Keywords: non-local damping, viscoelastic materials, Euler-Bernoulli beams, Galerkin method, damping capacity.

The dynamic response analysis of structures is important in many areas of engineering. Particularly, it is interesting to analyze the behaviour of damped structures with different damping models. The viscoelastic materials, as well as auxetic systems composed from different viscoelastic materials [1,2], due to its energy dissipating properties, are widely used to control vibrations in structures and machines and to reduce noise in acoustic engineering.

In this paper, the bending vibrations of Euler-Bernoulli beams treated with non-local viscoelastic damping patches have been studied. The non-local viscoelastic properties of the damping patch can be represented by a spatial kernel function and a relaxation function, respectively. In particular, the classical local and viscous behaviours can be reproduced with a correct choice of this functions. The practical needs of engineering problems motivate the study of non-local viscoelastic models.

The corresponding equation of motion has been solved using the Galerkin method [3] in the Laplace domain, following the procedure previously proposed by Lei et al. [4]. The bending vibrations of a homogeneous damped Euler-Bernoulli beam have been solved for different boundary conditions and several models of the spatial kernel function.

A parametric study has been conducted to analyze the influence of different parameters involved in the model. To do this, a new damping capacity function has been defined, which allows us to compare the solutions with different values of the parameters. The main conclusions are that the first mode is predominant, and higher values of damping capacity are obtained if the treatment is positioned in points with high transverse displacements. If the spatial kernel function is close to local behaviour, we obtain higher values of the damping capacity, in addition the relaxation function is close to the viscous damping behaviour.

1

L. Munteanu, P.P. Delsanto, D. Dumitriu, V. Mosnegutu, "On the characterization of auxetic materials", Research Trends in Mechanics, 2, Ed. Academiei, 2008.

2

V. Chiroiu, L. Munteanu, A.M. Mitu, C. Eisinger Borcea, "On the eigenvalues optimization of beams with auxetic patches", SISOM 2008 and Session of the Commission of Acoustics, Bucharest, 29-30 May 2008.

3

N.Y. Zhang, "On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type", Mathematics of Computation, 60, 133-166, 1993. doi:10.2307/2153159

4

Y. Lei, M.I. Friswell, S. Adhikari, "A Galerkin method for distributed systems with non-local damping", International Journal of Solids and Structures, 43, 3381-3400, 2006. doi:10.1016/j.ijsolstr.2005.06.058

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