Research on feature extraction method of spindle vibration detection of weak signals for rapier loom fault diagnosis in strong noise background

Abstract

Fault diagnosis of rapier loom is an inevitable requirement to meet the demand of intelligent manufacturing. Facing the strong noise interference caused by complex working environment, accurate and reliable vibration signal detection of blade loom spindle is the key to realize the rapier loom fault diagnosis. This paper proposes a method to extract the spindle vibration signal of the rapier loom by Adaptive Piecewise Hybrid Stochastic Resonance (APHSR) after the Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN). Firstly, ICEEMDAN is used to pre-process the weak vibration signal containing noise, decompose the signal into multiple IMF components and display the high and low frequency signal characteristics of the original signal. Then, the energy density method and the correlation coefficient method are used to remove high and low noise, respectively, to filter the optimal IMF components, and then the signal containing valid information is reconstructed. Finally, the reconstructed signal is input to APHSR for noise-assisted enhancement after scale transformation to restore the faint vibration signal feature frequencies and achieve effective feature extraction. Through the simulation experiment and the engineering fault experiment analysis, comparing ICEEMDAN-APHSR with CEEMDAN-SR, ICEEMDAN-SR, CEEMDAN-APHSR methods. The difference between the spectrum amplitude, the spectrum amplitude and the maximum noise and the maximum signal to noise ratio (SNR) of the fault feature frequency of the rapier loom spindle bearing increased by 3.3668 dB,1.7205 dB,2.3952 dB, respectively. The results show that ICEEMDAN-APHSR method can accurately extract the fault feature frequency of the spindle bearing of rapier loom, and effectively solves the problem of extracting the weak vibration signal feature of rapier loom in the background of strong noise. This method is beneficial to the future research of rapier loom fault diagnosis, and is of great significance to promote the maintenance of loom equipment and production safety and quality.

Keywords

Weak signal detection ICEEMDAN APHSR feature extraction rapier loom fault diagnosis

1 Introduction

1.1 Fault diagnosis technology

After several decades of development of mechanical fault diagnosis technology, it has produced a large number of diagnostic methods. It is mainly divided into model-based fault diagnosis, data-based fault diagnosis and knowledge-based fault diagnosis. Fault diagnosis mainly studies how to detect, separate and identify the faults in the system. It can be divided into actuator fault, sensor fault and component fault from the perspective of system structure. The article [1] considered actuator failures, sensor faults, and input interference. For a class of random Markovian jump system, the fault tolerance problem is solved by introducing a generalized sliding mode observer. An adaptive resilient control strategy, based on sliding mode control technique and projection operating technique is investigated for MJCPSs with state-dependent FDI attacks simultaneously launched in the channels of actuator and sensor [2]. H. Yang et al. [3] proposed a fault-tolerant controller based on an adaptive neural network to address the FTC problem that is investigated for MJS with simultaneous actuator faults and nonlinearity. However, such methods based on the analytical model rely on the precise mathematical model of the diagnosed object and have a large computational amount. Knowledge-based fault diagnosis is a qualitative model based on prior knowledge of a monitoring process. And its core is an expert system consisting of knowledge bases, databases, and interpretation components. This method does not require mathematical modeling of the system, but it is difficult to acquire knowledge, and the accuracy of fault diagnosis depends more on the richness of expert experience in the knowledge base. Data-based fault diagnosis does not require an accurate analytical model of the process, and directly analyzes and processes the process operation data. With the development of theories such as machine learning and deep learning, related methods have been gradually applied to various fields. In order to accurately predict the state of lithium-ion battery, S.Su et al. [4] proposes a novel capacity prediction method for SOH estimation based on the battery equivalent circuit model, deep learning, and transfer learning.

In order to simulate the battery behavior under different charge and discharge situations, on the basis of establishing the coupled thermoelectric aging battery model, the LHS method and ANN model are used to study the aging state of the battery [5]. This paper [6] realizes the fault diagnosis of wavelet packet decomposition and self-association neural network. This paper [7] uses the energy and entropy of the wavelet packet and ANN as the classifier.

1.2 Feature extration method

In the fault diagnosis of rotating machinery such as rapier loom, the strength and frequency change of a vibration signal can immediately reflect fault information, and the type and position of a fault can be identified by analyzing and processing a vibration signa [8]. However, due to the complex equipment environment and the complex vibrations among different components, the collected vibration signal often contains a large amount of interference noise. When the fault is not obvious, the fault characteristic vibration signal is often submerged by the noise and is difficult to separate, which makes it difficult to efficiently identify and eliminate a fault [9, 10]. In addition, the rapier weaving machine spindle is the power source of the weaving structure, bearing failure and other factors lead to excessive vibration will affect the rapier warp feeding action and weft beating action, further affecting the quality of the fabric. Therefore, it is of great importance to study the method of extracting the weak signal of rapier loom spindle vibration in strong noise background.

In the field of fault diagnosis, the steps of diagnostic method mainly consists of three parts: data collection, feature extraction and state classification. The most important step is the feature extraction of fault data. The quality of feature extraction determines whether the study of fault diagnosis can be further deepened.There are two main methods commonly used for feature extraction: one is to enhance the useful signal-to-noise ratio by removing and suppressing the noise in the original signal, such as wavelet noise reduction and empirical modal decomposition. The second is to enhance the energy of the output signal by using the noise energy to enhance the signal signal-to-noise ratio, such as random resonance theory [11].

Adaptive signal decomposition methods include Empirical Modal Decomposition (EMD), Variational Modal Decomposition (VMD), Singular Value Decomposition (SVD), etc. Among these techniques, EMD is more widely used in feature extraction. Empirical Mode Decomposition (EMD) is a method proposed by HUANG et al to deal with nonlinear and non-stationary signals for the purpose of reducing noise. According to the local characteristics of the vibration signal time scale, the adaptive time-frequency decomposition is carried out to obtain several intrinsic modal functions IMF and residuals. Then through the analysis of each mode component, the characteristic information of the original signal can be more accurately grasped [12]. Yi Zhang et al. [13] used EMD to preprocess the motor vibration signal and then analyzed the WP envelope spectrum of the decomposition result unimodal function, extracting certain nonsmooth motor signal features. Hongwei Lou et al. [14] effectively extracted the characteristics of the effective frequency component of the gearbox by combing EMD algorithm and fractal geometry algorithm, and realized the quantitative characterization and extraction of characteristic values of different gear faults. However, the EMD method also has some problems, such as modal overlap [15], boundary problem, endpoint effect, false component, overenvelope and underenvelope. Ensemble Empirical Mode Decomposition (EEMD) uses the principle of noise summation in the ensemble averaging process to add positive and negative white noise to the original signal to cancel each other out. However, this method has differences in the number of IMFs generated in the decomposition process, which leads to a more difficult matching of the number of IMFs in the calculation of the pooled average. The Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN) method is optimized for the noise problem in IMF component. In the process of CEEMDAN decomposition, white noise was added to the original signal to form a new input signal. In order to solve the problems of large component reconstruction error and inobvious fault feature extraction in EEMD. The CEEMDAN decomposition process adds white noise to the original signal to form a new input signal. The EMD decomposition is performed on the reconstructed signal, and the ensemble average calculation of the IMF components of each order is obtained as the first IMF of the CEEMDAN decomposition.White noise is added to the remaining residuals. This method reduces the number of screening. Qiang Zhu et al. [16] effectively extracted prominent fault features of rolling bearings by combining the adaptability and integrity of CCEMDAN decomposition signal with the efficient identification of fault areas by rapid spectrum kurtoir. Shipeng Chen et al. [17] decomposed the original vibration signal by CEEMDAN method, and extracted the proportion of the energy of the characteristic modal component to the total energy of the signal and the energy entropy as the fault characteristic component. Colominas et al. [18] proposed Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN). In this method, a special kind of white noise E_k (ω⁽ⁱ⁾) is added, that is, K IMF components obtained by Gaussian white noise through EMD decomposition. Haonan Wang et al. [19] applied ICEEMDAN and Support Vector Machines (SVM) to improve the accuracy of planetary gearbox composite fault classification.

However, the weak fault signal characterization of the rapier loom is not obvious, and the noise signal contains high energy when detecting the vibration signal of the spindle. A single noise reduction technology is not enough to effectively extract the signal features, affecting the fault detection effect, so the stochastic resonance is used to enhance the weak signal, to make the feature extraction more obvious. Stochastic resonance is a relatively common physical phenomenon generated by noise in a nonlinear system. It cannot be detected without three specific elements: a nonlinear system, a noise signal and a weak driving signal [20]. In a given nonlinear system, noise can act in conjunction with a useful periodic signal, allowing the weak signal to be augmented by the output. In mechanical system fault diagnosis, the value of vibration frequency, amplitude or noise intensity in the vibration signal may be much larger than 1, i.e., a large parameter signal. Therefore, it is important to study stochastic resonant systems with large parameter signals [21]. MitaimS et al. [22] proposed the concept of adaptive stochastic resonance by adjusting the magnitude of the input noise parameters of the stochastic resonance system. The measure is how strong the input noise of the random resonant system is when the system output signal can most clearly represent the characteristic frequency. Yi, Qin et al. [23] developed a new bistable model in order to improve the computational speed and the performance of weak feature detection. It is based on binary wavelet transform and least squares system parameter solution to implement an adaptive fast stochastic resonance iterative algorithm, and finally extract the fault characteristics of the rotor system. In terms of detection under interference, large parameters, and multiple signals, it is still difficult to extract weak signals effectively, and the application of adaptive stochastic resonance theory still has much to be improved.

According to the vibration characteristics and fault signal characteristics of rapier loom spindle bearings, we propose a method to extract the spindle vibration signal of the rapier loom by Adaptive Piecewise Hybrid Stochastic Resonance (APHSR) after the Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN). The ICEEMDAN-APHSR is based on the fault signal of the background of strong noise. The noise-containing signal is Hilbert transformed and input into the ICEEMDAN system for preprocessing, and the self-energy density method and correlation coefficient method are used to automatically obtain the optimal intrinsic mode function. The optimal IMF group is transformed into the APHSR system, and the feature signal is extracted to complete the detection of the spindle fault signal of the rapierloom.

The remainder of this paper is structured as follows: The second part analyzes the vibration characteristics of rapier looms. The third part proposes an ICEEMDAN-APHSR system for rapier loom spindle weak fault signal extraction based on the study of EMD theory and SR theory. In the fourth part, the proposed ICEEMDAN-APHSR system is simulated to verify the superiority of ICEEMDAN-APHSR in rapier weaving machine spindle rolling bearing fault feature extraction. In the fifth part, the proposed feature extraction method is experimentally validated and compared with other methods to illustrate the effectiveness of ICEEMDAN-APHSR feature extraction. The sixth part provides a summary andoutlook.

2 Analysis of the vibration characteristics of the rapier loom

In order to clarify the vibration components and vibration frequency range of rapier looms, we analyzed the vibration characteristics of rapier looms. The rapier loom is the most widely employed shuttleless loom. It can realize yarn weaving at high speeds. The main difference between the rapier loom and other looms is that the weave is mainly completed by the movement of a rapier with weft insertion. According to the number of rapiers, rapier looms can be divided into single rapier looms and double rapier looms. The vibration characteristics of rapier looms, from the mechanical structure and power source, are attributable to the shafting movement of the rotating shaft, the movement of the rapier and the vibration of the rolling bearing.

2.1 Analysis of the main structure of the rapier loom

At present, the main motion structure of rapier looms in the market mainly includes a spindle motion and weft insertion motion controlled by the main motor and a let-off motion and coiling motion controlled by the servo motor [24]. The structure of rapier loom is shown in Fig. 1. The main functions of each module are shown as follows:

Fig. 1

Structural diagram of rapier loom.

Spindle mechanism: The spindle mechanism is responsible for driving the spindle rotation, to complete the movement of each rotating axis. At the same time, the spindle structure drives the lever to pass the yarn through the spindle to complete the latitude introduction movement.

Weft insertion mechanism: The weft insertion mechanism itself does not have a separate control effect, but mainly depends on the power source of the spindle mechanism. It drives the double sword lever to pass the yarn through the spindle port.

Delivery mechanism: The delivery mechanism is a shaft used to drive the winding yarn. The servo motor drives the transmission mechanism to lead the yarn to the latitude mechanism.

Crimping mechanism: The crimping mechanism is responsible for pulling out the cloth formed by the interweaving of warp and weft and winding it onto the cloth winding rod to form a coiled cloth.

2.2 Dynamic model and vibration analysis of the rapier loom rotation axis

Although the rapier loom appears to be a complex system, the rapier loom is a multirotating shaft device. Vibration analysis is performed on the rotary axis of the rapier loom. Figure 2 shows a simplified model of the rotary axis of a rapier weaving machine. It mainly includes a winding shaft vibration model, a main shaft vibration model and a warp feeding shaft vibration model.The vibration system of the rotating shaft of a rapier weaving machine consists of three independent systems, namely, a spindle vibration model, a winding vibration model and a warp feeding vibration model. The body mass block in the system is in direct contact with the ground, so the effect on the warp yarn is slight. Therefore, the whole system model is considered as three independent rotating shaft vibration models. The independent rotary axis vibration models are investigated.

Fig. 2

Simplified diagram of rapier loom vibration model.

The conventional calculation methods for calculating the inherent frequency of a rotating shaft include Dunkley’s method, Ricker’s method and the self-zag method. Among the main parameters affecting the calculation are the equivalent mass and equivalent stiffness. The results of the equation for the equivalent mass are shown below: $\bar{m} = \sum_{i = 1}^{k} m_{i} {(\frac{V_{si}}{V_{B}})}^{2} + \sum_{i = 1}^{k} J_{si} {(\frac{W_{i}}{V_{B}})}^{2}$ (1)

Where W_i is the angular velocity of the i-th member, V_si is the velocity of the i-th component centroid si, m_i is the centroid mass of the i-th component, J_si is the inertia of rotation to the centroid axis, V_B is the central velocity.

The equivalence of stiffness is calculated as follows: $\bar{K} = [\begin{matrix} k_{1} + k_{2} & - k_{2} \\ - k_{2} & k_{2} + k_{3} \\ ⋱ \\ ⋱ & - k_{i} \\ - k_{i} & k_{i - 1} + k_{i} & - k_{i + 1} \\ - k_{i + 1} & k_{i} + k_{i + 1} \end{matrix}]$ (2)

Where each stiffness is calculated as: $k = \frac{48 EI}{l^{3}}$ (3)

Where EI is a constant, and l is the component length.

The equivalent mass and equivalent stiffness of the rapier loom calculated by Equations (1) and (2) are used to calculate the inherent frequency. $f = \frac{1}{2 π} \sqrt{\frac{k}{m}}$ (4)

Where, k is equivalent stiffness and m is equivalent mass.

The rapier loom model for the field experiment is QJH-910. Parameters such as the size, weight and assembly strength of the spindle of the field rapier loom are summarized according to parameters such as the size, weight and assembly strength of the spindle and warp beam of the field rapier loom. The structural parameters of the rapier loom spindles were obtained as shown in Table 1 below.

Table 1

Rapier loom spindle structure parameters

Parameter	Numerical value
Spindle diameter d/mm	100
Spindle length l/mm	2500
Spindle Density ρ/g/cm³	7.93
Modulus of elasticity E/Gpa	194.020
Poisson’ ratio	0.247

The warp beam material of the rapier weaving machine is aluminum alloy zl104. The structural parameters of the rapier weaving machine warp beam are listed in Table 2 below.

Table 2

Rapier loom spindle structure parameters

Parameter	Numerical value
Inner diameter/mm	152.7
Outer diameter/mm	300
Maximum outside diameter/mm	765
Outer shaft length/mm	2500
Inter shaft length/mm	2435
Warp Density/g/cm³	2.65
Modulus of elasticity/Gpa	69
Poission’s ratio	0.34

According to the actual measured dimensions of the spindle and warp beam, the calculated equivalent mass and equivalent stiffness are inserted into Equation (4), and the rapier loom spindle inherent frequency of 112.7 Hz and the warp beam inherent frequency of 71.2 Hz can be obtained.

2.3 Dynamic model and vibration analysis of the rapier loom rolling bearing

Rolling bearings are important support parts of the rapier loom spindle system as well as some of the rotary shafts. Fault of rolling bearing often occurs in the inner ring, outer ring, rolling body and cage. Each part of the fault occurs when the fault characteristics are different, according to the parameters of the rolling bearing can be calculated separately for each fault characteristic frequency [25]. The bearing detection uses the principle of periodic rotation and collision of the bearing. Thus, the periods corresponding to different types of faults and bearing resonance are different, while the rollers in the bearing have less irregular collision vibration than the balls, and the collision noise will be reduced after being lubricated by the grease in the bearing. Therefore, it is more difficult to extract an effective fault signal. It is necessary to observe the characteristic signal of each part of the bearing by various means, such as a spectrum diagram.

The failure frequency of the rolling bearing on the inner ring is: $BPFI = \frac{n}{2} (1 + \frac{d}{D} cos θ) f_{0}$ (5)

The failure frequency of the rolling bearing on the outer ring is: $BPFO = \frac{n}{2} (1 - \frac{d}{D} cos θ) f_{0}$ (6)

The failure frequency of the rolling element of the rolling bearing is: $BSFR = \frac{D}{2 d} f_{0}$ (7)

The failure frequency of the cage outer ring of the rolling bearing is: $BSFC = \frac{1}{2} (1 - \frac{d}{D} \cos θ) f_{0}$ (8) where D is the diameter of the bearing, d is the diameter of the rolling element, θ is the contact angle, and n is the individual of the rolling element and f₀ is the rotation frequency of the bearing. Table 3 shows the parameters of rolling bearings for rapier weaving machines.

Table 3

Parameters of rapier rolling bearing

Parameter	Numerical value
Bearing diameter/mm	39
Rolling element diameter/mm	7.938
Contact angle/°	0
Number of rolling elements	9
Bearing rotation frequency/Hz	10

Put the rapier loom rolling bearing parameters into formula 5, 6, 7, 8 to obtain the vibration frequency of the sword pole loom rolling bearing, as shown in Table 4.

Table 4

Vibration frequency of rapier loom rolling bearing

Parameter	Numerical value
Outer ring fault frequency/Hz	71.6
Inner ring fault frequency/Hz	108.32
Rolling element fault frequency/Hz	47.09
Cage failure frequency/Hz	7.91

2.4 Dynamic model and vibration analysis of rapier loom rapier

The rapier weaving machine is driven by the rapier. Therefore, studying the rapier is beneficial to ensure normal weaving efficiency and, according to the weft guiding mechanism, to establish the weft guiding process at any position. The mechanical model of the rapier is shown in Fig. 3.

Fig. 3

Mechanical model of rapier.

As is shown in Fig. 3, A is the end of the rapier, which is fixed with the mounting plate; C is the end of the rapier. The rapier is subjected to a distributed load of its own gravity of q, and the fraction of the yarn friction force in the direction of the vertical rapier at the rapier head is p. The deformation of the head of the rapier is:

$y = \frac{{pL}^{3}}{3 EI} + \frac{{qL}^{4}}{8 EI}$ (9)

Where L is the length of the pole; q is the distributed load of the blade to its own gravity; p is the force of the vertical direction of the blade; E is the elastic modulus of the blade; and I is the moment of inertia of the pole.

According to the principle of equal potential energy, the equivalent stiffness k of the rapier can be found at the time of latitude attraction, as shown in Fig. 4.

Fig. 4

Equivalent stiffness of rapier schematic diagram.

$\int_{0}^{y_{\max}} kydy = \int_{0}^{1} \frac{M_{1} + M_{2}}{2 EI} dx$ (10)

Where M₁ is the bending moment caused by the load p and M₂ is the bending moment caused by the distributed load q, E is the elastic modulus of the pole; I is the moment of inertia of the pole. $M_{1} = - p (L - x)$ (11) $M_{2} = - p {(L - x)}^{2} / 2$ (12)

Where, L is the sword pole length, and x is the moving distance.

By substituting M₁ and M₂ into the above equation, the integration is organized to obtain: $k = \frac{3 EI}{L^{3}}$ (13)

According to the vibration principle, the rapier vibration elicits a single-degree-of-freedom system vibration problem, whose vibration model is shown in Fig. 5, where k is the rapier equivalent stiffness, m is the mass of the rapier, and its rapier intrinsic frequency is w_n.

Fig. 5

Single degree of freedom vibration model.

As is shown in Fig. 5, the mechanical analysis shows that when the rapier mass is in equilibrium, only its own gravity and the spring force are received. Assume that the position of the rapier at this time is the initial position y₀. When the system equilibrium is destroyed, the system combined force is not zero. The rapier will be displaced by both the spring force and gravity, causing the rapier to vibrate freely. When the equilibrium is taken, the initial position of the rapier is set as the far point, the downward motion is positive, and the vibration displacement is y. The differential equation for the vibration of the single-degree-of-freedom system is shown in the following equation.

$m \ddot{y} - ky = 0$ (14)

Solving the differential Equation 14 yields $\ddot{x} + \frac{k}{m} x = 0$ (15) $w_{n} = \sqrt{\frac{k}{m}}$ (16)

Where k is the rapier equivalent stiffness, m is the mass of the rapier.

From the equation 16, it can be obtained that the inherent frequency of the rapier is related to the rapier’s own mass and stiffness coefficient.

3 Research on the feature extraction method of the weak fault signal of rapier loom spindle

Rapier loom signal is nonlinear and multivariable. In this chapter, empirical mode decomposition theory and stochastic resonance theory are studied. Through theoretical analysis and simulation comparison, it is concluded that iceemdan decomposition is relative to ceemdan decomposition. It runs faster, improves the problems of modal aliasing and false components, and has a higher degree of restoration of signal features. Aphsr solves the saturation problem of stochastic resonance and effectively extracts the characteristics of rapier loom vibration signals through adaptive parameter adjustment.

3.1 Empirical modal decomposition and the proposed improvement method

1) EMD THEORY

Because the characteristics of vibration signals are nonlinear and nonstationary, Huang et al. proposed the Hilbert–Huang transform to analyze vibration signals. The Hilbert–Huang transform combines two features—empirical mode decomposition and the Hilbert transform—to realize the adaptive parameter time-frequency analysis.

The EMD decomposes the time series signal into multiple intrinsic mode functions (IMFs) and a residual trend term according to the scale of its own signal fluctuation. The IMF of different layers represents the fluctuation and change in the signals in different frequency bands in the original signal, and the final residual trend term reflects the trend characteristics of slow change in the original signal. The eigenmode function generated in the decomposition process must meet the following two conditions: (1) In the whole data signal interval, the number of extreme points and zero crossings must be equal or have no more than one difference; (2) At any point, that is, at any time, the envelope defined by the local maximum point and the envelope defined by the local minimum value of the signal, that is, the mean value of the upper and lower envelope, is zero.

The decomposition process of EMD for the signal X(T) is:

Let h₀ (t) = X (t), The h₀ (t) signal is calculated to obtain the maximum value point to form a new function on the envelope e₊ (t) and obtain tne minimal value point to form a new function on the envelope e_- (t);

Averaging of the upper and lower envelopes to obtain m₀ (t);

Subtracting the envelope mean m₀ (t) from h₀ (t) to obtain the signal to be tested h₁ (t);

Check whether the signal h₁ (t) can be used as the IMF component of the EMD decomposition, and if it is not satisfied, treat h₁ (t) as the original input signal h₀ (t), repeat the process of steps (1), (2) and (3), we can obtain h₂ (t), h₃ (t), ... , h_i (t) and h_i (t) can be used as the IMF component. At this point, note that the first imf₁ = h_i (t);

Subtract the first IMF from the original signal to get the first residual;

Using the residual as the original signal for the decomposition h₀ (t) = r₁ (t). Repeat steps (1) to (5) until the standard deviation of two adjacent signals to be tested h_i-1 (t) and h_i (t) is less than a centain set value;

The final n IMFs and residuals r_n (t) are obtained.

2) Envelope Spectrum Analysis

Envelope spectrum analysis, also known as Hilbert demodulation, is a demodulation method for vibrational signals.The process of envelope analysis is shown in Fig. 6. The process is as follows: after EMD of the original signal, the Hilbert transform is utilized to transform the decomposed IMF components to obtain their instantaneous frequencies and then synthesize the instantaneous spectra of all IMF components. By analyzing the Hilbert spectrum, the transformation law of signal amplitude with time and frequency can be obtained.

Fig. 6

Envelope Spectrum Analysis Process.

For any time series, the Hilbert transform is defined as: $H [X (t)] = \frac{1}{π} P \int_{- \infty}^{+ \infty} \frac{X (τ)}{t - τ} d τ = X (t) * (\frac{1}{π t})$ (17)

where P represents the Cauchy principal value and * represents the convolution operation.

At this time, an analytical signal Y (t) can be constructed from X (t) and H [X (t)] $Y (t) = X (t) + iH [X (t)] = a (t) e^{i φ (t)}$ (18)

Where $a (t) = \sqrt{X^{2} (t) + {(H [X (t)])}^{2} (t)}$ (19) $φ (t) = \arctan [\frac{H [X (t)]}{X (t)}]$

Where a (t) is the instantaneous amplitude of X (t), i.e., the resolved signal; φ (t) is the instantaneous phase of X (t).

Instantaneous frequency is defined as the reciprocal of the analytical signal phase, and its physical meaning represents the speed of the vector amplitude. The instantaneous frequency is defined as: $f_{i} (t) = \frac{1}{2 π} ω_{i} (t) = \frac{1}{2 π} \frac{d φ_{i} (t)}{dt}$ (20)

The analyzed original signal X (t) can be expressed as: $X (t) = \sum_{i = 1}^{n} a_{i} e^{i φ_{i} (t)} = \sum_{i = 1}^{n} a_{i} (t) e^{i \int ω_{i} (t) dt}$ (21)

Through the above Hilbert transform of the original signal, the analytical signal is obtained, then the modulus of the analytical signal is obtained, and the envelope signal is obtained. Finally, the envelope spectrum is obtained by Fourier transform of the envelope signal.

3) The proposed ICEEMDAN principle and simulation verification

The ICEEMDAN method adds special white noise E_k (ω⁽ⁱ⁾), that is, the k-th IMF component of Gaussian white noise after EMD decomposition. The local mean of the signal plus noise is calculated for each modal component, and the decomposed IMF is defined as the difference between the residual signal and the local mean. The flow chart of the ICEEMDAN algorithm is shown in Fig. 7, where it is defined as the composite signal to be decomposed by X (n), andE_k (·) is defined as the k-th IMF obtained by EMD, representing a series of different Gaussian white noise elements, where M (·) represents envelope calculation.

Fig. 7

ICCEMDAN algorithm flow chart.

The ICEEMDAN algorithm is expressed as follows:

(1) Adding special white noise to the original signal E_k (ω⁽ⁱ⁾).

$x^{i} (n) = x (n) + β_{0} E_{1} (ω^{(i)}), i = 1, 2, 3 \dots, I$ (22)

(2) Calculate residual. $r_{1} = 〈 M (x^{i} (n)) 〉$ (23)

(3) Define the first modal as IMF₁ = x - r₁.

(4) Build new functions with engagement. $x^{i} (n) = r_{1} + β_{1} E_{2} (ω^{(i)}), i = 1, 2, 3 \dots, I$ (24)

(5) The second residual is calculated by Equation 3.21. $r_{2} = 〈 M (r_{1} + β_{1} E_{2} (ω^{(i)})) 〉$ (25)

(6) Calculate the second mode IMF₂ = r₁ - r₂. $r_{j} = 〈 M (r_{j - 1} + β_{j - 1} E_{j} (ω^{(i)})) 〉$ (26)

(7) For j = 3,4,⋯, J,respectively, the jth residual is removed, and obtain the jth mode IMF_j = r_j-1 - r_j.

Repeat step (7) until all IMF_s.

To verify the effectiveness of ICEEMDAN method for signal decomposition. The simulated signal S(t) consists of a continuous stationary signal S₁ (t) that is superimposed with the signal S₂ (t) with a higher frequency gap. The simulation formula of the vibration signal is expressed as follows: $S (t) = {\begin{matrix} S_{1} (t) = \sum \sin (2 π * 71 * k * t), 3 < t < 4 \\ S_{2} (t) = \sum \sin (2 π * 112 * k * t), 0 < t < 4 \end{matrix}$ (27)

Simulation research is carried out by software MATLAB. In the simulation system, the sampling frequency of the input signal is set to 2000 Hz, the sampling time is 4 S, and the number of sampling points is 8000. The time domain diagram of the simulated signal is shown in Fig. 8.

Fig. 8

Time domain diagram of the simulated signal.

The simulated signals are decomposed by CEEMDAN and ICEEMDAN respectively. The parameter settings are as follows: the noise weight ɛ is set to 0.2, the number of noise additions is 100, and the maximum number of iterations is 500. the sampling frequency was 1.

As shown in Fig. 9, CEEMDAN decomposes a total of 14 IMF components. There are IMF1 and IMF2 are false components, which can be decomposed into high-frequency signals in the third IMF. As shown in Fig. 10, ICEEMDAN decomposes a total of 12 IMF components. IMF2 can show the characteristics of high-frequency components in the original signal. It is faster than CEEMDAN decomposition and does not obtain false IMF, which improves the problem of modal aliasing. This shows that ICEEMDAN decomposition has a high degree of reduction for the vibration signal of rapier loom.

Fig. 9

IMF diagram obtained by ceemdan decomposition.

Fig. 10

IMF diagram obtained by iceemdan decomposition.

3.2 Adaptive piecewise hybrid stochastic resonance system theory

To overcome the output saturation phenomenon of the classical bistatic stochastic resonance (SR) method and the segmented hybrid stochastic resonance (PHSR) which may lead to inaccurate results when identifying the characteristic frequencies in a strong noise background. In this paper, we propose the adaptive segmented hybrid stochastic resonance method (APHSR). This method can improve the rapier weaving machine spindle weak signal feature extraction ability.The introduced parameter μ is optimized by adaptive parameter adjustment, and the signal-to-noise ratio is used as the evaluation criterion to obtain the best output of the system.

1) Random resonance theory

The principle of stochastic resonance is to input the noise signal and the weak signal to be measured together into the nonlinear system. Under the joint cooperation of the nonlinear system and the noise signal, the high-frequency part of the energy is continuously concentrated to the low-frequency useful signal, while the high-frequency energy is continuously cut and the low-frequency signal energy is continuously increased, so as to achieve a higher signal-to-noise ratio output.

In the SR system in the bistable stochastic resonance system, the Langzwan equation can be expressed as: $\frac{dx}{dx} = - dU (x) / dt + s (t) + n (t)$ (28) where U (x)is the potential function,s (t) = Acos(2πf₀t + φ) is the input signal of the random resonance, and A is the critical amplitude of the input signal, n (t) is a zero-mean Gaussian noise that also satisfies E (n (t) n (t + τ)) = 2Dδ [t - τ], where D is the noise intensity and E denotes expectation.

The potential function of the Langzwan equation in a bistable stochastic resonant system can be expressed as: $U (x) = - \frac{a}{2} x^{2} + \frac{b}{4} x^{4}$ (29)

To illustrate the saturation characteristics of SR, simulations were performed with the following values. Set the simulation input signal s (t) = Acos(2πf₀t), system parameter a = b = 1, the minimum detection resolution for signal frequency f₀ = 0.01Hz, set the noise intensity of the input signal to 0 V and 15 V in sequence, set the amplitude of the input signal to 0.4 V, 0.8 V, 1.2 V, 1.6 V, 2.0 V in sequence. As shown in Fig. 11 (a), the particles move back and forth around the steady state between a potential well and within a certain range. However, when A > 0.4 V, with increasing amplitude A, the output signal amplitude passing through the system hovers around 1.5 V. At this time, there is a small change in the amplitude of the output signal, which is known as the saturation phenomenon. As shown in Fig. 11 (b), although there is a large amount of noise in the graph, the saturation phenomenon that exists can still be observed.

Fig. 11

(a) The noise intensity of the 0 V SR diagram; (b) The noise intensity of the 15 V SR diagram; (c) The noise intensity of the 0 V PHSR diagram; (d) The noise intensity of the 15 V PHSR diagram.

To overcome the saturation phenomenon of SR, Wang et al. [26] proposed piecewise hybrid stochastic resonance, which is expressed as follows: $U (x) = {\begin{matrix} - \frac{\sqrt{a^{3} / b}}{4 (μ - 1)} (x + μ \sqrt{a / b}) x < - \sqrt{a / b} \\ - \frac{a}{2} x^{2} + \frac{b}{4} x^{4} - \sqrt{a / b ⩽} x ⩽ \sqrt{a / b} \\ \frac{\sqrt{a^{3} / b}}{4 (μ - 1)} (x - μ \sqrt{a / b}) x > \sqrt{a / b} \end{matrix}$ (30)

where the parameter in the μ piecewise hybrid stochastic resonance system is a > 0, b > 0, μ ≠ 1, and the slope of the potential function of the piecewise hybrid stochastic resonance system can be obtained from the following formula: $dU (x) / dt = {\begin{matrix} - \frac{\sqrt{a^{3} / b}}{4 (μ - 1)} x < - \sqrt{a / b} \\ - ax + b x^{3} - \sqrt{a / b ⩽} x ⩽ \sqrt{a / b} \\ \frac{\sqrt{a^{3} / b}}{4 (μ - 1)} x > \sqrt{a / b} \end{matrix}$ (31)

The same simulated signal is fed into the PHSR to illustrate its non-saturation characteristics and its drawbacks in a strong noise background. As shown in Fig. 11(c), the potential energy output via the PHSR system increases with the signal amplitude and there is no output saturation. As shown in Fig. 11(d), when the noise intensity of the PHSR system is D = 15 V, the contours are blurred by the noise although they can be recognized, which may lead to inaccurate rapier loom spindle signal feature extraction.

2) The proposed and applied aphsr method

Due to the high frequency of spindle failure signals on rapier weaving machines, it is necessary to scale it before entering the stochastic resonance system to convert the large parameter frequency signal into small parameter. Variable scale stochastic resonance is actually a linear transformation of the large frequency of the signal to be measured so that its frequency is compressed to meet the small parameter condition under adiabatic approximation theory. Then, the modified signal is input into the nonlinear system, after which the output signal of the stochastic resonance system is obtained. Finally, the same compression scale as before is used to restore the output signal.The system diagram of variable-dimension piecewise mixed stochastic resonance is shown in Fig. 12.

Fig. 12

Variable-dimension piecewise mixed stochastic resonance.

The original signal to be measured is first integrated at a variable scale, and a frequency compression scale ratio of k is set, after which a secondary sampling frequency is determined using k and the data sampling frequency, and the operation process is: $x_{2} (t) = Asin (2 π f_{0} \frac{t}{f_{s}}) = Asin (2 π (\frac{f_{0}}{k}) (\frac{t}{f_{s} / k}))$ (32)

Where f₀ is the signal frequency, A is the amplitude, f_s is the sampling frequency, and k is the frequency compression scale ratio.

Based on the above theory, the actual simulated signal is scaled compressed to achieve segmental hybrid random resonance method processing. The input signal to the simulation is set to be s (t) = Asin(2πf₀), where the frequency of the signal is f₀ = 150Hz and the amplitude is A = 0.1. The noise signal $n (t) = \sqrt{2 D} ɛ (t)$ , where D is the noise intensity and ɛ (t). is the Gaussian white noise. Figure 13 illustrates the time domain plot and spectrum of thissignal.

Fig. 13

Input signal time domain and frequency spectrum diagram.

If this large-frequency signal is directly processed by the segmented hybrid random resonance method, the system parameters are a = 1, b = 1, μ= 2, and the time domain and spectrum of the output signal are shown in Fig. 14.

Fig. 14

Segmented mixed stochastic resonance output signal in the time domain and frequency spectrum diagram.

It can be directly observed from Fig. 14, after being processed by the segmented hybrid stochastic resonance system, the change trend of the signal is increasing, and the signal characteristics are more divergent. At this time, it is meaningless to extract the characteristic frequency of the signal. This also proves that the segmented hybrid stochastic resonance system can not directly process the signal with excessive frequency.

The simulation signal of the same frequency is processed by the variable scale segmented hybrid stochastic resonance system, as shown in Fig. 15.

Fig. 15

Variable-dimension piecewise mixed stochastic resonance output time-domain diagram and spectrum diagram.

It can be seen from Fig. 15 that the signal shows periodic characteristics after variable scale segmented hybrid stochastic resonance processing. From the signal spectrum in the figure, it can be seen that relative to its surrounding characteristic frequency, when the characteristic frequency is 150hz, the signal has an obvious spike. It can be concluded that in order to extract the weak signal from the rapier loom spindle, a scale transformation of the signal is required first.

In the APHSR system, the signal-to-noise ratio is commonly used to evaluate the processing effect of random resonance. The signal-to-noise ratio (SNR) was first introduced by Fauve and Heslot in their experiments studying Schmitt trigger circuits. It represents the energy intensity, i.e., the ratio of power, of the active signal to the noisy signal in the overall signal. The unit is expressed in decibels (dB) [27]. In the study of bistable stochastic resonance systems, if the input signal is periodic, the most commonly used evaluation method is signal-to-noise ratio. The formula for calculating SNR is: ${SNR}_{out} = 10 {log}_{10} (\frac{S (f_{0})}{N (f_{0})})$ (33)

Where f₀ is the signal frequency, S (f₀) is the signal power, and N (f₀) is the noise power.

The system is adaptive because it automatically adjusts to obtain the best output during the analysis and processing. The most important thing in the adaptive system is the measurement standard of the optimal effect of the system and the adjustable parameters. The minimum mean square error method (LMS) is used to determine the minimum critical amplitude. The critical amplitudes of APHSR theory should satisfy: $A = {\begin{matrix} \frac{\sqrt{a^{3} / b}}{4 (μ - 1)}, x < - \sqrt{a / b} \\ \sqrt{\frac{{4 a}^{3}}{27 b}}, - \sqrt{a / b} ⩽ x ⩽ \sqrt{a / b} \\ \frac{\sqrt{a^{3} / b}}{4 (μ - 1)}, x > \sqrt{a / b} \end{matrix}$ (34)

The flow chart of APHSR is shown in Fig. 16, a, b, μis the parameter of piecewise hybrid stochastic resonance system. i, j, k are the cycle times of parameter a, b, μrespectively. SNR is the signal-to-noise ratio of the system output signal, SNR_max is the row maximum value of matrix SNR;A is the critical amplitude, A_min is the minimum critical amplitude of the system. The specific parameter adjustment process is as follows:The system first gives the cycle step size and sequence of parameters a, b, μ. during each calculation, fix the value of parameter u, adjust the value of parameters a and b, calculate the signal-to-noise ratio of the system output and store it in the three-dimensional matrix. When calculating each two-dimensional matrix, first fix the value of parameter a, adjust the value of parameter b, calculate the signal-to-noise ratio of the system output and store it in the two-dimensional matrix. When the parameter μ cycle is completed, a three-dimensional array is obtained. Then, calculate the maximum signal-to-noise ratio of each row in each two-dimensional value to form a new three-dimensional data, and calculate the maximum signal-to-noise ratio of each column in each two-dimensional value to form a new two-dimensional matrix. Then calculate the maximum signal-to-noise ratio of the column in the two-dimensional matrix and its corresponding critical amplitude, compare the magnitude of the critical amplitude in the column vector, and obtain the minimum critical amplitude A_min of the system. According to A_min, the system uses the reverse positioning method to determine the optimal parametersa, b, μ and SNR_max.

Fig. 16

Flowchart of the APHSR method.

3) An Iceemdan-APHSR system model for weak signal detection of loom spindle

The system model studies the vibration fault signal of the spindle under strong noise. In this paper, the optimal intrinsic mode function (in-trinsic modal function, IMF) group is automatically obtained by the self-energy density method and correlation coefficient method. After transforming the optimal IMF group, the input segment mixed random resonance system, extract the feature signal after the scale recovery, and complete the bearing fault signal detection. Maximum signal to noise ratio (signal to noise ratio, SNR) is the main reference to measure the quality of stochastic resonance, but a single pursuit of signal to noise ratio is the largest, cannot meet the parameter requirements of bistable system to achieve the best stochastic resonance. Therefore, by combining the critical amplitude as the common evaluation standard of the optimal parameters, the optimal parameters are automatically obtained and fed back to the system, so that the system is in the optimal stochastic resonance state. The frequency range of the bearing fault signal is usually greater than 1 Hz, and the frequency does not meet the adiabatic approximation condition, so the stochastic resonance cannot be realized. Therefore, this paper adopts the secondary sampling algorithm, and the signal frequency range is compressed to 0∼1 Hz, and then, the optimal output signal is obtained through the APHSRsystem.

The ICEEMDAN-APHSR method to extract the weak fault feature of the noise signal is shown in Fig. 17. The main process is as follows:

Fig. 17

ICEEMDAN-APHSR feature extraction model.

(1) ICEEMDAN system pre-processing: The envelope signal is first obtained by demodulating the Hilbert transform. Then ICEEMDAN is used to decompose the signal into multiple IMF components, showing the high and low frequency signal characteristics of the original signal. The energy density method and the correlation coefficient method are used to remove high and low noise, respectively, to filter the optimal IMF components, and to reconstruct the signal containing valid information.

(2) APHSR system feature extraction: Scale transform the reconstructed signal and input it into APHSR. The optimal parameters of the system are calculated using the signal-to-noise ratio as an evaluation criterion and the inverse localization method. After amplification and recovery of the characteristic frequency of the output signal, the weak fault characteristics of the rapier loom in the output signal are separated from the noise background to achieve effective fault feature extraction in a strong noise background.

4 The ICEEMDAN-APHSR simulation of rapier weaving machine spindle weak fault signal feature extraction

In order to verify the superiority of iceemdan-aphsr in fault feature extraction of rapier loom spindle rolling bearing, the fault model of rolling bearing is introduced in this simulation experiment. The fault model simulates the impact signal generated when the bearing fails, and adds white noise to simulate the strong noise in the vibration detection of rapier looms in complex factories. Its simulation signal can be expressed as: ${\begin{matrix} x (t) = s (t) + n (t) = \sum_{i} A_{i} h (t - iT) + n (t) \\ h (t) = \exp (- Ct) \cos (2 π f_{n} t) \\ A_{i} = 1 + A_{0} \cos (2 π f_{r} t) \end{matrix}$ (35)

Where x (t) is the simulation signal, s (t) is the periodic shock signal, and n (t) is the Gaussian white noise; A_i is the amplitude of the signal, f_n is the resonance frequency, and f_r is the frequency conversion.

This simulation experiment is based on the rolling bearing model in MATLAB. The parameter settings in the model are as follows: the sampling frequency is 16000 Hz, the number of sampling points is 4096, the conversion frequency is 20 Hz, the resonance frequency is 4000 Hz, the amplitude is 0.3 V, and the fault frequency is 70 Hz. According to formula (35), the bearing fault simulation signal is shown in Fig. 18.

Fig. 18

Original signal domain diagram and envelope spectrum diagram.

In order to simulate the high interference environment of rapier loom, Gaussian white noise with signal-to-noise ratio of -13db is added to the original simulation signal as external noise interference. Figure 19 simulates the real operating environment of a rapier loom, when the characteristic signals are drowned in noise, which affects the early diagnosis of faults.

Fig. 19

Noisy bearing fault time domain diagram and envelope spectrum diagram.

ICEEMDAN decomposition is carried out for the noisy fault signal to obtain the IMF component diagram as shown in Fig. 20, and the IMF characteristics as shown in Fig. 21 are obtained by analyzing each IMF.

Fig. 20

Analog signal decomposition based on ICEEMDAN.

Fig. 21

IMF characteristics.

As is shown in Fig. 21, after iceemdan decomposition of the simulation signal, 10 IMF are obtained, and the frequency characteristics of each IMF are analyzed. IMF1, IMF2 and IMF3 contain high-frequency components higher than 1 kHz, and the IMF containing high-frequency signals is filtered out during reconstruction. The reconstructed signal is shown in Fig. 22.

Fig. 22

Reconstruct the time domain diagram and envelope spectrum of the fault signal.

As shown in Fig. 22, by analyzing the envelope spectrum of the reconstructed signal, the amplitude of the characteristic frequency of the reconstructed signal is 0.025 V, and the characteristic frequency is completely integrated into its adjacent frequencies, so the characteristic frequency cannot be directly obtained. Finally, the reconstructed signal was input into APHSR system, and the adaptive algorithm was used to automatically obtain the following parameter values: a = 0.2, b = 0.1, μ = 1.1. At this time, the time domain diagram and envelope spectrum diagram of APHSR system output signal are shown in Fig. 23. As can be seen from the figure, after the signal passes through APHSR, the characteristic frequency amplitude is 0.09 V, which is 0.07 V higher than that of the reconstructed signal, and the SNR is 0.6896 dB. The characteristic frequency is obviously reduced, which is consistent with the theoretical value of bearing fault signal (70 Hz), indicating that the fault identification is correct and the effectiveness of the proposed method is verified.

Fig. 23

APHSR system output signal time domain diagram and envelope spectrum.

5 Feature extraction experiment of the weak fault signal of the rapier loom

In this section, the spindle bearing vibration signal is detected on an experimental platform of rapier loom in industrial field, and by comparing the signal-to-noise ratio and feature frequency amplitude of the output signals of four systems, CEEMDAN-SR, ICEEMDAN-SR, CEEMDAN-APHSR, and ICEEMDAN-APHSR, as well as analyzing the accuracy and success rate of feature extraction, it is concluded that the feature extraction model is more effective. The feature extraction experiment is mainly aimed at the spindle bearing of the rapier loom to verify the effectiveness of the method.

5.1 Experimental platform

(1) Rapier loom platform: The rapier loom equipment applied in the experimental test is shown in Fig. 24. Based on an on-site test of rapier loom performance, in the actual industrial environment, The spindle speed of rapier loom can reach 1200r/min at most, and the manufacturing weaving speed is 0.1 m/min, which can fully meet the experimental requirements.

Fig. 24

Field loom experimental platform.

The structure of the rapier loom vibration test-bed is shown in Fig. 25. The test-bed mainly includes the rapier loom spindle drive motor, belt pulley, spindle and rolling bearings installed at both ends of the spindle.

Fig. 25

Schematic diagram of experimental platform structure.

(2) Rapier loom control cabinet: the structure of the control cabinet is shown in Fig. 26. The control cabinet mainly includes the main circuit structures, such as the rapier loom main control system board, signal acquisition board and motor drive board, as well as the wiring between these circuit boards and various sensors, drivers, transformers and certain low-voltage appliances. Various buttons, switches and indicators are also built in the control cab inet to control and display the actual operation state of the rapier loom. This state information will also be fed back to the human–computer interaction system through the main control board to display specific parameter information.

Fig. 26

Rapier loom control cabinet.

(3) Sensors for detecting vibration signals: According to the analysis of the vibration dynamics model of rapier loom, the frequency range of the vibration signal is mainly distributed around 0∼1 kHz. The rapier weaving machine spindle vibration generates acceleration, and the vibration signal is collected by measuring the acceleration of the spindle bearing in the working condition, and then conditioning and A/D conversion of the signal. And considering the minimum resolution of 0.005 V for rapier weaving machine vibration detection system. In this paper, IEPE type acceleration sensor 333B50 is used, and a constant current source circuit with a constant current of 4 mA is used to drive the sensor to work properly. The main performance parameters of the IEPE sensor are shown in Table 5.

Table 5

IEPE sensor 333b50 parameters

Model number	33B50
Frequency accordingly	0.3∼15000
Frequency range	0.1–3kHz
Measurement Range	±5g
Sensitivity	1000 mV/g
Constant current	2mA∼20mA
Excitation voltage	24V

In order to ensure the accuracy of the collected weak fault signal, the transmission path between the selected acceleration sensor and the acquisition point should be as short as possible. The mounting position of the sensor is shown in Fig. 27. Sensor 1 is mounted on the left drive bearing base of the rapier weaving machine spindle to detect cage failure signals of the rapier weaving machine spindle bearings. Sensor 2 is installed at the base of the drive bearing on the right side of the rapier weaving machine spindle and is used to detect the outer ring failure signal of the rapier weaving machine spindle bearing.

Fig. 27

Rapier loom vibration field sites.

5.2 Fault diagnosis experiment of rapier loom spindle fault signal

In order to verify the effectiveness of the research method, the vibration of the outer ring of the rotor bearing of the rapier loom spindle is tested and analyzed. According to the theoretical analysis in Chapter 2, the outer ring fault frequency of the rapier loom rolling bearing is about 71.5hz. In fact, the fault of bearing outer ring collected is 76.5hz. The time domain waveform and envelope spectrum of the fault signal of the bearing outer ring are shown in Fig. 28.

Fig. 28

Original signal diagram of the rolling element fault.

In order to prove that the studied method has a filtering effect on the rapier loom vibration signal and at the same time the improved method is more advantageous. Therefore, the acquired signals are subjected to ICEEMDAN decomposition and CEEMDAN decomposition, respectively,as shown in Fig. 29.

Fig. 29

(a) ICEEMDAN exploded view; (b) CEEMDAN exploded view.

In order to prove that the studied method has filtering effect on rapier loom vibration signal and at the same time the improved method is more advantageous. So, the acquired signals are decomposed by ICEEMDAN and CEEMDAN respectively. As is shown in Fig. 29, we can see that firstly, from the number of decompositions,that ICEEMDAN decomposes into 12 IMFs and CEEMDAN decomposes into 14 IMFs, which shows that ICEEMDAN more effectively decomposes signals. This finding shows that ICEEMDAN has more advantages than CEEMDAN decomposition in terms of the filtering effect. Taking the energy density method and autocorrelation coefficient method as the evaluation criteria, the signal is reconstructed to obtain the reconstructed signal diagram, as shown in Fig. 30.

Fig. 30

(a) ICEEMDAN reconstruction diagram; (b) CEEMDAN reconstruction diagram.

As is shown in Fig. 30, The envelope curve of the signal reconstructed by CEEMDAN has no obvious prominent change in its envelope curve, and the difference between the amplitude of the characteristic frequency and the amplitude of the surrounding frequency is not obvious. The signal reconstructed by ICEEMDAN decomposition, on the other hand, has an obvious protrusion in the envelope curve at 76.5 Hz with an amplitude of 0.0182 V.

Extraction of rapier loom spindle fault signal features using ICEEMDAN-APHSR method. After ICEEMDAN preprocessing, the following parameters can be obtained: a = 0.5, b = 3.0, μ = 1.1. Input the parameters into the APHSR system to obtain the time domain diagram and envelope spectrum of the rapier weaving machine spindle bearing fault signal.

In order to compare the effectiveness of the proposed method with CEEMDAN-SR, ICEEMDAN-SR, and CEEMDAN APHSR in rapier weaving machine spindle fault signal extraction. The fault signals of these three methods are processed separately. The values of the parameters obtained are (a = 0.1, b = 2.7), (a = 0.1, b = 0.4) and (a = 0.2, b = 3.4, μ= 1.1), respectively.The time domain waveforms and envelope spectra are shown in Fig. 31 (a), (b), and (c), respectively. The eigenfrequency amplitudes of the three methods are only 0.014 V, 0.027 V and 0.041 V due to the saturation characteristics of SR and the problems of modal mixing in CEEMDAN; The difference in amplitude from the ambient frequency is 0.001 V, 0.008 V, and 0.02 V, respectively; The signal-to-noise ratios of the output signals are 0.0993 dB, 1.7456 dB, and 1.0709 dB, respectively.

Fig. 31

(a) CEEMDAN-SR system output diagram; (b) ICEEMDAN-SR system output diagram; (c) CEEMDAN-APHSR system output diagram; (d) ICCEMDAN-APHSR system output diagram.

Compared with these three methods, the eigenfrequency amplitude of ICEEMDAN-APHSR was improved by 3.79 times, 1.96 times and 1.29 times, respectively; The difference between the amplitude of the characteristic frequency and the maximum surrounding noise is improved by a factor of 30, 3.75 and 1.5, respectively; The output signal-to-noise ratio is 3.3668 dB, 1.7205 dB and 2.3952 dB respectively. The above data show that the proposed method has higher signal-to-noise ratio, higher feature frequency amplitude and better discrimination for rapier loom spindle vibration signal. The method is capable of detecting weak fault characteristics of rapier weaving machines in a strong noise background, and the proposed method outperforms the CEEMDAN-SR, ICEEMDAN-SR, and CEEMDAN-APHSR methods in identifying early spindle bearingfailures.

5.3 Comparison of the success rate and accuracy of the four methods

In order to further prove the effectiveness of the research method, this paper analyzes 100 sets of data collected by sensor 1 and sensor 2 respectively. Success and accuracy rates were analyzed for CEEMDAN-SR, ICEEMDAN-SR, CEEMDAN-APHSR and ICEEMDAN-APHSR, respectively.Success and accuracy rates were analyzed for CEEMDAN - SR, ICEEMDAN - SR, CEEMDAN - APHSR and ICEEMDAN - APHSR, respectively.

(1) Success rate: When the A_c proportion of the difference between the amplitude of the characteristic frequency and the amplitude of the surrounding frequency is higher than 1/3 of the amplitude of the characteristic frequency A_f, that is, A_c/A_f ⩾ 1/3, it indicates that the system has successfully extracted the characteristic frequency.

The success rate of four feature extraction systems in dealing with rapier weaving machine spindle bearing outer ring and cage faults is shown in Table 7. ICEEMDAN-APHSR system has a higher success rate compared to the other three systems.

Table 6
Analysis of the experimental results of the four methods

The parameter value Characteristic frequency amplitude Amplitude difference with surrounding ambient frequency Signal to Noise Ratio

CEEMDAN-SR a = 0.1b = 2.7 0.014 V 0.001 V 0.0993 dB

ICEEMDAN-SR a = 0.1b = 0.4 0.027 V 0.008 V 1.7456 dB

CEEMDAN-APHSR a = 0.2b = 3.4μ = 1.1 0.041 V 0.02V 1.0709 dB

ICEEMDAN-APHSR a = 0.5b = 3.0μ = 1.1 0.053V 0.03V 3.4661 dB

	The parameter value	Characteristic frequency amplitude	Amplitude difference with surrounding ambient frequency	Signal to Noise Ratio
CEEMDAN-SR	a = 0.1b = 2.7	0.014 V	0.001 V	0.0993 dB
ICEEMDAN-SR	a = 0.1b = 0.4	0.027 V	0.008 V	1.7456 dB
CEEMDAN-APHSR	a = 0.2b = 3.4μ = 1.1	0.041 V	0.02V	1.0709 dB
ICEEMDAN-APHSR	a = 0.5b = 3.0μ = 1.1	0.053V	0.03V	3.4661 dB

Table 7

Comparison of the success rate and accuracy of diagnostic results of four feature extraction systems

Four Methods	Number of groups successfully extracted by outer ring failure	Number of groups successfully extracted by cage failure	Success rate
CEEMDAN-SR	93	94	93.5%
ICEEMDAN-SR	93	95	94.0%
CEEMDAN - APHSR	94	₉₅	94.5%
ICEEMDAN - APHSR	98	98	98.0%

(2) Accuracy: The accuracy represents the error of the feature frequencies obtained after the feature extraction system compared to the feature frequencies in the original signal. The calculation process is shown in Equation (36). $η = 1 - \frac{| f_{1} - f_{2} |}{f_{2}}$ (36)

Where f₁ represents the characteristic frequency obtained by the feature extraction system, and f₂ represents the characteristic frequency in the original signal. The mean value of the outer circle fault characteristic frequency of the original signal is 76.1hz.

A comparison of the accuracy of the four feature extraction systems in dealing with outer ring and cage faults of rapier weaving machine spindle bearings is shown in Table 8. The accuracy of the system is the average of the cage fault accuracy and outer ring fault accuracy, and it can be seen from the table that the accuracy of ICEEMDAN-APHSR is higher.

Table 8

Comparison of the accuracy of diagnostic results of four feature extractionsystems

Four Methods	Mean value of outer ring fault characteristic frequency	Mean value of cage failure characteristic frequency	Outer ring failure accuracy	Cage failure accuracy	Accuracy
Original signal	76.1	11.2	100	100	100
CEEMDAN-SR	71.5	15.1	93.9	65.1	79.5
ICEEMDAN-SR	72.5	14.2	95.3	73.3	84.3
CEEMDAN - APHSR	72.0	_8.1	94.7	72.4	83.5
ICEEMDAN - APHSR	78.0	8.6	97.6	77.2	87.4

6 Conclusion

This paper presents a ICEEMDAN-APHSR method to improve the feature extraction ability of the weak fault signal of the blade loom spindle under the background of strong noise. Based on the theoretical and experimental findings of this study, the following conclusions are drawn in this paper:

(1) In this paper, we model the rotating shaft, rolling bearing and unique rapier of rapier loom, and clarify the vibration range of rapier loom and the failure characteristic frequency of main component rolling bearing. In this paper, the fault signal of the spindle bearing of the rapier weaving machine is processed into ICEEMDAN, and the optimal IMF group is screened by the energy density and correlation coefficient method. The optimal IMF group is scale transformed and input to the APHSR system, and the rapier loom spindle fault signal is extracted from the noise after scalerecovery.

(2) Compared to CEEMDAN-SR, ICEEMDAN-SR, and CEEMDAN-APHSR, the characteristic frequency amplitude of rapier weaving machine spindle bearings was improved by 3.79 times, 1.96 times, and 1.29 times by using ICEEMDAN-APHSR method, respectively, as verified by experiments on rapier weaving machine spindle bearings. The difference between the characteristic frequency amplitude and the maximum surrounding noise is improved by 30 times, 3.75 times and 1.5 times respectively; the output signal-to-noise ratio is 3.3668 dB, 1.7205 dB and 2.3952 dBrespectively.

The proposed method solves the coverage of noise to feature frequency in the fault signal of the main shaft bearing. Moreover, it uses the characteristics of the screening and filtering of noise by ICEEEMDAN and the transfer of noise energy to the signal, which greatly enhances the fault feature and improves the fault feature identification rate. It is beneficial to the fault diagnosis of rapier loom and has a broad application prospect. The limitation of this study is that only the vibration signal of the rapier loom spindle is analyzed for feature extraction. In the future, the multi-sensor fusion method can be used to study the fault diagnosis classification of rapier looms, and the intelligent algorithm can be combined to improve the efficiency of the feature extraction process.

Footnotes

Acknowledgment

This work was partially supported by the Natural Science Foundation of Hebei Province under Grant No. E2022202136.

References

, Gao

, Shi

and Zhao

, Fault-tolerant control of Markovian jump stochastic systems via the augmented sliding mode observer approach[J], Automatica 50(7) (2014), 1825–1834.

Yang

, Han

, Yin

, et al., Sliding mode-based adaptive resilient control for Markovian jump cyber-physical systems in face of simultaneous actuator and sensor attacks[J], Automatica (2022), 142.

Yang

, Yin

and Kaynak

, Neural Network-Based Adaptive Fault-Tolerant Control for Markovian Jump Systems With Nonlinearity and Actuator Faults, in IEEE Transactions on Systems, Man, and Cybernetics: Systems 51(6) (2021), pp. 3687–3698. doi: 10.1109/TSMC.2020.3004659.

, Li

, Mou

, Garg

, Gao

and Liu

, A Hybrid Battery Equivalent Circuit Model, Deep learning, and Transfer Learning for Battery State Monitoring, in IEEE Transactions on Transportation Electrification, 2022, doi: 10.1109/TTE.2022.3204843.

, Li

, Garg

, et al., An adaptive boosting charging strategy optimization based on thermoelectric-aging model, surrogates and multi-objective optimization[J], Applied Energy 312 (2022), 118795.

Sanz

, Perera

and Huerta

, Fault diagnosis of rotating machinery based on auto-associative neural networks and wavelet transforms [J], Journal of Sound and Vibration 302(4-5) (2007), 981–999.

Ekici

, Yildirim

and Poyraz

, Energy and entropy-based feature extraction for locating fault on transmission lines by using neural network and wavelet packet decomposition [J], Expert Systems with Applications 34(4) (2008), 2937–2944.

Ren

, Research on fault diagnosis of rotating machinery based on vibration signal analysis, M.S. Qingdao University, China, 2021.

Jiang

H.S.

, Noise vibration analysis and vibration reduction of rapier loom, Noise and vibration control (4) (1993), pp. 17–19+16, 12.

10.

Guo

J.J.

, Anti interference in mechanical vibration testing system, Modern machinery (5) (2007), pp. 19–21+33.

11.

Zhang

, Wavelet-EMD and stochastic resonance cascade weak signal detection, Journal of Electronic Measurement and Instrumentation 32(9) (2018), pp. 57–65, DOI. 10.13382/j.jemi.2018.01.008.

12.

L.F.

, Cao

and Zhang

T.Q.

, Stochastic resonance study of EMD noise reduction in Levy Noise, Electronic Measurement and Instrumentation 31(8) (2017), pp. 21–28. DOI. 10.13382/j.jemi.2017.01.004.

13.

Zhang

, Bearing fault feature extraction based on EMD and wavelet packet, Information and Electronic Engineering 10(5) (2012), pp. 330–333+338.

14.

Lou

H.W.

, Ma

Z.S.

and Sun

H.G.

, Research on gearbox feature extraction technology based on EMD and fractal, Machine Building 52(4) (2014), pp. 67–70.

15.

Cao

, Duan

Y.B.

and Liu

J.C.

, Modal asing problem in Hilbert-Huang transformation, Vibration Test and Diagnosis 36(8) (2016), pp. 518–523+605–606, DOI. 10.16450/j.cnki.issn.1004-6801.2016.03.018.

16.

Zhu

and Zhu

, CEEMDAN auxiliary fast spectrum cliff degree of rolling bearing fault diagnosis method, Light Machinery 40(7) (2022), pp. 74–79+84.

17.

Chen

S.P.

and Yang

Y.F.

, Rolling bearing troubleshooting based on CEEMDAN and SCA-MRVM, Bearing (7) (2021), pp. 53–59, DOI. 10.19533/j.issn1000-3762.2021.10.012.

18.

Colominas

M.A.

, Schlotthauer

and Torres

M.E.

, Improved complete ensemble EMD: A suitable tool for biomedical signal processing, Biomedical Signal Processing and Control 14 (2014), pp. 19–29.

19.

Wang

N.H.

and Cui

B.Z.

, Planetary gearbox fault diagnosis using ICEEMDAN and SVM, Mechanical Science and Technology (2022), pp. 1–7.

20.

Chen

X.-h.

, Cheng

, Shan

X.-l.

, Hu

, Guo

and Liu

H.-g.

, Research of weak fault feature information extraction of planetary gear based on ensemble empirical mode decomposition and adaptive stochastic resonance, Measurement 73 (2015), pp. 55–67.

21.

Qiao

, Lei

and Li

, Applications of stochastic resonance to machinery fault detection: A review and tutorial, Mechanical Systems and Signal Processing 122 (2019), pp. 502–536.

22.

Mitaim

and Kosko

, Adaptive stochastic resonance, Proceedings of the IEEE 86(11) (1998), pp. 2152–2183, DOI:10.1109/5.726785.

23.

Qin

, Tao

, He

and Tang

, Adaptive bistable stochastic resonance and its application in mechanical fault feature extraction, Journal of Sound and Vibration 333(26) (2014), pp. 7386–7400.

24.

Chen

Y.F.

and Hong

H.C.

, Principle and use of sword pole loom, China National Textile Press, 1994.

25.

Zhang

L.J.

, Rong

Y.L.

and L

, Condition warning and maintenance optimization of rotating machinery and equipment, Journal of Engineering Science 39(07) (2017), pp. 1094–1100. doi: 10.13374/j.issn2095-9389.2017.07.016.

26.

Wang

, Niu

, Guo

, Wang

and Han

, A Piecewise Hybrid Stochastic Resonance Method for Early Fault Detection of Roller Bearings, IEEE Access 8 (2020), pp. 73320–73329, DOI. 10.1109/ACCESS.2020.2987835.

27.

Fauve

and Heslot

, Stochastic resonance in a bistable system, Physics Letters A pp. 5–7.