Minimum-variance-response and irreversible energy confinement

This paper discusses the question of the energy confinement in mechanical structures in the light of the uncertainties affecting the natural frequencies of the system. More precisely, recent studies have shown that energy can be introduced to a linear system with near irreversibility, or energy within a system can migrate to a subsystem nearly irreversibly, even in the absence of dissipation, provided that the system has a particular natural frequency distribution. In this paper the case of uncertainty in the system's natural frequency is discussed and a remarkable statistical property of the natural frequency is derived for a permanent energy confinement within a part of the system. The results demonstrate the existence of a special class of linear non-dissipative dynamic systems that exhibit nearly-irreversible energy confinement-IEC if they satisfy a minimum-variance-response-MIVAR property. In this case, if the probability density function of the natural frequencies has a special shape, the conservative system shows an unexpected decaying impulse response.