Research ArticleInvestigation of Δ119864 Effect on Vibrational Behavior ofGiant Magnetostrictive Transducers

M Sheykholeslami1 Y Hojjat1 M Ghodsi2 K Kakavand1 and S Cinquemani3

1Faculty of Engineering Tarbiat Modares University Jalale-Ale-Ahmad Highway PO Box 14115143 Tehran Iran2College of Engineering Sultan Qaboos University PO Box 33 Al-Khod 123 Muscat Oman3Department of Mechanics Politecnico di Milano Via La Masa 1 20156 Milano Italy

Correspondence should be addressed to Y Hojjat yhojjatmodaresacir

Received 19 November 2014 Accepted 26 January 2015

Academic Editor Alicia Gonzalez-Buelga

Copyright copy 2015 M Sheykholeslami et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Resonant magnetostrictive transducers are used for generating vibrations in the sonic and ultrasonic range of frequency Asthe mechanical properties of magnetostrictive materials change according to different operating conditions (ie temperaturemechanical prestress and magnetic bias) the vibrational behavior of the transducer changes too Δ119864 effect is the change in theYoung modulus of the ferromagnetic material and it has to be considered as it leads to changes in the dynamics of the transducerThis paper deals with the study of such effect from both theoretical and experimental point of view Δ119864 effect on behavior of thetransducer based on Terfenol-D is analytically described as a function of different operating conditions focusing on effects onresonance frequency mode shape and moreover experimentally the quality factor Results of resonance frequency prediction havebeen validated with experiments and good agreement has been seen

1 Introduction

Magnetostrictive materials exhibit some interesting proper-ties such as the ability to change their shape when a magneticfield varies along the material itself (Joule effect) [1ndash4] Thiseffect has been exploited to realize transducers devoted togenerate vibration from low range of frequency to sonic andultrasonic range of frequency While low frequency range iscommonly used in active vibration control applications [5ndash7]sound [8] sonic and ultrasonic transducers are used as sonartransducers [9 10] machining transducers [11] and so forth

Despite benefits in using magnetostrictive materials intransducers mechanical properties of the material suchas Young modulus change according to the operationalcondition such as prestress magnetic bias and temperatureThis phenomenon is called Δ119864 effect Among ferromagneticmaterial Δ119864 effect in Terfenol-D has the highest value[12] Study on the Δ119864 effect in Terfenol-D and related

considerations has been presented in different papers [13ndash15] but some disagreements about results can be observedprobably due to interferences of various parameters in Youngmodulus measurement Some other works have been triedto use Δ119864 effect in Terfenol-D devices in passive operationsuch as tunable resonator [16] Some efforts have been doneto use this effect for modeling and optimizing nonresonantTerfenol-D actuator Yong and Lin [17] presented a dynamicmodel for giant magnetostrictive actuator considering Δ119864effect while Ueno et al [18] introduced magnetic designcircuit method for magnetic force control system with con-sideringΔ119864 effect for the Terfenol-D to improve the efficiencyof the device Dapino et al [19] introduced a model forΔ119864 effect in the magnetostrictive transducer In this modelthere is a discussion about Δ119864 in resonance frequency of thetransducer Changing resonance frequency in this transducerwas 140Hz for sweeping magnetic bias from the first regionof application to saturated region

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 478045 9 pageshttpdxdoiorg1011552015478045

2 Shock and Vibration

Notwithstanding these works there is no report aboutconsequences of Δ119864 effect on operational parameters ofTerfenol-D resonant transducers As a matter of fact reso-nance frequency mode shape and mechanical quality factorof the transducer depend on the Young modulus of themagnetostrictive material When resonance frequency ischanged due to Δ119864 effect the transducer starts working innonresonance region thus decreasing its performance Forthis reason it is important to predict the effect of thesechanges due to a change in theYoungmodulus of thematerialto adjust the functioning of the transducer accordingly

To achieve these goals a resonant Terfenol-D transducerhas been designed and fabricated Main parameters causingΔ119864 effect are measured to experimentally to find out theircorrelation with the change of the Terfenol-D Young modu-lus Vibrational behavior of the transducer is modeled analyt-ically and numerically using ANSYS12 commercial softwarePredictions results have been satisfactorily verified withexperiments Results can be helpful in using the Terfenol-D transducer in order to permanently adjust it in resonanceregion during operation Also effect of changing condition onperformance of the transducer can be predictable based onthese results

The paper is structured as follows Section 2 introducesan analytical model to take into account the effect of mainmechanical parameters of a resonant transducer and its oper-ating conditions on the resonance frequency mode shapeand the quality factor of the device Section 3 presents thetransducer used to validate the model and the experimentalset-up Section 4 collects results from both analytical andexperimental analysis A discussion of results is done andconclusions are drawn in Section 5

2 Δ119864 Effect in Resonant Transducers

Resonant Langevin transducers are used to generate sonicand ultrasonic vibration As suggested in their name thesedevices should work in resonance frequency to enhancemechanical vibration These devices are characterized by avery narrow bandwidth (especially those with high-qualityfactor) then a small change in the resonance frequency canreduce significantly the performance of the device Resonanttransducers have 3 main parts excitation part (based onpiezoelectric or magnetostrictive material) backing part(tail mass) and matching part (head mass) [20] Figure 1shows the layout of a magnetostrictive resonant transducerTerfenol-D should be suitablymagnetically andmechanicallypreloaded to increase the efficiency of the transducer [21]Backing in the transducer can be a part of magnetic circuitIn design of the transducer it is crucial to place the node ofvibration exactly where prestress and bias mechanisms act inorder to avoid disturbance in operation If this condition isnot guaranteed then the efficiency of the device falls down

21 Effects on Resonance Frequency Young modulus changeof Terfenol-D can change resonance frequency and modeshape For predicting this analytical and numerical methods

Preload mechanism Terfenol-DSpring

MatchingBacking

Figure 1 Layout of a magnetostrictive resonant transducer

Backing Terfenol-D Matching

x

L1 L2 L3 L4 L5

Figure 2 Main parts of the transducer

are exploited The vibrational behavior of the transducer canbe predicted using wave equation

1198892119906

1198891199092+1

119860

119889119860

119889119909

119889119906

119889119909+1205962

1198622119906 = 0 (1)

where 119906 is the amplitude of the vibration 119860 is the crosssection area 120596 is the circular frequency and 119862 is the soundspeed in the material Main parts of the transducer andcoordinate system are shown in Figure 2 If main parts ofthe transducer are divided into 5 parts (1) should be satisfiedalong them As each part has constant cross section area theterm 119889119860119889119909 is equal to zero Thus (1) can be reduced as

1198892119906

1198891199092+1205962

1198622sdot 119906 = 0 (2)

General solution for (2) along the 119894th parts

119906119894= 119896119894cos( 120596

119862119894

119909) + 119896119894+1

sin( 120596119862119894

119909) (3)

where 119896119894and 119896119894+1

are constants Diameter of each transducerpart is smaller than 120582

1198942 (120582119894is acoustic wavelength in each

part of the transducer) to prevent lateral mode in operation[20] Boundary conditions for solving this problem areNewton law and equivalency of displacement in interfaceareas of parts Also this transducer is free at two ends Hence

Shock and Vibration 3

forces at these points are equal to zero Equations (4) showboundary conditions for the two parts of the backing Thus

11986411198601

1205971199061

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199061(1198711) = 1199062(1198711)

11986411198601

1205971199061

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711

minus 11986421198602

1205971199062

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711

= 0

1199062(1198711+ 1198712) = 1199063(1198711+ 1198712)

11986411198602

1205971199062

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711+1198712

minus 11986431198603

1205971199063

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711+1198712

minus 119896screws1199062 (1198711 + 1198712) = 0

(4)

In (4) 119864 is the Young modulus and 119896 is stiffness Boundaryconditions for all the other parts can be easily obtainedaccordingly

Considering the whole system 10 boundary conditionscan be written and resumed into

119860 sdot 119862 = 0 (5)

where matrix119860 is the coefficient matrix (6)119862 is a rowmatrixof constant 119896

1to 11989610elements (8) and 0 is a null matrix (9)

Matrix 119860 is then

119860 =

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

[

0 11988610 0 0 0 0 0 0 0

11988711198872119887311988740 0 0 0 0 0

11988811198882119888311988840 0 0 0 0 0

0 0 11988911198892119889311988940 0 0 0

0 0 11989011198902119890311989040 0 0 0

0 0 0 0 11989111198912119891311989140 0

0 0 0 0 11989211198922119892311989240 0

0 0 0 0 0 0 ℎ1ℎ2ℎ3ℎ4

0 0 0 0 0 0 1198941119894211989431198944

0 0 0 0 0 0 0 0 11989511198952

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

]

(6)

whose element are

1198861=11986411198601120596

1198621

1198871= cos(1205961198711

1198621

)

1198872= sin(1205961198711

1198621

)

1198873= minus cos(1205961198711

1198622

) 1198874= minus sin(1205961198711

1198622

)

1198881= minus11988611198872 119888

2= 11988611198871

1198883=11986021198642120596

1198622

sin(12059611987111198622

) 1198884= minus

11986021198642120596

1198622

cos(12059611987111198622

)

1198891= cos(

120596 (1198711+ 1198712)

1198622

) 1198892= sin(

120596 (1198711+ 1198712)

1198622

)

1198893= minus cos(

120596 (1198711+ 1198712)

1198623

)

1198894= minus sin(

120596 (1198711+ 1198712)

1198623

)

1198901= minus

11986021198642120596

1198622

1198892minus 119896scr1198891 119890

2=11986021198642120596

1198622

1198891minus 119896scr1198892

1198903= minus

11986031198643120596

1198623

1198894 119890

4=11986031198643120596

1198623

1198893

1198911= cos(

120596 (1198711+ 1198712+ 1198713)

1198623

)

1198912= sin(

120596 (1198711+ 1198712+ 1198713)

1198623

)

1198913= minus cos(

120596 (1198711+ 1198712+ 1198713)

1198624

)

1198914= minus sin(

120596 (1198711+ 1198712+ 1198713)

1198624

)

1198921= minus

11986031198643120596

1198623

1198912 119892

2=11986031198643120596

1198623

1198911

1198923= minus

11986041198644120596

1198624

1198914 119892

4=11986041198644120596

1198624

1198913

ℎ1= cos(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198624

)

ℎ2= sin(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198624

)

ℎ3= minus cos(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198625

)

ℎ4= minus sin(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198625

)

1198942=11986041198644120596

1198624

ℎ1minus 119896spring ℎ2 119894

3= minus

11986051198645120596

1198625

ℎ4

1198944=11986051198645120596

1198625

ℎ3

1198951= minus

11986051198645120596

1198625

sin(120596 (1198711+ 1198712+ 1198713+ 1198714+ 1198715)

1198625

)

1198952=11986051198645120596

1198625

cos(120596 (1198711+ 1198712+ 1198713+ 1198714+ 1198715)

1198625

)

(7)

4 Shock and Vibration

Electrical branch Transducer Mechanical branch

ZeTem

Zm

I vFU

Tme

Figure 3 Model of ideal gyrator

Matrix 119862 is

119862 = [1198961 1198962 1198963 1198964 1198965 1198966 1198967 1198968 1198969 11989610]119879

(8)

As it is known necessary condition for the solution existenceof (5) is

det (119860) = 0 (9)

Roots of (9) are longitudinal resonance frequencies Thusresonance frequencies are function of each parameter ofmatrix 119860 including Young modulus of Terfenol-D (119864

3) This

relationship allows considering the effect of a change in theYoung modulus of the material on the behavior of the device

Analytical model is supported by a finite element modelof the device ANSYS12 commercial software has been usedto carry outmodal simulation to predict resonance frequencyand mode shape of the transducer

22 Effects on Quality Factor Amagnetostrictive transducercan be modeled with ideal gyrator as shown in Figure 3 [4]Equations of this gyrator are presented in (10a) and (10b) [14]

119880 = 119885e sdot 119868 + 119879em sdot V (10a)

119865 = 119879me sdot 119868 + 119885m sdot V (10b)

where 119880 is the applied voltage 119868 is the current and 119865 andV are mechanical force and speed respectively 119879em is theelectromotive force per velocity and 119879me is the mechanicalforce per current In free state condition of the transducerthere is no mechanical force By setting 119865 = 0 in (10a) andsubstituting the result in (10b) effective electrical impedanceof the transducer 119885ee can be calculated This parameter ispresented in [20]

119885ee = 119877ee + 119895 sdot 119909ee = 119885e +minus119879me sdot 119879em

119885m= 119885e + 119885mot (11)

where 119877ee and 119909ee are real and imaginary parts of thetransducer impedance 119885e is the electrical impedance and119885mot is motional impedance If 119885ee is measured in clampedstate 119885mot is vanished Thus 119885mot can be obtained bymeasuring impedance in free and clamped state condition

Real and imaginary parts of 119885mot form a circle inresonance region frequency This circle is tangent to originFor presented transducer this circle is depicted in Figure 4If the diameter between origin and the points of circle thatcorresponds to resonance frequency is drawn perpendicularline to this diameter crosses the circle at two points These

minus2 0 2 4 6 8 10 12minus8

minus6

minus4

minus2

0

2

4

6

Real part of motional impedance

Imag

e par

t of m

otio

nal i

mpe

danc

e

120596998400998400

120596

120596998400

Figure 4 Motional impedance locus

Figure 5 The transducer used for tests

points correspond to 1205961015840 and 12059610158401015840 12059610158401015840 minus 1205961015840 show bandwidthof the transducer [20] Thus quality factor can be calculatedusing (12) [22] In the following equation 120596

0is the circular

resonance frequency

119876119898=

1205960

12059610158401015840 minus 1205961015840 (12)

3 Experimental Validation

31 The Resonant Transducer To experimentally investigateΔ119864 effect resonant magnetostrictive transducers have beentested under different operating condition (Figure 5) Thetransducer has been designed for working in the secondmode (Figure 6) Resonance frequency for this transduceris 8252Hz Magnetic bias in the presented transducer is40 kAm and mechanical prestress is 1034MPa Magneticbias has been supplied by ceramic permanent magnet andprestress has been exerted by 4 screws and a bevel springMagnetic yoke was used to close magnetic flux path andminimize flux leakage Excitation coil has 1000 turns of wirewith 1mm2 cross section area In designing the transducerprestress and biasmechanisms are attached to nodes of vibra-tion in order to avoid disturbance in operation (Figure 6)Table 1 shows main dimensions of the device

32 Δ119864 Effect Measurement There are some methods forYoung modulus measurement Stress-Strain graph acoustic

Shock and Vibration 5

0 005 01 015 02 025 03 035 04 045 05

minus2

0

2

4

6

8

10

12

Mod

al am

plitu

de

Distance (m)

Figure 6 Modal shape of the transducer obtained with ANSYS12

Table 1 Transducer main dimensions

Main part Length [mm] Diameter [mm]Backing 150 50Terfenol-D 50 10

Matching 1575 231575 7

and impedance methods are the most important Stress-Strain graph has some intrinsic error such as data pickupHigh attenuation of transverse wave in Terfenol-D makessome problem for acoustic measurement Thus impedancemethod has been selected For using impedance method asearch coil is turned to Terfenol-D Resonance frequency ofTerfenol-D can be detected with measurement impedance ofthe coil In resonance band of frequency there is unusualchange in impedance response If constant voltage mode ofimpedance analyzer was used the maximum pick in thisregion would be resonance frequency and minimum pickwould be antiresonance frequency [20] Young modulus canbe calculated by

119864 = 2120587120588119891res1198972 (13)

33 Experimental Set-Up First tests for Δ119864 effect measure-ment have been done on the magnetostrictive rod Exper-imental set-up is shown in Figure 7 The Terfenol-D rodis wounded by an excitation coil to change the magneticfield and it is surrounded by a search coil The search coilis connected to the impedance analyzer To minimize fluxleakage a high permeability yoke made of pure iron has beenconsidered in the set-up

Tests have been carried out changing the magnetic biasand the mechanical prestress of Terfenol-D The change inthe first is obtained exploiting the excitation coil of the set-up Preload of themagnetostrictive rod can be easilymodifiedby means of 4 screws and can be measured using a load cellNecessary calibration with standard weights has been doneA temperature control circuit was used to keep temperatureof Terfenol-D constant during the test By using this set-up effect of the mentioned parameter in Terfenol-D Youngmodulus was measured precisely Results of measurementswould be input for FEM and analytical methods to predictmode shape and resonance frequency

a

b

c

d

e

f

g

Figure 7 The experimental set-up for measurement of Youngmodulus ((a) magnetic yoke (b) load cell (c) heater (d) powersupply (e) load indicator (f) temperature control unit and (g)impedance analyzer)

Figure 8 Experimental set-up for measuring changes in resonancefrequency

Supplementary tests are carried out to verify analyticaland numerical resonance frequency prediction The experi-mental set-up for these tests is shown in Figure 8 A searchcoil is attached to impedance analyzer excitation coil oftransducer is used as bias coil Mechanical preloading isexerted by screws and can be adjusted by a torque meterWith using this set-up effect of change in magnetic bias andprestress on resonance frequency of the transducer can beexperimentally studied Moreover according to (11) qualityfactor can be calculated

4 Results and Discussion

Figure 9 shows the relationship between the Young modulusof Terfenol-D rod and magnetic bias in different prestresslevels The temperature was kept at 20∘C Range of magneticfield in this figure is from 20 kAm to 100 kAmThese valuesfor prestress and magnetic field are common for Terfenol-D transducers According to this figure Young moduluschanged between 40 and 75GPa Figure 10 depicts depen-dency of Youngmodulus on temperature in 40 kAmbias and1034MPa prestress By changing temperature between 20∘Cand 80∘C Young modulus is changed about 5MPa

By applying Δ119864 effect to analytical and FEM modelresonance frequency and mode shape have been calculatedFigures 11 12 and 13 show effects of magnetic bias andmechanical prestress on the transducer resonance frequencyobtained fromFEMand analyticalmethod Results show thatin the transducer with the mentioned design the effect ofYoung modulus can change the resonance frequency about220Hz As this kind of device has a very narrow bandwidtha similar change in resonance frequency considerably reduces

6 Shock and Vibration

20 30 40 50 60 70 80 90 10040

45

50

55

60

65

70

75

H (kAm)

Youn

g m

odul

us (G

Pa)

689MPa1034MPa1389MPa

Figure 9 Young modulus versus magnetic bias field in differentprestress level in 20∘C

20 30 40 50 60 70 8040

41

415

42

425

43

435

44

445

45

455

Youn

g m

odul

us (G

Pa)

Temperature (∘C)

Figure 10 Young modulus versus temperature in 1034MPa pre-stress and 40 kAm magnetic bias field

the efficiency of the transducer This effect is more conspicu-ous for high-quality factor transducer

Table 2 shows effect of temperature on resonance fre-quency obtained from FEM and analytical method In bothmethods effect of temperature until 80∘C is less than 15Hz

Figure 14 shows effect of Youngmodulus change onmodeshape Node location is extremely important as they haveto correspond to places where both the spring and thepreloading mechanisms act A change in the position of thenode corresponds to a disturbance on the ideal workingcondition of the device Figure 15 shows changing first andsecond node location in different Young modulus In thepresented transducer maximum changing for node locationsis about 8mm

20 30 40 50 60 70 80 90 1008180

8190

8200

8210

8220

8230

8240

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 11 Analytical prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

20 30 40 50 60 70 80 90 1008330

8340

8350

8360

8370

8380

8390

8400

8410

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 12 FEM prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

Table 2 Resonance frequency in different temperature (1034MPaprestress 40 kAm magnetic bias)

119879 (∘C)Youngmodulus(GPa)

FEM prediction ofresonance frequency

(Hz)

Analytical predictionof resonance

frequency (Hz)20 405 8188 833140 424 8191 833560 444 8194 834080 452 8195 8342

Shock and Vibration 7

7 8 9 10 11 12 138150

8200

8250

8300

8350

8400

Prestress (MPa)

Freq

uenc

y (H

z)

FEMExperimentalAnalytical

Figure 13 Resonance frequency in different prestress in 40 kAmmagnetic bias

For experimental verification and calculating qualityfactor impedance response of the transducer in differentmagnetic bias has been studied Figure 16 shows impedanceresponse of the transducer in different magnetic bias field

Table 3 shows experimental verification of FEM and ana-lytical prediction for resonance frequency Good agreementcan be seen with near 97 accuracy It happened becauseprecise material properties are applied in both analytical andFEMmethods In Table 3 effect of changing in magnetic biason mechanical quality factor has been also reported In rangeof application mechanical quality factor varies much withmagnetic bias It can happen from changing node locationsand also variation inTerfenol-Ddamping coefficient Becauseof changing node locations a part of the transducer energy isconsumed for vibrating high stiffness prestress mechanismChanging quality factor reduces vibration amplitude of thetransducer

5 Conclusions

In this paper Δ119864 effect and its influence on vibrationalbehavior of the Terfenol-D transducer have been studiedFirst the effect of Young modulus change on resonance fre-quency and mode shape was studied through both analyticaland numerical methods Experimental tests on Terfenol-D have allowed measuring Δ119864 effect in different operatingconditions Then a resonant transducer has been testedto validate analytical and numerical predictions Results ofboth methods for resonance frequency were verified withexperiment in different operational condition In additioneffect of changing operational condition on quality factor wasstudied experimentally

Results show that with this type of transducer Δ119864 effectleads to a change in the resonance frequency about 4 percent

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

disp

lace

men

t

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(a)

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09N

orm

aliz

ed d

ispla

cem

ent

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(b)

Figure 14 Mode shape in different Young modulus (a) FEMsimulation (b) analytical method

Even if this change can be considered as small as the band-width of the transducer is very narrow the performance ofthe device can be affected negatively if this effect is neglectedFurthermore quality factor of this transducer considerablychanges and it can reduce the efficiency of the transducer Asa matter of fact quality factor falls down from 1121 to 3392

Changes in mode shape strongly affect the performanceof the device As bias mechanism of the transducer hasbeen designed to be attached on nodes of vibration thedisplacement of a node leads to a loss of isolation betweenthem and main parts of the transducer and energy cannot betransmitted efficiently Also the transducer is fixed throughnodes and changing node locations interferes in operation

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

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2 Shock and Vibration

Notwithstanding these works there is no report aboutconsequences of Δ119864 effect on operational parameters ofTerfenol-D resonant transducers As a matter of fact reso-nance frequency mode shape and mechanical quality factorof the transducer depend on the Young modulus of themagnetostrictive material When resonance frequency ischanged due to Δ119864 effect the transducer starts working innonresonance region thus decreasing its performance Forthis reason it is important to predict the effect of thesechanges due to a change in theYoungmodulus of thematerialto adjust the functioning of the transducer accordingly

To achieve these goals a resonant Terfenol-D transducerhas been designed and fabricated Main parameters causingΔ119864 effect are measured to experimentally to find out theircorrelation with the change of the Terfenol-D Young modu-lus Vibrational behavior of the transducer is modeled analyt-ically and numerically using ANSYS12 commercial softwarePredictions results have been satisfactorily verified withexperiments Results can be helpful in using the Terfenol-D transducer in order to permanently adjust it in resonanceregion during operation Also effect of changing condition onperformance of the transducer can be predictable based onthese results

The paper is structured as follows Section 2 introducesan analytical model to take into account the effect of mainmechanical parameters of a resonant transducer and its oper-ating conditions on the resonance frequency mode shapeand the quality factor of the device Section 3 presents thetransducer used to validate the model and the experimentalset-up Section 4 collects results from both analytical andexperimental analysis A discussion of results is done andconclusions are drawn in Section 5

2 Δ119864 Effect in Resonant Transducers

Resonant Langevin transducers are used to generate sonicand ultrasonic vibration As suggested in their name thesedevices should work in resonance frequency to enhancemechanical vibration These devices are characterized by avery narrow bandwidth (especially those with high-qualityfactor) then a small change in the resonance frequency canreduce significantly the performance of the device Resonanttransducers have 3 main parts excitation part (based onpiezoelectric or magnetostrictive material) backing part(tail mass) and matching part (head mass) [20] Figure 1shows the layout of a magnetostrictive resonant transducerTerfenol-D should be suitablymagnetically andmechanicallypreloaded to increase the efficiency of the transducer [21]Backing in the transducer can be a part of magnetic circuitIn design of the transducer it is crucial to place the node ofvibration exactly where prestress and bias mechanisms act inorder to avoid disturbance in operation If this condition isnot guaranteed then the efficiency of the device falls down

21 Effects on Resonance Frequency Young modulus changeof Terfenol-D can change resonance frequency and modeshape For predicting this analytical and numerical methods

Preload mechanism Terfenol-DSpring

MatchingBacking

Figure 1 Layout of a magnetostrictive resonant transducer

Backing Terfenol-D Matching

x

L1 L2 L3 L4 L5

Figure 2 Main parts of the transducer

are exploited The vibrational behavior of the transducer canbe predicted using wave equation

1198892119906

1198891199092+1

119860

119889119860

119889119909

119889119906

119889119909+1205962

1198622119906 = 0 (1)

where 119906 is the amplitude of the vibration 119860 is the crosssection area 120596 is the circular frequency and 119862 is the soundspeed in the material Main parts of the transducer andcoordinate system are shown in Figure 2 If main parts ofthe transducer are divided into 5 parts (1) should be satisfiedalong them As each part has constant cross section area theterm 119889119860119889119909 is equal to zero Thus (1) can be reduced as

1198892119906

1198891199092+1205962

1198622sdot 119906 = 0 (2)

General solution for (2) along the 119894th parts

119906119894= 119896119894cos( 120596

119862119894

119909) + 119896119894+1

sin( 120596119862119894

119909) (3)

where 119896119894and 119896119894+1

are constants Diameter of each transducerpart is smaller than 120582

1198942 (120582119894is acoustic wavelength in each

part of the transducer) to prevent lateral mode in operation[20] Boundary conditions for solving this problem areNewton law and equivalency of displacement in interfaceareas of parts Also this transducer is free at two ends Hence

Shock and Vibration 3

forces at these points are equal to zero Equations (4) showboundary conditions for the two parts of the backing Thus

11986411198601

1205971199061

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199061(1198711) = 1199062(1198711)

11986411198601

1205971199061

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711

minus 11986421198602

1205971199062

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711

= 0

1199062(1198711+ 1198712) = 1199063(1198711+ 1198712)

11986411198602

1205971199062

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711+1198712

minus 11986431198603

1205971199063

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711+1198712

minus 119896screws1199062 (1198711 + 1198712) = 0

(4)

In (4) 119864 is the Young modulus and 119896 is stiffness Boundaryconditions for all the other parts can be easily obtainedaccordingly

Considering the whole system 10 boundary conditionscan be written and resumed into

119860 sdot 119862 = 0 (5)

where matrix119860 is the coefficient matrix (6)119862 is a rowmatrixof constant 119896

1to 11989610elements (8) and 0 is a null matrix (9)

Matrix 119860 is then

119860 =

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

[

0 11988610 0 0 0 0 0 0 0

11988711198872119887311988740 0 0 0 0 0

11988811198882119888311988840 0 0 0 0 0

0 0 11988911198892119889311988940 0 0 0

0 0 11989011198902119890311989040 0 0 0

0 0 0 0 11989111198912119891311989140 0

0 0 0 0 11989211198922119892311989240 0

0 0 0 0 0 0 ℎ1ℎ2ℎ3ℎ4

0 0 0 0 0 0 1198941119894211989431198944

0 0 0 0 0 0 0 0 11989511198952

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

]

(6)

whose element are

1198861=11986411198601120596

1198621

1198871= cos(1205961198711

1198621

)

1198872= sin(1205961198711

1198621

)

1198873= minus cos(1205961198711

1198622

) 1198874= minus sin(1205961198711

1198622

)

1198881= minus11988611198872 119888

2= 11988611198871

1198883=11986021198642120596

1198622

sin(12059611987111198622

) 1198884= minus

11986021198642120596

1198622

cos(12059611987111198622

)

1198891= cos(

120596 (1198711+ 1198712)

1198622

) 1198892= sin(

120596 (1198711+ 1198712)

1198622

)

1198893= minus cos(

120596 (1198711+ 1198712)

1198623

)

1198894= minus sin(

120596 (1198711+ 1198712)

1198623

)

1198901= minus

11986021198642120596

1198622

1198892minus 119896scr1198891 119890

2=11986021198642120596

1198622

1198891minus 119896scr1198892

1198903= minus

11986031198643120596

1198623

1198894 119890

4=11986031198643120596

1198623

1198893

1198911= cos(

120596 (1198711+ 1198712+ 1198713)

1198623

)

1198912= sin(

120596 (1198711+ 1198712+ 1198713)

1198623

)

1198913= minus cos(

120596 (1198711+ 1198712+ 1198713)

1198624

)

1198914= minus sin(

120596 (1198711+ 1198712+ 1198713)

1198624

)

1198921= minus

11986031198643120596

1198623

1198912 119892

2=11986031198643120596

1198623

1198911

1198923= minus

11986041198644120596

1198624

1198914 119892

4=11986041198644120596

1198624

1198913

ℎ1= cos(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198624

)

ℎ2= sin(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198624

)

ℎ3= minus cos(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198625

)

ℎ4= minus sin(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198625

)

1198942=11986041198644120596

1198624

ℎ1minus 119896spring ℎ2 119894

3= minus

11986051198645120596

1198625

ℎ4

1198944=11986051198645120596

1198625

ℎ3

1198951= minus

11986051198645120596

1198625

sin(120596 (1198711+ 1198712+ 1198713+ 1198714+ 1198715)

1198625

)

1198952=11986051198645120596

1198625

cos(120596 (1198711+ 1198712+ 1198713+ 1198714+ 1198715)

1198625

)

(7)

4 Shock and Vibration

Electrical branch Transducer Mechanical branch

ZeTem

Zm

I vFU

Tme

Figure 3 Model of ideal gyrator

Matrix 119862 is

119862 = [1198961 1198962 1198963 1198964 1198965 1198966 1198967 1198968 1198969 11989610]119879

(8)

As it is known necessary condition for the solution existenceof (5) is

det (119860) = 0 (9)

Roots of (9) are longitudinal resonance frequencies Thusresonance frequencies are function of each parameter ofmatrix 119860 including Young modulus of Terfenol-D (119864

3) This

relationship allows considering the effect of a change in theYoung modulus of the material on the behavior of the device

Analytical model is supported by a finite element modelof the device ANSYS12 commercial software has been usedto carry outmodal simulation to predict resonance frequencyand mode shape of the transducer

22 Effects on Quality Factor Amagnetostrictive transducercan be modeled with ideal gyrator as shown in Figure 3 [4]Equations of this gyrator are presented in (10a) and (10b) [14]

119880 = 119885e sdot 119868 + 119879em sdot V (10a)

119865 = 119879me sdot 119868 + 119885m sdot V (10b)

where 119880 is the applied voltage 119868 is the current and 119865 andV are mechanical force and speed respectively 119879em is theelectromotive force per velocity and 119879me is the mechanicalforce per current In free state condition of the transducerthere is no mechanical force By setting 119865 = 0 in (10a) andsubstituting the result in (10b) effective electrical impedanceof the transducer 119885ee can be calculated This parameter ispresented in [20]

119885ee = 119877ee + 119895 sdot 119909ee = 119885e +minus119879me sdot 119879em

119885m= 119885e + 119885mot (11)

where 119877ee and 119909ee are real and imaginary parts of thetransducer impedance 119885e is the electrical impedance and119885mot is motional impedance If 119885ee is measured in clampedstate 119885mot is vanished Thus 119885mot can be obtained bymeasuring impedance in free and clamped state condition

Real and imaginary parts of 119885mot form a circle inresonance region frequency This circle is tangent to originFor presented transducer this circle is depicted in Figure 4If the diameter between origin and the points of circle thatcorresponds to resonance frequency is drawn perpendicularline to this diameter crosses the circle at two points These

minus2 0 2 4 6 8 10 12minus8

minus6

minus4

minus2

0

2

4

6

Real part of motional impedance

Imag

e par

t of m

otio

nal i

mpe

danc

e

120596998400998400

120596

120596998400

Figure 4 Motional impedance locus

Figure 5 The transducer used for tests

points correspond to 1205961015840 and 12059610158401015840 12059610158401015840 minus 1205961015840 show bandwidthof the transducer [20] Thus quality factor can be calculatedusing (12) [22] In the following equation 120596

0is the circular

resonance frequency

119876119898=

1205960

12059610158401015840 minus 1205961015840 (12)

3 Experimental Validation

31 The Resonant Transducer To experimentally investigateΔ119864 effect resonant magnetostrictive transducers have beentested under different operating condition (Figure 5) Thetransducer has been designed for working in the secondmode (Figure 6) Resonance frequency for this transduceris 8252Hz Magnetic bias in the presented transducer is40 kAm and mechanical prestress is 1034MPa Magneticbias has been supplied by ceramic permanent magnet andprestress has been exerted by 4 screws and a bevel springMagnetic yoke was used to close magnetic flux path andminimize flux leakage Excitation coil has 1000 turns of wirewith 1mm2 cross section area In designing the transducerprestress and biasmechanisms are attached to nodes of vibra-tion in order to avoid disturbance in operation (Figure 6)Table 1 shows main dimensions of the device

32 Δ119864 Effect Measurement There are some methods forYoung modulus measurement Stress-Strain graph acoustic

Shock and Vibration 5

0 005 01 015 02 025 03 035 04 045 05

minus2

0

2

4

6

8

10

12

Mod

al am

plitu

de

Distance (m)

Figure 6 Modal shape of the transducer obtained with ANSYS12

Table 1 Transducer main dimensions

Main part Length [mm] Diameter [mm]Backing 150 50Terfenol-D 50 10

Matching 1575 231575 7

and impedance methods are the most important Stress-Strain graph has some intrinsic error such as data pickupHigh attenuation of transverse wave in Terfenol-D makessome problem for acoustic measurement Thus impedancemethod has been selected For using impedance method asearch coil is turned to Terfenol-D Resonance frequency ofTerfenol-D can be detected with measurement impedance ofthe coil In resonance band of frequency there is unusualchange in impedance response If constant voltage mode ofimpedance analyzer was used the maximum pick in thisregion would be resonance frequency and minimum pickwould be antiresonance frequency [20] Young modulus canbe calculated by

119864 = 2120587120588119891res1198972 (13)

33 Experimental Set-Up First tests for Δ119864 effect measure-ment have been done on the magnetostrictive rod Exper-imental set-up is shown in Figure 7 The Terfenol-D rodis wounded by an excitation coil to change the magneticfield and it is surrounded by a search coil The search coilis connected to the impedance analyzer To minimize fluxleakage a high permeability yoke made of pure iron has beenconsidered in the set-up

Tests have been carried out changing the magnetic biasand the mechanical prestress of Terfenol-D The change inthe first is obtained exploiting the excitation coil of the set-up Preload of themagnetostrictive rod can be easilymodifiedby means of 4 screws and can be measured using a load cellNecessary calibration with standard weights has been doneA temperature control circuit was used to keep temperatureof Terfenol-D constant during the test By using this set-up effect of the mentioned parameter in Terfenol-D Youngmodulus was measured precisely Results of measurementswould be input for FEM and analytical methods to predictmode shape and resonance frequency

a

b

c

d

e

f

g

Figure 7 The experimental set-up for measurement of Youngmodulus ((a) magnetic yoke (b) load cell (c) heater (d) powersupply (e) load indicator (f) temperature control unit and (g)impedance analyzer)

Figure 8 Experimental set-up for measuring changes in resonancefrequency

Supplementary tests are carried out to verify analyticaland numerical resonance frequency prediction The experi-mental set-up for these tests is shown in Figure 8 A searchcoil is attached to impedance analyzer excitation coil oftransducer is used as bias coil Mechanical preloading isexerted by screws and can be adjusted by a torque meterWith using this set-up effect of change in magnetic bias andprestress on resonance frequency of the transducer can beexperimentally studied Moreover according to (11) qualityfactor can be calculated

4 Results and Discussion

Figure 9 shows the relationship between the Young modulusof Terfenol-D rod and magnetic bias in different prestresslevels The temperature was kept at 20∘C Range of magneticfield in this figure is from 20 kAm to 100 kAmThese valuesfor prestress and magnetic field are common for Terfenol-D transducers According to this figure Young moduluschanged between 40 and 75GPa Figure 10 depicts depen-dency of Youngmodulus on temperature in 40 kAmbias and1034MPa prestress By changing temperature between 20∘Cand 80∘C Young modulus is changed about 5MPa

By applying Δ119864 effect to analytical and FEM modelresonance frequency and mode shape have been calculatedFigures 11 12 and 13 show effects of magnetic bias andmechanical prestress on the transducer resonance frequencyobtained fromFEMand analyticalmethod Results show thatin the transducer with the mentioned design the effect ofYoung modulus can change the resonance frequency about220Hz As this kind of device has a very narrow bandwidtha similar change in resonance frequency considerably reduces

6 Shock and Vibration

20 30 40 50 60 70 80 90 10040

45

50

55

60

65

70

75

H (kAm)

Youn

g m

odul

us (G

Pa)

689MPa1034MPa1389MPa

Figure 9 Young modulus versus magnetic bias field in differentprestress level in 20∘C

20 30 40 50 60 70 8040

41

415

42

425

43

435

44

445

45

455

Youn

g m

odul

us (G

Pa)

Temperature (∘C)

Figure 10 Young modulus versus temperature in 1034MPa pre-stress and 40 kAm magnetic bias field

the efficiency of the transducer This effect is more conspicu-ous for high-quality factor transducer

Table 2 shows effect of temperature on resonance fre-quency obtained from FEM and analytical method In bothmethods effect of temperature until 80∘C is less than 15Hz

Figure 14 shows effect of Youngmodulus change onmodeshape Node location is extremely important as they haveto correspond to places where both the spring and thepreloading mechanisms act A change in the position of thenode corresponds to a disturbance on the ideal workingcondition of the device Figure 15 shows changing first andsecond node location in different Young modulus In thepresented transducer maximum changing for node locationsis about 8mm

20 30 40 50 60 70 80 90 1008180

8190

8200

8210

8220

8230

8240

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 11 Analytical prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

20 30 40 50 60 70 80 90 1008330

8340

8350

8360

8370

8380

8390

8400

8410

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 12 FEM prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

Table 2 Resonance frequency in different temperature (1034MPaprestress 40 kAm magnetic bias)

119879 (∘C)Youngmodulus(GPa)

FEM prediction ofresonance frequency

(Hz)

Analytical predictionof resonance

frequency (Hz)20 405 8188 833140 424 8191 833560 444 8194 834080 452 8195 8342

Shock and Vibration 7

7 8 9 10 11 12 138150

8200

8250

8300

8350

8400

Prestress (MPa)

Freq

uenc

y (H

z)

FEMExperimentalAnalytical

Figure 13 Resonance frequency in different prestress in 40 kAmmagnetic bias

For experimental verification and calculating qualityfactor impedance response of the transducer in differentmagnetic bias has been studied Figure 16 shows impedanceresponse of the transducer in different magnetic bias field

Table 3 shows experimental verification of FEM and ana-lytical prediction for resonance frequency Good agreementcan be seen with near 97 accuracy It happened becauseprecise material properties are applied in both analytical andFEMmethods In Table 3 effect of changing in magnetic biason mechanical quality factor has been also reported In rangeof application mechanical quality factor varies much withmagnetic bias It can happen from changing node locationsand also variation inTerfenol-Ddamping coefficient Becauseof changing node locations a part of the transducer energy isconsumed for vibrating high stiffness prestress mechanismChanging quality factor reduces vibration amplitude of thetransducer

5 Conclusions

In this paper Δ119864 effect and its influence on vibrationalbehavior of the Terfenol-D transducer have been studiedFirst the effect of Young modulus change on resonance fre-quency and mode shape was studied through both analyticaland numerical methods Experimental tests on Terfenol-D have allowed measuring Δ119864 effect in different operatingconditions Then a resonant transducer has been testedto validate analytical and numerical predictions Results ofboth methods for resonance frequency were verified withexperiment in different operational condition In additioneffect of changing operational condition on quality factor wasstudied experimentally

Results show that with this type of transducer Δ119864 effectleads to a change in the resonance frequency about 4 percent

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

disp

lace

men

t

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(a)

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09N

orm

aliz

ed d

ispla

cem

ent

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(b)

Figure 14 Mode shape in different Young modulus (a) FEMsimulation (b) analytical method

Even if this change can be considered as small as the band-width of the transducer is very narrow the performance ofthe device can be affected negatively if this effect is neglectedFurthermore quality factor of this transducer considerablychanges and it can reduce the efficiency of the transducer Asa matter of fact quality factor falls down from 1121 to 3392

Changes in mode shape strongly affect the performanceof the device As bias mechanism of the transducer hasbeen designed to be attached on nodes of vibration thedisplacement of a node leads to a loss of isolation betweenthem and main parts of the transducer and energy cannot betransmitted efficiently Also the transducer is fixed throughnodes and changing node locations interferes in operation

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 3

forces at these points are equal to zero Equations (4) showboundary conditions for the two parts of the backing Thus

11986411198601

1205971199061

120597119909

10038161003816100381610038161003816100381610038161003816119909=0

= 0

1199061(1198711) = 1199062(1198711)

11986411198601

1205971199061

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711

minus 11986421198602

1205971199062

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711

= 0

1199062(1198711+ 1198712) = 1199063(1198711+ 1198712)

11986411198602

1205971199062

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711+1198712

minus 11986431198603

1205971199063

120597119909

10038161003816100381610038161003816100381610038161003816119909=1198711+1198712

minus 119896screws1199062 (1198711 + 1198712) = 0

(4)

In (4) 119864 is the Young modulus and 119896 is stiffness Boundaryconditions for all the other parts can be easily obtainedaccordingly

Considering the whole system 10 boundary conditionscan be written and resumed into

119860 sdot 119862 = 0 (5)

where matrix119860 is the coefficient matrix (6)119862 is a rowmatrixof constant 119896

1to 11989610elements (8) and 0 is a null matrix (9)

Matrix 119860 is then

119860 =

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

[

0 11988610 0 0 0 0 0 0 0

11988711198872119887311988740 0 0 0 0 0

11988811198882119888311988840 0 0 0 0 0

0 0 11988911198892119889311988940 0 0 0

0 0 11989011198902119890311989040 0 0 0

0 0 0 0 11989111198912119891311989140 0

0 0 0 0 11989211198922119892311989240 0

0 0 0 0 0 0 ℎ1ℎ2ℎ3ℎ4

0 0 0 0 0 0 1198941119894211989431198944

0 0 0 0 0 0 0 0 11989511198952

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

]

(6)

whose element are

1198861=11986411198601120596

1198621

1198871= cos(1205961198711

1198621

)

1198872= sin(1205961198711

1198621

)

1198873= minus cos(1205961198711

1198622

) 1198874= minus sin(1205961198711

1198622

)

1198881= minus11988611198872 119888

2= 11988611198871

1198883=11986021198642120596

1198622

sin(12059611987111198622

) 1198884= minus

11986021198642120596

1198622

cos(12059611987111198622

)

1198891= cos(

120596 (1198711+ 1198712)

1198622

) 1198892= sin(

120596 (1198711+ 1198712)

1198622

)

1198893= minus cos(

120596 (1198711+ 1198712)

1198623

)

1198894= minus sin(

120596 (1198711+ 1198712)

1198623

)

1198901= minus

11986021198642120596

1198622

1198892minus 119896scr1198891 119890

2=11986021198642120596

1198622

1198891minus 119896scr1198892

1198903= minus

11986031198643120596

1198623

1198894 119890

4=11986031198643120596

1198623

1198893

1198911= cos(

120596 (1198711+ 1198712+ 1198713)

1198623

)

1198912= sin(

120596 (1198711+ 1198712+ 1198713)

1198623

)

1198913= minus cos(

120596 (1198711+ 1198712+ 1198713)

1198624

)

1198914= minus sin(

120596 (1198711+ 1198712+ 1198713)

1198624

)

1198921= minus

11986031198643120596

1198623

1198912 119892

2=11986031198643120596

1198623

1198911

1198923= minus

11986041198644120596

1198624

1198914 119892

4=11986041198644120596

1198624

1198913

ℎ1= cos(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198624

)

ℎ2= sin(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198624

)

ℎ3= minus cos(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198625

)

ℎ4= minus sin(

120596 (1198711+ 1198712+ 1198713+ 1198714)

1198625

)

1198942=11986041198644120596

1198624

ℎ1minus 119896spring ℎ2 119894

3= minus

11986051198645120596

1198625

ℎ4

1198944=11986051198645120596

1198625

ℎ3

1198951= minus

11986051198645120596

1198625

sin(120596 (1198711+ 1198712+ 1198713+ 1198714+ 1198715)

1198625

)

1198952=11986051198645120596

1198625

cos(120596 (1198711+ 1198712+ 1198713+ 1198714+ 1198715)

1198625

)

(7)

4 Shock and Vibration

Electrical branch Transducer Mechanical branch

ZeTem

Zm

I vFU

Tme

Figure 3 Model of ideal gyrator

Matrix 119862 is

119862 = [1198961 1198962 1198963 1198964 1198965 1198966 1198967 1198968 1198969 11989610]119879

(8)

As it is known necessary condition for the solution existenceof (5) is

det (119860) = 0 (9)

Roots of (9) are longitudinal resonance frequencies Thusresonance frequencies are function of each parameter ofmatrix 119860 including Young modulus of Terfenol-D (119864

3) This

relationship allows considering the effect of a change in theYoung modulus of the material on the behavior of the device

Analytical model is supported by a finite element modelof the device ANSYS12 commercial software has been usedto carry outmodal simulation to predict resonance frequencyand mode shape of the transducer

22 Effects on Quality Factor Amagnetostrictive transducercan be modeled with ideal gyrator as shown in Figure 3 [4]Equations of this gyrator are presented in (10a) and (10b) [14]

119880 = 119885e sdot 119868 + 119879em sdot V (10a)

119865 = 119879me sdot 119868 + 119885m sdot V (10b)

where 119880 is the applied voltage 119868 is the current and 119865 andV are mechanical force and speed respectively 119879em is theelectromotive force per velocity and 119879me is the mechanicalforce per current In free state condition of the transducerthere is no mechanical force By setting 119865 = 0 in (10a) andsubstituting the result in (10b) effective electrical impedanceof the transducer 119885ee can be calculated This parameter ispresented in [20]

119885ee = 119877ee + 119895 sdot 119909ee = 119885e +minus119879me sdot 119879em

119885m= 119885e + 119885mot (11)

where 119877ee and 119909ee are real and imaginary parts of thetransducer impedance 119885e is the electrical impedance and119885mot is motional impedance If 119885ee is measured in clampedstate 119885mot is vanished Thus 119885mot can be obtained bymeasuring impedance in free and clamped state condition

Real and imaginary parts of 119885mot form a circle inresonance region frequency This circle is tangent to originFor presented transducer this circle is depicted in Figure 4If the diameter between origin and the points of circle thatcorresponds to resonance frequency is drawn perpendicularline to this diameter crosses the circle at two points These

minus2 0 2 4 6 8 10 12minus8

minus6

minus4

minus2

0

2

4

6

Real part of motional impedance

Imag

e par

t of m

otio

nal i

mpe

danc

e

120596998400998400

120596

120596998400

Figure 4 Motional impedance locus

Figure 5 The transducer used for tests

points correspond to 1205961015840 and 12059610158401015840 12059610158401015840 minus 1205961015840 show bandwidthof the transducer [20] Thus quality factor can be calculatedusing (12) [22] In the following equation 120596

0is the circular

resonance frequency

119876119898=

1205960

12059610158401015840 minus 1205961015840 (12)

3 Experimental Validation

31 The Resonant Transducer To experimentally investigateΔ119864 effect resonant magnetostrictive transducers have beentested under different operating condition (Figure 5) Thetransducer has been designed for working in the secondmode (Figure 6) Resonance frequency for this transduceris 8252Hz Magnetic bias in the presented transducer is40 kAm and mechanical prestress is 1034MPa Magneticbias has been supplied by ceramic permanent magnet andprestress has been exerted by 4 screws and a bevel springMagnetic yoke was used to close magnetic flux path andminimize flux leakage Excitation coil has 1000 turns of wirewith 1mm2 cross section area In designing the transducerprestress and biasmechanisms are attached to nodes of vibra-tion in order to avoid disturbance in operation (Figure 6)Table 1 shows main dimensions of the device

32 Δ119864 Effect Measurement There are some methods forYoung modulus measurement Stress-Strain graph acoustic

Shock and Vibration 5

0 005 01 015 02 025 03 035 04 045 05

minus2

0

2

4

6

8

10

12

Mod

al am

plitu

de

Distance (m)

Figure 6 Modal shape of the transducer obtained with ANSYS12

Table 1 Transducer main dimensions

Main part Length [mm] Diameter [mm]Backing 150 50Terfenol-D 50 10

Matching 1575 231575 7

and impedance methods are the most important Stress-Strain graph has some intrinsic error such as data pickupHigh attenuation of transverse wave in Terfenol-D makessome problem for acoustic measurement Thus impedancemethod has been selected For using impedance method asearch coil is turned to Terfenol-D Resonance frequency ofTerfenol-D can be detected with measurement impedance ofthe coil In resonance band of frequency there is unusualchange in impedance response If constant voltage mode ofimpedance analyzer was used the maximum pick in thisregion would be resonance frequency and minimum pickwould be antiresonance frequency [20] Young modulus canbe calculated by

119864 = 2120587120588119891res1198972 (13)

33 Experimental Set-Up First tests for Δ119864 effect measure-ment have been done on the magnetostrictive rod Exper-imental set-up is shown in Figure 7 The Terfenol-D rodis wounded by an excitation coil to change the magneticfield and it is surrounded by a search coil The search coilis connected to the impedance analyzer To minimize fluxleakage a high permeability yoke made of pure iron has beenconsidered in the set-up

Tests have been carried out changing the magnetic biasand the mechanical prestress of Terfenol-D The change inthe first is obtained exploiting the excitation coil of the set-up Preload of themagnetostrictive rod can be easilymodifiedby means of 4 screws and can be measured using a load cellNecessary calibration with standard weights has been doneA temperature control circuit was used to keep temperatureof Terfenol-D constant during the test By using this set-up effect of the mentioned parameter in Terfenol-D Youngmodulus was measured precisely Results of measurementswould be input for FEM and analytical methods to predictmode shape and resonance frequency

a

b

c

d

e

f

g

Figure 7 The experimental set-up for measurement of Youngmodulus ((a) magnetic yoke (b) load cell (c) heater (d) powersupply (e) load indicator (f) temperature control unit and (g)impedance analyzer)

Figure 8 Experimental set-up for measuring changes in resonancefrequency

Supplementary tests are carried out to verify analyticaland numerical resonance frequency prediction The experi-mental set-up for these tests is shown in Figure 8 A searchcoil is attached to impedance analyzer excitation coil oftransducer is used as bias coil Mechanical preloading isexerted by screws and can be adjusted by a torque meterWith using this set-up effect of change in magnetic bias andprestress on resonance frequency of the transducer can beexperimentally studied Moreover according to (11) qualityfactor can be calculated

4 Results and Discussion

Figure 9 shows the relationship between the Young modulusof Terfenol-D rod and magnetic bias in different prestresslevels The temperature was kept at 20∘C Range of magneticfield in this figure is from 20 kAm to 100 kAmThese valuesfor prestress and magnetic field are common for Terfenol-D transducers According to this figure Young moduluschanged between 40 and 75GPa Figure 10 depicts depen-dency of Youngmodulus on temperature in 40 kAmbias and1034MPa prestress By changing temperature between 20∘Cand 80∘C Young modulus is changed about 5MPa

By applying Δ119864 effect to analytical and FEM modelresonance frequency and mode shape have been calculatedFigures 11 12 and 13 show effects of magnetic bias andmechanical prestress on the transducer resonance frequencyobtained fromFEMand analyticalmethod Results show thatin the transducer with the mentioned design the effect ofYoung modulus can change the resonance frequency about220Hz As this kind of device has a very narrow bandwidtha similar change in resonance frequency considerably reduces

6 Shock and Vibration

20 30 40 50 60 70 80 90 10040

45

50

55

60

65

70

75

H (kAm)

Youn

g m

odul

us (G

Pa)

689MPa1034MPa1389MPa

Figure 9 Young modulus versus magnetic bias field in differentprestress level in 20∘C

20 30 40 50 60 70 8040

41

415

42

425

43

435

44

445

45

455

Youn

g m

odul

us (G

Pa)

Temperature (∘C)

Figure 10 Young modulus versus temperature in 1034MPa pre-stress and 40 kAm magnetic bias field

the efficiency of the transducer This effect is more conspicu-ous for high-quality factor transducer

Table 2 shows effect of temperature on resonance fre-quency obtained from FEM and analytical method In bothmethods effect of temperature until 80∘C is less than 15Hz

Figure 14 shows effect of Youngmodulus change onmodeshape Node location is extremely important as they haveto correspond to places where both the spring and thepreloading mechanisms act A change in the position of thenode corresponds to a disturbance on the ideal workingcondition of the device Figure 15 shows changing first andsecond node location in different Young modulus In thepresented transducer maximum changing for node locationsis about 8mm

20 30 40 50 60 70 80 90 1008180

8190

8200

8210

8220

8230

8240

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 11 Analytical prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

20 30 40 50 60 70 80 90 1008330

8340

8350

8360

8370

8380

8390

8400

8410

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 12 FEM prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

Table 2 Resonance frequency in different temperature (1034MPaprestress 40 kAm magnetic bias)

119879 (∘C)Youngmodulus(GPa)

FEM prediction ofresonance frequency

(Hz)

Analytical predictionof resonance

frequency (Hz)20 405 8188 833140 424 8191 833560 444 8194 834080 452 8195 8342

Shock and Vibration 7

7 8 9 10 11 12 138150

8200

8250

8300

8350

8400

Prestress (MPa)

Freq

uenc

y (H

z)

FEMExperimentalAnalytical

Figure 13 Resonance frequency in different prestress in 40 kAmmagnetic bias

For experimental verification and calculating qualityfactor impedance response of the transducer in differentmagnetic bias has been studied Figure 16 shows impedanceresponse of the transducer in different magnetic bias field

Table 3 shows experimental verification of FEM and ana-lytical prediction for resonance frequency Good agreementcan be seen with near 97 accuracy It happened becauseprecise material properties are applied in both analytical andFEMmethods In Table 3 effect of changing in magnetic biason mechanical quality factor has been also reported In rangeof application mechanical quality factor varies much withmagnetic bias It can happen from changing node locationsand also variation inTerfenol-Ddamping coefficient Becauseof changing node locations a part of the transducer energy isconsumed for vibrating high stiffness prestress mechanismChanging quality factor reduces vibration amplitude of thetransducer

5 Conclusions

In this paper Δ119864 effect and its influence on vibrationalbehavior of the Terfenol-D transducer have been studiedFirst the effect of Young modulus change on resonance fre-quency and mode shape was studied through both analyticaland numerical methods Experimental tests on Terfenol-D have allowed measuring Δ119864 effect in different operatingconditions Then a resonant transducer has been testedto validate analytical and numerical predictions Results ofboth methods for resonance frequency were verified withexperiment in different operational condition In additioneffect of changing operational condition on quality factor wasstudied experimentally

Results show that with this type of transducer Δ119864 effectleads to a change in the resonance frequency about 4 percent

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

disp

lace

men

t

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(a)

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09N

orm

aliz

ed d

ispla

cem

ent

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(b)

Figure 14 Mode shape in different Young modulus (a) FEMsimulation (b) analytical method

Even if this change can be considered as small as the band-width of the transducer is very narrow the performance ofthe device can be affected negatively if this effect is neglectedFurthermore quality factor of this transducer considerablychanges and it can reduce the efficiency of the transducer Asa matter of fact quality factor falls down from 1121 to 3392

Changes in mode shape strongly affect the performanceof the device As bias mechanism of the transducer hasbeen designed to be attached on nodes of vibration thedisplacement of a node leads to a loss of isolation betweenthem and main parts of the transducer and energy cannot betransmitted efficiently Also the transducer is fixed throughnodes and changing node locations interferes in operation

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Shock and Vibration

Electrical branch Transducer Mechanical branch

ZeTem

Zm

I vFU

Tme

Figure 3 Model of ideal gyrator

Matrix 119862 is

119862 = [1198961 1198962 1198963 1198964 1198965 1198966 1198967 1198968 1198969 11989610]119879

(8)

As it is known necessary condition for the solution existenceof (5) is

det (119860) = 0 (9)

Roots of (9) are longitudinal resonance frequencies Thusresonance frequencies are function of each parameter ofmatrix 119860 including Young modulus of Terfenol-D (119864

3) This

relationship allows considering the effect of a change in theYoung modulus of the material on the behavior of the device

Analytical model is supported by a finite element modelof the device ANSYS12 commercial software has been usedto carry outmodal simulation to predict resonance frequencyand mode shape of the transducer

22 Effects on Quality Factor Amagnetostrictive transducercan be modeled with ideal gyrator as shown in Figure 3 [4]Equations of this gyrator are presented in (10a) and (10b) [14]

119880 = 119885e sdot 119868 + 119879em sdot V (10a)

119865 = 119879me sdot 119868 + 119885m sdot V (10b)

where 119880 is the applied voltage 119868 is the current and 119865 andV are mechanical force and speed respectively 119879em is theelectromotive force per velocity and 119879me is the mechanicalforce per current In free state condition of the transducerthere is no mechanical force By setting 119865 = 0 in (10a) andsubstituting the result in (10b) effective electrical impedanceof the transducer 119885ee can be calculated This parameter ispresented in [20]

119885ee = 119877ee + 119895 sdot 119909ee = 119885e +minus119879me sdot 119879em

119885m= 119885e + 119885mot (11)

where 119877ee and 119909ee are real and imaginary parts of thetransducer impedance 119885e is the electrical impedance and119885mot is motional impedance If 119885ee is measured in clampedstate 119885mot is vanished Thus 119885mot can be obtained bymeasuring impedance in free and clamped state condition

Real and imaginary parts of 119885mot form a circle inresonance region frequency This circle is tangent to originFor presented transducer this circle is depicted in Figure 4If the diameter between origin and the points of circle thatcorresponds to resonance frequency is drawn perpendicularline to this diameter crosses the circle at two points These

minus2 0 2 4 6 8 10 12minus8

minus6

minus4

minus2

0

2

4

6

Real part of motional impedance

Imag

e par

t of m

otio

nal i

mpe

danc

e

120596998400998400

120596

120596998400

Figure 4 Motional impedance locus

Figure 5 The transducer used for tests

points correspond to 1205961015840 and 12059610158401015840 12059610158401015840 minus 1205961015840 show bandwidthof the transducer [20] Thus quality factor can be calculatedusing (12) [22] In the following equation 120596

0is the circular

resonance frequency

119876119898=

1205960

12059610158401015840 minus 1205961015840 (12)

3 Experimental Validation

31 The Resonant Transducer To experimentally investigateΔ119864 effect resonant magnetostrictive transducers have beentested under different operating condition (Figure 5) Thetransducer has been designed for working in the secondmode (Figure 6) Resonance frequency for this transduceris 8252Hz Magnetic bias in the presented transducer is40 kAm and mechanical prestress is 1034MPa Magneticbias has been supplied by ceramic permanent magnet andprestress has been exerted by 4 screws and a bevel springMagnetic yoke was used to close magnetic flux path andminimize flux leakage Excitation coil has 1000 turns of wirewith 1mm2 cross section area In designing the transducerprestress and biasmechanisms are attached to nodes of vibra-tion in order to avoid disturbance in operation (Figure 6)Table 1 shows main dimensions of the device

32 Δ119864 Effect Measurement There are some methods forYoung modulus measurement Stress-Strain graph acoustic

Shock and Vibration 5

0 005 01 015 02 025 03 035 04 045 05

minus2

0

2

4

6

8

10

12

Mod

al am

plitu

de

Distance (m)

Figure 6 Modal shape of the transducer obtained with ANSYS12

Table 1 Transducer main dimensions

Main part Length [mm] Diameter [mm]Backing 150 50Terfenol-D 50 10

Matching 1575 231575 7

and impedance methods are the most important Stress-Strain graph has some intrinsic error such as data pickupHigh attenuation of transverse wave in Terfenol-D makessome problem for acoustic measurement Thus impedancemethod has been selected For using impedance method asearch coil is turned to Terfenol-D Resonance frequency ofTerfenol-D can be detected with measurement impedance ofthe coil In resonance band of frequency there is unusualchange in impedance response If constant voltage mode ofimpedance analyzer was used the maximum pick in thisregion would be resonance frequency and minimum pickwould be antiresonance frequency [20] Young modulus canbe calculated by

119864 = 2120587120588119891res1198972 (13)

33 Experimental Set-Up First tests for Δ119864 effect measure-ment have been done on the magnetostrictive rod Exper-imental set-up is shown in Figure 7 The Terfenol-D rodis wounded by an excitation coil to change the magneticfield and it is surrounded by a search coil The search coilis connected to the impedance analyzer To minimize fluxleakage a high permeability yoke made of pure iron has beenconsidered in the set-up

Tests have been carried out changing the magnetic biasand the mechanical prestress of Terfenol-D The change inthe first is obtained exploiting the excitation coil of the set-up Preload of themagnetostrictive rod can be easilymodifiedby means of 4 screws and can be measured using a load cellNecessary calibration with standard weights has been doneA temperature control circuit was used to keep temperatureof Terfenol-D constant during the test By using this set-up effect of the mentioned parameter in Terfenol-D Youngmodulus was measured precisely Results of measurementswould be input for FEM and analytical methods to predictmode shape and resonance frequency

a

b

c

d

e

f

g

Figure 7 The experimental set-up for measurement of Youngmodulus ((a) magnetic yoke (b) load cell (c) heater (d) powersupply (e) load indicator (f) temperature control unit and (g)impedance analyzer)

Figure 8 Experimental set-up for measuring changes in resonancefrequency

Supplementary tests are carried out to verify analyticaland numerical resonance frequency prediction The experi-mental set-up for these tests is shown in Figure 8 A searchcoil is attached to impedance analyzer excitation coil oftransducer is used as bias coil Mechanical preloading isexerted by screws and can be adjusted by a torque meterWith using this set-up effect of change in magnetic bias andprestress on resonance frequency of the transducer can beexperimentally studied Moreover according to (11) qualityfactor can be calculated

4 Results and Discussion

Figure 9 shows the relationship between the Young modulusof Terfenol-D rod and magnetic bias in different prestresslevels The temperature was kept at 20∘C Range of magneticfield in this figure is from 20 kAm to 100 kAmThese valuesfor prestress and magnetic field are common for Terfenol-D transducers According to this figure Young moduluschanged between 40 and 75GPa Figure 10 depicts depen-dency of Youngmodulus on temperature in 40 kAmbias and1034MPa prestress By changing temperature between 20∘Cand 80∘C Young modulus is changed about 5MPa

By applying Δ119864 effect to analytical and FEM modelresonance frequency and mode shape have been calculatedFigures 11 12 and 13 show effects of magnetic bias andmechanical prestress on the transducer resonance frequencyobtained fromFEMand analyticalmethod Results show thatin the transducer with the mentioned design the effect ofYoung modulus can change the resonance frequency about220Hz As this kind of device has a very narrow bandwidtha similar change in resonance frequency considerably reduces

6 Shock and Vibration

20 30 40 50 60 70 80 90 10040

45

50

55

60

65

70

75

H (kAm)

Youn

g m

odul

us (G

Pa)

689MPa1034MPa1389MPa

Figure 9 Young modulus versus magnetic bias field in differentprestress level in 20∘C

20 30 40 50 60 70 8040

41

415

42

425

43

435

44

445

45

455

Youn

g m

odul

us (G

Pa)

Temperature (∘C)

Figure 10 Young modulus versus temperature in 1034MPa pre-stress and 40 kAm magnetic bias field

the efficiency of the transducer This effect is more conspicu-ous for high-quality factor transducer

Table 2 shows effect of temperature on resonance fre-quency obtained from FEM and analytical method In bothmethods effect of temperature until 80∘C is less than 15Hz

Figure 14 shows effect of Youngmodulus change onmodeshape Node location is extremely important as they haveto correspond to places where both the spring and thepreloading mechanisms act A change in the position of thenode corresponds to a disturbance on the ideal workingcondition of the device Figure 15 shows changing first andsecond node location in different Young modulus In thepresented transducer maximum changing for node locationsis about 8mm

20 30 40 50 60 70 80 90 1008180

8190

8200

8210

8220

8230

8240

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 11 Analytical prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

20 30 40 50 60 70 80 90 1008330

8340

8350

8360

8370

8380

8390

8400

8410

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 12 FEM prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

Table 2 Resonance frequency in different temperature (1034MPaprestress 40 kAm magnetic bias)

119879 (∘C)Youngmodulus(GPa)

FEM prediction ofresonance frequency

(Hz)

Analytical predictionof resonance

frequency (Hz)20 405 8188 833140 424 8191 833560 444 8194 834080 452 8195 8342

Shock and Vibration 7

7 8 9 10 11 12 138150

8200

8250

8300

8350

8400

Prestress (MPa)

Freq

uenc

y (H

z)

FEMExperimentalAnalytical

Figure 13 Resonance frequency in different prestress in 40 kAmmagnetic bias

For experimental verification and calculating qualityfactor impedance response of the transducer in differentmagnetic bias has been studied Figure 16 shows impedanceresponse of the transducer in different magnetic bias field

Table 3 shows experimental verification of FEM and ana-lytical prediction for resonance frequency Good agreementcan be seen with near 97 accuracy It happened becauseprecise material properties are applied in both analytical andFEMmethods In Table 3 effect of changing in magnetic biason mechanical quality factor has been also reported In rangeof application mechanical quality factor varies much withmagnetic bias It can happen from changing node locationsand also variation inTerfenol-Ddamping coefficient Becauseof changing node locations a part of the transducer energy isconsumed for vibrating high stiffness prestress mechanismChanging quality factor reduces vibration amplitude of thetransducer

5 Conclusions

In this paper Δ119864 effect and its influence on vibrationalbehavior of the Terfenol-D transducer have been studiedFirst the effect of Young modulus change on resonance fre-quency and mode shape was studied through both analyticaland numerical methods Experimental tests on Terfenol-D have allowed measuring Δ119864 effect in different operatingconditions Then a resonant transducer has been testedto validate analytical and numerical predictions Results ofboth methods for resonance frequency were verified withexperiment in different operational condition In additioneffect of changing operational condition on quality factor wasstudied experimentally

Results show that with this type of transducer Δ119864 effectleads to a change in the resonance frequency about 4 percent

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

disp

lace

men

t

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(a)

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09N

orm

aliz

ed d

ispla

cem

ent

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(b)

Figure 14 Mode shape in different Young modulus (a) FEMsimulation (b) analytical method

Even if this change can be considered as small as the band-width of the transducer is very narrow the performance ofthe device can be affected negatively if this effect is neglectedFurthermore quality factor of this transducer considerablychanges and it can reduce the efficiency of the transducer Asa matter of fact quality factor falls down from 1121 to 3392

Changes in mode shape strongly affect the performanceof the device As bias mechanism of the transducer hasbeen designed to be attached on nodes of vibration thedisplacement of a node leads to a loss of isolation betweenthem and main parts of the transducer and energy cannot betransmitted efficiently Also the transducer is fixed throughnodes and changing node locations interferes in operation

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 5

0 005 01 015 02 025 03 035 04 045 05

minus2

0

2

4

6

8

10

12

Mod

al am

plitu

de

Distance (m)

Figure 6 Modal shape of the transducer obtained with ANSYS12

Table 1 Transducer main dimensions

Main part Length [mm] Diameter [mm]Backing 150 50Terfenol-D 50 10

Matching 1575 231575 7

and impedance methods are the most important Stress-Strain graph has some intrinsic error such as data pickupHigh attenuation of transverse wave in Terfenol-D makessome problem for acoustic measurement Thus impedancemethod has been selected For using impedance method asearch coil is turned to Terfenol-D Resonance frequency ofTerfenol-D can be detected with measurement impedance ofthe coil In resonance band of frequency there is unusualchange in impedance response If constant voltage mode ofimpedance analyzer was used the maximum pick in thisregion would be resonance frequency and minimum pickwould be antiresonance frequency [20] Young modulus canbe calculated by

119864 = 2120587120588119891res1198972 (13)

33 Experimental Set-Up First tests for Δ119864 effect measure-ment have been done on the magnetostrictive rod Exper-imental set-up is shown in Figure 7 The Terfenol-D rodis wounded by an excitation coil to change the magneticfield and it is surrounded by a search coil The search coilis connected to the impedance analyzer To minimize fluxleakage a high permeability yoke made of pure iron has beenconsidered in the set-up

Tests have been carried out changing the magnetic biasand the mechanical prestress of Terfenol-D The change inthe first is obtained exploiting the excitation coil of the set-up Preload of themagnetostrictive rod can be easilymodifiedby means of 4 screws and can be measured using a load cellNecessary calibration with standard weights has been doneA temperature control circuit was used to keep temperatureof Terfenol-D constant during the test By using this set-up effect of the mentioned parameter in Terfenol-D Youngmodulus was measured precisely Results of measurementswould be input for FEM and analytical methods to predictmode shape and resonance frequency

a

b

c

d

e

f

g

Figure 7 The experimental set-up for measurement of Youngmodulus ((a) magnetic yoke (b) load cell (c) heater (d) powersupply (e) load indicator (f) temperature control unit and (g)impedance analyzer)

Figure 8 Experimental set-up for measuring changes in resonancefrequency

Supplementary tests are carried out to verify analyticaland numerical resonance frequency prediction The experi-mental set-up for these tests is shown in Figure 8 A searchcoil is attached to impedance analyzer excitation coil oftransducer is used as bias coil Mechanical preloading isexerted by screws and can be adjusted by a torque meterWith using this set-up effect of change in magnetic bias andprestress on resonance frequency of the transducer can beexperimentally studied Moreover according to (11) qualityfactor can be calculated

4 Results and Discussion

Figure 9 shows the relationship between the Young modulusof Terfenol-D rod and magnetic bias in different prestresslevels The temperature was kept at 20∘C Range of magneticfield in this figure is from 20 kAm to 100 kAmThese valuesfor prestress and magnetic field are common for Terfenol-D transducers According to this figure Young moduluschanged between 40 and 75GPa Figure 10 depicts depen-dency of Youngmodulus on temperature in 40 kAmbias and1034MPa prestress By changing temperature between 20∘Cand 80∘C Young modulus is changed about 5MPa

By applying Δ119864 effect to analytical and FEM modelresonance frequency and mode shape have been calculatedFigures 11 12 and 13 show effects of magnetic bias andmechanical prestress on the transducer resonance frequencyobtained fromFEMand analyticalmethod Results show thatin the transducer with the mentioned design the effect ofYoung modulus can change the resonance frequency about220Hz As this kind of device has a very narrow bandwidtha similar change in resonance frequency considerably reduces

6 Shock and Vibration

20 30 40 50 60 70 80 90 10040

45

50

55

60

65

70

75

H (kAm)

Youn

g m

odul

us (G

Pa)

689MPa1034MPa1389MPa

Figure 9 Young modulus versus magnetic bias field in differentprestress level in 20∘C

20 30 40 50 60 70 8040

41

415

42

425

43

435

44

445

45

455

Youn

g m

odul

us (G

Pa)

Temperature (∘C)

Figure 10 Young modulus versus temperature in 1034MPa pre-stress and 40 kAm magnetic bias field

the efficiency of the transducer This effect is more conspicu-ous for high-quality factor transducer

Table 2 shows effect of temperature on resonance fre-quency obtained from FEM and analytical method In bothmethods effect of temperature until 80∘C is less than 15Hz

Figure 14 shows effect of Youngmodulus change onmodeshape Node location is extremely important as they haveto correspond to places where both the spring and thepreloading mechanisms act A change in the position of thenode corresponds to a disturbance on the ideal workingcondition of the device Figure 15 shows changing first andsecond node location in different Young modulus In thepresented transducer maximum changing for node locationsis about 8mm

20 30 40 50 60 70 80 90 1008180

8190

8200

8210

8220

8230

8240

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 11 Analytical prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

20 30 40 50 60 70 80 90 1008330

8340

8350

8360

8370

8380

8390

8400

8410

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 12 FEM prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

Table 2 Resonance frequency in different temperature (1034MPaprestress 40 kAm magnetic bias)

119879 (∘C)Youngmodulus(GPa)

FEM prediction ofresonance frequency

(Hz)

Analytical predictionof resonance

frequency (Hz)20 405 8188 833140 424 8191 833560 444 8194 834080 452 8195 8342

Shock and Vibration 7

7 8 9 10 11 12 138150

8200

8250

8300

8350

8400

Prestress (MPa)

Freq

uenc

y (H

z)

FEMExperimentalAnalytical

Figure 13 Resonance frequency in different prestress in 40 kAmmagnetic bias

For experimental verification and calculating qualityfactor impedance response of the transducer in differentmagnetic bias has been studied Figure 16 shows impedanceresponse of the transducer in different magnetic bias field

Table 3 shows experimental verification of FEM and ana-lytical prediction for resonance frequency Good agreementcan be seen with near 97 accuracy It happened becauseprecise material properties are applied in both analytical andFEMmethods In Table 3 effect of changing in magnetic biason mechanical quality factor has been also reported In rangeof application mechanical quality factor varies much withmagnetic bias It can happen from changing node locationsand also variation inTerfenol-Ddamping coefficient Becauseof changing node locations a part of the transducer energy isconsumed for vibrating high stiffness prestress mechanismChanging quality factor reduces vibration amplitude of thetransducer

5 Conclusions

In this paper Δ119864 effect and its influence on vibrationalbehavior of the Terfenol-D transducer have been studiedFirst the effect of Young modulus change on resonance fre-quency and mode shape was studied through both analyticaland numerical methods Experimental tests on Terfenol-D have allowed measuring Δ119864 effect in different operatingconditions Then a resonant transducer has been testedto validate analytical and numerical predictions Results ofboth methods for resonance frequency were verified withexperiment in different operational condition In additioneffect of changing operational condition on quality factor wasstudied experimentally

Results show that with this type of transducer Δ119864 effectleads to a change in the resonance frequency about 4 percent

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

disp

lace

men

t

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(a)

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09N

orm

aliz

ed d

ispla

cem

ent

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(b)

Figure 14 Mode shape in different Young modulus (a) FEMsimulation (b) analytical method

Even if this change can be considered as small as the band-width of the transducer is very narrow the performance ofthe device can be affected negatively if this effect is neglectedFurthermore quality factor of this transducer considerablychanges and it can reduce the efficiency of the transducer Asa matter of fact quality factor falls down from 1121 to 3392

Changes in mode shape strongly affect the performanceof the device As bias mechanism of the transducer hasbeen designed to be attached on nodes of vibration thedisplacement of a node leads to a loss of isolation betweenthem and main parts of the transducer and energy cannot betransmitted efficiently Also the transducer is fixed throughnodes and changing node locations interferes in operation

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Shock and Vibration

20 30 40 50 60 70 80 90 10040

45

50

55

60

65

70

75

H (kAm)

Youn

g m

odul

us (G

Pa)

689MPa1034MPa1389MPa

Figure 9 Young modulus versus magnetic bias field in differentprestress level in 20∘C

20 30 40 50 60 70 8040

41

415

42

425

43

435

44

445

45

455

Youn

g m

odul

us (G

Pa)

Temperature (∘C)

Figure 10 Young modulus versus temperature in 1034MPa pre-stress and 40 kAm magnetic bias field

the efficiency of the transducer This effect is more conspicu-ous for high-quality factor transducer

Table 2 shows effect of temperature on resonance fre-quency obtained from FEM and analytical method In bothmethods effect of temperature until 80∘C is less than 15Hz

Figure 14 shows effect of Youngmodulus change onmodeshape Node location is extremely important as they haveto correspond to places where both the spring and thepreloading mechanisms act A change in the position of thenode corresponds to a disturbance on the ideal workingcondition of the device Figure 15 shows changing first andsecond node location in different Young modulus In thepresented transducer maximum changing for node locationsis about 8mm

20 30 40 50 60 70 80 90 1008180

8190

8200

8210

8220

8230

8240

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 11 Analytical prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

20 30 40 50 60 70 80 90 1008330

8340

8350

8360

8370

8380

8390

8400

8410

H (kAm)

Freq

uenc

y (H

z)

689MPa1034MPa1389MPa

Figure 12 FEM prediction of resonance frequency in differentmagnetic bias in 1389MPa prestress

Table 2 Resonance frequency in different temperature (1034MPaprestress 40 kAm magnetic bias)

119879 (∘C)Youngmodulus(GPa)

FEM prediction ofresonance frequency

(Hz)

Analytical predictionof resonance

frequency (Hz)20 405 8188 833140 424 8191 833560 444 8194 834080 452 8195 8342

Shock and Vibration 7

7 8 9 10 11 12 138150

8200

8250

8300

8350

8400

Prestress (MPa)

Freq

uenc

y (H

z)

FEMExperimentalAnalytical

Figure 13 Resonance frequency in different prestress in 40 kAmmagnetic bias

For experimental verification and calculating qualityfactor impedance response of the transducer in differentmagnetic bias has been studied Figure 16 shows impedanceresponse of the transducer in different magnetic bias field

Table 3 shows experimental verification of FEM and ana-lytical prediction for resonance frequency Good agreementcan be seen with near 97 accuracy It happened becauseprecise material properties are applied in both analytical andFEMmethods In Table 3 effect of changing in magnetic biason mechanical quality factor has been also reported In rangeof application mechanical quality factor varies much withmagnetic bias It can happen from changing node locationsand also variation inTerfenol-Ddamping coefficient Becauseof changing node locations a part of the transducer energy isconsumed for vibrating high stiffness prestress mechanismChanging quality factor reduces vibration amplitude of thetransducer

5 Conclusions

In this paper Δ119864 effect and its influence on vibrationalbehavior of the Terfenol-D transducer have been studiedFirst the effect of Young modulus change on resonance fre-quency and mode shape was studied through both analyticaland numerical methods Experimental tests on Terfenol-D have allowed measuring Δ119864 effect in different operatingconditions Then a resonant transducer has been testedto validate analytical and numerical predictions Results ofboth methods for resonance frequency were verified withexperiment in different operational condition In additioneffect of changing operational condition on quality factor wasstudied experimentally

Results show that with this type of transducer Δ119864 effectleads to a change in the resonance frequency about 4 percent

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

disp

lace

men

t

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(a)

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09N

orm

aliz

ed d

ispla

cem

ent

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(b)

Figure 14 Mode shape in different Young modulus (a) FEMsimulation (b) analytical method

Even if this change can be considered as small as the band-width of the transducer is very narrow the performance ofthe device can be affected negatively if this effect is neglectedFurthermore quality factor of this transducer considerablychanges and it can reduce the efficiency of the transducer Asa matter of fact quality factor falls down from 1121 to 3392

Changes in mode shape strongly affect the performanceof the device As bias mechanism of the transducer hasbeen designed to be attached on nodes of vibration thedisplacement of a node leads to a loss of isolation betweenthem and main parts of the transducer and energy cannot betransmitted efficiently Also the transducer is fixed throughnodes and changing node locations interferes in operation

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

7 8 9 10 11 12 138150

8200

8250

8300

8350

8400

Prestress (MPa)

Freq

uenc

y (H

z)

FEMExperimentalAnalytical

Figure 13 Resonance frequency in different prestress in 40 kAmmagnetic bias

For experimental verification and calculating qualityfactor impedance response of the transducer in differentmagnetic bias has been studied Figure 16 shows impedanceresponse of the transducer in different magnetic bias field

Table 3 shows experimental verification of FEM and ana-lytical prediction for resonance frequency Good agreementcan be seen with near 97 accuracy It happened becauseprecise material properties are applied in both analytical andFEMmethods In Table 3 effect of changing in magnetic biason mechanical quality factor has been also reported In rangeof application mechanical quality factor varies much withmagnetic bias It can happen from changing node locationsand also variation inTerfenol-Ddamping coefficient Becauseof changing node locations a part of the transducer energy isconsumed for vibrating high stiffness prestress mechanismChanging quality factor reduces vibration amplitude of thetransducer

5 Conclusions

In this paper Δ119864 effect and its influence on vibrationalbehavior of the Terfenol-D transducer have been studiedFirst the effect of Young modulus change on resonance fre-quency and mode shape was studied through both analyticaland numerical methods Experimental tests on Terfenol-D have allowed measuring Δ119864 effect in different operatingconditions Then a resonant transducer has been testedto validate analytical and numerical predictions Results ofboth methods for resonance frequency were verified withexperiment in different operational condition In additioneffect of changing operational condition on quality factor wasstudied experimentally

Results show that with this type of transducer Δ119864 effectleads to a change in the resonance frequency about 4 percent

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

disp

lace

men

t

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(a)

0 005 01 015 02 025 03 035 04 045 05

0

01

02

03

04

05

06

07

08

09N

orm

aliz

ed d

ispla

cem

ent

minus01

x coordinate (m)

E = 40GPaE = 55GPaE = 75GPa

(b)

Figure 14 Mode shape in different Young modulus (a) FEMsimulation (b) analytical method

Even if this change can be considered as small as the band-width of the transducer is very narrow the performance ofthe device can be affected negatively if this effect is neglectedFurthermore quality factor of this transducer considerablychanges and it can reduce the efficiency of the transducer Asa matter of fact quality factor falls down from 1121 to 3392

Changes in mode shape strongly affect the performanceof the device As bias mechanism of the transducer hasbeen designed to be attached on nodes of vibration thedisplacement of a node leads to a loss of isolation betweenthem and main parts of the transducer and energy cannot betransmitted efficiently Also the transducer is fixed throughnodes and changing node locations interferes in operation

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Shock and Vibration

013 0134 0138 0142 0146 015

Nor

mal

ized

disp

lace

men

t

minus4

minus35

minus3

minus25

minus2

minus15

minus1

minus05

0

05

times10minus3

x coordinate (m)

(a)

034 0344 0348 0352 0356 036

0

001

002

003

Nor

mal

ized

disp

lace

men

t

minus004

minus003

minus002

minus001

x coordinate (m)

(b)

Figure 15 FEM prediction for changing node location in different Young modulus (a) first node (b) second node

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 90003

32

34

36

38

4

42

44

Frequency (Hz)

Impe

danc

e am

plitu

de (O

hm)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(a)

8000 8100 8200 8300 8400 8500 8600 8700 8800 8900 900030

31

32

33

34

35

36

37

Frequency (Hz)

Phas

e (de

g)

40kAm45kAm55kAm65kAm

75kAm85kAm95kAm

(b)

Figure 16 (a) Amplitude and (b) phase of the transducer in different magnetic bias in 1389MPa

Table 3 Experimental verification in 1389MPa prestress

119867

(kAm)Young modulus

(GPa)FEM prediction of

resonance frequency (Hz)Analytical prediction ofresonance frequency (Hz)

Experimental resonancefrequency (Hz) 119876

40 413 8189 8342 8330 5745 405 8188 8331 8325 1127155 517 8206 8356 8350 828065 606 8220 8376 8387 421475 676 8230 8391 8405 400285 707 8235 8398 8410 373995 737 8240 8403 8414 3392

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 9

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A E Clark Ferromagnetic Materials vol 1 North HollandAmsterdam The Netherlands 1980

[2] A Grunwald and A G Olabi ldquoDesign of a magnetostrictive(MS) actuatorrdquo Sensors and Actuators A Physical vol 144 no1 pp 161ndash175 2008

[3] J L Pons Emerging Actuator Technologies A MicromechanicalApproach 2005

[4] G Engdahl EdHandbook of Giant Magnetostrictive MaterialsRoyal Institute of Technology Stockholm Sweden 2000

[5] F Braghin S Cinquemani and F Resta ldquoA low frequencymagnetostrictive inertial actuator for vibration controlrdquo Sensorsand Actuators A Physical vol 180 pp 67ndash74 2012

[6] C May K Kuhnen P Pagliaruolo and H Janocha ldquoMagne-tostrictive dynamic vibration absorber (DVA) for passive andactive dampingrdquo in Proceedings of the Euronoise Naples Italy2003

[7] T Zhang C Jiang H Zhang and H Xu ldquoGiant magnetostric-tive actuators for active vibration controlrdquo Smart Materials andStructures vol 13 no 3 pp 473ndash477 2004

[8] F Braghin F Castelli-Dezza S Cinqueman and F RestaldquoA full-range hybrid device for sound reproductionrdquo SmartStructures and Systems vol 11 no 6 pp 605ndash621 2013

[9] S Deras and C Yuk ldquoUnderwater sonar transducer usingmagnetostrictive materialrdquo Journal of Magnetics vol 4 no 3pp 98ndash101 1994

[10] K R Dhilsha G Markandeyulu B V P SubrahmanyeswaraRao and K V S Rama Rao ldquoDesign and fabrication of a lowfrequency giant magnetostrictive transducerrdquo Journal of Alloysand Compounds vol 258 no 1-2 pp 53ndash55 1997

[11] A IMarkovUltrasonicMachining of IntractableMaterials IliffeBooks London UK 1966

[12] A G Olabi and A Grunwald ldquoDesign and application ofmagnetostrictive materialsrdquo Materials amp Design vol 29 no 2pp 469ndash483 2008

[13] H T Savage A E Clark and D Pearson ldquoThe stress depen-dence of theΔE effect in TerfenolminusDrdquo Journal of Applied Physicsvol 64 no 10 p 5426 1988

[14] Y Liang and X Zheng ldquoExperimental researches on magneto-thermo-mechanical characterization of Terfenol-Drdquo ActaMechanica Solida Sinica vol 20 no 4 pp 283ndash288 2007

[15] R Kellogg and A Flatau ldquoExperimental investigation ofterfenol-Drsquos elastic modulusrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 5 pp 583ndash595 2008

[16] R Kellogg and A Flatau ldquoWide band tunable mechanicalresonator employing the 998779E effect of Terfenol-Drdquo Journal ofIntelligentMaterial Systems and Structures vol 15 no 5 pp 355ndash368 2004

[17] Y Yong and L Lin ldquoDynamic model considering the ΔE effectfor giant magnetostrictive actuatorsrdquo in Proceedings of the IEEEInternational Conference on Control and Automation pp 667ndash672 Christchurch New Zealand December 2009

[18] T Ueno J Qiu and J Tani ldquoMagnetic circuit designmethod formagnetic force control systems using inverse magnetosrtictive

effect examination of energy conversation efficiency dependingon Δ119864 effectrdquo Electrical Engineering in Japan vol 140 no 1 pp8ndash15 2002

[19] M J Dapino R C Smith andA B Flatau ldquoModel for the delta-E effect in magnetostrictive transducersrdquo in Smart Structuresand Materials 2000 Smart Structures and Integrated Systemsvol 3985 of Proceedings of SPIE June 2000

[20] A Abdullah M Shahini and A Pak ldquoAn approach to designa high power piezoelectric ultrasonic transducerrdquo Journal ofElectroceramics vol 22 no 4 pp 369ndash382 2009

[21] R Frederick Ultrasonic Engineering Wiley New York NYUSA 1965

[22] G Mojtaba U Toshiyuki and M Mehdi ldquoQuality factor staticand dynamic responses of miniature galfenol actuator at widerange of temperaturerdquo International Journal of Physical Sciencesvol 6 no 36 pp 8143ndash8150 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of