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Analysis of the giant magnetostrictive actuator with strong bias magnetic field
Journal of Magnetism and Magnetic Materials 394 (2015) 416–421
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Analysis of the giant magnetostrictive actuator with strong bias magnetic field Guangming Xue n, Zhongbo He, Dongwei Li, Zhaoshu Yang, Zhenglong Zhao Vehicles and Electrical Engineering Department, Ordnance Engineering College, Heping West 97, Shijiazhuang 050003, China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 February 2014 Received in revised form 22 June 2015 Accepted 28 June 2015 Available online 8 July 2015
Giant magnetostrictive actuator with strong bias magnetic field is designed to control the injector bullet valve opening and closing. The relationship between actuator displacement amplitude and input signal direction is analyzed. And based on the approximate linearity of strain-magnetic field, second-order system model of the actuator displacement is established. Experimental system suitable for the actuator is designed. The experimental results show that, the square voltage amplitude being 12 V, the actuator displacement amplitude is about 17 μm with backward direction signal input while being 1.5 μm under forward direction signal. From the results, the suitable input direction is confirmed to be backward. With exciting frequncy lower than 200 Hz, the error between the model and experimental result is less than 1.7 μm. So the model is validated under the low-frequency signal input. The testing displacement-voltage curves are approximately straight lines. But due to the biased position, the line slope and the displacement-voltage linearity change as the input voltage changes. & 2015 Elsevier B.V. All rights reserved.
Keywords: Giant magnetostrictive actuator Strong bias magnetic field Displacement Square signals Sinusoidal signals
1. Introduction Giant magnetostrictive material (GMM) is a “smart” material. With high piezomagnetic coefficient, the material can generate relatively high output-force and displacement (compared with other smart materials). Hence the material has been widely used in many fields [1]. The giant magnetostrictive actuator (GMA), as a most widely application of the material, always finds its applications in a variety of ranges such as active isolation, precision machining, signal detection etc. [2–4]. A typical giant magnetostrictive actuator generally contains three parts including magnetic field generating part, preloading mechanism and cooling device. For the magnetic field generating part, an Alternating Current (AC) coil is always applied to exert driven magnetic field, while a permanent magnet (or a DC coil sometimes) is employed to bias the magnetostrictive material magnetically. These three parts could be changed in various forms for certain requirements. For examples, the cooling device could be removed to reduce the actuator’s volume, the permanent magnet is removed to utilize the double-frequency characteristics, the preloading spring selected as dishing etc. These actuators show different behaviors from traditional actuators [5,6]. The giant magnetostrictive actuator introduced in this paper, characterized by asymmetric bias magnetic field, is designed for n
Corresponding author. E-mail address: [email protected] (G. Xue).
http://dx.doi.org/10.1016/j.jmmm.2015.06.083 0304-8853/& 2015 Elsevier B.V. All rights reserved.
the bullet valve (normally closed) of an electrical injector. The output end of the actuator, touched with giant magnetostrictive material, is pushed against a fixed seat on the bullet valve. If the material were driven lengthening with supplied currents input, the output end would be moved and strikes the valve seat. For avoidance of this condition, the idea that giant magnetostrictive material shortens with exciting signals input is taken. The property could be reached from the actuator designed in this paper. The material, biased sharply by a strong permanent magnet, is guaranteed in a maximum length without exciting signal. So the bullet valve is closed in this situation. With exciting signal in certain direction, the material shortens and then the valve would open. The giant magnetostrictive actuator with strong bias magnetic field (strongly biased actuator) could find its displacement model at low frequency.
2. Operating principle of the strongly biased actuator The schematic diagram of the strongly biased giant magnetostrictive actuator is shown in Fig. 1. The magnetic field within the rod is generated not only by the magnetic coil but also the permanent magnet. The rod is mechanically compressed by the help of the cover and the disc spring. Compression of the rod is executed for the limited tensile strength of the material, and could obtain higher coupling factors. For a typical giant magnetostrictive actuator, the effect of the bias magnetic field could be explained in Fig. 2. The material with
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Fig. 4. Modified sinusoidal signal applied to the actuator.
Fig. 1. Schematic diagram of the actuator.
Fig. 2. Influence of bias magnetic field on GMM strain.
a bias magnetic field, shown as the dashed lines, could obtain a higher strain compared with the field without a bias. And the double-frequency phenomenon (the frequency of the rod strain becoming double to the frequency of the input signal), if not required, could be avoided [7]. Moreover, it can also be concluded from Fig. 2 that the behavior of the material differs with the bias magnetic field intensity changing. The property of the strongly biased actuator could be shown in Fig. 3. As the bias position is considerably high, Giant magnetostrictive rod is in a relatively long state initially (without input). With a “forward signal” input, the strain of the material changes little, and the corresponding displacement of the actuator is very small. While with the “backward current”, the magnetic field generated by the exciting coil within giant magnetostrictive rod, is in the opposite direction with the bias magnetic field. Hence the rod could shorten largely and the actuator delivers a considerable displacement. So a test of the direction of the input current is required for proper operation of the actuator.
It should be noted that, if the input current were in “backward” direction, the linearity between the rod strain and the exciting magnetic field is still approximately suitable. (The current and biased field value should be chosen properly, not too high.) For a considerable displacement, a common format is not suitable. That is because an ordinary sinusoidal signal is alternating and contains two directions values, one of which would be definitely cut off (not a double-frequency phenomenon). To avoid this situation, the input “sinusoidal signal” is a superposition of an ordinary sinusoidal current and a DC with equal amplitude to the sinusoidal one. The modified sinusoidal signal is shown in Fig. 4. From the analysis above, two targets could be achieved from these designs. An initial long state of the magnetostrictive rod is realized from a strong bias magnetic field, and a relatively high displacement without the signal cut-off is guaranteed by the modulated input format.
3. Displacement model of the actuator The displacement model of the actuator is established under these assumptions [8,9]: (1) The reluctance of the giant magnetostrictive rod occupies most of the total reluctances of the closed magnetic circuit. Hence a large proportion of the magnetomotive force generated by the magnetic coil, distributes on the rod. (2) The magnetic field H, axial strain ε and stress s distribute uniformly throughout the GMM rod, and the giant magnetostrictive rod could be seen as mass-spring-damper system. (3) The relationship between magnetic field and strain within the rod is approximately linear. Under assumption (2), the strongly biased actuator could be modeled as a linear time-invariant system just as shown in Fig. 5. The equation of motion of the actuator is presented as:
(m2 + m M ) x¨ + (c2 + c M ) ẋ + (k2 + k M ) x = F M
(1)
where mM, kM, cM are respectively the equivalent mass of giant magnetostrictive rod (one-third of the inertial mass of the rod),
Fig. 3. Property of the strongly biased actuator.
Fig. 5. Equivalent model of the actuator.
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G. Xue et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 416–421
equivalent mechanical stiffness, viscous damping, m2, k2, c2 are the inertial mass of the push rod, stiffness of the disc spring, the load damping (referring to the damping caused by the disc spring if without any load), x is the displacement of the inertial mass, FM is the magnetostrictive force generated by the rod. For giant magnetostrictive rod, the equivalent axial stiffness kM could be calculated by:
kM =
AE H (2)
l H
where E is the mechanical elastic modulus of the giant magnetostrictive material, A is the cross-section area of the rod and l is the length of the rod. The strain of giant magnetostrictive rod includes elastic and magnetostrictive strain. F is the force corresponding to the magnetostrictive strain and its relationship with λ is:
(3)
F = E H λA
For the relationship between strain λ and magnetic field H, nonlinear models like Jiles-Aiherton magnetization (combined with quadratic domain-transfer model) are always employed. These physics-based models can explain magnetization hysteresis of the material physically. However, they are limited by their strong nonlinearities, and require substantial knowledge which may be not easily obtained sometimes. For the strongly biased actuator, due to the linearity between the magnetostrictive strain and the magnetic field, a complex model is not necessary and concise compared to the linear model. So the expression of the strain λ and magnetic field H generated by the exciting coil can be offered by the linear equation:
λ = d33 H
(4)
where d33 is the magnetomechanical coupling coefficient. The magnetic circuit is analyzed as follow. The actuator has a closed magnetic circuit which includes the giant magnetostrictive rod, push rod, air gap, cover, crust and base. One branch of the loops is shown in Fig. 6. The flow of the magnetic field lines could be in the opposite direction with the one shown in this figure. In the magnetic circuit, the length of the air gap is so little that the magnetic field leakage could be neglected. As the magnetic flux flows in each part is equal, the magnetomotive force exerted by the coil is distributed to each magnetic segment according to the proportions of their magnetic reluctances. Magnetic reluctance is directly proportional to the length of magnetic part while inversely to the cross sectional area and permeability. For the actuator in this paper, the magnetic reluctances of the covers, push rod and crust are considerably small for their large permeability, while the reluctance of the air gap is small for its short length. So the magnetic reluctance of the magnetostrictive rod
occupies the whole reluctance of the loop and so does the magnetomotive force, which is described in assumption (1). Under assumption (1), the whole magnetomotive force is approximately equal to the magnetomotive force within the rod:
Hl ≈ NI
(5)
where N is the number of coil winding turns. For compensating magnetomotive force of other parts, a proportional coefficient C could be introduced to Eq. (5), and it becomes:
H=C
NI l
(6)
Impose m ¼m2 þmM, c ¼c2 þcM, k ¼k2 þkM, and substitut Eqs. (3), (4), (6) into Eq. (1), one gets:
mx¨ + cẋ + kx = Cd33 NIE HA/l
(7)
The output displacement under the mixed current shown in Fig. 4 is analyzed as follow. For a linear time-invariant model established in this section, the response under the mixed input current is the superposition of separate response caused by the corresponding sinusoidal current and DC respectively. Hence the steady-state displacement for the actuator, according to vibration theory, is composed by a sinusoidal and direct form. For the sinusoidal part, when the frequency of the input signal is far less than the resonance frequency of the actuator, the amplification of the output to the input is about 1. That is to say, the displacement amplitude is equal to the one caused by a constant force which could be calculated from Eq. (7) approximately as Cd33NIEHA/(lk). For the direct current part, the displacement is stimulated through a constant force, and its value is also equal to Cd33NIEHA/(lk). Compose both displacements and one gets: the steady-state response stimulated by the mixed input has the same format with the input.
4. Testing setup The testing system applied to the strongly biased actuator is shown in Fig. 7. Digital current signals, generated by the computer, could be transformed into analog forms by the help of the digital oscilloscope (PS2000). Then the analog signals would be transmitted to the coil after being amplified by the power amplifier (GF800). The actuator is driven and generates displacement signals which could be measured by the eddy current sensor (HN808). And the measured displacements could be transmitted back to the computer after a transformation to digital signals. Visual Basic 6.0 software is employed on the computer to generate or record the digital signals. The photo of strongly biased actuator is shown in Fig. 8 and parameters of the actuator are tabulated in Table 1.
5. Results and discussions 5.1. Responses of square wave signals
Fig. 6. Magnetic circuit of the actuator.
From the analysis in Section 2, the current signals in the same value while opposite directions would lead to different displacements of the giant magnetostrictive actuator. By this way, the “backward” direction of the input current could be recognized. Square wave signals with the same amplitude are generated to drive the actuator, and the alteration of the current directions is obtained by exchanging the connections between the magnetic coil and power amplifier. The square wave current at the
G. Xue et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 416–421
419
Fig. 7. Experimental system.
Fig. 9. Square wave responses of the actuator in different directions.
which could guarantee a relatively high output of the actuator. 5.2. Responses of sinusoidal wave signals The mixed signals shown in Fig. 4 are generated by PS2000 and amplified by the power amplifier. Adjust the signal voltage peak to peak values (up-p) separately to be 6, 12, 18, 24 V. The tests under each value would be repeated for different frequencies at 50, 200, 400, 800 Hz, a total of 16 trials. The testing results are shown in Fig. 10. The results show that the steady-state displacements of the actuator are approximately in the same formats, sinusoidal signals plus DCs with same amplitudes, as the input signals. That’s a characteristic of the strongly biased actuator, and then the conclusion on the output format in Section 3 is validated. 5.3. Analysis of the results Fig. 8. Object of the strongly biased actuator. (a) Without the crust, (b) With the crust. Table 1 Main parameters of the actuator. Length (l) [mm] Cross-section area of the rod (A) [mm2] Number of the winding turns (N) Elastic modulus (EH) [N/m2] Total damping (c) [N s/m] magnetomotive force coefficient (C) Total mass (m) [g] Stiffness of the disc spring (k2) [N/m] magnetomechanical coupling coefficient (d33) [m/A]
35 16 860 3 1010 3800 0.9 17 2.3 105 1.16 10-8
amplitude 2 A and frequency 50 Hz is generated to flow within the coil. Testing displacements are shown in Fig. 9. From the point of different directions of the current signals, the results are labeled respectively as “a” and “b”. Compare different square wave responses of the input currents in different directions, the amplitude in “a” direction (about 1.5 μm) is far less than the one in “b” direction (17 μm). Therefore, the current in “b” direction is in “backward” direction labeled in Fig. 3,
The displacement maximums are calculated as follows. For the model, the mechanical resonance frequency of the actuator could be calculated from Eq. (7) as (k/m)^(1/2) E28 kHz. As the current frequencies are no more than 800 Hz, far less than resonance frequency, the expression Cd33NIEHA/(lk) calculated in Section 3 is appropriate. Plus the DC response, the expression of maximum value of the calculated displacement is 2Cd33NIEHA/(lk). For the testing results, high-frequency noises, as components of the displacements, could be filtered through SPTOOL toolbox in MATLAB software. The low-limit frequencies of the designed lowpass filters are 2 times of the input frequencies. Then extracting the maximum values of five displacement cycles, the mean of them is chosen as the displacement value for this testing. The experimental and calculated results of the displacement peak to peak values (xp-p) are shown in Fig. 11. The experimental results are shown as the solid lines in Fig. 11. According to the tested results, xp-p changes little when the input signal frequency is no more than 200 Hz. When up-p values are respectively 6, 12, 18, 24 V, xp-p are 7.0, 17.7, 26.5, 35.2 μm. As the input signal frequencies become higher, xp-p reduces considerably. When the input signal frequency is 400 Hz, xp-p are about 3.5, 8.8, 13.2, 17.5 μm respectively, and when the input signal frequency is 800 Hz, xp-p are just about 1.1, 2.4, 3.5, 4.6 μm. The calculated results are shown as the dashed lines in Fig. 11.
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G. Xue et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 416–421
Fig. 11. Calculated and experimental results of GMA’s p-p displacements.
Fig. 12. xp-p-up-p curves at 50, 200, 400, 800 Hz.
occurs at up-p ¼6 V. While when the signal frequency is higher than 400 Hz, the error gets larger. It may be caused from that the model established in Section 2 does not take the eddy current loss of GMM into consideration. Therefore, this model is suitable to the low-frequency conditions. Fig. 11 also shows that, when the signal frequency is less than 200 Hz, with increase of the voltage amplitudes, the calculated displacements increase uniformly while the experimental results do not. The relationship between the peak to peak value of the displacement and voltage should be analyzed. The results are shown in Fig. 12. From Fig. 12, when up-p is lower than 24 V, with the increase of up-p, xp-p increase, and xp-p-up-p curves are approximately some straight lines across origin. When signal frequencies are lower than 200 Hz, the corresponding slopes are almost a constant about 1.47 μm/V. The slopes decrease obviously when the frequencies are higher than 400 Hz, as the slope is 0.74 μm/V at 400 Hz while 0.19 μm/V at 800 Hz. But the xp-p-up-p curves are not standard straight lines. No matter the signal frequencies are high or low, the data points corresponding to up-p ¼ 12 V, up-p ¼18 V, up-p ¼24 V can always fit
Fig. 10. Sinusoidal wave responses of the GMA. (a) up-p = 6 V, (b) up-p = 12 V, (c) up-p = 18 V and (d) up-p = 24 V.
Comparing the model and test results, if the input signal is in a low frequency (not more than 200 Hz), the error between the model and test result is acceptably low. The maximum error about 1.7 μm
Fig. 13. Relationship between xp-p and up-p.
G. Xue et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 416–421
in a straight line. While the up-p ¼ 6 V data points are always below the fitted line. This occurrence is caused by the strong bias magnetic field of the actuator and could be explained with the help of Fig. 13. From Fig. 13, the displacement p-p value increases with the voltage p-p value increasing, but the increasing rate is not a constant. When up-p increase respectively from 0 to 6 V, from 0 to 12 V, from 0 to 18 V, from 0 to 24 V, the displacement increasing rates are separately corresponding to the slope of OA, OB, OC, OD. The slope of OA or OB is obviously lower than the slope of OC or OD. That is to say, when up-p values get lower, the increasing rates of xp-p become lower. While when the voltages are higher (still smaller than 24 V), the increasing rates become higher and the linearity between xp-p and up-p becomes better at the same time.
421
than 1.7 μm, the model is verified. (4) The relationship between the peak to peak values of the displacement and voltage is analyzed. The displacement increases with the voltage increasing and the growth rate is nearly constant. While Due to the bias, the rate under low voltage input is a little smaller than the one with a higher voltage input. And with the voltage getting higher, the linearity between the displacement and the voltage becomes better.
Acknowledgments This work was supported by National Natural Science Foundation of China (No.51275525).
6. Conclusions References (1) A giant magnetostrictive actuator, characterized by a strong bias magnetic field, is designed. With the input current in a certain (“backward”) direction, mixed by a common sinusoid current and DC, the actuator could be driven. (2) Displacement model for the actuator is simplified to a single degree of freedom second-order system through reluctance theorem, approximate linearity between λ and H, linear system theory. This model doesn’t take eddy current losses of the giant magnetostrictive material in high frequencies into consideration. So the model is suitable for the condition that the input signal frequencies less than 200 Hz. (3) An experimental system is designed to measure the displacement of the actuator. Square voltage input is exerted for the direction reorganization and sinusoidal input for amplitude-frequency characteristics of the actuator. The displacement peak to peak value is introduced to analyze properties of the actuator. As the errors between measured and calculated results are no more
[1] A.G. Olabi, A. Grunwald, Design and application of magnetostrictive materials, Mater. Des. 22 (2) (2008) 469–483. [2] S.J. Moon, C.W. Lim, B.H. Kim, Y. Park, Structural vibration control using linear magnetostrictive actuators, J. Sound Vib. 302 (2007) 875–891. [3] C. Fan, Q. Yang, W. Yang, R. Yan, J. Sun, F. Liu, Giant magnetostrictive force sensor based on inverse magnetostrictive effect, Instrum. Tech. Sensor 4 (2007) 5–7. [4] R. Angara, L. Si, M. Anjanappa, A high speed magnetostrictive mirror deflector, Smart Mater. Struct. 18 (2009) 1–7. [5] Z. Cao, J. Cai, Design of a giant magnetostrictive motor driven by elliptical motion, Sens. Actuators A 118 (2005) 332–337. [6] X. Tan, J.S. Baras, Modeling and control of hysteresis in magnetostrictive actuators, Automatica 40 (2004) 1469–1480. [7] G. Engdahl, Handbook of Giant Magnetostrictive Materials, Academic press, 2000. [8] F. Braghin, S. Cinquemani, F. Resta, A model of magnetostrictive actuators for active vibration control, Sens. Actuators A 165 (2011) 342–350. [9] B. Yang, G. Meng, Z. Feng, D. Yang, Giant magnetostrictive clamping mechanism for heavy-load and precise positioning linear inchworm motors, Mechatronics 21 (2011) 92–99.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Analysis of the giant magnetostrictive actuator with strong bias magnetic field Guangming Xue n, Zhongbo He, Dongwei Li, Zhaoshu Yang, Zhenglong Zhao Vehicles and Electrical Engineering Department, Ordnance Engineering College, Heping West 97, Shijiazhuang 050003, China
art ic l e i nf o
a b s t r a c t
Article history: Received 27 February 2014 Received in revised form 22 June 2015 Accepted 28 June 2015 Available online 8 July 2015
Giant magnetostrictive actuator with strong bias magnetic field is designed to control the injector bullet valve opening and closing. The relationship between actuator displacement amplitude and input signal direction is analyzed. And based on the approximate linearity of strain-magnetic field, second-order system model of the actuator displacement is established. Experimental system suitable for the actuator is designed. The experimental results show that, the square voltage amplitude being 12 V, the actuator displacement amplitude is about 17 μm with backward direction signal input while being 1.5 μm under forward direction signal. From the results, the suitable input direction is confirmed to be backward. With exciting frequncy lower than 200 Hz, the error between the model and experimental result is less than 1.7 μm. So the model is validated under the low-frequency signal input. The testing displacement-voltage curves are approximately straight lines. But due to the biased position, the line slope and the displacement-voltage linearity change as the input voltage changes. & 2015 Elsevier B.V. All rights reserved.
Keywords: Giant magnetostrictive actuator Strong bias magnetic field Displacement Square signals Sinusoidal signals
1. Introduction Giant magnetostrictive material (GMM) is a “smart” material. With high piezomagnetic coefficient, the material can generate relatively high output-force and displacement (compared with other smart materials). Hence the material has been widely used in many fields [1]. The giant magnetostrictive actuator (GMA), as a most widely application of the material, always finds its applications in a variety of ranges such as active isolation, precision machining, signal detection etc. [2–4]. A typical giant magnetostrictive actuator generally contains three parts including magnetic field generating part, preloading mechanism and cooling device. For the magnetic field generating part, an Alternating Current (AC) coil is always applied to exert driven magnetic field, while a permanent magnet (or a DC coil sometimes) is employed to bias the magnetostrictive material magnetically. These three parts could be changed in various forms for certain requirements. For examples, the cooling device could be removed to reduce the actuator’s volume, the permanent magnet is removed to utilize the double-frequency characteristics, the preloading spring selected as dishing etc. These actuators show different behaviors from traditional actuators [5,6]. The giant magnetostrictive actuator introduced in this paper, characterized by asymmetric bias magnetic field, is designed for n
Corresponding author. E-mail address: [email protected] (G. Xue).
http://dx.doi.org/10.1016/j.jmmm.2015.06.083 0304-8853/& 2015 Elsevier B.V. All rights reserved.
the bullet valve (normally closed) of an electrical injector. The output end of the actuator, touched with giant magnetostrictive material, is pushed against a fixed seat on the bullet valve. If the material were driven lengthening with supplied currents input, the output end would be moved and strikes the valve seat. For avoidance of this condition, the idea that giant magnetostrictive material shortens with exciting signals input is taken. The property could be reached from the actuator designed in this paper. The material, biased sharply by a strong permanent magnet, is guaranteed in a maximum length without exciting signal. So the bullet valve is closed in this situation. With exciting signal in certain direction, the material shortens and then the valve would open. The giant magnetostrictive actuator with strong bias magnetic field (strongly biased actuator) could find its displacement model at low frequency.
2. Operating principle of the strongly biased actuator The schematic diagram of the strongly biased giant magnetostrictive actuator is shown in Fig. 1. The magnetic field within the rod is generated not only by the magnetic coil but also the permanent magnet. The rod is mechanically compressed by the help of the cover and the disc spring. Compression of the rod is executed for the limited tensile strength of the material, and could obtain higher coupling factors. For a typical giant magnetostrictive actuator, the effect of the bias magnetic field could be explained in Fig. 2. The material with
G. Xue et al. / Journal of Magnetism and Magnetic Materials 394 (2015) 416–421
417
Fig. 4. Modified sinusoidal signal applied to the actuator.
Fig. 1. Schematic diagram of the actuator.
Fig. 2. Influence of bias magnetic field on GMM strain.
a bias magnetic field, shown as the dashed lines, could obtain a higher strain compared with the field without a bias. And the double-frequency phenomenon (the frequency of the rod strain becoming double to the frequency of the input signal), if not required, could be avoided [7]. Moreover, it can also be concluded from Fig. 2 that the behavior of the material differs with the bias magnetic field intensity changing. The property of the strongly biased actuator could be shown in Fig. 3. As the bias position is considerably high, Giant magnetostrictive rod is in a relatively long state initially (without input). With a “forward signal” input, the strain of the material changes little, and the corresponding displacement of the actuator is very small. While with the “backward current”, the magnetic field generated by the exciting coil within giant magnetostrictive rod, is in the opposite direction with the bias magnetic field. Hence the rod could shorten largely and the actuator delivers a considerable displacement. So a test of the direction of the input current is required for proper operation of the actuator.
It should be noted that, if the input current were in “backward” direction, the linearity between the rod strain and the exciting magnetic field is still approximately suitable. (The current and biased field value should be chosen properly, not too high.) For a considerable displacement, a common format is not suitable. That is because an ordinary sinusoidal signal is alternating and contains two directions values, one of which would be definitely cut off (not a double-frequency phenomenon). To avoid this situation, the input “sinusoidal signal” is a superposition of an ordinary sinusoidal current and a DC with equal amplitude to the sinusoidal one. The modified sinusoidal signal is shown in Fig. 4. From the analysis above, two targets could be achieved from these designs. An initial long state of the magnetostrictive rod is realized from a strong bias magnetic field, and a relatively high displacement without the signal cut-off is guaranteed by the modulated input format.
3. Displacement model of the actuator The displacement model of the actuator is established under these assumptions [8,9]: (1) The reluctance of the giant magnetostrictive rod occupies most of the total reluctances of the closed magnetic circuit. Hence a large proportion of the magnetomotive force generated by the magnetic coil, distributes on the rod. (2) The magnetic field H, axial strain ε and stress s distribute uniformly throughout the GMM rod, and the giant magnetostrictive rod could be seen as mass-spring-damper system. (3) The relationship between magnetic field and strain within the rod is approximately linear. Under assumption (2), the strongly biased actuator could be modeled as a linear time-invariant system just as shown in Fig. 5. The equation of motion of the actuator is presented as:
(m2 + m M ) x¨ + (c2 + c M ) ẋ + (k2 + k M ) x = F M
(1)
where mM, kM, cM are respectively the equivalent mass of giant magnetostrictive rod (one-third of the inertial mass of the rod),
Fig. 3. Property of the strongly biased actuator.
Fig. 5. Equivalent model of the actuator.
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equivalent mechanical stiffness, viscous damping, m2, k2, c2 are the inertial mass of the push rod, stiffness of the disc spring, the load damping (referring to the damping caused by the disc spring if without any load), x is the displacement of the inertial mass, FM is the magnetostrictive force generated by the rod. For giant magnetostrictive rod, the equivalent axial stiffness kM could be calculated by:
kM =
AE H (2)
l H
where E is the mechanical elastic modulus of the giant magnetostrictive material, A is the cross-section area of the rod and l is the length of the rod. The strain of giant magnetostrictive rod includes elastic and magnetostrictive strain. F is the force corresponding to the magnetostrictive strain and its relationship with λ is:
(3)
F = E H λA
For the relationship between strain λ and magnetic field H, nonlinear models like Jiles-Aiherton magnetization (combined with quadratic domain-transfer model) are always employed. These physics-based models can explain magnetization hysteresis of the material physically. However, they are limited by their strong nonlinearities, and require substantial knowledge which may be not easily obtained sometimes. For the strongly biased actuator, due to the linearity between the magnetostrictive strain and the magnetic field, a complex model is not necessary and concise compared to the linear model. So the expression of the strain λ and magnetic field H generated by the exciting coil can be offered by the linear equation:
λ = d33 H
(4)
where d33 is the magnetomechanical coupling coefficient. The magnetic circuit is analyzed as follow. The actuator has a closed magnetic circuit which includes the giant magnetostrictive rod, push rod, air gap, cover, crust and base. One branch of the loops is shown in Fig. 6. The flow of the magnetic field lines could be in the opposite direction with the one shown in this figure. In the magnetic circuit, the length of the air gap is so little that the magnetic field leakage could be neglected. As the magnetic flux flows in each part is equal, the magnetomotive force exerted by the coil is distributed to each magnetic segment according to the proportions of their magnetic reluctances. Magnetic reluctance is directly proportional to the length of magnetic part while inversely to the cross sectional area and permeability. For the actuator in this paper, the magnetic reluctances of the covers, push rod and crust are considerably small for their large permeability, while the reluctance of the air gap is small for its short length. So the magnetic reluctance of the magnetostrictive rod
occupies the whole reluctance of the loop and so does the magnetomotive force, which is described in assumption (1). Under assumption (1), the whole magnetomotive force is approximately equal to the magnetomotive force within the rod:
Hl ≈ NI
(5)
where N is the number of coil winding turns. For compensating magnetomotive force of other parts, a proportional coefficient C could be introduced to Eq. (5), and it becomes:
H=C
NI l
(6)
Impose m ¼m2 þmM, c ¼c2 þcM, k ¼k2 þkM, and substitut Eqs. (3), (4), (6) into Eq. (1), one gets:
mx¨ + cẋ + kx = Cd33 NIE HA/l
(7)
The output displacement under the mixed current shown in Fig. 4 is analyzed as follow. For a linear time-invariant model established in this section, the response under the mixed input current is the superposition of separate response caused by the corresponding sinusoidal current and DC respectively. Hence the steady-state displacement for the actuator, according to vibration theory, is composed by a sinusoidal and direct form. For the sinusoidal part, when the frequency of the input signal is far less than the resonance frequency of the actuator, the amplification of the output to the input is about 1. That is to say, the displacement amplitude is equal to the one caused by a constant force which could be calculated from Eq. (7) approximately as Cd33NIEHA/(lk). For the direct current part, the displacement is stimulated through a constant force, and its value is also equal to Cd33NIEHA/(lk). Compose both displacements and one gets: the steady-state response stimulated by the mixed input has the same format with the input.
4. Testing setup The testing system applied to the strongly biased actuator is shown in Fig. 7. Digital current signals, generated by the computer, could be transformed into analog forms by the help of the digital oscilloscope (PS2000). Then the analog signals would be transmitted to the coil after being amplified by the power amplifier (GF800). The actuator is driven and generates displacement signals which could be measured by the eddy current sensor (HN808). And the measured displacements could be transmitted back to the computer after a transformation to digital signals. Visual Basic 6.0 software is employed on the computer to generate or record the digital signals. The photo of strongly biased actuator is shown in Fig. 8 and parameters of the actuator are tabulated in Table 1.
5. Results and discussions 5.1. Responses of square wave signals
Fig. 6. Magnetic circuit of the actuator.
From the analysis in Section 2, the current signals in the same value while opposite directions would lead to different displacements of the giant magnetostrictive actuator. By this way, the “backward” direction of the input current could be recognized. Square wave signals with the same amplitude are generated to drive the actuator, and the alteration of the current directions is obtained by exchanging the connections between the magnetic coil and power amplifier. The square wave current at the
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Fig. 7. Experimental system.
Fig. 9. Square wave responses of the actuator in different directions.
which could guarantee a relatively high output of the actuator. 5.2. Responses of sinusoidal wave signals The mixed signals shown in Fig. 4 are generated by PS2000 and amplified by the power amplifier. Adjust the signal voltage peak to peak values (up-p) separately to be 6, 12, 18, 24 V. The tests under each value would be repeated for different frequencies at 50, 200, 400, 800 Hz, a total of 16 trials. The testing results are shown in Fig. 10. The results show that the steady-state displacements of the actuator are approximately in the same formats, sinusoidal signals plus DCs with same amplitudes, as the input signals. That’s a characteristic of the strongly biased actuator, and then the conclusion on the output format in Section 3 is validated. 5.3. Analysis of the results Fig. 8. Object of the strongly biased actuator. (a) Without the crust, (b) With the crust. Table 1 Main parameters of the actuator. Length (l) [mm] Cross-section area of the rod (A) [mm2] Number of the winding turns (N) Elastic modulus (EH) [N/m2] Total damping (c) [N s/m] magnetomotive force coefficient (C) Total mass (m) [g] Stiffness of the disc spring (k2) [N/m] magnetomechanical coupling coefficient (d33) [m/A]
35 16 860 3 1010 3800 0.9 17 2.3 105 1.16 10-8
amplitude 2 A and frequency 50 Hz is generated to flow within the coil. Testing displacements are shown in Fig. 9. From the point of different directions of the current signals, the results are labeled respectively as “a” and “b”. Compare different square wave responses of the input currents in different directions, the amplitude in “a” direction (about 1.5 μm) is far less than the one in “b” direction (17 μm). Therefore, the current in “b” direction is in “backward” direction labeled in Fig. 3,
The displacement maximums are calculated as follows. For the model, the mechanical resonance frequency of the actuator could be calculated from Eq. (7) as (k/m)^(1/2) E28 kHz. As the current frequencies are no more than 800 Hz, far less than resonance frequency, the expression Cd33NIEHA/(lk) calculated in Section 3 is appropriate. Plus the DC response, the expression of maximum value of the calculated displacement is 2Cd33NIEHA/(lk). For the testing results, high-frequency noises, as components of the displacements, could be filtered through SPTOOL toolbox in MATLAB software. The low-limit frequencies of the designed lowpass filters are 2 times of the input frequencies. Then extracting the maximum values of five displacement cycles, the mean of them is chosen as the displacement value for this testing. The experimental and calculated results of the displacement peak to peak values (xp-p) are shown in Fig. 11. The experimental results are shown as the solid lines in Fig. 11. According to the tested results, xp-p changes little when the input signal frequency is no more than 200 Hz. When up-p values are respectively 6, 12, 18, 24 V, xp-p are 7.0, 17.7, 26.5, 35.2 μm. As the input signal frequencies become higher, xp-p reduces considerably. When the input signal frequency is 400 Hz, xp-p are about 3.5, 8.8, 13.2, 17.5 μm respectively, and when the input signal frequency is 800 Hz, xp-p are just about 1.1, 2.4, 3.5, 4.6 μm. The calculated results are shown as the dashed lines in Fig. 11.
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Fig. 11. Calculated and experimental results of GMA’s p-p displacements.
Fig. 12. xp-p-up-p curves at 50, 200, 400, 800 Hz.
occurs at up-p ¼6 V. While when the signal frequency is higher than 400 Hz, the error gets larger. It may be caused from that the model established in Section 2 does not take the eddy current loss of GMM into consideration. Therefore, this model is suitable to the low-frequency conditions. Fig. 11 also shows that, when the signal frequency is less than 200 Hz, with increase of the voltage amplitudes, the calculated displacements increase uniformly while the experimental results do not. The relationship between the peak to peak value of the displacement and voltage should be analyzed. The results are shown in Fig. 12. From Fig. 12, when up-p is lower than 24 V, with the increase of up-p, xp-p increase, and xp-p-up-p curves are approximately some straight lines across origin. When signal frequencies are lower than 200 Hz, the corresponding slopes are almost a constant about 1.47 μm/V. The slopes decrease obviously when the frequencies are higher than 400 Hz, as the slope is 0.74 μm/V at 400 Hz while 0.19 μm/V at 800 Hz. But the xp-p-up-p curves are not standard straight lines. No matter the signal frequencies are high or low, the data points corresponding to up-p ¼ 12 V, up-p ¼18 V, up-p ¼24 V can always fit
Fig. 10. Sinusoidal wave responses of the GMA. (a) up-p = 6 V, (b) up-p = 12 V, (c) up-p = 18 V and (d) up-p = 24 V.
Comparing the model and test results, if the input signal is in a low frequency (not more than 200 Hz), the error between the model and test result is acceptably low. The maximum error about 1.7 μm
Fig. 13. Relationship between xp-p and up-p.
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in a straight line. While the up-p ¼ 6 V data points are always below the fitted line. This occurrence is caused by the strong bias magnetic field of the actuator and could be explained with the help of Fig. 13. From Fig. 13, the displacement p-p value increases with the voltage p-p value increasing, but the increasing rate is not a constant. When up-p increase respectively from 0 to 6 V, from 0 to 12 V, from 0 to 18 V, from 0 to 24 V, the displacement increasing rates are separately corresponding to the slope of OA, OB, OC, OD. The slope of OA or OB is obviously lower than the slope of OC or OD. That is to say, when up-p values get lower, the increasing rates of xp-p become lower. While when the voltages are higher (still smaller than 24 V), the increasing rates become higher and the linearity between xp-p and up-p becomes better at the same time.
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than 1.7 μm, the model is verified. (4) The relationship between the peak to peak values of the displacement and voltage is analyzed. The displacement increases with the voltage increasing and the growth rate is nearly constant. While Due to the bias, the rate under low voltage input is a little smaller than the one with a higher voltage input. And with the voltage getting higher, the linearity between the displacement and the voltage becomes better.
Acknowledgments This work was supported by National Natural Science Foundation of China (No.51275525).
6. Conclusions References (1) A giant magnetostrictive actuator, characterized by a strong bias magnetic field, is designed. With the input current in a certain (“backward”) direction, mixed by a common sinusoid current and DC, the actuator could be driven. (2) Displacement model for the actuator is simplified to a single degree of freedom second-order system through reluctance theorem, approximate linearity between λ and H, linear system theory. This model doesn’t take eddy current losses of the giant magnetostrictive material in high frequencies into consideration. So the model is suitable for the condition that the input signal frequencies less than 200 Hz. (3) An experimental system is designed to measure the displacement of the actuator. Square voltage input is exerted for the direction reorganization and sinusoidal input for amplitude-frequency characteristics of the actuator. The displacement peak to peak value is introduced to analyze properties of the actuator. As the errors between measured and calculated results are no more
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