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Studies on prototypical systems that consist of a set of complex attachments, coupled to a primary structure characterized by a single degree of freedom system, have shown that vibratory energy can be transported away from the primary through use of complex undamped resonators. Properties and use of these subsystems as by energy absorbers have also been proposed, particularly using attachments that consist of a large set of resonators. These ideas have been originally developed for linear systems and they provided insight into energy sharing phenomenon in large structures like ships, airplanes, and cars, where interior substructures interact with a master structure, e.g., the hull, the fuselage, or the car body. This paper examines the effects of nonlinearities that develop in the attachments, making them even more complex. Specifically, two different nonlinearities are considered: (1) Those generated by impacts that develop among the attached resonators, and (2) parametric effects produced by time-varying stiffness of the resonators. Both the impacts and the parametric effects improve the results obtained using linear oscillators in terms of inhibiting transported energy from returning to the primary structure. The results are indeed comparable with those obtained using linear oscillators but with special frequency distributions, as in the findings of some recent papers by the same authors. Numerically obtained results show how energy is confined among the attached oscillators. [ABSTRACT FROM AUTHOR]

As reported in several recent publications, an undamped simple oscillator with a complex attachment that consists of a set of undamped parallel resonators can exhibit unusual energy sharing properties. The conservative set of oscillators of the attachment can absorb nearly all the impulsive energy applied to the primary oscillator to which it is connected. The key factor in the ability of the attachment to absorb energy with near irreversibility correlates with the natural frequency distribution of the resonators within it. The reported results also show that a family of optimal frequency distributions can be determined on the basis of a variational approach, minimizing a certain functional related to the system response. The present paper establishes a link between these optimal frequency distributions and the energy equipartition principle: optimal frequency distributions are those that spread the injected energy as uniformly as possible over the degrees of freedom or over the modes of the system. Theoretical as well as numerical results presented support this point of view. [ABSTRACT FROM AUTHOR]

Pseudo-damping is a counter-intuitive phenomenon observed in a special class of linear structures that exhibit an impulse response characterized by a decaying amplitude, even in the absence of any dissipation mechanism. The conserved energy remains within but designated parts of the system. Pseudo-damping develops when the natural frequency distribution of the system includes condensation points. The recently formulated theoretical foundation of this phenomenon, based on mathematical properties of special trigonometric series, makes it possible to describe a class of mechanical systems capable of displaying pseudo-damping characteristics. They include systems with discrete oscillators and one-dimensional continuous beamlike structures already reported by the authors in recent studies. This paper examines development of pseudo-damping phenomenon in two-dimensional structures, using plates and shells as examples, and shows how a preloaded plate on an elastic foundation can lead to pseudo-damping. Moreover, in the case of curved shell elements examined here, pseudo-damping can result due to the curvature of the structure, which naturally introduces condensation points in the modal density. [ABSTRACT FROM AUTHOR]

The complex envelope displacement analysis (CEDA) is a procedure to solve high frequency vibration and vibro-acoustic problems, providing the envelope of the physical solution. CEDA is based on a variable transformation mapping the high frequency oscillations into signals of low frequency content and has been successfully applied to one-dimensional systems. However, the extension to plates and vibro-acoustic fields met serious difficulties so that a general revision of the theory was carried out, leading finally to a new method, the complex envelope vectorization (CEV). In this paper the CEV method is described, underlying merits and limits of the procedure, and a set of applications to vibration and vibro-acoustic problems of increasing complexity are presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Through two complementary approaches, using modal response and wave propagation, the analyses presented here show the conditions under which a decaying impulse response, or a nearly irreversible energy trapping, takes place in a linear conservative continuous system. The results show that the basic foundation of near-irreversibility or apparent damping rests upon the presence of singularity points in the modal density of dynamic systems or, analogously, in the wave-stopping properties associated with these singularities. To illustrate the concept of apparent damping in detail, a simple undamped beam is modified to introduce a singularity point in its modal density distribution. Simulations show that a suitable application of a compressive axial force to an undamped beam placed on an elastic foundation attenuates its impulse response with time and develops the characteristics of a nearly irreversible energy trap. [ABSTRACT FROM AUTHOR]

This paper describes a new concept referred to here as “energy sinks” as an alternative to conventional methods of vibration absorption and damping. A prototypical energy sink envisioned here consists of a set of oscillators attached to, or an integral part of, a vibrating structure. The oscillators that make up an energy sink absorb vibratory energy from a structure and retain it in their phase space. In principle, energy sinks do not dissipate vibratory energy as heat in the classical sense. The absorbed energy remains in an energy sink permanently (or for sufficiently long durations) so that the flow of energy from the primary structure appears to it as damping. This paper demonstrates that a set of linear oscillators can collectively absorb and retain vibratory energy with near irreversibility when they have a particular distribution of natural frequencies. The approach to obtain such a frequency distribution is based on an optimization that minimizes the energy retained by the structure as a function of frequency distribution of the oscillators in the set. The paper offers verification of such optimal frequency spectra with numerical simulations and physical demonstrations. [ABSTRACT FROM AUTHOR]

This paper presents experiments that demonstrate the concept of energy sinks, which when attached to a vibrating structure can absorb most of its energy. Energy sinks consist of a set of undamped linear oscillators and, in principle, do not require presence of damping in the classical sense. The set of undamped oscillators that make up an energy sink collectively absorb the vibratory energy and retain it in their phase space. Earlier optimization studies by the authors have shown the feasibility of vibration absorption and energy retention by energy sinks if the set of oscillators have a particular frequency distribution. Experimental results presented in this paper support the concept of energy sinks. Different physical realizations of energy sinks demonstrate the significance of frequency distributions and the ability of energy sinks to reduce vibration amplitude of a primary structure to which they are attached. [ABSTRACT FROM AUTHOR]

In this paper we examine the conditions that influence the return time, the time it takes before energy returns from a set of satellite oscillators attached to a primary structure. Two methods are presented to estimate the return time. One estimate is based on an analysis of the reaction force on a rigid base by a finite number of oscillators as compared with an infinite number of continuously distributed oscillators. The result gives a lower-bound estimate for the return time. A more accurate estimation results from considering the dynamic behavior of a set of oscillators as waves in a waveguide. Such an analogy explains energy flow between a primary structure and the oscillators in terms of pseudowaves and shows that a nonlinear frequency distribution of the oscillators leads to pseudodispersive waves. The resulting approximate expressions show the influence of the natural frequency distribution within the set of oscillators, and of their number, on the return time as compared with the asymptotic case of a continuous set with infinite oscillators. In the paper we also introduce a new method based on a Hilbert envelope to estimate the apparent damping loss factor of the primary structure during the return time considering transient energy flow from the primary structure before any energy reflects back from the attached oscillators. The expressions developed for return time and damping factor show close agreement with direct numerical simulations. The paper concludes with a discussion of the return time and its relation to apparent damping and optimum frequency distribution within a set of oscillators that maximize these quantities. [ABSTRACT FROM AUTHOR]

The complex envelope displacement analysis (CEDA) is a procedure to solve high frequency vibration and vibro-acoustic problems, providing the envelope of the physical solution. CEDA is based on a variable transformation mapping the high frequency oscillations into signals of low frequency content and has been successfully applied to one-dimensional systems. However, the extension to plates and vibro-acoustic fields met serious difficulties so that a general revision of the theory was carried out, leading finally to a new method, the complex envelope vectorization (CEV). In this paper the CEV method is described, underlying merits and limits of the procedure, and a set of applications to vibration and vibro-acoustic problems of increasing complexity are presented.