# Analysis of the Nonlinear Response of Piezo-Micromirrors with the Harmonic Balance Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{®}Realsense LiDAR Camera™ (Santa Clara, CA, USA), Microsoft

^{®}Hololens2™ (Redmond, WA, USA) augmented reality glasses, consumer pico projectors [1], automotive Head-Up-Displays, as well as medical applications [2].

## 2. Devices and Experimental Data

^{®}using monocrystalline silicon with [110] orientation [24] aligned with the ${\mathbf{e}}_{1}$ axis. The sol-gel PZT patches, the passivation layers and the electrodes were deposited with the P$\epsilon $tra Thin-Film Piezoelectric technology developed by STMicroelectronics

^{®}.

## 3. Model for Material and Geometrical Nonlinearities

#### Numerical Solution

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Image of mirror M1 (

**a**); and schematic representation of the device geometry (

**b**). The silicon reflective surface is coloured in light yellow. Anchor points are coloured in green. Piezoelectric patches appear in orange in the picture. The numbers over the piezoelectric patches highlight the actuation groups. (

**c**,

**d**) Displacement field associated to the torsional mode of the structure.

**Figure 2.**Image of mirror M2 (

**a**); and schematic representation of the device geometry (

**b**). The silicon reflective surface is coloured in light blue. Anchor points are coloured in green. Piezoelectric patches appear in dark blue in the picture. The numbers over the piezoelectric patches highlight the actuation groups. (

**c**,

**d**) Displacement field associated to the torsional mode of the structure.

**Figure 4.**(

**a**,

**b**) Polarisation data measured during unipolar cycles for mirrors M1 and M2, respectively. (

**c**,

**d**) Polarisation data measured during bipolar cycles for mirror M1 and M2, respectively.

**Figure 6.**Comparison between experimental data and numerical results for mirror M1 for ${V}_{0}$ values of 10 V (

**a**) 15 V (

**b**), and 20 V (

**c**).

**Figure 7.**Comparison between experimental data and numerical results for mirror M2 for ${V}_{0}$ values of 20 V (

**a**) 25 V (

**b**), and 30 V (

**c**).

**Figure 8.**Comparison between numerical results obtained for a piezoelectric force formulated within the small displacements assumption (Infinitesimal) and the one computed within the framework of large transformations (Finite).

**Table 1.**Quality factors obtained from Equation (14) of [15] under the assumption of linear damping.

M1 | M2 | |||||
---|---|---|---|---|---|---|

V${}_{0}$ [V] | 10 | 15 | 20 | 20 | 25 | 30 |

Q [-] | 186 | 143 | 105 | 1386 | 1200 | 1092 |

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**MDPI and ACS Style**

Opreni, A.; Boni, N.; Carminati, R.; Frangi, A.
Analysis of the Nonlinear Response of Piezo-Micromirrors with the Harmonic Balance Method. *Actuators* **2021**, *10*, 21.
https://doi.org/10.3390/act10020021

**AMA Style**

Opreni A, Boni N, Carminati R, Frangi A.
Analysis of the Nonlinear Response of Piezo-Micromirrors with the Harmonic Balance Method. *Actuators*. 2021; 10(2):21.
https://doi.org/10.3390/act10020021

**Chicago/Turabian Style**

Opreni, Andrea, Nicolò Boni, Roberto Carminati, and Attilio Frangi.
2021. "Analysis of the Nonlinear Response of Piezo-Micromirrors with the Harmonic Balance Method" *Actuators* 10, no. 2: 21.
https://doi.org/10.3390/act10020021