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

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.