# Shannon Entropy in Stochastic Analysis of Some Mems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations of the Problem

**Problem**

**1.**

_{1}, k

_{2}and k

_{3}are first, second and third-order stiffness coefficients related to various physical fields and sources, F and$\omega $are the amplitude and frequency of the modulation signal. External forcing has been chosen as perfectly periodic at the initial stage to verify how stationary signals affect the structural response of the MEMS. It also enables for some verification of the output signals and is frequently chosen in various theoretical and numerical studies. Initial equation of motion is solved with traditional initial conditions equivalent to static equilibrium in the undeformed configuration:

## 3. Mems Device Description

_{m}and k

_{e}to the overall stiffness of the micro-resonator are represented by the following stiffness coefficients k

_{i}, i = 1, 2, 3, which in turn are taken as

_{1}= (k

_{m1}− k

_{e1}) = (0.829 − 0.068) = 0.761 N/m,

_{2}= 0 N/m

^{2},

_{3}= (k

_{m3L}− k

_{e3}) = (1.45 × 10

^{11}− 2.2 × 10

^{10}) = 12.3 × 10

^{10}N/m

^{3}.

_{2}simply vanishes for the perfect micro-resonators, where the gaps in between the resonator and driving, as well as sensing electrode are equal to each other. In this specific case, Equation (10) becomes the Duffing equation, whose further numerical solution enables stochastic response analysis. Some other physical phenomena appearing in the micro-resonators together with their mathematical consequences have been reported in detail in [40]. The effective mass of the micro-resonator was calculated in [41] from its length L = 400 μm, width t = 1.2 μm, out of the plane thickness w = 15 μm, the silicon mass density $\rho =2330\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$. The value m = 0.3965 × M = 6.65 × 10

^{−12}kg is obtained, using the formula that gives the equivalent mass m as a fraction of the total beam mass M = 16.78 × 10

^{−12}kg. The mean value of the damping coefficient has been proposed using the formula $c=\frac{1}{Q}\sqrt{km}\left[\frac{\mathrm{Nsec}}{\mathrm{m}}\right]$, where k includes all the stiffnesses introduced in Equation (1); the expected value of damping parameter c has been taken as equal to E [c] = [0.00394] x 10

^{−6}[Nsec/m] and the coefficient of variation of this physical parameter is taken further from the interval $\alpha \left(c\right)\in \left[0.00,0.20\right]$. The harmonic external force representing electrostatic actuation is introduced as

_{p}is the bias voltage, ε

_{0}is the absolute vacuum permittivity constant, d means the gap between the oscillating beam and the electrode, v

_{a}(t) is the actuation voltage, usually modulated at the mechanical frequency of the oscillating beam ω. Finally, the external force has the following multiplier: $F=56.7\times {10}^{-10}[N]$, while a frequency ω has been adopted as 10

^{3}.

## 4. Numerical Simulation and Discussion

^{6}, it is even shorter than statistical estimation of the first two probabilistic moments of the structural response. If one could estimate the first four probabilistic characteristics (including also skewness and kurtosis [41]), then the entropy calculus brings definitely more information about uncertainty propagation with definitely smaller computer effort. This is seen for the single-degree-of-freedom system, whereas the Stochastic Finite Element Method analysis of large scale systems would show huge disproportion of computer power consumption. Therefore, a proper calculation and interpretation of probabilistic entropy may bring essential time savings while using the Monte Carlo simulation approach.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**The expected values of the MEMS device vibration (

**left**) and its coefficient of variation (

**right**).

**Figure 8.**Shannon entropies logarithm time fluctuations for uniformly distributed structural damping.

**Figure 10.**Shannon entropies logarithm time fluctuations for triangularly distributed structural damping.

Polynomial Order | Correlation | RMS Error | Squares Sum | Fitting Variance |
---|---|---|---|---|

2 | 0.944648 | 6.50235 × 10^{−12} | 4.65126 × 10^{−22} | 4.65228 × 10^{−23} |

3 | 0.999390 | 7.22270 × 10^{−13} | 5.76870 × 10^{−24} | 5.86760 × 10^{−25} |

4 | 0.999571 | 6.09214 × 10^{−13} | 4.11380 × 10^{−24} | 2.90649 × 10^{−24} |

5 | −0.594698 | 7.08217 × 10^{−10} | 5.51726 × 10^{−18} | 2.45674 × 10^{−19} |

6 | −0.924625 | 8.02584 × 10^{−9} | 7.08554 × 10^{−16} | 4.98402 × 10^{−16} |

7 | 0.936636 | 8.11866 × 10^{−7} | 7.25038 × 10^{−12} | 3.53456 × 10^{−12} |

8 | −0.942430 | 8.61679 × 10^{−5} | 8.16740 × 10^{−8} | 1.25495 × 10^{−6} |

9 | −0.937340 | 6.38454 × 10^{−4} | 4.48386 × 10^{−6} | 6.55977 × 10^{−7} |

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Kamiński, M.; Corigliano, A.
Shannon Entropy in Stochastic Analysis of Some Mems. *Energies* **2022**, *15*, 5483.
https://doi.org/10.3390/en15155483

**AMA Style**

Kamiński M, Corigliano A.
Shannon Entropy in Stochastic Analysis of Some Mems. *Energies*. 2022; 15(15):5483.
https://doi.org/10.3390/en15155483

**Chicago/Turabian Style**

Kamiński, Marcin, and Alberto Corigliano.
2022. "Shannon Entropy in Stochastic Analysis of Some Mems" *Energies* 15, no. 15: 5483.
https://doi.org/10.3390/en15155483