Shock and Vibration

Volume 2015 (2015), Article ID 808457, 11 pages

http://dx.doi.org/10.1155/2015/808457

## Study on Fault Diagnosis of Rolling Bearing Based on Time-Frequency Generalized Dimension

^{1}State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710054, China^{2}School of EMU Application & Maintenance Engineering, Dalian Jiaotong University, Dalian 116028, China

Received 27 August 2014; Accepted 13 November 2014

Academic Editor: Jiawei Xiang

Copyright © 2015 Yu Yuan 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

The condition monitoring technology and fault diagnosis technology of mechanical equipment played an important role in the modern engineering. Rolling bearing is the most common component of mechanical equipment which sustains and transfers the load. Therefore, fault diagnosis of rolling bearings has great significance. Fractal theory provides an effective method to describe the complexity and irregularity of the vibration signals of rolling bearings. In this paper a novel multifractal fault diagnosis approach based on time-frequency domain signals was proposed. The method and numerical algorithm of Multi-fractal analysis in time-frequency domain were provided. According to grid type *J* and order parameter *q* in algorithm, the value range of *J* and the cut-off condition of *q* were optimized based on the effect on the dimension calculation. Simulation experiments demonstrated that the effective signal identification could be complete by multifractal method in time-frequency domain, which is related to the factors such as signal energy and distribution. And the further fault diagnosis experiments of bearings showed that the multifractal method in time-frequency domain can complete the fault diagnosis, such as the fault judgment and fault types. And the fault detection can be done in the early stage of fault. Therefore, the multifractal method in time-frequency domain used in fault diagnosis of bearing is a practicable method.

#### 1. Introduction

Recently, modern industry is gradually developing in the direction of large-scale, continuous, high speed, and artificial intelligence, with the main advantage of improving productivity, reducing the rejection rate, and ensuring quality of products. But on the other hand, once there is some fault happening on modern sophisticated equipment or structure, the maintenance costs would be much increased and may even lead to major accident [1].

Rolling bearings have been widely used in various rotating machinery and play an important role in rotating machinery, which is easy to go wrong. With the improvement of automation equipment and device complexity, as well as the wide usage of large-scale rotating machinery in engineering, high security and advanced fault prediction capability for the devices and the new fault diagnosis methods are required. Therefore, the fault diagnosis analysis of rolling bearing, especially the correct detection of the early failure has practical value in extending service life and reducing cost. There is a wide range of needs in the exploration and application of bearing fault diagnostic. It has practical significance, broad market prospect, and economic value in the social development [2, 3].

As a newly arisen subject, fractal theory is especially suitable for analyzing complex system. Fractal theory is used in the area of fault analyzing of mechanical system as a current trend in academia. Based on the fractal theory, the vibration signals of mechanical system are analyzed, and fractal dimension is extracted as the feature information, and then the running state of the system can be analyzed not qualitatively but also quantitatively. Used with fractal theory, the faults of complex machinery system can be diagnosed, which can improve the fault identification and analysis ability. It is a practical and promising signal analyzing method for machinery devices. Based on the results of previous studies, this paper presents some research on the vibration signals in time-frequency domain by fractal theory. According to the simulation and experimental data, the result has shown that the analysis result varies with different vibration signals. Compared with the obvious differences in dimensions and parameters for different bearing failures, the method can be used for fault diagnosis.

#### 2. Empirical Mode Decomposition and Time-Frequency Transform

In essence, the decomposition process for a signal based on EMD is a smooth processing for a signal. This decomposition will gradually decompose a complex nonlinear and nonstationary signal into different components of intrinsic mode function (IMF) and the remaining final trend. Each basic component has different characteristic scales, and the component of low-frequency limit stands for the trend and DC component of the original signal. As each intrinsic mode function stands for different local features of the original signal, different local features of the original signal can be acquired by separately analyzing the intrinsic mode function [4].

By EMD decomposition, another expression of time series can be acquired [5, 6]: where is the intrinsic mode function and is the residual component.

The signal acquired is still a time-series signal, and the basic parameters to express signal feature are time and frequency [7]. The time characteristic scale describes the variations of signal with time, and the other signal parameter is frequency characteristic scale. The common Fourier transform can express the frequency characteristic in overall scale. The instantaneous frequency can express the frequency characteristic and time characteristic. So the instantaneous frequency can be more convincing in expressing signals [8, 9].

For signal , through Hilbert transform, can be acquired. Its analytic signal is

Instantaneous frequency is

It can be transformed into difference format, and the frequency can be expressed as:

If sample frequency is , instantaneous frequency can be defined as follows according to discrete-time signal :

Phase can be acquired by arctan function, which has the characteristic of periodicity. The value of phase is between and . Therefore, phase distribution has discontinuous characteristic on the change point. When the value of phase extends to , phase aberration can be produced.

Phase unwrapping algorithm is to compare the principle value of adjacent points. Consider

If exceeds some threshold (such as ), then there is jump in the phase. According to positive-negative sign of , phase jump value can be adjusted. If , ; if , . And is the practical phase.

#### 3. Numerical Algorithm of Generalized Dimension in Time-Frequency Domain

Currently, the fractal theory which is used in feature extraction of vibration signals is restricted in time domain. For the nonlinear and nonstationary signal, it is not enough to only analyze in the aspect of time domain. Combining the advantages of time domain and frequency domain, time-frequency analysis method is more suitable for the nonlinear and nonstationary vibration signal analysis of rotating machinery [10]. For time-varying characteristics of nonstationary signals, the one-dimensional signal is extended to two-dimensional time-frequency plane for calculation. The purpose is to describe the frequency components of signal or the variations of energy distribution. Using EMD decomposition for the original signal, various basic mode components and the corresponding instantaneous frequencies can be obtained with high time-frequency concentration and signal analyzing capability [11].

Multi-fractal in time-frequency domain is a method which describes the complexity of energy distribution. The traditional time domain fractal of vibration signal is based on geometric measures. By analyzing the energy distribution variations of the signals measured, Multi-fractal analysis in time-frequency domain can extract the quantitative characteristics of vibration signals. The numerical calculating method of generalized dimension of vibration signals in time-frequency domain is shown as follows.

As time series is and is carried on EMD decomposition and instantaneous frequency transform, the formula is obtained as follows:

Based on the formula above, amplitude matrix was obtained by normalization. Each point in matrix corresponds to one time and one instantaneous frequency. The squared value of the points stands for normalization energy in time-frequency domain.

Similar to sampling space division of time-domain signal [12], stands for grid width, and stands for the number of data points of grid generation, which is called grid type. The number of rows and columns is separately set to , . If the grid of row and column is , the energy of grid is , and the energy distribution probabilities of grid is

According to the above formula, can be calculated.

Define , , and ; is the slope of characteristic straight line (generalized dimensions) and is offset and the function can be defined as follows:

According to least square method, the condition of getting the minimum value for the function is [13]

Then the following equation can be obtained through calculation:

When , it stands for box dimension; when , it stands for information dimension; when , it stands for correlation dimension. According to the rule, we can obtain . When different failures happen in rolling bearing, there are differences between each dimension. Particularly in the condition of high dimensions, the failure type of bearing can be judged by the differences.

#### 4. Chosen Principle of Grid Type and Order Parameter

Through the experimental comparison analysis for the grid generation of time-frequency signal, the change and stability of the multifractal is different when selecting different grid type , which can be shown in Figure 1. How to select grid type is studied from the aspect of fractal geometry and the multifractal original definition to know the character of under different grid type .