An investigation on energy dynamics of complex resonators

In the last decades the design process of engineering structures was characterized by an increasing demanding of the prediction capabilities at the design stage, which would have allowed to asses not only the performance and the integrity of the structure but also the vibro-acoustic performances. The more realistic is the model, the greater the opportunity to produce optimal design; this implies the need of using a model with a very high number of degrees of freedom. Predicting the response of a complex structural-acoustic system presents some difficulties. In fact, direct numerical solution of the governing equations of the system, while possible in principle, can be so computationally demanding as to be impractical. Furthermore, the application of deterministic methods of analysis to the prediction of its vibrational response would be not appropriate. In fact, in a complex system the presence of a high number of heterogeneous subsystems (beams, plates, acoustic cavities, etc.) and, above all, the high number of joints connecting the components are sources of a certain degree of variability of the system. However, geometric and fabrication tolerances, variation in material properties, structural irregularities, non uniform damping distribution are some of the cause of variability that lead to different dynamical behaviours of samples of a set of nominally identical manufactures. Moreover, many important engineering problems involve high frequency vibrations. In fact, especially in recent years, the use of light structures in aircraft and aerospace engineering, broad band excitations due to engines of increasing power and the increasing interest for the high speeds in ship, automotive and train engineering, created a general attention in the analysis of the high frequency vibration and of the acoustic problems. Approaching the high frequency problem both the difficulties above presented meet two serious limitations. In fact, high frequency problems involve vibrations characterized by short wavelengths compared with a typical length of the system. In conventional vibrational analysis based on a discretization of the continuum domains into small elements, being the size of these elements dependent on the minimum wavelength of interest, the computational time demanding could be so high that it would become prohibitive. Besides the problem of high time demanding, which could be in principle overcome with the growth of the computer power, the use of a very fine mesh also implies an accurate modelling of the system details. Obviously, such a modelling is difficult to obtain and it also introduces a certain degree of variability. On the other hand, at high frequency the response of the system becomes increasingly sensitive to small perturbations of its parameters: even small variations in geometrical component dimensions, material properties and assembly tolerances imply large variations in the mid- and high- frequency responses. In fact, while low order eigenvalues of the system are lightly affected by small variation of the system parameters, the more the order of the eigenvalues increases the more their values are modified, such that the behaviour of the structure becomes unpredictable. These considerations led to introduce a different way of tackling dynamic problems, based on a thermodynamic analogy. It is in fact in that frame that the problem related with system characterized by a high number of degrees of freedom (atoms) was studied for the first time. The necessity to reduce the computational cost has led to introduce global parameter as descriptor of the systems, rather than the local characterization, i.e. the punctual displacement. Further, to overcome the second problem, i.e. the inherent uncertainties of the system paramete