A new improved Kurtogram and its application to planetary gearbox degradation feature analysis

Because of various advantages of planetary transmission system, it has been widely used in modern industry. And study on planetary gearbox degradation feature analysis method has important significance for mechanical system prognostics and health management (PHM). In order to analysis the degradation characteristic of planetary gearbox, Energram is proposed in this paper based on Kurtogram. Kurtogram is used for finding the optimal frequency band to rotating machinery fault diagnosis by calculating kurtosis. Similarly, Energram is used to show the energy trend of each frequency band by calculating energy, and arithmetic Energram is used to show the change of frequency band energy. The principle and application of Energram and arithmetic Energram are described by experimental data examples in this paper. A detailed study of planetary gearbox degradation characteristics is expressed in case study, which including Energram, arithmetic Energram and four particular comparative analyses. And the conclusions of each comparative analysis are given.


Introduction
As the advantages of strong load-bearing capacity, large transmission ratio, etc., planetary transmission systems are widely used in complex mechanical equipment, such as wind turbine, helicopters and heavy trucks.Once planetary transmission systems severe degradations occur they may cause machines to malfunction and even fail, which leads to financial losses and even fatal incidents.Moreover, planetary transmission system significantly differs from fixed-axis transmission system because of its unique structure [1,2].As a result, it is very important to research on health state assessment method and degradation analysis approach of planetary transmission systems.
Health state evaluation of planetary transmission system is a hot research topic in recent years, many scholars study on this aspect.For example, Chaari [3,4] investigated the effects of planetary gearbox gear fault on vibration responses through dynamics modeling and analysis.In order to calculate both local and distributed fault frequencies, Feng and Zuo [5] proposed planetary gearbox vibration signal models and deprived equations.Considering the working environment of wind turbines was easy to change, Chen and Feng [2,6] studied on planetary gearbox fault diagnosis and condition monitoring methods under nonstationary conditions.Lei [7] and Bartelmus [8,9] respectively put forward feature indices for planetary gearboxes condition monitoring under constant and nonstationary operations.Some transmission systems have more than primary planet gears, Lei [10,11] researched on health condition identification of multi-stage planetary gearboxes.He also summarized the research and development of planetary gearboxes condition monitoring and fault diagnosis [12].
Some scholars have focused on the degradation analysis and fault prediction of planetary gearbox.For instance, Marcos and George [13] investigated the prediction of axial crack growth in an UH-60 planetary carrier plate.Cheng and Hu [14,15] researched on pitting damage level estimation and quantitative damage detection of planetary gear sets based on simulations and physical models.Ni [16] used state-space model to estimate remaining useful life of planetary

Spectral kurtosis and Kurtogram
It is critical to grasp the basic principle of SK for its application in fault diagnosis.This article will describe the basic theory of SK and Kurtogram, and illustrate their application by examples using open experimental data.
According to Wold-Cramer representation, any stochastic nonstationary process can be decomposed into a causal, linear and time-varying system [20,33]: where , is the complex envelope of (the time varying transfer function of the system) at frequency , and is a spectral increment.Then, the SK can be clearly expressed as the fourth-order normalized [20,33]: where the 2 -order spectral moments are expressed as: Spectral cumulants of order 2 ≥ 4 have the interesting property that is non-zero for non-Gaussian processes.
In general, the vibration signal corrupted with noise, = + , is stationary noise.And SK can be described by: where is the noise-to-signal between and : Moreover, when is a stationary Gaussian noise independent of , the SK can be simplified as: It is not difficulty to found that the basic idea behind the SK is to get a high value when the signal is transient, and will be zero when the signal is stationary Gaussian [34].
Antoni [20] proposed SK on the basis of a series of digital filtering, and he made detailed research on it.The mainly harvest in computing speed is the SK calculation method based on binary decomposition, which is very similar with the FFT algorithm.In the calculation algorithm, the frequency bandwidth is equal to half of the frequency bandwidth in previous stage.And the calculation algorithm is known as binary tree.Moreover, there is also a 1/3-binary tree, all combinations of center frequency and bandwidth for the 1/3-binary tree Kurtogram are shown in Fig. 1 and Table 1 ( is the signal sampling frequency).
Take bearing failure data of Case Western Reserve University (CWUR) as an example to illustrate the application of Kurtogram method.The test bearing with 7 mils single point fault in bearing outer raceway located 6 o'clock position.The test stand speed is 1730 rpm and load is 3 hp.The sampling frequency is 12 kHz and the Nyquest frequency is 6000 Hz.According to experience, the signal components within 30 times rotating frequency are mainly frequency components generated by shaft and gear, which tends to be ignored in analysis.Therefore, this paper only analysis the frequency band in range [1000, 6000] Hz.The SK is calculated by 1/3-binary tree Kurtogram and the output Kurtogram is shown in Fig. 2. It is easy to find the optimum resonance frequency band in 6-th level, and the frequency band interval is [3083.5,3500.2]Hz.Then, envelope analysis of signal within [3083.5, 3500.2]Hz is implemented to check fault characteristic frequency of bearing outer ring.The outer ring fault feature frequency (BPFO) is obviously in Fig. 3.As a result, Kurtogram is effective for fault diagnosis.

Energram
Previous studies have shown that kurtosis can be effective used in rotating machinery fault diagnosis.Meanwhile, energy can rise gradually as system degradation increase and it has been widely used in rotating machinery degradation research.
The idea of Energram is put forward based on Kurtogram.In order to reflect the system degradation degree, energy is calculated as the characteristic index instead of kurtosis.In Kurtogram method, kurtosis is extracted (as shown in Eq. ( 7)) after the frequency bands decomposed.Similarly, energy is calculated as Eq. ( 8) in Energram.As a result, the Energram method can be used for degradation research.The flow chart of Energram method is presented in Fig. 4: where is discrete vibration signal of the time series over the time interval [1, ], and ̅ is the mean value of discrete vibration signals. .The data set interval -axis is = 3. Fig. 7 is the total energy of the (850-984)th set of data.Contrast Fig. 7 and Fig. 8, it can be found that Energram color depth is changing along with the change of bearing total energy, and the energy is mainly concentrated in frequency band [625, 1250] Hz.

Arithmetic Energram
In general, system runs the longer, degradation degree is the larger, and mean vibration signal energy is the higher.However, as the energy value will fluctuate, not every energy value is absolute growth.
The system total running time is , is the system energy at running time = 1,2, … , , the interval time Δ = − , the energy difference between and is Δ = − , namely Δ is the energy increment during Δ .As the energy is not absolute growth, Δ may be greater than zero and may also be less than zero.Of course, Δ is greater than zero in most cases.The flow chart of arithmetic Energram method is presented in Fig. 5.
The arithmetic Energram of the (850-984)th set of data is shown in Fig. 9.It can be found that the arithmetic Energram color changes only when total energy fluctuates is intense, and the energy fluctuation is also mainly focused on frequency band [625, 1250] Hz.

Planetary gearbox degradation analysis
A case study of planetary gearbox degradation process is carried out to validate the effectiveness of the proposed method.

Planetary gearbox degradation experiment
The degradation process data of planetary gearbox is collected from a life-cycle experiment.The planetary gearbox experiment rig is shown in Fig. 10.The experiment rig consists of a test planetary gearbox, a drive motor, a speed and torque sensor, and a magnetic powder brake.The test gearbox is a one-stage planetary gearbox, as shown in Fig. 11 and Fig. 12, it comprises one ring gear, three planet gears and one sun gear.The gear parameters of single stage planetary gearbox are listed in Fig. 12 and Table 2.
In the life-cycle experiment, the input speed of planetary gearbox is about 1005 rpm (as shown in Fig. 13), the load provided by magnetic powder brake is about 340 Nm, and the vibration signal sampling frequency is 20 kHz for duration of 12 seconds.There are four accelerometers fitted onto the casing of planetary gearbox to record vibration signal, as shown in Fig. 11.The total experimental time is 1003 hours.As shown in Fig. 14, the abrasions of gears are very obvious after experiment.As shown in Fig. 15, in the first stage, the energy trends of sensor 1# and 3# is exactly similar, numerical difference is small and contact ratio is high.In the second stage, the energy increase speed of sensor 3# is much larger than sensor 1#, there are no longer overlap between the energies of sensor 1# and 3#.In the third stage, the energy fluctuation range of sensor 3# is also much larger than sensor 1#, the phenomenon shows that the energy of sensor 3# instantaneous increase and reduce is very obvious.It can be found from -axis in Fig. 16 that evident color part focuses on frequency band [625, 3750] Hz.This phenomenon shows that the energy in frequency band [625, 3750] Hz is higher than other frequency bands.In the Energram for whole degradation process (Fig. 16(a)), the color in frequency band [625, 3750] Hz is more and more deeply over time.This suggests that the energy is gradually rising with running time increase.The color of last 5 Δ in the Fig. 16(a) y-axis is most evident, and color depth fluctuates slightly.This phenomenon is completely corresponding to the third stage (the intense fluctuation stage) in energy trend.

Energram of sensor 3#
Fig. 17 is the Energram of sensor 3#, the relevant data is similar to Fig. 16.It can be found from -axis in Fig. 17    As a result, the conclusion can be gained that main frequency band contains most of total energy.Moreover, the trends of main frequency band energy and total energy are similarly, it also can be used for degradation prediction.As analyzed in section 4.2, the main frequency band of sensor 1# is [625, 3750] Hz and the main frequency band of sensor 3# is [1250, 5000] Hz (the low frequency band [0, 625] Hz is neglected).Take the intersection interval of main frequency bands [1250,3750] Hz to analysis.

Arithmetic Energram of sensor 3#
The energies of frequency band [1250, 3750] Hz for different sensor are shown in Fig. 22. Obviously, the energy values and energy trends of sensor 1# and 3# are so close.The only difference is that energy value of sensor 3# is a little bit more than sensor 1# in the third stage, but the energy trends and fluctuation ranges of both sensor 1# and 3# are the same.In contrast, as shown in Fig. 15, the increase amplitude of sensor 3# total energy is much more than sensor 1# total energy in the second stage.And the fluctuation range of sensor 3# total energy is significantly greater than sensor 1# total energy in the third stage.These phenomena show that the energy value and the energy trend of main frequency band energy are significant difference from total energy in planetary gearbox degradation process.
Therefore, the conclusion can be obtained that the values and trend of degradation feature index (such as energy) in a specific frequency band are very close, even if the sensor position is different.As shown in the energy trend of [1250, 1875] Hz (Fig. 23), it can be found that: (a) Although the energy fluctuation is large, there is not obvious acceleration stage in energy trend, and the energy trend in whole degradation process appears as linear.This phenomenon is closer to the physical truth of planetary gearbox degradation process.(b) Energy of sensor 1# is greater than sensor 3#, this phenomenon is inconsistent to the appearance that total energy of sensor 1# is smaller than sensor 3#.Through statistics found that a total of 4 groups of frequency bands appear this phenomenon in the 16 groups of frequency bands between [0, 10000] Hz, the other 3 groups of frequency bands are [7500, 8125] Hz, [8750, 9375] Hz and [9375, 10000] Hz, respectively.As a result, the conclusions can be gained that even in the same system degradation process, the change trends of different frequency band energies also differ.In the degradation prediction, we should choose the frequency band energy that reflects the system physical deterioration process better.
Xianglong Ni is the major authors.Jianmin Zhao, Qiwei Hu and Xinghui Zhang are advisers, they give many suggestions in data analysis.Haiping Li is the cooperator of planetary gearbox experiment.

Conclusions
As the planetary transmission system is widely used in modern industrial equipment, this paper is meant to investigate degradation feature analysis method for planetary transmission system.To this purpose, the new improved Kurtogram method (Energram and arithmetic Energram) for degradation analysis is put forward and researched.A detailed study of planetary gearbox degradation characteristics is expressed in this paper, which including Energram, arithmetic Energram and four particular comparative analyses.
The planetary gearbox degradation feature analysis based on new improved Kurtogram, which is proposed in this study, shows that Energram and arithmetic Energram can effective analysis the planetary gearbox degradation process from frequency domain and time domain at the same time.
Moreover, there are several valuable points for engineering can be obtained: (a) The meaning of degradation trend prediction.Fully understand the change trends of each frequency band energy, it is useful for finding and extracting appropriate feature index to respond system degradation trend, and it is conducive to remaining useful life prediction.For example, linear trend energy is more befitting than three stages trend energy for system degradation prediction.
(b) The meaning of sensor layout in condition monitoring.The feature indexes in a certain frequency band for different position sensor signals have highly similar trends, such as Fig. 22 shows.It helps to solve some problems in sensor installation.For instance, in the engineering, the key position of a component cannot install sensor, position can install sensor easily, and the trends of feature indexes for position signal and position signal are very close in a certain frequency band, so feature index trend of position signal can be used to instead of feature index trend of position signal in the certain frequency band.The problem is that we should find the certain frequency band by Energram and arithmetic Energram at first.
(c) The meaning of the requirements for hardware equipment in condition monitoring.If the suitable frequency band [ , ] is known, which can be used to extract appropriate feature index to respond system degradation trend, the sensor performance parameter range must cover the suitable frequency band [ , ] in hardware selection.And the sampling frequency is not necessary to set too large in condition monitoring, just makes sure that the sampling frequency is greater than double (set as Nyquest frequency).As a result, this reduces the hardware performance parameter requirements in signal storage and signal analysis, and it is very meaningful for engineering.

Fig. 4 . 5 .
Fig. 4. Flow chart of Energram Fig. 5. Flow chart of arithmetic EnergramTake the second group of Intelligent Maintenance Systems Center (IMS) bearing life-cycle data as an example to illustrate the generation process of Energram.This group of bearing whole life data is a total of 984 set data, and the sampling frequency is 20 kHz for duration of 1.024 seconds, namely = 20 kHz.The Energram of the 850th set of data is shown in Fig.6.According to Table1, the frequency domain [0, /2] is divided into 16 groups of frequency bands at the 7th level, and the frequency bandwidth is 625 Hz.Extracting all the 7th level Energram of the (850-984)th set of data to form

Fig. 17 .
Fig.17is the Energram of sensor 3#, the relevant data is similar to Fig.16.It can be found from -axis in Fig.17that evident color part focuses on frequency band [0, 625] Hz and [1250, 5000] Hz.This phenomenon shows that the energy is mainly distributed in frequency band [0, 625] Hz and [1250, 5000] Hz.Compared with sensor 1#, the significant difference of sensor 3# energy is the appearance in low frequency band [0, 625] Hz.Moreover, color depth changes in frequency band [0, 625] Hz is the biggest as time goes on.It shows that energy fluctuation in this frequency band is much bigger than any other frequency band in the whole degradation process.

Fig. 19 Fig. 19 . 4 . 4 . 2 .
Fig. 19 is the arithmetic Energram of sensor 3#.It can be found from Fig. 19(b) that the energy fluctuation in the first stage is mainly caused by the energy fluctuations in frequency band [1250, 3750] Hz.As shown in Fig.19(c) and Fig.19(d), it is obviously that the energy fluctuation in the second stage and the third stage is primary because of the energy fluctuations in frequency band [0, 625] Hz.The change of energy in other frequency band is much smaller.That is to say, if filter out the [0, 625] Hz signal components from original signal, the energy fluctuation range will be reduced and the energy stability characteristics will be improved.

Table 2 .
Gear parameters of single stage planetary gearbox Gear Sun gear Ring gear Planet gear Planet gear number