A Two-Step Hybrid Approach for Modeling the Nonlinear Dynamic Response of Piezoelectric Energy Harvesters (original) (raw)
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Coupled Electromechanical Numerical Modelling of Piezoelectric Vibration Energy Harvesters
DAAAM Proceedings, 2018
Energy harvesting is the process of collecting low-level ambient energy and converting it into electrical energy to be used for powering miniaturized autonomous devices, sensor networks, wearable electronics or Internet-of-Things components. The use of the pervasive kinetic energy, converted into electrical energy, is of special interest in this frame. The possibility to use bimorph piezoelectric cantilevers to convert ambient vibrations to electrical energy is therefore thoroughly analyzed in this work. A reliable modelling tool for optimizing the design of the miniature harvesters to be used in a broad frequency range, while maximizing the obtained powers, is hence needed. The problem complexity is induced by the necessity to simulate the dynamic response of the considered harvesting devices via a coupled electromechanical model. The recently developed comprehensive coupled analytical model based on distributed parameters is thus used as a benchmark to verify and tune suitable finite element (FE) numerical models. Modal (allowing to determine the mechanical dynamic response and the respective eigenfrequencies), harmonic (resulting in coupled frequency response functions) as well as linear and nonlinear transient FE analyses (resulting in dynamic responses under forced excitation at discrete time steps, including geometric nonlinearities) are therefore performed and complex dynamics effects are observed.
Modeling of geometric, material and damping nonlinearities in piezoelectric energy harvesters
The performance of piezoelectric energy harvesters (PEHs) operating in the anticipated vibration environments depends upon various nonlinearities existing in electromechanical dynamic system. In this paper, the influence of geometric, material and damping nonlinearities on the dynamic response of PEH is investigated. So far, the nonlinear geometric finite element analysis has not been used for analyzing PEHs. Moreover, the criterion for inclusion/exclusion of geometric nonlinearity in analyzing PEHs is not studied in detail. In this paper, firstly, the finite element modeling is used for analyzing the effect of geometric nonlinear-ity for low-frequency vibration sources. Simulations are carried out using ANSYS for different PEH configurations to assess the influence of geometric non-linearity on harvester's performance, and a parameter is proposed to determine the inclusion/exclusion of geometric nonlinearity in the analyzing PEHs. Subsequently , a nonlinear electromechanical model considering material and damping nonlinearities is derived for a cantilever-type PEH which uses macro-fiber composite (MFC) for power generation. In the earlier works on PZT-5A and PZT-5H, the nonlinear elastic and damping coefficients were identified by matching the analytical responses (e.g., voltage or displacement) with a set of experimental responses, which is an indirect method and might result in erroneous estimation of unknown coefficients as pointed out in the present work. In this study, the material behavior of MFC is directly obtained from tensile tests. The observed nonlinear stress–strain behavior of MFC is included in the nonlinear model to study the dynamics of the harvester. Energy harvesting experiments are conducted, and the harvester's response is compared with the predictions from the proposed nonlinear model. The comparison shows very good agreement between the experimental and predicted responses. Moreover, the damping parameters are identified using the energy balance method, and it is shown that nonlinear damping is essential in accurate modeling of PEHs.
Issues in mathematical modeling of piezoelectric energy harvesters
Smart Materials and Structures, 2008
The idea of vibration-to-electric energy conversion for powering small electronic components by using the ambient vibration energy has been investigated by researchers from different disciplines in the last decade. Among the possible transduction mechanisms, piezoelectric transduction has received the most attention for converting ambient vibrations to useful electrical energy. In the last five years, there have been a considerable number of publications using various models for the electromechanical behavior of piezoelectric energy harvester beams. The models used in the literature range from elementary single-degree-of-freedom (SDOF) models to approximate distributed parameter models as well as analytical distributed parameter solution attempts. Because of the diverse nature of researchers working in energy harvesting (including electrical, mechanical and materials engineers), several oversimplified and incorrect physical assumptions have been propagated in the literature. Issues of the correct formulation for piezoelectric coupling, correct physical modeling, use of low fidelity models, incorrect base motion modeling, and the use of static expressions in a fundamentally dynamic problem are discussed and clarified here. These common indiscretions, which have been repeated in the existing piezoelectric energy harvesting literature, are addressed and clarified with improved models, and examples are provided. This paper aims to provide corrections and necessary clarifications for researchers from different engineering disciplines interested in electromechanical modeling of piezoelectric energy harvesters.
Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation
Nonlinear Dynamics
A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.
Shock and Vibration
A practical piezoelectric vibration energy harvesting (PVEH) system is usually composed of two coupled parts: a harvesting structure and an interface circuit. Thus, it is much necessary to build system-level coupled models for analyzing PVEH systems, so that the whole PVEH system can be optimized to obtain a high overall efficiency. In this paper, two classes of coupled models are proposed by joint finite element and circuit analysis. The first one is to integrate the equivalent circuit model of the harvesting structure with the interface circuit and the second one is to integrate the equivalent electrical impedance of the interface circuit into the finite element model of the harvesting structure. Then equivalent circuit model parameters of the harvesting structure are estimated by finite element analysis and the equivalent electrical impedance of the interface circuit is derived by circuit analysis. In the end, simulations are done to validate and compare the proposed two classes ...
Equivalent circuit modeling of piezoelectric energy harvesters
Last decade has seen growing research interest in vibration energy harvesting using piezoelectric materials. When developing piezoelectric energy harvesting systems, it is advantageous to establish certain analytical or numerical model to predict the system performance. In the last few years, researchers from mechanical engineering established distributed models for energy harvester but simplified the energy harvesting circuit in the analytical derivation. While, researchers from electrical engineering concerned the modeling of practical energy harvesting circuit but tended to simplify the structural and mechanical conditions. The challenges for accurate modeling of such electromechanical coupling systems remain when complicated mechanical conditions and practical energy harvesting circuit are considered in system design. In this article, the aforementioned problem is addressed by employing an equivalent circuit model, which bridges structural modeling and electrical simulation. First, the parameters in the equivalent circuit model are identified from theoretical analysis and finite element analysis for simple and complex structures, respectively. Subsequently, the equivalent circuit model considering multiple modes of the system is established and simulated in the SPICE software. Two validation examples are given to verify the accuracy of the proposed method, and one further example illustrates its capability of dealing with complicated structures and non-linear circuits.
Institute of Physics
A new electromechanical finite element modelling of a vibration power harvester and its validation with experimental studies are presented in this paper. The new contributions for modelling the electromechanical finite element piezoelectric unimorph beam with tip mass offset under base excitation encompass five major solution techniques. These include the electromechanical discretization, kinematic equations, coupled field equations, Lagrangian electromechanical dynamic equations and orthonormalized global matrix and scalar forms of electromechanical finite element dynamic equations. Such techniques have not been rigorously modelled previously by other researchers. There are also benefits to presenting the numerical techniques proposed in this paper. First, the proposed numerical techniques can be used for applications in many different geometrical models, including micro-electro-mechanical system power harvesting devices. Second, applying tip mass offset located after the end of the piezoelectric beam length can result in a very practical design, which avoids direct contact with piezoelectric material because of its brittle nature. Since the surfaces of actual piezoelectric material are covered evenly with thin conducting electrodes for generating single voltage, we introduce the new electromechanical discretization, consisting of the mechanical and electrical discretized elements. Moreover, the reduced electromechanical finite element dynamic equations can be further formulated to obtain the series form of new multimode electromechanical frequency response functions of the displacement, velocity, voltage, current and power, including optimal power harvesting. The normalized numerical strain node and eigenmode shapes are also further formulated using numerical discretization. Finally, the parametric numerical case studies of the piezoelectric unimorph beam under a resistive shunt circuit show good agreement with the experimental studies.
Intrinsic electromechanical dynamic equations for piezoelectric power harvesters
Springer-Verlag, 2017
Acta Mechanica, vol. 228 (2), pp 631–650, 2017 This paper discusses, compares and contrasts two important techniques for formulating the electromechanical piezoelectric equations for power harvesting system applications. It presents important additions to existing literature by providing intrinsic formulation techniques of the harvesting system for the two different electromechanical dynamic equation-based voltage and charge-type systems associated with the standard AC–DC circuit interface developed using the extended Hamiltonian principle. The derivations of the two analytical methods rely on the fundamental continuum thermopiezoelectricity concepts of the electrical enthalpy energy and Helmholtz free energy. The benefit of using analytical charge-type modelling is that the technique shows more compact formulation for developing simultaneous derivations by coupling the mechanical and electromechanical systems of the piezoelectric devices and electronic system so that the frequency response functions (FRFs) and time wave form systems can be formulated. On the other hand, the analytical voltage-type modelling is obviously convenient but can show tedious derivation process for joining with the electronic circuit part. To tackle this situation, the analytical voltage type with mechanical and electromechanical forms of the piezoelectric structure can be derived separately from the electronic system where they can be combined together after applying further derivations to formulate the FRFs. In this paper, the two analytical techniques also show particular benefit and even further development of how to model the power harvesting scheme with the combinations of piezoelectric structure and electronic system. Moreover, validations of the two analytical methods show good agreement with previous authors’ electromechanical finite element analysis and experimental works. Further parametric electromechanical energy harvesting behaviours have been explored to study the system responses.
A dimensionally reduced order piezoelectric energy harvester model
Energy, 2018
Presently reduced power requirement for small electronic components have been the main motivation for developing vibration based energy harvesting. The ultimate objective in this research field is to provide an easy, sustainable and efficient technology to power such small electronic devices from the unused vibrational energy available in the environment. A comprehensive, reliable mathematical technique is thus in high demand which can model a piezoelectric energy harvester, predict its coupled dynamics (structural and electromechanical) accurately. The present work focuses on developing a mathematical model for a slender, piezoelectric energy harvester based on Variational Asymptotic Method, a dimensional reduction methodology. Variational Asymptotic Method approximates the 3D electromechanical enthalpy as an asymptotic series to formulate an equivalent 1D electromechanical enthalpy functional to perform a systematic dimensional reduction. For validation purpose, we have picked up experimental results for a bimorph PZT harvester, available in the literature. We have studied the extension-bending structural coupling along with the parameter dependence of the voltage, power output from the harvester and validated with the experiments. The present study provides an unique, accurate modelling technique which is capable of capturing material anisotropy, structural coupling and can analyse arbitrary cross section, surface mounted as well as embedded piezo layered energy harvester.
Advanced Model for Fast Assessment of Piezoelectric Micro Energy Harvesters
Frontiers in Materials, 2016
The purpose of this work is to present recent advances in modeling and design of piezoelectric energy harvesters, in the framework of micro-electro-mechanical systems (MEMS). More specifically, the case of inertial energy harvesting is considered, in the sense that the kinetic energy due to environmental vibration is transformed into electrical energy by means of piezoelectric transduction. The execution of numerical analyses is greatly important in order to predict the actual behavior of MEMS devices and to carry out the optimization process. In the common practice, the results are obtained by means of burdensome 3D finite element analyses (FEA). The case of beams could be treated by applying 1D models, which can enormously reduce the computational burden with obvious benefits in the case of repeated analyses. Unfortunately, the presence of piezoelectric coupling may entail some serious issues in view of its intrinsically threedimensional behavior. In this paper, a refined, yet simple, model is proposed with the objective of retaining the Euler-Bernoulli beam model, with the inclusion of effects connected to the actual three-dimensional shape of the device. The proposed model is adopted to evaluate the performances of realistic harvesters, both in the case of harmonic excitation and for impulsive loads.