With the development of multi-signal monitoring technology, the research on multiple signal analysis and processing has become a hot subject. Mechanical equipment often works under variable working conditions, and the acquired vibration signals are often non-stationary and nonlinear, which are difficult to be processed by traditional analysis methods. In order to solve the noise reduction problem of multiple signals under variable speed, a COT-DCS method combining the Computed Order Tracking (COT) based on Chirplet Path Pursuit (CPP) and Distributed Compressed Sensing (DCS) is proposed. Firstly, the instantaneous frequency (IF) is extracted by CPP, and the speed is obtained by fitting. Then, the speed is used for equal angle sampling of time-domain signals, and angle-domain signals are obtained by COT without a tachometer to eliminate the nonstationarity, and the angledomain signals are compressed and reconstructed by DCS to achieve noise reduction of multiple signals. The accuracy of the CPP method is verified by simulated, experimental signals and compared with some existing IF extraction methods. The COT method also shows good signal stabilization ability through simulation and experiment. Finally, combined with the comparative test of the other two algorithms and four noise reduction effect indicators, the COT-DCS based on the CPP method combines the advantages of the two algorithms and has better noise reduction effect and stability. It is shown that this method is an effective multi-signal noise reduction method.

The rotating machinery is an important mechanical equipment. As the core component of transmission machinery, the gearbox is vulnerable to damage, resulting in machine damage and accidents. Gearboxes are mostly in variable working conditions and are often used in situations with high working intensity and complex speed changes [

The shaft, gear and bearing in the gearbox will generate vibration during operation, and the vibration signal is the carrier of gearbox fault features. Therefore, vibration signals processing methods have become a common method of mechanical condition monitoring and diagnosis [

For the variable speed signal, it is an important processing step to obtain the speed signal to observe the speed change. When analyzing and processing the signal under time-varying working conditions, it is necessary to extract the IF or speed from the signal firstly. For the variable speed gearbox, the IF of the signal is also the meshing frequency of the gear. The speed can be obtained by dividing IF by the number of gear teeth. Therefore, it is necessary to extract the IF curve through some algorithms from the vibration signal under variable working conditions. In the early stage of speed tracking and extraction, the digital differential five-point formula method of numerical analysis was generally used to calculate the speed directly, but the accuracy of the speed by this method was very poor. At present, the commonly used method is to obtain the speed indirectly by obtaining the IF. In order to get IF, an IF estimation method based on the peak search was proposed by Guo Yu and their team [

The CPP algorithm which was put forward by Candès et al. [

In addition to proposing the algorithm, Candès and his team also detect highly oscillatory signals by CPP to verify the feasibility of the algorithm. In addition, Luo et al. [

After being proposed and applied for an invention patent by Potter of HP in 1989, the COT was used to realize equal angle resampling instead of Hardware Order Tracking (HOT) which realizes equal angle sampling through analog equipment. The COT is a method to transform non-stationary signals in the time domain into stationary signals in the angle domain, there are two kinds of converted signals which are stationary and cyclostationary. At present, order tracking has been widely used in various fields, the monitoring of ducted fans and turbomachinery in the aerospace industry is a typical application scenario of this method. Truong et al. [

The signal under variable speed conditions is non-stationary and nonlinear. Therefore, algorithms such as the Fourier transform cannot be directly used. The COT is an effective method for processing variable speed signals. The angle domain signal obtained after conversion by COT is stable so that some traditional signal processing methods can continue to be used. Different from HOT, the COT only needs to use the device to collect the speed pulse of the reference shaft to calculate the equal angle time, but sometimes it does not have the conditions to collect the pulse signal. Therefore, the COT is generally divided into COT with and without a tachometer whose difference is whether the speed pulse of reference shaft is collected. Li et al. [

In the field of multi-signal collection and data fusion, Multiple Measurement Vectors (MMV) models is usually used to process multiple signals. He et al. [

Torkamani et al. [

The multi-scale chirplet primitive function library adopted by the CPP algorithm is as follows:

The idea of CPP is to use chirplet atoms in the primitive function library to approximate the time domain analysis signal, use the linear combination of these chirplet atoms to estimate the signal, and connect the corresponding lines of each atom segment by segment to approximate the IF curve. According to this theory, we can find the core problem of the CPP algorithm: the connection of chirplet atoms and the selection of chirplet atoms. Among them, the selection of chirplet atoms can be realized by drawing a chirplet diagram. The chirplet diagram obtained by calculating a signal mixed with colored noise is shown in

Chirplet atoms have linear IF

(1) Initialization.

(2) For each element

There is

COT is essentially to convert the time-domain signal sampled by Shannon-Nyquist sampling theorem into angle-domain signal. Different from HOT, the equal angle sampling for COT calculation only needs a tachometer. The equipment collects the speed pulse signal of the axis with equal time interval to calculate the equal angle time, then the interpolation algorithm is used to interpolate and fit the resampling time to obtain the final angle domain signal.

The calculation formula of order ratio

The order tracking calculation requires a total of two filters, anti-aliasing filter and low-pass filter. Since the definition of order is obtained from frequency, anti-aliasing filter is to use Shannon-Nyquist sampling theorem to process firstly to prevent frequency aliasing. The angle domain resampling for COT is similar to the time domain signal sampling, and the angle domain sampling theorem is also similar to the time domain sampling theorem, corresponding to the time domain sampling frequency, the angle domain sampling theorem is:

The maximum order of the signal can be limited by low-pass filtering, and the highest order is determined by the cut-off frequency

The angle sampling rate can be calculated from the maximum order of the signal. If the reference axis is uniformly accelerated for a short time period, the angle of rotation

The three coefficients

By using the coefficient, we can obtain the equal angle time

CS is a new sampling method, which is different from the traditional Shannon-Nyquist sampling. The principle is: as long as the signal is sparse on the sparse basis, the signal can be projected from high dimension to low dimension space through the measurement matrix, and the original signal can be reconstructed from the projection by solving the optimization problem in the end. It can be seen that the three important steps of compressed sensing are: sparse representation of signal, selection of measurement matrix and solution of optimization problem.

The sparsity or compressibility of the signal is the premise of the application of the compression sensing theory. Therefore, compressed sensing is completed based on the signal sparsity theory. If a signal

If the original signals

If the sparse bases of the original signals whose sparsity are all K are the same, then the JSM-2 model can be used:

Same as JSM-1 model, except that

The residual

Due to the good data processing ability of DCS, it is introduced into multi-signal noise reduction. However, mechanical equipment often operates under variable speed conditions, the collected signals are also non-stationary and nonlinear due to the impact of speed, which affects the noise reduction effect. To eliminate this impact, preprocessing is carried out by COT before DCS processing. However, to calculate order tracking, it is necessary to calculate the pulse trigger time through the tacho pulse to obtain the resampling time. Therefore, CPP is used to extract the speed from the original signal and use it for COT calculation. The steps of the algorithm which is COT-DCS based on CPP method are:

(1)To prevent too much signal data from taking too long to calculate the sparse matrix, a segment of the vibration signal data is intercepted as the original signal in the time domain.

(2)Pre analyze one of the signals to determine

(3)The speed pulse is obtained through CPP, and the angle resampling time is calculated by angle domain sampling rate

(4)The obtained equal angle time sequence is interpolated with cubic spline to obtain angle domain signal.

(5)Perform steps (1) to (4) for other time domain signals to obtain respective angle domain signals.

(6)The JSM-2 model of DCS is used to compress the angle domain signal and obtain the measured value, which is then transmitted to the decoder.

(7)The SOMP algorithm is used to complete joint decoding at the decoding end, and the noise-reduced signal is successfully reconstructed.

In order to study the effect of CPP algorithm, a simple sine simulation signal

Then try to extract the original sine signal from the mixed signal by using the CPP method. The best path pursuit algorithm of CPP calculates that the best path is the 8th path, and then the chirplet diagram obtained after fitting the IF is shown in

It can be seen from the figure that the signal extracted by CPP is close to the original signal in shape, and the maximum and minimum values of the sine signal can also be accurately reflected, which can indicate the effectiveness of the CPP algorithm. However, because the algorithm is more suitable for extracting the IF of non-stationary signals, the extraction effect of mixed noise sine signal is worse than that of non-stationary signal.

With the development of non-stationary signal fault diagnosis under variable speed condition, the International Conference on Condition Monitoring of Machinery in Non-stationary Operations (CMMNO) was held this years, and there is a contest from CMMNO14 in 2014 to make the most relevant diagnosis of a wind turbine operating. In order to study the processing effect of CPP method on non-stationary signals, the IF will be extracted from the sound signal of the contest under variable working conditions provided by Leclère et al. [

The signal is a non-stationary signal whose sampling frequency is 5000 Hz. Its time-domain signal image is shown in

The speed of the signal will be tracked through CPP. Firstly, the chirplet diagram which contains the IF information is obtained, and then the IF curve is redrawn. In the competition, the unit of speed is rpm, so it is necessary to convert the amplitude of the IF diagram, and then divide it by the number of teeth to get the final speed diagram, as shown in

In order to study the accuracy of the speed curve obtained by CPP, it is compared with the real speed which is shown in

In

In order to further study the effect of CPP algorithm, it is compared with several existing methods of IF and speed extraction by the signal in

Then the IF is obtained by local peak search based on STFT. This method first obtains the time-frequency spectrum by STFT, and then extracts the IF curve from the graph by peak search algorithm. The speed is obtained by converting units and dividing by the number of teeth, as shown in

Viterbi algorithm is a maximum likelihood decoding algorithm proposed by Viterbi for Hidden Markov Model (HMM). The algorithm is applied to IF extraction, and the speed is obtained through processing which is shown in

After comparing the results of the above three algorithms with that of

In order to study the effect of the COT algorithm, a simulation signal defined

Then the signal in

It can be seen from the simulation and experimental results that after the time domain signal is converted to the angle domain, the signal becomes more stable, and many unstable factors are removed. The non-stationary signals that cannot be directly processed by Fourier transform can continue to be processed by Fourier transform after using COT. It shows the superiority of COT in processing non-stationary signals.

In order to verify the effectiveness of COT-DCS method, the open-source data about gearbox vibration signals collected by Southeast University on the Drivetrain Dynamic Simulator (DDS) [

Where: ① Tachometer ② Induction motor ③ Bearing ④ Shaft ⑤ Load ⑥ Drive belt ⑦ Data Acquisition board ⑧ Bevel gear reducer ⑨ Magnetic load ⑩ Crank mechanism ⑪ Transmission ⑫ Current probe. The data platform simulated five bearing operation states under two working conditions, namely, speed 20 Hz (1200 rpm)-no-load 0 V (0 N·m) and speed 30 Hz (1800 rpm)-loaded 2 V (7.32 N·m). The data set of bearing operation is divided into normal operation state, ball fault, inner race fault, outer race fault and inner and outer race compound fault, which respectively represent healthy operation state, ball crack state, inner race crack, outer race crack and inner race outer race crack. The signals of each fault state are collected by two acceleration sensors from x, y and z directions, and the sampling frequency is 20 KHz.

The COT can be divided into two types: COT with tachometer and COT without tachometer, the former can directly collect the speed through tachometer, this method has high reliability and is also widely used in industry. However, in the collection of multiple signals, tachometer equipment for synchronous acquisition of rotational speed is sometimes not installed. In view of this situation, the idea of simulating the speed signal in reference [

Then the speed is used to calculate the order tracking of time domain signals in three directions. In order to show the advantages of COT-DCS algorithm, the original signal is also directly denoised using the DCS algorithm. The processing results of using COT and DCS separately are shown in

In

Compared with the time-domain signal, it can be seen from

Comparing the three groups of the noise reduction results of experimental data, SVD-GLCT, EMD-MPE and COT-DCS have all removed part of the noise. Since the three algorithms involve signal reconstruction, in order to compare and analyze the noise reduction effects of the three methods, four new indicators, Reconstructed Standard Deviation (RSD), Root Mean Square Error (RMSE), Root of Variance Ratio (RVR) and Noise Rejection Ratio (NRR), are calculated respectively, as shown in

Original signal | Method | RSD | RMSE | RVR | NRR |
---|---|---|---|---|---|

SVD-GLCT | 0.0038 | 0.0038 | 0.1945 | 5.7397 | |

EMD-MPE | 0.0044 | 0.0044 | 0.0564 | 4.3040 | |

COT-DCS | 0.0043 | 0.0033 | 0.0159 | 9.4581 | |

SVD-GLCT | 0.0054 | 0.0054 | 0.1364 | 9.3126 | |

EMD-MPE | 0.0060 | 0.0060 | 0.0276 | 14.4765 | |

COT-DCS | 0.0061 | 0.0051 | 0.0239 | 17.7354 | |

SVD-GLCT | 0.0044 | 0.0044 | 0.3939 | 5.1207 | |

EMD-MPE | 0.0055 | 0.0055 | 0.0458 | 11.3996 | |

COT-DCS | 0.0057 | 0.0037 | 0.0455 | 14.6048 |

From the RSD, since it is used to evaluate signal fidelity, the smaller the result is, the more signal components are retained. The RSD of the three algorithms are all between 0.0038 and 0.0064, which is at a low level, indicating that the fidelity of the results obtained by the three algorithms is good. The RMSE reflects the similarity between the signal after noise reduction and the original signal, the more obvious the value is, the more similar the two signals are, and the better the noise reduction effect is. Compared with this evaluation index, the RMSE of SVD-GLCT is slightly less than EMD-MPE, while that of COT-DCS is the smallest. RVR reflects the degree of signal smoothness, the smaller the value is, the smoother the signal is. The value processed by COT-DCS method is relatively minimum, closer to 0, and the signal is smoother. The NRR is used to evaluate the ability of signal to suppress noise. The larger the value is, the higher the signal-to-noise ratio will be, and the better the noise reduction effect will be. The NRR of COT-DCS is greater than that of the other two methods, indicating that this method has a strong ability to suppress noise.

To test the treatment effect of the COT-DCS method on different non-stationary working conditions, the bearing data set to collect the inner race fault data with a speed load configuration of 30-2 is used. The results of COT-DCS, SVD-GLCT and EMD-MPE are shown in

The four new indicators of them under the working condition are shown in

Original signal | Method | RSD | RMSE | RVR | NRR |
---|---|---|---|---|---|

SVD-GLCT | 0.0090 | 0.0084 | 0.1339 | 6.1415 | |

EMD-MPE | 0.0088 | 0.0086 | 0.0581 | 13.4596 | |

COT-DCS | 0.0076 | 0.0075 | 0.0363 | 15.4258 | |

SVD-GLCT | 0.0079 | 0.0086 | 0.0936 | 7.8542 | |

EMD-MPE | 0.0085 | 0.0090 | 0.0687 | 10.5623 | |

COT-DCS | 0.0076 | 0.0073 | 0.0078 | 13.4869 | |

SVD-GLCT | 0.0073 | 0.0084 | 0.2876 | 7.4565 | |

EMD-MPE | 0.0082 | 0.0085 | 0.0989 | 16.8574 | |

COT-DCS | 0.0072 | 0.0080 | 0.0320 | 20.4838 |

Due to the addition of a 2 V load and an increase in speed to 30 Hz in the second working condition, non-stationary effect of the signal is greater in this working condition. Comparing the processing results of the same fault under different working conditions, the results of comparison of

Therefore, from the above noise reduction results and noise reduction indicators, the noise reduction effect and the ability to suppress non-stationary of the COT-DCS method is better than the other two algorithms, and it can better filter the noise in the signal while retaining the effective information in the original signal and reflecting its advantages in the noise reduction of multiple signals.

As a key variable speed transmission component of a mechanical system, the smooth operation of the gearbox is related to the safe operation of the entire machinery. With the development of multi-signal monitoring technology, the analysis, processing and denoising of multiple signals under variable working conditions have become an important issue. In order to solve the problem, a COT-DCS method based on CPP is proposed. The IF of the variable speed signal is obtained through the CPP algorithm, and then the speed information is fitted. Through simulation, experimental study and comparison with other three methods which include digital differentiation, local peak search and Viterbi algorithm, it is shown that the IF extracted by CPP is closest to the real speed. Aiming at the non-stationary and nonlinear of the variable speed signal, the time-domain signal is converted into an angle-domain signal through COT, and the simulated and experimental study shows that the algorithm has a good ability to deal with the non-stationary signal. The COT-DCS based on the CPP method is proposed and verified by the actual data. This algorithm combines the advantages of the two algorithms, which not only eliminates the nonstationarity, but also effectively removes the noise embedded in the signal. Compared with using COT and DCS alone, the COT-DCS method has a better noise reduction effect. Moreover, the processing results of this algorithm are compared with SVD-GLCT and EMD-MPE algorithms. The three algorithms all have certain noise reduction effects and signal fidelity, the noise reduction results of the COT-DCS are more stable than those of the others, and the noise reduction effect indicators are also better than SVD-GLCT and EMD-MPE. This shows the superiority and good effect of COT-DCS on multi-signal noise reduction.

The authors are grateful for the financial support provided by the National Natural Science Foundation of Hebei Province, Meanwhile, sincere thanks are given to the previous researchers who contributed to the development of deep learning technology, which lay the foundation for this research. Moreover, the authors would like to thank the anonymous reviewers for providing their invaluable comments.

The authors are grateful for the financial support of this work by the National Natural Science Foundation of Hebei Province China under Grant E2020208052.

Study conception and design: Guangfei Jia, Zhe Wu, Fengwei Guo; data collection: Zhe Wu, Fengwei Guo; analysis and interpretation of results: Guangfei Jia, Fengwei Guo; draft manuscript preparation: Fengwei Guo, Suxiao Cui, Jiajun Yang. All authors reviewed the results and approved the final version of the manuscript.

Readers can get the data in the website shown in references [

The authors declare that they have no conflicts of interest to report regarding the present study.