Parametric Instabilities in Spatially Continuous Systems: New Approaches and Phenomena

La missione è stata regolarmente autorizzata dal Consiglio di Facoltà di Ingegneria nel mese di dicembre 2004 o nel gennaio 2005 anche se non è possibile al momento fornire la data precisa.

Parametrically forced structures and systems, governed by Mathieu-Hill's equations, are pervasive in mechanics (e.g., dynamic buckling of columns, rings and shells, stability of general motions, water waves in vertically forced containers) as well as in various areas of physics (propagation of electromagnetic waves in media with a periodic structure, motions of electrons in a crystal lattice). Although a vast amount of literature has been devoted to the theory of parametrically excited linear discrete systems, only a few works have treated parametrically excited nonlinear structural systems taking into account inertia, geometric and material nonlinearities. The overall objective of the project is to provide rigorous and sound analyses of the large motions of nonlinearly elastic and viscoelastic structures (mostly, rings and shells) subject to parametric excitations and to investigate new phenomena associated with the onset of the parametric dynamic instability. Further, the interesting study of the effect of high-frequency parametric excitations on the motions is expected to shed light onto new aspects of the nonlinear behavior of important structural members subject to fast parametric forcing. The objective of this project is to give illuminating and rigorous analyses of the large motions of nonlinearly elastic and viscoelastic structures subject to parametric excitation. The first problem to be treated is that of the planar motion of a nonlinearly elastic and viscoelastic circular ring subject to a time-periodic pressure. The ring can suffer flexure, longitudinal and transverse extension, and shear (Antman, 2005). A preliminary analysis shows that even the trivial problem for the unbuckled ring has a rich behavior depending critically on the constitutive properties. Further problems to be studies are spatial motions of straight rods subject to periodic terminal thrusts and torques, and spherical/cylindrical shells, subject to periodic pressure. For certain ranges of data, such problems seem amenable to the method of multiple scales, which is equivalent to the method of averaging. A variety of global methods from Antman and Seidman (2005) are availbale to treatd data in other regimes. Instabilities including finite-time blowup will be correlated with material response with the aim of discovering new phenomena. From a practical point of view, parametrically forced rings and shells are important in engineering applications such as aircraft fuselages and turbo machineries. Hence, accurate predictions based on refined theories for the onset of parametric instabilities and for the behavior of post-critical solutions exhibiting new phenomena would be both scientifically and technologically valuable.