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Stress corrosion cracks (SCC) in low-pressure steam turbine discs are serious hidden dangers to production safety in the power plants, and knowing the orientation and depth of the initial cracks is essential for the evaluation of the crack growth rate, propagation direction and working life of the turbine disc. In this paper, a method based on phased array ultrasonic transducer and artificial neural network (ANN), is proposed to estimate both the depth and orientation of initial cracks in the turbine discs. Echo signals from cracks with different depths and orientations were collected by a phased array ultrasonic transducer, and the feature vectors were extracted by wavelet packet, fractal technology and peak amplitude methods. The radial basis function (RBF) neural network was investigated and used in this application. The final results demonstrated that the method presented was efficient in crack estimation tasks.

Low-pressure steam turbine discs are critical components in power plants which rotate at high speed throughout the year. With the increase of usage time, stress corrosion cracking may occur in the blade attachment region of the turbine discs, leading to heavy financial losses, and even severe accidents [

There are some traditional methods for the estimation of flaw size in ultrasonic non-destructive testing. When the flaw size is smaller than the ultrasonic beam diameter, the amplitude-equivalent method is most often used, including the equivalent test specimen method and the AVG curve method; to the contrary, when the flaw size is larger than the ultrasonic beam diameter, the length testing method is always used which includes the 3, 6 and 12 dB methods. However, none of the methods mentioned above take the orientations of the flaws into account, which is an obstacle to the testing of the cracks with different orientations. To solve this problem, crack tip diffraction signals are generally used to detect the location of the crack tip, and then the depth and orientation information can be obtained simultaneously [

In recent years, artificial neural networks have been proved to be effective for flaw identification and evaluation in ultrasonic non-destructive testing. For instance, in respect of qualitative analysis, neural networks can be used to identify the different flaw types such as cracks, pores and slags in metal welds [

In this paper, based on a phased array ultrasonic transducer and artificial neural network, a method to estimate both the depth and orientation of initial cracks in fir-tree type turbine discs, which are the most prevalent in low-pressure turbine rotors, is proposed. In the following sections, firstly, the data collection system was built and the experimental data is described. The A-scan echo signals from cracks with different depths and orientations were collected through a phased array ultrasonic transducer, and the feature vectors were extracted by the wavelet packet, fractal technology and peak amplitude methods. Then, the RBF neural network was investigated for this estimation work. Finally, the estimation results of the neural network were analyzed. Our results showed that the proposed method can efficiently estimate both the depth and orientation of initial cracks in turbine discs.

The experimental data was obtained using the pulse-echo inspection method. The data collection system consists of the following main components: M2M ultrasonic detector, linear phased array ultrasonic transducer with a 36° wedge, the specimen with artificial defects, and a PC. The detector sampling rate is 100 MHz, and a phased array ultrasonic transducer with a center frequency of 5 MHz served as the transmitter and receiver, the number of elements is 64, element width is 0.49 mm, element length is 10 mm and the inter-spacing between centers of adjacent elements is 0.59 mm. For our study, we machined a small part of the turbine disc which is made of steel and has three hooks as the experimental specimen. The whole experimental set up is shown in

In order to reduce the transducer movement, for this inspection we chose a sector scan, which can scan all the three hooks without movement in radial direction of the disc, then the A-scan signals from cracks in each hook can be obtained from this sector scan data. The steering beams were not focused because the three hooks are in different depths. The sector scan in the specimen is shown in

Feature extraction means capturing information that is relevant to different depths and directions of cracks from the ultrasonic echo signals, which is the critical task for the crack estimation. Ultrasonic echo signals always have the characteristics of high level non-linearity, non-stationarity and transient nature. Wavelet and fractal have been proven to be excellent methods for processing this kind of signals. In [

During the detection process of the ultrasonic phased array, the transducers are excited by narrow pulses which are similar to function

The wavelet packet is an extension of wavelet analysis which can provide more fineness in signal decomposition than the wavelet. This analysis can decompose the ultrasonic signals into independent frequency bands completely, and then the features of different echo signals can be characterized by the frequency band energy.

The wavelet transform decomposes a signal into a set of basic functions which are obtained from the mother wavelet. A mother wavelet is a function _{a,b}_{a}_{,}_{b}

In practice, for the convenience of processing the data by computer, the signals are always a discrete series; therefore, one prefers to write the signal as a discrete superposition of a discrete set of continuous wavelets, which is called Discrete Wavelet Transform (DWT). In the transform, the scaling and shifting factors are discretized and given by _{0}^{j}_{0}_{0}^{j}_{0} and _{0} are always set to 2 and 1, respectively. The function family with discretized factors becomes:

Then the wavelet decomposition can be expressed as:
_{j}_{,}_{k}

Mallat has extensively studied this discrete wavelet transform [

In the multiresolution analysis, the scaling subspaces _{j}_{j}_{2}⊂ _{1}⊂ _{0}⊂ _{−1} ⊂_{−2}⊂…⊂^{2}. The scaling function _{j,k}_{j}_{j,k}_{j}_{j}_{j}_{− 1},

Then the original signal _{j}_{j}

To construct the mother wavelet ^{n}h

Then the Mallat algorithm for the computation of the decomposition coefficients can be summarized by the following equations:

The reconstruction of _{j}_{− 1}(

The wavelet packet method is a generalization of wavelet decomposition that can provide more sophisticated analysis. Wavelet analysis decomposes signals into two parts: low-frequency and high-frequency. During the course of decomposition, only the low-frequency part is decomposed into two parts and this decomposition can be continued to a number of deeper levels. It can be seen that, in the wavelet decomposition, the frequency resolution reduces in higher frequency. Wavelet packet transform is more accurate in signal decomposition, as it decomposes signals not only in the low-frequency part, but also in the high-frequency part.

For the convenience of discussion, in wavelet packet analysis, scaling subspace _{j}_{j}

Then the formula

Define the subspace
_{n}_{0}(_{1}(^{k}h

Set

Wavelet packet decomposition algorithm is described as:

Wavelet packet reconstruction algorithm is described as:

Energy spectrum feature extraction by wavelet packet can be described as four steps:

Decompose the ultrasonic echo signal.

We decompose the ultrasonic echo signals into four layers, and the decomposition coefficients can be acquired from low frequency to high frequency of 16 sub-bands in the fourth layer. The wavelet packet decomposition tree of four levels is shown in ^{th}^{th}_{j}_{0}, _{j}_{1}, _{j}_{2} … _{j(2j−1)} (

Reconstruct decomposition coefficients.

Reconstruct the wavelet packet decomposition coefficients, and extract the frequency range signal _{4}_{i}

Calculate energy of each frequency band.

The energy of signal _{4}_{i}_{4}_{i}_{4}_{ik}^{th}

Construct feature vectors.

The energy spectrum feature vector of the 4

In order to accommodate the analyze model in the following section, the energy spectrum vectors are normalized as follows:

To illustrate this abstract feature extraction method, the wavelet packet energy spectrum of one set of data is shown in

In the level 4, the main frequency bands which concentrate most energy of the crack echo signal,

Fractal theory has been developed into a powerful tool in dealing with non-linear problems in natural science and engineering. In nonlinear signal processing, fractal dimensions can be used to quantitatively analyze the signal irregularity and complexity. The combination of wavelet and fractal is proposed according to the unity that the multi-scale decompositions and self-similarity in both wavelet transform and fractal theory possess, that is, the ability to analyze the signal information from low-resolution to high-resolution of the wavelet transform is consistent with the fractal method which gets more and more abundant details through transforming the signal from big scale to small scale.

In the fractal analysis, the more tortuous, convoluted and richer in detail, the higher the fractal dimensions. The box-counting dimension is one of the best known fractal dimensions which can be easily defined and obtained numerically. In this study, fractal box-counting dimensions are combined with wavelet packet transform and extracted as features for crack echoes.

Fractal theory was introduced by Mandelbrot [

The box dimension of a set ^{n}^{n}_{δ}

In actual fact, the box dimensions of the discrete time domain signals cannot be obtained under the condition of _{l}_{Δ}(

According to formula _{B}_{B}_{l}_{Δ}(_{s}_{e}_{s}_{e}

So, the box dimension estimating method for the discrete time domain ultrasonic signals can be summarized as follows:

Generate the meshes. According to the approximate method, take

Cover the signal with boxes whose side-length are _{l}_{Δ}(

Plot _{l}_{Δ}(_{B}

In our present study, box-counting dimensions were calculated for the frequency band signals extracted in Section 3.1., _{4}_{i}

To show the identifiability of this impalpable feature, the fractal features of one set of crack signals are shown in

Peak amplitude of an ultrasonic echo signal is a typical feature which is easily interpreted visually and widely used in quantitative detections. Besides the frequently-used flaw echo amplitude, echo amplitude from bottom under the flaw is also an effective characteristic parameter. The bottom echo can assist to perceive larger flaws which have weak echos because of the influence of geometry, reflectivity or the reflective surface direction, so this feature should be considered in the present issue. The crack echo wave and bottom wave are shown in

Up to now, a total of eight features are extracted by wavelet packet, fractal and echo amplitude, then the feature vector F can be constructed as:

This 8-dimensional vector will be the input of the estimation model proposed in the next section.

In this work, the RBF neural network is used for the estimation. The RBF neural network is an important supervised learning tool of machinery learning technology which can perform arbitrary nonlinear mapping from the input space ^{d}^{n}

The RBF neural network is a multi-input, multi-output forward networks model which has three layers consisting of an input layer, a hidden layer, and an output layer. In this work, the 8-dimensional feature vector F constructed in Section 3 is the input of the neural network, and the crack orientation angle and depth constitute the 2-dimensional output, then the structure of the RBF neural network is shown in

The input layer sends the input variables to each neuron in the hidden layer. The activation function applied to neuron in hidden layer is radial basis function. In RBF neural network, the Gaussian function is the most common radial basis function, so the activation function of the ^{th}_{i}_{i}^{th}_{i}_{i}

The output layer is a linear combination of the hidden layer output with associative weights and biases, and the ^{th}_{ij}^{th}^{th}_{j}^{th}

The presented estimate method can be summarized by the flowchart illustrated in

In this section, we take the crack echo signals in the 2nd hook as an example to analyze the performance of the proposed model. The 240 available data samples were divided into 80% for training and 20% for testing the networks. The partial test results of the model for estimation of the crack orientation angle and depth are given in

The estimation results show that the proposed method can evaluate the crack orientation angle and depth with a reasonable level of accuracy. Especially for the depth estimation, fairly small errors occur. Orientation angle estimation results show greater errors than the depth estimation, but the error level is still within an acceptable range. In order to analyze the errors accurately, the root mean square errors (RMSE) of the testing data are calculated as:
_{i}_{,}_{t}_{i}_{,}_{e}

Stress corrosion cracks in low-pressure steam turbine discs are serious hidden dangers for production safety in power plants, and the initial crack inspection and the forecast of their propagation are essential to the safe operation of the turbine discs. In order to estimate the crack orientation and depth at an early stage, a method based on a phased array ultrasonic transducer and an artificial neural network was proposed in our study. The A-scan echo signals from cracks with different depths and orientations were collected in the lab using a phased array ultrasonic transducer, and the feature vectors were extracted by the wavelet packet, fractal technology and peak amplitude methods. Then, a RBF neural network was investigated for this estimation work and the estimation results were analyzed. The test results showed that the proposed model was a useful tool to estimate both the depth and orientation of initial cracks in turbine discs. At the present stage, all the research is still in the laboratory and the experimental data is collected from the machined specimen which is not enough in practice. In our future work, a field data acquisition system will be developed and the field data will be collected and analyzed using our model.

This research is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20100032110057). The authors also acknowledge the vital contributions made by Bin Xue, who was always willing to help during the research and provided much instructive advice.

The authors declare no conflict of interest.

Data collection system.

(

One set of A-scan echo signals from the cracks in the specimen.

The relevant frequency spectra of the A-scan echo signals.

Wavelet packet decomposition tree of four levels.

Wavelet packet energy spectrum of one set of echo signals.

Crack echo wave and bottom echo wave.

Structure of the RBF neural network.

The flowchart of the presented estimate method.

The test results of the model. (

Orientation and depth of crack in the specimen.

1 | 5 | 0.5 |

2 | 5 | 1.0 |

3 | 5 | 1.5 |

4 | 5 | 2.0 |

5 | 25 | 0.5 |

6 | 25 | 1.0 |

7 | 25 | 1.5 |

8 | 25 | 2.0 |

9 | 45 | 0.5 |

10 | 45 | 1.0 |

11 | 45 | 1.5 |

12 | 45 | 2.0 |

The fractal features of one set of crack signals.

Band 1 | 0.167 | 0.184 | 0.165 | 0.175 | 0.164 | 0.178 | 0.165 | 0.205 | 0.155 | 0.167 | 0.247 | 0.191 |

Band 2 | 0.245 | 0.217 | 0.246 | 0.280 | 0.314 | 0.206 | 0.236 | 0.246 | 0.325 | 0.261 | 0.237 | 0.220 |

Band 3 | 0.395 | 0.382 | 0.354 | 0.396 | 0.418 | 0.348 | 0.295 | 0.382 | 0.404 | 0.331 | 0.302 | 0.311 |