Shock and Vibration

Volume 2016, Article ID 2375859, 17 pages

http://dx.doi.org/10.1155/2016/2375859

## Multicrack Localization in Rotors Based on Proper Orthogonal Decomposition Using Fractal Dimension and Gapped Smoothing Method

^{1}School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China^{2}School of Engineering, University of Liverpool, Liverpool L69 3GH, UK^{3}The State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, Liaoning 116023, China

Received 30 May 2016; Revised 2 August 2016; Accepted 4 September 2016

Academic Editor: Marcello Vanali

Copyright © 2016 Zhiwen Lu 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

Multicrack localization in operating rotor systems is still a challenge today. Focusing on this challenge, a new approach based on proper orthogonal decomposition (POD) is proposed for multicrack localization in rotors. A two-disc rotor-bearing system with breathing cracks is established by the finite element method and simulated sensors are distributed along the rotor to obtain the steady-state transverse responses required by POD. Based on the discontinuities introduced in the proper orthogonal modes (POMs) at the locations of cracks, the characteristic POM (CPOM), which is sensitive to crack locations and robust to noise, is selected for cracks localization. Instead of using the CPOM directly, due to its difficulty to localize incipient cracks, damage indexes using fractal dimension (FD) and gapped smoothing method (GSM) are adopted, in order to extract the locations more efficiently. The method proposed in this work is validated to be effective for multicrack localization in rotors by numerical experiments on rotors in different crack configuration cases considering the effects of noise. In addition, the feasibility of using fewer sensors is also investigated.

#### 1. Introduction

Rotors are one of the most important components of rotating machines, which are widely used in many engineering fields, such as turbines, generators, and aeroengines. Cracks in rotors are the most critical and fundamental damage which may lead to a sudden and catastrophic failure of equipment. So it is of vital significance to identify these cracks, in order to reduce maintenance cost and avoid failure of a rotating machine. In view of the importance, crack identification in rotors has been the focus of many investigations in recent decades and numerous papers have been published [1–9], but crack localization is still a challenge for operating rotor systems.

A brief review of the relevant studies of crack localization in rotors is given firstly. The localization methods can be classified as model-based and non-model-based methods. It should be noted that model-based methods defined here are the methods which require a mathematical representation of the system under study, for example, a partial differential equation of motion of a rotor as a beam or mass and stiffness matrices of a finite element model of a rotor.

When it comes to model-based methods in crack identification, approaches based on equivalent crack forces will be mentioned at the first beginning. Equivalent crack force methods consider the effects of cracks as equivalent forces applied in intact systems. They have been adopted to identify the location and depth of cracks in rotors by many researchers, such as Pennacchi et al. [10], Lees et al. [11], and Sekhar [12]. There are also some other model-based methods. Dong et al. [13] presented a method based on intersection of the contour curves of the first three natural frequencies obtained from a rotor modelled by wavelet finite element method to determine the crack location and depth. Seibold and Weinert [14] proposed a method in time domain based on a bank of Extended Kalman Filters to realize the crack location and depth identification for an operating rotor. Methods based on pattern recognition can be model-based when the samples for training are obtained from mathematical models rather than experiments. Their main ideas are extracting features sensitive to crack parameters, which then are trained by artificial intellectual methods or machine learning methods to obtain the relationship between the crack parameters and features, and then matching the measured features with the established relationship to determine the crack parameters. Zapico-Valle et al. [15] adopted the artificial neural network which trained by samples gathered from a finite element model to identify crack location and depth of rotors. Söffker et al. [16] compared a modern model-based technique based on a proportional-integral observer with a signal-based technique based on support vector machine using features extracted from wavelets to identify a crack in an operating rotor. Methods based on optimization are also often model-based, because a large number of iterations are required, and it is almost impossible if there is no model. Genetic algorithm was adopted by Saridakis et al. [17], Xiang et al. [18], and He et al. [19] to minimize the difference between real outputs and model outputs to determine the location and depth of a crack in a rotor-bearing system. Cavalini Jr. et al. [20] put forward a crack identification methodology using external diagnostic forces at certain frequencies to obtain the nonlinear combinational resonances which were used as the objective function of a differential evolution optimization code to determine the crack location and depth minimizing the difference between the measured and modelled rotor system.

In contrast with model-based methods, some crack localization methods do not need a mathematical model of the system under study and are thus called non-model-based methods which often just need inputs and outputs of the system, or even outputs only. Rubio et al. [21] used changes in resonant and antiresonant frequencies to detect crack locations in a two-cracked torsional shaft. Rahman et al. [22] utilized the changes in phase angle of frequency response function to identify the location and depth of a rotor with an open crack. Seo et al. [23] proposed a method for open crack localization by comparing the map of the modal constants of the reverse directional frequency response functions with the reference map of the uncracked model. These methods are based on changes of natural dynamic characteristics; though no mathematical model of the system is needed, they often require information from intact systems which sometimes is not convenient to obtain. There are also some non-model-based methods which do not need reference information from intact systems. ODS measured by sensors distributed along rotors was used for crack localization in rotors by Saravanan and Sekhar [24] and Zhang et al. [25]. Babu and Sekhar [26] proposed a modified ODS method called amplitude deviation curve to identify cracks in rotor-bearing systems. A residual ODS based method considering higher harmonic components of exciting frequency was developed to localize cracks in a rotor by Asnaashari and Sinha [27]. Singh and Tiwari [28] proposed an algorithm for crack localization based on the fact that cracks cause slope discontinuities in the shaft deflection. Due to the lower sensitivity of ODS to incipient cracks, some after-treatment techniques were developed, such as wavelets [29], FD [30, 31], and GSM [32].

Both model-based and non-model-based methods have their advantages and disadvantages. In this paper, a non-model-based method based on POD is proposed to realize multicrack localization for operating rotors. The POD is a multivariate statistical method which aims at obtaining a compact representation of vibration data [33, 34]. Galvanetto and Violaris [35] investigated the feasibility of POD to localize a crack in beams. Shane and Jha [36] applied POD to detect damage in composite beams. POD combined with radial basis functions was proposed by Benaissa et al. [37] to identify a crack in plate structures. However, to the authors’ best knowledge, the POD has not been used for multicrack localization in operating rotor systems.

Due to the breathing phenomenon of cracks during rotation and the difficulty to generate breathing cracks in rotors, a model that reflects the essential behaviour of a crack is vitally important to get the response of cracked rotors more accurately. There are many methods to model a crack. A nonlinear 3D finite element method was adopted to model a breathing crack in a rotor in [38], which may be the most accurate model, but the computation workload is very heavy. In [39], a rigid finite element method was put forward to model a cracked rotor, which also had good accuracy to model a breathing crack. Papadopoulos’ review paper [4] elaborated the approach for modelling cracks in rotors based on SERR and showed that the model put forward by Darpe [40] which was also based on SERR could characterise a breathing crack in rotors more accurately, and it had the advantages of allowing general excitations without assuming that the gravitational force was dominant, and the behaviour of the breathing crack was response-dependent instead of being rotation-dependent. So, considering the complexity in computation and accuracy in modelling in relation to other methods, Darpe’s method is adopted to model a cracked rotor in this investigation.

In this work, the feasibility of multicrack localization based on POD using FD and GSM in operating rotor systems is validated by numerical investigation. The proposed method is a kind of non-model-based approach and it does not need the knowledge of the undamaged rotors. The rest of the paper is organized as follows. In Section 2, the model of a two-disc rotor-bearing system considering the static unbalance of discs with response-dependent breathing cracks is established by the finite element method. Section 3 presents the multicrack localization method based on POD using FD and GSM. In Section 4, numerical experiments are carried out for the multicracked rotor with different crack configuration cases. Finally, conclusions are drawn.

#### 2. Cracked Rotor Modelling

A finite element model of the cracked rotor considering bending-torsion coupling introduced by static unbalance is established in this work. A generalized breathing crack model which can represent any crack angles and any types of excitations applied to the rotor is adopted. To model the cracked rotor, the key point is to simulate the crack appropriately and calculate the stiffness matrix of the crack element. After that, through assembling the cracked and uncracked elements, the finite element model of the rotor can be obtained.

##### 2.1. Model of a Cracked Shaft Element

Figure 1 shows a cracked shaft element of length and radius . are the loads acting on the 12 degrees of freedom of the two nodes in the element coordinate system . The local coordinate system is defined on the flat crack face to describe the crack cross-section. is the crack angle between the crack face and the shaft centre line (formed by the negative -axis turning to the negative -axis in the counter-clockwise direction); is the location of the crack centre in the element coordinate system. CCL is an imaginary line that separates the open and closed parts of the crack which will be used to simulate the breathing of crack. The hatched area corresponds to the open area of the crack.