Research On The Vibration Band Gaps Of Periodic Beams And Plates Based On The Theory Of Phononic Crystals

Phononic crystalsare periodic composites with elastic waves band gaps within which the propagation of vibration and sound are forbidden. Phononic crystals are the counterparts of photonic crystals in elasticity. It becomes a hotspot topic in condensed matter physics recently. Phononic crystals exhibit rich new physics and promising potential applications, which have attracted much attention from various disciplines. Many important progresses in formation mechanism and calculation methods have been achieved in last decade. But its application is just to get under the way.Controlling vibrations in structures has been one of the pop research topics in academic and engineering for a long time. If the philosophy of Phononic crystals is introduced into their design, engineering structures will exhibit frequency band gaps and attenuate their vibration, which provides a new approach for vibration control. Under this circumstance, funded by the State Key Development Program for Basic Research of China (973 program), this dissertation addresses the vibration band gaps in periodic structures using the Phononic crystals theory. With the calculation methods of Phononic crystals and their extensions, together with finite element method and experimental tests, the vibration band gaps in periodic beams and plates, which are applied widely in engineering, are investigated deeply and systemically. The main work and achievements are as follows:1. The calculation methods of Phononic crystals are modified and improved to deal with different periodic beam or plate structures, which are validated by experiments to be powerful numerical tools for investigating vibration band gaps of periodic structures.(1) The transfer matrix method is extended to deal with the coupling effects of flexural and torsional vibration in beams. It is illustrated that the extension can calculate locally resonant band structures of shafts or beams effectively.(2) The plane wave expansion method is improved to calculate vibration band gaps in the periodic grid structure, as well as by changing the expansion form of Fourier series so that it can handle locally resonant gaps in two-dimensional plate structures with better convergence.2. The vibration gaps with Bragg scattering mechanism in periodic beams and plates are studied deeply and systemically, which enrich the elastodynamics of periodic structures. (1) Different vibration gaps are found theoretically and experimentally in the different periodic structures, such as longitudinal vibration gap in rods, torsional vibration gap in shafts, flexural vibration gap in beams, longitudinal and flexural vibration gaps in two-dimensional thin plates and grids.(2) It is found that surface localized modes are the essential reason for the transmission peaks in one-dimensional periodic rods. It is found that when the longitudinal velocity of the material with free surface is less than that of the other material, there will exist surface localized modes in the free surface of periodic rods.(3) The coupled flexural and torsional vibration gaps in periodic beams are studied deeply. Particularly, for the coupled vibration gaps in thin-walled beams, it is found that the lattice constant is an important factor that affects the normalized gap width in addition to the contrast of Young's modulus.(4) The differences between the longitudinal vibration gap in two-dimensional thin plates and the XY mode gaps in two-dimensional Phononic crystals are illustrated. The effects of rotary inertia and shear deformation on gaps in two-dimensional periodic plates are investigated and summarized.(5) It is shown that complete flexural vibration gaps exist in periodic grids. The vibration attenuation in the complete gaps is stronger than that in the directional gaps.3.The locally resonant gap mechanism of Phononic crystals is introduced in the design of periodic structures. Several small size periodic structures with different locally resonators are designed and tested. Wide gaps in low frequency with strong attenuation are observed in vibration experiments. Further theoretical and experimental studies opened out some valuable phenomena, which provide new idea for the applications of PCs in low-frequency vibration/noise control. All these related works expand the study field of periodic structures.(1) It is illustrated that low frequency torsional gap exists in theshaft with periodic cylindrical locally resonators. It is found that torsional stiffness of the shaft has important influence on the propagating attenuation in the frequency ranges of gaps.(2) The locally resonator with two-degree-of-freedom, i.e. the resonator with vertical and rotational vibration, is invented to obtain wide gaps in low frequency with strong attenuation in beams. Experimental results confirm the theoretical design reasonably. Such an idea is extremely promising for the design of locally resonant structures, as well as their applications in the vibration/noise control.(3) Theoretical and experimental results show there are locally resonant gaps of longitudinal vibration in an optimized periodic plate. But it is impossible to obtain the gaps of longitudinal vibration in low frequency with strong attenuation in such plates.(4) It is found that locally resonant gaps of flexural vibration not only exist in the two-dimensional ternary thin plates, but also in binary thin plates. But it is noteworthy that gaps in binary thin plate are much narrower than that in its ternary counterpart, and the attenuation in gaps are much less either.(5) The effects of the damping on the width of the locally resonant gap and its attenuation are investigated. It is shown that damping can widen the frequency range while bate its attenuation.In summary, the theories and calculation methods of PCs are introduced to the research of vibration gaps in periodic structures. The vibration gaps with Bragg scatter mechanism and locally resonant mechanism in periodic beams and plates are studied theoretically and experimentally. These systematic studies are valuable for periodic structure in deepening the theory, enriching the calculation methods, and expanding the study field. The research results in this dissertation are meaningful in the theory of the PCs as well as its application to vibration/noise control.