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Research about influence of the tension forces, asymmetrical tensioning and filling rate of pipe conveyor belt filled with the material on the contact forces of idler rolls in hexagonal idler housing
Measurement 156 (2020) 107598
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Research about influence of the tension forces, asymmetrical tensioning and filling rate of pipe conveyor belt filled with the material on the contact forces of idler rolls in hexagonal idler housing Beata Stehlikova a, Vieroslav Molnar b,⇑, Gabriel Fedorko a, Peter Michalik b, Alena Paulikova c a b c
Faculty of Mining, Ecology, Process Control and Geotechnologies of the Technical University of Kosice, Letna 9, 042 00 Kosice, Slovak Republic Faculty of Manufacturing Technologies of Technical University in Kosice with a Seat in Presov, Bayerova 1, 080 01 Presov, Slovak Republic Slovak University of Technology in Bratislava, Faculty of Materials Science and Technology in Trnava, Paulínska 16, 917 24 Trnava, Slovak Republic
a r t i c l e
i n f o
Article history: Received 9 December 2016 Received in revised form 17 September 2019 Accepted 6 February 2020 Available online 11 February 2020 Keywords: Contact force Rubber-textile conveyor belt Idler roll Regression models
a b s t r a c t Reliable operation of the pipe conveyors requires continuous monitoring and evaluation of their selected indicators. Asymmetrical tensioning of the conveyor belt is the serious negative and undesirable operational situations. There is still remaining without answer one important question, namely it is the question whether such improper kind of the tensioning is able to cause changes of the contact forces and in such a way also to influence negatively the processes of conveyor belt wearing or if the asymmetrical tensioning has a relevant impact on the contact force values. The main task of this article is to analyse relations among the tension force values, asymmetrical tensioning and filling rate in the cross-sectional area of the piped belt, which is filled with the material, with regard to the values of the contact forces that are acting on the individual idler rolls arranged in the hexagonal idler housing of the pipe conveyor. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction The pipe belt conveyors represent progressive ecological transport equipment specified for continuous conveying of the raw materials in the area of intra-plant transport. Thanks to the undisputed advantages of these conveyors the clients from the area of bulk materials transport are more and more requiring applications of the pipe conveyors. The main advantages of the pipe conveyors are: transportation in vertical and horizontal curves, ecological and dust-free conveying up to the long distances as well as a possibility to transport the material in both strands of the pipe conveyor [1– 3]. Research and development of the pipe conveyors is still continuing and improving itself [4] and in this way it enables to look for the answers concerning the unanswered questions relating to the systematic, technological and constructional aspects. The most important part of the pipe conveyor is the conveyor belt, because all the advantages of the pipe conveyors are determined by a special construction of the conveyor belt. That’s why the producers of the conveyor belts are trying to develop new and more perfect constructions. Nowadays, there is mainly applied in this development process such approach, which is using the empirical knowledge without a deeper understanding to the ⇑ Corresponding author. E-mail address: [email protected] (V. Molnar). https://doi.org/10.1016/j.measurement.2020.107598 0263-2241/Ó 2020 Elsevier Ltd. All rights reserved.
theory of deformations and contact pressures in the individual layers of the conveyor belt construction. Zamiralova and Lodewijks [5] measured stiffness of the conveyor belt by means of a special sixpoint measuring equipment. The obtained results confirmed a fact that higher diameters of the piped belt require also a higher flexural rigidity in order to ensure stability of the conveyor belt shaped into the pipe form. Fedorko and Molnár [6] created a simulation model using the software product Abaqus. This model is suitable for simulation of the experimental measurements realised by means of a special testing equipment, which was designed for testing of the pipe conveyors [7]. Petríková et al. [8] presented the experimental and numerical analyses of the mechanical characteristics describing the specimens separated from the rubber-textile conveyor belt. The mechanical properties and the friction coefficient were investigated using the tensile tests. Research of the transition section in the pipe conveyor is just the area with many unanswered questions. The existing theories that are describing the shaping process of the transition section for the conveyor belt, which is installed in the pipe conveyor, are limited. Weigang [9] developed a relatively simple method, which enables to define the cross-sectional curves of the individual parts in the transition section of the pipe conveyor belt. Zhang et al. [10] and Zhipping [11] determined length of the transition section so that it equals to 25 diameters in the case of the rubber-textile conveyor belt and it equals to 50 diameters in the case of the steel rope
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conveyor belt. Maton [12] investigated a possibility whether the elliptic cross-section of the pipe conveyor belt is more suitable in comparison with the standard circular shape of the cross-section. He developed a methodology for a configuration of the conveyor belt profile or the conveyor belt curves in order to analyse and to evaluate them. Fedorko et al. [13] analysed the transition section of the pipe conveyor belt during transformation of it from the flat shape into the pipe shape, using the finite element method (FEM). The correct dimensioning of the individual constructional parts of the pipe conveyor, which is based on calculation of the main parameters, improves efficiency of the continuous belt transport [14]. Marasová et al. [15] used a modification of the main parameter calculations for the classic belt conveyor in order to determine the basic parameters for the pipe conveyor. Egorov [16] performed calculation of several parameters, for example the belt elasticity coefficient, belt width, fictive belt density, belt speed, conveyor belt stress and distances between the idler housings. Zamiralova and Lodewijks [17] presented a similar approach to calculation of the rolling resistances and to the viscoelastic behaviour of the rubber belt. The conveyor belt was simulated in the form of a generalised three-dimensional Maxwell model with various parameters. Kinoshita et al. [18] proposed a simple method in order to estimate the total energy losses of the conveyor belt moving on the carrying idler rolls. There were investigated effects of the normal forces acting on the idler rolls and influence of the driving pulley on the resistance forces. This investigation confirmed a fact that the resistance force of the idler roll is rising if the idler roll speed is increasing. Molnar et al. [19] dealt with determination of the contact force values for a certain type of the conveyor belt. The contact forces can be estimated in the intersection of axis at the positions of the individual idler rolls in dependence on the tension force as well as in dependence on the filling rate of the crosssectional area of the piped belt filled with the material, taking into consideration the performed experimental measurements. A properly scheduled replacement of the worn conveyor belt is a decisive factor with regard to minimization of idle times and losses in the continual belt transport, whereas in some of the cases there is required a special procedure for replacement of the used belt or a specific maintenance system should be applied for a new belt [20]. Modeling and simulation of static behaviour of the pipe conveyor belt, using the finite element method (FEM) and comparison of the computed results with the real values obtained from the experimental measurements, can be utilized for proposals of the new technical standards valid in the area of the pipe conveyor system design [21]. Yan and He [22] applied a new technology of virtual prototype, which was developed during the last years, in order to create a digital model of the belt conveyor specified for performing of dynamic analyses. This new procedure offers an innovative methodology oriented to the safety analysis and belt conveyor design. This article deals with a new research of influence of the tension forces, asymmetry in tensioning of the pipe conveyor belt and filling rate of the cross-sectional area of the pipe shaped belt filled with the material on the value of the contact forces acting on the individual idler rolls, which are situated in the hexagonal idler housings of the pipe conveyor. There were realised several series of measurements at the real type of the rubber-textile conveyor belt using a special testing equipment [7] developed at the Technical University of Košice. This testing stand enables to perform simulation of a real transition section, which is forming the conveyor belt into the pipe shape. The length of this testing equipment is 8 m and it allows application of various tension force values.
2. Material and methods The operational area of a feed chute in the case of the pipe conveyors is the most critical point within the pipe conveyor transport system because there are concentrated in this section several key factors, which are significantly influencing operation and functioning of the whole technical equipment. The first important factor is the process of closing (i.e. forming) of the conveyor belt into the pipe shape. This process is connected with the intensive force impacts caused due to transformation of the conveyor belt shape and due to following interaction of the conveyor belt with the forming idler rolls. The second relevant factor is the transported material, namely the force effects caused by the weight of material. In this way the transported material increases and markedly influences the already existing forces. The third factor is impact of the tension force. These three factors together are affecting operation of the pipe conveyor as well as value of the motional resistances. The motional resistance value is a decisive criterion, which is important with regard to the consumption of electric energy and the motional resistances are also critical regarding to wearing of the belt bottom cover. It is important to investigate how the above-mentioned three factors are influencing values of the contact forces and the motional resistances. Monitoring and measuring of the whole process during current operation of the conveyor is not an easy task and often it is also complicated due to asymmetry of the conveyor belt tensioning. It is very important to choose a suitable method in order to perform a scientific investigation of the given problems. Anyway, realisation of an experimental measurement is necessary in this case. However, the experimental measurement is very demanding with regard to realisation of the measuring process itself and it is almost impossible to perform it during standard operation of the pipe conveyor. Therefore, we decided to apply the mathematical models. The mathematical model cannot be utilised without the experiment. Thus, it is necessary to combine the experimental methodology with the mathematical methods within the research area in order to define such relations, which are able to describe behaviour of the whole system and interactions among the individual parameters. Taking into consideration all the circumstances it is possible to say that a methodology of the system dynamics is such scientific tool, which is suitable for the detailed research of the given problems. It was already mentioned a fact that the research concerning the contact force values requires the experimental measurements, however realisation of the experiments is almost impossible during real operation of the pipe conveyor. A possible solution offers application of an appropriate physical model, which is representing the monitored section of the pipe conveyor in order to obtain relevant data that are necessary for the scientific investigation in the given research area. The used physical model of the feed chute section for the pipe conveyor was created in the form of an original testing stand. Fig. 1 illustrates the feed chute of the pipe conveyor with the marked area, which was replaced by the physical model, whereby the physical model is represented with the experimental testing equipment according to Fig. 2. The physical simulation model (the testing equipment) enabled to induce the tension forces during the experimental measurements. The tension force values TF = 4000–28,000 N were induced in the conveyor belt by means of a couple of the tensioning screws ID23, ID24, Fig. 3. The filling rate A of the cross-sectional area of the piped belt filled with the material was simulated by means of four sandbags, Fig. 4. The asymmetry AS was created by a different ratio
B. Stehlikova et al. / Measurement 156 (2020) 107598
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Fig. 1. Feed chute of the pipe conveyor (without the conveyor belt).
Fig. 2. Experimental testing equipment installed at the TU of Košice.
Fig. 3. Marking of the sensors on the idler housing No.2.
of the tension forces TF induced in the above-mentioned tensioning screws ID23, ID24, Fig. 3. The physical simulation model was created as an identical copy of the feed chute part of the pipe conveyor with regard to the dimensions. Rotation of the guiding rolls does not influence the contact
force values. Therefore, the idler rolls were applied only in a fixed position in order to simplify the measuring process, however without any misinterpretation or inaccuracy of the obtained results. Taking into consideration principles of the system dynamics the physical model consists of the next components: the tensioning
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Fig. 4. Cross-section of conveyor belt with the sandbags and illustration of the filling rate A.
system component, the system component of the guiding rolls, the material system component and the system component of the conveyor belt (Fig. 3). The system component of the guiding rolls contains three subcomponents, which are representing the individual idler housings, whereby there are identified 6 partial parts in the form of the fixed guiding rolls within the idler housing subcomponent. The main attention was paid to the idler housing No.2, because it corresponds to the major part of the conveyor trajectory and to the related contact force values. The greatest problem, which occurred during creation of the physical model, was simulation related to filling of the piped conveyor belt with the transported material. Aspect of the filling process simulation is very important because it is necessary to simulate the real operational conditions of the pipe conveyor as realistic as it is possible. Therefore, it was chosen an original system of the sandbags. This solution allowed to control the filling shape of the conveyor belt as well as amount and weight of the transported material. The research was performed using a sample of the rubbertextile conveyor belt. The basic characteristics of the rubbertextile conveyor belt, experimental testing equipment and transported material are summarized in Table 1. The conveyor belt is tensioned using the experimental testing equipment in a horizontal plane linearly, i.e. without curves. The given sample of the rubber-textile conveyor belt represents the most often used type of the conveyor belt, which is applied in the real operation and therefore the obtained results will be applicable in a wide range of the belt conveyor operation. The experimental method, which was applied during the performed
analysis, is divided into two steps, according to the block scheme presented in Fig. 5. In the first step, the methodology of Molnar et al. [23] has been used. The main goal of the performed research was focused on the next two hypotheses, which are resulting from the actual sum of knowledge and from a requirement to obtain new information for a better understanding to the investigated problems. The 1st Hypothesis: ‘‘There is existing a direct dependence among the contact forces and the tension forces, together with the weight of transported material.” The 2nd Hypothesis: ‘‘It is possible to create a mathematical model based on the experimental measurements, which will describe the above-mentioned dependence.”
3. Theory/calculation There were applied the mathematical-statistical methods and the method of regression model in order to verify the individual hypotheses. Certain tests of the hypotheses relating to the model parameters during solution of the regression problems are helpful for measuring of the model usefulness [24,25]. Evaluation process of the regression models is utilizing a sum of squares and decomposition of it. The total sum of the squares SST consists of the sum of the squares SSR, which includes the regression model (mathematical relation among the inputs and outputs) and the sum of squares SSE created by the random deviations:
SST ¼ SSR þ SSE
ð1Þ
Table 1 Basic characteristics of the conveyor belt, experimental testing equipment and transported material. Type of the conveyor belt
EP500/3 HP 5 + 3 D
Polyester rubber-textile conveyor belt Strength of the belt Pipe shaped belt Top cover thickness Bottom cover thickness Category: transport of materials with the recommended belt surface temperature + 150 °C Length of the conveyor belt Width of the conveyor belt Thickness of the conveyor belt Diameter of the pipe shaped conveyor belt Distance between the idler housings Diameter of the idler rolls Bulk density of transported dry sand Angle of repose for dry sand Maximal angle of incline for dry sand
EP 500 HP 5 3 D 8 000 800 20 200 1000 60 1650 15 25
N mm1 mm mm mm mm mm mm mm mm kg m3 ° °
B. Stehlikova et al. / Measurement 156 (2020) 107598
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1. Equalizing of the contact forces CF according to the required tension forces TF Using the methodology of Molnar et al. [23]
2. Simulaon of the contact forces CF as a result of the next factors: tension force value TF, filling rate A of the cross-seconal area of the piped belt filled with the material, asymmetry AS a) Model proposal b) Calculaon of the model regressors and evaluaon of a stascal significance for the model; Minimisaon of count of regressors in the model c) Graphical interpretaon of applicaon possibilies for obtained informaon
Fig. 5. Block scheme of the analysis procedure [23].
whereas the individual members are given according to the relation (2) using the empirical values, together with the model values: n n n X X X 2 2 yi y ¼ Yi y þ ðY i yi Þ2 i¼1
i¼1
ð2Þ
i¼1
where:
r2 ¼
Pn
yi Þ2 np1
i¼1 ðY i
Extra sum of the squares method is used in order to consider if the last regressors, which are indexed (p + 1), increase significantly the sum of squares of the model. The hypotheses are:
H0 : bpþ1 ¼ 0
Yi are the regression model values, yi are the empirical values, n is dimension of the vector Yi and vector yi, p is count of regressors.
H1 : bpþ1 –0 The test statistic for this hypothesis is the FE (extra):
Pn Test for significance of the regression appropriate hypotheses are:
H0 : b1 ¼ b2 ¼ ¼ bp ¼ 0
Rejection of the hypothesis H0 implies a fact that at least one of the regressor variables contributes significantly to the model. The test procedure for the H0 is specified in order to compute the testing criterion F:
ð3Þ
and to reject the H0 if F exceeds the F crit ¼ F a;k;nk1 . Test on the individual regression coefficients, where the hypotheses are:
H1 : bj –0 The test statistic for this hypothesis is:
^ b j ^ se b j
qffiffiffiffiffiffiffiffiffiffiffi ^ ¼ r ^ 2 C jj se b j
i¼1 ðY i
2 P 2 yÞpþ1 ni¼1 ðY i yÞp ðn p 1Þ Pn 2 1 i¼1 ðY i yi Þpþ1
ð7Þ
FE ¼
SSRðpþ1Þ SSRðpÞ ðn p 1Þ 1 SSEðpþ1Þ
ð8Þ
where (p) and (p + 1) are the last added regressor and the H0 is rejected if the FE exceeds F E crit ¼ F a;1;ðnp1Þ . The validity range of the results, which are obtained using the described experimental testing equipment, is from the interval of the tension forces TF = 4000–28,000 N. 4. Results The choice of model with a suitable number of the regressors was based on the estimated polynomial dependence (of the third degree maximally) on the tension force value TF, together with a linear dependence on the filling rate A of the cross-sectional area of the piped belt filled with the material and applying a polynomial dependence on the asymmetry AS.
H0 : bj ¼ 0
t0 ¼
FE ¼
and computation of the testing criterion, using the sum of squares, is according to (8):
H1 : bj –0 for at least one j
Pn 2 ðY i yÞ ðn p 1Þ SSR ðn p 1Þ ¼ F ¼ Pni¼1 2 ðpÞ ðpÞ SSE i¼1 ðY i yi Þ
ð6Þ
ð4Þ
Y ¼ b0 þ b1 TF þ b2 TF 2 þ b3 TF 3 þ b4 A þ b5 AS þ b6 AS2 ð9Þ
ð5Þ
1 where the Cjj are the diagonal parameters from the matrix X T X and X is matrix of the input variables Xij and first columns, all the values of matrix at the position Xi1 = 1 and there is:
This model was consequently modified for each of the idler roll positions in order to take into consideration the regressors b for those effects that are statistically significant with regard to formulation of variability for the results. The SSR is an additive composition of the components in the sum of squares for the effects TF; TF 2 ; TF 3 ; A; AS; AS2 .
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B. Stehlikova et al. / Measurement 156 (2020) 107598 Table 2 Test for significance of regression.
The effect with the lowest sum of squares was gradually eliminated from the model, whereas this sum of squares was replaced by the value 0 in the calculations and the value SSE was increased by the previous value of the sum of squares. There was calculated for the modified model the significance test for regression and it was also calculated the extra sum of squares if the last regressor, which is indexed (p + 1), significantly increased the sum of squares in the model. The obtained results are in Table 2. If the test criterion value is higher than the critical value so it means that the last regressor under consideration (the sum of squares of this criterion was replaced by the zero value) represents a considerable contribution in order to explain variability of the results and therefore this regressor has to remain in the model. The sums of squares of the resulting model are highlighted in Table 2 and the zero values of the sums of squares of the effects are marked by the red colour. The yellow colour means such value of the test criterion FE, which is higher than the critical value FE crit. In our case this fact means that the last added regressor of the model (the previous line of sum of squares) is contributing significantly to the description of variability and therefore it is not suitable to remove this regressor from the model. It is sufficient to take into consideration a linear dependence of the contact force CF on the tension force TF in order to explain the contact force value CF on the idler roll position ID7. The model could not be simplified on the idler roll position ID8, i.e. all the regressors were important in this case in order to explain the contact force value CF on this idler roll position. It was possible to eliminate an irrelevant effect of the component TF3 in the case of all other idler roll positions, i.e. to simplify the polynomial dependence of the contact force CF on the tension force TF using the polynomial form of the second degree. Elimination of the linear effect concerning the asymmetry AS, together with keeping of the effect AS2 in the model on the idler roll positions ID9–ID12 was an interesting finding. This situation
means that influence of asymmetric tensioning on these positions does not depend on the kind of asymmetry, but it depends on absolute value of the asymmetry only. Contribution of the individual effects, together with the absolute member I, to the contact force CF on the individual idler roll positions was analysed according to the value and to the sign of the regressors b0–b6. The values of the regressors b0–b6 are presented in Table 2. The positive values are highlighted by the red colour and the negative values are green. An example of graphical interpretation related to the influence of the tension force TF value, the filling rate A of the cross-sectional area of the piped belt filled with the material and the asymmetry AS on the contact force CF of the idler rolls ID8–ID12 is presented in Fig. 2. The calculations were performed according to the regressors from Table 3, using the values TF = 28,000 N; A = 0–0.75; AS = 2000 N to 2000 N. The positive or negative value of the regressor b0 means that the testing equipment causes the non-zero values of the contact forces CF even without acting of the tension forces TF and without the filling rate A of the cross-sectional area of the piped belt. However, these values cannot be taken into consideration, because they are out of the result validity ranges. The positive value of the regressor b1 confirms an accordance of the results with the physical assumptions.
Table 3 Values of the regressors b0–b6.
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B. Stehlikova et al. / Measurement 156 (2020) 107598 ID8
47 46 45
CF [N] 44 43 42 2000 1000 [ AS
0
N]
0.6
-1000
TF = 28000N A = 0 ÷ 0.75; AS = -2000N ÷ 2000N
0.4
-20000.0
ID9
0.2
A [-]
ID10
130 220
125
200
CF [N]
CF [N] 180
120 2000
2000
1000
1000
[ AS
0 0.6
[ AS
N]
0
N]
-1000 -20000.0
-1000
0.6 0.4 0.2
-20000.0
A [-]
ID11
0.4 0.2
A [-]
ID12
240 220
220
200
CF [N]
180
200
CF [N]
160
180
140 2000
2000
1000
1000
0
0.2
A [-]
0.6
N]
N]
-20000.0
0.4
[ AS
[ AS
0
0.6
-1000
-1000 -20000.0
0.4 0.2
A [-]
Fig. 6. Example of graphical interpretation describing influence of the tension force TF value, the filling rate A of the cross-sectional area of the piped belt filled with the material and the asymmetry AS on the contact force CF in the idler rolls ID8–ID12, calculated for the TF = 28,000 N; A = 0–0.75; AS = 2000 N to 2000 N.
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The regressor b2 is forming the polynomial in a dependence on the tension forces TF and it also represents an influence of the material elasticity (negative value) in the case of the idler roll positions ID9–ID12. There is a special position of the idler roll ID8 due to overlapping of the side edges in the conveyor belt. The regressors b2 and b3 are forming the polynomial on the idler roll position ID8 and there is always a positive increment of the contact force CF within the validity range of the results. The regressor b4 describes an influence of the filling rate A of the cross-sectional area of the piped belt filled with the material. Taking into consideration a fact that only two values were used in the measuring process, namely the values 0 and 0.75, so it was applied the linear dependence. However, it is quite possible that a form of this dependence will be actualised and modified according to the future measured results. The sign of the regressor confirms the accordance with the physical assumptions. The value of the contact force CF is reduced in the idler roll positions ID8 and ID9 due to influence of the filling rate A of the cross-sectional area of the piped belt filled with the material and the contact force value CF is increased in the idler roll positions ID10, ID11 and ID12. The regressors b5 and b6 in the idler roll position ID8 are causing a displacement of the axis of symmetry. The regressor b5 determines an intensity of the linear dependence on the asymmetry AS and it is intended for the idler roll position ID8. Effect of this regressor reduces the contact force value CF. The regressor b6 describes a quadratic dependence on the asymmetry AS. This regressor describes behaviour of dependence, which is symmetric with the axis of asymmetry for the asymmetry value AS = 0 in the idler roll positions ID9–ID12. The regressor b6 causes increasing of the contact force CF in the idler roll position ID9 and
it reduces the contact force CF in the idler roll positions ID10, ID11 and ID12. Fig. 6 presents a dependence of the contact force CF on the filling rate A of the cross-sectional area of the piped belt filled with the material and on the asymmetry AS in the idler roll positions ID8– ID12 for one chosen tension force value TF = 28,000 N, which is close to the working-operational tension force TF. The graphs create an areal formation shaped into the form of U-profile. This shape can be defined by means of an incline, which is given by the line slope, i.e. by the regressor b4 situated near the factor filling rate A of the cross-sectional area of the piped belt filled with the material. The graphical interpretation corresponds to the physical assumptions concerning a fact that the fulfilment of the conveyor belt with the material causes reduction of the contact force CF in the upper part of the idler housing No.2 on the idler roll positions ID8 and ID9 as well as it causes increasing of the contact force CF in the bottom part of the equipment on the idler roll positions ID10, ID11 and ID12. The second attribute of the U-profile is a height of the arc shape. This attribute is resulting from the value of the regressor b6, which is situated near the factor of asymmetry AS2. The height of arc for the idler roll positions ID9–ID12 can be determined from Fig. 6. It is the value of the regressor b6 multiplied by the value 20002. The U-profile also has another attribute on the idler roll position ID8, namely this attribute is a displacement caused by the regressor b5. The axis of symmetry is situated in the value of asymmetry AS = 1000 N. 5. Discussion There is presented in Fig. 7 quantification of influence of the relevant factors, i.e. the tension force TF, the filling rate A of the
Fig. 7. Influences of the values TF, A and AS on the contact force CF in the idler rolls ID7–ID12, calculated for the TF = 28,000 N; A = 0.75; AS = 2000 N.
B. Stehlikova et al. / Measurement 156 (2020) 107598
cross-sectional area of the piped belt filled with the material and the asymmetry AS on the contact force CF. The final contact force value CF is generated by means of an additive method on each of the idler roll positions ID. A positive linear dependence of the contact force CF on the tension force TF was indicated for the idler roll position ID7. The regressor b1 has a positive value at the tension force value TF. It was not indicated a statistically significant influence of the changed filling rate A and asymmetry AS on the change of CF. The results for the idler roll position ID8, where the edges of the conveyor belt are overlapped each other, are as follows: all the considered factors are statistically significant with regard to description of the contact force CF on this idler roll position. Influence of the changed TF on the change of CF is described by a polynomial dependence, namely by a polynomial of the third degree, whereby the last member at TF3 is negative, what reduces the value of CF. The CF value, which depends on TF, is always a positive number within the validity range of the results. Influence of the filling rate A of the conveyor piped belt on the change of CF is described by means of a linear negative dependence, what means that increase of the material filling rate causes decrease of CF. Influence of the asymmetry AS on the change of CF is described by a polynomial of the second degree with the positive regressor at AS2 and with the negative regressor at AS. The final value of the additive CF can be either a positive or a negative number due to influence of AS. Dependence of CF on TF is described using a polynomial of the second degree for the idler roll positions ID9, ID10, ID11 and ID12. The value of CF, which is caused by TF, is always a positive number. Influence of the filling rate A is statistically significant on the above-mentioned idler roll positions and it is described by a linear function. Higher value of the filling rate A of the cross-sectional area of the piped belt filled with the material on the position ID9 means a decrease of the contact force CF. The contact force CF is a positive number for higher value of the filling rate A on the idler roll positions ID10, ID11 and ID12. The effect of asymmetry was verified in the case of all these idler roll positions for the member AS2, what means that influence of the asymmetrical tensioning depends on the absolute value of the asymmetry and it does not depend on the sign of asymmetry, i.e. on the direction. There was indicated, in the case of the idler roll position ID9, an increase of CF due to asymmetry and it was recorded, in the case of the positions ID10, ID11 and ID12, a decrease of CF caused by the asymmetry. 6. Conclusions The asymmetric tensioning of the rubber-textile conveyor belt creates an undesirable operational state of the pipe belt conveyor, which causes not only belt drift from the end pulleys, but it also causes changes of the contact forces, whereas this phenomenon is resulting in an excessive wear of the conveyor belt. The main reason of the above-mentioned negative occurrences is a fact that due to the asymmetric tensioning of the rubber-textile conveyor belt the mutual interactions among the guiding roll surfaces and the bottom cover of the conveyor belt are changing. These negative facts should be analysed in the case of the conveyor belt operation with the transported material and also without this material. There was applied in this article an originally modified regression model in order to obtain and to compare the differences relating to the values of the contact forces on the idler rolls. It was confirmed a validity of the stated two research hypotheses: – the 1st Hypothesis: ‘‘There is existing a direct dependence among the contact forces and the tension forces, together with the weight of transported material.”
9
– the 2nd Hypothesis: ‘‘It is possible to create a mathematical model based on the experimental measurements, which will describe the above-mentioned dependence.” The regression models confirmed a possibility to simulate dependence of the contact force CF on the tension force TF, on the filling with material A and on the asymmetry of tensioning AS in the case of the individual idler roll positions. The corresponding calculated parameters are summarized in Table 3. The resulting models and their parameters were determined using a significance test relating to the added regressor. This test eliminated, for each of the positions, such factors that did not caused a statistically significant improvement of the model as well as did not describe a statistically relevant share of variability concerning the measured values. Table 2 presents an importance of the individual considered factors TF, A and AS using a decomposition of the variability to the particular components related to these factors. The regression models were calculated for each of the positions ID and the statistical significance was evaluated for the individual regressors. The results are as follows: the dependence of the contact force CF on the tension force TF was confirmed for all the monitored idler roll positions ID7–ID12. The statistically significant dependence of CF on the factor ‘‘filling of belt with material” A and on the factor ‘‘asymmetry” AS was confirmed in the case of all other positions ID8–ID12. The dependence of CF on A and AS for the constant value of TF is presented in Fig. 6, where is TF = 28,000 N. Filling of the conveyor belt with the transported material on the positions ID8 and ID9 reduces the resulting CF, whereby the same factor on the positions ID10, ID11 and ID12 increases the value of CF. In such a situation when the resulting tension force is generated by the asymmetrical tension forces, the influence of CF on the individual ID positions is following: the CF value is increased in the case of ID8 and ID9; namely the increment on the position ID8 could be up-to 6% of the original CF value and the increment on the position ID9 is 2.3% of the original CF value. The asymmetrical tensioning of the conveyor belt causes a reduction of the resulting CF in the case of the positions ID10, ID 11 and ID12; namely this decrement is up-to 13% of the original CF value. This phenomenon seems to be applicable in order to reduce loading of ID on these positions, however nowadays there are not investigated the processes occurring inside the conveyor belt material that are caused due to the asymmetric tensioning. Results of the analysis, using the numerical values and signs of the regressors b1–b6, are helpful for improvement of the knowledge concerning influence of the factors TF, A and AS on CF for the individual idler roll positions. The obtained knowledge defines the kind and the value of the individual effects of the tension forces TF, the tensioning asymmetry AS and the filling rate of the piped belt filled with the material A on the contact forces of the idler rolls in the hexagonal idler housing.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements This work is a part of these projects VEGA 1/0332/20, VEGA 1/0600/20, VEGA 1/0045/18, VEGA 1/0403/18, KEGA 012TUKE4/2019, KEGA 013TUKE-4/2019, SK-SRB-18-0053.
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Measurement journal homepage: www.elsevier.com/locate/measurement
Research about influence of the tension forces, asymmetrical tensioning and filling rate of pipe conveyor belt filled with the material on the contact forces of idler rolls in hexagonal idler housing Beata Stehlikova a, Vieroslav Molnar b,⇑, Gabriel Fedorko a, Peter Michalik b, Alena Paulikova c a b c
Faculty of Mining, Ecology, Process Control and Geotechnologies of the Technical University of Kosice, Letna 9, 042 00 Kosice, Slovak Republic Faculty of Manufacturing Technologies of Technical University in Kosice with a Seat in Presov, Bayerova 1, 080 01 Presov, Slovak Republic Slovak University of Technology in Bratislava, Faculty of Materials Science and Technology in Trnava, Paulínska 16, 917 24 Trnava, Slovak Republic
a r t i c l e
i n f o
Article history: Received 9 December 2016 Received in revised form 17 September 2019 Accepted 6 February 2020 Available online 11 February 2020 Keywords: Contact force Rubber-textile conveyor belt Idler roll Regression models
a b s t r a c t Reliable operation of the pipe conveyors requires continuous monitoring and evaluation of their selected indicators. Asymmetrical tensioning of the conveyor belt is the serious negative and undesirable operational situations. There is still remaining without answer one important question, namely it is the question whether such improper kind of the tensioning is able to cause changes of the contact forces and in such a way also to influence negatively the processes of conveyor belt wearing or if the asymmetrical tensioning has a relevant impact on the contact force values. The main task of this article is to analyse relations among the tension force values, asymmetrical tensioning and filling rate in the cross-sectional area of the piped belt, which is filled with the material, with regard to the values of the contact forces that are acting on the individual idler rolls arranged in the hexagonal idler housing of the pipe conveyor. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction The pipe belt conveyors represent progressive ecological transport equipment specified for continuous conveying of the raw materials in the area of intra-plant transport. Thanks to the undisputed advantages of these conveyors the clients from the area of bulk materials transport are more and more requiring applications of the pipe conveyors. The main advantages of the pipe conveyors are: transportation in vertical and horizontal curves, ecological and dust-free conveying up to the long distances as well as a possibility to transport the material in both strands of the pipe conveyor [1– 3]. Research and development of the pipe conveyors is still continuing and improving itself [4] and in this way it enables to look for the answers concerning the unanswered questions relating to the systematic, technological and constructional aspects. The most important part of the pipe conveyor is the conveyor belt, because all the advantages of the pipe conveyors are determined by a special construction of the conveyor belt. That’s why the producers of the conveyor belts are trying to develop new and more perfect constructions. Nowadays, there is mainly applied in this development process such approach, which is using the empirical knowledge without a deeper understanding to the ⇑ Corresponding author. E-mail address: [email protected] (V. Molnar). https://doi.org/10.1016/j.measurement.2020.107598 0263-2241/Ó 2020 Elsevier Ltd. All rights reserved.
theory of deformations and contact pressures in the individual layers of the conveyor belt construction. Zamiralova and Lodewijks [5] measured stiffness of the conveyor belt by means of a special sixpoint measuring equipment. The obtained results confirmed a fact that higher diameters of the piped belt require also a higher flexural rigidity in order to ensure stability of the conveyor belt shaped into the pipe form. Fedorko and Molnár [6] created a simulation model using the software product Abaqus. This model is suitable for simulation of the experimental measurements realised by means of a special testing equipment, which was designed for testing of the pipe conveyors [7]. Petríková et al. [8] presented the experimental and numerical analyses of the mechanical characteristics describing the specimens separated from the rubber-textile conveyor belt. The mechanical properties and the friction coefficient were investigated using the tensile tests. Research of the transition section in the pipe conveyor is just the area with many unanswered questions. The existing theories that are describing the shaping process of the transition section for the conveyor belt, which is installed in the pipe conveyor, are limited. Weigang [9] developed a relatively simple method, which enables to define the cross-sectional curves of the individual parts in the transition section of the pipe conveyor belt. Zhang et al. [10] and Zhipping [11] determined length of the transition section so that it equals to 25 diameters in the case of the rubber-textile conveyor belt and it equals to 50 diameters in the case of the steel rope
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conveyor belt. Maton [12] investigated a possibility whether the elliptic cross-section of the pipe conveyor belt is more suitable in comparison with the standard circular shape of the cross-section. He developed a methodology for a configuration of the conveyor belt profile or the conveyor belt curves in order to analyse and to evaluate them. Fedorko et al. [13] analysed the transition section of the pipe conveyor belt during transformation of it from the flat shape into the pipe shape, using the finite element method (FEM). The correct dimensioning of the individual constructional parts of the pipe conveyor, which is based on calculation of the main parameters, improves efficiency of the continuous belt transport [14]. Marasová et al. [15] used a modification of the main parameter calculations for the classic belt conveyor in order to determine the basic parameters for the pipe conveyor. Egorov [16] performed calculation of several parameters, for example the belt elasticity coefficient, belt width, fictive belt density, belt speed, conveyor belt stress and distances between the idler housings. Zamiralova and Lodewijks [17] presented a similar approach to calculation of the rolling resistances and to the viscoelastic behaviour of the rubber belt. The conveyor belt was simulated in the form of a generalised three-dimensional Maxwell model with various parameters. Kinoshita et al. [18] proposed a simple method in order to estimate the total energy losses of the conveyor belt moving on the carrying idler rolls. There were investigated effects of the normal forces acting on the idler rolls and influence of the driving pulley on the resistance forces. This investigation confirmed a fact that the resistance force of the idler roll is rising if the idler roll speed is increasing. Molnar et al. [19] dealt with determination of the contact force values for a certain type of the conveyor belt. The contact forces can be estimated in the intersection of axis at the positions of the individual idler rolls in dependence on the tension force as well as in dependence on the filling rate of the crosssectional area of the piped belt filled with the material, taking into consideration the performed experimental measurements. A properly scheduled replacement of the worn conveyor belt is a decisive factor with regard to minimization of idle times and losses in the continual belt transport, whereas in some of the cases there is required a special procedure for replacement of the used belt or a specific maintenance system should be applied for a new belt [20]. Modeling and simulation of static behaviour of the pipe conveyor belt, using the finite element method (FEM) and comparison of the computed results with the real values obtained from the experimental measurements, can be utilized for proposals of the new technical standards valid in the area of the pipe conveyor system design [21]. Yan and He [22] applied a new technology of virtual prototype, which was developed during the last years, in order to create a digital model of the belt conveyor specified for performing of dynamic analyses. This new procedure offers an innovative methodology oriented to the safety analysis and belt conveyor design. This article deals with a new research of influence of the tension forces, asymmetry in tensioning of the pipe conveyor belt and filling rate of the cross-sectional area of the pipe shaped belt filled with the material on the value of the contact forces acting on the individual idler rolls, which are situated in the hexagonal idler housings of the pipe conveyor. There were realised several series of measurements at the real type of the rubber-textile conveyor belt using a special testing equipment [7] developed at the Technical University of Košice. This testing stand enables to perform simulation of a real transition section, which is forming the conveyor belt into the pipe shape. The length of this testing equipment is 8 m and it allows application of various tension force values.
2. Material and methods The operational area of a feed chute in the case of the pipe conveyors is the most critical point within the pipe conveyor transport system because there are concentrated in this section several key factors, which are significantly influencing operation and functioning of the whole technical equipment. The first important factor is the process of closing (i.e. forming) of the conveyor belt into the pipe shape. This process is connected with the intensive force impacts caused due to transformation of the conveyor belt shape and due to following interaction of the conveyor belt with the forming idler rolls. The second relevant factor is the transported material, namely the force effects caused by the weight of material. In this way the transported material increases and markedly influences the already existing forces. The third factor is impact of the tension force. These three factors together are affecting operation of the pipe conveyor as well as value of the motional resistances. The motional resistance value is a decisive criterion, which is important with regard to the consumption of electric energy and the motional resistances are also critical regarding to wearing of the belt bottom cover. It is important to investigate how the above-mentioned three factors are influencing values of the contact forces and the motional resistances. Monitoring and measuring of the whole process during current operation of the conveyor is not an easy task and often it is also complicated due to asymmetry of the conveyor belt tensioning. It is very important to choose a suitable method in order to perform a scientific investigation of the given problems. Anyway, realisation of an experimental measurement is necessary in this case. However, the experimental measurement is very demanding with regard to realisation of the measuring process itself and it is almost impossible to perform it during standard operation of the pipe conveyor. Therefore, we decided to apply the mathematical models. The mathematical model cannot be utilised without the experiment. Thus, it is necessary to combine the experimental methodology with the mathematical methods within the research area in order to define such relations, which are able to describe behaviour of the whole system and interactions among the individual parameters. Taking into consideration all the circumstances it is possible to say that a methodology of the system dynamics is such scientific tool, which is suitable for the detailed research of the given problems. It was already mentioned a fact that the research concerning the contact force values requires the experimental measurements, however realisation of the experiments is almost impossible during real operation of the pipe conveyor. A possible solution offers application of an appropriate physical model, which is representing the monitored section of the pipe conveyor in order to obtain relevant data that are necessary for the scientific investigation in the given research area. The used physical model of the feed chute section for the pipe conveyor was created in the form of an original testing stand. Fig. 1 illustrates the feed chute of the pipe conveyor with the marked area, which was replaced by the physical model, whereby the physical model is represented with the experimental testing equipment according to Fig. 2. The physical simulation model (the testing equipment) enabled to induce the tension forces during the experimental measurements. The tension force values TF = 4000–28,000 N were induced in the conveyor belt by means of a couple of the tensioning screws ID23, ID24, Fig. 3. The filling rate A of the cross-sectional area of the piped belt filled with the material was simulated by means of four sandbags, Fig. 4. The asymmetry AS was created by a different ratio
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Fig. 1. Feed chute of the pipe conveyor (without the conveyor belt).
Fig. 2. Experimental testing equipment installed at the TU of Košice.
Fig. 3. Marking of the sensors on the idler housing No.2.
of the tension forces TF induced in the above-mentioned tensioning screws ID23, ID24, Fig. 3. The physical simulation model was created as an identical copy of the feed chute part of the pipe conveyor with regard to the dimensions. Rotation of the guiding rolls does not influence the contact
force values. Therefore, the idler rolls were applied only in a fixed position in order to simplify the measuring process, however without any misinterpretation or inaccuracy of the obtained results. Taking into consideration principles of the system dynamics the physical model consists of the next components: the tensioning
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Fig. 4. Cross-section of conveyor belt with the sandbags and illustration of the filling rate A.
system component, the system component of the guiding rolls, the material system component and the system component of the conveyor belt (Fig. 3). The system component of the guiding rolls contains three subcomponents, which are representing the individual idler housings, whereby there are identified 6 partial parts in the form of the fixed guiding rolls within the idler housing subcomponent. The main attention was paid to the idler housing No.2, because it corresponds to the major part of the conveyor trajectory and to the related contact force values. The greatest problem, which occurred during creation of the physical model, was simulation related to filling of the piped conveyor belt with the transported material. Aspect of the filling process simulation is very important because it is necessary to simulate the real operational conditions of the pipe conveyor as realistic as it is possible. Therefore, it was chosen an original system of the sandbags. This solution allowed to control the filling shape of the conveyor belt as well as amount and weight of the transported material. The research was performed using a sample of the rubbertextile conveyor belt. The basic characteristics of the rubbertextile conveyor belt, experimental testing equipment and transported material are summarized in Table 1. The conveyor belt is tensioned using the experimental testing equipment in a horizontal plane linearly, i.e. without curves. The given sample of the rubber-textile conveyor belt represents the most often used type of the conveyor belt, which is applied in the real operation and therefore the obtained results will be applicable in a wide range of the belt conveyor operation. The experimental method, which was applied during the performed
analysis, is divided into two steps, according to the block scheme presented in Fig. 5. In the first step, the methodology of Molnar et al. [23] has been used. The main goal of the performed research was focused on the next two hypotheses, which are resulting from the actual sum of knowledge and from a requirement to obtain new information for a better understanding to the investigated problems. The 1st Hypothesis: ‘‘There is existing a direct dependence among the contact forces and the tension forces, together with the weight of transported material.” The 2nd Hypothesis: ‘‘It is possible to create a mathematical model based on the experimental measurements, which will describe the above-mentioned dependence.”
3. Theory/calculation There were applied the mathematical-statistical methods and the method of regression model in order to verify the individual hypotheses. Certain tests of the hypotheses relating to the model parameters during solution of the regression problems are helpful for measuring of the model usefulness [24,25]. Evaluation process of the regression models is utilizing a sum of squares and decomposition of it. The total sum of the squares SST consists of the sum of the squares SSR, which includes the regression model (mathematical relation among the inputs and outputs) and the sum of squares SSE created by the random deviations:
SST ¼ SSR þ SSE
ð1Þ
Table 1 Basic characteristics of the conveyor belt, experimental testing equipment and transported material. Type of the conveyor belt
EP500/3 HP 5 + 3 D
Polyester rubber-textile conveyor belt Strength of the belt Pipe shaped belt Top cover thickness Bottom cover thickness Category: transport of materials with the recommended belt surface temperature + 150 °C Length of the conveyor belt Width of the conveyor belt Thickness of the conveyor belt Diameter of the pipe shaped conveyor belt Distance between the idler housings Diameter of the idler rolls Bulk density of transported dry sand Angle of repose for dry sand Maximal angle of incline for dry sand
EP 500 HP 5 3 D 8 000 800 20 200 1000 60 1650 15 25
N mm1 mm mm mm mm mm mm mm mm kg m3 ° °
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1. Equalizing of the contact forces CF according to the required tension forces TF Using the methodology of Molnar et al. [23]
2. Simulaon of the contact forces CF as a result of the next factors: tension force value TF, filling rate A of the cross-seconal area of the piped belt filled with the material, asymmetry AS a) Model proposal b) Calculaon of the model regressors and evaluaon of a stascal significance for the model; Minimisaon of count of regressors in the model c) Graphical interpretaon of applicaon possibilies for obtained informaon
Fig. 5. Block scheme of the analysis procedure [23].
whereas the individual members are given according to the relation (2) using the empirical values, together with the model values: n n n X X X 2 2 yi y ¼ Yi y þ ðY i yi Þ2 i¼1
i¼1
ð2Þ
i¼1
where:
r2 ¼
Pn
yi Þ2 np1
i¼1 ðY i
Extra sum of the squares method is used in order to consider if the last regressors, which are indexed (p + 1), increase significantly the sum of squares of the model. The hypotheses are:
H0 : bpþ1 ¼ 0
Yi are the regression model values, yi are the empirical values, n is dimension of the vector Yi and vector yi, p is count of regressors.
H1 : bpþ1 –0 The test statistic for this hypothesis is the FE (extra):
Pn Test for significance of the regression appropriate hypotheses are:
H0 : b1 ¼ b2 ¼ ¼ bp ¼ 0
Rejection of the hypothesis H0 implies a fact that at least one of the regressor variables contributes significantly to the model. The test procedure for the H0 is specified in order to compute the testing criterion F:
ð3Þ
and to reject the H0 if F exceeds the F crit ¼ F a;k;nk1 . Test on the individual regression coefficients, where the hypotheses are:
H1 : bj –0 The test statistic for this hypothesis is:
^ b j ^ se b j
qffiffiffiffiffiffiffiffiffiffiffi ^ ¼ r ^ 2 C jj se b j
i¼1 ðY i
2 P 2 yÞpþ1 ni¼1 ðY i yÞp ðn p 1Þ Pn 2 1 i¼1 ðY i yi Þpþ1
ð7Þ
FE ¼
SSRðpþ1Þ SSRðpÞ ðn p 1Þ 1 SSEðpþ1Þ
ð8Þ
where (p) and (p + 1) are the last added regressor and the H0 is rejected if the FE exceeds F E crit ¼ F a;1;ðnp1Þ . The validity range of the results, which are obtained using the described experimental testing equipment, is from the interval of the tension forces TF = 4000–28,000 N. 4. Results The choice of model with a suitable number of the regressors was based on the estimated polynomial dependence (of the third degree maximally) on the tension force value TF, together with a linear dependence on the filling rate A of the cross-sectional area of the piped belt filled with the material and applying a polynomial dependence on the asymmetry AS.
H0 : bj ¼ 0
t0 ¼
FE ¼
and computation of the testing criterion, using the sum of squares, is according to (8):
H1 : bj –0 for at least one j
Pn 2 ðY i yÞ ðn p 1Þ SSR ðn p 1Þ ¼ F ¼ Pni¼1 2 ðpÞ ðpÞ SSE i¼1 ðY i yi Þ
ð6Þ
ð4Þ
Y ¼ b0 þ b1 TF þ b2 TF 2 þ b3 TF 3 þ b4 A þ b5 AS þ b6 AS2 ð9Þ
ð5Þ
1 where the Cjj are the diagonal parameters from the matrix X T X and X is matrix of the input variables Xij and first columns, all the values of matrix at the position Xi1 = 1 and there is:
This model was consequently modified for each of the idler roll positions in order to take into consideration the regressors b for those effects that are statistically significant with regard to formulation of variability for the results. The SSR is an additive composition of the components in the sum of squares for the effects TF; TF 2 ; TF 3 ; A; AS; AS2 .
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B. Stehlikova et al. / Measurement 156 (2020) 107598 Table 2 Test for significance of regression.
The effect with the lowest sum of squares was gradually eliminated from the model, whereas this sum of squares was replaced by the value 0 in the calculations and the value SSE was increased by the previous value of the sum of squares. There was calculated for the modified model the significance test for regression and it was also calculated the extra sum of squares if the last regressor, which is indexed (p + 1), significantly increased the sum of squares in the model. The obtained results are in Table 2. If the test criterion value is higher than the critical value so it means that the last regressor under consideration (the sum of squares of this criterion was replaced by the zero value) represents a considerable contribution in order to explain variability of the results and therefore this regressor has to remain in the model. The sums of squares of the resulting model are highlighted in Table 2 and the zero values of the sums of squares of the effects are marked by the red colour. The yellow colour means such value of the test criterion FE, which is higher than the critical value FE crit. In our case this fact means that the last added regressor of the model (the previous line of sum of squares) is contributing significantly to the description of variability and therefore it is not suitable to remove this regressor from the model. It is sufficient to take into consideration a linear dependence of the contact force CF on the tension force TF in order to explain the contact force value CF on the idler roll position ID7. The model could not be simplified on the idler roll position ID8, i.e. all the regressors were important in this case in order to explain the contact force value CF on this idler roll position. It was possible to eliminate an irrelevant effect of the component TF3 in the case of all other idler roll positions, i.e. to simplify the polynomial dependence of the contact force CF on the tension force TF using the polynomial form of the second degree. Elimination of the linear effect concerning the asymmetry AS, together with keeping of the effect AS2 in the model on the idler roll positions ID9–ID12 was an interesting finding. This situation
means that influence of asymmetric tensioning on these positions does not depend on the kind of asymmetry, but it depends on absolute value of the asymmetry only. Contribution of the individual effects, together with the absolute member I, to the contact force CF on the individual idler roll positions was analysed according to the value and to the sign of the regressors b0–b6. The values of the regressors b0–b6 are presented in Table 2. The positive values are highlighted by the red colour and the negative values are green. An example of graphical interpretation related to the influence of the tension force TF value, the filling rate A of the cross-sectional area of the piped belt filled with the material and the asymmetry AS on the contact force CF of the idler rolls ID8–ID12 is presented in Fig. 2. The calculations were performed according to the regressors from Table 3, using the values TF = 28,000 N; A = 0–0.75; AS = 2000 N to 2000 N. The positive or negative value of the regressor b0 means that the testing equipment causes the non-zero values of the contact forces CF even without acting of the tension forces TF and without the filling rate A of the cross-sectional area of the piped belt. However, these values cannot be taken into consideration, because they are out of the result validity ranges. The positive value of the regressor b1 confirms an accordance of the results with the physical assumptions.
Table 3 Values of the regressors b0–b6.
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B. Stehlikova et al. / Measurement 156 (2020) 107598 ID8
47 46 45
CF [N] 44 43 42 2000 1000 [ AS
0
N]
0.6
-1000
TF = 28000N A = 0 ÷ 0.75; AS = -2000N ÷ 2000N
0.4
-20000.0
ID9
0.2
A [-]
ID10
130 220
125
200
CF [N]
CF [N] 180
120 2000
2000
1000
1000
[ AS
0 0.6
[ AS
N]
0
N]
-1000 -20000.0
-1000
0.6 0.4 0.2
-20000.0
A [-]
ID11
0.4 0.2
A [-]
ID12
240 220
220
200
CF [N]
180
200
CF [N]
160
180
140 2000
2000
1000
1000
0
0.2
A [-]
0.6
N]
N]
-20000.0
0.4
[ AS
[ AS
0
0.6
-1000
-1000 -20000.0
0.4 0.2
A [-]
Fig. 6. Example of graphical interpretation describing influence of the tension force TF value, the filling rate A of the cross-sectional area of the piped belt filled with the material and the asymmetry AS on the contact force CF in the idler rolls ID8–ID12, calculated for the TF = 28,000 N; A = 0–0.75; AS = 2000 N to 2000 N.
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The regressor b2 is forming the polynomial in a dependence on the tension forces TF and it also represents an influence of the material elasticity (negative value) in the case of the idler roll positions ID9–ID12. There is a special position of the idler roll ID8 due to overlapping of the side edges in the conveyor belt. The regressors b2 and b3 are forming the polynomial on the idler roll position ID8 and there is always a positive increment of the contact force CF within the validity range of the results. The regressor b4 describes an influence of the filling rate A of the cross-sectional area of the piped belt filled with the material. Taking into consideration a fact that only two values were used in the measuring process, namely the values 0 and 0.75, so it was applied the linear dependence. However, it is quite possible that a form of this dependence will be actualised and modified according to the future measured results. The sign of the regressor confirms the accordance with the physical assumptions. The value of the contact force CF is reduced in the idler roll positions ID8 and ID9 due to influence of the filling rate A of the cross-sectional area of the piped belt filled with the material and the contact force value CF is increased in the idler roll positions ID10, ID11 and ID12. The regressors b5 and b6 in the idler roll position ID8 are causing a displacement of the axis of symmetry. The regressor b5 determines an intensity of the linear dependence on the asymmetry AS and it is intended for the idler roll position ID8. Effect of this regressor reduces the contact force value CF. The regressor b6 describes a quadratic dependence on the asymmetry AS. This regressor describes behaviour of dependence, which is symmetric with the axis of asymmetry for the asymmetry value AS = 0 in the idler roll positions ID9–ID12. The regressor b6 causes increasing of the contact force CF in the idler roll position ID9 and
it reduces the contact force CF in the idler roll positions ID10, ID11 and ID12. Fig. 6 presents a dependence of the contact force CF on the filling rate A of the cross-sectional area of the piped belt filled with the material and on the asymmetry AS in the idler roll positions ID8– ID12 for one chosen tension force value TF = 28,000 N, which is close to the working-operational tension force TF. The graphs create an areal formation shaped into the form of U-profile. This shape can be defined by means of an incline, which is given by the line slope, i.e. by the regressor b4 situated near the factor filling rate A of the cross-sectional area of the piped belt filled with the material. The graphical interpretation corresponds to the physical assumptions concerning a fact that the fulfilment of the conveyor belt with the material causes reduction of the contact force CF in the upper part of the idler housing No.2 on the idler roll positions ID8 and ID9 as well as it causes increasing of the contact force CF in the bottom part of the equipment on the idler roll positions ID10, ID11 and ID12. The second attribute of the U-profile is a height of the arc shape. This attribute is resulting from the value of the regressor b6, which is situated near the factor of asymmetry AS2. The height of arc for the idler roll positions ID9–ID12 can be determined from Fig. 6. It is the value of the regressor b6 multiplied by the value 20002. The U-profile also has another attribute on the idler roll position ID8, namely this attribute is a displacement caused by the regressor b5. The axis of symmetry is situated in the value of asymmetry AS = 1000 N. 5. Discussion There is presented in Fig. 7 quantification of influence of the relevant factors, i.e. the tension force TF, the filling rate A of the
Fig. 7. Influences of the values TF, A and AS on the contact force CF in the idler rolls ID7–ID12, calculated for the TF = 28,000 N; A = 0.75; AS = 2000 N.
B. Stehlikova et al. / Measurement 156 (2020) 107598
cross-sectional area of the piped belt filled with the material and the asymmetry AS on the contact force CF. The final contact force value CF is generated by means of an additive method on each of the idler roll positions ID. A positive linear dependence of the contact force CF on the tension force TF was indicated for the idler roll position ID7. The regressor b1 has a positive value at the tension force value TF. It was not indicated a statistically significant influence of the changed filling rate A and asymmetry AS on the change of CF. The results for the idler roll position ID8, where the edges of the conveyor belt are overlapped each other, are as follows: all the considered factors are statistically significant with regard to description of the contact force CF on this idler roll position. Influence of the changed TF on the change of CF is described by a polynomial dependence, namely by a polynomial of the third degree, whereby the last member at TF3 is negative, what reduces the value of CF. The CF value, which depends on TF, is always a positive number within the validity range of the results. Influence of the filling rate A of the conveyor piped belt on the change of CF is described by means of a linear negative dependence, what means that increase of the material filling rate causes decrease of CF. Influence of the asymmetry AS on the change of CF is described by a polynomial of the second degree with the positive regressor at AS2 and with the negative regressor at AS. The final value of the additive CF can be either a positive or a negative number due to influence of AS. Dependence of CF on TF is described using a polynomial of the second degree for the idler roll positions ID9, ID10, ID11 and ID12. The value of CF, which is caused by TF, is always a positive number. Influence of the filling rate A is statistically significant on the above-mentioned idler roll positions and it is described by a linear function. Higher value of the filling rate A of the cross-sectional area of the piped belt filled with the material on the position ID9 means a decrease of the contact force CF. The contact force CF is a positive number for higher value of the filling rate A on the idler roll positions ID10, ID11 and ID12. The effect of asymmetry was verified in the case of all these idler roll positions for the member AS2, what means that influence of the asymmetrical tensioning depends on the absolute value of the asymmetry and it does not depend on the sign of asymmetry, i.e. on the direction. There was indicated, in the case of the idler roll position ID9, an increase of CF due to asymmetry and it was recorded, in the case of the positions ID10, ID11 and ID12, a decrease of CF caused by the asymmetry. 6. Conclusions The asymmetric tensioning of the rubber-textile conveyor belt creates an undesirable operational state of the pipe belt conveyor, which causes not only belt drift from the end pulleys, but it also causes changes of the contact forces, whereas this phenomenon is resulting in an excessive wear of the conveyor belt. The main reason of the above-mentioned negative occurrences is a fact that due to the asymmetric tensioning of the rubber-textile conveyor belt the mutual interactions among the guiding roll surfaces and the bottom cover of the conveyor belt are changing. These negative facts should be analysed in the case of the conveyor belt operation with the transported material and also without this material. There was applied in this article an originally modified regression model in order to obtain and to compare the differences relating to the values of the contact forces on the idler rolls. It was confirmed a validity of the stated two research hypotheses: – the 1st Hypothesis: ‘‘There is existing a direct dependence among the contact forces and the tension forces, together with the weight of transported material.”
9
– the 2nd Hypothesis: ‘‘It is possible to create a mathematical model based on the experimental measurements, which will describe the above-mentioned dependence.” The regression models confirmed a possibility to simulate dependence of the contact force CF on the tension force TF, on the filling with material A and on the asymmetry of tensioning AS in the case of the individual idler roll positions. The corresponding calculated parameters are summarized in Table 3. The resulting models and their parameters were determined using a significance test relating to the added regressor. This test eliminated, for each of the positions, such factors that did not caused a statistically significant improvement of the model as well as did not describe a statistically relevant share of variability concerning the measured values. Table 2 presents an importance of the individual considered factors TF, A and AS using a decomposition of the variability to the particular components related to these factors. The regression models were calculated for each of the positions ID and the statistical significance was evaluated for the individual regressors. The results are as follows: the dependence of the contact force CF on the tension force TF was confirmed for all the monitored idler roll positions ID7–ID12. The statistically significant dependence of CF on the factor ‘‘filling of belt with material” A and on the factor ‘‘asymmetry” AS was confirmed in the case of all other positions ID8–ID12. The dependence of CF on A and AS for the constant value of TF is presented in Fig. 6, where is TF = 28,000 N. Filling of the conveyor belt with the transported material on the positions ID8 and ID9 reduces the resulting CF, whereby the same factor on the positions ID10, ID11 and ID12 increases the value of CF. In such a situation when the resulting tension force is generated by the asymmetrical tension forces, the influence of CF on the individual ID positions is following: the CF value is increased in the case of ID8 and ID9; namely the increment on the position ID8 could be up-to 6% of the original CF value and the increment on the position ID9 is 2.3% of the original CF value. The asymmetrical tensioning of the conveyor belt causes a reduction of the resulting CF in the case of the positions ID10, ID 11 and ID12; namely this decrement is up-to 13% of the original CF value. This phenomenon seems to be applicable in order to reduce loading of ID on these positions, however nowadays there are not investigated the processes occurring inside the conveyor belt material that are caused due to the asymmetric tensioning. Results of the analysis, using the numerical values and signs of the regressors b1–b6, are helpful for improvement of the knowledge concerning influence of the factors TF, A and AS on CF for the individual idler roll positions. The obtained knowledge defines the kind and the value of the individual effects of the tension forces TF, the tensioning asymmetry AS and the filling rate of the piped belt filled with the material A on the contact forces of the idler rolls in the hexagonal idler housing.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements This work is a part of these projects VEGA 1/0332/20, VEGA 1/0600/20, VEGA 1/0045/18, VEGA 1/0403/18, KEGA 012TUKE4/2019, KEGA 013TUKE-4/2019, SK-SRB-18-0053.
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