Some research has been devoted in modelling the cracks as always open during vibrations. In these works crack location and amplitude may be detected from alterations in natural frequencies and modes of vibrations as well as amplitude of forced vibrations [1,2]. However the cracks usually exhibit non-linear behaviour. In fact, if an insufficient static prelude is present, they open and close depending on the vibration direction causing the variation of the physical system parameters such as the stiffness.

The importance of considering the crack closure has been widely recognised [3,4], putting on evidence the change on the modal parameters due to the presence of a cross-sectional crack and investigating the response of a cantilevered beam with a breathing crack to harmonic force. In both these studies the measured parameters are related to deterministic setting.

Other techniques may assess the presence of a breach investigating the probabilistic characteristics of the response when the beam is subject to a random excitation. In fact, if the beam is undamaged (linear state) and a Gaussian load is applied, the structural response will be Gaussian too. On the contrary as soon as non-linearity arises due to the appearance of a crack the response becomes non- Gaussian. Inspection of the integrity of the structural element may be performed by estimating the non-Gaussianity of the response, through the evaluation of higher order statistics. The non-Gaussianity may be revealed in several ways. For example the kurtosis or skewness coefficient for one or more degree of freedom may be adopted as a measure of the non-Gaussianity of the system. Moreover time or frequency domain analysis can be performed. If the crack extension is important (let say greater than half of the beam height) the non-linearity is evident and any of the such measures is suitable. However in practical application the crack dimension is usually quite small and to assess the presence of the crack is much more cumbersome. This is due to the slight non-linearity of the system whose behaviour is very close to the linear one (the beam without any crack). In this case the choice of the measure to detect the non-Gaussianity is a crucial point to give clear information on the beam condition.

In this paper the stochastic dynamic analysis of a cracked cantilever beam under white noise is addressed in the time domain. The beam is discretised by finite elements in which a so-called closing crack model, with fully open or fully closed crack, is used to describe the damaged element. Once the mathematical model of the beam is defined, the dynamic response is evaluated by applying a numerical procedure based on the philosophy of structural systems with dynamic modification [5]. The statistics of the stochastic response are evaluated by means of Monte Carlo simulation. Both kurtosis and skewness coefficient are evaluated in correspondence of nodal degree of freedoms. A remarkable variability of the skewness at the rotational degree of freedom for different crack location and depth is encountered, assessing the capability of this measure to detect both the presence and the position of the crack along the beam even in the case the crack is small.

1

M.M.F. Yuen, "A numerical study of the eigenparameters of a damaged cantilever", Journal of Sound and Vibration 103(3), 301-310, doi:10.1016/0022-460X(85)90423-7

2

M. Kisa, J. Brandon, Topcu M., "Free vibration analysis of a cracked beams by a combination of finite elements and component mode synthesis methods", Computer & Structures 67, 215-223, 1998. doi:10.1016/S0045-7949(98)00056-X

3

G.L. Qian, S.N. Gu, Jiang J.S., "The dynamic behaviour and crack detection of a beam with a crack" Journal of Sound and Vibration 138(2), 233-243, 1990. doi:10.1016/0022-460X(90)90540-G

4

R. Ruotolo, C. Surace, P. Crespo, D. Storer, "Harmonic Analysis of the vibrations of a cantilevered beam with a closing crack", Computer & Structures, 6, 1057-1074, 1996. doi:10.1016/0045-7949(96)00184-8

5

G. Muscolino, "Dynamically modified linear structures: deterministic and stochastic response", Journal of Engineering Mechanics Division, ASCE, 122(11), 1044-1051, 1996. doi:10.1061/(ASCE)0733-9399(1996)122:11(1044)

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