Crack identification in elastically restrained vibrating rods

Abstract

In this paper we consider the problem of identifying an open crack in a longitudinally vibrating rod with smooth variable profile by minimal eigenfrequency data. The crack is assumed to be open during vibration and it is modelled by an elastic spring acting along the beam axis. Most, if not all, the results available in the literature for this inverse problem refer to ideal end conditions, that is the rod is either under free or supported end conditions. As an example of almost optimal result, it is known that the knowledge of the fundamental (positive) natural frequency of the rod under free-free and cantilever end conditions allows for the unique determination of the crack, without any restriction on the damage severity. In this paper we show that the analysis of the analogous crack identification problem for rods under elastically restrained end conditions leads to different results and that, in general, the knowledge of the fundamental frequency belonging to two spectra associated to different end conditions is not sufficient for the uniqueness of the solution. The method we used to solve the inverse problem is of constructive type and it is based on general properties of Preprint submitted to International Journal of Non-Linear Mechanics December 17, 2016 the eigenfrequencies as functions of the position and severity of the crack. The identification procedure has been tested numerically on rods under various damage scenarios. Numerical results agree well with the theory, even in presence of noisy input data.

Natural frequencies of parabolic arches with a single crack on opposite cross-section sides

“…Many other contributions focus on damage identification, for instance by evaluating structural static response; this is done in Morassi (2007, 2011), Greco and Pau (2011), Greco et al (2018b), just to quote a few relatively recent examples. Very promising, damage identification techniques still debated among researchers, rely on the analysis of dynamic response; relatively recent instances of contributions in this field are those by Eroglu and Tufekci (2016), Dilena et al (2017), andFernandez-Saez et al (2017).…”

On the use of a roving body with rotary inertia to locate cracks in beams

“…In particular, in this section a numerical investigation of a beam with multiple cracks in the presence of a roving body is conducted based on the explicit solution formulated in Eq. (28).…”

“…In addition, the observed jumps normalized with respect to the average frequency value defined as ( ) ( ) ( ) ( ) Table 2, are larger than the usual frequency measurement precision [28]. Table 2, also shows that the sudden frequency shifts generally increase with mode number.…”

Crack identification in rods and beams under uncertain boundary conditions

“…In fact, this would avoid the introduction of truncation errors in the Taylor's series of the eigenvalues. The use of the Lambda Curve Method, however, is not trivial and, at present, its application to beams with elastically restrained end conditions is known only for bars in axial vibration with known boundary flexibility [7].…”