The use of quasi-isospectral operators for damage detection in rods

Abstract

We consider the inverse problem of reconstructing the axial stiffness of a damaged rod from the knowledge of a finite number of resonant frequencies of the free axial vibration under supported end conditions. The damage is described as a reduction of the axial stiffness, and the undamaged and damaged configurations of the rod are assumed to be symmetric. The method is based on repeated determination of quasi-isospectral rod operators, that is rods which have the same spectrum of a given rod with the exception of a single resonant frequency which is free to move in a prescribed interval. The reconstruction procedure is explicit and it is numerically implemented and tested for the identification of single and multiple localized damages. The sensitivity of the technique to the number of frequencies used and to the shape, intensity and position of the damages, as well as to the presence of noise in the data, is evaluated and discussed. The effect of suitable filtering of the results based on a priori information on the physics of the problem is proposed. An experimental application to the identification of localized damage in a free-free steel rod is also presented.

Computation of brittle fracture propagation in strain gradient materials by the FEniCS library

“…Continuum mechanics has been widely used to study damage phenomena [1–13]. In continuum damage models, the studied system is described not only by a displacement field, but also by a damage field.…”

Isospectrals of non-uniform Rayleigh beams with respect to their uniform counterparts

“…Also, recent advancements in machining techniques—such as additive manufacturing and rapid prototyping [20,21]—facilitate the manufacturing of beams with known breadth, height and material property variation. Structural identification issues, such as damage and blockage identification, have been solved with the help of quasi-isospectral operators for rods and symmetric ducts [22,23]. We can develop similar transformation for various versions of beams which along with the quasi-isospectral operators are an integral part of damage and blockage identification.…”

Evolution of bone tissue based on angiogenesis as a crucial factor: New mathematical attempt

“…the theoretical description of overloaded dying bone cells could be extended with an extra killing effect of the damaged tissue [54, 55].…”

“…• more biochemical factors should be considered such as influence of extracellular matrix or other types of bone cells which affect the process of bone creation; • all theoretical parameters should be identified and replaced by parameters determined experimentally [47][48][49][50][51][52][53]; • cell death calculations should be included in the formula for the changes in the bone cell densities; • the formulation should be extended to include dynamic effects as it is in a real state; • the theoretical description of overloaded dying bone cells could be extended with an extra killing effect of the damaged tissue [54,55].…”