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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Planetary gearboxes exhibit complicated dynamic responses which are more difficult to detect in vibration signals than fixed-axis gear trains because of the special gear transmission structures. Diverse advanced methods have been developed for this challenging task to reduce or avoid unscheduled breakdown and catastrophic accidents. It is feasible to make fault features distinct by using multiwavelet denoising which depends on the feature separation and the threshold denoising. However, standard and fixed multiwavelets are not suitable for accurate fault feature detections because they are usually independent of the measured signals. To overcome this drawback, a method to construct customized multiwavelets based on the redundant symmetric lifting scheme is proposed in this paper. A novel indicator which combines kurtosis and entropy is applied to select the optimal multiwavelets, because kurtosis is sensitive to sharp impulses and entropy is effective for periodic impulses. The improved neighboring coefficients method is introduced into multiwavelet denoising. The vibration signals of a planetary gearbox from a satellite communication antenna on a measurement ship are captured under various motor speeds. The results show the proposed method could accurately detect the incipient pitting faults on two neighboring teeth in the planetary gearbox.

Accurate fault detection of planetary gearboxes is important to reduce unscheduled machine downtime and avoid catastrophic accidents [

With a special gear transmission structure, planetary gearboxes exhibit complicated dynamic responses which are more difficult to detect than fixed-axis gear trains [

Up to now, researchers have proposed a few interesting methods based on advanced signal processing techniques for detecting planetary gearbox faults. Blunt and Keller [

In summary, researches on planetary gearbox fault diagnosis have only focused on the condition monitoring and fault classifications. Studies on weak feature detections of incipient faults are rare and these weak features are always immersed in noises generated by the equipment and the surrounding environment. It is significant to detect weak fault features as early as possible, which is a complicated and challenging task that requests advanced analytical methods with high reliability, high accuracy and high efficiency.

The emerging notion of multiwavelet transform (MWT), which uses vector-valued scaling and wavelet functions, is an important development of the wavelet theory. Multiwavelets possess excellent properties of orthogonality, symmetry, compact support and high vanishing moments simultaneously [

To overcome the limitations of standard or fixed MWTs, integrating multiwavelets with lifting schemes (LS) is an exciting motivation to construct customized multiwavelets with desired properties. LS, introduced by Sweldens [

It is a challenging task to detect weak features of incipient faults, which are always immersed in heavy noises generated by the surrounding environment or the equipment. Multiwavelet denoising plays an important role in eliminating noise as much as possible. Its effect mainly depends on the feature separation by using multiwavelets and the threshold denoising. A redundant multiwavelet possesses the time invariant property [

In this paper, a method which incorporates the customized multiwavelets and INC is proposed for fault detections of planetary gearboxes. The experimental results show that the proposed method is effective and promising to detect these weak impulse features. The rest of the paper is organized as follows: The theory of multiwavelets and the symmetric lifting schemes are briefly introduced in Section 2. In Section 3, the redundant symmetric lifting scheme is proposed to construct customized multiwavelets and the improved neighboring coefficients is introduced into multiwavelets denoising. In Section 4 experimental results are performed. The conclusions are summarized in Section 5.

Like scalar wavelets, the theory of multiwavelets is based on multiple resolution analysis (MRA) [_{1}_{2}_{r}^{T}, where _{j}

The vector functions _{1},_{2}, …, _{r}]^{T} are multiwavelet functions, _{k}_{k}_{1}_{j},_{k}_{2}_{j},_{k}^{T} and (_{1}_{j},_{k}_{2}_{j},_{k}^{T} can be obtained:

Similarly, multiwavelet reconstruction can be obtained by:
_{k}_{k}_{k}_{k}

Decomposition and reconstruction of MWT can be represented in _{1,0,}_{k}_{1,−1,}_{k}_{1,}_{j}_{0,}_{k}_{1,}_{j}_{0,}_{k}_{2,0,}_{k}_{2,−1,}_{k}_{2,}_{j}_{0,}_{k}_{2,}_{j}_{0,}_{k}

Different from scalar wavelets, multiwavelets require two or more input streams because of their matrix filter banks. Usually, there is only one input stream _{k}

Although multiwavelets have several advantages over scalar wavelets, fixed or standard multiwavelets are usually not the optimal for specified applications because they are usually independent of a measured signal [

Lifting scheme [

[_{new}_{new}_{new}_{new}_{new}^{2}

According to the two-scale relations and

Lifting scheme can be used to improve the existing multiwavelets. The obtained multiwavelets are considered ideal since they have a desirable vanishing moment. The key of this method is the design of the lifting matrices

The ^{n}dx^{n}dx

Hence, the construction of a wavelet with specified vanishing moments is now a straightforward procedure. An initial wavelet _{0}(_{0}(_{1} or _{2}) is chosen and a set of _{1}(_{k}_{0}(

One of the most attractive advantages in multiwavelet lifting is that the functions using to construct new multiwavelets are more than scalar wavelets. When a scalar wavelet is lifted, _{i}_{1}, _{i}_{2},_{1},_{2}} and for _{2}, _{i}_{1},_{2}}. Obviously, there are more kinds of “brick” in constructing new wavelets. In other words, there are more degrees of freedom in multiwavelet lifting scheme.

Supposing that the vanishing moment

The integrals can be calculated through _{i}

Symmetry is an important property for multiwavelets. It could ensure the filter banks possess linear phase or generalized linear phase, which avoids the reconstruction error. However, the algorithm described above could not ensure the symmetry of multiwavelet lifting scheme. To guarantee the symmetry, a “symmetric selection” method is adopted to select the translation k of multiwavelet functions. Suppose the multiscaling functions _{1},_{2} and multiwavelets _{1},_{2} are symmetric or anti-symmetric about the points _{ϕ,}_{1},_{ϕ}_{2},_{Ψ,}_{1},_{Ψ}_{2} respectively. Thus, the “symmetric selection” is shown as below [

Take _{1} for example:

Suppose _{ϕ}_{1}, _{ϕ}_{2}, _{Ψ}_{1}, _{Ψ}_{2} = ±1 (1 means symmetry and -1 means anti-symmetry) stand for the symmetry of multiple scaling functions and multiwavelets respectively. ^{n}dx^{n}dx

Let _{B}_{Ψ1}_{ϕi}_{,1}, …,_{Ψ1}_{Ψ2}_{,1},… ]^{T} and _{Ψ}_{i}_{i}_{i}^{T}, then:

The solutions of _{1}. The lifting of _{2} is similar to _{1} except adopting _{1} and _{2} only. The lifting coefficients are substituted into the “lifting coefficients equation”. Then the equation is performed z-transform and the presentation of the lifting scheme is obtained as follows:

New symmetric biorthogonal multiwavelets are constructed with the help of the lifting matrices _{B}C_{ψ}

Discrete multiwavelet transform (DMWT) is essentially a decimated multiwavelet transform. It is an ideal tool for non-stationary signal processing, while there are still several limitations. First, the decomposition results of DMWT are time-variant due to down sampling. A forward or backward translation of the original signal will generate different decomposition results [

RMWT is time-invariant, which is beneficial to the feature extraction of periodic impulses. Moreover, RMWT supply more abundant features and more precise frequency localizations, which are beneficial for mechanical fault detections. The decomposition of RMWT is shown in ^{j}^{j}_{i}_{i}

Then the filters of redundant multiwavelet can be calculated by the following equations:

Cai and Silverman [_{0} is a constant, it should be selected according to the signal duration of features and the support of wavelet filters.

The INC algorithm is shown in _{j}_{j}

In this paper, redundant symmetric lifting scheme is applied to construct customized multiwavelets with specified properties for specific signals. Then the lifting coefficients of

We construct new multiwavelet starting from Hermite splines [

There are _{f}_{B}

Kurtosis is widely used for fault feature detections because it is sensitive to sharp variant structures, such as impulses. The bigger the impulses in signals, the larger the kurtosis [_{i}_{p}

In dynamic response signals, the mechanical fault often expresses as periodic impact features which can be detected through the envelope spectrum of the vibration signal. Hence, the envelope spectrum entropy is selected as an evaluation indicator to obtain the customized multiwavelets.

Assume {_{i}_{i}_{= 1}_{,}_{…,}_{M}_{i}_{i}_{= 1}_{,}_{…,}_{M}

The multiwavelet entropy _{mwt}

According to the information theory, the most uncertain probability distribution has the maximum entropy value, and the entropy value reflects the uniformity of the probability distribution. So _{mwt}_{mwt}_{mwt}

In order to improve the limitation of kurtosis, the proposed method chooses

Genetic algorithms (GAs) are based on the idea of natural selection. The major advantages are their flexibility and robustness as an adaptive global search method. GAs can deal with highly nonlinear problems and non-differentiable functions, as well as functions with multiple local optima. They are parallel implementation in nature. Thus, they are utilized as the tool to optimize free parameters. According to our experimental experience, GA parameters are set as follows: arithmetic crossover and non-uniform mutation operators are adopted, the range of the parameter is chosen to be [–3,3] except 0, the population scale is set to 50, the number of iteration to 30, the probability of crossover to 0.6 and the probability of mutation to 0.05.

Preprocessing method is performed to translate the one-stream input signal into multiple streams.

The multiple streams are decomposed by using the customized multiwavelets.

Apply INC to shrink the multiwavelet coefficients.

The thresholded multiwavelet coefficients are reconstructed.

Post-processing method is performed to translate the multiple streams into one-stream. The denoising result is obtained to detect the fault features.

In the process of the customized multiwavelet cnostructions, optimization is an important step for modifying the coefficients to match the signal. Suppose the free parameters are {_{1},_{2},…_{Nf}

Initialize the free parameters {_{1},_{2},…_{Nf}

Substitute {_{1},_{2},…_{Nf}_{B}_{B}

Compute the lifting coefficients and get

Decompose the signal with new multiwavelet.

Compute

Generate the more optimal values of the free parameters and return to step 2), otherwise finish the computation.

Planetary gearboxes play an important role in the transmission train of a satellite communication antenna or a telemetry, tracking, and command (TT&C) antenna for the aerospace industry. The aerospace measurement ship is mainly responsible for the maritime measurement and control, communication, salvage and recovery of spacecrafts. Satellite communication antennas (SCA) are critical devices of a measurement ship to support voice, data, fax and video integration services. An SCA comprises three axes, which are the azimuth axis, the pitching axis and the crossing axis. Searching satellite is that makes an SCA circumrotate the azimuth axis, the pitching axis and the crossing axis with controllers to change these axis angles, so the antenna can point itself to different satellite in the light of demands. If the SCA direction departs from a satellite or the satellite makes an excursion, it can adjust the SCA to track the satellite signal automatically.

The testing framework is shown in

The planetary gearbox of the azimuth axis transmission train is a two-stage gearbox. The carrier of the first stage works as the input shaft of the second stage. The parameters of the planetary gearbox are shown in

With the special gear transmission structure, the transmission ratio is calculated using the primary principle of the conversion mechanism method. The first stage in the planetary gearbox has a transmission ratio 8. The meshing frequency of the first stage is calculated by using _{mesh}_{in}_{out}_{sun}_{in}

Vibration signals of a normal planetary gearbox in the left transmission train of the azimuth axis were acquired. They are applied to verify the effectiveness of the propose method. The sampling frequency and signal length were 12.8 kHz and 5,760, respectively. The rotating speed of the motor was 150 r/m. The signal in time domain is shown in

The proposed method chose the customized multiwavelet by using redundant symmetric lifting scheme and the optimal threshold by using INC and it was applied to the measured signal. The signal was decomposed into four levels. As shown in

According to the kinetic principles of planetary gearboxes, both contact conditions by means of rolling and sliding exist on the gear tooth when meshing. The force of sliding friction changed its direction at the meshing points, which caused the shocking line impacts. Moreover, when each pair of gear tooth got in or out of contact, the load and deformation of each gear increased or decreased suddenly, causing the meshing shocks. The dynamic load consisted of shocking line impacts and meshing shocks, which caused the meshing vibration of gears. Therefore, the meshing vibration in

The Kurtogram [

The left transmission train of the azimuth axis had a slight abnormal sound when the measurement ship was sailing in the sea. The vibration signals were measured at a sampling frequency of 12.8 kHz from the measuring points on the planetary gearbox by using ICP acceleration sensors. The AC motor was running at 255 r/m. The signal in time domain is shown in

The denoising result of the proposed method is shown in

^{2}. The field inspection verifies the fault diagnosis conclusion.

The Kurtogram and the resulting signals of spectral kurtosis are illustrated in

From the planetary gearbox fault detections, it can be seen that multiwavelet denoising methods can effectively eliminate noise and improve the signal-to-noise ratio so as to outstand the fault features. Besides, the denoising methods using customized multiwavelets [shown in

The computing times of these methods in planetary gearbox fault detections are listed in

As shown in

As key components of the transmission train, planetary gearboxes play an important role in guaranteeing the normal operation of the satellite communication antenna. With their special gear transmission structure, planetary gearboxes exhibit complicated dynamic responses, which increase the difficulty of fault feature extractions for planetary gearboxes. It is proved that the customized multiwavelets which are similar to fault features and have excellent properties can achieve a good result in fault feature detections. The redundant symmetric lifting scheme is applied to produce customized multiwavelet functions. Moreover, the quotient of kurtosis and entropy is proposed to choose the optimal multiwavelets. On the basis of the local concentrated energy, the improved neighboring coefficients choose variant thresholds and sizes of neighbors at different decomposition levels. The proposed method incorporated customized multiwavelets and INC threshold. It was applied to the planetary gearbox fault detections. Experimental results showed that the proposed method could detect the pitting fault features on two neighboring teeth of the sun gear in a planetary gearbox.

This research is financially supported by the Project of National Natural Science Foundation of China (No. 51275384, 51035007), National Basic Research Program of China (No. 2009CB724405), Research Fund for the Doctoral Program of Higher Education of China (No. 20110201130001), Important National Science and Technology Specific Projects (No. 2010ZX04014-016) and Program for Changjiang Scholars and Innovative Research Team in University.

Schematic of an elementary planetary gear set having three planet gears.

(

The decomposition of redundant multiwavelet transform. (

The algorithm of improved neighboring coefficients.

Hermite spline multiscaling functions and multiwavelet functions. (

The flow chart of the proposed method.

The flow chart of the customized multiwavelets.

The transmission mechanism of the azimuth axis.

The testing framework of the SCA of a measurement ship.

(

The denoising results of the normal planetary gearbox signal using the customized multiwavelet. (

The denoising results of the normal planetary gearbox signal using GHM multiwavelet. (

The analyzed results of the normal planetary gearbox signal using spectral kurtosis. (

(

The denoising results of the vibration signal using the proposed method.

The denoising results of the planetary gearbox signal. (

The defects in the sun gear of the planetary gearbox.

The analyzed results of the defective planetary gearbox signal using spectral kurtosis. (

The Fourier spectrum of the denoising result using the proposed method.

The parameters of acceleration sensors.

333B32 | 100 mv/g | ±50,g pk | 0.5–3,000 Hz | ≤1% | Caremic |

The parameters of the planetary gearbox.

Type | The first stage | Sun gear tooth number | 12 |

Planet gear tooth number | 36 | ||

Ring gear tooth number | 84 | ||

PLS142-32 | The second stage | Sun gear tooth number | 28 |

Planet gear tooth number | 28 | ||

Ring gear tooth number | 84 |

The computing time of the methods in planetary gearbox fault detections.

Customized multiwavelets with INC | 156.05 ( |
156.05 ( |

Customized multiwavelet with NC | 155.18 ( |
155.18 ( |

GHM multiwavelet with INC | 3.79 | 3.79 |

GHM multiwavelet with NC | 3.71 | 3.71 |

The time in italics represents the cost for the optimization of the customized multiwavelets.