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Mathematical Problems in Engineering
Volume 2012, Article ID 454568, 16 pages
Research Article

Fractal Dimension Analysis of Higher-Order Mode Shapes for Damage Identification of Beam Structures

1Department of Engineering Mechanics, College of Water-Conservancy and Civil Engineering, Shandong Agricultural University, Taian 271000, China
2Department of Engineering Mechanics, Hohai University, Nanjing 210098, China
3Institute of Fluid Flow Machinery, Polish Academy of Sciences, 80-952 Gdańsk, Poland
4Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 23 April 2012; Accepted 27 June 2012

Academic Editor: Ivan Bartoli

Copyright © 2012 Runbo Bai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fractal dimension analysis is an emerging method for vibration-based structural damage identification. An unresolved problem in this method is its incapability of identifying damage by higher-order mode shapes. The natural inflexions of higher-order mode shapes may cause false peaks of high-magnitude estimates of fractal dimension, largely masking any signature of damage. In the situation of a scanning laser vibrometer (SLV) providing a chance to reliably acquire higher-order (around tenth-order) mode shapes, an improved fractal dimension method that is capable of treating higher-order mode shapes for damage detection is of important significance. This study proposes a sophisticated fractal dimension method with the aid of a specially designed affine transformation that is able to obviate natural inflexions of a higher-order mode shape while preserving its substantial damage information. The affine transformed mode shape facilitates the fractal dimension analysis to yield an effective damage feature: fractal dimension trajectory, in which an abruptly risking peak clearly characterizes the location and severity of the damage. This new fractal dimension method is demonstrated on multiple cracks identification in numerically simulated damage scenarios. The effectiveness of the method is experimentally validated by using a SLV to acquire higher-order mode shapes of a cracked cantilever beam.