Shock and Vibration

Volume 2015, Article ID 832738, 17 pages

http://dx.doi.org/10.1155/2015/832738

## Detection on Structural Sudden Damage Using Continuous Wavelet Transform and Lipschitz Exponent

^{1}Key Laboratory of Roadway Bridge and Structural Engineering, Wuhan University of Technology, Mail Box No. 219, No. 122 Luoshi Road, Wuhan 430070, China^{2}Guangdong Power Grid Corporation Co. Ltd., Guangzhou 510080, China

Received 3 August 2014; Accepted 12 October 2014

Academic Editor: Ting-Hua Yi

Copyright © 2015 Bo Chen 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.

#### Abstract

The degradation of civil engineering structures may lead to a sudden stiffness reduction in a structure and such a sudden damage will cause a discontinuity in the dynamic responses. The detection on structural sudden damage has been actively carried out in this study. The signal singularity of the acceleration responses with sudden stiffness reduction is characterized by the coefficients of continuous wavelet transform with fine scales. A detection approach based on the CWT is proposed in terms of the decomposed detail coefficients of continuous wavelet transform to detect the damage time instant and location. The Lipschitz exponent is mathematically used to estimate the local properties of certain function and is applied to reflect the damage severity. Numerical simulation using a five-story shear building under different types of excitation is carried out to assess the validity of the proposed detection approach for the building at different damage levels. The sensitivity of the damage index to the intensity and frequency range of measurement noise is also investigated. The effects of both measurement noise intensity and frequency range on the damage detection are numerically investigated.

#### 1. Introduction

The wavelet transform is an extension of the traditional Fourier transform with adjustable window location and size. Wavelet analysis combines both time and frequency analysis, which allows it to zoom in on time without any loss of scale resolution. The wavelet transform has recently emerged as a promising tool for structural health monitoring and it is an ideal tool in addressing the issue of time locality of structural damages [1–4]. The earliest work of applying wavelet analysis in structural health monitoring dated back to the work of Masuda and his group in 1995 [5, 6]. Faults in gear systems were detected using wavelet approaches and some results were verified by an inspection [7]. Hong et al. [8] studied the effectiveness of the wavelet transform in detecting structural cracks. In their analysis, the magnitude of the Lipschitz exponent is used as a useful indicator of the damage extent. The detection results from both numerical simulation and experiment prove the efficiency of wavelet based damage detection approaches.

The degradation of civil engineering structures may lead to a sudden stiffness reduction in a structure associated with the events such as weld fracture, column buckling, and brace breakage [9]. Such a sudden damage of stiffness in a structure will cause a discontinuity in acceleration responses and can be detected by using signal based detection approaches such as wavelet transform, empirical mode decomposition, time series, and time-frequency analysis. Hou et al. [10, 11] proposed a wavelet-based approach to identify the damage time instant and damage location of a simple structural model with breakage springs. Sohn et al. [12] used wavelet transform and the Holder exponent to detect the time varying nature of discontinuities. Their experimental results demonstrated that the Holder exponent could be an effective tool for identifying certain types of events that introduce discontinuities in the measured dynamic response data. The same idea for detecting sudden damage was adopted by Vincent et al. [13] and Yang et al. [9, 14] but using empirical mode decomposition to decompose the vibration signal to capture the signal discontinuity. In addition to the above-mentioned numerical studies, Xu and Chen [15] carried out experimental studies on the applicability of empirical mode decomposition for detecting structural damage caused by a sudden change of structural stiffness. Chen and Xu [16] proposed two online detection approaches for the sudden damage detection.

The sudden stiffness loss of structural components may induce the signal discontinuity in the acceleration responses close to the damage location at the damage time instant. It is reported that the time instant and location of the sudden stiffness loss can be detected by using the discrete WT. However, the severity of different sudden damage events cannot be estimated directly by the WT. In this regard, the investigation of detection on sudden damage event of building structures has been actively carried out in this study. The signal feature of the structural acceleration responses of an example building is examined. Three types of dynamic loading, sinusoidal, seismic, and impulse excitations are taken as the inputting excitations. The signal singularity of the acceleration responses with sudden stiffness reduction is characterized by the coefficients of continuous wavelet transform with fine scales. A detection approach based on the CWT is proposed in terms of the decomposed detail coefficients of continuous wavelet transform to detect the damage time instant and location. The Lipschitz exponent is mathematically used to estimate the local properties of certain function and is applied to reflect the damage severity. Numerical simulation using a five-story shear building under different types of excitation is carried out to assess the validity of the proposed detection approach for the building at different damage levels. The sensitivity of the damage index to the intensity and frequency range of measurement noise is also investigated. The effects of both measurement noise intensity and frequency range on the damage detection are numerically investigated. The made observations demonstrate that the proposed approach can accurately identify the damage events and the damage severity can be estimated by the Lipschitz exponent.

#### 2. Wavelet Transform

Morlet and Grossmann initially proposed wavelet theory and Meyer developed the mathematical foundations of wavelets. The two America-based researchers Daubechies [17] and Mallat [18, 19] changed this by defining the connection between wavelets and digital signal processing. Wavelets have been applied to a number of areas, including data compression, image processing, and time-frequency spectral estimation. A mother wavelet is a waveform that has limited duration and an average value of zero. Based on this mother wavelet, the wavelet kernel can be expressed bywhere and are dilation and translation parameters, respectively. Both are real numbers and must be positive. Similar to the Short Time Fourier Transform, one can analyze square-integrable function with wavelet transform, which decomposes a signal in the time domain into a two-dimensional function in the time-scale plane where denotes complex conjugation. The term frequency instead of scale has been used in order to aid in understanding, since a wavelet with large-scale parameter is related to low-frequency content component and vice versa. The mother wavelet should satisfy the following admissibility condition to ensure existence of the inverse wavelet transform such aswhere is the Fourier transform of . The existence of the integral in (3) requires thatThe signal can be reconstructed by an inverse wavelet transform of as defined by

The calculating wavelet coefficients at every possible scale will generate a lot of redundant data. In some practical signal processing cases, the discrete version of the wavelet is often utilized by discretizing the dilation parameter and the translation parameter . The procedure becomes much more efficient if dyadic values of and are used; that is,where is the set of positive integers. For some special choices of , the corresponding discrete wavelets can be written to constitute an orthonormal basisUsing the orthonormal basis, the wavelet expansion of a function and the coefficients of the wavelet expansion are defined as

#### 3. Signal Feature due to Sudden Damage

The signal feature due to a sudden stiffness reduction is firstly investigated by taking a five-story shear building as an example structure (Figure 1). The elevation of the shear building is displayed in Figure 2. The building is subject to different types of external excitations and a sudden stiffness loss occurs in the first story. The mass and horizontal stiffness of the undamaged building are uniform for all stories and the floor mass and stiffness are 1.3 × 10^{6} kg and 4.0 × 10^{9} N/m, respectively. The Rayleigh damping assumption is adopted to construct the structural damping matrix with the damping ratios in the first two vibration modes being set as 0.05. The original building is supposed to suffer a sudden 20% stiffness reduction in the first story while the horizontal stiffness in other stories remains unchanged. The frequency reduction due to 20% stiffness reduction in the first story is small with a maximum reduction no more than 5% in the first natural frequency. The sinusoidal excitation, seismic excitation, and impulse excitation are utilized, respectively, to calculate the acceleration responses of the example building to examine the signal features due to the sudden stiffness reduction. The seismic excitation used is the first 10 second portion of the El-Centro 1940 earthquake ground acceleration (S-N component) with a peak amplitude 1.0 m/s^{2}. A sinusoidal excitation expressed by (9) with 10 second duration is assumed to act on each floor of the building An impulse excitation represented by 0.1 m/s initial velocity is supposed to occur on the first floor of the building. The damage time instant of the building is set as 6.0 s for seismic excitation and sinusoidal excitation and 0.2 s for impulse excitation. The dynamic responses under each type of external excitation are computed by using the Newmark- method with a time interval of 0.002 s. The two factors in the Newmark- method are selected as and .