# Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Fractional Order Viscoelasticity

- The left Riemann–Liouville fractional derivative of order $\alpha $ is of the form:$${}_{a}{D}_{t}^{\alpha}x\left(\tau \right)=\frac{1}{\Gamma \left(1-\alpha \right)}\frac{d}{dt}\underset{a}{\overset{t}{\int}}\frac{x\left(\tau \right)}{{\left(t-\tau \right)}^{\alpha}}d\tau ,t\in \left[a,b\right]$$
- The right Riemann–Liouville fractional derivative of order $\alpha $ is of the form:$${}_{t}{D}_{b}^{\alpha}x\left(\tau \right)=\frac{1}{\Gamma \left(1-\alpha \right)}\left(-\frac{d}{dt}\right)\underset{t}{\overset{b}{\int}}\frac{x\left(\tau \right)}{{\left(\tau -t\right)}^{\alpha}}d\tau ,t\in \left[a,b\right]$$

#### 2.2. Nonlocal Theory

_{0}is a material constant that can be determined from molecular dynamics simulations or by using the dispersive curve of the Born–Karman model of lattice dynamics. Later, Eringen [2] proposed a differential form of the constitutive relation with an appropriate kernel function as:

## 3. Governing Equation of the Nanobeam Resting on the Fractional Order Viscoelastic Foundation

#### Solution of the Governing Equation

## 4. Numerical Results

#### Validation Study

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CNT | Carbon nanotube |

SDCNT | Single-walled carbon nanotube |

DWCNT | Double-walled carbon nanotube |

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**Figure 1.**Boundary conditions for different beam supports. (

**a**) Simple-simple case and (

**b**) clamped-clamped case.

**Figure 4.**First three modes of the fractional nonlinear frequency versus nonlocality $\eta $ ($\alpha =0.5$, $K=5$, $Kp=2$, $Cp=0.001$).

**Figure 5.**First three modes of the fractional nonlinear frequency versus amplitude ($\alpha =0.5$, $K=5$, $Kp=2$, $Cp=0.001$, $\eta =0.5$).

**Figure 6.**Fractional nonlinear frequency versus amplitude for different values of $Kp$ ($\alpha =0.5$, $K=100$, $Cp=0.001$, $\eta =0.5$).

**Figure 7.**Fractional nonlinear frequency versus amplitude for different values of K ($\alpha =0.5$, $Kp=5$, $Cp=0.001$, $\eta =0.5$).

**Figure 8.**Fractional nonlinear frequency versus amplitude for different values of $Cp$ ($\alpha =0.5$, $Kp=5$, $K=100$, $\eta =0.5$).

**Figure 9.**Frequency-response curves versus amplitude for different values of $\chi $ ($\alpha =1$, $C=0.025$, $Cp=0.025$, $wo=1$, $\overline{F}=0.2$).

**Figure 10.**Fractional contribution frequency versus stiffness K and nonlocality $\eta $ ($\alpha =0.2$).

**Figure 11.**Fractional contribution frequency versus stiffness K and nonlocality $\eta $ ($\alpha =0.5$).

**Figure 12.**Fractional contribution frequency versus stiffness K and nonlocality $\eta $ ($\alpha =0.8$).

**Figure 13.**Fractional contribution frequency versus stiffness $Kp$ and nonlocality $\eta $ ($\alpha =0.2$).

**Figure 14.**Fractional contribution frequency versus stiffness $Kp$ and nonlocality $\eta $ ($\alpha =0.5$).

**Figure 15.**Fractional contribution frequency versus fractional damping coefficient $Cp$ and nonlocality $\eta $ ($\alpha =0.2$).

**Figure 16.**Fractional contribution frequency versus fractional damping coefficient $Cp$ and nonlocality $\eta $ ($\alpha =0.5$).

**Figure 17.**Fractional contribution frequency versus fractional damping coefficient $Cp$ and nonlocality $\eta $ ($\alpha =1$).

**Table 1.**The first five non-dimensional natural frequencies of a local Euler–Bernouilli beam resting on a Winkler–Pasternak foundation for the simple-simple boundary condition ($\eta =0$, $\delta =0$, $K=25$, ${K}_{p}=25)$.

Mode | Present | Ref. [28] | Ref. [17] | Ref. [43] |
---|---|---|---|---|

1 | 19.2133 | 19.2133 | 19.2178 | 19.21 |

2 | 50.7002 | 50.7002 | 50.7804 | 50.71 |

3 | 100.6767 | 100.677 | - | - |

4 | 170.0281 | 170.028 | - | - |

5 | 258.9868 | 258.987 | - | - |

**Table 2.**The first five non-dimensional natural frequencies of a local Euler–Bernouilli beam resting on a Winkler–Pasternak foundation for the simple-simple boundary condition ($\eta =0$, $\delta =0$, $K=36$, ${K}_{p}=36)$.

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**MDPI and ACS Style**

Eyebe, G.J.; Betchewe, G.; Mohamadou, A.; Kofane, T.C.
Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations. *Fractal Fract.* **2018**, *2*, 21.
https://doi.org/10.3390/fractalfract2030021

**AMA Style**

Eyebe GJ, Betchewe G, Mohamadou A, Kofane TC.
Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations. *Fractal and Fractional*. 2018; 2(3):21.
https://doi.org/10.3390/fractalfract2030021

**Chicago/Turabian Style**

Eyebe, Guy Joseph, Gambo Betchewe, Alidou Mohamadou, and Timoleon Crepin Kofane.
2018. "Nonlinear Vibration of a Nonlocal Nanobeam Resting on Fractional-Order Viscoelastic Pasternak Foundations" *Fractal and Fractional* 2, no. 3: 21.
https://doi.org/10.3390/fractalfract2030021