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CivilComp Proceedings
ISSN 17593433 CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 74
Forced NonLinear Vibration of Damped Sandwich Beams by the Harmonic Balance  Finite Element Method N. Jacques^{1}, E.M. Daya^{2} and M. PotierFerry^{2}
^{1}Laboratory of Mechanics of Naval and Offshore Structures, Brest, France
N. Jacques, E.M. Daya, M. PotierFerry, "Forced NonLinear Vibration of Damped Sandwich Beams by the Harmonic Balance  Finite Element Method", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 74, 2006. doi:10.4203/ccp.83.74
Keywords: nonlinear vibration, sandwich beams, viscoelastic damping, harmonic balance, finite element.
Summary
This study deals with the development of a numerical tool for computing the
nonlinear forced response of sandwich beams subjected to harmonic excitations. It is
now well known that large vibration amplitude of beamlike structures may induce a
dynamic behaviour significantly different from the behaviour predicted by the linear
theories. Nevertheless, they are only few works dealing with nonlinear 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
nonlinearities 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 nonlinear
response of structures subjected to harmonic forces should be predicted with use of
direct timeintegration 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
frequencydomain. 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
nonlinear 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 nonlinear vibration of
viscoelastically damped sandwich beams is proposed.
Threelayer 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 crosssectional behaviour of such beams is inadequate because it would underestimate the shear strain in the core. Thus, the proposed model uses a ZigZag kinematics for constructing the displacement field. This laminate theory is based on the EulerBernoulli kinematical assumption for the face layers and on the Timoshenko kinematical assumption for the core layer. The geometrical nonlinearity 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 nonlinear 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 frequencydependant 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 nonlinear 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. References
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