# Modelling the Periodic Response of Micro-Electromechanical Systems through Deep Learning-Based Approaches

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*Actuators*in 2022)

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. POD-DL-ROM Technique: Outline and Critical Issues

## 4. Periodic DL-ROM Technique

## 5. Frequency Response Function Modelling: Arch Length Abscissa

## 6. Application: Electromechanical Disk Resonating Gyroscope

^{TM}7.1 using one of the templates available in the software itself, slightly modified for the purposes of this work [48,49,50].

#### 6.1. Problem Description

#### 6.2. Hyperparameters and Training

#### 6.3. Latent Coordinates and Frequency Features

#### 6.4. Results

#### 6.5. Key Advantages and Comparison with POD-DL-ROM

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Electromechanical problem reference quantities describing the PDE problem in Equations (1)–(8).

**Figure 2.**POD-DL-ROM architecture. Starting from the FOM solution $\mathbf{U}(t;\mathit{\mu})$, the POD subspace coordinates ${\mathbf{u}}_{N}(t;\mathit{\mu})={\mathbf{V}}_{N}^{T}\mathbf{U}(t;\mathit{\mu})$ are computed. The neural network provides their approximation ${\tilde{\mathbf{u}}}_{N}(t;\mathit{\mu})$ as an output. The reconstructed solution $\mathbf{U}(t;\mathit{\mu})$ is then recovered through the basis matrix ${\mathbf{V}}_{N}$.

**Figure 4.**Arc length abscissa. (

**a**) Arc length abscissa s and (

**b**) angular frequency $\omega $ as a function of the arc length abscissa s in the case of a autoparametric resonance like those reported in Section 6 e.g., Figure 6.

**Figure 5.**Disk resonating gyroscope. (

**a**) Geometry of the device, (

**b**) close-up of the electrodes, (

**c**) drive mode and (

**d**) sensing modes.

**Figure 6.**Training. (

**a**) FRCs reporting the sampling points used in the training process. The displacements refer to the point highlighted in the figure. (

**b**) Loss function evolution during the training process considering the training and the validation dataset.

**Figure 7.**Latent coordinates (

**a**–

**d**) and frequency function (

**e**) interpolations. The black dots mark the reference values given by the training datasets and the interpolated surface is in red.

**Figure 8.**Frequency response curves reconstructed during the testing phase. (

**a**) Envelope of the FRCs of the drive motion ${u}_{DRIVE}$. (

**b**,

**c**) Enlarged views of the FRC peak and main resonance region, respectively. (

**d**) Envelope of the FRCs of the sense motion ${u}_{SENSE}$. The sense motion has been normalised with respect to the maximum amplitude of drive motion, i.e., for ${V}_{AC}=9.5$ V and $max\left({u}_{DRIVE}\right)=0.5019$$\mathsf{\mu}$, to better highlight the strength of the autoparametric resonance when the drive mode is excited. (

**e**) Enlarged view of the FRC peak and main resonance region. (

**f**) Points selected as representative of the drive and sense motion.

**Table 1.**Features of convolutional and dense layers in the encoder ${\mathbf{f}}_{n}^{E}$. The total number of parameters is 6651. The stride was always set to one and a padding equal to zero was used. There are two input channels because the FOM deploys Euler–Bernulli beams, so displacements and rotations DOFs are kept separated. The encoding process reduces the input matrix with a shape of $N,{N}_{F},2$ towards a bottleneck representation with a size n, with N number of POMs, ${N}_{F}$ number of harmonic terms, and n latent dimensions.

Layer | Input | Output | Kernel | # of Filters |
---|---|---|---|---|

Dimension | Dimension | Size | ||

1 | [N, ${N}_{F}$, 2] | [$N-1$, ${N}_{F}-1$, 3] | [2, 2] | 3 |

2 | [$N-1$, ${N}_{F}-1$, 3] | [$N-2$, ${N}_{F}-2$, 5] | [5, 5] | 5 |

3 | [$N-2$, ${N}_{F}-2$, 5] | [$N-3$, ${N}_{F}-3$, 5] | [5, 5] | 5 |

5 | $(N-3)({N}_{F}-3)5$ | 10 | ||

6 | 10 | n |

**Table 2.**Features of dense and transposed convolutional layers in the decoder ${\mathbf{f}}_{N}^{D}$. The total number of parameters is 7284. The stride was always set to one and a padding equal to zero was used. There are two input channels because the FOM utilises Euler–Bernulli beams with both displacements and rotations DOFs. The decoding process expands the latent representation matrix with a size n towards a matrix with a shape $N,{N}_{F},2$, with N number of POMs, ${N}_{F}$ number of harmonic terms, and n latent dimensions.

Layer | Input | Output | Kernel | # of Filters |
---|---|---|---|---|

Dimension | Dimension | Size | ||

1 | n | 10 | ||

2 | 10 | $(N-3)({N}_{F}-3)5$ | ||

3 | [$N-3$, ${N}_{F}-3$, 5] | [$N-2$, ${N}_{F}-2$, 5] | [2, 2] | 5 |

4 | [$N-2$, ${N}_{F}-2$, 5] | [$N-1$, ${N}_{F}-1$, 3] | [2, 2] | 3 |

5 | [$N-1$, ${N}_{F}-1$, 3] | [N, ${N}_{F}$, 2] | [2, 2] | 2 |

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**MDPI and ACS Style**

Gobat, G.; Baronchelli, A.; Fresca, S.; Frangi, A.
Modelling the Periodic Response of Micro-Electromechanical Systems through Deep Learning-Based Approaches. *Actuators* **2023**, *12*, 278.
https://doi.org/10.3390/act12070278

**AMA Style**

Gobat G, Baronchelli A, Fresca S, Frangi A.
Modelling the Periodic Response of Micro-Electromechanical Systems through Deep Learning-Based Approaches. *Actuators*. 2023; 12(7):278.
https://doi.org/10.3390/act12070278

**Chicago/Turabian Style**

Gobat, Giorgio, Alessia Baronchelli, Stefania Fresca, and Attilio Frangi.
2023. "Modelling the Periodic Response of Micro-Electromechanical Systems through Deep Learning-Based Approaches" *Actuators* 12, no. 7: 278.
https://doi.org/10.3390/act12070278