A comparative study of pick degradation prediction based on grey prediction and Bayesian algorithm

Abstract

The performance of the pick gradually degrades under the cyclic complex load, so it is difficult to realize the state prediction, in order to be able to accurately predict the pick degradation state, this paper develops a test system for the wear degradation of pick, the method of experimental analysis and numerical simulation is adopted, Multi-signal fusion model of vibration and acoustic emission was constructed, the grey prediction and the Gamma method of Bayesian parameter updating are used to realize the application of the degenerate data, the results prove that the relative error of grey prediction under vibration signal is only 0.45%, the relative error of the Gamma model was 0.57%, the relative error of the Gamma model under Bayesian updating was 0.22%, the three models have good prediction accuracy, the prediction error of the Gamma model under Bayesian updating is minimal.

Keywords

Pick wear pick degradation grey prediction gamma process

1 Introduction

Coal is an important energy source in the world. With the rapid development of artificial intelligence today, it is an inevitable trend for coal mining to improve coal mining technology and reduce personnel and increase efficiency. The cutting tooth, as an important mining tool, is widely used in mining machinery [1]. Because the environment of coal mine is bad, the working environment of cutting tooth as cutting tool is evident. As a result, the truncated tooth is in a state of high impact stress for a long time, which is easy to fail. In the process of work by pressure, friction, temperature and corrosion and other factors, wear and failure rate rapidly increased [2], the proportion of wear failure is as high as 75% ∼90% in the failure form of pick [3].

In recent years, scholars have conducted a series of researches on the degradation models of cut teeth and tool wear. Chu kaiyu et al. [4] realized the on-line monitoring process of tool wear by using fusion wavelet analysis and neural network. Chen xiaoyu et al. [5] determined the tool wear state by observing the tool surface deformation with an optical image measuring instrument. Zhang qiang et al. [6 –10] aimed at the problem of pick degree identification, the methods of neural network and multi-information fusion were adopted to realize the pick degree identification and online monitoring. Park et al. [11] established the accelerated degradation model of the system based on the Gamma process and obtained its degradation probability density distribution. By using OPCUA communication technology to collect and store information, lu zhiyuan et al. [12] established a tool wear state recognition model by using convolutional neural network. He guangchun [13] concluded that the wear states of different tool surfaces were different by observing and detecting tool wear states with instruments. Xiao zhongyue et al. [14] proposed a tool wear detection technology based on the maximum entropy of information theory and the theory of cross entropy, aiming at the problem of signal processing in the cutting process.

Combined with the above domestic and foreign research status analysis research findings, at present, the research on tool wear, degradation theory, degradation model and solving algorithm is in the stage of rapid development, more and more scholars are working on it, however, there is little research on the degradation law of truncated tooth wear. Therefore, in this paper, according to the tool degradation model and the solution method, the paper proposes the research on the truncated tooth degradation state based on a variety of models to make up for the shortage of the research on the truncated tooth wear degradation state.

2 Extraction of vestigial vibration signal of pick

2.1 Establishment of experimental system for monitoring of pick wear

The experimental system mainly consists of two parts: the pick and the multi-sensor testing subsystem. The pick consists of cutting mechanism, walking mechanism and control cabinet; the multi-sensor test system consists of vibration signal acquisition system, acoustic emission signal acquisition system and current signal acquisition system, its main form of composition is shown in Fig. 1.

Fig.1

Pick wear monitoring experimental system.

2.2 Pick processing

In order to obtain the characteristic signal of pick with different degree of wear, before the experiment, the process from pick to failure was divided into 6 stages, new pick, slight wear pick, medium wear pick, large wear pick, severe wear pick and failure pick six wear state. The relationship between the wear degree and parameters of the pick was established and the experiment was carried out. The definition of different degrees of pick is based on weight change and size change, according to the actual situation of the project, the partial wear of the tooth body and the complete alloy head will be regarded as the failed pick. The weighing test was carried out for 6 kinds of pick with wear degree, obtain the weight interval corresponding to each pick state, the quality difference is used as a hierarchical description to evaluate the wear state of pick.

Select an electronic balance with an accuracy of 0.1 g to measure the mass, each group was measured three times and averaged. Select vernier calipers with an accuracy of 0.01 mm to measure dimensions. Each group was measured three times and averaged. The measurement results are shown in Table 1.

Table 1
Degree of pick wear

Attrition rate New pick Slight wear Medium wear Large wear Severe wear Failure pick

Poor quality of pick/g 0∼50 50∼100 100∼150 150∼200 200∼250 250∼300

Measurement discrepancy/mm 0∼4.5 4.5∼9.0 9.0∼13.5 13.5∼18 18∼22.5 22.5∼27

Attrition rate	New pick	Slight wear	Medium wear	Large wear	Severe wear	Failure pick
Poor quality of pick/g	0∼50	50∼100	100∼150	150∼200	200∼250	250∼300
Measurement discrepancy/mm	0∼4.5	4.5∼9.0	9.0∼13.5	13.5∼18	18∼22.5	22.5∼27

2.3 Establish characteristic signal sample database

The time domain signal of vibration acceleration in the pick process of 6 different wear degrees (A1∼A6) was decomposed by three wavelet packets, the vibration energy of 6 kinds of pick with different wear degrees in the same frequency band tends to increase gradually, among them, the energy values in frequency bands of 50∼62.5 Hz, 62.5∼75 Hz, 75∼87.5 Hz and 87.5∼100 Hz are relatively high, and the energy values show an obvious upward trend with the aggravation of the degree of gear pick wear, therefore, the energy sum of these four frequency bands is selected as the sample data, Some data samples are shown in Table 2.

Table 2
Sum of energy different picks’ cutting vibration acceleration signal characteristics sample/mV²

Number A1 A2 A3 A4 A5 A6

1 105.23 118.70 134.19 144.55 162.71 177.76

2 105.52 118.93 134.40 145.40 162.96 178.13

3 105.78 119.28 134.65 145.63 163.25 178.61

4 106.21 119.67 134.83 146.04 163.64 178.97

5 106.58 120.01 135.04 146.75 164.10 179.34

6 106.96 120.70 135.26 147.47 164.47 179.73

7 107.15 121.19 135.41 147.78 164.81 180.23

8 107.33 121.60 135.69 148.52 165.14 180.67

9 107.68 121.98 135.99 148.89 165.61 181.11

10 108.19 122.32 136.18 149.26 165.97 181.34

... ... ... ... ... ... ...

46 120.43 134.59 145.10 163.04 178.80 194.98

47 120.56 134.93 145.35 163.49 179.17 195.25

48 120.69 135.28 145.63 163.78 179.40 195.49

49 120.83 135.86 145.96 164.05 179.58 195.83

50 120.95 136.09 146.38 164.50 179.65 196.19

Number	A1	A2	A3	A4	A5	A6
1	105.23	118.70	134.19	144.55	162.71	177.76
2	105.52	118.93	134.40	145.40	162.96	178.13
3	105.78	119.28	134.65	145.63	163.25	178.61
4	106.21	119.67	134.83	146.04	163.64	178.97
5	106.58	120.01	135.04	146.75	164.10	179.34
6	106.96	120.70	135.26	147.47	164.47	179.73
7	107.15	121.19	135.41	147.78	164.81	180.23
8	107.33	121.60	135.69	148.52	165.14	180.67
9	107.68	121.98	135.99	148.89	165.61	181.11
10	108.19	122.32	136.18	149.26	165.97	181.34
...	...	...	...	...	...	...
46	120.43	134.59	145.10	163.04	178.80	194.98
47	120.56	134.93	145.35	163.49	179.17	195.25
48	120.69	135.28	145.63	163.78	179.40	195.49
49	120.83	135.86	145.96	164.05	179.58	195.83
50	120.95	136.09	146.38	164.50	179.65	196.19

It can be seen from Table 2, with the increase of the degree of pick wear, the energy of the selected frequency range of the vibration acceleration signal of A1∼A6 pick is increasing, but it may be influenced by the environment and other factors, part of the data samples have intersection, in the wavelet packet analysis, the acceleration energy and value of the vibration signal spectrum diagram (50∼100 kHz) and acoustic emission signal spectrum diagram (12.5∼50 kHz) were obtained as the characteristic samples, build the data sample library.

3 Grey prediction model was established

3.1 Construction of grey prediction model

You have the original data column Y⁽⁰⁾ = (Y⁽⁰⁾ (1) , Y⁽⁰⁾ (2) , …, Y⁽⁰⁾ (n)), n is the number of data. The Y⁽⁰⁾ data column is used as the input to build the model, and then the future data calculation and prediction can be realized step by step. The basic steps are as follows:

The data samples are accumulated to solve the problem of sample data instability. The first data of the generated column is the first data of the original sequence; The second data of the generated column is the sum of the second and first data of the original data; The third data of the generated column is the sum of the third and second data of the original data; By analogy, the new data sequence can be obtained: $Y^{(1)} = (Y^{(1)} (1), Y^{(1)} (2), \dots, Y^{(1)} (n))$ (1) $Y^{(1)} = \sum_{k = 1}^{t} Y^{(0)} (k), t \sim 1, 2, \dots, n$ (2)

A first order linear differential equation is established for Y⁽¹⁾ (t), modelG (1, 1); $\frac{{dy}^{(1)}}{dt} + {ay}^{(1)} = u$ (3)

In the formula 3, ais called the development coefficient, a ∈ (- 2, 2), u is called the grey action. The matrix determined by a, uis grey parameter $\hat{a} = (\begin{matrix} a \\ u \end{matrix})$ . So if we solve for a, u, we get Y⁽¹⁾ (t), then calculate the predicted value of Y⁽⁰⁾.

Add up the generated data, so let’s take the mean of B and the constant term vector Y_n, as follow; $B = [\begin{matrix} - \frac{1}{2} (Y^{(1)} (1) + Y^{(1)} (2)) & 1 \\ - \frac{1}{2} (Y^{(1)} (2) + Y^{(1)} (3)) & 1 \\ \dots \\ - \frac{1}{2} (Y^{(1)} (n - 1) + Y^{(1)} (n)) & 1 \end{matrix}]$ (4) $Y_{n} = (\begin{matrix} y^{(0)} (2) \\ y^{(0)} (3) \\ \dots \\ y^{(0)} (n) \end{matrix})$ (5)

Grey parameter is solved by least square method $\hat{a}$ , so; $\hat{a} = {(B^{T} B)}^{- 1} B^{T} Y_{n}$ (6)

Substitute grey parameter $\hat{a}$ into formula (3), to solve the, so you get; ${\hat{Y}}^{(1)} (t + 1) = (Y^{(0)} (1) - \frac{u}{a}) e^{- at} + \frac{u}{a}$ (7)

Because $\hat{a}$ itself is only an approximation, All ${\hat{Y}}^{(1)} (t + 1)$ is are just approximations.

Discrete processing of ${\hat{Y}}^{(1)} (t + 1)$ and ${\hat{Y}}^{(1)} (t)$ , and you do the subtraction, finally, restore Y⁽⁰⁾ original sequence, the approximate data sequence ${\hat{Y}}^{(1)} (t + 1)$ is as follows; ${\hat{Y}}^{(0)} (t + 1) = {\hat{Y}}^{(1)} (t + 1) - {\hat{Y}}^{(1)} (t)$ (8)

The final prediction data sequence is:

$\begin{matrix} {\hat{Y}}^{(0)} = {{\hat{Y}}^{(0)} (1), {\hat{Y}}^{(0)} (2), \dots, {\hat{Y}}^{(0)} (n), \\ {\hat{Y}}^{(0)} (n + 1), \dots, {\hat{Y}}^{(0)} (n + m)} \end{matrix}$ (9)

3.2 Grey prediction model testing

The smoothness test of the original sequence and the series test of the accumulative sequence are also needed before the model is determined, verify the established grey model, the residual, relative error and posterior difference ratio of the two are calculated, the results should satisfy the series requirement.

Calculate the residual of e⁽⁰⁾ (t) between Y⁽⁰⁾ (t) and ${\hat{Y}}^{(0)} (t)$ ; $e^{(0)} (t) = Y^{(0)} (t) - {\hat{Y}}^{(0)} (t)$ (10)

Calculate the relative error between Y⁽⁰⁾ (t) and ${\hat{Y}}^{(0)} (t)$ by q⁽⁰⁾ (t); $q^{(0)} (t) = e^{(0)} (t) / - Y^{(0)} (t)$ (11)

Posterior difference ratio C; $C = \frac{q^{(0)} (t)}{e^{(0)} (t)}$ (12)

4 GAMMA process modeling

4.1 Gamma process principle

The change of the wear signal energy sum at time t is defined as X (t), X (t) goes up and up as tgoes up; When the failure, X (t) = r₀, it reaches the maximum, which is consistent with the characteristics of the Gamma process. In general, the Gamma process {X (t) t⩾ 0 } has the following properties;

Random increment, X (t_i) - X (t_i-1) is distributed as Gamma, is X (t_i) - X (t_i-1) ∼ Ga (v (t_i) - v (t_i-1) , u). Where, Ga is the Gamma function; v (t) > 0 is the shape parameter, u > 0 is the scale parameter, t_i for the moment, i = 1, 2, ⋯ , n.

The GAMMA process function has independent increments, and the random parameters on the disjoint interval are independent of each other.

X (0) = 0, Probability is 1.

Then, under the modeling framework of Gamma process, based on the first arrival time, the remaining life of the device at time t_k is L_k: $\begin{matrix} L_{k} = inf {t : X (t_{k} + l_{k}) ⩾ ω | X (t_{k}) < ω} \\ = {t : X (t_{k} + l_{k}) ⩾ ω | X (t_{k}) < ω} \end{matrix}$ (13)

Where, ω is the failure threshold.

Since the energy and increment of truncated tooth wear is non-negative, the wear process is a stable Gamma process, as Γ (ct). Therefore, assume v (t) = ct, c > 0, (cis constant) then, The probability density function of X (t) can be written as F_X(t) (x); $F_{X (t)} (x) = Ga (\begin{matrix} \begin{matrix} x | ct, & u \end{matrix} \end{matrix}) = \frac{u^{ct}}{Γ (ct)} x^{ct - 1} exp (- ux)$ (14)

You get the expectation of X (t) $E (X (t)) = \int_{0}^{\infty} {xf}_{X (t)} (x) d x = \frac{ct}{u}$ (15)

The variance of $Var (X (t)) = E ({(X (t) - E (X (t)))}^{2}) = \frac{c t}{u^{2}}$ (16)

4.2 Process parameter estimation method based on gamma

If at time t_i, 0 = t₀ < t₁ < t₂ < ⋯ < t_n, corresponding degenerate data x_i, and 0 = x₀ ⩽ x₁ ⩽ x₂ ⩽ ⋯ ⩽ x_n. Considering the increment of δ_i = x_i - x_i-1, i = 1, 2, ⋯ , n of the monitoring signal, the likelihood function is $L (δ_{1}, \dots, δ_{n} | c, u) = \prod_{i = 1}^{n} \frac{u^{c (t_{i} - t_{i - 1})}}{Γ (c (t_{i} - t_{i - 1}))} δ_{i}^{c (t_{i} - t_{i - 1}) - 1} \exp (- u δ_{i})$ (17) Take the derivative of cand u, you can get an expression for the estimated values of $\hat{c}$ and $\hat{u}$ . ${\begin{matrix} \hat{u} = \frac{{\hat{ct}}_{n}}{x_{n}} \\ \sum_{i = 1}^{n} [t_{i} - t_{i - 1}] {φ (\hat{c} [t_{i} - t_{i - 1}]) - lg δ} = t_{n} lg (\frac{{\hat{ct}}_{n}}{x_{n}}) \end{matrix}$ (18) Solve for $\hat{c}$ and $\hat{u}$ , from degradation data x_i, i = 1, 2, ⋯ , n, the estimated value of ${\hat{μ}}_{i}$ , ${\hat{σ}}_{i}^{2}$ degradation quantity of each monitoring time can be obtained first, then from ${\begin{matrix} {\hat{μ}}_{i} = \frac{ct}{u} \\ {\hat{σ}}_{i}^{2} = \frac{ct}{u^{2}} \end{matrix}$ (19) After the cal $\hat{u}$ culation of equation (18), the estimated values of unknown parameters c and u, $\hat{c}$ and, are obtained by means of linear regression.

4.3 Gamma process model construction

Set T_{r
_o}to represent the moment when the gear cutting degradation quantity X (t)first reaches the wear threshold value r₀, according to the Gamma process, the distribution function of the failure time of truncated tooth was obtained; $\begin{matrix} F (t) = \int_{x = r_{0}}^{\infty} f_{X (t)} (x) dx \\ = \int_{x = r_{0}}^{\infty} \frac{u^{ct}}{Γ (ct)} x^{ct - 1} exp (- ux) dx = \frac{Γ (ct, r_{0} u)}{Γ (ct)} \end{matrix}$ (20) Where, at different times, the reliability of pick wear can be obtained by determining $\hat{c}$ and $\hat{u}$ . If the distribution function F (t) of the failure time of pick is differentiable, the probability density of the failure time of pick can be obtained, is $f (t) = \frac{\partial}{\partial t} [\frac{Γ (v (t), r_{0} u)}{Γ (v (t))}] = \frac{\partial}{\partial \bar{v}} {[\frac{Γ (\bar{v}, r_{0} u)}{Γ (\bar{v})}]}_{\bar{v} = v (t)} v^{'} (t)$ (21) If v (t)is differentiable, then $f (t) = \frac{v^{'} (t)}{Γ (v (t))}$ (22) Because in the modeling process, the first arrival time of the failure threshold of pick wear cannot be accurately obtained, so for validation purposes, in this paper, the method with a given reliability threshold of 0.95 is adopted, then the monitoring values were predicted by modeling the pick wear failure in the Gamma process.

4.4 Bayesian updating method based on gamma parameter

At the beginning of the test, the amount of wear data was too small, if the parameters cand uof the new Gamma process are calculated according to the original method, because the process of demanding a solution is too complex, and the data is small, the error is large. Therefore, it is considered to use the existing sample data as a prior distribution of the truncated tooth wear in the ongoing test, and update the parameters c and u of the Gamma process by Bayesian method.

The basic process of bayes is described from the perspective of probability density function;

Write the probability density function that depends on the unknown parameterz θ as p (x|θ), it represents the conditional distribution of the population indicator X when a random variable θ is given a value;

According to the prior information of θ, determine its prior distribution π (θ);

It takes two steps to solve for x = (x₁, x₂, …, x_n). First, parameter θ is obtained from prior distribution π (θ), and then I’m going to set θ, you get a sample x = (x₁, x₂, …, x_n) from population distribution p (x|θ). Thus, the likelihood function L (θ) is obtained, namely, $L (θ) = \prod_{i = 1}^{n} p (x_{i} | θ)$ (23)

The joint distribution of sample x and parameter θ is: $h (x, θ) = p (x | θ) π (θ)$ (24)

In the absence of sample information, an inference is first made to θ through prior distribution (θ). After the sample observations are available, θ is inferred from the joint distribution. Therefore, h (x, θ) is decomposed as follows: $h (x, θ) = h (θ | x) m (x)$ (25) Where m (x) is the marginal density function of x, namely $m (x) = \int_{h (x, θ) d θ = \int_{p (x | θ) π (θ) d θ}}$ (26)m (x) contains no information about parameter θ. Θ represents the value space of θ. Therefore, only the conditional distribution h (θ, x) can be inferred from θ, and its calculation formula is as follows: $h (θ, x) = \frac{h (x, θ)}{m (x)} = \frac{p (x | θ) π (θ)}{\int_{p (x | θ) π (θ) d θ}}$ (27) The above formula is the probability density form of the Bayesian formula. In the case of a given sample x, the posterior distribution of the conditional distribution of θ is obtained. A posterior distribution is the result of centralizing everything that is θ and excluding irrelevant information.

When θ is a random variable, the prior distribution π (θ_i) , i = 1, 2, ⋯ can be expressed as a formal prior distribution column. And then the posterior distribution is also zero: $h (θ_{i}, x) = \frac{p (x θ_{i}) π (θ_{i})}{\sum_{i} p (x | θ_{i}) π (θ_{i})}, i = 1, 2, \dots$ (28)

According to bayes’ principle: $\begin{matrix} π (c, u | δ_{1}, \dots, δ_{n}) \\ = \frac{L (δ_{1}, \dots, δ_{n} | c, u) π (c, u)}{\int_{0}^{\infty} \int_{0}^{\infty} L (δ_{1}, \dots, δ_{n} | c, u) π (c, u) dcdu} \end{matrix}$ (29) Where, when (c, u) is known to be the likelihood function of degenerate increment δ₁, ⋯ , δ_n; A priori distribution where π (c, u) is (c, u); π (c, u|δ₁, ⋯ , δ_n) is a posterior distribution of (c, u). According to the Bayesian formula, when the wear monitoring data is updated, the posterior distribution of (c, u) can be updated according to the new degradation data, so as to obtain the real-time reliability assessment of the cutting tooth wear under test. For the prior distribution π (c, u), you can think about fixing c first. And assume that the prior distribution of u, π (u|c), is the Gamma distribution with the shape parameter of α and the scale parameter of β, according to the distribution property of Gamma, it can be seen that: the posterior distribution of u is the Gamma distribution with shape parameter (α + ct_n) and scale parameter (β + x_n). Thus, the Bayesian formula can be rewritten as: $\begin{matrix} π (c, u | δ_{1}, \dots, δ_{n}) = π (u, c | δ_{1}, \dots, δ_{n}) π (c | δ_{1}, \dots, δ_{n}) \\ \propto \prod_{i = 1}^{n} \frac{u^{c [t_{i} - t_{i - 1}]}}{Γ (c [t_{i} - t_{i - 1}])} δ_{i}^{c [t_{i} - t_{i - 1}] - 1} • \\ exp (- u δ_{i}) \times \frac{β^{α}}{Γ (α)} x^{α - 1} exp (- u β) π (c) \end{matrix}$ (30) When parameter c is unknown, the prior distribution π (u|c) of u can be considered as the Gamma distribution with shape parameter α (c) and scale parameter β (c). Since the prior distribution and the likelihood function meet the conjugate conditions, the posterior mean value of ucan be written as: $E_{1} (U | c, δ_{1}, \dots, δ_{n}) = \frac{α (c) + {ct}_{n}}{β (c) + x_{n}}$ (31) According to the estimated values of cand uupdated by Bayesian formula, $\hat{c}$ and $\hat{u}$ , the reliability function of truncated tooth wear can be obtained as: $R (t) = 1 - \frac{Γ (\hat{ct}, r_{0} \hat{u})}{Γ (\hat{ct})}$ (32)

5 Application instance

5.1 Data prediction based on grey prediction process

The first 50 groups of data of the new tooth (A1) under the cutting vibration acceleration energy and state were selected as samples. A total of 50 data were collected for GM segment prediction under the new tooth state cycle, as shown in Fig. 2.

Fig.2

Fitting diagram of A1 state prediction data of vibration signal.

The grey parameter a = -0.0031, u = 105.0746 is solved to obtain the expression of the grey model.

After verification, it can be seen from Table 4, the accuracy level of the model is 1, and the prediction effect is very good.

Table 3

Precision rating Evaluation of Prediction Model

	Posterior difference ratio C	Probability of error
Level 1 (ok)	<0.35	≥0.95
Level 2 (qualified)	<0.5	≥0.8
Level 3 (barely)	<0.65	≥0.7
Level 4 (unqualified)	≥0.65	<0.7

Table 4

Results of various test index values for the model

Designation	Residual	Relative error	Posterior difference ratio C	Probability of error
numerical value	0.943	0.077	0.0817	1

The subsequent 250 sets of data are predicted under this model, and the initial values are substituted into the grey model to calculate the subsequent data in an iterative manner. As shown in Fig. 3.

Fig.3

Fitting diagram of A2∼A6 state prediction data of vibration signal.

Error calculation and analysis were carried out on these 250 predicted data, and the results are shown in Table 5.

Table 5

Comparison of grey predictive value and real value of vibration signal

Number	True value	Predicted value	Relative error /%	Number	True value	Predicted value	Relative error /%
51	120.95	120.16	0.46	176	155.54	154.61	0.70
52	118.70	120.40	1.01	177	156.15	154.92	0.70
53	118.93	120.64	1.02	178	156.47	155.23	0.71
54	119.28	120.88	0.95	179	156.80	155.54	0.88
55	119.67	121.12	0.86	180	157.50	155.86	0.89
...	....	...	...	...	...	...	...
171	153.11	152.77	0.16	296	194.98	196.57	0.74
172	153.61	153.07	0.25	297	195.25	196.97	0.72
173	153.83	153.38	0.21	298	195.49	197.36	0.76
174	154.15	153.69	0.21	299	195.83	197.76	0.82
175	154.73	154.00	0.34	300	196.19	198.15	0.84
				mean value			0.45
				root-mean-square			0.61

As can be seen from the above table, the average relative error of the grey prediction model is 0.45%, and the root mean square of the relative error is 0.61%, indicating that the model has a high prediction accuracy and can be used for the prediction of tooth cutting degradation data.

5.2 Data prediction based on gamma process

According to the signal collection and processing method in chapter 1, the vibration acceleration energy and samples of different cutting teeth in the same environment and of the same type under the new tooth wear state were extracted, as shown in Table 6:

Table 6
Sum of energy different picks’ cutting vibration acceleration signal characteristics sample/mV²

Number A1-1 A1-2 A1-3 Number A1-1 A1-2 A1-3

1 105.23 96.75 105.82 26 113.52 109.4 113.16

2 105.52 100.7 105.48 27 113.94 109.77 113.69

3 105.78 101.8 105.87 28 114.33 109.86 114.18

4 106.21 102.32 106.35 29 114.78 110.4 114.34

5 106.58 102.35 106.81 30 115.23 110.76 115.09

... .... ... ... ... ... ... ...

21 111.36 108.72 111.38 46 120.43 114.41 120.57

22 111.77 108.73 111.61 47 120.56 115.36 120.64

23 112.22 108.92 112.65 48 120.69 116.02 121.39

24 112.66 109.39 112.75 49 120.83 116.72 121.65

25 113.08 96.75 112.82 50 120.95 120.83 121.65

Number	A1-1	A1-2	A1-3	Number	A1-1	A1-2	A1-3
1	105.23	96.75	105.82	26	113.52	109.4	113.16
2	105.52	100.7	105.48	27	113.94	109.77	113.69
3	105.78	101.8	105.87	28	114.33	109.86	114.18
4	106.21	102.32	106.35	29	114.78	110.4	114.34
5	106.58	102.35	106.81	30	115.23	110.76	115.09
...	....	...	...	...	...	...	...
21	111.36	108.72	111.38	46	120.43	114.41	120.57
22	111.77	108.73	111.61	47	120.56	115.36	120.64
23	112.22	108.92	112.65	48	120.69	116.02	121.39
24	112.66	109.39	112.75	49	120.83	116.72	121.65
25	113.08	96.75	112.82	50	120.95	120.83	121.65

Because the experimental data are selected from different cutting teeth, they are physically independent of the data, and the initial quantity meets the Gamma definition, the KS test in MATLAB is used to carry out Gamma distribution test on the wear increment data. It has been proved that each set of data conforms to the Gamma distribution, which means that the tooth wear process meets the requirements of the Gamma process.

Where, A1 wear follows Ga (x|512.647 0.221) distribution, and the fitting results corresponding to Gamma are shown in Figs. 4 and 5.

Fig.4

Gamma distribution density function of cutting pick wear monitoring data.

Fig.5

Incremental frequency map of degradation of cutter wear monitoring data.

Using this method, the parameter estimates of each set of data A1-1, A1-2 and A1-3 were calculated, as shown in Table 7.

Table 7

Parameter estimates

Number	$\hat{c}$	$\hat{u}$
1	512.647	0.221
2	447.967	0.244
3	495.413	0.229
Mean value	485.342	0.231

Based on the calculation results in Table 7, taking the mean value of $\hat{c}$ , 485.342, as the initial value of c, the estimated value of Gamma process parameter u of 3 groups of data $\hat{u}$ , was updated by Bayesian method respectively, as shown in Table 8.

Table 8

Update values for Gamma procedure parameter $\hat{u}$

Group	1	2	3
$\hat{u}$	0.227	0.238	0.230

On this basis, the reliability of sample data is calculated according to Equation (31). The reliability curves of new teeth, slight wear, medium wear, large wear, severe wear, and failure section are shown in Fig. 6.

Fig.6

Reliability curves under different wear conditions.

In Fig. 6, R1–R6 respectively represent the reliability curves of vibration signals of the pick under different degrees of wear. As can be seen from the figure, R1 represents the reliability curve when 50 sets of data are known, Similarly, R2–R6 represents the reliability curve of 100, 150, 200, 250 and 300 groups respectively. With the increasing of the wear data, especially when the data reaches 300 sets, the reliability curve gets closer and closer to the true value, so it meets the reliability assessment of the pick.

According to the reliability curve in Fig. 6, under the given threshold value of 0.95, the predicted values of the pick at different wear degrees can be obtained. In order to prove that the prediction effect of the Gamma model under Bayesian updating is better, the prediction data under parameter updating and without updating are given in Table 9.

Table 9

Comparison of the predicted and true values of the Gamma model

Number	True value	Parameter doesn’t update the predicted value	Parameter updates the predicted value	Relative error without parameter update /%	Relative error under parameter update /%
51	118.7	118.17	118.91	1.63	1.19
52	118.93	118.39	119.04	0.18	0.20
53	119.28	119.64	119.35	0.42	0.25
54	119.67	119.88	119.86	0.36	0.34
55	120.01	120.21	120.92	0.32	0.74
...	...	...	...	...	...
171	153.61	153.84	153.72	0.34	0.28
172	153.83	154.55	153.92	0.43	0.14
173	154.15	154.84	154.36	0.46	0.24
174	154.73	155.36	154.80	0.56	0.30
175	155.31	155.80	155.48	0.49	0.34
176	156.15	156.02	156.07	0.06	0.04
177	156.47	156.12	156.29	0.16	0.08
178	156.80	156.54	156.84	0.12	0.02
179	157.50	157.26	157.57	0.11	0.03
180	157.85	157.48	157.70	0.17	0.07
...	...	...	...	...	...
296	194.53	203.35	194.37	3.21	0.06
297	194.98	204.18	196.26	3.34	0.46
298	195.25	206.12	196.59	3.94	0.49
299	195.49	207.29	197.74	4.27	0.81
300	195.83	209.33	198.32	4.87	0.90
			mean value	0.57	0.22
			root-mean-square	0.86	0.28

It can be seen from Table 9 that the relative error and mean square error of the updated predicted value of the lower pick are 0.57% and 0.86% respectively. The relative error and mean square error of Bayesian updating were 0.22% and 0.28% respectively. It can be seen that the Gamma model can realize the life prediction of the pick, and the prediction effect is better under the Bayesian parameter update.

6 Conclusion

Considering the application of gray prediction, a gray prediction model based on the change of degenerated data is established, and the prediction accuracy of the model is good, which can realize the prediction of degenerated teeth. The results show that the relative error of the predicted value is only 0.45% under the vibration signal, the prediction accuracy is high, and the prediction effect is very good.

Based on the characteristics of non-negative increment of truncated data and stable change after dryness, the Gamma degradation prediction model was established, and the Bayesian updating parameter method was used to update the prediction accuracy. Based on the sample data, the results show that the relative error and mean square error of the energy and predicted value of the vibration signal acceleration frequency band are 0.57% and 0.86% respectively when the parameters are not updated. The relative error and mean square error of Bayesian updating were 0.22% and 0.28% respectively.

By comparing the three prediction models, it can be seen that the gray prediction model has the largest error, the Gamma model has higher accuracy, and the Gamma model has the last prediction effect under the Bayesian parameter update.

References

Shang

H.L.

, Influence analysis and Countermeasures on failure condition of shearer drum Cutting teeth, Coal Science and Technology40(8) (2012), 75–77.

TIRYAKIB, Insitustudies on service life and pick consumption character is tics of shearer drums, Journal of the South African Institute of Mining and Metallurgy31(3) (2006), 380–385.

Zhu

, Wu

Z.H.

, Li

, et al., Study on wear failure of coal machinery, Acta Coal Sinica31(3) (2006), 380–385.

Chu

K.Y.

, Zhou

J.H.

, Zhang

, Wang

, Application of Multisensor Fusion technology based on wavelet analysis and Neural Network in Tool wear monitoring of Machine Tools, Coal Mine Machinery33(09) (2012), 261–263.

Chen

X.Y.

, Xue

K.H.

, Yao

, Study on wear law of micro cemented carbide milling cutters for milling coated sand, Tool technology (2019), 1–4.

Zhang

, Zhang

, Tian

, Liu

Z.H.

, Identification of tooth wear based on LVQ neural network, Journal of Sensing Technology31(11) (2018), 1721–1726.

Hang

, Gu

J.Y.

, Liu

J.M.

, Liu

Z.H.

, Tian

, Identification of gear wear State based on Wavelet Packet and SOM Neural network, Acta Coal Sinica43(07) (2018), 2077–2083.

Zhang

, Liu

Z.H.

, Wang

H.J.

, Zhang

H.Z.

, Research on multi-feature Signal Fusion recognition of truncated tooth wear, Journal of Engineering Design25(03) (2018), 278–287.

Zhang

, Liu

Z.H.

, Dai

X.D.

, Huang

C.H.

, Zhang

S.L.

, Tian

, Research on identification of pick based on vibration and acoustic Emission signals, Mechanical Design and Research34(01) (2018), 126–132+136.

10.

Zhang

, Liu

Z.H.

, Wang

H.J.

, Gu

J.Y.

, Tian

, Zhong

C.H.

, Research on Identification of Cutting Tooth wear Based on multi-feature Information Fusion, Journal of Electronic Measurement and Instrumentation31(12) (2017), 1974–1983.

11.

Park

, Padgett

, Stochastic degradation models with several accelerating variables, IEEE Transactions on Reliability55(2) (2006), 379–390.

12.

Z.Y.

, Ma

P.F.

, Xiao

J.L.

, Wang

M.Q.

, Tang

X.Q.

, Online tool wear monitoring based on machine tool information, China Mechanical Engineering (02) (2019), 1–5.

13.

G.C.

, Research on PCBN tool wear mechanism of high-speed hard Cutting AISIH13, Machine Tools and Hydraulics47(01) (2019), 185–188.

14.

Xiao

Z.Y.

, Zhang

W.M.

, Liu

Z.H.

, Tool Wear Detection Technology based on Maximum entropy and Cross entropy theory, Machine Tools and Hydraulics46(22) (2018), 89–93.

A comparative study of pick degradation prediction based on grey prediction and Bayesian algorithm

Abstract

Keywords

1 Introduction

2 Extraction of vestigial vibration signal of pick

2.1 Establishment of experimental system for monitoring of pick wear

Table 1 Degree of pick wear Attrition rate New pick Slight wear Medium wear Large wear Severe wear Failure pick Poor quality of pick/g 0∼50 50∼100 100∼150 150∼200 200∼250 250∼300 Measurement discrepancy/mm 0∼4.5 4.5∼9.0 9.0∼13.5 13.5∼18 18∼22.5 22.5∼27

3.1 Construction of grey prediction model

4.1 Gamma process principle

5.1 Data prediction based on grey prediction process

References

Table 1
Degree of pick wear

Attrition rate New pick Slight wear Medium wear Large wear Severe wear Failure pick

Poor quality of pick/g 0∼50 50∼100 100∼150 150∼200 200∼250 250∼300

Measurement discrepancy/mm 0∼4.5 4.5∼9.0 9.0∼13.5 13.5∼18 18∼22.5 22.5∼27