Keywords: non-linear vibration, sandwich beams, viscoelastic damping, harmonic balance, finite element.

This study deals with the development of a numerical tool for computing the non-linear forced response of sandwich beams subjected to harmonic excitations. It is now well known that large vibration amplitude of beam-like structures may induce a dynamic behaviour significantly different from the behaviour predicted by the linear theories. Nevertheless, they are only few works dealing with non-linear vibration in which damping effects are taken into account. Moreover, most of these studies are based on a one mode Galerkin approach [1,2]. So, results of these studies are likely to be accurate only near a resonance frequency. Moreover, effects of geometrical non-linearities on the mode shape are disregarded. In order to overcome these limitations, numerical techniques, sucha as the finite element method, have to be used. The non-linear response of structures subjected to harmonic forces should be predicted with use of direct time-integration analyses. However, this method is not very appropriate to investigate forced vibration within a large range of excitation frequencies. The use of the harmonic balance method with finite element approximations in space has proved to be a versatile and efficient technique to compute the response in the frequency-domain. Nevertheless, to the knowledge of the authors, there are only a few works dealing with the use of the finite element - harmonic balance technique for non-linear damped vibration [3,4]. In these studies, only viscous damping (where the damping forces are proportional to the absolute velocity), which is not representative of the damping properties of structures with viscoelastic parts, is considered. In this paper, a new computational approach dedicated to the non-linear vibration of viscoelastically damped sandwich beams is proposed.

Three-layer beams with a soft viscoelastic core are considered. This sandwich construction induces shearing motions in the viscoelastic layer, which is a very effective damping mechanism. The use of a unique laminate model to describe the cross-sectional behaviour of such beams is inadequate because it would underestimate the shear strain in the core. Thus, the proposed model uses a Zig-Zag kinematics for constructing the displacement field. This laminate theory is based on the Euler-Bernoulli kinematical assumption for the face layers and on the Timoshenko kinematical assumption for the core layer. The geometrical non-linearity is taken into account with use of the classical theory of finite deflections and moderate rotations.

The essence of the proposed procedure is to perform a harmonic balance reduction followed by a finite element discretization. The harmonic balance method allows the derivation of approximate equations of motion in the frequency domain: the displacements are assumed to be harmonics in the time domain and expanded with a Fourier series. By integrating over a period, a new variational equation, governing the harmonic motions of the beam, is obtained. This equation does not depend on the time, its unknowns are the coefficient of the Fourier expansion. After that, finite element approximations in space are used. Note that, in the current practice in non-linear vibration, the harmonic balance reduction is generally performed after the finite element discretization [3,4]. Nevertheless, this way leads to difficulties when the core material behaviour is modeled with use of a general frequency-dependant viscoelastic constitutive law. The finite element based on the proposed method have been implemented in the commercial software ABAQUS via a UEL subroutine. The Riks method is used to predict unstable paths of the frequency response curve.

For validation purposes, results derived from this new approach are compared to results of fully non-linear dynamic analyses using direct time integration. In this case, the beam is modelled with plane stress continuum elements. A good agreement is found and validates both the beam kinematics assumptions and the harmonic balance approximations.

1

E.J. Kovac, W.J. Anderson, R.A. Scott, "Forced non-linear vibrations of damped sandwich beam", Journal of Sound and Vibration, 17(1), 25-39, 1971. doi:10.1016/0022-460X(71)90131-3

2

E.M. Daya, L. Azrar, M. Potier-Ferry, "An amplitude equation for the non-linear vibration of viscoelastically damped sandwich beams", Journal of Sound and Vibration, 271(3-5), 789-813, 2004. doi:10.1016/S0022-460X(03)00754-5

3

V.P. Iu, Y.K. Cheung, S.L. Lau, "Non-linear vibration analysis of multilayer beams by incremental finite elements, Part II: Damping and forced vibrations", Journal of Sound and Vibration, 100(3), 373-382,1985. doi:10.1016/0022-460X(85)90293-7

4

P. Ribeiro, M. Petyt, "Non-linear vibration of plates by the hierarchical finite element and continuation method", International Journal of Mechanical Sciences, 41(4-5), 437-459, 1999. doi:10.1016/S0020-7403(98)00076-9

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