POWER DENSITY OPTIMIZATION OF AN ARRAY OF PIEZOELECTRIC HARVESTERS USING A GENETIC ALGORITHM

Transcription

1 International Worksho SMART MATERIALS, STRUCTURES & NDT in AEROSPACE Conference NDT in Canada 11-4 November 11, Montreal, Quebec, Canada POWER DENSITY OPTIMIZATION OF AN ARRAY OF PIEZOELECTRIC HARVESTERS USING A GENETIC ALGORITHM S. PAQUIN and Y. ST-AMANT LSMI, Mechanical Engineering Deartment, Université Laval Yves.St-Amant@gmc.ulaval.ca ABSTRACT In the last decade, several studies have shown an increasing interest in harvesting energy from vibrations. A common way to roceed consists of using integrated iezoelectric ceramics into a vibrating structure excited by a harmonic source. However, literature has shown that this kind of harvester is only effective under a narrow bandwidth of excitation frequency. If the excitation frequency shifts from this band, the ower density of the harvester will significantly decrease. Different strategies have been used to enhance the harvesting erformances when the vibration source has a larger frequency bandwidth. One of them is to use an array of harvesters which consists of multile harvesters having different resonance frequencies in order to increase the harvested ower on a wider frequency bandwidth. However, this leads to a harvester having a higher volume, which decreases its ower density (mw.cm -3 ). Design otimization can hel to attenuate this decrease, but it has never been investigated to our knowledge. In this aer, we investigate the ossibility to maximize the ower density for a vibration source having a wider frequency bandwidth, using an array of two iezoelectric energy harvesters. It is shown that the otimization rocess can limit the decrease of the mean ower density to 6% by adding a second harvester while increasing the mean ower by 75%. Keywords: Piezoelectric energy harvester, wideband vibrations, array of harvesters, genetic algorithm otimization, Rayleigh-Ritz method, ower density. 11 CANSMART CINDE IZFP

2 INTRODUCTION Mechanical energy harvesting from vibrating structures using iezoelectricity has been studied in several works of the last decade. It has been shown that a vibration energy harvester is only effective when the first resonance frequency of the structure matches the frequency of a harmonic vibration source. Different strategies have been roosed to increase the harvesting erformances over a wider frequency bandwidth. For instance, Eichhorn et al. [1] have roosed to aly a mechanical stress to the converter in order to tune its resonance frequency. Moss et al. [] have designed a harvester that u-converts a low-frequency excitation source to its resonance frequency by using imacts. Ferrari et al. [3] have roosed an aroach based on a bistable nonlinear oscillator in order to use its instability state to widen the harvester s effective bandwidth. Xue et al. [4] have roosed a multi-mode structure comosed of ten iezoelectric bimorhs with different resonance frequencies. This structure increases the harvested ower over a wide range of excitation frequencies, but decreases the ower density of the harvester. This aer investigates the design otimization of an array of two iezoelectric harvesters by using a genetic algorithm. An electromechanical model is first described. This model redicts the ower density of an energy harvester using a semi-analytical formulation based on Rayleigh-Ritz method. The model is next integrated into a genetic algorithm in order to maximize this ower density. Otimization roblems are then formulated and solved for a single harvester (SH) and for an array of two harvesters (AH). Electromechanical model PIEZOELECTRIC ENERGY HARVESTING MODEL The model detailed in this study is based on the work of Sodano et al. [5] who established the constitutive equations of a iezoelectric bimorh cantilever beam like the one shown in Figure 1. Fig. 1: Electromechanical model of a iezoelectric energy harvester. A base acceleration a(t) roduces flexural vibrations to the bimorh while the iezoelectric ceramics generate a charge flow q& ( t). The harvested ower P is modeled by the ower dissiated across a resistive load R. One can show that the electromechanical model of this structure can be reresented by the following set of differential equations: Mh &&(t) + Ch &(t) + Kh (t) +θrq& ( t ) = Da(t) θh(t)- C Rq& ( t) - q( t ) = (1) 11 CANSMART CINDE IZFP

3 where h(t) is the modal time resonse vector. In these equations, M is the modal mass matrix, K is the modal stiffness matrix, C is the modal daming matrix, θ is the electromechanical couling vector, D is the inut mass vector and C is the caacitance of the iezoelectric volume. These arameters are defined by: T T T T M = ρ φ (x) φ (x) dv + ρ φ (x) φ (x) d V + ρ y φ '(x) φ '(x) d V + ρ y φ '(x) φ '(x) dv b b b b Vb V Vb V T T K = E y φ ''(x) φ ''(x) d V + E y φ ''(x) φ ''(x) dv Vb C = α M + β K θ = d E y φ'' (x) ψ (y) dv 31 b b V V D = ρ φ (x) dv + C Vb S = ε ψ (y) dv ρ φ (x) dv b b V V () where ρ b, ρ are material densities, E b, E are modulus of elasticity, V b, V are the volumes, d 31 is the iezoelectric charge constant, ε S is the ermittivity constant, ψ(y) secifies the electrical field across the iezoelectric volume, φ(x) is the transverse dislacement distribution vector and ( ' ) denotes the first derivative with resect to x. α and β are the mechanical daming coefficients which reresent the roortional daming in the structure. It should be noted that the subscrits b and refer resectively to the beam and the iezoelectric ceramics. By assuming a harmonic base acceleration having an amlitude A and a circular frequency ω, one can solve the differential equations system (1) and obtain the following steady-state solution: h(t) H = sin( ωt φ ) (3) q(t) Q where H is the amlitude vector of each modal time resonse, Q is the amlitude of the charge function and ϕ is the hase angle vector. One can then determine the harvested ower with: R( ωq) P = (4) In order to determine the harvested ower, only the transverse dislacement distribution vector φ(x) still remains to be determined from the mechanical vibrations. A semi-analytical aroach is resented in the following. 11 CANSMART CINDE IZFP

4 Semi-analytical mechanical model Figure shows the harvester under study. It is comosed of a monolithic cantilever beam and two iezoelectric ceramics. L, b and t are resectively the length, the width and the thickness while the subscrits 1, and refer resectively to the first section of the beam, the second section of the beam and one of the two iezoelectric ceramics. Fig. : Mechanical model of the iezoelectric energy harvester. Using Bernouilli s assumtions, the harvester s dislacement field u can be defined as: ux y w'( x, t) u = u y = w( x, t) (5) u z where w(x,t) is the transverse dislacement along the beam. Using Rayleigh-Ritz aroach, one can aroximate w(x,t) as: T { } { } w( x, t) = f ( x) g ( t) = f ( x) f ( x)... f ( x) g ( t) g ( t)... g ( t) (6) 1 N 1 where f(x) and g(t) are resectively the trial functions vector and the time functions vector. Using this formulation, the harvester is then aroximated as a N degree-of-freedom system. The trial functions set used in this study is a hierarchical trigonometric set defined as: N T ai x ci x fi ( x) = sin + bi sin + di L L i = 1,,..., N (7) This set has numerous advantages (e.g. numerically more stable) which are well discussed in [6]. Mode shaes can then be found using Lagrange s equations with the generalized coordinates g i (t): 11 CANSMART CINDE IZFP

5 d T T U + = dt g& i gi gi i = 1,,.., N (8) where T and U are resectively the kinetic energy and otential energy. T and U can be defined as: 1 L w( x, t) L w( x, t) T = ρ ( ) d ( ) d b Ab x x + ρ A x x t t L w( x, t) L w( x, t) + ρ ( ) d ( ) d bib x x + ρ I x x x t x t 1 L w( x, t) L w( x, t) U = E ( ) d ( ) d bib x x E + I x x x x (9) where A b (x), A (x) are the cross-section area functions and I b (x), I (x) are the area moment of inertia functions. Next, (9) is inserted in (8) to determine a differential equations system. These equations can be reresented using a matrix formulation, i.e.: Mg %&&(t) + Kg % (t) = (1) where %M and % K are resectively the mass and stiffness matrices. The eigenvalue roblem can be solved in order to find the mode shaes [ 1... N ] of the structure. Knowing the mode shaes of the harvester, it is ossible to determine the transverse dislacement distribution vector φ(x) of the system which is defined by: [ ] T φ 1 N ( x) =... f ( x) (11) This model has been reviously validated in [7] for a similar harvesting structure. GENETIC ALGORITHM OPTIMIZATION Genetic algorithm is an otimization method based on biological evolution which can otimize highly comlex cost functions. The otimization roblem solution is obtained by an iterative rocess by using the concets of natural selection and mutation of individuals forming a oulation. The rocess stos according to a defined criterion of convergence. Figure 3 shows the flowchart of a tyical genetic algorithm. 11 CANSMART CINDE IZFP

6 Fig. 3: Flowchart of a tyical genetic algorithm. The genetic algorithm rocess works according to the following rocedure: Each harvester constitutes an individual of a oulation and is reresented by a chromosome comosed of genes which deend on the number of variables. Each gene is reresented by a binary number comosed of a secific number of bits. The initial oulation is randomly created by a secific number of individuals, their cost functions are evaluated and the individuals with the higher cost function are selected to comose the first generation. Poulation ranking is then executed in regard of the evaluation cost of each individual to aly a natural selection. The airing of arents is determined in order to roceed to the mating which is the first rocess of the algorithm to exlore the cost function. Offsring of the arents are introduced into the oulation. The second rocess to exlore the cost function is the mutation which changes the genetic code of some individual. This rocess consists of randomly ermute a ercentage of bits of the oulation genetic code. At the end of this rocess, cost functions of the muted individuals are evaluated. This iterative rocess is reeated until the cost function reached a criterion of convergence. For a more detailed analysis of this rocess, the reader is referred to Haut and Haut [8]. Otimization simulation arameters POWER DENSITY MAXIMIZATION Some arameters of the harvester are set to be constant: The harvester is comosed of a brass beam and two PZT-5H ceramics. The mechanical daming coefficients α and β are chosen to set the first daming ratio to %, which is a tyical value. 11 CANSMART CINDE IZFP

7 The harvester is driven by a base acceleration amlitude A of 1g, which is an arbitrary value frequently used in the literature. Material roerties and constant geometric dimensions are gathered in Table 1. Table 1: Material roerties and constant geometric dimensions. Harvester roerties Symbol Value Brass density ρ b 841 kg.m -3 PZT-5H density ρ 78 kg.m -3 Brass modulus of elasticity E b 13.4 GPa PZT-5H modulus of elasticity E 6 GPa Permittivity of ceramics ε S 7.3 nf.m -1 Piezoelectric charge constant d 31-3 C.N -1 Width of the iezoelectric ceramic b 5 mm Length of the iezoelectric ceramic L 5 mm Thickness of the iezoelectric ceramic t.7 mm Width of the first section of the beam b 1 5 mm Width of the second section of the beam b 4 mm Length of the first section of the beam L 1 51 mm Thickness of the second section of the beam t 5 mm In regards of the rocess of the genetic algorithm discussed reviously, Table shows the otimization arameters used for the simulations. In ractice, a lot of simulations have been erformed and these arameters have led to a good trade-off between otimization comuting time and accuracy. Table : Simulation arameters for the genetic algorithm otimization. Simulation arameters Value Number of individuals in a oulation 1 Number of bits by genes 8 Proortion of mutation.5 Number of generation having the same maximum 15 Two otimizations roblems are formulated in the following. Otimization I Mean ower density otimization The cost function of this first otimization roblem is to maximize the mean ower density γ m over a certain frequency bandwidth. The ower density γ is defined as the ratio of the harvested ower (see equation (4)) and the harvester volume (V = b 1 L 1 t 1 + b L t + b L t ). The mean ower density γ m over a secified bandwidth [f 1, f ] is then defined by: γ γ = = m R ( ωq) f f df df f1 f 1 f (1) f 1 ( f f1 )( b1 L1t 1 + b Lt + b Lt ) 11 CANSMART CINDE IZFP

8 In this work, f 1 and f are arbitrary set to 9 and 11 Hz. For the otimization of the SH, the length L, the thickness t 1 and the resistive load R have to be otimized by the genetic algorithm. The otimization roblem can then be summarized as follows: Find the maximum of: γ m (µ 1 ) (13) with µ 1 = [L, t 1, R] Subject to: 1 L 4 [mm].75 t 1. [mm] 1 R 1 [kω] In this aer, ranges for variables have been arbitrary set. For an AH, reliminary simulations have shown that the otimization simly leads to two identical otimal SHs. This otimal AH allows harvesting of twice as much ower than the otimal SH and it reresents the same ower density. However, the bandwidth of excitation frequency where the harvester is effective is still narrow and the ower distribution is not wider than the otimal SH. So, another otimization aroach is hereafter roosed to widen the ower distribution. Otimization II Weighted mean ower density otimization In order to obtain a more uniform ower density distribution in the frequency domain, we roose to modify the revious cost function by adding a weight factor corresonding to the ratio of the mean ower density γ m and the maximum ower density γ M of the distribution, i.e. γ m /γ M. For the otimization of the AH, six variables must be otimized and the otimization roblem can be summarized as follow: Find the maximum of: γ m (µ ) γ m (µ ) / γ M (µ ) (14) with µ = [L A, t 1A, R A, L B, t 1B, R B ] Subject to: 1 L A 4 [mm].75 t 1A. [mm] 1 R A 1 [kω] 1 L B 4 [mm].75 t 1B. [mm] 1 R B 1 [kω] Otimizations results Table 3 summarizes the results of both otimizations. The design obtained by otimization I is a SH having a volume of 5.7 cm 3 while the design obtained with otimization II is an AH having a volume of 1.7 cm 3. The volume of the otimal AH is thus 88% higher than the one of the otimal SH. 11 CANSMART CINDE IZFP

9 Table 3: Results of both otimization roblems. Otimisation I Otimisation II Variable Symbol Single Harvester Harvester harvester A B Thickness of the first section of the beam t mm 1.8 mm 1.5 mm Length of the second section of the beam L mm 15.9 mm mm Resistive load R 6.39 kω 6.8 kω 6.36 kω Figure 4 (a) comares the harvested ower for an otimal SH and an otimal AH as a function of the excitation frequency. The mean harvested ower of the SH is 9.3 mw while the one of the AH is 16.3 mw. It means that the AH is able to harvest a ower 75% higher than the one of the SH. Furthermore, the minimal ower harvested by the AH is 364% higher than the SH (8.41 mw versus.31 mw), which is an advantage if a continuous minimal harvested ower is required for an alication. Figure 4 (b) shows the ower density of the otimal SH and the otimal AH as a function of the excitation frequency. The mean ower density of the SH is 1.6 mw.cm -3 while that of the AH is 1.5 mw.cm -3. It reresents a decrease of 6% of the mean ower density. However, one can observe that the ower density is higher for the AH when the excitation frequency is between 9 and 94.4 Hz and, between 1.4 and 11 Hz, which reresents 6% of the frequency bandwidth under study. Fig. 4: (a) Harvested ower and (b) ower density as a function of the excitation frequency of the otimal SH (solid line) and the otimal AH (dashed line). 11 CANSMART CINDE IZFP

10 CONCLUSION The main objective of this aer was to maximize the ower density of an array of two iezoelectric harvesters. An electromechanical model using a semi-analytical mechanical model was first described to redict the erformances of a iezoelectric energy harvester. This model was then used to maximize the ower density by using a genetic algorithm. This otimization method was next described and otimization roblems for a single harvester and an array of harvesters were formulated and solved. By using an otimal array of two harvesters, we have shown that it is ossible to increase the mean harvested ower by 75% while limiting the decrease of the mean ower density to 6%. Furthermore, the ower density for the otimal array of two harvesters is higher than that of the otimal single harvester over 6% of the bandwidth of interest. In future works, we will exerimentally validate the erformances of these harvesters. ACKNOWLEDGEMENTS This work was suorted by the Fond Québécois de la Recherche sur la Nature et les Technologies (FQRNT) and Natural Sciences and Engineering Research Council (NSERC). REFERENCES 1. Eichhorn C., Goldschmidtboeing F. and Woias P., Bidirectional Frequency Tuning of a Piezoelectric Energy Converter Based on a Cantilever Beam, J. Micromech. Microeng., Vol 19, 946, 9.. Moss S., Barry A., Powlesland I., Galea S. and Carman G.P., A Broadband Vibro-imacting Power Harvester with Symmetrical Piezoelectric Bimorh-stos, Smart Mater. Struct., Vol, 4513, Ferrari M., Ferrari V., Guizzetti M., Andò B., Baglio S. and Trigona C., Imroved Energy Harvesting from Wideband Vibrations by Nonlinear Piezoelectric Converters, Sens. Actuat. A: Phys, Vol 16, , Xue H., Hu Y., and Wang Q.M., Broadband Piezoelectric Energy Harvesting Devices Using Multile Bimorhs with Different Oerating Frequencies, IEEE Trans. Ultrason. Ferr., Vol 55, , Sodano H.A., Park G. and Inman D.J., Estimation of Electric Charge Outut for Piezoelectric Energy Harvesting, J. Strain, Vol 4, , Beslin O. and Nicolas J., A Hierarchical Functions set for Predicting Very High Order Plate Bending Modes with Any Boundary Conditions, J. Sound Vib., Vol, , Paquin S. and St-Amant Y., Imroving the Performance of a Piezoelectric Energy Harvester Using a Variable Thickness Beam, Smart Mater. Struct., Vol 19, 15, Haut R. L. and Haut S.E., Practical Genetic Algorithms, Wiley Inter-Science, CANSMART CINDE IZFP