Antonio Carcaterra

Transient energy exchange between a primary structure and a set of oscillators: Return time and apparent damping

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.

Energy sinks: Vibration absorption by an optimal set of undamped oscillators

This presentation offers the concept of 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 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 response 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.

Near-irreversibility in a conservative linear structure with singularity points in its modal density

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.

Energy equipartition and frequency distribution in complex attachments

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.

Vibration absorption using non-dissipative complex attachments with impacts and parametric stiffness

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.

Experiments on vibration absorption using energy sinks

This presentation describes experiments that demonstrate the concept of energy sinks where a set of multiple undamped linear oscillators attached to a vibrating structure can absorb most of its energy. In principle, energy sinks do not require presence of damping in the classical sense. A 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 retention by energy sinks if the set of oscillators have a particular frequency distribution. Experimental results 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.

Prediction of the compressible stage slamming force on rigid and elastic systems impacting on the water surface

In this paper some models focusing on hydrodynamic and elasticforces arising during the impact of rigid and elastic systems on thewater surface are investigated. In particular, the supersoniccompressible stage of the impact is considered by modelling the slammingphenomenon through the Skalk–Feit acoustic approximation. The dynamicequations of the dropping system are coupled to those of the fluid and anonlinear fluid-solid interaction problem is stated. Generalrelationships between the bodys shape, slamming force and body motionare determined. These equations are applied to the wedge water entrycases, and a closed-form expression for the maximum hydrodynamic forceis found. Moreover the theoretical correlation between the hydrodynamicforce and the body geometry allows us to control the inverse problem andthe shape associated to a constant slamming force is determined.Due to some simplifications allowed in the supersonic compressibleimpact, the results of the hydrodynamic analysis hold in closed form.This permits us to focus on the basic result of the paper addressed to asystematic correlation between hydrodynamic and elastic maximum forcesin terms of some characteristic dimensionless quantities involved influid-solid interaction.In particular, ‘critical conditions’corresponding to those hydorelastic parameters combinations areinvestigated, leading to severe elastic response of the impactingsystem.

Space average and wave interference in vibration conductivity

Plane wave interference and space averages play a significant role in the derivation of some vibration conductivity equations that are becoming more and more popular in modelling vibroacoustic problems. Particularly, the thermal approach and the modified vibration conductivity equations are here considered with the aim of establishing similarities and/or differences between them and stating relevant consequences. It is shown by formal developments that the thermal equation is obtained under the assumption of performing appropriate space averages, while the modified vibration conductivity equation does not need, in some cases, this condition. It is discussed, however, that in practical applications the conditions of validity of both approaches are quite similar.

Fractional dissipation generated by hidden wave-fields

This paper considers the damping induced on a single degree-of-freedom system when it is coupled at one end of a waveguide in which waves are radiated producing an energy loss in the oscillator motion that appears as a damping effect. In general, the whole system is described by the equation of the motion of the harmonic oscillator coupled with the wave equation of the propagation field. Hiding the variable that describes the wave motion and expressing it in terms of the oscillator vibration, a new equation for the oscillator is determined. In general, a nonconventional damping effect is born from the coupling terms. This paper examines cases in which the induced damping effect is of fractional-derivative type. This point of view produces physical examples of the way that simple mechanical structural systems, familiar to engineers, can exhibit fractional damping, a concept that does not always have a clear physical interpretation.

High frequency vibration analysis by the complex envelope vectorization

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.