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Computational Science, Engineering & Technology Series
ISSN 1759-3158
Edited by: B.H.V. Topping, L.F. Costa Neves, R.C. Barros
Chapter 13

Toward Overcoming the Concept of Effective Stiffness and Damping in the Dynamic Analysis of Structures with Viscoelastic Components

A. Palmeri

School of Engineering, Design and Technology, University of Bradford, United Kingdom

Full Bibliographic Reference for this chapter
A. Palmeri, "Toward Overcoming the Concept of Effective Stiffness and Damping in the Dynamic Analysis of Structures with Viscoelastic Components", in B.H.V. Topping, L.F. Costa Neves, R.C. Barros, (Editors), "Trends in Civil and Structural Engineering Computing", Saxe-Coburg Publications, Stirlingshire, UK, Chapter 13, pp 267-292, 2009. doi:10.4203/csets.22.13
Keywords: additional internal variable, cable beam, dynamics of structures, energy dissipation, Laguerre's polynomial approximation technique, modal strain energy method, numerical scheme, relaxation function, state-space formalism, transition matrix, vibration analysis, viscoelastic damping.

In the current state-of-practice, the time-domain dynamic analysis of structures incorporating viscoelastic members is generally carried out through the modal strain energy (MSE) method, or other procedures somehow based on the quite simplistic idea of substituting the actual viscoelastically damped structure with an equivalent system featuring a pure viscous damping.

This crude approximation in civil engineering applications is very often encouraged by manufacturers of the viscoelastic devices themselves, whose interest is to simplify as much as possible the design procedures for structures embedding their products. As an example, elastomeric seismic isolators are generally advertised and sold with a table listing the equivalent values of elastic stiffness and viscous damping ratio for different amplitudes of vibration. Unfortunately, many experimental and analytical studies confirm that the real dynamic behaviour of such devices is much more complicated, and cannot be bent to the interests of manufacturers and designers.

Despite the advances in the field made in the last two decades, two well-established beliefs continue to underpin use and abuse of the concepts of effective stiffness and damping for viscoelastically damped structures: first, MSE method and similar procedures are unconditionally assumed to provide good approximations, which are acceptable for design purposes; second, the implementation of more refined approaches is thought to be computationally too expensive, and hence suitable just for a few very important constructions.

In this chapter, as a further contribution to overcoming these popular beliefs, a novel time-domain numerical scheme of dynamic analysis is presented and numerically validated. After a brief review of the LPA (Laguerre's polynomial approximation) technique for one-dimensional viscoelastic members of known relaxation function, the state-space equations of motion for linear structures with viscoelastic components are derived in the modal space. Aimed at making the proposed approach more general, the distribution of the viscoelastic components is allowed to be non-proportional to mass and elastic stiffness, thus removing the most severe limitation of previous formulations. Then, a cascade scheme is derived by decoupling in each time step traditional state variables (i.e. modal displacements and velocities) and additional internal variables. The joint use of modal analysis and improved cascade scheme permits reducing the size of the problem and keeping the computational burden low. The illustrative application to the small-amplitude vibration of a cable beam made of different viscoelastic materials demonstrates the versatility of the proposed approach. The numerical results confirm a superior accuracy with respect to the classical MSE method, whose underestimate in the low-frequency range can be as large as 75%.

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