Noureddine Bouhaddi

Enhancement of the performance of a hybrid nonlinear vibration energy harvester based on piezoelectric and electromagnetic transductions

A multiphysics model of a hybrid piezoelectric–electromagnetic vibration energy harvester (VEH), including the main sources of nonlinearities, is developed. The continuum problem is derived on the basis of the extended Hamilton principle, and the modal Galerkin decomposition method is used in order to obtain a reduced-order model consisting of a nonlinear Duffing equation of motion coupled with two transduction equations. The resulting system is solved analytically using the method of multiple time scales and numerically by means of the harmonic balance method coupled with the asymptotic numerical continuation technique. Closed-form expressions for the moving magnet critical amplitude and the critical load resistance are provided in order to allow evaluation of the linear dynamic range of the proposed device. Several numerical simulations have been performed to highlight the performance of the hybrid VEH. In particular, the power density and the frequency bandwidth can be boosted, by up to 60% and 29% respectively, compared to those for a VEH with pure magnetic levitation thanks to the nonlinear elastic guidance. Moreover, the hybrid transduction permits enhancement of the power density by up to 84%.

Multi-modal vibration energy harvesting approach based on nonlinear oscillator arrays under magnetic levitation

We propose a multi-modal vibration energy harvesting approach based on arrays of coupled levitated magnets. The equations of motion which include the magnetic nonlinearity and the electromagnetic damping are solved using the harmonic balance method coupled with the asymptotic numerical method. A multi-objective optimization procedure is introduced and performed using a Non-dominated Sorting Genetic Algorithm (NSGA) for the cases of small magnet arrays in order to select the optimal solutions in term of performances by bringing the eigenmodes close to each other in terms of frequencies and amplitudes. Thanks to the nonlinear coupling and the modal interactions even for only three coupled magnets, the proposed method enable harvesting the vibration energy in the operating frequency range of 4.6 − 14.5 Hz, with a bandwidth of 190 % and a normalized power of 20.2 mW cm −3 g −2 .

Robust Design in Structural Mechanics

Many of the multi-objective optimization problems are often subject to parameters with uncertainties and noises. In such cases, to identify the robust solutions we generally add small amounts of noise and evaluate them with Monte Carlo simulation. In this paper, we suggest a new methodology for solving these types of multi-objective optimization problems. This methodology consists of increasing the objective function space with robustness functions in order to find robust and optimal solutions. The multi-objective optimization problem is solved with an evolutionary algorithm. A neural network is used to significantly reduce the computational time, in particular for the robustness function evaluations.

Use of Metamodels in the Multi-Objective Optimization of Mechanical Structures with Uncertainties

In this paper, we propose a method which takes into account the propagation of uncertainties in the finite element models in a multi-objective optimization procedure. This method is based on the coupling of the Stochastic Response Surface Method (SRSM) and a genetic algorithm of NSGA type (Non-dominated Sorting Genetic Algorithm). The SRSM is based on the use of Stochastic Finite Element Method (SFEM) via the use of the perturbation method. Thus, we can avoid the use of Monte Carlo simulation, whose cost is prohibitive in the optimization problems, especially when the finite element models are large and with a considerable number of design parameters. The objective of this study is, on the one hand, to quantify efficaciously the effects of these uncertainties on the variability of responses which we wish to optimize, and on the other hand, to calculate solutions which are both optimal and robust resulting from the numerical simulation. At the end of a multi-objective optimization procedure, the space of optimal solutions is generally of a large dimension. The solutions obtained are practically non-exploitable by the designer. To facilitate this interpretation, a study of sensitivity a posteriori can be exploited in order to eliminate the non-significant design parameters. The use of the clusters resulting from the Self-Organizing Maps of Kohonen (SOM) is also suggested for a rational management of the design space. The importance of the methodology that we have used along with suggestions for its performances are highlighted by two numerical examples. The criterion of quality selected consists in obtaining the best compromise: the minimal computing time versus the maximum precision of results.

Prediction of the dynamic response of a plate treated by particle impact damper

In this paper, an experimental characterisation of a particle impact damper (PID) under periodic excitation is investigated. The developed method allows the measurement of damping properties of PID without the supplementary use of a primary structure. The passive damping of PID varies with the excitation frequency and its design parameters. The nonlinear damping of PID is then interpreted as an equivalent viscous damping to be introduced in a finite element model of a structure to predict its dynamic response. The results of numerical simulations are in good agreement with those of experiment and show the relevance of the developed method to predict the dynamic behaviour of a structure treated by PID’s.

A TIME-DOMAIN FINITE ELEMENT MODEL REDUCTION METHOD FOR VISCOELASTIC LINEAR AND NONLINEAR SYSTEMS

Many authors have shown that the effective design of viscoelastic systems can be conveniently carried out by using modern mathematical models to represent the frequency- and temperature-dependent behavior of viscoelastic materials. However, in the quest for design procedures of real-word engineering structures, the large number of exact evaluations of the dynamic responses during iterative procedures, combined with the typically high dimensions of large finite element models, makes the numerical analysis very costly, sometimes unfeasible. It is especially true when the viscoelastic materials are used to reduce vibrations of nonlinear systems. As a matter of fact, which the resolution of the resulting nonlinear equations of motion with frequency- and temperature-dependent viscoelastic damping forces is an interesting, but hard-to-solve problem. Those difficulties motivate the present study, in which a time-domain condensation strategy of viscoelastic systems is addressed, where the viscoelastic behavior is modeled by using a four parameter fractional derivative model. After the discussion of various theoretical aspects, the exact and reduced time responses are calculated for a three-layer sandwich plate by considering nonlinear boundary conditions.

Stochastic modeling of surface viscoelastic treatments combined with model condensation procedures

Engineering structures incorporating viscoelastic materials are characterized by inherent uncertainties affecting the parameters that control the efficiency of the viscoelastic dampers. In this context, the handling of variability in viscoelastic systems is a natural and necessary extension of the modeling capability of the present techniques of deterministic analysis. Among the various methods devised for uncertainty modeling, the stochastic finite element method has received major attention, as it is well adapted for applications to complex engineering systems. In this paper, the stochastic finite element method applied to a structural three-layer sandwich plate finite element containing a viscoelastic layer, with random parameters modelled as random fields, is presented. Accounting for the dependence of the behaviour of the viscoelastic materials with respect to frequency and temperature, using the concepts of complex modulus and shift factor, the uncertainties are modelled as homogeneous Gaussian stochastic fields and are discretized according to the spectral method, using Karhunen-Loeve expansions. The modeling procedure is confined to the frequency domain, and the dynamic responses are characterized by frequency response functions (FRFs). Monte Carlo Simulation (MCS) combined with Latin Hypercube Sampling is used as the stochastic solver. The typically high dimensions of finite element models of viscoelastic systems combined with the large number of Monte Carlo samples to be computed make the evaluation of the FRFs variability computer intensive. Those difficulties motivate the use of condensation methods specially adapted for viscoelastic systems, in order to alleviate the computational cost. After the presentation of the underlying formulation, numerical applications of moderate complexity are presented and discussed aiming at demonstrating the main features and, particularly, the computation cost savings provided by the association of MCS with the suggested condensation procedure.

Parameterized Reduced Models for Efficient Optimization of Structural Dynamic Behavior

Substructure synthesis is a method of model order reduction which is generally more efficient, computationally speaking, than analyzing the complete structural system. However, these methods are not necessarily well adapted for use within an optimization process since they do not preserve the fidelity of the reduced models when structural modifications within the reduced substructure are introduced. As a result, a costly model reduction must be performed at each iteration step. This paper presents a new method to improve standard reduction methods by taking into account a priori knowledge of the potential structural modifications. Indeed, this information proves to be salutary in creating a single enriched model reduction transformation that preserves the precision of the reduced substructure model throughout the optimization process. The proposed approach consists in extending the standard transformation matrix by a set of static residual vectors which are optimized with respect to the design variables to be modified. The proposed method can be used with a variety component mode synthesis approaches with any type of substructure natural modes: free-free, cantilever or hybrid modes. The proposed methodology is illustrated on the basis of a simulated test case taken from the aero-engine industry. Introduction The ever increasing demand for faster engineering analysis in the design process has resulted in a substantial amount of research and development on faster and more accurate approximate reanalysis method. The difficulty of any model reduction procedure lies in how to complete the representation basis in order to reduce truncation effects. Two main classes of methodologies can be found in the literature. The first class seeks to generate a set of Ritz vectors capable of representing with precision the structural behavior under a wide variety of structural modifications. For example, Balmes 1 10 studied the possibility of using a constant basis of Ritz vectors to create parametric families of reduced models whereas Bouazzouni A., et al 2 9 developed a method for optimally constructing additional vectors by using the dynamic behavior of the structure before modification combined with the a priori knowledge of the design variables. Both of these approaches have already been used effectively in an industrial context. The second class of reduction methods are based on a high order polynomial expansions of model responses about a nominal point in parameter space. This approach proves to be particularly interesting for small number of design variables and can be used for topological optimization. In this paper, we will extend the approach developed in 2 for use with substructure synthesis techniques. The latter represent a economic means for evaluating the structural behavior of complex mechanical assemblies and are known collectively as component mode synthesis methods. This approach represents a structure as an assembly of individually reduced substructures called superelements. For example, the Craig-Bampton (CB) method is one of many techniques of component mode synthesis used intensively in the aerospace industry 3 4 . However, these methods are not always well adapted to the industrial problem. This is especially true when parametric studies are to be performed with respect to design variables contained within individual superelements. The designer has the choice of either re-using the nominal model reduction transformation or performing a new superelement analysis for each modified component. The first option generally leads to inaccurate results while the latter is often impracticable due to cost considerations. We propose in this paper a new method to improve the standard superelement reduction methods by taking into account an a priori knowledge of the potentially modifiable design variables with the objective of constructing a unique reduction transformation matrix which will preserve Copyright  2002 The American Institute of Aeronautics and Astronautics Inc. All rights rserved. 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado AIAA 2002-1392 Copyright

Nonlinear 2-DOFs Vibration Energy Harvester Based on Magnetic Levitation

The nonlinear dynamics of a two-degree-of-freedom (2-DOFs) vibrating energy harvester (VEH) based on magnetic levitation is modeled and investigated. The equations of motion have been derived while taking into account the magnetic nonlinearity and the electro-magnetic damping. The associated linear eigenvalue problem has been analyzed and optimality conditions have been expressed in term of distance minimization between the two eigenfrequencies of the considered system. The resulting optimal design parameters have been substituted into the coupled nonlinear equations of motion which have been numerically solved. It is shown that the performances of a classical single degree of freedom VEH can be significantly enhanced up to 270 % in term of power density, up to 34 % in term of frequency bandwidth and up to 10 % in term of resonance frequency attenuation.

Estimation of Modal Damping for Structures with Localized Dissipation

Damping plays an important role in bolted joints of assembled structures due to their significant capacity to dissipate energy. The underlying mechanisms of these dissipative phenomena are generally poorly understood and result from contact and friction effects within the joint interfaces. In order to provide useful virtual prototyping tools for reducing response levels, accurate model-based estimation of modal damping is required. The present study employs an energetic method to calculate the loss factor associated with the localized dissipative interfaces of a global linear structure. This method is based on the concept of the dissipated energy in the interfaces for which the closed-form expression of the loss factor is the ratio between dissipated energy and maximal potential energy, over a cycle of periodic vibration. The aim of this work is to investigate the advantages and drawbacks of this approach for particular conditions such as: modal projection, localized damping level and model density. Simulated academic examples, where accurate estimations of the exact solutions are available, will be used to illustrate the methodology and to explore the potential difficulties that may arise in more complex industrial applications.