Fault diagnosis of gearbox based on improved polynomial adaptive chirp mode decomposition algorithm

: The adaptive chirp mode decomposition (ACMD) has good time-frequency representation results in analyzing chirp signals, while there is a time-frequency ambiguity problem in the analysis of variable speed planetary gearbox vibration signals. To address this problem, a planetary gearbox fault diagnosis method based on improved polynomial adaptive chirp mode decomposition wavelet is proposed (IPACMD). Using Adaptive chirp mode decomposition, the amplitude and instantaneous frequency of multiple signal components are estimated; To avoid overdecomposition to generate spurious signal components, the similarity conditional entropy is used to optimize the adaptive chirp mode decomposition threshold ;The polynomial chirp transform (PCT) using a polynomial function instead of the linear chirp kernel in the chirp transform to improve the time-frequency aggregation of the instantaneous frequency curve of each signal component and output high-resolution time-frequency representation results. Compared with the original method, the proposed method has better time-frequency aggregation and is more effective for the analysis of variable speed planetary gearbox vibration signals. The simulation and experimental study results show that the method can effectively identify the frequency components and time-frequency characteristics of the variable-speed planetary gearbox vibration signal and realize the fault diagnosis of the planetary gearbox.


Introduction
In the fault diagnosis of variable speed planetary gearboxes, the extraction of the fault characteristic frequency of the signal and its amplitude variation is the key to fault diagnosis. As the operating conditions, load and speed parameters of the planetary gearbox change, the fault characteristic frequency and amplitude of the planetary gearbox also change with the load and speed, and the mechanical system generates a non-stationary signal of the fault characteristics over time [1] [2] . Niu et al. [3,4] have effectively improved the machining accuracy of machine tools by investigating the effect of interference factors inside the planetary gearbox. In addition, the early fault features are weak and easily disturbed by environmental noise, and it is difficult to extract the fault feature frequencies by traditional time domain and frequency domain methods. Therefore, it is important to study the time-varying working condition planetary gearbox fault feature extraction method. 2 Time-frequency analysis uses a combined time-domain and frequency-domain distribution to describe the frequency components and time-varying trends of a signal. Huang E [5] proposed the Empirical mode decomposition (EMD) method, which uses a recursive screening algorithm to find each signal component with adaptive decomposition characteristics. Lei Y et al [6] applied EMD to rolling bearing fault diagnosis, but EMD is an empirical algorithm with problems such as modal confusion and endpoint effects [7] , which affects the fault diagnosis effect. To avoid the limitations of EMD, many alternative methods are generated. Feng et al [8] used Ensemble Empirical mode decomposition (EEMD) to extract the features of the modulated timevarying signals. The modulated time-varying signal and successfully identified the failure of the sun gear in the planetary gear system. Liu et al [9] used the Local Mean Decomposition (LMD) method to decompose the multicomponent signals and successfully extracted the fault characteristics of the planetary gear system. Empirical Wavelet Transform (EWT) [10,11] decomposed the multicomponent time-varying signal into single frequency components and successfully accomplished the fault diagnosis of bearings. Variational mode decomposition (VMD) [12] was used to extract faulty bearing signal features in wind turbines. In addition, researchers used an iterative generalized synchronous compression transform, which draws the advantages of the synchronous compression transform and makes the time-frequency features more readable [13] . The High-Order Synchrosqueezing Transform (HSST) uses an accurate instantaneous frequency approximation to obtain a clear time-frequency signature. However, these methods still have some drawbacks in gearbox fault diagnosis; EEMD based on empirical mode decomposition suffers from modal aliasing [14] ; LMD is less effective in dealing with narrowband signals; the processing effectiveness of EWT and VMD depends largely on the choice of parameters, which is difficult to be applied in practice; HSST is proposed on the basis of synchronous compression transform when evaluating strong time-varying modulated signals in noisy environments, a high timefrequency resolution is required, which makes the calculation process complicated and affects the practical application of engineering. To address the shortcomings of the above time-frequency analysis methods, Chen et al [15] proposed the variational nonlinear chirp mode decomposition (VNCMD) method, which not only extracts all signal components of the non-stationary vibration signal, but also accurately estimates the instantaneous frequency. And further proposed a more adaptive and stable method called Adaptive chirp mode decomposition (ACMD) [16,17] . This method can effectively extract all signal components of the non-stationary vibration signal and has good advantages in non-stationary vibration signal decomposition and instantaneous frequency estimation. However, in variable speed gearbox fault diagnosis, ACMD has the phenomenon of over-decomposition, which is difficult to handle signals with many signal components and time-varying frequencies, and the obtained results have low time-frequency aggregation, which is difficult to meet the requirements of variable speed gearbox fault diagnosis.
In order to solve the limitations of the traditional time-frequency analysis method for processing variable speed gearbox signals, an improved polynomia adaptive frequency modulation mode decomposition (IPACMD) method is proposed in this 3 paper. The similarity conditional entropy alternative to the residual energy to the original signal energy ratio is introduced as a new decomposition termination threshold condition to optimize the ACMD algorithm and solve the problem of signal overdecomposition; the polynomial chirplet transform (PCT) method is applied to replace the linear chirp of the traditional chirp transform by constructing a polynomial matching transform kernel function. The polynomial chirplet transform (PCT) method is applied to replace the linear chirp function of the traditional chirp transform by constructing a polynomial matching transform kernel function, which achieves higher time-frequency aggregation and improves the initial instantaneous frequency extraction accuracy. The IPACMD method is verified by simulation and experimental signals, which can effectively avoid the over-decomposition phenomenon, extract the key feature components, construct the time-frequency curve with high time-frequency aggregation and clearly represent the vibration signal fault characteristics.

The principle of ACMD
This paper focuses on the analysis of the gearbox vibration signal, and the gearbox signal contains multiple non-stationary vibration signal components, and the multicomponent non-stationary signal is modeled as: where K is the number of signal components, where ̃ is the target frequency function and ( ) and ( ) represent the two chirp signals. ACMD uses a matching tracking algorithm to adaptively extract the components of the target signal, and the model for the kth component sought as: where is ( ) the input signal, ( ) is the target signal component that should be extracted, ‖ * ‖ 2 represents the 2 norm, ( * ) ′′ represents the second-order derivative, and represents the penalty parameter. Assume that the signal is discretized into N points, i.e., t=t_0:t_(N-1). The objective function matrix can be obtained by bringing Eq.2 into Eq.4: , and and satisfy the condition that: . In order to solve Eq. (5) minimally and optimally, the target signal and the frequency function ̃( ) are updated alternately by means of iterations. Then, after the mth iteration, the target signal is represented as : the signal components sought as: the transformation of frequency can be obtained as: then the required instantaneous frequency as: where is the unit matrix and is the weighting factor. The individual signal components of ( ) can be obtained after several iterations.

Polynomial chirp wavelet transform
According to the non-stationary vibration signal model established in the previous section, polynomial matching kernel parameters are constructed [18] , and when the kernel parameters and the signal model are in better agreement, then high timefrequency aggregation can be obtained. The time-varying speed gearbox signal has the characteristics of nonlinear frequency modulation, and the polynomial frequency modulation wavelet transform kernel parameter signal model can be constructed as: where is the polynomial coefficient and is the number of polynomials, the instantaneous frequency is: (11) construct the polynomial transformation kernel function ( ) as: (12) where is the polynomial coefficient, then the polynomial chirp wavelet transform is: where 0 and t represent the window of the time span of the sliding analysis centered on the time span of the analysis and the window of the constant centered on 0 .
( ) is a Gaussian window function: the PCT algorithm is able to concentrate energy along each signal instantaneous frequency and has good interference immunity, we use the PCT algorithm to provide the initial instantaneous frequency for the ACMD algorithm. 3.2 Similarity conditional entropy termination criterion he iteration stopping threshold in the ACMD algorithm is based on stopping when the ratio of the residual signal energy to the original signal falls below : However, this criterion fails in some cases of low signal amplitude energy, where the energy ratio of the decomposed signal to the original signal is as high as 0.1361 ( ≤ 0.1), as shown in Figure 1. At this time, ACMD has completed the decomposition of the signal to extract the signal frequency and amplitude information, which, according to the original stopping criterion, cannot properly stop the ACMD decomposition, leading to over-decomposition. In order to avoid this situation, this paper introduces the similarity conditional entropy to replace the original stopping criterion of the remaining signal energy ratio. The similarity conditional entropy can effectively measure the interdependence between two variables and identify the degree of correlation. The mutual correlation information of two discrete random variables X and Y can be defined as: (16) where H(Y) is the similar conditional entropy of variable Y and H(Y|X) is the uncertainty of variable Y conditional on variable X.
When X is known, the weaker the correlation between X and Y, the larger H(Y|X) is. Therefore, ( , ) is small when the correlation between X and Y is weak.
Whether the residual signal contains important information of the original vibration signal should be the stopping criterion for ACMD decomposition. The original gearbox signal is preprocessed using ACMD, and the preprocessing result is denoted as , which contains much less noise than the original signal. Assuming that k signal components ( = 1, … , )are obtained, the residual signal is represented as: (18) The mutual information between the residual signals and is ( , ). Set 0.02 as the stopping threshold of decomposition, when ( , ) is lower than 0.02, the residual signal is considered to contain little useful information of the gearbox signal. By the similarity conditional entropy termination criterion, the ACMD decomposition can stop the decomposition in time to avoid the overdecomposition phenomenon.

Improved polynomial adaptive chirp mode decomposition
In the application of variable speed gearbox fault diagnosis, the fault characteristic frequency of vibration signal changes with the speed, the vibration shock is not periodic, and the traditional time domain and frequency domain methods are difficult to reflect its non-stationary and time-varying characteristics. For gearbox vibration signals, this paper proposes an improved polynomial adaptive chirp mode decomposition method with the following procedure.
(1) The signal is STFT transformed to obtain ridge 1, and the initial transform kernel parameters are obtained by fitting a polynomial to ridge 1 using the least squares 7 method.
(2) The signal is polynomially wavelet chirp transformed according to the initial transform kernel parameters to obtain ridge 2, and a polynomial fit to ridge 2 is performed by the least squares method to obtain parameter 2 replacing parameter 1.
(3) Determine whether the transformation kernel parameter 2 is optimal, if it is optimal, end the process and output the instantaneous frequency result, if it is not optimal, set parameter 2 as the initial transformation kernel parameter and repeat step (2).
(4) After the initial instantaneous frequency is obtained, the target signal function and signal components are iteratively updated according to equations (6) and (7). (5) The instantaneous frequency increment ∆̃( ) and the instantaneous frequency +1 are calculated according to equations (8) and (9).
(6) When the similarity condition entropy stop condition is satisfied, i.e., even the residual signal basically does not contain fault characteristic information, the iteration stops and the signal component and instantaneous frequency are output; otherwise, return to step (4) to continue the cyclic process, and the decomposition process is shown in Fig.2  The IPACMD for non-stationary signals is investigated in this paper, and the simulated signals are composed of two non-stationary signals that are very close in the time-frequency plane. the instantaneous frequencies of the two signal modes are : 1 = 130 + 300 + 10 cos 20 , 2 = 80 + 300 + 10 cos 20 , the signal waveform is shown in Fig.3 Using the traditional time-frequency analysis method for initial instantaneous frequency prediction of the signals, the extracted time-frequency result ridges cannot reveal the correct instantaneous frequencies due to the interference between the two signal frequencies that are relatively close and affected by noise, as shown in Figure 4. By comparing the time-frequency results obtained by ACMD and IPACMD, it can be observed that ACMD has serious energy diffusion and loss in the interference region of the two signal components, as shown in Fig. 5, which cannot clearly represent the time-frequency structure of the signal. On the contrary, the time-frequency image obtained by IPACMD has good time-frequency aggregation, as shown in Figure 6, which can effectively resolve the two close signal components and clearly reveal the time-varying instantaneous amplitude and instantaneous frequency.The time-domain waveform corresponding to each item is shown in Figure 3, and the time-domain waveform and envelope spectrum of the original simulation signal are shown in Figure  4.
The results show that IPACMD can further optimize the instantaneous frequency and extract the signal components with closer frequencies, and well balance the signal energy of these two close components, so as to accurately estimate the instantaneous frequency and reconstruct the signal components. It has a good effect on the extraction of signal components.

Analysis of fault simulation signal of variable speed planetary gearbox
When the variable speed planetary gearbox has a local wear fault of the sun wheel, the vibration signal has a steady state chirp signal with the engagement frequency as the carrier frequency and the gear rotation frequency and its multiplier as the modulation frequency, and there is also a periodic shock amplitude modulation signal.
According to the characteristics of the vibration signal, the simulation signal is constructed as: The simulated signal waveform, spectrum and time-frequency spectrum of Eq. (21) are shown in Figs. 7, 8 and 9. From Fig. 8, it can be seen that the speed variation leads to the blurring phenomenon of frequencies in the spectrum plot and the confusion of the engagement frequency and its side frequencies. As shown in Figure 9, the timefrequency spectrum obtained by STFT is not able to represent these signal components clearly due to the interference of noise and poor resolution.
Therefore, using the IPACMD to process the simulated signal, as shown in Fig. 10(b), the proposed method can clearly identify the fault feature components, and the timefrequency image obtained by the IPACMD method correctly reveals the instantaneous frequencies and energies that shift with time compared to the time-frequency ridges obtained by the ACMD algorithm (as shown in Fig. 10(a)). According to the characteristic frequencies given in the simulated signals, the three and their combined frequencies of sun wheel wear fault characteristic frequency, gear meshing frequency and sun wheel absolute rotation frequency are successfully separated and extracted in the time-frequency map obtained by the proposed method, as shown by the logo in the figure. The ability of the method to handle the time-varying features of the variable speed gearbox fault signal under strong noise conditions is demonstrated.
Therefore, IPACMD results can be applied to the separation and analysis of variable speed gearbox fault signal feature extraction. The simulation results show that the method can effectively extract the time-varying features of the non-stationary vibration signal under strong noise conditions.

Experimental Signal Analysis of Planetary Gearbox Sun Wheel Failure
In this section, IPACMD is used to analyze the experimental signals of localized faults in the sun wheel of the planetary gearbox. Figure 11 shows the planetary gear train test bench, and the gearbox parameters are shown in Table 2. The experimental procedure was performed by testing the vibration signal through the acceleration sensor on the top of the gearbox case with a sampling frequency of 20 kHz, and the signal acquisition was performed during the growth of the motor speed from 12 Hz to 20 Hz. In order to simulate gear failure, spalling damage was machined on a certain gear of the 1st stage sun wheel.
Since the characteristic frequency changes with time, it is difficult to determine the fault characteristics from the waveform and frequency peaks. As shown in Figure 15, the STFT, due to the fixed window function, although the trend of frequency is well represented at 0.5-1s, there is obvious mixing and interference between frequencies at 0-0.5s, where the frequencies are closer, and the obtained time-frequency images are not able to clearly represent these signal components.
IPACMD can effectively deal with the closer frequency components and can effectively avoid the mutual influence between adjacent components, as shown in Figure 16(b). IPACMD separates the time-varying frequency components and obtains the results with high time-frequency accuracy. While the ACMD algorithm obtains time-frequency ridges (as shown in Fig. 16(a)), although it has better components for signal components in the 0.5-1s part, due to the reason of insufficient accuracy of instantaneous frequency extraction, there is obvious mixing between frequency components in the part with small frequency interval of 0-0.5s, and the frequency components of the signal cannot be accurately identified.
Therefore, IPACMD solves the problem of frequency component crossover and interference, and combines the PCT method with similarity conditional entropy to solve the problem of insufficient accuracy and over-decomposition of initial instantaneous frequency acquisition. It can effectively analyze the complex time-frequency structure of the planetary gearbox under time-varying speed, as shown by the logo in Fig. 16(b). Among them, the main frequency is the difference between the meshing frequency and the solar path fault characteristic frequency − , whose energy is significantly enhanced in 0~1s. In addition, the sum of the engagement frequency and the fault characteristic frequency + , the engagement frequency , the difference between the engagement frequency and the solar wheel rotation frequency − , and the associated combined frequencies are also present. These features prove the existence of a localized fault in the sun wheel, which is consistent with the experimental reality.
In summary, IPACMD method can more clearly and accurately represent the vibration signal components of gearboxes at time-varying speeds than the ACMD algorithm, and the experimental data test results show that the method can effectively extract the fault characteristics of variable speed gearboxes

Conclusions
The results of simulation and experimental analysis show that IPACMD for gearbox fault feature extraction has good time-frequency aggregation. The main conclusions obtained are as follows.
IPACMD time-frequency analysis method is proposed, and the similarity conditional entropy is used to optimize the adaptive chirp mode decomposition threshold, which solves the problem of over-decomposition of the original method for variable speed gearbox signals. IPACMD method uses the PCT method for initial instantaneous frequency estimation. Compared with the traditional time-frequency detection method, PCT can effectively solve the problem of interference between the vibration signal components of gearboxes when analyzing the signal components of time-varying frequencies.
According to the simulation and experimental analysis results, IPACMD can effectively extract the time-varying fault characteristics and generate high-resolution time-frequency representation results, clearly represent the fault-related characteristic components in the vibration signal, and effectively identify the local faults in planetary gearboxes.

Prospects
The method proposed in this paper can distinguish different types of health conditions, but it is still unable to identify specific fault types. Therefore, the deep learning algorithm will be combined to achieve qualitative analysis of signals in the future.
Funding: This research is funded by the National Natural Science Foundation of China (Grant Nos. 52075008).
Conflicts of interest/Competing interests：The content of this paper does not involve any conflict of interest. Availability of data and material：The data used in this paper are all owned by the lab of the research group. As the research is still continuing, the data involved in this paper is not publicly available. Code availability：The algorithm involved in this paper is still being studied by the research group, so it is not publicly disclosed. Ethics approval：The team affirmed that all the research was original research of the research group without any bad behavior. Consent to participate：The methods involved in this paper are agreed to be shared. Consent for publication：The content of this thesis is agreed to be published in the journal.
Authors' contributions：Y,C. P analyzed the data; L,L. C. and T,T. L provided guidance and recommendations for the research. Y,C. P contributed to the contents and writing of the manuscript. All authors have read and approved the final manuscript.