International Journal of Rotating Machinery

Volume 2017 (2017), Article ID 7218646, 11 pages

https://doi.org/10.1155/2017/7218646

## Study on Frequency Characteristics of Rotor Systems for Fault Detection Using Variational Mode Decomposition

^{1}Key Laboratory of High Performance Ship Technology (Wuhan University of Technology), Ministry of Education, Wuhan 430063, China^{2}Reliability Engineering Institute, School of Energy and Power Engineering, Wuhan University of Technology, Wuhan 430063, China^{3}School of Machinery and Automation, Wuhan University of Science and Technology, Wuhan 430081, China

Correspondence should be addressed to Kai Chen

Received 28 April 2017; Revised 10 July 2017; Accepted 16 July 2017; Published 14 August 2017

Academic Editor: Adam Glowacz

Copyright © 2017 Kai 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

Due to the complicated structure, vibration signal of rotating machinery is multicomponent with nonstationary and nonlinear features, so it is difficult to diagnose faults effectively. Therefore, effective extraction of vibration signal characteristics is the key to diagnose the faults of rotating machinery. Mode mixing and illusive components existed in some conventional methods, such as EMD and EEMD, which leads to misdiagnosis in extracting signals. Given these reasons, a new fault diagnosis method, namely, variation mode decomposition (VMD), was proposed in this paper. VMD is a newly developed technique for adaptive signal decomposition, which can decompose a multicomponent signal into a series of quasi-orthogonal intrinsic mode functions (IMFs) simultaneously, corresponding to the components of signal clearly. To further research on VMD method, the advantages and characteristics of VMD are investigated via numerical simulations. VMD is then applied to detect oil whirl and oil whip for rotor systems fault diagnosis via practical vibration signal. The experimental results demonstrate the effectiveness of VMD method.

#### 1. Introduction

For a long time, faults of rotating machinery were mainly diagnosed by spectrum analysis method of vibration signal, which determine the failure by analyzing the frequency spectrum or frequency characteristics of vibration signals [1]. Practically, the vibration signal is generally nonstationary and nonperiodic when the fault occurs, so the traditional FFT cannot meet the needs of the time-varying and nonstationary signals detection and time-frequency analysis [2].

Like many fault signals, the vibration signal of rotor system is a typical nonlinear and nonstationary time-varying signal whose frequency components change over time [3]. The feature extracting of fault signal not only affects the accuracy of fault diagnosis, but also may lead to misdiagnosis. Therefore, effective extraction of vibration signal characteristics is the key to diagnose the faults of rotating machinery [4].

Analysis, processing, and feature extraction of nonlinear and nonstationary signal always are one of the hot topics concerning engineers and researchers [5–7]. EMD is an effective signal analysis method which is suitable for dealing with nonlinear and nonstationary signals [8]. It consists in a local and fully data-driven separation of a signal with fast and slow oscillations. However, EMD experiences some problems, such as end effect and mode mixing [9]. To overcome shortcomings, the ensemble empirical mode decomposition (EEMD) was proposed [10], which can suppress the appearance of modal mixing. But both of them lack theoretical support. Besides being recursive decomposition, error of envelope line will be spread; thus they cannot eliminate the problem of modal mixing completely.

Dragomiretskiy and Zosso [11] have proposed a new adaptive decomposition method called variational mode decomposition (VMD) in 2014, which can nonrecursively decompose a signal into a number of intrinsic mode functions (IMFs). In the process of decomposition, by using alternate direction multipliers method (ADMM), the center frequency and bandwidth of each modal were updated, to search the optimal solution of the signal decomposition. The updating process of each mode is carried out synchronously and makes the component of each modal in the corresponding baseband, finally, to achieve the effective decomposition of the signal. Compared with EMD and EEMD, VMD does not strip the signal step by step, but decomposes the signal synchronously. It is nonrecursive and reduces the spread of error. This method also has a good theoretical basis which is easy to understand. Besides, it has stronger robustness to noise [12, 13]. With its advantages, VMD is a wide application in the research field of signal denoising [14, 15], feature extraction [16, 17], image processing [18], energy and economics price forecasting [19, 20], speech signals detection [21], and especially in fault diagnosis [22–25]. However, these literatures mainly focus on the advantages of VMD comparing with the EMD and EEMD, rather than investigating the own capabilities of the VMD in signal analysis. The research of VMD needs to be conducted in depth. For this reason, a further investigation of the VMD is conducted in this paper via both numerical simulation and experimental approaches.

In this paper, the characteristics of VMD are investigated based on the theoretical of VMD, as well as the advantages of VMD. And then, VMD is applied to detect the fault of oil whirl and oil whip in rotor systems. This provides an effective solution for fault diagnosis of rotor system.

#### 2. Variational Mode Decomposition

##### 2.1. EMD and EEMD Method

EMD is a time-frequency signal analysis method for nonlinear signals, which can decompose the data adaptively and obtain a series of IMFs [9]. These IMFs reflect the characteristics of the signal itself. EMD algorithm is very suitable for analyzing the nonstationary signal.

Each signal could be decomposed into a number of intrinsic mode functions (IMFs), each of which must satisfy the following definition [9]:

(1) In the whole data set, the number of extrema and the number of zero-crossings must either be equal or differ at most by one.

(2) At any point, the mean value of the envelope defined by local maxima and the envelope defined by the local minima is zero.

With the definition, any signal can be decomposed as follows [9]:

(1) Identify all local maxima and all local minima in the signal .

(2) Connect all local maxima and all local minima by using a cubic spline line as the upper envelope and the lower envelope , respectively. The mean of upper and low envelope is calculated by using the equation

(3) Calculate the equationand examine whether the resultant is an IMF satisfying the two aforementioned conditions. If is not an IMF, regard as original signal and repeat steps (1)–(3) until is an IMF.

(4) Separate the first IMF from and then get the residual component :

Treat as original data and repeat the above processes; therefore the second IMF is obtained. Iterate the previous calculations times and IMFs of the signal could be got. Then

The decomposition process can be stopped when the final residual component is a monotonic function or a constant from which no more IMF can be extracted.

(5) The original signal finally can be expressed as

Thus, one can achieve a decomposition of the signal into -empirical modes, and a residue , which is the mean trend of . The IMFs include different frequency bands ranging from high to low.

EMD is a self-adaptive signal decomposition method, and the obtained IMFs have the advantages of mutual orthogonality. However, practice has disclosed that the EMD also shows the following defects in signal processing and feature extraction.

*(1) No Mathematical Foundation*. Without strict mathematical proof and mathematical model, EMD lacks theoretical support. And there is no valid way to stop the decomposition process in the sifting process of EMD.

*(2) End Effect*. In the decomposition process, endpoint will be used as an extreme point for fitting the upper and lower envelopes. But if the endpoint is not an extreme point, the fitted curve using cubic spline line will have some error. This error will impact on each subsequent decomposition step; the error will accumulate gradually and then is likely to pollute the internal decomposed results from both ends.

*(3) Mode Mixing*. Mode mixing occurs in EMD; when steps change, pulse interference and outside noise will show in the signal. There are two forms of mode mixing; one form is a single IMF including oscillations of dramatically disparate scales and the other is a component of a similar scale residing in different IMFs. Once the mode mixing is generated, subsequent decomposition step will be influenced, even making IMFs meaningless in physical interpretation.

To alleviate the mode mixing problem occurring in EMD, an ensemble empirical mode decomposition (EEMD) is presented [26]. The essential essence of EEMD algorithm is decomposing the original signal added Gaussian noise repeatedly by EMD method, and the original vibration data is decomposed into a series of IMFs with different scales and continuous characteristics because of the characteristic of frequency uniform distribution of Gaussian white noise, which can suppress the appearance of modal mixing.

##### 2.2. VMD Method

VMD is a new signal analysis method for nonlinear and nonstationary signal, which aims to decompose the signal into different discrete modes [11]. The VMD has a solid mathematical foundation, which is able to decompose a signal into an ensemble of band-limited intrinsic mode functions (IMFs) simultaneously. IMFs are redefined as amplitude modulated frequency modulated (AM-FM) signals, written aswhere is IMF, is nonnegative envelope, and is the phase and a nondecreasing function.

The core of VMD is to construct and solve the variational problem, the decomposition process of VMD algorithm is the solution of the variational problem. There are three important concepts in the signal processing, Wiener Filtering, Hilbert transform, and frequency mixing, which constitute the building blocks of VMD model.

Assume each mode is band-limited and compactly distributed around with a center frequency , so the solution of variational problem turns to seeking for modes to make the sum of the bandwidth minimized, while the constraint condition is the sum of modes equal to the input signal . The principle of the VMD algorithm is the following:

(1) The analytic signal of mode was computed by means of Hilbert transform, so obtain the unilateral spectrum of the analytic signal.

(2) Shift the frequency spectrum of each mode to the respective estimated central frequency.

(3) The bandwidth is estimated through the Gaussian smoothness of the demodulated signal, that is, the squared -norm of the gradient. The resulting constrained variational problem is given by the following: where and are the identified set of modes and their central frequencies.

(4) A quadratic penalty term and a Lagrangian multiplier factor are introduced in order to render the problem unconstrained. The quadratic penalty is a classic way to encourage reconstruction fidelity, and the Lagrangian multipliers are a common way of enforcing constraints strictly. The augmented Lagrangian is shown as follows:

(5) An alternate direction multipliers method (ADMM) is applied to solve the original minimization problem, to find the saddle point of the augmented Lagrange expression via updating , , and alternately. So the signal was decomposed into different discrete modes adaptively.

The flowchart of VMD methods is shown in Figure 1, and the detailed steps are as follows.