Keywords: generalised Maxwell model, Laguerre polynomial approximation technique, modal strain energy method, random vibration, state-space formalism, viscoelastic devices.

Viscoelastic dampers have been successfully used with the aim of mitigating the structural vibrations induced by natural actions, such as ground shaking, gusting winds, or ocean waves (for example [1,2]). Continuous improvements in the techniques of identification and analysis, in fact, paralleled by noticeable refinements of device hardware, made the use of viscoelastic dampers completely suitable for consideration in both new or retrofitted constructions.

Vast effort has been devoted to developing effective tools for the dynamic analysis of viscoelastically damped structures. Even under the assumption of a linear behaviour, in fact, this is not an easy task because motion is ruled in the time domain by linear integro-differential equations of the Volterra type [3]. Depending on the mathematical description of the viscoelastic behaviour, however, alternative state-space models have been proposed recently in order to reduce the computational burden (for example [4,5,6]). In these models the viscoelastic behaviour is taken into account by appending to the array of `classical' state variables, that is, displacements and velocities of the structure, a suitable number of additional `internal' variables, for which alternative definitions can be thought up. The forces experienced by the viscoelastic components are then expressed as linear combinations of these new state variables, which are ruled by linear differential equations. As a result, the integro-differential equations of motion are turned into a set of linear differential equations, of greater order, but easier to solve.

As opposed to the large number of papers dealing with the response to deterministic excitations, only a few works are devoted to the stochastic analysis of structures provided by viscoelastic devices (references, for instance, can be found in the paper by Palmeri et al. [7]). Nevertheless, it is worth noting that state-space formalism lends itself to be directly extended to the point where the excitation is properly modelled as a Gaussian process, and allows us to measure the reliability of the system probabilistically under random loading.

The aim of this lecture is to summarise the main findings of the authors in the deterministic and stochastic analysis of vibrating systems provided with viscoelastic devices. For the purposes of simplicity, only the case of SDoF oscillators is dealt with here, even if the extension to MDoF structures is quite straightforward when the viscoelastic components are distributed proportionally to the inertia and, or the stiffness of the system [8]. Moreover, two novel integration schemes are proposed: the first one for the dynamic analysis of viscoelastically damped structures under deterministic loading; the second one in the case of excitations modelled as random processes. The accuracy and efficiency of the proposed approaches are demonstrated through numerical applications. It is also shown that the use of `equivalent' Kelvin-Voigt models, with a simpler viscous damping, may be excessively un-conservative.

1

B. Samali, K.C.S. Kwok, "Use of viscoelastic dampers in reducing wind-induced and earthquake-induced motion of building structures", Engineering Structures, 17, 639-54, 1995. doi:10.1016/0141-0296(95)00034-5

2

H.H. Lee, "Stochastic analysis for offshore structures with added mechanical dampers", Ocean Engineering, 24, 817-34, 1997. doi:10.1016/S0029-8018(96)00039-X

3

I. Patlashenko, D. Givoli, P. Barbone, "Time-stepping schemes for systems of Volterra integro-differential equations", Computer Methods in Applied Mechanics and Engineering, 190, 5691-718, 2001. doi:10.1016/S0045-7825(01)00192-X

4

L.X. Yuan, O.P. Agrawal, "A numerical scheme for dynamic systems containing fractional derivatives", Journal of Vibration and Acoustics - Transactions of the ASME, 124, 321-4, 2002. doi:10.1115/1.1448322

5

A. Palmeri, F. Ricciardelli, A. De Luca, G. Muscolino, "State space formulation for linear viscoelastic dynamic systems with memory", Journal of Engineering Mechanics - ASCE, 129, 715-24, 2003. doi:10.1061/(ASCE)0733-9399(2003)129:7(715)

6

N. Wagner, S. Adhikari, "Symmetric state-space method for a class of nonviscously damped systems", AIAA Journal, 41, 951-6, 2003. doi:10.2514/2.2032

7

A. Palmeri, F. Ricciardelli, G. Muscolino, A. De Luca, "Random vibration of systems with viscoelastic memory", Journal of Engineering Mechanics - ASCE, 130, 1052-61, 2004. doi:10.1061/(ASCE)0733-9399(2004)130:9(1052)

8

A. Palmeri, F. Ricciardelli, G. Muscolino, A. De Luca, "Effects of viscoelastic memory on the buffeting response of tall buildings", Wind and Structures, 7, 89-106, 2004.

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