Design,Modeling,andSimulationofTwo-PieceTrapezoidal ...

Research ArticleDesign Modeling and Simulation of Two-Piece TrapezoidalPiezoelectric Devices for Sensing and Energy Harvesting

Nan Chen 1 and Vishwas Bedekar 2

1Department of Computational Science Middle Tennessee State University Murfreesboro 37132 TN USA2Department of Engineering Technology Middle Tennessee State University Murfreesboro 37132 TN USA

Correspondence should be addressed to Vishwas Bedekar vishwasbedekarmtsuedu

Received 13 June 2019 Revised 28 December 2019 Accepted 14 January 2020 Published 11 February 2020

Academic Editor Antonio Boccaccio

Copyright copy 2020 Nan Chen and Vishwas Bedekar +is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

+e objective of the research is to design a high power energy harvester device through a two-piece trapezoidal geometryapproach +e performance of the composite two-piece trapezoidal piezoelectric PZT-PZN polycrystalline ceramic material issimulated using COMSOL Multiphysics Results are analysed using the series configuration of a two-piece trapezoidal compositebimorph cantilever which vibrates at the first fundamental frequency+e two-piece trapezoidal composite beam designs resultedin a full-width half-maximum electric power bandwidth of 25Hz while providing an electric power density of 1681mWcm3 witha resistive load of 008MΩ+e authors believe that these results could help design a piezoelectric energy harvester to provide localenergy source which provides high electric power output

1 Introduction

+e recent microsensors and energy harvesters are made ofsmart materials such as the piezoelectric materials andmagnetoelectric materials as devices made of these materialsconvert the mechanical energy from surrounding ambientenvironment to the electric energy to provide the electricpower to devices However magnetostrictive energy har-vester not only requires the vibration in an environment butalso demands the magnetic field in an environment Pie-zoelectric energy harvester can provide low power at mil-liwatts level to the small electronic devices Traditionallythese small electric devices were powered by batteries Whena battery is depleted the old battery needs to be replacedwith a new battery +e periodic battery replacements notonly interrupt the operation of these devices but also poserisks when the devices are taken out of the service for batteryplacement +erefore these small electric devices call for adifferent power source other than the traditional chemicalbatteries +e piezoelectric material is one of the promisingmaterials in harvesting the ambient mechanical vibrationenergy due to the high-voltage and milliwatt-level power

density output +ere have been two different focuses toenhance the performance of a vibration energy harvester(VEH) the electric power [1 2] and the bandwidth [2] of theVEHs Zhang et al claimed that the higher change rate of thesection area of the VEH contributes to the higher poweroutput of the VEH [2] Zhang et al reported that adding theexternal magnets and decreasing the distance between theVEH and the external magnets increase the bandwidth of theVEH [2] Benasciutti et al reported that the trapezoidalshaped VEHs produce more electric power density than therectangular shaped VEHs due to the nonlinearly distributedstress of the trapezoidal VEH design [3] However Benas-ciutti et al did not define the degree of the stress uniformitynor compare the bandwidth difference between the trape-zoidal and the triangular shaped VEHs Muthalif andNordin reported that the electric output voltage and themechanical resonance frequency increase when the width ofthe free end of the composite cantilever approaches to 0mmbecause they suggested the uniform strain of the triangularVEH contributed to the higher voltage output [4] butneither the power density nor the power density bandwidthwas investigated Muthalif and Nordin proposed an

HindawiAdvances in Materials Science and EngineeringVolume 2020 Article ID 9743431 14 pageshttpsdoiorg10115520209743431

analytical solution for modes of the resonance frequencyof the trapezoidal and the triangular composite piezoelectriccantilever beam [4] Reilly et al classified five broadbandapplications out of fourteen ambient vibrational applica-tions [5] and the authors reported that the 50 of theelectric power (2mW4mW) bandwidth is 8Hz (95ndash103Hz)of their proposed composite trapezoidal piezoelectric VEHs[5] Compared with Reilly et alrsquos work our firsttrapezoidal bimorph design reported 50 electric power (786mW1571 mW) with a bandwidth of 25 Hz (195ndash220 Hz)+is work addresses the goal on designing a two-piece trap-ezoidal composite bimorph piezoelectric energy harvesteroperating in the transverse mode with the aim to enhance themaximum real power density in an electric circuit with anoptimal resistor as well as the full-width half-maximum(FWHM) power density-mechanical vibration broadbandperformance compared them with the results of the one-piecetrapezoidal beam design based on the PZT-PZN-Scheme 4polycrystalline ceramic piezoelectric material 08 [Pb (Zr052Ti048)O3]ndash02 [Pb (Zn13 Nb23)O3] from our previous work onthe one-piece trapezoidal beam model [6 7] +e PZT-PZN-Scheme 4 material was reported because of its superiorstructural power density (01713mWcm3 measured) 2097higher than that of PZT-ZNN-Scheme 2 (01416mWcm3)1538 higher (495mWcm3 measured) in the electric poweroutput and 3113 higher (0499mWcm3 measured) in thepiezoelement power density as the ldquotwo-step sinteringrdquomethod was used to reduce grain size and increase densitywhich leads to higher relative dielectric (εr is 1588 295sim423times higher than those of PZT-ZNNs) and the piezoelectricproperty of PZN-PZN-Scheme 4 material (the charge constantin the thickness mode d33 is 400 pCN 24sim26 times higherthan that of PZT-ZNNs +e charge constant of PZT-PZN inthe transversemode d31 is 15373 pCN 275sim307 times higherthan those of PZT-ZNNs) [8] Not only are the power densitiesof the PZT-PZN-Scheme 4 higher than its PZT-ZNN com-petitors but also is PZT-PZN-Scheme 4rsquos quality factor Qsignificantly lower (787) than that of PZN-ZNN (780) whichmakes PZT-PZN-Scheme 4 a better material choice for a highpower energy harvester as the PZT-PZN-Scheme 4 is a low-quality factor material [2] Yuan et al derived a formula for thevoltage sensitivity of a triple-layer trapezoidal piezoelectricbeam which is useful for sensor applications [9] Many re-searchers reported that the electric power and voltages of atrapezoidal piezoelectric beam are higher than those of arectangle beam given the same volume of PZT [9 10]Benasciutti et al studied the voltage and the electric powercharacteristics of a one-piece trapezoidal and the reversed one-spice trapezoidal shapes of the piezoelectric bimorphs +emaximum power generated was reported to be 650μW at50Hz excitation frequency [10] Yuan et al reported that therectangular piezoelectric beamrsquos maximum electric power isabout 86mW at the operating frequency of 180Hz whereasthe trapezoidal piezoelectric beamrsquosmaximumelectric power is242mW at the operating frequency of 130Hz [9] Shachteleet al proposed a two-ported model to describe a trapezoidalpiezoelectric beam using an admittance matrix +e admit-tance matrix describes the linear relations between the electriccharge Q the tip deflection δ and the voltage V and the force F

at the tip [11] Yet the admittance is used to determine theinternal inductance of a given trapezoidal bimorph in ourresearch Hosseini and Hamedi also reported improvement inenergy harvesting using V-shaped piezoelectric bimorphs It isshown that increasing the width W2 (at free end) lowers theresonance frequency which agrees with our results It was alsoshown that the trapezoidal shape generates higher voltagecompared to the simple rectangular cantilever beam bimorphwhich matches our findings in previous works [6 12] In thispaper we explore a unique design of 2-piece trapezoidalshaped piezoelectric bimorph to understand its effects onharvested power power density and bandwidth +e 2-piecetrapezoidal geometry which is explained later has not beenexplored in the literature so far for energy harvesting appli-cations +e material property values of PZT-PZN-Scheme 4material are tabulated in Table 1

2 Materials and Methods

+e authors used the technique to convert the piezoelectricand the mechanical material property values from Table 1 tothe compliance matrix (a tensor of 4th order) of the PZT-PZN-Scheme 4 piezoelectric material from our previouswork [7]

+e procedure used in seeking themaximum electric realpower output is through finding the first resonance fre-quency of the two-piece piezoelectric beam once the firstresonance frequency of the beam is known through runningeigenfrequency study on the model then we vary the loadingresistor to find the optimal resistance where the powerreaches maximum the external resistance was varied from001MΩ to 02MΩ (an arbitrarily large range) with 001MΩresolution to match the internal impedance of the beam001MΩ interval resolution is arbitrarily chosen to limit thecomputation time to a reasonable level +e optimal resis-tance is found when a peak of the electric power appears inthe scanning rangeWhen the optimal resistance is found fora given model we then vary the vibrational frequencysymmetrically around the first eigenfrequency to see howwide the vibration frequency can get from the peak poweroutput to 50 of the peak power output

+e equivalent electric circuit is presented below inFigure 1 +e piezoelectric energy harvester can be simplymodeled as a series LRC electric circuit with an alternativevoltage source V in a circuit L represents the internal in-ductance r represents the internal resistance (damping ef-fect) and C represents the internal capacitance +e letter Zin the equivalent electric circuit represents the externalimpedance When the external impedance Z matches theinternal impedance Zint of LRC circuit

(Zext Zint

r2 + (Xc minus XL)21113969

) where Xc is the internalcapacitive reactance the bimorph and XL is the internalinductance of the bimorph the electric power output reachesmaximum (widely known as the rule of impedancematching) Due to the current plan of research we onlyconsider the external loading resistor Z When the externalload impedance Zext matches the internal impedance Zintnumerically (Zext Zint) the electric circuit delivers the

2 Advances in Materials Science and Engineering

maximum electric power to the external load resistor Zext Inthis case when the external loading resistance matches theimpedance of the magnetite of the internal impendence thebimorph generates the maximum electric power as thefollowing equation shows

Z Rloading

r2 + XL minus Xc( 11138572

1113969

(1)

Taking the square on both sides of equation (1) weobtain equation (2) where f is the frequency of the oscil-lating electric signal in the equivalent electric circuit whichis presented in Figure 1

R2loading +r

2+ 2πfL minus

12πfC

1113888 1113889

2

(2)

In equation (2) the internal resistance r the internalinductance L and the internal capacitance C of a bimorphare fixed values therefore the frequency f of the oscillatingsignal in the equivalent electric circuit will be affected by thevalue of the external loading resistor +e internal resistancer the internal inductance L and the internal capacitance Cneed to be found before modeling the electric characteristicsof a piezoelectric bimorph cantilever beam +e internalresistance r can be found by applying Kirchhoffrsquos voltage law

(KVL) and Ohmrsquos law in a closed electric equivalent circuitas shown in Figure 1 as it is expressed in the followingequation

ε Ir + IR + IZc + IZL (3)

where ε is the electromotive force (EMF) I is the electriccurrent R is the external loading resistor Zc is the im-pedance of the capacitor and ZL is the impedance of theinductor in the closed electric equivalent circuit as shown inFigure 1 We can then take two measurements by using twodifferent external loading resistors in the simulation andrewriting Kirchhoffrsquos voltage law (KVL) and Ohmrsquos law in apair of the following equations

U1 + I1 XL minus XC( 1113857 minus rI1 + ε (4)


where U1 is the voltage across the external loading resistorR1 I1 is the current going through the external resistor onthe first measurement U2 is the voltage across a differentexternal loading external resistor R2 and I2 is the currentgoing through the external resistor R2 in the secondmeasurement As U1 U2 I1 I2XL and Xc are knownparameters as they can be found in the simulations wehave two equations and two unknowns (r ε) and theinternal resistor r can be easily found by solving the pair ofequations (4) and (5) +us the pair of equations (4) and(5) can be simplified to equation (6) where Z1 is the vectorsum of the impedance of the inductor and the capacitor inthe first measurement with one external loading resistorR1 and Z2 is the vector sum of the impedance of theinductor and the capacitor in the second measurementwith one external loading resistor R2

r minus U2 minus U1( 1113857 + I2 Z2

11138681113868111386811138681113868111386811138681113868 minus I1 Z1

111386811138681113868111386811138681113868111386811138681113872 11138731113960 1113961

I2 minus I1 (6)

However it would be impossible to find the internalresistor r of a given bimorph without knowing the values ofthe internal inductance L and the internal capacitance C of abimorph in the closed electric equivalent circuit as shown inFigure 1 +e inductance L and the capacitance C in theclosed electric equivalent circuit can be found in theCOMSOL simulation +e reactance of the equivalentelectric circuit can be found by the following equations

Z11113868111386811138681113868

1113868111386811138681113868 XL1 minus XC11113868111386811138681113868

1113868111386811138681113868 2πf1L minus1

2πf1C

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (7)

Z21113868111386811138681113868

1113868111386811138681113868 XL2 minus XC21113868111386811138681113868

1113868111386811138681113868 2πf2L minus1

2πf2C

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)

+e frequencies f1 and f2 of the oscillating electric signalin two different resistive loading are different due to theloading resistance difference which was expressed inequation (2) therefore |Z1| ne |Z2| Equations (7) and (8) aresubstituted into equation (6) and the internal resistance r isexpressed in the following equation

Table 1 PZT-PZN-Scheme 4 and brass material property namesand values [8]

Value Namek31 032 Electro-mach coupling factor of PZT-PZNd31 15373 Piezoelectric charge constant pCNg31 0011 Piezoelectric voltage constant VmNkp 059 +e coupling coefficient of PZT-PZNε r 1588 +e dielectric constant of PZT-PZNQm 787 Mechanical quality factorρ 7879 +e density of PZT-PZN (kgm3)ζ 0017 Mechanical damping ratioTemp 1100 Sintering temperature (degC)T 3 Sintering time (hours)Mt 0 Tip mass (g)d33 400 Piezoelectric charge constant pCNg33 0028 Piezoelectric voltage constant VmNρ s 8800 +e density of brass UNS C22000 (kgm3)νs 0307 Poissonrsquos ratio of the brass layerε rs 4500 +e dielectric constant of the brass layerEs 110 Youngrsquos modulus of the brass layer (GPa)E 66 Youngrsquos modulus of PZT-PZN (GPa)

C

Z

R L

V

Figure 1 An equivalent LRC electric circuit of a piezoelectricbimorph energy harvester

Advances in Materials Science and Engineering 3

r minus U2 minus U1( 1113857 + I2 2πf2L minus 12πf2C

11138681113868111386811138681113868111386811138681113868 minus I1 2πf1L minus 12πf1C

111386811138681113868111386811138681113868111386811138681113872 11138731113960 1113961

I2 minus I1

(9)

where the two distinctive oscillating frequencies f1 and f2 ofthe electric signal in two different resistive loadings can befound in simulations +e internal resistance r of thebimorph can be found by taking two different measurementsas shown in equation (9) Equation (9) can be simplified toequation (10) once the inductance L and the capacitance C inthe closed equivalent electric circuit are known through thesimulation as two oscillating frequencies f1 and f2 are set tothe resonance frequencies of the equivalent electric circuit

r minus U2 minus U1( 1113857

I2 minus I1 f f r (10)

+e internal resistance r of the bimorph can be found byjust taking two different voltage and current measurements(U1 U2 I1 I2 U1prime U2prime I1prime I2prime) with two different externalresistive loads as shown in equation (10) given the value ofthe inductance L and the capacitance C which are to befound in two separate simulations +e capacitance C is theratio between the electric chargeQ(71028 times 10minus 6 C 83341 times 10minus 6 C) on the upper surface ofthe bimorph and the voltage U (100V for both designs)between the surfaces of the bimorph as shown in equation(11) which can be found by looking up the component of thecapacitance matrixes C11 in the ldquoDerived Valuesrdquo section ina stationary study where Cd1 is the capacitance of the firsttrapezoidal bimorph beam design when the shorter widthW1 is 18mm and the longer width W2 is 40mm Cd2 is thecapacitance of the second trapezoidal bimorph beam designwhen the longer shorter W1 is 18mm and the longer widthW2 is 52mm +e inductance L of the bimorph can becalculated by equation (13) where mef Y11 is the admittanceand mefomega is the angular frequency of the electric signalin the AC equivalent electric circuit both of which can befound in the simulation

Cd1 Q

U71028 times 10minus 6 C

100V 71028 pF

Ld1 776H

f r11

2πLC

radic 214Hz

(11)

Cd2 Q


100V 83341 pF

Ld2 643H

f r21

2πLC

radic 217Hz

(12)

L imaginary 1

mef middot Y11mef middot iomega1113888 1113889 (13)

+e internal resistances r1 and r2 are calculated byequation (10) under the resonance frequencies fr1 and fr2 of

the electric signal in the equivalent circuit by taking twodifferent voltage and current measurements (U1 U2 I1 I2U1prime U2prime I1prime I2prime) with two different external resistive loads R1(001MΩ) and R2 (002MΩ) which are arbitrarily chosen

r1 minus U2 minus U1( 1113857

I2 minus I1

minus (2295V minus 11393V)

00011475A minus 00011393A 14MΩ

(14)

r2 minus U2prime minus U1prime( 1113857

I2prime minus I1prime


00015768A minus 00017844A 65 kΩ

(15)

+us far the internal resistances (r1 and r2) the ca-pacitance and the inductance of two bimorphs can beobtained through calculation

+e values of the internal inductance L the internalcapacitance C and the internal resistance r of the bimorph inan equivalent circuit together determine the value of thequality factor Q by equation (16) by using equations(10)ndash(13)

Q 1r

L

C

1113970

(16)

Alternatively the quality factor of a structure can also becalculated by equation (17) which is also adopted byCOMSOL f is the resonance frequency of the vibration Dueto the mechanical damping the complex value of resonancefrequency has an imaginary part (complex number) Forinstance the complex resonance frequency of the two-piecetrapezoidal bimorph (in the first design the length of thesingle plate is 60mm W1 is 40mm and W2 is 2mm)145 + 02i Hz in a COMSOL eigenfrequency study +equality factor is 28645 can be obtained by the followingequation

Q abs (f)

2lowast imaginary( f) (17)

In such a way we can plot the quality factor Q for allpermutations of each bimorph geometry of two trapezoidaldesigns in Figures 2 and 3 +e quality factor Q increaseswhen the longer length W2 increases for both trapezoidaldesigns Yet the shorter widthW1 does not have a significanteffect on the quality factor Q

To investigate the quality factor Q we need to under-stand the complex nature of the eigenfrequency +e reasonfor the imaginary component wi of the angular frequency isthat it determines if the amplitude (ewit) of the oscillation ofthe VEH grows or shrinks in time as equation (18) indi-cated A(t) is a general time-dependent damped oscillationfunction eminus jw

rt is a regular oscillation term which can be

further expanded by Eulerrsquos formula +e real part (wr) ofthe angular frequency determines the physical oscillationfrequency +e positive real part (ewit) of the angular fre-quency indicates the amplitude of the oscillation that growswith time +e negative real part (ewit) of the angular fre-quency indicates the amplitude of oscillation that shrinkswith time [13]


A(t) ewit eminus jwrt

w wi + wr(18)

+e damping ratio ζ is defined by COMOSL in thefollowing equation

ζ imaginary (f)

abs(f) (19)

+e damping ratio ζ (0017) can be obtained by takingout the real part and the imaginary part out of the complex

40 50 60

Qua

lity

fact

or

W1 = 2 (mm)

W2 (mm)

2864

2865

2866

2867

(a)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 6 (mm)

2863

28635

2864

28645

(b)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 10 (mm)

28624

28626

28628

2863

28632

(c)Q

ualit

y fa

ctor

40 50 60

W1 = 14 (mm)

W2 (mm)

2862

28622

28624

(d)Q

ualit

y fa

ctor

40 50 6028617

28618

28619

2862

W1 = 18 (mm)

W2 (mm)

(e)

Figure 2 +e quality factor of the first trapezoidal bimorph design+e structure loss factor is 0025+e damping ratio is 0017+e lengthof the beam L is 60mm the thickness of Tpiezo is 03mm and the thickness of the brass layer Ts is 005mm

Qua

lity

fact

or

W1 = 2 (mm)

4006

40065

4007

40075

50 6040W2 (mm)

(a)

Qua

lity

fact

or

W1 = 6 (mm)

4005

40052

40054

40056

50 6040W2 (mm)

(b)

Qua

lity

fact

orW1 = 10 (mm)

40046

40048

4005

40052

50 6040W2 (mm)

(c)

Qua

lity

fact

or

W1 = 14 (mm)

40046

40047

40048

50 6040W2 (mm)

(d)

Qua

lity

fact

or

W1 = 18 (mm)

40045

40046

40047

50 6040W2 (mm)

(e)

Figure 3 +e quality factor of the second trapezoidal bimorph design +e structure loss factor is 0025 +e damping ratio is 0017 +elength of the beam L is 60mm the thickness of Tpiezo is 03mm and the thickness of the brass layer Ts is 005mm


eigenfrequency and plug them into equation (19) +ere isvery little damping (ζ is 0017) when the two-piece trape-zoidal bimorph (in the first design the length of the singleplate is 60mm W1 is 40mm and W2 is 2mm) vibratesunder its first resonance frequency 145Hz +e dampingratio or the structural loss factor (0015) has no effect on theresonance frequency of the bimorphs A high damping ratioζ has a negative impact on the power output of a trapezoidalcomposite bimorph due to a high loss factor

+e quality factorQ and the resonance frequency fr of thetwo-piece trapezoidal beam both contribute to the resonancewidth +e resonance bandwidth Δf of an oscillator can bedefined by the full-width at half-maximum of its power atthe resonance vibrational frequency A piezoelectric energyharvester with a low-quality factor has a wider resonancebandwidth as the resonance bandwidth Δf (FWHM) ispositively proportional to the resonance frequency fr and isnegatively proportional to the quality factor+e relation canbe expressed in equation (20) fr is the resonance frequencyof the trapezoidal beam

Δf fr

Q (20)

+e resonance frequency fr of a series electric circuit canbe found when the capacitive reactance and the inductivereactance are equal (Xc XL and 2πfL 12πC) +ereforethe resonance oscillating frequency fr is commonlyexpressed by equation (21) where L is the inductance and Cis the capacitance of a given bimorph

f r 1

2πLC

radic (21)

Equations (16) and (21) can be substituted into equation(20) therefore the FWHM resonance bandwidth Δf can beexpressed in a relation (22) where r is the internal resistanceof the bimorph and L is the internal inductance of theequivalent electric circuit +e resistivity of the compositematerials (PZTPZN brass) and the internal inductance L ofthe bimorph both contribute to the length of the bandwidthof a bimorph which can be expressed in the followingequation

Δf simr

2πL (22)

+e bandwidth of the resonance frequency Δf is foundonce the quality factor Q and the resonance frequency fr areknown As we can see from equations (20) and (22) a higherequivalent internal resistance r (series-connected bimorph)andor a lower equivalent internal inductance L in anelectrical circuit will contribute to a lower quality factor of aseries LRC system which ultimately leads to a wider reso-nance bandwidth Δf of a system Connecting two unimorphsin series to make one bimorph helps to widen the bandwidthresponse as the series connection of bimorph has the highertotal resistance and the lower total inductance compares itwith that of the parallel connection as the total internalresistance r and the total internal inductance L are expressedin equations (23) and (24) r1 is the resistance of the upperPZTPZN layer L1 is the inductance of the upper PZTPZN

layer r2 is the resistance of the lower PZTPZN layer L2 is theinductance of the lower PZTPZN layer r3 is the resistance ofthe middle brass layer Many researchers modeled the pi-ezoelectric energy harvesters with similar equivalent LRCcircuit [9 14]

r r1 + r2 + r3 (23)

1L

1L1

+1L2

(24)

+e resonance frequency can be expressed by theanalytical computation and the eigenfrequency analysis+e analytical resonance frequency is calculated byYoungrsquos modulus E the rotational momentum I (momentof inertia around the axis of rotation) the length of thebeam L (60mm) the mass of beam m and tip mass Mt(0g) as explained in our previous work [7] +e analyticalformula of the resonance vibrational frequency can beexpressed in equation (25) for the transversal vibration Asthe tip mass m increases the resonance angular frequencyω decreases k is the stiffness of the beam+erefore the tipmass is often used to fine-tune the resonance frequency ofa beam

ω

km

1113970

3EIL3

(33140)mL + Mt

1113971

f 2πω (25)

+e principal axis of the two-piece trapezoidal bimorphdesigns is along the edge of the fixed end of the beam duringvibration Let us name that the fixed end of the beam in thex-direction +e rotational momentum Ix at the center of thetwo-piece bimorph can be calculated by the definition of therotational momentum of a rigid body in equation (26)

Ix 1113946 r2 dm (26)

where mi is the mass of an infinitesimally small volume inthe two-piece trapezoidal bimorph domain r is the distancefrom that region to the axis dm can be calculated by findingthe product of the density of the composite materials ρ andthe infinitesimally small volume dv

dm ρdv (27)

+e infinitesimally small volume dv is the product of thesurface area dA of that small region and the thickness dt ofthat small region on the bimorph domain

dv dA dt (28)

+e surface area dA of the small region is the product ofthe width dw and the height dl

dA dw dt (29)

Equation (28) is substituted into equation (27) +ere-fore equation (26) is rewritten into equation (30)

dm ρ dA dt ρ dw dl dt (30)

Equation (30) is substituted into equation (26) +ere-fore equation (26) can be rewritten to the integral formula in


equation (31) where w1 is the shorter width of the bimorphand w2 is the longer width of the bimorph

Ix 21113946w2

w11113946

L

01113946

t

01113946

L

01113946 r

2ρ dw dl dt dr (31)

+e multiplier ldquo2rdquo in the equation is accounted for theldquotwordquo-piece composite bimorph in equation (31) Equation(31) can be simplified to equation (32) which is used toapproximate the rotational momentum of a compositebimorph with the rotational axis along the x-axis throughthe centroid of the beam +e density of PZT-PZN and thedensity of brass are similar numerically and the thickness ofthe brass layer is very thin (005mm) t is the total thicknessof the composite bimorph (065mm) +erefore equation(32) is derived to approximate the rotational momentum ofany trapezoidal shaped composite bimorph beam with a thinsubstrate along the center principle axis

Ix 2t W2 minus W1( 1113857L4

3 (32)

+e rotational momentum Ixrsquo along the fixed edge in thex-direction can be obtained by applying the Parallel Axis+eorem in equation (32) where Ix is the rotational mo-mentumwith the rotational axis along the x-axis through thecentroid of the composite trapezoidal bimorphM is the totalmass of the trapezoidal d is the distance of the translationfrom the original axis to the new axis which is the length ofthe single trapezoidal plate L (60mm)

Ixprime Ix + Md2 (33)

Equation (32) is substituted into equation (33) +erotational momentum Ixrsquo along the fixed edge in the x-di-rection can be obtained as follows

Ix 2tρ W2 minus W1( 1113857L4

3+ ML2 (34)

+us far we can calculate the rotational momentum ofany one end-free and one end-fixed composite trapezoidalbimorph cantilever with a thin substrate in the middle like asandwiched structure by using equation (34) For exampleequation (34) can be used to calculate the rotational mo-mentum of a composite trapezoidal bimorph cantilever witha longer with W2 (60mm) a shorter width W1 (2mm) thelength of L (60mm) and total thickness of 065mm +edensity ρ of the PZT-PZN is 7869 kgm3 +e rotationalmomentum is 685 times10minus 5 kgmiddotm2 Table 2 shows the resonancefrequency when the widths of the bimorph change

For the first design of the two-piece trapezoidal beam(Figure 4(a)) the volume of the two-piece trapezoidal beamranges from 1638 cm3 to 3042 cm3 as the widthsW1 andW2grow +e real electric power density of the beam reaches amaximum of 19595mWcm3 (38044mW) when theshorter width W1 is 2mm and the longer width W2 is56mm+e beam vibrates at the first resonance frequency at137Hz with the optimal resistor of 014MΩ connected inthe series configuration +e maximum full-width half-maximum (FWHM) bandwidth of the real electric powerdensity is found by iterating 30 geometries and scanning the

vibration frequency around the first resonance frequency frof 200Hz of the two-piece trapezoidal beam (W1 40mm andW2 18mm) with an optimal resistor the scan ranges from09 fr (180Hz 01Hz interval) to 11 fr (220Hz 01Hzinterval) with an optimal resistor of 008MΩ +e beamvibrates at the minimum half-real electric power density of36327mWcm3 (1517mW power) when vibrating between195Hz and 220Hz it reaches peak real power 1517mW ata frequency close to first resonance frequency at 20767Hztherefore the real power density FWHM bandwidth of thetwo-piece trapezoidal beam is 25Hz +e voltage-frequencyplot and power-frequency plot are shown in Figure 5

For the second design of the two-piece trapezoidal beam(Figure 4(b)) the volume of the two-piece trapezoidal beamranges from 1638 cm3 to 3042 cm3 as the widthsW1 andW2grow +e real power density of the beam has a maximum of4352mWcm3 (97136mW) when the shorter width of thebimorph W1 is 2mm and the longer width of the bimorphW2 is 60mm +e beam vibrates at the first resonancefrequency fr at 151Hz with an optimal resistor of 007MΩconnected in the series +e maximum full-width half-maximum (FWHM) bandwidth of the real electric powerdensity is revealed by iterating 30 geometries and scanningthe vibration frequency around the first resonance frequencyfr of 151Hz of the two-piece trapezoidal beam (W1 52mmand W2 18mm) with an optimal resistor the scan rangesfrom 09 fr (136Hz) to 11 fr (166Hz) with an optimalresistor of 004MΩ +e beam has the minimum half-realelectric power density of 8408mWcm3 (42376mWpower)when vibrating between 195Hz and 206Hz it reaches peakreal power 42376mW at a frequency close to first resonancefrequency at 200Hz therefore the real electric powerdensity FWHM bandwidth of the two-piece trapezoidalbeam is 11Hz +e voltage-frequency plot and power-fre-quency are shown in Figure 6

Although the FWHM bandwidth of the one-piecetrapezoidal composite beams shows its results in the de-signing of broadband energy harvesters in the previous work

Table 2 Resonance frequency when the widths of the bimorphchange

Width W1 Width W2 +ickness 2 Tp +Ts Resonance fr2mm 40mm 065mm 145Hz6mm 40mm 065mm 174Hz10mm 40mm 065mm 187Hz14mm 40mm 065mm 195Hz18mm 40mm 065mm 200Hz2mm 44mm 065mm 143Hz6mm 44mm 065mm 171Hz10mm 44mm 065mm 185Hz14mm 44mm 065mm 194Hz18mm 44mm 065mm 199Hz

Note the length of the beam is 60mm (the first trapezoidal design) +evariable widths thickness and the first real resonance frequency of the two-piece trapezoidal composite beams with no tip mass (two out of six iter-ations of W1 and W2 incremental pattern the total of ten permutations ofW1 and W2 out of 30 geometry permutations) +e imaginary part of theresonance fr is ignored as the imaginary part indicates the damping of thestructure


[6] the two-piece trapezoidal beam design has the potentialto increase the energy harvesterrsquos power output and thefrequency bandwidth +e authors proposed two new two-piece trapezoidal beam designs which are shown in Figure 7

For the first design the average resonance frequency ofthirty permutations of the two-piece trapezoidal beams is176Hz For the second design the mean resonance fre-quency of 30 permutations of the two-piece trapezoidalbeams is 179Hz Tp is the thickness of one PZT-PZN layerTs is the thickness of the UNS C22000 Brass layer 2Tp +Ts isthe total thickness of the composite bimorph beam as shownin Figure 8

3 Results and Discussion

For both trapezoidal composite bimorph cantilever beamdesigns the resonance frequency increases as the shorterwidth W1 of the beam increases as the pattern is shown inFigures 9 and 10 Among thirty permutations of geometriesfor each two-piece bimorph design the average resonancefrequency of the first trapezoidal beam design is 176Hz andthe average resonance frequency of the second trapezoidalbeam design is 179Hz It indicates that the first trapezoidalbeam design is suitable for harvesting lower vibration fre-quency applications while the second trapezoidal beam

Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)

Figure 4 +e top-down view of the two-piece piezoelectric trapezoidal beam designs (a) Subplot of the first two-piece trapezoidal beamdesign and (b) subplot of the second two-piece trapezoidal beam design W1 is the shorter width W2 is the longer width and L is the lengthof one single plate

14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)

Real electric power vs frequency

(a)

35

30

25

20

15

18 20 22f (Hz)

Voltage vs frequency

Volta

ge (V

)

(b)

Figure 5 Electric power output vs vibration frequency of the first trapezoidal bimorph two-piece design (a) Voltage vs vibration frequencyof the first trapezoidal bimorph two-piece design (b) W1 is 40mm and W2 is 18mm +e full-width half-maximum is 25Hz +e qualityfactor is 28 +e structure loss factor is 0025 +e damping ratio is 0017 +e optimal resistance is 008MΩ +e length of the beam L is60mm the thickness of Tpiezo is 03mm and the thickness of the brass layer Ts is 005mm


design is suitable for harvesting higher vibration frequencyapplications From Figures 9 and 10 we can also see the firstresonance frequency of the two-piece trapezoidal compositebeam decreases linearly as the longer width W2 increasesfrom 40mm to 60mm due to the increase in the mass of thebeam

+e maximum real electric power density is defined bythe unit real electric energy consumed in an electric circuitBoth two-piece trapezoidal designs revealed two differentpatterns as shown in Figures 11 and 12 For the secondbimorph beam design the electric power density (y-axis)increases as the shorter width W1 increases (x-axis)However the electric power density (y-axis) increases as the

Eigenfrequency = 19646 + 00014194i HzVolume total displacement (m) 465 times 10ndash4

times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)

Eigenfrequency = 45309 ndash 52912i HzVolume total displacement (m)

005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)

Figure 7 +e displacement of two different two-piece trapezoidal piezoelectric bimorph beam which vibrates at its first resonancefrequency (a)+e left subplot is the stationary displacement (without deformation) of the first design (b)+e right subplot is the stationarydisplacement (without deformation) of the second design +e shorter width W1 is 18mm and the longer width W2 is 40mm for both twobimorph designs +e length of the beam L is 60mm the thickness of Tpiezo is 03mm and the thickness of the brass layer Ts is 005mm

PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm

Figure 8 +e thickness view of the composite two-piece trape-zoidal beamrsquos upper and lower PZT-PZN-Scheme 4 layer thicknessfor both two bimorph designs +e length of the beam L is 60mmthe thickness of Tpiezo is 03mm and the thickness of the brass layerTs is 005mm


Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)

Figure 6 Electric power output vs vibration frequency of the second trapezoidal bimorph two-piece design (a) Voltage vs vibrationfrequency of the second trapezoidal bimorph two-piece design (b) W1 is 40mm and W2 is 18mm +e full-width half-maximum is 25Hz+e quality factor is 28 +e structure loss factor is 0025 +e damping ratio is 0017 +e optimal resistance is 008MΩ +e length of thebeam L is 60mm the thickness of Tpiezo is 03mm and the thickness of the brass layer Ts is 005mm


W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)

Figure 9 +e first resonance frequency vs 30 permutations of the two-piece trapezoidal piezoelectric composite bimorph cantilever beam(first design) +e length of the beam L is 60mm the thickness of Tpiezo is 03mm and the thickness of the brass layer Ts is 005mm

40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)

Figure 10 +e first resonance frequency of 30 permutations of the two-piece trapezoidal piezoelectric composite bimorph beam (seconddesign) +e length of the beam L is 60mm the thickness of Tpiezo is 03mm and the thickness of the brass layer Ts is 005mm


5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)

Figure 11 +e maximum electrical power density of 30 various geometries of the two-piece trapezoidal composite piezoelectric bimorphbeam (first design) in a closed circuit with the optimal resistor load+e length of the beam L is 60mm the thickness of Tpiezo is 03mm andthe thickness of the brass layer Ts is 005mm

W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)

Figure 12 +e maximum electrical power density of 30 various geometries of the trapezoidal two-piece composite piezoelectric bimorphbeam (second design) in a closed circuit with the optimal resistor load +e length of the beam L is 60mm the thickness of Tpiezo is 03mmand the thickness of the brass layer Ts is 005mm


longer width W2 increases (x-axis) in Figure 12 In orderwords for the second trapezoidal bimorph design theelectric power density increases when the longer width of thebeam increases at the fixed and free end the electric powerdensity increases when the shorter width of the beam de-creases at the midpoint the length of the two-piece trape-zoidal beams is fixed at 60mm and the total thickness T ofthe composite bimorph cantilever is 065mm +e maxi-mum electric real power densities are compared betweenboth two-piece trapezoidal piezoelectric bimorph compositebeam designs in Table 3

+e full-width half-maximum bandwidth of the realelectrical power is evaluated on both two-piece trapezoidalbeam designs +e length L of both two-piece trapezoidalbeams is fixed at 60mm +e total composite thickness T is065mm+e full-width half-maximum bandwidth of the realelectrical power is found by scanning the vibrational frequencynear the first resonance frequency with an optimal resistor

For the first two-piece trapezoidal design the mean valueof the real electric power FWHMdensity bandwidth of thirty

geometry permutations is 176Hz +e maximum FWHMbandwidth is 25Hz (minimum power density is 363mWcm3 the maximum power density is 727mWcm3 W1 is40mmW2 is 18mm the structural volume is 2262 cm3 andthe output electric power is 727mW) as they are shown inFigure 13

For the second two-piece trapezoidal design the averagereal electric power FWHM density of thirty geometrypermutations is 179Hz +e maximum FWHM bandwidthis 11Hz (the minimum power density is 1866mWcm3 themaximum power density is 3732mWcm3 W1 is 18mmW2 is 52mm the structural volume is 273 cm3 and theelectric power is 9404mW) and the electric power densityincreases FWHM bandwidth when the shorter width W1 atthe center of the bimorph as they are shown in Figure 14

+e comparison between the two-piece and the one-piece trapezoidal beam designs is tabulated in Table 3 +eone-piece trapezoidal bimorph design has the same di-mension (length width and thickness) of the two-piecetrapezoidal bimorph for the comparison +e two-piece

Table 3 FWHM bandwidth the minmax real power density of various trapezoidal bimorph beams [6 8] +e one-piece trapezoidalbimorph design has the same dimension of the two-piece trapezoidal bimorph for the comparison

Design Max FWHM bandwidth Min structural power density Max structural power density1st trapezoidal (one-piece) 29Hz 518mWcm3 1037mWcm3

2nd trapezoidal (one-piece) 56Hz 211mWcm3 422mWcm3

1st trapezoidal (two-piece) 25Hz 363mWcm3 726mWcm3

2nd trapezoidal (two-piece) 11Hz 840mWcm 3 1681mWcm 3

W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )

Figure 13 Real electrical power density FWHM bandwidth of 30 geometries of the two-piece trapezoidal composite piezoelectric bimorphbeam (first design) in a closed circuit with corresponding optimal resistor load fixed +e length of the beam L is 60mm the thickness ofTpiezo is 03mm and the thickness of the brass layer Ts is 005mm


trapezoidal beam (second design) has the maximum structureelectric power density of 1681mWcm3 and the highestminimum structure electric power density of 841mWcm3)+e two-piece trapezoidal beam (first design) has a broaderFWHM power frequency bandwidth of 25Hz

4 Conclusions

In this paper the authors conclude that newer geometry anddesign can result in enhanced power density of bimorphpiezoelectric energy harvester +e bandwidth of the har-vester was found to be decreased with 2-piece trapezoidaldesign however the minimum structure power densityincreased by 131 to 840mWcm3 from 518mWcm3 aswell as the maximum structural power density improved by1315 from 1037mWcm3 to 1681mWcm3 Authorsbelieve these results would help researchers design highpower high sensitivity bimorph harvesters that can provideonboard battery solutions to sensor networks

Data Availability

+e numerical data used to support the findings of this studyare included within the article

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+e authors would like to thank the support from theComputational Science Program the Department of Engi-neering Technology and Graduate Studies at Middle Ten-nessee State University

References

[1] Z S Chen Y M Yang and G Q Deng ldquoAnalytical andexperimental study on vibration energy harvesting behaviorsof piezoelectric cantilevers with different geometriesrdquo inProceedings of the 2009 Sustainable Power Generation andSupply SUPERGEN rsquo09 International Conference Nan JingChina 2009

[2] G Zhang S Gao H Liu and S Niu ldquoA low frequency pi-ezoelectric energy harvester with trapezoidal cantilever beamtheory and experimentrdquo Microsystem Technologies vol 23no 8 pp 3457ndash3466 2017

[3] D Benasciutti L Moro S Zelenika and E Brusa ldquoVibrationenergy scavenging via piezoelectric bimorphs of optimizedshapesrdquoMicrosystem Technologies vol 16 no 5 pp 657ndash6682010

[4] A GMuthalif and N D Nordin ldquoOptimal piezoelectric beamshape for single and broadband vibration energy harvestingmodeling simulation and experimental resultsrdquo MechanicalSystems and Signal Processing vol 54-55 pp 417ndash426 2014

[5] E K Reilly F Burghardt R Fain and P Wright ldquoPowering awireless sensor node with a vibration-driven piezoelectricenergy harvesterrdquo Smart Materials and Structures vol 20no 12 p 125006 2011

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )

Figure 14 Real electrical power density FWHM bandwidth of 30 various geometries of the trapezoidal composite piezoelectric bimorphbeam (second design) in a closed circuit with an optimal resistor load fixed +e length of the beam L is 60mm the thickness of Tpiezo is03mm and the thickness of the brass layer Ts is 005mm


[6] N Chen and V Bedekar ldquoModeling simulation and opti-mization of piezoelectric bimorph transducer for broadbandvibration energy harvesting in multi-beam and trapezoidalapproach (accepted)rdquo Journal of Materials Science Researchvol 10 2017

[7] N Chen and V Bedekar ldquoModeling simulation and opti-mization of piezoelectric bimorph transducer for broadbandvibration energy harvestingrdquo Journal of Material ScienceResearch vol 6 no 4 2017

[8] V Bedekar J Oliver and S Priya ldquoDesign and fabrication ofbimorph transducer for optimal vibration energy harvestingrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 57 no 7 pp 1513ndash1523 2010

[9] J-B Yuan T Xie and W-S Chen ldquoEnergy harvesting withpiezoelectric cantileverrdquo in Proceedings of the IEEE Interna-tional Ultrasonics Symposium Proceedings Beijing China2008

[10] D Benasciutti E Brusa1 L Moro and S Zelenika ldquoOpti-mised piezoelectric energy scavengers for elder carerdquo inProceedings of the Euspen International Conference ZurichSwitzerland 2008

[11] J Schachtele E Goll P Muralt and D Kaltenbacher ldquoAd-mittance matrix of a trapezoidal piezoelectric heterogeneousbimorphrdquo IEEE Transactions on Ultrasonics Ferroelectricsand Frequency Control vol 59 no 12 pp 2765ndash2776 2012

[12] R Hosseini and M Hamedi ldquoImprovements in energy har-vesting capabilities by using different shapes of piezoelectricbimorphsrdquo Journal of Micromechanics and Microengineeringvol 25 no 12 Article ID 125008 2015

[13] Y D Chong ldquoMH2801 Complex methods for the sciencerdquo2016

[14] D I S Priya ldquoEnergy harvesting technologiesrdquo SpringerBerlin Germany 2009


analytical solution for modes of the resonance frequencyof the trapezoidal and the triangular composite piezoelectriccantilever beam [4] Reilly et al classified five broadbandapplications out of fourteen ambient vibrational applica-tions [5] and the authors reported that the 50 of theelectric power (2mW4mW) bandwidth is 8Hz (95ndash103Hz)of their proposed composite trapezoidal piezoelectric VEHs[5] Compared with Reilly et alrsquos work our firsttrapezoidal bimorph design reported 50 electric power (786mW1571 mW) with a bandwidth of 25 Hz (195ndash220 Hz)+is work addresses the goal on designing a two-piece trap-ezoidal composite bimorph piezoelectric energy harvesteroperating in the transverse mode with the aim to enhance themaximum real power density in an electric circuit with anoptimal resistor as well as the full-width half-maximum(FWHM) power density-mechanical vibration broadbandperformance compared them with the results of the one-piecetrapezoidal beam design based on the PZT-PZN-Scheme 4polycrystalline ceramic piezoelectric material 08 [Pb (Zr052Ti048)O3]ndash02 [Pb (Zn13 Nb23)O3] from our previous work onthe one-piece trapezoidal beam model [6 7] +e PZT-PZN-Scheme 4 material was reported because of its superiorstructural power density (01713mWcm3 measured) 2097higher than that of PZT-ZNN-Scheme 2 (01416mWcm3)1538 higher (495mWcm3 measured) in the electric poweroutput and 3113 higher (0499mWcm3 measured) in thepiezoelement power density as the ldquotwo-step sinteringrdquomethod was used to reduce grain size and increase densitywhich leads to higher relative dielectric (εr is 1588 295sim423times higher than those of PZT-ZNNs) and the piezoelectricproperty of PZN-PZN-Scheme 4 material (the charge constantin the thickness mode d33 is 400 pCN 24sim26 times higherthan that of PZT-ZNNs +e charge constant of PZT-PZN inthe transversemode d31 is 15373 pCN 275sim307 times higherthan those of PZT-ZNNs) [8] Not only are the power densitiesof the PZT-PZN-Scheme 4 higher than its PZT-ZNN com-petitors but also is PZT-PZN-Scheme 4rsquos quality factor Qsignificantly lower (787) than that of PZN-ZNN (780) whichmakes PZT-PZN-Scheme 4 a better material choice for a highpower energy harvester as the PZT-PZN-Scheme 4 is a low-quality factor material [2] Yuan et al derived a formula for thevoltage sensitivity of a triple-layer trapezoidal piezoelectricbeam which is useful for sensor applications [9] Many re-searchers reported that the electric power and voltages of atrapezoidal piezoelectric beam are higher than those of arectangle beam given the same volume of PZT [9 10]Benasciutti et al studied the voltage and the electric powercharacteristics of a one-piece trapezoidal and the reversed one-spice trapezoidal shapes of the piezoelectric bimorphs +emaximum power generated was reported to be 650μW at50Hz excitation frequency [10] Yuan et al reported that therectangular piezoelectric beamrsquos maximum electric power isabout 86mW at the operating frequency of 180Hz whereasthe trapezoidal piezoelectric beamrsquosmaximumelectric power is242mW at the operating frequency of 130Hz [9] Shachteleet al proposed a two-ported model to describe a trapezoidalpiezoelectric beam using an admittance matrix +e admit-tance matrix describes the linear relations between the electriccharge Q the tip deflection δ and the voltage V and the force F

at the tip [11] Yet the admittance is used to determine theinternal inductance of a given trapezoidal bimorph in ourresearch Hosseini and Hamedi also reported improvement inenergy harvesting using V-shaped piezoelectric bimorphs It isshown that increasing the width W2 (at free end) lowers theresonance frequency which agrees with our results It was alsoshown that the trapezoidal shape generates higher voltagecompared to the simple rectangular cantilever beam bimorphwhich matches our findings in previous works [6 12] In thispaper we explore a unique design of 2-piece trapezoidalshaped piezoelectric bimorph to understand its effects onharvested power power density and bandwidth +e 2-piecetrapezoidal geometry which is explained later has not beenexplored in the literature so far for energy harvesting appli-cations +e material property values of PZT-PZN-Scheme 4material are tabulated in Table 1

2 Materials and Methods

+e authors used the technique to convert the piezoelectricand the mechanical material property values from Table 1 tothe compliance matrix (a tensor of 4th order) of the PZT-PZN-Scheme 4 piezoelectric material from our previouswork [7]

+e procedure used in seeking themaximum electric realpower output is through finding the first resonance fre-quency of the two-piece piezoelectric beam once the firstresonance frequency of the beam is known through runningeigenfrequency study on the model then we vary the loadingresistor to find the optimal resistance where the powerreaches maximum the external resistance was varied from001MΩ to 02MΩ (an arbitrarily large range) with 001MΩresolution to match the internal impedance of the beam001MΩ interval resolution is arbitrarily chosen to limit thecomputation time to a reasonable level +e optimal resis-tance is found when a peak of the electric power appears inthe scanning rangeWhen the optimal resistance is found fora given model we then vary the vibrational frequencysymmetrically around the first eigenfrequency to see howwide the vibration frequency can get from the peak poweroutput to 50 of the peak power output

+e equivalent electric circuit is presented below inFigure 1 +e piezoelectric energy harvester can be simplymodeled as a series LRC electric circuit with an alternativevoltage source V in a circuit L represents the internal in-ductance r represents the internal resistance (damping ef-fect) and C represents the internal capacitance +e letter Zin the equivalent electric circuit represents the externalimpedance When the external impedance Z matches theinternal impedance Zint of LRC circuit

(Zext Zint

r2 + (Xc minus XL)21113969

) where Xc is the internalcapacitive reactance the bimorph and XL is the internalinductance of the bimorph the electric power output reachesmaximum (widely known as the rule of impedancematching) Due to the current plan of research we onlyconsider the external loading resistor Z When the externalload impedance Zext matches the internal impedance Zintnumerically (Zext Zint) the electric circuit delivers the



Z Rloading

r2 + XL minus Xc( 11138572

1113969

(1)


R2loading +r

2+ 2πfL minus

12πfC

1113888 1113889

2

(2)









11138681113868111386811138681113868111386811138681113868 minus I1 Z1

111386811138681113868111386811138681113868111386811138681113872 11138731113960 1113961

I2 minus I1 (6)


Z11113868111386811138681113868

1113868111386811138681113868 XL1 minus XC11113868111386811138681113868

1113868111386811138681113868 2πf1L minus1

2πf1C

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (7)

Z21113868111386811138681113868

1113868111386811138681113868 XL2 minus XC21113868111386811138681113868

1113868111386811138681113868 2πf2L minus1

2πf2C

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)




C

Z

R L

V





111386811138681113868111386811138681113868111386811138681113872 11138731113960 1113961

I2 minus I1

(9)





Cd1 Q


100V 71028 pF

Ld1 776H

f r11

2πLC

radic 214Hz

(11)

Cd2 Q


100V 83341 pF

Ld2 643H

f r21

2πLC

radic 217Hz

(12)

L imaginary 1





I2 minus I1


00011475A minus 00011393A 14MΩ

(14)




00015768A minus 00017844A 65 kΩ

(15)



Q 1r

L

C

1113970

(16)


Q abs (f)








w wi + wr(18)


ζ imaginary (f)

abs(f) (19)


40 50 60

Qua

lity

fact

or

W1 = 2 (mm)

W2 (mm)

2864

2865

2866

2867

(a)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 6 (mm)

2863

28635

2864

28645

(b)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 10 (mm)

28624

28626

28628

2863

28632

(c)Q

ualit

y fa

ctor

40 50 60

W1 = 14 (mm)

W2 (mm)

2862

28622

28624

(d)Q

ualit

y fa

ctor

40 50 6028617

28618

28619

2862

W1 = 18 (mm)

W2 (mm)

(e)


Qua

lity

fact

or

W1 = 2 (mm)

4006

40065

4007

40075

50 6040W2 (mm)

(a)

Qua

lity

fact

or

W1 = 6 (mm)

4005

40052

40054

40056

50 6040W2 (mm)

(b)

Qua

lity

fact

orW1 = 10 (mm)

40046

40048

4005

40052

50 6040W2 (mm)

(c)

Qua

lity

fact

or

W1 = 14 (mm)

40046

40047

40048

50 6040W2 (mm)

(d)

Qua

lity

fact

or

W1 = 18 (mm)

40045

40046

40047

50 6040W2 (mm)

(e)





Δf fr

Q (20)


f r 1

2πLC

radic (21)


Δf simr

2πL (22)



r r1 + r2 + r3 (23)

1L

1L1

+1L2

(24)


ω

km

1113970

3EIL3

(33140)mL + Mt

1113971

f 2πω (25)


Ix 1113946 r2 dm (26)


dm ρdv (27)


dv dA dt (28)


dA dw dt (29)






Ix 21113946w2

w11113946

L

01113946

t

01113946

L

01113946 r



Ix 2t W2 minus W1( 1113857L4

3 (32)





3+ ML2 (34)














Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)


14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)


(a)

35

30

25

20

15

18 20 22f (Hz)


Volta

ge (V

)

(b)






times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )














Z Rloading

r2 + XL minus Xc( 11138572

1113969

(1)


R2loading +r

2+ 2πfL minus

12πfC

1113888 1113889

2

(2)









11138681113868111386811138681113868111386811138681113868 minus I1 Z1

111386811138681113868111386811138681113868111386811138681113872 11138731113960 1113961

I2 minus I1 (6)


Z11113868111386811138681113868

1113868111386811138681113868 XL1 minus XC11113868111386811138681113868

1113868111386811138681113868 2πf1L minus1

2πf1C

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (7)

Z21113868111386811138681113868

1113868111386811138681113868 XL2 minus XC21113868111386811138681113868

1113868111386811138681113868 2πf2L minus1

2πf2C

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)




C

Z

R L

V





111386811138681113868111386811138681113868111386811138681113872 11138731113960 1113961

I2 minus I1

(9)





Cd1 Q


100V 71028 pF

Ld1 776H

f r11

2πLC

radic 214Hz

(11)

Cd2 Q


100V 83341 pF

Ld2 643H

f r21

2πLC

radic 217Hz

(12)

L imaginary 1





I2 minus I1


00011475A minus 00011393A 14MΩ

(14)




00015768A minus 00017844A 65 kΩ

(15)



Q 1r

L

C

1113970

(16)


Q abs (f)








w wi + wr(18)


ζ imaginary (f)

abs(f) (19)


40 50 60

Qua

lity

fact

or

W1 = 2 (mm)

W2 (mm)

2864

2865

2866

2867

(a)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 6 (mm)

2863

28635

2864

28645

(b)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 10 (mm)

28624

28626

28628

2863

28632

(c)Q

ualit

y fa

ctor

40 50 60

W1 = 14 (mm)

W2 (mm)

2862

28622

28624

(d)Q

ualit

y fa

ctor

40 50 6028617

28618

28619

2862

W1 = 18 (mm)

W2 (mm)

(e)


Qua

lity

fact

or

W1 = 2 (mm)

4006

40065

4007

40075

50 6040W2 (mm)

(a)

Qua

lity

fact

or

W1 = 6 (mm)

4005

40052

40054

40056

50 6040W2 (mm)

(b)

Qua

lity

fact

orW1 = 10 (mm)

40046

40048

4005

40052

50 6040W2 (mm)

(c)

Qua

lity

fact

or

W1 = 14 (mm)

40046

40047

40048

50 6040W2 (mm)

(d)

Qua

lity

fact

or

W1 = 18 (mm)

40045

40046

40047

50 6040W2 (mm)

(e)





Δf fr

Q (20)


f r 1

2πLC

radic (21)


Δf simr

2πL (22)



r r1 + r2 + r3 (23)

1L

1L1

+1L2

(24)


ω

km

1113970

3EIL3

(33140)mL + Mt

1113971

f 2πω (25)


Ix 1113946 r2 dm (26)


dm ρdv (27)


dv dA dt (28)


dA dw dt (29)






Ix 21113946w2

w11113946

L

01113946

t

01113946

L

01113946 r



Ix 2t W2 minus W1( 1113857L4

3 (32)





3+ ML2 (34)














Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)


14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)


(a)

35

30

25

20

15

18 20 22f (Hz)


Volta

ge (V

)

(b)






times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )















111386811138681113868111386811138681113868111386811138681113872 11138731113960 1113961

I2 minus I1

(9)





Cd1 Q


100V 71028 pF

Ld1 776H

f r11

2πLC

radic 214Hz

(11)

Cd2 Q


100V 83341 pF

Ld2 643H

f r21

2πLC

radic 217Hz

(12)

L imaginary 1





I2 minus I1


00011475A minus 00011393A 14MΩ

(14)




00015768A minus 00017844A 65 kΩ

(15)



Q 1r

L

C

1113970

(16)


Q abs (f)








w wi + wr(18)


ζ imaginary (f)

abs(f) (19)


40 50 60

Qua

lity

fact

or

W1 = 2 (mm)

W2 (mm)

2864

2865

2866

2867

(a)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 6 (mm)

2863

28635

2864

28645

(b)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 10 (mm)

28624

28626

28628

2863

28632

(c)Q

ualit

y fa

ctor

40 50 60

W1 = 14 (mm)

W2 (mm)

2862

28622

28624

(d)Q

ualit

y fa

ctor

40 50 6028617

28618

28619

2862

W1 = 18 (mm)

W2 (mm)

(e)


Qua

lity

fact

or

W1 = 2 (mm)

4006

40065

4007

40075

50 6040W2 (mm)

(a)

Qua

lity

fact

or

W1 = 6 (mm)

4005

40052

40054

40056

50 6040W2 (mm)

(b)

Qua

lity

fact

orW1 = 10 (mm)

40046

40048

4005

40052

50 6040W2 (mm)

(c)

Qua

lity

fact

or

W1 = 14 (mm)

40046

40047

40048

50 6040W2 (mm)

(d)

Qua

lity

fact

or

W1 = 18 (mm)

40045

40046

40047

50 6040W2 (mm)

(e)





Δf fr

Q (20)


f r 1

2πLC

radic (21)


Δf simr

2πL (22)



r r1 + r2 + r3 (23)

1L

1L1

+1L2

(24)


ω

km

1113970

3EIL3

(33140)mL + Mt

1113971

f 2πω (25)


Ix 1113946 r2 dm (26)


dm ρdv (27)


dv dA dt (28)


dA dw dt (29)






Ix 21113946w2

w11113946

L

01113946

t

01113946

L

01113946 r



Ix 2t W2 minus W1( 1113857L4

3 (32)





3+ ML2 (34)














Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)


14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)


(a)

35

30

25

20

15

18 20 22f (Hz)


Volta

ge (V

)

(b)






times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )














w wi + wr(18)


ζ imaginary (f)

abs(f) (19)


40 50 60

Qua

lity

fact

or

W1 = 2 (mm)

W2 (mm)

2864

2865

2866

2867

(a)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 6 (mm)

2863

28635

2864

28645

(b)

Qua

lity

fact

or

40 50 60W2 (mm)

W1 = 10 (mm)

28624

28626

28628

2863

28632

(c)Q

ualit

y fa

ctor

40 50 60

W1 = 14 (mm)

W2 (mm)

2862

28622

28624

(d)Q

ualit

y fa

ctor

40 50 6028617

28618

28619

2862

W1 = 18 (mm)

W2 (mm)

(e)


Qua

lity

fact

or

W1 = 2 (mm)

4006

40065

4007

40075

50 6040W2 (mm)

(a)

Qua

lity

fact

or

W1 = 6 (mm)

4005

40052

40054

40056

50 6040W2 (mm)

(b)

Qua

lity

fact

orW1 = 10 (mm)

40046

40048

4005

40052

50 6040W2 (mm)

(c)

Qua

lity

fact

or

W1 = 14 (mm)

40046

40047

40048

50 6040W2 (mm)

(d)

Qua

lity

fact

or

W1 = 18 (mm)

40045

40046

40047

50 6040W2 (mm)

(e)





Δf fr

Q (20)


f r 1

2πLC

radic (21)


Δf simr

2πL (22)



r r1 + r2 + r3 (23)

1L

1L1

+1L2

(24)


ω

km

1113970

3EIL3

(33140)mL + Mt

1113971

f 2πω (25)


Ix 1113946 r2 dm (26)


dm ρdv (27)


dv dA dt (28)


dA dw dt (29)






Ix 21113946w2

w11113946

L

01113946

t

01113946

L

01113946 r



Ix 2t W2 minus W1( 1113857L4

3 (32)





3+ ML2 (34)














Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)


14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)


(a)

35

30

25

20

15

18 20 22f (Hz)


Volta

ge (V

)

(b)






times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )















Δf fr

Q (20)


f r 1

2πLC

radic (21)


Δf simr

2πL (22)



r r1 + r2 + r3 (23)

1L

1L1

+1L2

(24)


ω

km

1113970

3EIL3

(33140)mL + Mt

1113971

f 2πω (25)


Ix 1113946 r2 dm (26)


dm ρdv (27)


dv dA dt (28)


dA dw dt (29)






Ix 21113946w2

w11113946

L

01113946

t

01113946

L

01113946 r



Ix 2t W2 minus W1( 1113857L4

3 (32)





3+ ML2 (34)














Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)


14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)


(a)

35

30

25

20

15

18 20 22f (Hz)


Volta

ge (V

)

(b)






times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )














Ix 21113946w2

w11113946

L

01113946

t

01113946

L

01113946 r



Ix 2t W2 minus W1( 1113857L4

3 (32)





3+ ML2 (34)














Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)


14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)


(a)

35

30

25

20

15

18 20 22f (Hz)


Volta

ge (V

)

(b)






times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )

















Fixed end

L = 60mm

W1 = 18mm

W2 = 40mm

(a)

Fixed end

W1 = 18mm

W2 = 40mm

L = 60mm

(b)


14

12

10

8

6

4

20ndash1

ndash06

06

ndash04

04

ndash08

1

08

ndash02

02

0

f (Hz)

Real

elec

tric

pow

er (m

W)


(a)

35

30

25

20

15

18 20 22f (Hz)


Volta

ge (V

)

(b)






times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )
















times10ndash5

45

40

35

30

25

20

15

10

5

00

ndash2

ndash0050

m005 004

0

m

zyx

(a)


005

0

ndash005

ndash001

002

m

m

zy x

1 times 10ndash9

2 times 10ndash3 m

times10ndash10

1098765432100

(b)


PZT-PZN layer 03mm

PZT-PZN layer 03mm

Brass layer 005mm



Real

elec

tric

pow

er (m

W)

908070605040302010

18 20f (Hz)

(a)


Volta

ge (V

)

60555045403530252015

18 20f (Hz)

(b)



W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )













W1 = 2 (mm)

136

138

14

142

144

146Re

sona

nce f

requ

ency

(Hz)

50 6040W2 (mm)

(a)

W1 = 6 (mm)

164

166

168

17

172

174

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(b)

W1 = 10 (mm)

178

18

182

184

186

188

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(c)

W1 = 14 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

50 6040W2 (mm)

(d)

W1 = 18 (mm)

194

196

198

20

202

Reso

nanc

e fre

quen

cy (H

z)50 6040

W2 (mm)

(e)


40 50 60

W1 = 2 (mm)

W2 (mm)

15

152

154

156

158

16

Reso

nanc

e fre

quen

cy (H

z)

(a)

40 50 60

W1 = 6 (mm)

W2 (mm)

17

172

174

176

178

18

Reso

nanc

e fre

quen

cy (H

z)

(b)

40 50 60

W1 = 10 (mm)

W2 (mm)

18

182

184

186

188

19

Reso

nanc

e fre

quen

cy (H

z)

(c)

W1 = 14 (mm)

40 50 60W2 (mm)

186

188

19

192

194

196

Reso

nanc

e fre

quen

cy (H

z)

(d)

W1 = 18 (mm)

40 50 60W2 (mm)

19

192

194

196

198

20

Reso

nanc

e fre

quen

cy (H

z)

(e)



5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )













5

6

7

8

9

10

11El

ectr

ic p

ower

den

sity

(mW

cm

3 )

W1 = 2 (mm)

45 50 55 6040W2 (mm)

(a)

W1 = 6 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(b)

W1 = 10 (mm)

0

10

20

30

40

50

60

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(c)

695

7

705

71

715

72

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

W1 = 14 (mm)

45 50 55 6040W2 (mm)

(d)

W1 = 18 (mm)

5

10

15

20

25

30

35

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

45 50 55 6040W2 (mm)

(e)


W1 = 2 (mm)

415

42

425

43

435

44

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(a)

W1 = 6 (mm)

384

386

388

39

392

394

396

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(b)

W1 = 10 (mm)

376

378

38

382

384

386

388

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(c)

W1 = 14 (mm)

372

374

376

378

38

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

50 6040W2 (mm)

(d)

W1 = 18 (mm)

50 6040W2 (mm)

3715

372

3725

373

3735

Elec

tric

pow

er d

ensit

y(m

Wc

m3 )

(e)














W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )
























W2 = 40 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(a)

W2 = 44 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(b)

W2 = 48 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(c)

W2 = 52 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

(d)

W2 = 56 (mm)

10 200W1 (mm)

05

1

15

2

25

FWH

M b

andw

idth

(Hz)

(e)

W2 = 60 (mm)

10 200W1 (mm)

1

15

2

25

FWH

M b

andw

idth

(Hz)

(f )




4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )














4 Conclusions


Data Availability




Acknowledgments


References






06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 40 (mm)

10 200W1 (mm)

(a)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 44 (mm)

10 200W1 (mm)

(b)

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 48 (mm)

10 200W1 (mm)

(c)

06

08

1

12

FWH

M b

andw

idth

(Hz)

10 200W1 (mm)

W2 = 52 (mm)

(d)

06

07

08

09

1FW

HM

ban

dwid

th (H

z)

W2 = 56 (mm)

10 200W1 (mm)

(e)

06

07

08

09

1

FWH

M b

andw

idth

(Hz)

W2 = 60 (mm)

10 200W1 (mm)

(f )













Design,Modeling,andSimulationofTwo-PieceTrapezoidal ...

Documents

Transcript of Design,Modeling,andSimulationofTwo-PieceTrapezoidal ...