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 timefrequency domain signals was proposed. The method and numerical algorithm of Multifractal analysis in timefrequency domain were provided. According to grid type
Recently, modern industry is gradually developing in the direction of largescale, 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 [
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 largescale 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 [
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 timefrequency 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.
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 lowfrequency 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 [
By EMD decomposition, another expression of time series can be acquired [
The signal acquired is still a timeseries signal, and the basic parameters to express signal feature are time and frequency [
For signal
Instantaneous frequency
It can be transformed into difference format, and the frequency can be expressed as
If sample frequency is
Phase
Phase unwrapping algorithm is to compare the principle value of adjacent points. Consider
If
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, timefrequency analysis method is more suitable for the nonlinear and nonstationary vibration signal analysis of rotating machinery [
Multifractal in timefrequency 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, Multifractal analysis in timefrequency domain can extract the quantitative characteristics of vibration signals. The numerical calculating method of generalized dimension of vibration signals in timefrequency domain is shown as follows.
As time series is
Based on the formula above, amplitude matrix
Similar to sampling space division of timedomain signal [
According to the above formula,
Define
According to least square method, the condition of getting the minimum value for the function is [
Then the following equation can be obtained through calculation:
When
Through the experimental comparison analysis for the grid generation of timefrequency signal, the change and stability of the multifractal
The generalized dimension curve of different grid types.
Fractal is an expression of object selfsimilar, so it can indicate the complexity of the measured object. While when the observation scale of the measured object is different, its complexity is not the same also. For example, when the map scale changed from big to small, the geomorphic characteristics are more and more complex. That means that the more precise the map is divided, the higher the degree of complexity is. And that is the same for signal, which is the signal observed under highresolution that is different from that under low resolution. And the Nyquist sampling theorem restricts the sampling rate in the process of signal acquisition.
In fractal theory, this theorem can be interpreted that the measured object should be divided into two parts at least for the selfsimilarity comparison. Moreover, the amount of information of every part should be same and without overlap to do the selfsimilarity comparison. So the largest segmentation scale of the measured object is half of itself. There are two scales of timefrequency for the signal in time and frequency domain, so the amount of information cannot be larger than a quarter of the whole information. The higher limit value of grid type
For the discrete signal, it is the direct sampling value with some loss and error during the process of sampling, processing, and calculation. To eliminate the error, longer time series is used in digital signal processing. So the grid type
The following is the definition of the generalized dimension, which is introduced by Renyi for the first time and found in the singular collection research by Junsheng et al. [
Most of the fractal dimensions in the selfsimilar fractal theory are included in
the sum of generalized fractal dimension
The grid type
From a large number of simulations and experiments, when the information distribution of the measured object is different, the lower limit of grid type
The value size distribution can affect the lower limit grid type
The range and density of the data distribution can affect the lower limit of grid type
There is identifiable information in other extracted parameters in different degrees. But it does not mean that the more the characteristics are extracted, the more the useful information there will be. When the amount of characteristics reaches a certain limit, the correlation will be enhanced inevitably. But it is useless for increasing useful classified information, even the identified performance may be weakened in some cases because the important characteristics may be submerged in the unimportant characteristics.
The range of order parameter
In the real calculation, the value of
Two groups of simulation signals generated by LabVIEW are compared to validate the feasibility of the theory as in Table
Components of two simulation signals.
Simulation signal  Frequency 1 (Hz)  Amplitude 1  Frequency 2 (Hz)  Amplitude 2 

Signal 1  80  10  200  10 
Signal 2  80  10  200  5 
The intensity maps are shown in Figures
Intensity matrix of Signal 1.
Intensity matrix of Signal 2.
From the intensity matrixes shown in Figures
From the curves in Figure
Components of four simulation signals.
Simulation signal  Frequency 1 (Hz)  Frequency 2 (Hz)  Frequency 3 (Hz) 

Signal 1  50  100  null 
Signal 2  50  200  null 
Signal 3  50  100  200 
Signal 4  50  200  400 
The generalized dimension curve of two simulation signals.
The intensity matrixes can be got after doing timefrequency analysis for the four signals shown in Figures
Intensity matrix of Signal 1.
Intensity matrix of Signal 2.
Intensity matrix of Signal 3.
Intensity matrix of Signal 4.
The curves can be got after doing generalized dimension calculation for the four signals, respectively, when grid type
The generalized dimension curve (
The generalized dimension curve (
The generalized dimension curve (
The generalized dimension curve (
From the graphs, comparing between Signal 1 and Signal 2, Signal 1 does not meet the requirement when
From the simulation experiments, the generalized dimension of different signals has good differentiation and stability while
The generalized dimension curve of exceeding higher limit
Therefore, the lower limit of grid type is almost constant when the measured object is constant. To distinguish the difference of the generalized dimension when the fault happens, the grid type
From the simulation experiments, the generalized dimension can distinguish the different signals, while different signal components are generated when different faults happen to bearing, so this method can be applied in the bearing fault diagnosis. The experiment has been done in this paper to test the feasibility of the timefrequency domain dimension method further.
Doing timefrequency domain generalized dimension analysis for the real bearing measurement signal which is got by the acceleration sensor, the experimental process is shown in Figure
Experimental flow charts.
We use the method on draught fan bearing fault detection. Table
The bearing information of the equipment.
Equipment number  Bearing model  Rotating speed (rpm) 

1#  22320CA  780 
2#  22320CA  985 
3#  22320CA  780 
4#  22320CA  985 
5#  22320CA  780 
Figure
The schematic figure of equipment under test and measurement layout.
The information in detail of each measuring point is given in Table
Measuring point layout information.
Measurement point  Measurement position  Measurement direction 

①  
1  Belt wheel side bearing 
Horizontal 
2  Belt wheel side  Vertical 
②  
3  Bearing box shell 
Horizontal 
4  Bearing box shell 
Vertical 
③  
5  Fan side bearing 
Horizontal 
6  Fan side bearing 
Vertical 
④  
7  Stents 
Horizontal 
8  Stents 
Vertical 
⑤  
9  Motor output end bearing 
Horizontal 
10  Motor output end bearing 
Vertical 
⑥  
11  Fan bearing box shell side 
Axial 
The analysis frequency of the measurement is 1 k Hz, and the sampling point is 1 kb. Data acquisition equipment information is given in Table
Data acquisition equipment information.
Equipment  Types  Model 

Sensor  ICP accelerometer  YD84DV 
Acquisition equipment  Data acquisition card  MPS060602 
We obtained large fault data after long period measurement. Those data could be classified to four types: the faultfree signal, bearing cages fault signal, inner ring fault signal, and outer ring fault signal. Those signals are selected to be analyzed.
The decomposition results of the four signals by EMD are shown in Table
The results of EMD.
Signal  Faultfree  Inner ring fault  Bearing cages fault  Outer ring fault 

Number of IMFs  4  5  6  6 
The first four IMFs of every signal are selected to be compared as in Figures
Faultfree signal and IMFs.
Inner ring fault signal and IMFs.
Bearing cages fault signal and IMFs.
Outer ring fault signal and IMFs.
From the EMD results, the four signals have different signal components. To know the characteristics of the signal components, doing the instantaneous frequency calculation and energy calculations for the four signals, respectively, the energy intensity matrix can be got as in Figures
Intensity matrix of faultfree signal.
Intensity matrix of inner ring fault signal.
Intensity matrix of bearing cages fault signal.
Intensity matrix of outer ring fault signal.
The energy distributions of the four signals are different at individual frequency scale. To show the difference clearly, making the grid type
The generalized dimension value of faultfree signal.

0  1  2  3  4 

2.10271  1.15764  0.68751  0.54359  0.47031 

5  6  7  8  9 

0.43111  0.40787  0.39288  0.38257  0.37513 
The generalized dimension value of inner ring fault signal.

0  1  2  3  4 

2.10094  1.08670  0.89636  0.83366  0.78689 

5  6  7  8  9 

0.75352  0.72955  0.71193  0.69860  0.68826 
The generalized dimension value of bearing cages fault signal.

0  1  2  3  4 

2.09131  1.08977  0.81367  0.71429  0.65361 

5  6  7  8  9 

0.61586  0.59109  0.57394  0.56149  0.55209 
The generalized dimension value of outer ring fault signal.

0  1  2  3  4 

2.10112  1.23425  1.08282  1.04355  1.00652 

5  6  7  8  9 

0.97820  0.95706  0.94106  0.92867  0.91884 
The curves for the four groups of data are shown in Figure
The generalized dimension curve of four experimental signals.
From Figure
Generalized dimension spectrum has been obtained after the signals are calculated by doing timefrequency domain generalized dimension method; slight difference in different singles can be distinguished based on different values of generalized dimension spectrum. And the specific algorithm and experimental analysis method have been provided. The correctness of the timefrequency domain generalized dimension method has been verified by simulation signals analysis and experimental signals analysis. The result shows that the numerical distribution of fractal dimension is different and has obvious separability under different fault modes. This indicates the correctness and feasibility of estimating the bearing fault condition qualitatively and quantitatively by timefrequency domain generalized dimension. For single or various fault types of rolling bearing, the fault mode can be confirmed by comparing the generalized fractal dimension spectrum value with the generalized fractal dimension value in the sample databases.
The working feature of bearing can be extracted by timefrequency generalized fractal dimension method. It can be applied to the diagnosis of the rolling bearing fault, and it is a simple and effective method to recognize accurately the fault models, which opens a new way for the rolling bearing fault diagnosis.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The project is supported by the National High Technology Research and Development Program (2013AA041108), the Research Foundation of Education Bureau of Liaoning Province, China (L2012166), the Foundation of State Key Laboratory (sklms2012006 and sklms2012003), and the National Natural Science Foundation of China (51475065).