Signal de-noising in gear pitting fault identification by an improved singular value decomposition method

Gears are critical transmission components in rotating machinery with extensive application field especially in modern industrial areas. The whole machinery’s performance and production efficiency are directly affected by the working state of gears [1]. The stable operation is directly affected by the state of gear motion whether the gear run normally or not [2, 3]. Therefore, it is very crucial to identify the early faults before the gears develop into serious state. However, in practical applications, it is difficult to predict the formation and development of gear fault in advance because while the vibration around the gearbox is perceived, the gear fault has been a severe condition. Up to now, the research of the gear failure mechanism in early and middle condition is not mature. For the practical applications, it is important to develop the research of gear failure on vibration characteristics of gear box.

Recently, many theoretical types of research are developed on the vibration of gearbox by gear failure, few type are based on practical application. And there is a certain gap between theory and practical application. Generally, fault detection of gear teeth is the collection of fault signal samples site, and the identification of the gear fault type according to the information by the signal collection [4]. Since the measured signal is interfered with strong noise background, de-noising for the extraction of useful characteristics from the original signal is the first important stage. At present, the common methods of signal de-noising are various, such as Empirical mode decomposition method (EMD) [5, 6], Local mean decomposition method (LMD) [7, 8], Singular value decomposition method (SVD) [9‐11] wavelet analysis method [12‐14]. In practical, the methods are limited by some problems. EMD is affected by mode aliasing which is occurred frequently in the decomposition process, and the quality of signal decomposition is reduced in the condition [15, 16]. LMD also generates mode aliasing which makes the decomposition and extraction of fault features incomplete, and lead to decomposition failure on the whole time scale [17‐19]. The effect of Wavelet de-noising method is depended on the calculation and selection of reasonable parameters [20, 21].

Beside the de-noising methods above, the singular value decomposition (SVD) is a better method to deal with problems of mode aliasing and endpoint effect. The singular value of effective feature is distinguished from original signal by SVD which is based on the singular value matrix of the characteristic signal. In the method, characteristic information of the noise signal is eliminated, the useful characteristic signal is reconstructed. As it is invariant, stable and effective for the de-noising, SVD is applied to the filtering of nonlinear signals, such as practical application of de-noising in fault identification of gear fault [22].

However, the accuracy of de-noising in fault identification by SVD is depended on the calculations of τ and m, which is highly subjective and unreliable in theory at present. Although suitable parameters of τ and m are calculated by a new method which is called k-SVD, the determination of k is restricted by other conditions, and affects the de-noising effect of SVD [23, 24].

It is urgency to reasonably determinate the values of τ and m in SVD respectively. For the value of τ, although is appropriate to any value in theory, the determination in practical applications is directly affected by the vibration signal. In the case that τ is too small, the correlation is concentrated to between different elements of the delay vector, and the reconstructed phase space is meaningless. Contrary, while τ is too large, it leads to loss of noise delay coordinates of mutual information elements, and the reconstructed phase space is unable to reflect the initial dynamic behavior. The parameter m controls the phase space, determines the accuracy of de-noising and identification. While m is large, the phase space is too large, and makes the calculation tedious and complex. Contrary, while m is small, the reconstructed information is unrealizable for the extraction from mutation signals [25, 26].

In order to improve the SVD for de-noising, an improved singular value decomposition (ISVD) is developed based on autocorrelation function with Cao’s algorithm for the parameter of τ and m. For the determination of τ, the autocorrelation function is closely related to the shape of the attractor in the reconstructed phase space and approximated the principal element direction of the signal system. Empirically, while the decay of autocorrelation function is approximated to 1/e, the optimal delay time is obtained [35]. For the determination of m, Cao’s algorithm which is based on the False Nearest Neighbor (FNN) method, makes the calculation of m only related to τ, and overcomes the influence of various thresholds on m [27, 28].

In this paper, ISVD is proposed for the vibration signal de-noising of gear pitting fault identification. Firstly, the parameter of τ and m of the Hankel matrix for SVD are optimized by autocorrelation function and Cao’s algorithm respectively. Secondly, simulation of ISVD by autocorrelation function and Cao’s algorithm is conducted to verify the proposed method. Thirdly, the experiment of tooth surface pitting failure is conducted by ISVD method to verify the effectiveness of the proposed method.

Fig. 1 shows the flowchart of ISVD for de-noising. In the flow, the optimal τ is calculated by the autocorrelation function method of fault signal sequence; m is calculated by the Cao’s algorithm of fault signal phase space, based on them, the phase space of the effective characteristic signals and noise signals are obtained respectively; then the optimal de-nosing order is calculated by the singular value difference spectrum (DSSV) method of the characteristic signal; finally, the inverse operation of Hankel matrix is used to reconstruct the characteristic signal after de-noising. It retains the effective signal value and rid the noise signal value, then reconstructs the effective signal in a suitable order. Essentially, SVD is a method of matrix orthogonalization in the mathematical view, which decomposes the given matrix into two matrixes U^m×m and V^n×n, as shown: [29‐32].where S is the singular value matrix of the real matrix H, in S, σ_i is the singular value of the real matrix H, and \(\sigma _{1}\geq \sigma _{2}\geq \cdot \cdot \cdot \geq \sigma _{i}\geq \sigma _{r}\geq 0\), where r is the rank of the matrix H.

For the signals \(x=(x_{1},x_{2},\cdot \cdot \cdot ,x_{m})\), the de-noising is described as follows:

where m is the embedded dimension, τ is delay time, N is the length of the signal, and n = N − (m − 1) × τ.

k is investigated by Singular values difference spectrum (DSSV) D and the elements in D is d = σ_i − σ_i+1, where i = r − 1. The DSSV sequence D_i is constructed shown in Eq. 3.

While the difference spectrum of singular value is positioned on the mutation point, the spectrum value d_i is the maximum in the whole sequence D_i. It is expressed by:

Based on of Eq. 4, the sequence number corresponding to the maximum SVDS d_max = Max (D_i) is calculated of the signal. According to the above principle, the new singular value matrix S_New is obtained from the effective characteristic signal, as shown in Eq. 5:where, σ^H is the singular value left over.where, \(u_{i}^{H}\) is element of the matrix U, \(v_{i}^{H}\) is element of the matrix V.

In this section, the two parameters of τ and m are optimized by the combination of autocorrelation and the advantages of Cao’s algorithm.

Fig. 2 is flowchart of the simulation for the signal de-noising. Firstly, the model of signal is based on the noise-free signal f_th(t), the coupling signal f₀(t) is formed by the combination of a random noise signal ξ_G‑noise(t) and impulse signal ξ_G‑pulse(t). Fig. 3 is the obtained time-domain diagram of the coupling signal after fitting simulations. And the expression of the simulation signal is shown as

The other parameters are shown in Table 1.

Base on the signal, the operation of de-nosing by the simulation is listed as:

The signals are decomposed of f₀(t) by the principle of the SVD method, and the σ₁ to σ₄ are obtained. Combined with Cao’s algorithm, the optimum parameters are m = 5 and τ = 8. Finally, as shown in Fig. 8, the inverse solution of singular value is used to reconstruct the effective characteristic signal after de-noising. From the figure the reconstructed effective signal is proximity to the pure signal, which indicates that ISVD has the potential for de-noising.

As shown in Table 2, after the de-noising by ISVD. And the SNR = 31.31 dB, RMSE = 0.34 are acquired by Eqs. 15 and 16. It reminds ISVD has a good de-noising potential.

Fig. 9a is shown the experiment layout. Fig. 9b is the photograph of experiment. On the outer side of the gearbox, four measurement points are set as Points I, II, III and IV with four uniaxial acceleration sensors for the collection of vibration signal.

The speed of input is 1500 r/min; the input current of the magnetic powder is P = 38 W; the modulus of gear is m_o = 2 mm. The gear material is 18CrNiMo7 steel.

In this research, a new method of ISVD is proposed for the vibration signal de-noising of gear pitting fault identification. Simulation and experiments have been conducted to prove the efficiency of ISVD. The conclusions are listed as follows: