Measurement 47 (2014) 794–803

Contents lists available at ScienceDirect

Measurement

journal homepage: www.elsevier .com/ locate /measurement

Mathematical models for indirect measurement of contactforces in hexagonal idler housing of pipe conveyor

0263-2241/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.measurement.2013.10.012

⇑ Corresponding author. Tel.: +421 556023147; fax: +421 556028023.E-mail address: vieroslav.molnar@tuke.sk (V. Molnár).

Vieroslav Molnár a,⇑, Gabriel Fedorko a, Beáta Stehlíková a, Peter Michalik b, Melichar Kopas c

a Faculty of Mining, Ecology, Process Control and Geotechnology, Technical University of Kosice, Park Komenskeho 14, 042 00 Kosice, Slovak Republicb Faculty of Manufacturing Technologies of Technical University in Kosice with a Seat in Presov, Bayerova 1, 080 01 Presov, Slovak Republicc Faculty of Mechanical Engineering, Technical University of Kosice, Letna 9, 042 00 Kosice, Slovak Republic

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 August 2013Received in revised form 25 September2013Accepted 4 October 2013Available online 16 October 2013

Keywords:Indirect measurementModelContact forceHexagonal idler housingPipe conveyor

This article presents creation and verification of linear regression models intended for pre-diction of the pipe conveyor belt contact forces on the idler rolls in the hexagonal idlerhousing. This problem is solved for using the method of indirect measurement, whichenables to specify other required parameters with certain desired calculation accuracyon the basis of one monitored parameter. There is defined an initial hypothesis, which con-sists in assumption that it is possible to determine the contact forces in all other guide idlerrolls according to detection of the contact force in one guide idler roll. The obtained modelsare corresponding with real operational conditions. The main advantage of these modelsconsists in a fact that they have been created according to the real experimental measure-ment. The results presented in this paper are useful for the real practice because theyenable to arrange such position of the guide idler rolls in the hexagonal idler housing,which is easy accessible not only with regard to the mounting of it, but also taking intothe consideration safety, as well as which is suitable for installation and service of the mea-suring equipment.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Belt conveyors are one of the most popular methods ofmaterial transport in many branches of industry, especiallyin mining [1]. In the case of ‘‘classic’’ belt conveyors therollers used to support the conveyor belt deteriorate asthe result of a three-body wear process, in which ore par-ticles become trapped between the belt and roller to createabrasive wear under slight stress. Fiset and Dussault [2]compared the wear resistance of a low carbon steel andpolyester matrix composites containing hard particles.For all tests, the loss of weight converted into loss of vol-ume was used to determine wear resistance. Lodewijks[3] analysed the rolling resistance of belt conveyors.Fedorko and Ivanco [4] deals with analysis of forceconditions in a belt of classic belt conveyor by FEA.

Analysis determines rolls contact forces acting on the beltat idler station. Wang et al. [5] researched the dynamicanalysis and imitation of belt conveyor with verticalcurves. The dynamic model solution method was pre-sented by fully taking the variable load of conveyor intoaccount.

With over 600 installations worldwide the pipe con-veyor is a modern mode of bulk materials transport whereroute planning, problematic materials or environmentaleffects are in a projects considerations [6]. New develop-ments in the pipe conveyor and belt are advanced toaddress the growth in the capacity, length and complexityof pipe conveyor systems. Improved belt construction canoffer better stability during horizontal curves and resis-tance to twist [7]. Obtaining of information about contactforces acting on the idler rolls in the idler housing of thepipe conveyors is an important factor for research of pipeconveyor’s belt properties. Many authors were dealingwith this topic worldwide. Intensity of the contact forces

V. Molnár et al. / Measurement 47 (2014) 794–803 795

depends on various factors (pipe rotation, guide idler rollsrotation, mass of transported material, rolling resistancepipe, lateral forces in curves, resistance to twist). Dmitrievand Efimov [8] considered the possible approach to solu-tion of some problems controlled with a rotary movementof a belt of a tubular conveyor and way of decrease of belt’sangular deflection of tubular conveyor at rotary move-ment. Due to the fact that the belt width is larger thanthe circumference of the pipe, the pipe conveyor has anoverlap. This overlap ensures that the bulk solid materialinside is protected from the ambient conditions and thatthe ambience is protected from the bulk solid material.To ensure proper performance of the conveyor belt in thecarry flight, the overlap needs to stay on top of the pipebelt. However, due to forces that act on the pipe, inparticular the lateral forces in curves, the pipe belt tendsto rotate. This rotation needs to be limited as much as pos-sible or, if it occurs, it needs to be corrected. Lodewijkset al. [9] discussed ways to correct pipe rotation and thelimitations of controlling pipe orientation. Bartelmuset al. [10] investigated the vibration measurements whichare presented in the form of water fall spectrums and spec-trums of vibration for an idler rotated at its rated rotationspeed. Dmitriev and Sergeeva [11] presented calculation ofindividual contact forces on the guide idler rolls in the idlerhousing caused by a passive loading from the transportedmaterial of belt tubular conveyors. Zamiralova and Lode-wijks [12] presented determination of the normal contactforces as concentrated load forces, exerted on each roll ofan idler set and load forces depend on the mass of thetransported material, the filling ratio of the pipe, and themass and stiffness of the belt. Marasova et al. [13] investi-gated the design of analysis model for determination of themagnitude of the contact loads which are induced by 7 theinteraction of the pair of tube-shaped conveyor belt – car-rier rolls. Obtained results were the basis of research,which is focused on the search of tube conveyor beltdynamic wear creation.

Maton [14] investigated the theoretical approach toanalyse an existing tubular pipe conveyor and hence re-view the possible advantages and disadvantages of install-ing a tubular pipe conveyor with an elliptical shaped tubeor pipe and also refers to the importance of understandinghow belt specification and construction influence theoperational characteristics of a tubular conveyor such astwisting.

Contact form forces, resulting from the bending stiff-ness, keep the belt in its closed form. If these form forcesare too small, the edges of the belt could collapse. A safetransport of bulk solids could not be ensured. If contactform forces are too high, the pipe belt aspires to open itselfbetween the hexagonal idler housings. In this case, the fric-tion among the edges of the belt in the overlap leads to ahigher rolling resistance and a reduced energy-efficientmode of the pipe conveyor. Moreover, a minimum bendingstiffness of the belt is necessary, to ensure the requiredform stability of the belt and to prevent the belt from col-lapsing or buckling in curves. From the demands for a min-imum running resistance, a good shaping behaviour and ahigh form stability, divergent requirements result in thequantity of the bending stiffness [15]. Molnár et al. [16]

presented verification of the effect of emitted tension forceof conveyor belt to the size of contact forces which are in-duced by the closed conveyor belt on the guide idlers in thehexagonal idler housing of pipe conveyor. The measureddata are presented by usual descriptive statistic character-istics and compared by nonparametric statistical methods.For comparison of position and tension force Nonparamet-ric Friedman test and Shapiro–Wilk Normality Test wereused.

The friction of rubber belt on pulley under cold condi-tions was examined by Chen et al. [17]. A test rig was con-structed to study the friction and noise behaviour of rubberbelt with interfacial ice film under low temperatures.

Many authors investigated the indirect measurementmethods. Author Tien [18] investigated a relationshipbetween the tensile strength and Brinell hardness numberof a material. Gross et al. [19] used mathematicalmodelling of indirect measurements in scatterometry.Camas-Anzueto et al. [20] exploited novel approach toindirect measurements of alternating current based onthe interrogation of an all-fibber laser. Strafford and Audy[21] suggested the assist in the optimisation of machiningprocesses and provide indirect monitoring of the machin-ability of a work material. Kim and Kim [22] used theindirect cutting force measurements for adaptive cuttingforce control of machining centre.

Based on the measured values of the contact forces aris-ing in the guide idler rolls of the pipe conveyor belt, it ispossible to find the correlation models of one selected idlerroll, which will be able to predict the pipe conveyor beltcontact forces for all other guide idler rolls. There areseveral approaches how to solve this task.

The pipe conveyor’s belt is subjected to the loading con-tinuously during its current operation due to a repeatingprocess of gradual bending of the belt into the final pipeshape and next gradual opening of it to the initial flatshape. It is necessary to ensure a correct behaviour of theseprocesses thanks to a suitably proposed design of thetransport belt, which is specified for pipe conveyor applica-tion. The second important factor, which is substantial forthe correct bending and opening of the conveyor belt, isdesign of the guide idler rolls.

The guide idler rolls are situated in the idler housingsin order to bend and to open the pipe conveyor’s beltgradually into the required shape. There are arising themoving resistances during operation of the conveyor asa result of a mutual interaction between the runningbelt and guide idler rolls. The moving resistances ofthe pipe conveyors consist of several individual compo-nents and just the interaction between the running beltand guide idler rolls is one of the most important ofthem.

The moving resistance, which is caused by the interac-tion between the running belt and guide idler rolls, influ-ences durability of the pipe conveyor’s belt, operationalsafety of the whole conveyor as well as consumption ofthe driving energy. It is evident, taking into considerationthe above-mentioned facts, that knowledge about interac-tion between the belt and guide idler rolls is very impor-tant information for designers and users of the pipeconveyors.

796 V. Molnár et al. / Measurement 47 (2014) 794–803

The up to now applied methods for determination ofthe interaction between the belt and guide idler rolls arebased on empirical knowledge and theoretical calculations.The experimental measuring performed during the pipeconveyor operation is very complicated nowadays, becausethis kind of conveyor contains a large amount of the guideidler rolls. Thus, data monitoring and recording during themeasuring process is demanding from various points ofviews.

However, this problem can be solved using the methodof indirect measuring. The indirect measuring methodenables to determine other required parameters withcertain desired calculation accuracy on the basis of onemonitored parameter. In this way it can be defined aninitial hypothesis, which consists in assumption that it ispossible to specify the contact forces in all other guide idlerrolls according to detection of the contact force in oneguide idler roll. In order to verify this hypothesis it is nec-essary to select such guide idler roll, which is the mostsuitable for application of the indirect measuring method.There is needed also determination and verification ofmathematical methods for prediction of the contact forceson other guide idler rolls in the hexagonal idler housing.

The main goal of the paper is to create assumption forindirect measuring of the contact forces in hexagonal idlerhousing of pipe conveyor. The introductory part describesthe measuring process. The measured statistical data are

Fig. 1. Test equipment with the designated idler hous

Fig. 2. Identification of the contact forces in the measurement places on the id

analysed in the next chapter and the final postulates aredetermined in the following part of this article. The result-ing postulates are based on the statistical data and they en-able to apply the indirect measurement methods.

2. Materials and methods

There were realised a series of experimental measure-ments in order to investigate the contact force correlationsbetween the reference guide idler roll and other guide idlerrolls in the hexagonal idler housing. The proposal of propermeasuring conception required performing of themeasurement for three kinds of the hexagonal idlerhousing.

The first type of idler housing is that, which causes totalzipping of the belt, i.e. formation of the belt into the pipeshape. This idler housing is loaded more than most of otheridler housings in the pipe conveyor.

The second kind of the idler housing represents such id-ler housings that are supporting and guiding the shapedpiped belt (closed belt), i.e. it represents most of the idlerhousings along the whole conveyor trajectory.

The third type of the idler housing is that, which is thelast one before opening of the belt into the open, flat shape.

The whole measurement was realised by means ofequipment specified for the static measuring test. Its basic

ings Nos. 1–3 and the tension forces ID23, ID24.

ler rolls of the test equipment in the hexagonal idler housings Nos. 1–3.

Fig. 3. Overlapping of the piped belt in the idler housing No. 1.

V. Molnár et al. / Measurement 47 (2014) 794–803 797

conception is developed from the real design of the pipeconveyor. This testing equipment is able to measure andto record the contact forces on the idler rollers of the pipeconveyor belt in different places, even during formation of

Fig. 4. Block diagram describing analysis pro

the pipe and for the selected width range of the rubber–textile conveyor belts in the static position. The conveyorbelt is tensioned by means of two tensioning screws thatare producing tension forces in the positions ID23, ID24according to Fig. 1. Identification of the contact force mea-surement places on the idler rolls for the hexagonal idlerhousings Nos. 1–3 is presented in Fig. 2. Overlapping ofthe piped conveyor belt in the idler housing No. 1 is inFig. 3. The overall dimensions of the test equipment are9.5 � 1.5 � 1.7 m.

3. Theory/calculation

The main goal of this presented research work is analy-sis of a linear dependence between the contact forces andtension forces. The methodology of this analysis is showedin the block diagram in Fig. 4. The analyses and calcula-tions were performed using the project R. The graphicalpresentation of the results uses surroundings R Projectfor Statistical Computing [23] and MS Excel.

cess of the contact and tension forces.

Fig. 6. The time behaviour of the contact and tension forces in the idler housing No. 2 and the radar scheme of the average contact force values at thepositions ID7–ID12 with the defined tension forces ID23 and ID24.

Fig. 5. The time behaviour of the contact and tension forces in the idler housing No. 1 and the radar scheme of the average contact force values at thepositions ID1–ID6 with the defined tension forces ID23 and ID24.

798 V. Molnár et al. / Measurement 47 (2014) 794–803

The block diagram describes (in direction from the leftto the right) the partial tasks of the analysis according totheir realisation sequence. The objective or the result ofthis analysis is described in the left side of the block dia-gram. The tasks are determined in the middle columnand the dominant element of the partial task is expressedin the right column after a logical connection betweenthe middle and right block.

1. The first step is selection of the measurement ser-ies. These series have to be representative. Fromthis reason the time behaviours of the individualforces will be reviewed visually and the recordsfrom measuring will be controlled. The most rele-vant is measuring in such case, when the highestnumber of set-up levels characterizes the tensionforces.

2. The second step is creation of a pair of graphs forevery idler housing. The first graph describes timebehaviour of the contact and tension forces in theindividual positions. The second graph (radargraph) represents the average values of the contactforces in the individual positions for the givenrequired values of the tension force.

3. There will be created and analysed the correlationmatrixes of the measured system according to theresults obtained from the task No. 2.

4. The regressive models of the analysed system willbe calculated in this step, together with determina-tion of criteria for selection of a suitable regressivemodel.

5. A determination possibility of the reference mea-suring position and applicability of this positionaccording to the results from the task No. 4 willbe discussed in the last step.

4. Results

4.1. The 1st group of results from the block diagram

The tensioning was realised by means of gradualincreasing of the tension forces with the constant incre-mental value 1000 N in each of the positions ID23 andID24; thus the total incremental step of the tension forceincreasing was 2000 N. The measuring process started atthe zero level of the tension force and there were applied12 steps overall, up to the final value of the tension force24,000 N.

4.2. The 2nd group of results from the block diagram

It is possible to evaluate a synchronisation according tothe graphical presentation of the forces in the individualidler housings Nos. 1–3 (Figs. 5–7) and in the ID positions,

Table 1Correlation matrix for all ID positions calculated for all used tension forces.

ID24 ID23 ID18 ID17 ID16 ID15 ID14 ID13 ID12 ID11 ID10 ID9 ID8 ID7 ID6 ID3 ID2 ID1

ID24 1.000 0.998 -0.781 0.848 0.042 0.996 0.995 0.995 0.997 0.979 0.998 0.996 0.983 0.996 0.997 0.996 0.987 0.988

ID23 0.998 1.000 -0.747 0.839 0.001 0.998 0.997 0.998 0.996 0.988 0.998 0.999 0.987 0.993 0.995 0.999 0.991 0.991

ID18 -0.781 -0.747 1.000 -0.857 -0.613 -0.758 -0.729 -0.729 -0.756 -0.658 -0.746 -0.748 -0.683 -0.808 -0.756 -0.733 -0.678 -0.687

ID17 0.848 0.839 -0.857 1.000 0.393 0.852 0.826 0.832 0.824 0.788 0.830 0.844 0.779 0.865 0.820 0.828 0.781 0.783

ID16 0.042 0.001 -0.613 0.393 1.000 0.023 -0.023 -0.021 -0.002 -0.107 -0.008 0.007 -0.094 0.095 -0.007 -0.018 -0.109 -0.099

ID15 0.996 0.998 -0.758 0.852 0.023 1.000 0.996 0.997 0.992 0.987 0.997 0.999 0.983 0.993 0.993 0.997 0.987 0.988

ID14 0.995 0.997 -0.729 0.826 -0.023 0.996 1.000 0.997 0.993 0.992 0.998 0.997 0.988 0.989 0.995 0.997 0.994 0.994

ID13 0.995 0.998 -0.729 0.832 -0.021 0.997 0.997 1.000 0.993 0.992 0.998 0.998 0.986 0.989 0.994 0.998 0.993 0.993

ID12 0.997 0.996 -0.756 0.824 -0.002 0.992 0.993 0.993 1.000 0.978 0.996 0.992 0.987 0.990 0.997 0.997 0.991 0.991

ID11 0.979 0.988 -0.658 0.788 -0.107 0.987 0.992 0.992 0.978 1.000 0.989 0.990 0.985 0.971 0.982 0.989 0.992 0.993

ID10 0.998 0.998 -0.746 0.830 -0.008 0.997 0.998 0.998 0.996 0.989 1.000 0.998 0.988 0.992 0.998 0.997 0.994 0.995

ID9 0.996 0.999 -0.748 0.844 0.007 0.999 0.997 0.998 0.992 0.990 0.998 1.000 0.985 0.993 0.993 0.998 0.990 0.990

ID8 0.983 0.987 -0.683 0.779 -0.094 0.983 0.988 0.986 0.987 0.985 0.988 0.985 1.000 0.972 0.987 0.989 0.990 0.991

ID7 0.996 0.993 -0.808 0.865 0.095 0.993 0.989 0.989 0.990 0.971 0.992 0.993 0.972 1.000 0.991 0.990 0.976 0.977

ID6 0.997 0.995 -0.756 0.820 -0.007 0.993 0.995 0.994 0.997 0.982 0.998 0.993 0.987 0.991 1.000 0.994 0.992 0.995

ID3 0.996 0.999 -0.733 0,828 -0.018 0.997 0.997 0.998 0.997 0.989 0.997 0.998 0.989 0.990 0.994 1.000 0.993 0.992

ID2 0.987 0.991 -0.678 0.781 -0.109 0.987 0.994 0.993 0.991 0.992 0.994 0.990 0.990 0.976 0.992 0.993 1.000 0.998

ID1 0.988 0.991 -0.687 0.783 -0.099 0.988 0.994 0.993 0.991 0.993 0.995 0.990 0.991 0.977 0.995 0.992 0.998 1.000

Table 2Correlation matrix for the positions ID4 and ID5 calculated for the tension forces up to the value 12,000 N.

ID24 ID23 ID18 ID17 ID16 ID15 ID14 ID13 ID12 ID11 ID10 ID9 ID8 ID7 ID6 ID5 ID4 ID3 ID2 ID1

ID5 0.999 0.995 -0.798 0.856 0.070 0.995 0.993 0.993 0.994 0.976 0.996 0.995 0.978 0.997 0.996 1.000 0.999 0.992 0.982 0.985

ID4 1.000 0.998 -0.774 0.845 0.033 0.997 0.996 0.996 0.997 0.982 0.999 0.997 0.984 0.996 0.997 0.999 1.000 0.997 0.988 0.990

Fig. 7. The time behaviour of the contact and tension forces in the idler housing No. 3 and the radar scheme of the average contact force values at thepositions ID13–ID18 with the defined tension forces ID23 and D24.

V. Molnár et al. / Measurement 47 (2014) 794–803 799

as well as to analyse symmetry in relation to the verticalline by means of the radar graphs.

There are following changes of the contact forces con-sidering the mutual synchronisation as well as synchroni-sation with the tension forces ID23 and ID24:

(a) the idler housing No. 1 – synchronous, however theposition ID4 is synchronous only up to the tensionforce value 12,000 N and the position ID5 is synchro-nous up to the tension force value 16,000 N (Fig. 5),

(b) the idler housing No. 2 – synchronous in the wholerange (Fig. 6),

(c) the idler housing No. 3 – synchronous, except thepositions ID16, ID17, ID18, (Fig. 7).

Therefore the correlation matrix will be analysed for alltension forces without the positions ID4 and ID5. Thesewill be analysed only up to the tension force level12,000 N.

The radar graphs (Figs. 5–7) are assembled in such way,so that positions of axis in the graphs are corresponding tothe real placement of the guide idler rolls in the hexagonalidler housing. The contact force values are situated on theaxis of the radar graph in a connection with the individual

contact force ID10 [N], n= 3

cont

act f

orce

ID

7 [N

]

input is contact force ID10 [N], response is ID7

re

sidu

esof

mod

el [N

]

contact force ID10 [N], n= 3

cont

act f

orce

ID

8 [N

]

input is contact force ID10 [N], response is ID8

contact force ID10 [N], n= 5

cont

act f

orce

ID

9 [N

]

input is contact force ID10 [N], response is ID9

contact force ID10 [N], n= 5

cont

act f

orce

ID

11 [N

]

input is contact force ID10 [N], response is ID11

contact force ID10 [N], n= 5

cont

act f

orce

ID

12 [N

]

0 50 100 150 2000

4080

-22

6

-20

020

-20

2

050

150

-40

4

5015

0

-20

-55

010

0

-10

10

input is contact force ID10 [N], response is ID12

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200 0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

0 50 100 150 200

re

sidu

esof

mod

el [N

]

resi

dues

of m

odel

[N]

re

sidu

esof

mod

el [N

]

resi

dues

of m

odel

[N]

Fig. 8. Illustration of models for the referential position ID10.

800 V. Molnár et al. / Measurement 47 (2014) 794–803

tension forces. Such representation enables to evaluate thesymmetry of positions in relation to the vertical line.

The evaluation of symmetry is resulting in the nextfacts:

(d) at the idler housing No. 1 are evaluated the positionsID1 with ID3 and ID4 with ID6 mutually; the asym-metry is caused due to the belt position before pipeformation,

(e) at the idler housing No. 2 is the asymmetry reducedvisually and it is evaluated between the positionsID7 with ID9 and ID10 with ID12. It is caused dueto the belt overlapping; the inner part is at the sideof the position ID9 and the exterior part is at theside of the position ID7, according to the Fig. 2,

(f) at the idler housing No. 3 are analysed the positionsID16 with ID18 and ID15 with ID13 and theasymmetry is caused due to the tensioning systemis applied for tensioning of the transport belt,probably.

4.3. The 3rd group of results from the block diagram

The correlation matrix (Table 1) confirms someassumptions resulting from the graphical evaluation ofthe contact and tension force time behaviours. The posi-tions of placement in the individual idler housings areframed in colour. The orientation and intensity of depen-dence is expressed by means of the blue-red colouredscale. The strong negative correlation is formatted usingthe dark blue colour of the cell. The strong positive corre-lation is formatted using the dark red colour of the cell.The reduced intensity is described by means of a less inten-sive grade of colour, up to the white background, whichrepresents a format of independency. The correlationmatrix, which is created for all applied tension forces, isin Table 1.

a. The idler housing No. 1 – the green frame: there is astrong positive correlation among the all positionsmutually, also with the tension forces ID23 and

V. Molnár et al. / Measurement 47 (2014) 794–803 801

ID24, as well as with other positions from the secondidler housing for the positions ID1, ID2, ID3 and ID6.The positions ID4 and ID5 have a strong positivecorrelation with the tensioning positions for the ten-sion forces up to 12,000 N and with all positionsfrom the 1st and 2nd idler housing.

b. The idler housing No. 2 – the yellow frame: it is astrong positive correlation among the contact forcesin all positions ID7–ID12, as well as with the ten-sioning positions ID23 and ID24.

c. The idler housing No. 3 – the red frame: in this caseis visible a strong positive correlation also with thetensioning positions, however only in the positionsID13, ID14 and ID15. The position ID16 is distin-guished by a high level of negative correlation coef-ficient with the other positions, except the positionID18. This fact can be caused due to changing ofthe tension forces according to the Fig. 7. The corre-lation of the position ID18 is negative in relation toother positions. The correlation with other positionsis high and positive, except the positions ID16 andID18.

0 50 100 150contact force ID12 [N], n= 3

contact force ID12 [N], n= 3

contact force ID12 [N], n= 5

010

020

0

contact force ID12 [N], n= 1

cont

act f

orce

ID

10 [N

]

contact force ID12 [N], n= 2

cont

act f

orce

ID

7 [N

]0

4080

cont

act f

orce

ID

8 [N

]-2

00

20

cont

act f

orce

ID

9 [N

]0

5015

0

cont

act f

orce

ID

11 [N

]50

150

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

Fig. 9. Illustration of models for th

One part of the correlation matrix, which is analysed fortwo positions ID4, ID5 and for the tension force values upto 12,000 N, is in Table 2.

4.4. The 4th group of results from the block diagram

Calculation of the regressive models for the individualpositions was performed for the positions in the idlerhousing No. 2. Despite of a high level of the correlationcoefficient, the linear regressive model cannot be appliedfor all pairs in order to describe relations between them.The two-dimensional points that are situated in the coordi-nate system x (one position)/y (second position) are creat-ing an accumulation of points in the graph. The calculatedmodel relation is a line passing through the graph accord-ing to Figs. 8 and 9 in the left column. Taking into consid-eration behaviour of the given system it can be supposedrelation in the polynomial form:

Y ¼ a0 þ a1x1 þ a2x2 þ . . .þ anxn ð1Þ

The polynomial degree may be from 1 to n.

input is contact force ID12 [N], response is ID7

input is contact force ID12 [N], response is ID8

input is contact force ID12 [N], response is ID9

input is contact force ID12 [N], response is ID10

-10

05

-40

4-1

00

10-2

00

20-3

00

20

input is contact force ID12 [N], response is ID11

re

sidu

esof

mod

el [N

]

resi

dues

of m

odel

[N]

re

sidu

esof

mod

el [N

]

resi

dues

of m

odel

[N]

re

sidu

esof

mod

el [N

]

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

0 50 100 150

e referential position ID12.

Table 3Linear regressive models.

Autonomousvariable x

Dependentvariable y

Model

V7 V8 Y = �1.949E + 01 + (8.121E � 02)x^1 + (7.561E � 03)x^2 + (�3.195E � 05)x^3V9 Y = �2.224E � 02 + (9.763E � 01)x^1 + (5.204E � 02)x^2 + (�7.770E � 04)x^3+(3.393E � 06)x^4V10 Y = 2.297E + 00 + (1.655E + 00)x^1 + (2.901E � 02)x^2 + (�2.386E � 04)x^3V11 Y = 3.096E + 01 + (�8.106E � 02)x^1 + (8.040E � 02)x^2+(�1.017E � 03)x^3+(4.180E � 06)x^4V12 Y = �1.517E + 01+(1.960E + 00)x^1+(�3.510E � 02)xx^2+(1.973E � 03)x^3+(�3.014E � 05)x^4+(1.402E � 07)x^5

V8 V7 Y = 5.230E + 01+(1.649E + 00)x^1+(�2.980E � 02)x^2+(8.817E � 04)x^3V9 Y = 1.056E + 02+(3.366E + 00)x^1+(�7.446E � 02)x^2+(1.330E � 03)x^3V10 Y = 1.326E + 02+(4.534E + 00)x^1+(�8.249E � 02)x^2+(1.049E � 03)x^3V11 Y = 1.292E + 02+(4.122E + 00)x^1+(�6.172E � 02)x^2+(�9.251E � 05)x^3+(3.250E � 05)x^4V12 Y = 9.946E + 01+(3.921E + 00)x^1+(�7.684E � 02)x^2+(9.729E � 04)x^3

V9 V7 Y = 5.955E � 02+(7.561E � 01)x^1+(�6.511E � 03)x^2+(4.740E � 05)x^3+(�9.613E � 08)x^4V8 Y = �1.947E + 01+(1.141E � 01)x^1+(2.296E � 04)x^2+(4.411E � 06)x^3V10 Y = 2.500E + 00+(1.744E + 00)x^1+(�2.688E � 02)x^2+(4.173E � 04)x^3+(�2.531E � 06)x^4+(5.353E � 09)x^5V11 Y = 3.145E + 01+(9.202E � 02)x^1+(2.333E � 02)x^2+(�2.723E � 04)x^3+(1.534E � 06)x^4+(�3.180E � 09)x^5V12 Y = �1.603E + 01+(2.024E + 00)x^1+(�4.685E � 02)x^2+(7.420E � 04)x^3+(�4.687E � 06)x^4+(1.033E � 08)x^5

V10 V7 Y = �1.434E + 00+(5.785E � 01)x^1+(�2.480E � 03)x^2+(8.353E � 06)x^3V8 Y = �1.996E + 01+(1.169E � 01)x^1+(�8.615E � 05)x^2+(2.551E � 06)x^3V9 Y = �1.127E + 00+(3.965E � 01)x^1+(1.589E � 02)x^2+(�1.914E � 04)x^3+(9.155E � 07)x^4+(�1.535E � 09)x^5V11 Y = 3.119E + 01+(�2.064E � 01)x^1+(2.589E � 02)x^2+(�2.685E � 04)x^3+(1.219E � 06)x^4+(�1.979E � 09)x^5V12 Y = �1.799E + 01+(1.124E + 00)x^1+(�9.720E � 03)x^2+(1.284E � 04)x^3+(�6.756E � 07)x^4+(1.221E � 09)x^5

V11 V7 Y = �3.511E + 01+(1.579E + 00)x^1+(�1.438E � 02)x^2+(7.330E � 05)x^3+(�1.292E � 07)x^4V8 Y = �2.214E + 01+(7.968E � 02)x^1+(7.873E � 04)x^2+(�3.636E � 07)x^3V9 Y = �4.140E + 01+(1.558E + 00)x^1+(�4.061E � 03)x^2+(6.459E � 06)x^3V10 Y = �5.128E + 01+(2.184E + 00)x^1+(�1.185E � 02)x^2+(6.299E � 05)x^3+(�1.310E � 07)x^4V12 Y = �5.389E + 01+(1.427E + 00)x^1+(�1.791E � 03)x^2

V12 V7 Y = 9.318E + 00+(5.371E � 01)x^1+(�2.331E � 03)x^2+(1.192E � 05)x^3V8 Y = �1.759E + 01+(1.260E � 01)x^1+(1.218E � 04)x^2+(4.010E � 06)x^3V9 Y = 8.645E + 00+(9.384E � 01)x^1+(1.689E � 02)x^2+(�3.820E � 04)x^3+(2.835E � 06)x^4+(�6.903E � 09)x^5V10 Y = 1.972E + 01+(1.148E + 00)x^1V11 Y = 4.070E + 01+(7.351E � 01)x^1+(1.538E � 03)x^2

Table 4Adjusted R-squared of the model relevancy.

y

x V7 V8 V9 V10 V11 V12

V7 0.9961 0.9984 0.9983 0.9953 0.9937V8 0.9941 0.9958 0.9967 0.9914 0.9934V9 0.9982 0.997 0.9994 0.9953 0.9961V10 0.9984 0.9972 0.9994 0.9962 0.995V11 0.9914 0.9894 0.9942 0.9947 0.9822V12 0.9916 0.9918 0.9967 0.9945 0.9807

Table 5The polynomial degree of model.

y

x V7 V8 V9 V10 V11 V12

V7 3 4 3 4 5V8 3 3 3 4 3V9 4 3 5 5 5V10 3 3 5 5 5V11 4 3 3 4 2V12 3 3 5 1 2

802 V. Molnár et al. / Measurement 47 (2014) 794–803

The following criteria were defined in order to selectthe best model:

I. In the case of each tension force it has to be reachedsuch situation that the line is passing through theinside of the point accumulation (Figs. 8 and 9, leftcolumn).

II. The graph of model residues should be arranged as aformless cloud situated around the x-axis, with thezero value (Figs. 8 and 9, right column).

III. The polynomial degree n should be as low aspossible.

IV. Statistical criterion for the Student’s test of impor-tance of the regressive model parameters: t criterionof model coefficient relevancy – all parameters arerelevant at the relevancy level a = 0.001.

The linear regressive models obtained according to thedefined criteria are presented in Table 3, where V7–V12are variables in accordance with identification of the con-tact forces in the measurement places on the idler rolls ofthe test equipment in the hexagonal idler housings No. 2.

It is possible to say, with regard to Table 4, that themodels are describing a high percentage of the measureddata for all other positions using the input data from oneposition. However, there is a disadvantage of high polyno-mial degree of the models with regard to the simulationpossibility of relations among the contact forces. It is

visible from Table 3 that in the case of higher power valuesof the autonomous variable x, the parameter of regressivemodel is a low-level value, what means that the polyno-mial is only ‘‘tuning’’ a difference from the linearity. Thepolynomial degree of model is presented in Table 5.

V. Molnár et al. / Measurement 47 (2014) 794–803 803

4.5. The 5th group of results from the block diagram

The optimal practical position for measuring of the con-tact force is the position ID10 and ID12. The figures Figs. 8and 9 are presenting models for calculation of the contactforces at other positions for the idler housing No. 2. (This isvalid for the models with inputs obtained from the posi-tions ID10 and ID12.)

5. Conclusions

The simulation possibilities of the linear dependencebetween the contact and tension forces in the hexagonalidler housing of the pipe conveyor that are presented inthis article, will be very useful for the practical applica-tions. In order to perform a measuring in real conditionsit is necessary to choose such position of the guide idlerroll only, which is easy accessible with regard to theassembling or mounting, as well as which is safe andsuitable for installation and service of the applied mea-suring equipment. The obtained results can be appliedin order to increase the pipe conveyor operational reli-ability taking into consideration prediction of failureoccurrence and transport belt damage. Determination ofthe corresponding monitored values with the requiredaccuracy in the other measuring points will be possibleby monitoring of only one measuring point. Thus, themeasuring process is simplified markedly and it can beapplied not only for the new transport systems, but alsofor already installed transport equipment. The on-linemeasuring performed during the current operation ofthe pipe conveyor enables to predict a potential accidentsituation. The user can intervene on the right time in thisway and to minimise other factors, for example financialcosts, idle time, damage of the transport belt and others.It is possible to say, according to Figs. 8 and 9 that at thepositions ID7, ID8 and ID9 cannot be applied a simplerkind of relations for the whole range of the tension forceswith regard to the positions ID10 and ID12. However, itcould be possible for the positions ID10, ID11 and ID12mutually and this fact can be helpful for proposal andcreation of the indirect measuring system of the contactforces. It could be also suitable to perform a verificationof application possibility of linear relation among thecontact forces in the range of the tension forces. Thereare arising also another questions concerning influenceof the belt overlapping orientation (left/right), as wellas influence of the belt relaxation and asymmetry of thetension forces.

The obtained results are widening the online monitor-ing possibilities of the pipe conveyor operation. It will bepossible to exploit these results in practice by the usersand designers of the pipe conveyors. The results usefulfor producers of the conveyor belts will be also presented.So, the producers have a possibility to apply the obtainedconclusions in a development process of the new transportbelt constructions in order to optimise the conveyor beltparameters taking into consideration the given realconditions.

Acknowledgments

This work is a part of research project VEGA 1/0922/12Research of effect of material characteristics and techno-logical parameters of conveyor belts on size of contactforces and resistance to motion in pipe conveyors withexperimental and simulation methods.

References

[1] R. Blazej, L. Jurdziak, R. Zimroz, Novel approaches for processing ofmulti-channels NDT signals for damage detection in conveyor beltswith steel cords, Key Engineering Materials 569–570 (2013) 978–985.

[2] M. Fiset, D. Dussault, Laboratory simulation of the wear process ofbelt conveyor rollers, Wear 162–164 (1993) 1012–1015.

[3] G. Lodewijks, The rolling resistance of belt conveyors, bulk solidshandling, Trans Tech Publications 15 (1995) 15–22.

[4] G. Fedorko, V. Ivanco, Analysis of force ratios in conveyor belt ofclassic belt conveyor, Procedia Engineering 48 (2012) 123–128.

[5] J. Wang, M. Guoying, X. Cheng, Dynamic analysis and imitation ofbelt conveyor with vertical curves, in: Mechanic Automation andControl Engineering, Wuhan, 2010, pp. 6013–6016.

[6] P. Holt, Enclosed bulk transport – tubular/pipe belt conveyor, BulkSolids Handling 21 (2001) 347–349.

[7] Y. Zhang, R. Steven, Pipe conveyor and belt: belt construction, lowrolling resistance and dynamic analysis, in: 2012 SME AnnualMeeting and Exhibit, 2012, pp. 616–619.

[8] V.G. Dmitriev, M.S. Efimov, Estimation and way of decrease of beltsangular deflection of tubular conveyor at rotary movement, MiningInformational and Analitical Bulletin (2009) 31–53.

[9] G. Lodewijks, K.F. Drenth, P.S. van der Mel, The Rotation of PipeConveyors, in: Powder & Bulk Solids India 2010, Mumbai, India,2010, pp. 1–13.

[10] W. Bartelmus, L. Gladysiewicz, W. Sawicki, Test rig for qualitycontrol of conveyor belt idlers, in: H.X. T.S. Golosinski (Eds.), MiningScience and Technology ’99, 1999, pp. 671–676.

[11] V.G. Dmitriev, N.V. Sergeeva, Tractive calculation of belt tubularconveyors, Mining Informational and Analitical Bulletin (2009) 144–169.

[12] M.E. Zamiralova, G. Lodewijks, Energy consumption of pipe beltconveyors: indentation rolling resistance, Transactions_FME 40(2012) 171–176.

[13] D. Marasova, G. Fedorko, K. Kubin, V. Molnar, N. Husakova, Analysismodel for determination of contact loads between tube-shapedconveyor belt of a pipe conveyor and carrier rolls, Cuprum:Czasopismo Naukowo-techniczne Górnictwa Rud 3 (2010) 71–76.

[14] A.E. Maton, Tubular pipe conveyor design: a review of cross sectionand belt selection, Bulk Solids Handling 21 (2001) 179–182.

[15] S. Hotte, L. Overmeyer, T. Wennekamp, Research on the Form ForceBehaviour of a Pipe Conveyor in Different Curve Radii Research onthe Form Force Behaviour of Pipe Conveyors in Different, Bulk SolidsHandling, 2011.

[16] V. Molnár, G. Fedorko, B. Stehlíková, L. Kudelás, N. Husáková,Statistical approach for evaluation of pipe conveyor’s belt contactforces on guide idlers, Measurement 46 (2013) 3127–3135.

[17] G. (Sheng) Chen, J.H. Lee, V. Narravula, T. Kitchin, Friction and noise ofrubber belt in low temperature condition: the influence of interfacialice film, Cold Regions Science and Technology 71 (2012) 95–101.

[18] T.-L. Tien, The indirect measurement of tensile strength for a highertemperature by the new model IGDMC(1, n), Measurement 41(2008) 662–675.

[19] H. Gross, R. Model, M. Bär, M. Wurm, B. Bodermann, A. Rathsfeld,Mathematical modelling of indirect measurements in scatterometry,Measurement 39 (2006) 782–794.

[20] J.L. Camas-Anzueto, J.B. Roblero-Galdámez, R. Vázquez-Sánchez, C.García-Lara, G. Anzueto-Sánchez, M. Pérez-Patricio, et al., Novelapproach to indirect measurements of alternating current based onthe interrogation of an all-fiber laser, Measurement 46 (2013) 4108–4113.

[21] K.N. Strafford, J. Audy, Indirect monitoring of machinability incarbon steels by measurement of cutting forces, Journal of MaterialsProcessing Technology 67 (1997) 150–156.

[22] T.-Y. Kim, J. Kim, Adaptive cutting force control for a machiningcenter by using indirect cutting force measurements, InternationalJournal of Machine Tools and Manufacture 36 (1996) 925–937.

[23] The R Project for Statistical Computing, (n.d.).