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A novel standing wave linear piezoelectric actuator using the longitudinal-bending coupling mode
Sensors and Actuators A 251 (2016) 119–125
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
A novel standing wave linear piezoelectric actuator using the longitudinal-bending coupling mode Yingxiang Liu ∗ , Shengjun Shi, Chunhong Li, Weishan Chen, Junkao Liu State Key Laboratory of Robotics and System, Harbin Institute of Technology No.92, West Da-Zhi Street, Harbin, 150001, Heilongjiang Province, China
a r t i c l e
i n f o
Article history: Received 22 June 2016 Received in revised form 14 September 2016 Accepted 11 October 2016 Available online 13 October 2016 Keywords: Piezoelectric actuator Longitudinal vibration Bending vibration Vibration coupling Boundary condition
a b s t r a c t A novel standing wave linear piezoelectric actuator is proposed and tested by using a sandwich transducer operated in longitudinal-bending coupling mode. The vibration mode used in this work is neither longitudinal mode nor bending mode, but a longitudinal-bending hybrid one, which is generated simultaneously by only one group of PZT ceramic elements. The exciting principle of this coupling mode is discussed, and then realized by designing a standing wave piezoelectric actuator. When the 1 st longitudinal and 3rd bending modes of the transducer have colse resonance frequencies and unsymmetrical boundary condition is applied, the desired longitudinal-bending coupling mode can be generated by a sine signal, which finally produces oblique elliptical movement on the end tip of the transducer. The design and analysis work is accomplished by finite elements method (FEM), and verified by a scanning laser Doppler vibrometer after the fabrication of a prototype. The proposed standing wave linear piezoelectric actuator achieves maximum no-load speed and thrust force of about 891.3 mm/s and 39.2 N under voltage of 400 VP-P and working frequency of 29.4 kHz. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Piezoelectric actuators usually push the runners by the resonant movements on the surface particles; these movements should have transverse and normal components simultaneously: the normal one is used to overcome the preload between the interface, whereas the driving force is produced by the transverse vibration [1–3]. According to the vibration mode utilized, piezoelectric actuators can be classified into travelling wave ones [4–6], standing wave ones [7–9] and mode-hybrid ones [10–15] up to the present. For the travelling wave piezoelectric actuators, two flexual standing waves with the same resonance frequency, a temporal shift of 90◦ and a spatial distance of quarter wavelength should be generated in a ring or a disk to compose the travelling wave, which means that they need two phase of exciting signals to produce the two independent standing waves. Most of the mode-hybrid piezoelectric actuators also need two phase of exciting signals as they operate by the hybrid of two different vibration modes. Theoretically speaking, the standing wave piezoelectric actuators have
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Liu), [email protected] (S. Shi), [email protected] (W. Chen), [email protected] (J. Liu). http://dx.doi.org/10.1016/j.sna.2016.10.015 0924-4247/© 2016 Elsevier B.V. All rights reserved.
merits of simple structure and simple control performance as they only need one exciting signal. For example, a novel standing wave piezoelectric actuator using the bending mode of a plate had been proposed and tested by He et al.; the 3rd and 4th bending modes of the plate were used for the rightward and leftward linear driving, respectively [16]. Chen et al. proposed a linear standing wave piezoelectric actuator using a sandwich type bending transducer; their prototype achieved maximum speed and thrust force of 180 mm/s and 14 N, respectively [17]. Park et al. proposed a standing wave square tubular piezoelectric actuator, their prototype achieved the no-load speed of 1000 rpm and the maximum torque of 0.37 mN m under a size of 3.975 mm × 3.975 mm × 16 mm[18]. In these works, the driving tips of the actuator moved with oblique linear trajectories, which had both transverse and normal components. Genernally speaking, the standing wave piezoelectric actuator only use one vibration mode to obtain the desired oblique linear movements. In this paper, a novel standing wave linear piezoelectric actuator is proposed and tested, in which a new longitudinal-bending coupling mode is generated in a sandwich transducer to produce oblique elliptical movement on the driving tip. In this new design, the longitudinal and bending modes are generated simultaneously by only one group of PZT ceramic elements; in other words, the vibration mode used by the proposed actuator is neither pure longitudinal mode nor pure bending mode, but a longitudinal-bending
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Fig 1. Longitudinal-bending coupling mode of a square beam: (a) the longitudinal mode under free boundary condition, (b) the bending mode under free boundary condition, (c) the longitudinal-bending coupling mode under partial fixed boundary condition.
hybrid one. Section 2 explains the principle of the longitudinalbending coupling mode, Section 3 describes the operating principle of the proposed standing wave piezoelectric actuator, Section 4 shows the design process by finite element method (FEM), Section 5 shows the experimental testing and results, which is followed by a summary in Section 6. 2. Longitudinal-bending coupling mode of a beam Generally speaking, an elastic beam has three types of basic vibration modes, which are longitudinal mode, bending mode and torsional mode. Therefore, sandwich type piezoelectric transducer can operate under these three modes separately or their hybrid. Usually, these vibration modes are generated by separated PZT elements as the problem of the independence of vibration modes: longitudinal mode is excited by whole pieces of PZT plates vibrate in axial direction, bending mode is always generated by half pieces of PZT plates with reverse polarizations, whereas PZT elements deform in circumferential direction are needed for the excitation of the torsional mode. Thus, for the generation of a hybrid vibration in a beam, can either be longitudinal-bending hybrid or longitudinal-torsional hybrid, two separated groups of PZT elements are necessary.
Fig. 2. The structure of the proposed standing wave linear piezoelectric actuator.
However, the longitudinal-bending coupling vibration mode propose in this work is different with the traditional hybrid vibration as the longitudinal and bending vibrations are generated simultaneously by only one group of PZT elements; in other words, we can see this coupling vibration as a particular type of mode that contains of both longitudinal and bending components. Fig. 1 gives a specific illustration of the longitudinal-bending coupling mode of a square beam, which is obtained by FEM modal analysis. A duralumin alloy square beam with cross-section of 10.8mm × 10.8 mm and length of 100 mm is analyzed, whose vibration shapes of longitudinal and bending modes under free boundary condition are shown by Fig. 1(a) and (b); and these two modes can produce axial and transverse displacements on the beam end, respectively. On the other side, Fig. 1(c) gives the vibration shape of the desired longitudinal-bending coupling mode, which is obtained under fixed boundary condition on the middle part of the downside surface. This coupling mode is a superposition of the longitudinal and bending ones, which can produce axial and transverse displacements on the beam end synchronously. In summary, there are two necessary conditions for the generating of the longitudinal-bending coupling mode: the first one is that the longitudinal mode must have close resonance frequency with the bending mode, whereas the other one is that unsymmetrical boundary condition must be applied. Under these conditions, we
Fig. 3. The operating sequence of the transducer in one circle.
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Fig. 6. Motion trajectories of the two feet.
3. Operating principle of the proposed piezoelectric actuator
Fig. 4. The longitudinal and bending modes of the transducer under free boundary condition: (a) the FEM model, (b) the longitudinal mode, (c) the bending mode.
The proposed standing wave piezoelectric actuator using the above longitudinal-bending coupling mode is designed as shown in Fig. 2. It has a similar basic structure with the traditional sandwich transducer operated in pure longitudinal mode: eight pieces of PZT plates with outer diameter of 30 mm, inner diameter of 14 mm and thickness of 2 mm are clamped between two horns by a bolt, beryllium bronze plates are clamped between the PZT elements to serve as the electrodes, and a flange is set in the middle for the fixing. The horns are machined to cone shapes to magnify the vibration amplitudes [12,19–21]; qualitatively speaking, the coneshape horn can decrease the moment of inertia of area, which can improve the vibration amplitude as the momentum balance. The proposed transducer has a total length of 125 mm. The different point is that the flange located in the middle has an unsymmetrical structure; the end tip of this flange can be used for the fixing, which can bring in the desired unsymmetrical boundary condition. The details about the materials of the elements are: horns of duralumin alloy, flange and bolts of steel and ceramics of PZT-4. The physical parameters of the PZT ceramic are:
⎡
d=⎣
0
0
0
5 0
0
0
0
5
0
0 ⎦ × 10−10 C/N
0
0
0
−1.6 −1.6 3.3
⎡
⎤
0
15
8.4
6.8
0
0
⎢ 8.4 15 6.8 0 0 ⎢ ⎢ 6.8 6.8 12.9 0 0 cE = ⎢ ⎢ 0 0 0 3.3 0 ⎢ ⎣ 0 0 0 0 2.8 ⎡
0 8.1
εT = ⎣ 0 0
Fig. 5. The longitudinal-bending coupling mode of the transducer: (a) the FEM model, (b) the longitudinal-bending coupling mode.
can generate a longitudinal-bending coupling vibration in a beam by only one group of PZT elements vibrated in axial direction; and this coupling vibration can form produce oblique trajectories on the beam end, which can be used for the design of a standing wave piezoelectric actuator.
0
0
0
0
8.1
0
0
6.7
⎤
0
0
⎦ × 10−9 F/m
0
(1)
⎤
⎥ ⎥ ⎥ ⎥ × 1010 N/m2 0 ⎥ ⎥ 0 ⎦ 0 0
(2)
2.8
(3)
where d, cE , and εT are the piezoelectric matrix, the stiffness matrix and the dielectric matrix, respectively. Fig. 3 shows the operating sequence of the transducer in one circle, which can give a very clear illustration for the linear driving. Step 1 is the initial vibration state, which is the original vibration shape of the transducer, the transducer extends to its maximum length and bends to the upside extreme in step 2, which is followed by another initial vibration state shown in step 3, and step 4 shows the other extreme shape contains both shorten and downward bending. From, step 1 to step 4, the two end tips of the transducer
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Fig. 7. The prototype of the proposed piezoelectric actuator.
will vibrate in oblique elliptical trajectories, which can be used to drive runner linearly. In this longitudinal-bending coupling mode, the longitudinal component will overcome the preload between the driving tip and the runner, whereas the driving force is generated by the bending component. It should be noted that the two end tips of the transducer both can be used to push one runner independently, and the output thrust force will be doubled if the two runners are linked together. 4. Motions of the driving feet
During the modal analysis, two FEM models were built and calculated: the first one had symmetrical structure and free boundary condition, whereas the other one had unsymmetrical flange as shown in Fig. 2 and fixed boundary condition was applied on the end of the flange. The vibration shapes and the corresponding resonance frequencies of the two models gained by the modal analysis are shown in Figs. 4 and 5, , respectively. For the first transducer with free boundary condition, the 1 st longitudinal and 3rd bending modes are found independently, whose resonance frequencies are 28.553 kHz and 28.562 kHz, respectively. The desired longitudinal-bending coupling mode is gained in the second transducer under resonance frequency of 28.887 kHz, which can be seen as a hybrid mode of the 1 st longitudinal and 3rd bending modes; there is no independent 1 st longitudinal or 3rd bending mode anymore. The comparison between these two models indicates that the unsymmetrical boundary condition not only causes the coupling of the two modes with close frequencies, but also results in a minute increase on the resonance frequency, which can be explained as that the partial fixing enhances the stiffness. Then, transient analysis was accomplished on the second FEM model by applying a sine voltage with frequency of 28.887 kHz and amplitude of 100 Vrms , the motion trajectories of the two driving tips in one circle are plotted as shown in Fig. 6. It is found that the two tips vibrate under symmetrical oblique elliptical trajectories, and the maximum axial displacement is about 6.4 um, while the transverse one is about 6.2 um. These trajectories can verify the above operating principle shown in Fig. 3.
FEM (ANSYS software) was used to accomplish the modal and transient analysis of the proposed linear piezoelectric actuator.
Fig. 8. Vibration scanning results of the transducer: (a) schematic diagram for the measurement of the end of the horn, (b) schematic diagram for the measurement of side surface of the transducer, (c) vibration shape of the horn end and the corresponding of the vibration velocity response spectrum, (d) vibration shape of the side surface and the corresponding of the vibration velocity response spectrum.
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5. Experiments A prototype of the proposed piezoelectric actuator was fabricated, as shown in Fig. 7. Firstly, we used a scanning laser Doppler vibrometer (PSV-400-M2, Polytec, Germany) to measure the longitudinal-bending coupling mode and the corresponding resonance frequency. To obtain a clear illustration of the coupling mode, two areas were selected to accomplish the vibration measurement: the first one was the end surface of the horn, whereas the other one was the side surface of the whole transducer. During the measurement, the prototype was fixed on a flat tongs by its flange, which provided the unsymmetrical boundary condition. The tested vibration shapes and the corresponding vibration velocity response spectrums are shown in Fig. 8. The vibration shapes shown in Fig. 8 states that an up-down movement is tested on the end of the horn, while the side surface vibrates under a bending movement with four wave nodes; in other words, normal and transverse displacements of the driving tip are generated together. These tested vibration shapes verify that the mode generated in the transducer is neither pure longitudinal mode nor pure bending mode, but a longitudinal-bending coupling one. The resonance frequency of this coupling mode is tested to be about 28.897 kHz by the vibrometer, which is very close to the FEM calculated value; the good agreement between the tested frequency and the calculated one states that we have designed and fabricated the piezoelectric actuator accurately. Then, the mechanical output performances of the proposed standing wave piezoelectric actuator were tested under a platform shown in Fig. 9. The transducer was clamped by a flat tongs firstly, and then the flat tongs was fixed on a base, a runner limited by a guider was pressed onto the end tip of the horn by a screw-spring system, an encoder was contacted with the runner by a wheel on its shaft to get the linear speed, the thrust force was added on the runner by linking weights with string through a glide-wheel. Firstly, the effect of the boundary condition on the output speed of the actuator was measured under exciting frequency of 29.4 kHz, voltage of 400 VP-P and preload of 200 N, and the no-load speed was
Fig. 9. The experiment set-up of the prototype.
tested by adjusting the dimension of the flange clamped by the flat tongs. The clamping part of the flange was change from 6 mm to 20 mm, and it was found that the no-load speed had minute change from 6 mm to 12 mm, and increased quickly from 12 mm to 16 mm, but decreased sharply from 16 mm to 20 mm, as shown in Fig. 10(a). This phenomenon states that the boundary condition has a quite sensitive effect on the no-load speed; in other words, the boundary condition can change the proportion of the longitudinal and bending components in the coupling mode. In the following experiments, the clamping dimension of the flange was set as 16 mm. Then, the exciting frequency was changed to obtain its effect on the no-load speed under preload of 200N, as shown in Fig. 10(b); the piezoelectric actuator achieves maximum no-load speed of 262.8 mm/s, 602.9 mm/s and 788.5 mm/s under voltage of 200 VP-P , 300 VP-P and 400 VP-P , respectively. During the measurement, a matching inductance was linked with the transducer in series to improve the mechanical output performance.
Fig. 10. The tested mechanical performances of the prototype: (a) plots of the speed under different boundary conditions, (b) plots of the speed under different exciting frequencies, (c) plots of the speed versus the exciting voltage, (d) plots of the speed versus the thrust force under different preloads.
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Table 1 Comparison between the proposed standing wave linear piezoelectric actuator and a previous work. Parameters
The previous actuator by Chen [17]
The proposed actuator
Exciting frequency (kHz) Weight (kg) Maximum speed (mm/s) Maximum thrust force (N) Force weight ratio (N/kg)
25.1 0.592 180 14 23.65
29.4 0.258 891.3 39.2 151.9
Innovative Research Groups of the National Natural Science Foundation of China (Grant No.51521003), and in part by the Fok Ying Tung Education Foundation (No. 151053).
References
The comparison between Fig. 10(a) and (b) indicates that the prototype increase to be about 788.5 mm/s from 566.8 mm/s by using the matching inductance. The optimal exciting frequency is tested to be about 29.4 kHz, which is higher than the one got by the vibrometer as the applied preload can increase the resonance frequency. Fig. 10(c) lists the effect of the voltage on the no-load speed under preload of 200 N and frequency of 29.4 kHz, the start voltage of the piezoelectric actuator is found to be about 100 VP-P , and the voltage has a nearly linear effect on the no-load speed. At last, the output speed under different thrust forces were measured, as shown in Fig. 10(d). The prototype can produce maximum thrust forces of 13.7 N, 27.5 N and 39.2 N under preload of 100 N, 200 N and 300 N, respectively; and the corresponding maximum no-load speed are tested to be about 891.3 mm/s, 788.5 mm/s and 396.8 mm/s. The force weight ratio is a key parameter to evaluate a piezoelectric actuator. Table 1 gives a detail comparison between the proposed standing wave linear piezoelectric actuator and a previous work [17], from which we can see that the proposed actuator achieves a large improvement on the force weight ratio under a small size, which illustrate the merit of the proposed linear piezoelectric actuator operated under longitudinal-bending coupling mode. 6. Conclusion A longitudinal-bending coupling mode of beam was proposed and discussed in this work, which had intrinsic difference with the pure longitudinal mode or the pure bending mode. By applying unsymmetrical boundary condition, longitudinal and bending modes with close resonance frequncies were combined together to become a special coupling mode; and the independence of different vibration modes was broken. A standing wave piezoelectric actuator using the longitudinal-bending coupling mode was designed and tested, whose unique feature was that only one group of PZT ceramic elements excited by a sine voltage generated axial and transverse vibrations on the two tip ends of a sandwich transducer. Oblique elliptical movements were produced on the two tip ends under the coupling mode, which were symmetrical in space. The working frequency proposed piezoelectric actuator was designed to be about 28.887 kHz, and tested to be 28.875 kHz. Both longitudinal and bending vibrations were measured on the transducer by a vibrometer. The prototype achieved maximum no-load speed of 891.3 mm/s under preload of 100N, and a maximum thrust force of 39.2 N was obtained under preload of 300N. This work not only gives a new configuration of linear standing wave piezoelectric actuator, but also provides a new idea and discussion about the coupling effect of different vibration modes in an elastic beam. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (No. 51475112 and No. 51622502), in part by the Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 201428), in part by the Foundation for
[1] C.S. Zhao, Ultrasonic Motors Technologies and Applications, Science Press, Beijing, 2010. [2] S.Y. He, P.R. Chiarot, S. Park, A single vibration mode tubular piezoelectric ultrasonic motor, IEEE Trans. Ultrason. Ferroelect. Freq. Control. 58 (5) (2011) 1049–1061. [3] Y.X. Liu, W.S. Chen, X.H. Yang, J.K. Liu, A T-shape linear piezoelectric motor with single foot, Ultrasonics 56 (2) (2015) 551–556. [4] Y. Ting, L.C. Chen, C.C. Li, J.L. Huang, Traveling-wave piezoelectric linear motor part I: The stator design, IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 54 (4) (2007) 847–853. [5] X.L. Lu, J.H. Hu, L. Yang, C.S. Zhao, Principle and experimental verification of novel dual driving face rotary ultrasonic motor, Chin. J. Mech. Eng. 26 (5) (2013) 1006–1012. [6] S.Y. Li, W.C. Ou, M. Yang, C. Guo, C.Y. Lu, J.H. Hu, Temperature evaluation of traveling-wave ultrasonic motor considering interaction between temperature rise and motor parameters, Ultrasonics 57 (3) (2015) 159–166. [7] Z.J. Chen, X.T. Li, P.H. Ci, G.X. Liu, S.X. Dong, A standing wave linear ultrasonic motor operating in in-plane expanding and bending modes, Rev. Sci. Instrum. 86 (3) (2015) 035002. [8] Y.X. Liu, W.S. Chen, J.K. Liu, S.J. Shi, A cylindrical standing wave ultrasonic motor using bending vibration transducer, Ultrasonics 51 (5) (2011) 527–531. [9] Y.L. Shi, C. Chen, C.S. Zhao, Transient response model of standing wave piezoelectric linear ultrasonic motor, J. Wuhan Univ. Technol. -Mat. Sci. Edit. 27 (6) (2012) 1188–1192. [10] Y.X. Liu, W.S. Chen, X.H. Yang, J.K. Liu, A rotary piezoelectric actuator using the third and fourth bending vibration modes, IEEE Trans. Ind. Electron. 61 (8) (2014) 4366–4373. [11] Z.J. Chen, X.T. Li, G.X. Liu, S.X. Dong, A two degrees-of-freedom piezoelectric single-crystal micromotor, J. Appl. Phys. 116 (22) (2014) 224101. [12] Y.L. Shi, C.S. Zhao, W.Q. Huang, Linear ultrasonic motor with wheel-shaped stator, Sens. Actuators A 161 (2010) 205–209. [13] K. Asumi, R. Fukunaga, T. Fujimura, M.K. Kurosawa, Miniaturization of a V-shape transducer ultrasonic motor, Jpn. J. Appl. Phys. 48 (2009) 07GM02. [14] Y.X. Liu, W.S. Chen, J.K. Liu, X.H. Yang, A high-power linear ultrasonic motor using bending vibration transducer, IEEE Trans. Ind. Electron. 60 (11) (2013) 5160–5166. [15] H. Al-Budairi, M. Lucas, P. Harkness, A design approach for longitudinal-torsional ultrasonic transducers, Sens. Actuators A. 198 (2013) 99–106. [16] S.Y. He, W.S. Chen, T. Xie, Z.L. Chen, Standing wave bi-directional linearly moving ultrasonic motor, IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 45 (5) (1998) 1133–1139. [17] W.S. Chen, S.J. Shi, A bidirectional standing wave ultrasonic linear motor based on langevin bending transducer, Ferroelectrics 350 (2007) 102–110. [18] S. Park, S.Y. He, Standing wave brass-PZT square tubular ultrasonic motor, Ultrasonics 52 (2012) 880–889. [19] Y.X. Liu, D.M. Xu, J.P. Yan, X.H. Ni, W.S. Chen, Design of a bending vibrations hybrid type ultrasonic motor with a bonded-type structure, The 2015 Symposium on Piezoelectricity, Acoustic Waves, and Device Applications (2015) 50–54. [20] X.H. Yang, Q. Zhang, X.Y. Zhou, Ultrasonic motor using bending modes with single foot, Adv. Mech. Eng. (2013) 747583. [21] J.H. Oh, H.S. Yuk, K.J. Lim, Study on the ring type stator design technique for a traveling wave rotary type ultrasonic motor, Jpn. J. Appl. Phys. 51 (9) (2012) 09MD13.
Biographies
Yingxiang Liu was born in Hebei province, China, in 1982. He received the B.S., M.S. and Ph.D. degrees from the School of Mechatronics Engineering at Harbin Institute of Technology, China, in 2005, 2007 and 2011, respectively. He is currently a professor of the School of Mechatronics Engineering at the Harbin Institute of Technology. He is the vice director of the Department of Mechatronic Control and Automation. He is also a member of the State Key Laboratory of Robotics and System at Harbin Institute of Technology. He joined the School of Mechatronics Engineering, Harbin Institute of Technology in 2011, where he has been a professor since December 2013. He was a Visiting Scholar at the Mechanical Engineering Department, University of California, Berkeley, from August 2013 to August 2014. His research interests include piezoelectric actuating, ultrasonic motor, piezoelectric actuator, precision actuating, piezoelectric micro jet, bionic robot, fish robot and soft robot.
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Shengjun Shi was born in Heilongjiang province, China, in 1974. He received the B.S. degree in aircraft manufacturing engineering from Northwestern Polytechnical University in 1997. He received his M.S. and Ph.D. degrees from the School of Mechatronics Engineering, Harbin Institute of Technology, China, in 2003 and 2007 respectively. He is currently an associate professor in the Harbin Institute of Technology, China. His research interests include ultrasonic motor, ultrasonic application etc.
Weishan Chen was born in Hebei, China, in 1965. He received his B.S. and the M.S. degrees in precision instrumentation engineering, and the Ph.D. degree in Mechatronics engineering from Harbin Institute of Technology, China, in 1986, 1989, and 1997, respectively. Since 1999, he is a professor with the School of Mechatronics Engineering, Harbin Institute of Technology. His research interests include ultrasonic driving, smart materials and structures, bio-robotics.
Chunhong Li was born in Yunnan province, China, in 1991. He received the B.S. degree from the School of Mechanical Engineering, Tongji University, China in 2014. He is currently a M.S. degree candidate in the Harbin Institute of Technology, China. His research interests include ultrasonic motor and piezoelectric actuating.
Junkao Liu was born in Hebei, China, in 1973. He received his B.S. and Ph.D. degrees from the School of Mechatronics Engineering, Harbin Institute of Technology, China, in 1995 and 2001, respectively. Since 2011, he is a professor with the School of Mechatronics Engineering, Harbin Institute of Technology. His research interests include ultrasonic driving, biomimetic robots and simulation of multi-degree of freedom parallel mechanism.
Contents lists available at ScienceDirect
Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna
A novel standing wave linear piezoelectric actuator using the longitudinal-bending coupling mode Yingxiang Liu ∗ , Shengjun Shi, Chunhong Li, Weishan Chen, Junkao Liu State Key Laboratory of Robotics and System, Harbin Institute of Technology No.92, West Da-Zhi Street, Harbin, 150001, Heilongjiang Province, China
a r t i c l e
i n f o
Article history: Received 22 June 2016 Received in revised form 14 September 2016 Accepted 11 October 2016 Available online 13 October 2016 Keywords: Piezoelectric actuator Longitudinal vibration Bending vibration Vibration coupling Boundary condition
a b s t r a c t A novel standing wave linear piezoelectric actuator is proposed and tested by using a sandwich transducer operated in longitudinal-bending coupling mode. The vibration mode used in this work is neither longitudinal mode nor bending mode, but a longitudinal-bending hybrid one, which is generated simultaneously by only one group of PZT ceramic elements. The exciting principle of this coupling mode is discussed, and then realized by designing a standing wave piezoelectric actuator. When the 1 st longitudinal and 3rd bending modes of the transducer have colse resonance frequencies and unsymmetrical boundary condition is applied, the desired longitudinal-bending coupling mode can be generated by a sine signal, which finally produces oblique elliptical movement on the end tip of the transducer. The design and analysis work is accomplished by finite elements method (FEM), and verified by a scanning laser Doppler vibrometer after the fabrication of a prototype. The proposed standing wave linear piezoelectric actuator achieves maximum no-load speed and thrust force of about 891.3 mm/s and 39.2 N under voltage of 400 VP-P and working frequency of 29.4 kHz. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Piezoelectric actuators usually push the runners by the resonant movements on the surface particles; these movements should have transverse and normal components simultaneously: the normal one is used to overcome the preload between the interface, whereas the driving force is produced by the transverse vibration [1–3]. According to the vibration mode utilized, piezoelectric actuators can be classified into travelling wave ones [4–6], standing wave ones [7–9] and mode-hybrid ones [10–15] up to the present. For the travelling wave piezoelectric actuators, two flexual standing waves with the same resonance frequency, a temporal shift of 90◦ and a spatial distance of quarter wavelength should be generated in a ring or a disk to compose the travelling wave, which means that they need two phase of exciting signals to produce the two independent standing waves. Most of the mode-hybrid piezoelectric actuators also need two phase of exciting signals as they operate by the hybrid of two different vibration modes. Theoretically speaking, the standing wave piezoelectric actuators have
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Liu), [email protected] (S. Shi), [email protected] (W. Chen), [email protected] (J. Liu). http://dx.doi.org/10.1016/j.sna.2016.10.015 0924-4247/© 2016 Elsevier B.V. All rights reserved.
merits of simple structure and simple control performance as they only need one exciting signal. For example, a novel standing wave piezoelectric actuator using the bending mode of a plate had been proposed and tested by He et al.; the 3rd and 4th bending modes of the plate were used for the rightward and leftward linear driving, respectively [16]. Chen et al. proposed a linear standing wave piezoelectric actuator using a sandwich type bending transducer; their prototype achieved maximum speed and thrust force of 180 mm/s and 14 N, respectively [17]. Park et al. proposed a standing wave square tubular piezoelectric actuator, their prototype achieved the no-load speed of 1000 rpm and the maximum torque of 0.37 mN m under a size of 3.975 mm × 3.975 mm × 16 mm[18]. In these works, the driving tips of the actuator moved with oblique linear trajectories, which had both transverse and normal components. Genernally speaking, the standing wave piezoelectric actuator only use one vibration mode to obtain the desired oblique linear movements. In this paper, a novel standing wave linear piezoelectric actuator is proposed and tested, in which a new longitudinal-bending coupling mode is generated in a sandwich transducer to produce oblique elliptical movement on the driving tip. In this new design, the longitudinal and bending modes are generated simultaneously by only one group of PZT ceramic elements; in other words, the vibration mode used by the proposed actuator is neither pure longitudinal mode nor pure bending mode, but a longitudinal-bending
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Fig 1. Longitudinal-bending coupling mode of a square beam: (a) the longitudinal mode under free boundary condition, (b) the bending mode under free boundary condition, (c) the longitudinal-bending coupling mode under partial fixed boundary condition.
hybrid one. Section 2 explains the principle of the longitudinalbending coupling mode, Section 3 describes the operating principle of the proposed standing wave piezoelectric actuator, Section 4 shows the design process by finite element method (FEM), Section 5 shows the experimental testing and results, which is followed by a summary in Section 6. 2. Longitudinal-bending coupling mode of a beam Generally speaking, an elastic beam has three types of basic vibration modes, which are longitudinal mode, bending mode and torsional mode. Therefore, sandwich type piezoelectric transducer can operate under these three modes separately or their hybrid. Usually, these vibration modes are generated by separated PZT elements as the problem of the independence of vibration modes: longitudinal mode is excited by whole pieces of PZT plates vibrate in axial direction, bending mode is always generated by half pieces of PZT plates with reverse polarizations, whereas PZT elements deform in circumferential direction are needed for the excitation of the torsional mode. Thus, for the generation of a hybrid vibration in a beam, can either be longitudinal-bending hybrid or longitudinal-torsional hybrid, two separated groups of PZT elements are necessary.
Fig. 2. The structure of the proposed standing wave linear piezoelectric actuator.
However, the longitudinal-bending coupling vibration mode propose in this work is different with the traditional hybrid vibration as the longitudinal and bending vibrations are generated simultaneously by only one group of PZT elements; in other words, we can see this coupling vibration as a particular type of mode that contains of both longitudinal and bending components. Fig. 1 gives a specific illustration of the longitudinal-bending coupling mode of a square beam, which is obtained by FEM modal analysis. A duralumin alloy square beam with cross-section of 10.8mm × 10.8 mm and length of 100 mm is analyzed, whose vibration shapes of longitudinal and bending modes under free boundary condition are shown by Fig. 1(a) and (b); and these two modes can produce axial and transverse displacements on the beam end, respectively. On the other side, Fig. 1(c) gives the vibration shape of the desired longitudinal-bending coupling mode, which is obtained under fixed boundary condition on the middle part of the downside surface. This coupling mode is a superposition of the longitudinal and bending ones, which can produce axial and transverse displacements on the beam end synchronously. In summary, there are two necessary conditions for the generating of the longitudinal-bending coupling mode: the first one is that the longitudinal mode must have close resonance frequency with the bending mode, whereas the other one is that unsymmetrical boundary condition must be applied. Under these conditions, we
Fig. 3. The operating sequence of the transducer in one circle.
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Fig. 6. Motion trajectories of the two feet.
3. Operating principle of the proposed piezoelectric actuator
Fig. 4. The longitudinal and bending modes of the transducer under free boundary condition: (a) the FEM model, (b) the longitudinal mode, (c) the bending mode.
The proposed standing wave piezoelectric actuator using the above longitudinal-bending coupling mode is designed as shown in Fig. 2. It has a similar basic structure with the traditional sandwich transducer operated in pure longitudinal mode: eight pieces of PZT plates with outer diameter of 30 mm, inner diameter of 14 mm and thickness of 2 mm are clamped between two horns by a bolt, beryllium bronze plates are clamped between the PZT elements to serve as the electrodes, and a flange is set in the middle for the fixing. The horns are machined to cone shapes to magnify the vibration amplitudes [12,19–21]; qualitatively speaking, the coneshape horn can decrease the moment of inertia of area, which can improve the vibration amplitude as the momentum balance. The proposed transducer has a total length of 125 mm. The different point is that the flange located in the middle has an unsymmetrical structure; the end tip of this flange can be used for the fixing, which can bring in the desired unsymmetrical boundary condition. The details about the materials of the elements are: horns of duralumin alloy, flange and bolts of steel and ceramics of PZT-4. The physical parameters of the PZT ceramic are:
⎡
d=⎣
0
0
0
5 0
0
0
0
5
0
0 ⎦ × 10−10 C/N
0
0
0
−1.6 −1.6 3.3
⎡
⎤
0
15
8.4
6.8
0
0
⎢ 8.4 15 6.8 0 0 ⎢ ⎢ 6.8 6.8 12.9 0 0 cE = ⎢ ⎢ 0 0 0 3.3 0 ⎢ ⎣ 0 0 0 0 2.8 ⎡
0 8.1
εT = ⎣ 0 0
Fig. 5. The longitudinal-bending coupling mode of the transducer: (a) the FEM model, (b) the longitudinal-bending coupling mode.
can generate a longitudinal-bending coupling vibration in a beam by only one group of PZT elements vibrated in axial direction; and this coupling vibration can form produce oblique trajectories on the beam end, which can be used for the design of a standing wave piezoelectric actuator.
0
0
0
0
8.1
0
0
6.7
⎤
0
0
⎦ × 10−9 F/m
0
(1)
⎤
⎥ ⎥ ⎥ ⎥ × 1010 N/m2 0 ⎥ ⎥ 0 ⎦ 0 0
(2)
2.8
(3)
where d, cE , and εT are the piezoelectric matrix, the stiffness matrix and the dielectric matrix, respectively. Fig. 3 shows the operating sequence of the transducer in one circle, which can give a very clear illustration for the linear driving. Step 1 is the initial vibration state, which is the original vibration shape of the transducer, the transducer extends to its maximum length and bends to the upside extreme in step 2, which is followed by another initial vibration state shown in step 3, and step 4 shows the other extreme shape contains both shorten and downward bending. From, step 1 to step 4, the two end tips of the transducer
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Fig. 7. The prototype of the proposed piezoelectric actuator.
will vibrate in oblique elliptical trajectories, which can be used to drive runner linearly. In this longitudinal-bending coupling mode, the longitudinal component will overcome the preload between the driving tip and the runner, whereas the driving force is generated by the bending component. It should be noted that the two end tips of the transducer both can be used to push one runner independently, and the output thrust force will be doubled if the two runners are linked together. 4. Motions of the driving feet
During the modal analysis, two FEM models were built and calculated: the first one had symmetrical structure and free boundary condition, whereas the other one had unsymmetrical flange as shown in Fig. 2 and fixed boundary condition was applied on the end of the flange. The vibration shapes and the corresponding resonance frequencies of the two models gained by the modal analysis are shown in Figs. 4 and 5, , respectively. For the first transducer with free boundary condition, the 1 st longitudinal and 3rd bending modes are found independently, whose resonance frequencies are 28.553 kHz and 28.562 kHz, respectively. The desired longitudinal-bending coupling mode is gained in the second transducer under resonance frequency of 28.887 kHz, which can be seen as a hybrid mode of the 1 st longitudinal and 3rd bending modes; there is no independent 1 st longitudinal or 3rd bending mode anymore. The comparison between these two models indicates that the unsymmetrical boundary condition not only causes the coupling of the two modes with close frequencies, but also results in a minute increase on the resonance frequency, which can be explained as that the partial fixing enhances the stiffness. Then, transient analysis was accomplished on the second FEM model by applying a sine voltage with frequency of 28.887 kHz and amplitude of 100 Vrms , the motion trajectories of the two driving tips in one circle are plotted as shown in Fig. 6. It is found that the two tips vibrate under symmetrical oblique elliptical trajectories, and the maximum axial displacement is about 6.4 um, while the transverse one is about 6.2 um. These trajectories can verify the above operating principle shown in Fig. 3.
FEM (ANSYS software) was used to accomplish the modal and transient analysis of the proposed linear piezoelectric actuator.
Fig. 8. Vibration scanning results of the transducer: (a) schematic diagram for the measurement of the end of the horn, (b) schematic diagram for the measurement of side surface of the transducer, (c) vibration shape of the horn end and the corresponding of the vibration velocity response spectrum, (d) vibration shape of the side surface and the corresponding of the vibration velocity response spectrum.
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5. Experiments A prototype of the proposed piezoelectric actuator was fabricated, as shown in Fig. 7. Firstly, we used a scanning laser Doppler vibrometer (PSV-400-M2, Polytec, Germany) to measure the longitudinal-bending coupling mode and the corresponding resonance frequency. To obtain a clear illustration of the coupling mode, two areas were selected to accomplish the vibration measurement: the first one was the end surface of the horn, whereas the other one was the side surface of the whole transducer. During the measurement, the prototype was fixed on a flat tongs by its flange, which provided the unsymmetrical boundary condition. The tested vibration shapes and the corresponding vibration velocity response spectrums are shown in Fig. 8. The vibration shapes shown in Fig. 8 states that an up-down movement is tested on the end of the horn, while the side surface vibrates under a bending movement with four wave nodes; in other words, normal and transverse displacements of the driving tip are generated together. These tested vibration shapes verify that the mode generated in the transducer is neither pure longitudinal mode nor pure bending mode, but a longitudinal-bending coupling one. The resonance frequency of this coupling mode is tested to be about 28.897 kHz by the vibrometer, which is very close to the FEM calculated value; the good agreement between the tested frequency and the calculated one states that we have designed and fabricated the piezoelectric actuator accurately. Then, the mechanical output performances of the proposed standing wave piezoelectric actuator were tested under a platform shown in Fig. 9. The transducer was clamped by a flat tongs firstly, and then the flat tongs was fixed on a base, a runner limited by a guider was pressed onto the end tip of the horn by a screw-spring system, an encoder was contacted with the runner by a wheel on its shaft to get the linear speed, the thrust force was added on the runner by linking weights with string through a glide-wheel. Firstly, the effect of the boundary condition on the output speed of the actuator was measured under exciting frequency of 29.4 kHz, voltage of 400 VP-P and preload of 200 N, and the no-load speed was
Fig. 9. The experiment set-up of the prototype.
tested by adjusting the dimension of the flange clamped by the flat tongs. The clamping part of the flange was change from 6 mm to 20 mm, and it was found that the no-load speed had minute change from 6 mm to 12 mm, and increased quickly from 12 mm to 16 mm, but decreased sharply from 16 mm to 20 mm, as shown in Fig. 10(a). This phenomenon states that the boundary condition has a quite sensitive effect on the no-load speed; in other words, the boundary condition can change the proportion of the longitudinal and bending components in the coupling mode. In the following experiments, the clamping dimension of the flange was set as 16 mm. Then, the exciting frequency was changed to obtain its effect on the no-load speed under preload of 200N, as shown in Fig. 10(b); the piezoelectric actuator achieves maximum no-load speed of 262.8 mm/s, 602.9 mm/s and 788.5 mm/s under voltage of 200 VP-P , 300 VP-P and 400 VP-P , respectively. During the measurement, a matching inductance was linked with the transducer in series to improve the mechanical output performance.
Fig. 10. The tested mechanical performances of the prototype: (a) plots of the speed under different boundary conditions, (b) plots of the speed under different exciting frequencies, (c) plots of the speed versus the exciting voltage, (d) plots of the speed versus the thrust force under different preloads.
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Table 1 Comparison between the proposed standing wave linear piezoelectric actuator and a previous work. Parameters
The previous actuator by Chen [17]
The proposed actuator
Exciting frequency (kHz) Weight (kg) Maximum speed (mm/s) Maximum thrust force (N) Force weight ratio (N/kg)
25.1 0.592 180 14 23.65
29.4 0.258 891.3 39.2 151.9
Innovative Research Groups of the National Natural Science Foundation of China (Grant No.51521003), and in part by the Fok Ying Tung Education Foundation (No. 151053).
References
The comparison between Fig. 10(a) and (b) indicates that the prototype increase to be about 788.5 mm/s from 566.8 mm/s by using the matching inductance. The optimal exciting frequency is tested to be about 29.4 kHz, which is higher than the one got by the vibrometer as the applied preload can increase the resonance frequency. Fig. 10(c) lists the effect of the voltage on the no-load speed under preload of 200 N and frequency of 29.4 kHz, the start voltage of the piezoelectric actuator is found to be about 100 VP-P , and the voltage has a nearly linear effect on the no-load speed. At last, the output speed under different thrust forces were measured, as shown in Fig. 10(d). The prototype can produce maximum thrust forces of 13.7 N, 27.5 N and 39.2 N under preload of 100 N, 200 N and 300 N, respectively; and the corresponding maximum no-load speed are tested to be about 891.3 mm/s, 788.5 mm/s and 396.8 mm/s. The force weight ratio is a key parameter to evaluate a piezoelectric actuator. Table 1 gives a detail comparison between the proposed standing wave linear piezoelectric actuator and a previous work [17], from which we can see that the proposed actuator achieves a large improvement on the force weight ratio under a small size, which illustrate the merit of the proposed linear piezoelectric actuator operated under longitudinal-bending coupling mode. 6. Conclusion A longitudinal-bending coupling mode of beam was proposed and discussed in this work, which had intrinsic difference with the pure longitudinal mode or the pure bending mode. By applying unsymmetrical boundary condition, longitudinal and bending modes with close resonance frequncies were combined together to become a special coupling mode; and the independence of different vibration modes was broken. A standing wave piezoelectric actuator using the longitudinal-bending coupling mode was designed and tested, whose unique feature was that only one group of PZT ceramic elements excited by a sine voltage generated axial and transverse vibrations on the two tip ends of a sandwich transducer. Oblique elliptical movements were produced on the two tip ends under the coupling mode, which were symmetrical in space. The working frequency proposed piezoelectric actuator was designed to be about 28.887 kHz, and tested to be 28.875 kHz. Both longitudinal and bending vibrations were measured on the transducer by a vibrometer. The prototype achieved maximum no-load speed of 891.3 mm/s under preload of 100N, and a maximum thrust force of 39.2 N was obtained under preload of 300N. This work not only gives a new configuration of linear standing wave piezoelectric actuator, but also provides a new idea and discussion about the coupling effect of different vibration modes in an elastic beam. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (No. 51475112 and No. 51622502), in part by the Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 201428), in part by the Foundation for
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Biographies
Yingxiang Liu was born in Hebei province, China, in 1982. He received the B.S., M.S. and Ph.D. degrees from the School of Mechatronics Engineering at Harbin Institute of Technology, China, in 2005, 2007 and 2011, respectively. He is currently a professor of the School of Mechatronics Engineering at the Harbin Institute of Technology. He is the vice director of the Department of Mechatronic Control and Automation. He is also a member of the State Key Laboratory of Robotics and System at Harbin Institute of Technology. He joined the School of Mechatronics Engineering, Harbin Institute of Technology in 2011, where he has been a professor since December 2013. He was a Visiting Scholar at the Mechanical Engineering Department, University of California, Berkeley, from August 2013 to August 2014. His research interests include piezoelectric actuating, ultrasonic motor, piezoelectric actuator, precision actuating, piezoelectric micro jet, bionic robot, fish robot and soft robot.
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Shengjun Shi was born in Heilongjiang province, China, in 1974. He received the B.S. degree in aircraft manufacturing engineering from Northwestern Polytechnical University in 1997. He received his M.S. and Ph.D. degrees from the School of Mechatronics Engineering, Harbin Institute of Technology, China, in 2003 and 2007 respectively. He is currently an associate professor in the Harbin Institute of Technology, China. His research interests include ultrasonic motor, ultrasonic application etc.
Weishan Chen was born in Hebei, China, in 1965. He received his B.S. and the M.S. degrees in precision instrumentation engineering, and the Ph.D. degree in Mechatronics engineering from Harbin Institute of Technology, China, in 1986, 1989, and 1997, respectively. Since 1999, he is a professor with the School of Mechatronics Engineering, Harbin Institute of Technology. His research interests include ultrasonic driving, smart materials and structures, bio-robotics.
Chunhong Li was born in Yunnan province, China, in 1991. He received the B.S. degree from the School of Mechanical Engineering, Tongji University, China in 2014. He is currently a M.S. degree candidate in the Harbin Institute of Technology, China. His research interests include ultrasonic motor and piezoelectric actuating.
Junkao Liu was born in Hebei, China, in 1973. He received his B.S. and Ph.D. degrees from the School of Mechatronics Engineering, Harbin Institute of Technology, China, in 1995 and 2001, respectively. Since 2011, he is a professor with the School of Mechatronics Engineering, Harbin Institute of Technology. His research interests include ultrasonic driving, biomimetic robots and simulation of multi-degree of freedom parallel mechanism.