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Meccanica
Erschienen in: Meccanica 13/2017

29.08.2017 | New Trends in Dynamics and Stability

Invariant subspace reduction for linear dynamic analysis of finite-dimensional viscoelastic structures

verfasst von: Angelo Luongo, Francesco D’Annibale

Erschienen in: Meccanica | Ausgabe 13/2017

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Abstract

The linear dynamics of finite-dimensional viscoelastic structures is addressed in this paper. The equations of motion of a general, discrete or discretized, dynamical system, made of elements behaving as multiparameter viscoelastic solid models, are formulated in terms of internal variables, whose evolution is ruled by flow laws. The classical elastic-viscoelastic Principle of Correspondence is discussed in conjunction with the Fourier transform, and a new strategy, which leaves the system in the time-domain is proposed. By exploiting the fact that the spectrum of the system is well-separated if damping is small, a Center Manifold-like reduction is performed, which eliminates the internal variables, lowering the dimensions to those of the corresponding elastic system, thus filtering the fast dynamics. The order of magnitude of the error related to the reduction is investigated. A comparison with the popular Kelvin–Voigt model is performed for homogeneous structures. Examples on sample structures are worked out, namely the one degree-of-freedom viscoelastic system and the discretized elastic beam on viscoelastic soil. In all the examples the (3-Parameters) Standard Model is adopted.
Vorheriger Artikel Regular and singular kernel problems in magneto-viscoelasticity
Nächster Artikel Weakly nonlinear dynamics of taut strings traveled by a single moving force

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Fußnoten
1
For example, for the 5-Parameter Model, the internal power reads (apex j omitted):
$$\begin{aligned} \begin{aligned}\mathcal {P}_{int}:= \,&\sigma \left( \frac{\dot{\sigma }}{E_{0}}+\dot{\varepsilon }_{v1}+\dot{\varepsilon }_{v2}\right) \\ =&\frac{d}{dt}\left( \frac{\sigma ^{2}}{2E_{0}}+\frac{1}{2}E_{v1}\varepsilon _{v1}^{2}+\frac{1}{2}E_{v2}\varepsilon _{v2}^{2}\right) +\eta _{1}\dot{\varepsilon }_{v1}^{2}+\eta _{2}\dot{\varepsilon }_{v2}^{2} \end{aligned} \end{aligned}$$
in which use has been made of Eq. (2-a) differentiated with respect to the time, and of the flow laws (2-b), each solved with respect to \(\sigma \). It is, therefore, \(\mathcal {P}_{int}=\dot{\psi }+\delta \), with \(\psi \) the elastic energy of the three springs and \(\delta \) the dissipation (see, e.g., [5, 6, 40]). Positive viscous coefficients \(\eta _{k}\) assure that \(\delta \ge 0\), and, therefore, dissipation of the mechanical work.
 
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Metadaten
Titel
Invariant subspace reduction for linear dynamic analysis of finite-dimensional viscoelastic structures
verfasst von
Angelo Luongo
Francesco D’Annibale
Publikationsdatum
29.08.2017
Verlag
Springer Netherlands
Erschienen in
Meccanica / Ausgabe 13/2017
Print ISSN: 0025-6455
Elektronische ISSN: 1572-9648
DOI
https://doi.org/10.1007/s11012-017-0741-y

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