# Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

## 3. Influence of the Magnetic Field

## 4. Application of the Ritz Method

## 5. Solution by the Navier Method

## 6. Results and Discussions

## 7. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NEMS | nano-electro mechanical systems |

MEMS | micro-electro mechanical systems |

SSSS | all edges are simply supported |

CCCC | all edges are clamped |

SSSC | three edges are simply supported and one is clamped |

SCSC | two opposite sides are simply supported, the other two ones are clamped |

SCSF | two opposite sides are simply supported, one is clamped and one is free |

SSSF | the sides except one are simply supported and one is free |

CSCS | two opposite sides are clamped, the other two ones are simply supported |

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**Figure 2.**Dimensionless frequencies $\Omega $ in terms of $MP$ and $l/h$ with SSSS boundary conditions.

**Figure 3.**Dimensionless frequencies $\Omega $ in terms of $MP$ and $l/h$ with CCCC boundary conditions.

**Figure 4.**Dimensionless frequencies $\Omega $ depending on Winkler modulus for different thickness ratios $l/h$ and SSSS boundary conditions ($b/a=1.5,KG=0,MP=0$).

**Figure 5.**Dimensionless frequencies $\Omega $ depending on shear modulus for different thickness ratios $l/h$ and SSSS boundary conditions ($b/a=1.5,KW=100,MP=0$).

**Figure 6.**Dimensionless frequencies $\Omega $ for various types of the boundary conditions ($b/a=1.5,KW=100,KG=10,MP=10$).

**Table 1.**Dimensionless frequencies $\Omega $ of isotropic square plate for various foundation parameters and boundary conditions.

Method | $\mathit{KW}$ | $\mathit{KG}$ | SSSS | SSSC | SCSC | SCSF | SSSF |
---|---|---|---|---|---|---|---|

[36] | 0 | 0 | 19.737 | 23.643 | 28.944 | 12.693 | 11.69 |

[37] | 19.735 | 23.659 | 28.995 | - | 11.677 | ||

RS | 19.739 | 23.646 | 28.949 | 12.687 | 11.418 | ||

[36] | 100 | 48.615 | 51.318 | 54.674 | 37.977 | 37.152 | |

[37] | 48.547 | 51.253 | 54.617 | - | 37.102 | ||

RS | 48.615 | 51.323 | 54.679 | 37.981 | 37.129 | ||

[36] | 1000 | 141.873 | 144.2 | 146.719 | 112.481 | 111.745 | |

[37] | 140.182 | 142.439 | 144.877 | - | 110.424 | ||

RS | 141.873 | 144.479 | 146.74 | 112.672 | 111.746 | ||

[36] | 100 | 0 | 22.126 | 25.671 | 30.623 | 16.149 | 15.383 |

[37] | 22.125 | 25.687 | 30.672 | - | 15.373 | ||

RS | 22.127 | 25.673 | 30.628 | 16.155 | 15.178 | ||

[36] | 100 | 49.633 | 52.283 | 55.581 | 39.272 | 38.474 | |

[37] | 49.566 | 52.22 | 55.524 | - | 38.426 | ||

RS | 49.633 | 52.289 | 55.586 | 39.276 | 38.453 | ||

[36] | 1000 | 142.225 | 144.547 | 147.06 | 112.925 | 112.192 | |

[37] | 140.538 | 142.789 | 145.222 | - | 110.876 | ||

RS | 142.225 | 144.824 | 147.081 | 113.115 | 112.193 | ||

[36] | 1000 | 0 | 37.276 | 39.483 | 42.869 | 34.075 | 33.714 |

[37] | 37.274 | 39.493 | 42.902 | - | 33.708 | ||

RS | 37.277 | 39.485 | 42.873 | 34.073 | 33.621 | ||

[36] | 100 | 57.995 | 60.278 | 63.160 | 49.419 | 48.789 | |

[37] | 57.936 | 60.222 | 63.109 | - | 48.749 | ||

RS | 57.995 | 60.283 | 63.165 | 49.420 | 48.771 | ||

[36] | 1000 | 145.355 | 147.627 | 150.088 | 116.842 | 116.134 | |

[37] | 143.704 | 145.906 | 148.288 | - | 114.862 | ||

RS | 145.355 | 147.899 | 150.109 | 117.026 | 116.135 |

**Table 2.**Dimensionless frequencies $\Omega $ of the orthotropic square small-scale plate for different thickness ratios.

Method | $\mathit{l}/\mathit{h}$ | ${\mathit{\omega}}_{1}$ | ${\mathit{\omega}}_{2}$ | ${\mathit{\omega}}_{3}$ |
---|---|---|---|---|

[21] | 0 | 17.543 | 36.034 | 45.660 |

[38] | 17.860 | 36.295 | 45.683 | |

RS | 17.880 | 36.299 | 45.704 | |

[21] | 0.1 | 18.432 | 37.326 | 47.920 |

RS | 18.687 | 37.441 | 47.734 | |

[21] | 0.2 | 20.697 | 40.921 | 53.832 |

RS | 20.802 | 40.629 | 53.060 | |

[21] | 0.3 | 23.745 | 46.228 | 62.111 |

RS | 23.717 | 45.350 | 60.470 | |

[21] | 0.4 | 27.258 | 52.708 | 71.983 |

RS | 27.114 | 51.122 | 69.194 |

**Table 3.**Dimensionless frequencies $\Omega $ of the orthotropic rectangular small-scale plate for different thickness ratios and magnetic parameters, SSSS.

Method | MP | $\mathit{l}/\mathit{h}$ | |||||
---|---|---|---|---|---|---|---|

0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 | ||

RS | 0 | 21.0203 | 21.8251 | 24.0784 | 27.4257 | 31.5203 | 36.1088 |

NS | 21.0205 | 21.8253 | 24.0788 | 27.4262 | 31.5209 | 36.1096 | |

RS | 25 | 24.0608 | 24.767 | 26.7739 | 29.8201 | 33.6244 | 37.9594 |

NS | 24.0612 | 24.7674 | 26.7744 | 29.8207 | 33.6251 | 37.9602 | |

RS | 50 | 26.7581 | 27.3948 | 29.2218 | 32.0361 | 35.6043 | 39.7239 |

NS | 26.7585 | 27.3953 | 29.2223 | 32.036 | 35.6051 | 39.7248 |

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**MDPI and ACS Style**

Mazur, O.; Awrejcewicz, J.
Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field. *Symmetry* **2020**, *12*, 515.
https://doi.org/10.3390/sym12040515

**AMA Style**

Mazur O, Awrejcewicz J.
Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field. *Symmetry*. 2020; 12(4):515.
https://doi.org/10.3390/sym12040515

**Chicago/Turabian Style**

Mazur, Olga, and Jan Awrejcewicz.
2020. "Ritz Method in Vibration Analysis for Embedded Single-Layered Graphene Sheets Subjected to In-Plane Magnetic Field" *Symmetry* 12, no. 4: 515.
https://doi.org/10.3390/sym12040515