents and conclusions include:(1) A two-and three-dimensional locally resonant PC or PC plate with various resonant units is designed by drilling different holes in a homonegeous solid. Effects of the geometry parameters of the hole’s cross-sections are discussed. Combined with the analysis of the vibration modes at the bandgap edges, the mechanism of the bandgap generation are explained, and equivalent spring-mass/pendulum models are developed to predict the bandgap edges. The numerical results show that PCs and PC plates with resonat units are easy to generate wide complete bandgaps (but at relatively high frequencies), and the bandgaps are induced by the local resonance of the resonant unit. The bandgap properties can be tuned by changing the geometry parameters of the resonant unit. The results predicted by the equivalent models are generally in good agreement with the numerical ones.(2) Zigzag honeycomb PCs and grid PCs are proposed by introducing the bending slender connector. Effects of the geometry parameters on the bandgp and passing band properties are discussed. Combined with the iso-frequency contours and the transient displacement distribution, the directional properties of elastic wave propagation are investigated. The results show that no complete bandgap is found for honeycomb PCs and grid PCs with only straight arms; and by introducing zigzag arms, the degeneracy of some bands are seperated and give rise to multiple bandgaps. The symmetry of the system has a strong influence on the bandgap properties. Generally speaking, symmetrical systems have wide bandgaps at relatively high frequencies, while asymmetrical systems can generate bandgaps at low frequencies but with narrow widths. The bandgap edges are very senstive to the structural parameters, and wide complete bandgaps at low frequencies can be obtained by changing the structural parameters (including the geometry parameters, bending distance, bending angle, and bending types). Directional propagation of ealstic waves in different directions and at different frequencies can be relaized by a careful design of zigzag honeycomb PC. The symmetry of the excitation sources has effects on the properties of the directional wave propagation. Different excitation positions can affect the directional propagation pattern, but have no influence on the propagation directions.(3) Two-dimensional ternary locally resonant PC is proposed by introducing a comblike coating. Effects of the geometry parameters on the bandgaps are discussed. The mechanism of the bandgap generation is investigated by analyzing the vibration modes at the band edges, and the equivalent spring-mass/pendulum models are developed to predict the eigenfrequencies of the band edges. The effective acoustic properties of the system with a chiral comblike coating are also studied. The results show that a complete bandgap at a relatively lower frequency can be obtained by introducing the comblike coating (together with the decrease of the bandgap). The bandgap properties, as well as the effective acoustic properties, can be tuned by changing the geometry parameters of the comblike coating. The band edges of the systems can be well predicted by the equivalent models.(4) Numerical method for the calculation of the complex band structures are developed by using the PDE module of COMSOL. The physical meaning of the complex bands is explained by analyzing the vibration modes; and effects of the viscoelasticity on the complex band structures of two-dimensional ternary locally resonant PC are studied. To investigate the coupling effects of the evanescent waves and propagating waves, one-dimensional locally resonant PCs made of acoustic resonators grafted onto a waveguide are frabricated to analyze the difference of the complex bands between the Bragg and locally resonant bandgaps, and to discuss the dependence of the locally resonant bandgaps on the length of the resonators and the lattice constants; explicit expression of the complex bands is also developed. The results show that the diffraction orders of the complex bands for the mixed modes are determined by the distribution of the dominant displacement component, while that of the other component is one-order higher. Viscoelasticity has a strong influence on the complex bands. The sharp corners at the high symmetry points of the corresponding elastic system without viscosity become rounded, and the attenuation in the passing band is strengthed. The degenercy of the complex bands are separated, and the attenuation at the transmission dips inside the bandgap are decreased to some extent. The smallest imaginary part of the complex band for the Bragg bandgap is continous and smooth, while that for the locally resonant bandgap is composed of two crossing bands forming a cusp. The attenuation is stronger inside the locally resonant bandgap than in the Bragg bandgap. Different resonators correspond to different locally resonant bandgaps. For a big lattice constant, a periodic array of the resonators can increase the attenuation effects; and the complex band structures obtained by the explicit expression are in excellent agreement with the FEM results. While in the subwavelength case, the periodic effect is not so clear, and may also result in the disappearence of the bandgap; and the complex band structures obtained by the explicit expression disagree with the FEM results to some extent, because the interaction between resonators is not considered in the explicit expression.Finally, the present analysis shows that PCs of the same type generally have the commom topological characteristics to exhibit low and wide bandgaps and that bandgap properties can be tuned by changing the structural parameters for a PC with particular topology. This is relevant to the bandgap engineering. For example, a PC formed by etching holes in an homogeneous solid, which exhibit wide complete bandgaps at relatively low frequencies, should be so designed that the system consist of resonant units with large lumps and narrow connectors. For a PC with a high porosity, large lumps are not easily formed. In this case, multiple directional bandgaps or even complete bandgaps can be found by introducing bending slender connectors. A complete bandgap at a relatively lower frequency can be obtained by designing a comblike coating in a conventional two-dimensional ternary locally resonant PC. Moreover, our study also shows that complex band structures can manifest more comprehensive properties of the wave propagation, which is helpful to the analysis of the bandgap properties of the viscoelastic PCs, as well as the mechanism of the coupling between evanescent waves generated by the local resonance and propagating waves.