# Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Nonlocal Elasticity Theory

#### Hamilton’s Principle

## 3. Analytical Solution

## 4. Results and Discussion

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**TEM image of a platinum nanowire with an elliptical cross section on the MgO (110) substrates [90].

**Figure 5.**Variation of the nondimensional frequency ratio with the geometrical ratio $b/a$ for a C-C elliptical nanorod.

**Figure 6.**Variation of the nondimensional frequency ratio with the geometrical ratio $b/a$ for a C-F elliptical nanorod.

**Figure 7.**Variation of the nondimensional frequency ratio with $\mu $ and ${k}_{b}$. $a=0.4\mathrm{nm}$, $b=0.2\mathrm{nm}$, and $l=10\mathrm{nm}$.

**Table 1.**Comparative evaluation of the first four dimensionless natural frequencies for a C-C circular nanorod with respect to the literature ($a=b=1\mathrm{nm}$, $\mu /l=0.1$, $l=20\mathrm{nm}$ ).

n = 1 | n = 2 | n = 3 | n = 4 | |
---|---|---|---|---|

Present | 2.9971 | 5.3201 | 6.8586 | 7.8247 |

Ref. [95] | 2.9971 | 5.3201 | 6.8586 | 7.8247 |

**Table 2.**Free torsional vibration of C-C elliptical nanorod with a varying $b/a$ ratio and nonlocal parameter $\mu $.

$\mathit{\mu}\times {10}^{-9}$ | $\mathit{b}/\mathit{a}=0.1$ | $\mathit{b}/\mathit{a}=0.2$ | $\mathit{b}/\mathit{a}=1$ | $\mathit{b}/\mathit{a}=5$ | $\mathit{b}/\mathit{a}=10$ |
---|---|---|---|---|---|

0 | 0.6220 | 1.2082 | 3.1415 | 1.2062 | 0.6208 |

1 | 0.6145 | 1.1935 | 3.1035 | 1.1916 | 0.6133 |

1.5 | 0.6055 | 1.1760 | 3.0578 | 1.1741 | 0.6043 |

2 | 0.5934 | 1.1526 | 2.9971 | 1.1508 | 0.5923 |

**Table 3.**Free torsional vibration of C-F elliptical nanorod with a varying $b/a$ ratio and nonlocal parameter $\mu $.

$\mathit{\mu}\times {10}^{-9}$ | $\mathit{b}/\mathit{a}=0.1$ | $\mathit{b}/\mathit{a}=0.2$ | $\mathit{b}/\mathit{a}=1$ | $\mathit{b}/\mathit{a}=5$ | $\mathit{b}/\mathit{a}=10$ |
---|---|---|---|---|---|

0 | 0.3110 | 1.2565 | 1.5708 | 1.2564 | 0.3108 |

1 | 0.3100 | 1.2527 | 1.5659 | 1.2525 | 0.3099 |

1.5 | 0.3089 | 1.2479 | 1.5600 | 1.2478 | 0.3087 |

2 | 0.3072 | 1.2413 | 1.5517 | 1.2412 | 0.3071 |

**Table 4.**Free torsional vibration of a C-T elliptical nanorod for a varying $\mu $ and ${k}_{b}$. $a=0.4\mathrm{nm}$, $b=0.2\mathrm{nm}$, and $l=20\mathrm{nm}$.

$\begin{array}{l}{\mathit{k}}_{\mathit{b}}\times {10}^{-18}\\ \left(\mathbf{GPa}\times {\mathbf{nm}}^{3}\right)\end{array}$ | $\mathit{\mu}\times {10}^{-9}$ | |||
---|---|---|---|---|

0 | 1 | 1.5 | 2 | |

0.01 | 1.2815 | 1.2774 | 1.2723 | 1.2653 |

0.1 | 1.4690 | 1.4628 | 1.4552 | 1.4448 |

1 | 2.1225 | 2.1040 | 2.0817 | 2.0515 |

10 | 2.4638 | 2.4351 | 2.4006 | 2.3547 |

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**MDPI and ACS Style**

Khosravi, F.; Hosseini, S.A.; Hamidi, B.A.; Dimitri, R.; Tornabene, F.
Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions. *Vibration* **2020**, *3*, 189-203.
https://doi.org/10.3390/vibration3030015

**AMA Style**

Khosravi F, Hosseini SA, Hamidi BA, Dimitri R, Tornabene F.
Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions. *Vibration*. 2020; 3(3):189-203.
https://doi.org/10.3390/vibration3030015

**Chicago/Turabian Style**

Khosravi, Farshad, Seyyed Amirhosein Hosseini, Babak Alizadeh Hamidi, Rossana Dimitri, and Francesco Tornabene.
2020. "Nonlocal Torsional Vibration of Elliptical Nanorods with Different Boundary Conditions" *Vibration* 3, no. 3: 189-203.
https://doi.org/10.3390/vibration3030015