Low-frequency magnetostrictive inertial actuator

F. Braghin, S. Cinquemani, F. Resta

Mechanical Engineering Department, Politecnico di Milano,Campus Bovisa Sud, via La Masa 34, 20156, Milano, Italy

ABSTRACT

The present paper presents the design of an innovativelow-frequency magnetostriuctive inertial actuator that isable to multiply by almost a factor of 10 the amplitudeof striction-elongation amplitude of the magnetostrictivebar, thus leading to an increase in the generated force am-plitude, and to obtain a working frequency from 30Hz up,i.e. well below the working frequencies of traditional mag-netostrictive inertial actuators. Moreover, the design hasbeen optimized through an analitical model and a finiteelement model taking into account all design parameters.The optimised low-frequency magnetostriuctive inertial ac-tuator has then be produced and its frequency responsecompared to that of a traditional magnetostrictive actuatormade up of the same components (except for the support-ing structure).

1. INTRODUCTION

Magnetostrictive inertial actuators are able to effectivelygenerate an actuation force only above their first eigen-frequency that is determined by the inertial mass and thestiffness along the actuation direction ( [1]). In traditionalmagnetostrictive actuators the inertial mass is directly con-nected to the actuator foot through the bar of magnetostric-tive material (plus a rigid spring that is required for ap-plying the necessary mechanical pre-load to the magne-tostrictive bar, [2]). Thus, the stiffness along the actua-tion direction is strictly related to Young’s modulus of themagnetostrictive material. To obtain low-frequency actu-ators one could therefore either increase the inertial mass(e.g. by adding more weight on top of the actuator) or de-crease the cross-section of the magnetostrictive bar (thusreducing its axial stiffness). There is however a limit tothe increase in inertial mass and/or decrease in the bar’scross-section imposed by Euler’s critical load (buckling ofthe magnetostrictive bar) and by the frailty of the actua-tor (magnetostrictive materials usually have a fragile be-haviour, [3]). This leads to traditional magnetostrictiveactuators that have their first eigenfrequency well above100Hz ( [4]).

To obtain a low-frequency magnetostrictive inertial ac-tuator (with the first eigenfrequency below 100Hz) it isnecessary to re-design the actuator in order to decouple thestiffness along the actuation direction from Young’s mod-

ulus of the magnetostrictive material. In the present pa-per an innovative design of the magnetostrictive actuatoris proposed that uses the same components of traditionalmagnetostrictive actuators but with a different layout: themagnetostrictive bar is mounted orthogonally with respectto the direction of actuation and is connected to the inertialmass through a special deformable structure having lowstiffness along the direction of actuation. This innovativeconfiguration allows to improve two performances:

- the up and down motion of the inertial mass is notequal to the amplitude of striction-elongation (about1% of the length of the magnetostrictive bar) but canbe multiplied by almost a factor of 10 thus leadingto an increase in the force amplitude that the inertialactuator is able to generate (at equal inertial mass);

- the eigenfrequency of the actuator is determined bythe deformability of the supporting structure and notto the axial stiffness of the magnetostrictive bar thusallowing to reduce the working frequency from about30Hz on (i.e. significantly extending the bandwidthof the inertial actuator towards low frequencies).

Moreover, the proposed configuration allows to greatlyreduce the overall height of the actuator thus allowing tomount it even in narrow cavities.

The design of the low-frequency magnetostrictive in-ertial actuator has been optimized both through analyt-ical models and through finite element models in orderto achieve the best compromise between force amplitude,bandwidth, stresses in the supporting structure, height andcomplexity of the assembly/final cost of the device. Thecomparison between the developed models (simple ana-lytical and finite element ones) shows that the analyticalmodel is able to correctly estimate the motion amplifi-cation of the innovative device while the finite elementmodel is required for correctly assessing to increase inbandwidth and for determining the stresses the supportingstructure will be subjected to. Optimization parametersare all the design parameters of the supporting structure,i.e. its lengths, widths, radii, points of connection to theinertial mass and to the magnetostrictive bar, etc. Also thematerial of the supporting structure has been consideredin this optimization phase although, at the end, the mostbounding requirement was that of the cost of the materialof the supporting structure.

Once optimized, the low-frequency magnetostrictiveinertial actuator has been produced (the supporting struc-ture has been obtained through water-jet technology, andits performances compared to those of a traditional mag-netostrictive inertial actuator made up of the same compo-nents (except for the supporting structure) and having thesame inertial mass ( [5]). This comparison has shown tocorrectness of the design approach and that, through theaddition of just the deformable structure, performance in-dexes are greatly improved.

2. DESIGN

As already observed, magnetostrictive inertial actuatorsare not able to generate a significant force below theirfirst (axial) eigenfrequency ( [1]) and this eigenfrequencyis well abobe 100Hz due to the fact that their stiffness isassociated to the magnetostrictive bar:

ω0 =

√ksm

=

√A

sHLm(1)

m being the inertial mass fixed on top of the magnetostric-tive bar and ks being its axial stiffness equal to A/sHL(where A = 9mm2 is the cross-section area of the mag-netostrictive bar, sH = 3.3 ∗ 10−11 m2/N its mechanicalcompliance when a constant magnetic field is applied andL its length). Figure1 shows the eigenfrequency of themagnetostrictive inertial actuator evaluated through equa-tion 1 as a function of the suspended mass and the lengthof the magnetostrictive bar. It can be seen that, to obtain amagnetostrictive inertial actuator working at low frequen-cies, a very slender bar with high suspended mass is nec-essary. However, such configuration is impossible to ob-tain practically due to the reaching of Euler’s critical load(buckling of the magnetostrictive bar).

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163.1615174.5907 massa inerziale [kg]

5 10 15 20 25 305

10

15

20

25

30

Figure 1: Eigenfrequency of a magnetostrictive inertialactuator as a function of the suspended mass m and barlength L (A=9mm2).

In order to develop a low-frequency magnetostrictiveinertial actuator, it is therefore necessary to design a mech-anism that is able to decouple the stiffness of the actuatorfrom the stiffness of the axial magnetostrictive bar and, ifpossible, to amplify the deformations of the bar that areusually about 1% of the bar length itself. The power bal-ance of suche decoupled magnetostrictive inertial actuatorsounds:

my · y + ksx · x = 0 (2)

where x is the elongation of the terfenol-D bar and y is thedisplacemnt of the inertial mass. The transmission ratio isequal to the ration between the output speed, i.e. y, andthe the input speed, i.e. x:

τ =y

x(3)

Substituting the definition of τ in equation 2 we get:

mx · xτ2 + kx · x = 0 (4)

and thus the eigenfrequency of the system becomes:

ω =

√k

mτ−2 =

ω0

τ(5)

It is therefore clear that, by conveniently designing thesupport structure of the magnetostrictive bar, the eigenfre-quency of the actuator can be reduced by a factor equal tothe transmission ratio τ .

The proposed mechanism exploits a double motion am-plification as schematically shown in figure 2: both thelever mechanism and elastic deformation of the arc/ cross-bow cooperate to amplify the deformation of the magne-tostrictive bar.

Fmag Fmag

x

y

O1 O2

A BA′ B′

Figure 2: Kinematic scheme of the motion amplificationmechanism

3. OPTIMIZATION

In order to design a low-frequency magnetostrictive iner-tial actuator that is able to maximise the transmission ra-tio τ as well as the force transmitted tio the inertial masskeeping the stresses of the supporting structure as low aspossible, thus guaranteeing an infinite fatigue life, severaldifferent configurations were taken into account. Figure 3shows the main configurations considered.

Figure 3: Main configurations considered for the low-frequency magnetostrictive inertial actuator

For each considered configuration, the lever effect, thegenerated force, the eigenfrequency and the maximum stres-ses in the supporting structure have been deteremined. Froma preliminary analysis, it has been found that the mostpromising, as well as the easiest one to produce, config-uration is the last one shown in figure 3. The parameteroptimization procedure has therefore been applied to thislast configuration.

At first, a simple kinematic model has been adoptedfor the optimization procedure to limit the parameter spacetaking into account just the lever effect. Then, a detailed fi-nite element model has been adopted in order to maximisenot only the lever effect but also the generated force aswell as to minimize the eigenfrequency and the maximumstresses in the supporting structure.

Analytical Model

In figure 4 the undeformed kinematic model adopted isshown while in figure ?? the deformed kinematic modeladopted is siplayed.

The boundary conditions of the analytical model arethe following:

m

H

L

l

R

α

Figure 4: Undeformedconfiguration of the an-alytical kinematic modeladopted

m

R’

β

α'

x

n

e

y

Figure 5: Deformed con-figuration of the ana-lytical kinematic modeladopted

• the maximum dimensions of the support structure(H e m) are fixed,

• the point of application of the displacemnet imposedby the magnetostrictive bar (l) is fixed,

• the amplitude of half the elonfation of the magne-tostrictive bar (x) is imposed,

• the length of the arc of the crossbow remains con-stant (Rα = R′α′)

The equations that govern the kinematics of the analyt-ical model are the following:

α = a sin(mR

)L = H −R (1− cosα)

β = a sin(xl

)n = L sin (β)

Rα = R′α′

m+ n = R′ sinα′

e = R′ (1− cosα′)

y = H − L cos (β)− e

(6)

The reference configuration of the supporting structureis characterized by a height H = 15mm, by a width 2m =30mm and by a point of application of the displacemnetimposed by the magnetostrictive bar l = 3mm from thebasis.

The model results are shown in figure 7. As can beclearly seen, the crossbow lever effect (equal to the trans-mission ratio τ ) significantly varies as a function of thecrossbow curvature radius and reaches a maximum valueof 7.4 for a curvature radius equal to 39.2mm.

Also the effect of other parameters has on the levereffect has been investigated. Figure ?? shows the influ-ence of the width m of the crossbow: at increasing width,higher lever effects are reached but for greater crossbowcurvature radii (with 2m = 40mm the lever effect be-comes equal to 8.3 for a curvature radius equal to 60.9mm).

20 40 60 80 100 120 140 160 180 2000.5

1

1.5

2

2.5

3

Raggio di curvatura [mm]

Spostamento verticale [mm]

Allungamento imposto del magnetostrittivo pari a 0.2

Figure 6: Vertical displacement y as a function of thecrossbow curvature radius

Reducing the distance between the hinge and the pointof application of the imposed displacement l (figure 9) asignificant increase in the lever effect for slightly greatercurvature radii is obtained (with l = 2mm the lever ef-fect reaches a value of 9.5 for a curvature radius equal to52.2mm). Figure 8, instead, shows how the increase inthe height H of the crossbow determines a significant in-crease in the lever effect (with only a slight increase in thecrossbow curvature radii): for H = 20mm the lever ratiobecomes 9.2 while the curvature radius becomes equal to53.0mm (for a distance between the hinge and the point ofapplication of the imposed displacement l = 4mm).

Finite Element Model

The optimization of the geometry of the spproting struc-ture as thus been carried out using a detailed finite ele-ment model that allows to predict not only the lever ef-fect but also the reaction forces, the eignefrequency of thestructure as well as the stresses inside the structure. Thedeveloped finite element model of the structure has the fol-lowing characteristics:

• approx. 7000 linear 8-node brick elements (charac-teristcis dimensions of the mesh equal to 0.5mm),

• approx. 25000 degrees of freedom,

• lower surface bounded to the ground,

• inertial mass equal to 0.3kg,

• imposed deformation fo the magnetostrictive bar and

• linear elastic material for the supproting structure(ρ = 7800kg/m3, E = 210000MPa, ν = 0.3).

Figures 10, 11 and 12 show, respectively, the verticaldisplacement y of the inertial mass (in mm) for an elenga-

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0.5

1

1.5

2

2.5

3

3.5

Raggio di curvatura [mm]

Spostamento verticale [mm]

Allungamento imposto del magnetostrittivo pari a 0.2

m ↑

Figure 7: Vertical dis-placement y as a func-tion of the crossbow cur-vature radius. Effect ofthe crossbow width m(H = 15mm, m =20mm, l = 3mm)

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1

1.5

2

2.5

3

3.5

4

Raggio di curvatura [mm]

Spostamento verticale [mm]

Allungamento imposto del magnetostrittivo pari a 0.2

H ↑

Figure 8: Vertical dis-placement y as a func-tion of the crossbow cur-vature radius. Effect ofthe crossbow height H(H = 20mm, m =20mm, l = 4mm)

20 40 60 80 100 120 140 160 180 2000.5

1

1.5

2

2.5

3

3.5

4

Raggio di curvatura [mm]

Spostamento verticale [mm]

Allungamento imposto del magnetostrittivo pari a 0.2

l ↑

Figure 9: Vertical displacement y as a function of thecrossbow curvature radius. Effect of the distance be-tween the hinge and the point of application of the im-posed displacement l (H = 15mm, m = 20mm,l = 2mm)

tion of the magnetostrictive bar x equal to 0.4mm, the iner-tia force generated by the magnetostrictive actuator (in N)for the same elongation of the magnestostrictive bar con-sidered before and the first (vertical) eigenfrequency of theactuator (in Hz) as a function of the crossbow curvatureradius and width. Comparing the vertical displacementy obtained from the analytical model and the one deter-mined through the finite element model, a good agreementcan be found. Moreover, for a curvature radius of 50mmand a crossbow width of 0.3mm the maximum lever effectis reached. The force generated by the magnetostrictiveactuator, instead, does not significantly vary in the param-eter range considered (it varies from 370N a 420N) andincreases at increasing crossbow width and curvature ra-dius. Finally, the eigenfrequency decreases with the cross-bow width and increases with the curvature radius. For theconfiguration that maximizes the lever ratio,the eigenfre-quency is lower than 60Hz. It has therefore been decidedto adopt the configuration that maximizes the lever ratio.

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0.35

0.4

0.45

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0.65

0.7

Curvatura [mm]

Spessore [mm]

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Figure 10: Vertical displacement y of the inertial mass (inmm) for an elengation of the magnetostrictive bar x equalto 0.4mm as a function of the curvature radius and corss-bow width

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0.55

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0.65

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Curvatura [mm]

Spessore [mm]

370

380

390

400

410

420

Figure 11: Inertia force generated by the magnetostrictiveactuator (in N) for an elengation of the magnetostrictivebar x equal to 0.4mm as a function of the curvature radiusand corssbow width

4. PRODUCTION

Figure 13 shows the section of the 3D model of the low-frequency magnetostrictive inertial actuator prototype. Thecentral part is the magnetostrictive bar and at its esxtrem-ities two Neodymius-Iron-Boron magnet discs are places.The supporting structure also supports the windings thatgenerate the magnetic field inside the magnetostrictive ma-terial. Table 1 reports the most significant quantities of thedesigned low-frequency magnetostrictive inertial actuator.

Figure 14 shows a photograph of the produced proto-type.

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0.45

0.5

0.55

0.6

0.65

0.7

Curvatura [mm]

Spessore [mm]

60

80

100

120

140

160

180

200

220

Figure 12: First (vertical) eigenfrequency of the actuator(in Hz) as a function of the curvature radius and corssbowwidth

Table 1: Most significant quantities of the designed low-frequency magnetostrictive inertial actuator

Quantity Value

Inertial mass 220 gMaterial of supporting structure & mass steelMaterial of magnetostrictive bar Terfenol-DLength of magnetostrictive bar 20 mmStiffness of magnetostrictive bar 1.4·107 N/mNumber of windings 400Total height 25 mmExternal diameter 54 mm

Figure 13: Section of the 3D model of the low-frequencymagnetostrictive inertial actuator

5. EXPERIMENTAL RESULTS

The test bench adopted for assessing the low-frequencymagnetostrictive inertial actuator prototype characteristics

Figure 14: Low-frequency magnetostrictive inertial actua-tor prototype

is shown in figure 15.

Figure 15: Set up sperimentale

The inertial mass is free to vibrate and the actuator isdriven through a time varying supply current I . Both ran-dom excitation and sweep sine excitation in the frequencyrange 1-2000 Hz were adopted. During the test both sup-ply current and voltage are acquired as well as the baseand the inertial mass accelerations (through piezolectricaccelerometers) and the force transmitted to the ground(through a piezolectric load cell). Figure 16 shows themeasured transfer function between supply current I andforce transmitted to the ground FT . It can be seen thatthe first (vertical) eignefrequency is equal to ω = 167Hzwell below the eigenfrequency of the magnetostrictive in-ertial actuator without the designed supporting structure(ω0 = 1260Hz). Moreover, the measured transmissionratio is equal to 7.5 quite close to the expected one.

Figure 16: Experimental transfer function G =FT

I

6. CONCLUSIONS

In the present paper an innovative supporting structure forobtaining a low-frequency magnetostrictive inertial actu-ator has been designed and optimised trhough a simplekinematic analytical model and through a complex finiteelement model. Thus, a prototype has been produced andtested showing a good agreement with the simulated re-sults.

REFERENCES

[1] Braghin F., Cinquemani S., Resta F., “A model ofmagnetostrictive actuators for active vibration control” Sensors and Actuators A: Physical 165(2), 342-350,(2011)

[2] Engdahl G., “Handbook of Giant MagnetostrictiveMaterials”, Academic press, (2000).

[3] Clark A.E., Savage H.T., Spano M.L., “Effect ofStress on Magnetostriction and Magnetization ofSingle Crystal of Tb27Dy73Fe2”, IEEE Transactionson magnetics, 20(5), 1443-1445, (1984).

[4] Dhilsha K.R., Markandeyulu G., SubrahmanyeswaraRao B.V.P., Rama Rao K.V.S., “Design and fabricationof a low frequency giant magnetostrictive transducer”Journal of Alloys and Compounds, 258, 53-55, (1997).

[5] Pons J.L., “Emerging Actuator Technologies: AMicromechatronic Approach”, John Wiley and SonsLtd, (2005).