Shock and Vibration

Volume 2018, Article ID 4830391, 13 pages

https://doi.org/10.1155/2018/4830391

## Damage Identification in Structures Based on Energy Curvature Difference of Wavelet Packet Transform

Institute of Solid Mechanics, Beihang University (Beijing University of Aeronautics and Astronautics), Beijing 100191, China

Correspondence should be addressed to Pengbo Wang; nc.ude.aaub@gnawobgnep

Received 24 November 2017; Revised 25 March 2018; Accepted 31 March 2018; Published 28 May 2018

Academic Editor: Chao Tao

Copyright © 2018 Pengbo Wang and Qinghe Shi. 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

Damage identification is of tremendous significance in engineering structures. One key issue in damage identification is to determine an index that is sensitive to the structural damage. Current damage identification indices are generally focused on dynamic characteristics such as the natural frequencies, modal shapes, frequency responses, or their mathematical combinations. In this study, based on the wavelet packet transform, we propose a novel index, the energy curvature difference (ECD) index, to identify the damage in structures. The ECD index is the summation of component energy curvature differences after a signal is decomposed using WPT. Moreover, two numerical examples are used to demonstrate the feasibility and validity of the proposed ECD index for damage identification. Stiffness reduction is employed to simulate the structural damage. The damage can be identified by the ECD index curve plot. The results of the examples indicate that the proposed ECD index is sensitive to low damage levels because even 5% stiffness reduction can be apparently identified. The proposed ECD index can be employed to effectively identify structural damage.

#### 1. Introduction

Damage is a serious threat to structures during their service life, and a damage identification technology is of tremendous significance in engineering applications. Once a structure is damaged, the dynamic characteristics of the structure, such as the modal shapes, natural frequencies, and frequency responses, will change [1]. Various methods have been proposed for the use of dynamic responses for damage identification [2–6]. These methods are regarded as vibration-based damage identification methods. Among these methods, the approach based on wavelet transform is playing an increasingly important role because it can focus on any detail of a signal in the time or frequency domain [7, 8].

The wavelet transform is an extension of the traditional Fourier transform and is capable of performing local and self-adaptive time-frequency analyses. Therefore, the wavelet transform can reveal some hidden phenomena of a signal that other signal processing techniques fail to observe [9, 10]. This property has been introduced in the damage identification application. Numerous studies have been published regarding the use of wavelet transform for damage identification [11–15]. However, an obvious disadvantage of the wavelet transform is that the frequency resolution in the higher frequency domain is not high. It may be difficult to differentiate a signal containing high-frequency components that are very close to each other. Thus, scholars developed the wavelet packet transform (WPT) technique.

The WPT can be viewed as an extension of the traditional wavelet transform. It is capable of executing a complete level-by-level decomposition on a signal [16]. By performing the WPT on a signal, the signal can be decomposed into a series of wavelet packet components with a certain decomposition level, and the component energies can be obtained. The energy of the original signal is the summation of the component energies corresponding to different frequency bands. The WPT has been used for the damage identification of structures [17–20]. Wavelet packet energy can be obtained in different frequency bands after a signal is decomposed using WPT. The wavelet packet energy has been employed for the damage identification in ancient wood structures, beams, cable-stayed bridges, and so on [21–25]. Curvature difference can reflect the small changes in a function or a signal and has been introduced in damage identification together with wavelet packet energy. Some researchers have used curvature difference based on wavelet packet energy to locate the damage in ancient wood structures, concrete frames, and bridges [26–29].

One key issue in damage identification is to determine an index that is sensitive to the structural damage. Current damage identification indices are generally focused on dynamic characteristics such as the natural frequencies, modal shapes, frequency responses, or their mathematical combinations. In this study, based on the WPT, we propose a novel index, the energy curvature difference (ECD) index, to identify the damage in structures. The ECD index is the summation of component energy curvature differences after a signal is decomposed using WPT. It takes into account the spatial distribution of the collected signals and is sensitive to low damage levels because even a 5% stiffness reduction can be apparently identified. Moreover, two numerical examples are used to demonstrate the feasibility and validity of the proposed ECD index for damage identification. Mutation on the ECD index curve can identify the damage accurately.

The remainder of this paper is organised as follows. In Section 2, some basics of the WPT are briefly introduced. In Section 3, an ECD index based on the WPT is proposed for damage identification. In Section 4, two numerical examples are provided to illustrate the applicability of the proposed ECD index. Section 5 presents the conclusions of this study.

#### 2. Basics of Wavelet Packet Transform

The WPT is capable of accomplishing a complete level-by-level decomposition of the signals. Wavelet packet can be expressed as follows: where indicates the modulation parameter, indicates the scale parameter, and indicates the translation parameter. The wavelet packet functions possess the orthogonality property, i.e.,

The first wavelet function is the mother wavelet function, i.e.,

Wavelet functions can be calculated using the recursive equation where and are the quadrature mirror filters. is related to the scaling function and is related to the mother wavelet function.

Assume that is a time signal. If is decomposed to the th level, it can be expressed as where represents the th order wavelet packet component signal at level . can be expressed by a linear combination of wavelet packet functions in the following manner: where represents the wavelet packet coefficient, expressed as

WPT is a generalization of wavelet decomposition that offers a richer range of possibilities for signal analysis. In wavelet transform, a signal is split into an approximation and a detail. The approximation is then itself split into a second-level approximation and detail, and the process is repeated. In WPT, the details as well as the approximations can be split. This yields more different ways to encode the signal. The decomposition tree of WPT is plotted in Figure 1, where A represents the approximations and D represents the details, and denotes the decomposition level.