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The vibrational characteristics of a microbeam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical study of the modal analysis of microbeams partially immersed in a viscous fluid. A fixed-free microbeam vibrating in a viscous fluid is modeled using the Euler-Bernoulli equation for the beams. The unsteady Stokes equations are solved using a Helmholtz decomposition technique in a two-dimensional plane containing the microbeams cross sections. The symbolic software Mathematica is used in order to find the coupled vibration frequencies of beams with two portions. The frequency equation is deduced and analytically solved. The finite element method using Comsol Multiphysics software results is compared with present method for validation and an acceptable match between them was obtained. In the eigenanalysis, the frequency equation is generated by satisfying all boundary conditions. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy.

The objective of this paper is to provide an analytical method to calculate the coupled frequencies of vibration of microbeams partially immersed in a viscous fluid. The microbeams are clamped on one edge while the other edge is free.

The motivation of this work is to provide a theoretical model that can be used in the design and interpretation of density and viscosity sensors.

Due to their size and potential for highly sensitive and low cost compact device applications, microstructures are becoming increasingly attractive for sensing applications and have been studied extensively in recent years. Microstructures are commonly used in atomic force microscopy (AFM) to probe surface properties and to measure interfacial forces [

For microstructures, fluid viscosity can greatly affect their frequency response. Reference [

The frequency analysis of a microbeam can be dramatically affected by the properties of the fluid in which it is immersed. Whereas calculation of the natural frequencies in vacuum can be performed routinely, analysis of the effects of immersion in fluid poses a formidable challenge. The modal response of an immersed microbeam can be considerably affected by the properties of fluid. The added mass effect due to the fluid structure interaction can, however, cause considerable variations in natural frequencies. The knowledge and understanding of this viscous fluid-structure coupling are lacking at present.

In this contribution, we investigate the vibrational behavior of microbeams partially immersed in a viscous fluid, which describes the interrelation between the fluid’s density and viscosity. For a viscous fluid problem, the analytical formulation is based upon a convenient decomposition of the velocity field into two contributions, one being related to the scalar potential and the other being the vector potential. The solutions of the differential equations of motion turn out to be complex and can be conveniently treated with the aid of the symbolic software Mathematica. Furthermore, this work investigates the influence of the fluid’s viscosity on the vibrational behavior of the microbeams.

Modal analysis of elastic immersed structures is needed in every modern construction and should have wide engineering application. In this study, modal analysis is important to predict the dynamic behavior of the submerged beams. It is well known that the natural frequencies of the submerged elastic structures are different from those in vacuum. The effect of fluid forces on the immersed beam decreases the natural frequencies from those that would be measured in the vacuum. This decrease in the natural frequencies is caused by the increase of the kinetic energy of the fluid-beams system without a corresponding increase in strain energy. The Euler-Bernoulli beam is partially immersed inside rectangular fluid domain (Figure ^{−2} [m], width 4.6·10^{−3} [m], and thickness 1.27·10^{−4} [m] as shown in Figure

Sketch of beam composed of 2 uniform beam segments (denoted by

In this section we present the general theory for the dynamic deflection of beams partially submerged in a viscous fluid. A schematic depiction of beams partially submerged in viscous fluid is displayed in Figure

For a beam moving in a viscous fluid, the applied load

Consider time harmonic motion throughout with angular frequency

The general solutions of the ordinary differential equations (

The nine constants

The model that was developed in Section ^{9} [Pa], Poisson’s ratio being ^{−3}] and dynamic’s viscosity of

For coupled vibration analysis of beams partially immersed in a viscous fluid, the accuracy of the present method has been compared with the results obtained with Comsol Multiphysics FEM Simulation Software. The FEM used model had

In this paper the comparison of the values of the angular frequency parameter

The first 24 coupled natural angular frequencies

Order | Present | FEM | Difference |
---|---|---|---|

1 | 4.90 | 4.86 | 0.82 |

2 | 18.18 | 17.91 | 1.50 |

3 | 20.26 | 20.01 | 1.24 |

4 | 21.11 | 20.86 | 1.19 |

5 | 24.68 | 24.50 | 0.73 |

6 | 30.48 | 30.20 | 0.92 |

7 | 31.59 | 31.44 | 0.47 |

8 | 39.24 | 38.91 | 0.84 |

9 | 44.68 | 44.51 | 0.38 |

10 | 47.65 | 47.31 | 0.71 |

11 | 50.20 | 49.72 | 0.96 |

12 | 54.68 | 54.02 | 1.22 |

13 | 56.10 | 55.33 | 1.39 |

14 | 61.47 | 61.25 | 0.35 |

15 | 61.90 | 61.36 | 0.88 |

16 | 64.57 | 64.15 | 0.65 |

17 | 67.29 | 66.72 | 0.85 |

18 | 73.47 | 72.99 | 0.65 |

19 | 79.63 | 79.21 | 0.53 |

20 | 80.06 | 79.42 | 0.80 |

21 | 85.67 | 85.41 | 0.30 |

22 | 87.58 | 87.09 | 0.56 |

23 | 91.33 | 90.85 | 0.52 |

24 | 97.44 | 97.10 | 0.35 |

With the derived eigenfrequency equations, natural frequencies

Table

The two possible first coupled mode shapes are represented in Figure

Two possible first coupled mode shapes.

Figures

The influence of fluid’s viscosity on the first five odd natural frequencies.

The influence of fluid’s viscosity on the first five even natural frequencies.

Figures

View of displacement filled of beams and Stokes eddies, showing various mode shapes for the coupled vibration.

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

View of displacement filled of beams and Stokes eddies, showing various mode shapes for the coupled vibration.

Mode 7

Mode 8

Mode 9

Mode 10

Mode 11

Mode 12

View of displacement filled of beams and Stokes eddies, showing various mode shapes for the coupled vibration.

Mode 13

Mode 14

Mode 15

Mode 16

Mode 17

Mode 18

Mode 19

Mode 20

Mode 21

Mode 22

Mode 23

Mode 24

This paper has presented an analytical method to understand the modal behavior of beams partially immersed in a viscous fluid. The validity of the present solution is solidly confirmed numerically. The main application of the proposed method is in characterization of rheological properties of viscous materials [

In this paper the interface

We postulate the following solution of (

For nontrivial solution, the determinant of the matrices

The coefficients

The matrix

The authors declare that there is no conflict of interests regarding the publication of this paper.