Nonlinear normal modes for damage detection

The sensitivity of the nonlinear dynamic response to damage is investigated in simply supported beams by computing the nonlinear normal modes (NNMs) and exploiting the rich information they unfold about the nonlinear system behavior. Damage is introduced as a reduction of the flexural stiffness within a small segment of the beam span. The problem is formulated in a piece-wise fashion and analytically tackled by using the method of multiple scales to compute the NNMs and the backbone curves. The latter describe the frequency versus free oscillation amplitude for each mode and as such they represent the skeleton of the nonlinear frequency response around each beam modal frequency. The bending of the backbones is regulated by the so-called effective nonlinearity coefficients associated with each mode. The comparison between the damaged and undamaged beams shows a sensitivity of the effective nonlinearity coefficients higher than the sensitivity of the linear natural frequencies. Moreover, the nonlinear frequency trends unfold an interesting dependence of the nonlinear free response on the stiffness reduction at the damage site. Such dependence yields useful information about the damage position. An effective identification strategy for the damage position is proposed by computing the damage-induced discontinuities in the second derivative of the NNMs (i.e., first order estimate of the bending curvature), without resorting to a baseline model of the undamaged beam. This goal is achieved by computing a high order derivative of the Nadaraya–Watson kernel estimator, which is evaluated from a finite set of sampled beam deflections associated with each individual NNM. The results show that the second derivatives of higher NNMs are more sensitive to damage-induced discontinuities than the linear modal curvatures. The robustness against noise of the damage identification process is also discussed.

[1]  Irène Gijbels,et al.  Data-Driven Discontinuity Detection in Derivatives of a Regression Function , 2005 .

[2]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[3]  C. Ratcliffe.  DAMAGE DETECTION USING A MODIFIED LAPLACIAN OPERATOR ON MODE SHAPE DATA , 1997 .

[4]  Bogdan I. Epureanu,et al.  Damage detection in nonlinear systems using system augmentation and generalized minimum rank perturbation theory , 2005 .

[5]  Walter Lacarbonara,et al.  Nonlinear Structural Mechanics: Theory, Dynamical Phenomena and Modeling , 2013 .

[6]  Timothy M. Whalen,et al.  Experimental validation of the higher‐order derivative discontinuity method for damage identification , 2008 .

[7]  Cecilia Surace,et al.  On the use of the polynomial annihilation edge detection for locating cracks in beam-like structures , 2013 .

[8]  Giuseppe Quaranta,et al.  On the reliability of a PCA-based method for structural diagnosis in bridge structures with environmental disturbances , 2012 .

[9]  Walter Lacarbonara,et al.  Nonlinear normal modes of structural systems via asymptotic approach , 2004 .

[10]  Daniele Dessi,et al.  Damage identification techniques via modal curvature analysis: Overview and comparison , 2015 .

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

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

[13]  R. M. Rosenberg.  On Normal Vibrations of a General Class of Nonlinear Dual-Mode Systems , 1961 .

[14]  Francesco Benedettini,et al.  Vibration analysis and structural identification of a curved multi-span viaduct , 2015 .

[15]  Jorge Daniel Riera,et al.  Damage detection by means of structural damping identification , 2008 .

[16]  Fabrizio Vestroni,et al.  Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics , 2006 .

[17]  J. Hensman,et al.  Damage localisation in delaminated composite plates using a Gaussian process approach , 2015 .

[18]  Fabrizio Vestroni,et al.  Nonstandard Bifurcations in Oscillators with Multiple Discontinuity Boundaries , 2004 .

[19]  Walter Lacarbonara,et al.  Non-linear cancellation of the parametric resonance in elastic beams: Theory and experiment , 2007 .

[20]  Fabrizio Vestroni,et al.  Dynamical tests and analysis for the assessment of structural conditions of bridges , 2010 .

[21]  Y. Chu,et al.  Analysis of forced bilinear oscillators and the application to cracked beam dynamics , 1992 .

[22]  Walter Lacarbonara,et al.  A three-dimensional continuum approach to the thermoelastodynamics of large-scale structures , 2012 .

[23]  Giuseppe Quaranta,et al.  Damage detection by modal curvatures: numerical issues , 2016 .

[24]  Fabrizio Vestroni,et al.  A perturbation method for evaluating nonlinear normal modes of a piecewise linear two-degrees-of-freedom system , 2008 .

[25]  Yong Xia,et al.  Experimental Investigation on Statistical Moment-based Structural Damage Detection Method , 2009 .

[26]  S. Mukherjee,et al.  MODAL ANALYSIS OF A CRACKED BEAM , 1997 .