MODELLING AND OPTIMISATION OF A BIMORPH PIEZOELECTRIC CANTILEVER BEAM IN AN ENERGY HARVESTING APPLICATION

Transcription

1 Joural of Egieerig Sciece ad Techology Vol. 11, No. (016) 1-7 School of Egieerig, Taylor s Uiversity MODELLING AND OPTIMISATION OF A BIMORPH PIEZOELECTRIC CANTILEVER BEAM IN AN ENERGY HARVESTING APPLICATION CHUNG KET THEIN 1, *, BENG LEE OOI, JING-SHENG LIU 3, JAMES M. GILBERT 3 1 School of Egieerig, Taylor s Uiversity, Taylor's Lakeside Campus, No. 1 Jala Taylor's, 47500, Subag Jaya, Selagor DE, Malaysia Faculty of Itegrative Scieces ad Techology, Quest Iteratioal Uiversity Perak, Plaza Teh Teg Seg (Level ), 7, Jala Permaisuri Baiu, 3050 Ipoh, Perak, Malaysia 3 School of Egieerig, Uiversity of Hull, HU6 7RX, Hull, UK *Correspodig Author: ckthei@gmail.com Abstract Piezoelectric materials are excellet trasducers i covertig vibratioal eergy ito electrical eergy, ad vibratio-based piezoelectric geerators are see as a eablig techology for wireless sesor etworks, especially i selfpowered devices. This paper proposes a alterative method for predictig the power output of a bimorph catilever beam usig a fiite elemet method for both static ad dyamic frequecy aalyses. Experimets are performed to validate the model ad the simulatio results. I additio, a ovel approach is preseted for optimisig the structure of the bimorph catilever beam, by which the power output is maximised ad the structural volume is miimised simultaeously. Fially, the results of the optimised desig are preseted ad compared with other desigs. Keywords: Piezoelectric, Multi-discipliary optimisatio, Shape optimisatio, Eergy harvestig, Bimorph catilever beam. 1. Itroductio Recet treds i electroic techology have eabled a decrease i both the size ad power cosumptio of complex digital systems, meaig that wireless sesor etworks are ow poised to be a sigificat eablig techology i may fields. It 1

2 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam Nomeclatures Strai related to vertical displacemet of the beam c Dampig coefficiet C p Capacitace of the piezoelectric device, F d Piezoelectric strai coefficiet k Couplig coefficiet l b Legth of base, mm l f Legth of clamp, mm l m Legth of tip mass, mm m eff Effective mass, kg m tip Tip mass, kg t c Thickess of the piezoelectric material, mm t sh Thickess of the shim material, mm V s Structural volume, mm 3 Y Youg s modulus, GPa Youg s Modulus for the piezoelectric material, GPa b * Y c Greek Symbols ε Vertical displacemet at the tip ed, m is highly desirable for wireless sesor odes to be self-powered. There are may potetial power sources for wireless sesor odes, especially ambiet vibratios aroud the ode [1, ]. It is possible to covert part of the ambiet eergy aroud the ode ito electrical eergy usig various methods, icludig the use of piezoelectric beams. Piezoelectric materials are physically deformed i the presece of a electric field, ad coversely, produce a electrical charge whe deformed. Whe mechaical stress is applied to a piezoelectric material, a ope-circuit voltage (a charge separatio) appears across the material. Likewise, if a voltage is placed across the material, mechaical stress develops i the material. I this paper, a 31-mode piezoelectric material, mouted as a catilever beam, is ivestigated ad optimised. The 31-mode material is able to create relatively large deflectios, takes up less space, ad has lower resoat frequecy tha material used i 33-mode [3]. May recet studies have focused o the performace of a catilever beam with various geometries, i order to idetify desig geometries that maximise the scavegig performace i terms of output power desity [4-6]. Sodao et al. [7, 8] performed experimets to ivestigate a piezoelectric composite actuator for power geeratio. Three differet materials were assessed for their effectiveess i power-harvestig applicatios: Quick Pack, piezoelectric material (PZT) (lead zircoate titaate), ad MFC (micro-fibre composite) were mouted to a catilever beam which was tested at 1 differet resoat frequecies. PZT was show to be more effective i the radom vibratio eviromets that are usually ecoutered whe dealig with ambiet vibratios. Miller et al. [9] reported a icrease i the weighted strai of a catilever with the additio of a slit through the middle of the beam, which yielded a weighted strai that is more tha twice that of a rectagular catilever. Hece, the authors cocluded that a typical solid rectagular catilever beam is o-optimized for micro-scale eergy scavegig. Mateu ad Moll [10] performed a aalytical compariso betwee rectagular ad triagular catilevers i which they assumed

3 14 Chug K. Thei et al. uiform stress across the width of the catilever. This revealed that a triagular catilever with the same beam volume as a rectagular beam has a higher average strai ad larger deflectio for a give load, thereby producig more power per uit volume. Simo ad Yves [11] showed that the tapered beam with 0.3 slope agle could icrease the eergy harvested by 69%. These proved that the geometry of the catilever beam affects the power output. Roudy also reported that the power desity of a beam ca be icreased by usig a smaller volume, ad that the strai is distributed more evely i the case of a trapezoidal catilever beam, which geerates more tha twice the eergy of a rectagular beam for a give volume [1]. Dhakar et al. [13] demostrated that by reducig the resoat frequecy of the trapezoidal catilever beam the overall power output could sigificatly be icreased. All the above metioed studies suggest that optimisig the structural details of the trapezoidal catilever beam may further icrease the power output. The power output of a catilever beam is directly related to the shape. What is the best desig of a trapezoidal catilever beam that geerates the maximum power desity? To aswer this questio, the sesitivity of power desity to beam should be examied. I this paper, a multi-objective method, MOST (multifactor optimisatio of structures techique) [14-15], is exteded to automatically accommodate ad execute problems related to eergy-harvestig optimisatio. The MOST techique utilizes commercially available fiite elemet codes (e.g., ANSYS) ad combies static aalysis, dyamics aalysis (for vibratio frequecy), ad a uique optimisatio techique, with the aim of simultaeously icreasig both the power output ad the power desity. The MOST optimisatio system ca efficietly ad systematically solve complex egieerig-desig problems, which may have multiple objectives ad ivolve multiple disciplies, by performig a parameter profile aalysis [16], thereby seekig the optimum solutio. This method icorporates a assessmet system which brigs the scores ad merit idices ito a defied rage (i this case 0 10) for all performace ad loadig cases. These features make MOST a powerful, cost-effective, ad reliable tool with which to optimise complex structural systems. This paper proposes a alterative method of predictig the power output of a piezoelectric catilever beam by static ad dyamic (modal) aalyses usig the fiite elemet method. The power outputs of the proposed method are verified by compariso with the results of both experimetal ad aalytical aalyses (the Roudy method [3]). A ew method is preseted i optimisig a bimorph piezoelectric catilever beam i a eergy harvestig applicatio, with the aim of simultaeously maximisig power output ad miimisig structural volume, while also satisfyig the stregth ad stiffess requiremets of the structure. The performace of the optimised desig is compared with triagular ad rectagular shapes.. Predictig the output power of piezoelectric geerator desigs with differet geometrical shapes usig fiite elemet aalysis (FEA) The costitutive equatios for a piezoelectric material are as follows [17]: δ =σ Y + de D = εe + dσ (1)

4 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam where δ is mechaical strai, σ is mechaical stress, Y is the modulus of elasticity, d is the piezoelectric strai coefficiet, E is the electric field, D is electric displacemet, ad ε is the dielectric costat of the piezoelectric material. Roudy [3] proposed that the magitude of voltage trasferred to the load for a piezoelectric beder ca be give as follows (assumig that the drivig frequecy is ot matched with the atural frequecy): V = 1 RC p 1 ω RC p * Yc dtcb jω ε + ζω ω + jω ω ( 1 + k ) A ζω + ω RC p where V is the geerated voltage from the piezoelectric material, ω is the drivig frequecy, Y c is Youg s Modulus for a piezoelectric material, d is the piezoelectric strai coefficiet, t c is the thickess of the piezoelectric material, b * is strai related to vertical displacemet of the beam, ε is the dielectric costat of the piezoelectric material, R is the load resistace, C p is the capacitace of the piezoelectric device, ω is the atural frequecy of the system, ζ is the mechaical dampig ratio, k is a couplig coefficiet, ad A i is the magitude of the iput acceleratio. This sectio focuses o the developmet of the power equatio of a piezoelectric beder. Several terms from Eq. () eed to be redefied to accommodate the results of the FEA. Figure 1 shows a schematic of a piezoelectric beder. i () Fig. 1. Schematic piezoelectric beder. I Fig. 1, l b is the legth of the base, l m is the legth of the tip mass, z t is the vertical deflectio of the catilever beam tip, ad w(y) is the width of the piezoelectric material i terms of the electrode legth (l e ). First, the mechaical dampig ratio of the system ca be stated as: c ζ = (3) ω m eff

5 16 Chug K. Thei et al. where m eff is the effective mass ad c is a dampig coefficiet. The effective mass ad atural frequecy ca be foud from the FEA. Secod, the capacitace of the beam is defied as: C p = le 0 cεw( y) dx t c (4) where c is the umber of piezoelectric layers. Usig Hooke s law, b * i Eq. () ca be related to the average elemet stress (σ ave ) as follows: 1 c σ ave = σ c (5) * σ ave b = Y z c c= 1 c t The power trasferred to the load is simply V /R. Equatio () ca be further simplified if the atural frequecy (ω ) is assumed that it is equivalet to the drivig frequecy (ω). This is because the geerated power is maximised whe the vibratio frequecy is equal to the resoat frequecy. Thus, the atural frequecy ad the eviromet frequecy must be very similar [3, 18]. The power output (P) of the beam ca the be formulated as follows: 4 ( 4ζ + k )( RC ω ) + 4ζk ( RC ω ) ( ζ ) dtcσ ave RC p 1 εz t P = A i (7) ω + p The optimum resistace ca be foud by differetiatig Eq. (7) with respect to R, settig the result equal to zero ad solvig for R. The optimum resistace (R o ) is as follows: Ro = ω C p ζ 4ζ + k 4 p (6) (8) 3. Verificatio of ANSYS Simulatios, Experimetal ad Theoretical Results 3.1. Desig costraits, load, ad ANSYS simulatio setup A catilever beam is modelled ad aalysed. The ANSYS SOLID9 elemet is used to geerate the model rather tha the SOLID98 elemet, although both elemets are 10-ode tetrahedral shapes suitable for large deflectio ad stressstiffeig behaviour. The SOLID9 elemet adapts well to the free meshig of irregular shapes. Both static aalysis ad dyamic (modal) aalysis (for vibratio frequecy) are performed i the aalysis. The catilever beam (kow as a bimorph system) is composed of two layers of piezoelectric materials ad a layer of shim material. The iitial model cosists of 960 elemets (both piezoelectric ad shim elemets) with a uiform elemet legth of 0.8 mm. Figure shows a schematic diagram of the desig domais, geometric costraits,

6 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam load, ad boudary coditios of the desig. Poits A ad D are fixed at three coordiates (x, y, ad z). A cocetrated pressure is applied at the free ed of the catilever beam (betwee B ad C). The iitial dimesios of the beam are listed i Table 1. The values of the mechaical ad electric properties of the piezoelectric material (PZT-5A4E) ad brass shim are give i Table [19, 0]. Fig.. Load ad boudary coditios of the simulatio setup. Table 1. Dimesios of the iitial desig. Parameters Iitial value (mm) Thickess of piezoelectric material (t c ) 0.19 Thickess of shim material (t sh ) 0.13 Legth of base (l b ) 1.5 Legth of tip mass (l m ).00 Effective legth of piezoelectric material PZT (l e = l m + l b ) 3.5 Width (w(y)) 1 w(y) 1.7 mm 1.7 Table. Mechaical ad electrical properties used i the FEA. Property Piezoelectric material Brass shim Youg s modulus (GPa) Yield stress (MPa) 4* 00 Maximum deflectio (µm) Poisso s ratio Desity (kg/m 3 ) Relative dielectric costat d31 (m/v) *Dyamic peak tesile stregth [1] 3.. Experimet setup A experimetal validatio of the model based o Eq. () for a rectagular catilever beam has bee coducted. A vertical vibratio geerated from a shaker (model Number LDS-V406/8) was used to excite the catilever ad the vibratio was also moitored usig a accelerometer (MTN1800. The voltage geerated by the piezoelectric material for a give load resistace was captured by a oscilloscope (Agilet MSO-6054A), alog with the accelerometer sigal). The vibratio is i the periodic behaviour []. The obtaied data from the oscilloscope were trasferred to the MATLAB workspace via a USB flash drive. I the

7 18 Chug K. Thei et al. MATLAB workspace, the data were subjected to a fast Fourier trasform, ad the frequecy spectrum of the geerated voltage ad the vibratio source plotted. The piezoelectric catilever (PZT-5A4E) has effective dimesios of mm with a tip mass of 4.1 g (Fig. 3). Because the tip mass is located off-cetre at the free ed of the catilever beam, it itroduces a torque ad force which cause the material to bed. The structure is excited by a siusoidal wave with a acceleratio magitude of ms i the frequecy rage of Hz. The mechaical ad electric properties of the piezoelectric ad shim materials are give i Table. Fig. 3. Schematic diagram of the experimet setup (uits i mm) Verificatio of Experimetal, Theoretical, ad ANSYS Simulatio Results Figure 4 compares 14 sets of data (k = 1,, 3,...,14). All the experimetal results show a resoat frequecy (f re ) at aroud 73 Hz [3]; however, for clarity of presetatio, they are separated by a offset frequecy (f off = 10 Hz) i order to plot them o a sigle graph. Hece the resistace-offset frequecies (f k ) o the horizotal axis for each set of data ca be expressed as follows: f = f + f k (9) k re off I this research, the mai aim is to maximise the power output, which is calculated by usig Eq. (7). However, this equatio must be verified before proceedig with the aalysis. Therefore, a compariso is made betwee the existig techique (the Roudy method [3]), experimetal results, ad the proposed techique. Uder the same setup i each case, but varyig the resistace, Eq. () is calculated ad plotted i Fig. 4 ( theory ). To calculate the power output usig the proposed method, a fiite elemet model of the rectagular piezoelectric catilever beam was developed for use i predictig the behaviour of the beam uder a cocetrated load at the free ed, as show i Fig.. The average elemet stress ad the vertical deflectio are obtaied i the aalysis. The obtaied values are substituted ito Eq. (7) ad the results are plotted i Fig. 4 as ANSYS simulatio. The maximum power is produced whe the resistace is i the rage kω (Fig. 4). The results of the ANSYS simulatio ad the theoretical calculatio

8 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam differ by approximately 4.9%, whereas the experimet results are markedly differet from these two sets of results. May factors that may affect the experimet result, icludig the surroudig eviromet, the positio of the tip mass, ad the sesitivity of the apparatus. Hece, the error i the experimet is about 13% 45% compared with the FEA. However, the frequecy respose for each load resistace ad the effect of the load do show similar treds to both theory ad simulatio. Table 3 shows that the percetage of discrepacy of resistace-offset frequecy. Fig. 4. Compariso of theoretical calculatios, ANSYS simulatio, ad experimetal results for output power. Resistace (kω) Table 3. Percetage of discrepacy of theory, experimet ad ANSYS simulatio. Theoretical ad Experimet Percetage discrepacy ANSYS simulatio ad Experimet ANSYS simulatio ad Theoretical 4.9

9 0 Chug K. Thei et al Optimisatio methodology multifactor optimisatio of structures techique The requiremets for a complex structural desig dictate that the optimisatio must ivolve multiple objectives, multiple disciplies, ad a large umber of desig variables. A m matrix (d ij ) the so-called performace data matrix (PDM) is defied by a set of performace parameters P i (i = 1,,, m) ad loadig case parameters C j (j = 1,,, ), respectively. The PDM is a schematic represetatio of a collectio of data as show i Table 4. Thus, the data poit d ij is the i-th performace P i of the structure at the loadig case C j. I this case the data poits of the matrix are obtaied by a fiite elemet aalysis of the structure. The matrix lists every loadig case as well as every performace parameter relevat to the idividual loadig cases. Table 4. Performace data matrix. C 1 C C P 1 d 11 d 1 d 1 P d 1 d d P m d m1 d m d m A parameter profile matrix (PPM) is created to review the profile of the performaces for differet loadig cases (Table 5). To simplify the calculatios, the values of the performace idices are ormalised to the rage This eables differet loadig cases ad parameters to be compared, i order to gai a overall perspective of the characteristics of the system. The PPM assesses the character of the structure with respect to the actual performaces at their worst acceptable limits ad the best expected values of the performaces. Table 5. Parameter profile matrix. C 1 C C P 1 D 11 D 1 D 1 P D 1 D D P m D m1 D m D m The data poit D ij for oe acceptable limit (e.g., lower limit) is calculated as follows: dij lij D ij = 10 (10) b l ij ij where d ij is the actual value of the performace obtaied from the PDM, ad l ij ad b ij are the lower acceptable limit ad the best expected value, respectively. Eq. (10) is valid for l ij < d ij < b ij ; for d ij > b ij, D ij = 10; ad for d ij < l ij, D ij = 0. The

10 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam data poit for the cases of acceptable upper limit ad double acceptable limits ca be calculated i a similar way. The mea ad stadard deviatio (SD) are calculated for each parameter ad loadig case i each colum ad row i the PPM. A well-desiged system should have low SDs ad high mea values (close to 10). The existece of high SDs sigifies that the system is likely to have sigificat problematic areas. Therefore, a high SD for a row idicates variable system performace at differet loadig cases for a particular parameter. Coversely, a high SD for a colum idicates the system is likely to have sigificat problematic performace for the specific loadig case. The system ca be further aalysed usig a parameter performace idex (PPI) ad a case performace idex (CPI), which are defied as follows: PPI CPI i = 1, i 1,, L, m j = D 1 ij m = ad = j 1,, L, j m 1 i = 1 Dij = (11) Whe i-th parameter is very vulerable, some data poits D ij of the PPM will have values close to 0 ad hece the PPI i will also close to 0. Similarly, whe the system is vulerable at the j-th loadig case, CPI j will be close to 0. The highest values for PPI ad CPI are 10. PPI ad CPI values close to 10 idicate good desig, whereas values close to zero idicate poor desig. The mea values, CPIs, PPIs, ad SDs provide a overall performace assessmet for the system ad loadig cases. These idices are calculated by summig the iverse of the data poits as a performace ratig to avoid the effect associated with low scores beig hidde by high scores. The mea values are ot used directly to rate the performace. The system may be reviewed by usig the iformatio i the idices, as follows: A compariso of PPIs idicates whether the system performs better with respect to some performaces tha to others. A compariso of CPIs shows whether the system performs better uder certai loadig cases tha uder others. Accordig to the matrix profile aalysis, PPI ad CPI are measures of the vulerability of each performace parameter ad each loadig case, respectively. Hece, the itegratio of PPI ad CPI idicates the vulerability of a particular parameter/loadig case combiatio. A overall performace idex (OPI) is used to develop the overall objective fuctio. The OPI, which takes the form of a qualitative score, ca be established for the system by cosiderig all the performaces ad all the loadig cases. The OPI fuctio lies i the rage of The OPI ca be expressed as follows: 100 m OPI = W m i= 1 j= 1 Pi PPI W i C j CPI j where Wp i ad WC j are weightig factors i the rage of 0 1 reflectig the preferece for each performace parameter ad each loadig case. The OPI ca be used to compare the performaces of differet desigs. The higher the OPI (1)

11 Chug K. Thei et al. score, the more reliable the desig would be. The objective fuctio is maximised usig the effective zero-order method, employig cojugate search directios [13]. A effective polyomial iterpolatio ui-dimesioal search method is also used i the algorithm. This optimisatio techique has the advatage of forcig the performaces to approach their optimal values. The earer the performaces to the acceptable limits, the stricter the puishmet (pealties) will be. 4. Electrical Eergy ad Structural Optimisatio of a Bimorph Piezoelectric Catilever Beam The power output of a catilever beam is directly related to its shape. This study cosiders the sesitivity of the power to the shape of the catilever. The mai objective is to fid the optimum geometrical shape of a bimorph catilever beam that yields the maximum power ad has the miimum structural volume. I additio, the beam must satisfy the stregth ad stiffess requiremets. A dyamic aalysis (for the vibratio frequecy) is also required i the power calculatio as idicated i Eq. (7) Formulatio of the optimisatio problem I this study, the optimal shape of the bimorph catilever is determied by MOST, which is used i cojuctio with the ANSYS fiite elemet software. The desig problem is therefore to maximise the power output ad the average elemet stress, ad simultaeously to miimise the structural volume, subject to the desig costraits. The optimisatio to be solved is stated as follows: fid X = (x 1, x,, x k ) mi {V s (X)} ad max {P j (X) ad σ ave,j (X)} s.t. {P j P ii,j ; V s V s,ii ; σ ave,j σ ii,j ; σ max,j σ y ; δ ii,j δ j δ lim, j }ad { x x x, i = 1,, K, k} mi i j = 1,,, i max i where k is the umber of desig variables, V s is the structural volume (excludig the volume of the tip mass), σ ave is the average elemet stress of the structure, δ is the displacemet of poit E (see Fig. ), σ max is the maximum Vo Mises stress of the structure, ad P is the power output. The subscript ii idicates the iitial value for the structure (here, the iitial iteratio whe i = 0), ad is the umber of loadig cases (here, = 1). The subscript lim idicates a specified performace limit for the structure. I this research, the catilever beam is optimised to carry a tip mass of 4.1 g with a maximum vertical displacemet of δ lim = 300 µm at ay ode, satisfyig a maximum stregth of σ y = 4 MPa (see mi max Table 1). x ad i x are the lower ad upper bouds of the desig variables i of x i, respectively. There are eight desig variables i the structural model, which represet the width of the catilever beam. I this case, the lower ad upper bouds are set to 1 ad 15 mm, respectively.

12 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam Optimisatio results The shape optimisatio of the catilever beam required i = 7 iteratios to coverge, as show i Figs. 5 ad 6, which depicts the evolutio of the structural volume, the power output ad the power desity. A sharp icrease i the output power is see up to i = 5, because the width at the free ed of the beam (where the tip mass is located) is reduced to a teth of its origial size. This is followed by a sharp decrease i power output up to i = 8, due to the removal of material from the structure ad chagig values of the atural frequecy, dampig ratio, resistace, ad capacitace. Subsequetly, the power output fluctuates before covergig to a optimal solutio at i = 7. The opposite tred is observed for the structural volume. The distributio of Vo Mises stress for the iitial ad optimised desigs is show i Fig. 7. The tip mass is ot show i Fig. 7 but is followed the ANSYS simulatio setup as show i Fig.. The attributes of the iitial ad optimised desigs are give i Table 6. The resoat frequecy of the iitial ad optimised desigs are 113 Hz ad 9 Hz respectively, as obtaied from the FEA. The frequecy is affected by the positio ad geometry (i.e., same volume but differet shape) of the tip mass. More specifically, the atural frequecy of the catilever beam is affected by the height of the tip mass. Thus, appropriate dimesios should be chose to suit the vibratio desig. I this case, the fiite elemet models ad the experimets models are similar except for the shape of the tip mass (see Figs. ad 4, respectively) Roudy [1] reported that a triagular beam produces more tha twice the power desity of a rectagular beam. Table 7 lists the power desity (per uit volume of piezoelectric material) ad the dimesios of a rectagular beam, triagular beam, ad the optimised shape of the preset study (Fig. 8). Uder the same costraits (i.e., the width of the free ed is the same for both optimised desig ad triagular shape), the power desity of the optimised desig is superior to that of the triagular ad rectagular shapes. Fig. 5. Optimisatio covergece history of the structural volume ad power.

13 4 Chug K. Thei et al. Fig. 6. Optimisatio covergece history of the power desity. Fig. 7. Distributio of Vo Mises stress for the iitial desig (left) ad the optimised desig (right) (Pa). Table 6. Desig attributes of the iitial ad optimised desigs of a catilever beam. Iitial desig Optimised desig Power (mw) Volume of piezoelectric material (mm 3 ) Volume of shim material (mm 3 ) Power desity (µw/ mm 3 )* Maximum Vo Mises stress (MPa) Average elemet stress (MPa) 0.4 Maximum vertical displacemet (µm) 8.48 Capacitace (F) Frequecy (Hz) *per uit volume of piezoelectric material

14 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam Table 7. Compariso of power desity amog beams with differet shapes. Rectagular shape Triagular shape Optimised desig Volume of piezoelectric material (mm 3 ) Power desity (µw/mm 3 ) Fig. 8. Dimesio of (a) rectagular, (b) triagular, ad (c) optimised shapes (i ratio). 5. Coclusios The output power is icreased from approximately 0.6 to 0.8 mw, which correspods to a icrease of approximately 5% compared with the iitial desig. The structural volume (piezoelectric ad shim material combied) is reduced sigificatly from 15.1 to mm 3, represetig a 46.% savig i materials. The power desity of the optimised desig is more tha twice that of the iitial desig, ad the vertical deflectio of the catilever beam is icreased by about 47%, from 8.48 to 1.50 µm. The maximum Vo Mises stress shows a icrease from 0.89 to 1.00 MPa. These results are well withi the stiffess ad stregth costraits (δ lim = 300 µm) ad yield stress (σ y = 4 MPa). The results show good agreemet betwee the fiite elemet simulatio ad the theoretical results, which differs by approximately 4.5% i terms of the maximum power output. The maximum power output differs by ~0% betwee the fiite elemet results ad the experimetal results, reflectig the fact that the experimetal results were affected by various coditios (e.g., the eviromet effect). Therefore, the fiite elemet simulatio yields more accurate ad reliable results compared with theoretical values. Thus, Eq. (7) preseted i this paper is a ovel developmet i estimatig the power output of piezoelectric catilever beams of various sizes ad shapes by meas of fiite elemet aalysis. I the secod part of this paper, a shape optimisatio of the piezoelectric catilever beam was preseted. Simulatio results idicate that the optimised desig ca geerate 4.6 µw/mm 3 for a piezoelectric volume of mm 3.

15 6 Chug K. Thei et al. Future research will focus o maximisig the power desity by seekig the optimum topology of a bimorph catilever beam. The results demostrate the efficiecy of the MOST techique. The results obtaied for the bimorph catilever beam demostrate that the proposed method was successful i idetifyig the optimum desig, resultig i improved performace i terms of power output ad power desity. I future work, the effect of stochastic forcig case o the optimised geometry may be cosidered. Refereces 1. Gilbert, J.M.; ad Balouchi, F. (008). Compariso of eergy harvestig systems for wireless sesor etworks. Iteratioal Joural of Automatio ad Computig, 5(4), Roudy, S.; Steigart, D.; Frechette, L.; Wright, P.; ad Rabaey. J. (004). Power source for wireless sesor etworks. Wireless sesor etwork-lecture Notes i Computer Sciece, Spriger-Verlag, 90, Roudy, S. (003). Eergy Scavegig for wireless sesor odes with a focus o vibratio to electricity coversio. PhD thesis, Uiversity of Califoria Berkely. 4. Goldschmidtboeig, F.; ad Woias, P. (008). Characterizatio of differet beam shapes for piezoelectric eergy harvestig. Joural of Micromechaics ad Microegieerig, 18(10), Beasciutti, D.; Moro, L.; ad Zeleika, S. (010). Vibratio eergy scavegig via piezoelectric bimorphs of optimized shapes. Microsystem Techologies, 16(5), Dietl, J.M.; ad Garcia, E. (010). Beam shape optimisatio for power harvestig. Joural of Itelliget Material Systems ad Structures, 1(6), Sodao, H.A.; Ima, D.J.; ad Park, G. (005). Compariso of Piezoelectric Eergy Harvestig Devices for Rechargig Batteries. Joural of Itelliget Material Systems ad Structures, 16(10), Sodao, H.A.; Ima, D.J.; ad Park, G. (005). Geeratio ad Storage of Electricity from Power Harvestig Devices. Joural of Itelliget Material Systems ad Structures, 16(1), Miller, L.M.; Emley, N.C.; Shafer, P.; ad Wright, P.K. (008). Strai Ehacemet withi Catilevered, Piezoelectric MEMS Vibratioal Eergy Scavegig Devices. Advaces i Sciece ad Techology, Smart Materials ad Mico/Naosystems, 54, Mateu, L.; ad Moll, F. (005). Optimum piezoelectric bedig beam structures for eergy harvestig usig shoe iserts. Joural of Itelliget Material Systems Structures, 16, Simo, P.; ad Yves, S.A. (009). Electromechaical Performaces of Differet Shapes of Piezoelectric Eergy Harvesters. Iteratioal Workshop Smart Materials ad Structures, -3 Oct 009, Motreal, Caada

16 Modellig ad Optimisatio of a Bimorph Piezoelectric Catilever Beam Roudy, S. (005). O the effectiveess of vibratio-based eergy harvestig. Joural of Itelliget Material Systems Structures, 16, Dhakar, L.; Liu, H.; Tay, F.E.H.; ad Lee, C. (013). A ew eergy harvester desig for high power output at low frequecies. Sesors ad Actuators, 199, Liu, J.S.; ad Hollaway, L. (000). Desig optimisatio of composite pael structures with stiffeig ribs uder multiple loadig cases. Computers ad Structures, 78(4), Liu, J.S.; ad Lu, T.J. (004). Multi-objective ad multi-loadig optimizatio of ultralight weight truss materials. Iteratioal Joural of Solids ad Structures, 41, Liu, J.S.; ad Thompso, G. (1996). The multi-factor desig evaluatio of atea structures by parameters profile aalysis. Proceedigs of the Istituitio of Mechaical Egieers, Part B: Joural of Egieerig Maufacture, 10(5), Ikeda, T. (1996). Fudametals of piezoelectricity. Oxford sciece publicatios, Oxford. 18. Shu, Y.C.; ad Lie, I.C. (006). Aalysis of power output for piezoelectric eergy harvestig systems. Smart Materials ad Structures, 15, Gallas, Q.; Wag. G.; Papila, M.; Sheplak, M.; ad Cattafesta, L. (003). Optimizatio of sythetic jet actuators. 41 st AIAA Aerospace Scieces Meetig ad Exhibit, Reo, NV, USA, AIAA Piezo System (009), Ic. CATALOG #7C (008). Retrieved July 31, 009, from 1. Bert, C.W.; ad Birma, V. (1998). Effects of stress ad electric field o the coefficiets of piezoelectric materials: oe-dimesioal formulatio. Mechaics Research Commuicatios, 5(), Friswell, M.I.; Ali, S.F.; Bilge, O.; Adhikari, S.; Lees, A.W.; ad Litak, G. (01). No-liear piezoelectric vibratio eergy harvestig from a vertical catilever beam with tip mass. Joural of Itelliget Material Systems ad Structures, 3(13), Ooi, B.L. (010). Optimisatio ad frequecy tuig cocepts for a vibratio eergy harvester. PhD Thesis, Uiversity of Hull.