| Mechanical vibration is a widespread phenomenon in physics and engineering, inhibition of harmful vibration has been the urgent need to address the problems in the physics and engineering. In engineering, the structures which is treated as periodic structures have mechanical vibration band gaps, i.e.. when the elastic waves propagate in the periodic structures, the vibration can not be within a certain frequency range.Periodic structure has been widely carried out in the physical realm, electronic crystals, photonic crystals, phononic crystals (PCs). PCs are the periodic structural materials with acoustic or elastic band gaps. Acoustic waves and vibrations in the frequency range of the phononic band gaps (PBGs) are not allowed through. The PBGs theory and algorithm have made important progress, but the application of exploration has just begun.In this background, we separated compression and shear waves in the elastic waves by divergence and curl operators. Based on the elastic waves separation theory of PCs, we conduct in-depth research of compression and shear waves vibrating bandgaps characteristics of the periodic structure.First, from periodic structures and photonic crystals, we introduced the PCs concept, properties, mechanism of PBGs, incomplete periodic PCs, band structures calculation method, applications test and research significance.Next, we proceed from the elastic wave equation of ideal elastic medium and separate of longitudinal and transverse waves using the divergence operator and curl operator. We have given the PCs common arrangement, the derivation of the eigenvalue equation of the plane wave expansion method of solid-solid and liquid-liquid (gas-gas) system, and several common unit structure functions and rotating structure functions. Details of the transfer matrix method in the one-dimensional PCs are described.Third, we study the two-dimensional solid-liquid and solid-solid systems arc-shaped PCs. For steel and water systems, we have studied the low-frequency band structure; the relationship of the upper edge in the first band gap and the modes; the thickness of the steel layer to affect the first PBGs. For steel and epoxy systems, we have studied the longitudinal and transverse waves low frequency complex band structures, respectively; found the localized flat bands with local frequency corresponds to the elastic parameters; the thickness of the steel layer to affect the locally resonant frequency. The results showed that the first PBGs start with zero Hz with low modes. The locally resonant gaps are obtained with higher-order rotation symmetry, for locally resonant frequencies corresponding to the speeds of acoustic waves in the background materials. These properties can be efficiently used in a structure for low frequencies that are forbidden, or in a device that permits the propagation of signals within a narrow window of frequencies.Fourth, by the plane wave expansion method based on the decomposition of elastic waves, a study on P-wave band structures in2D solid-solid PCs revealed multiple flat bands at low frequency. We considered the systems which comprising a circular scatterer in the square lattice and a rectangular scatterer in the square lattice. An analysis of the pressure field distribution of the localised modes at Γ points in the lowest bands of the systems shows that a lower scatterer symmetry are more effective in strengthening localisation. The number of localisations is reduced, and the vibration amplitudes increase. The localisation phenomena are more evident. Vibrations in the multi-channels of the localised modes are stronger. The proposed property has potential applications in the design of waveguides.Fifth, we study the shear wave band structures of a square lattice2D PCs comprising the lead cylinders embedded in the Magneto-rheological elastomers matrix. Shear wave band gaps (SBGs) are sensitive to the length of the cylinders, such that the SBGs opened up only within small ranges of crystal length, increasing rapidly and finally disappearing; the widths of the SBG frequency and the middle frequency of gaps increased linearly along with the external magnetic magnitude. And the old gaps shifted toward new gaps; when30°<θ<45°the SBGs are almost linearly related to the external rotational angle of the magnetic field. The results show that by introduction of an external contactless magnetic field which can be rotated or has a variable magnitude, the SBGs can be obtained and tuned as frequency-selective filters. These methods can be potential applications in the design of SBG-tunable devices.Finally, we study the point group symmetry of two-dimensional PCs effecting on the irreducible Brillouin zone. Additional atom in the unit cell, designing it in kinds of structures with different operation number of symmetry, we calculated eigen-frequencies of the five lowest bands in the Brillouin zone, the results show that:in the calculation of2D PCs band structure, if the PCs point group symmetry operation number is n, the irreducible Brillouin zone should be1/n of the whole first Brillouin zone.In this paper, the PCs theory and algorithms introduced to the research of periodic structure vibration bandgap characteristics, with the application of theoretical analysis and simulation combining research methods, further deepen the vibration bandgap theory of the periodic structure, especially through the elastic wave separation method, it provide new ideas and technical approaches to the low-frequency noise and vibration reduction for the periodic structure. These studies have important theoretical significance and value of engineering, promoting the engineering application of the theory of PCs in the field of noise and vibration reduction. |
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