Study of the Resonant Frequency of Unimorph Triangular V-shaped Piezoelectric Cantilever Energy Harvester

Transcription

1 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December Study o te Resonant Frequency o Unimorp Triangular V-saped Piezoelectric Cantilever nergy arvester R. osseini* Young Researcers and lite Club, Sout Teran Branc, Islamic Azad University, Teran, Iran -mail: R.osseini.mec@gmail.com *Corresponding autor M. amedi Department o Mecanical ngineering, University o Teran, Iran -mail: mamedi@ut.ac.ir Received: May 5, Revised: 4 August 5, Accepted: September 5 Abstract: Te main aim o te vibration energy arvesters is to locally power autonomous devices suc as wireless sensors. Generally, power levels are low and te environmental beneit o te tecnology is to replace batteries rater tan saving energy per se. Piezoelectric vibrational energy arvesters are usually inertial mass based devices, were a cantilever beam wit a piezoelectric outer layer is ecited into resonance by a mecanical vibration source at te root o te cantilever beam. owever, te geometry o a piezoelectric cantilever beam will greatly aect its vibration energy arvesting ability. Tis paper deduces a remarkably precise analytical ormula or calculating te undamental resonant requency o unimorp V-saped cantilevers using Rayleig-Ritz metod. Tis analytical ormula, wic is convenient or mecanical energy arvester design based on piezoelectric eect, is ten validated by ABAQUS simulation. Tis ormula raises a new perspective tat, among all te unimorp V-saped cantilever beams and in comparison wit rectangular one, te simplest tapered cantilever can lead to igest resonant requency and maimum sensitivity. Keywords: Finite lement, Mecanical nergy arvester, Piezoelectric, Unimorp V-saped Cantilever, Resonant Frequency Reerence: osseini, R., amedi, M., Study o te Resonant Frequency o Unimorp Triangular V-saped Piezoelectric Cantilever nergy arvester, Int J o Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4, 5, pp Biograpical notes: R. osseini received is PD in Mecanical ngineering rom University o Teran. e is currently premier member o Young Researcers and lite Club at Islamic Azad University Sout Teran branc. is current researc interest includes nergy arvesting. M. amedi is Proessor o Mecanical engineering at te University o Teran. 5 IAU, Majlesi Branc

2 76 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December 5 INTRODUCTION nergy arvesting is used to describe te scavenging o ambient energy in te environment tat would oterwise be wasted. To eed te world s needs or energy, macro scale energy arvesting tecnologies ave successully establised. On te oter and, te lack o cables induces a constraint on power supply or eac low powered wireless electronic sensors. Batteries wear out wit time, tus regular replacement is an integral and inevitable part o maintenance. As te dense network is employed in structures, replacing batteries becomes a major time-consuming task tat is uneconomical and unmanageable, ironically contradictory to te original objective o structural ealt monitoring. Scavenging energy rom ambient vibrations, wind, eat or ligt could enable smart sensors to be unctional indeinitely. Tree mecanisms are available or vibration energy arvesting; using electrostatic devices, electromagnetic ield and utilizing piezoelectric based materials. Te leibility associated wit piezoelectric materials makes tem very attractive or power scavenging. Te perormance o piezoelectric vibration energy arvesters is more oten tan oter metods. Piezoelectric materials possess a large amount o mecanical energy tat can be converted into electrical energy, and tey can witstand large strain magnitude. Vibration rom engines can stimulate piezoelectric materials, as can te eel o a soe, or te pusing o a button. Compared to oter structural orms o beams, a cantilever beam can obtain te maimum deormation and strain under te same conditions. Te larger delection leads to more stress, strain, and consequently a iger output voltage and power. Tereore te vast majority o piezoelectric vibration energy arvesting devices use a cantilever beam structure. [-4]. A cantilever-type energy arvester as been intensively studied. Te cantilever geometrical structure plays an important role in improving te arvester s eiciency and a triangular tapered cantilever as been ound to be te optimum design [5], [6], because it ensures a large constant strain in te piezoelectric layer resulting in iger power output compared wit te rectangular beam wit te widt and lengt equal to te base and eigt o te corresponding triangular tapered cantilever beam. Most o te previous researc works ocused on designing a linear vibration resonator, wic as maimum output power wen reacing resonance requency. Tereore te practical applications o tese devices are limited due to narrow bandwidt as well as small power density. I te ecitation requency sligtly sits, te perormance o te arvester will dramatically decrease. Since in te majority o practical cases, te vibration in te environment is requencyvarying or totally random wit te energy distributed in a wide spectrum, ow to broaden te bandwidt o arvesters becomes one o te most callenging issues beore teir practical deployment. en a arvester operates in an environment wit multi-requency spectra, it is desirable to design te arvester wit a tailorable operating requency band [7]. In practice, te energy arvester is a multi-degree-o-reedom system or a distributed parameter system. Certain vibration mode can be ecited wen te driving requency approaces one natural requency o te arvester. To date, one o te most important strategies to widen te bandwidt includes using a generator array consisting o small generators wit dierent resonant requencies. Multiple cantilever energy arvesters wit dierent resonant requencies can be connected in series or parallel to widen te operating requency bandwidt o a arvesting structure. I multiple vibration modes o te arvester structure are utilized, useul power can be arvested over multiple requency spectra, tat is, wider bandwidt can be covered or eicient energy arvesting. Rater tan discrete bandwidt due to te multiple modes o a single beam, multiple cantilevers or cantilever array integrated in one energy arvesting device can provide continuous wide bandwidt, i te geometric parameters o te arvester are appropriately selected. Power spectrum o a generator array is a combination o te power spectra o eac small generator [7-9]. Accordingly, by division o a triangular unimorp piezoelectric beam into some V-saped unimorp beams wit dierent dimensions and mass and ence dierent resonant requencies, can be ound in an array o beams tat can cover a wider range o requencies (Fig. ) []. Also in a new design, pizza model can be used to make te array o energy arvesters. Te main advantage o tis sceme is optimum use o space and to create iger power density (Fig. ). Fig. Division o a triangular beam into some V-saped beams [] A systematic procedure or designing mecanical bandpass ilters to meet a desired requency bandwidt is given in []. Suc a researc about rectangular sapes sows tat by using some cantilevers in series 5 IAU, Majlesi Branc

3 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December connection, not only te output power increases wit te use o more cantilevers, but also te requency band is widened []. ρ and ρ, tickness and, and Young's modulus and or substrate and piezoelectric layers, respectively. Also te total cross-sectional area moment o inertia is I z. Fig. Te Pizza model o semi-triangular cantilever energy arvesters Te geometry o a piezoelectric cantilever beam will greatly aect its vibration energy arvesting ability. Te sensitivity o resonant cantilever piezoelectric energy arvesters is directly proportional to te resonant requency. So ar, te calculation o resonant requency o unimorp V-saped cantilevers as not been reported in te literature and te calculations are only or a simple V-saped cantilever beam []. In order to calculate te resonant requency o V-saped cantilevers, tis paper deduces a igly precise analytical ormula using Rayleig-Ritz metod, and ten introduces te optimization metod or enancing te resonant requency wit tis ormula. Tis useul analytical ormula is conirmed by simulation results in ABAQUS 4. sotware, and presents a strong potential to be used in te design and optimization o triangular V-saped cantilever unimorp piezoelectric energy arvesters. It is noteworty tat a cantilever beam can ave many dierent modes o vibration wit a dierent resonant requency. Te irst mode o vibration as te lowest resonant requency, and typically provides te most delection and tereore electrical energy. Accordingly, energy arvesters are generally designed to operate in te irst resonant mode. Tis researc proposes a new design or a cantilever-type unimorp piezoelectric energy arvester called V-saped cantilever and te main ocus o tis paper is to study te resonant requency o te new design in piezoelectric mecanical energy arvester. Fig. Te scematic drawing o a cantilever beam For beam cross-sections tat are not symmetric about te z-ais wit regard to eiter geometry or te variation o elasticity modulus (), a convenient metod or treating bending problems is provided by te concept o te transormed section. I we coose a certain value o as a reerence value and call it re, ten we can deine a transormed section and transormed widt n, were n /. In te case we assume tat re =. Te line o action o an aial orce produces purely aial deormation, tereore passes troug te centroid o te transormed section. In te case o bending wit no aial orce, te neutral ais also passes troug tis point. In tis case we assume tat te location o te eective centroid is determined by (Fig. 4 and Fig. 5) [4]. Fig. 4 Te cross section o unimorp cantilever beam TORTICA ANAYSIS.. Delection Function o Rectangular Unimorp Cantilevers Fig. sows te structure o unimorp piezoelectric rectangular cantilever wit lengt, widt, density Fig. 5 Te transormed section o unomorp cantilever bram determines te neutral ais location and can be epressed as; 5 IAU, Majlesi Branc

4 78 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December 5 n ) ( n ) n n Also te total cross-sectional area moment o inertia relation is; n I z n ) ( ) en applying a normal orce F at te ree end o te cantilever, te dierential equation o te cantilever can be epressed as [5]; d z F F ( ) ( ) ( ) d I I ere is te distance rom te ied end. As one end o te cantilever is ied, te corresponding boundary conditions are; () () () z () (4) and dz( ) d Te solution o ()- (5) can be epressed as; F ( ) ( ) ( ) z A 6 I (5) (6) Tis is te delection unction along te lengt direction were A is a constant... Resonant Frequency o Cantilevers wit Arbitrary Sapes en considering te resonant beavior o a unimorp cantilever beam wit an arbitrary sape wose widt unction is (), te delection unction o (6) can be used as te mode sape, and te vibration displacement at eac position can be written as []; z (, t ) A ( )sin( t ) (7) ere A and α are constants, t is te time, and ω=π is te angular requency. Te kinetic energy o te system is [6]; T ( ) ( ) d ( ) A cos ( t ) 4 ( ) ( ) d So te maimum kinetic energy o te system is; Tma ( ) A 4 ( ) ( ) d Te potential energy o te system is [6]; V V ere; z, z, dv z dv Accordingly; V ( ) z dz ( ) z ) ( ) dz z t ( ) ) ) d ) ) ) ( ) 6 A ( ) ) d sin( t ) ( ) ) 8A sin ( t ) ( ) ( )( ) d ( ) d (8) (9) () () () 5 IAU, Majlesi Branc

5 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December Tereore, te maimum potential energy o te system is; ) V 8 A ( ) ma ( )( ) d ) ().. Resonant Frequency o unimorp triangular V- saped Cantilevers Fig. 6(a) sows tat a typical unimorp triangular V- saped cantilever can be treated as te dierence between two unimorp triangular cantilevers, wit lengts and, and wit widts and respectively. It can be easily conirmed by (5), tat due to te mirror symmetry o unimorp triangular V- saped cantilever, we need only analyze al o it, wic is a quadrilateral cantilever as sown in Fig. 6(b). According to conservation law o mecanical energy; T V (4) ma ma ence, te resonant requency can be obtained as; ( ( )) ) ( ) ) (5) ( )( ) d 4 ( ) ( ) d In particular, or te case o a rectangular cantilever wit lengt and widt, te resonant requency can be deduced rom (5); rect ) ( ) ) ( ) d 4 ( ) ) ( ) ) 5 rect d 55 ) ) ) (6) Fig. 6 Sape and dimension o (a) unimorp V-saped cantilever (b) al o te unimorp V-saped cantilever (c) triangular tapered cantilever [] Obviously, te widt unction o te quadrilateral cantilever is a piecewise-continuous unction o, tat is; ( ), [, ], [, ] (7) For calculation convenience, it is reasonable to deine te widt ratio u and te lengt ratio v o te two unimorp tapered cantilevers; u, v (8) Substituting (7) and (8) into (5), te resonant requency ormula o te quadrilateral cantilever (just te resonant requency o unimorp triangular V- saped cantilever) is obtained. 5 IAU, Majlesi Branc

6 8 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December 5 ( ( )) ) ( ) ) reac te maimum resonant requency and tus te igest sensitivity. ( )( ) d 4 ( ) ( ) d ) ( ) ) ) 6 7 ( ) ) 7( 6 4 ) ( ) ) 6 4 ( ) ) uv uv uv uv 4uv 5uv (9) Fig. 7 Te unction image o g(u,v) VRIFICATION BY SIMUATION RSUTS In order to assess te accuracy o (), relative error δ is introduced to compare te calculation results using tis ormula wit te corresponding simulation results. () In order to represent te relationsip between te resonant requency and te two ratios u and v, we can deine a caracteristic unction; 6uv 4uv uv g ( u, v ) 49 84uv 4uv 5uv u [,], v [,] Tus, te resonant requency o V-saped cantilever is; 7 ( ( )) g ( u, v ) ) ( ) ) () () As sown in Fig. 7, g(u,v) reaces te maimum value 7.474, wen v= or v= or u=. Tat means unimorp V-saped cantilever acieves maimum resonant requency only wen = or = or =. Apparently, wen = or =, te V-saped cantilever turns into a tapered cantilever as sown in Fig. 6(c). en =, te unimorp V-saped cantilever turns into two side by side unimorp triangular tapered cantilevers, owever, tis peculiar sape is diicult to carry out in practice. Anyway, triangular tapered cantilever, a special kind o V-saped cantilever and easy or micro-abrication, can ere reers to te calculation results wit (), and reers to simulation results wit ABAQUS modal analysis. Consider a unimorp rectangular cantilever beam, assuming ρ =874 kg/m, ρ =78 kg/m, =9.7 Pa, =6.6 Pa, =mm, =mm, =8mm and =mm. Te requency calculation according to (6) is 6.4 z and te corresponding simulation result wit ABAQUS is 6. z. ence te relative error is only.66% and an ecellent agreement is obtained between te calculation results and te simulation results, yielding little relative error. Te simulated sape is sown in Fig. 8. Fig. 8 Deormed saped or te irst vibration mode o unimorp piezoelectric cantilever Also consider a series o V-saped cantilevers wit dierent sapes, assuming, ρ =874 kg/m, ρ =78 kg/m, =9.7 Pa, =6.6 Pa, =.6 mm, =.4 mm, =8 mm, =4mm, =mm and canging, te calculation according to () and te corresponding simulation results wit ABAQUS are listed in Table. 5 IAU, Majlesi Branc

7 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December Table Te comparison between te calculation results and te simulation results o te resonant requencies o unimorp triangular V-saped cantilevers (mm) (z) (z) δ It can be seen rom Table tat, a very good agreement is obtained between te calculation results and te simulation results, yielding little relative error (less tan 8.%). en =7mm, te simulated sape is sown in Fig. 9. tap ) ( ) 7 ) ) g ( u,) 7 ( ) ) 7 ) ( ) ) ) ( ) ) () Apparently, te resonant requency o a unimorp tapered cantilever is unrelated to its widt. It is necessary to point out tat, or a tapered cantilever, wen increasing and keeping oter parameters ied, its resonant requency will remain constant. It is wort comparing (6) and (), and we can get te resonant requency ratio o unimorp tapered cantilever and unimorp rectangular cantilever. Fig. 9 Deormed saped or te irst vibration mode o unimorp piezoelectric cantilever 4 APPICATION ) ( ).659 ) tap rect ) ( ).78 ).4 (4) Te resonant requency ormula presented in tis paper is useul or many applications. First, tis simple ormula can be eectively used to determine te resonant requency o unimorp triangular V-saped cantilevers o any dimensions and material properties. Anoter signiicant application is te optimization o unimorp V-saped cantilever vibration energy arvesters. Te sensitivity o resonant cantilever vibration energy arvesters is directly proportional to te resonant requency, and te resonant requency is a key parameter to design a mecanical energy arvester. As mentioned above, wit given lengt, given widt, given tickness and and given material properties,, ρ and ρ, triangular tapered cantilever-a special kind o V-saped cantilevers can reac te maimum resonant requency and igest sensitivity. For a triangular tapered cantilever, substituting v = into (), te maimum resonant requency is obtained. ence, te unimorp tapered cantilevers can lead to muc iger resonant requency and iger sensitivity tan tat o unimorp rectangular cantilevers. 5 CONCUSION nergy arvesters provide a very small amount o power or low-energy electronics. Vibration energy arvesters are generally designed to operate in te irst resonant mode. Te piezoelectric eect converts mecanical vibration strain into electric current or voltage. Tis paper deduces a igly precise eplicit ormula to calculate te undamental resonant requency o unimorp V-saped piezoelectric cantilever beams based on Rayleig-Ritz metod. it tis analytical ormula, te calculation results are in perect agreement wit te simulation results, yielding little relative error (less tan 8.%). Tis error or a 5 IAU, Majlesi Branc

8 8 Int J Advanced Design and Manuacturing Tecnology, Vol. 8/ No. 4/ December 5 unimorp rectangular cantilever reduces to only.66%. Triangular tapered cantilever, a special kind o V- saped cantilever and easy or micro-abrication, can reac te maimum resonant requency and tus te igest sensitivity. In te irst mode o vibration, te eact sape o te cantilever is not identical to te static delection proile. Accordingly te velocity distribution is not eactly proportional to te static delection proile. Tis is wy te natural requency estimates are sligtly dierent rom te simulation values. Because o simplicity o te derived ormula, it is an easily learned and easily applied procedure or approimately calculating or recalling some value, or or making some determination. Finally, an application or calculating requency o unimorp V-saped cantilever energy arvesters, is presented wit tis ormula in order to acieve a Multi-Modal energy arvester. Tis ormula can be commonly used in te design and optimization o vibration energy arvesters. perimental analysis can validate te results and epand te researc or output voltage and power density. ACKNODGMNTS Tis researc was partially supported by Young Researcers and lite Club at Islamic Azad University Sout Teran Branc. e tank our colleagues rom tis club wo provided inancial support tat greatly assisted te researc. RFRNCS [] Anderson, T. A., Seton, D.., A Vibration nergy arvesting Sensor Platorm or Increased Industrial iciency, in Smart structures and materials, 6, pp. 674Y-674Y-9. [] Beeby, S. P., Tudor, M. J., and ite, N., nergy arvesting Vibration Sources or Microsystems Applications, Measurement science and tecnology, Vol. 7, pp. R75, 6. [] rturk, A., Inman, D. J., Piezoelectric nergy arvesting, Jon iley & Sons,. [4] Priya, S., Inman, D. J., nergy arvesting Tecnologies, Vol. : Springer, 9. [5] Mutali, A. G., Nordin, N. D., Optimal Piezoelectric Beam Sape or Single and Broadband Vibration nergy arvesting: Modeling, Simulation and perimental Results, Mecanical Systems and Signal Processing, Vol. 54, 5, pp [6] Cen, Z., Yang, Y., and Deng, G., Analytical and perimental Study on Vibration nergy arvesting Beaviors o Piezoelectric Cantilevers wit Dierent Geometries, in Sustainable Power Generation and Supply, 9. SUPRGN'9. International Conerence on, 9, pp. -6. [7] Tang,., Yang, Y., and So, C. K., Toward Broadband Vibration-based nergy arvesting, Journal o Intelligent Material Systems and Structures,Vol.,, pp [8] Saruz, S., imits o Perormance o Mecanical Band-pass Filters used in nergy Scavenging, Journal o sound and vibration, Vol. 9, 6, pp [9] Yang, Z., Yang, J., Connected Vibrating Piezoelectric Bimorp Beams as a ide-band Piezoelectric Power arvester, Journal o Intelligent Material Systems and Structures, Vol., 9, pp [] osseini, R., amedi, M., An investigation into resonant requency o trapezoidal V-saped cantilever piezoelectric energy arvester, Microsystem Tecnologies, pp. -8, 5/6/4 5. [] Saruz, S., Design o Mecanical Band-pass Filters or nergy Scavenging, Journal o Sound and Vibration, Vol. 9, 6, pp [] Xue,., u, Y., and ang, Q. -M., Broadband Piezoelectric nergy arvesting Devices using Multiple Bimorps wit Dierent Operating Frequencies, Ultrasonics, Ferroelectrics, and Frequency Control, I Transactions on, Vol. 55, 8, pp [] Yang, K., i, Z., Jing, Y., Cen, D., and Ye, T., Researc on te Resonant Frequency Formula o V- saped Cantilevers, in 9 4t I International Conerence on Nano/Micro ngineered and Molecular Systems, 9, pp [4] ubliner, J., Papadopoulos, P., Introduction to Solid Mecanics, An Integrated Approac, Springer Science & Business Media,. [5] Senturia, S. D., Microsystem Design, Vol. : Kluwer academic publisers Boston,. [6] Rao, S. S., Vibration o continuous systems, Jon iley & Sons, 7. 5 IAU, Majlesi Branc