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In order to study the dynamic force identification method of an end-plate pick of shearer, a dynamic force identification technique based on interval theory was proposed. The dynamic force interval identification model is established by describing and quantifying the identified parameters. By using the interval analysis method of the first-order Taylor expansion, the dynamic force identification is transformed into two kinds of deterministic inverse problems at the midpoint of the uncertain parameter and its gradient identification. The Tikhonov regularization method is used to solve two kinds of deterministic problems, and the upper and lower boundaries of dynamic force of the end-plate pick are determined. The results show that the deviations between the identified dynamic force and the actual dynamic force are basically within 2% and 5%, and the average uncertainties are up to 7% and 10%. Therefore, the proposed method can effectively determine the upper and lower boundaries of dynamic force of the end-plate pick, improve the solving efficiency, and provide a new research method for studying the coal rock mechanism of the pick cutting load.

In addition to the structural parameters and motion parameters of the pick, the load of coal and rock is also affected by uncertain parameters such as coal quality parameters and geological conditions. As a result, the established forward load identification model and the actual load inevitably have certain approximation and uncertainty, which makes the uniqueness, existence, and stability of the load identification solution impossible to adopt certainty [

At present, there are mainly probability methods, such as fuzzy method and interval method, for describing uncertain information. Among them, it is most common to transform uncertain parameters into mathematical expressions of random variables and analyze uncertainties based on probability and statistics theory. However, for most practical engineering problems, due to the limitations of measurement technology, economy, and other practical conditions, it is often difficult to obtain enough samples, which makes it difficult for this method to obtain satisfactory identification load. For the method of fuzzy theory, in practical problems, the degree of fuzzy membership is determined by the experience of limited samples and decision-makers, which has a lot of subjective factors, leading to the use of the fuzzy method to identify whether loads will bring greater errors. For practical engineering problems, it is usually much easier to obtain the possible bounds of system uncertainties than its statistics. In the study of Liu et al. [

Domestic and foreign scholars have made some achievements in the research of load identification based on the interval analysis method. Wang and Matthies [

To sum up, the research of the interval theory analysis method is not perfect in practical engineering application. The main problems at this stage are described as follows: the quantification and propagation methods of uncertainties based on poor information are still unclear, and the inversion effect of the interval inversion algorithm is different for different research objects, and there is no unified theoretical model. In the field of mining machinery, its interval theory method is still in the stage of exploration and research, especially the research on load identification of end-plate pick is rarely reported at home and abroad. Because of the bad arrangement of end-plate pick and cutting environment, the current research on end-plate pick load is not mature and perfect [

When some parameters of the structure are uncertain, the convolution integral of the dynamic response of the structure can be written as follows [

Since the solution of formula (

Interval numbers are defined as a pair of ordered real numbers [

For load identification of uncertain structures, the uncertain vector

For the same measurement response, since all possible values of structural uncertainties belong to intervals

According to the theory of interval mathematics, formula (

The level of uncertainty in the interval

According to formulas (

Assuming that the uncertainties of all variables in

To extend the restricted applicability of the traditional perturbation method with small uncertainty level, reference [

Therefore, the interval model of dynamic force identification, and the upper and lower bounds of dynamic force identification, is obtained from equation (

From formulas (

Considering the influence of noise, formula (

Singular value decomposition (SVD) of the kernel function matrix is carried out, and the result is obtained:

Therefore, the identified dynamic force can be expressed in the following form:

By using the Tikhonov method, the following formulas can be obtained:

Combining formulas (

Therefore, the dynamic force

The interval theory method is applied in the identification of dynamic force of end-plate pick, and the upper and lower bounds of identification dynamic force are obtained, so as to provide theoretical reference and method for the study of drum load deduction.

The test system for dynamic force of coal and rock cut by pick is shown in Figure

Test system.

The load testing system of the pick is composed of the force measuring device, pressure sensor, signal amplifier, Dasp V10 intelligent data acquisition, and signal processing system. The parameters are as follows: the rated power of the frequency conversion motor is 55 kW, the cutting device can simulate the speed range of the shearer drum which is 0–48 r/min, the force sensor is 0–5000 N, the measuring range of the torque is 0–22000 N·m, the cutting diameter is 1200–2000 mm, and the traction speed is 0.5–2 m/min. In the process of rotary cutting, the dynamic force on the pick is converted into electrical signals through the deformation of five pressure sensors, and the signals are transmitted to Dasp V10 intelligent data acquisition and signal processing system through multiple sliding rings.

The experimental conditions are as follows: the arrangement of the end-plate pick is chessboard, the installation angle of pick is 40°, the axial inclination angle is 10°, the secondary rotation angle is 0°, the maximum cutting thickness is 20 mm, the speed of the cutting arm is 40.8 r/min, and the traction speed is 0.82 m/min. The measured axial load of the pick is shown in Figure

Cutting load.

Measurement response.

Kernel function.

Because of the complex genesis of coal and rock, uneven medium, and anisotropy, the density of coal and rock is uncertain. Therefore, the density of coal and rock is regarded as an uncertain parameter, in which the point value is 1 g/cm^{3}, and the intervals of four uncertainty levels, such as 2%, 5%, 7%, and 10%, are taken as shown in Table

Interval range of uncertain parameters.

Uncertainty parameter | 2% | 5% | 7% | 10% |
---|---|---|---|---|

Density (g/cm^{3}) |
[0.98,1.02] | [0.95,1.05] | [0.93,1.07] | [0.90,1.10] |

In addition, in order to quantitatively judge the effect and quality of dynamic force identification, the following evaluation indexes are defined [

Firstly, the cutting load is identified by taking the midpoint value of the uncertainty parameter and using the regularization method. As shown in Figure

Identified result at the midpoint of interval.

Secondly, in order to obtain the sensitivity curve of the cutting random load to uncertain parameters (coal rock density), the central difference method is used to solve the problem. The step length is 5% of the midpoint value, and the sensitivity curve of parameters (coal rock density) is obtained, as shown in Figure

Sensitivity curve of the load on cutting impedance.

It can be seen from the figure that the cutting random load is relatively sensitive to the density of coal and rock. Loads are calculated by formulas (

The higher the uncertainty level, the greater the radius of the interval. The uncertainty parameters can take a wider sample point, so the upper and lower bounds of the identified cutting load will surround a wider region. Therefore, the identification of upper and lower bounds of cutting loads is studied under four uncertainties. The upper and lower bounds of identified results are shown in Figures

Identified results of different uncertainty levels: (a) 2%, (b) 5%, (c) 7%, and (d) 10%.

From the identification results of four uncertain horizontal cutting loads, the upper and lower bounds of identification loads are expanding with the increase of uncertainty level. Because the interval analysis method is limited to a small level of uncertainty, the upper and lower bounds of the identified cutting load are increasingly unsatisfactory with the increase of the level of uncertainty.

Figures

Figures

The evaluation index.

Uncertainty level | 2% | 5% | 7% | 10% |
---|---|---|---|---|

Interval analysis | 0.02 | 0.05 | 0.07 | 0.10 |

From Table

In summary, the identification effect of the cutting load is better when the uncertainty level is 2%. That is to say, the interval theory analysis method has better identification effect and higher identification efficiency at low uncertainty level. At the same time, it provides a theoretical reference for the study of drum load identification.

An effective method for identifying dynamic force of the end-plate pick of shearer is proposed. The identified parameters are described and quantified by intervals. Using the interval analysis method of the first-order Taylor expansion, the dynamic force identification is transformed into two kinds of deterministic inverse problems of midpoint and gradient identification of uncertain parameters, and then the mathematical model of dynamic force interval identification is established.

When the uncertainty level is 2% and 5%, the identification dynamic force index values of the interval analysis identification algorithm are 0.02 and 0.05, respectively. When the uncertainty level is more than 5%, the index value increases with the increase of the uncertainty level. When the uncertainty level is 7% and 10%, the difference between the index value and 2% is twice. Therefore, when the uncertainty level is 2%, the effect of dynamic force identification is better.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This work was supported by the Chinese National Natural Science Foundation (contract nos. 51674106 and 51274091).