dynamic modeling and characteristic analysis of cantilever...

Research ArticleDynamic Modeling and Characteristic Analysis of CantileverPiezoelectric Bimorph

Heng Chen Jun-shanWang Chao Chen Shi-xiang Liu and Hai-peng Chen

State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics amp AstronauticsNanjing 210016 China

Correspondence should be addressed to Chao Chen chaochennuaaeducn

Received 26 November 2018 Accepted 5 February 2019 Published 3 March 2019

Academic Editor Gareth A Vio

Copyright copy 2019 Heng Chen et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The analytical model of an axially precompressed cantilever bimorph is established using the Hamiltonrsquos principle in this study andthe static characteristics are obtained The dynamic equations of the cantilever bimorph in generalized coordinates are establishedusing a numerical method and the dynamic characteristics are analyzed Finally simulations are performed and experiments areconducted to verify the validity of the theory The results show that increase of axial force has significant amplification effects onthe steady-state response amplitude of the displacement and it reduces the resonance frequency The response time is still in themillisecond range under a large axial force which indicates that the bimorph has excellent dynamic characteristics as an actuator

1 Introduction

With the development of microelectromechanical systemsand flight control technology the miniaturization and intel-lectualization of unmanned aerial vehicles (UAVs) haveattracted increasing attention from researchers [1] As animportant part of the aircraft the actuator determines theperformance of the flight control system [2] The traditionalelectromagnetic actuator has many problems such as anarrow control band low precision low energy density andit has gradually been unable to satisfy the requirements ofdesign and application [3 4] Therefore a new driving modeof control surface is required to solve the problem

With the rapid development of intelligent materialsresearchers are increasingly attempting to apply them to thedesign and manufacture of aircraft [5 6] The energy densityand control bandwidth are two important criteria for theactuator of micro air vehicles Compared with shapememoryalloys andmagnetostrictivematerials piezoelectricmaterialsowing to many advantages such as wide band high energydensity and high conversion rate ofmechanical and electricalenergy have become the ideal material for the design ofaircraft actuator [7] As early as 1998 Wlezien R W of theNASA Langley Research Center proposed that piezoelectricmaterials would be the best choice for the actuator of variant

aircraft but they must overcome the limitation of their smalldeformation [8]

Piezoelectric bimorph has advantages such as low weightlow power consumption simple structure and absence ofelectromagnetic interference which is very suitable for theactuator of micro aircraft and intelligent ammunition [9ndash11] Owing to the small excitation strain of piezoelectricmaterials [12] piezoelectric bimorph has the disadvantage ofsmall displacement output In order to solve this problemresearchers have conducted extensive studies to investigatethe characteristics of the bimorph [13 14] and have proposedvarious solutions including various kinds of mechanicalmagnification mechanisms such as lever mechanism planarfour bar mechanism and triangular displacement amplifi-cation mechanism [15 16] In 1997 Lesieutre proposed theconcept of PBP (postbuckled precompressed) [17 18] whichshowed that the displacement output and electromechani-cal conversion efficiency will increase upon applying axialprecompression to a piezoelectric bimorph In theory theelectromechanical coupling factor can reach 100 percentwhen the axial force is equal to the buckling load Basedon this theory researchers at Delft University of Technologysuccessfully applied piezoelectric bimorph to the design andmanufacture of various aircraft such as morphing wingUAVs [19] and XQ138 [20] Moreover they also used PBP

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3926906 9 pageshttpsdoiorg10115520193926906

2 Mathematical Problems in Engineering

actuators to optimize the design of the piezoelectric flightcontrol surface [21] In order to analyze the static anddynamic characteristics of a piezoelectric bimorph underaxial precompression Giannopoulos et al of the Civil andMaterials Engineering Department at the Royal MilitaryAcademy used nonlinear mechanics to establish a theoreticalmodel for bimorphs and verified the validity of the analyticalmodel through experiments and simulations [22 23]

For actuators driven by axially precompressed piezoelec-tric bimorphs the characteristics of the bimorphs determinethe performance of the actuators Therefore it is necessaryto definitively describe the static and dynamic characteristicsof axially precompressed bimorphs However the dynamicexpression of the existing precompressed bimorph is not per-fect and the established kinetic model has been significantlysimplified Most of the researchers took the effect of thepiezoelectricity as the bending moment when establishingthe statics model [19 20] For the establishment of thekinetic model they simplified the bimorph into a single-degree-of-freedom system to obtain the first-order naturalfrequency [21] but this method could not get the outputcharacteristics of the system at other excitation frequenciesIn this paper the rigid-flexible coupling dynamics modelof the bimorph is established based on the finite elementmethod which can be used to analyze various dynamiccharacteristics such as amplitude-frequency characteristicsof the system The validity of the model is verified throughsimulations and experiments thus providing a theoreticalbasis for the application of the piezoelectric bimorph

2 Establishment of Static and Dynamic Model

21 Driving Principle of Piezoelectric Bimorphs as ActuatorsPiezoelectric bimorph consists of two piezoelectric layers andone substrate layer As shown in Figure 1 the polarizationdirection of the two piezoelectric materials is the same Thesame voltage signal is connected to the upper and lowersurfaces and the substrate is groundedThe two piezoelectriclayers extend and contract respectively so that the bimorphcan deform in two directions If a mechanical device is addedat the end of the structure the bimorph will become anactuator of the servo system

22 StaticModel of Axially Precompressed Cantilever BimorphThe bimorph is simplified to a three-layer Euler beam modelwithout considering the kinetic energy of infinitesimal rota-tion [24] The kinetic energy potential energy and the workdone by the axial force and the electric field are calculatedThe following equations can be obtained by consideringvariation of the energy functionals

120575119879119887119894 = (120588A) int1198710120575119889119909

120588A = 2120588119901119860119901 + 120588s119860 119904(1)

where 120575119879119887119894 is the variation of the total kinetic energy and 120588119860represents the linear density of the bimorph The subscript 119904and 119901 indicate the substrate and piezoelectricity respectively

120588119904 and 120588119901 are the corresponding density 119860 119904 and 119860119901 are thecorresponding cross-sectional area L represents the lengthof the bimorph is the first derivative of the neutral layerdeflection versus time

120575119880119887119894 = 119864119868int11987101199081015840101584012057511990810158401015840119889119909 + 1198721198902 int119871

012057511990810158401015840119889119909

119872119890 = 119887119864311988931119864119901ℎ119901 (ℎ119904 + ℎ119901)119864119868 = 119864s119868119904 + 2119864119901119868119901

(2)

120575119880119887119894 is the variation of the total potential energy andEI represents the bending stiffness of the bimorph 119864119904 and119864119901 represent the elastic moduli of the substrate and thepiezoelectric layer respectively 119868119904 and 119868119901 are the correspond-ing inertias ℎ119904 and ℎ119901 are the corresponding thicknessesb represents the width of the bimorph 11990810158401015840 is the secondderivative of the neutral layer deflection versus the x-axis 1198643represents the electric field strength along the z-axis 11988931 isthe piezoelectric constant in the corresponding directionThefollowing conclusion can be obtained from (2)

120575119880119887119894 = 119864119868(11990810158401015840 1205751199081015840100381610038161003816100381610038161198710 minus 119908101584010158401015840120575119908100381610038161003816100381610038161198710 + int119871

0119908(4)120575119908119889119909)

+ 12119872119890 120575119908101584010038161003816100381610038161003816119871

0

(3)

Assuming that the beam is not extensible and the axialpressure 119865 is applied to the beam the total work of the systemcan be obtained The following formula can be achieved byconsidering the variation of the total work

120575119882 = minus12119872119890 120575119908101584010038161003816100381610038161003816119871

0+ 1198651199081015840 120575119908|1198710 minus 119865int119871

011990810158401015840120575119908119889119909 (4)

From the Hamiltonrsquos principle the stationary point of thefunctional is obtained by

120575int11990521199051

(119879119887119894 minus 119880119887119894 +119882)119889119905 = 0 (5)

According to the partial differential equations obtainedabove it can be observed that the kinetic equation of thebimorph is consistent with that of the ordinary homogeneousbeamThe converse piezoelectric effect is not reflected in theequation but in the boundary condition

120588119860 + 119864119868119908(4) + 11986511990810158401015840 = 0 (6)

Assuming that the axial force is always horizontal theboundary conditions of the cantilever bimorph can beobtained as follows

11990810158401015840 = minus119872119890119864119868 119909 = 119871119864119868119908101584010158401015840 + 1198651199081015840 = 0 119909 = 119871

1199081015840 = 0 119909 = 0119908 = 0 x = 0

(7)

Mathematical Problems in Engineering 3

(a)

PZT

PZTSubstrate

(b)

Substrate PZT

(c)

Figure 1 Piezoelectric bimorphs (a) Deformation modes (b) Connection mode (c) Structure

The deflection of the cantilever bimorph is obtained bysolving the above formula as follows

119908119904 = 119872119890 [cos (120582119909) minus 1]119865 cos (120582119871)120582 = radic 119865119864119868

(8)

When the denominator of formula (8) is zero the first-order buckling critical load of the cantilever bimorph isobtained as

119865119888119903 = 119864119868 120587241198712 (9)

23 Rigid-Flexible Coupling Dynamic Model of CantileverBimorph For the cantilever bimorph it is difficult to obtainthe analytical solution Therefore the bimorph can be dis-cretized along the x-axis and the deflection and rotationangles at both ends of the beam element can be selectedas the generalized coordinates The deflection is assumed toobtain the shape function matrix of the element and the lin-ear differential equations and element characteristic matrixbased on the generalized coordinates are obtained using theHamiltonrsquos principle Then the integral characteristic matrixof the bimorph is obtained

The bimorph is dispersed along the x-axis into 119899 unitsand the length of each unit is 119909 = 119871119899 The unit deflectionis assumed as [24]

119908 = [119891]119879 [119886] = [119873] [120575] (10)

Here

[119891] = [1 119909 1199092 1199093]119879[119886] = [1198861 1198862 1198863 1198864]119879

(11)

[120575] =[[[[[[

119908111990810158401119908211990810158402

]]]]]]=[[[[[[

1 0 0 00 1 0 01 Δ119909 Δ1199092 Δ11990930 1 2Δ119909 3Δ1199092

]]]]]]

[[[[[[

1198861119886211988631198864

]]]]]]

= [Λ] [119886]

(12)

From (11) and (12) the shape function matrix of the unitcan be calculated as

[119873] = [119891]119879 [Λ]minus1 (13)

Therefore the deflection119908 can be expressed as a functionof the generalized coordinates 120575 and the bimorph is sim-plified as a multi-degree-of-freedom system Subsequentlythe dynamic equation of the unit is built using the Lagrangeequation The kinetic energy strain energy potential energyand work done by the axial force are expressed as a functionof the generalized coordinate 120575 and themassmatrix stiffnessmatrix and excitation vector are obtained Here the kineticenergy of the bimorph element is expressed as

119879 = 119879119904 + 2119879119901 = 12120588119860intΔ119909

0

120575119879119873119879119873 120575119889119909 (14)

Here120588A andEI have the same formas formula (6) exceptthat the length 119871 is replaced by x The expression of thestrain energy and electric potential energy of the piezoelectric


material layer can be obtained from the piezoelectric equa-tion The total strain energy of the unit is as follows

119880 = 119880119904 + 2119880119901= 12119864119868int

Δ119909

01205751198791198731015840101584011987911987310158401015840120575119889119909 + 1198721198902 [0 minus1 0 1] 120575 (15)

where 119872119890 is the equivalent of 119872119890 in (2) Subsequently theexpression of the effect of voltage and axial force on thegeneralized coordinates is calculated The electric potentialenergy is

119882119864 = minus1198721198904 [0 minus1 0 1] 120575 minus 119887Δ119909ℎ119901119864119901119864231198892312+ 119887Δ119909ℎ11990111986423120576332

(16)

The effect of axial force is

119882119865 = 119865intΔ1199090

119908101584022 119889119909 = 1198652 intΔ119909

012057511987911987310158401198791198731015840120575119889119909 (17)

The total work is

119882 = 119882119865 + 2119882119864 (18)

The linear differential equations are established using theLagrange equation as follows

119889119889119905 120597119879120597 120575 +120597119880120597120575 = 120597119882120597120575 (19)

The dynamic equation of the element is given by

[119872] 120575 + [119870] 120575 = 119872119890 [0 1 0 minus1]119879 (20)

The characteristic matrix of the element is obtained bycalculating the integrals of the shape function and the massmatrix is as follows

[119872] = 120588119860intΔ1199090119873119879119873119889119909

= 120588119860[[[[[[[[[[[[

13Δ11990935 11Δ1199092210 9Δ11990970 minus13Δ119909242011Δ1199092210 Δ1199093105 13Δ1199092420 minusΔ11990931409Δ11990970 13Δ1199092420 13Δ11990935 minus11Δ1199092210minus13Δ1199092420 minusΔ1199093140 minus11Δ1199092210 Δ1199093105

]]]]]]]]]]]]

(21)

Thus the stiffness matrix of the unit can be obtainedaccording to the above calculation For the case of additionalelastic elements in the unit (such as springs) this method is

also applicable by simply adding the required stiffness factorto the corresponding position in the stiffness matrix

[119870] = 119864119868

[[[[[[[[[[[[[

12Δ1199093 6Δ1199092 minus12Δ1199093 6Δ11990926Δ1199092 4Δ119909 minus6Δ1199092 2Δ119909minus12Δ1199093 minus6Δ1199092 12Δ1199093 minus6Δ11990926Δ1199092 2Δ119909 minus6Δ1199092 4Δ119909

]]]]]]]]]]]]]

minus 119865

[[[[[[[[[[[[[

65Δ119909 110 minus65Δ119909 110110 2Δ11990915 minus110 minusΔ11990930minus65Δ119909 minus110 65Δ119909 minus110110 minusΔ11990930 minus110 2Δ11990915

]]]]]]]]]]]]]

(22)

The characteristicmatrix of the bimorph can be describedusing an iterative method When n=i+1 the mass matrix ofthe bimorph is

[119872]119887119894 = [[119872119894]2(119894+1)times2(119894+1) 02(119894+1)times202times2(119894+1) 02times2 ]

+ [02119894times2119894 02119894times404times2119894 [119872]4times4]

(23)

For the cantilever boundary conditions the first two rowsand the first two columns in the above matrix are removedand the final mass matrix is obtained The stiffness matrixis homogeneous Moreover the external excitation vectorinduced by the voltage is in the following form

119887119864311988931119864119901ℎ119901 (ℎ119904 + ℎ119901) [0 1 0 sdot sdot sdot 0 minus1]1times2(119899+1)119879 (24)

3 Establishment of Finite Element Simulation

Combining the dynamics equation of the bimorph given bythe finite element method the characteristic matrix of theunit is assembled into the totalmatrixThen the first two rowsand the first two columns are removed from the total matrixTherefore the first-order natural frequency of the bimorphcan be achieved which is shown in Figure 2 The parametersof the bimorph can be obtained from [25]

Figure 3 shows the first-order natural frequency ofthe cantilever bimorph under different axial forces In thenumerical method the bimorph is equally divided into 5elements The ANSYS simulation results are consistent withthe numerical results

In order to get the amplitude-frequency characteristicsof the cantilever bimorph the total characteristic matrix isconstructed For the cantilever condition the final stiffnessmatrix [119870]119887119894 and themassmatrix [119872]119887119894 are obtainedwhen the


1 2 3 4 5 6 7 8 90 10

Number of elements divided

1075

1076

1077

1078

1079

1080

1081

1082

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)

Figure 2 Convergence of first-order natural frequency

Numerical solutionANSYS solution

40

50

60

70

80

90

100

110

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)

02 04 06 08 1000

Normalized axial force

Figure 3 First-order natural frequency under different axial forces

first two rows and the first two columns of the characteristicmatrix are removed Similarly the first two rows in theexcitation vector are removedThus the following expressionis achieved

119872ebi = 119887119864311988931119864119901ℎ119901 (ℎ119904 + ℎ119901) [0 sdot sdot sdot 0 minus1]119879 (25)

The vibration equation of the cantilever bimorph is asfollows

[119872]119887119894 120575 + [119870]119887119894 120575= 11988711988931119864119901119881(ℎ119904 + ℎ119901) [0 sdot sdot sdot 0 minus1]119879

(26)

Suppose the voltage is 119881 = V sin(120596119905) the steady-stateresponse can be obtained by solving (26)

120575 = 11988711988931119864119901119881(ℎ119904 + ℎ119901) [119867] [0 sdot sdot sdot 0 minus1]119879 (27)

30 60 90 120 150

Frequency (Hz)

12 N Numerical solution109 N Numerical solution206 N Numerical solution12 N ANSYS solution109 N ANSYS solution206 N ANSYS solution

00

01

02

03

04

05

06

Am

plitu

de (m

m)

Figure 4 Amplitude-frequency characteristics under different axialforces

The frequency response function is as follows

[119867] = (minus1205962 [119872]119887119894 + [119870]119887119894)minus1 (28)

Finally the amplitude of the cantilever bimorph isachieved

119860 = [0 sdot sdot sdot 0 1 0 ] 11988711988931119864119901V (ℎ119904 + ℎ119901)sdot [119867] [0 sdot sdot sdot 0 minus1]119879 (29)

From formulae (27) (28) and (29) the amplitude of thefree end at the voltage of 1 V is calculated The results arecompared with the ANSYS simulation results as shown inFigure 4 It can be observed from the diagram that the firstnatural frequency of the bimorph decreases with the increaseof axial force whereas the amplitude gradually increases

In order to analyze the transient characteristics of thebimorph it is assumed that the voltage is a step signal

119881 = 0 119905 lt 0V 119905 gt 0 (30)

The bimorph starts moving from the static state andthe unit impulse response of (26) can be obtained from thevibration theory as follows

[ℎ (119905)] = [Φ] diag( sin120596119903119905120596119903 ) [Φ]119879 1205750 (31)

where [B] is the mode matrix normalized by the main massand120596119903 is the rth natural frequency of the bimorph If dampingis added the unit impulse response can be expressed as


30

25

20

15

10

5

0

Am

plitu

de (u

m)

0 20 40 60 80 100

Time (ms)

12N ANSYS solution12N Numerical solution109N ANSYS solution109N Numerical solution

Figure 5 Step response of the bimorph under different axial forces

[ℎ (119905)] = [Φ] diag(119890minus120577120596119903119905 sinradic1 minus 1205772120596119903119905120596119903radic1 minus 1205772 ) [Φ]119879 1205750 (32)

where 120577 is the damping ratio The zero-initial state stepresponse of the undamped bimorph can be obtained usingDuhamelrsquos integral

120575 (119905) = int1199050[ℎ (119905)] 119872119890119887119894 119889119905

= 11988711988931119864119901V (ℎ119904 + ℎ119901) [Φ] diag(1 minus cos1205961199031199051205962119903 )sdot [Φ]119879 [0 sdot sdot sdot 0 minus1]119879

(33)

Further the step response of a damped zero-initial stateis

120575 (119905) = 11988711988931119864119901V (ℎ119904 + ℎ119901) [Φ] diag (119881119903)sdot [Φ]119879 [0 sdot sdot sdot 0 minus1]119879 (34)

where

119881119903 (119905) = radic1 minus 1205772 minus 119890minus120577120596119903119905 (120577 sin (radic1 minus 1205772120596119903119905) + radic1 minus 1205772 cos (radic1 minus 1205772120596119903119905))

1205962119903radic1 minus 1205772 (35)

By solving the diagonal matrix diag(120596119903) constructedusing generalized eigenvalues and the characteristic matrix[B] the following equations are established Subsequently[B] is normalized by the main mass and the unit matrix[B]119879[119872]119887119894[B] is obtained

[119870]119887119894 [Φ] = [119872]119887119894 [Φ] diag (120596119903) (36)

The step response of the bimorph at different axial forcesis calculated from (34) The transient dynamic analysis withprecompression is performed using ANSYS software and theequivalent torque is added in the simulation

119872119890 = 11988711988931119864119901V (ℎ119904 + ℎ119901) (37)

The voltage is set to 1 V and the amplitude of the freeend can be obtained under different axial forces as shownin Figure 5 It can be seen that the ANSYS simulationresults are consistent with the numerical calculation resultsand the response time in the no-load state is of the orderof milliseconds The axial force can apparently enlarge thedisplacement output but has little effect on the response timeof the bimorph

4 Experiments of Piezoelectric Bimorph

41 Static Experiment of Piezoelectric Bimorph As shown inFigure 6 a clamping device is designed for the installation

of the piezoelectric bimorph The axial force was changedby increasing or decreasing the weights in the experimentThe deflection of the bimorph was tested using the laserdisplacement sensor

In the experiment the voltage of the upper and lower sur-faces was supplied by the DC power and the substrate layerwas grounded The buckling critical load of the cantileverbimorph was calculated to be 36 N by using formula (9) Themaximum static deflection under the axial forces of 0 N and16 N are shown in Figure 7

The analytical solution in Figure 7 is obtained from(8) It can be seen that when the axial force is constantthe deformation is linear with the voltage The axial forcecan evidently amplify the displacement output and themaximum deflection of the bimorph under an axial force of16 N has increased by approximately 30 The experimentalresults and the numerical results obtained using formula (8)are very close when no axial force is applied Even underlarge axial force the theoretical and experimental results arebasically the same

42 Modal Experiment of Piezoelectric Bimorph The modalexperiment of the piezoelectric bimorph mainly tests itsamplitude-frequency characteristics natural frequencies andnatural modes under different axial forces The amplitude ofthe piezoelectric material is in the micron range and theone-dimensional out-of-plane vibration is often measuredusing a laser Doppler vibrometerTherefore the PSV-300F-B


w

PBP

xy

DC Power

PC

Laser sensor

(a)

Piezoelectric bimorph

Bearing

Guiding device

Sliding table

Pedestal

(b)

DC PowerLaser sensor


(c)

Figure 6 Static experiment of precompressed cantilever piezoelectric bimorph (a) experiment system (b) clamping device (c) experimentaldevices

Table 1 First-order natural frequency of the cantilever bimorphswith different axial forces

Axial force (N) 0 2 10Numerical results (Hz) 144 138 125Experimental results (Hz) 150 129 96

laser scanning Doppler vibration measurement was used toconduct the modal experiments on the bimorph

In the experiment the excitation voltage amplitude was30 V and the axial forces were 0 N 2 N and 10 N Theexperimental and numerical results are shown in Table 1As the damping of the bimorph cannot be ignored theproportional dampingmatrix (calculated using (38)) is addedto the numerical calculation and the damping ratio is set to120577=002

[119862]119887119894 = ([Φ]119879119887119894)minus1 diag (2119872119903120596119903120577) [Φ]minus1119887119894 (38)

From the experimental results it can be observed that thenatural frequency of the bimorph gradually decreases withthe increase in the axial force but its bandwidth can stillreach 96Hz with an axial prepressure of 10 N which hasa significant advantage over the traditional electromagneticactuator

The resonant frequency of the bimorph calculated usingthe numerical method is very similar to the experimental

result when the axial force is not applied With the increaseof the axial force the experimental results of the first-ordernatural frequency are relatively smaller than the numericalcalculation results whichmay be related to additional factorsgenerated by the instruments in the experimentHowever thetrends in the results are the same

5 Conclusion

In this study the rigid-flexible coupling dynamic model ofthe piezoelectric bimorph is established by using the the-oretical tools of analytical mechanics structural dynamicsand finite element method The force-displacement charac-teristics under different axial forces are analyzed and thefrequency response function is given The steady-state andtransient responses of the bimorph are analyzed Simul-taneously according to the theoretical analysis above thecorresponding code is written using MATLAB softwareFinally the experimental platform is set up to study the staticand dynamic characteristics of the axially precompressedbimorph The results show that increase of axial force has asignificant amplification effect on the steady-state responseamplitude of the displacement and it reduces the resonancefrequency However the resonance frequency can still reach60Hz and the displacement output is amplified by nearly30 while 06 times the buckling critical load is applied Theresponse time is still in the millisecond range under a large


700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80

Voltage (V)0 N Analytical solution0 N Measured data16 N Analytical solution16 N Measured data

Figure 7 Deflection of bimorph under different voltages and axialforces

axial force which indicates that the bimorph has excellentdynamic characteristics as an actuatorThis article provides asolid theoretical basis for the design of the PBP servo system

Data Availability

All the data used to support the findings of this study areincluded within the article and are reflected in the form offigures or tables

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grant no 51575259) National BasicResearch Program (973 Program) (2015CB057500) theFunding of Jiangsu Innovation Program for Graduate Edu-cation (nos KYLX 0236 and KYLX15 0226) and the Fun-damental Research Funds for the Central Universities (noNS2015004)

References

[1] M Rushdi T Tennakoon K Perera A Wathsalya and S RMunasinghe ldquoDevelopment of a small-scale autonomous UAVfor research and developmentrdquo in Proceedings of the 2016 IEEEInternational Conference on Information and Automation forSustainability (ICIAfS) pp 1ndash6 Galle Sri Lanka December2016

[2] O Bilgen andM I Friswell ldquoPiezoceramic composite actuatorsfor a solid-state variable-camber wingrdquo Journal of IntelligentMaterial Systems and Structures vol 25 no 7 pp 806ndash817 2014

[3] C Zhao Ultrasonic Motors Science Press 2010[4] A Henke M A Kummel and J Wallaschek ldquoA piezoelectri-

cally driven wire feeding system for high performance wedge-wedge-bonding machinesrdquo Mechatronics vol 9 no 7 pp 757ndash767 1999

[5] J C Gomez and E Garcia ldquoMorphing unmanned aerialvehiclesrdquo Smart Materials and Structures vol 20 no 10 ArticleID 103001 2011

[6] D Karagiannis D Stamatelos V Kappatos and T Spathopou-los ldquoAn investigation of shape memory alloys as actuatingelements in aerospace morphing applicationsrdquo Mechanics ofAdvanced Materials and Structures vol 24 no 8 pp 647ndash6572017

[7] R Barrett ldquoAdaptive fight control actuators andmechanisms formissiles munitions and uninhabited aerial vehicles (UAVs)rdquo inAdvances in Flight Control Systems InTech 2011

[8] R Wlezien G Horner A McGowan et al The AircraftMorphing Program 98-1927 1998

[9] P C Chen E Sulaeman D D Liu and L M Auman ldquoBody-flexure control with smart actuation for hypervelocity missilesrdquoin Proceedings of the 44th AIAAASMEASCEAHSASC Struc-tures Structural Dynamics and Materials Conference pp 5157ndash5167 USA April 2003

[10] X Zhang W Wang and G H Zong ldquoThe application ofpiezoelectric actuator for small aircraftrdquo Advanced MaterialsResearch vol 998-999 pp 674ndash677 2014

[11] RM Barrett andP Tiso ldquoActuatorrdquo 1Mar 2011 USPat 7898153[12] E F Crawley ldquoIntelligent structures for aerospace a technology

overview and assessmentrdquo AIAA Journal vol 32 no 8 pp1689ndash1699 1994

[13] F S Huang Z H Feng Y T Ma et al ldquoHigh-frequency per-formance for a spiral-shaped piezoelectric bimorphrdquo ModernPhysics Letters B Condensed Matter Physics Statistical PhysicsAtomic Molecular and Optical Physics vol 32 no 10 Article ID1850111 2018

[14] A Erturk and D J Inman ldquoAn experimentally validatedbimorph cantilever model for piezoelectric energy harvestingfrom base excitationsrdquo Smart Materials and Structures vol 18no 2 Article ID 025009 2009

[15] S R Hall and E F Prechtl ldquoDevelopment of a piezoelectricservoflap for helicopter rotor controlrdquo Smart Materials andStructures vol 5 no 1 pp 26ndash34 1996

[16] R J Wood S Avadhanula E Steltz et al ldquoDesign fabricationand initial results of a 2g autonomous gliderrdquo in Proceedingsof the IECON 2005 31st Annual Conference of IEEE IndustrialElectronics Society pp 1870ndash1877 November 2005

[17] G A Lesieutre and C L Davis ldquoCan a coupling coefficient of apiezoelectric device be higher than those of its active materialrdquoJournal of Intelligent Materials Systems and Structures vol 8 pp859ndash867 1997

[18] G A Lesieutre and C L Davis ldquoTransfer having a couplingcoefficient higher than its active materialrdquo 22 May 2011 US Pat6236143

[19] R Vos and R Barrett ldquoPost-buckled precompressed (PBP)piezoelectric actuators for UAV flight controlrdquo in Proceedingsof SPIE - e International Society for Optical Engineering vol6173 pp 1ndash12 2006


[20] R Barrett R McMurtry R Vos P Tiso and R De BreukerldquoPost-buckled precompressed piezoelectric flight control actu-ator design development and demonstrationrdquo Smart Materialsand Structures vol 15 no 5 pp 1323ndash1331 2006

[21] R De Breuker P Tisot R Vos and R Barrett ldquoNonlinear semi-analytical modeling of Post-Buckled Precompressed (PBP)piezoelectric actuators for UAV flight controlrdquo in Proceedingsof the 47th AIAAASMEASCEAHSASC Structures StructuralDynamics and Materials Conference vol 1795 pp 2461ndash2473USA May 2006

[22] G Giannopoulos J Monreal and J Vantomme ldquoSnap-throughbuckling behavior of piezoelectric bimorph beams I analyticaland numerical modelingrdquo Smart Materials and Structures vol16 no 4 pp 1148ndash1157 2007

[23] G Giannopoulos J Monreal and J Vantomme ldquoSnap-throughbuckling behavior of piezoelectric bimorph beams II experi-mental verificationrdquo Smart Materials and Structures vol 16 no4 pp 1158ndash1163 2007

[24] D V Hutton Fundamentals of Finite Element Analysis NewYork NY USA 2004

[25] H Chen J Wang C Chen S Liu and H Chen ldquoDynamicmodeling and characteristic analysis of piezoelectric rudderactuatorrdquo Review of Scientific Instruments vol 90 no 1 ArticleID 016102 2019

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actuators to optimize the design of the piezoelectric flightcontrol surface [21] In order to analyze the static anddynamic characteristics of a piezoelectric bimorph underaxial precompression Giannopoulos et al of the Civil andMaterials Engineering Department at the Royal MilitaryAcademy used nonlinear mechanics to establish a theoreticalmodel for bimorphs and verified the validity of the analyticalmodel through experiments and simulations [22 23]

For actuators driven by axially precompressed piezoelec-tric bimorphs the characteristics of the bimorphs determinethe performance of the actuators Therefore it is necessaryto definitively describe the static and dynamic characteristicsof axially precompressed bimorphs However the dynamicexpression of the existing precompressed bimorph is not per-fect and the established kinetic model has been significantlysimplified Most of the researchers took the effect of thepiezoelectricity as the bending moment when establishingthe statics model [19 20] For the establishment of thekinetic model they simplified the bimorph into a single-degree-of-freedom system to obtain the first-order naturalfrequency [21] but this method could not get the outputcharacteristics of the system at other excitation frequenciesIn this paper the rigid-flexible coupling dynamics modelof the bimorph is established based on the finite elementmethod which can be used to analyze various dynamiccharacteristics such as amplitude-frequency characteristicsof the system The validity of the model is verified throughsimulations and experiments thus providing a theoreticalbasis for the application of the piezoelectric bimorph

2 Establishment of Static and Dynamic Model

21 Driving Principle of Piezoelectric Bimorphs as ActuatorsPiezoelectric bimorph consists of two piezoelectric layers andone substrate layer As shown in Figure 1 the polarizationdirection of the two piezoelectric materials is the same Thesame voltage signal is connected to the upper and lowersurfaces and the substrate is groundedThe two piezoelectriclayers extend and contract respectively so that the bimorphcan deform in two directions If a mechanical device is addedat the end of the structure the bimorph will become anactuator of the servo system

22 StaticModel of Axially Precompressed Cantilever BimorphThe bimorph is simplified to a three-layer Euler beam modelwithout considering the kinetic energy of infinitesimal rota-tion [24] The kinetic energy potential energy and the workdone by the axial force and the electric field are calculatedThe following equations can be obtained by consideringvariation of the energy functionals

120575119879119887119894 = (120588A) int1198710120575119889119909

120588A = 2120588119901119860119901 + 120588s119860 119904(1)

where 120575119879119887119894 is the variation of the total kinetic energy and 120588119860represents the linear density of the bimorph The subscript 119904and 119901 indicate the substrate and piezoelectricity respectively

120588119904 and 120588119901 are the corresponding density 119860 119904 and 119860119901 are thecorresponding cross-sectional area L represents the lengthof the bimorph is the first derivative of the neutral layerdeflection versus time

120575119880119887119894 = 119864119868int11987101199081015840101584012057511990810158401015840119889119909 + 1198721198902 int119871

012057511990810158401015840119889119909

119872119890 = 119887119864311988931119864119901ℎ119901 (ℎ119904 + ℎ119901)119864119868 = 119864s119868119904 + 2119864119901119868119901

(2)

120575119880119887119894 is the variation of the total potential energy andEI represents the bending stiffness of the bimorph 119864119904 and119864119901 represent the elastic moduli of the substrate and thepiezoelectric layer respectively 119868119904 and 119868119901 are the correspond-ing inertias ℎ119904 and ℎ119901 are the corresponding thicknessesb represents the width of the bimorph 11990810158401015840 is the secondderivative of the neutral layer deflection versus the x-axis 1198643represents the electric field strength along the z-axis 11988931 isthe piezoelectric constant in the corresponding directionThefollowing conclusion can be obtained from (2)

120575119880119887119894 = 119864119868(11990810158401015840 1205751199081015840100381610038161003816100381610038161198710 minus 119908101584010158401015840120575119908100381610038161003816100381610038161198710 + int119871

0119908(4)120575119908119889119909)

+ 12119872119890 120575119908101584010038161003816100381610038161003816119871

0

(3)

Assuming that the beam is not extensible and the axialpressure 119865 is applied to the beam the total work of the systemcan be obtained The following formula can be achieved byconsidering the variation of the total work

120575119882 = minus12119872119890 120575119908101584010038161003816100381610038161003816119871

0+ 1198651199081015840 120575119908|1198710 minus 119865int119871

011990810158401015840120575119908119889119909 (4)

From the Hamiltonrsquos principle the stationary point of thefunctional is obtained by

120575int11990521199051

(119879119887119894 minus 119880119887119894 +119882)119889119905 = 0 (5)

According to the partial differential equations obtainedabove it can be observed that the kinetic equation of thebimorph is consistent with that of the ordinary homogeneousbeamThe converse piezoelectric effect is not reflected in theequation but in the boundary condition

120588119860 + 119864119868119908(4) + 11986511990810158401015840 = 0 (6)

Assuming that the axial force is always horizontal theboundary conditions of the cantilever bimorph can beobtained as follows

11990810158401015840 = minus119872119890119864119868 119909 = 119871119864119868119908101584010158401015840 + 1198651199081015840 = 0 119909 = 119871

1199081015840 = 0 119909 = 0119908 = 0 x = 0

(7)


(a)

PZT

PZTSubstrate

(b)

Substrate PZT

(c)




(8)


119865119888119903 = 119864119868 120587241198712 (9)



119908 = [119891]119879 [119886] = [119873] [120575] (10)

Here

[119891] = [1 119909 1199092 1199093]119879[119886] = [1198861 1198862 1198863 1198864]119879

(11)

[120575] =[[[[[[

119908111990810158401119908211990810158402

]]]]]]=[[[[[[

1 0 0 00 1 0 01 Δ119909 Δ1199092 Δ11990930 1 2Δ119909 3Δ1199092

]]]]]]

[[[[[[

1198861119886211988631198864

]]]]]]

= [Λ] [119886]

(12)


[119873] = [119891]119879 [Λ]minus1 (13)


119879 = 119879119904 + 2119879119901 = 12120588119860intΔ119909

0

120575119879119873119879119873 120575119889119909 (14)




119880 = 119880119904 + 2119880119901= 12119864119868int

Δ119909

01205751198791198731015840101584011987911987310158401015840120575119889119909 + 1198721198902 [0 minus1 0 1] 120575 (15)



(16)


119882119865 = 119865intΔ1199090

119908101584022 119889119909 = 1198652 intΔ119909

012057511987911987310158401198791198731015840120575119889119909 (17)

The total work is

119882 = 119882119865 + 2119882119864 (18)


119889119889119905 120597119879120597 120575 +120597119880120597120575 = 120597119882120597120575 (19)


[119872] 120575 + [119870] 120575 = 119872119890 [0 1 0 minus1]119879 (20)


[119872] = 120588119860intΔ1199090119873119879119873119889119909

= 120588119860[[[[[[[[[[[[


]]]]]]]]]]]]

(21)



[119870] = 119864119868

[[[[[[[[[[[[[


]]]]]]]]]]]]]

minus 119865

[[[[[[[[[[[[[


]]]]]]]]]]]]]

(22)




(23)








1 2 3 4 5 6 7 8 90 10


1075

1076

1077

1078

1079

1080

1081

1082

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)



40

50

60

70

80

90

100

110

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)

02 04 06 08 1000







(26)



30 60 90 120 150

Frequency (Hz)


00

01

02

03

04

05

06

Am

plitu

de (m

m)



[119867] = (minus1205962 [119872]119887119894 + [119870]119887119894)minus1 (28)





119881 = 0 119905 lt 0V 119905 gt 0 (30)


[ℎ (119905)] = [Φ] diag( sin120596119903119905120596119903 ) [Φ]119879 1205750 (31)



30

25

20

15

10

5

0

Am

plitu

de (u

m)

0 20 40 60 80 100

Time (ms)





120575 (119905) = int1199050[ℎ (119905)] 119872119890119887119894 119889119905


(33)



where


1205962119903radic1 minus 1205772 (35)


[119870]119887119894 [Φ] = [119872]119887119894 [Φ] diag (120596119903) (36)


119872119890 = 11988711988931119864119901V (ℎ119904 + ℎ119901) (37)









w

PBP

xy

DC Power

PC

Laser sensor

(a)


Bearing

Guiding device

Sliding table

Pedestal

(b)



(c)










5 Conclusion



700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80




Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









(a)

PZT

PZTSubstrate

(b)

Substrate PZT

(c)




(8)


119865119888119903 = 119864119868 120587241198712 (9)



119908 = [119891]119879 [119886] = [119873] [120575] (10)

Here

[119891] = [1 119909 1199092 1199093]119879[119886] = [1198861 1198862 1198863 1198864]119879

(11)

[120575] =[[[[[[

119908111990810158401119908211990810158402

]]]]]]=[[[[[[

1 0 0 00 1 0 01 Δ119909 Δ1199092 Δ11990930 1 2Δ119909 3Δ1199092

]]]]]]

[[[[[[

1198861119886211988631198864

]]]]]]

= [Λ] [119886]

(12)


[119873] = [119891]119879 [Λ]minus1 (13)


119879 = 119879119904 + 2119879119901 = 12120588119860intΔ119909

0

120575119879119873119879119873 120575119889119909 (14)




119880 = 119880119904 + 2119880119901= 12119864119868int

Δ119909

01205751198791198731015840101584011987911987310158401015840120575119889119909 + 1198721198902 [0 minus1 0 1] 120575 (15)



(16)


119882119865 = 119865intΔ1199090

119908101584022 119889119909 = 1198652 intΔ119909

012057511987911987310158401198791198731015840120575119889119909 (17)

The total work is

119882 = 119882119865 + 2119882119864 (18)


119889119889119905 120597119879120597 120575 +120597119880120597120575 = 120597119882120597120575 (19)


[119872] 120575 + [119870] 120575 = 119872119890 [0 1 0 minus1]119879 (20)


[119872] = 120588119860intΔ1199090119873119879119873119889119909

= 120588119860[[[[[[[[[[[[


]]]]]]]]]]]]

(21)



[119870] = 119864119868

[[[[[[[[[[[[[


]]]]]]]]]]]]]

minus 119865

[[[[[[[[[[[[[


]]]]]]]]]]]]]

(22)




(23)








1 2 3 4 5 6 7 8 90 10


1075

1076

1077

1078

1079

1080

1081

1082

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)



40

50

60

70

80

90

100

110

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)

02 04 06 08 1000







(26)



30 60 90 120 150

Frequency (Hz)


00

01

02

03

04

05

06

Am

plitu

de (m

m)



[119867] = (minus1205962 [119872]119887119894 + [119870]119887119894)minus1 (28)





119881 = 0 119905 lt 0V 119905 gt 0 (30)


[ℎ (119905)] = [Φ] diag( sin120596119903119905120596119903 ) [Φ]119879 1205750 (31)



30

25

20

15

10

5

0

Am

plitu

de (u

m)

0 20 40 60 80 100

Time (ms)





120575 (119905) = int1199050[ℎ (119905)] 119872119890119887119894 119889119905


(33)



where


1205962119903radic1 minus 1205772 (35)


[119870]119887119894 [Φ] = [119872]119887119894 [Φ] diag (120596119903) (36)


119872119890 = 11988711988931119864119901V (ℎ119904 + ℎ119901) (37)









w

PBP

xy

DC Power

PC

Laser sensor

(a)


Bearing

Guiding device

Sliding table

Pedestal

(b)



(c)










5 Conclusion



700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80




Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018










119880 = 119880119904 + 2119880119901= 12119864119868int

Δ119909

01205751198791198731015840101584011987911987310158401015840120575119889119909 + 1198721198902 [0 minus1 0 1] 120575 (15)



(16)


119882119865 = 119865intΔ1199090

119908101584022 119889119909 = 1198652 intΔ119909

012057511987911987310158401198791198731015840120575119889119909 (17)

The total work is

119882 = 119882119865 + 2119882119864 (18)


119889119889119905 120597119879120597 120575 +120597119880120597120575 = 120597119882120597120575 (19)


[119872] 120575 + [119870] 120575 = 119872119890 [0 1 0 minus1]119879 (20)


[119872] = 120588119860intΔ1199090119873119879119873119889119909

= 120588119860[[[[[[[[[[[[


]]]]]]]]]]]]

(21)



[119870] = 119864119868

[[[[[[[[[[[[[


]]]]]]]]]]]]]

minus 119865

[[[[[[[[[[[[[


]]]]]]]]]]]]]

(22)




(23)








1 2 3 4 5 6 7 8 90 10


1075

1076

1077

1078

1079

1080

1081

1082

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)



40

50

60

70

80

90

100

110

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)

02 04 06 08 1000







(26)



30 60 90 120 150

Frequency (Hz)


00

01

02

03

04

05

06

Am

plitu

de (m

m)



[119867] = (minus1205962 [119872]119887119894 + [119870]119887119894)minus1 (28)





119881 = 0 119905 lt 0V 119905 gt 0 (30)


[ℎ (119905)] = [Φ] diag( sin120596119903119905120596119903 ) [Φ]119879 1205750 (31)



30

25

20

15

10

5

0

Am

plitu

de (u

m)

0 20 40 60 80 100

Time (ms)





120575 (119905) = int1199050[ℎ (119905)] 119872119890119887119894 119889119905


(33)



where


1205962119903radic1 minus 1205772 (35)


[119870]119887119894 [Φ] = [119872]119887119894 [Φ] diag (120596119903) (36)


119872119890 = 11988711988931119864119901V (ℎ119904 + ℎ119901) (37)









w

PBP

xy

DC Power

PC

Laser sensor

(a)


Bearing

Guiding device

Sliding table

Pedestal

(b)



(c)










5 Conclusion



700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80




Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









1 2 3 4 5 6 7 8 90 10


1075

1076

1077

1078

1079

1080

1081

1082

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)



40

50

60

70

80

90

100

110

Firs

t-ord

er b

endi

ng fr

eque

ncy

(Hz)

02 04 06 08 1000







(26)



30 60 90 120 150

Frequency (Hz)


00

01

02

03

04

05

06

Am

plitu

de (m

m)



[119867] = (minus1205962 [119872]119887119894 + [119870]119887119894)minus1 (28)





119881 = 0 119905 lt 0V 119905 gt 0 (30)


[ℎ (119905)] = [Φ] diag( sin120596119903119905120596119903 ) [Φ]119879 1205750 (31)



30

25

20

15

10

5

0

Am

plitu

de (u

m)

0 20 40 60 80 100

Time (ms)





120575 (119905) = int1199050[ℎ (119905)] 119872119890119887119894 119889119905


(33)



where


1205962119903radic1 minus 1205772 (35)


[119870]119887119894 [Φ] = [119872]119887119894 [Φ] diag (120596119903) (36)


119872119890 = 11988711988931119864119901V (ℎ119904 + ℎ119901) (37)









w

PBP

xy

DC Power

PC

Laser sensor

(a)


Bearing

Guiding device

Sliding table

Pedestal

(b)



(c)










5 Conclusion



700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80




Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









30

25

20

15

10

5

0

Am

plitu

de (u

m)

0 20 40 60 80 100

Time (ms)





120575 (119905) = int1199050[ℎ (119905)] 119872119890119887119894 119889119905


(33)



where


1205962119903radic1 minus 1205772 (35)


[119870]119887119894 [Φ] = [119872]119887119894 [Φ] diag (120596119903) (36)


119872119890 = 11988711988931119864119901V (ℎ119904 + ℎ119901) (37)









w

PBP

xy

DC Power

PC

Laser sensor

(a)


Bearing

Guiding device

Sliding table

Pedestal

(b)



(c)










5 Conclusion



700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80




Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









w

PBP

xy

DC Power

PC

Laser sensor

(a)


Bearing

Guiding device

Sliding table

Pedestal

(b)



(c)










5 Conclusion



700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80




Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









700

600

500

400

300

200

100

0

Defl

ectio

n (u

m)

0 10 20 30 40 50 60 70 80




Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018






















Journal of












Journal of







Volume 2018






Volume 2018








dynamic modeling and characteristic analysis of cantilever...

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