Crack identification of beam structures using homotopy continuation algorithm

Abstract An approach based on homotopy continuation algorithm is presented to identify the parameters of a cracked beam. Euler–Bernoulli finite beam element with a fully opened crack model is adopted to establish the dynamic equation of the structural system. In the inverse problem, the homotopy equation is derived from minimizing the error between the calculated and the simulated measured acceleration responses. The range of homotopy parameter is divided into a number of divisions. Newton iterative method is employed to estimate the solution at each of these division points. The solution at the last division point corresponds to the homotopy equation matching the objective function. Numerical simulations with a simply supported beam and two-span beam show that the proposed method is very accurate compared to an existing method for both single and multiple cracks identification. The effects of type of excitation, division of the homotopy parameter and measurement noise on the identified results are discussed. It is noted that there is no need for an accurate set of initial values with the proposed approach.

[1]  Nikos A. Aspragathos,et al.  Identification of crack location and magnitude in a cantilever beam from the vibration modes , 1990 .

[2]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[3]  J. K. Liu,et al.  A two-step approach for crack identification in beam , 2013 .

[4]  Q. W. Yang,et al.  Damage identification by the eigenparameter decomposition of structural flexibility change , 2009 .

[5]  A. Morassi,et al.  Detecting cracks in pipes filled with fluid from changes in natural frequencies , 2011 .

[6]  K. Iba,et al.  Calculation of critical loading condition with nose curve using homotopy continuation method , 1991 .

[7]  G. De Roeck,et al.  Damage detection in bridges using modal curvatures: application to a real damage scenario , 1999 .

[8]  F. J. Gould,et al.  Homotopy methods and global convergence , 1983 .

[9]  Jyoti K. Sinha,et al.  SIMPLIFIED MODELS FOR THE LOCATION OF CRACKS IN BEAM STRUCTURES USING MEASURED VIBRATION DATA , 2002 .

[10]  S. S. Law,et al.  Dynamic condition assessment of a cracked beam with the composite element model , 2009 .

[11]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[12]  Maosen Cao,et al.  Crack detection in beams in noisy conditions using scale fractal dimension analysis of mode shapes , 2014 .

[13]  A. Barr,et al.  One-dimensional theory of cracked Bernoulli-Euler beams , 1984 .

[14]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[15]  Xueguan Song,et al.  Crack location in beams by data fusion of fractal dimension features of laser-measured operating deflection shapes , 2014 .

[16]  J. Yorke,et al.  The homotopy continuation method: numerically implementable topological procedures , 1978 .

[17]  Marek Krawczuk,et al.  Hexahedral finite element with an open crack , 1993 .

[18]  Daniel J. Inman,et al.  TIME DOMAIN ANALYSIS FOR DAMAGE DETECTION IN SMART STRUCTURES , 1997 .

[19]  Neil L. Book,et al.  A robust path tracking algorithm for homotopy continuation , 1996 .

[20]  A. K. Pandey,et al.  Damage Detection in Structures Using Changes in Flexibility , 1994 .

[21]  S. Chinchalkar.  DETERMINATION OF CRACK LOCATION IN BEAMS USING NATURAL FREQUENCIES , 2001 .

[22]  A. Trochidis,et al.  A method for determining the location and depth of cracks in double-cracked beams , 2004 .