Abstract: This is the first of two companion articles addressing an integrated study on the mathematical modeling and assessment of the efficiency of surface mounted or embedded viscoelastic damping treatments, typically used to reduce structural vibration and/or noise radiation from structures, incorporating the adequate use and development of viscoelastic (arbitrary frequency dependent) damping models, along with their finite element (FE) implementation, and the experimental identification of the constitutive behavior of viscoelastic materials. This first article (Part I) is devoted to the development of mathematical descriptions of material damping to represent the linear viscoelastic constitutive behavior, their implementation into FE formulations and the use of the underlying different solution methods. To this end, internal variables models, such as the Golla-Hughes-McTavish (GHM) and anelastic displacement fields
(ADF) models, and other methods such as the direct frequency response (DFR), based on the complex modulus approach (CMA), iterative modal strain energy (IMSE) and an approach based on an iterative complex eigensolution (ICE) are described and implemented at the global FE model level. The experimental identification of viscoelastic materials properties and the aforementioned viscoelastically damped FE modeling approaches are assessed and validated
in the companion article [Vasques, C.M.A. et al., Viscoelastic damping technologiesâ€“Part II: Experimental identification procedure and validation, Journal of Advanced Research in Mechanical Engineering 1(2): 96-110 (2010)].

Abstract: This is the second of two companion articles addressing an integrated study on the mathematical modeling and assessment of the efficiency of surface mounted or embedded viscoelastic damping treatments, typically used to reduce structural vibration and/or noise radiation from structures, incorporating the adequate use and development of viscoelastic (arbitrary frequency dependent) damping models, along with their finite element (FE) implementation, and the experimental identification of the constitutive behavior of viscoelastic materials. In the first article [Vasques, C.M.A. et al., Viscoelastic damping technologiesâ€“Part I: Modeling and finite element implementation, Journal of Advanced Research in Mechanical Engineering 1(2): 76-95 (2010)] viscoelastic damping has been tackled from a mathematical point of view and the implementation, at the global FE model level, of time and frequency domain methods, namely the internal variables models, Golla-Hughes-McTavish (GHM) and anelastic displacement fields (ADF), and the complex modulus approach based ones, direct frequency response (DFR), iterative modal strain energy (IMSE) and an iterative complex eigensolution (ICE), respectively, were described and formulated. This second article is a natural extension of the first one. It presents a generic methodology to identify the complex shear modulus of viscoelastic materials. In this case, the complex shear modulus of the well-known viscoelastic material 3M ISD112 is identified and up-to-date values for this material are used and curve-fitted in order to obtain the modeling parameters of the GHM and ADF models. Afterward, a viscoelastic sandwich (three-layered) plate specimen and the correspondent FE model are considered numerically and experimentally. Measured and predicted frequency response functions (FRFs) are compared with the purpose of assessing the performance of the damping models presented in the companion article. The analysis allows to assess the validity of the methodology to determine the frequency dependent complex modulus, the GHM and ADF parameters identification procedure and the outcomes and drawbacks of the DFR, IMSE, ICE, GHM and ADF viscoelastic damping modeling strategies and their FE implementations, with the aim of assisting structural designers in the selection of the most appropriate viscoelastic damping modeling approach for their specific needs.

Abstract: This article concerns the adaptive feedforward control of vibration of a freely supported beam with two distinct surface mounted hybrid activeâ€“passive damping treatments. The first configuration concerns the use of an Active Constrained Layer Damping (ACLD) patch alone, where the piezoelectric constraining layer is actively utilized to increase the shear deformation of the sandwiched passive viscoelastic layer and at the same time to apply forces and moments into the structure, which will balance the power flows into the structure, and is denoted by ACLD configuration. The second configuration regards the use, as an active element in the control, of the piezoelectric patch alone, denoted by Active Damping (AD), and since the constraining layer of the ACLD treatment also bonded on the other side of the beam is not actively utilized, a Passive Constrained Layer Damping (PCLD) treatment is utilized in combination with an AD one, yielding an AD/PCLD configuration. A finite element model of the beam with the damping treatments is used for the simulation of the adaptive feedforward
controller which is also implemented and tested in real-time. The aims are to compare the predicted and measured damping performances of the two treatments in terms of vibration reduction, control effort, stability and robustness, when a filtered-reference LMS algorithm is used to cancel the effects of a broadband voltage disturbance applied into a third surface
mounted piezoelectric patch which is used to excite the beam.

Abstract: This paper concerns the numerical simulation of feedback, adaptive feedforward and hybrid (combined feedback/feedforward) control systems on the active control of vibrations of beams with Active Constrained Layer Damping (ACLD) treatments. For the simulation a 1-D beam finite element (FE) model with an arbitrary number of elastic, piezoelectric and viscoelastic layers attached to both sides of the beam is utilized. The damping behavior of the viscoelastic layers is considered by a Laplace transformed Anelastic Displacement Fields (ADF) method. The analyzed case study regards the disturbance rejection of an aluminium beam with a pair of surface mounted ACLD patches. In the design and simulation of the control system a Single-Input Single-Output (SISO) configuration with the output being the velocity at one point of the beam and the input being the control voltage applied into the piezoelectric constraining layers is considered. The case study allows to assess and discuss the outcomes and drawbacks of the feedback and feedforward controllers when used individually and the advantages of the hybrid controller.

Abstract: A velocity feedback control system is evaluated in the active control of vibrations of a smart beam with a pair of surface mounted piezoelectric ceramic patches, and finite element (FE) model results are validated against measured ones. To this end, a three-layered smart beam FE model is utilized, where a partial layerwise theory and a fully coupled electro-mechanical theory are considered for the formulation of the displacement field and electric potential, respectively. Regarding the test rig, it consists of a cantilever smart aluminum beam with two piezoelectric patches mounted close to the clamped end. One of the piezoelectric patches is utilized to excite the beam while the other is utilized as an actuator in the feedback control loop. The control voltage applied to the actuator is proportional to the transverse velocity at the free end, which is measured by a laser vibrometer. First, the quasi-static actuation capacity of the piezoelectric patches is evaluated. Next, the free and forced velocity responses to an initial displacement field and harmonic excitation are analyzed. The capacity to predict instabilities and the accuracy of the FE model are demonstrated and the applicability and functionality of the velocity feedback vibration control system are discussed

Abstract: This paper concerns the analytical formulation and finite element modelling of arbitrary active constrained layer damping (ACLD) treatments applied to beams. A partial layerwise theory is utilized to define the displacement field of beams with an arbitrary number of elastic, viscoelastic and piezoelectric layers attached to both surfaces, and a fully coupled electro-mechanical theory is considered for modelling the behavior of the piezoelectric layers. The damping of the viscoelastic layers is modelled by the complex modulus approach. The weak forms of the analytical formulation, governing the motion and electric charge equilibrium, are presented. Based on the weak forms, a one-dimensional finite element (FE) model is developed, with the nodal mechanical degrees of freedom being the axial displacement, transverse displacement and the rotation of the mid-plane of the host beam and the rotations of the individual layers, and the electrical elemental degrees of freedom being the electrical potential difference of each piezoelectric layer. Frequency response functions were measured experimentally and evaluated numerically for a freely suspended aluminium beam with an ACLD patch. In order to validate the FE model the results are presented and discussed.

Abstract: This paper presents a numerical study concerning the active vibration control of smart piezoelectric beams. A comparison between the classical control strategies, constant gain and amplitude velocity feedback, and optimal control strategies, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) controller, is performed in order to investigate their effectiveness to suppress vibrations in beams with piezoelectric patches acting as sensors or actuators. A one-dimensional finite element of a three-layered smart beam with two piezoelectric surface layers and metallic core is utilized. A partial layerwise theory, with three discrete layers, and a fully coupled electro-mechanical theory is considered. The finite element model equations of motion and electric charge equilibrium are presented and recast into a state variable representation in terms of the physical modes of the beam. The analyzed case studies concern the vibration reduction of a cantilever aluminum beam with a collocated asymmetric piezoelectric sensor/actuator pair bonded on the surface. The transverse displacement time history, for an initial displacement field and white noise force disturbance, and point receptance at the free end are evaluated with the open- and closed-loop classical and optimal control systems. The case studies allow the comparison of their performances demonstrating some of their advantages and disadvantages.

Abstract: This paper presents a theoretical and finite element (FE) formulation of a three-layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed- or open-circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three-layered smart beam with different EBC is proposed considering a fully coupled electro-mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi-static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC.

Abstract: This paper concerns the mathematical modeling and finite element (FE) solution of general anisotropic shells with hybrid active-passive damping treatments. A fully-coupled piezo-visco-elastic mathematical model of the shell (host structure) and segmented arbitrarily stacked layers of damping treatments is considered. A discrete layer approach is employed in this work, and the weak form of the governing equations is derived for a single generic layer of the multilayer shell using Hamilton's principle and a mixed (displacement/stresses) definition of the displacement field. First, a fully refined deformation theory of the generic layer, based on postulated out-of-plane shear stress definitions and in the in-plane stresses obtained with a Reissner-Mindlin type shell theory, is outlined. A semi-inverse procedure is used to derive the layer mixed non-linear displacement field, in terms of a blend of the generalized displacements of the Love-Kirchhoff and Reissner-Mindlin theories and of the stress components at the generic layer interfaces. No assumptions regarding the thinness of the shell are considered. Regarding the definition of the electric potential, the direct piezoelectric effects are condensed into the model through effective stiffness and strains definitions, and the converse counterpart is considered by the action of prescribed electric potential differences in each piezoelectric layer. Then, the weak forms of a partially refined theory, where only the zero-order term of the non-linear fully refined transverse displacement is retained, are derived for an orthotropic doubly-curved piezo-elastic generic shell layer. Based on the weak forms a FE solution is initially developed for the single layer. The degrees of freedom (DoFs) of the resultant four-noded generic piezo-elastic single layer FE are then "regenerated" into an equivalent eight-node 3-D formulation in order to allow through-the-thickness assemblage of displacements and stresses, yielding a partially refined multilayer FE assuring displacement and shear stress interlayer continuity and homogeneous shear stress conditions at the outer surfaces. The shear stresses DoFs are dynamically condensed and the FE is reduced to a displacement-based form. The viscoelastic damping behavior is considered at the global FE model level by means of a Laplace transformed ADF model. The active control of vibration is shortly discussed and a set of indices to quantify the damping performance and the individual contributions of the different mechanisms are proposed.

Abstract: A phenomenological electromechanical analytical model of beams with piezoelectric transdu-cers shunted with a passive electrical network with general impedance is presented and dis-cussed. A case study is considered and the damping performance of a cantilever beam with a piezoelectric patch shunted with a resistance tuned to damp the first mode is investigated.

Abstract: In this article, modal sensing via spatially shaped distributed piezoelectric transducers is inves-tigated for beams. A simple beam model considering the electromechanical coupling effects is presented and the spatially distribution of modal sensors is discussed and assessed.

Abstract: In the last decade the interest demonstrated, mainly by public and private companies from the
aerospace and space engineering sectors, in the development and use of high performance
structures, in general, and adaptive structures, in particular, comprising piezo-visco-elastic
damping treatments as technological solutions to tackle vibration- and noise-related problems
has grown. In the meantime, a relative maturity in the adequate modeling and design of piezovisco-
elastic damping treatments has also been achieved. However, with the ever increasing
strong demands of lighter, stiffer, lower cost and more efficient and reliable structures, engineers
and scientists are faced nowadays with the requirement of having at their disposal
refined and more accurate multiphysics models of those complex multilayer structures and
damping technologies. That problem pushes the underlying complexity of more representative
models to higher levels and poses a sustained and application-motivated challenge to the
adaptive structures research community.
In that context, this dissertation addresses issues on the vibration control of adaptive structures,
tentatively covering the design, analysis and application of the emerging piezo-viscoelastic
damping technologies focusing, in general, main five research fields/topics. (1) MODELING:
new refined finite element (FE) models of multilayer piezo-visco-elastic beam, plate
and shell structures with a high degree of accuracy and a suitable trade-off between accuracy
and complexity are developed; (2) DAMPING: viscoelastic (frequency-dependent) damping
modeling and solution approaches are developed, assessed and implemented into FE modeling;
(3) CONTROL: passive, active and hybrid damping treatments and control strategies
considering both feedback and adaptive feedforward algorithms used separately or blended
are developed, simulated, assessed and implemented in real-time control; (4) DESIGN: the efficiency
of the damping treatments/technologies in terms of material properties, location, geometric
configuration, thicknesses of the layers and control law is analyzed and discussed, and
a general design methodology to assess their design is proposed and demonstrated; (5) APPLICATION:
FE routines devoted to hybrid damping design and simulation, real-time test
and analysis, validation and application of hybrid (with multi-algorithms and multi-damping
materials) damping technologies are developed and implemented into a FE modeling and programing
environment.

Abstract: In this dissertation a numerical and experimental study about active control of
vibrations of beams with piezoelectric sensors and actuators is presented. A revision about
intelligent materials, focusing the piezoelectric materials, is realized and their constitutive
laws identified. A precise theoretical formulation of an intelligent beam with piezoelectric
sensors and actuators is also presented. The established theoretical model is discritized by the
finite element method. The layerwise theory is considered in the formulation of the
mechanical model in order to approximate the displacement field of the beam and
piezoelectric layers. An electric model for different electric boundary conditions is
considered, namely, electroded layers with the electrodes left open or closed, with the
equipotencial area condition verified and the electric potentials prescribed, and non-electroded
layers. Using a variational formulation, the direct piezoelectric effect, for the different electric
boundary conditions, is incorporated into the beam theory via suitable approximations for the
axial and transverse electric fields. In that way, a finite element model of a beam taking into
account the electromechanical coupling by means of effective stiffness parameters is
proposed. The displacement or velocity feedback control theories, at constant amplitude or
constant gain, and the optimal control theories LQR and LQG, are presented and evaluated on
their ability to control the vibration. For the analysis of the free and forced response of the
static action of the piezoelectric actuators, frequency response functions and active control of
vibrations of beams with continuous or segmented piezoelectric layers, several case studies
and solutions obtained with the proposed finite element model are presented. Comparisons
with numerical and approximated analytical solutions have shown the reliability and
robustness of the developed theory in applications with thin and thick beams. Finally, an
experimental study of the static action of the piezoelectric actuators and active control of
vibrations with velocity feedback, using a laser transducer to measure the velocity, or
displacement feedback, using a piezoelectric sensor, is presented and a comparison with
numerical results is performed for a beam with piezoelectric sensors and actuators.