Elastic wave propagation in periodic structures

Abstract

The 'band gap' phenomenon, which is a range of frequencies or wavenumbers in which waves cannot propagate in a wave guide, has been studied for a long time in physics, especially in solid-state physics. Wave dispersion and interference properties are used to create 'band gaps' in artificially structured materials called 'photonic crystals'. In these structures, the dispersion and interference properties of electromagnetic waves are modified by the crystal's refraction index periodicity, thus not allowing light to propagate at some frequencies. This phenomenon inspired physicists to search for an equivalent for mechanical waves, called 'phononic crystals'. A 'band gap' in a phononic crystal can be caused by the 'Bragg scattering' phenomenon in a waveguide structure that has periodic properties such as density, elasticity modulus, or geometry of cross sectional area (material or geometry periodicity). In this case, the 'non-allowed frequencies' are inversely proportional to the wavenumber, which is of the same order of magnitude as the unit cell's length. As a result, the 'band gaps' usually occur at high frequencies, because a 'bang gap' at a low frequency would require a large unit cell length, which is only possible for very large structures.Another mechanism by which a mechanical 'band gap' can be created is through the periodic insertion of resonators or elastic supports in the structure (called an acoustic metamaterial), generating a 'band gap' in the frequency the resonators were designed for (local resonance) or starting at zero frequency (elastic support ). In such cases, the 'band gap' may occur at much lower frequencies, which is desirable in most vibration and noise control applications.In recent years, both Bragg scattering and local resonance phenomena have been investigated in research related to noise and vibration attenuation in structures. By using analytical techniques (such as the spectral element method) it is possible to study the elastic 'band gaps' in simple structures that allow an analytical description, such as rods, beams and plates. The study of structures with complex geometries becomes possible by a combination of these analytical techniques with the finite element method.In this project a theoretical and experimental study of both Bragg scattering and local resonance 'band gap' phenomena is proposed. For that purpose, first the analysis of elastic wave propagation in homogeneous rods and beams will be investigated using analytical and numerical models. Then, rods and beams with some kind of periodicity, so that 'band gap' phenomenon occurs, will be formulated. The spectral element method implemented in MATLAB® will be used to obtain the dispersion relations for the periodic cell and the forced response of the assembled structure. Periodic bars and beams will be manufactured using 3D printing technology (in collaboration with CTI Renato Archer) in order to investigate experimentally some of the results obtained using analytical and numerical analyses.