Modal Approach for Forced Vibration of Beams with a Breathing Crack

This paper presents a method for the vibration of a beam with a breathing crack under harmonic excitation. The infinitely thin crack is characterised by a parameter that takes into account the shape and the depth of the crack. The closed- and open-crack states are both modelled by a modal approach: two sets of equations of motion cast in the modal coordinates of their individual mode shapes. The state change (from closed to open or vice versa) involves the calculation of the modal coordinates associated with the new state from the modal coordinates of the previous state. By imposing the continuity of displacement and velocity the beam at the instant of the state change, the matrix that transforms the modal coordinates from one state to the other is determined and proved to be the Modal Scale Factor matrix. This analytical approach takes advantage of exact nature and mathematical convenience of beam modes and is time-efficient. Forced vibration at various values of crack parameter is determined. It is found that as decreases (crack length increases) the vibration becomes increasingly erratic and finally chaotic.

[1]  Liming Dai,et al.  Bifurcation and chaotic response of a cracked rotor system with viscoelastic supports , 2007 .

[2]  Yuan-Di Zhao,et al.  Structural power flow analysis of Timoshenko beam with an open crack , 2006 .

[3]  H. Baruh,et al.  ON-LINE IDENTIFICATION AND LOCATION OF ROTOR CRACKS , 1996 .

[4]  Y. M. Huang,et al.  Vibration And Stability Of A Rotating Shaft Containing A Transverse Crack , 1993 .

[5]  Michael J. Brennan,et al.  Numerical evaluation of high-order modes of vibration in uniform Euler–Bernoulli beams , 2007 .

[6]  W. Ostachowicz,et al.  Analysis of the effect of cracks on the natural frequencies of a cantilever beam , 1991 .

[7]  E Luzzato.  Approximate computation of non-linear effects in a vibrating cracked beam , 2003 .

[8]  I. Ballo,et al.  NON-LINEAR EFFECTS OF VIBRATION OF A CONTINUOUS TRANSVERSE CRACKED SLENDER SHAFT , 1998 .

[9]  M. Kısa,et al.  THE EFFECTS OF CLOSURE OF CRACKS ON THE DYNAMICS OF A CRACKED CANTILEVER BEAM , 2000 .

[10]  Matjaž Skrinar,et al.  New finite element for transversely cracked slender beams subjected to transverse loads , 2007 .

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

[12]  Arthur W. Lees,et al.  A non-linear study of a cracked rotor , 2007 .

[13]  Shun-Chang Chang,et al.  Forced responses of cracked cantilever beams subjected to a concentrated moving load , 2006 .

[14]  S. S. Law,et al.  Dynamic behavior of damaged concrete bridge structures under moving vehicular loads , 2004 .

[15]  John E. T. Penny,et al.  Crack Modeling for Structural Health Monitoring , 2002 .

[16]  K. Weiss.  Vibration Problems in Engineering , 1965, Nature.

[17]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

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

[19]  Shuncong Zhong,et al.  Response-only method for damage detection of beam-like structures using high accuracy frequencies with auxiliary mass spatial probing , 2008 .

[20]  Pierfrancesco Cacciola,et al.  Numerical and experimental verification of a technique for locating a fatigue crack on beams vibrating under Gaussian excitation , 2007 .

[21]  H. P. Lee,et al.  Dynamic response of a cracked beam subject to a moving load , 1994 .

[22]  T. Chondros,et al.  VIBRATION OF A BEAM WITH A BREATHING CRACK , 2001 .

[23]  K. Worden,et al.  Past, present and future of nonlinear system identification in structural dynamics , 2006 .

[24]  Paul R. White,et al.  The detection of cracks in beams using chaotic excitations , 2007 .

[25]  Hai-Ping Lin,et al.  Direct and inverse methods on free vibration analysis of simply supported beams with a crack , 2004 .

[26]  P. Casini,et al.  Non-linear dynamics of a cracked cantilever beam under harmonic excitation , 2007 .

[27]  Mohamed S. Gadala,et al.  Dynamic behavior analysis of cracked rotor , 2008 .