## Doctoral thesis defence of Jacques Cuenca

**Doctoral thesis defence of Jacques Cuenca**

Tuesday 20th October 2009 at 2 P.M. Salle de conférences, LAUM

For obtaining the degree of Doctor of Philosophy

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

Submitted to the examining committee:

W. Desmet Professor, Dept. of Mechanical Engineering, K.U. (Louvain) Reviewer

B.R. Mace Professor, ISVR (Southampton) Reviewer

J.R.F Arruda Professor, Faculdade de Engenharia Mecânica, Unicamp (Campinas) Examinor

E. Foltête Professor, FEMTO-ST (Besançon) Examinor

V.V. Krylov Professor, Dept. of Aeronautical and Automotive Engg. (Loughborough) Examinor

V. Martin CNRS Research Director, IJLRDA (Paris) Examinor

C. Pezerat Professor, LAUM (Le Mans) Examinor

F. Gautier Professor, LAUM (Le Mans) Supervisor

L. Simon Professor, LAUM (Le Mans) Co-supervisor

Abstract:

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 of crucial importance are the development of accurate tools for the prediction and analysis of vibrations and eﬃcient vibration damping.

In the ﬁrst part of the thesis, 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. The developed approach allows to predict the vibrations of individual plates and plate assemblies of arbitrary convex polygonal geometry and having arbitrary boundary conditions. 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.

The second part of the thesis concerns vibration damping using the acoustic black hole eﬀect. It is weel-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 stops travelling, without being reﬂected back. Such is the principle of the so-called acoustic black hole eﬀect. A model of the ﬂexural vibrations of such proﬁle is proposed, allowing to determine optimal geometrical and material properties in order to maximise vibration damping. Simulated and 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, by using a shape-memory polymer subjected to a thermal load, which leads to similar vibration reduction levels. Finally, combining the thermoelastic and geometrical approaches leads to signiﬁcant vibration damping.