Near-irreversibility in a conservative linear structure with singularity points in its modal density
ABSTRACT 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.
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ABSTRACT: In this paper the algebraic structure of the isotropic nth-order gradient elasticity is investigated. In the classical isotropic elasticity it is well-known that the constitu-tive relation can be broken down into two uncoupled relations between elementary part of the strain and the stress tensors (deviatoric and spherical). In this paper we demonstrate that this result can not be generalized and since 2nd-order isotropic elasticity there exist couplings between elementary parts of higher-order strain and stress tensors. Therefore, and in certain way, nth-order isotropic elasticity have the same kind of algebraic structure as anisotropic classical elasticity. This structure is investigated in the case of 2nd-order isotropic elasticity, and moduli characterizing the behavior are provided.Mathematics and Mechanics of Solids 05/2013; 20(5). DOI:10.1177/1081286513507941 · 0.86 Impact Factor
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ABSTRACT: The most general and elegant axiomatic framework on which continuum mechanics can be based starts from the Principle of Virtual Works (or Virtual Power). This Principle, which was most likely used already at the very beginning of the development of mechanics (see e.g. Benvenuto (1981), Vailati (1897), Colonnetti (1953), Russo (2003)), became after D’Alembert the main tool for an efficient formulation of physical theories. Also in continuum mechanics it has been adopted soon (see e.g. Benvenuto (1981), Salençon (1988), Germain (1973), Berdichevsky (2009), Maugin (1980), Forest (2006)). Indeed the Principle of Virtual Works becomes applicable in continuum mechanics once one recognizes that to estimate the work expended on regular virtual displacement fields of a continuous body one needs a distribution (in the sense of Schwartz). Indeed in the present paper we prove, also by using concepts from differential geometry of embedded Riemanniam manifolds, that the Representation Theorem for Distributions allows for an effective characterization of the contact actions which may arise in N-th order strain-gradient multipolar continua (as defined by Green and Rivlin (1964)), by univocally distinguishing them in actions (forces and n-th order forces) concentrated on contact surfaces, lines (edges) and points (wedges). The used approach reconsiders the results found in the pioneering papers by Green and Rivlin (1964)-(1965), Toupin (1962), Mindlin (1964)-(1965) and Casal (1961) as systematized, for second gradient models, by Paul Germain (1973). Finally, by recalling the results found in dell’Isola and Seppecher (1995)-(1997), we indicate how Euler-Cauchy approach to contact actions and the celebrated tetrahedron argument may be adapted to N-th order strain-gradient multipolar continua.
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ABSTRACT: We show that the steady state response of a primary system under harmonic excitation with an attached nonlinear energy sink (NES) exhibits not only common steady — state and weakly modulated responses, but also strongly modulated responses (SMRs), which may be regarded as the extension of the targeted energy transfer (TET) phenomenon to structures under periodic excitation. SMRs are related to relaxation oscillations of the corresponding averaged dynamical flows, and can be regarded as repetitive TET under persisting periodic forcing. The possible application of NESs as strongly nonlinear vibration absorbers is then discussed, and it is shown that the efficiency of the NESs can far exceed that of properly tuned linear absorbers (or tuned mass dampers — TMDs).01/1970: pages 53-128;