Shock and Vibration

Volume 2017 (2017), Article ID 5687837, 20 pages

https://doi.org/10.1155/2017/5687837

## Reconstructed Phase Space-Based Damage Detection Using a Single Sensor for Beam-Like Structure Subjected to a Moving Mass

^{1}College of Science & Engineering, Jinan University, Guangzhou, Guangdong, China^{2}State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an, Shanxi, China^{3}Department of Infrastructure Engineering, The University of Melbourne, Melbourne, VIC, Australia^{4}Dongguan University of Technology, Dongguan 523000, China^{5}College of Civil Engineering, Qinghai University, Xining, China

Correspondence should be addressed to Hongwei Ma

Received 17 June 2016; Revised 31 October 2016; Accepted 24 November 2016; Published 28 February 2017

Academic Editor: Ivo Caliò

Copyright © 2017 Zhenhua Nie 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

This paper presents a novel damage detection method based on the reconstructed phase space of vibration signals using a single sensor. In this approach, a moving mass is applied as excitation source, and the structure vibration responses at different positions are measured using a single sensor. A Moving Filter Function (MFF) is also presented to be used to separate and filter the responses before phase space reconstruction. Using the determined time delay and embedding dimensions, the responses are translated from time domain into the spatial domain. The index CPST (changes of phase space topology) values are calculated from the reconstructed phase space and used to identify structural damage. To demonstrate the method, six analysis scenarios for a beam-like structure considering the moving mass magnitude, damage location, the single sensor location, moving mass velocity, multiple types of damage, and the responses contaminated with noise are calculated. The acceleration and displacement responses are both used to identify the damage. The results indicate that the proposed method using displacement response is more sensitive to damage than that of acceleration responses. The results also proved that the proposed method can use a single sensor installed at different location of the beam to locate the damage/much damage reliably, even though the responses are contaminated with noise.

#### 1. Introduction

Structural Health Monitoring (SHM) is a practical tool in the prediction of safety level and system performance of in-service structures, while damage detection is an important task in SHM and condition assessment of structures. Vibration-based damage detection methods are based on the fact that any change introduced in a structure results in changes in its dynamic behavour. Thus, theoretically, occurrence of even a small damage will change the physical characteristics of a structure (its mass, stiffness, and damping characteristics), which in turn will affect its vibration response and change its dynamic characteristics. Vibration-based damage detection methods are especially attractive because they are global monitoring methods in the sense that none a priori information for the location of the damage is needed and/or immediate access to the damaged part is not required for structural damage detection [1].

The most commonly used vibration parameters for structural damage detection are frequencies [2, 3], mode shapes, and mode shape related parameters such as MAC, COMAC [4], flexibility [5], mode shape derivatives [6–8], and frequency response functions (FRF) based parameters [9]. An overview of the vibration-based methods for structural damage detection can be found in [10, 11]. Although the theory of vibration-based methods is straightforward and are generally accepted for the purposes of structural damage detection, they still pose a number of problems and challenges in practical application, such as the low sensitivity, relying on an accurate structure model, and so on. Particularly, modal based methods need enough sensors to insure the accuracy of the mode. The future development direction of structural health monitoring is how to deal with the real engineering and solve the specific problems. At present, an important reason that the health monitoring technology is difficult to be applied in real engineering is that it requires a large number of sensors, as well as the transmission, recording, and processing equipment. It is impractical and uneconomical to install sensors on the whole structure in all degrees of freedom. Therefore, we must develop the methods using less sensors to solve this problem.

In recent years, many researchers have sought to use a small amount of sensors, even a single, to detect structural damage, which becomes a hot issue in the research. The typical method is applying a moving load (vehicle load in bridge structure) as excitation source, and using a single sensor to measure the structure response at a certain position to identify the damage location. This approach is closely related to the practical engineering, does not find the problem of optimizing the allocation of sensors, and is a qualitative leap in the sensor consumption. This idea was put forward early in 2006 by Zhu and Law [12]. The authors proposed a method using the response obtained at a single measuring point of a beam structure and analyzed by Continuous Wavelet Transform (CWT) and the location of the cracks was estimated. The locations of the cracks were determined from the sudden changes in the spatial variation of the transform responses. Hester and González [13] using the acceleration signal and employing a vehicle-bridge finite element interaction model developed a wavelet based approach using wavelet energy content at each bridge section, which was proved to be more sensitive to damage than a wavelet coefficient. Nguyen and Tran [14] proposed a method that the dynamic response of the bridge-vehicle system measured directly from the moving vehicle was analyzed by wavelet transform. The locations of the cracks were pinpointed by positions of peaks of the transform responses. Khorram et al. [15] compared two wavelet based damage detection approaches to find the location and the size of a crack in a beam subjected to a moving load, one of which uses the time varying deflection attributed to the beam at midspan, while in another the sensor is attached to the moving load. It is found that the moving sensor approach is more effective than the fixed sensor. Later, Khorram et al. [16] proposed another method of multiple cracks detection in a simply supported beam subjected to a moving load based on CWT combined with factorial design. The similar wavelet based methods can be found in [17, 18].

In addition to the methods of wavelet transform, in 2012, Roveri and Carcaterra [19] proposed a novel HHT-based method for damage detection of bridge structures under a travelling load. The technique uses a single point measurement and is able to identify the presence and the location of the damage along the beam. The measured data is processed by the HHT technique. And none a priori information is needed about the response of the undamaged structure. Damage location is revealed by direct inspection of the first instantaneous frequency, which presents a sharp crest in correspondence with the damaged section. The advantage of this method is that it needs no baseline data, but it shows no sensitivity enough as the crack cannot be identified when the crack depth ratio is 20%. Li and Law [20] presented a substructural damage identification approach under moving vehicular loads based on a dynamic response reconstruction technique. The effectiveness of this method was proved by an experiment [21]. Cavadas et al. [22] presented a damage detection method using data-driven methods applied to moving load responses, which focuses on two data-driven methods: moving principal component analysis (MPCA) and robust regression analysis (RRA). But to locate the damage, the proposed method also needs a large amount of data measured by enough sensors. Zhang et al. [23] presented a local damage detection method for beam and plate like structures based on operating deflection shape curvature extracted from dynamic response of a passing vehicle.

Actually, to the best of authors’ knowledge, the articles with the idea that using the response measured with single sensor to locate structural damage are relatively rare in SHM (generally the above-mentioned methods). All these approaches have something in common; the damage is visualised through the appearance of a singularity in a processed signal supposed to be smooth in a healthy case. This paper focuses on using the phase space reconstructed method to detect the damage with single sensor. Phase space approach is a novel signal preceding method which is proved sensitive to damage [24–28]. This approach transfers the response from time domain into the spatial domain to analyze the vibration properties. Dynamics are most easily understood when viewed from a phase space perspective. Therefore, phase space-based classification for the response of a structural system is employed for structural damage detection because the effect of damage alters the behavior of phase trajectory [29–31]. In the previous study, Nie et al. [32] proved that the damage index extracted from reconstructed phase space using ambient excitation is significantly more sensitive to damage than modal based methods but relatively insensitive to noise and used this index in the experimental study of a two-span RC slab (6 m length) for damage detection, which proved that it is a good candidate for continuous structural health monitoring [1]. This approach also requires enough number of sensors to locate the damage. However, Nie et al. [33] also proved that the reconstructed phase space of the response measured at a single point of the beam, compared to the undamaged condition, would warp during the time of moving load passing by the damage zone. This phenomenon indicates that reconstructed phase space can be used as a candidate for damage localization using a single sensor approach.

To the best of authors’ knowledge, all the methods use a single point mass/load to generate the vibration data and, there are no literature studies on the application of phase space-based techniques for the health monitoring of civil structures under moving mass/load. Therefore, an innovative technique, phase space-based using a single sensor, for damage detection of a bridge structure excited by a moving mass is here proposed, and the damage index extracted from the response phase space is presented.

#### 2. Dynamic Theory of Simple Beam Subject to a Moving Mass

The uniform Euler-Bernoulli beam with moving mass is considered as shown in Figure 1. A moving mass travels along the beam with a constant velocity . The bending stiffness, and the mass per unit length of the beam , is assumed to be constant. Let denote the deflection of the beam with and representing the position of a point in the beam and the time ( as the moving mass enters the beam from left to right), respectively. For a one-dimensional beam problem, the position vector will represent the *-*axis with its origin coincided on the left support. The governing differential equation for vibration of the beam within the domain , neglecting the damping, the rotary inertia, and shearing force effects can be written aswhereThe term denotes the interaction force between moving mass and beam (assuming continuous contact between the mass and beam), and is the gravitational acceleration and is Dirac delta function that represents the location of the moving mass with the speed of at time . In (3), the differential term can be expanded asHence (3) can be rewritten asUsing the eigenfunction expansion, deflection can be considered aswhere is the th orthogonal modal shape of the beam, is time-dependent modal amplitudes function, and denotes the total number of considered modes. Substituting above expression in (5), the result isSubstituting (7) and (6) into the beam equation of motion (1), multiplying by , and integrating for the entire beam yieldwhere Using the orthogonality condition of eigenfunctions and rearranging (8), the results arewhere