Bond Graph Model of a SHM Piezoelectric Energy Harvester

Transcription

1 COVER SHEET NOTE: This coversheet is intended for you to list your article title and author(s) name only this page will not appear in the book or on the CD-ROM. Title: Authors: Bond Graph Model of a SHM Piezoelectric Energy Harvester Sainthuile Thomas Delebarre Christophe Grondel Sébastien Paget Christophe PAPER DEADLINE: May 5, 20 PAPER LENGTH: INQUERIES TO: 8 PAGES (Maximum) Sainthuile Thomas IEMN-DOAE UMR CNRS 8520 UVHC, Le Mont-Houy, 5933 Valenciennes Cedex 9, France thomas.sainthuile@univ-valenciennes.fr Please submit your paper in Microsoft Word format or PDF if prepared in a program other than MSWord. We encourage you to read attached Guidelines prior to preparing your paper this will ensure your paper is consistent with the format of the articles in the CD-ROM. NOTE: Sample guidelines are shown with the correct margins. Follow the style from these guidelines for your page format. Hardcopy submission: Pages can be output on a high-grade white bond paper with adherence to the specified margins (8.5 x inch paper. Adjust outside margins if using A4 paper). Please number your pages in light pencil or non-photo blue pencil at the bottom. Electronic file submission: When making your final PDF for submission make sure the box at Printed Optimized PDF is checked. -- Also in Distiller make certain all fonts are embedded in the document before making the final PDF.

2 (FIRST PAGE OF ARTICLE) ABSTRACT This paper presents a Bond Graph model of a Structural Health Monitoring (SHM) piezoelectric sensor used as a power harvester. A Finite Element Method (FEM) model of the sensor bonded onto a structure subjected to low frequency mechanical vibration has been developed. The forces applied to the sensor and measured from the FEM model have been implemented in the Bond Graph model. An analytical model, providing the voltage response of the sensor subjected to an in-plane displacement field and a simplified FEM model representing the sensor alone have been used to check the Bond Graph model. The Bond Graph model provides good correlation with the analytical and simplified FE model. Bond Graph models also describe the power distribution inside a system. As a result it has been possible with this model to estimate the power harvesting capabilities of different piezoelectric sensors under natural mechanical vibrations. INTRODUCTION Nowadays, SHM has become a part of new avionic systems which will form future aircrafts. To allow their deployment throughout the aircraft, they have to be autonomous. Vibration Energy Harvesting is a promising solution to provide energy to such systems. The concept developed herein uses the SHM piezoelectric transducers to convert the mechanical vibration of the structure they are bonded on into electrical power []. This SHM system aims to have a double functionality since it will carry out classical SHM tasks but will also be fully autonomous. An analysis of the piezoelectric harvester itself is of real importance to estimate both the amount of power that can be effectively harvested but also to obtain a better comprehension of the whole process of piezoelectric energy conversion [2, 3, 4]. Consequently, in this paper, an original Bond Graph model of the SHM piezoelectric energy harvester is proposed. Thomas Sainthuile, Christophe Delebarre, Sébastien Grondel, IEMN-DOAE UMR CNRS 8520 UVHC, Le Mont-Houy, 5933 Valenciennes, France. thomas.sainthuile@univ-valenciennes.fr Christophe Paget, Airbus Operations Limited, Bristol BS99 7AR, UK. christophe.paget@airbus.com

3 First, the general configuration is presented. Then the model of the piezoelectric energy harvester is built. Furthermore, a verification of the model is done with FEM and analytical models comparison. Finally, a study of the power transfer using this model is carried out. GENERAL CONFIGURATION The aim of this study is to determine the behaviour of piezoelectric SHM sensors integrated in the aircraft and submitted to low frequency mechanical vibrations. One has to know both the voltage response of a piezoelectric harvester and its power harvesting capabilities. To carry out this study, a FEM model of an aluminium specimen instrumented with a piezoelectric SHM sensor has been developed using Comsol Multiphysics. The plate has one fixed edge while the others remain free, resulting in a cantilever configuration. The plate is subjected to low frequency mechanical vibrations, as shown in Figure. The sensor length is noted L, its width W and its thickness T. The aluminium plate is 500mm long, 300mm wide and 3mm thick. The working frequency has been chosen as 589Hz to generate a flexion mode. This ensures having the displacement along the length only x-axis dependant. For the piezoelement considered herein, electric charges are only present on the electrodes and only the electric field and electrical displacement components parallel to the polarization direction are non null i.e. D =D 2 =0 and E =E 2 =0. The thickness of the piezoelectric sensor is small compared to its length and width and also small compared to the structure thickness. As a result, one can assume that the deformation of the sensor will be mainly radial [5, 6]. The FEM model shows that the sensor is subjected to compression along its length, noted u(x), and, due to Poisson s effect, there is also an expansion of its width, noted v(y). Moreover, at 589Hz, the deformation wavelength is equal to 230mm. Consequently, the displacement applied to the sensor can be considered as linear along its length and its width. The displacement measured from this FEM model will be used in the analytical model. The equivalent force applied to the piezoelement will also be determined from this measure of displacement and used as the input of the Bond Graph model. Piezoelectric sensor Force Fixed edge 2,y 3,z,x Figure : aluminium specimen instrumented with a piezoelectric SHM power harvester sensor

4 MODELLING OF THE PIEZOELECTRIC ENERGY HARVESTER The piezoelectric harvester will be modelled as a single degree-of-freedom oscillator in its length direction and another one in its width direction to take into account both the compression along x-axis and expansion along y-axis. The mechanical part of the piezoelectric energy harvester is modelled as a {Mass + Spring + Damper} system as shown in Figure 2 [2, 4, 7]. The mass M represents the mass of the piezoelectric sensor, the spring (stiffness k x ) represents the storage of mechanical energy while the damper (η x ) relates to the structural damping of the piezoelectric sensor. The electromechanical conversion is modelled considering an ideal transformer. The electrical behaviour of the piezoelectric energy harvester is also characterized by the piezoelement static capacitance. Equivalent circuit modelling has been extensively used to represent the piezoelectric energy harvester. Based on this technique, a Bond Graph model of the piezoelectric energy harvester is proposed. The Bond Graph approach is well suited to this lumped parameters modelling since it will be possible to investigate the behaviour of each lumped parameter distinctively. This model will be checked with the FEM model of the plate and piezoelement. An analytical model will be used to obtain the voltage response of the thin piezoelectric sensor and will be compared to the Bond Graph model. As for equivalent circuit, the bond graph model is based on the constitutive piezoelectric equations that can be written as: i ij T ij D = ε E + d T T = C E ijkl j S kl ikl e kl kij E k () Under plane stress condition and in open circuit condition, the analytical model provides the voltage response of the sensor [5, 6]: L W T V = e S 3 22 ε LW ( S + S ) dx dy (2) with u S = L v S 22 = (3) W Figure 2: model of a piezoelectric sensor under mechanical vibration

5 The equivalent circuit and the Bond Graph of Figure 2 are presented in Figures 3 and 4. To represent both the compression and expansion phenomenons, two branches have been modelled. They are connected in parallel since they are equivalent to current sources after electromechanical transformation. The voltage source corresponds to the force applied to the piezoelement. Bond Graph I-elements represent the kinetic energy of the structure, modelled by the mass M of the piezoelement. The R element represents the mechanical energy dissipation modelled by the mechanical damping η and η 2 of the piezoelement. In this context of natural low frequency vibrations, one can neglect this contribution. These elements will still be implemented in the model but with a null value. The C-elements represent the potential energy, modelled by the inverse of the mechanical stiffness k, and k 2 of the structure along its length and width, respectively. The electromechanical energy conversion is modelled with two TF elements for the two branches. The static capacitance C s is the electrical parameter. The test being conducted in harmonic domain, one has to use a modulated source represented by the sinusoidal shape and the MSe element which provides a modulated effort. The resistive load appears in these models and will be developed in the next section. mechanical part electromechanical conversion electrical part f I R C θ f S F e C S e S R load f 2 I 2 R 2 C 2 θ 2 F 2 e 2 Figure 3: equivalent circuit representation of the piezoelectric power harvester R C R C Sine e f MSe MSe θ TF I e s C C S I 0 f s R C P R 2 C 2 R load R Sine 2 e 2 f 2 MSe MSe 2 TF θ 2 I I 2 Figure 4: Bond Graph model of the piezoelectric power harvester

6 From this Bond Graph model, one obtains the following equations, similar to the equivalent circuit ones. Only one branch will be considered to describe the parameters. with u the displacement of the sensor. df ( t) + R f ( t) + f ( t) dt + Θ e ( t) F ( t) (4.a) I s = dt C des ( ) ( t) Θ f t C (4.b) = s dt du f = (5) dt The density of the piezoelement is noted ρ and the stiffness matrix is (Cij) <i,j<6 The I element is the mass of the piezoelement. I = ρ LWT (6) The mechanical stiffness is k C W T k = = (7) L C The static capacitance is C S Equation (2) can be written as and (4.b) gives C T LW = (8) T S ε 33 T u = es = e (9) S ε L V 3 33 θ V = u (0) Cs T Te3C S e3ε 33 W = () S S ε 33 L ε 33 θ = From (4.a), the force F becomes 2 F I ² u C C θ = ω + (2) s The same calculations have been performed to obtain k 2, θ 2 and F 2.

7 MODEL VERIFICATION To check the Bond Graph model, several sensors lengths have been tested. They vary from 0mm to 30mm. The thickness of the sensor is 0.5mm and its width is 3mm. An harmonic excitation of 0N has been applied to the plate at its free edge, as shown in Figure. The displacements have been measured with the FE model and applied to the analytic model which gives the voltage response of a sensor submitted to displacement along its length and its width, as shown in Equation 2. A simplified FEM model of the sensor alone has also been built and provides the voltage response of the sensor subjected to the same displacements. The forces determined with equations 2 and 5, have been applied to the Bond Graph model as the input forces F and F 2. The Bond Graph parameters have been calculated for the different sensors dimensions. Figure 5 presents the voltage across the Cs element, i.e. across the piezoelement without external load for the different models. This result shows that the Bond Graph model correctly simulates the behaviour of the piezoelectric harvester. The model has been able to predict the voltage variation depending on the sensor dimensions and the displacements applied to the sensors. Moreover, the voltage from the Bond Graph is close to the analytic results and simplified FE model. The analytical model and the simplified FEM model give almost the same voltage since curves are superimposed. The amplitude difference between the Bond Graph model and the complete FE model may be due to the displacement estimation. The plane stress assumption is not completely verified and the displacements along the width and thickness have been approximated as linear. These simplifications induce errors in the displacement determination and create this difference, as shown in Figure 5. However, with the Bond Graph model, one can rapidly know which dimensions ensure the best voltage response for a given set-up. The Bond Graph model is an efficient tool to work on design optimization. Figure 5: voltage response of PZT harvester using various models

8 POWER TRANSFER ESTIMATION A major feature of Bond Graph is that it also brings the possibility of measuring energy exchange between elements. As a result, one can estimate the power harvesting capabilities of the piezoelectric element. To do so, a resistive load and a power sensor have been added to the previous model to measure the amount of power harvested as shown in Figure 4. Simulations have been run for different load values, as shown in Figure 6. For this configuration, the sensor can harvest and transfer a maximum of 86µW to a 0kΩ load. With the Bond Graph approach, one also has the possibility to monitor the power distribution inside the model, as shown in Figure 7. It shows that the displacement resulting from the force applied to the sensor is mainly converted into potential energy due to the mechanical stiffness of the sensor. It also shows that the mass does not have a significant effect since the working frequency is far below the sensor resonant frequency. The Bond Graph model provides an idea of the electromechanical efficiency of the piezoelectric power harvester Figure 6: power from Bond Graph model versus resistive load. Sensor length: 22.5mm Figure 7: power distribution throughout the conversion process

9 CONCLUSION: A numerical model representing a piezoelectric element subjected to a two dimensional deformation field has been proposed. By considering a thin sensor, it has been possible to measure the displacement along the piezoelement length and width with a FEM model. Several lengths have been tested to validate the model. The measured displacements have been applied to the analytic model to obtain the voltage response of the sensor in open-circuit conditions. A simplified FEM model of the sensor itself has been developed. The voltage response of the sensor subjected to the measured displacement has been determined. The corresponding forces determined analytically have been used in the Bond Graph model. A good correlation has been obtained between all models behaviours. There is still an amplitude difference between the approximated models and the complete FE model. Assumptions made for the displacement measures may explain this difference. Further work will focus on the improvement of these measures to match the results and will include an experimental verification. Nevertheless, the Bond Graph model provides the voltage level for a given mechanical excitation and also provides the amount of harvested power from mechanical vibrations for all sensors tested. Moreover, it has been possible to monitor the power distribution and highlight the effects of the geometric and mechanical parameters on the power transmitted to a resistive load. As a result, one can estimate the efficiency of the piezoelectric conversion for various mechanical excitation levels and this Bond Graph model may serve as a tool to find an efficient geometrical configuration to optimize the level of harvested energy. The Bond Graph model is a good candidate to optimize a power harvesting piezoelectric sensor. REFERENCES. Sainthuile, T. Delebarre, C. Grondel, S and Paget, C Vibrational Power Harvesting for Wireless PZT-based SHM applications. Proceedings of the 5th European Workshop on Structural Health Monitoring, Sorrento, Italy, Roundy, S and Wright, P.K A piezoelectric vibration based generator for wireless electronics. Smart Materials And Structures, 3: Yang, Y and Tang, L Equivalent Circuit Modeling of Piezoelectric Energy Harvesters. J. of Intelligent Material, Systems and Structures, 20: Elvin, N.G. Elvin, and A.A A General Equivalent Circuit Model for Piezoelectric Generators. J. of Intelligent Material, Systems and Structures, 20: Di Scalea, S.L. Matt, H. and Bartoli, I The response of rectangular piezoelectric sensors to Rayleigh and Lamb ultrasonic waves. J. Acoust. Soc. Am, : Chapuis, B. Contrôle Santé Intégré par méthode ultrasonore des réparations composites collées sur des structures métalliques PhD thesis. Université Denis Diderot - Paris VII. 7. Lefeuvre, E. Badel,, A. Richard, C. Petit, L. and Guyomar, D A comparison between several piezoelectric generators for standalone vibration-powered systems. Sensors and Actuators, A 26: