Modelling and identification of structures with rate-independent linear damping

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.

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

$$\begin{aligned} K_j^*(s) = k_{0j}^* \bigg (1 + \sum _{l=1}^n \beta _l \frac{s}{s+\mu _l}\bigg ) \end{aligned}$$

(47)

in the Laplace domain and

$$\begin{aligned} K_j^*(\omega ) = k_{0j}^* \left[ 1 + \sum _{l=1}^n \beta _l \bigg ( \frac{\omega ^2}{\omega ^2 + \mu _l^2} + i\frac{\omega \mu _l}{\omega ^2 + \mu _l^2}\bigg )\right] \end{aligned}$$

(48)

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 $K_j^*(\omega )$

$$\begin{aligned} \eta _j^*(\omega ) = \frac{{\mathfrak{I}}[K_j^*(\omega )]}{{\mathfrak{R}}[K_j^*(\omega )]} = \frac{k_{0j}^*\,\bigg (\sum _{l=1}^n \beta _l \frac{\omega \mu _l}{\omega ^2 + \mu _l^2}\bigg )}{k_{0j}^*\,\bigg (1 + \sum _{l=1}^n \beta _l \frac{\omega ^2}{\omega ^2 + \mu _l^2}\bigg )} \end{aligned}$$

(49)

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

$$\begin{aligned} \eta _j^*(\omega ) = \frac{\sum _{l=1}^n \beta _l \frac{\omega \mu _l}{\omega ^2 + \mu _l^2}}{1 + \sum _{l=1}^n \beta _l \frac{\omega ^2}{\omega ^2 + \mu _l^2}} = \eta ^*(\omega ) \end{aligned}$$

(50)

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

$$\begin{aligned} {\mathbf{G}}(s) = {\mathbf{M}} \sum _h ({\mathbf{M}}^{-1} {\mathbf{K}}_0)^h \chi _h(s) \end{aligned}$$

(51)

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

$$\begin{aligned} {\mathbf{K}}(s) = {\mathbf{K}}_0 + s\,\mathbf{G}(s) = {\mathbf{K}}_0 + s\,{\mathbf{M}}\sum _h ({\mathbf{M}}^{-1} {\mathbf{K}}_0)^h \chi _h(s) \end{aligned}$$

(52)

Accounting for the relationship

$$\begin{aligned} {\varvec{\phi}}_j^{\text{T}} {\mathbf{M}}({\mathbf{M}}^{-1}{\mathbf{K}}_0)^h {\varvec{\phi}}_k = m_j^*\,\omega _{0j}^{2h}\delta _{jk} \end{aligned}$$

(53)

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

$$\begin{aligned} K_j^*(s) = {\varvec{\phi}}_{j}^{\text{T}} {\mathbf{K}}(s) {\varvec{\phi}}_j = k_{0j}^* + s\,m_j^*\sum _h \omega _{0j}^{2h}\, \chi _h(s) \end{aligned}$$

(54)

in the Laplace domain, and subsequently as

$$\begin{aligned} K_j^*(\omega ) = k_{0j}^* + i\omega \,m_j^*\sum _h \omega _{0j}^{2h}\, \chi _h(\omega ) \end{aligned}$$

(55)

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

$$\begin{aligned} \eta _j^*(\omega )&= \frac{{\mathfrak{I}}[K_j^*(\omega )]}{{\mathfrak{R}}[K_j^*(\omega )]} \\&= \frac{\omega \,m_j^*\sum _h\omega _{0j}^{2h}\;{\mathfrak{R}}[\chi _h(\omega )]}{k_{0j}^* - \omega \,m_j^*\sum _h\omega _{0j}^{2h}\;{\mathfrak{I}}[\chi _h(\omega )]} \\&= \frac{\omega \sum _h \omega _{0j}^{2(h-1)}\,{\mathfrak{R}}[\chi _h(\omega )]}{1 - \omega \sum _h \omega _{0j}^{2(h-1)}\,{\mathfrak{I}}[\chi _h(\omega )]} \end{aligned}$$

(56)

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

$$\begin{aligned} {\mathfrak{R}}[\chi _h(\omega )] = {\mathfrak{I}}[\chi _h(\omega )] = 0 \qquad \forall \; h \ne 1 \end{aligned}$$

(57)

from which it follows

$$\begin{aligned} \eta _j^*(\omega ) = \frac{\omega \,{\mathfrak{R}}[\chi _1(\omega )]}{1 - \omega \,{\mathfrak{I}}[\chi _1(\omega )]} = \eta ^*(\omega ) \end{aligned}$$

(58)

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

$$\begin{aligned} \sum _{l=1}^n {\mathbf{C}}_l \mu _l \frac{1}{s + \mu _l} = {\mathbf{K}}_0 \chi _1(s) \end{aligned}$$

(59)

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

$$\begin{aligned} {\mathbf{C}}_l \mu _l \frac{1}{s + \mu _l} = {\mathbf{K}}_0 \chi _{1l}(s), \qquad l = 1,\,2,\, \ldots ,\, n \end{aligned}$$

(60)

from which it is obtained

$$\begin{aligned} {\mathbf{C}}_l = {\mathbf{K}}_0 \frac{s + \mu _l}{\mu _l} \chi _{1l}(s) = {\mathbf{K}}_0 \frac{\beta _l}{\mu _l} \end{aligned}$$

(61)

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 $s$.