This paper presents a finite element model for adaptive sandwich beams to deal with either extension or shear actuation mechanism. The former corresponds to an elastic core sandwiched beam between two transversely polarized active surface layers; whereas, the latter consists of an axially polarized core, sandwiched between two elastic surface layers. For both configurations, an electric field is applied through thickness of the piezoelectric layers. The mechanical model is based on Bernoulli-Euler theory for the surface layers and Timoshenko beam theory for the core. It uses three variables, through-thickness constant deflection, and the mean and relative axial displacements of the core's upper and lower surfaces. Augmented by the bending rotation, these are the only nodal degrees of freedom of the proposed two-node adaptive sandwich beam finite element. The piezoelectric effect is handled through modification of the constitutive equation, when induced electric potential is taken into account, and additional electric forces and moments. The proposed finite element model is validated through static and dynamic analysis of extension and shear actuated, continuous and segmented, cantilever beam configurations. Finite element results show good comparison with those found in the literature, and indicate that the newly defined shear actuation mechanism presents several promising features over conventional extension actuation mechanism, particularly for brittle piezoceramics use and energy dissipation purposes.

This paper presents a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material using fractional derivative constitutive equations. The sandwich configuration is composed of a viscoelastic core (based on Timoshenko theory) sandwiched between elastic faces (based on EulerBernoulli assumptions). The viscoelastic model used to describe the behavior of the core is a four-parameter fractional derivative model. Concerning the parameter identification, a strategy to estimate the fractional order of the time derivative and the relaxation time is outlined. Curve-fitting aspects are focused, showing a good agreement with experimental data. In order to implement the viscoelastic model into the finite element formulation, the Grünwald definition of the fractional operator is employed. To solve the equation of motion, a direct time integration method based on the implicit Newmark scheme is used. One of the particularities of the proposed algorithm lies in the storage of displacement history only, reducing considerably the numerical efforts related to the non-locality of fractional operators. After validations, numerical applications are presented in order to analyze truncation effects (fading memory phenomena) and solution convergence aspects. IntroductionMany investigations have demonstrated the potential of viscoelastic materials to improve the dynamics of lightly damped structures. There are numerous techniques to incorporate these materials into structures. The constrained layer passive damping treatment is already largely used to reduce structural vibrations, especially in conjunction with active control [2, 13]. One of the crucial questions is how to quantify such a material damping if the viscoelastic solid has a weak frequency dependence on its dynamic properties over a broad frequency range. Classical linear viscoelastic models, using integer derivative operators, convolution integral or internal variables, become cumbersome due to the high quantity of parameters needed to describe the material behavior. In order to overcome these difficulties, fractional derivative operators acting on both, strain and stress can be employed.Until the beginning of the 80s, the concept of fractional derivatives associated to viscoelasticity was regarded as a sort of curve-fitting method. Later, Bagley and Torvik [1] gave a physical justification of this concept in a thermodynamic framework. Their fractional model has become a reference in literature. Special interest is today dedicated to the implementation of fractional constitutive equations into FE formulations. In this context, the numerical methods in the time domain are generally associated with the Grünwald formalism for the fractional order derivative of the stress-strain relation in conjunction with a time discretization scheme. Padovan [7] derived several implicit, explicit and predictor-corrector type algorithms. Escobedo-Torres and Ricles [6] analyzed a numerical procedure based on the central difference method and its ...

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SUMMARYThe finite element method is used for the computation of the variational modes of the system composed of an elastic tank partially filled with a compressible liquid. We propose, on the one hand, a direct approach based on a three field mixed variational formulation, and, on the other hand, a variational modal interaction scheme allowing the use of the acoustic eigenmodes of the liquid in a rigid motionless enclosure and the hydroelastic modes of the enclosure. Numerical results show the advantage of the second procedure.

This work intends to compare two viscoelastic models, namely ADF and GHM, which account for frequency dependence and allow frequency and time-domain analysis of hybrid active-passive damping treatments, made of viscoelastic layers constrained with piezoelectric actuators. A modal strain energy (MSE) based iterative model is also considered for comparison. As both ADF and GHM models increase the size of the system, through additional dissipative coordinates, and to enhance the control feasibility, a modal reduction technique is presented for the first time for the ADF model and then applied to GHM and MSE ones for comparison. The resulting reduced systems are then used to analyze the performance of a segmented hybrid damped cantilever beam under parameters variations, using a constrained input optimal control algorithm. The open loop modal damping factors for all models match well. However, due to differences between the modal basis used for each model, the closed loop ones were found to be different.

SUMMARYThis work, in two parts, proposes, in this ÿrst part, an electromechanically coupled ÿnite element model to handle active-passive damped multilayer sandwich beams, consisting of a viscoelastic core sandwiched between layered piezoelectric faces. The latter are modelled using the classical laminate theory, whereas the face=core=face system is modelled using classical three-layers sandwich theory, assuming Euler-Bernoulli thin faces and a Timoshenko relatively thick core. The frequency-dependence of the viscoelastic material is handled through the anelastic displacement ÿelds (ADF) model. To make the control system feasible, a modal reduction is applied to the resulting ADF augmented system. Validation of the approach developed in this part is presented in Part 2 of the paper together with the hybrid damping performance analysis of a cantilever beam.

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