Failure Feature Identification of Vibrating Screen Bolts under Multiple Feature Fusion and Optimization Method

Abstract

Strong impacts and vibrations exist in various structures of rice combine harvesters in harvesting, so the bolt connection structure on the harvesters is prone to loosening and failure, which would further affect the service life and working efficiency of the working device and structure. In this paper, based on the vibration signal acquisition experiment on the bolt and connection structure of the vibrating screen on the harvester, failure feature identification is studied. According to the sensitivity analysis results and the primary extraction of the time-frequency feature, most features have limitations on the identification of failure features of vibrating screen bolts. Therefore, based on the establishment of a high-dimensional feature matrix and multivariate fusion feature matrix, the validity of the feature set was verified based on the whale optimization algorithm. And then, based on the SVM method and high-dimensional mapping of the kernel functions, the high-dimensional feature matrix is trained by the LIBSVM classification decision model. The identify success rates of time domain feature matrix A, frequency domain feature matrix B, WOA-VMD energy entropy matrix C, and normalized multivariate fusion feature matrix G are 64.44%, 74.44%, 81.11%, and more than 90%, respectively, which can reflect the applicability of the failure state identification of the normalized multivariate fusion feature matrix. This paper provided a theoretical basis for the identification of a harvester bolt failure feature.

1. Introduction

The combine harvester has many working devices with a complex working structure, and a vibrating screen is one of them [1,2]. The bolts are the connecting elements of each device of the harvester, and the bolts will be impacted by the vibration of each device but also by the load force in different directions [3,4]. The impact and imbalance of the bolts at the working parts and connections are the most intense, and the instantaneous impact and alternating load of the bolts in the working process can best cause the instantaneous fracture and fatigue failure of the bolts [5]. The vibration of the working device will not only affect the normal work of each device but also improve the mechanical harvest loss rate and impurity content rate of the overall harvester to the crop and then affect the harvest efficiency and harvest quality of the harvester [6,7]. The further superposition of the vibration will make the overall combined harvester produce stronger vibrations, causing machine resonance and fatigue failure of the bolt structural parts, which seriously affect the working performance and service life of the working device of the combined harvester [8,9]. In addition, the strong vibration will not only affect the driving experience of the driver but also create safety risks for the working device, which may affect the safety of the driver [10]. In recent years, with the development of the combine harvester’s high speed, high efficiency, and high intelligence, the bolt vibration characteristics and an identification method of the combine harvester for vibration reduction and the structure optimization of the combine harvester are very urgent [11,12].

At present, domestically, the harvester cutter, vibrating screen, engine, and other devices and structures must be disassembled to carry out vibration tests to obtain the vibration signals of the working devices one by one [13,14,15]. However, with advancements in microscopic friction contact and macroscopic dynamics models, it is now feasible to study the excitation characteristics and structural failure of harvesters by analyzing the bolt structure’s macroscopic dynamics under actual load conditions [16,17]. By measuring the vibration signals of each component of the harvester, the response signals can be analyzed to study the signal characteristics of each device [18]. The processing of vibration signals using both standard and improved algorithms, validated through simulation analysis and experimental verification, can significantly enhance the efficiency and accuracy of signal processing and identification [19]. For the problem of a single feature dimension and low accuracy in assessing the structural state, the optimization algorithm can extract the time-domain features of the vibration signal to develop an optimized algorithm model [20]. And the optimization algorithm is widely used for signal processing, which can obtain the optimal data parameters by the processing and optimization of signal data characteristics [21]. The various algorithms have advantages in extracting the vibration signal features of the combine harvester, which can provide a theoretical basis for the optimization of the harvester structure [22]. Algorithm optimization and optimization methods are widely used in system diagnosis, and their performance has been significantly improved compared with the traditional fault diagnosis model [23,24]. Although optimization algorithm and optimization methods have been widely used in system diagnosis, there are few multivariate feature extraction and combination analyses of bolt-loosening state identification for combine harvesters, and there are few studies on the optimization training of multivariate feature datasets of vibrating screens.

Based on the different signal acquisitions of the bolt structures, the different loosening states and signal characteristics of bolts are extracted in this paper. The feature dataset was established according to the establishment of a high-dimensional feature matrix and a multivariate fusion feature matrix. The optimal fitness values were found, and the effectiveness of the feature set was verified based on the whale optimization algorithm. Then, based on the support vector machine method and the introduction of kernel function for high-dimensional mapping, the high-dimensional feature matrix of the bolt failure state is trained and tested by the LIBSVM classification decision model of the one-versus-one classification method. This paper provided a theoretical basis for the identification of harvester bolt failure features.

2. Materials and Methods

2.1. The Method of Obtaining the Loosening Signals of the Vibrating Screen Bolts

The vibration response signals acquisition equipment for different failure states of vibrating screen bolt structures include the DH5902 signal acquisition instrument, the DHDAS dynamic signal acquisition system, the 1A312E three-way acceleration sensor, and a torque wrench (Suzhou, Jiangsu, China). The acquisition process of the failure signals is shown in Figure 1. The acceleration sensors were installed at the measuring position of the vibrating screen bolt structure first. The DH5902 signal acquisition instrument, DHDAS dynamic signal acquisition system, and 1A312E three-way acceleration sensors are connected by custom wires, and the sensor installation position is at the bolt connection on the edge of the vibrating screen, as shown in Figure 2. The response signals of bolt structure failure were collected after starting the combine harvester to a stable working state. According to the requirement that the amount of data of the bolt structure failure state identification needs to be enough to support the validity of the results, the system sampling frequency was set to 2 kHz and sampling was continued for five minutes. The pretension state of the bolts needs to be adjusted using the torque wrench after each round of response-signal acquisition. The states of the signals were set to complete the pre-tightening state, the loosening state, and the complete loosening state, so the signals of the three states were collected and saved.

2.2. Sensitivity Analysis and Extraction Method of the Time-Frequency Feature

The time-frequency response curves of the three loosing states of the bolt structure are shown in Figure 3 in the main vibration direction of the vibrating screen. Under a fully pre-tightened state, the phenomenon of normal clearance or tangential displacement beyond the small displacement assumption did not occur at each contact surface of the bolt structure. As a result, the vibration amplitude of the curves is the smallest and exhibits distinct impact features.

The amplitude of the random signal is much higher than that of the working signal, causing the working signal of the vibrating screen to be submerged by the random loosening signal, which exhibits strong irregularity. Based on time-domain analysis, the two states of loosening and tightening can be clearly distinguished. But the two degrees of loosening in the loosening state are not easily distinguished, so the phenomenon of a large negative bias of the response signal occurring in the complete loosening state can be seen.

The frequency spectrogram in Figure 4 can be obtained through the fast Fourier transform of signals. The bolt structure includes only the operating frequency and the engine rotation frequency, but the amplitude and distribution of frequency components during loosening have changed greatly. The signal energy surge caused by wear and collision is shown in the amplitude, and the contact surface gap will lead to the increases in collisions between the bolt and the clamp, the clamp and the clamp, and the clamp and the nut. When the degree of loosening increases, the frequency component becomes more and more complex, and the amplitude will increase accordingly. Compared with the time-domain curve, the frequency-domain curve is easier for distinguishing the degrees of loosening.

Bolt structure loosening will cause the change in the response signal amplitude and increase the randomness of signal data. The time-domain feature can obviously distinguish between the loosening and pre-tightening states of the bolt structure, but it is difficult to distinguish the different loosening degrees of the bolt structure. The primary extraction of the loosening feature progressed through a method in which the frequency-domain feature is mainly used, supplemented by the time-domain feature. All the time-domain features more commonly used are signal statistics, and some of the calculation formulas are shown in

Table 1. The calculation formulas of time-domain feature.

Item	Formula	Item	Formula
Mean square values	$Rms=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{x}_i^2}$	Kurtosis factor	$Ku={\displaystyle \frac{\frac{1}{N}{\sum }_{i=1}^N{(\left|x_i\right|-\mu )}^4}{{\left(\frac{1}{N}{\sum }_{i=1}^N{\left(\left|x_i\right|-\mu \right)}^2\right)}^2}}$
Skewness factor	$Skew={\displaystyle \frac{\frac{1}{N}{\sum }_{i=1}^N{(\left|x_i\right|-\mu )}^3}{{\left(\frac{1}{N}{\sum }_{i=1}^N{\left(\left|x_i\right|-\mu \right)}^2\right)}^3}}$	Margin factor	$C_e={\displaystyle \frac{x_{peak}}{{\left(\frac{1}{N}{\sum }_{i=1}^N\sqrt{\left|x_i\right|}\right)}^2}}$
Standard deviation	$S_t=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^N{(x_i-\mu )}^2}{N}}}$	Average value	$\mu ={\displaystyle \frac{{\sum }_{i=1}^N{x}_i}{N}}$
Pulse factor	$I_m={\displaystyle \frac{x_{peak}}{\frac{1}{N}{\sum }_{i=1}^N\left|x_i\right|}}$	Waveform factor	$S_f={\displaystyle \frac{Rms}{\frac{1}{N}{\sum }_{i=1}^N\left|x_i\right|}}$

.

The relationship between the amplitude and the frequency components of the signal can be obtained through fast Fourier transform. Because bolt loosening will inevitably cause a change in the frequency composition, the focused research on the frequency domain feature can improve the accuracy of the failure-state identification of the bolt structure. In addition to extracting the statistics of the frequency spectrum curve, three important frequency-domain features (center frequency, frequency variance, and mean square frequency) are added into the analysis. The three features are mainly used to reflect the main signal frequency band and have a strong differentiation between the signal frequency components in different loosening states.

$$FC={\displaystyle \frac{{\sum }_{i=1}^N(f_i·s(f_i\left)\right)}{{\sum }_{i=1}^{N}s\left(f_i\right)}}$$

(1)

$$MSF={\displaystyle \frac{{\sum }_{i=1}^N(f_i^2·s(f_i\left)\right)}{{\sum }_{i=1}^{N}s\left(f_i\right)}}$$

(2)

$$VF={\displaystyle \frac{{\sum }_{i=1}^N{(f_i-FC)}^2·s\left(f_i\right)}{{\sum }_{i=1}^{N}s\left(f_i\right)}}$$

(3)

In the formula: $f_i$—the frequency of signals; $s\left(f_i\right)$—the amplitude of signals.

2.3. The Methods of Secondary Feature Analysis under Optimization of the VMD

The collected bolt-loosening signals show the features of nonlinearity and non-stationarity. To better represent the degree of confusion of each frequency component in different loosening states and extract useful information from the signals in a more discriminative manner, the secondary feature analysis method of VMD (variational mode decomposition) energy entropy is adopted to analyze the loosening signals [25]. In order to make the resulting sub-signal contain as much loosing state information as possible and for the interference factors, such as noise, to be weakened as much as possible, the bolted loose response signals were processed by decomposition. The VMD method can decompose the original signals into a series of IMF components, and certain constraints are added to the components. Therefore, the component that satisfies these constraints can be used as the effective component of the intrinsic mode function (IMF). 1. The number of peak points and the number of over zeros must be equal or no more than one within the whole data segment obtained by the decomposition (the curve, once through the zero axis, has only one maximum or minimum value). 2. The average values of the upper envelope line formed by the local maximum points and the lower envelope line formed by the local minimum points are zero at any moment (the upper and lower envelope lines are locally symmetrically distributed relative to the time axis).

The VMD method decomposes the response signals into multiple single-component IMF signals at one time, thus avoiding problems such as the endpoint effects and mode aliasing encountered in the iterative process, which can be understood as the construction of a variational problem and the solving of the optimal solution process. The central frequency of each IMF component is ${\omega }_{\left(t\right)}$ under the assumption that any complex signal is composed of k finite bandwidth modal components, ${\nu }_k\left(t\right)$. The constraint condition is that the sum of the k components and the one residue is equal to the input signal. The analysis signal ${\nu }_k\left(t\right)$ is obtained through the Hilbert transform method, and the central band of ${\nu }_k\left(t\right)$ is modulated to the corresponding band based on the one-sided spectrum being multiplied by the operator $e^{-j^{\omega }kt}$:

$$\left[\left(\delta \left(t\right)+{\displaystyle \frac{J}{\pi t}}\right)\ast {\nu }_k(t)\right]e^{-j^{\omega }kt}$$

(4)

In the formula: *—convolution operation; J—demodulating the gradient with the squared norm ${\mathrm{L}}^2$. The estimation method of the bandwidth of each modal component, as the constraints in the derivation process are X(t) = ${\sum }_{i=1}^n{IMF}_n\left(t\right)+res$, is as follows:

$$\left\{\begin{array}{c}\underset{\left\{{\nu }_k\right\},\left\{{\omega }_k\right\}}{\mathit{min}}\left\{\sum _k{‖{\zeta }_t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\ast {\nu }_k(t)\right]e^{-j^{\omega }kt}‖}^2\right\}\\ s.t.\hspace{1em}\sum _k{\nu }_k\left(t\right)=s\end{array}\right.$$

(5)

In the formula: $\left\{{\nu }_k\right\}$—IMF component; $\left\{{\omega }_k\right\}$—center frequency of the IMF component. In order to find the optimal solution to the constrained variational problem, the Lagrange multiplier τ(t) and the second-order penalty factor α are introduced to convert the constrained variational problem to an unconstrained variational problem:

$$L\left(\left\{{\nu }_k\right\},\left\{{\omega }_k\right\},\tau \right)=\alpha {\sum }_k{‖{\zeta }_t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\ast {\nu }_k(t)\right]e^{-j^{\omega }kt}‖}^2+{‖s\left(t\right)-{\sum }_k{\nu }_k\left(t\right)‖}^2+〈\tau \left(t\right),s\left(t\right)-{\sum }_k{\nu }_k\left(t\right)〉$$

(6)

Each component and the center frequencies are updated by the alternating direction method of multipliers, so the saddle points of unconstrained model will be obtained (the optimal solution to the constrained variational problem). All the equations of the components are obtained according to the frequency domain space:

$${\widehat{\nu }}_k^{n+1}\left(\omega \right)={\displaystyle \frac{\widehat{s}\left(\omega \right)-\sum _{i\ne k}{\widehat{\nu }}_i+\frac{\widehat{\tau }\left(\omega \right)}{2}}{1+2\alpha {(\omega -{\omega }_k)}^2}}$$

(7)

In the formula: ${\widehat{\nu }}_k^{n+1}\left(\omega \right)$, $\widehat{s}\left(\omega \right)$, $\widehat{\tau }\left(\omega \right)$—the Fourier transform of the corresponding signals; ${\widehat{\nu }}_k^{n+1}\left(\omega \right)$—the residual amount of $\widehat{s}\left(\omega \right)-\sum _{i\ne k}{\widehat{\nu }}_i$ after the Wiener filter. And then, the center frequency from the center of gravity of the power spectrum of each component is estimated, so ${\{\widehat{\nu }}_k^1\}$, ${\{\widehat{\omega }}_k^1\}$, ${\{\widehat{\tau }}_k^1\},$ and n are initialized to start cycle calculation. The values of ${\omega }_k$ and ${\tau }_k$ are gradually updated:

$${\omega }_k^{n+1}={\displaystyle \frac{{\int }_0^{\infty }\omega {\left|{\nu }_k^{n+1}\left(\omega \right)\right|}^{2}d\omega }{{\int }_0^{\infty }{\left|{\nu }_k^{n+1}\left(\omega \right)\right|}^{2}d\omega }}$$

(8)

$${\widehat{\tau }}^{n+1}\left(\omega \right)={\widehat{\tau }}^n\left(\omega \right)+\tau (\widehat{s}\left(\omega \right)-\sum _{i\ne k}{\widehat{\nu }}_k^{n+1}\left(\omega \right))$$

(9)

Finally, the k module components are obtained when the iterative stop condition is satisfied. The judgment conditions are provided as follows:

$$\sum _k{\displaystyle \frac{{‖{\widehat{\nu }}_k^{n+1}\left(\omega \right)-{\widehat{\nu }}_k^n\left(\omega \right)‖}_2^2}{{‖{\widehat{\nu }}_k^n\left(\omega \right)‖}_2^2}}<\epsilon$$

(10)

The above analysis methods are obviously affected by the number of modes k, the second-order penalty factor α, the fidelity coefficient τ, and the judgment convergence value ε. The fidelity coefficient τ and the judgment convergence value ε are used as the judgment criterion parameters, so the two factors were set to the default value. Therefore, the second-order penalty factor α and the number of modes k of the VMD algorithm need to be analyzed. The number of modes k has the greatest influence, due to the smaller set of k causing multiple modal components in one IMF component, and the higher set of k will cause the same modal component to be distributed in multiple IMF components; the value of k may cause a modal stacking problem. The second-order penalty factor α has the smaller influence. The smaller set of α will cause a larger bandwidth of the IMF component center frequency, and the more redundant frequency components and the higher set of α will cause a smaller bandwidth of the IMF component center frequency and fewer useful frequency components.

The most appropriate values of k and α need to be analyzed. The value of k is determined by the traditional center-frequency close method, and the specific steps are described as follows:

• Initialize the secondary weight parameters of the VMD algorithm: each parameter is set up (α = 1000, τ = 0, number of initial decomposition layers k = 3) based on relevant experience.
• Run the VMD algorithm and export the IMF component data tables: the VMD algorithm is run under the initial number of the number of modes k, and then, the spectrum curves after fast Fourier transform algorithm of the k IMF component data tables, k central frequency values of the IMF components, and the k IMF components of the response signal are obtained.
• Determine the VMD signal decomposition quality and whether to skip the loop: if $f_{FC}\left(k+1\right)-f_{FC}\left(k\right)\ll f_{FC}\left(k\right)-f_{FC}(k-1)$, the value of $f_{FC}\left(k+2\right)-f_{FC}(k+1)$ and $f_{FC}\left(k+1\right)-f_{FC}\left(k\right)$ will be compared after k + 1. If $f_{FC}\left(k+1\right)-f_{FC}\left(k\right)\ll f_{FC}\left(k\right)-f_{FC}(k-1)$ and $f_{FC}\left(k+2\right)-f_{FC}(k+1)\ll f_{FC}\left(k+1\right)-f_{FC}\left(k\right)$, the loop end and the value of the number of modes k are obtained.

The above cycles were performed by MATLAB R2021b software, and the central frequency iteration values of three sets of bolt-loosening response signals are shown in

Table 2. The k values of VMD decomposition were determined by the central frequency method.

Complete Loosening	Number of Initial Decomposition Layers	The Center Frequency of IMF Component/2 kHz
		${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(1\right)$	${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(2\right)$	${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(3\right)$	${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(4\right)$	${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(5\right)$	${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(6\right)$	${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(7\right)$	${\mathbf{f}}_{\mathbf{F}\mathbf{C}}\left(8\right)$
Loosening	k = 3	0.0780	0.2207	0.2878
	k = 4	0.0761	0.2054	0.2697	0.3698
	k = 5	0.0735	0.1552	0.2220	0.2842	0.3720
	k = 6	0.0697	0.1556	0.2209	0.2800	0.3400	0.4173
	k = 7	0.0534	0.0840	0.1558	0.2213	0.2835	0.3677	0.4217
Complete pre-tightening	k = 3	0.0860	0.1520	0.3382
	k = 4	0.0849	0.1514	0.2665	0.3979
	k = 5	0.0822	0.1554	0.2601	0.3413	0.4114
	k = 6	0.0755	0.1199	0.1538	0.2643	0.3357	0.4048
Complete loosening	k = 3	0.1062	0.1821	0.3054
	k = 4	0.1028	0.1820	0.3035	0.3774
	k = 5	0.0730	0.1362	0.1834	0.3027	0.3756
	k = 6	0.0688	0.1350	0.1832	0.2923	0.3223	0.3800
	k = 7	0.0869	0.1781	0.2513	0.3056	0.3653	0.4134	0.4664
	k = 8	0.0564	0.0984	0.1398	0.1834	0.2876	0.3146	0.3752	0.4502

. It is seen that the number of decomposition k layers for the three sets of bolt-loosening response signals are 6, 5, and 7, respectively.

The k values of VMD decomposition were determined by the central frequency method.

The spectrum curves of the three sets of IMF components were obtained by activating the main program of VMD decomposition, as shown in Figure 5. The horizontal axis represents the frequency (HZ), and the vertical axis represents the acceleration signal (m/s²).

The above method to obtain the central frequency of each IMF component has some limitations. It can be seen that significant overlap exists in the IMF component bands and multiple-spikes in each IMF component spectrum curve, which indicates that the IMF component is not thoroughly decomposed. Although the initial value of the penalty factor α is set as reasonably as possible, the bandwidth of the partial IMF component cannot be well limited. Therefore, the classic method of fool search k does not perform well for affecting the signal decomposition results. The classic approach only targets the central frequency search for k while ignoring the effects of other parameters; there is no one other criterion other than the subjective criterion of whether $f_{FC}(k+1)$ is close to $f_{FC}\left(k\right),$ which will directly lead to the result of incomplete decomposition and the modal aliasing problem. In order to find the optimal parameter combination and complete the signal feature extraction, the optimization algorithm and the entropy value theory are used to optimize the VMD parameters.

2.4. The Identification Method of Bolt-Loosening-State-Based SVM Method

The support vector machine (SVM) is a supervised learning method, which is a linear separable second-classification method based originally on the principle of the maximal interval of feature space [26]. The nonlinear kernel technique makes SVM actually become a nonlinear classifier later, and the fundamental purpose is to find an optimal hyperplane in the space that can divide up all the data samples and make the distance from all the data in the sample set to this hyperplane the shortest. Due to the final decision function of the SVM being determined by only a handful of support vectors and to the computational complexity depending on the number of support vectors rather than the dimensionality of the sample space, the dimension disaster can be avoided. The optimal hyperplane ${\omega }^{T}x+b=0$ and classification decision function $f\left(x\right)=sign({\omega }^{T}x+b)$ can be obtained by training and studying the dataset though SVM.

The function interval and the geometric interval need to be defined first:

$$\tilde{\gamma }=y({\omega }^{T}x+b)$$

(11)

$${\gamma }_i=y_i({\displaystyle \frac{{\omega }^T}{‖\omega ‖}}{x}_i+{\displaystyle \frac{b}{‖\omega ‖}})$$

(12)

In the formula: $(x_i,y_i)$—sample points for hyperplane (ω, b) in the training set; x—training set feature; y—training results label; i—the i th sample.

And then, the objective function of the maximum interval classifier is defined:

$$y_i\left({\omega }^T{x}_i+b\right)=\widetilde{{\stackrel{~}{\gamma }}_i}\ge \tilde{\gamma }$$

(13)

The training sample points (support vector) that are closest to the hyperplane make the equality hold. The distance of the support vector to the optimal hyperplane is $1/\omega$. So, the above-mentioned objective function is converted into:

$$max{\displaystyle \frac{1}{‖\omega ‖}}, s.t. y_i\left({\omega }^T{x}_i+b\right)\ge 1$$

(14)

The above maximum interval solving problem is transformed into a convex second-order optimization problem. In order to solve it conveniently, the kernel function needs to be introduced. If one Lagrange multiplier is added to each constraint, the Lagrangian function of the linear separable problem can be obtained:

$$L\left(\omega ,b,\alpha \right)={\displaystyle \frac{1}{2}}{‖\omega ‖}^2-\sum _{i=1}^n{\alpha }_i(y_i\left({\omega }^T{x}_i+b\right)-1)$$

(15)

$$\theta \left(\omega \right)=max L\left(\omega ,b,\alpha \right),{\alpha }_i\ge 0$$

(16)

The original problem can be transformed into a dual problem of the original problem through the duality of the Lagrangian function:

$$\underset{\omega ,b}{\mathit{min}}\theta \left(\omega \right)=\underset{\omega ,b}{\mathit{min}}\underset{{\alpha }_i\ge 0}{\mathit{max}}L\left(\omega ,b,\alpha \right)=p*$$

(17)

$$\underset{{\alpha }_i\ge 0}{\mathit{max}}\underset{\omega ,b}{\mathit{min}}L\left(\omega ,b,\alpha \right)=d*$$

(18)

The relationship between the original problem optimal solution p* and the dual problem optimal solution d* is $d*\le p*$. When the conditions are satisfied, the optimal solution to the dual problem is easier to solve and is equivalent to the original problem, so the optimal solution of the dual problem can be used directly instead of the original problem optimal solution. The original problem can be reduced to solve the above equivalent problem:

$$\underset{\alpha }{\mathit{min}}L\left(\alpha \right)={\displaystyle \frac{1}{2}}{\sum }_{i,j=1}^n{\alpha }_i{\alpha }_j{y}_i{y}_j{x_i}^T{x}_j-{\sum }_{i=1}^n{\alpha }_i$$

(19)

$${\sum }_{i=1}^n{\alpha }_i{y}_i=0,{\alpha }_i\ge 0$$

(20)

The following equation can be obtained by the Karush–Kuhn–Tucker and SMO algorithms:

$$\omega *={\sum }_{\mathrm{i}=1}^{\mathrm{n}}{\alpha }_{\mathrm{i}}^{*}{\mathrm{x}}_{\mathrm{i}}{\mathrm{y}}_{\mathrm{i}}$$

(21)

$$\mathrm{b}*={\mathrm{y}}_{\mathrm{j}}-{\sum }_{\mathrm{i},\mathrm{j}=1}^{\mathrm{n}}{\alpha }_{\mathrm{i}}^{*}{\mathrm{y}}_{\mathrm{i}}{{\mathrm{x}}_{\mathrm{i}}}^{\mathrm{T}}{\mathrm{x}}_{\mathrm{j}}$$

(22)

Therefore, the classification decision function is:

$$f\left(x\right)=sign(\sum _{i=1}^n{\alpha }_i^{*}{y}_i(x•x_i)+b*)$$

(23)

The Langrangian functions that can be constructed for the data have defect points that will lead to a linear inseparable situation:

$$L\left(\omega ,b,\epsilon ,\alpha ,\gamma \right)={\displaystyle \frac{1}{2}}{‖\omega ‖}^2+C{\sum }_{i=1}^n{\epsilon }_i-{\sum }_{i=1}^n{\alpha }_i[\left(y_i\left({\omega }^T{x}_i+b\right)-1\right)+{\epsilon }_i]-{\sum }_{i=1}^n{{\gamma }_i\epsilon }_i$$

(24)

In the formula: ε—the relaxation variables introduced by the defect points, which can represent the deviation value of the functional interval allowed by the defect point from the optimal hyper plane. Therefore, the outlier point can also be considered a point in the optimal hyper plane within the allowable range of the offset. If ε→∞, all values within the dataset can be considered to fall within the optimal hyper plane. The constant C is then introduced to constrain the relaxation variable, so that the value of the function $1/2{‖\omega ‖}^2+C{\sum }_{i=1}^n{\epsilon }_i$ can always be taken as the minimum value. The classification decision function of SVM can be obtained by the Karush–Kuhn–Tucker and the SMO algorithms, which is the same as the linear separable problem, but the penalty factor is restricted ($0\le {\alpha }_i\le C$).

Through the preliminary analysis of the obtained feature matrix, it can be seen that the large probability of the vibrating screen bolt failure state feature matrix belongs to the category of being linear-inseparable. In order to obtain a better decision function of 2D classification, the kernel function needs to be inducted to perform high-dimensional mapping, which allows the data to obtain a discriminative optimal hyper plane in high-dimensional space to complete the identification of the bolt failure states. The distributabilities of the multivariate fusion feature matrix for the response signal of the vibrating screen bolt loosening state are shown in Figure 6. It can be seen that the data are nonlinear for the multivariate fusion feature matrix, which does not meet the above linear-separable situation. In addition, the data do not have obvious distribution features and show strong randomness. It is found that there is some local reparability in the mapping of feature space from 3D to 2D in Figure 7, but the accuracy of state identification still follows the probability distribution of the data from a global perspective, which is a reference method for feature data processing.

The kernel function is induced to discriminate the SVM state for a multivariate fusion feature matrix for nonlinear bolt-loosening states, which makes this formula, $K\left(v_1, v_2\right)={(<v_1, v_2>)}^2,$ hold true. Linear non-separable data are mapped to a high-dimensional space; at this point, the optimal hyper plane is really an optimal hyper plane. Due to the vector inner product of the two data points after the mapping being equal to the square of the vector inner product before the mapping, the calculation of the kernel function is equivalent to the mapped vector inner product. Therefore, the linearly non-separable SVM classification decision function can be obtained:

$$f\left(x\right)=sign(\sum _{i=1}^n{\alpha }_i^{*}{y}_{i}K(x•x_i)+b*)$$

(25)

The applicability and classification of kernel functions are shown in

Table 3. The applicability and classification of kernel functions.

Order	Kernel Function	Advantage	Shortcoming
1	linearity	The characteristic space dimension is equal to the input space dimension, less parameters, fast speed	Only can solve the linear separable problem
2	multinomial	Power numbers can be set to realize the summary and prediction, can solve the nonlinear problems	More parameters, calculate slow
3	Gaussian	Can map to the infinite dimensions, diverse decision-making boundaries, only one parameter	Calculate slow, easy to appear overfit
4	Sigmoid	Difficult to appear overfit	Global rather than local optima

.

The probability distribution of the global data is a positively skewed Gaussian distribution and forms the global feature of the feature data, which further demonstrates that the response signal after bolt loosening is highly random. The box distribution regions, the upper and lower limits of the single group feature, also occupy the absolute probability interval of the overall data, and the outlier points beyond the boundary become very few. Overall consideration, both of the global data distribution and the local single-group data distribution, can do well to comply with the Gaussian distribution feature of the data. Therefore, the Gaussian kernel is chosen as the kernel function for the mapping of the feature matrix to the higher-dimensional kernel space.

2.5. The Dimension Reduction and Optimization Method of Characteristic Matrix

The quality of the data structure of the high-dimensional feature matrix fundamentally affects the success rate of the failure-state identification. In order to achieve comprehensive data capture and hold the original feature of the feature dataset, all of the characteristic values were directly combined. Because the domain features are not sensitive to the bolt failure state, the less sensitive data will generate cumulative classification errors. In order to reduce the effect of outlier data in the multivariate fusion feature matrix on the global classification properties, the different dimension-reduction methods need to be chosen for the most appropriate restructuring of the matrix data structure. It can be argued that there exists a nonlinear manifold with a lower dimensional manifold than the bolt failure feature matrix that can reflect the data structure of the high-dimensional matrix while completely retaining the global feature, and the dimension of the lower-dimensional manifold was defined as the intrinsic dimension. At present, the estimation problem of this intrinsic dimension is mainly divided into global estimation and local estimation, and the local estimation of the intrinsic dimension can improve the authenticity of the data structure. The two common local methods are mainly the correlation dimension method (CorrDim method) and the maximum likelihood estimation method (MLE method) [27].

The CorrDim method calculates the number of data points in the hypersphere with radius r, and the relative quantity $C\left(r\right)$ can be obtained according to the relationship between the number of data points in the hypersphere with radius r, which is proportional to $r^d$.

$$C\left(r\right)={\displaystyle \frac{2}{n(n-1)}}{\sum }_{i=1}^n{\sum }_{j=i+1}^{n}c$$

(26)

$$c=\left\{\begin{array}{c}1,‖x_i-x_j‖\le r\\ 0,‖x_i-x_j‖>r\end{array}\right.$$

(27)

Because $C\left(r\right)$ is proportional to $r^d$, $C\left(r\right)$ can be used to estimate the intrinsic dimension:

$$d=\underset{r\to 0}{\mathit{lim}}{\displaystyle \frac{logC\left(r\right)}{logr}}$$

(28)

The $C\left(r\right)$ can be estimated according to radius $r_1$ and $r_2$.

$$d=\underset{r\to 0}{\mathit{lim}}{\displaystyle \frac{log(C\left(r_1\right)-C(r_2\left)\right)}{log(r_1-r_2)}}$$

(29)

The data point in the hypersphere with radius r of the MLE method was estimated by constructing a Poisson model, and the data coverage rate of the intrinsic dimension d is expressed as:

$$\lambda \left(t\right)={\displaystyle \frac{f(x){\pi }^{\frac{d}{2}}d{t}^{d-1}}{\Gamma (\frac{d}{2}+1)}}$$

(30)

In the formula: $f\left(x\right)$—sampling density; $\Gamma \left(x\right)$—gamma function. Based on the Poisson model and the k neighbor points, the intrinsic dimension d of the data point $x_i$ is as follows:

$${\widehat{d}}_k\left(x_i\right)={\left({\displaystyle \frac{1}{k-1}}{\sum }_{j=1}^{k-1}log{\displaystyle \frac{T_k\left(x_i\right)}{T_j\left(x_i\right)}}\right)}^{-1}$$

(31)

In the formula: $T_k\left(x_i\right)$—the radius of the minimal hypersphere covering k neighbor points centered around $x_i$.

3. Results and Discussion

3.1. The Analysis of Primary Extraction of the Time-Frequency Domain Feature

The procedure was programmed on MATLAB software according to the sensitivity analysis and extraction method of the time-frequency feature, and the procedure divides each segment of data collected into 300 groups with a total of 900 groups. Groups 1 to 300 were divided as the complete loosening group, groups 301 to 600 were divided as the loosening group, and groups 601 to 900 were divided as the complete pre-tightening group, as shown in

Table 4. Time-domain feature of the bolt failure signal.

State	Order	Time Domain Feature Set
		Average Value	Standard Deviation	Skewness Factor	Kurtosis Factor	Maximal Value	Minimum Value	Peak Value	Mean Square Values	Pulse Factor	Waveform Factor	Impact Factor	Margin Factor	Vibration Energy
Complete loosening	1	0.57	31.74	−0.08	3.96	129.37	−127.23	256.60	31.74	4.08	1.31	5.33	0.22	2,014,806.25
	2	0.48	26.46	0.04	3.31	104.09	−84.29	188.38	26.45	3.93	1.27	5.00	0.24	1,399,520.25
	3	0.43	34.90	−0.01	3.87	151.80	−160.05	311.85	34.89	4.35	1.30	5.66	0.21	2,434,780.61
	…	…	…	…	…	…	…	…	…	…	…	…	…	…
	299	0.32	27.33	−0.10	3.59	101.16	−111.97	213.13	27.32	3.70	1.29	4.77	0.23	1,492,985.86
	300	0.33	27.80	−0.13	3.56	96.20	−128.69	224.89	27.80	3.46	1.28	4.44	0.20	1,545,213.23
Loosening	1	−0.10	22.42	0.02	3.36	94.68	−98.24	192.92	22.41	4.22	1.27	5.35	0.30	1,004,771.78
	2	−0.11	28.43	0.16	4.25	131.33	−127.92	259.25	28.42	4.62	1.30	6.03	0.28	1,615,899.78
	3	−0.10	26.08	0.01	3.40	111.93	−106.03	217.96	26.07	4.29	1.27	5.46	0.27	1,359,484.86
	…	…	…	…	…	…	…	…	…	…	…	…	…	…
	299	−0.16	26.37	0.14	3.20	125.53	−74.12	199.65	26.36	4.76	1.26	5.99	0.29	1,389,953.20
	300	−0.20	24.70	0.07	3.12	88.89	−78.86	167.75	24.69	3.60	1.25	4.51	0.23	1,219,567.43
Complete pre-tightening	1	−0.25	23.59	−0.08	3.47	78.67	−87.22	165.89	23.59	3.34	1.28	4.25	0.23	1,112,740.70
	2	−0.32	28.94	0.16	3.92	133.96	−104.64	238.59	28.93	4.63	1.29	5.99	0.27	1,673,967.63
	3	−0.28	24.02	0.16	3.40	105.09	−78.78	183.86	24.01	4.38	1.27	5.54	0.29	1,153,217.03
	…	…	…	…	…	…	…	…	…	…	…	…	…	…
	299	−0.25	30.01	0.13	4.13	139.97	−126.36	266.33	30.01	4.66	1.30	6.08	0.26	1,800,708.01
	300	−0.29	26.55	−0.01	3.08	103.49	−92.11	195.59	26.54	3.90	1.25	4.89	0.23	1,408,760.97

.

The sensitivities of the three states of the time-domain characteristic were obtained though the analysis of MATLAB software, as shown in Figure 8.

Although the time-domain index cannot be used as the main feature index of bolt structure loosening-state identification, the accurate analysis of the sensitivity of the time-domain feature to loosening and the extraction of the corresponding feature both play an important role in improving the accuracy of the bolt loosening state identification.

It can be seen that most features have limitations in the distinguishing effect of the loosening state and the degree of loosening, and only a few features can complete the state identification of the three states without the identification of other classification methods. Low-order center distance, origin moment, and peak values have the main distinguishing effects, which directly reflect the magnitude or trend of the data. The ratio of higher-order data statistics actually weakens the influence of the data exception points, so some features are not effective for data trend classification.

The analysis of the frequency domain feature set is similar to the time-domain feature set, and the frequency curve statistics are shown in

Table 5. Frequency-domain feature of the bolt failure signal.

Loosing State	Order	Frequency-Domain Feature Set
		Average Value	Standard Deviation	Skewness Factor	Kurtosis Factor	Maximal Value	Minimum Value	Peak Value	Mean Square Values	Pulse Factor	Waveform Factor	Impact Factor	Margin Factor	Vibration Energy
Complete loosening	1	1.10	0.89	1.45	5.28	5.09	0.00	5.08	1.42	3.58	1.29	4.61	4.18	4029.61
	2	0.91	0.75	1.64	6.46	4.65	0.01	4.64	1.18	3.93	1.29	5.09	5.57	2799.04
	3	1.22	0.97	1.30	4.84	5.96	0.01	5.94	1.56	3.82	1.27	4.86	3.97	4869.56
	…	…	…	…	…	…	…	…	…	…	…	…	…	…
	299	0.96	0.75	1.37	5.50	5.57	0.02	5.55	1.22	4.56	1.27	5.77	5.99	2985.97
	300	0.97	0.78	1.35	4.88	4.47	0.01	4.47	1.24	3.60	1.28	4.60	4.74	3090.43
Loosening	1	0.80	0.60	1.42	6.29	4.38	0.00	4.38	1.00	4.37	1.25	5.47	6.83	2009.54
	2	1.02	0.75	1.73	8.98	6.77	0.01	6.76	1.27	5.32	1.24	6.61	6.46	3231.80
	3	0.94	0.69	1.19	4.57	3.87	0.00	3.86	1.17	3.32	1.24	4.10	4.34	2718.97
	…	…	…	…	…	…	…	…	…	…	…	…	…	…
	299	0.95	0.70	1.14	4.40	4.13	0.02	4.11	1.18	3.50	1.24	4.34	4.56	2779.91
	300	0.90	0.65	1.16	4.50	3.87	0.02	3.86	1.10	3.51	1.23	4.32	4.83	2439.13
Complete pre-tightening	1	0.86	0.61	1.24	4.78	3.62	0.01	3.62	1.05	3.43	1.23	4.22	4.92	2225.48
	2	1.05	0.76	1.25	4.97	4.57	0.04	4.53	1.29	3.53	1.23	4.36	4.15	3347.94
	3	0.86	0.64	1.52	6.24	4.34	0.03	4.31	1.07	4.04	1.24	5.03	5.83	2306.43
	…	…	…	…	…	…	…	…	…	…	…	…	…	…
	299	1.07	0.81	1.50	6.23	5.59	0.02	5.57	1.34	4.16	1.26	5.23	4.90	3601.42
	300	0.96	0.70	1.30	5.30	4.52	0.01	4.51	1.19	3.81	1.24	4.71	4.91	2817.52

.

The characteristic data of center frequency, frequency variance, and mean square frequency are shown in

Table 6. Frequency feature of bolt failure signals.

Loosening State	Order	Frequency-Domain Feature Set
		Center Frequency	Frequency Variance	Mean Square Frequency
Complete loosening	1	506.34	31,843.23	288,219.59
	2	502.85	33,217.48	286,080.40
	3	509.23	33,176.29	292,493.05
	…	…	…	…
	299	497.61	34,155.43	281,774.23
	300	504.59	32,248.29	286,862.79
Loosening	1	492.48	46,514.56	289,054.60
	2	506.22	49,295.62	305,559.34
	3	496.24	48,697.87	294,948.13
	…	…	…	…
	299	484.82	47,391.68	282,442.35
	300	498.11	45,658.78	293,774.00
Complete pre-tightening	1	560.15	49,696.03	363,465.36
	2	550.03	48,386.07	350,920.99
	3	557.62	48,949.41	359,894.80
	…	…	…	…
	299	555.48	44,860.79	353,419.69
	300	565.47	46,735.68	366,492.60

.

The frequency-domain characteristic trends of the bolt failure signals are shown in Figure 9.

It can be seen that the signal classification ability of most frequency domain characteristic values are only limited to the single aspect of the loosening state or to the degree of loosening. Therefore, the two degrees of signals are unable to be completely distinguished through a single feature or simply a combination of single features. Although center frequency, frequency variance, and mean square frequency cannot accurately distinguish between the three states, the difference between each of the two states is obvious enough.

3.2. The Second-Order Feature Optimization of Optimizing the VMD Energy Entropy

The whale optimization algorithm (WOA) is a relatively novel group-intelligence optimization algorithm used to simulate the hunting behavior of marine humpback whales proposed in recent years. After years of optimization and improvement, the WOA has achieved good adaptability in performance. The algorithm can complete the balance between global optimization and local optimality, and the main principle and process are shown in Figure 10. The size of whale populations is first initialized during the VMD decomposition process, then the parameter dimension is optimized, and finally, the upper and lower limits of the optimization parameters need be set. The optimal individual position vector and the infinite optimal parameter judgment value are initialized randomly, and the current fitness function value will be calculated in circulating iteration to find an excellent process.

Envelope entropy (EP) is used as the optimal fitness function, which can represent the sparse nature of the signal. If there is less noise and less feature information in the IMF component, the value of EP is smaller. In order to decompose the VMD as thoroughly as possible (the IMF components contain the fewest frequency characteristics), the envelope entropy minimum (EPMin) is used as the adaptation value. The mathematical principles of the EP are as follows: the sequence $\left\{x_i\right\}$ obtained by the Hilbert envelope was normalized to obtain $\left\{p_i\right\}$, and the EP of the IMF component was obtained according to the formula $E_p=-{\sum }_{i=1}^N{p}_{i}lg{p}_i$. The EPMin was obtained after making the following comparisons:

The current EPMin was compared with the optimal parameter judgment value in the “surround the prey” phase, and also, the individual variable values $D=CX*-X$ and population optimum position $X\left(t+1\right)=X*\left(t\right)-AD$ were updated (A and C are the calculated coefficients). Entering the “prey capture” phase, the optimization path is optimized along the “spiral path $X\left(t+1\right)=X*\left(t\right)-\left|X*\left(t\right)-X\left(t\right)\right|e^{bl}cos2\pi l$”, at which time the individual whales can move between their current positions and the upper optimal positions along the spiral path. The purpose of the stage of “hunting prey” is to search for better solutions. The contractile mechanism was established through the method of the whale individuals being restricted to the local optimal range and randomized. And then, the probability value p of the whale individual was established. The optimal position was updated according to the values of p and A until the end of the iteration (Figure 11, Figure 12 and Figure 13).

The parameters and the central frequency values of the IMF components obtained by the above optimization process are shown in

Table 7. The WOA-VMD optimization parameters of three states.

Loosening State	EPMin	Optimal α	Optimal k	Center Frequency of the IMF Components/2 kHz
				k = 1	k = 2	k = 3	k = 4	k = 5	k = 6	k = 7	k = 8
Complete loosening	16.6095	1079.1	7	0.0535	0.0841	0.1565	0.2221	0.2856	0.3669	0.4225	-
Loosening	16.6422	1430.2	6	0.0763	0.1244	0.1538	0.2642	0.3307	0.4051	-	-
Complete pre-tightening	16.6425	1500	8	0.0551	0.098	0.14	0.1839	0.2875	0.3153	0.3764	0.4398

.

The resulting VMD decomposition results are shown in Figure 14; the horizontal axis represents the frequency (HZ), and the vertical axis represents the acceleration signal (m/s²). Compared with the classic method, it can be found that the IMF components obtained by WOA optimization are actually extracting the frequency components of large amplitude values, while the frequency components of medium and smaller values are placed as residual terms. This method filters out the minor components in the signals, reduces the signal interference, and defines the main frequency of the vibration response signal in the vibrating screen bolt-loosening state. However, the low-frequency operating frequency is also attributed to the minor components, which indicates that the high-frequency and intermediate-frequency signals caused by the bolt failure will completely drown out the low-frequency working signals.

The energy entropy can reflect the uncertainty of the distribution of the vibration energy in the vibrating screen bolt failure response in different frequency bands. The signal is more complex with larger IMF component amplitude and numbers of bolt structural failure responses, and also, the degree of bolt failure is more obvious.

The energy entropy is calculated ($HE=-{\sum }_{j=1}^K{P}_{j}lg{P}_j$) by calculating the IMF energy density ($P_j={\sum }_{i=1}^N{x}_{IMFj}^2/{\sum }_{i=1}^N{x}_i^2$), and the results are shown in

Table 8. The VMD energy entropy feature in the three states.

Loosening State	The Energy Entropy of the IMF Component
	IMF1	IMF2	IMF3	IMF4	IMF5	IMF 6	IMF7	IMF8
Complete loosening	0.0547	0.0667	0.1258	0.2207	0.2116	0.116	0.0864	-
Loosening	0.0780	0.0723	0.1583	0.1673	0.1039	0.1302	-	-
Complete pre-tightening	0.0415	0.0517	0.0679	0.1489	0.1226	0.1389	0.1318	0.0787

.

As can be seen by combining Table 7 and Table 8, the energy entropy values of different loosening degrees obtained under similar frequency components basically meet the hierarchical relationship of HE_{Complete loosening} < HE_loosening < HE_{complete loosening}, which indicates that this feature set meets the requirements of the feature sets for the identification of the vibrating screen bolt structure and has higher sensitivity to the degree of loosening.

The multidimensional dataset cannot be used directly for model training and state identification because of the roughness of the initial data and the different dimensional phenomena of the multidimensional dataset. Therefore, the multidimensional datasets need to be combined and reconstructed to train the multivariate fusion feature high-dimensional matrix. The main combination methods and steps are shown as follows:

First, the high-dimensional feature matrices A and B are constructed from the time-frequency feature dataset, which was extracted based on the time-frequency feature. And then, according to the results in Table 7 and Table 8, the energy entropy values of IMF 1 to IMF 6 in the complete pre-tightening state are extracted to form the characteristic set C₁, the energy entropy values of IMF 1 to IMF 6 in the loosening state are extracted to form the characteristic set C₂, and the energy entropy values of IMF 1 to IMF 6 in the complete loosening state are extracted to form the characteristic set C₃.

Third, C₁, C₂, and C₃ are increased to the dimension and combined into second-order feature extraction matrix C. And then, the multivariate fusion feature matrix G is established, although the matrices A, B, and C were combined and normalized:

• Step 1:

  $$A={\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right]}_{m\times n}$$

  (32)
• Step 2:

  $$B={\left[\begin{array}{ccc}{b}_{11}& \cdots & b_{1k}\\ ⋮& \ddots & ⋮\\ b_{m1}& \cdots & b_{mk}\end{array}\right]}_{m\times k}$$

  (33)
• Step 3:

  $$C=vertcat(repmat\left(C_1,m,1\right),repmat\left(C_2,m,1\right),repmat\left(C_3,m,1\right))$$

  (34)
• Step 4:

  $$Q_{m\times (n+k+j)}=horzcat(A,B,C)$$

  (35)
• Step 5:

  $$L=\left(L_1\cdots L_N\right)=(norm\left(Q\left(:,1\right)\right),\cdots ,norm\left(Q\left(:,n\right)\right))$$

  (36)

  $$S=Q/\left(repmat\right(M, m, 1\left)\right)$$

  (37)

  $$G=mapminmax\left(Q\right)$$

  (38)

In the formula: m—classification group number of the feature set of high dimension; n—feature number of feature set of high dimension; A—dataset of time-domain feature; B—dataset of frequency-domain feature; C—the VMD energy entropy feature set; L—the norm vector of column vectors of high-dimensional feature dataset; S—normal-normalized high-dimensional feature data matrix; Q—high-dimensional feature dataset; G—normalized high-dimensional feature data matrix; $Vertcat\left(X_1,\cdots ,X_n\right)$—vertical connection of n same-dimensional matrices; $Horzcat\left(X_1,\cdots ,X_n\right)$—horizontal connection of n same-dimensional matrices;$Repmat\left(X, n, dim\right)$—the matrix forming by copying the matrix X of n times by rows (dim = 1) or by columns (dim = 2); Norm(X)—the second norm of the vector X;$Mapminmax\left(X\right)$—the normalization of the vector X.

3.3. The Comparison and Optimization of Bolt-Loosening-State-Based SVM Method

The classic methods cannot complete the identification of the bolt failure states according to the classification process analysis of the SVM method. With the continuous improvement of classification methods, the SVM methods have also been expanded into multi-classification SVM, which can solve the multi-classification state identification in the field of fault diagnosis. The multi-classification method of SVM is divided into the overall optimization method and the combination learning method. Due to the few categories of bolt failure states, the combinatorial learning method was selected for the research. The combination learning method can be divided into “one-versus-one (OVO-SVMs)”, “one-versus-rest (OVR-SVMs)”, “binary tree-based classification (BTA-SVMs)”, etc. The multi-classification algorithm structures are shown in Figure 15.

The OVR-SVMs method can complete the classification by constructing n SVM models for the n-class data samples. For the bolt failure state identification, each SVM classified the sample of a specific failure state as a class and classified the remaining state bolt failure state features into a class. Finally, n training results are obtained, and the classification decision function of each result can be expressed as $f\left(x\right)=sign\left({\sum }_{i=1}^n{\alpha }_i^{*}{y}_{i}K\right(x•x_i)+b*)$. The test set data were calculated n times in parallel, and the classification decision function of n times results is compared to find $f\left(x\right)=MAX\left(sign\right({\sum }_{i=1}^n{\alpha }_i^{*}{y}_{i}K(x•x_i)+b*)$). This method has obvious disadvantages in that distinguishing a certain state single-feature set from all remaining state feature sets will cause a “One against Others” unbalanced training sample. Therefore, the training results are biased, and the use of down-sampling or up-sampling will lead to a model global feature reduction or over-fitting.

The BTA-SVMs method is similar to the OVR-SVMs method. The “One against Others” unbalanced training sample problem also exists in a branch with an odd number of classification states. First, the class k combined training sample and the (n − k) class combined training sample are classified, and then, all the above steps are repeated for different training samples within their combinations until all states are finally distinguished. However, the selection of the initial k samples needs human intervention. If the selected feature combination is less differentiated, it will not achieve the effect of classification in the process.

The OVO-SVMs method needs to construct the n(n − 1)/2 SVM classifiers of binary classification, and the classification output is made in the “voting” form. The OVO-SVMs method does not have the “One against Others” unbalanced training sample problem, and the pattern classification through “voting results” greatly weakens the classification error in the classification process and improves the classification accuracy. However, with the increase in the failure state and degree n, the number of classifiers required will also increase, which causes the surge of classification calculations leading to a decrease in classification speed and the real-time classification of the bolt failure state.

The classification states of the multi-fusion feature matrix of the bolt structure failure state are “Tighten and Fail”, “Loosening”, and “Complete loosening” (n = 3), so the OVO-SVMs method was selected as the multi-state classification method for bolt failure. Based on the results of the kernel function selection and the SVM method, the LIBSVM classification decision model of the OVO-SVMs method, with the kernel function as a Gaussian kernel function, was used to train and test the high-dimensional feature matrix of the bolt failure state. Feature matrix A, feature matrix B, feature matrix C, and normalized multivariate fusion feature matrix G of the second-order feature extraction of VMD energy entropy were trained and tested to verify the sensitivity of the different features to the bolt failure state.

According to the comparison of identification success rates of different test sets, not only can the optimal feature matrix combination pattern be obtained, but the sensitivity analysis of the previous features can also be verified. Therefore, the most obvious direction of the response feature change in the loosening state can be determined, providing insights for the simple identification of the bolt failure state. The bolt failure state identification process is shown in Figure 16.

The high-dimensional feature matrix data need to be added to label the LIBSVM training. So, the 1~300 data were set to label 1, the 301~600 data were set to label 2, and the 601~900 data were set to label 3. In order to make the number of training samples as large as possible to meet the requirement of the global optimization of the model, 90% of the samples were used as a training set (10% of the samples were included as the testing set). The identification success rates of the four groups’ feature matrices are shown in Figure 17.

The identification success rates of time-domain feature matrix A, frequency-domain feature matrix B, WOA-VMD energy entropy matrix C, and normalized multivariate fusion feature matrix G are 64.444%, 74.444%, 81.111%, and 90%, respectively. The conclusions that the discriminating capacity of the frequency-domain feature is bigger than the time-domain characteristic and that the single feature cannot accurately complete the bolt failure state identification were verified. It can be seen that the multiple fusion feature matrix is more suitable to identify the failure state of the bolt structure, but the model parameter selection limits the identification success rate, only keeping it at a high level. The miscalculation rates of the classification state with inhomogeneity are still at a high level.

It can be seen that the relatively low success rate of bolt failure state identification is mainly because only the kernel function is set in the LIBSVM model; the penalty coefficient c and the kernel function parameter g of LIBSVM model need to be strictly constrained when Gaussian kernel functions for high-dimensional kernel space mapping of a nonlinear feature matrix are introduced. If ε is leveled off to ∞, the optimal hyper plane will lose affect. Therefore, the penalty coefficient c is inducted to meet the requirement of the minimization of ${‖\omega ‖}^2$ and ${\sum }_{i=1}^n{\epsilon }_i$. The kernel function parameter g, which can affect the number of support vectors in the optimal hyper plane, is a hyper-parameter of the Gaussian kernel function. A higher number of support vectors will generate worse generalization of the model, and a smaller g will lose the ability to classify the nonlinear feature matrix. The process of finding the parameter optimal solution is mainly undertaken by the optimization algorithm, and the dimension and computational amount of the high-dimensional feature matrix also have a great influence on the identification process. Therefore, in order to obtain a bolt failure state identification model with a higher success rate, the optimization treatment of the model needs to be finished by the data dimensionality reduction method and optimization algorithm.

3.4. The Data Dimension Reduction Optimization of the Failure Feature Matrix

The numbers of the nearest neighbor points in the MLE method are set to k₁ = 6 and k₂ = 12, according to testing, and the intrinsic dimensions of the two methods are ${\mathrm{d}}_1=3$ and ${\mathrm{d}}_2=7$. In order to obtain the merits of the two results, the identification success rates of the different dimensional reduction methods need to be analyzed. The representative data dimension reduction method was combined with the obtained Eigen dimension for combined verification, and the methods and properties of the data dimension reduction are shown in

Table 9. Typical reduction methods and properties.

Method	Linearity	Feature	Properties	Supervise
Principal Component Analysis (PCA)	Yes	Global	Largest global variance	No
Locally Linear Embedding (LLE)	No	Local	Local linear reconstruction	No
Kernel Principal Component Analysis (KPCA)	No	Global	High dimensional structure mapping	No
Linear Discriminant Analysis (LDA)	No	Global	Intra-class variance was minimal	Yes
Local Tangent Space Arrangement (LTSA)	No	Local and Global	Local coordinates are indicated	No

.

The resulting Eigen dimension d₁ and d₂ are reduced by five typical dimension reduction methods to train and test under the LIBSVM model, and the kernel function of the LIBSVM model is the Gaussian kernel function. The data structures of the five dimension reduction methods obtained when d₁ = 7 are used as the intrinsic dimensions are shown in Figure 18. It can be seen that the LDA method and the LTSA method have the highest data discrimination; although there are still admixed regions of the data, there are already better distinguishing features overall. Other dimension reduction methods only have the trend of data identification; compared with matrix G, other methods have a certain identification capacity but still have limitations.

The data structures of the five dimension reduction methods obtained when d₂ = 3 is used as the intrinsic dimensions are shown in Figure 19. The data structures have no big differences, and only the data structures of the LLE method and the LTSA method have somewhat changed after dimension reduction, which is mainly because these two methods reduce the data dimension based on local features, so the perception of the intrinsic dimension is obvious.

The feature matrix after dimension reduction through the LIBSVM model was used to identify the bolt failure states to obtain the identification capacity of the feature matrix (90% of data are the training set; 10% of data are the testing set). The identification success rates are shown in

Table 10. The identification success rate of LIBSVM model under different combinations of intrinsic dimension and dimension reduction methods.

Order	Dimensional Reduction Method	Intrinsic Dimension
		${\mathbf{d}}_1=3$	${\mathbf{d}}_2=7$
1	PCA	86.667%	97.778%
2	LLE	82.222%	95.556%
3	KPCA	86.667%	95.556%
4	LDA	80.000%	91.111%
5	LTSA	95.556%	97.778%

.

It is clear that the identification success rate of ${\mathrm{d}}_2=3$ is significantly weaker than ${\mathrm{d}}_1=7$, which may be because the global data features of the original data are destroyed in the process of dimension reduction, making the data space “overlap” between different data spaces, so the optimal hyper plane during training cannot be found. The intrinsic dimension d = 7 is more appropriate to the dimensional reduction in the high-dimensional feature matrix in the bolt failure state of the vibrating screen, which also verified the universality of the MLE method. It can be seen that in the intrinsic dimension d₁ = 3, the LTSA method can also obtain a 95.556% success rate, which is significantly higher than other dimension reduction methods, which reflects the good universality of the LTSA method to high-dimensional data.

3.5. The Parameter Optimization of LIBSVM Model Based on Optimization Algorithm

The optimization experiment of the penalty factor c and the kernel function parameter g of the LIBSVM model was established to study the effect of the hyper-parameters on the identification accuracy. Ant colony optimization (ACO) is a classic biomimetic optimization algorithm [28]. The optimal solution of the desired parameters can be obtained by constructing paths and updating pheromones. The main steps of the identification are as follows:

• Initializing the parameters. The values of populations (P), iterations (M), variables (D), volatilization coefficient of pheromone (R), and the transition probability constant (p₀) are 30, 50, 2, 0.9, and 0.2, respectively. The ranges of the lower limits of variables (LB) and upper limits of variables (UB) are (0.1, 0.01) and (2000, 100), respectively. The pheromone recorder is named as the τ.
• Random initialization of the ant positions. The random position of the values of the two optimization variables c and g between the upper and lower limits was calculated by the pseudo-random number function, and the identification success rate of all bolt states based on the initial random positions of c and g was obtained by the LIBSVM model. And then, the maximum value was assigned to the pheromone recorder τ, and the ant position was updated the initialized to the pheromone recorder τ.
• Enter the parameter optimization main cycle. The $P_i$ was obtained according to $P_i=\left|{\tau }_{i-1}-{\tau }_i/{\tau }_{i-1}\right|,$ and the local population coefficient is λ = 1/P. If $P_i<{\mathrm{p}}_0$, a local search will start; if $P_i>{\mathrm{p}}_0,$ a global search will start. The new upper limits are obtained of the global/local population optimization parameters c and g. The new upper limit and feature data were imported into LIBSVM to recalculate the identification success rate (${\tau }_i$) of the current-position bolt states, and then, the ${\tau }_{\mathrm{i}}$ was compared with the initialized pheromone marker position τ. If ${\tau }_i>\tau$, the optimized parameters c and g will be updated.

Step 3 was cycled until the maximum number of iterations was reached. The values of the optimization parameters (c and g) and the optimization process curve are shown in Figure 20. The identification success rate of current bolt state identification is shown in Figure 21 and Figure 22. It can be seen that the best combination of the penalty factor and the kernel function hyper-parameters is “Best c = 11.87 and Best g = 3.4106”. The identification success rate was 98.8889%, and the best fitness was reached at the number of iterations of 20 times under this combination. The identification success rate of the “complete pre-tightening” state has reached 100% best results, but one test sample was misjudged in each of the “complete loosening” and “loosening” states.

Teaching–learning-based optimization (TLBO) is a new optimization algorithm simulating class teaching, and the optimization process is mainly divided into a “teaching stage” and a “learning stage” [29].

In the “teaching stage”, the new random lists were obtained by combining c and g randomly within the upper and lower limits. All combinations of the random list were imported into the LIABSVM model for training and testing to obtain the identification success rates of all combinations. The maximum success rate was assigned to $X_T$ as the “teacher” in the first stage. The average assignment value was assigned to ${\mathrm{M}}_{\mathrm{T}}$ as the “Average class score”. If we want the “student” grades to be as “teacher” as possible, the optimization method is $X_{(i, new)}=X_{(i, old)}+r_i\left(X_T-T_F{M}_T\right)$ and the teaching factors $T_F$ ($T_F=1$ or $T_F=2$) mainly affect the $M_T$. The $r_i$ is the original state of the “student”, which is a random number within (0, 1). The values were updated by comparing $X_{(i, new)}$ and $X_T$.

In the “learning stage”, the process of students learning from each other after class was simulated. The data iteration process results i and j are the combination of random numbers generated in the first stage. The identification success rates under the two combined parameters obtained by the LIBSVM model are compared, and the corresponding parameter combination of the high identification success rate is used to subtract the parameter combinations of the identification success rate. And then, the optimal solution completion parameters can be obtained by iterating following the equation:

$$X_{i,new}=\left\{\begin{array}{c}{X}_{i,old}-r_i\left(X_i-X_j\right),f\left(X_i\right)<f\left(X_j\right)\\ X_{i,old}-r_i\left(X_j-X_i\right),f\left(X_i\right)>f\left(X_j\right)\end{array}\right.$$

(39)

In the formula: $f\left(X_i\right)$—fitness values of the i th combination. The evolutionary curves of the optimized parameters c and g are shown in Figure 23. The identification success rate under the optimal parameter combination was obtained, as shown in Figure 24 and Figure 25. The best parameter combination of the penalty factor c and the kernel function parameter g is “Best c = 86.5759 and Best g = 1.3871”. The identification success rate of all failure states was 100% under this combination, which indicated the adaptability of this method to the identification of the bolt failure state and which met the high precision requirements of bolt failure state identification. The identification success rate of the bolt failure state is gradually increased, and the overall method of the failure state identification method of the vibrating screen bolt structure has been completed. Therefore, the bolt failure state identification method of the TLBO method of LIBSVM, based on the LTSA method of high-dimensional data spatial dimension reduction in the intrinsic dimension (d = 7), was obtained.

4. Conclusions

• According to the sensitivity analysis and extraction of the time-frequency feature of vibrating screen bolt structure, most features have limitations on the distinguishing effect of the loosening states and the degree of loosening. The main identification effects of the data are the low order center distance, the origin moment, and the peak value. The signal classification ability of most frequency-domain features is only limited to identifying the single aspect of the loosening state or loosening degree, and the two levels of signals cannot be completely distinguished by a single feature or a simple combination of features.
• High-dimensional feature matrices A and B are constructed according to the time-frequency feature dataset initially extracted from the time-frequency feature. And then, the second-order feature extraction matrix C was obtained from the analysis of the second-order feature of the VMD energy entropy. The best fitness value of EPMin was obtained based on the whale optimization algorithm (WOA) to verify that the selected feature set is satisfactory for the identification requirement of the feature. The high-dimensional feature matrix of the bolt failure state was trained and tested through the LIBSVM classification model of the OVO-SVMs method. The identifying success rates of time-domain feature matrix A, frequency-domain feature matrix B, WOA-VMD energy entropy matrix C, and normalized multivariate fusion feature matrix G are 64.44%, 74.44%, 81.11%, and more than 90%, respectively, which can reflect the applicability of the failure-state identification of the normalized multivariate fusion feature matrix.
• The Eigen dimension d = 7 is more suitable for reducing the dimension of the high-dimensional feature matrix of the bolt failure state of the vibrating screen, according to the success rate of different Eigen dimensions. The best parameter combination of penalty factor c and kernel function parameter g for the LIBSVM model are “Best c = 86.5759 and Best g = 1.3871”, as based on ant colony optimization and teaching–learning-based optimization, and the identification success rate of all failure states was 100% under this combination. Therefore, the bolt failure state identification method of the TLBO method of LIBSVM, based on the LTSA method of high-dimensional data spatial dimension reduction in intrinsic dimensions, was obtained.