Shock and Vibration

Volume 2016, Article ID 2086274, 10 pages

http://dx.doi.org/10.1155/2016/2086274

## Free Vibration of a Perfectly Clamped-Free Beam with Stepwise Eccentric Distributed Masses

^{1}Department of Mechanical Engineering, Faculty of Engineering and Management, “Eftimie Murgu” University of Resita, P-ta Traian Vuia 1-4, 320085 Resita, Romania^{2}Soete Laboratory, Faculty of Engineering and Architecture, Ghent University, Technologiepark Zwijnaarde 903, 9052 Zwijnaarde, Belgium^{3}BMW AG, Hufelandstrasse 4, 80937 München, Germany

Received 18 October 2015; Revised 18 January 2016; Accepted 19 January 2016

Academic Editor: Tai Thai

Copyright © 2016 Gilbert-Rainer Gillich 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

A direct approach for the calculation of the natural frequencies and vibration mode shapes of a perfectly clamped-free beam with additional stepwise eccentric distributed masses is developed, along with its corresponding equations. Firstly there is contrived influence of a mass, located on a given position along the beam, upon the modal energies, via an energy analysis method. Secondly, the mass participation coefficient is defined as a function of the mass location and the bending vibration mode number. The proposed coefficient is employed to deduce the mathematical relation for the frequencies of beams with supplementary eccentric loads, generally available for any boundary conditions. The accuracy of the obtained mathematical relation was examined in comparison with the numerical simulation and experimental results for a cantilever beam. For this aim, several finite element models have been developed, individualized by the disturbance extent and the mass increase or decrease. Also, one real system was tested. The comparisons between the analytically achieved results and those obtained from experiments proved the accuracy of the developed mathematical relation.

#### 1. Introduction

During operation, engineering structures are subjected to loads, often produced by supplementary masses. These structures can suffer various damage types, including corrosion due to the interaction with the environment, which produce alterations of the geometry and the mechanical/physical properties [1]. All these alterations from the original state affect the structure’s modal parameters. Some typical damage geometries are known. Among cracks, one can distinguish between breathing cracks, producing just a stiffness decrease, and open cracks, which additionally generate a decrease in mass. Regarding corrosion, a conventional criterion to classify this phenomenon is concerned with the appearance of the corroded area. The two basic forms of corrosion are defined as (a) generalized corrosion, when the components surface is affected at the same rate on a large area, and (b) localized corrosion, which is restricted to compact areas [2].

Development of damage detection methods has drawn the interest of numerous researchers [3–8]. Even if most papers presented in literature tackle open cracks, the influence of the loss of mass is neglected, introducing this way inaccuracy in the damage detection methods. This fact is also valid for the detection of corrosion in structures. Another aspect concerns the design of damage detection methods for structures loaded by supplementary masses. Thus, deriving simple mathematical relations, which are able to predict the frequency changes due to mass alteration, is an actual challenge.

Herein, the cases of mass increase and decrease are equally treated, for different beam segment lengths, affected by these mass distortions. The idea was to find a reliable equation, which is able to indicate the natural frequencies, valid for beams irrespective of their support type and for any mass change scenario.

In prior research devoted to the analysis of beams with open cracks (see [9, 10]), slim beam regions were considered affected by stiffness and mass changes. It was proved that the superposition principle can be applied to find the natural frequencies of beams with open cracks. As a consequence, the problem of stiffness and mass variation can be approached separately. This paper introduces a behavioral model and the subsequent mathematical relation for beams with uniform stiffness but variable distributed mass, for which the accuracy is proved by comparison with numerical simulations and experimental test results.

#### 2. Analytical Investigation

The aim of this section is to introduce a mathematical relation contrived by the authors, which predicts the frequency changes due to mass variation. In order to illustrate the case, a cantilever beam is used. In this example, the geometrical asymmetry assures an unequivocal link between the mass disturbance, defined by position and intensity, and the frequency changes. The analyzed prismatic beam of steel, illustrated in Figure 1, has length , width , and thickness . As a consequence, the beam has a cross-sectional area and a moment of inertia . The mechanical parameters involved in this analysis are the longitudinal elasticity modulus , the volumetric mass density , and Poisson’s ratio *μ*. In addition, the earth gravity is considered. On the cantilever beam only the dead mass is considered.