# Electrochemical Coupled Analysis of a Micro Piezo-Driven Focusing Mechanism

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{−5}rad. Using traditional proportional integral differential control (PID) and fuzzy control, Pan et al. [25] proposed the parallel switching control strategy of the electric objective lens driven by an ultrasonic motor. The predictive control method of macro and micro phase fusion is established to realize the high precision and fast switching control of the objective lens converter. Addditionally, the repeated positioning error of the electric objective lens converter is less than 0.015°. For the drive system of the piezoelectric objective lens in digital confocal microscopy, Chen et al. [26] analyzed the control performance of the fuzzy PID controller and proposed the idea of off-line optimization. The optimized initial parameters are configured in the fuzzy PID control, the system is adjusted online and in real time, and the control system is driven by step positioning, which improves the response speed of the system.

## 2. Operating Principle of the Piezo-Driven Focusing Mechanism

^{3}piezoelectric actuators. In Ref. [27], the largest output torque of a harmonic piezo-drive motor with eight 5 × 5 × 50 mm

^{3}piezoelectric actuators is only 0.75 Nm. Thus, the torque density of the proposed piezoelectric focusing mechanism is more than three times larger than that of the harmonic piezo-drive motor.

_{1}and O

_{2}are in the near-focus and far-focus images, and the light flux in the two quadrants is not equal.

_{1}, U

_{2}, U

_{3}, U

_{4}are the voltage that is applied to the piezoelectric actuators; U

_{p}is the peak value of the voltage; f is the driving frequency.

_{s}+r

_{p}, r

_{s}and r

_{p}are the radius of the wave generator and the movable tooth; i

_{cp}is the transmission ratio of the movable tooth system; α

_{j}is the angle between center of the j-th movable tooth and geometric center of the center gear.

## 3. Electrochemical Coupled Models and Equations

_{33}(piezoelectric strain constant), so the working mode of d

_{33}is adopted in this paper. Except the d

_{33}mode can meet the work requirements, other modes of piezoelectric actuators can also meet the requirements. Here, only one working mode is utilized for analysis. The output strain of each piezoelectric actuator varies with time, its expression is:

_{33}is the elastic flexibility coefficient of the piezoelectric actuator; P

_{1}, P

_{2}, P

_{3}and P

_{4}are preloads of the piezoelectric actuators; S

_{p}is the sectional area of the piezoelectric actuator; h

_{p}is thickness of the piezoelectric layer.

_{33}is elastic modulus of the piezoelectric actuator.

_{pi}is the output force of the piezoelectric actuators, F

_{hi}is the force that applied to the wave generator, M

_{i}

_{1}and M

_{i}

_{2}are torsional moment of the flexible hinge, d

_{i}

_{1}and d

_{i}

_{2}are length dimension, and i = 1, 2, 3, 4. According to theoretical mechanics, establishing the force balance equation as follows:

_{i}can be expressed by:

_{1}to D

_{1}; n is the number of the piezoelectric layer for the actuator.

_{i}can be expressed by:

_{i}to E

_{i}; ${d}_{i5}$ is the length from D

_{i}to E

_{i}.

_{c}and O

_{s}represent the center of the central gear and harmonic generator, γ

_{j}is the angle between the tangent line of the center track of the movable tooth and the x-axis, F

_{cj}, F

_{rj}and F

_{sj}are the forces of the movable tooth that are applied by the central gear, tooth carrier and harmonic generator, respectively.

_{q}and F

_{t}are axial pushing force and horizontal pushing force, supporting force F

_{N}and frictional force F

_{f}constitute the opposite force F

_{R}, Ψ and ζ are the helix angle and frictional angle of the screw drive.

_{t}and T

_{r}can be written as:

_{m}is the effective diameter of the thread.

## 4. Results and Discuss

#### 4.1. Output Forces of Piezoelectric Actuator

^{3}of the piezoelectric actuator is selected to drive the focusing mechanism. Meanwhile, a 100 N preload is applied to the piezoelectric actuator. Applying those conditions and Equation (6), the output forces of the piezoelectric actuator are investigated as shown in Figure 8, where sub-figure a shows the output forces of the different piezo-actuator, whereas Figure 8b–f represent the changes of output forces with different peak voltage, piezoelectric strain constant, elastic modulus of piezoelectric actuator, thickness of piezoelectric layer and sectional area of piezoelectric actuator. Based on Figure 8, one can draw the conclusions:

_{p}increases, the output forces of the piezoelectric actuator increase linearly. Therefore, the output force of the piezoelectric actuator can be adjusted by changing the driving voltage.

_{33}, c

_{33}and S

_{p}is similar to that with U

_{p}. The output force is proportional to the change of parameters.

#### 4.2. Forces of Movable Tooth Drive

_{sj}, the central rotation force F

_{cj}and the tooth carrier force F

_{rj}. With the change of the rotating angle of the tooth carrier, the force of each movable tooth will change correspondingly, but its maximum force maintains constant.

_{sj}, F

_{cj}and F

_{rj}of a single movable tooth change significantly with the voltage. When the movable gear rotates at an angle of π/58, the force exerted on the movable tooth reaches the maximum value, and the force exerted on the movable tooth is most sensitive to the change of voltage. Forces F

_{sj}, F

_{cj}and F

_{rj}show the tendency of increasing first and then decreasing with the increase of the rotational angle of the tooth carrier.

_{cj}is the largest, whereas the peak value of F

_{sj}is the smallest. Therefore, the meshing force between the center gear and the movable teeth is the largest. At the same time, the magnitude of force F

_{cj}directly affects the magnitude of F

_{rj}.

#### 4.3. Thrust Force of Piezo-Driven Focusing Mechanism

_{p}. Figure 14, Figure 15, Figure 16 and Figure 17 depict the output torque and thrust force change with the piezoelectric strain constant d

_{33}, elastic modulus of piezoelectric actuator c

_{33}, thickness of piezoelectric layer h

_{p}and sectional area of piezoelectric actuator S

_{p}. From Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, it can be concluded that:

_{p}= 150 V, the output torque of the movable tooth drive system and thrust force of moving lens are 1.16 Nm and 562.5 N under the driven by four 5 × 5 × 20 mm

^{3}piezoelectric actuators. In Ref. [23], the largest output torque of a harmonic piezo-drive motor with eight 5 × 5 × 50 mm

^{3}piezoelectric actuators is only 0.75 Nm. Thus, the torque density of the proposed piezoelectric focusing mechanism is more than three times larger than that of the harmonic piezo-drive motor.

_{33}, c

_{33}and S

_{p}is similar to that with driving voltage U

_{p}. That is, the output torque and thrust force increase in direct proportion to those parameters.

_{p}increases, the output torque and thrust force decrease nonlinearly. Therefore, the smaller the thickness of the piezoelectric layer, the better the performance of the piezo-driven focusing mechanism.

#### 4.4. Finite Element Analysis

^{3}, while the stiffness matrix

**c**, dielectric constant matrix

**ε**and piezoelectric stress constant

**e**matrix are respectively as follows:

^{3}, the Young’s modulus and Poisson’s ratio are 2 × 10

^{11}Pa and 0.3, respectively. In addition, the compressive yield strength and tensile ultimate strength of this material are 2.5 × 10

^{8}Pa and 4.6 × 10

^{8}Pa, respectively.

#### 4.5. Principium Experiment

#### 4.6. Future Work Prospects

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**Working principle of the harmonic generator. (

**a**) Initial position and 360° position; (

**b**) 90° position; (

**c**) 180° position; (

**d**) 270° position.

**Figure 8.**Output forces of the piezoelectric actuator. (

**a**) Force changes with different piezo-actuator; (

**b**) force changes with U

_{p}; (

**c**) force changes with d

_{33}; (

**d**) force changes with c

_{33}; (

**e**) force changes with h

_{p}; (

**f**) force changes with S

_{p}.

**Figure 9.**Harmonic force of the wave generator. (

**a**) Harmonic under different input signals; (

**b**) harmonic force varies with input voltage and time.

**Figure 10.**The force of movable tooth F

_{sj}. (

**a**) The force varies with rotational angle and voltage; (

**b**) force sectional view at U

_{p}= 150 V.

**Figure 11.**The force of movable tooth F

_{cj}. (

**a**) The force varies with rotational angle and voltage; (

**b**) force sectional view at U

_{p}= 150 V.

**Figure 12.**The force of movable tooth F

_{rj}. (

**a**) The force varies with rotational angle and voltage; (

**b**) force sectional view at U

_{p}= 150 V.

**Figure 19.**Finite element analysis results of piezoelectric actuator. (

**a**) Voltage load applied on the piezoelectric actuator; (

**b**) deformation displacement of the piezoelectric actuator; (

**c**) transient response at 1000 Hz of the exciting frequency; (

**d**) transient response at 5713 Hz of the exciting frequency.

**Figure 20.**Stress analysis results at different times. (

**a**) At t = 0 s; (

**b**) at t = 0.01 s; (

**c**) at t = 0.025 s; (

**d**) at t = 0.05 s; (

**e**) at t = 0.075 s; (

**f**) at t = 0.1 s.

**Figure 21.**Modal analysis results of the piezoelectric focusing mechanism. (

**a**) First-order mode; (

**b**) second-order mode; (

**c**) third-order mode.

**Figure 22.**Output displacement test of the piezoelectric actuator. (

**a**) Experimental device structure; (

**b**) test results.

**Figure 23.**Principium experiment of the piezo-driven focusing mechanism. (

**a**) Exploded view of the principium prototype; (

**b**) test scheme.

$\mathbf{Rotational}\text{}\mathbf{Angle}\text{}\mathbf{of}\text{}\mathbf{Tooth}\text{}\mathbf{Carrier}\text{}\mathit{\phi}$ (rad) | Number of Meshing Movable Teeth | $\mathbf{Rotational}\text{}\mathbf{Angle}\text{}\mathbf{of}\text{}\mathbf{Tooth}\text{}\mathbf{Carrier}\text{}\mathit{\phi}$ (rad) | Number of Meshing Movable Teeth |
---|---|---|---|

0–π/870 | 2–16 | π/29–31π/870 | 17–30, 1 |

π/870–π/435 | 2–17 | 31π/870–16π/435 | 17–30, 1–2 |

π/435–π/290 | 3–17 | 16π/435–11π/290 | 18–30, 1–2 |

π/290–2π/435 | 3–18 | 11π/290–17π/435 | 18–30, 1–3 |

2π/435–π/174 | 4–18 | 17π/435–7π/174 | 19–30, 1–3 |

π/174–π/145 | 4–19 | 7π/174–6π/145 | 19–30, 1–4 |

π/145/–7π/870 | 5–19 | 6π/145–37π/870 | 20–30, 1–4 |

7π/870–4π/435 | 5–20 | 37π/870–19π/435 | 20-30, 1–5 |

4π/435–3π/290 | 6–20 | 19π/435–39π/870 | 21–30, 1–5 |

3π/290–π/87 | 6–21 | 39π/870–4π/87 | 21–30, 1–6 |

π/87–11π/870 | 7–21 | 4π/87–41π/870 | 22–30, 1–6 |

11π/870–2π/145 | 7–22 | 41π/870–7π/145 | 22–30, 1–7 |

2π/145–13π/870 | 8–22 | 7π/145–43π/870 | 23–30, 1–7 |

13π/870–7π/435 | 8–23 | 43π/870–22π/435 | 23–30, 1–8 |

7π/435–π/58 | 9–23 | 22π/435–3π/58 | 24–30, 1–8 |

π/58–8π/435 | 9–24 | 3π/58–23π/435 | 24–30, 1–9 |

8π/435–17π/870 | 10–24 | 23π/435–47π/870 | 25–30, 1–9 |

17π/870–3π/145 | 10–25 | 47π/870–8π/145 | 25–30, 1–10 |

3π/145–19π/870 | 11–25 | 8π/145–49π/870 | 26–30, 1–10 |

19π/870–2π/87 | 11–26 | 49π/870–5π/87 | 26–30, 1–11 |

2π/87–7π/290 | 12–26 | 5π/87–17π/290 | 27–30, 1–11 |

7π/290–11π/435 | 12–27 | 17π/290–52π/870 | 27–30, 1–12 |

11π/435–23π/870 | 13–27 | 52π/870–53π/870 | 28–30, 1–12 |

23π/870–4π/145 | 13–28 | 53π/870–9π/145 | 28–30, 1–13 |

4π/145–5π/174 | 14–28 | 9π/145–11π/174 | 29–30, 1–13 |

5π/174–13π/435 | 14–29 | 11π/174–28π/435 | 29–30, 1–14 |

13π/435–9π/290 | 15–29 | 28π/435–57π/870 | 30, 1–14 |

9π/290–14π/435 | 15–30 | 57π/870–π/15 | 30, 1–15 |

14π/435–π/30 | 16–30 | π/15–59π/870 | 1–15 |

π/30–π/29 | 16–30, 1 | 59π/870–2π/29 | 1–16 |

Voltages | 100 V | 110 V | 120 V | 130 V | 140 V | 150 V |
---|---|---|---|---|---|---|

f = 0.5 Hz | 0 | 0.492 | 0.594 | 0.612 | 0.625 | 0.625 |

f = 1 Hz | 0 | 1.017 | 1.132 | 1.176 | 1.188 | 1.200 |

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## Share and Cite

**MDPI and ACS Style**

Li, C.; Liang, K.; Zhong, W.; Fang, J.; Sun, L.; Zhu, Y.
Electrochemical Coupled Analysis of a Micro Piezo-Driven Focusing Mechanism. *Micromachines* **2020**, *11*, 216.
https://doi.org/10.3390/mi11020216

**AMA Style**

Li C, Liang K, Zhong W, Fang J, Sun L, Zhu Y.
Electrochemical Coupled Analysis of a Micro Piezo-Driven Focusing Mechanism. *Micromachines*. 2020; 11(2):216.
https://doi.org/10.3390/mi11020216

**Chicago/Turabian Style**

Li, Chong, Kang Liang, Wei Zhong, Jiwen Fang, Lining Sun, and Yong Zhu.
2020. "Electrochemical Coupled Analysis of a Micro Piezo-Driven Focusing Mechanism" *Micromachines* 11, no. 2: 216.
https://doi.org/10.3390/mi11020216