Balance Sparse Decomposition Method with Nonconvex Regularization for Gearbox Fault Diagnosis

Abstract

As an important part of rotating machinery, gearboxes often fail due to their complex working conditions and harsh working environment. Therefore, it is very necessary to effectively extract the fault features of the gearboxes. Gearbox fault signals usually contain multiple characteristic components and are accompanied by strong noise interference. Traditional sparse modeling methods are based on synthesis models, and there are few studies on analysis and balance models. In this paper, a balance nonconvex regularized sparse decomposition method is proposed, which based on a balance model and an arctangent nonconvex penalty function. The sparse dictionary is constructed by using Tunable Q-Factor Wavelet Transform (TQWT) that satisfies the tight frame condition, which can achieve efficient and fast solution. It is optimized and solved by alternating direction method of multipliers (ADMM) algorithm, and the non-convex regularized sparse decomposition algorithm of synthetic and analytical models are given. Through simulation experiments, the determination methods of regularization parameters and balance parameters are given, and compared with the L1 norm regularization sparse decomposition method under the three models. Simulation analysis and engineering experimental signal analysis verify the effectiveness and superiority of the proposed method.

1 Introduction

As an important power transmission component in industrial systems and rail transit systems, gearboxes play the role of transmitting power and torque, and their operating conditions are directly related to the health of industrial systems and the service performance of high-speed trains. Due to the complex processing technology, high assembly precision requirements and harsh working environment, if the gearbox fails, it will cause damage to the entire rotating mechanical system, and even cause greater economic losses and casualties [1, 2]. Therefore, under the condition of strong noise and other related disturbances, it is of great significance to study and realize the effective diagnosis of gearbox faults for the condition monitoring of rotating mechanical systems. Many scholars have done a lot of research and proposed a series of fault feature extraction methods for gearbox vibration signals, mainly including wavelet transform [3, 4], empirical mode decomposition [5], spectral kurtosis [6], and deep learning [7,8,9], etc. The sparse decomposition method utilizes the sparse feature of the fault signal, thereby promoting the reliability of fault feature extraction, and succinctly representing the fault signal with fewer sparse coefficients. In recent years, sparse decomposition method has been widely studied in gearbox fault feature extraction.

The sparse decomposition method can obtain the most concise representation of the signal by using the sparse performance of the fault characteristic signal, which can further mine the characteristics of the signal, and provide a potential solution for the analysis of the gearbox fault vibration signal [10,11,12,13]. Compared with the other methods mentioned above, the sparse decomposition method can optimize and match the intrinsic structural features of the fault signal to design a sparse dictionary and penalty function that are consistent with the signal, and show good anti-noise performance [12, 14, 15]. At present, researchers have done a lot of work on sparse decomposition methods. Fan et al. [16] constructed an overcomplete dictionary based on the Laplace wavelet basis, and used the L1 norm as a regular term to effectively extract the weak fault features of bearings. Huang et al. [17] proposed to construct a transformation dictionary based on discrete Fourier transform (DFT) and short-time Fourier transform (STFT), and used the MC non-convex penalty function as the penalty term to accurately extract the transient components of gear faults in gearbox faults. Wang et al. [18] proposed to combine the TQWT dictionary with the generalized minimax-concave (GMC) penalty function to realize bearing fault diagnosis and achieve good extraction results. Huang et al. [19] proposed a multi-source sparse decomposition method and successfully applied it to gearbox fault diagnosis, which can accurately achieve fault extraction without prior knowledge of the type and number of gearbox faults. Zhang et al. [20] proposed a generalized logarithmic regularization sparse decomposition method, which was realized the accurate extraction of bearing fault features under the background of strong noise.

The above studies show that the sparse decomposition dictionary and penalty function of the signals play key roles in the modeling of sparse signals. In the construction of sparse decomposition model, according to the difference of sparse prior, we can divide into synthesis prior model, analysis prior model and balance prior model [21, 22]. In the past studies, sparse decomposition methods based on synthesis models have received extensive attention, and most of the above studies are constructed based on synthesis model. However, there are few studies based on analysis model and balance model. In recent years, analysis models have gradually gained attention. Wang et al. [23] discussed the performance of synthesis and analysis models in mechanical fault diagnosis, and the results show that the synthesis model has better extraction performance than the analytical model. At present, the balance model has not been studied in mechanical fault diagnosis, but the effectiveness of the balance method has been verified in image processing [22, 24].

In this paper, we investigate the performance of the balance model in gearbox fault diagnosis. Based on the balance model and the arctangent nonconvex penalty function method, a balance nonconvex regularized sparse decomposition method is proposed. We investigate TQWT decomposition dictionaries that satisfy the compact frame condition and do not involve matrix inversion when optimizing interpretation. Based on ADMM algorithm, a balanced non-convex regularized sparse decomposition algorithm is obtained, and a non-convex regularized sparse decomposition algorithm under synthesis and analysis models are given. Based on multiple sets of simulation signals, a strategy for determining regularization parameters and balance parameters is proposed. Simulation and engineering experimental analysis verify the effectiveness and generality of the proposed method.

2 Balance Sparse Decomposition Model

For gearbox fault vibration signals, the measured vibration signals $y \in R^{N}$ often contain various components, such as transient impact components, harmonic components and noise interference components [25]. Therefore, the gearbox vibration signals $y$ can be described as

$$y = t + h + e,$$

(1)

where $t \in R^{N}$ is the transient impulse component of the fault, $h \in R^{N}$ is the meshing component and its harmonic components, $e \in R^{N}$ represents the noise component. Assuming that there is a sparse coefficient vector that satisfies the condition $t = Dc$, then the signal $y$ is said to have sparse characteristics on the known frame $D \in R^{N \times M}$, where $c$ is called the frame coefficient. In this paper, a redundant and normalized tight frame is used, i.e., $DD^{{\text{T}}} = I$, where $I$ is the identity matrix.

In this case, the vibration signal $y$ can be expressed as

$$y\,{ = }\,DD^{{\text{T}}} y = D\left( {D^{{\text{T}}} y} \right),$$

(2)

where $D$ is the synthesis operator, $D^{{\text{T}}}$ is the analysis operator, $D^{{\text{T}}} y$ is the canonical coefficients. Since $D$ is a redundant tight frame, the sparse representation coefficient $c$ is not unique. In order to extract the transient feature component $t$ from the noisy signals $y$, the common way is to penalize the sparsity of the frame coefficients $c$ or the canonical coefficients $D^{{\text{T}}} y$. A sparse least-squares cost function is constructed, and the above problem is transformed into an optimization problem of least-squares function.

When the frame coefficients are penalized, the corresponding regularized least squares problem can be expressed as

$$\mathop {\min }\limits_{{c_{1} ,c_{2} }} \left\{ {\frac{1}{2}\left\| {y - D_{1} c_{1} - D_{2} c_{2} } \right\|_{2}^{2} + \sum\limits_{i = 1}^{2} {\lambda_{i} \varphi \left( {c_{i} } \right)} } \right\},$$

(3)

where $\lambda_{i}$ are regularization parameters, and balance the relationship between the fidelity term and the penalty term, $\varphi \left( \cdot \right)$ is the regularization function, and penalized frame coefficients $c$. The notation D₁ and D₂ represents two dictionaries which can steer the interested signal to the corresponding sparse coefficients vectors c₁, c₂. The above formula is a sparse regularization problem based on the synthetic model. By minimizing the above formula to obtain the reconstructed frame coefficient $\hat{c}$, the fault transient component can be reconstructed as $\hat{t} = D\hat{c}$.

When the canonical coefficients of the analytical operators are penalized, the regularized least squares problem is expressed as

$$\mathop {\min }\limits_{t,h} \left\{ {\frac{1}{2}\left\| {y - t - h} \right\|_{2}^{2} + \lambda_{1} \varphi \left( {D_{{1}}^{{\text{T}}} t} \right) + \lambda_{2} \varphi \left( {D_{{2}}^{{\text{T}}} h} \right)} \right\},$$

(4)

where the canonical coefficients $D_{{1}}^{{\text{T}}} t$ and $D_{2}^{{\text{T}}} h$ are penalized. The above problem is known as the sparse regularization problem based on analysis models.

The balanced model penalizes not only the frame coefficients, but also the distance between the frame coefficients and the canonical coefficients. The sparse regularized least squares problem is expressed as

$$\begin{gathered} \mathop {\min }\limits_{{c_{1} ,c_{2} }} = \frac{1}{2}\left\| {y - D_{1} c_{1} - D_{2} c_{2} } \right\|_{2}^{2} \hfill \\ \, + \sum\limits_{i = 1}^{2} {\frac{{\beta_{i} }}{2}\left\| {\left( {I - D_{i}^{{\text{T}}} D_{i} } \right)c_{i} } \right\|_{2}^{2} + \sum\limits_{i = 1}^{2} {\lambda_{i} \varphi \left( {c_{i} } \right)} }, \hfill \\ \end{gathered}$$

(5)

where $\beta_{i}$ are non-negative balance parameters. For an orthogonal frame, i.e., $D^{{\text{T}}} D{ = }I$, the above three models are equivalent. For redundant frameworks, the three models cannot be derived from each other.

When the balance parameters $\beta_{i} { = }0$, the balance model is simplified to a synthetic model. When the balance parameters $\beta_{i} { = }\infty$, if there is a solution to Eq. (5), then $\left\| {\left( {I - D_{{\text{i}}}^{{\text{T}}} D_{i} } \right)c_{i} } \right\|_{2}^{2}$ must be 0. This means that the frame coefficient $c_{i}$ are affine of $D_{i}^{{\text{T}}}$, i.e., $c = D_{{}}^{{\text{T}}} t$. So, Eq. (5) can be expressed as

$$\begin{gathered} \mathop {\min }\limits_{{c \in Range(D^{{\text{T}}} )}} \left\{ {\frac{1}{2}\left\| {y - D_{1} c_{1} - D_{2} c_{2} } \right\|_{2}^{2} + \sum\limits_{i = 1}^{2} {\lambda_{i} \varphi \left( {c_{i} } \right)} } \right\} \hfill \\ \, = \mathop {\min }\limits_{t,h} \left\{ {\frac{1}{2}\left\| {y - t - h} \right\|_{2}^{2} + \lambda_{1} \varphi \left( {D_{1}^{{\text{T}}} t} \right) + \lambda_{2} \varphi \left( {D_{{2}}^{{\text{T}}} h} \right)} \right\}. \hfill \\ \end{gathered}$$

(6)

In this way, the balance model is transformed into an analysis model. Through the above analysis, we can see that the balance model connects the synthetic model and the analytical model. By adjusting the balance parameters $\beta_{i}$, the balance model can be approximated to the other two models. Figure 1 shows the relationship between these three models.

Figure 1

The relation of analysis, synthesis, and balanced models

3 Balance Nonconvex Regularized Sparse Decomposition Optimization Method

This paper mainly analyzes the performance of the balance method in fault component extraction. In this part, we first introduce the relevant principles of TQWT, and then use the non-convex penalty function to construct a sparse decomposition model. Combined with the balanced model and arctangent penalty function, a balanced non-convex regularization sparse optimization method is proposed, and the solution flow of the other two models is given.

3.1 TQWT

Ideally, in the field of signal processing, the Q factor of the wavelet transform should be determined by the oscillation characteristics of the signal. Specifically, when using wavelet transform to analyze signals with high oscillation characteristics (harmonics, speech signals), the wavelet transform should select a high Q factor; when using wavelets to analyze signals with low or no oscillation characteristics (instantaneous state pulse signal), the wavelet transform should choose a lower Q factor. However, with the exception of continuous wavelet transforms, most wavelet transforms do not provide the ability to adjust the wavelet Q-factor.

This section presents a TQWT designed for discrete-time signals whose properties are determined by the parameter Q-factor and the oversampling rate (redundancy). TQWT is proposed to be applied to discrete-time signal analysis, which can be efficiently implemented with FFT, and moderate oversampling rate (3–4 times) can make the analysis performance of TQWT well exhibited [26, 27]. TQWT is close to the traditional binary discrete wavelet transform, with perfect reconstruction characteristics and moderate overcompleteness, and is a simple and reversible transform. TQWT satisfies the tight frame condition, which designed based on an iterative two-channel filter, and implemented using DFT. It consists of an iterative two-channel filter bank consisting of a decomposition filter and a synthesis filter, as shown in Figure 2. It is a TQWT with a decomposition level of 3, consisting of three filter banks. The filter is further decomposed into two groups: a high pass filter and a low pass filter in the low-pass channel after each decomposition level. w{1}, w{2}, and w{3} are low-pass channels, while w{4} is a high pass channel. The main parameters of TQWT are: Q factor, which affects the wavelet oscillation characteristics, redundancy $r$ and the number of stages of decomposition $J$.

Figure 2

Decomposition level 3 TQWT

The above three parameters are the main parameters of TQWT. The Q factor is a parameter to measure the number of wavelet oscillations, which is the ratio of the center frequency and the bandwidth, and is specifically described as

$$Q = \frac{{f_{0} }}{BW}.$$

(7)

The redundancy $r$ is the ratio of the sum of the wavelet coefficients to the length of the signal processed with TQWT. Redundancy must be greater than 1. The wavelet decomposition level $J$ represents the number of filter banks. Each output signal will constitute a self-contained wavelet transform, so there will be J+1 subbands. The specific value of J can be set as

$$J_{\max } = \left\lfloor {\frac{{\lg \left[ {N/4\left( {Q + 1} \right)} \right]}}{{\lg \left[ {\left( {Q + 1} \right)/\left( {Q + 1 - 2/r} \right)} \right]}}} \right\rfloor,$$

(8)

where $N$ is the signal length, $\left\lfloor \cdot \right\rfloor$ is the round down symbol.

3.2 Sparse Decomposition Optimization Algorithm

As mentioned in the previous section, TQWT is a redundant linear transform in a tight frame. This paper studies TQWT as frame $D$, which can realize the derivation between the three models.

A balance non-convex regularized sparse decomposition cost function based on the balance model and arctangent non-convex penalty function is constructed for the characteristics of the gearbox composite fault signal. The cost function expression is as follows:

$$\begin{gathered} L\left( {c_{1} ,c_{2} } \right) = \frac{1}{2}\left\| {y - D_{1} c_{1} - D_{2} c_{2} } \right\|_{2}^{2} \hfill \\ \, + \sum\limits_{i = 1}^{2} {\frac{{\beta_{i} }}{2}\left\| {\left( {I - D_{i}^{{\text{T}}} D_{i} } \right)c_{i} } \right\|_{2}^{2} } + \sum\limits_{i = 1}^{2} {\lambda_{i} \varphi \left( {c_{i} ;a_{i} } \right)}, \hfill \\ \end{gathered}$$

(9)

where $\beta_{i}$ represent the balance parameters of different components, $\varphi \left( {c_{i} ;a_{i} } \right)$ represents the arctangent non-convex penalty function [28], defined as

$$\varphi \left( {c;a} \right) = \frac{2}{{a\sqrt 3 }}\,\left( {{\text{arc}}\tan \,\left( {\frac{{1 + 2a\left| x \right|}}{{\sqrt 3 }}} \right) - \frac{\pi }{6}} \right),$$

(10)

where $a_{i}$ is the non-convexity parameter of the arctangent and satisfies the following conditions

$$0 \le a_{i} \le \frac{1}{{\lambda_{i} }}.$$

(11)

Next, we utilize the alternating direction method of multipliers (ADMM) algorithm to optimize this nonconvex regularized sparse decomposition problem. For the minimization optimization solving problem of Eq. (9), first use the variable separation technique to rewrite Eq. (9) as a constrained optimization problem

$$\begin{gathered} \arg \mathop {\min }\limits_{{c_{1} ,c_{2} ,u_{1} ,u_{2} }} g_{1} \left( {c_{1} ,c_{2} } \right) + g_{2} \left( {u_{1} ,u_{2} } \right), \hfill \\ \begin{array}{*{20}c} {} & {} & \text{s.t.}, \\ \end{array} \begin{array}{*{20}c} { \, c_{1} = u_{1} ,c_{2} = u_{2} }, & {} \\ \end{array} \hfill \\ \end{gathered}$$

(12)

where $g_{1} \left( {c_{1} ,c_{2} } \right)$ and $g_{2} \left( {u_{1} ,u_{2} } \right)$ are defined as

$$g_{1} \left( {c_{1} ,c_{2} } \right){ = }\frac{1}{2}\left\| {y - D_{1} c_{1} - D_{2} c_{2} } \right\|_{2}^{2} + \sum\limits_{i = 1}^{2} {\frac{{\beta_{i} }}{2}\left\| {\left( {I - D_{{\text{i}}}^{{\text{T}}} D_{i} } \right)c_{i} } \right\|_{2}^{2} },$$

(13)

$$g_{2} \left( {u_{1} ,u_{2} } \right){ = }\sum\limits_{i = 1}^{2} {\lambda_{i} \varphi \left( {c_{i} ;a_{i} } \right)}.$$

(14)

Then, the ADMM algorithm is used to obtain the iterative solutions of the following sub-problems

$$c_{1} ,c_{2} \leftarrow \arg \mathop {\min }\limits_{{c_{1} ,c_{2} }} g_{1} \left( {c_{1} ,c_{2} } \right) + \sum\limits_{i = 1}^{2} {\frac{\mu }{2}\left\| {u_{i} - c_{i} - d_{i} } \right\|_{2}^{2} },$$

(15)

$$u_{1} ,u_{2} \leftarrow \arg \mathop {\min }\limits_{{u_{1} ,u_{2} }} g_{2} \left( {u_{1} ,u_{2} } \right) + \sum\limits_{i = 1}^{2} {\frac{\mu }{2}\left\| {u_{i} - c_{i} - d_{i} } \right\|_{2}^{2} },$$

(16)

$$d_{i} \leftarrow d_{i} - \left( {u_{i} - c_{i} } \right),i = 1,2,$$

(17)

where $\mu$ satisfies $\mu > 1/\rho$. To optimize the minimization problem of $c_{1}$ and $c_{2}$ in Eq. (15), the partial derivatives of $c_{1}$ and $c_{2}$ can be obtained respectively, and we get

$$\begin{gathered} c_{1} = \left[ {\frac{1}{{\mu + \beta_{1} }}I - \frac{{1 - \beta_{1} }}{{\left( {\mu + \beta_{1} } \right)\left( {1 + \mu } \right)}}D_{1}^{{\text{T}}} D_{1} } \right] \hfill \\ \, \times \left[ {D_{1}^{{\text{T}}} \left( {y - D_{2} c_{2} } \right) + \mu \left( {u_{1} - d_{1} } \right)} \right], \hfill \\ \end{gathered}$$

(18)

$$\begin{gathered} c_{2} = \left[ {\frac{1}{{\mu + \beta_{2} }}I - \frac{{1 - \beta_{2} }}{{\left( {\mu + \beta_{2} } \right)\left( {1 + \mu } \right)}}D_{2}^{{\text{T}}} D_{2} } \right] \hfill \\ \, \times \left[ {D_{2}^{{\text{T}}} \left( {y - D_{1} c_{1} } \right) + \mu \left( {u_{2} - d_{2} } \right)} \right]. \hfill \\ \end{gathered}$$

(19)

Eq. (16) is separable for the minimization problem of $u_{1}$ and $u_{2}$, and the minimization problem of $u_{1}$ and $u_{2}$ is solved by the proximity operator, we can get

$$u_{1} = thresh\_\arctan \left( {c_{1} + d_{1} ;\lambda_{1} /\mu ,a_{1} } \right),$$

(20)

$$u_{2} = thresh\_\arctan \left( {c_{2} + d_{2} ;\lambda_{2} /\mu ,a_{2} } \right),$$

(21)

where $thresh\_\arctan \left( \cdot \right)$ is the approximation operators for arctangent nonconvex penalty function.

To sum up, we can obtain the balance non-convex regularized sparse decomposition algorithm flow for extracting composite fault features of the gearbox, as shown in Figure 3.

Figure 3

Balance nonconvex regularized sparse decomposition algorithm

It can be seen from the solution process of the above algorithm that $D^{{\text{T}}} y$ can be regarded as a pre-calculation of an overall term. In this paper, TQWT is used as the transformation dictionary. When the balance parameters $\beta_{i} { = }0$, the above algorithm is simplified to the non-convex penalty function sparse decomposition method under the synthesis model, and the algorithm flow is shown in Figure 4. When the balance parameters $\beta_{i} { = }\infty$, the non-convex regularized sparse decomposition method under the analysis model cannot be directly obtained, and the above solution steps need to be re-derived. For analysis non-convex regularized sparse decomposition algorithms, the signal itself $t^{k + 1}$ needs to be updated. According to the above solution ideas, the analysis non-convex regularized sparse decomposition algorithm is derived, as shown in Figure 5.

Figure 4

Synthesis nonconvex regularized sparse decomposition algorithm

Figure 5

Analysis nonconvex regularized sparse decomposition algorithm

Furthermore, for verification the superiority of the proposed non-convex regularized sparse decomposition algorithm, this paper will compare with the L1 norm regularized sparse decomposition algorithm. Since the proximity operator corresponding to the L1 norm is the soft threshold function, it is only necessary to replace the arctangent threshold function in the above three algorithms with the soft threshold function to obtain the balance, synthesis and analysis of the L1 norm regularization sparse decomposition algorithm.

4 Simulation Analysis and Parameter Determination

In order to verify the performance of the balance non-convex regularized sparse decomposition method proposed in this paper, this section will analyze the simulation signal and discuss the setting method of relevant parameters. Finally, the performances of the three models are compared, and a horizontal comparative analysis is carried out with the three models of the traditional L1 norm regularization method.

4.1 Simulation Signals Construction

Considering the local fault response signal of the gearbox, simulation signals of the gearbox composed of the fault transient component $t$, the harmonic component $h$ generated by gear meshing and the noise interference component $e$ are constructed.

$$\begin{gathered} y = t + h + e \hfill \\ { = }\sum\limits_{k} {E_{1}^{k} } s\left( {t - k/f_{c} - \tau_{k} } \right) + E_{2} o\left( t \right) + e\left( t \right), \hfill \\ \end{gathered}$$

(22)

$$s\left( t \right) = \left\{ {\begin{array}{*{20}c} {\exp \left( {\frac{{ - \zeta_{L} }}{{\sqrt {1 - \zeta_{L}^{2} } }}\left( {2\pi f_{n} t} \right)^{2} } \right)\cos \left( {2\pi f_{n} t} \right)}, & {t < 0}, \\ {\exp \left( {\frac{{ - \zeta_{R} }}{{\sqrt {1 - \zeta_{R}^{2} } }}\left( {2\pi f_{n} t} \right)^{2} } \right)\cos \left( {2\pi f_{n} t} \right)}, & {t \ge 0}, \\ \end{array} } \right.$$

(23)

$$\begin{gathered} o\left( t \right){ = }\cos \left( {2\pi f_{m} t + 0.2\cos \left( {2\pi f_{2} t} \right)} \right) \hfill \\ \, \times \left( {1.5 + 0.2\cos \left( {2\pi f_{1} t} \right)} \right), \hfill \\ \end{gathered}$$

(24)

where $E_{1}^{k}$ is the amplitude of the kth fault transient characteristic component, set to $E_{1}^{k} { = }1$; $f_{c}$ is the fault characteristic frequency, set as $f_{c} { = }100\,{\text{Hz}}$; $E_{2}$ is the amplitude of the harmonic component, set to $E_{2} { = }0.5$; the left and right damping ratios are set to $\zeta_{L} { = }0.01$, $\zeta_{R} { = }0.005$; $\tau_{k}$ represents the random sliding parameter of the fault transient component, set to $\tau_{k} { = }0.005$; other relevant frequency modulation parameters are set as: $f_{1} {\text{ = 65 Hz}}$, $f_{2} {\text{ = 15 Hz}}$, $f_{m} {\text{ = 650 Hz}}$; the noise standard deviation is set to $\sigma_{n} { = 0}{\text{.6}}$. In addition, the sampling frequency of the simulated signal is 20.48 kHz and the signal-to-noise ratio of the simulation signal is calculated as $- 1{1}.{3}7{\text{ dB}}$.

Figure 6 shows the above gearbox simulation signals. Figure 6(a) is the time domain waveform of the fault component of the simulated signal; Figure 6(b) is the time domain waveform of the harmonic component of the simulated signal; and Figure 6(c) is the composite signal of the simulated signal components, harmonic components and noise components; Figure 6(d) is the spectrum of the simulated synthesized signal. This fault characteristic component is difficult to identify from Figure 6(c), and the fault component is covered by the disturbance component.

Figure 6

Gearbox composite fault simulation signal

In order to accurately measure the similarity between the reconstructed signal and the original signal, the root-mean-square-error (RMSE) is used as the evaluation index, and the specific definition is as follows:

$${\text{RMSE = }}\sqrt {\frac{1}{N}\left\| {x - \hat{x}} \right\|_{2}^{2} },$$

(25)

where $x$ represents the original signal, $\hat{x}$ represents the reconstructed signal, and $N$ is the signal length. The smaller the RMSE value, the more similar the reconstructed signal is to the original signal.

4.2 Parameters Determination

First, we analyze the regularization parameter and the balance parameter determination problem in the balanced nonconvex regularized sparse decomposition method.

In the algorithm of gearbox fault feature extraction, the regularization parameter is defined as

$$\lambda_{i} \left( {j_{i} } \right) = \theta \left\| {\varphi_{i} \left( {j_{i} } \right)} \right\|_{2} ,i = 1,2,$$

(26)

where $\varphi_{1} \left( {j_{1} } \right)$ represents the wavelet basis of the TQWT with low Q factor in the $j_{1}$ layer, and $\varphi_{2} \left( {j_{2} } \right)$ represents the wavelet basis of the TQWT with high Q factor in the $j_{2}$ layer. It can be seen from the above Eq. (26) that the regularization parameter is related to the subband. When analyzing the signal, once the TQWT is determined, the parameter $\theta$ is the only factor that affects the regularization parameter. So just study the relationship between the parameter $\theta$ and the noise standard deviation. In the balanced non-convex regularized sparse decomposition algorithm, in addition to the regularization parameters, there are also balance parameters that need to be determined. Therefore, when studying one parameter setting, the other parameter should be kept constant. First, the balance parameters are fixed, and the performance of the algorithm to analyze signals with different noise levels under different parameters $\theta$ is analyzed. Figure 7 shows the average RMSE values of the reconstructed transient components for the balanced non-convex regularized sparse decomposition algorithm for different parameters $\theta$ and noise standard deviation. We can get that the relationship between the regularization parameter and the noise standard deviation can be roughly described as

$$l\left( {{\text{B}}\arctan } \right):\theta = 2.21\sigma + 0.347.$$

(27)

Figure 7

The relationship between $\sigma$ and $\theta$

Similarly, using the above method, the relationship between the noise level and the parameter $\theta$ of other sparse decomposition methods can be obtained:

$$l\left( {{\text{S}}\arctan } \right):\theta = 1.37\sigma + 2.31,$$

(28)

$$l\left( {{\text{A}}\arctan } \right):\theta = 1.38\sigma + 0.41,$$

(29)

$$l\left( {{\text{BL}}1} \right):\theta = 1.78\sigma - 0.02,$$

(30)

$$l\left( {{\text{SL}}1} \right):\theta = 1.95\sigma + 0.35,$$

(31)

$$l\left( {{\text{AL}}1} \right):\theta = 1.29\sigma + 0.04.$$

(32)

Then, we fix the regularization parameters and analyze the effect of the balanced parameters in the balance nonconvex regularization and balance L1 norm regularization sparse decomposition algorithm on the performance of the algorithm through three sets of signals. The average RMSE values of the reconstructed transient components obtained by using the two balance algorithms under different balance parameter settings are shown in Figure 8. It can be concluded from Figure 8 that when the balance parameter is larger than a certain value, the RMSE hardly changes. According to Figure 8, the balance parameters are respectively set as $\beta_{1} { = }0.5,\,\beta_{2} { = }1$.

Figure 8

Barctan sparse decomposition algorithm and BL1 sparse decomposition algorithm balance parameter selection

4.3 Comparative Analysis of Simulation Results

Based on the parameter setting strategy obtained above, we use the above six algorithms to analyze the gearbox fault simulation signal. First, the arctangent non-convex regularized sparse decomposition method is analyzed under three different models, and the results are shown in Figure 9. Figure 10 shows the analysis results of the L1-norm regularized sparse decomposition method under three different models. First, we conduct a horizontal comparison, that is, under the same model, the effect of different regularization functions on the results. It can be seen from Figure 9 that the amplitudes of the transient components reconstructed by the arctangent non-convex regularization algorithm are well preserved. However, in the reconstruction results of the L1 regularization algorithm in Figure 10, there is a serious problem of underestimating the amplitude. We then conduct a longitudinal comparison, that is, compare the results of three different models under the same regularization method. It can be seen from Figure 9 that among the RMSE values of the analysis results of the three models, the reconstruction accuracy of the balance model and the analysis model is higher, and the algorithms have better performance. As can be seen in Figure 10, although there is a problem of underestimating the amplitude, the overall effect of the synthesis model is better, followed by the balance model.

Figure 9

Arctangent nonconvex regularized sparse decomposition results under three models

Figure 10

L1-norm regularized sparse decomposition results under three models

To further verify the performance of different methods, we use RMSE as an evaluation metric to obtain the RMSE of the reconstructed transient components under different parameters $\theta$. Figure 11 shows the average RMSE of the reconstructed transient components for the six methods with noise standard deviations of 0.5 and 0.6, respectively. Comparing the RMSE of the methods in the Figure 11 under the optimal parameters, we can get that the balance non-convex regularized sparse decomposition method proposed in this paper always has excellent fault extraction performance and high extraction accuracy.

Figure 11

RMSE of reconstructed transient components with different parameters

5 Analysis

The simulation signal analysis preliminarily verified the effectiveness of the balance non-convex regularized sparse decomposition method proposed in this paper. In this section, the engineering experimental signals will be further analyzed to verify the applicability and superiority of the proposed method.

5.1 Case 1

First, the vibration signal of an automobile transmission gearbox is analyzed. The structure of the gearbox and the placement of the sensors are shown in Figure 12. The gearbox consists of 5 forward gears and 1 reverse gear, the specific model is LC5T81. The sampling frequency of the accelerometer is 3000 Hz and the speed of the gearbox is 1600 r/min. The working parameters of the third gear are shown in Table 1. After calculation and analysis, the characteristic frequency of the third gear failure of the test gearbox is $f_{1} = 20{\text{ Hz}}$. Figure 13 shows the time-domain waveform of this signal with length 2048 and its spectrum. As can be seen in Figure 13, it is very difficult to identify its fault characteristic components directly from the test signal.

Figure 12

Test rig for gearbox

Table 1 Working parameters

Figure 13

Gearbox measured signal and its spectrum

The above gearbox fault signals are analyzed by Barctan and BL1 sparse decomposition algorithms respectively, and the analysis results are shown in Figure 14. The fault transient component can be clearly identified from Figure 14(a), and the fault characteristic frequency is also significant from Figure 14(b). In Figure 14(c) and (d), it can be seen that the fault characteristic component is relatively weak, and the amplitude of the fault characteristic frequency is also very low, which makes it difficult to identify. Further, the gearbox signal is analyzed using Sarctan, SL1, Aarctan and AL1 sparse decomposition algorithms, and the results are shown in Figures 15 and 16. We can see that the arctangent non-convex penalty function sparse method can better identify the gearbox fault feature components, while the L1 norm regularization method extracts a larger error in the transient components.

Figure 14

Analysis results of Barctan method and BL1 method

Figure 15

Analysis results of Sarctan method and SL1 method

Figure 16

Analysis results of Aarctan method and AL1 method

5.2 Case 2

Next, the proposed method will be used to analyze the vibration signal of the wind turbine gearbox. The data comes from the Gearbox Reliability Collaborative (GRC) Project run by the National Renewable Energy Laboratory's Wind Energy Technology Center. The transmission ratio of the wind turbine gearbox is 1:81.49, and its structure diagram is shown in Figure 17. During a field test of the GRC project, the gearbox under test suffered oil losses twice, resulting in damage to the internal bearings and gears. A later investigation revealed that most of the gears had varying degrees of wear, as well as damage to the inner ring of bearing D, as shown in Figure 18. Vibration signal data was acquired using an NI high-speed data acquisition system at a sampling frequency of 40 kHz.

Figure 17

Structure diagram of wind turbine gearbox

Figure 18

Schematic diagram of faulty gear and bearing

In this part of the case study, the vibration signal of a gearbox measured with an accelerometer AN6, which is close to bearing D, is analyzed. The vibration signal is collected under the condition that the high-speed shaft speed is 1200 r/min. First, we use the proposed Barctan method to analyze the vibration signal of the wind turbine gearbox. The extraction result of the gearbox signal is shown in Figure 19. From Figure 19(a), the gear fault characteristic frequency $f_{g}$ and bearing D fault characteristic component $f_{{\text{b}}}$ can be identified at the same time. But from Figure 19(b), the gear fault characteristic frequency $f_{g}$ can be effectively identified, but the bearing fault characteristic frequency $f_{{\text{b}}}$ is difficult to identify.

Figure 19

Analysis results of Barctan method and BL1 method

Further, the gearbox signals are analyzed using the Sarctan, SL1, Aarctan and AL1 methods, and the results are shown in Figure 20 and 21. The arctangent nonconvex regularization reconstructs the fault characteristic frequency of the transient components more pronounced than the L1 norm regularization method. And the analysis results of the balance model have higher characteristic amplitudes, which can increase the reliability and accuracy of gearbox fault diagnosis.

Figure 20

Analysis results of Sarctan method and SL1 method

Figure 21

Analysis results of Aarctan method and AL1 method

6 Conclusions

• (1) According to the sparse representation model of the gearbox composite fault, three classical sparse decomposition models based on balance, synthesis and analysis are given, and the relationship between the three models is studied.

• (2) In terms of sparse decomposition dictionary, a linear transformation dictionary based on TQWT is studied. TQWT satisfies the tight frame condition, is a redundant linear transformation, and does not involve the inversion operation of high-dimensional matrices, which can achieve efficient and fast solution. By adjusting the Q factor, a dictionary matching the fault signal can be constructed separately.

• (3) Based on arctangent nonconvex penalty function and balanced model, a balanced nonconvex regularized sparse decomposition method, namely Barctan method, is proposed. And use ADMM algorithm to optimize it. The non-convex regularized sparse decomposition algorithm flow under synthetic and analytical models is given, namely Sarctan and Aarctan methods.

• (4) The performance of different algorithms is analyzed and compared through simulation signals and engineering experimental signals. In the simulation part, the determination strategies of regularization parameters and balance parameters are given. The vibration signals of automobile transmission gearbox and wind turbine generator gearbox are analyzed. The superiority of the proposed balanced non-convex regularized sparse decomposition method in gearbox fault feature extraction is verified.