Theoretical foundations of apparent-damping phenomena and nearly irreversible energy exchange in linear conservative systems.

This paper discusses a class of unexpected irreversible phenomena that can develop in linear conservative systems and provides a theoretical foundation that explains the underlying principles. 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. The present work introduces a general theory that provides a mathematical foundation and a physical explanation for the near irreversibility phenomena observed and reported in previous publications. Inspired by the properties of probability distribution functions, the general formulation developed here is based on particular properties of harmonic series, which form the common basis of linear dynamic system models. The results demonstrate the existence of a special class of linear nondissipative dynamic systems that exhibit nearly irreversible energy exchange and possess a decaying impulse response. In addition to uncovering a new class of dynamic system properties, the results have far-reaching implications in engineering applications where classical vibration damping or absorption techniques may not be effective. Furthermore, the results also support the notion of nearly irreversible energy transfer in conservative linear systems, which until now has been a concept associated exclusively with nonlinear systems.

[1]  Richard L. Weaver,et al.  Equipartition and mean-square responses in large undamped structures , 2001 .

[2]  Richard L. Weaver The effect of an undamped finite degree of freedom ‘‘fuzzy’’ substructure: Numerical solutions and theoretical discussion , 1996 .

[3]  Adnan Akay,et al.  Dissipation in solids: Thermal oscillations of atoms , 1999 .

[4]  B. O. Koopman On distributions admitting a sufficient statistic , 1936 .

[5]  T. Petrosky Chaos and irreversibility in a conservative nonlinear dynamical system with a few degrees of freedom , 1984 .

[6]  Raymond J. Nagem,et al.  VIBRATION DAMPING BY A CONTINUOUS DISTRIBUTION OF UNDAMPED OSCILLATORS , 1997 .

[7]  M. Strasberg,et al.  Vibration damping of large structures induced by attached small resonant structures , 1993 .

[8]  Antonio Carcaterra,et al.  Energy sinks: Vibration absorption by an optimal set of undamped oscillators , 2005 .

[9]  Victor W. Sparrow,et al.  Fundamental Structural-Acoustic Idealizations for Structures with Fuzzy Internals , 1995 .

[10]  A Carcaterra,et al.  Near-irreversibility in a conservative linear structure with singularity points in its modal density. , 2006, The Journal of the Acoustical Society of America.

[11]  Antonio Carcaterra,et al.  Transient energy exchange between a primary structure and a set of oscillators: return time and apparent damping. , 2004, The Journal of the Acoustical Society of America.

[12]  Datta,et al.  Energy transport in one-dimensional harmonic chains. , 1995, Physical review. B, Condensed matter.

[13]  Antonio Carcaterra,et al.  An Entropy Formulation for the Analysis of Energy Flow Between Mechanical Resonators , 2002 .

[14]  Antonio Carcaterra,et al.  Experiments on vibration absorption using energy sinks , 2005 .

[15]  Vulpiani,et al.  Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model. , 1985, Physical review. A, General physics.

[16]  Antonio Carcaterra,et al.  Ensemble energy average and energy flow relationships for nonstationary vibrating systems , 2005 .

[17]  E. Pitman,et al.  Sufficient statistics and intrinsic accuracy , 1936, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  G. Maidanik INDUCED DAMPING BY A NEARLY CONTINUOUS DISTRIBUTION OF NEARLY UNDAMPED OSCILLATORS: LINEAR ANALYSIS , 2001 .