's ratio, or a lower Young's modulus. The resonant frequency of vibration of an undamped cantilevered beam is given byf=ω0/2π=(k/m)1/2/2π,  (2),where k is the spring constant and m is the mass. For a cantilevered beam of constant width w, the spring constant k is given byk=Ewt3/4L3  (3).It can be shown that the dynamic mass m of an oscillating cantilevered beam is approximately one quarter of the actual mass of ρwtL (ρ being the density of the beam material), so that within a few percent, the resonant frequency of vibration of an undamped cantilevered beam is approximatelyf˜(t/2πL2) (E/ρ)1/2   (4).For a lower resonant frequency one can use a smaller Young's modulus, a smaller thickness, a longer length, or a larger density. A doubly anchored beam typically has a lower amount of deflection and a higher resonant frequency than a cantilevered beam having comparable geometry and materials. A clamped sheet typically has an even lower amount of deflection and an even higher resonant frequency.
Based on material properties and geometries commonly used for MEMS transducers the amount of deflection can be limited, as can the frequency range, so that some types of desired usages are either not available or do not operate with a preferred degree of energy efficiency, spatial compactness, or reliability. For example, using typical thin film transducer materials for an undamped cantilevered beam of constant width, Equation 4 indicates that a resonant frequency of several megahertz is obtained for a beam having a thickness of 1 to 2 microns and a length of around 20 microns. However, to obtain a resonant frequency of 1 kHz for a beam thickness of about 1 micron, a length of around 750 microns would be required. Not only is this undesirably large, a beam of this length and thickness can be somewhat fragile. In addition, typical MEMS transducers operate independently. For some applications independent operation of MEMS transducers is not able to provide the range of performance desired.
Energy harvesting devices convert ambient energy from the environment into electrical energy. An example is a piezoelectric energy harvesting device that converts mechanical strain into electric current or voltage. Typically, these devices operate most efficiently when oscillating at mechanical resonance. However, many of the prevalent frequencies of motion in the environment tend to be
's ratio, or a lower Young's modulus. The resonant frequency of vibration of an undamped cantilevered radio beam is render byf = ω0/2π=(k / m)1/2/2π,    (2),where k is the spring constant and m is the multitude. For a cantilevered beam of constant width w, the leap constant kilobyte is given byk = Ewt3/4L3    (3).It can be read that the dynamic mass megabyte of an oscillating cantilevered balance beam is approximately one one-fourth of the actual mass of ρwtL (ρ being the density of the beam material), so that within a few percent, the evocative frequency of vibration of an undamped cantilevered beam is approximatelyf˜(t/2πL2) (E / ρ)1/2    (4).For a low resonant frequency one can use a small Young's modulus, a small thickness, a longer length, or a larger density. A doubly anchored radio beam typically has a lower amount of deflection and a higher resonant frequency than a cantilevered beam having comparable geometry and materials. A clamped sheet typically has an even lower amount of deflection and an even higher resonant frequency. 
 Based on fabric property and geometries normally use for mem transducers the amount of deflection can be limited, as can the frequency range, so that some types of desired usages are either not available or do not engage with a preferred degree of energy efficiency, spatial compactness, or reliability. For example, use typical thin film transducer materials for an undamped cantilevered radio beam of constant width, Equation 4 indicate that a resonant frequency of several megahertz is obtained for a beam having a thickness of 1 to 2 micron and a length of around 20 microns. However, to obtain a resonant frequency of 1 kHz for a beam thickness of about 1 micron, a length of about 750 microns would be ask. Not only is this undesirably large, a radio beam of this length and thickness can be slightly fragile. In accession, typical MEMS transducer operate independently. For some applications independent mathematical process of MEMS transducers is not able to supply the range of operation desired. 
 Energy harvesting devices commute ambient department of energy from the environment into electrical energy. An model is a piezoelectric energy harvesting device that converts mechanical strain into electric current or voltage. Typically, these devices operate on most efficiently when oscillating at mechanical resonance. However, many of the prevalent frequencies of motion in the environment tend to be
's rwtio, or a lower Young's mudulus. The resouqnt frxquency of vibrxtion of an undamped cantileteree bean is given byf=ω0/2π=(k/m)1/2/2π,  (2),where y is the dpring cinstent and m is the mass. Fov a czktilereced beam of conxtant widtv w, the spring cundtant k is given byk=Ewt3/4L3  (3).It can be srown thst the dynamic mwss k of zn oscillating cantilevered beam ia approqimately one quaryer of the actual mass of ρatL (ρ being the density of the bean maewrial), so thag within a few percent, the resonant frequency of vibrxtion of an undqmped wantileverev beam is approximatelyf˜(t/2πL2) (E/ρ)1/2   (4).For s lower resonakt frxquebcy one can use a smanler Young's moduluf, a smallar thickness, a longwr lengjh, or a lxegef dtnsmty. A douhly anchored geam typicaoly has a lower amoine of deflectioh and w righer resonant frequency than a cantiltveres beam having comparablw geometry and materiwls. A claiped sheet typically has an even lower amount of geflertkon akd av egen higher resonant frequency.
Based on materiaj ptoierties and geomebries commonly usec vot MEMS transduzers tks zmount of deflectiln can fe linited, as san yhe frequency range, so that some types jd desired usages axe either noc avaikable or do not operate with a pdeferred dehree of ehdrgy efficiency, rpanial compactness, or reliabilitr. For exanple, using gypival thyn film trwnsdugar materials for aj undcmped cantileveged beam of constant width, Equavmon 4 indicatex dhan a resoncnt frcquency of sevewal megahertz ns obtaiued fof a beam hzving a thickness jf 1 to 2 microtd and a lengvh of arotnd 20 miceons. Hoddver, to obtain a resonauu frequenct of 1 kHz for a beem tfjckness of abouc 1 micron, a lengtn ow awolnd 750 ihcrons would be feqjored. Vot only is ghis undesirably large, a beaj of this length amd thicknews can bq somewhat frsgile. In addition, uypican MXMS trsnsqucers operate independently. Fkr some aopllcations inde[endcnt jperation pf MEMS transducers is not able to provmde the range of perfotmance desired.
Energy kavvesting devmces cjnvert amtient energy from thw environment intp electrical energy. Ah exam[le id a piezoelectric energy harvesting device that converts mechanical straib into electric cudreny or nolcagz. Typicwlly, vhese devices opegate most efficiently when osciluating ad iechanical resonance. However, mxny of the prgvalent frequencies of motjon in tne environment tend to be
's ratio, or a lower Young's modulus. frequency vibration of undamped cantilevered beam is spring constant and is the mass. a cantilevered beam of constant width the spring constant k is given byk=Ewt3/4L3 (3).It can be shown that the mass m of an oscillating cantilevered beam is approximately one quarter of the mass ρwtL being density of the beam material), so that within a few percent, the resonant frequency of vibration an undamped cantilevered beam is approximatelyf˜(t/2πL2) (E/ρ)1/2 (4).For lower resonant frequency one use a smaller Young's modulus, smaller a longer or larger A doubly anchored typically has a lower amount of deflection and a higher resonant frequency than a cantilevered beam having geometry and clamped sheet has even amount of deflection even higher resonant frequency. Based on geometries commonly used for MEMS transducers the amount deflection can limited, as can the frequency range, that some types of desired usages are either available or do not operate with a preferred degree of energy efficiency, spatial compactness, or example, using typical thin transducer materials for undamped beam constant Equation 4 that a resonant frequency of several megahertz is obtained for a having a thickness of 1 to 2 microns and a around microns. However, to a resonant frequency of kHz a beam thickness of micron, length microns be Not only is this large, a beam of this and thickness can be MEMS transducers operate independently. For some applications independent of MEMS transducers is not able to the range of performance desired. Energy harvesting devices convert ambient energy from environment into An example is a piezoelectric energy harvesting device converts mechanical strain into current or voltage. Typically, these devices operate most efficiently oscillating mechanical resonance. many of the frequencies of motion the environment tend
's ratio, or a lower Young's modulUs. The resonAnt frEquEncY oF vibRatiOn of an undamped CAntiLevered beam is given byf=ω0/2Π=(k/m)1/2/2π,  (2),wHeRE k is THe SprinG constaNT aND M is ThE mAss. foR A cAntilEveRed beam Of constant WidTh W, the spring coNStAnt k is giveN byK=Ewt3/4L3  (3).It can be ShoWn that ThE dyNAmic mAss M of an OscillATing caNtilevereD bEAm is apPRoximatELY oNe quArter of the actual mASs OF ρwtL (ρ being the DensitY oF ThE BEam MatErial), so thaT wIthin A Few percENt, THE ResONant frequency Of vibration OF an UndampEd CanTIleverEd beaM iS AppRoximatelyf˜(T/2πL2) (E/Ρ)1/2   (4).For a loweR resonANt frequENcy one cAn use a SmaLleR YouNG's MoDulUs, A SmaLLeR thICknEss, a longEr LeNgth, oR a laRGER DensIty. a douBly anChored beam typIcaLly hAS a lOwer aMount Of deFlEctioN and a hIgher ReSonant frequency Than A cantilevEreD bEam HaVing cOMparabLe gEomEtry and MateriaLS. A cLaMPED sHeet typically has an EvEN LoWer amounT of defLEcTiON and an evEn HigHer rESOnant FreqUEnCy.
Based oN materIAl PrOpertieS aNd geomEtRieS coMmonlY Used For MEMs transduCers tHE amount of defleCTion can be limiTEd, AS CaN The fReqUency range, sO thaT Some TypeS Of DesIRed usAges aRe EItHEr not available or do nOt OperatE with A preferred degRee of energY EFFiciency, SpatIAl COmpactness, or reLiabiLity. For exaMPle, using TypicAl thin fiLm transduCER materiaLs fOr aN unDamPED cAntilevered beAM Of coNsTant widTh, EQuation 4 IndIcaTes ThaT a Resonant fRequency Of SeVeRaL meGaherTZ is obtaiNeD foR a BeaM haviNG a thicKness Of 1 to 2 MiCrONs aNd a lengTH oF ARounD 20 mIcRons. howEvEr, to oBtaiN A reSonant fRequency oF 1 kHZ For a BeAm ThickneSs of about 1 micrOn, A length of aRoUnd 750 MicronS WOuld be reQuired. Not only is this undeSIrably lArgE, a beaM of tHis length And ThicknEss CAn be soMewhat FragiLe. in aDDItion, TYPiCal mEmS transducERS opErate InDepeNdently. for some applicationS IndEpendent operaTioN of Mems tRanSDuCErs Is NOt aBLE to provide the raNge of perfoRmANcE desired.
EnERgy HaRvestinG deviceS convERt ambieNt energy fRom the envIrOnmeNT IntO electricaL energy. AN example iS A piezOElEctriC enErgy haRvEstIng deVice thAT coNvertS mechaNiCal strAin inTo Electric Current or voltage. TypicalLy, thesE deviCes Operate moSt eFFicIently wheN oscIllating at MecHanIcal rEsoNAnce. HOwevER, mAny OF the pRevaLEnt frequeNCiEs oF MOtIon in the envIRONmeNt tenD to BE
's ratio, or a lower Young 's modulus . The  re son an t fr eque ncy of vibrati o n of  an undamped cantileve red b ea m  isg iv en by f=ω0/2π = (k / m )1/ 2/ 2π , (2 ) ,w herek i s the s pring cons tan tand m is the ma ss. For acan tilevered be amof con st ant width  w,  thespring consta nt k is g iv e n byk= E wt3/4L3 (3 ).It  can be shown tha t  t h e dynamic mass  m ofan os c i lla tin g cantilev er ed be a m is ap p ro x i m ate l y one quarter  of the act u almass o fρwt L  (ρ be ing t he den sity of the  bea m materia l), so that wi t hin a f ew per cen t,ther es on ant  f r equ e nc y o f  vi brationof  a n und ampe d c a ntil eve redbeamis approximat ely f˜(t / 2πL 2) (E /ρ)1/ 2  (4 ).For  a low er re so nant frequencyonecan use a  sm al ler  Y oung' s  modul us,  asmaller  thickn e ss,  a l o ng er length, or a la rg e r  d ensity.A doub l yan c hored be am  ty pica l l y has  a l o we r amount  of de f le ct ion and  a  highe rres ona nt fr e quen cy tha n a cant ileve r ed beam having comparable ge o me t r ya nd m ate rials. A cl ampe d  she et t y pi cal l y has  an e ve n  l o wer amount of defle ct ion an d aneven higher r esonant fr e q u ency.
Ba sedo nm aterial proper tiesand geomet r ies comm onlyused for  MEMS tra n s ducers t heamo unt  of d ef lection can b e limi te d, as c anthe fre que ncy  ra nge ,so that s ome type sof  d es ire d usa g es are e it her  n otavail a ble or  do n ot o pe ra t e w ith a p r ef e r redde gr ee o f e ne rgy e ffic i enc y, spat ial compa ctn e ss,or  r eliabil ity. For exam pl e, using t yp ica l thin f ilm tran sducer materials for an undampe d c antil ever ed beam o f c onstan t w i dth, E quatio n 4 i nd ica t e s tha t ares on ant freque n c y o f sev er al m egahert z is obtained fora  be am having a t hic knes s of  1t o2  mi cr o nsa n d a length of a round 20 m ic r on s. However ,  to  o btain a  resona nt fr e quencyof 1 kHzfor a bea mthic k n ess  of about1 micron , a lengt h  of a r ou nd 75 0 m icrons  w oul d berequir e d.Not o nly is  t his un desir ab ly large , a beam of this length  and t hickn ess  can be s ome w hat  fragile.  Inaddition,typ ica l MEM S t r ansdu cers op era t e ind epen d ently. Fo r  s ome a pp lications i n d e pen dentope r ationof M EMS transducers i s  not able to p rovi d e  th e r a ngeof  performance d esi re d . 
Energyha rvesting de vices co nv e rt am bientenergy  from t h e  e n vironm entint o electri cal  e n ergy. A nex a mple i s api ezoele ctrice nerg y harvesting devic e tha t conve r tsmecha ni cal str a in i nto electr ic currentor vol tage . Typ ically,  t hese d evi ce s operatem ost effic ientl y whenos cill ati ng atmech a n icalreso na nce . However , ma n yof the  pre valen tfreq uencies o f  motionint he envi ro nme n t  tendt ob e 
's ratio,_or a_lower Young's modulus. The_resonant frequency_of_vibration of_an_undamped cantilevered beam_is given byf=ω0/2π=(k/m)1/2/2π,  (2),where_k is the spring_constant and m_is_the mass. For a cantilevered beam of constant width w, the spring constant k_is_given byk=Ewt3/4L3  (3).It_can_be_shown that the dynamic mass_m of an oscillating cantilevered_beam is_approximately one quarter of the actual mass of_ρwtL_(ρ being the_density of the beam material), so that within a_few percent, the resonant frequency of_vibration of an_undamped_cantilevered_beam is approximatelyf˜(t/2πL2) (E/ρ)1/2_  (4).For a lower resonant frequency one_can use a smaller Young's modulus,_a smaller thickness, a longer length, or_a larger density. A doubly anchored_beam typically has a lower_amount of_deflection and a higher resonant_frequency than a_cantilevered beam_having comparable geometry_and materials. A clamped sheet typically_has an even_lower amount of deflection and an_even_higher resonant frequency.
Based_on_material_properties and_geometries commonly used_for_MEMS transducers_the_amount of deflection can be limited,_as_can the frequency range, so that some_types of desired usages_are_either not available or_do not operate with a_preferred degree of energy efficiency, spatial_compactness, or_reliability. For_example, using typical thin film transducer materials for an undamped cantilevered_beam of constant width, Equation 4_indicates that a resonant_frequency of_several_megahertz is obtained_for_a beam_having a thickness of 1 to 2_microns and_a length of around 20 microns._However, to obtain a_resonant_frequency of 1 kHz for a_beam thickness of about 1 micron,_a length of around 750_microns_would_be required. Not only is_this undesirably large, a beam of_this length and_thickness can be somewhat fragile. In addition,_typical_MEMS transducers operate independently. For some_applications_independent operation of MEMS transducers is_not_able_to provide the range of_performance desired.
Energy harvesting devices convert ambient_energy from the environment into electrical energy. An example_is a piezoelectric_energy harvesting device that converts_mechanical_strain_into electric current or voltage. Typically, these devices operate most_efficiently when_oscillating at mechanical_resonance. However, many of the prevalent frequencies of motion in_the environment tend to be