Nonlinear piezoelectric behavior of ceramic bending mode actuators under strong electric fields

JOURNAL OF APPLIED PHYSICS VOLUME 86, NUMBER 6 15 SEPTEMBER 1999 Nonlinear piezoelectric behavior of ceramic bending mode actuators under strong electric fields Qing-Ming Wang, a) Qiming Zhang, Baomin Xu, Ruibin Liu, and L. Eric Cross Intercollege Materials Research Laboratory, The Pennsylvania State University, University Park, Pennsylvania 16802-4801 Received 18 December 1998; accepted for publication 9 June 1999 The nonlinear electromechanical behavior of cantilevered piezoelectric ceramic bimorph, unimorph, and reduced and internally biased oxide wafer actuators is studied in a wide electric field and frequency range. It is found that under quasistatic condition, linear relationships between actuator tip displacement-electric field, and blocking force-electric field are only valid under weak field driving. With increasing the driving field, electromechanical nonlinearity begins to contribute significantly to the actuator performance because of ferroelectric hysteresis behavior associated with piezoelectric lead zirconate titanate PZT -type ceramic materials. The bending resonance frequencies of all these actuators vary with the magnitude of the electric field. The decrease of resonance frequency with electric field is explained by the increase of elastic compliance of PZT ceramic due to elastic nonlinearity. Mechanical quality factors of the actuators also depend on the magnitude of electric field strength. No significant temperature increase is observed when actuators are driven near resonance frequency under high electric field. 1999 American Institute of Physics. S0021-8979 99 00518-6 I. INTRODUCTION Actuators, sensors, or transducers are integral parts of smart systems or structures which play the key roles in making the system smart or adaptive. A material which can sense and response to one or more than one of such stimuli as pressure, temperature, voltage, electric and magnetic fields, chemicals, nuclear radiation, etc. can be called smart material. The sense or response of the smart materials to external stimuli is primarily done by changing their physical properties or modifying one or more than one property coefficients of the smart materials. 1,2 Sensors are usually specifically designed and fabricated from a certain type of smart materials to more effectively sense certain external stimuli usually by converting the stimuli into electrical signals thus sensitivity and selectivity are the primary concerns in practical applications; while actuators are designed to respond usually a mechanical response such as shape change, mechanical motion, force, displacement, etc. to system stimuli, response speed and output force or displacement are concerns in general. Smart materials and systems have been gaining worldwide attention over the past few years because of their applications in almost every branch of engineering. 3 These applications range from smart skins for submarines to controlling large structures in space. Active acoustic and vibration control is one area, which benefits the most from the development of smart materials and undergoes great change due to new concepts and emerging technologies. For example, absorbing the vibration at some points of the structure a Author to whom correspondence should be addressed; Currently affiliated with Lexmark International, Inc., Lexington, KY 40550; electronic mail: qmwang@lexmark.com by active control is more effective when the vibration field is sensed at the same location. 4 This is now possible by using smart materials. With the development of actuators and sensors, new techniques are being developed for active acoustic and vibration control where these actuators and sensors can be used. 5 7 There are several classes of active materials which have been extensively studied in recent years for actuator and sensor applications. Examples of active materials include piezoelectric and electrostrictive ceramics, crystals, and polymers; magnetostrictive materials; shape memory alloys; electrorheological fluids; and optical fibers, etc. Detailed discussion on these materials and their operational mechanism for actuator and sensor applications have been reviewed elsewhere. 1,8 10 Among these materials, piezoelectric ceramics demonstrate unique electromechanical properties which are especially suitable for various smart system applications. Advantages of piezoelectric actuators include being capable of large stresses, high response speed, reasonable efficiency, and also being inexpensive, light-weight, space efficient, easily shaped, and bonded to or embedded in a variety of surfaces. Furthermore, piezoelectric actuators possess distributed character which allows them to be tailored to selectively reduce structural modes with little control spillover. Therefore, piezoelectric ceramic sensors, actuators, and transducers are receiving much attention for smart structure and systems applications such as active noise control and active vibration damping technologies. Piezoelectric materials belong to a class of materials without crystallographic centrosymmetry such that electrical charge or voltage can be generated when they are subject to external stresses direct effect. 11 A converse effect also exists in these materials, i.e., mechanical deformation occurs when piezoelectric materials are driven by an electric field. 0021-8979/99/86(6)/3352/9/$15.00 3352 1999 American Institute of Physics

J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. 3353 FIG. 1. Schematical drawing of cantilever mounted piezoelectric bimorph, unimorph, and RAINBOW actuators. The principle of operation of piezoelectric actuators is accordingly, the converse piezoelectric effect. With piezoelectric actuators, the electrical energy is converted into mechanical energy. On the other hand, piezoelectric sensors are based on a direct piezoelectric effect, through which mechanical energy is converted into electrical energy. 8 For linear piezoelectrics, strain is linearly related to the electric field. Therefore, the displacement produced by an actuator can be easily controlled by changing the applied electric field. According to their displacement mechanism, piezoelectric ceramic actuators can be classified into three categories: bimorph-type bending actuator, multilayer stacked actuator, and moonie-type flextensional actuators. 2,10,12 Bimorph-type bending mode actuators can produce the largest displacement but the generative force is very small. With multilayer actuators, the largest generative force can be produced but a displacement level is usually much lower than bending mode actuators. The flex tensional piezoelectric actuator was developed to bridge the gap between multilayer actuators and bending mode actuators, 2,9 i.e., to provide a relatively large displacement and an intermediate level of generative force. All these actuators have been used in a variety of applications, among which bending mode actuators are most widely used in smart materials and structure applications because of their excellent electromechanical properties, easy fabrication, and design flexibility. Usually, a high electric field is applied to the actuators to generate sufficient displacement or force in practical applications. Under these circumstances, material properties or parameters provided by manufacturers are no longer applicable to describe actuator performance since they were measured at a weak signal level. Recently, a new type of air acoustic transducer was developed as an auxiliary sound source for low frequency active noise systems, which utilizes bimorph or unimorph bending actuators as the driving elements. 13,14 Up to 300 m displacement amplitude has been achieved by combining both bending and flextensional mechanisms in a transducer design to obtain sufficient sound pressure level and suitable radiation characteristics at a low frequency range for active noise cancellation. To optimize the actuator or transducer performance, a detailed study on the performance of the bending mode actuators, especially under high field driving, is necessary. In this article, the performance of cantilever mounted piezoelectric bending actuators will be analyzed with an emphasis on the electromechanical nonlinear behavior of actuators since in most practical applications, actuators are driven under high electric fields where a piezoelectric response may be significantly different with the case of low field driving. II. PERFORMANCE ANALYSIS OF BENDING MODE ACTUATORS For piezoelectric actuators, three characteristic parameters, i.e., displacement, generative force, and response speed resonance frequency, are of primary concern in practical applications. Largest possible generative displacement and force are desired. However, the direct piezoelectric strain is often quite small, in the range of 10 4 to 10 3, even under a high electric field for materials with high piezoelectric coefficients such as PZT s, which is often insufficient for applications requiring large mechanical strain or displacement level. While extensive effort is conducted for searching new or improved materials with an even higher piezoelectric response, strain/displacement amplification mechanisms are also developed in the actuator design. A bending actuation mechanism is an effective way to achieved large displacement in piezoelectric actuator design. Bimorph and unimorph are the typical examples of bending mode actuators, which are constructed by bonding together two thin piezoelectric plates or one piezoelectric plate and one elastic plate in such a manner that when the actuators are driven, the piezoelectric transverse strain will be converted to large bending displacement in the perpendicular direction due to the constraining of each component in the actuator structure. The structures of cantilever mounted bimorph and unimorph

3354 J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. actuators are schematically shown in Fig. 1. Two different electrical connections are usually used in bimorph fabrication: one is a series connection in which two piezoelectric plates have opposite polarization direction and the actuator is driven by applying an electric field between the top and bottom electrodes Fig. 1 a ; the other is a parallel connection in which two piezoelectric plates are with the same polarization direction and the actuator is driven by applying an electric field between surface electrodes and the bonding layer Fig. 1 b. In the latter case, two ceramic plates are electrically connected parallel and driving voltage is applied across half the actuator thickness, thus enabling half driving voltage to achieve the same electric field as in the series case. Usually, a metallic sheet called a metal shim is sandwiched between the two piezoelectric plates to increase the reliability and mechanical strength. In a unimorph actuator, one metal layer and one ceramic layer are bonded together, when an electric field is applied on the ceramic layer, a bending deformation will also be generated as in the case of the bimorph actuator Fig. 1 c. It should be noted that in the actuator design, large displacement is usually achieved with sacrificing generative force. Therefore, a compromise has to be made in designing an actuator to meet the requirements of specific applications. The tip deflection, blocking force at static condition, and fundamental bending resonance frequency of cantilever mounted bimorph and unimorph actuators have been derived in earlier articles to describe their electromechanical performance. 15 18 For a bimorph with a sandwiched elastic layer, tip deflection, blocking force, and resonance frequency can be written as and 3L2 2t 1 B 2B 1 AB 3 3B 2 3B 1 d 31E 3, F bl 3wt2 E p 2B 1 8L B 1 d 31E 2 3, f r i 2 t 4 L 2 E p 3 p 1 2 1 3 1 2B 2 4AB 3 1/2, 3 4 1 B 2 BC 1 where L, w, and t are actuator length, width, and total thickness, A E m /E p s E 11 /s m 11 is Young s modulus ratio of elastic metal layer and ceramic layer, B t m /2t p is thickness ratio, and C m / p is density ratio; d 31 is transverse piezoelectric coefficient and E 3 is applied electric field. In Eq. 3, i is the eigen value where i is an integer that describes the resonance mode number; for the first and second modes, 1 1.875 and 2 4.694. Usually the polymeric epoxy bonding layer is very thin, in the range of 5 15 m. Its effect on actuator performance can be neglected, therefore, t m 2t p t. When no elastic layer is used, i.e., t m 0, B 0, and t 2t p, the above equation can be simplified to 3L2 d 2t 31 E 3, 4 F bl 3wt2 E p d 8L 31 E 3, and f r 3.52t E p. 6 4 L 2 3 p For a unimorph actuator consisting of a piezoelectric layer and an elastic layer, tip deflection, blocking force and resonance frequency can be written as and 3L2 2t F bl 3wt2 E p 8L 2AB 1 B 2 A 2 B 4 2A 2B 3B 2 2B 3 1 d 31E 3, 2AB AB 1 1 B d 31E 3, f r i 2 t E p 4 L 2 3 p A2 B 4 2A 2B 3B 2 2B 3 1 1 BC AB 1 1 B 2 5 7 8 1/2, 9 where B t m /t p is the thickness ratio of the piezoelectric layer and elastic layer in the unimorph actuator. Obviously, the difference in performance between a bimorph and a unimorph actuator comes from the nondimensional terms which are functions of Young s modulus ratio, thickness ratio, and density ratio of piezoelectric and elastic layers. It can be verified that if the total thickness of an actuator is a constant, maximum tip displacement and blocking force of a unimorph actuator can be obtained by choosing a suitable thickness ratio 18 B max 1 A. 10 Under this condition, the neutral axis or plane of the unimorph actuator is at the interface of piezoelectric layer and elastic layer, and the maximum tip displacement is half the value of that of a bimorph actuator with the same geometrical dimension. The use of a stiff elastic layer leads to large blocking for unimorph actuators. A reduced and internally biased oxide wafer RAINBOW actuator, shown in Fig. 1 d, is a recently developed unimorph-type actuator fabricated by a special high temperature reduction processing 19,20 in which a piezoelectric lead zirconate titanate PZT or lanthanum doped lead zirconate titanate PLZT ceramic wafer is directly placed on a flat graphite plate and treated at approximately 1000 C for a period of time. As a consequence of the high temperature chemical reduction processing, a conducting and electromechanically inert layer is formed in the piezoelectric ceramic wafer. Internal thermal stresses are also developed after cooling down to room temperature due to a thermal expansion coefficient mismatch between the reduced layer and the remaining piezoelectric layer. Equations 7 9 are applicable to the cantilever piezoelectric RAINBOW actuator. However, the value of d 31 coefficient of the remaining piezoelectric layer could be slightly different because of the effect of

J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. 3355 high temperature treatment and the existence of internal stresses. The Young s modulus of the reduced layer has been determined for optimizing actuator performance. 21 The tip deflection and blocking force of the RAINBOW actuators have been studied under a quasistatic condition. 22 As described in above equations, the tip deflection and blocking force of cantilever bending actuators are proportional to the transverse piezoelectric coefficient d 31 of the piezoelectric ceramic layer and the driving electric field E 3, and are also functions of Young s modulus ratio and thickness ratio of the two composed layers. The analysis on experimental results can be made based on these relationships. FIG. 2. Setup for the measurement of actuator tip displacement and blocking force. III. EXPERIMENTAL PROCEDURES A. Sample preparation Cantilever bimorph, unimorph, and RAINBOW actuators with the same geometrical dimensions are prepared using soft PZT ceramics PZT 3203 HD, Motorola, Albuquerque, NM. For bimorph actuator fabrication, PZT bulk ceramics are cut into thin plates 40.0 mm 7.0 mm 0.5 mm. Gold electrodes are then sputtered on the major surfaces. These plates are poled along the thickness direction under a dc electric field of 2.0 kv/mm, temperature of 80 C in poling oil for 5 min. A series-type configuration is adopted in bimorph actuator fabrication in which two PZT plates with antiparallel polarization direction are bonded using epoxy resin. For unimorph actuator fabrication, a polished stainless steel plate 0.38 mm in thickness is bonded using epoxy resin with a poled ceramic plate with dimensions of 40.0 mm 7.0 mm 0.68 mm. The Young s modulus ratio A of stainless steel and soft PZT ceramics is about 3.14. The selection of thickness of the stainless steel and ceramic plates is based on Eq. 10 in order to achieve maximum tip deflection and generative force. After bonding, both bimorph and unimorph are cured at 80 C for 24 h. The thickness of the epoxy bonding layer is approximately 20 m or less, measured by a micrometer and an optical microscopy. For RAINBOW actuator fabrication, a rectangular ceramic plate with 1.02 mm thickness is placed on a flat graphite block. This assembly is increased to a temperature of 975 C for 5 h, where reduction reaction occurs at the ceramic graphite interface. As a result, a monolithic composite structure is formed with an electrically conducting but electromechanically inert layer formed at the bottom side of PZT ceramic plate. After rapidly cooling down to room temperature, bending deformation is developed due to the internal thermal stresses. A RAINBOW sample 40.0 mm in length and 7.0 mm in width is then cut from the reduced ceramic plate. A sharp interface is observed and a reduced layer thickness t r 0.30 mm is measured by using an optical microscopy. Because the Young s modulus ratio of the reduced layer and the remaining piezoelectric PZT layer is about 0.85, which was determined previously, 21 the reduced layer thickness here is less than the value for maximum tip deflection or generative force. Gold electrodes are sputtered on the surfaces and the sample is then poled under a dc electric field of 2.0 kv/mm. B. Measurements The bending resonance frequencies of the actuators under weak field excitation are measured by using HP 4194 impedance/phase analyzer, while under high electric field driving, resonance frequency is determined by measuring the variation of actuator tip displacement with frequency. At resonance frequencies, drastic changes can be observed for electric impedance and its phase angle, as well as tip displacement. Therefore, the resonance frequencies can be precisely determined. For actuator tip displacement measurement, an MTI- 2000 Fotonic sensor systems MTI Instrument, Lathem, New York is used. A small thin glass coated with gold was attached on the actuator tip as a mirror for reflecting the perpendicularly incident light from optic fiber sensor. When actuator vibrates under ac electric field driving, the distance between probe and mirror changes periodically, the reflected light is detected by the sensor and transferred into voltage signal. The sensor output voltage is proportional to the vibration displacement in its linear working range. Three probe modules, 2032RX, 2032R and 2062R are used in a measurement, which covers a wide measurement range from 2.5 Å to more than 1 mm. The measurement setup is schematically shown in Fig. 2. The sensor probe is attached on a micropositioner Ealing Electro-Optics, Inc., Auburn, California which enables three-dimensional adjustment of the probe. A function generator Model DS345, Stanford Research Systems, Inc., Sunnyvale, California is used to generate a small ac signal which is then amplified through a power amplifier 790 series, PCB Piezotronics, Depew, New York. The output of the amplifier is applied on the actuator. A lock-in amplifier Model SR830 DSP, Stanford Research Systems, Inc., Sunnyvale, California which synchronized with the driving signal frequency is used to measure the output signal from the MTI sensor. The blocking force is measured by using a force load cell Entran PS-15, Entran Devices, Inc., Fairfield, NJ which has a sensitivity of 2.90 mv/n for 15.0 V driving. The force probe is bonded on the tip of the actuator by using a fast bonding epoxy. When the actuator is driven by the electric field, no tip displacement is observed. The output signal of force cell is the measure of blocking force.

3356 J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. FIG. 4. Tip displacement of a RAINBOW actuator is plotted against driving frequency at several driving field. FIG. 3. Electrical impedance/phase spectra for bimorph and unimorph actuators under weak field excitation. IV. RESULTS AND DISCUSSIONS A. Frequency characteristics of the bending mode actuators At weak signal excitation, the resonance characteristic of bending actuators are measured using HP4194 impedance/ phase analyzer. The impedance/phase-frequency spectra of bimorph and unimorph actuators are shown in Fig. 3, in which both the fundamental and the second harmonic modes are shown. At bending resonance frequency, changes in impedance and phase can be observed. The impedance minimum peak corresponds to the resonance frequency, while impedance maximum corresponds to antiresonance frequency. The fundamental and second harmonic bending resonances are 314 and 2033 Hz for the bimorph actuator, and 413 and 2700 Hz for the unimorph actuator. The dimensions of cantilever mounted bimorph and unimorph actuators used here are 38.0 mm 7.0 mm 1.02 mm, and 38.0 mm 7.0 mm 1.08 mm. These results agree well with theoretical calculation. Obviously, the bending resonance frequency can be adjusted by changing the actuator length and thickness, as well as the Young s modulus and thickness ratio of elastic layer and ceramic layer for the unimorph actuator, as discussed in Sec. II. Since in most cases, bimorph and unimorph actuators are operated near the fundamental resonance frequency of bending vibrations, resonance characteristics under high field are also important. The values of the fundamental resonance frequency and the mechanical quality factor are usually used to characterize resonance properties. The bending resonance of an actuator under high field condition is observed by monitoring tip deflection varying with frequency under different electric field level. In Fig. 4, the tip deflection of a RAINBOW actuator is plotted against driving frequency at several driving field levels. It is found the bending resonant frequency f r is strongly dependent on the magnitude of the driving field. With electric field increasing, the fundamental bending resonance frequency decreases. Similar results are observed for both bimorph and unimorph actuators. In Fig. 5, the fundamental bending resonance frequencies are plotted as a function of electric field for bimorph, unimorph, and RAINBOW actuators. From Eqs. 3, 6, and 9, the fundamental bending resonance frequency of a cantilever actuator is a function of actuator thickness, length, density, and elastic compliance. FIG. 5. Bending resonant frequencies decrease with electric field for bimorph, unimorph, and RAINBOW actuators.

J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. 3357 When actuators are deformed under various electric fields, the actuator length, thickness and density will be forced to change their values. However, the tiny changes in these parameters seem not likely to significantly decrease the resonance frequency. Therefore, the decrease of f r with driving field can be mainly attributed to the variation of the elastic compliance with driving field, i.e., the nonlinear elastic behavior of ferroelectric PZT ceramics. In Eqs. 3, 6, and 9, s E 11 ( 1/E p ) is the linear elastic compliance. When a piezoelectric ceramic actuator is driven under a high electric field, nonlinear elastic coefficients have to be taken into account. In the case of narrow cantilever bending actuators such as bimorph, unimorph, and RAINBOW, the electric field E 3 is applied along the thickness polar axis, E 1 E 2 0, and the mechanical stress acts only along the length, i.e., X 1 0, the strain can be expressed as: 23 x 1 s E 11 X 1 d 31 E 3 s E 111 X 2 2 1 2d 311 X 1 E 3 R 331 E 3 s E 111 X 3 1..., 11 where high order elastic, piezoelectric and electrostrictive terms are included. Equation 11 can be rewritten as: x 1 s E 11 E,X X 1 d 31 E,X E 3. 12 Obviously, for nonlinear materials, s E 11 (E,T) is dependent on electric fields and stresses. The nonlinear elastic coefficients are directly related to and account for the shift of resonant frequency under various electric fields. Actually, previous research also demonstrated that the effective elastic compliance s E 11 of ferroelectric PZT ceramics increases with electric field, i.e., the ferroelectric material becomes softer when the higher field is applied. 24 If an oscilloscope is used to monitor the frequency spectrum of actuator tip displacement, a small peak can be observed at f r /2. This is an evidence of the contribution of higher order elastic coefficients to the performance of the piezoelectric actuator under high electric field, as demonstrated by Bryant and Keltie. 25,26 From Fig. 5, it is found that the resonance frequency of bimorph actuators decreases faster than unimorph and RAINBOW actuators. This observation is reasonable since the elastic compliance of the stainless steel layer in the unimorph and the reduced layer in the RAINBOW actuator do not change with electric field. As laminate composite structures, the effective elastic compliance of the unimorph and RAINBOW actuators increase more slowly with electric field than that of the bimorph actuator. Therefore, the resonance frequency of bimorph actuator decreases faster than both unimorph and RAINBOW actuators. The mechanical quality factor, Q, governing the sharpness and magnitude of the resonance of the cantilever bimorph, unimorph, and RAINBOW actuators can be calculated by Q f r, 13 f r1 f r2 where f r is the resonant frequency, f r1 and f r2 are the frequency at which the deflection magnitude drops of 0.707 of its peak values. 25 As determined from the deflectionfrequency spectra, Q is plotted as a function of electric field FIG. 6. Mechanical Q factors decrease with electric field for bimorph, unimorph, and RAINBOW actuators. for bimorph and unimorph in Fig. 6. Q factor decreases with electric field, indicating that large mechanical loss resulted in under a high field, which can be attributed the increase in air damping effect as well as actuator internal friction. B. Tip displacement and blocking force The tip displacement as a function of electric field for bimorph, unimorph, and RAINBOW actuators at quasistatic condition far off resonance frequency, f 10 Hz is shown in Figs. 7 a 7 c. It should be noted here that the rootmean-square rms values of the actuator tip displacements are used in these plots. Instead of a linear relationship as expected from Eqs. 4 and 7, a nonlinear relationship between tip displacement and electrical field is observed, i.e., as electrical field increases, actuator tip displacement increases rapidly. When the magnitude of the applied field is low, the experimental results agree well with theoretical calculation using Eqs. 4 and 7. However, as the magnitude of the electric field strength increases, the tip displacement deviates from the linear function of electric field, and the difference between measured results and linear theoretical calculation becomes larger at a higher field. This can be attributed to the nonlinear electromechanical response of PZT ceramics under a high field, i.e., the effective piezoelectric coefficient d 31 increases with electric field. The contribution of a higher order piezoelectric coefficient and elastic compliance under high field leads to the deviation of actuator performance from the linear theoretical calculation results, as indicated by Eq. 11. Previous studies have demonstrated that this nonlinearity is due to the extrinsic piezoelectric response related to ferroelectric non-180 domain wall and phase boundary motion 24 when actuators are driven under a high field. This nonlinear behavior is more significant in soft type PZT ceramics than undoped and hard PZT ceramics. Only when a weak electric field is applied for example, when E 3 5 V/mm, tip displacement is approximately a linear function of electric field, since under a weak field only intrinsic linear piezoelectric response contributes to piezoelectric strain and the input electric energy is not sufficient to generate extrinsic contribution.

3358 J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. FIG. 8. Tip displacement as a function of electric field near resonance frequency for a RAINBOW actuator. FIG. 7. Tip deflection is plotted against electric field for a bimorph, b unimorph, and c RAINBOW actuators. Obviously, bending actuators can produce a large tip displacement, and the nonlinear electromechanical responses of ferroelectric PZT ceramics significantly contribute to the generative displacement of the piezoelectric actuators. The tip displacement of RAINBOW actuator as a function of electric field near resonance frequency is shown in Fig. 8. Since resonance frequency decreases with driving field, the tip displacement at each electric field level is measured by adjusting signal frequency to obtain the maximum value. Some resonance frequency values at several driving field levels are also indicated in Fig. 8. It is found that tip displacement becomes gradually saturated with increasing magnitude of the electric field strength. This is because a higher driving field results in larger mechanical loss such as air damping and larger internal heat generation caused by ferroelectric hysteresis behavior. It is also noted that electric impedance of the actuator decreases with increasing driving field, which further facilitates the generation of joule heat. Similar results are observed in unimorph and bimorph actuator. The blocking force of the cantilever bimorph, unimorph, and RAINBOW actuators is depicted in Figs. 9 a 9 c. Similar to the tip displacement, the blocking force of these bending actuators also shows a nonlinear relationship with electric field. Again, this behavior is due to the piezoelectric nonlinearity of the ferroelectric PZT ceramics under strong field. Obviously, the force level which can be achieved by the cantilever bending mode actuators are rather low. In other word, the large displacement is achieved by sacrificing the force capability in these actuators. It should be noted that among all types of piezoelectric actuators, bending mode actuators have low electromechanical coupling, low output mechanical energy, and low energy transmission efficiency, since the generative force and displacement or bending moment is obtained by converting transverse piezoelectric stress or strain into bending deformation in the perpendicular direction through internal constraining mechanism. A large amount of mechanical energy is stored in the actuators themselves. Discussions on these issues are given in a separate articles 27 and are beyond the scope of this article. Actuator reliability is always of a concern in practical applications, especially in the case of bimorph and unimorph actuators where epoxy bonding is used in the actuator fabrication. In our experiments, no obvious performance degradation is observed under off-resonance driving condition up to 10 6 cycles for unimorph, bimorph, and RAINBOW actuators. Moreover, when actuators are driven near resonance under a high electric field bipolar sinusoidal wave, 150

J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. 3359 area and volume ratio (A/ ). The fundamental bending resonance of the cantilever actuators are in the range of 150 400 Hz in our experiments, depending on actuator dimension. In this low frequency range, heat generation and heat dissipation in the actuator can easily reach equilibrium without significant temperature rise. No mechanical failure delamination and fracture was observed in all actuators even under a near-resonance high field driving, indicating the actuators have good reliability. It should be pointed out that the reliability of bimorph and unimorph is largely dependent on the selection of the epoxy, the uniformity of the bonding layer, and the cleanliness of the surfaces of the ceramic or elastic plate before bonding. V. SUMMARY In this article, the bending resonance frequency, tip displacement, and blocking force of the cantilever mounted piezoelectric bimorph, unimorph, and RAINBOW actuators have been studied in a wide electric field and frequency range. It is found that with the same geometrical dimension and under the same applied electric field strength, cantilever bimorph actuator demonstrates larger tip displacement and blocking force than both unimorph and RAINBOW actuators. Under quasistatic conditions, a linear relationship between mechanical displacement and driving field is only valid under weak driving field. As the magnitude of electrical field strength increases, because of ferroelectric hysteresis behavior associated with PZT-type ceramic materials, electromechanical nonlinearity begins to contribute significantly to the actuator performance. The bending resonance frequencies of all these bending actuators vary with the magnitude of electrical field. The decrease of resonance frequency with electrical field is explained by the increase of elastic compliance of PZT ceramic due to elastic nonlinearity. Mechanical quality factor of actuators is also dependent on driving field. No significant temperature increase is observed when actuators are driven at resonance frequency under high electrical field because of large surface area and volume ratio in the bending actuator. FIG. 9. Blocking force is plotted against electric field for a bimorph, b unimorph, and c RAINBOW actuators. V/mm, their temperature increases only slightly about 5 10 C and then reaches thermal equilibrium. The heat generation is mainly due to mechanical loss and ferroelectric hysteresis loss. 28,29 Heat dissipation is proportional to the actuator surface area, while heat generation is usually proportional to actuator effective volume. For bending actuators, temperature rise is not significant because of large surface 1 R. E. Newnham and G. R. Ruschau, J. Am. Ceram. Soc. 74, 463 1991. 2 R. E. Newnham, Q. C. Xu, and S. Yoshikawa, US Patent No. 999,819 12 March 1991. 3 H. B. Strock, Am. Ceram. Soc. Bull. 75, 71 1996. 4 H. Hui-xiong Law, a Ph.D. thesis, Monash University, Australia, 1994. 5 T. Bailey and J. E. Hubbard, AIAA J. Guid. Control Dynamics 6, 606 1985. 6 H. S. Tzou, G. C. Wan, and C. I. Tseng, IEEE International Conference on Robotics and Automation, 14 19 May 1989 Scottsdale, AZ, unpublished, pp. 1716 1721. 7 E. Crawley and J. de Luis, AIAA J. 25, 1373 1985. 8 D. J. Taylor, Ph.D. thesis, The Pennsylvania State University, 1992. 9 A. Dogan, Ph.D. thesis, The Pennsylvania State University, 1994. 10 K. Uchino, Piezoelectric Actuator and Ultrasonic Motors Kluwer Academic, Boston, MA, 1996. 11 B. Jaffe, W. R. Cook, and H. Jaffe, Piezoelectric Ceramics Academic, London, 1971. 12 S. Takahashi, Jpn. J. Appl. Phys. 24, 41 1985. 13 B. Xu, Q. M. Zhang, V. D. Kugel, and L. E. Cross, Proc. SPIE 271, 388 1996. 14 Q.-M. Wang and L. E. Cross, Presented at the 98th Annual Meeting of the American Ceramic Society, Indianapolis, IN, 1996 unpublished. 15 J. G. Smits, S. I. Dalke, and T. K. Cooney, Sens. Actuators A 28, 41 1991a.

3360 J. Appl. Phys., Vol. 86, No. 6, 15 September 1999 Wang et al. 16 J. G. Smits and W. Choi, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 38, 256 1991b. 17 M. R. Steel, F. Harrison, and P. G. Harper, J. Phys. D: Appl. Phys. 11, 979 1978. 18 Q.-M. Wang and L. E. Cross, Ferroelectrics 215, 187 1998. 19 G. H. Haertling, Am. Ceram. Soc. Bull. 73, 93 1994a. 20 G. H. Haertling, Ferroelectrics 154, 101 1994b. 21 Q.-M. Wang and L. E. Cross, J. Appl. Phys. 83, 5358 1998. 22 Q.-M. Wang and L. E. Cross, J. Am. Ceram. Soc. 82, 103 1999. 23 H. Beige and G. Schmidt, Ferroelectrics 41, 39 1982. 24 S. Li, W. Cao, and L. E. Cross, J. Appl. Phys. 69, 7219 1991. 25 M. D. Bryant and R. F. Keltie, Sens. Actuators 9, 95 1986. 26 M. D. Bryant and R. F. Keltie, Sens. Actuators 9, 105 1986. 27 Q.-M. Wang, X. Du, B. Xu, and L. E. Cross, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 46, 638 1999. 28 J. Zheng, S. Takahashi, S. Yoshikawa, and K. Uchino, J. Am. Ceram. Soc. 79, 3193 1996. 29 S. Hirose, S. Takahashi, K. Uchino, M. Aoyagi, and Y. Tomikawa, Mater. Res. Soc. Symp. Proc. 360, 15 1995.

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