research article modal analysis of 27mm piezo electric...

Hindawi Publishing CorporationJournal of EngineeringVolume 2013 Article ID 549865 10 pageshttpdxdoiorg1011552013549865

Research ArticleModal Analysis of 27 mm Piezo Electric Plate for Small-ScaleUnderwater Sonar-Based Navigation

M O Afolayan D S Yawas C O Folayan and S Y Aku

Mechanical Engineering Department Ahmadu Bello University Samaru Zaria 810006 Nigeria

Correspondence should be addressed to M O Afolayan tunde afolayanyahoocom

Received 14 September 2012 Revised 15 January 2013 Accepted 12 February 2013

Academic Editor Jie Zhou

Copyright copy 2013 M O Afolayan et alThis 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

This work presents progress towards the development of a small-scale purely sonar-based navigation device for a robotic fish(sim394mm long) Aperture overloading of small (5mm diameter) ultrasonic transmitters does not allow them to be used effectivelyinside water A test on a 27mm diameter buzzer piezo plate shows promising performance under water at frequencies from45 kHz to 80 kHz ANSYS-based simulation was therefore used to find modal frequencies at higher frequencies so as to optimizethis encouraging result The simulation process also discovered several antiresonant frequencies such as 385 kHz 54 kHz and575 kHz All frequencies above the 8th harmonic (1058902Hz) are out of phase with the input load except a resonance frequencyof 425 kHz and an antiresonance frequency of 565 kHz Also the first harmonic (164873Hz) is the only frequency that gave anodal deformation

1 Introduction

Many marine organisms such as dolphin navigate usingultrasonic means Artificial systems such as autonomousunderwater vehicles (AUV) and remotely operated system(ROV) commonly use acoustic means in their navigationobject detections and avoidance control and communica-tion Underwater modems such as those used by Eustice et al[1] are often based on acoustic links that transmit at a very lowfrequency to extremely high frequencies while bearing datain the kilobit (kb) range as exemplified by Kumagai et al [2]work According to Eustice et al [1] ldquofew techniques exist forreliable three-dimensional position sensing for underwatervehicles Depth altitude heading and rollpitch attitudecan all be instrumented with high-bandwidth internal sen-sorsrdquo Many AUV are equipped with multiple numbers ofdead-reckoning sensors such as Doppler velocity logs andinertial measurement systems or magnetic compasses toestimate vehicle position In contrast the exact coordinateof the robots remains difficult to instrument and is normallymeasured acoustically in oceanographic and commercialapplications and even in submarines [1 3 4] Other methodsused on the surface such as inertia navigation and accumulate

error that is further aggravated by water waves and currents[5] Sonar systems have been very attractive for underwaterimagerymdashbeing capable of longer range and in a variety ofwater conditions such as poor visibility and lighting [6ndash8]The lower frequencies are even more effective for longerranges as depicted in Table 1 they decay extremely slowlyin water [5] In robotics sonar systems are often used forranging due to their low cost and small size The signal issent out continuously or pulsed The pulsed mode is usedfor eliminating frequent misreading caused by crosstalk orexternal sources operating nearby [9]

Safety and reliability are factors universally desired inany system design Navigation based on acoustic has its owninherent problems also High-power ultrasonic systems havebeen known to negatively affect underwater ecosystem [14]Also very powerful low-frequency and activated sonars (andmidfrequency sonar) have been claimed to also affect marinelife [15]

Another problem is that acoustic degrades in clutteredenvironments such as reefs or close to the seafloor andsubsea structures [8] In the presence of air bubbles thesignature of object to be detected is adversely affected [16]although algorithms exist for overcoming such limitation

2 Journal of Engineering

Table 1 Sonar frequency and its range

Frequency (kHz) Range (m)Low frequency (LF) 8ndash16 gt10000Medium frequency (MF) 18ndash36 2000ndash3000High frequency (HF) 30ndash60 1500Extra high frequency (EHF) 50ndash110 lt1000Very high frequency (VHF) 200ndash300 lt100Sources [10ndash13]

Ultrasonic sensors

39401

1956 19841

498

2

Figure 1 The biomimetic robotic fish that was being investigatedThe receiver is at the tip and the transmitter is at the bottom

Also without side lobe transmission attenuation there willbe multiple-object detection that is not directly ahead of thesonar transmitter

In some situations the problems associated with sonarnavigation or object detection are augmented for by othersensing methods for example Asakawa et al [17] AUV hadthe underwater cable to be located and carried a signaturein the form of 16ndash25Hz electric current which enabled it toautomatically locate it Another augmentation is in the formof bump switch to assist in navigating very tight environmentwhere sonar ranging will be too close or ineffective [18]Also Lygouras et al [19] used terrain structure to augmentthe sonar navigation processes which fall into the categoriesreferred to as baseline navigation systems [20 21]

2 The Motivation

Our team built a biomimetic robotic fish named BlueMac(Figure 1)mdasha 1 1 scale model of Mackerel fish to investigatethe use of rubber for building planar hyperredundant robotjoint [18] The robot is meant to swim in a low pressure headenvironment such as shallow lake pond or stream wherewater flow is not very strong or turbulent It was meantto use commercially available 40 kHz ultrasonic transmit-terreceiver combination for its navigation among obstaclefields but it failed to work in all the possible configurationswe tried except out of water The sensors were sealed againstwater with materials of various thicknesses such as polytheneand oil-impregnated paper In one of the experiments low-densitymineral oil (used for sewingmachine lubrication)wasused to fill the void in front of the sensors to improve contactwith the water boundary As a last attempt the sensors

(transmitter and receiver) were immersed in water withoutany insulationmdashbased on the premises that water is a poorconductor compared to the metallic conductor the sensorusesmdashthis still yielded no result Our conclusion was thatthe transmitter must have been overloaded for the followingreasons

(1) The piezo crystal plates drive the medium they are inforward and backward at the applied frequency

(2) This action is not a problem when the fluid it is inter-acting with is air with average density of 1184 kgm3(259∘C mean temperature)

(3) If the fluid is water the density becomes approxi-mately 1000 kgm3 which is 847-fold increment ofmass of fluid to move when compared to air Thisseems to be an overload for the small-aperture sonarsensor

Exhaustive literature search gave no clue on how tosolve this problem except that Jindong and Huosheng [22ndash24] seemed to have encountered such problem also Theycreated an ultrasonic-based model for their robotic fish butimplemented it with infrared light sensor There are attemptsby some researchers to imitate fish lateral lines [25ndash28] butsmall-scale purely sonar-based underwater navigation byrobots that are less than 05m in length is unknown to theauthors as of this writing and this has prompted us to do athorough investigation about it at this scale

3 Objectives of This Study

The goal of this study is to setup a simulation environmentin ANSYSMultiphysics software to find out the mode shapesof audible range piezo crystal (PZT-5H) material (commonlyused in 27mm diameter commercial buzzer alarms) at itsharmonics We also aim at finding possible nodal shapewithin the audible range (0ndash22 kHz)which implies absence ofcompressive and tensile stress simultaneously on one surfaceof the piezoelectric biomorph plate Furthermore we aim totest the piezo plates at higher frequencies so as to find outtheir real responses

4 Theoretical Background

A piezoelectric material will generate voltages when sub-jected to tension or compression The voltage generated isproportional to the compressive or tension load which ismathematically represented as [29 30]

119878 = [119904

119864] 119879 + [119889

119905] 119864

119863 = [119889] 119879 + [120598

119879] 119864

(1)

where 119878 is strain vector 119879 is the vector of stresses 119863 is thedielectric displacement vector 119864 is the electric field vector119878

119864 is the compliance matrix evaluated at constant electricfield 119889 is the piezoelectric strain coefficients 119889

119905is the mode

of operation (compressive or transverse mode) and 120598119879 is

Journal of Engineering 3

Log

of im

peda

nce

Frequency

119891119898

119891119899

Figure 2 The signal response of matched piezo crystal transmitter and receiver

Figure 3 Watch buzzer crystal plates

AB

C

D

Receivingcomputer

Sendingcomputer

Figure 4 The preliminary experimental setup (A) is the woodenbox tank (without water) (B) is the obstacle place (C) is the enlargeview of the piezo crystal plates and (D) with sinker

the dielectric constant matrix evaluated at constant stressThese equations (called ldquocoupledrdquo equations) reduce to thewell-known stress-strain relationship at zero electric field andthe electric field and charge displacement relationship at zerostressThe poling direction is a plane perpendicular to the flatside of the piezoelectric plate

Furthermore when piezoelectric plates are subjected tocyclic loading electrically the usual response is shown inFigure 2 As the cyclic input frequency increases there existvalues at which the piezo element exhibits low impedance(119891119898) and resonate As the cyclic load frequency increases

there exists also another frequency at which it has the highestimpedance referred to as antiresonance frequency (119891

119899) Piezo

plates working in pair (one acting as a transmitter and theother as a receiver) are selected in such a manner that the

transmitter resonant frequency coincides with the receiverantiresonance frequency

5 Methodology

51 Preliminary Experimentation A preliminary experimentwas performed to find out the potential of using ordinarybuzzer plate (Figure 3) for underwater navigation using theidea from [31]The following list of material was used for thispreliminary experiment

(1) two Intel-based computer systems (Dell Latitude cp120591with Celeron 500MHz and and IBM clone with IntelPentium 4)

(2) spectraPlus 5 software (shareware) installed on thetwo computers

(3) two 2 cm back plate old wrist watch buzzer(4) water tank 50 cm times 50 cm times 121 cm (width height

length)(5) plain cable (wire)(6) obstaclesmdashplywood (635mm) flat PVC (sim15mm)

iron sheet (sim1mm) and asbestos (sim2mm) thick-nesses

The SpectraPlus 5 software in the Dell Latitude laptop(Figure 4) acts as the precision transmitter while the one inthe other computer acts as a receiver and spectrum analyzerThe line-out and line-in ports of the computers were used todrive the piezo plates in the sending and receiving computersrespectively The experiment was started by first finding outthe frequency between 0Hz to 22 kHz that will give peakresponse (at the receiver end) The frequency that gave peakoutput was found to be 45 kHzThis value was therefore usedfor the underwater transmission preliminary tests The testsinvolve spacing the plates at distances of 30 cm and 60 cm inthe air andwaterwhile obstacles were placed in between themat equal distances from both the receiver and transmitterplates

52 Mode Shape Simulation Processes and the HarmonicAnalysis The mode shape refers to the bending or contourpattern of the piezo crystal plates at its harmonics ANSYS


0145mm brass plate0016mm PZT5H

120601188mm

12060127mm

Figure 5 Dimension of plate

Figure 6 The simulation environment constraints and the meshpattern as used for the simulation

10 Multiphysics software was used for all the simulationswhile Autodesk Inventor software was used for all thegeometrical drawingsThe harmonic analysis was performedfrom 0Hz to 22 kHz and then from 20 kHz to 100 kHzusing harmonic tools built into ANSYS 10 MultiphysicsThe analyses performed are (a) directional deformation (b)frequency response and (c) phase response all were at a planeperpendicular to the piezo crystal plate surface

53The Simulation Environment Frequency finder tool builtinto ANSYS Multiphysics was used with shape-optimizedmeshing The simulation was performed for the 0Hz to22 kHz range A dummy input load of 01 N (normal to theplate assembly) was used For the higher frequencies 4060 80 and 100 kHz the surface deformation was estimatedat these frequencies using harmonic tool directional defor-mation command The dimension of the piezo crystal plateassembly used is shown in Figure 5 The constraints appliedare shown in Figure 6 with the optimized mesh pattern InFigure 6 a 01 Ndummy load is shown acting perpendicularlyto the piezo plate surface this is to ensure a degree ofdeformation Choosing a higher value will result in greaterdeformation which is equivalent to increasing the voltage atthe piezo plate terminals Higher or lower dummy load hasno effect on the plate frequency response

54 Experimental Verification of Transmission at 45 10 2040 60 80 and 100 kHz Frequencies An experiment wassetup to find out the wave pattern as the frequency increasesusing setup similar to Figure 4 The idea here is to find outif the piezo plate arrangement can follow the input signaland what the distortion in the output signal will be as thefrequency increases

This experiment uses Microchip PIC18F4520 microcon-troller to generate a precise 5V p-p square wave signal atselected frequencies of 45 10 20 40 60 80 and 100 kHzThe output of themicrocontroller was used to drive one of the

piezo crystal plates while the other plate acts as the receiverBoth plateswere connected to an oscilloscope formonitoringThe plates are 50mm apart and immersed inside water atroom temperature of 28∘C

Square wave signal was used as the input signal The riseand fall (of the signal edge) approximate an impulsive load-ing If the piezo plate assembly is approximated by a linearlydamped mass-spring body (voigt body) the impulsive inputimplies that its response will give its dynamic characteristic[32 33] For linear time-invariant system (for which is anapproximation in this scenario) the impulsive response isgiven by

119862 (119904) = 119866 (119904) 119877 (119904) = 119866 (119904) (2)

where 119871120575(119905) = 1 = 119877(119904) and transfer function = 119862(119904)119877(119904) =119866(119904) therefore 119888(119905) = 119871minus1119866(119904) = 119892(119905) and using convolutionintegral

119888 (119905) = int

119905

0

119892 (119905 minus 120591) 119903 (120591) 119889120591 (3)

the systemrsquos response can be found to any input [32 33]Summarily driving the sample with a square wave signalallows us to have an idea about its realistic response whichwill be a function of its nature

6 Results Obtained

61 The Result of the Preliminary Experiment The resultsobtained in the preliminary experiment are shown in Figure 7and Table 2 Table 2 also shows the relative absorption valuewhich we hereby define as the difference in magnitudes inpeak response in air and water with the same piezo platespacing specifically as illustrated in (4)

1003816

1003816

1003816

1003816

119877water1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119877air1003816

1003816

1003816

1003816

at equal spacing (4)

62 The Result of Mode Shape Simulation Processes and theHarmonic Analysis The 15 harmonics existing between 0Hzand 22 kHz as simulated byANSYS 10Multiphysics are shownin Figure 8 Each pair of adjacent pictures shows plan viewand side or isometric view of themode shape at the harmonicfrequencies indicated below them When the simulationinput frequencies were set at 40 60 80 and 100 kHz that isabove the audible range the mode shapes were as shown inFigure 9 for each input frequency

Furthermore the frequency response curves for 0 to22 kHz and 20 kHz to 100 kHz input ranges are plotted in


The observed frequency The observed frequency

30 cm apart 60 cm apart

Figure 7 Spectrums obtained when transmitter and receiver were 30 and 60 cm apart in water and no object in between them

Table 2 Summary of the peak response with and without obstacles in the air and water

Material used Relative amplitude (dB) in air at Relative amplitude (dB) in water at Relative absorption at Relative absorption at30 cm 60 cm 30 cm 60 cm 30 cm 60 cm

No object minus7129 minus8147 minus1113 minus2356 minus6016 minus5791Plywood minus8895 minus9068 minus3411 minus3263 minus5484 minus5805Asbestos minus8312 minus8822 minus2740 minus3094 minus5572 minus5728PVC plastics minus7895 minus8136 minus2476 minus3021 minus5419 minus5115Iron sheet minus8617 minus8912 minus2063 minus2333 minus6554 minus6579

164873Hz 381296Hz 430708Hz

696674Hz 739168Hz 938413Hz

1043579Hz 1058902Hz 1332319Hz

1394286Hz 1439747Hz 1560828Hz

1841116Hz 1872452Hz 2008707Hz

Figure 8 Mode shapes at various harmonics as indicated at thebottom of each pair of view

40kHz 60kHz 80kHz 100 kHz

Figure 9 Mode shapes at fixed frequencies indicated

Figures 10 and 11 respectively The maximum and minimumdeformation at axis perpendicular to the plate face is shown

in Figure 12(a) along with the total displacement at each har-monic (Figure 12(b)) Table 3 is the maximum deformationfor the indicated frequencies The total displacement plotwithout the first natural frequency of 164873Hz is shown inFigure 13 Phase response analysis for the frequency range of100 to 20 kHz and 20 kHz to 100 kHz gave the plots of Figures14 and 15 respectively Figure 16 is the frequency responseplot for 20 kHz to 100 kHz input frequency range The insetin Figure 16 has frequency range of 52 kHz to 67 kHz

63 Results of the Experimental Verification of Transmissionat 45 10 20 40 60 80 and 100 kHz Frequencies Theexperimental verification at 45 10 20 40 60 80 and 100 kHzfrequencies is shown in Figure 17The upper trace is the inputsignal and the lower trace is the output signal on a dual-traceoscilloscope

7 Discussion of the Results

71 The Preliminary Experiment The frequency of interestis 45 kHz (Figure 7) the other frequencies indicated areartifacts due to the use of unshielded cable (since this is apreliminary investigation) Also from Table 2 the followingscan be observed

(1) The signal strength drops with distance between thetransmitter plate and receiver plate (while in the airand water medium)

(2) With obstacles in between the plates the plywoodinfluence on the signal strength is opposite to that of


216119890 minus 5

16119890 minus 5

12119890 minus 5

8119890 minus 6

4119890 minus 6

733119890 minus 9

100 25119890 + 3 5119890 + 3 75119890 + 3 1119890 + 4 2119890 + 4

Am

plitu

de (m

)

Frequency (Hz)

Frequency (Hz)

12119890 minus 6

8119890 minus 7

4119890 minus 7

207119890 minus 9

5119890 + 3 6119890 + 3 8119890 + 37119890 + 3 9119890 + 3 1119890 + 4 11119890 + 4 12119890 + 4408119890 + 3

125119890 + 4 15119890 + 4 175119890 + 4

Figure 10 Frequency response plot for 100Hzndash20 kHz range The inset is from 408 kHz to 12 kHz

385 kHz54 kHz

575 kHz

59 kHz 66 kHz

425 kHz343119890 minus 8174119890 minus 8885119890 minus 945119890 minus 9228119890 minus 9116119890 minus 9589119890 minus 10299119890 minus 10152119890 minus 10

2119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Am

plitu

de (m

)

Frequency (Hz)

Figure 11 Frequency response plot for 20 kHzndash100 kHz showing ageneral drop with spikes and deeps corresponding to resonance andantiresonance respectively

the others when the experiment was conducted insidewatermdashthe signal strength is less affected by it

(3) The relative absorption at 60 cm is less than that at30 cm (as defined in (4)) generally except for theplywood obstacle

(4) Generally the signal strength for the experimentinside water is always higher than that conducted inthe air by approximately 4 times We attribute this tobetter coupling of solid-liquid interfaces than solid-air (gas) interface

72 Mode Shape Simulation Processes and the HarmonicAnalysis There are 15 harmonics (Figure 8) between 0Hzand 22 kHz as simulated by ANSYS 10Multiphysics Only thefirst harmonic exhibits nodal shape while some of the otherharmonics are inverse of the subsequent harmonic frequencyThe frequencies used in Figure 9 were selected to be 20 kHzapart and are not harmonics (theywere checkedusingANSYSmultiphysics) None of the simulation in Figure 9 has nodal

Table 3 Maximum deformation for indicated frequencies

Frequency(Hz)

Maximumsurface

deformation(m)

Remark

(1) 164873 553119864 minus 02

(2) 381296 315119864 minus 05

(3) 430708 453119864 minus 05

(4) 696674 235119864 minus 03

(5) 739168 185119864 minus 03

(6) 938413 732119864 minus 03

(7) 1043579 489119864 minus 05

(8) 1058902 352119864 minus 03

The first 15 harmonicsoccurring between0ndash22 kHz

(9) 1332319 607119864 minus 05

(10) 1394286 864119864 minus 05

(11) 1439747 424119864 minus 06

(12) 1560828 231119864 minus 06

(13) 1841116 212119864 minus 05

(14) 1872452 314119864 minus 05

(15) 2008707 471119864 minus 04

119891

14000000 133119864 minus 08

Higher frequencies119891

26000000 123119864 minus 08

119891

38000000 911119864 minus 09

119891

410000000 260119864 minus 09

shape they all seems to have the same area of crests andvalleys

The plots of Figure 11 further show that there are lots ofresonance frequencies (peeks) and antiresonance frequencies


1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

MinMax

minus001

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Uni

t vol

ume s

wep

t (m

m3)

(b)

Figure 12 Plot of maximum andminimum deformation in119885 direction (perpendicular to the plate surface) (a) and total displacement by theplate at various harmonics (b)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0002

0004

0006

0008

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

minus0006

minus0004

minus0002

MinMax

Figure 13 Plot of maximum and minimum deformation in 119885direction the first harmonics is ignored in this plot

(deeps) however the 0ndash22 kHz (audio range) of Figure 10does not exhibit such pattern Also the frequency responsegenerally drops as the frequency increases for the ranges (0ndash22 kHz and 20ndash100 kHz) combined

The maximum and minimum deformation at axis per-pendicular to the plate face is shown in Figures 12 and13 A closer study of Figures 8 12 and 13 shows that the2nd and 3rd natural frequencies have crests and valleys thatoccupy about the same area The same applies to the 7119905ℎand 9thndash14th harmonics The 1st 4th 5th 6th 8th and 15thharmonics have unequal crest and valley areas This cancelsout each other displacement This is more glaring with the1st harmonic that has zero valley areas The 40 60 80 and100 kHz displacements are much lower in comparison tothose of the natural frequencies occurring in the 0ndash22 kHzrange as shown in Table 3 The implication is that much

100

25119890+3

5119890+3

75119890+3

1119890+4

2119890+4

Frequency (Hz)

10589kHz

180

150125100

755025

0

Phas

e ang

le (∘

)

125119890+4

15119890+4

175119890+4

(8th harmonic)

Figure 14 Phase response plots for 100Hzndash20 kHz range

180150125100

755025

0

Phas

e ang

le (∘

)

2119890+4

25119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Frequency (Hz)

425 kHz 575kHz

Figure 15 Phase response plots for 20 kHzndash100 kHz range

energymust be used to increase the piezo plate displacementsat these higher frequencies

One lesson that can be learned from this behavior isthat the harmonics 1 4 5 6 8 and 15 are to be investedin as they sweep more volume of the media than other


575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9

Figure 16 Frequency response plot for 20 kHzndash100 kHz range The inset is from 52 kHz to 67 kHz

45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz

Figure 17 The transmission and reception degrade in quality as the square wave input frequency was increased For each output the toptrace is the input and the lower trace is the response of the receiving piezo plate

harmonics For practical purposes and where a bit audibleoutput is not disturbing the 1st harmonic is the most idealbut where audibility is undesired higher harmonics suchas the 15th harmonic should be used We are not affirmingthat frequencies above 22 kHz will not work but as expectedmore energy will have to go in to drive the plates as the outputwill be very low

A phase response analysis for the harmonics Figure 14shows that lower frequencies up to 164873Hz (first har-monic) and then at 696674Hz (fourth harmonic) and938413Hz (sixth harmonic) are in phase (0∘) At highfrequencies Figure 15 the frequencies 425 kHz and 575 kHzare also in phase with the input but the 575 kHz is anantiresonance (a deep-lowered amplitude) as shown in theinset of Figure 16 The deep is a trap to be watched for whenoptimizing the transmitter output

73 The Experimental Verification of Transmission at 45 1020 40 60 80 and 100 kHz Frequencies At higher frequen-cies the receiver output drops and are derivative (spikes)of the square wave input especially for the 100 kHz testfrequency Also the transmitter output depreciates from aperfect square wave as frequency increases with discontinuity

in the oscilloscope plot It is also noticeable from Figure 17that this piezo plate assembly is able to follow the input betterat the test signals of 40 kHz and 60 kHz compared to the testsignals of 80 kHz and 100 kHz

8 Conclusion

The immediate implication of this work is that we can useordinary buzzer piezo plate for underwater navigation (atleast for a short range of about 2m or less we usedmaximumof 06m in this work) by transmitting at a selected higherfrequency In selecting a higher frequency above audio wecan take the advantage of the knowledge that 40 kHz to60 kHz test input signal can work as earlier explained andcombine that knowledgewith the fact that there is a harmonicnearby that is 425 kHz which we can tune into and anantiresonance frequency nearby such as 575 kHz which willgive a much lowered output and should not be used

As shown in this paper this common buzzer workedwell as an underwater transmitter-cum-receiver at audiblerange of 0ndash22 kHz which implies that a short range-low-power underwater navigation device can be realized using


buzzer plates A carefully selected resonance frequency canbe used with a matching receiver piezo plate that is tuned toits resonant frequency for reflective sonar sensing

For the robotic fish we developed short-range communi-cation is desired For example when investigating clutteredenvironment such as underwater crevices short range andreliable ultrasonic sensor will work well and especially if thewater is mucky

Disclosure

The authors want to recommend that a realistic sonar-basednavigation object detection and avoidance using this typeof piezo plate described in this work should be developedand tested using higher frequencies (above 22 kHz) that areresonance Though not discussed in this work the issue ofside lobe transmission and wave guide or signal concentratorneeds to be investigated for low-power small-scale underwa-ter robots based on the buzzer piezo crystal plate

References

[1] R M Eustice H Singh and L L Whitcomb ldquoSynchronous-clock one-way-travel-time acoustic navigation for underwatervehiclesrdquo Journal of Field Robotics vol 28 no 1 pp 121ndash1362011

[2] M Kumagai T Ura Y Kuroda and R Walker ldquoA newautonomous underwater vehicle designed for lake environmentmonitoringrdquo Advanced Robotics vol 16 no 1 pp 17ndash26 2002

[3] G Grenon P E An S M Smith and A J Healey ldquoEnhance-ment of the inertial navigation system for the Morpheusautonomous underwater vehiclesrdquo IEEE Journal of OceanicEngineering vol 26 no 4 pp 548ndash560 2001

[4] X Yun E R Bachmann S Arslan K Akyol and R BMcGhee ldquoAn inertial navigation system for small autonomousunderwater vehiclesrdquo Advanced Robotics vol 15 no 5 pp 521ndash532 2001

[5] H Bo Z Hongjin L Chao Z Shujing L Yan and Y TianhongldquoAutonomous navigation for autonomous underwater vehiclesbased on information filters and active sensingrdquo Sensors vol 11pp 10958ndash10980 2011

[6] E Dura J Bell and D Lane ldquoReconstruction of texturedseafloors from side-scan sonar imagesrdquo IEE Proceedings vol 151no 2 pp 114ndash126 2004

[7] C G Capus A C Banks E Coiras I Tena Ruiz C JSmith and Y R Petillot ldquoData correction for visualisation andclassification of sidescan SONAR imageryrdquo IET Radar Sonarand Navigation vol 2 no 3 pp 155ndash169 2008

[8] A Negre C Pradalier andM Dunbabin ldquoRobust vision-basedunderwater homing using self-similar landmarksrdquo Journal ofField Robotics vol 25 no 6-7 pp 360ndash377 2008

[9] T Kodama K Nakahira Y Kanaya and T Yoshikawa ldquoAppli-cation of digital polarity correlators in a sonar ranging systemrdquoElectronics and Communications in Japan vol 91 no 4 pp 20ndash26 2008

[10] T C Austin ldquoThe application of spread spectrum signalingtechniques to underwater acoustic navigationrdquo in Proceedingsof the IEEE Symposium on Autonomous Underwater VehicleTechnology pp 443ndash449 Boston Mass USA July 1994

[11] K Vickery ldquoAcoustic positioning systemsmdasha practical overviewof current systemsrdquo in Proceedings of the IEEE Symposium onAutonomous Underwater Vehicle Technology pp 5ndash17 Cam-bridge Mass USA August 1998

[12] T C Austin and R Stokey ldquoRelative acoustic trackingrdquo SeaTechnology vol 39 no 3 pp 21ndash27 1998

[13] S M Smith and D Kronen ldquoExperimental results of aninexpensive short baseline acoustic positioning system for AUVnavigationrdquo inProceedings of theOceansMTSIEEEConferencepp 714ndash720 Boca Raton Fla USA October 1997

[14] B Pierce ldquoProtecting Whales from Dangerous Sonarrdquo 2008httpwwwnrdcorgwildlifemarinesonarasp

[15] L J Jennifer ldquoSonar ban sounded good A skeptical analysisrdquoSkeptic vol 10 no 4 pp 14ndash15 2004

[16] T G Leighton ldquoFrom seas to surgeries from babbling brooks tobaby scans the acoustics of gas bubbles in liquidsrdquo InternationalJournal of Modern Physics B vol 18 no 25 pp 3267ndash3314 2004

[17] K Asakawa J Kojima Y Kato et al ldquoDesign concept andexperimental results of the autonomous underwater vehicleAQUA EXPLORER 2 for the inspection of underwater cablesrdquoAdvanced Robotics vol 16 no 1 pp 27ndash42 2002

[18] M O Afolayan D S Yawas C O Folayan and S Y AkuldquoMechanical description of a hyper-redundant robot jointmechanism used for a design of a biomimetic robotic fishrdquoJournal of Robotics vol 2012 Article ID 826364 16 pages 2012

[19] J Lygouras V Kodogiannis T Pachidis and P Liatsis ldquoTerrain-based navigation for underwater vehicles using an ultrasonicscanning systemrdquo Advanced Robotics vol 22 no 11 pp 1181ndash1205 2008

[20] J N Lygouras C M Dimitriadis M C Tsortanidis G CBakos and P G Tsalides ldquoDigital ultrasonic scanning systemfor positioning underwater remotely operated vehiclesrdquo Inter-national Journal of Electronics vol 76 no 3 pp 541ndash550 1994

[21] J N Lygouras K A Lalakos and P G Tsalides ldquoTHETISan underwater remotely operated vehicle for water pollutionmeasurementsrdquo Microprocessors and Microsystems vol 22 no5 pp 227ndash237 1998

[22] L Jindong and H Huosheng ldquoBuilding a simulation environ-ment for optimising control Parameters of an AutonomousRobotic Fishrdquo in Proceedings of the 9th Chinese Automationamp Computing Society Conference in the UK Luton EnglandSeptember 2003

[23] J Liu and H Hu ldquoBuilding a 3D simulator for autonomousnavigation of robotic fishesrdquo in Proceedings of the IEEERSJInternational Conference on Intelligent Robots and Systems(IROS rsquo04) pp 613ndash618 Sendai Japan October 2004

[24] J Liu and O Hu ldquoA methodology of modelling fish-like swimpatterns for robotic fishrdquo in Proceedings of the IEEE Interna-tional Conference onMechatronics and Automation (ICMA rsquo07)pp 1316ndash1321 Harbin China August 2007

[25] F Zhifang C Jack Z Jun B David and L Chang ldquoDesignand fabrication of artificial lateral line flow sensorsrdquo Journalof Micromechanics and Microengineering vol 12 pp 655ndash6612002

[26] Y Yang J Chen J Engel et al ldquoDistant touch hydrodynamicimaging with an artificial lateral linerdquo Proceedings of theNational Academy of Sciences of the United States of Americavol 103 no 50 pp 18891ndash18895 2006

[27] J C Liao ldquoThe role of the lateral line and vision on bodykinematics and hydrodynamic preference of rainbow trout inturbulent flowrdquo Journal of Experimental Biology vol 209 part20 pp 4077ndash4090 2006


[28] Y Yang N Nguyen N Chen et al ldquoArtificial lateral line withbiomimetic neuromasts to emulate fish sensingrdquo Bioinspirationand Biomimetics vol 5 no 1 Article ID 016001 2010

[29] J Sirohi and I Chopra ldquoFundamental understanding of piezo-electric strain sensorsrdquo Journal of Intelligent Material Systemsand Structures vol 11 no 4 pp 246ndash257 2000

[30] IEEE Standard on Piezoelectricity ldquoStandard Committee ofthe IEEE Ultrasonics Ferroelectrics and Frequency ControlSocietyrdquo ANSIIEEE Std 176-1987

[31] M O Afolayan ldquoPotential of watch buzzer as underwaternavigation device in shallow water streamsrdquo Research Journal ofApplied Sciences Engineering and Technology vol 2 no 5 pp436ndash446 2010

[32] O KatsuihikoModern Control Engineering Pearson Education4th edition 2005

[33] I J Nagrath and M Gopal Control Systems EngineeringPublisher New Age International Publishers 4th edition 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Table 1 Sonar frequency and its range

Frequency (kHz) Range (m)Low frequency (LF) 8ndash16 gt10000Medium frequency (MF) 18ndash36 2000ndash3000High frequency (HF) 30ndash60 1500Extra high frequency (EHF) 50ndash110 lt1000Very high frequency (VHF) 200ndash300 lt100Sources [10ndash13]

Ultrasonic sensors

39401

1956 19841

498

2

Figure 1 The biomimetic robotic fish that was being investigatedThe receiver is at the tip and the transmitter is at the bottom

Also without side lobe transmission attenuation there willbe multiple-object detection that is not directly ahead of thesonar transmitter

In some situations the problems associated with sonarnavigation or object detection are augmented for by othersensing methods for example Asakawa et al [17] AUV hadthe underwater cable to be located and carried a signaturein the form of 16ndash25Hz electric current which enabled it toautomatically locate it Another augmentation is in the formof bump switch to assist in navigating very tight environmentwhere sonar ranging will be too close or ineffective [18]Also Lygouras et al [19] used terrain structure to augmentthe sonar navigation processes which fall into the categoriesreferred to as baseline navigation systems [20 21]

2 The Motivation

Our team built a biomimetic robotic fish named BlueMac(Figure 1)mdasha 1 1 scale model of Mackerel fish to investigatethe use of rubber for building planar hyperredundant robotjoint [18] The robot is meant to swim in a low pressure headenvironment such as shallow lake pond or stream wherewater flow is not very strong or turbulent It was meantto use commercially available 40 kHz ultrasonic transmit-terreceiver combination for its navigation among obstaclefields but it failed to work in all the possible configurationswe tried except out of water The sensors were sealed againstwater with materials of various thicknesses such as polytheneand oil-impregnated paper In one of the experiments low-densitymineral oil (used for sewingmachine lubrication)wasused to fill the void in front of the sensors to improve contactwith the water boundary As a last attempt the sensors

(transmitter and receiver) were immersed in water withoutany insulationmdashbased on the premises that water is a poorconductor compared to the metallic conductor the sensorusesmdashthis still yielded no result Our conclusion was thatthe transmitter must have been overloaded for the followingreasons

(1) The piezo crystal plates drive the medium they are inforward and backward at the applied frequency

(2) This action is not a problem when the fluid it is inter-acting with is air with average density of 1184 kgm3(259∘C mean temperature)

(3) If the fluid is water the density becomes approxi-mately 1000 kgm3 which is 847-fold increment ofmass of fluid to move when compared to air Thisseems to be an overload for the small-aperture sonarsensor

Exhaustive literature search gave no clue on how tosolve this problem except that Jindong and Huosheng [22ndash24] seemed to have encountered such problem also Theycreated an ultrasonic-based model for their robotic fish butimplemented it with infrared light sensor There are attemptsby some researchers to imitate fish lateral lines [25ndash28] butsmall-scale purely sonar-based underwater navigation byrobots that are less than 05m in length is unknown to theauthors as of this writing and this has prompted us to do athorough investigation about it at this scale

3 Objectives of This Study

The goal of this study is to setup a simulation environmentin ANSYSMultiphysics software to find out the mode shapesof audible range piezo crystal (PZT-5H) material (commonlyused in 27mm diameter commercial buzzer alarms) at itsharmonics We also aim at finding possible nodal shapewithin the audible range (0ndash22 kHz)which implies absence ofcompressive and tensile stress simultaneously on one surfaceof the piezoelectric biomorph plate Furthermore we aim totest the piezo plates at higher frequencies so as to find outtheir real responses

4 Theoretical Background

A piezoelectric material will generate voltages when sub-jected to tension or compression The voltage generated isproportional to the compressive or tension load which ismathematically represented as [29 30]

119878 = [119904

119864] 119879 + [119889

119905] 119864

119863 = [119889] 119879 + [120598

119879] 119864

(1)

where 119878 is strain vector 119879 is the vector of stresses 119863 is thedielectric displacement vector 119864 is the electric field vector119878

119864 is the compliance matrix evaluated at constant electricfield 119889 is the piezoelectric strain coefficients 119889

119905is the mode

of operation (compressive or transverse mode) and 120598119879 is


Log

of im

peda

nce

Frequency

119891119898

119891119899



AB

C

D

Receivingcomputer

Sendingcomputer





119899) Piezo



5 Methodology











120601188mm

12060127mm









119862 (119904) = 119866 (119904) 119877 (119904) = 119866 (119904) (2)


119888 (119905) = int

119905

0

119892 (119905 minus 120591) 119903 (120591) 119889120591 (3)


6 Results Obtained


1003816

1003816

1003816

1003816

119877water1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119877air1003816

1003816

1003816

1003816











164873Hz 381296Hz 430708Hz

696674Hz 739168Hz 938413Hz

1043579Hz 1058902Hz 1332319Hz

1394286Hz 1439747Hz 1560828Hz

1841116Hz 1872452Hz 2008707Hz












216119890 minus 5

16119890 minus 5

12119890 minus 5

8119890 minus 6

4119890 minus 6

733119890 minus 9

100 25119890 + 3 5119890 + 3 75119890 + 3 1119890 + 4 2119890 + 4

Am

plitu

de (m

)

Frequency (Hz)

Frequency (Hz)

12119890 minus 6

8119890 minus 7

4119890 minus 7

207119890 minus 9

5119890 + 3 6119890 + 3 8119890 + 37119890 + 3 9119890 + 3 1119890 + 4 11119890 + 4 12119890 + 4408119890 + 3

125119890 + 4 15119890 + 4 175119890 + 4


385 kHz54 kHz

575 kHz

59 kHz 66 kHz


2119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Am

plitu

de (m

)

Frequency (Hz)







Frequency(Hz)

Maximumsurface

deformation(m)

Remark

(1) 164873 553119864 minus 02

(2) 381296 315119864 minus 05

(3) 430708 453119864 minus 05

(4) 696674 235119864 minus 03

(5) 739168 185119864 minus 03

(6) 938413 732119864 minus 03

(7) 1043579 489119864 minus 05

(8) 1058902 352119864 minus 03


(9) 1332319 607119864 minus 05

(10) 1394286 864119864 minus 05

(11) 1439747 424119864 minus 06

(12) 1560828 231119864 minus 06

(13) 1841116 212119864 minus 05

(14) 1872452 314119864 minus 05

(15) 2008707 471119864 minus 04

119891

14000000 133119864 minus 08


26000000 123119864 minus 08

119891

38000000 911119864 minus 09

119891

410000000 260119864 minus 09




1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

MinMax

minus001

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Uni

t vol

ume s

wep

t (m

m3)

(b)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0002

0004

0006

0008

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

minus0006

minus0004

minus0002

MinMax




100

25119890+3

5119890+3

75119890+3

1119890+4

2119890+4

Frequency (Hz)

10589kHz

180

150125100

755025

0

Phas

e ang

le (∘

)

125119890+4

15119890+4

175119890+4

(8th harmonic)


180150125100

755025

0

Phas

e ang

le (∘

)

2119890+4

25119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Frequency (Hz)

425 kHz 575kHz





575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9


45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz






8 Conclusion






Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Log

of im

peda

nce

Frequency

119891119898

119891119899



AB

C

D

Receivingcomputer

Sendingcomputer





119899) Piezo



5 Methodology











120601188mm

12060127mm









119862 (119904) = 119866 (119904) 119877 (119904) = 119866 (119904) (2)


119888 (119905) = int

119905

0

119892 (119905 minus 120591) 119903 (120591) 119889120591 (3)


6 Results Obtained


1003816

1003816

1003816

1003816

119877water1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119877air1003816

1003816

1003816

1003816











164873Hz 381296Hz 430708Hz

696674Hz 739168Hz 938413Hz

1043579Hz 1058902Hz 1332319Hz

1394286Hz 1439747Hz 1560828Hz

1841116Hz 1872452Hz 2008707Hz












216119890 minus 5

16119890 minus 5

12119890 minus 5

8119890 minus 6

4119890 minus 6

733119890 minus 9

100 25119890 + 3 5119890 + 3 75119890 + 3 1119890 + 4 2119890 + 4

Am

plitu

de (m

)

Frequency (Hz)

Frequency (Hz)

12119890 minus 6

8119890 minus 7

4119890 minus 7

207119890 minus 9

5119890 + 3 6119890 + 3 8119890 + 37119890 + 3 9119890 + 3 1119890 + 4 11119890 + 4 12119890 + 4408119890 + 3

125119890 + 4 15119890 + 4 175119890 + 4


385 kHz54 kHz

575 kHz

59 kHz 66 kHz


2119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Am

plitu

de (m

)

Frequency (Hz)







Frequency(Hz)

Maximumsurface

deformation(m)

Remark

(1) 164873 553119864 minus 02

(2) 381296 315119864 minus 05

(3) 430708 453119864 minus 05

(4) 696674 235119864 minus 03

(5) 739168 185119864 minus 03

(6) 938413 732119864 minus 03

(7) 1043579 489119864 minus 05

(8) 1058902 352119864 minus 03


(9) 1332319 607119864 minus 05

(10) 1394286 864119864 minus 05

(11) 1439747 424119864 minus 06

(12) 1560828 231119864 minus 06

(13) 1841116 212119864 minus 05

(14) 1872452 314119864 minus 05

(15) 2008707 471119864 minus 04

119891

14000000 133119864 minus 08


26000000 123119864 minus 08

119891

38000000 911119864 minus 09

119891

410000000 260119864 minus 09




1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

MinMax

minus001

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Uni

t vol

ume s

wep

t (m

m3)

(b)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0002

0004

0006

0008

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

minus0006

minus0004

minus0002

MinMax




100

25119890+3

5119890+3

75119890+3

1119890+4

2119890+4

Frequency (Hz)

10589kHz

180

150125100

755025

0

Phas

e ang

le (∘

)

125119890+4

15119890+4

175119890+4

(8th harmonic)


180150125100

755025

0

Phas

e ang

le (∘

)

2119890+4

25119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Frequency (Hz)

425 kHz 575kHz





575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9


45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz






8 Conclusion






Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











120601188mm

12060127mm









119862 (119904) = 119866 (119904) 119877 (119904) = 119866 (119904) (2)


119888 (119905) = int

119905

0

119892 (119905 minus 120591) 119903 (120591) 119889120591 (3)


6 Results Obtained


1003816

1003816

1003816

1003816

119877water1003816

1003816

1003816

1003816

minus

1003816

1003816

1003816

1003816

119877air1003816

1003816

1003816

1003816











164873Hz 381296Hz 430708Hz

696674Hz 739168Hz 938413Hz

1043579Hz 1058902Hz 1332319Hz

1394286Hz 1439747Hz 1560828Hz

1841116Hz 1872452Hz 2008707Hz












216119890 minus 5

16119890 minus 5

12119890 minus 5

8119890 minus 6

4119890 minus 6

733119890 minus 9

100 25119890 + 3 5119890 + 3 75119890 + 3 1119890 + 4 2119890 + 4

Am

plitu

de (m

)

Frequency (Hz)

Frequency (Hz)

12119890 minus 6

8119890 minus 7

4119890 minus 7

207119890 minus 9

5119890 + 3 6119890 + 3 8119890 + 37119890 + 3 9119890 + 3 1119890 + 4 11119890 + 4 12119890 + 4408119890 + 3

125119890 + 4 15119890 + 4 175119890 + 4


385 kHz54 kHz

575 kHz

59 kHz 66 kHz


2119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Am

plitu

de (m

)

Frequency (Hz)







Frequency(Hz)

Maximumsurface

deformation(m)

Remark

(1) 164873 553119864 minus 02

(2) 381296 315119864 minus 05

(3) 430708 453119864 minus 05

(4) 696674 235119864 minus 03

(5) 739168 185119864 minus 03

(6) 938413 732119864 minus 03

(7) 1043579 489119864 minus 05

(8) 1058902 352119864 minus 03


(9) 1332319 607119864 minus 05

(10) 1394286 864119864 minus 05

(11) 1439747 424119864 minus 06

(12) 1560828 231119864 minus 06

(13) 1841116 212119864 minus 05

(14) 1872452 314119864 minus 05

(15) 2008707 471119864 minus 04

119891

14000000 133119864 minus 08


26000000 123119864 minus 08

119891

38000000 911119864 minus 09

119891

410000000 260119864 minus 09




1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

MinMax

minus001

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Uni

t vol

ume s

wep

t (m

m3)

(b)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0002

0004

0006

0008

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

minus0006

minus0004

minus0002

MinMax




100

25119890+3

5119890+3

75119890+3

1119890+4

2119890+4

Frequency (Hz)

10589kHz

180

150125100

755025

0

Phas

e ang

le (∘

)

125119890+4

15119890+4

175119890+4

(8th harmonic)


180150125100

755025

0

Phas

e ang

le (∘

)

2119890+4

25119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Frequency (Hz)

425 kHz 575kHz





575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9


45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz






8 Conclusion






Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation
















164873Hz 381296Hz 430708Hz

696674Hz 739168Hz 938413Hz

1043579Hz 1058902Hz 1332319Hz

1394286Hz 1439747Hz 1560828Hz

1841116Hz 1872452Hz 2008707Hz












216119890 minus 5

16119890 minus 5

12119890 minus 5

8119890 minus 6

4119890 minus 6

733119890 minus 9

100 25119890 + 3 5119890 + 3 75119890 + 3 1119890 + 4 2119890 + 4

Am

plitu

de (m

)

Frequency (Hz)

Frequency (Hz)

12119890 minus 6

8119890 minus 7

4119890 minus 7

207119890 minus 9

5119890 + 3 6119890 + 3 8119890 + 37119890 + 3 9119890 + 3 1119890 + 4 11119890 + 4 12119890 + 4408119890 + 3

125119890 + 4 15119890 + 4 175119890 + 4


385 kHz54 kHz

575 kHz

59 kHz 66 kHz


2119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Am

plitu

de (m

)

Frequency (Hz)







Frequency(Hz)

Maximumsurface

deformation(m)

Remark

(1) 164873 553119864 minus 02

(2) 381296 315119864 minus 05

(3) 430708 453119864 minus 05

(4) 696674 235119864 minus 03

(5) 739168 185119864 minus 03

(6) 938413 732119864 minus 03

(7) 1043579 489119864 minus 05

(8) 1058902 352119864 minus 03


(9) 1332319 607119864 minus 05

(10) 1394286 864119864 minus 05

(11) 1439747 424119864 minus 06

(12) 1560828 231119864 minus 06

(13) 1841116 212119864 minus 05

(14) 1872452 314119864 minus 05

(15) 2008707 471119864 minus 04

119891

14000000 133119864 minus 08


26000000 123119864 minus 08

119891

38000000 911119864 minus 09

119891

410000000 260119864 minus 09




1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

MinMax

minus001

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Uni

t vol

ume s

wep

t (m

m3)

(b)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0002

0004

0006

0008

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

minus0006

minus0004

minus0002

MinMax




100

25119890+3

5119890+3

75119890+3

1119890+4

2119890+4

Frequency (Hz)

10589kHz

180

150125100

755025

0

Phas

e ang

le (∘

)

125119890+4

15119890+4

175119890+4

(8th harmonic)


180150125100

755025

0

Phas

e ang

le (∘

)

2119890+4

25119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Frequency (Hz)

425 kHz 575kHz





575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9


45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz






8 Conclusion






Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










216119890 minus 5

16119890 minus 5

12119890 minus 5

8119890 minus 6

4119890 minus 6

733119890 minus 9

100 25119890 + 3 5119890 + 3 75119890 + 3 1119890 + 4 2119890 + 4

Am

plitu

de (m

)

Frequency (Hz)

Frequency (Hz)

12119890 minus 6

8119890 minus 7

4119890 minus 7

207119890 minus 9

5119890 + 3 6119890 + 3 8119890 + 37119890 + 3 9119890 + 3 1119890 + 4 11119890 + 4 12119890 + 4408119890 + 3

125119890 + 4 15119890 + 4 175119890 + 4


385 kHz54 kHz

575 kHz

59 kHz 66 kHz


2119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Am

plitu

de (m

)

Frequency (Hz)







Frequency(Hz)

Maximumsurface

deformation(m)

Remark

(1) 164873 553119864 minus 02

(2) 381296 315119864 minus 05

(3) 430708 453119864 minus 05

(4) 696674 235119864 minus 03

(5) 739168 185119864 minus 03

(6) 938413 732119864 minus 03

(7) 1043579 489119864 minus 05

(8) 1058902 352119864 minus 03


(9) 1332319 607119864 minus 05

(10) 1394286 864119864 minus 05

(11) 1439747 424119864 minus 06

(12) 1560828 231119864 minus 06

(13) 1841116 212119864 minus 05

(14) 1872452 314119864 minus 05

(15) 2008707 471119864 minus 04

119891

14000000 133119864 minus 08


26000000 123119864 minus 08

119891

38000000 911119864 minus 09

119891

410000000 260119864 minus 09




1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

MinMax

minus001

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Uni

t vol

ume s

wep

t (m

m3)

(b)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0002

0004

0006

0008

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

minus0006

minus0004

minus0002

MinMax




100

25119890+3

5119890+3

75119890+3

1119890+4

2119890+4

Frequency (Hz)

10589kHz

180

150125100

755025

0

Phas

e ang

le (∘

)

125119890+4

15119890+4

175119890+4

(8th harmonic)


180150125100

755025

0

Phas

e ang

le (∘

)

2119890+4

25119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Frequency (Hz)

425 kHz 575kHz





575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9


45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz






8 Conclusion






Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

MinMax

minus001

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

001

002

003

004

005

006

Harmonics

Uni

t vol

ume s

wep

t (m

m3)

(b)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0

0002

0004

0006

0008

Harmonics

Def

orm

atio

n pe

ak p

erpe

ndic

ular

to

the p

late

(mm

)

minus0006

minus0004

minus0002

MinMax




100

25119890+3

5119890+3

75119890+3

1119890+4

2119890+4

Frequency (Hz)

10589kHz

180

150125100

755025

0

Phas

e ang

le (∘

)

125119890+4

15119890+4

175119890+4

(8th harmonic)


180150125100

755025

0

Phas

e ang

le (∘

)

2119890+4

25119890+4

375119890+4

5119890+4

625119890+4

75119890+4

875119890+4

1119890+5

Frequency (Hz)

425 kHz 575kHz





575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9


45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz






8 Conclusion






Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










575kHz

343119890 minus 8

3119890 minus 8

2119890 minus 8

15119890 minus 8

1119890 minus 8

5119890 minus 9

152119890 minus 10

2119890 + 4 375119890 + 4 5119890 + 4 625119890 + 4 75119890 + 4 875119890 + 4 1119890 + 5

Frequency (Hz)

Frequency (Hz)

Am

plitu

de (m

) 25119890 minus 82119890 minus 9

1119890 minus 9

152119890 minus 10

528119890 + 4 55119890 + 4 575119890 + 4 6119890 + 4 625119890 + 4 65119890 + 4 672119890 + 4

3119890 minus 9


45kHz 10kHz 20kHz

40kHz 60kHz 80kHz

100 kHz






8 Conclusion






Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Disclosure


References





































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article modal analysis of 27mm piezo electric...

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