Vibration-based classification of centrifugal pumps using support vector machine and discrete wavelet transform

2.1.1. Wavelet transform (WT)

The use of a wavelet transform (WT) was very popular for two decades, due to advantageous properties of this transformation and availability of computer software. The WT is used in different fields of science, like medicine, biology and engineering; it is also employed to process signals and images [26]. Generally, conventional data processing is computed in a time or frequency domain. Wavelet processing combines both time and frequency. In simple language, we use the term ‘time-frequency’ analysis. A wavelet is a basis function characterized by two aspects; one is its shape and amplitude that is chosen by the user and the other is its scale (frequency) and time (location) relative to the signal [27]. The wavelet transform of signal $f\left(t\right)\in L^2\left(R\right)$ is defined by the inner-product between ${\psi }_{ab}\left(t\right)$ and $f\left(t\right)$ as [28]:

where $a$ and $b$ are the scaling (dilation) and translation (time shift) constants respectively, and $\psi$ is the wavelet function that may not be real as assumed in the above equation for simplicity. The choice of the wavelet function (mother wavelet) is flexible provided that it satisfies the so called admissibility conditions [28]. The discrete wavelet transform (DWT) is derived from the discretization of continuous wavelet transform (CWT). The most common discretization is dyadic. The DWT is given by:

where, the parameters $a$ and $b$ in Eq. (1) are replaced by $a_0^m$ and $k{a}_0^m$, respectively, with $k$ and $m$ being integer variables [28].

The discrete signal is passed through a high pass filter (H) and a low pass filter (L), resulting in two vectors at the first level; approximation coefficient (A1) and detail coefficient (D1). Application of the same transform on the approximation (A1) causes it to be decomposed further into approximation (A2) and detail (D2) coefficients at the second level. Finally, the signal is decomposed at the expected level. The approximations are the high-scale low-frequency components of the signal, whereas details are its low-scale, high-frequency components. The wavelet decomposition for level 3 is illustrated in Fig. 1 [29].

Each vector $A_j$ includes approximately $N/2^j$coefficients, where $N$ is the number of data points in the input signal $s$, and provides information about a frequency band [0, $Fs/2^{j+1}$], where $Fs$ is the sampling frequency. In Fig. 1, H and L represent the decomposition filters, and ↓2 denotes a down sampling by a factor of 2. An important property of the DWT is [29]:

Fig. 1The principle of the DWT decomposition [29]

2.2.1. Mathematical formulation: primal

The data for training is a set of points (vectors) $x_i$ along with their categories $y_i$. For some dimension $d$, $x_i∊R^d$, and $y_i=$ ±1. The equation of a hyperplane is $〈w,x〉+b=$ 0, where $w∊R^d$, $〈w,x〉$ is the inner product of $w$ and $x$, and $b$ is real. The following problem defines the best separating hyperplane. Find w and b that minimize $‖w‖$ such that for all data points ($x_i,y_i$), $y_i\left(〈w,x_i〉+b\right)\ge$ 1. The support vectors are $x_i$ on the boundary, those for which $y_i\left(〈w,x_i〉+b\right)=$ 1. For mathematical convenience, the problem is usually given as the equivalent problem of minimizing $〈w,w〉/2$. This is a quadratic programming problem. The optimal solution w, b enables classification of a vector z as follows [31]:

Class ($z$) = sign $\left(〈w,z〉+b\right)$.

2.2.2. Mathematical formulation: dual

It is computationally simpler to solve the dual quadratic programming problem. To obtain the dual, take positive Lagrange multipliers ${\alpha }_i$ multiplied by each constraint, and subtract from the objective function [31, 32]:

where you look for a stationary point of $L_P$ over $w$ and $b$. Setting the gradient of $L_P$ to 0, you get $w=\sum _i{\alpha }_i{y}_i{x}_i$, $\sum _i{\alpha }_i{y}_i=$ 0, and substituting into $L_P$, you get the dual $L_D$[31]:

Which you maximize over ${\alpha }_i\ge 0$. In general, many ${\alpha }_i$ are 0 at the maximum. The nonzero ${\alpha }_i$ in the solution to the dual problem defines the hyperplane, as seen in Eq. (4), which gives $w$ as the sum of ${\alpha }_i{y}_i{x}_i$. The data points $x_i$ corresponding to nonzero ${\alpha }_i$ are the support vectors. The derivative of $L_D$ with respect to a nonzero ${\alpha }_i$ is 0 at an optimum. This gives $y_i\left(〈w,x_i〉+b\right)-1=$ 0.

In particular, this gives the value of $b$ at the solution, by taking any $i$ with nonzero ${\alpha }_i$. The dual is a standard quadratic programming problem.

2.2.3. Non-separable data

Your data might not allow for a separating hyperplane. In that case, the SVM can use a soft margin, meaning a hyperplane that separates many, but not all data points. There is a standard formulation of soft margins which involves adding slack variables $s_i$ and a penalty parameter $C$.

The $L^1$-norm problem is:

Such that $y_i(〈w,x_i〉+b)\ge 1-s_i,s_i\ge 0$.

In these formulations, you can see that increasing $C$ places more weight on the slack variables $s_i$, meaning that the optimization attempts to make a stricter separation between classes. Equivalently, reducing $C$ towards 0 makes misclassification less important.

For easier calculations, consider the $L^1$ dual problem to this soft-margin formulation. Using Lagrange multipliers ${\mu }_i$, the function to minimize for the $L^1$-norm problem is [31]:

where you look for a stationary point of $L_P$ over $w$, $b$ and positive $s_i$. Setting the gradient of $L_P$ to 0, you get $b=\sum _i{\alpha }_i{y}_i{x}_i,\sum _i{\alpha }_i{y}_i=0$, ${\alpha }_i=C-{\mu }_i$. These equations lead directly to the dual formulation [31, 32]:

Subject to the constraints $\sum _i{\alpha }_i{y}_i=\mathrm{0,0}\le {\alpha }_i\le C$.

The final set of inequalities, $0\le {\alpha }_i\le C$, shows why $C$ is sometimes called a box constraint. $C$ keeps the allowable values of the Lagrange multipliers ${\alpha }_i$ in a “box”, a bounded region. The gradient equation for b gives the solution b in terms of the set of nonzero ${\alpha }_i$, which corresponds to the support vectors.

2.2.4. Nonlinear transformation with Kernels

Some binary classification problems do not have a simple hyperplane as a useful separating criterion. For those problems, there is a variant of the mathematical approach that retains nearly all the simplicity of an SVM separating hyperplane. This approach uses these results from the theory of reproducing kernels [31]:

There is a class of functions $K\left(x,y\right)$ with the following property. There is a linear space $S$ and a function $\phi$ mapping $x$ to $S$ such that:

The dot product takes place in the space $S$.

This class of functions includes:

Polynomials: For some positive integer $d$:

Radial basis function: For some positive number $\sigma$:

In this study, different parameters including degree of the polynomial ($d$), the RBF kernel and the penalty parameter ($C$) were considered to find the best SVM model.

3.1.1. Fault in impeller

Two impellers ($d=$ 220 mm) were used. One impeller was assumed to be free from defect. The other impeller was prepared from an out-of-service pump. This impeller was rusted and the corrosion existed at the surface and the eye of the impeller as shown in Fig. 5.

Fig. 3Experimental setup

Fig. 4Location of the sensor

Fig. 5a) Faulty seal, b) faulty impeller

a)

b)

3.1.2. Fault in sealing system

Packing’s are one of the oldest sealing devices, named for the way they perform the sealing function. Packing is a rope-like material that is inserted into the cylindrical opening in the rear of the pump casing where the shaft passes through it. This soft material, usually with lubricant in it, consists of a number of rings that wrap around the shaft and compress the inner sides of the stuffing box. This effectively limits leakage out of the pump. The bronze component that separates the two sets of packing is called a lantern ring. The bronze lantern ring serves a number of functions inside the pump packing. The lantern ring is grooved with an oil groove and drilled with holes to allow lubrication to reach the packing material. The lantern ring is also used to distribute cooling water to all packing rings as well as keep the stuffing box clean of containments. Faulty packing was used to reduce the pressure of the pumped liquid and maximize leakage. Sometimes gland packing slightly swells on contact with water. This must be considered and an appropriate clearance should be provided. Otherwise, it would seize the shaft leading to overheating and burning of the packing. In some cases, the shafts have been even broken due to excessively tightened packing. Also, wear in the internal surface of the lantern ring makes friction and heat between the pump shaft and lantern ring, and destroys them. Heat and friction destroys pump shaft and lantern ring. Thus, in this study, we used a burned gland packing and a worn lantern ring, as shown in Fig. 5.

3.1.3. Cavitation

When a pump is under low pressure or high vacuum conditions, suction cavitation occurs. Thus, to simulate the cavitation, the valve at the inlet of the pump is used to make the pressure drop between the suction and the eye of the impeller. The valve control system is used to adjust the flow at the inlet and outlet of the pump. The control of priming and leakage of the pump was done. After closing the delivery valve, the pump started. Then the delivery valve was opened fully and the valve at the suction side was gradually closed. For more information on how to make cavitation, refer to [19].