Physics Research International

Volume 2016, Article ID 6761372, 14 pages

http://dx.doi.org/10.1155/2016/6761372

## Vibration Analysis of Euler-Bernoulli Beams Partially Immersed in a Viscous Fluid

^{1}Arts et Métiers ParisTech, ENSAM Angers, 2 boulevard du Ronceray, 49035 Angers, France^{2}IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

Received 17 September 2015; Revised 20 January 2016; Accepted 27 January 2016

Academic Editor: Israel Felner

Copyright © 2016 Wafik Abassi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1–7], in biological and chemical sensors [8–10]. A precise modeling of the solid-fluid interaction and the determination of the frequency response enable the measurement of the density and the rheological behavior of fluids [11–16]. Reference [16] uses finite element analysis (FEA) method in order to predict the dynamic response of the cantilever beam. This method can be easily applied to the measurement of the fluid viscosity.

For microstructures, fluid viscosity can greatly affect their frequency response. Reference [7] presented a rigorous theoretical model for the frequency response of cantilever beams that are undergoing flexural vibrations and immersed in viscous fluids, which is of particular relevance to applications of the AFM. The knowledge and understanding of the frequency analysis of microbeams are of fundamental practical importance in application to the AFM.

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.

#### 2. Modal Analysis of Beams and Frequency Equation

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 1). Consider a beam of length 3.06·10^{−2} [m], width 4.6·10^{−3} [m], and thickness 1.27·10^{−4} [m] as shown in Figure 1, which corresponds to a model developed in [16]. The interaction between the fluid and the Euler-Bernoulli beams is taken into account to calculate the natural frequencies and mode shapes of the coupled system. The dynamics of each beam portion are treated separately. It is assumed that the beam has aligned neutral axis.