Pedestal looseness extent recognition method for rotating machinery based on vibration sensitive time-frequency feature and manifold learning

2.1. The characterization method of pedestal looseness extent based on vibration

It is bound to change the structure dynamic characteristics of the rotating machinery when the pedestal is loose as in [3], and the change of structure dynamic characteristics is reflected as the change of vibration characteristics. So, the pedestal looseness extent of rotating machinery can be characterized by vibration characteristics.

Fig. 1The waveform and spectrum of 5 kinds of looseness extents a) all bolts tightened; b) 1 bolt looseness; c) 2 bolts looseness; d) 3 bolts looseness; e) 4 bolts looseness

The vibration signals are collected in different pedestal looseness extents, such as T1 (all bolts tightened), T2 (1 bolt looseness), T3 (two bolts looseness), T4 (3 bolts looseness) and T5 (4 bolts looseness). When the number of loose bolts increases, the pedestal looseness extent strengthens. Fig. 1 shows the collected vibration signals and its spectrum of the five kinds of looseness extents.

Fig. 1 shows that there is difference among the vibration signal and its spectrum, for example the amount of spectrum energy, the position change of main frequency band and the decentralization or centralization degree of the spectrum. Fig.1 shows that the dominant frequency is 175 Hz, 156.3 Hz, 943.8 Hz, 425 Hz and 175 Hz respectively, and the dominant frequency amplitude is 0.677 m/s², 0.802 m/s², 0.759 m/s², 0.793 m/s² and 0.872 m/s² respectively, the amplitude at 287.5 Hz is 0.247 m/s², 0.369 m/s², 0.596 m/s², 0.401 m/s² and 0.203 m/s² respectively.

Fig. 2The probability density function in 5 kinds of looseness extent

In order to further observe the characteristics of different looseness extents, the probability density function is calculated as shown in Fig. 2. The waveform of probability density function is different with different looseness ex-tents of pedestal, for example degree of asymmetry of probability density function, the peak value of density distribution curve at the average and so on.

Overall, the characteristics of vibration signal are changed with the change of the pedestal looseness extent, and the different pedestal looseness extents are characterized by different characteristics of vibration signal and its spectrum.

2.2. Looseness extent feature extraction and looseness extent feature set construction

The pedestal looseness extent of rotating machinery can be visibly characterized by time-frequency characteristics of vibration signal, but it difficult to automatic recognition by pattern recognition algorithm. In order to realize quantitative characterization of pedestal looseness extent and easily implement intelligent recognition, multiple features are extracted to construct origin looseness extent feature set. To reflect the change of the characteristic of the vibration signal as comprehensive as possible, integrated vibration signal time and frequency domain information, 29 time-frequency parameters are extracted. 15 time-domain parameters are extracted from vibration time-domain signal such as maximum amplitude, average amplitude, root mean square, shape factor, kurtosis index, crest factor, impulse factor, clearance factor, skewness index and so on. And, 14 frequency-domain parameters are extracted from frequency spectrum such as average frequency amplitude, centroid frequency, mean-square frequency, frequency variance, root mean square frequency and so on. Finally, we construct 15 feature parameters in time-domain and 14 feature parameters in frequency-domain as provided in Table 1, and further combine them into a 29-dimensional time-frequency feature parameter set in order to fully and reliably obtain the pedestal looseness extent features. Also, the origin looseness extent feature set is obtained.

In Table 1, $x\left(n\right)$ represents the time series of a signal, $n=$1, 2,…, $N$, where $N$ is a sampling number. $s\left(k\right)$ represents the frequency spectrum of $x\left(n\right)$, $k=$1, 2,…, $K$, where $K$ is spectrum line number. $f_k$ is the frequency of the $k$-th spectrum line. $u$ represents the mean value of the $x\left(n\right)$. Time domain feature parameters $P_1$-$P_6$ represent the amplitude and energy of the time domain signal; $P_7$-$P_{15}$ denote the distribution situation of time series of the signal. Frequency domain feature parameters $P_{16}$ characterizes the spectrum energy; $P_{17}$-$P_{20}$ characterize the position change of main frequency band; $P_{21}$-$P_{29}$ characterize the decentralization or centralization degree of frequency spectrum.

Compared with single domain signal feature extraction, the 29-dimensional time-frequency domain feature set has three advantages as explained below: 1) it is more benefit for pattern recognition than classic spectrum analysis method. 2) It is more comprehensive than single time-domain or single frequency-domain feature extraction method to reflect the characteristics of looseness extent. 3) It is more sensitive to the change of looseness extent characteristics than STFT, WVD, wavelet decomposition and EMD.

2.3. Looseness extent sensitive feature selection and the looseness extent sensitive feature set construction

The looseness extent sensitivity of each feature from origin looseness extent feature set is different, the non-sensitive feature and poor sensitivity feature are not only bound to affect the looseness characterization capabilities of the feature set but also interfere with the recognition results. Therefore, the sensitive feature for pedestal looseness extent must be selected to construct the looseness extent sensitive feature set, which characterization capabilities is stronger than origin looseness extent feature set, and the recognition accuracy will be improved.

The scatter matrix includes between-class scatter matrix and within-class scatter matrix [23, 24], the between-class scatter value and within-class scatter value could be calculated by the two scatter matrix. The identifiability of different feature classes is reflected by the between-class scatter value, the larger the between-class scatter value is, the better identifiability of the feature is. The clustering characteristic of the same feature classes is reflected by the within-class scatter matrix, the smaller the within-matrix scatter value is, the better clustering characteristic of the feature is. If the between-class scatter value is larger and the within-class scatter value is smaller for the feature, the identifiability of feature will be better for different looseness extent feature sets, and the clustering characteristic of the feature will be better too for the same looseness extent feature set, then, feature’s recognition capability will be stronger for different looseness extents of pedestal, and the looseness sensitivity of the feature will be better. Therefore, the looseness sensitivity index algorithm is designed based on between-class scatter matrix and within-class scatter matrix, according to the feature’s characteristics of reflecting the identification and the degree of clustering. At the same time the looseness extent sensitive features are able to select from origin looseness extent feature set to construct the looseness extent sensitive feature set.

There are class $C$ looseness samples of different looseness extents, including $N_i$ sample number each class. For origin high-dimensional feature set, $X=\left\{x_1,x_2,\dots ,x_D\right\}$, $D$ is number of dimension.

The within-scatter matrix $S_B$ as follows:

where $u_i$ is mean of the $i$th class, $u_0$ is the mean of total samples.

The between-scatter matrix $S_W$ as follows:

where $x_i^j$ is the $i$th eigenvalues of $j$th class.

Then, the $tr\left\{S_W\right\}$ and $tr\left\{S_B\right\}$ are calculated, $tr\left\{·\right\}$ is the trace of the matrix. $tr\left\{S_W\right\}$ is average measurement of characteristic variance for all feature classes, $tr\left\{S_B\right\}$ is one of average measurement of average distance between global mean value and mean of each class.

Different looseness samples can be identified because they are located in different regions of the feature space. The larger the distance of these regions is, the better identifiability is. So, the algorithm of the looseness sensitivity index $J$ is designed as follows:

Obviously, the value of $J$ will be larger, if value of $tr\left\{S_W\right\}$ is larger or value of $tr\left\{S_B\right\}$ is smaller. The looseness sensitivity index $J$ reflect the feature’s recognition capability. The larger the value $J$ is, the stronger the recognition capability is. Also the smaller the value $J$ is, the weaker the recognition capability is.

The looseness extent sensitive feature is selected by the value of $J$. The information is more overall if more features are selected, and it is useful to improve the recognition accuracy. But, the non-sensitive feature or poor sensitivity feature will be brought into feature set if the too many features are selected, the recognition accuracy will be affected. So, it is very important to decide how many features are selected. First of all, the looseness sensitivity index of each feature is calculated, $J_i$ ($i=$1, 2,…, $D$). Secondly, $u_J$, the mean of all looseness sensitivity index is calculated. Finally, the looseness extent sensitive features are selected which sensitivity index $J_i\ge u_J$, and the looseness extent sensitive feature set is constructed.

3.1. Problem description

Consider a looseness samples set ${\mathbf{X}}_{ORG}=\left[{\mathbf{x}}_{org1},{\mathbf{x}}_{org2},\dots ,{\mathbf{x}}_{orgN}\right]$ ($N$ is the number of all looseness samples) for training and test with noise from ${\mathbf{R}}^m$ which exists an underlying nonlinear manifold ${\mathbf{M}}^d$ (${\mathbf{M}}^d\subset {\mathbf{R}}^d$) of dimension $d$. Moreover, suppose ${\mathbf{M}}^d$isembedded in the ambient Euclidean space $R^m$, where $d<m$. The problem that dimension reduction with LLTSA solves is finding a transformation matrix $\mathbf{A}$ which maps the looseness sample set ${\mathbf{X}}_{ORG}=\left[{\mathbf{x}}_{org1},{\mathbf{x}}_{org2},\dots ,{\mathbf{x}}_{orgN}\right]$ in ${\mathbf{R}}^m$ to the set $\mathbf{Y}=\left[{\mathbf{y}}_1,{\mathbf{y}}_2,\dots ,{\mathbf{y}}_N\right]$ in ${\mathbf{R}}^d$ as follows:

where ${\mathbf{H}}_N=\mathbf{I}-\mathbf{e}{\mathbf{e}}^T/N$ is the centering matrix which is used to improve the aggregation of $k$ nearest neighbors set ${\mathbf{X}}_i=\left[{\mathbf{x}}_{i1},{\mathbf{x}}_{i2},\dots ,{\mathbf{x}}_{ik}\right]$ of each sample ${\mathbf{X}}_i$, and $\mathbf{I}$ is the identity matrix, $\mathbf{e}$ is an $N$-dimensional column vector of all ones, $\mathbf{Y}$ is the underlying $d$-dimensional nonlinear manifold ${\mathbf{M}}^d$ of ${\mathbf{X}}_{ORG}$.

3.2. The algorithm of LLTSA

Given the data set $\mathbf{X}$ obtained by treating ${\mathbf{X}}_{ORG}$ with Principal Component Analysis, for each point ${\mathbf{X}}_i\in \mathbf{X}$, its $k$ nearest neighbors by a matrix ${\mathbf{X}}_i=\left[{\mathbf{x}}_{i1},{\mathbf{x}}_{i2},\dots ,{\mathbf{x}}_{ik}\right]$. In order to ensure the local structure of each ${\mathbf{X}}_i$, local linear approximation of the data points in ${\mathbf{X}}_i$ is calculated with tangent space as follows:

where ${\mathbf{H}}_k=\mathbf{I}-\mathbf{e}{\mathbf{e}}^T/k$, $Q_i$ is an orthonormal basis matrix of the tangent space and has $d$ eigenvalues of ${\mathbf{X}}_i{\mathbf{H}}_k$corresponding to its $d$ largest eigenvalues, and ${\mathrm{\Theta }}_i$ is defined as:

where ${\theta }_j^{\left(i\right)}$ is the local coordinate corresponding to the basis ${\mathbf{Q}}_i$. Now, we can construct the global coordinates ${\mathbf{y}}_i$, $i=$1, 2,…, $N$, in ${\mathbf{R}}^d$ based on the local coordinate ${\theta }_j^{\left(i\right)}$, which represents the local geometry as follows:

where ${\stackrel{-}{y}}_i$ is the mean of the various ${\mathbf{y}}_{ij}$, ${\mathbf{L}}_i$ is a local affine transformation matrix that needs to be determined, and ${\epsilon }_j^{\left(i\right)}$ is the local reconstruction error. Letting ${\mathbf{Y}}_{\mathbf{i}}=\left[{\mathbf{y}}_{i1},{\mathbf{y}}_{i2},\dots ,{\mathbf{y}}_{ik}\right]$ and $E_i=\left[{\epsilon }_1^{\left(i\right)},{\epsilon }_2^{\left(i\right)},{\epsilon }_3^{\left(i\right)},\cdots ,{\epsilon }_k^{\left(i\right)}\right]$, we have:

To preserve as much of the local geometry as possible in the low dimensional feature space, we intend to find ${\mathbf{y}}_i$ and ${\mathbf{L}}_i$ to minimize the reconstruction errors ${\epsilon }_j^{\left(i\right)}$ as follows:

Therefore, the optimal affine transformation matrix $L_i$ has the form $L_i=Y_i{H}_k{\mathrm{\Theta }}_i^{+}$, and $E_i=Y_i{H}_k(I-{\mathrm{\Theta }}_i^{+}{\mathrm{\Theta }}_i)$, where ${\mathrm{\Theta }}_i^{+}$ is the Moor-Penrose generalized inverse of ${\mathrm{\Theta }}_i$.

Let $\mathbf{Y}=\left[{\mathbf{y}}_1,{\mathbf{y}}_2,\dots ,{\mathbf{y}}_N\right]$ and ${\mathbf{S}}_i$ be the 0-1 selection matrix, such that $\mathbf{Y}{\mathbf{S}}_i={\mathbf{Y}}_i$, then the objective function is converted to this form:

where $\mathbf{S}=\left[{\mathbf{S}}_1,{\mathbf{S}}_2,{\mathbf{S}}_3,\dots ，{\mathbf{S}}_n\right]$, and $\mathbf{W}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}（{\mathbf{W}}_1,{\mathbf{W}}_2,{\mathbf{W}}_3,\dots ,{\mathbf{W}}_n）$ with $W_i=H_k(I-{\mathrm{\Theta }}_i^{+}{\mathrm{\Theta }}_i)$ According to the numerical analysis in [17], Wi can also be written as follows:

where ${\mathbf{V}}_i$ is the matrix of $d$ eigenvectors of ${\mathbf{X}}_i{\mathbf{H}}_k$ corresponding to its $d$ largest eigenvalues. To uniquely determine $\mathbf{Y}$, we impose the constraint $\mathbf{Y}{\mathbf{Y}}^T={\mathbf{I}}_d$. Considering the map $\mathbf{Y}={\mathbf{A}}^T\mathbf{X}{\mathbf{H}}_N$, the objective function has the ultimate form:

where $\mathbf{B}=\mathbf{S}\mathbf{W}{\mathbf{W}}^T{\mathbf{S}}^T$. It is likely to be proved that the above minimization problem can be converted to solving a generalized eigenvalue problem as follows:

Let the column vectors $a_1$, $a_2$,…, $a_d$ be the solutions of Eq. (13), ordered according to the eigenvalues, ${\lambda }_1<{\lambda }_2<\dots <{\lambda }_d$. Thus, the transformation matrix ${\mathbf{A}}_{LLTSA}$ which minimizes the objective function is as follows:

In the practical problems, one often encounters the difficulty that $\mathbf{X}{\mathbf{H}}_N{\mathbf{X}}^T$ is singular. This stems from the fact that the number of data points is much smaller than the dimension of the data. To attach the singularity problem of $\mathbf{X}{\mathbf{H}}_n{\mathbf{X}}^T$, we use the PCA to project the data set to the principal subspace. In addition, the preprocessing using PCA can reduce the noise. ${\mathbf{A}}_{PCA}$ is applied to denote the transformation matrix of PCA.

Therefore, the ultimate transformation matrix is as follows: $\mathbf{A}={\mathbf{A}}_{PCA}{\mathbf{A}}_{LLTSA}\text{,}$ and $\mathbf{X}\to \mathbf{Y}={\mathbf{A}}^T{\mathbf{X}}_{OGR}{\mathbf{H}}_N$.

Because of LLTSA’s good clustering performance, $d$-dimensional looseness feature set $\mathbf{Y}={\mathbf{A}}^T{\mathbf{X}}_{OGR}{\mathbf{H}}_N={\left({\mathbf{A}}_{PCA}{\mathbf{A}}_{LLTSA}\right)}^T{\mathbf{X}}_{OGR}{\mathbf{H}}_N$ outputted by LLTSA from high-dimensional looseness extent sensitive feature set ${\mathbf{X}}_{ORG}$. The low-dimensional looseness extent feature set $\mathbf{Y}$ has better identifiability than high-dimensional looseness extent sensitive set ${\mathbf{X}}_{ORG}$, and the features of $\mathbf{Y}$ are mutually independent, so the set $\mathbf{Y}$ is good for pattern recognition. The set $\mathbf{Y}$ is inputted into Weight $K$ nearest neighbor classifier for looseness extent recognition.

4.1. Looseness extent recognition by WKNNC

In order to finally realize pedestal looseness extent automatic recognition, the low-dimensional looseness extent sensitive feature set output by LLTSA serves should be recognized by classifier. The simplicity of $K$ nearest neighbor classifer (KNNC) algorithm makes it easy to implement. The KNNC has no complex training process, compared with other classification algorithms such as support vector machine, neural network and decision tree. It directly uses the local information of the classified data to form the final classification decision, which improves its efficiency of computation, makes it easy to realize and be widely used. But, it suffers from poor classification performance when samples of different classes overlap in some regions in the feature space, and its result is affected by the neighborhood size of $k$, it has poor stability. Therefore, the Weight $K$ nearest neighbor classifier (WKNNC) [21] is useful for overcoming the shortcomings of KNNC. The WKNNC is an improvement of KNNC, it not only has the all advantages of KNNC, but also is insensitive the neighbor size of $k$ and better robustness. So, the WKNNC is used for pedestal looseness extent recognition.

Given a training set $\mathbf{X}=\left\{\left(x_i,l_i\right),x_i\in {\mathbf{R}}^m，i=\mathrm{1,2},\dots ,n\right\}$, consisting of $n$ samples, the class label $l_i$ of each sample $x_i$ is known, $l_i\in \left\{l_1,l_2,\dots ,l_r\right\}$. The class label $l_t$ of testing sample $x_t$ need to be identified. The general idea of WKNNC is summarized as follows: for given testing sample $x_t$, its $k$ nearest neighbors are selected from the training samples. The class label $l_t$ of testing sample $x_t$ is identified based on the class label of the $k$ nearest neighbors. The target of classification is to make the classification error $M$ minimum. And for each value $l_j$, the classification error $M\left(l_j\right)$ is as follows:

where $P\left(l_i|x_t\right)$ is the probability of $x_t$ classified as $l_i$, $R\left(l_i,l_j\right)$ is error of class label $l_i$ classified as $l_j$. The all misclassification errors set by WKNNC are same as follows:

Then, the calculation steps of the WKNNC are described as follows:

Step 1: The Euclidean distance $d\left(x_t,x_i\right)$ between the testing sample $x_t$ and the training sample $x_i$ is defined as:

The $k+1$ nearest neighbor samples are selected from training samples based on the value of $d\left(x_t,x_i\right)$, denoted as $x_{t,1}$, $x_{t,2}$,…, $x_{t,k+1}$_.

Step 2: The sample $x_{t,k+1}$ is selected from the $k+1$ nearest neighbor samples, which Euclidean distance with $x_t$ is maximum, the maximum Euclidean distance is denoted $d\left(x_t,x_{t,k+1}\right)$. Then, the $d\left(x_t,x_{t,k+1}\right)$ is used to standardize the Euclidean distance between the other $k$ nearest neighbor samples and $x_t$ as follows:

where $D\left(x_t,x_i\right)$ is the standard Euclidean distance.

Step 3: Using Gauss kernel function, the $D\left(x_t,x_i\right)$ are transformed into similar probability $p\left(x_i|x_t\right)$ between $x_t$ and $x_i$ as follows:

Step 4: The posterior probability $P\left(l_i|x_t\right)$ that $x_t$ belongs to the class label $l_i$ ($i=$ 1, 2,…, $r$) is calculated based on the similar probability $p\left(x_i|x_t\right)$ between $x_t$ and the $k$ nearest neighbor samples, the $P\left(l_i|x_t\right)$ is as follows:

then, the most likely classification results $KNN\left(x_t\right)$ are obtained as follows:

According to the similarity degree between the nearest neighbor samples and the testing sample $x_t$, the WKNNC gives different weights to the nearest neighbor samples, which makes the classification results of the testing sample more close to the similar degree of training sample. Therefore, the WKNNC is insensitive the neighbor size of $k$, and the robustness of the results for the looseness extent identification is better.

4.2. Looseness extent recognition process

According to above preparation, the flowchart of the proposed method is shown in Fig. 3, the steps of the method are described as follows:

Step 1: data acquisition. The vibration signal is collected, and corresponding frequency spectrum is calculated, the training samples and test samples are obtained.

Step 2: extracted original features and constructed the original looseness extent feature set. 15 time-domain parameters are extracted from vibration signal, and 14 frequency-domain characteristic parameters are extracted from frequency spectrum, then 29-dimensional time-frequency looseness extent feature set is constructed by fusing the 14 time-domain parameters and 15 frequency-domain parameters.

Step 3: selected looseness sensitive feature and constructed looseness extent sensitive feature set. The looseness sensitivity index of each feature is calculated, $J_i$ ($i=$ 1, 2,…, $D$). Secondly, $u_J$, the mean of all looseness sensitivity index is calculated. Then, the looseness features are selected which sensitivity index $J_i\ge u_J$, and the looseness extent sensitive feature set is constructed.

Step 4: dimension reduction. The high-dimensional looseness extent sensitive feature set are compressed to low-dimensional looseness extent sensitive feature set with LLTSA. The low dimension and good classification performance feature set is obtained.

Step 5: recognition of looseness extents and output the recognition result. The low-dimensional feature set is inputted into WKKNC for looseness recognition, and the pedestal looseness extent recognition for rotating machinery is realized.

Fig. 3The flowchart of the proposed method

5.1. Experiment set up and signal acquisition

In this study, a pedestal looseness experiment of rotating machinery is performed to verify the effectiveness of the proposed looseness extent recognition method. The test rig is constructed as shown in Fig. 4.

Fig. 4Field installation diagram and test diagram. (Notes:1, 2,…, 10 are the serial number of the connecting bolt for embedded steel plate)

The embedded steel plate is fixed on the top of the tunnel by 10 bolts, and the installing bracket of fan is installed in the embedded steel plate. Pedestal looseness extent is simulated by different number of loose bolts, when the number of loose bolts increases, the pedestal looseness extent strengthens. At the test point 1 and test point 2, the speed of fan is 1500 r/min, the vibration signals of 5 kinds of looseness extent are collected, such as T1 (all bolts tightened), T2 (1 bolt looseness), T3 (two bolts looseness), T4 (3 bolts looseness) and T5 (4 bolts looseness), are shown in Table 2. Fig. 1 shows the collected vibration signal and its spectrum of the five kinds of looseness extents, it only shows time domain waveform of first 4096 points and corresponding spectrum in the frequency range of 0-5000 Hz, collected at the test point 1. The complete sensor selection and distribution scheme are shown in Table 3.

5.2. Experimental results and analysis

2048 continuous data were intercepted from each sample as time series for looseness extent, then, we can obtain 100 samples in each of looseness extent. We randomly select just 20 samples out of the 100 acquired samples as the training samples, and randomly select 20 samples out of the remaining 80 samples as testing samples. Application of the proposed method to recognize the looseness extent of pedestal.

In order to compare the dimension reduction and redundant treatment effect of PCA, the LPP, the LDA, and the LLTSA method are used to reduce the dimension for origin looseness extent feature set. To be comparable, the dimensions of PCA, LPP, LDA and LLTSA are set to 3. The comparison of dimension reduction results is shown in Fig. 5(a)-(d).

By comparing Figs. 5(a)-(d) we can find that the PCA-based data dimension reduction method cannot effectively separate the high dimension looseness feature set, the looseness extent of T2 is separated, but there is still serious aliasing, which will affect the accuracy of the WKNNC looseness recognition effect. The LPP-based data dimension reduction method can partly separate the different looseness feature sets, there are still some data mixed together, the looseness extent of T1, looseness extent of T3 and looseness extent of T4 mixed together. The LDA-based data dimension reduction method can’t separate feature sets from the looseness extent of T1, the looseness extent of T2 and the looseness extent of T3. The LLTSA-based data dimension reduction method works better than the PCA, LPP and LDA methods, but the looseness extent of T3 and the looseness extent T4 don’t completely separated, and the clustering performance is not good enough.

The looseness sensitivity index of 20 features is calculated as shown in Table 4, and the mean of the sensitivity index $u_J=$12.352. Then, 9 features (bolded part of the Table 4) that the sensitivity index $J_i\ge u_J$ are selected to construct the looseness sensitive feature set.

In order to compare the characterization capabilities between the origin looseness extent feature set and the looseness extent sensitive feature set, the looseness extent sensitive feature set is inputted into LLTSA to reduce the dimension, the dimensions of LLTSA is set to 3. The result is shown in Fig. 6.

By comparing Fig. 5(d) and Fig. 6 we can find that the result of Fig. 6 is better than result of Fig. 5(d). The 5 different looseness extents are completely separated in Fig. 6, and the better clustering performance is obtained.

In order to compare the recognition accuracy of origin looseness extent feature set and looseness extent sensitive feature set, and the recognition accuracy of dimension reduction methods such as PCA, LPP, LDA and LLTSA, the origin looseness extent feature set and looseness extent sensitive feature set are respectively inputted PCA, LPP, LDA and LLTSA to reduce dimension. The results of dimension reduction are respectively inputted WKNNC to recognize the looseness extent. The dimensions are set to 3, and the nearest neighbor $k$ is set to 9. The measurement of computation time was made in the follows computer configuration environment: 10G RAM, 3.4 GHz Inter Core i7-3770 CPU, 64-bit Operating System and MATLAB R2015b.

Fig. 5The comparison of dimension reduction results for origin looseness extent feature set

a) The dimension reduction results of PCA

b) The dimension reduction results of LPP

c) The dimension reduction results of LDA

d) The dimension reduction results of LLTSA

The comparison results are shown in Table 5. In Table 5, the recognition accuracy rate ${\eta }_i$ is defined as:

where $N_{}^{T_i}$ represents the total number of testing samples, $N_C^{T_i}$ represents the number of testing samples correctly recognized, $T_i$ represents the pedestal looseness extent of the testing samples, $i=$1, 2,…, 5.

The average recognition accuracy rate $\eta$ can be expressed as:

Fig. 6The dimension reduction results of LLTSA for looseness extent sensitive feature set

Table 5 indicates that the average recognition accuracy rate $\eta$ of test samples achieved by looseness extent sensitive feature set is 97 %, which is higher than that of the origin looseness extent feature set, when the dimension reduction method remains unchanged. The reason for the looseness extent sensitive feature set don’t include the non-sensitive feature and poor sensitivity feature, which enhanced the ability to characterize the pedestal looseness, and improved the recognition accuracy rate. Also, we can see that after the LLTSA-based dimension reduction method, the accuracy of looseness extent recognition improved significantly, much higher than the other dimension reduction methods, because LLTSA has advantage of suitable nonlinear reduction and more superior clustering performance, it is conducive to the recognition. The computation time of LLTSA is 0.46 s, which is more other dimension reduction methods, but it also very fast, doesn’t affect the engineering applications. Also, we can find that the computation time is reduced, when using looseness extent sensitive feature set to recognize and the dimension reduction method remain unchanged.

Next, recognition accuracy of WKNNC is compared with KNNC, where the nearest neighbor size of WKNNC and KNNC is also set as $k=$3-20, the dimension reduction method is LLTSA. Also, the origin looseness extent feature set and the looseness extent sensitive feature set are used to compare. The recognition accuracy rate curve with the nearest neighbor size $k$ are as shown in Fig. 7, then, for each curve, the average recognition accuracy rate, standard deviation and peak-peak value are calculated as shown in Table 6.

Fig. 7The comparison of recognition results

Fig. 7 and Table 6 show that the recognition accuracy rate of WKNNC is higher than KNNC, the standard deviation and peak-peak value of WKNNC are less than KNNC, because the nearest neighbor samples are given different weights, which makes the classification results of the testing sample more close to the similar degree of training sample, which also prove that the WKNNC is insensitive the neighbor size of $k$, having better stability and robustness than KNNC. The recognition accuracy rate based-looseness extent sensitive feature set is higher than the based-origin looseness extent feature set when the pattern recognition algorithm is same, which confirm again that the characterization capabilities of looseness extent sensitive feature set is stronger than origin looseness extent feature set. The standard deviation and peak-peak value based-looseness extent sensitive feature set is smaller than the based-origin looseness extent feature set when the pattern recognition algorithm is same, which confirm that the stability of characterization method based-looseness extent sensitive feature set is better than based- origin looseness extent feature set, and the clustering performance of looseness extent sensitive feature set is better too. Table 5, Fig. 7 and Table 6 show that the recognition accuracy rate of the proposed method is higher than other method, and it insensitive the neighbor size of $k$, having better stability and robustness.

This application example demonstrates well the performance of the proposed method which comprehensively uses characteristics of vibration signal to characterize the pedestal looseness extent, extracts time-frequency feature to construct origin looseness extent feature set, obtains looseness extent sensitive feature set to enhance the characterization capabilities, reduce dimension with LLTSA, and recognize looseness extent with WKNNC. The results conform the proposed method can recognize the pedestal looseness extent. Also, the feasibility and validity of the proposed method are verified by experimental results.