A Linear Model of Magnetostrictive Actuators for Active Vibration Control

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1 A Linear Model of Magnetotrictive Actuator for Active Vibration Control Franceco BRAGIN, Simone CINQUEMANI, Ferruccio RESTA Mechanical Engineering Department, Politecnico di Milano, Campu Bovia Sud, via La Maa 1, 0156, Milano, Italy ABSTRACT If there i one actuator technology that i almot excluively linked to a ingle application, that i the magnetotrictive actuator, the application i active tructural vibration control (AVC). Almot all the application decribed in the literature on magnetotrictive actuator are related in one way or another to vibration uppreion mechanim. Magnetotrictive actuator (MA) deliver high-output force and relatively high diplacement (compared to other emerging actuator technologie) and can be driven at high frequencie. Thee characteritic make them uitable for a variety of vibration control application. The ue of thi technology, however, require an accurate knowledge of the dynamic of uch actuator. The paper introduce a linear model of magnetotrictive actuator hold in a range of frequencie below kz ueful in real time application a AVC. The hypotei upporting the linearity of the ytem are dicued and the theoretical model i preented. Finally the model i validated by teting two different model of magnetotrictive actuator and comparing experimental reult with the theoretical one. 1. MAGNETOSTRICTIVE ACTUATORS The magnetotrictive effect The term magnetotriction i a ynonym for magnetically induced deformation and refer to the capacity, that mot of ferromagnetic material have, to change their ize when the level of magnetization of the material itelf change. The magnetization variation i a reult of a re-orientation of magnetic domain within the material. Thi change of direction can be obtained by ubjecting the material to change in magnetic field, tre or temperature. Although mot of ferromagnetic material having magnetotrictive propertie, only thoe that contain element known a rare earth are able to develop thee propertie ignificantly. Such material are called Giant Magnetotrictive Material. Thi effect i made poible by a high level of magnetomechanical coupling due to reorientation of the magnetic domain within the material itelf. When a magnetic field i applied to a magnetotrictive material, the magnetic domain rotate in the direction of the magnetic field producing a deformation in the material tructure and a deformation of the material itelf. The o-called poitive magnetotriction produce an elongation and a pinch of the material, converely, the negative magneto-triction i characterized by a hortening of the material and a correponding increae in ection (Fig.1). The change in the tate of magnetization can be reverible or irreverible. Reverible change are energy conervative and are obervable with mall variation in the magnetic field. Figure 1: Poitive (a) and negative (b) magnetotriction Under thee condition, when the magnetic field return to it initial value, the material return to the original tate of magnetization. Converely, when applied magnetic field are too high, change in the tate of magnetization are irreverible [1-]. Functioning principle In practice the functioning of a M.A. can be decribed with reference to the cheme in Fig.: a bar of magnetotrictive material i placed inide a coil and it i ubjected to a magnetic field generated by permanent magnet poitioned outide the coil itelf. Feeding the olenoid, a variation of the electric field that pae through the magnetotrictive material produce a change in the oppoite magnetic field with the ubequent alignment of magnetic domain. Thi phenomenon lead to a hift actuator and to the generation of a very high force. Actuator Spring Coil Figure : Layout of a M.A.. TE MODEL Permanent Magnet Terfenol-D bar Inertial ma The ue of thi technology, however, require an accurate knowledge of the dynamic of uch actuator.

2 The paper introduce a linear model of magnetotrictive actuator hold in a range of frequencie below kz ueful in real time application a (AVC). Linear contitutive equation The behaviour of magnetotrictive material i complex but, under appropriate condition, it can be approached a linear [10-14]. Main hypothei conit in: low working frequencie, reverible procee of magnetotriction (no power loe), tre and train uniform in all the ection of the magnetotrictive rod.. Under thee condition the coupling between the mechanical train and the magnetization of the material i repreented by the linear magnetomechanical coupling equation [10]: S = T + d (1) T B = dt + µ () with train S, tre T, mechanical compliance at contant applied magnetic-field trength, linear piezomagnetic crocoupling coefficient d and d*, magnetic permeability at a contant tre µ T, and magnetic-flux denity B within the material. B,, and µ T all have a tatic bia and a timevarying term e jω t (for example, T(t) = T 0 +Te jω t ), which are not to be diplayed here. If the magnetotrictive proce i aumed to be reverible, then d*=d. Thi would be normally true for low-level driving force or field, but for high-level driving they may be different. Main feature on magnetotrictive material Terfenol-D are [10]: mechanical compliance = m /N linear piezomagnetic crocoupling coefficient d = m/a magnetic permeability µ T /µ 0 = 1 The ytem Referring to the model reported in Fig.3, two dinamical equation can be written, repectively related to: mechanic: m & x + rx& = F mag (3) electric: dφ V = R 0 I + n (4) dt The force exerted by the magnetotrictive actuator : F mag = TA i a function of T a in eq. (1): S + d T = (5) The applied magnetic-field trength i: n n = I = I (6) δ L l + x Figure 3 Sytem model where n i the number of the winding turn, l i the length of the magnetotrictive bar, L i the length of the bar when = 0 and T=T 0, I i the current flowing in the winding and δ i a coefficient to take into account the effective length of field line along which perform the circulation (δ ). Then, ubtituting eq.(6) in eq.(5), one get: S nd I x / l nd I x / l nd I T = + = + + δ ( l + x) δ ( l + x) δl (7) The force exerted by the magnetotrictive actuator i: nda I Fmag = TA kx + (8) δl where: A k = (9) L i the mechanical tiffne of the magnetotrictive bar. A reported in Fig.3 force tranmitted to the tructure, and then the force available to control and uppre vibration i: = Fmag rx& (10) F T = mx & (11) Tranfer function Let uppoe to control the device upplying a known current I. It allow to diregard the electromagnetical dynamic and taking into account jut the mechanical behaviour of the ytem. Comparing eq. (3), (11) the tranfer function between force exerted by the device and the force tranmitted to the tructure (ground) i: mx && G1 = = (1) Fmag mx && + rx& and through Laplace' tranformation: G1 ( ) = = (13) Fmag r + m

3 Let derive with repect to time eq.(1) with the condition l l+x. Subtituting the reult in eq.(3) one get: dnl dnl m LT&& + m && I r LT& + r I& = TA (14) δl δl And then the tranfer function: Fmag + ( r / m) G ( ) = = C (15) I + ( r / m) + ( k / m) where: nda C = (16) δl Combining eq. (14), (16) the tranfer function G 3 () between the current I upplied to the actuator and the force F T tranmitted to the vibrating tructure i: G3( ) = = G1 ( ) G( ) = C (17) I + r / m + k / m ( ) ( ) 3. EXPERIMENTAL TESTS Magnetotrictive actuator teted To validate the model, two different magnetotrictive actuator are teted (Fig.4) whoe mot ignificant mechanical feature are: M.A. E75W A = 9e -6 [m] Area of a bar ection L = 64 [mm] Bar lenght n = 1550 no. of winding turn m = [kg] inertial ma A 6 k = = 4.6e [ N / m] mechanical tiffne (eq.(9)) L M.A. E30W A = 9e -6 [m] Area of a bar ection L = 43 [mm] Bar lenght n = 1550 no. of winding turn m = 0.74 [kg] inertial ma A 6 k = = 6.34e [ N / m] mechanical tiffne (eq.(9)) L Figure 5 Theoretical tranfer function of M.A. teted (magnitude and phae) Work bench The work-bench ued for dynamical tet i hown in Fig.6 and Fig.7. The MA are placed on a load cell, fixed to the ground, in order to have their inertial ma free and they are fed with a known current I that varie over time. During the tet, ignal of voltage (V) and upply current (I) are acquired, a the acceleration of the bae (a b ) and the one of the inertial ma (a m ) (via uniaxial piezoelectric accelerometer). The force tranmitted to the ground (F T ) i acquired through a load cell. a m F T a b V I Power upply Figure 6 Meaure ytem layout Figure 4 Magnetotrictive actuator E75W and E30W Theoretical tranfer function G 3 () decribed in eq.(17) are depicted in Fig.5 for both the magnetotrictive actuator teted. Figure 7 Tet : load cell (a) and unidirectional accelerometer (b)

4 Experimental reult Experimental tranfer function G 3 () between upplied current and tranmitted force are obtained for both the actuator teted. Device are excited firt with an harmonic input current (from 10z to 000z), econdly with a weep ine excitation in the ame range of frequencie. tet are repeated for different current gain. Fig.8 and Fig.9 repectively how the experimental tranfer function obtained for the E.30W magnetotrictive actuator, while Fig.10 and Fig.11 are related to the model E.75W. All the graph how amplitude and phae of the tranfer function and the coherence between the input and the output ignal. Let remember coeherence i defined a: S AB ( f ) γ AB = 0 γ AB 1 S AA ( f ) S BB ( f ) where S AA (f)$ and S BB (f) are the auto-pectrum repectively of the input and output ignal, while S AB (f) i their cropectrum. The coherence function i a calar and it aume a unit value only in cae of perfect correlation between the two ignal. In other word, it identifie the preence or abence of a caue-and-effect relationhip between the input ignal and output, at different frequencie. For thi reaon the experimental tranfer function can be acceptable only in the range of frequencie where coherence i near the unit value and ha to be neglected elewhere. Figure 9 - E.30W MA. Experimental tranfer function G 3 ()=F T /I obtained with weep-ine excitation for different value of current. Figure 10 E.75W MA. Experimental tranfer function G 3 ()=F T /I obtained with harmonic excitation for different value of current. Figure 8 E.30W MA. Experimental tranfer function G 3 ()=F T /I obtained with harmonic excitation for different value of current. Figure 11 - E.75W MA. Experimental tranfer function G 3 ()=F T /I obtained with weep-ine excitation for different value of current.

5 The experimental tranfer function are well defined for frequency higher than 40 z, where the force exerted by the actuator i o mall that the tranducer can't appreciate it. Thi effect i evident looking the coherence between the input and output acquired ignal reported in graph. Tet with harmonic excitation allow to reach a better reult, epecially in low frequencie operating field. 4. TEORETICAL AND EXPERIMENTAL COMPARISON The comparion between the theoretical tranfer function and the experimental one are hown in figure For each actuator teted, both the function uniquely identify the reonant frequency of the ytem and properly decribe it dynamical behaviour in the range of frequency of interet. A mall hift-effect in experimental tranfer function i aociated to current gain. Thi effect can't be evaluated in the model but it i really negligible. Figure 14 E.75W Magnetotrictive actuator: comparion between experimental and theoretical tranfer function G 3 =F T /I (Amplitude) Figure 1 E.30W Magnetotrictive actuator: comparion between experimental and theoretical tranfer function G 3 =F T /I (Amplitude) Figure 15 E.75W Magnetotrictive actuator: comparion between experimental and theoretical tranfer function G 3 =F T /I (Phae) 5. CONCLUSIONS Magnetotrictive actuator are a promiing technology in active vibration control field. The paper invetigate the opportunity of modelize uch device with a linear model to correctly decribe their mechanical behaviour. A tranfer function between the upplied current and the force uefull to controll a vibrating ytem ha been introduced. The model ha been validated teting two different actuator in a range of frequencie between z. The comparion between theoretical tranfer function and experimental one confirm the goodne of the propoed model. REFERENCES Figure 13 E.30W Magnetotrictive actuator: comparion between experimental and theoretical tranfer function G 3 =F T /I (Phae) [1] G. Engdahl, andbook of Giant Magnetotrictive Material, Academic pre, 000 [] J.L. Pon, Emerging Actuator Technologie: A Micromechatronic Approach, John Wiley & Son Ltd, 005

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