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.
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 [
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 [
Analysis, processing, and feature extraction of nonlinear and nonstationary signal always are one of the hot topics concerning engineers and researchers [
Dragomiretskiy and Zosso [
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.
EMD is a time-frequency signal analysis method for nonlinear signals, which can decompose the data adaptively and obtain a series of IMFs [
Each signal could be decomposed into a number of intrinsic mode functions (IMFs), each of which must satisfy the following definition [
(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
(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
(3) Calculate the equation
(4) Separate the first IMF from
Treat
The decomposition process can be stopped when the final residual component
(5) The original signal
Thus, one can achieve a decomposition of the signal into
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.
To alleviate the mode mixing problem occurring in EMD, an ensemble empirical mode decomposition (EEMD) is presented [
VMD is a new signal analysis method for nonlinear and nonstationary signal, which aims to decompose the signal into different discrete modes [
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
(1) The analytic signal of mode
(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
(4) A quadratic penalty term
(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
The flowchart of VMD methods is shown in Figure
Flowchart of VMD.
(1) Initialize
(2)
(3) Update
(4) Repeat steps (2) and (3), until the following convergence condition is satisfied:
To show the advantage of VMD, the simulated signal in (
Comparison results of EMD, EEMD, and VMD.
Simulated signal
EMD results
EEMD results
VMD results
It can be seen from Figure
Compared to EMD and EEMD, VMD has advantages in solving the problem of mode mixing and end effect. What is more, according to the principle and decomposition steps of VMD, the following characteristic of VMD can be obtained.
Reconfigurability of VMD.
Original signal
Reconstructed signal
Error
The filtering characteristics of VMD.
Original signal
Added noise signal
Reconstructed signal
By comparing the original signal and the reconstructed signal, the added noise was eliminated, and the characteristic peaks of spectrum were more prominent and obvious after reconstitution. And, in the time domain, fluctuating character of the reconstructed signal is more clear than the signal added noise, which is hard to identify. VMD does have the smoothing filter characteristic.
This can also be deduced from the VMD algorithm in theory. It is observed from formula (
To evaluate the performance of VMD method, experimental analysis of a rotor system has been carried out in a rotor test rig. As shown in Figure
Experimental setup and sensor arranged.
A phase sensor closed to the motor is fixed to measure the rotating speed. A velocity sensor (ZA-HV-2-5) is fixed on bearing seat in the middle of test rig to measure the vibration of bearing seat. Two eddy current sensors, fixed on sensor support closed to oil film bearing, are used to pick up horizontal and vertical directions displacement of rotor.
The oil film instability of the rotor system usually consists of two stages, oil whirl and oil whip. The whirl is a form of motion in which the rotor rotates around its own axis, while the axis rotates around the center of the bearing. When the rotational speed of the rotor reaches a certain speed, the oil whirl is generated, and the whirling frequency is about half of that speed. When the rotational speed is up to 2 times of the first-order critical speed of rotor system, the oil whirl turns into the oil whip, the amplitude of the rotor increases, and the whip frequency stabilizes in a certain value (first-order critical speed of rotor system). The fault feature extraction of oil whirl and oil whip is the main basis to diagnose these type faults.
In the process of oil film instability, the oil whirl occurs near the speed of 3100 r/min. When the rotational speed is up to 4500 r/min, the rotor system test rig shows bigger vibration because of the oil whip.
Time Domain waveform and spectrum of oil whirl signal.
Five IMFs decomposed by VMD for oil whirl signal are shown in Figure
VMD, EEMD, and EMD decomposed results of oil whirl signal.
VMD decomposed results
EEMD decomposed results
EMD decomposed results
For comparison, IMFs analyzed by EEMD and EMD are shown in Figures
Furthermore, the time-frequency spectrum of the VMD, EEMD, and EMD results was analyzed. The Hilbert spectrum of each IMF decomposed by the VMD, EEMD, and EMD is shown in Figure
Hilbert spectrum of the VMD, EEMD, and EMD results of oil whirl signal.
Hilbert spectrum of the VMD results
Hilbert spectrum of the EEMD results
Hilbert spectrum of the EMD results
Time domain waveform and spectrum of oil whip signal.
Four IMFs obtained by VMD for oil whip signal are illustrated in Figure
VMD, EEMD, and EMD decomposed results of oil whip signal.
VMD decomposed results
EEMD decomposed results
EMD decomposed results
The Hilbert spectrum of each IMF decomposed by VMD, EEMD, and EMD is shown in Figure
Hilbert spectrum of the VMD, EEMD, and EMD results of oil whip signal.
Hilbert spectrum of the VMD results
Hilbert spectrum of the EEMD results
Hilbert spectrum of the EMD results
The analysis results in Figures
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 simultaneously. Unlike EMD and EEMD, VMD has theoretical support and can solve the problem of mode mixing and end effect more effectively. It also has the characteristic of adaptivity, reconfigurability, smoothing filtering, and orthogonality.
VMD is proposed to determine the fault of rotor systems, and an experimental analysis has been carried out. Comparative analysis results show that VMD method is more effective than EEMD and EMD in fault feature extraction of oil whirl and oil whip, so VMD method has important practical application value for rotor fault diagnosis and should be further considered in the future.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This project is sponsored by the grants from the Ministry of Transport Applied Basic Research Project (no. 2013329811360), the National Natural Sciences Foundation of China (no. 51139005, no. 51509194), and the Fundamental Research Funds for the Central Universities (no. 2014-yb-018).