Eigenvalue based inverse model of beam for structural modification and diagnostics: theoretical formulation

Abstract

In the work, the problems of the beam structural modification through coupling the additional mass or elastic support, as well as the problem of diagnostics of the beam cracks, are discussed. The common feature for both problems is that the material parameters in each of the discussed cases change only in one point (additional mass, the support in one point, the crack described by the elastic joint). These systems, after determination of the value of additional element and its localization, should have a given natural vibration frequency. In order to solve the inverse problem, i.e. the problem of finding values of the additional quantities (mass, elasticity), the beam inverse model was proposed. Analysis of this model allows finding such a value of additional mass (elasticity) as a function of its localization so that the system has the free vibration frequency, which is desired in the modification problem or measured on the object in the diagnostics.

Mass sensor using mode localization in two weakly coupled MEMS cantilevers with different lengths: Design and experimental model validation

“…On the long cantilever, we also add a discrete point mass located at a distancex m from the fixed end in order to consider the mass perturbation. According to the equation of beams carrying lumped element [31,32,33], the equations governing the bending vibration of the model shown in Fig. 1b is given by…”

Modes Orthogonality of the Mechanical System Simple Supported Beam-Actuators-Concentrated Masses

“…This problem corresponds to the one given in previous subsection; it is not the same, however, there are many similarities. Let a few masses be attached to the beam [22,1]. They are marked by {m r }, r = 1, 2, .…”

“…To solve it, some methods may be applied. One of them is presented in [1,4,18]; another attitude may be found in [19] and it is applied here.…”

“…The beam is a structural element carrying masses elastically or stiy attached to it; this is the natural task of the beam. The concentrated masses are also used to the modication of the structure dynamics [1,2]. The free vibration of these systems are solved in many papers, but only natural frequencies or/and modes have usually been obtained by means of an exact analysis [2,3], as well as by employing some approximate methods [4,5].…”

An Optimal Distribution of Actuatorsin Active Beam Vibration – Some Aspects, Theoretical Considerations

“…(1) must be rounded out. First of all, to obtain asymmetric modes and consistently asymmetric general vibration, a few concentrated masses are added to the beam (Low & Naguleswaran, 1998;Majkut, 2010;Naguleswaran, 1999). They are marked by { } r m , and their distribution is described with set of coordinates { } r…”