## Abstract

In the present paper, a linear model for multi-degree-of-freedom systems with rate-independent damping is proposed to the purposes of dynamic response prediction and identification. A viscoelastic model with memory, equivalent to the ideal hysteretic model as for the energy dissipation properties, but causal and physically consistent in both the time and the frequency domain, is developed by adopting the Maxwell–Wiechert kernel function and by requiring the loss modulus to be substantially independent of frequency in a specified range of interest. The finite element model of the equivalent viscoelastic system is constructed and its equations of motion are shown to be uncoupled, in terms of modal coordinates, by the real-valued eigenvectors of the conservative system. An augmented state-space formulation, which encompasses, besides the customary displacements and velocites, a number of internal variables devoted to represent the viscoelastic memory, is then provided for the sake of system identification. Mechanical and modal properties of the equivalent viscoelastic model are finally illustrated by means of numerical examples.

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## Acknowledgments

Anna Reggio is beneficiary of an AXA Research Post-Doctoral Grant, whose generous support is gratefully acknowledged. Maurizio De Angelis thanks the Italian Ministry of Education, University and Research (PRIN Grant 2010MBJK5B).

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## Appendix

### Appendix

It will be shown that, if a \(N\)-DoF proportionally damped viscoelastic system is modelled by adopting the Maxwell-Wiechert kernel function matrix (20), then (24) is necessary and sufficient condition for (25).

### 1.1 Sufficiency

The sufficiency of (24) is demonstrated by considering that, as a result of this assumption, the equations of motion of the \(N\)-DoF system can be uncoupled into a set of \(N\) modal equations of the kind (32). Complex stiffness in the \(j\)-th normal mode is therefore

in the Laplace domain and

in the frequency domain. The \(j\)-th modal loss factor \(\eta _j^*(\omega )\) is computed as the ratio between the imaginary and the real part of \(K_j^*(\omega )\)

By dropping the modal stiffness \(k_{0j}^*\), it is obtained

The variation of each \(\eta _j^*\) versus frequency \(\omega \) is hence described by the same function \(\eta ^*(\omega )\).

### 1.2 Necessity

The general representation of the matrix of kernel functions \(\mathbf{G}(s)\) for a \(N\)-DoF proportionally damped viscoelastic system is given by

where \(\chi _h(s)\) is a complex-valued scalar function of the Laplace variable \(s\). This expression is equivalent, in the Laplace domain, to those reported by Inaudi and Kelly [28] in the frequency domain and by Palmeri and Muscolino [29] in the time domain. It follows the complex stiffness matrix

Accounting for the relationship

that results from the orthogonality properties (30), the complex stiffness in the \(j\)-th normal mode is determined as

in the Laplace domain, and subsequently as

in the frequency domain. Taking the ratio between the imaginary and the real part of \(K_j^*(\omega )\) yields the \(j\)-th modal loss factor

In order for \(\eta _j^*(\omega )\) to be independent of modal frequencies \(\omega _{0j}\), it must hold

from which it follows

where \(\eta^*(\omega)\) is a function of frequency \(\omega \) independent of modal quantities.

By substituting into (51) the expression (22) of \({\mathbf{G}}(s)\) prescribed by the Maxwell-Wiechert model and taking into account (57) yields

By writing \(\chi _1(s) = \sum _{l=1}^n \chi _{1l}(s)\) in (59), it follows

from which it is obtained

as was to be shown. In (61), \(\beta _l = (s + \mu _l)\chi _{1l}(s) = {\text{cost}}\) since matrices \({\mathbf{C}}_l\) and \({\mathbf{K}}_0\) are constant with Laplace variable \(s\).

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Reggio, A., De Angelis, M. Modelling and identification of structures with rate-independent linear damping.
*Meccanica* **50**, 617–632 (2015). https://doi.org/10.1007/s11012-014-0046-3

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DOI: https://doi.org/10.1007/s11012-014-0046-3