Nonlinear Vibrations of Thin Rectangular Plates: A Numerical Investigation with Application to Wave Turbulence and Sound Synthesis

Abstract : Thin plate vibrations display a rich and complex dynamics that ranges from linear to strongly nonlinear regimes when increasing the vibration amplitude with respect to the thickness. This thesis is concerned with the development of a numerical code able to simulate without restrictions this large spectrum of dynamical features, described by the von Kármán equations, in the case of flat, homogeneous plates presenting a rectangular geometry. The main application of such a code is to produce gong-like sounds, in the context of sound synthesis by physical modelling. For that, a modal approach is used, in order to reduce the original Partial Differential Equations to a set of couped Ordinary Differential Equations. An energy-conserving, second-order accurate time integration scheme is developed in order to yield a stability condition. The most appealing features of the modal scheme are its accuracy and the possibility of implementing a rich loss mechanism by selecting an appropriate damping factor for each one of the modes. The sound produced by the numerical code is systematically compared to another numerical technique based on Finite Difference techniques. Fundamental aspects of the physics of nonlinear vibrations are also considered in the course of this work. When a plate vibrates in a weakly nonlinear regime, modal couplings produce amplitude-dependent vibrations, internal resonances, instabilities, jumps and bifurcations. The modal scheme is used to construct and analyse the nonlinear response of the plate in the vicinity of its first eigenfrequencies, both in free and forced-damped vibrations, showing as a result the effect of damping and forcing on the nonlinear normal modes of the underlying Hamiltonian system. When plates vibrate in a strongly nonlinear regime, the most appropriate description of the dynamics is given in terms of the statistical properties of the system, because of the vast number of interacting degrees-of-freedom. Theoretically, this framework is offered by the Wave Turbulence theory. Given the large amount of modes activated in such vibrations, a Finite Difference, energy-conserving code is preferred over the modal scheme. Such a scheme allows to produce a cascade of energy including thousands of modes when the plate is forced sinusoidally around one of its lowest eigenfrequencies. A statistical interpretation of the outcome of the simulation is offered, along with a comparison with experimental data and other numerical results found in the literature. In particular, the effect of the pointwise forcing as well as geometrical imperfections of the plates are analysed.
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Submitted on : Thursday, September 25, 2014 - 1:16:29 PM
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Michele Ducceschi. Nonlinear Vibrations of Thin Rectangular Plates: A Numerical Investigation with Application to Wave Turbulence and Sound Synthesis. Vibrations [physics.class-ph]. ENSTA ParisTech, 2014. English. ⟨pastel-01068284⟩

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