# Non-Reciprocal MEMS Periodic Structure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Parametric Analysis for Harmonic Modulation

#### 3.1. Dispersion Relation for Rod on Elastic Foundations

#### 3.2. Dispersion Analysis

**L**are

#### 3.3. From Rod to Spring-Mass Chain

## 4. Numerical Study of the MEMS Device

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Unitary cell composed of three sub-cells. The masses (gray) are connected to each other through folded beams (blue) and to the ground (red). The electrodes (yellow) are fixed and placed in the hollowed masses. While the structure is set to a constant voltage ${V}_{0}$, an alternate voltage ${V}_{i}\left(t\right)$ is imposed to each electrode i.

**Figure 2.**Sub-cell scheme considering a mass displaced by x. The area of interest for electrostatic forces is highlighted in light green.

**Figure 3.**Dispersion relations for a rod on elastic suspensions with space–time modulation. Numerical results (black lines) are compared to the analytic results (colored lines), and key properties are shown on the dispersion diagrams for completeness. (

**a**) Case with no amplitude modulation (${\gamma}_{m}=0$); (

**b**) with constant properties in time (${\omega}_{m}=0$); (

**c**) with both modulations active.

**Figure 4.**Comparison between the dispersion diagrams of a rod (black line) and of its spring-mass chain discretization (blue line), using $R=3,\phantom{\rule{0.166667em}{0ex}}6,\phantom{\rule{0.166667em}{0ex}}10$ masses (

**a**–

**c**). Directional band-gaps predicted with the rod model are highlighted in green and yellow, while the band-gaps of the spring-mass chain system are enclosed in dashed red boxes.

**Figure 5.**Scheme of the geometry of the COMSOL model (only 2 sub-cells are shown). The dashed-dotted line denotes a symmetry axis.

**Figure 6.**Numerical dispersion, obtained with the COMSOL Multiphysics model, vs. analytical prediction (black line), obtained with the spring-mass chain model and the BBP (

**right**) and spectrum of the input tone burst (

**left**).

${L}_{g}=150$ | ${L}_{i}=150$ | ${L}_{x}=100$ | ${L}_{y}=200$ |

${L}_{ex}=5$ | ${L}_{ey}=100$ | ${L}_{hx}=7$ | ${L}_{hy}=102$ |

${x}_{0}=1$ | ${w}_{b}=2$ | ${w}_{c}=8$ |

$E=148$ [GPa] | silicon elastic modulus |

$\rho =2330$ [kg/m${}^{3}$] | silicon density |

$t=10$ [$\mathsf{\mu}$m] | silicon wafer thickness |

${\lambda}_{m}=312$ [$\mathsf{\mu}$m] | cell spatial period |

$\epsilon =8.854\times {10}^{-12}$ [F/m] | vacuum permittivity |

${V}_{a}=2$ [V] | AC voltage amplitude |

${V}_{0}=14$ [V] | DC voltage amplitude |

${\omega}_{m}=12,500$ [rad/s] | voltage modulation frequency |

$k=3.51$ [N/m] | stiffness between masses |

${k}_{g}=7.02$ [N/m] | ground stiffness |

${k}_{E0}=3.51$ [N/m] | constant electrostatic stiffness |

${k}_{E1}=0.99$ [N/m] | time-modulated electrostatic stiffness |

$m=44.81$ [ng] | mass value |

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**MDPI and ACS Style**

Marconi, J.; Enrico Quadrelli, D.; Braghin, F.
Non-Reciprocal MEMS Periodic Structure. *Actuators* **2023**, *12*, 161.
https://doi.org/10.3390/act12040161

**AMA Style**

Marconi J, Enrico Quadrelli D, Braghin F.
Non-Reciprocal MEMS Periodic Structure. *Actuators*. 2023; 12(4):161.
https://doi.org/10.3390/act12040161

**Chicago/Turabian Style**

Marconi, Jacopo, Davide Enrico Quadrelli, and Francesco Braghin.
2023. "Non-Reciprocal MEMS Periodic Structure" *Actuators* 12, no. 4: 161.
https://doi.org/10.3390/act12040161