# Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Timoshenko Beam Model for Vibration of Curved Single-Walled Carbon Nanotubes (SWCNTs)

#### 2.2. Nonlocal Theory of Nano-Beams

#### 2.3. Derivation of Nonlocal Governing Equations of Motion

#### 2.4. Differential Quadrature Solution Procedure

## 3. Numerical Results and Discussions

#### 3.1. Results Verification

_{s}= 0.563 [37]. As expected, increasing the nonlocal effect causes a decrease in dimensionless frequency. A noteworthy agreement is observed between the present results and those reported by Yang et al. [22]. Good agreement is observed between the results of current study and those by Wang et al. [37]. However, because they neglected the nonlocal terms in shearing force relation, the discrepancy between the results of two methods increases for higher values of nonlocal parameter.

_{s}= 0.563. Nonlocal parameter, $\mu $, and nonlinear vibration amplitude, ${w}_{max}^{*}$, are set to 0.15 and 0.4, respectively. For both linear and nonlinear frequencies, good agreement exists between the results of current study and those reported in [22].

#### 3.2. Geometric Imperfection Function

_{1}” to “L

_{3}” types. It is seen that the imperfection is exactly located at the center of SWCNT for L

_{3}type while for the L

_{1}type the imperfection is the closest case to the end.

#### 3.3. Nonlinear Vibration of Curved SWCNTs

_{1},” “L

_{2}” and “L

_{3}” are local imperfections with maximum amplitude of η = 0.1. For all local imperfect cases, the hard-spring behavior is observed. It is seen that the local imperfection has the largest effect on the nonlinear frequency ratio when it is located in the center of the single-walled carbon nanotube (L

_{3}) and its effect is decreased when the imperfection recedes from the center to the boundary (L

_{1}).

_{3}” is selected as the geometric imperfection with maximum amplitude of η = 0.1. Unlike straight SWCNTs (perfect case), curved nanotubes (imperfect cases) do not have an overall increasing trend for nonlinear frequency ratio while the nonlocal parameter μ raises, and there is a certain value of amplitude vibration ${w}_{max}^{*}$ in which the effect of nonlocal parameter gets reversed. Furthermore, it is observed that SWCNTs with C–C boundary condition have the highest nonlinear frequency and the H–H SWCNTs have the lowest, as expected.

_{2}) geometry is illustrated in Figure 8. Like the “G” type imperfection, the imperfection sensitivity ${S}_{w}$ increases when the maximum imperfection amplitude η is raised. For small values of imperfection amplitude η, all values of the nonlocal parameter almost have the same behavior, but after a certain value of η, a sudden large increase in imperfection sensitivity value occurs for higher values of nonlocal parameter μ. It depicts that for the case of local imperfection with large values of nonlocal parameter, initial geometric imperfections may cause considerable change in the fundamental frequency which cannot be neglected.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CNTs | Carbon Nano Tubes |

DQ | Differential Quadrature |

MD | Molecular Dynamics |

NEMS | Nano Electro Mechanical System |

SWCNTs | Single-Walled Carbon Nano Tubes |

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**Figure 4.**Variation of nonlinear vibration frequency ratio for different combinations of imperfection amplitude η and nonlocal parameter μ for a “G” type imperfection for (

**a**) clamped-clamped (

**b**) clamped-hinged and (

**c**) hinged-hinged endpoint conditions.

**Figure 5.**Variation of nonlinear vibration frequency ratio for (

**a**) clamped-clamped and (

**b**) hinged-hinged endpoint conditions of a nonlocal nano-beam for various “L” types imperfection.

**Figure 6.**Effects of nonlocal parameter μ on the nonlinear vibration frequency ratio of nano-beam with “L

_{3}” type of initial imperfection.

**Figure 7.**Effect of nonlocal parameter μ on sensitivity indicator of a nano-beam with “G” type initial imperfection with hinged-hinged endpoint conditions.

**Figure 8.**Effect of nonlocal parameter μ on sensitivity indicator of a nano-beam with “L

_{2}” type initial imperfection for (

**a**) clamped-clamped (

**b**) clamped-hinged and (

**c**) hinged-hinged endpoint conditions.

**Table 1.**Comparison of nonlinear frequency ratio of a Timoshenko straight beam for different vibration amplitudes.

$\frac{{\mathit{w}}_{\mathbf{\text{max}}}}{\sqrt{{\mathit{I}}_{\mathbf{2}}\mathbf{/}\mathcal{A}}}$ | H–H | C–C | C–H | ||||||
---|---|---|---|---|---|---|---|---|---|

Ref. [35] | Ref. [36] | Present | Ref. [35] | Ref. [36] | Present | Ref. [35] | Ref. [36] | Present | |

1.0 | 1.118 | 1.118 | 1.1181 | 1.0295 | 1.0283 | 1.0295 | 1.0641 | 1.0582 | 1.0593 |

2.0 | 1.4141 | 1.4135 | 1.4143 | 1.1127 | 1.1105 | 1.1128 | 1.2318 | 1.215 | 1.2182 |

3.0 | 1.8026 | 1.8027 | 1.8029 | 1.2377 | 1.2336 | 1.2378 | 1.4603 | 1.4368 | 1.4416 |

4.0 | 2.2359 | 2.2361 | 2.2363 | 1.3920 | 1.3856 | 1.3921 | 1.7210 | 1.6822 | 1.7026 |

5.0 | 2.6923 | 2.6925 | 2.6928 | 1.5659 | 1.5574 | 1.5660 | 1.9995 | 1.9180 | 1.9862 |

**Table 2.**Comparison of dimensionless linear frequency ($\omega {l}^{2}\sqrt{\rho \mathcal{A}/E{I}_{2}}$) of a straight nano-beam with different nonlocal parameter µ and various endpoint conditions.

μ | H–H | C–C | C–H | ||||||
---|---|---|---|---|---|---|---|---|---|

Ref. [37] | Ref. [22] | Present | Ref. [37] | Ref. [22] | Present | Ref. [37] | Ref. [22] | Present | |

0.0 | 3.0929 | – | 3.0929 | 4.4491 | – | 4.4491 | 3.7845 | – | 3.7844 |

0.1 | 3.0243 | 3.0210 | 3.0210 | 4.3471 | 4.3269 | 4.3269 | 3.6939 | 3.6849 | 3.6849 |

0.3 | 2.6538 | 2.6385 | 2.6385 | 3.7895 | 3.7032 | 3.7032 | 3.2115 | 3.1724 | 3.1724 |

0.5 | 2.2867 | 2.2665 | 2.2665 | 3.2420 | 3.1372 | 3.1371 | 2.7471 | 2.6982 | 2.6980 |

0.7 | 2.0106 | – | 1.9899 | 2.8383 | – | 2.7327 | 2.4059 | – | 2.3569 |

**Table 3.**Comparison of linear and nonlinear dimensionless frequency of a straight nonlocal nano-beam with μ = 0.15 and ${w}_{max}^{*}=0.4$.

Frequency | H–H | C–C | C–H | |||
---|---|---|---|---|---|---|

Ref. [22] | Present | Ref. [22] | Present | Ref. [22] | Present | |

${\omega}_{l}^{*}$ | 0.4233 | 0.4233 | 0.8055 | 0.8055 | 0.6052 | 0.6052 |

${\omega}_{nl}^{*}$ | 0.4405 | 0.4435 | 0.8188 | 0.8219 | 0.6197 | 0.6236 |

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**MDPI and ACS Style**

Eshraghi, I.; Jalali, S.K.; Pugno, N.M.
Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory. *Materials* **2016**, *9*, 786.
https://doi.org/10.3390/ma9090786

**AMA Style**

Eshraghi I, Jalali SK, Pugno NM.
Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory. *Materials*. 2016; 9(9):786.
https://doi.org/10.3390/ma9090786

**Chicago/Turabian Style**

Eshraghi, Iman, Seyed K. Jalali, and Nicola Maria Pugno.
2016. "Imperfection Sensitivity of Nonlinear Vibration of Curved Single-Walled Carbon Nanotubes Based on Nonlocal Timoshenko Beam Theory" *Materials* 9, no. 9: 786.
https://doi.org/10.3390/ma9090786