El Mostafa Daya

Iterative algorithms for non-linear eigenvalue problems. Application to vibrations of viscoelastic shells

In this paper, two numerical iterative algorithms are developed for the vibrations of damped sandwich structures. These methods associate homotopy, asymptotic numerical techniques and Pade approximants. The first one is a sort of high order Newton method and the second one uses a more or less arbitrary matrix. So one can determine the natural frequencies and the loss factors of viscoelastically damped sandwich structures. To assess their efficiency, a few sandwich beams and plates have been considered. The techniques can be applied to large scale structures, to large damping and to strongly non-linear viscoelastic modulus.

An amplitude equation for the non-linear vibration of viscoelastically damped sandwich beams

Abstract An elementary theory for non-linear vibrations of viscoelastic sandwich beams is presented. The harmonic balance method is coupled with a one mode Galerkin analysis. This results in a scalar complex frequency–response relationship. So the non-linear free vibration response is governed by only two complex numbers. This permits one to recover first the concept of linear loss factor, second a parabolic approximation of the backbone curve that accounts for the amplitude dependence of the frequency. A new amplitude–loss factor relationship is also established in this way. The forced vibration analysis leads to resonance curves that are classical within non-linear vibration theory. They are extended here to any viscoelastic constitutive behaviour. This elementary approach could be extended to a large class of structures and in a finite element framework. The amplitude equation is obtained in closed form for a class of sandwich beams. The effects of the boundary conditions and of the temperature on the response are discussed.

A shell finite element for viscoelastically damped sandwich structures

In this paper, a shell finite element is proposed for viscoelastically damped sandwich structures, in which a thin viscoelastic layer is sandwiched between identical elastic layers. The sandwich finite element is obtained by assembling three elements throughout the thickness of the sandwich structure. Using specific assumptions and displacement continuity at the interfaces, one reduces to eight the number of degrees of freedom per node that are the longitudinal displacements of the elastic layers, the deflection and three rotations. The finite element computations have been compared with known analytical, numerical and experimental data concerning the vibrations of sandwich beams, plates and shells.

Evaluation of Kinematic Formulations for Viscoelastically Damped Sandwich Beam Modeling

For efficiently simulating static and dynamic behaviors of sandwich structures, an accurate kinematic model is essential. This study presents analytical and numerical evaluations of kinematics and theories proposed in the literature. Several types of assumed displacement fields are considered. This article compares and addresses the efficiency, the applicability, and the limits of classical models, higher order models (CLT, FSDT, and HSDT), and zig-zag theories. To achieve this, a comparative study with a finite element based solution free of any kinematic assumptions as well as a qualitative and quantitative assessment of displacement, stress fields, and modal parameters (natural frequency and loss factor) are conducted. The results are presented and discussed for several sandwich beam configurations where the faces and the cores are both isotropic. For these purposes, static (three-point bending test) and dynamic (free vibration) problems are considered.

A Shell Finite Element for Active-Passive Vibration Control of Composite Structures with Piezoelectric and Viscoelastic Layers.

This paper presents an accurate shell finite element (FE) formulation to model composite shell structures with embedded viscoelastic and piezoelectric layers and an integrated active damping control mechanism. The five-layered finite element introduced in this paper uses the first order shear deformation theory in the viscoelastic core and Kirchoff theory for the elastic and piezoelectric layers. The corresponding coupled FE formulation is derived starting from the shell kinematic and electromechanical governing equations. Assuming a linear strain field through each layer and exactly the same transverse displacement and the rotations in the elastic and piezoelectric layers, the number of degree of freedoms (dof) per node is reduced to 8. All the eight of these dofs are mechanical in nature. Constant velocity and constant displacement feedback control algorithms are used to actively control the dynamic response of the adaptive structure. Based on this formulation, a finite element code is implemented and the obtained results are compared to those in the literature analytical model and to the numerical results obtained using a commercial finite element code.

Vibration Modeling of Multilayer Composite Structures with Viscoelastic Layers.

This work deals with the vibration of orthotropic multilayer sandwich structures with viscoelastic core. A finite element model is derived from a classical zigzag model with shear deformation in the viscoelastic layer. The aim of the present work is to establish numerical models and develop numerical tools to design multilayer composites structures with high damping properties. To fulfill this purpose, a finite element model has been developed for vibration analysis of a sandwich plate (elastic orthotropic)/(viscoelastic orthotropic)/(elastic orthotropic). A numerical study from the variation of the damping properties of the structures was performed according to the faces materials fibers orientation.

An Approximated Harmonic Balance Method for Nonlinear Vibration of Viscoelastic Structures

In this paper, we deal with the nonlinear vibration of viscoelastic shell structures. Coupling an approximated harmonic balance method with one mode Galerkin s procedure, one obtains an amplitude equation depending on two complex coefficients. The latter are determined by solving a classical eigenvalue problem and two linear ones. To show the applicability and the validity of our approach, the amplitude-frequency and the amplitude loss factor relationships are illustrated for a sandwich plate with a viscoelastic central layer and a viscoelastic circular ring.

Forced harmonic response of viscoelastic sandwich beams by a reduction method

ABSTRACT The aim of this article is to develop a reduction method to determine the forced harmonic response of viscoelastic sandwich structures at a reasonable computational cost. The numerical resolution is based on the asymptotic numerical method. This type of problem is complex, and its number of degrees of freedom is double the number of the undamped structure, leading to a high computational time. To address the problem, three reduction methods are evaluated, which differ from the projection operator. Numerical tests have been performed in the case of cantilever sandwich beams. The comparison of the results obtained by the reduction order resolution with those given by the full order resolution shows both a good agreement and a significant reduction in computational cost.

A two scale method for modulated vibration modes of large, nearly repetitive, structures

Abstract Nearly repetitive structures can present at least two kinds of vibration modes: localized modes and modulated ones. In this Note, the multiple scale method is applied to characterize a packet of modulated modes. In this respect, only small size problems are to be solved: periodic problems posed on a few basic cells and amplitude equations, which define a sort of homogenized model for this type of modes. It is established that the influence of the non-repetitive part of the structure is accounted by a boundary condition. To cite this article: E.M. Daya et al., C. R. Mecanique 331 (2003).

Linear and quadratic solid–shell finite elements SHB8PSE and SHB20E for the modeling of piezoelectric sandwich structures

ABSTRACT In this article, hexahedral piezoelectric solid–shell finite element formulations with linear and quadratic interpolation, denoted by SHB8PSE and SHB20E, respectively, are proposed for the modeling of piezoelectric sandwich structures. Compared to conventional solid and shell elements, the solid–shell concept reveals to be very attractive, due to a number of well-established advantages and computational capabilities. More specifically, the present study is devoted to the modeling and analysis of multilayer structures that incorporate piezoelectric materials in the form of layers or patches. The interest in this solid–shell approach is shown through a set of selective and representative benchmark problems. These include numerical tests applied to various configurations of beam, plate and shell structures, both in static and vibration analysis. The results yielded by the proposed formulations are compared with those given by state-of-the-art piezoelectric elements available in ABAQUS, in particular, the C3D20E quadratic hexahedral finite element with piezoelectric degrees of freedom.