Shock and Vibration

Volume 2018 (2018), Article ID 5382398, 14 pages

https://doi.org/10.1155/2018/5382398

## A SVDD and -Means Based Early Warning Method for Dual-Rotor Equipment under Time-Varying Operating Conditions

College of Mechanical and Electrical Engineering, Beijing University of Chemical Technology, Beijing, China

Correspondence should be addressed to Kun Feng

Received 15 August 2017; Revised 4 December 2017; Accepted 12 December 2017; Published 4 January 2018

Academic Editor: Sandris Ručevskis

Copyright © 2018 Zhinong Jiang 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

Under frequently time-varying operating conditions, equipment with dual rotors like gas turbines is influenced by two rotors with different rotating speeds. Alarm methods of fixed threshold are unable to consider the influences of time-varying operating conditions. Hence, those methods are not suitable for monitoring dual-rotor equipment. An early warning method for dual-rotor equipment under time-varying operating conditions is proposed in this paper. The influences of time-varying rotating speeds of dual rotors on alarm thresholds have been considered. Firstly, the operating conditions are divided into several limited intervals according to rotating speeds of dual rotors. Secondly, the train data within each interval is processed by SVDD and the allowable ranges (i.e., the alarm threshold) of the vibration are determined. The alarm threshold of each interval of operating conditions is obtained. The alarm threshold can be expressed as a sphere, whose controlling parameters are the coordinate of the center and the radius. Then, the cluster center of the test data, whose alarm state is to be judged, can be extracted through -means. Finally, the alarm state can be obtained by comparing the cluster center with the corresponding sphere. Experiments are conducted to validate the proposed method.

#### 1. Introduction

Gas turbines, representative equipment of dual rotors, are the key power equipment in aviation, shipping, electric power, petroleum, and so on. Once a gas turbine undergoes a fault or accident, the relevant production and management will suffer a lot. And even worse, those problems will probably lead to fatal disasters [1, 2]. Therefore, it is very important to ensure the efficient and normal operation of a gas turbine [3].

Vibration monitoring is one of the main methods for mechanical faults monitoring. An early warning of gas turbines’ state can be realized through a vibration alarm before a serious fault occurs. To create an alarm for the vibration conditions of certain equipment, the main steps include collecting vibration signals in the main parts of the equipment, computing the vibration overall amplitude, and setting a fixed alarm threshold. The fixed threshold alarm is usually able to ensure the safety and reliable operation for the equipment to some extent. However, this alarm method is mainly suitable for single-rotor equipment under steady operating conditions. This method applies the same alarm threshold under every operating condition and cannot deal with problems that are caused by variable operating conditions. Because the mentioned characteristic of a fixed threshold can easily lead to missing alarms under low operating conditions and false alarms under high operating conditions, this method is not applicable in the warning for the vibration state for a gas turbine under time-varying operating conditions.

Unlike general equipment with one rotor, dual-rotor equipment has two rotors, whose rotational speeds are usually different. A complete description of the operating conditions of rotating machinery should include the speed and the load. The object of this research, however, is dual-rotor equipment including gas turbines and aircraft engines. As for the equipment, the load is the output power or the thrust. Two speeds and load are generated when the rotor is shocked by high temperature and high pressure gas. There is a positive correlation between the speed set and the load. The load does not change when the speed set is constant. The load factor is included in the speed factor in this research, so only speed set is analyzed to study the operating conditions for dual-rotor equipment. In engineering applications, the operating conditions of the dual-rotor equipment are usually characterized by rotating speeds. In conclusion, different speeds can be used for characterizing the time-varying operating conditions due to the special research object. Namely, vibration is affected by two different and variable rotational speeds simultaneously, and thus an effective warning for early faults in this type of equipment cannot be realized through the fixed alarm threshold. As previously mentioned, missing and false alarms occur when variable operating conditions are not considered, making traditional alarm methods unable to warn against early faults for this type of equipment effectively. It is necessary to consider the following two aspects of alarm thresholds for dual-rotor equipment: One is to learn alarm threshold values and identify alarm status. And the other is to consider variable operating conditions influenced by two varying rotating speeds.

To solve problems concerning the computation of alarm thresholds and the identification of alarm state, a great number of researches have been carried out for improving the algorithm based on Support Vector Machine (SVM) or Artificial Neural Network (ANN). Empirical Mode Decomposition (EMD) has been applied to obtain the feature set [4]. Then, the ANN has been trained and tested for warning against bearing faults. In recent years, a huge number of scholars have utilized SVM to achieve fault alarms and identify faults. The principle of SVM is that two types of data are separated by finding the optimal hyperplane that has the same distance from itself to both types of data. This means that SVM can warn against faults better than ANN does [5]. SVM, a classifier based on statistical learning theory, was initially proposed to deal with problems when the number of fault samples is not enough [6]. At present, SVM has been used to monitor equipment of various kinds and has been increasingly improved. SVM, whose kernel function is Gaussian, has been utilized to solve classification issues for nonlinear datasets [7], but how to set parameters was still unclear. Immune algorithms that imitate the artificial immune system have been used to optimize parameters of SVM, which plays a significant role in obtaining a classifier with better performance [8]. Several SVMs have been utilized simultaneously to warn against and identify various faults [9]. Then, outputs of each SVM on the basis of the inference of the case database have been obtained. Next, all kinds of the outputs have been compared through the same standard, and the SVM model, suitable for identifying the corresponding fault, has been selected. Those efforts have made great contributions to warning against and identifying faults of bearings. From the above descriptions, it is clear that SVM can achieve higher accuracy and better generalization ability in mechanical fault alarm and recognition [10].

To solve issues caused by variable operating conditions for dual-rotor equipment, a large number of researches focus on how to extract fault features from nonstationary signals in time domain [11–13]. So far, those researches have brought about great benefits. Feature extraction is not concerned in this paper. Nevertheless, there are seldom published researches on early warning methods, which consider varying alarm thresholds caused by operating conditions in dual-rotor equipment. The gearbox used in wind turbines is currently mainly studied for the early warning method under time-varying operating conditions. Considering the gearbox under time-varying operating conditions, Ren et al. have obtained influences of speeds and loads on the vibration signal through a large number of experiments, but they have not solved fault warning problems under time-varying operating conditions [14]. Using order tracking and feature extraction in the angle domain, Gu et al. have obtained the vibration features and load index. Then, they have set up different relevant index models under different operating conditions to recognize gearbox faults, realizing the fault warning under time-varying operating conditions [15]. After extracting the features of the vibration signal at different speeds when the equipment was under normal operating conditions, Lin and Makis have determined the time series models under normal operating conditions. Then, these models have been compared with real-time data to define the status of the equipment. Additionally, load intervals have been divided based on the torque and speed. Then, the gearbox faults were recognized by the Bayesian model [16]. For solving problems of the constantly changeable vibration signal component of the gearbox under time-varying operating conditions, Shao et al. have used an autoregressive model and hypothetical test method to warn against faults in the equipment [17]. Kouadri et al. proposed a method based on the statistical test. This method can define the status of the gearbox by comparing the confidence intervals of the vibration signal between normal and fault experiments. The validity of fault warning for the gearbox under time-varying operating conditions was proved through experimental data [18]. The fault mechanism of dual-rotor equipment is different from that of gearboxes, so the above methods cannot be directly applied to dual-rotor equipment.

This paper proposes an early warning method for dual-rotor equipment under time-varying operating conditions using support vector domain description (SVDD) and -means algorithm. To solve problems caused by variable conditions, the range of operating conditions is divided into finite intervals, with each interval considered as a steady operating condition. Alarm thresholds of all intervals are computed one by one. Because the two rotating speeds affect the vibration value simultaneously when the dual-rotor equipment works, SVDD is used to decide the allowable ranges of vibration for the equipment under normal conditions, determining the alarm threshold under each operating condition. Meanwhile, -means clustering algorithm is used to obtain the cluster center of vibration data whose alarm state is uncertain. The early warning of vibration state can be realized by comparing the cluster center with the alarm threshold under the corresponding operating condition.

The rest of this paper is organized as follows. Section 2 introduces the theoretical backgrounds of this work, including the SVDD, the -means, and the parameter optimization algorithm. Section 3 describes this proposed early warning method. The experimental results are presented to verify this proposed method in Section 4. The conclusions are drawn in Section 5. Section 6 presents some discussions about this study.

#### 2. Basic Theory

##### 2.1. Support Vector Domain Description

SVM, a statistical learning theory based on machine learning method, can classify data according to structural risk minimization [5]. SVM is mainly utilized for data classification and regression prediction, and this paper only studies the former. SVM is suitable for small sample data, so it is used in this paper to classify data.

Classification problems in engineering practice can be divided into two categories: relatively simple linearly separable problems and linearly inseparable nonlinear problems. SVM initially solves problems when an optimal separating hyperplane is computed in linear separable problems [19]. SVM projects nonlinear separable data onto a high dimension through a nonlinear kernel function to make it a linearly separable problem. Thus, linear distinction of nonlinear data in high dimension will be realized [20].

As a derivative of SVM, one-class SVM is different from binary classification SVM, as it only has one class of data [21]. Currently, there are two types of one-class SVM: one-class-SVM and support vector domain description (SVDD). All of them can be used to distinguish abnormal data from normal one [22]. As for SVDD, it is used to discriminate the data by constructing a hypersphere in a high-dimensional space. The center and radius of the hypersphere can be obtained by using the penalty parameter [23]. In this method, spatial features of vibration data under multivariable factors can be shown better. Therefore, SVDD is applied to classify data in this paper.

A training vector is known and there is no class label; the optimization objective of SVDD is to obtain an optimal hypersphere [24], whose center is and radius is . It can be expressed asAs it is shown in (2), the fixed proportional training data points are included in this sphere.where represents the relaxation variable. To set is to prevent interference of individual outliers in the hypersphere. If there is no relaxation variable, the hypersphere will be worse because of few outliers. is used to adjust the influences of . If is larger, more outliers will be included. On the contrary, if is smaller, it is likely that no outliers will be here. Therefore, optimization of is of great significance. This problem will be explained in Section 2.3.

The hypersphere can solve nonlinear problems. For this purpose, data points must be projected onto high-dimensional space for finding the optimal hyperplane, that is, a kernel function , which can satisfy the following equation:where and denote coordinates of the data and represents the function which can project the coordinate onto high-dimensional space. Thus, the optimization problem is shown as follows:which is subjected to in (4)-(5) denotes the number of training samples. The dual form of (5) is shown as The constraint of (6) iswhere denotes the coefficient of Lagrange function. In solving the above optimization problem, most is zero. If is nonzero, it is a support vector, which decides the shape and size of the hypersphere. Based on all the support vectors, , the center of the sphere, is shown as follows:where represents the set of support vectors. As for support vector with , satisfies The radius of the sphere can be obtained from the above equation.

As mentioned earlier, the kernel function can be divided into two types: linear and nonlinear. Nonlinear kernel functions include polynomial, Gaussian, sigmoid, and self-defined types.** The **Gaussian kernel function**, **one of the most commonly used kernel functions, can be used easily and nearly without problems of numerical solutions [25]. Thus, the Gaussian kernel function is used as a mapping function in high dimension in this paper, as it is shown inwhere denotes the sample point, represents the center decided by the sample points, and denotes the parameter which decides the change rate of the kernel function. When becomes larger, the corresponding parting surface will be more complex. On the contrary, when is less, the relevant parting surface will become smoother. Therefore, choosing the value of is also very important. This problem will be explained in Section 2.3.

##### 2.2. -Means Clustering Algorithm

The clustering algorithm is to determine the distribution of data through a statistical method. The distribution can be regarded as a certain kind of geometry, and the cluster center is the barycenter of the geometry [26, 27]. Actually, it is difficult to find this center in many data. Meanwhile, the dataset must be divided into several different classes. -means clustering algorithm has been initially introduced and has been applied widely into various research fields [28]. In this paper, the cluster center of test data can be found in three-dimensional spaces via the -means clustering algorithm. When compared with the distribution of the data points, the cluster center of data is a more stable feature and can characterize the state of equipment better [29]. This is because misjudgment due to individual points is avoided by the cluster center.

is a -dimensional dataset including samples, where . in the -means algorithm represents the notion that is divided into subclasses. Each subclass is expressed as (), and every has its own cluster center . Firstly, select elements randomly to be the initial cluster centers of subclasses. Then, compute the distance from each data point to every subclass center. The initial classification can be realized according to the criterion of the shortest distance. Next, calculate the average Euclidean distance of each subclass, update the cluster center on account of this average, and renew the classification according to the principle of the shortest distance. Equations (11) and (12) are, respectively, the quadratic sum of distance from the data point in each subclass to the corresponding class center and the quadratic sum of the total distance of all classes [30].where

-means clustering algorithm aims at minimizing the quadratic sum of distance of all classes. Update the cluster centers and classifications constantly according to the above steps when the quadratic sum is convergent. The flow chart of the -means clustering algorithm is shown in Figure 1.