Research on gearbox compound fault diagnosis method and system development based on entire gearbox health maintenance

Abstract

In general, gearbox is prone to occur compound fault due to its harsh working environment and its fault vibration signal contains multi-components which correspond to each gearbox parts. As the multi-components are often coupled with each other and accompanied by strong noise which brings great difficulties to diagnose fault, however, the existing diagnosis methods are mainly applied on single fault rather than the entire gearbox health maintenance, therefore, this paper presents a gearbox compound fault diagnosis method and develops a diagnosis system which has potential value for gearbox health maintenance. In specific, on account of the morphological difference between multi-components, this paper uses resonance sparse signal decomposition (RSSD) to decompose the fault vibration signal into high and low resonance components respectively for achieving gearbox compound fault separation. Furthermore, as for low resonance component containing rolling bearing fault information, a weak fault feature extraction algorithm based on singular value decomposition (SVD) and cepstrum pre-whitening stochastic resonance is proposed, besides, aiming at the high resonance component containing gear fault information, an early gear fault warning algorithm based on local mean decomposition and two-dimensional approximate entropy of chaotic oscillator is also given. Finally, a gearbox fault diagnosis system, which has the ability such as the gearbox vibration signal acquisition, fault indicator warning, health status evaluation, fault signal storage is developed. Simulation validation and comparison prove the effectiveness of proposed method in this paper.

Introduction

Gearbox generally acting as an important power transmission structure in rotating machinery results in the failure of entire rotating machinery system ¹ due to complex working environment. In actual, as the gearbox contains gear, rolling bearing and shaft, and so on, its corresponding fault vibration signal often presents polymorphic property. For example, the rolling bearing fault shows transient impact appearance, gear fault reflects side frequency modulation phenomenon related to the shaft speed. ² As a result, a series of gearbox compound fault detection methods have been developed such as priori guided matching filtering ³ and data-driven intelligent learning. ⁴ However, under the entire gearbox health maintenance condition, developing a gearbox compound fault detection method has significantly value.

In general, the prior guided matching filter is based on the feature extraction technology. Firstly, the dynamic fault response of gearbox is derived through the fault mechanism, and then the feature information reflecting the fault state is obtained from the fault vibration signal, so as to identify and confirm the compound fault.^5,6 Generally, wavelet transform, ⁷ time-frequency representation technology, ⁸ adaptive decomposition technology^3,9 and its derivative algorithm,^1,6 improved envelope spectrum algorithm, ⁴ Kurtosis technique, ¹⁰ sparse model optimization characterization,^11,12 and other techniques are often applied to extract weak feature information which also provides the basis for subsequent fault mode classification, health monitoring, reliability evaluation, and life prediction. On the other hand, data-driven intelligent diagnosis technology ¹³ is a new developed subject in recent years. Its advantage is that it integrates the expert experience in many fields, improves the diagnostic accuracy ¹⁴ which possesses the engineering advantage. ¹² In recent years, artificial intelligence diagnosis technology^15,16 has developed vigorously, such as support vector machine^7,15 and deep learning.^13,17 Besides, higher-order statistics^18,19 is a powerful tool for gearbox fault diagnosis.

However, the classical gearbox fault diagnosis methods still have the following problems. Firstly, the lack of multi-component fault vibration signal separation which can diagnose compound fault. Secondly, the deficiency of weak fault identification on the key components of gearbox. For example, periodic transients in fault vibration signal which discriminate incipient rolling bearing fault are often disturbed by noise and require enhancement. Nevertheless, the sideband of modulation component in fault vibration signal reflects the gear’s deterioration degree and by monitoring its energy quantitatively, it achieves early fault warning. Finally, the existing diagnosis system in the market has no ability to evaluate gearbox health status with a reasonable cost.

In summary, this paper proposes a novel gearbox compound fault diagnosis method and develops a diagnostic system. Firstly, it separates fault vibration signal into multi-components, Furthermore, by processing multi-components, it achieve the rolling bearing weak fault enhancement and gear early fault warning respectively. Finally, experiment proves the feasibility of proposed method and combination Labview and Matlab, this paper develops a gearbox compound fault diagnosis system.

Gearbox compound fault diagnosis method

Diagnosis principle

When rolling bearing and gear occur fault in gearbox, its fault vibration signal often presents center frequency overlap phenomenon, ²⁰ when using classical diagnosis algorithm decomposes fault signal, it will obtain irrelevant signal components, thus affecting the correctness of fault diagnosis result.

In order to solve this problem, considering the fault vibration signal presents morphological multi-component property, this paper decouples fault vibration signal, diagnoses rolling bearing weak fault, achieves early gear fault warning as shown in Figure 1.

Figure 1.

Flowchart of gearbox complex fault diagnosis.

Gearbox fault vibration signal decomposition based on resonance sparse signal decomposition (RSSD)

In actual working condition, the gearbox vibration signal often presents multi-component property, such as periodic impacts, frequency doubling modulation, and so on. Although sparse decomposition has been extensively applied on detecting rotating machinery fault, the classical sparse decomposition method lack signal fidelity ability, it is not proper to directly detect gearbox compound weak fault consequently. As a result, this paper uses the RSSD which has significant advantage on penalty function and over-complete dictionary ²¹ to propose a gearbox multi-component fault vibration signal separation method.

Assuming that the gearbox fault vibration signal symbolizes X, in order to automatically match the periodic transients X_L and the harmonic modulation component X_H respectively that presents variously oscillating morphologies, RSSD uses Tunable-Q Wavelet Transform (TQWT) to construct the basis functions S₁ and S₂ with low quality factor and high quality factor respectively. On this basis, with the help of morphological component analysis (MCA), by constructing the penalty function of L1 norm, as shown in formula (1), the sparse representation of the two signal components X_L and X_H is realized, W₁ and W₂ represent the transformation coefficients of signal components X_L and X_H under the basis functions S₁ and S₂, respectively, λ₁ and λ₂ are regularization parameters. Finally, by using split augmented lagrangian shrinkage algorithm (SALSA) to compute iteratively formula (1), and achieve the minimum value of objective function J so as to obtain the estimated values $X_{\mathrm{LE}}$ and $X_{\mathrm{HE}}$ as shown in formula (2) corresponding to two signal components X_L and X_H, ${W_1}^{*}$ and ${W_2}^{*}$ also are the estimated value corresponding to transformation coefficients.

$$J\left(W_1,W_2\right)=\left|\right|X-S_1{W}_1-S_2{W}_2|{|}_2^2+{\lambda }_1|\left|W_1\right|{|}_1+{\lambda }_2\left|\right|W_2|{|}_1$$

(1)

$$X_{\mathrm{LE}}=S_1{W}_1^{*},X_{\mathrm{HE}}=S_2{W}_2^{*}$$

(2)

Weak rolling bearing fault enhancement based on SVD and CEP-SR

In general, although the extracted periodic transients $X_{\mathrm{LE}}$ contains numerous rolling bearing fault information, considering the coupling effect between multi-components, the periodic transient X_LE is inevitably interfered by harmonic modulation and random noise ²² which induces that it is necessary to strengthen weak fault. In order to solve this problem, a rolling bearing weak fault enhancement method based on singular value decomposition and inverted spectrum pre-whitening-stochastic resonance is proposed. Firstly, this paper uses SVD to reduce noise and obtains the processed signal $\overline{X_{\mathrm{LE}}}$ . Then, the proposed method uses the inverted spectrum pre-whitening algorithm to process the signal $\overline{X_{\mathrm{LE}}}$ for suppressing the harmonic modulation interference, highlighting the periodic impacts property, and obtaining the corresponding whitening signal $\overline{X_{\mathrm{whiten}}}$ which is convenient to act as ideal input of stochastic resonance. Finally, the whitening signal $\overline{X_{\mathrm{whiten}}}$ inputs the SR combination with scale transformation and its output is analyzed by using Hilbert envelope spectrum which enhances the rolling bearing weak fault. The specific process is shown in following.

• Step 1: According to the phase-space reconstruction theory, the extracted periodic transients $X_{\mathrm{LE}}=(X_1,X_2,\dots \dots ,X_N)$ are constructed into a $a\times b$ order Hankel matrix H as shown in formula (3). N represents the signal length, $N=a+b-1$ and $a\ge b$ .

$$H=\left(\begin{array}{ccc}{X}_1\hfill & \cdots \hfill & X_b\hfill \\ \multicolumn{1}{c}{⋮}& \ddots \hfill & ⋮\hfill \\ \multicolumn{1}{c}{X_a}& \cdots \hfill & X_N\hfill \end{array}\right)$$

(3)

Furthermore, the Hankel matrix H is organized in the form of $H=D+W$ , matrix D corresponds to the effective signal component in the reconstruction space and matrix W symbolizes the interference component. As a result, this paper obtains the best approximation matrix $\overline{H}$ corresponding to matrix H for eliminating the influence of matrix W.

• Step 2: This paper performs SVD on matrix H as shown in formula (4). U represents a $a\times b$ order matrix, $V^T$ is a $b\times b$ order matrix, H is a $a\times b$ order diagonal matrix, and its main diagonal element is ${\lambda }_i(i=1,2,\cdots ,k),k=\min (p,q),S=\mathrm{diag}({\lambda }_1,{\lambda }_2,\cdots {\lambda }_k)$ , ${\lambda }_1,{\lambda }_2,\cdots {\lambda }_k$ is the singular value of matrix H, and ${\lambda }_1\ge {\lambda }_2\ge ,\cdots ,\ge {\lambda }_k$ , U, V ^T represents the left singular array and the right singular array respectively.

$$H=\mathrm{US}{V}^T$$

(4)

• Step 3: In order to reduce the noise, this paper uses the difference value between adjacent singular value ${\lambda }_1,{\lambda }_2,\cdots {\lambda }_k$ to form a q−1 vector, and obtains the differential spectrum corresponding to singular value. According to the differential spectrum, we extract specific larger singular values containing rolling bearing fault information, remove the smaller singular values containing noise interference. Then, the proposed method uses the inverse transformation of SVD to get the best approximation matrix $\overline{H}$ and the signal $\overline{X_{\mathrm{LE}}}$ .

• Step 4: Furthermore, the proposed method gets the real reciprocal spectrum C as shown in formula (5). By using the real reciprocal spectrum C, a whitening signal $\overline{X_{\mathrm{whiten}}}$ is obtained which retains the periodic transients and white noise, and its resonant frequency band is equally important everywhere, there is no need to select the optimal resonant frequency band consequently.

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}X\left(f\right)=\mathrm{FFT}\left(\overline{X_{\mathrm{LE}}}\right)\hfill \\ \multicolumn{1}{c}{X\left(f\right)=A\left(f\right)e^{jØ\left(f\right)}}\\ \multicolumn{1}{c}{C=\mathrm{FF}{T}^{-1}\left[\ln \left(A\left(f\right)\right)\right]}\end{array}}\end{array}$$

(5)

• Step 5: Finally, the SR is introduced to process the whitening signal $\overline{X_{\mathrm{whiten}}}$ which acts as the input of SR model as shown in formula (6) and the $X_{\mathrm{whiten}\_\mathrm{out}}$ symbolizes output signal which can highlight the weak rolling bearing fault frequency by transferring the energy of the white noise and extract the fault frequency by envelope spectrum analysis. a and b are non-zero system parameters.

$$\frac{d{X}_{\mathrm{whiten}\_\mathrm{out}}}{\mathrm{dt}}=a{X}_{\mathrm{whiten}\_\mathrm{out}}-b{X_{\mathrm{whiten}\_\mathrm{out}}}^3-\overline{X_{\mathrm{whiten}}}$$

(6)

Early gear fault warning based on chaotic oscillators and two-dimensional approximate entropy

In general, the high resonance component X_HE which corresponds to gear vibration often presents modulation property. However, when the gear occurs early fault, the amplitude of its side frequency band tends to increase. As a result, the key for incipient gear fault warning is to dynamically monitor the energy varying law of side frequency band. In order to solve this problem, firstly, local mean decomposition (LMD) decomposes the high resonance component X_HE into a series of product functions (PF). Furthermore, by observing corresponding spectrum, some considerable PF components with obvious fault information construct a new signal X_{HE_RC}, and the duffing chaotic oscillators are used to monitor whether the reconstructed signal X_{HE_RC} has a side frequency band by observing its phase plane trajectory diagram. Finally, in order to identify the energy varying law of side frequency band and realize the gear early fault warning, a two-dimensional approximate entropy is used to quantitatively measure the confusion degree of trajectory map in phase plane. The specific process is shown as follows.

• Step 1: Firstly, it decomposed signal X_HE into a series of PF components as shown in formula (7). Furthermore, a reconstruction signal X_{HE_RC} is obtained by combining the PF components whose spectrum exhibits side frequency band modulation property.

$$X_{\mathrm{HE}}=\sum _{k=1}^{N}P{F}_k+U$$

(7)

• Step 2: Furthermore, it uses Holmes-type duffing oscillator to identify the chaotic state of phase plane trajectory map corresponding to signal X_{HE_RC} and its initial equation is shown in formula (8). In the formula, c symbolizes damping coefficient, $F_0\cos w_{0}t$ is the harmonic force inside the oscillator, F₀ represents the amplitude, and as the varying energy of side frequency band can directly affect the chaotic state of phase plane trajectory map, w₀ is gear meshing frequency, y is the output of the oscillator equation, and the sequence of phase plane trajectory is $[y,\stackrel{·}{y}/w_0]$ . In specific, firstly, by setting a reasonable value F₀ and giving an amplitude threshold value F_b, the initial state of formula (8) is in a large-scale periodic state. Secondly, the signal X_{HE_RC} is input into the oscillator equation, as shown in formula (9), when gear occurs incipient fault, the phase plane trajectory will vary from original large-scale periodic state to chaotic state until it is completely under chaotic state which reflects the varying energy of side frequency band corresponding to meshing frequency w₀.

$$-\mathrm{c}{\mathrm{w}}_0\stackrel{·}{\mathrm{y}}+{{\mathrm{w}}_0}^2\mathrm{y}-\stackrel{··}{\mathrm{y}}-{{\mathrm{w}}_0}^2{\mathrm{y}}^3+{{\mathrm{w}}_0}^2{\mathrm{F}}_0\cos {\mathrm{w}}_0\mathrm{t}=0$$

(8)

$$\begin{array}{c}-\mathrm{c}{\mathrm{w}}_0\stackrel{·}{\mathrm{y}}+{{\mathrm{w}}_0}^2\mathrm{y}-\stackrel{··}{\mathrm{y}}-{{\mathrm{w}}_0}^2{\mathrm{y}}^3+{{\mathrm{w}}_0}^2{\mathrm{F}}_0\cos {\mathrm{w}}_0\mathrm{t}\hfill \\ +{{\mathrm{w}}_0}^2{\mathrm{X}}_{\mathrm{HE}\_\mathrm{RC}}=0\hfill \end{array}$$

(9)

• Step 3: Considering two-dimensional approximate entropy AE2d can describe the irregularity and complexity of phase plane trajectory, it can be used to achieve incipient gear fault warning quantitatively. In specific, it gives the entropy’s threshold value A_t and sets the embedding dimension $\mathrm{m}=2$ , the sequence $[y\left(i\right),\stackrel{.}{y}\left(i\right)/w_0]\left(i=1,2,\cdots ,N\right)$ is composed into $m$ dimension vectors $\mathrm{O}\left(\mathrm{i}\right)$ , as shown in the formula (10). Furthermore, it calculates the distance between $O\left(i\right)$ and $O\left(j\right)$ as shown in formula (11), and the number of $d[O\left(i\right),O\left(j\right)]<r$ is counted, r represents the given threshold. On this basis, it needs to compute the ${C_i}^m\left(r\right)$ which is shown in formula (12), $\mathrm{\Theta }\left(\mathrm{x}\right)=\{\begin{array}{c}1,x\ge 0\hfill \\ \multicolumn{1}{c}{0,x<0}\end{array}$ . Besides, it needs to obtain ${\varnothing }^m\left(r\right)$ and ${Ø}^{m+1}\left(r\right)$ as shown in formula (13). Finally, the two-dimensional approximate entropy AE2d is calculated as shown in formula (14).

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{O}\left(\mathrm{i}\right)=[(\mathrm{y}\left(\mathrm{i}\right),\stackrel{·}{\mathrm{y}}\left(\mathrm{i}\right)/{\mathrm{w}}_0),\cdots ,(\mathrm{y}(\mathrm{i}+\mathrm{m}-1),\stackrel{·}{\mathrm{y}}(\mathrm{i}+\mathrm{m}-1)/{\mathrm{w}}_0)],}\\ \mathrm{i}=1,2,\cdots ,\mathrm{N}-\mathrm{m}+1\hfill \end{array}$$

(10)

$$\begin{array}{c}\mathrm{d}[\mathrm{O}\left(\mathrm{i}\right),\mathrm{O}\left(\mathrm{j}\right)]=\max \hfill \\ \left(\sqrt{{(\mathrm{y}(\mathrm{i}+\mathrm{k})-\mathrm{y}(\mathrm{j}+\mathrm{k}))}^2+{(\stackrel{·}{\mathrm{y}}(\mathrm{i}+\mathrm{k})-\stackrel{·}{\mathrm{y}}(\mathrm{j}+\mathrm{k}))}^2/{{\mathrm{w}}_0}^2}\right),\hfill \\ \mathrm{k}=0,1,\cdots ,\mathrm{m}-1\hfill \end{array}$$

(11)

$${C_i}^m\left(r\right)=\sum _{i\ne j}\mathrm{\Theta }\{\mathrm{r}-\mathrm{d}[\mathrm{O}\left(\mathrm{i}\right),\mathrm{O}\left(\mathrm{j}\right)]\}/(N-m+1)$$

(12)

$${Ø}^m\left(r\right)=\frac{1}{N-m+1}\sum _{i=1}^{N-m+1}\ln {C_i}^m\left(r\right)$$

(13)

$$\mathrm{AE}2\mathrm{d}(\mathrm{m},\mathrm{r})={Ø}^m\left(r\right)-{Ø}^{m+1}\left(r\right)$$

(14)

Validation

In this paper, the fault vibration signal $s\left(\mathrm{t}\right)$ is used to verify the effectiveness of proposed method as shown in formula (15). In this formula, $i\left(\mathrm{t}\right)$ represents the periodic transients generated by rolling bearing fault, $m\left(\mathrm{t}\right)$ represents the side frequency modulation component caused by gear fault, and $h\left(\mathrm{t}\right)$ represents the harmonic component. $\mathrm{noise}\left(\mathrm{t}\right)$ represents random white Gaussian noise with −8 dB intensity.

$$s\left(t\right)=i\left(t\right)+m\left(t\right)+h\left(t\right)+\mathrm{noise}\left(t\right)$$

(15)

As for periodic transients $s\left(t\right)$ , it is derived from outer ring fault signal generated by the XJTU-SY rolling bearing accelerated life test bench, School of Mechanical Engineering, Xi’an Jiaotong University. The test bench contains digital display, motor speed controller, rotary shaft, AC motor, support bearing, hydraulic loading system, vertical acceleration sensor, horizontal acceleration sensor and experimental rolling bearing, dynamic signal collector. Its sampling frequency is 25.6 kHz, and the number of signal samples is $16384\times 6$ . Besides, the diameter of outer ring raceway is $D_1=39.80\mathrm{mm}$ , diameter of inner ring raceway is $D_2=29.30\mathrm{mm}$ , middle diameter of rolling bearing is $D_D=34.55\mathrm{mm}$ , ball diameter is $\mathrm{d}=7.92\mathrm{mm}$ , number of ball is $\mathrm{Z}=8$ , contact Angle is $\mathrm{\alpha }=0^{\mathrm{o}}$ , speed of outer ring is $n_a=2100\mathrm{r}/\min$ , speed of inner ring is $n_b=2400\mathrm{r}/\min$ . Finally, its outer ring fault frequency $f_a=107.8\mathrm{Hz}$ is calculated by formula (16), and rolling bearing parameters are shown in Table 1.

$$f_a=\frac{n_a}{2\times 60}Z(1-\frac{d}{D_1}\cos \mathrm{\alpha })$$

(16)

Table 1.

Bearing parameter table.

Diameter of outer ring raceway	Diameter of bearing inner ring raceway	Rolling bearing middle diameter	Ball diameter	Number of ball	Contact angle	Speed of outer ring	Speed of inner ring
39.80 mm	29.30 mm	34.55 mm	7.92 mm	8	0	2100 r/min	2400 r/min

As for side frequency modulation component $m\left(t\right)$ , it is an simulation signal generated by the dynamic signal processing system. The dynamic signal processing system is composed of two DG1022U dual-channel signal generators and industrial computer. Its sampling frequency is also 25.6 kHz, and the number of signal samples is also $16384\times 6$ . Here, assuming that the number of driving gear teeth is $Z_1=19$ , and the number of passive gear teeth is $Z_2=29$ , the fault tooth is the driving tooth and with fault rolling bearing is located on a same drive shaft. Since the inner ring speed is $n_b=2400\mathrm{r}/\min$ , its rotation frequency is $f_b=40\mathrm{Hz}$ , the corresponding meshing frequency is $f_m=760\mathrm{Hz}$ . During the whole life cycle, gearbox transits from the health state into the early degradation state, and then gradually into the deterioration state. Therefore, signal $s\left(\mathrm{t}\right)$ should present a time-varying property, and energy of side frequency modulation $m\left(\mathrm{t}\right)$ gradually increases with the deterioration of fault degree synchronously. As a result, signal $m\left(\mathrm{t}\right)$ is constructed as shown in formula (17)–(20) which presents time-varying feature. Given a series of parameters $A_{m1},A_{m2},A_{m3},B_{m1}$ , $B_{m2}$ is shown as follows, $t\in (0,0.64)$ , $A_{m1}=0.5,A_{m2}=0.2,A_{m3}=0.1,B_{m1}$  = 0.5, $B_{m2}$  = 0.2. $t\in (0.64,1.28)$ , $A_{m1}=0.55,A_{m2}=0.22,A_{m3}=0.12,B_{m1}$  = 0.52, $B_{m2}$  = 0.22. $t\in (1.28,1.92)$ , $A_{m1}=0.6,A_{m2}=0.24,A_{m3}=0.14,B_{m1}$  = 0.54, $B_{m2}$  = 0.24. $t\in (1.92,2.56)$ , $A_{m1}=0.65,A_{m2}=0.26,A_{m3}=0.16,B_{m1}$  = 0.56, $B_{m2}$ =0.26. $t\in (2.56,3.20)$ , $A_{m1}=0.7,A_{m2}=0.28,A_{m3}=0.18,B_{m1}$  = 0.58, $B_{m2}$  = 0.28. $t\in (3.20,3.84)$ , $A_{m1}=0.75,A_{m2}=0.30,A_{m3}=0.20,B_{m1}$  = 0.6, $B_{m2}$  = 0.3.

$$m\left(t\right)=m_1\left(t\right)+m_2\left(t\right)+m_3\left(t\right)$$

(17)

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\mathrm{m}}_1\left(\mathrm{t}\right)=0.15((1+{\mathrm{a}}_1\left(\mathrm{t}\right))\times \cos (2\mathrm{\pi }{\mathrm{f}}_{\mathrm{m}}\mathrm{t}+{\mathrm{b}}_1\left(\mathrm{t}\right)))\hfill \\ \multicolumn{1}{c}{{\mathrm{m}}_2\left(\mathrm{t}\right)=0.15((1+{\mathrm{a}}_2\left(\mathrm{t}\right))\times \cos (4\mathrm{\pi }{\mathrm{f}}_{\mathrm{m}}\mathrm{t}+{\mathrm{b}}_2\left(\mathrm{t}\right)))}\\ \multicolumn{1}{c}{{\mathrm{m}}_3\left(\mathrm{t}\right)=0.15((1+{\mathrm{a}}_3\left(\mathrm{t}\right))\times \cos (6\mathrm{\pi }{\mathrm{f}}_{\mathrm{m}}\mathrm{t}+{\mathrm{b}}_2\left(\mathrm{t}\right)))}\end{array}}\end{array}$$

(18)

$$\begin{array}{c}\multicolumn{1}{c}{\{\begin{array}{c}{\mathrm{a}}_1\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{m}1}\times \cos \left(2\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)\hfill \\ \multicolumn{1}{c}{{\mathrm{a}}_2\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{m}1}\times \cos \left(2\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)+{\mathrm{A}}_{\mathrm{m}2}\times \cos \left(4\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)}\\ \multicolumn{1}{c}{{\mathrm{a}}_3\left(\mathrm{t}\right)={\mathrm{A}}_{\mathrm{m}1}\times \cos \left(2\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)+{\mathrm{A}}_{\mathrm{m}2}\times \cos \left(4\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)}\\ \multicolumn{1}{c}{+{\mathrm{A}}_{\mathrm{m}3}\times \cos \left(6\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)}\end{array}}\end{array}$$

(19)

$$\{\begin{array}{c}{\mathrm{b}}_1\left(\mathrm{t}\right)={\mathrm{B}}_{\mathrm{m}1}\times \sin \left(2\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)\hfill \\ \multicolumn{1}{c}{{\mathrm{b}}_2\left(\mathrm{t}\right)={\mathrm{B}}_{\mathrm{m}1}\times \sin \left(2\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)+{\mathrm{B}}_{\mathrm{m}2}\times \sin \left(2\mathrm{\pi }{\mathrm{f}}_{\mathrm{r}}\mathrm{t}\right)}\end{array}$$

(20)

As for the harmonic component $h\left(t\right)$ , it simulates the rotational frequency component generated by the gear during operation, as shown in formula (21).

$$h\left(t\right)=0.2\times \cos (2\pi \times 2\times 40t)+0.2\times \cos (2\pi \times 3\times 40t)$$

(21)

Finally, the experimental bench used is shown in Figure 2, experimental validation signal $\mathrm{s}\left(\mathrm{t}\right)$ , periodic transients $i\left(t\right)$ , side frequency modulation component $\mathrm{m}\left(\mathrm{t}\right)$ are Figures 3 to 5 respectively.

Figure 2.

Experimental test bench: (a) time domain diagram of gearbox fault vibration verification signal and (b) frequency domain diagram of gearbox fault vibration verification signal.

Figure 3.

Gearbox compound fault vibration validation signal: (a) time domain diagram of gearbox fault vibration verification signal and (b) frequency domain diagram of gearbox fault vibration verification signal.

Figure 4.

Periodic transients corresponding to rolling bearing outer ring fault: (a) bearing outer ring fault vibration signal time domain diagram and (b) bearing outer ring fault vibration signal frequency domain diagram.

Figure 5.

Side frequency modulation component corresponding to gear fault: (a) time domain diagram of simulation vibration signal of simulated fault gear and (b) frequency domain diagram of simulation vibration signal of simulated fault gear.

Firstly, RSSD is performed on the validation signal $\mathrm{s}\left(\mathrm{t}\right)$ to obtain the corresponding high-resonance component and low-resonance component respectively, as shown in Figure 6. It can be seen intuitively that Figure 6(a) reflects the harmonic modulation property, and Figure 6(b) shows impact transients phenomenon. Therefore, it can be seen that RSSD can indeed achieve the multi-oscillatory components separation. However, due to the influence of interference components, each signal component still have a certain oscillatory aliasing, so it is necessary to further process the high and low resonance components to highlight the fault information corresponding to gear and rolling bearing respectively.

Figure 6.

Signal decomposition based on RSSD: (a) high resonance component, (b) low resonance component, and (c) residual component.

In order to warn the incipient gear fault, we firstly perform local mean decomposition (LMD) on the high resonance component, and the its PF components is shown in Figure 7, and the corresponding spectrum is shown in Figure 8. It can be clearly seen from Figure 8 that PF3 component has obvious side frequency modulation feature corresponding to meshing frequency f_m, while PF4 and PF5 components present significantly harmonic property. As a result, the proposed method uses PF3 component to combine a reconstructed signal to highlight gear fault information as shown in Figure 9.

Figure 7.

PF components corresponding to high resonance component.

Figure 8.

PF components spectrum.

Figure 9.

Reconstructed signal: (a) reconstruct the signal time domain diagram and (b) reconstruct the signal frequency domain diagram.

Finally, by inputting the reconstructed signal into Duffing chaotic oscillator and combining the two-dimensional approximate entropy, it can quantitatively identify the varying law of side frequency band and detect incipient gear fault consequently. Besides, the Duffing chaotic oscillator under a stable large-scale periodic state infers the gear works in normal operation as shown in Figure 10. However, when the gear overcomes the early fault degradation degree, accompanying with the gear fault deterioration deepening, the influence of side frequency band on the oscillator is strengthened which induces the state of oscillator gradually varies from large-scale periodic state to completely chaotic state (Figure 11). In this process, the two-dimensional approximate entropy is used to evaluate quantitatively fault degree. Finally, as shown in Figure 12, it can see that this healthy indicator plays a good role in gear early fault warning.

Figure 10.

The oscillator phase plane corresponding to the gear healthy running state.

Figure 11.

Variation diagram of oscillator chaos: (a) oscillator diagram of early failure of gear Ae2d = 0.0083, (b) gearbox oscillator diagram for normal operation Ae2d = 0.0087, (c) gearbox oscillator diagram for normal operation Ae2d = 0.0097, (d) gearbox oscillator diagram for normal operation Ae2d = 0.0114, (e) gearbox oscillator diagram for normal operation Ae2d = 0.0234, and (f) gearbox oscillator diagram for normal operation Ae2d = 0.0223.

Figure 12.

Two-dimensional approximate entropy varying law.

Simultaneously, SVD is used to improve the SNR of low resonance component containing rolling bearing fault information, and its singular value difference spectrum is shown in Figure 13. Considering the amplitude peak in singular value difference spectrum may be caused by rolling bearing fault, in order to avoid the interference, the first seven singular values are used to reconstruct the low-resonance component, as shown in Figure 14. By comparing Figure 14(b) and (c), it can see intuitively that the SNR of reconstructed signal is effectively improved by using SVD, while by analyzing the envelope spectrum of reconstructed signal, it is still affected by interference. As a result, the fault frequency still needs to be enhanced.

Figure 13.

Singular value distribution curve and difference spectrum: (a) singular value distribution curve and (b) singular value difference spectrum.

Figure 14.

Reconstructed low resonance component based on SVD noise reduction: (a) signal time domain diagram after denoising by singular value decomposition, (b) signal frequency domain diagram after denoising by singular value decomposition, (c) signal frequency domain diagram before denoising by singular value decomposition, and (d) Hilbert envelope spectrum of signal before denoising by singular value decomposition.

Therefore, cepstral pre-whitening is performed on the reconstructed low resonance component as shown in Figure 15 and Figure 15(a) shows the real cepstrum. Furthermore, spectrum editing combination with whitening processing is carried out by using the signal’s original phase, Figure 15(b) shows the time domain waveform by using whitening processing. It can be seen that the periodic transients are significantly enhanced, considering each frequency is equally important in the whole spectrum, so there is no optimal resonance band, and envelope analysis can be conducted directly. Therefore, the extracted envelope can input into stochastic resonance operator to identify fault frequency and its multiple frequencies, as shown in Figure 16. The actual fault frequency 106.9 Hz and its doubling frequency can be identified, which is consistent with the theoretically fault frequency $f_a=107.8\mathrm{Hz}$ .

Figure 15.

Real cepstrum and whitened signal: (a) real cepstrum and (b) Albino signal.

Figure 16.

Stochastic resonance outputs the Hilbert envelope spectrum.

For comparison on gearbox compound fault diagnose effect, discrete wavelet transform (DWT) in the literature ⁷ is used to analyze validation signal $\mathrm{s}\left(\mathrm{t}\right)$ , its result is shown in Figure 17. It can be seen that by using DWT, the gear meshing frequency is further extracted, but its side frequency band containing the gear fault information is not obvious, and rolling bearing fault frequency is also can be identified which validates the effectiveness of proposed method.

Figure 17.

Frequency spectrum of validation signal $\mathrm{s}\left(\mathrm{t}\right)$ based on DWT.

In order to further prove the accuracy and feasibility of proposed method quantitatively, it introduces kurtosis and correlation coefficient to evaluate the rolling bearing fault diagnose effect and gear fault diagnose effect respectively which is shown in Tables 2 and 3 respectively. Obviously, this result proves the superiority of proposed method.

Table 2.

Kurtosis index corresponding to different methods.

Time (s)	0–0.64	0.64–1.28	1.28–1.92	1.92–2.56	2.56–3.20	3.20–3.84
RSSD	3.9209	3.9881	3.8760	3.9034	3.8967	3.5171
SVD	5.1726	5.1828	5.1267	5.2375	5.0325	5.2389
SVD-CEP	3.6865	3.7283	3.9575	4.1603	4.2913	3.8010
DWT	7.4958	7.8987	7.4934	7.6024	7.5547	7.6211
Proposed method	988.7545	990.1279	980.9763	981.0986	978.7846	991.3847

Table 3.

Correlation coefficient index corresponding to different methods.

Time (s)	0–0.64	0.64–1.28	1.28–1.92	1.92–2.56	2.56–3.20	3.20–3.84
RSSD	0.2435	0.2401	0.2511	0.2513	0.2678	0.2325
LMD	0.3744	0.3984	0.3869	0.3810	0.3943	0.3894
DWT	0.0244	0.0124	0.0275	0.0194	0.0215	0.0188
Proposed method	0.4361	0.4162	0.4031	0.4092	0.4013	0.4101

System implementation

Gearbox compound fault diagnosis system mainly composes hardware device and software system. The hardware device contains industrial computer, data acquisition card, vibration acceleration sensor, and so on, as for software system, combination with LabVIEW and MATLAB, a software system based on the proposed gearbox compound fault diagnosis method is developed in this paper.

Hardware device

The hardware device composes IEPE vibration acceleration sensor, industrial computer, vibration signal acquisition card, as shown in Figure 18 which can realize four-channel vibration acceleration acquisition which varies from 0 to 50g and one channel rotating speed signal synchronously.

Figure 18.

Schematic diagram of the hardware device.

Software system

The software system mainly focuses on evaluating the running state of gearbox, early fault warning, and key parts of gearbox fault identification. In specific, the running state of gearbox is evaluated by using vibration intensity and other indicators. In addition, in order to facilitate system modification and expansion in future, the software system is designed into four functional modules, namely, the login module, the signal acquisition module, the online monitoring module, and the fault diagnosis module and they do not interfere with each other, as shown in Figure 19.

Figure 19.

Software system flow chart.

Login interface of the system (main interface)

In order to prevent data leakage and ensure the system security, the login module is designed. If there is an input error in the account number or password, the gearbox composite fault diagnosis system will pop up a prompt “The account number or password is entered incorrectly, please enter it again!” as shown in Figure 20.

Figure 20.

Login interface of this system (main interface).

Signal acquisition module

In the software system, the acquisition module is completed through LabVIEW programming, which uses DAQmx to create virtual channels, DAQmx timing and other sub-VI settings acquisition channels, input voltage range, terminal configuration, sampling frequency, and other parameters. Besides, signal is stored in the format of TDMS file storage, as shown in Figure 21.

Figure 21.

Acquisition module interface.

On-line monitoring module

The online monitoring module mainly covers time domain analysis, frequency domain analysis, and data preservation. As for time domain analysis, it contains gearbox fault early warning by using a series of indicators which are very sensitive and more intuitive to the gearbox health state as shown in Figure 22.

Figure 22.

Online monitoring system front panel.

On the other hand, as for frequency domain analysis, it mainly includes amplitude spectrum, envelope spectrum, time-frequency analysis, Hilbert spectrum, and vibration severity. Especially vibration severity is the most important indicator to judge whether the gearbox occurs fault which is calculated from frequency $f_{\min }$ to frequency $f_{\max }$ as shown in formula (22). In the formula, $f_s$ represents the sampling frequency, N corresponds to sampling number, $k_a$ is the lower limit value corresponding to frequency $f_{\min }$ , and $k_b$ is the supper limit value corresponding frequency $f_{\max }$ . $\mathrm{X}\left(\mathrm{k}\right)$ represents the fast Fourier transform. Finally, compared with the reference value given in the national standard GB/T11348.1-1999 and the Canada gearbox maintenance vibration limit standard respectively, this indicator can evaluate the gearbox healthy state properly.

$$\mathrm{VS}=\frac{1}{2\sqrt{2}\mathrm{\pi }{\mathrm{f}}_{\mathrm{s}}}\times \sqrt{\sum _{\mathrm{k}={\mathrm{k}}_{\mathrm{a}}}^{{\mathrm{k}}_{\mathrm{b}}}{\left(\frac{\left|\mathrm{X}\left(\mathrm{k}\right)\right|}{\mathrm{k}}\right)}^2},\mathrm{k}=0,1,\cdots ,\mathrm{N}/2$$

(22)

The data preservation function mainly stores the original fault vibration signal and a series of healthy indicators in the ACCESS database as shown in Figure 23.

Figure 23.

Data preservation function.

Fault diagnosis module

The fault diagnosis module mainly uses the proposed method in this paper to decompose gearbox compound fault vibration signal, weak rolling bearing fault enhancement, early gear fault warning, as shown in Figure 24.

Figure 24.

Fault diagnosis module.

Conclusion and prospect

Aiming at gearbox compound fault diagnosis, the proposed method can decompose fault vibration signal, enhance weak rolling bearing fault and achieve early gear fault warning, its feasibility and superiority also are verified by experiment. Finally, a diagnosis system is developed based on the proposed method ultimately, and a series of conclusions are obtained as follows.

• (1) A gearbox compound fault diagnosis method is proposed. Firstly, it uses RSSD to decompose the fault vibration signal into signal components which contain rolling bearing fault information and gear fault information respectively. Then, weak rolling bearing fault is enhanced based on singular value decomposition combination with cepstral pre-whitening and stochastic resonance. Finally, by using local mean decomposition and Duffing chaotic oscillator, it can warn incipient gear fault and two-dimensional approximate entropy is used to quantitatively evaluate its fault deterioration degree.

• (2) A gearbox compound fault diagnosis system is designed based on proposed method. This system uses LabVIEW and MATLAB software environment to develop a set of software and hardware which can realize the gearbox healthy status on-line monitoring and key components fault diagnosis and it can solve the problem that the existing fault diagnosis system can only be applied on a certain part.

For subsequent research, it is necessary to improve the signal decomposition efficiency based on RSSD, the software recently runs slowly when the signal number is large (the number of sampling points is more than 10 M). Secondly, based on the oscillator chaos combination with two dimensional approximate entropy, it can warn gear early fault based on analyzing side frequency modulation, while as the gear vibration signal in healthy state also presents side frequency modulation property, therefore, how to confirm the approximate entropy threshold is very important which still needs further research in future.