A Piezoelectric Linear Actuator Controlled by the Reversed-Phase Connection of Two Bimorphs

A linear actuator controlled by the reversed-phase connection of two piezoelectric bimorphs is proposed in this study to solve the problem of drawback in the existing piezoelectric linear actuators. The actuator is driven by one excitation signal, the two piezoelectric bimorphs will always bend in opposite directions by the reversed-phase connection, so the directions of the force output by the two piezoelectric bimorphs are opposite. Because of the length difference of the clamping blocks, the two forces outputted by each bending is different in magnitude, and always have a resultant force in the same direction to make the actuator move forward two steps in one cycle without drawback. A series of experiments were conducted to evaluate the performance of the actuator provided. The starting voltage is 40 <inline-formula> <tex-math notation="LaTeX">$\text{V}_{\mathrm {p-p}}$ </tex-math></inline-formula>, resolution can reach <inline-formula> <tex-math notation="LaTeX">$1.16~\mu \text{m}$ </tex-math></inline-formula> without load. The load capacity of the actuator is 450 g with a 100 <inline-formula> <tex-math notation="LaTeX">$\text{V}_{\mathrm {p-p}}$ </tex-math></inline-formula> voltage, a 2-Hz frequency, and the average step displacement in this case is <inline-formula> <tex-math notation="LaTeX">$0.725~\mu \text{m}$ </tex-math></inline-formula>. The prototype has high linearity and good repeatability. Experiments have proved that the actuator controlled by reversed-phase connection can eliminate drawback in principle, and can move forward two steps in one cycle. The resolution of the prototype by the reversed-phase connection is much higher than the in-phase connection, and it is a new method to improve the driving performance of piezoelectric actuators.

Inertial piezoelectric actuators can be divided into signalcontrol type (SCT), mechanical-control type (MCT) and friction-control type (FCT) according to the control method. The signal control type is to pass an asymmetric signal to the piezoelectric element to make the vibration speed different in two directions, so that the actuator moves different distances in the two directions [13]- [16]. Li et al. proposed a parallelogram bending flexure hinge mechanism, which can output a minimum displacement of 0.04 µm with a sawtooth wave [13]. SCT has the advantage of high positioning The associate editor coordinating the review of this manuscript and approving it for publication was Hui Xie . accuracy, but it involves a signal generator to generate the asymmetric signal. Similar to SCT, MCT also moves the actuator with asymmetric driving force. But the travel difference comes from the asymmetric structure [17]- [21]. Shen et al. presented an inertial piezoelectric actuator based on asymmetric clamping materials. A minimum step displacement of the actuator under 2 µm can be achieved under a voltage of 15 V [18]. Bao et al. presented an inertial piezoelectric actuator equipped with different combinations of asymmetrical clamping structures and bias unit. At that a voltage of 50 Vp-p, a maximum angular velocity of 18.88 mrad/s and a minimum resolution of 2.8 mrad can be obtained [19]. Instead of asymmetric driving force, FCT actively produces asymmetric friction in the forward and backward direction to generate the travel difference [22]- [25], [31]. Wen et al. proposed a new inertial piezoelectric actuator based on the control of the friction force between the rotor and the supporting surface through the use of a triangular block. At that a frequency of 10 Hz and a volage of 15 V p−p , the prototype reaches the minimum step angle displacement, the minimum step angle displacement is 10 µrad [24].
A common problem of the above inertial piezoelectric actuators is drawback [26]- [28] which reduces the stability of actuators and brings limitations to their applications in critical situations like cell puncture. Theoretically, drawback can be eliminated if friction is greater than the driving force in backward direction. Researchers have originated many ways to eliminate or reduce drawback [29]- [36]. Some researchers adjust the driving force of backward to match to the friction by changing the voltage [22], [24], [29], [30]. Zhang et al. adjusted structural parameters and voltage to match friction, which also achieved the goal of no drawback [29]. This method can achieve no backward only when the friction force and the driving force are well adapted, but cannot be universally applied to various actuators. Another way to reduce drawback is to optimize the waveform of the drive signal [25], [31], [32]. Wang et al. proposed a driving method realized by a composite waveform which consists of a sawtooth driving wave and a sinusoidal friction driving wave for backward motion restraint of the actuator [31]. Cheng et al. proposed an actuator using the hybrid excitation. Compared with saw-tooth signal excitation, the step efficiency of the prototype was increased from 36.9% to 91.2% under the resonant/off-resonant hybrid excitation [25]. But the optimized waveform is often synthesized from a variety of simple signals, which is more complicated. Complex signals increase the cost and operation difficulty of the drive. In addition to the above two methods, in recent years, researchers have also used smart materials to reduce drawback [33], [34]. Wang et al. proposed an inertial piezoelectric actuator that used magnetorheological fluid to adjust friction, achieving the goal of no drawback, the prototype resolution reached 0.0204 µm [33]. This type of actuator also requires complex control signals, and the performance of the smart material itself needs to be studied. The sedimentation of the magnetorheological fluid also has an impact on the output performance of this type of actuator.
The above methods eliminate drawback only under certain working conditions. Aiming for a universal way of drawback control, this paper present a piezoelectric linear actuator controlled by the reversed-phase connection of two bimorphs. Two bimorphs of the actuator are asymmetrically clamped and connected in reversed-phase, so that the actuator always receives net force in the same direction. This method eliminates the backward phenomenon in principle, so there will be no backward under any frequency and voltage.

II. WORKING PRINCIPLE A. STRUCTURE DESIGN
The structure of the piezoelectric actuator recommended in this paper is shown in Fig. 1. The actuator is mainly composed of a guide, a base plate, a load plate, two support frames, four short-clamping, two piezoelectric bimorphs and some bolts and nuts. The two piezoelectric bimorphs are fixed to the support frames with four short clamping plates by bolts and nuts.

B. MOVEMENT PROCESS
Piezoelectric actuators usually use one piezoelectric bimorph or two piezoelectric bimorphs that provide torque in the same direction. The actuator recommended in this article was connected by reversed-phase circuit. The positive pole of PZT1 and the negative pole of PZT2 are connected to the same end of the power supply, while the negative pole of PZT1 and the positive pole of PZT2 are connected to the other end of the power supply, as shown in Fig. 2(a). In this way, the two piezoelectric bimorphs will always bend in opposite directions. The excitation signal input is shown in Fig. 2(b). In Fig. 2(c), the two bimorphs are marked PZT1 (left) and PZT2 (right), respectively. A symmetric square wave is shared by the two reversed-phase connected bimorphs, which provides the reversed-phase exciting signal, as shown in Fig. 2(a). The vibration state of the two piezoelectric bimorphs is shown in Fig. 2(c). There are four stages in a cycle: Holding stage I: The signal goes from ''a'' to ''b'' and the bimorphs hold the initial state.
Bending stage I: The signal goes from ''b'' to ''c''. Both bimorphs bend to the opposite position. According to previous studies [27], [29], the force pointing to the long-clamping direction is greater than that to the short-clamping direction.  Then F 1 is greater than F 2 . The net force F 1 − F 2 moves the actuator a displacement x 1 to the right, as shown in Fig. 2(c).
Holding stage II: The signal goes from ''c'' to ''d'' and the bimorphs hold the previous state.
Bending stage II: The signal goes from ''d'' to ''e''. Similar to bending stage I, the net force F 1 -F 2 moves the actuator a displacement x 1 to the right, as shown in Fig. 2(c).
The actuator moves two steps in a cycle consisting of the above four stages. Drawback is theoretically eliminated as F 1 is greater than F 2 . In case of bidirectional motion, another pair of piezoelectric bimorphs with short clampings on the right could be fixed on the same slider in parallel. This new pair of piezoelectric bimorphs will move the actuator to the opposite direction. Since the motion of the new pair works the same way to its counterpart, bidirectional motion is not further discussed in this study.  displacement of the prototype is measured by the laser sensor and further processed with LK-Navigator software.

A. EXPERIMENTAL SYSTEM
A zoomed picture of the prototype is shown in Fig. 3. The dimensional parameters and materials of the components are listed in Table 1. The load plate, the support frame, the base plate and the four short-clampings are all made of acrylic. The clamping difference between the long-clamping and short-clamping is 5 mm. The bolts and nuts are made of steel, and piezoelectric bimorphs are composed of PZT-5H and copper substrate. PZT 5H coupling factor: K p = 0.66,

B. FREQUENCY CHARACTERISTIC
Resonance frequency is an essential parameter to piezoelectric actuators. Resonant actuators work at resonance frequency for high transmission efficiency and output performance. Stepping actuators, usually non-resonant ones, try to keep away from the resonance frequency for sake of positioning precision. In order to find the resonance frequency of the prototype before frequency characteristic tests, an impedance analyzer was used to measure the impedance (Z ) and the phase angle (PH) of the prototype at each frequency. Fig. 4 shows the impedance and phase angle of the prototype at each frequency. Due to the lagging characteristic of the capacitance response, the steady response phase angle of the prototype is around −87 • . When the frequency is 462 Hz, the response phase angle suddenly changes to about −10 • , and the transmission efficiency reaches the maximum. It can be seen that the first order resonance frequency of the prototype is 462 Hz. Frequency characteristic of the actuator displacement peaks to 5.72 µm at 12 Hz. And the  average step at 80 Vp-p is illustrated in Fig. 5. It can be seen that the average step displacement reaches the minimum of 1.44 µm at 16 Hz. As frequency increases, there is no obvious law between the average step displacement and frequency. The stable range of frequency is from 2 to 7 Hz where the step displacements are all around 4 µm. Working frequency of the proposed actuator is recommended to select from this range.  step displacement of in-phase connection is larger than that of single bimorph, and reversed-phase connection shows the smallest average step displacement. In addition, when the voltage is more than 80 V p−p , the moment of inertia produced with in-phase connection far exceeds the moment of gravity, which causes the prototype to vibrate violently on the track, basically no forward displacement. The starting voltage of reversed-phase connection of reversed phase connection is 40 V p−p , which is higher than the staring voltage of in-phase connection and single bimorph. The minimum average step displacement is 1.16 µm without load.

D. LOAD PERFORMANCE
In order to measure the output performance of the actuator, an investigation on the bearing capacity of the piezoelectric actuators was conducted. Fig. 7 shows the load characteristic of the piezoelectric actuator under the frequency of 2 Hz and the driving voltage of 100 V p−p . Obviously, the average step displacement decreases as load is increased and the actuator cannot work with the load above 450 g. The minimum average step displacement of the actuator controlled by reversed-phase connection is 0.725 µm by 450-g load.

E. OUTPUT STEPPING CHARACTERISTIC
Stepping characteristic test aims to verify the basic features of reversed-phase connection excitation: two steps each cycle and zero drawback. This experiment is conducted at a frequency of 2 Hz, a voltage range of 40-80 V p−p and no load. Fig. 8 shows the output stepping characteristic of the actuator. This figure illustrates that the actuator moves two steps forward regularly without drawback, and the output movement is relatively stable. Fig. 9 shows the stepping characteristic curve of the actuator at the frequency of 1 Hz, 2 Hz and 3 Hz under 80 V p−p .  In order to compare the linearity of the stepping curves, linear fitting was conducted for both the start and end point of each step. The dotted line below the stepping curve represents the fitting line of start points, marked R 2 s, and the dotted line above the stepping curve represents the fitting line of end points, marked R 2 e. R-squares of linear fittings of all three frequencies are above 0.99, demonstrating a good linearity. This indicates that the actuator works quite stable at the selected test conditions. The maximum R-square is 0.99984 happening at 2 Hz.

G. REPEATABILITY
Repeatability is an important indicator to evaluate the stability of piezoelectric. Two indexes, fluctuation range and standard deviation, are selected to verify the repeatability of the proposed actuator. Fluctuation range is calculate d with equation 1: where x max and x min are the maximum and minimum value of the 10 repeat tests.   Standard deviation S is defined by equation 2: where N denotes the number of repetitions, x i is the average step displacement of the i th experiment, and x is the average step displacement of N experiment times. The experiment repeated 10 times with voltages of 40, 60, 80 V p−p and the frequency of 2 Hz. Each test recorded 20 steps started from the same position. As shown in Fig. 10, both fluctuation range and standard deviation decrease as the voltage goes lower. The standard deviation is reduced from 0.09 to 0.07 µm and the fluctuation range is reduced from 0.24 to 0.21 µm.

A. COMPARISION OF IN-PHASE CONNECTION AND REVERSED-PHASE CONNECTION
In order to better understand the influence of bimorph connections, the output stepping characteristics of in-phase connection, reversed-phase connection and single bimorph were tested under the frequency of 2 Hz and the voltage of 40 V p−p . As shown in Fig. 11, both in-phase connection and single bimorph move a big step forward and a small step backward, whereas reversed-phase connection moves two steps forward. Reversed-phase connection eliminates drawback. The drawback rate, formula is as follows: where x 1 represents the forward average step displacement, x 2 is the average step displacement of drawback. The drawback rate of actuator by one single bimorph is 34.62%, and the drawback rate of actuator by in-phase connection is 21.76%. In addition, the resolution of actuator by reversed-phase connection is higher than that of in-phase connection and single bimorph, which makes reversed-phase connected actuators suitable for precise positioning. Table 2 shows the performance comparison of the proposed and other actuators. The actuator proposed in this paper can motion without drawback in any state by a simple control signal, and it moves forward two steps in a cycle. In addition, the actuator proposed in this article is also competitive in minimum average step displacement. Although reversed-phase connection demonstrates good performance for asymmetric clamping piezoelectric actuators, it is not a universal way of eliminating drawback for inertial piezoelectric actuators. The key factor of reversed-phase connection is to neutralize the backward force by the forward force, which demands simultaneous actions of the two piezoelectric elements. For stick-slip actuators, e.g. sawtooth ones, the forward impact force happens instantly, but the backward impact force lasts for a short period. Neutralization in this case can hardly be fully reached. Thus it is not applicable to every piezoelectric actuator.

V. CONCLUSION
A piezoelectric linear actuator with two asymmetrically clamped bimorphs in reversed-phase connection is presented in this paper. Characteristic tests were conducted on a prototype with the proposed connection. Comparison tests of in-phase connection, reversed-phase connection and single bimorph were also provided. The following conclusions are verified by experiments: 1) The piezoelectric actuator driven by reversed-phase connection can move forward two steps in a straight line each cycle without drawback.
2) The maximum load of the actuator is 450 g with a 100-V p−p voltage, a 2-Hz frequency, and the average step displacement in this case is 0.725 µm.
3) The proposed actuator shows preferable repetition in terms of fluctuation range and standard deviation.