Design and Analysis of a Micro Displacement Actuator Based on Stick-slip Principle Using Piezoelectric Ceramic

This work introduces a triangular shape piezoelectric actuator using the principle of piezoelectric ceramic bending vibration and stick-slip motion. The lateral displacement of the driving foot is used to press the slider, while the vertical displacement is used to drive the slider. The structural dimensions are determined by simulation considering the required parameters. Finite element software is used to analyze the static performance. The dynamic performance of the actuator are simulated and analyzed, including its response to different input voltages and frequencies. Dynamic model provides a comprehensive understanding of the behaviour of the actuator during operation and further reveals the principle. The actuator combines piezoelectric bending vibration and stick-slip motion with the merits of compactness, high accuracy and fast response.


Introduction
Recently, there have been remarkable advancements in the field of piezoelectric actuators, driven by their distinct properties and wide-ranging applications [1].Piezoelectric materials, known for their ability to convert electrical energy into mechanical energy and vice versa, have garnered significant attention across various domains such as micro/nano manufacturing, biomedical engineering, robotics, and precision positioning [2].The realm of micro/nano manufacturing demands exceptional precision and accuracy, often at sub-micron or nanometer scales, relying heavily on the performance of actuation systems [3].In this regard, piezoelectric actuators prove to be a fitting choice due to their remarkable attributes, including high-resolution positioning, rapid response times, and precision at the nanometer level [4].Among the commonly employed driving mechanisms in piezoelectric actuators, both resonant and non-resonant actuators hold prominence [5].Resonant actuators harness the power of vibration resonance to achieve swift and efficient motion, whereas non-resonant actuators offer precise and controlled movement, independent of resonance effects [6].In the realm of non-resonant piezoelectric actuators, researchers have extensively focused on the development of stick-slip actuators [7].These actuators exploit the longitudinal expansion of piezoelectric stacks to induce deformation in the stator, thereby propelling the slider [8].However, a major limitation of stick-slip actuators lies in the bonding of piezoelectric stacks using adhesive materials [9].This bonding technique restrains the actuator's ability to withstand bending loads, consequently leading to fatigue damage over time, thereby reducing their suitability for practical applications [10].In order to surmount these limitations, scholars have turned to the adoption of bending piezoelectric ceramic actuators.
This work proposes a triangular shape piezoelectric actuator using stick-slip driving principle.The second section presents the structure and principle of the proposed actuator and determines the dimensions through simulations.The third section analyzes the static performance of the actuator.In the fourth section, the actuator is dynamically modeled.The fifth section concludes the paper.

Structure and Principle
The structure of the designed actuator is shown in Figure 1, which mainly consists of a driving foot and piezoelectric ceramic.The driving foot has internal threads, which cooperates with the bolts to realize the clamping and mounting of piezoelectric ceramic.The base has two through holes, the bolts go through the piezoelectric ceramic and the base to connect with the driving foot.The piezoelectric ceramics contain two polarization partitions with opposite directions.Two adjacent partitions of the piezoelectric ceramic are polarized in opposite directions.When the same electrical signal is applied to the multilayer piezoelectric ceramic, one side of the polarization partition becomes thicker and the other side becomes thinner, resulting in the overall bending vibration of the actuator.(2) t=t 0 -t 1 : with the slow increase of the driving voltage, the piezoelectric ceramic become thinner on one side and thicker on the other, and bending deformation occurs.As a result, the driving foot is displaced both laterally and vertically, and the lateral displacement is used to press the slider.The slider is driven by the vertical displacement of the driving foot.Due to static friction, the slider is driven forward to produce displacement d 1 as shown in Figure 2(b).
(3) t=t 1 -t 2 : The voltage decreases rapidly.The bending piezoelectric ceramic quickly returns to the initial state.Since the driving foot quickly returns to the original position, the friction between the driving foot and the slider is regarded as sliding friction, and the slider is driven backward by displacement d 2 , as shown in Figure 2(c).
At the end of one cycle, the slider moves downward by one effective step d (d 1 -d 2 ).In addition, when a sawtooth signal is continuously applied to the piezoelectric ceramic, the slider is driven by the actuator to achieve a large travel displacement.

Design and Analysis
The overall dimensions of the actuator are shown in Figure 3, with length of 40mm, width of 22mm, and height of 53mm.The effects of triangle angle and height L on the actuator output displacement are discussed.Changing the angle from 30° to 120°, the displacement results in both the pressure and driving directions are shown in Figure 4(a).It can be seen that the maximum displacement occurs in driving direction at angle of 60°, while the displacement change is not significant in pressure direction.The angle is set to 60° and the displacement results in both pressure and driving directions at L=26mm to 31mm are shown in Figure 4(b).It can be seen that the maximum displacement in driving direction occurs at L=29mm.Thus, the final structure is determined as L=29mm and an angle of 60°.

Theoretical Modeling
A dynamic model of the actuator is built to study its motion characteristics in Figure .6.The built dynamic system consists of piezoelectric ceramics, a stator and a slider.
where m P , k P and c P are the mass, stiffness and damping coefficients of the piezoelectric ceramic, m s1 , c s1 and k s1 is the mass, damping coefficient and stiffness of the stator, respectively.m s2 is the mass of the slider.F P is the output force of the piezoelectric ceramic.N is the contact force between the piezoelectric ceramic and the stator.f is the friction force between the stator and the slider.x s1 is the displacement of the stator, x s2 is the displacement of the slider.
The force F P caused by the piezoelectric ceramic can be calculated according to equation ( where n, d 33 and k P represent the layer number, piezoelectric coefficient and stiffness of the piezoelectric ceramic, respectively.U P is the applied voltage to the piezoelectric ceramic.Equation ( 1) is added to Equation (2).Equation ( 5) can be expressed as follows: The input voltage of the piezoelectric ceramic can be obtained by equation (6).
where R denotes the resistance of the driving circuit, k A is the amplification ratio of input voltage for the piezoelectric ceramic, C is the capacitance of the piezoelectric ceramic and U 0 stands for the initial input voltage.The Laplace transform is used to determine the relationship between x s1 (s) and U 0 (s) as shown in the following equation.7, has been developed and analyzed utilizing MATLAB-Simulink.The respective values for each parameter can be found in Table 1, where the operating conditions and significant parameters have been referred to from previous literature.The motion of the slider, depicted in Figure 8, reflects a sawtooth pattern for input voltages of 100V and 120V, with a frequency of 1Hz.From the simulation results, it can be seen that the displacements are in the form of stepwise type, which further reveals the working principle.

Conclusion
In this study, a triangular-shaped piezoelectric actuator utilizing the stick-slip driving principle was proposed, and its structure and principle were introduced.The dimensions of the actuator were determined through simulations, and its static performance was analyzed.A dynamic model of the actuator was developed, and its dynamic performance was investigated.The proposed actuator holds promising applications in various fields such as precision positioning, micro/nano manufacturing, and robotics.Future research efforts should focus on prototyping, control strategies, and practical applications.By further advancing these aspects, the full potential of the actuator can be realized.

Figure 1 .
Figure 1.Configuration of the piezoelectric actuator The working principle is shown in Figure 2 when a sawtooth wave voltage is applied.(1) t=t 0 : the whole actuator becomes stationary, as shown in Figure 2(a).(2)t=t 0 -t 1 : with the slow increase of the driving voltage, the piezoelectric ceramic become thinner on one side and thicker on the other, and bending deformation occurs.As a result, the driving foot is displaced both laterally and vertically, and the lateral displacement is used to press the slider.The slider is driven by the vertical displacement of the driving foot.Due to static friction, the slider is driven forward to produce displacement d 1 as shown in Figure2(b).(3) t=t 1 -t 2 : The voltage decreases rapidly.The bending piezoelectric ceramic quickly returns to the initial state.Since the driving foot quickly returns to the original position, the friction between the driving foot and the slider is regarded as sliding friction, and the slider is driven backward by displacement d 2 , as shown in Figure2(c).At the end of one cycle, the slider moves downward by one effective step d (d 1 -d 2 ).In addition, when a sawtooth signal is continuously applied to the piezoelectric ceramic, the slider is driven by the actuator to achieve a large travel displacement.

Figure 3 .
Figure 3. Size of the stator.

Figure 4 .
Figure 4. Structural design.(a)Effect of different angles on displacement.(b)Effect of different lengths on displacement.Static analysis is carried out by using finite element software to verify the design result.The material is AL7075, and the density, modulus of elasticity and Poisson's ratio are 2810 kg m -3 , 71.7 GPa and 0.33.The constraints are applied to the two holes of the flexure stator.The voltage was 200 V.The results are shown in Figure5, and the lateral and longitudinal displacements of the driving foot were obtained as 1.68 μm and 1.46 μm, respectively.The results show that the driving foot generates vertical and horizontal deformations under the bending deformation.The simulation results further validate the the working principle of the proposed piezoelectric actuator.

Figure 5 .
Figure 5. Static simulation results of the actuator.

Figure 6 .
Figure 6.Schematic diagram of the dynamic model.The mechanics equation can be described as:

Figure 7 .
Figure 7. Dynamic model of the actuator in Simulink.

Figure 8 .
Figure 8. Simulation results by Simulink of the actuator.The dynamic model, as depicted in Figure7, has been developed and analyzed utilizing MATLAB-Simulink.The respective values for each parameter can be found in Table1, where the operating conditions and significant parameters have been referred to from previous literature.The motion of the slider, depicted in Figure8, reflects a sawtooth pattern for input voltages of 100V and 120V, with a frequency of 1Hz.From the simulation results, it can be seen that the displacements are in the form of stepwise type, which further reveals the working principle.

Table 1
Parameters in Simulink simulation