WAVE MODELS FOR THE FLEXURAL VIBRATIONS OF THIN PLATES - Model of the vibrations of polygonal plates by the image source method - Vibration damping using the acoustic black hole effect

Flexural vibrations of thin structures are strongly related to sound radiation and structural damage, for which they deserve careful attention in many domains of science and engineering. Two aspects that are of crucial importance are accurate tools for the prediction and analysis of vibrations, which require appropriate modelling methods and numerical tools, and efficient vibration damping. The first aim of this thesis is to develop a model for predicting the flexural vibrations of thin polygonal plates of arbitrary convex shape in the medium and high frequency ranges. The second aim is to contribute to the development and understanding of the acoustic black hole effect as an alternative technique for passive vibration damping. Those two topics are developed in two separate parts. In the first part of the dissertation, a model of the flexural vibrations of thin convex polygonal plates based on the image source method is presented. Considering a polygonal plate excited by a harmonic point source, the image source method consists in describing the successive wave reflections on the boundaries of the plate as contributions from virtual sources, obtained by successive symmetries of the original source with respect to the boundaries. First, the approach is applied to simply supported polygonal plates, which present the particularity that the reflection coefficient of the boundaries does not depend on the angle of incidence of waves. Then, the approach is generalised to the case of arbitrary boundary conditions in the case of individual plates and plate assemblies. This is achieved by representing the contributions of the different image sources as continuous sums of elementary waves. The relative weights of such elementary waves are determined by the reflection and transmission matrices of the boundaries, obtained from a state vector approach. It is shown that the method is particularly suitable for mid- and high-frequency dynamics, in that its accuracy is improved with an increase in frequency or structural damping. A tool for estimating the Young's modulus and structural damping ratio of highly damped flat panels is also proposed, using the image source method as a means of separately identifying the influence of stiffness and damping in the response. The second part of the dissertation is an investigation on vibration damping using the acoustic black hole effect. It is well known that a flexural wave travelling in a thin plate or beam slows down in a zone of decreasing thickness. Thus, if the thickness decreases sufficiently smoothly to zero, the wave velocity reaches zero and the wave stops travelling, without being reflected back. Such is the principle of the so-called acoustic black hole effect. Because of technological difficulties in achieving such a thickness profile, a thin damping layer partially covering the structure allows to compensate for non-zero reflection, leading to efficient vibration damping with a low amount of additional mass. In the present study, a model of the flexural vibrations of such profile is proposed,allowing to determine optimal geometrical and material properties for the damping layer allowing to maximise vibration damping. Experimental studies are then carried on plates whose boundaries induce a focusing of waves towards a zone treated with the thickness profile. The measured responses show a reduction of vibration level up to 20 decibels. An alternative implementation of the acoustic black hole effect is investigated, consisting in decreasing wave velocity near the edge of a beam by decreasing its Young's modulus. This is achieved by using a temperature gradient in a shape-memory polymer, leading to similar results to those obtained by a geometrical control of thickness. Finally, combining the thermoelastic and geometrical approaches leads to significant vibration damping.