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Time-domain averaging (TDA) is an effective signal processing technique in fault diagnosis that can extract the periodic components of interest from signals mixed with noise interference while suppressing other irrelevant periodic signals. However, there are two obvious shortcomings to TDA: first, the acquisition of keyphasor signals is often restricted by the application environment and conditions. Even if the signal is obtained by TDA, owing to the existence of periodic truncation errors, satisfactory results cannot be obtained. Second, due to the velocity fluctuation, the actual mechanical signal is easy to produce a large error in TDA stacking. To solve the above challenges, first, based on the disadvantage of using traditional resampling to solve the TDA synchronization problem, this paper proposes a new method of subsection resampling, which improves the analysis effect of the traditional TDA. Second, to further expand the range of the practical applications, according to the amplitude-frequency map of TDA, a method for realizing TDA function by using the FIR multiband filter is proposed. This approach effectively avoids the requirement of traditional methods to collect the keyphasor signals and broadens the application in practical engineering. Finally, the improved TDA method is compared with the filter implementation technology, and their respective application conditions are given.

Vibration signal processing has been an effective means to monitor mechanical equipment for many years [

Time-domain averaging (TDA) or time synchronization averaging (TSA) is widely used in practice with its simple algorithm. It mainly uses the principles of random statistics to stack and average the original signal for many times to eliminate or reduce the noise influence from other sources, to further enhance the signal components of interest [

In summary, the research studies in this domain have enhanced the analysis effect of TDA from different aspects, but most of the research studies only focus on improving the traditional methods. For the problem that it is very difficult to obtain the keyphasor signal in complex and harsh environment, further research is still required. In order to successfully apply the TDA in complex environment, this paper first studies the traditional TDA and points out the shortcoming of using resampling to correct the synchronization problem of TDA. On the basis of the traditional TDA, we propose a subsection resampling method to improve this shortcoming. The simulation and experimental results shows that the segmented resampling can effectively improve the analysis accuracy and has well application value. Secondly, to solve the difficulties of acquisition keyphasor signal in complex environment, a filter implementation technique is proposed to realize the TDA function without the keyphasor signal. The simulation and experimental results show the feasibility and effectiveness of the filter implementation technology. Finally, the improved TDA is compared with the method using the filter implementation technology, and their application conditions are given.

TDA not only needs to pick up the signal to be analyzed but also needs to acquire the timing pulse of the rotation axis to lock the starting point of each signal segment. Since the signal is digital on average, the same number of points is required for each segment of data. Because the sampling frequency of A/D converter is constant once set and the timing pulse frequency and the period of the signal to be analyzed also change at any time due to the change of mechanical speed even if the small change, it is impossible to guarantee the equal number of data of each segments by conventional sampling. The method to solve this problem is to use frequency tracking technology so that the actual sampling frequency can track the rotation frequency in real time and equal to the integer multiple of it.

The key of TDA is signal synchronization. If the signal is asynchronous, the useful components of the signal after the TDA processing will be attenuated, leading to the preset purpose not being achieved.

At present, frequency tracking technology is generally used to solve the problem of signal asynchronization caused by various factors [

Specific approach of the traditional resampling:

The analyzed signal

Assume that the number of points of a data segment sampled for the first time is

According to the uniform sampling number

The sampling interval

Software implementation technology of frequency tracking mainly refers to the subsequent processing of the collected asynchronous signals, which mainly uses a programming language to implement certain algorithms to synchronize the collected signals, and frequency tracking is realized by the data resampling technique.

The sampling interval adopted by traditional data resampling technology is carried out under the default constant speed condition within a section of the data signal, that is, the sampling interval is the same for the data within a section, and its procedure is shown in Figure

Traditional process.

Improved process.

The only difference between the improved segmented resampling technique and the traditional resampling technique is that the sampling interval of the segmented resampling of the whole truncated data is determined according to the situation. The specific method of segmented resampling is as follows: after the long signal is truncated, each segment signal to be superposed is divided into four parts at phases of 0,

The implementation flow chart. T is the rotation period of rotary machinery, which is generally used as the truncation period.

When the rotation frequency is not available, which means the exact period of the periodic component of the signal is not clear, the period can be determined by obtaining the maximum value of the autocorrelation coefficient [

Both the cointegration method and the multiple regression method study the correlation between variables. Cointegration method is mostly used to study the relationship between economic variables [

A composite signal

Simulation signal. (a) The simulated impact signal

The composite signal was processed according to Figure

Comparison of treatment results of different methods. (a) Traditional processing results, (b) improved treatment results, and (c) ideal synchronization results.

In this section, the improved TDA, the segmented resampling, is applied to the actual vibration signal of the bearing rotation collected in the small rotor test bed under the condition of misalignment. The BK3000XL type eddy current sensor with a probe of

The picture of the test rig.

The parameters of the small rotor test bed are set as follows: the sampling frequency is 1600

Time-domain waveform of the acquisition signal.

The collected signals are analyzed and processed by traditional resampling and the segmented resampling proposed in this paper, and the results are presented in Figures

Traditional TDA processing results. (a) Synchronous correction data segment before improvement and (b) stacking average result.

Improved TDA processing results. (a) Improved synchronous correction data segment and (b) stacking average result.

Obtained from the TDA formula, the amplitude-frequency response is shown in equation (

Different segment numbers

Amplitude-frequency response of TDA.

From Figure

In fact, there are many kinds of signal denoising methods, such as wavelet analysis and empirical mode decomposition. Wavelet analysis (WV) [

The selection of the filter is critical for the realization of the TDA function. There are various kinds of filters, among which digital filters are mainly classified into FIR and IIR filters. The biggest advantage of FIR filter is that it has a linear phase and stable system structures and can realize zero phase shift filtering. To obtain a better filtering characteristic curve, the filtering order is usually selected to be larger. However, the disadvantages of FIR filter are the relatively large amount of computation required and the poor real-time performance of signal processing. The greatest advantage of IIR filter is that it can obtain accurate passband and stopband edge frequencies and effective passband and stopband attenuation. However, the disadvantage is that the system has a stability problem, and the linear phase relationship cannot be guaranteed for the processed signals. Since the TDA highly requires phase, to ensure that there is no phase shift between the fundamental frequency of a specific periodic component and its frequency doubling, a zero phase shift bandpass filter is designed from the FIR digital filtering technology to realize the TDA function [

A finite length sequence is sought as the unit impulse response

The design idea of multiband filter.

From Figure

Determining the ideal frequency characteristic of the multiband filter

The multiband filter needs to retain the frequencies around the fundamental frequency and its octave, i.e. That is

Unit sampling response:

where the symbol

Selection of window function

Under the condition that the stopband attenuation meets the requirements, the window function with the narrow main lobe and small side lobe peak should be selected as far as possible. Compared with rectangular window, Hamming window and Hanning window are commonly used and have slightly wider main lobes, but smaller side lobes and larger attenuation speeds [

Calculating the unit impact response:

Namely:

where the symbol

Response curve of multiband filter. (a) Impact response and (b) frequency characteristic.

One of the key technologies involved in the construction of multiband filter is the determination of the fundamental frequency and its octave frequency. The error of fundamental frequency

For discrete signal

Interpolation correction diagram.

Set

Equation (

After the fundamental frequency of the period is determined, the frequency multiplication is obtained by equation (

The damage signal of the gear tooth surface is simulated in this paper. When the gear tooth surface is evenly worn, which causes the tooth gap to increase, or even there is cracking, pitting, peeling, and other damage on the tooth surface, the impact vibration will be generated, and the gear will be excited to vibrate according to its natural frequency. In the frequency component, the second and third harmonics or the frequency division component of 1/2, 1/3, etc., of the meshing frequency are generated. Suppose the analog discrete signal is composed of a periodic signal and random white noise, as follows:

Simulation signal analysis results. (a) Original waveform and spectrum and (b) waveform and spectrum processed by the multiband filter.

From Figure

Figure

The specific methods are as follows.

For signal

Considering the sampling frequency range and the frequency components of interest, the number of passband is set as

The frequency response function of the comb filter [

Original signal. (a) Time-domain waveform of the original signal and (b) original signal spectrum.

Processed signal. (a) Time-domain waveform of the processed signal and (b) processed signal spectrum.

The signal to be analyzed is filtered through the constructed multiband filter, and the filtering results are shown in Figure

After processing, the waveform signals with serious noise interference, as shown in Figure

The gear signals obtained from the experiment in Section

Comparison results of two methods of gear signal. The red line is the result of traditional TDA, and the blue line is the result of multiband filter.

Two methods were used to analyze and compare the bearing colliding fault signals obtained from the tests in Section

Comparison results of two methods for processing the impact wear signal.

In this paper, the principle of TDA is described, and an improved resampling method, called the segmented resampling method, is proposed to address the synchronicity problem. The effectiveness and practicability of the proposed method are demonstrated by analyzing the simulation signal and the actual signal, respectively. Under the condition of velocity fluctuation, the segmented resampling method broadens the application range of TDA and improves the analysis precision. Then, based on the TDA amplitude-frequency map, this paper proposes a method to realize the TDA function with a multiband filter. The selection, construction, and parameter optimization of the TDA filter are studied, and the feasibility of the method is verified by analyzing and processing the simulated signal and the actual signal. The experimental results of the filter implementation technology show that this method can effectively retain the fundamental frequency and the octave frequency of the periodic signal of interest, filter out the irrelevant interference components, and realize the TDA function.

In this section, the calculation complexity is compared. On the premise of reaching a similar result accuracy, the traditional TDA method needs to iterate

The mechanical vibration signal data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The work was financially supported by the basic scientific research fees of Zhejiang Ocean University (2019JZ00004), provincial first-class discipline construction project supported by Young and Middle-Aged Discipline Leaders of Zhejiang Province, and Science and Technology Plan Project of Zhoushan Science and Technology Bureau (2018C21014).