Abstract
In this paper, an effective method for identification of multiple cracks is presented based on discrete wavelet transform in a cracked beam. First, a compactly supported semi-orthogonal B-spline wavelet on interval (BSWI) is employed to construct Euler beam-bending element for free vibration analysis of cracked beams. Next, the construction of general order one-dimensional B-spline wavelets is presented and applied for damage identification in a cantilever beam modeled by wavelet-based elements. Also, principles of an appropriate wavelet selection are presented. Natural vibration modes of a cantilever beam with three cracks are analyzed using one-dimensional fourth-order B-spline wavelet. The results illustrate that BSWI elements can be used in determining the un-cracked and cracked beam natural frequencies with a high accuracy and efficiency. Moreover, the applicability of the presented method in crack identification is studied by numerical examples under several situations, such as in the presence of random noises, and the efficiency of B-spline wavelets in damage prognosis is compared with other types of wavelet functions. The obtained results show the effectiveness of B-spline wavelets in modeling of the damaged beam and identifying multiple crack locations in a free baseline scheme.
Similar content being viewed by others
References
Mallat S.G.: A Wavelet Tour of Signal Processing. Academic, New York (1999)
Daubechies I.: Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996 (1988)
Daubechies I.: Ten Lectures on Wavelets. Society of Industrial and Applied Mechanics (SIAM), Philadelphia (1992)
Cohen A., Daubechies I., Feauveau J.C.: Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)
Chui C.K., Wang J.: A general framework of compactly supported splines and wavelets. J. Approx. Theory 71, 54–68 (1992)
Katunin A.: Construction of high-order B-spline wavelets and their decomposition relations for faults detection and localization in composite beams. Acta Mech. Autom. 46, 43–59 (2011)
Timofiejczuk, A.: Methods of Analysis of Non-Stationary Signals. Gliwice Publ House, Gliwice (in Polish) (2004)
Katunin A.: Damage identification in composite plates using two-dimensional B-spline wavelets. Mech. Syst. Signal Process 25, 3153–3167 (2011)
Dahmen W.: Wavelet methods for PDEs- some recent developments. J. Comput. Appl. Math. 128, 133–185 (2001)
Ma J.X., Xue J.J., Yang S.J., He Z.J.: A study of the construction and application of a Daubechies wavelet-based beam element. Finite Elem. Anal. Des. 39, 965–975 (2003)
Chen X.F., Yang S.J., Ma J.X., He Z.J.: The construction of wavelet finite element and its application. Finite Elem. Anal. Des. 40, 541–554 (2004)
Chen W.H., Wu C.W.: Extension of spline wavelets element method to membrane vibration analysis. Comput. Mech. 18, 46–54 (1996)
Cohen A.: Numerical Analysis of Wavelet Method. Elsevier, Amsterdam (2003)
Xiang J.W., Chen X.F., He Z.J.: The construction of plane elastomechanics and Mindlin plate elements of B-spline wavelet on the interval. Finite Elem. Anal. Des. 42, 1269–1280 (2006)
Xiang J.W., Chen X.F., He Z.J. et al.: The construction of 1D wavelet finite elements for structural analysis. Comput. Mech. 40(2), 325–339 (2007)
Xiang J.W., Chen X.F., He Z.J. et al.: A new wavelet-based thin plate element using B-spline wavelet on the interval. Comput. Mech. 41(2), 243–255 (2008)
Xiang J.W., Chen X.F., Yang L.F. et al.: A class of wavelet-based flat shell elements using B-spline wavelet on the interval and its applications. Comput. Model. Eng. Sci. 23(1), 1–12 (2008)
Douka E., Loutridis S., Trochidis A.: Crack identification in double-cracked beams using wavelet analysis. J. Sound Vib. 277, 1025–1039 (2004)
Wang Q., Deng X.: Damage detection with spatial wavelets. Int. J. Solids Struct. 36, 3443–3468 (1999)
Hong J.C., Kim Y.Y., Lee H.C., Lee Y.W.: Damage detection using the Lipschitz exponent estimated by the wavelet transform: applications to vibration modes of a beam. Int. J. Solids Struct. 39, 1803–1816 (2002)
Gentile A., Messina A.: On the continuous wavelet transforms applied to discrete vibrational data for detecting open cracks in damaged beams. Int. J. Solids Struct. 40, 295–315 (2003)
Mallat S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989)
Katunin A.: Identification of multiple cracks in composite beams using discrete wavelet transform. Sci. Problems Mach. Oper. Maint. 45, 41–52 (2010)
Fan W., Qiao P.: A 2-D continuous wavelet transform of mode shape data for damage detection of plates structures. Int. J. Solids Struct. 46, 4379–4395 (2009)
Unser M., Aldroubi A., Eden M.: On the asymptotic convergence of B-spline wavelets to Gabor functions. IEEE Trans. Inform. Theory 38, 864–872 (1992)
Ueda M., Lodha S.: Wavelets: An Elementary Introduction and Examples. University of California, Santa Cruz (1995)
Nandwana B.P., Maiti S.K.: Detection of the location and size of a crack in stepped cantilever beams based on measurements of natural frequencies. J. Sound Vib. 203(3), 435–446 (1997)
Yoona H.-I., Sona I.-S., Ahn S.-J.: Free vibration analysis of Euler–Bernoulli beam with double cracks. J. Mech. Sci. Technol. 21, 476–485 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Amiri, G.G., Jalalinia, M., Hosseinzadeh, A.Z. et al. Multiple crack identification in Euler beams by means of B-spline wavelet. Arch Appl Mech 85, 503–515 (2015). https://doi.org/10.1007/s00419-014-0925-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-014-0925-z