# Free Vibrations of Bernoulli-Euler Nanobeams with Point Mass Interacting with Heavy Fluid Using Nonlocal Elasticity

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*Nanomaterials*

**2022**,

*12*(15), 2676; https://doi.org/10.3390/nano12152676 (registering DOI)

## Abstract

**:**

## 1. Introduction

## 2. Equilibrium Equations and Boundary Conditions

_{y}is the axial stress and A is area of the cross-section.

#### 2.1. Separation of Variables Method

_{1}is the first ordinal number n of a wave number $\widehat{\kappa}$ for which condition (27) is fulfilled, and G

_{n}are real constants.

_{z}is the second moment of the cross-sectional area:

#### 2.2. Application of Nonlocal PurelySDM Theory

_{λ}is the special convolution kernel

_{c}is the characteristic nanobeam length defined by the expression.

_{j}of Equation (49) are:

_{n}in the particular solution are determined from the fluid-structure interaction condition (13), which, using the definition of fluid pressures (28) and beam displacements (15), has the following form:

_{k}is the number of used pressure field mode shapes. With the growth of n

_{k}the solution converges.

_{n}, and later on ${\tilde{E}}_{n}$, is used:

_{m}is the dimensionless ratio of the concentrated mass to the mass of the beam.

#### 2.3. Definition of the Eigenfrequencies of the Beam-Fluid System

_{i}and unknown eigenfrequency $\overline{\omega}$. These algebraic equations can be rewritten in the matrix form as the product of a quadratic matrix of linearly independent equation solutions, A($\overline{\omega}$), and the vector of constants C

_{i}, p:

## 3. Examples

#### 3.1. Convergence and the Influence of Nonlocal Parameter

_{k}) needed for the convergence of the results. The first part of this example deals with this issue. The beam length is L = 100 nm, the thickness F = 3.85 nm, the mass density ${\rho}_{s}$ = 2600 kg/m

^{3}and density of fluid is ${\rho}_{f}$ = 1000 kg/m

^{3}(water), so dimensionless parameter mass ratio of water to beam is γ = 10. Width of the beam is W = 10 nm and modulus of elasticity is E = 160 GPa. The linear free surface waves are not considered, i.e., p(x,y = H,t) = 0. Water is compressible and the speed of sound is c = 1439 m/s. The beam is completely immersed in water and there is no tip mass. Solutions in this and other examples were obtained by the aid of Wolfram Mathematica software. Possible numerical ill-conditioning didn’t appear [23].

_{k}= 4, so calculations will be performed with this value. Also, it can be seen that frequency ${\overline{\omega}}_{2}$ calculated with local theory, which is defined as the eigenfrequency of water domain in the rigid beam case [19], do not change its value with growth of n

_{k}. But, it can be also seen, that second frequency calculated with nonlocal theory changes with growth of n

_{k}which points out that frequencies connected with fluid are influenced by nonlocal theory. The reason for this new effect can be explained with the expression for fluid standing wave (28, 34) where coefficient G

_{n}is now a function of λ. If the fluid standing wave changes then its eigenfrequency also changes.

#### 3.2. The Influence of Tip Point Mass, Water and Nano-Scale Effects

_{m}= 0, 1, 2 and 3. The number of used pressure field mode shapes is n

_{k}= 4. The results are shown in Figure 4 and Table 3.

## 4. Conclusions

- an increase of the nonlocal parameter of PurelySDM method leads to an increase of the eigenfrequencies for all tested boundary conditions of the system beam-water, indicating a higher stiffness,
- the increase in point mass leads to a decrease in eigenfrequencies of a nanobeam for both local and nonlocal theory,
- the calculated eigenfrequencies of the coupled system with the local and nonlocal theory (corresponding to frequencies of the water domain) are equal for various tip point masses, but growth of eigenfrequencies occurs with the growth of the nonlocal parameter of PurelySDM method,
- when the beam is immersed in water, the main effect of the tip point mass (decrease of the eigenfrequency) is reduced for local and nonlocal theories. This effect can have a crucial impact on functionality of nanosensors.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Influence of the dimensionless nonlocal parameter growth on the eigenfrequencies of dry nanobeam and nanobeam immersed in water.

n_{k} | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

local theory | |||||

${\overline{\omega}}_{1}$ | 2.3456 | 1.9889 | 1.9418 | 1.9247 | 1.9169 |

${\overline{\omega}}_{2}$ | 2.4674 | 2.4674 | 2.4674 | 2.4674 | 2.4674 |

nonlocal PurelySDM theory, λ = 0.1 | |||||

${\overline{\omega}}_{1}$ | 2.4936 | 2.2502 | 2.1919 | 2.1712 | 2.1615 |

${\overline{\omega}}_{2}$ | 2.6732 | 2.4967 | 2.4977 | 2.4977 | 2.4977 |

λ | 0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 |

n_{k} = 0 | ||||||

${\overline{\omega}}_{1}$ | 3.5160 | 3.5515 | 3.5877 | 3.6246 | 3.6621 | 3.7002 |

${\overline{\omega}}_{1}$ [10] | 3.516013 | 3.551528 | 3.587734 | 3.624609 | 3.662122 | 3.700236 |

${\overline{\omega}}_{2}$ | 22.0345 | 22.2764 | 22.5608 | 22.8868 | 23.2523 | 23.6552 |

n_{k} = 4 | ||||||

${\overline{\omega}}_{1}$ | 1.9247 | 1.9482 | 1.972 | 1.9961 | 2.0206 | 2.0452 |

${\overline{\omega}}_{1}$ [13] ^{1} | 1.9047 | - | - | - | - | - |

${\overline{\omega}}_{2}$ | 2.4674 | 2.4677 | 2.4686 | 2.4701 | 2.4723 | 2.475 |

${\overline{\omega}}_{3}$ | 12.1148 | 12.2681 | 12.4436 | 12.6404 | 12.8574 | 13.0929 |

${\overline{\omega}}_{3}$ [13] ^{1} | 12.5670 | - | - | - | - | - |

${\overline{\omega}}_{4}$ | 22.2066 | 22.2313 | 22.305 | 22.4274 | 22.5977 | 22.8147 |

**Table 3.**Influence of tip point mass, water and nano-scale effects (linear free surface waves not considered).

r_{m} | 0 | 1 | 2 | 3 |
---|---|---|---|---|

local theory, no fluid | ||||

${\overline{\omega}}_{1}$ | 3.5160 | 1.5573 | 1.1582 | 0.9628 |

${\overline{\omega}}_{2}$ | 22.0345 | 16.2501 | 15.8609 | 15.7198 |

PurelySDM, λ = 0.04, no fluid | ||||

${\overline{\omega}}_{1}$ | 3.6621 | 1.609 | 1.1957 | 0.9937 |

${\overline{\omega}}_{2}$ | 23.2523 | 17.1006 | 16.6962 | 16.5498 |

local theory; fluid, n_{k} = 4 | ||||

${\overline{\omega}}_{1}$ | 1.9247 | 1.2982 | 1.0404 | 0.8923 |

${\overline{\omega}}_{2}$ | 2.4674 | 2.4674 | 2.4674 | 2.4674 |

${\overline{\omega}}_{3}$ | 12.1148 | 7.8824 | 7.2903 | 7.0561 |

${\overline{\omega}}_{4}$ | 22.2066 | 22.2066 | 22.2066 | 22.2066 |

PurelySDM, λ = 0.04; fluid, n_{k} = 4 | ||||

${\overline{\omega}}_{1}$ | 2.0206 | 1.3491 | 1.0781 | 0.9235 |

${\overline{\omega}}_{2}$ | 2.4723 | 2.4723 | 2.4723 | 2.4723 |

${\overline{\omega}}_{3}$ | 12.8574 | 8.3265 | 7.7124 | 7.4708 |

${\overline{\omega}}_{4}$ | 22.5977 | 22.5977 | 22.5977 | 22.5977 |

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**MDPI and ACS Style**

Barretta, R.; Čanađija, M.; Marotti de Sciarra, F.; Skoblar, A.
Free Vibrations of Bernoulli-Euler Nanobeams with Point Mass Interacting with Heavy Fluid Using Nonlocal Elasticity. *Nanomaterials* **2022**, *12*, 2676.
https://doi.org/10.3390/nano12152676

**AMA Style**

Barretta R, Čanađija M, Marotti de Sciarra F, Skoblar A.
Free Vibrations of Bernoulli-Euler Nanobeams with Point Mass Interacting with Heavy Fluid Using Nonlocal Elasticity. *Nanomaterials*. 2022; 12(15):2676.
https://doi.org/10.3390/nano12152676

**Chicago/Turabian Style**

Barretta, Raffaele, Marko Čanađija, Francesco Marotti de Sciarra, and Ante Skoblar.
2022. "Free Vibrations of Bernoulli-Euler Nanobeams with Point Mass Interacting with Heavy Fluid Using Nonlocal Elasticity" *Nanomaterials* 12, no. 15: 2676.
https://doi.org/10.3390/nano12152676