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Determination of Dynamic Loads of Sprocket Drum Teeth and Seats by Means of a Mathematical Model of the Longwall Conveyor / Wyznaczenie Obciążeń Dynamicznych Zębów I Gniazd Bębna Łańcuchowego Za Pomocą Modelu Matematycznego Przenośnika Ścianowego

Determination of Dynamic Loads of Sprocket Drum Teeth and Seats by Means of a Mathematical Model... Arch. Min. Sci., Vol. 57 (2012), No 4, p. 1101­1119 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.2478/v10267-012-0073-7 MARIAN DOLIPSKI*, ERYK REMIORZ*, PIOTR SOBOTA* DETERMINATION OF DYNAMIC LOADS OF SPROCKET DRUM TEETH AND SEATS BY MEANS OF A MATHEMATICAL MODEL OF THE LONGWALL CONVEYOR WYZNACZENIE OBCIE DYNAMICZNYCH ZBÓW I GNIAZD BBNA LACUCHOWEGO ZA POMOC MODELU MATEMATYCZNEGO PRZENONIKA CIANOWEGO Scraper conveyors are one of the key machines forming part of mechanised longwall systems. They are currently the only means of transporting the mined rock from longwalls in hard coal mines. The hauling force caused by the drive is transmitted onto a link chain through drive wheels with their external shape corresponding to a geometric polygon. The number of teeth (seats) in such wheels ranges between 5 and 8. The horizontal links running on the drum are arranged in the drive wheel seats and are meshing with the teeth segments. The geometric relationships between the sprocket drum and the links are decisive for the position of the chain links in the seats. The abrasive wear of the chain parts and of the drive drum parts occurring due to conveyor operation is increasing the chain pitch and decreasing the wheel pitch. The position of a link in the seats changes as a result along with the load on the sprocket drum teeth and seats. Sprocket drums are the weakest element in longwall conveyors. It is, therefore, urgently necessary to determine the dynamic loads of such drums' teeth and seats. The article presents a physical model and a mathematical model of a longwall conveyor created for the purpose of determination of dynamic loads of the sprocket drum teeth and seats. The results of computer simulations are also presented (dynamic loads: in chains, dynamic loads of sprocket drums and dynamic loads of sprocket drums' teeth and seats) carried out using the created mathematical model for a 350 m long face conveyor. Keywords: scraper conveyor, sprocket drum, mathematical and physical model, dynamic loads, teeth, seats Koncentracja produkcji wgla kamiennego wymusza potrzeb prowadzenia intensywnych bada maszyn górniczych w aspekcie zwikszenia ich niezawodnoci i ywotnoci. Jedn z podstawowych maszyn wchodzcych w sklad cianowych kompleksów zmechanizowanych s przenoniki zgrzeblowe. Przenoniki zgrzeblowe cianowe s obecnie jedynymi rodkami odstawy urobku z wyrobisk cianowych w kopalniach wgla kamiennego. W czasie swojego rozwoju wyposaane byly w róne typy lacuchów pocigowych, z których najlepszym okazal si lacuch ogniwowy. Przenoniki cianowe mog by wyposaone w jeden lacuch, dwa lacuchy skrajne, trzy lacuchy lub dwa lacuchy rodkowe, przy czym ostatnie rozwizanie stosowane jest najczciej. * INSTITUTE OF MINING MECHANISATION, FACULTY OF MINING AND GEOLOGY, SILESIAN UNIVERSITY OF TECHNOLOGY, AKADEMICKA 2, 44-100 GLIWICE, POLAND Sila ucigu wywolana napdem przekazywana jest lacuchowi poprzez kola napdowe, które maj posta geometryczn wieloboku i wyposaone s najczciej w 5÷8 zbów (gniazd). Ogniwa poziome nabiegajce na bben ukladaj si w gniazdach kola napdowego i wchodz w zazbienie z segmentami zbów. O poloeniu ogniw lacucha w gniazdach decyduj relacje geometryczne pomidzy bbnem lacuchowym a ogniwami. Zuycie cierne elementów lacucha i bbna napdowego nastpujce na skutek eksploatacji przenonika powoduje zwikszenie podzialki lacucha i zmniejszenie podzialki kola. W efekcie zmienia si zarówno poloenie ogniw w gniazdach jak i obcienie zbów i gniazd bbna lacuchowego. Obecnie najslabszym elementem w przenonikach cianowych s bbny lacuchowe. Zachodzi zatem pilna potrzeba poznania obcie dynamicznych zbów i gniazd tych bbnów. Dla potrzeb wyznaczania obcie dynamicznych zbów i gniazd bbna lacuchowego zostal rozbudowany model fizyczny i matematyczny przenonika cianowego o elementy zazbienia lacuchowego. Dyskretny model fizyczny i matematyczny przenonika cianowego zbudowano wczeniej i wielokrotnie zweryfikowano go dowiadczalnie. Po rozbudowaniu o elementy zazbienia lacuchowego model fizyczny przyjmuje posta jak na rysunku 1. Ruch w tym rozbudowanym modelu fizycznym opisuje uklad nieliniowych równa róniczkowych zwyczajnych drugiego rzdu (wzory 1, 2 i 3). Podczas wspóldzialania bbna lacuchowego o wymiarach normowych z lacuchem o wydluonej podzialce nabiegajce ogniwo poziome nie styka si z dnem gniazda na calej swej dlugoci. Ten wariant zazbienia charakteryzuje si tym, e ogniwa poziome lacucha znajdujce si na kole gniazdowym o liczbie zbów z s nachylone wzgldem den gniazd pod ktem tak, e ich torusy przednie stykaj si dnami gniazd a torusy tylne stykaj si z bokami roboczymi segmentów zbów kola o kcie pochylenia wzgldem dna gniazda . W celu jednoznacznego opisu poloenia ogniw lacucha w gniazdach kola wyznaczono parametry , u i u (rys. 2). Przy analizowaniu wspóldzialania bbna lacuchowego z lacuchem ogniwowym uwzgldniono zjawisko ruchliwoci ogniw w przegubach podczas wzajemnego przechylania ogniw, którego nastpstwem jest przemieszczanie si punktu styku ogniw. Przechylaniu ogniwa poziomego wzgldem ogniwa pionowego towarzyszy toczenie si ogniwa poziomego wzgldem ogniwa pionowego w wyniku panujcego w przegubie tarcia lub polizg ogniw w przegubie w zalenoci od wartoci modulu przegubu m p i wartoci wspólczynnika tarcia w przegubie µ p. Podczas toczenia ogniwa poziomego w przegubie nastpuje przemieszczanie si punktu styku ogniw w przegubie, za podczas polizgu ogniwa poziomego poloenie punktu styku w przegubie ogniwa pionowego pozostaje bez zmian. Ze wzgldu na powtarzalno poloenia ogniw w gniazdach kola lacuchowego o liczbie zbów z podczas ich nabiegania nastpuje cykliczne obcianie kolejnych den gniazd, flanek zbów i ogniw lacucha silami w czasie obrotu bbna lacuchowego o kt podzialowy = 2/z. Podczas analizy obcienia elementów bbna lacuchowego przyjto zmienno kta obrotu bbna od chwili zetknicia si torusa przedniego nabiegajcego ogniwa poziomego z dnem gniazda ( = 0) do chwili zetknicia si torusa przedniego kolejnego ogniwa poziomego z dnem nastpnego gniazda ( = 2/z). W zakresie obrotu bbna lacuchowego o kt podzialowy wyróniono trzy przedzialy charakteryzujce si odmiennym sposobem obcienia elementów bbna lacuchowego (P1, P2 i P3 na rys. 1). Wzory od (4) do (39) opisuj obcienia dna gniazda i flanki zba bbna lacuchowego w tych przedzialach. Utworzony model matematyczny pozwolil na komputerowe wyznaczenie obcie dynamicznych lacuchów, bbnów napdowych oraz zbów i gniazd bbnów lacuchowych w przenoniku cianowym o dlugoci 350 m (rys. 3÷8). W czasie bada symulowano stan nieluzowania lacuchów i stan stalego luzowania. Na podstawie przeprowadzonych bada komputerowych ruchu ustalonego cianowego przenonika zgrzeblowego, wyposaonego w bbny lacuchowe o liczbie zbów z = 8, obcionego urobkiem wglowym na calej dlugoci sformulowano nastpujce wnioski: 1. Wydluenie podzialki lacucha, w praktyce spowodowane glównie zuyciem ciernym przegubów ogniw, powoduje osiadanie torusa tylnego ogniwa poziomego coraz wyej na flance zba (wzrost wartoci któw oraz u ). Prowadzi to do skracania czasu od chwili zetknicia si torusa przedniego ogniwa poziomego z dnem gniazda do chwili zetknicia si torusa tylnego tego ogniwa z flank zba. Konsekwencj tego jest zmniejszanie si wartoci maksymalnej obcienia dna gniazda w punkcie styku z torusem przednim ogniwa oraz wzrost maksymalnej wartoci wymaganej sily tarcia zapobiegajcej polizgowi torusa tylnego po flance zba zarówno w stanie stalego luzowania jak i w stanie nieluzowania lacucha. 2. Stosunek maksymalnej wartoci sily obciajcej flank zba w punkcie styku z torusem tylnym ogniwa do maksymalnej wartoci sily obciajcej dno gniazda w punkcie styku z torusem przednim ogniwa oraz stosunek maksymalnej wartoci wymaganej sily tarcia zapobiegajcej polizgowi torusa tylnego po flance zba do maksymalnej wartoci sily obciajcej dno gniazda w punkcie styku z torusem przednim ogniwa rosn nieliniowo ze wzrostem wydluenia podzialki ogniw. Wzrosty te przebiegaj niemal identycznie dla stanu stalego luzowania i stanu nieluzowania lacucha. 3. Zwikszenie podzialki lacucha od 1% do 4% spowodowalo ponad czterokrotny wzrost wartoci maksymalnej sily tarcia zapobiegajcej polizgowi torusa tylnego ogniwa poziomego po flance zba w stron dna gniazda. Jeeli warto sily tarcia rozwinitego wywolanego sil nacisku torusa tylnego ogniwa poziomego na flank zba jest co najmniej równa wartoci rozpatrywanej sily tarcia to uklad sil jest w równowadze. Jeli natomiast sila tarcia pochodzca od nacisku torusa tylnego na flank zba jest mniejsza od wartoci tej sily tarcia to nastpuje polizg torusa tylnego po flance zba w stron dna gniazda. Z tego wzgldu due wartoci rozwaanej sily tarcia w miejscu styku torusa tylnego ogniwa poziomego z flank zba s niekorzystne, gdy zwikszaj moliwo wystpienia polizgu ogniwa po flance zba co powoduje zwikszenie zuycia ciernego flanki zba obniajc trwalo bbna lacuchowego i powodujc zmniejszenie sprawnoci zazbienia lacuchowego. Slowa kluczowe: przenonik cianowy, bben lacuchowy, model fizyczny i matematyczny, obcienia dynamiczne zbów, obcienia dynamiczne gniazd 1. Introduction The concentration of hard coal production necessitates the intensive tests of mining machines in the context of improving their reliability and life (Dolipski, 1997; Hoseinie et al., 2011; Krauze et al., 2009; Krauze, 2004; Langosch & Ruppel, 2008). Scraper conveyors are one of the key machines forming part of mechanised longwall systems. Longwall scraper conveyors are currently the only means of transporting the mined rock from longwalls in hard coal mines. The conveyors, throughout their development, have been equipped with the different types of drive chains, with a link chain turning out to be the best. Its essential advantages include simple production technology, the links can be rotated relatively in the horizontal and vertical plane, high tensile strength, easy installation and the simple and fast linking of the broken sections using connecting links. A mining plain link chain consists of alternate horizontal (active) and vertical (passive) links. The chains are made of a steel rod with the d diameter and are shaped as flat rings with a front torus, rear torus and two drums. The hauling force caused by the drive is transmitted onto a link chain through drive wheels with their external shape corresponding to a geometric polygon. The number of teeth (seats) in such wheels ranges between 5 and 8. The horizontal links running on the drum are arranged in the drive wheel seats and are meshing with the teeth segments. The geometric relationships between the sprocket drum and the links are decisive for the position of the chain links in the seats. The abrasive wear of the chain parts and of the drive drum parts occurring due to conveyor operation is increasing the chain pitch and decreasing the wheel pitch. As a result the position of links in the seats changes along with the load on the sprocket drum teeth and seats. Investigations into sprocket drums have been concentrated to date on the meshing geometry and on the calculation of static loads (Dolipski, 1997; Strümpfel, 1989; Uhr, 1993), and the investigations aimed at determining dynamic loads in longwall conveyors were concentrated on chains and couplings (Dolipski, 1997, 2001; Kallrath & Brychta, 1986; Wölfe & Flöte, 2000; Ziegler et al., 2007). Sprocket drums are currently the weakest element in longwall conveyors. It is, therefore, urgently necessary to determine the dynamic loads of such drums' teeth and seats. 2. Discrete physical and mathematical model High-capacity longwall scraper conveyors with two central chains are used most often these days in longwall faces. The hauling force caused by the drive is transmitted onto a scraper chain by means of a sprocket drum equipped with two seat wheels. A physical and mathematical longwall conveyor model has been extended with the chain meshing elements for the purpose of dynamic loads determination of the sprocket drum teeth and seats. A discrete physical and mathematical longwall conveyor model was built earlier (Dolipski, 1997) and verified experimentally many times. The physical model, after expanding it with the chain meshing elements, assumes its form as in Fig. 1. Motion in this expanded physical model is described with the following system of ordinary nonlinear second-order differential equations: ×× × × m1k1 × q1k 1 + Z1k1 × H [ ] × é k1k B × (q1k 1 - jB × R0k B ) + h1k B × (q1k1 - jB × R0k B ) + S1k1ù + ë û × -q × - Z1k 2 × H [ ] × ék1k1 × ( q1k 2 - q1k1) + h1k1 × ( q1k 2 1k1) + S1k1ù + W1k1 = 0 ë û ............................................................................................................................... ×× × × m1k i × q1k i + Z1k i × H [ ] × ék1k (i -1) × (q1k i - q1k (i -1) ) + h1k (i -1) × ( q1k i - q1k (i -1)) + S1k iù + ê ú ë û × × - Z1k (i +1) × H [ ] × é k1ki × (q1k (i +1) - q1k i) + h1k i × (q1k (i +1) - q1k i ) + S1k iù + W1k i = 0 ê ú ë û ............................................................................................................................... × × ×× m1k j × q1k j + Z1k j × H [ ] × ék1k ( j -1) × (q1k j - q1k ( j -1)) + h1k ( j -1) × (q1k j - q1k ( j -1) ) + S1k j ù + ê ú ë û × ×R -q × - Z1k A × H [ ]× ék1k A × (jA × R0k A - q1k j ) + h1k A × (jA 0k A 1k j ) + S1k jù + W1k j = 0 ë û H H × IA × jA + Z11 A × H [ ] × éS11Aù × R11 A + Z × H [ ]× éSù × R12A + ë û ë û × -j × R - Z211 × H [ ] × ék21 A (q211 - jA × R01 A ) + h21A × (q211 ×A 01A ) + S21Aù × R21 A + ë û × - j ×R × ù - Z 221 × H [ ] × ék 22 A × (q221 - jA × R02 A) + h22 A × (q221 A 02 A) + S22 Aû × R22 A = ë × × k × j -j + h × j - j + Z ×S ×R + Z ×S ×R + 11 A 11A 11A - Z 21A × H [ ] × [S21A ] × R21 A - Z22 A × H [ ] × [S22 A ] × R22 A × × ×× × × I× j + k × (j - jA) + h× (j - jA) + kA 2 × (j - jA2 ) + hA2 × (j - jA 2) + Z11A × S11A × R11A + Z12A × S × R - Z21A × H [ ]× [S21 A ] × R21A - Z22 A × H [ ] × [S22 A] × R22A = 0 ............................................................................................................................... × × ×× IA4 × jA4 + kA4 × (jA4 - jA3) + hA4 × (jA4 - jA3) + Z11A × S11A × R11A + Z12A × S × R12A + - Z21A × H [ ] × [S21A ] × R21A - Z22 A × H [ ] × [S22A ] × R22A = MA × × ×× m2k1 × q1k1 + Z 2k1 × H [ ] × ék 2k A × (q2k1 - jA × R0k A ) + h2k A × (q2k1 - jA × R0k A ) + S 2k1ù + ë û × × - Z2k 2 × H [ ] × ék 2k1 × (q2k 2 - q2k1 ) + h2k1 × (q2k 2 - q2k1) + S 2k1ù + W2k 1 = 0 ë û ............................................................................................................................... × × ×× m2k i × q2k i + Z 2ki × H [ ] × ék 2k (i -1) × (q2k i - q2k (i -1)) + h2k (i -1) × (q2k i - q2k (i -1)) + S2k iù + ê ú ë û × × - Z2k (i +1) × H [ ] × ék 2ki × (q2k (i +1) - q2k i) + h2k i × (q2k (i +1) - q2k i) + S2k iù + W2k i = 0 ê ú ë û ............................................................................................................................... ×× × × m2k j × q2k j + Z 2k j × H [ ] × ék 2k ( j-1) × (q2k j - q2k ( j -1) ) + h2k ( j -1) × (q2k j - q2k ( j -1)) + S 2k j ù + ê ú ë û × ×R -q × ù - Z2k B × H [ ] × ék2 k B × (jB × R0kB - q2k j ) + h2k B × (jB 0k B 2k j ) + S2k jû + W2k j = 0 ë ×× H H IB × jB + Z × H [ ] × éS ù × R + Z22 B × H [ ]× éS 22B ù × R22B + ë û ë û × -j ×R - Z111 × H [ ] × ék11B × (q111 - jB × R01B ) + h11B × (q111 × B 01B ) + S11Bù × R11B + ë û × × - Z121 × H [ ] × ék12 B × (q121 - jB × R02B ) + h12 B × (q121 - jB × R02B ) + S12 Bù × R12 B = ë û × × -j + Z × S ×R +Z ×S ×R + k × (j - j ) + h × j B1 B1 B B1 B1 22B 22 B 22B - Z11B × H [ ] × [S11B ] × R11B - Z12B × H [ ] × [S12 B ] × R12B × × × × × IB1 × jB1 + kB1 × ( B1 - jB ) + hB1 × (jB1 - jB) + kB 2 × (jB1 - jB2) + hB2 × (jB1 - jB 2) + Z11B × S11B × R11B j + Z12B × S12B × R12B - Z × H [ ] × [S ] × R - Z22B × H [ ] × [S22B ] × R22B = 0 ............................................................................................................................... × × × IB4 × jB 4 + kB4 × (jB 4 - jB3) + hB 4 × (jB 4 - jB3) + Z × S × R + Z22B × S22B × R22B + - Z11B × H [ ] × [S11B ] × R11B - Z12B × H [ ] × [S12B] × R12B = MB (1) × × H S1k A = k1k A × (jA × R0k A - q1k j ) + h1k A (jA × R0k A - q1k j) + S1k A × × H S2kB = k 2k B × (jB × R0k B - q2k j ) + h2kB × (jB × R0k B - q2k j ) + S2kB (2) (3) i = 2, 3, ... , j ­ 1 = 1, 2 where: h -- a substitute damping coefficient of additional longitudinal chain dampers, h, hB1 -- a substitute damping coefficient of sprocket drum in the main drive (index A) and auxiliary drive (index B), hA2, hB2 -- a substitute damping coefficient of reducer in the main drive (index A) and auxiliary drive (index B), hA3, hB3 -- a substitute damping coefficient of coupling in the main drive (index A) and auxiliary drive (index B), hA4, hB4 -- a substitute damping coefficient of the driving motor in the main drive (index A) and auxiliary drive (index B), H [ ] -- Heaviside's function (square bracket means the content of the Heaviside's function argument), IA, IB -- moment of inertia of sprocket drum in the main drive (index A) and auxiliary drive (index B), I, IB1 -- moment of inertia of reducer reduced onto the sprocket drum shaft in the main and auxiliary drive, IA2, IB2 -- moment of inertia of the output member of the coupling reduced onto the sprocket drum shaft in the main and auxiliary drive, IA3, IB3 -- moment of inertia of the input member of the coupling reduced onto the sprocket drum shaft in the main and auxiliary drive, IA4, IB4 -- moment of inertia of the motor rotor reduced onto the sprocket drum shaft in the main and auxiliary drive, k -- specific rigidity of the chain's flexible bond, k, kB1 -- specific torsional rigidity of the drive drum reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), kA2, kB2 -- specific torsional rigidity of the reducer reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), kA3, kB3 -- specific torsional rigidity of the coupling reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), kA4, kB4 -- specific torsional rigidity of the asynchronous motor rotor reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), m -- substitute vibrating mass, MA, MB -- driving torque of the asynchronous motor in the main and auxiliary drive reduced onto the sprocket drum shaft, q -- translation coordinates, R11A, R12A -- functions of radiuses of the chain No. 1 and No. 2 running on the sprocket drum of the main drive, R, R22B -- functions of radiuses of the chain No. 1 and No. 2 running on the sprocket drum of the auxiliary drive, R21A, R22A -- functions of radiuses of the chain No. 1 and No. 2 running off the sprocket drum of the main drive, R11B, R12B -- functions of radiuses of the chain No. 1 and No. 2 running off the sprocket drum of the auxiliary drive, R01A, R02A -- pitch radiuses of sprocket wheels of the drive A, R01B, R02B -- pitch radiuses of sprocket wheels of the drive B, S -- static load in the chain (being the function of resistance of motion, residual initial chain tension and power distribution factor), W -- force of external friction in the place of arranging the substitute vibrating mass, Z -- the chain break coefficient (Z = 0 for broken chain, Z = 1 if chain breakage does not occur), -- rotation coordinates. When a sprocket drum with standard dimensions is co-working with a chain with the elongated pitch, the running-on horizontal link does not contact the seat bottom along its entire length. This meshing variant is characterised by the fact that the horizontal chain links located on the seat wheel with the number of teeth z are inclined relative to the seat bottoms under the angle so that their front tori contact the seat bottoms and the rear tori contact the working sides of the wheel teeth segments with the inclination angle relative to the seat bottom . The following parameters are determined in order to clearly describe the location of the chain links in the wheel seats (Fig. 2) (Dolipski et al., 2010): - the links' inclination angle relative to the bottoms of the wheel seats ; - the distance between the centre of the joint with the front torus of the horizontal link from the beginning of the side of a regular polygon u; - the rotation angle of the vertical link relative to the preceding horizontal link in the middle of the joint with the horizontal link rear torus u. The longer relative elongation of the pitch the greater values are achieved by the parameters describing the location of links in the sprocket wheel seats (, u and u ). The mobility effect of links in joints, when the links are tilting mutually, has been considered when analysing the co-working of the sprocket drum with the link chain, the result of which is the displacement of the contact point of the links. The tilting of the horizontal link relative to the vertical link is accompanied by the rolling of the horizontal link relative to the vertical link as a result of the friction existing in the joint or by the slip of links in the joint depending on the value of the joint module m p and the value of the friction coefficient in the joint µ p. When the horizontal link is rolling, the contact point of the links is displacing in the joint, and the position of the contact point in the vertical link joint remains unchanged during the slip of the horizontal link. Due to the repeatability of the links' location in the sprocket wheel seats with the number of teeth z, when the teeth are running on, the relevant seats bottoms, teeth flanks and chain links are loaded with forces in a cyclic manner when the sprocket drum is rotated by the pitch angle of = 2/z. It was assumed when analysing the loading of the sprocket drum elements that the drum rotation angle is changing from the moment the front torus of the running-on horizontal link is contacting the seat bottom ( = 0) until the front torus of another horizontal link is contacting the bottom of the next seat ( = 2/z). Three ranges, characterised by the different loading of the sprocket drum parts, are differentiated for sprocket drum rotation by the pitch angle. The first range (P1 in Fig. 1) lasts from the moment the front torus of the horizontal link contacts the seat bottom until the rear torus of the horizontal link contacts the tooth flank and includes drum rotation by the angle changing within the range of: 0£ j £ 2×p -au +l z (4) The horizontal link that is meshed is loaded with the run-on force SH (determined with the formula (2) and the system of equations (1) in the main drive and with the formula (3) and the system of equations (1) in the auxiliary drive), with the reactive force N between the front torus of the horizontal link and the seat bottom, whereas the slip of the front torus on the seat bottom Fig. 1. Physical model of a longwall scraper conveyor with the sprocket drums marked is accompanied by the friction force, perpendicular to the reaction, dependent upon the value of reaction and friction coefficient on the seat bottom µ g and the force S V transmitted from the horizontal link on the preceding vertical link (P1 in Fig. 1). The tilting of the horizontal link relative Fig. 2. Location of chain links in the sprocket wheel seats to the preceding vertical link is accompanied by link rolling or slip in the joint resulting in the displacement of the contact point of the links by the angle p. The mobility of links in the joint of the horizontal link and of the preceding vertical link, and the friction force at the seat bottom are forcing the tilting of the running-on vertical link from the axis of the meshing horizontal link by the deflection angle of and this is accompanied by the rolling or slip of the link in the joint causing the displacement of the contact point of the links by the angle t. The equations of balances of forces acting on the horizontal link are then assuming the following form: S × cos(n - j) - S × cos(n ) + N × m = 0 SH × sin(n - j) - SV × sin(n ) + N (5) (6) (7) S × where: d d ù g d H é × sin(g p ) + N × m × - S × ê( p + d ) × sin(l ) + × sin(g t )ú = 0 2 2 2 ë û 2 ×p +en - au z n= (8) The forces S V and N in the function of the run-on force S H can be determined from the equations recorded with the relationships (5) and (6): SH cos(n - j) - m × sin(n - j) cos(n ) - m g × sin(n ) (9) N S sin(j) cos(n ) - m g × sin(n ) (10) and the deviation angle can be determined from the equation (7) according to the mobility of links in the joint. Links are rolling for the following range of the angle of rotation: 0 ( ­ ) gr (11) j gr = (1 - m ) mp × arctan(m ) (12) gp= (1 - m p ) × (j - l ) (13) The slip of links in the joint occur for the following range of the angle of rotation: j gr £ j £ 2 ×p -a u + l z (14) p = arctan ( p) (15) Similarly, the mobility range of the deviation angle of the vertical link in the axis of the horizontal link should consider the rolling or slip of links in the joint. Links are rolling for the following range: 0 gr (16) l gr = (1 - m ) mp × arctan ( m p ) (17) gt = mp (1 - m ) ×l (18) meanwhile the slip of links in the joint occur for the following angle of rotation: > gr t = arctan (µ p ) (20) (19) The second range (P2 in Fig. 1) lasts from the moment the rear torus of the horizontal torus contacts the tooth flank until the reaction N reaches a zero value and includes drum rotation by the angle changing within the following range: 2 ×p - a u + l £ j £ j N0 z (21) A horizontal link is loaded with the run-on force S H, with the force of reaction N between the front torus of the horizontal link and the seat bottom, with the force of reaction F in the contact point of the rear torus of the horizontal link with the tooth flank and with the force SV transmitted from the horizontal link onto the preceding vertical link (P2 in fig. 1). The equations of balances of forces acting on the horizontal link are then assuming the following form: S × cos(j - n ) - S × cos(n ) - F × sin(b ) = 0 - S × sin(j - n ) - S × sin(n ) + F × cos( b ) + N = 0 d H é n t ù V d p n S × ê( p + d ) × sin(j - n + e ) + × sin( g )ú - S × × sin(g ) - F × ( p + d ) × cos(b - e ) = 0 2 2 ë û H V (22) (23) (24) The forces S V, F and N in the function of the run-on force S H can be determined from the equation notated with the relationships (22), (23) and (24): SV SH = é2 × ( p + d ) × sin(j -n + e n ) + d × sin(g t )ù × sin(b ) - 2 × ( p + d ) × cos (j - n ) × cos(b - e n ) ë û d × sin( b ) × sin(g p ) - 2 × ( p + d ) × cos (b - e n ) × cos (n ) (25) F SH cos(j -n ) sin( b ) S V cos(n ) × S H sin( b ) (26) N S = sin(j -n ) + SV S × sin(n ) - F SH × cos( b ) (27) The variability range of the vertical link rotation angle relative to the preceding horizontal link has to take into account either the rolling of the vertical link relative to the horizontal link as a result of the friction existing in the joint or the slip of links in the joint. Links' rolling occurs already before contacting the rear torus of the horizontal link within the range of the angle of rotation and lasts until the limit angle is reached gr, whereas gr (1 - m p ) × arctan( m p ) + 2 ×p - au z (28) gt = 2 ×p æ æ × çj + a uç z è (1 - m ) è m (29) The slip of links in the joint occurs for the following range of the angle of rotation: j gr £ j £ (30) t = arctan (µ p ) (31) As the value of the drum rotation angle is rising so is decreasing the values of the force in the preceding vertical link S V and of the reaction between the front torus of the link and the bottom of the seat N while the value of the reaction between the rear torus of the horizontal link and the tooth flank F is rising. The values of forces up to the rotation angle N 0 at which the value of reaction N falls to zero have been determined from the above dependencies. The third range (P3 in Fig. 1) starts from this moment lasting from the time the reaction N reaches a zero value until the front torus contacts the next horizontal link with the bottom of the next seat: j N0 £ j £ (32) The system of forces acting on the horizontal link changes for the angle > N 0 and the force T occurs on the tooth flank necessary for maintaining the balance of the horizontal link preventing the slip of the rear torus of the horizontal link towards the seat bottom at the tooth flank (P3 in Fig. 1). Equations for the balance of the horizontal link are then assuming the following form: S × cos(j - n ) - S × cos(n ) - F × sin(b ) + T × cos(b ) = 0 H V (33) (34) (35) - S × sin(j - n ) - S × sin (n ) + F × cos( b ) + T × sin(b ) = 0 d d ù V é n H d - S × ê( p + d ) × sin(n - e ) + × sin(g p )ú + S × × sin(g t ) - T × = 0 2 2 2 ë û The forces S V, F and T as the function of the run-on force S H can be determined from the equation notated with the relationships (33), (34) and (35): T S SV SH SV SH × cos( b - n ) - cos ( b -n + j) (36) T SH × cos( b - n ) - cos ( b -n + j) (37) F S cos(j - n ) sin (b ) SV S cos(n ) sin( b ) T S tan(b ) (38) If the reaction N does not reach the zero value in the second range until the moment the front torus of the next horizontal link contacts the bottom of the next seat, i.e. for: j N0 ³ (39) then the third range of loading the sprocket drum elements does not occur. 3. Computer tests A longwall scraper conveyor with the length of L = 350 m equipped with a single main drive and a single auxiliary drive was subjected to computer tests. The driving systems with asynchronous motors with the power of 400 kW each were equipped with sprocket drums with the number of teeth of z = 8 and the pitch angle of 45°. The drums were co-working with two central chains sized 34 ×126 with the joint module of m p = 0,85. The upper conveyor branch was loaded with the mined coal with the intensity of 300 kg/m. The conditions of non-slackened chains and permanently slackened chains were simulated during the tests. A non-slackened chain condition refers to the longwall conveyor's such dynamic condition where no interlink slackening exists in the chain (Dolipski, 1997). This means that the static and dynamic elastic elongation of the chain was compensated by initial tension. In the condition of constant slackening, interlink slackening occurs constantly in the run off point from the drive sprocket drum. In the case where the condition of constant chain slackening occurred in the tested longwall conveyor, interlink slackening was accumulated in the run off point from the sprocket drum in the auxiliary drive (Fig. 3). For the steady-state motion of the investigated conveyor (in the condition of constant chains` slackening), the maximum dynamic load value in the chain in the place it runs on the sprocket drum in the main drive was S11A H ,Umax = 416,2 kN. The sprocket drum in the main = 195,5 kNm (Fig. 3). drive was loaded with the maximum moment of K,Umax MA In the case where the elongation of the chain pitch was p/p = 1%, the maximum value of the force of reaction in the contact point of the rear torus of the horizontal link with the sprocket drum tooth flank in the main drive F11 A was 388,5 kN (Fig. 4). On the other hand, the maximum value of the force of reaction between the front torus of the horizontal link and the bottom of the sprocket drum seat in the main drive for the conveyor's steady-state motion N11 A the tooth flank towards the bottom seat (where N = 0) T11 A was 124,7 kN and the maximum value of the force preventing the slip of the rear torus of the horizontal link at was 17,2 kN. The value of the chains' initial tension was increased in the next stage of the computer tests (in practise this is done when the longwall conveyor is at standstill). As a result, the elastic elongations of the chain were completely compensated and the condition of non-slackened chains occurred in the conveyor. A higher value of the set initial tension of the chains, in the steady-state motion of the conveyor, increased a dynamic load in the chain in the place where it runs on the sprocket drum in the main drive to S11A was K,Umax MA H ,Umax = 532,6 kN. The maximum load of the drive drum = 195,5 kNm (Fig. 5). & AW B &W S t11A S11 tB Fig. 3. Start-up and steady-state motion of the longwall scraper conveyor in the condition of constant slackening of chains: a) rotational velocity of sprocket drums, b) dynamic loads in the chain, c) dynamic loads of sprocket drums The examples of curves for dynamic loads for the sprocket drum teeth and seats in the main drive in the steady-state motion of the investigated longwall conveyor are presented in Fig. 6. In the case where the elongation of the link chain pitch was p/p = 1%, the maximum values of the loads were: N11 A = 160,2 kN, F11 A = 493,2 kN and T11 A = 22,7 kN. E & d H S11A t Fig. 4. Dynamic loads of teeth and bottoms of sprocket drum seats in the main drive in the steady-state motion of the conveyor in the condition of constant slackening of chains for the chain pitch elongation of p/p = 1% The abrasive wear of the links joints, being the main cause of the higher pitch of the chain links, is decisive for the position of links in the sprocket drum seats, causing the rear torus of the horizontal link to be settled higher and higher on the tooth flank. This shortens the time from the moment the front torus of the horizontal link contacts the seat bottom until the rear torus contacts this link with the tooth flank and extends the time from the moment this reaction reaches a zero value, in the contact point of the front torus of the horizontal link with the seat bottom, until the front torus of the next horizontal link contacts the bottom of the next seat. An increase in pitch p/p within the examined range is affecting adversely the maximum value of the force T (it occurs when N = 0 and is perpendicular to the reaction F) existing in the steady-state motion of the conveyor. In the condition of a non-slackened chain, its value grew from 22,7 kN to 98,7 kN, and between 17,2 kN to 80,0 kN (Fig. 7) in the condition of constant slackening. It should be emphasised that the increased value of the required friction force at the tooth flank increases the probability of link slip at the tooth flank causing the faster abrasive wear of the teeth and effecting adversely a sprocket wheel's efficiency. A relationship between the maximum value of the force loading the tooth flank in the contact point of the rear link torus and the maximum value of the force loading the seat bottom in the contact point with the front torus of the link (F/N) is growing nonlinearily as the elongation of the links pitch is growing from 3,1 for pitch growth by 1% to 11,5 for pitch growth by 4%. As the pitch of the links is growing so is growing the relationship between the maximum value of the required friction force preventing the slip of the rear torus at the tooth flank and the & A W &W B d tH A S11 H tB S11 Fig. 5. Start-up and steady-state motion of the longwall scraper conveyor in the condition of non-slackening of chains: a) rotational velocity of sprocket drums, b) dynamic loads in the chain, c) dynamic loads of sprocket drums maximum value of the force loading the bottom of the seat in the contact point with the front torus of the link (T/N) from 0,1 for pitch growth by 1% to 2,4 for pitch growth by 4% (Fig. 8). Growth in the relationships of the forces is progressing almost identically for the condition of constant chain slackening and non-slackening. E & d H S11A t Fig. 6. Dynamic loads of teeth and bottoms of sprocket drum seats in the main drive in the steady-state motion of the conveyor in the condition of constant non-slackening of chains for the chain pitch elongation of p/p = 1% Fig. 7. Impact of the chain pitch elongation p/p on the maximum value of the friction force T preventing the slip of the rear torus of the horizontal link on the tooth flank towards the seat bottom in the steady-state motion of the longwall conveyor dE dE &E &E d E & E Fig. 8. Impact of the chain pitch elongation p/p on the relationship between the maximum values of the forces F/N and T/N 4. Summary An expanded physical and mathematical model of the longwall scraper conveyor allows to determine dynamic loads not only of sprocket drums but also their teeth and seats during start-up and in the steady-state motion. A characteristic feature of a scraper conveyor is the occurrence of high dynamic loads not only during start-up but also for the entire duration of steady-state motion. The following conclusions have been drawn on the basis of the computer tests of the steady-state motion of the longwall scraper conveyor equipped with sprocket drums with the number of teeth z = 8, loaded with the mined rock at its entire length: 1. The elongation of the chain pitch, in practise caused mainly by the abrasive wear of the joints links, is causing the rear torus of the horizontal link to settle higher and higher at the tooth flank (higher values of angles and u ). This shortens the time from the moment the front torus of the horizontal link contacts the bottom of the seat until the moment the rear torus of this link contacts the tooth flank. The consequence of this fact is a smaller maximum value of the load on the seat bottom in the contact point with the front torus of the link and a larger maximum value of the required friction force preventing the slip of the rear torus at the tooth flank both, in the condition of constant slackening as well as in the condition of an non-slackened chain. 2. The relationship between the maximum value of the force loading the tooth flank in the contact point with the rear torus of the link and the maximum value of the force loading the seat bottom in the contact point with the front torus of the link and the relationship between the maximum value of the required friction force preventing the slip of the rear torus at the tooth flank to the maximum value loading the seat bottom in the contact point with the front torus of the link are rising in a non-linear manner as the elongation of the links pitch is growing. Growth in the relationships is progressing almost identically for the condition of constant chain slackening and chain non-slackening. 6. If the chain pitch is increased between 1% to 4%, the maximum value of the friction force preventing the slip of the rear torus of the horizontal link at the tooth flank towards the seat bottom is growing over four times. If the value of the expanded friction force induced by the pressing force of the rear torus of the horizontal link on the tooth flank is at least equal to the value of the considered friction force, then the system of forces is balanced. If, however, the friction force coming from the pressure of the rear torus on the tooth flank is smaller than the value of this friction force, then the slip of the rear torus on the tooth flank towards the seat bottom occurs. For this reason, the high values of the considered friction force in the contact place of the rear torus of the horizontal link with the tooth flank are disadvantageous, as they increase a possibility of link slip occurring on the tooth flank causing the greater abrasive wear of the tooth flank and deteriorating the durability of the sprocket drum and compromising chain meshing efficiency. The assignment carried out under the development project No. N R09 0026 06/2009 financed by the Ministry of Science and Higher Education under decision No. 0481/R/ T02/2009/06 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archives of Mining Sciences de Gruyter

Determination of Dynamic Loads of Sprocket Drum Teeth and Seats by Means of a Mathematical Model of the Longwall Conveyor / Wyznaczenie Obciążeń Dynamicznych Zębów I Gniazd Bębna Łańcuchowego Za Pomocą Modelu Matematycznego Przenośnika Ścianowego

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de Gruyter
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Copyright © 2012 by the
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0860-7001
DOI
10.2478/v10267-012-0073-7
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Abstract

Arch. Min. Sci., Vol. 57 (2012), No 4, p. 1101­1119 Electronic version (in color) of this paper is available: http://mining.archives.pl DOI 10.2478/v10267-012-0073-7 MARIAN DOLIPSKI*, ERYK REMIORZ*, PIOTR SOBOTA* DETERMINATION OF DYNAMIC LOADS OF SPROCKET DRUM TEETH AND SEATS BY MEANS OF A MATHEMATICAL MODEL OF THE LONGWALL CONVEYOR WYZNACZENIE OBCIE DYNAMICZNYCH ZBÓW I GNIAZD BBNA LACUCHOWEGO ZA POMOC MODELU MATEMATYCZNEGO PRZENONIKA CIANOWEGO Scraper conveyors are one of the key machines forming part of mechanised longwall systems. They are currently the only means of transporting the mined rock from longwalls in hard coal mines. The hauling force caused by the drive is transmitted onto a link chain through drive wheels with their external shape corresponding to a geometric polygon. The number of teeth (seats) in such wheels ranges between 5 and 8. The horizontal links running on the drum are arranged in the drive wheel seats and are meshing with the teeth segments. The geometric relationships between the sprocket drum and the links are decisive for the position of the chain links in the seats. The abrasive wear of the chain parts and of the drive drum parts occurring due to conveyor operation is increasing the chain pitch and decreasing the wheel pitch. The position of a link in the seats changes as a result along with the load on the sprocket drum teeth and seats. Sprocket drums are the weakest element in longwall conveyors. It is, therefore, urgently necessary to determine the dynamic loads of such drums' teeth and seats. The article presents a physical model and a mathematical model of a longwall conveyor created for the purpose of determination of dynamic loads of the sprocket drum teeth and seats. The results of computer simulations are also presented (dynamic loads: in chains, dynamic loads of sprocket drums and dynamic loads of sprocket drums' teeth and seats) carried out using the created mathematical model for a 350 m long face conveyor. Keywords: scraper conveyor, sprocket drum, mathematical and physical model, dynamic loads, teeth, seats Koncentracja produkcji wgla kamiennego wymusza potrzeb prowadzenia intensywnych bada maszyn górniczych w aspekcie zwikszenia ich niezawodnoci i ywotnoci. Jedn z podstawowych maszyn wchodzcych w sklad cianowych kompleksów zmechanizowanych s przenoniki zgrzeblowe. Przenoniki zgrzeblowe cianowe s obecnie jedynymi rodkami odstawy urobku z wyrobisk cianowych w kopalniach wgla kamiennego. W czasie swojego rozwoju wyposaane byly w róne typy lacuchów pocigowych, z których najlepszym okazal si lacuch ogniwowy. Przenoniki cianowe mog by wyposaone w jeden lacuch, dwa lacuchy skrajne, trzy lacuchy lub dwa lacuchy rodkowe, przy czym ostatnie rozwizanie stosowane jest najczciej. * INSTITUTE OF MINING MECHANISATION, FACULTY OF MINING AND GEOLOGY, SILESIAN UNIVERSITY OF TECHNOLOGY, AKADEMICKA 2, 44-100 GLIWICE, POLAND Sila ucigu wywolana napdem przekazywana jest lacuchowi poprzez kola napdowe, które maj posta geometryczn wieloboku i wyposaone s najczciej w 5÷8 zbów (gniazd). Ogniwa poziome nabiegajce na bben ukladaj si w gniazdach kola napdowego i wchodz w zazbienie z segmentami zbów. O poloeniu ogniw lacucha w gniazdach decyduj relacje geometryczne pomidzy bbnem lacuchowym a ogniwami. Zuycie cierne elementów lacucha i bbna napdowego nastpujce na skutek eksploatacji przenonika powoduje zwikszenie podzialki lacucha i zmniejszenie podzialki kola. W efekcie zmienia si zarówno poloenie ogniw w gniazdach jak i obcienie zbów i gniazd bbna lacuchowego. Obecnie najslabszym elementem w przenonikach cianowych s bbny lacuchowe. Zachodzi zatem pilna potrzeba poznania obcie dynamicznych zbów i gniazd tych bbnów. Dla potrzeb wyznaczania obcie dynamicznych zbów i gniazd bbna lacuchowego zostal rozbudowany model fizyczny i matematyczny przenonika cianowego o elementy zazbienia lacuchowego. Dyskretny model fizyczny i matematyczny przenonika cianowego zbudowano wczeniej i wielokrotnie zweryfikowano go dowiadczalnie. Po rozbudowaniu o elementy zazbienia lacuchowego model fizyczny przyjmuje posta jak na rysunku 1. Ruch w tym rozbudowanym modelu fizycznym opisuje uklad nieliniowych równa róniczkowych zwyczajnych drugiego rzdu (wzory 1, 2 i 3). Podczas wspóldzialania bbna lacuchowego o wymiarach normowych z lacuchem o wydluonej podzialce nabiegajce ogniwo poziome nie styka si z dnem gniazda na calej swej dlugoci. Ten wariant zazbienia charakteryzuje si tym, e ogniwa poziome lacucha znajdujce si na kole gniazdowym o liczbie zbów z s nachylone wzgldem den gniazd pod ktem tak, e ich torusy przednie stykaj si dnami gniazd a torusy tylne stykaj si z bokami roboczymi segmentów zbów kola o kcie pochylenia wzgldem dna gniazda . W celu jednoznacznego opisu poloenia ogniw lacucha w gniazdach kola wyznaczono parametry , u i u (rys. 2). Przy analizowaniu wspóldzialania bbna lacuchowego z lacuchem ogniwowym uwzgldniono zjawisko ruchliwoci ogniw w przegubach podczas wzajemnego przechylania ogniw, którego nastpstwem jest przemieszczanie si punktu styku ogniw. Przechylaniu ogniwa poziomego wzgldem ogniwa pionowego towarzyszy toczenie si ogniwa poziomego wzgldem ogniwa pionowego w wyniku panujcego w przegubie tarcia lub polizg ogniw w przegubie w zalenoci od wartoci modulu przegubu m p i wartoci wspólczynnika tarcia w przegubie µ p. Podczas toczenia ogniwa poziomego w przegubie nastpuje przemieszczanie si punktu styku ogniw w przegubie, za podczas polizgu ogniwa poziomego poloenie punktu styku w przegubie ogniwa pionowego pozostaje bez zmian. Ze wzgldu na powtarzalno poloenia ogniw w gniazdach kola lacuchowego o liczbie zbów z podczas ich nabiegania nastpuje cykliczne obcianie kolejnych den gniazd, flanek zbów i ogniw lacucha silami w czasie obrotu bbna lacuchowego o kt podzialowy = 2/z. Podczas analizy obcienia elementów bbna lacuchowego przyjto zmienno kta obrotu bbna od chwili zetknicia si torusa przedniego nabiegajcego ogniwa poziomego z dnem gniazda ( = 0) do chwili zetknicia si torusa przedniego kolejnego ogniwa poziomego z dnem nastpnego gniazda ( = 2/z). W zakresie obrotu bbna lacuchowego o kt podzialowy wyróniono trzy przedzialy charakteryzujce si odmiennym sposobem obcienia elementów bbna lacuchowego (P1, P2 i P3 na rys. 1). Wzory od (4) do (39) opisuj obcienia dna gniazda i flanki zba bbna lacuchowego w tych przedzialach. Utworzony model matematyczny pozwolil na komputerowe wyznaczenie obcie dynamicznych lacuchów, bbnów napdowych oraz zbów i gniazd bbnów lacuchowych w przenoniku cianowym o dlugoci 350 m (rys. 3÷8). W czasie bada symulowano stan nieluzowania lacuchów i stan stalego luzowania. Na podstawie przeprowadzonych bada komputerowych ruchu ustalonego cianowego przenonika zgrzeblowego, wyposaonego w bbny lacuchowe o liczbie zbów z = 8, obcionego urobkiem wglowym na calej dlugoci sformulowano nastpujce wnioski: 1. Wydluenie podzialki lacucha, w praktyce spowodowane glównie zuyciem ciernym przegubów ogniw, powoduje osiadanie torusa tylnego ogniwa poziomego coraz wyej na flance zba (wzrost wartoci któw oraz u ). Prowadzi to do skracania czasu od chwili zetknicia si torusa przedniego ogniwa poziomego z dnem gniazda do chwili zetknicia si torusa tylnego tego ogniwa z flank zba. Konsekwencj tego jest zmniejszanie si wartoci maksymalnej obcienia dna gniazda w punkcie styku z torusem przednim ogniwa oraz wzrost maksymalnej wartoci wymaganej sily tarcia zapobiegajcej polizgowi torusa tylnego po flance zba zarówno w stanie stalego luzowania jak i w stanie nieluzowania lacucha. 2. Stosunek maksymalnej wartoci sily obciajcej flank zba w punkcie styku z torusem tylnym ogniwa do maksymalnej wartoci sily obciajcej dno gniazda w punkcie styku z torusem przednim ogniwa oraz stosunek maksymalnej wartoci wymaganej sily tarcia zapobiegajcej polizgowi torusa tylnego po flance zba do maksymalnej wartoci sily obciajcej dno gniazda w punkcie styku z torusem przednim ogniwa rosn nieliniowo ze wzrostem wydluenia podzialki ogniw. Wzrosty te przebiegaj niemal identycznie dla stanu stalego luzowania i stanu nieluzowania lacucha. 3. Zwikszenie podzialki lacucha od 1% do 4% spowodowalo ponad czterokrotny wzrost wartoci maksymalnej sily tarcia zapobiegajcej polizgowi torusa tylnego ogniwa poziomego po flance zba w stron dna gniazda. Jeeli warto sily tarcia rozwinitego wywolanego sil nacisku torusa tylnego ogniwa poziomego na flank zba jest co najmniej równa wartoci rozpatrywanej sily tarcia to uklad sil jest w równowadze. Jeli natomiast sila tarcia pochodzca od nacisku torusa tylnego na flank zba jest mniejsza od wartoci tej sily tarcia to nastpuje polizg torusa tylnego po flance zba w stron dna gniazda. Z tego wzgldu due wartoci rozwaanej sily tarcia w miejscu styku torusa tylnego ogniwa poziomego z flank zba s niekorzystne, gdy zwikszaj moliwo wystpienia polizgu ogniwa po flance zba co powoduje zwikszenie zuycia ciernego flanki zba obniajc trwalo bbna lacuchowego i powodujc zmniejszenie sprawnoci zazbienia lacuchowego. Slowa kluczowe: przenonik cianowy, bben lacuchowy, model fizyczny i matematyczny, obcienia dynamiczne zbów, obcienia dynamiczne gniazd 1. Introduction The concentration of hard coal production necessitates the intensive tests of mining machines in the context of improving their reliability and life (Dolipski, 1997; Hoseinie et al., 2011; Krauze et al., 2009; Krauze, 2004; Langosch & Ruppel, 2008). Scraper conveyors are one of the key machines forming part of mechanised longwall systems. Longwall scraper conveyors are currently the only means of transporting the mined rock from longwalls in hard coal mines. The conveyors, throughout their development, have been equipped with the different types of drive chains, with a link chain turning out to be the best. Its essential advantages include simple production technology, the links can be rotated relatively in the horizontal and vertical plane, high tensile strength, easy installation and the simple and fast linking of the broken sections using connecting links. A mining plain link chain consists of alternate horizontal (active) and vertical (passive) links. The chains are made of a steel rod with the d diameter and are shaped as flat rings with a front torus, rear torus and two drums. The hauling force caused by the drive is transmitted onto a link chain through drive wheels with their external shape corresponding to a geometric polygon. The number of teeth (seats) in such wheels ranges between 5 and 8. The horizontal links running on the drum are arranged in the drive wheel seats and are meshing with the teeth segments. The geometric relationships between the sprocket drum and the links are decisive for the position of the chain links in the seats. The abrasive wear of the chain parts and of the drive drum parts occurring due to conveyor operation is increasing the chain pitch and decreasing the wheel pitch. As a result the position of links in the seats changes along with the load on the sprocket drum teeth and seats. Investigations into sprocket drums have been concentrated to date on the meshing geometry and on the calculation of static loads (Dolipski, 1997; Strümpfel, 1989; Uhr, 1993), and the investigations aimed at determining dynamic loads in longwall conveyors were concentrated on chains and couplings (Dolipski, 1997, 2001; Kallrath & Brychta, 1986; Wölfe & Flöte, 2000; Ziegler et al., 2007). Sprocket drums are currently the weakest element in longwall conveyors. It is, therefore, urgently necessary to determine the dynamic loads of such drums' teeth and seats. 2. Discrete physical and mathematical model High-capacity longwall scraper conveyors with two central chains are used most often these days in longwall faces. The hauling force caused by the drive is transmitted onto a scraper chain by means of a sprocket drum equipped with two seat wheels. A physical and mathematical longwall conveyor model has been extended with the chain meshing elements for the purpose of dynamic loads determination of the sprocket drum teeth and seats. A discrete physical and mathematical longwall conveyor model was built earlier (Dolipski, 1997) and verified experimentally many times. The physical model, after expanding it with the chain meshing elements, assumes its form as in Fig. 1. Motion in this expanded physical model is described with the following system of ordinary nonlinear second-order differential equations: ×× × × m1k1 × q1k 1 + Z1k1 × H [ ] × é k1k B × (q1k 1 - jB × R0k B ) + h1k B × (q1k1 - jB × R0k B ) + S1k1ù + ë û × -q × - Z1k 2 × H [ ] × ék1k1 × ( q1k 2 - q1k1) + h1k1 × ( q1k 2 1k1) + S1k1ù + W1k1 = 0 ë û ............................................................................................................................... ×× × × m1k i × q1k i + Z1k i × H [ ] × ék1k (i -1) × (q1k i - q1k (i -1) ) + h1k (i -1) × ( q1k i - q1k (i -1)) + S1k iù + ê ú ë û × × - Z1k (i +1) × H [ ] × é k1ki × (q1k (i +1) - q1k i) + h1k i × (q1k (i +1) - q1k i ) + S1k iù + W1k i = 0 ê ú ë û ............................................................................................................................... × × ×× m1k j × q1k j + Z1k j × H [ ] × ék1k ( j -1) × (q1k j - q1k ( j -1)) + h1k ( j -1) × (q1k j - q1k ( j -1) ) + S1k j ù + ê ú ë û × ×R -q × - Z1k A × H [ ]× ék1k A × (jA × R0k A - q1k j ) + h1k A × (jA 0k A 1k j ) + S1k jù + W1k j = 0 ë û H H × IA × jA + Z11 A × H [ ] × éS11Aù × R11 A + Z × H [ ]× éSù × R12A + ë û ë û × -j × R - Z211 × H [ ] × ék21 A (q211 - jA × R01 A ) + h21A × (q211 ×A 01A ) + S21Aù × R21 A + ë û × - j ×R × ù - Z 221 × H [ ] × ék 22 A × (q221 - jA × R02 A) + h22 A × (q221 A 02 A) + S22 Aû × R22 A = ë × × k × j -j + h × j - j + Z ×S ×R + Z ×S ×R + 11 A 11A 11A - Z 21A × H [ ] × [S21A ] × R21 A - Z22 A × H [ ] × [S22 A ] × R22 A × × ×× × × I× j + k × (j - jA) + h× (j - jA) + kA 2 × (j - jA2 ) + hA2 × (j - jA 2) + Z11A × S11A × R11A + Z12A × S × R - Z21A × H [ ]× [S21 A ] × R21A - Z22 A × H [ ] × [S22 A] × R22A = 0 ............................................................................................................................... × × ×× IA4 × jA4 + kA4 × (jA4 - jA3) + hA4 × (jA4 - jA3) + Z11A × S11A × R11A + Z12A × S × R12A + - Z21A × H [ ] × [S21A ] × R21A - Z22 A × H [ ] × [S22A ] × R22A = MA × × ×× m2k1 × q1k1 + Z 2k1 × H [ ] × ék 2k A × (q2k1 - jA × R0k A ) + h2k A × (q2k1 - jA × R0k A ) + S 2k1ù + ë û × × - Z2k 2 × H [ ] × ék 2k1 × (q2k 2 - q2k1 ) + h2k1 × (q2k 2 - q2k1) + S 2k1ù + W2k 1 = 0 ë û ............................................................................................................................... × × ×× m2k i × q2k i + Z 2ki × H [ ] × ék 2k (i -1) × (q2k i - q2k (i -1)) + h2k (i -1) × (q2k i - q2k (i -1)) + S2k iù + ê ú ë û × × - Z2k (i +1) × H [ ] × ék 2ki × (q2k (i +1) - q2k i) + h2k i × (q2k (i +1) - q2k i) + S2k iù + W2k i = 0 ê ú ë û ............................................................................................................................... ×× × × m2k j × q2k j + Z 2k j × H [ ] × ék 2k ( j-1) × (q2k j - q2k ( j -1) ) + h2k ( j -1) × (q2k j - q2k ( j -1)) + S 2k j ù + ê ú ë û × ×R -q × ù - Z2k B × H [ ] × ék2 k B × (jB × R0kB - q2k j ) + h2k B × (jB 0k B 2k j ) + S2k jû + W2k j = 0 ë ×× H H IB × jB + Z × H [ ] × éS ù × R + Z22 B × H [ ]× éS 22B ù × R22B + ë û ë û × -j ×R - Z111 × H [ ] × ék11B × (q111 - jB × R01B ) + h11B × (q111 × B 01B ) + S11Bù × R11B + ë û × × - Z121 × H [ ] × ék12 B × (q121 - jB × R02B ) + h12 B × (q121 - jB × R02B ) + S12 Bù × R12 B = ë û × × -j + Z × S ×R +Z ×S ×R + k × (j - j ) + h × j B1 B1 B B1 B1 22B 22 B 22B - Z11B × H [ ] × [S11B ] × R11B - Z12B × H [ ] × [S12 B ] × R12B × × × × × IB1 × jB1 + kB1 × ( B1 - jB ) + hB1 × (jB1 - jB) + kB 2 × (jB1 - jB2) + hB2 × (jB1 - jB 2) + Z11B × S11B × R11B j + Z12B × S12B × R12B - Z × H [ ] × [S ] × R - Z22B × H [ ] × [S22B ] × R22B = 0 ............................................................................................................................... × × × IB4 × jB 4 + kB4 × (jB 4 - jB3) + hB 4 × (jB 4 - jB3) + Z × S × R + Z22B × S22B × R22B + - Z11B × H [ ] × [S11B ] × R11B - Z12B × H [ ] × [S12B] × R12B = MB (1) × × H S1k A = k1k A × (jA × R0k A - q1k j ) + h1k A (jA × R0k A - q1k j) + S1k A × × H S2kB = k 2k B × (jB × R0k B - q2k j ) + h2kB × (jB × R0k B - q2k j ) + S2kB (2) (3) i = 2, 3, ... , j ­ 1 = 1, 2 where: h -- a substitute damping coefficient of additional longitudinal chain dampers, h, hB1 -- a substitute damping coefficient of sprocket drum in the main drive (index A) and auxiliary drive (index B), hA2, hB2 -- a substitute damping coefficient of reducer in the main drive (index A) and auxiliary drive (index B), hA3, hB3 -- a substitute damping coefficient of coupling in the main drive (index A) and auxiliary drive (index B), hA4, hB4 -- a substitute damping coefficient of the driving motor in the main drive (index A) and auxiliary drive (index B), H [ ] -- Heaviside's function (square bracket means the content of the Heaviside's function argument), IA, IB -- moment of inertia of sprocket drum in the main drive (index A) and auxiliary drive (index B), I, IB1 -- moment of inertia of reducer reduced onto the sprocket drum shaft in the main and auxiliary drive, IA2, IB2 -- moment of inertia of the output member of the coupling reduced onto the sprocket drum shaft in the main and auxiliary drive, IA3, IB3 -- moment of inertia of the input member of the coupling reduced onto the sprocket drum shaft in the main and auxiliary drive, IA4, IB4 -- moment of inertia of the motor rotor reduced onto the sprocket drum shaft in the main and auxiliary drive, k -- specific rigidity of the chain's flexible bond, k, kB1 -- specific torsional rigidity of the drive drum reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), kA2, kB2 -- specific torsional rigidity of the reducer reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), kA3, kB3 -- specific torsional rigidity of the coupling reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), kA4, kB4 -- specific torsional rigidity of the asynchronous motor rotor reduced onto the sprocket drum shaft in the main drive (A) and auxiliary drive (B), m -- substitute vibrating mass, MA, MB -- driving torque of the asynchronous motor in the main and auxiliary drive reduced onto the sprocket drum shaft, q -- translation coordinates, R11A, R12A -- functions of radiuses of the chain No. 1 and No. 2 running on the sprocket drum of the main drive, R, R22B -- functions of radiuses of the chain No. 1 and No. 2 running on the sprocket drum of the auxiliary drive, R21A, R22A -- functions of radiuses of the chain No. 1 and No. 2 running off the sprocket drum of the main drive, R11B, R12B -- functions of radiuses of the chain No. 1 and No. 2 running off the sprocket drum of the auxiliary drive, R01A, R02A -- pitch radiuses of sprocket wheels of the drive A, R01B, R02B -- pitch radiuses of sprocket wheels of the drive B, S -- static load in the chain (being the function of resistance of motion, residual initial chain tension and power distribution factor), W -- force of external friction in the place of arranging the substitute vibrating mass, Z -- the chain break coefficient (Z = 0 for broken chain, Z = 1 if chain breakage does not occur), -- rotation coordinates. When a sprocket drum with standard dimensions is co-working with a chain with the elongated pitch, the running-on horizontal link does not contact the seat bottom along its entire length. This meshing variant is characterised by the fact that the horizontal chain links located on the seat wheel with the number of teeth z are inclined relative to the seat bottoms under the angle so that their front tori contact the seat bottoms and the rear tori contact the working sides of the wheel teeth segments with the inclination angle relative to the seat bottom . The following parameters are determined in order to clearly describe the location of the chain links in the wheel seats (Fig. 2) (Dolipski et al., 2010): - the links' inclination angle relative to the bottoms of the wheel seats ; - the distance between the centre of the joint with the front torus of the horizontal link from the beginning of the side of a regular polygon u; - the rotation angle of the vertical link relative to the preceding horizontal link in the middle of the joint with the horizontal link rear torus u. The longer relative elongation of the pitch the greater values are achieved by the parameters describing the location of links in the sprocket wheel seats (, u and u ). The mobility effect of links in joints, when the links are tilting mutually, has been considered when analysing the co-working of the sprocket drum with the link chain, the result of which is the displacement of the contact point of the links. The tilting of the horizontal link relative to the vertical link is accompanied by the rolling of the horizontal link relative to the vertical link as a result of the friction existing in the joint or by the slip of links in the joint depending on the value of the joint module m p and the value of the friction coefficient in the joint µ p. When the horizontal link is rolling, the contact point of the links is displacing in the joint, and the position of the contact point in the vertical link joint remains unchanged during the slip of the horizontal link. Due to the repeatability of the links' location in the sprocket wheel seats with the number of teeth z, when the teeth are running on, the relevant seats bottoms, teeth flanks and chain links are loaded with forces in a cyclic manner when the sprocket drum is rotated by the pitch angle of = 2/z. It was assumed when analysing the loading of the sprocket drum elements that the drum rotation angle is changing from the moment the front torus of the running-on horizontal link is contacting the seat bottom ( = 0) until the front torus of another horizontal link is contacting the bottom of the next seat ( = 2/z). Three ranges, characterised by the different loading of the sprocket drum parts, are differentiated for sprocket drum rotation by the pitch angle. The first range (P1 in Fig. 1) lasts from the moment the front torus of the horizontal link contacts the seat bottom until the rear torus of the horizontal link contacts the tooth flank and includes drum rotation by the angle changing within the range of: 0£ j £ 2×p -au +l z (4) The horizontal link that is meshed is loaded with the run-on force SH (determined with the formula (2) and the system of equations (1) in the main drive and with the formula (3) and the system of equations (1) in the auxiliary drive), with the reactive force N between the front torus of the horizontal link and the seat bottom, whereas the slip of the front torus on the seat bottom Fig. 1. Physical model of a longwall scraper conveyor with the sprocket drums marked is accompanied by the friction force, perpendicular to the reaction, dependent upon the value of reaction and friction coefficient on the seat bottom µ g and the force S V transmitted from the horizontal link on the preceding vertical link (P1 in Fig. 1). The tilting of the horizontal link relative Fig. 2. Location of chain links in the sprocket wheel seats to the preceding vertical link is accompanied by link rolling or slip in the joint resulting in the displacement of the contact point of the links by the angle p. The mobility of links in the joint of the horizontal link and of the preceding vertical link, and the friction force at the seat bottom are forcing the tilting of the running-on vertical link from the axis of the meshing horizontal link by the deflection angle of and this is accompanied by the rolling or slip of the link in the joint causing the displacement of the contact point of the links by the angle t. The equations of balances of forces acting on the horizontal link are then assuming the following form: S × cos(n - j) - S × cos(n ) + N × m = 0 SH × sin(n - j) - SV × sin(n ) + N (5) (6) (7) S × where: d d ù g d H é × sin(g p ) + N × m × - S × ê( p + d ) × sin(l ) + × sin(g t )ú = 0 2 2 2 ë û 2 ×p +en - au z n= (8) The forces S V and N in the function of the run-on force S H can be determined from the equations recorded with the relationships (5) and (6): SH cos(n - j) - m × sin(n - j) cos(n ) - m g × sin(n ) (9) N S sin(j) cos(n ) - m g × sin(n ) (10) and the deviation angle can be determined from the equation (7) according to the mobility of links in the joint. Links are rolling for the following range of the angle of rotation: 0 ( ­ ) gr (11) j gr = (1 - m ) mp × arctan(m ) (12) gp= (1 - m p ) × (j - l ) (13) The slip of links in the joint occur for the following range of the angle of rotation: j gr £ j £ 2 ×p -a u + l z (14) p = arctan ( p) (15) Similarly, the mobility range of the deviation angle of the vertical link in the axis of the horizontal link should consider the rolling or slip of links in the joint. Links are rolling for the following range: 0 gr (16) l gr = (1 - m ) mp × arctan ( m p ) (17) gt = mp (1 - m ) ×l (18) meanwhile the slip of links in the joint occur for the following angle of rotation: > gr t = arctan (µ p ) (20) (19) The second range (P2 in Fig. 1) lasts from the moment the rear torus of the horizontal torus contacts the tooth flank until the reaction N reaches a zero value and includes drum rotation by the angle changing within the following range: 2 ×p - a u + l £ j £ j N0 z (21) A horizontal link is loaded with the run-on force S H, with the force of reaction N between the front torus of the horizontal link and the seat bottom, with the force of reaction F in the contact point of the rear torus of the horizontal link with the tooth flank and with the force SV transmitted from the horizontal link onto the preceding vertical link (P2 in fig. 1). The equations of balances of forces acting on the horizontal link are then assuming the following form: S × cos(j - n ) - S × cos(n ) - F × sin(b ) = 0 - S × sin(j - n ) - S × sin(n ) + F × cos( b ) + N = 0 d H é n t ù V d p n S × ê( p + d ) × sin(j - n + e ) + × sin( g )ú - S × × sin(g ) - F × ( p + d ) × cos(b - e ) = 0 2 2 ë û H V (22) (23) (24) The forces S V, F and N in the function of the run-on force S H can be determined from the equation notated with the relationships (22), (23) and (24): SV SH = é2 × ( p + d ) × sin(j -n + e n ) + d × sin(g t )ù × sin(b ) - 2 × ( p + d ) × cos (j - n ) × cos(b - e n ) ë û d × sin( b ) × sin(g p ) - 2 × ( p + d ) × cos (b - e n ) × cos (n ) (25) F SH cos(j -n ) sin( b ) S V cos(n ) × S H sin( b ) (26) N S = sin(j -n ) + SV S × sin(n ) - F SH × cos( b ) (27) The variability range of the vertical link rotation angle relative to the preceding horizontal link has to take into account either the rolling of the vertical link relative to the horizontal link as a result of the friction existing in the joint or the slip of links in the joint. Links' rolling occurs already before contacting the rear torus of the horizontal link within the range of the angle of rotation and lasts until the limit angle is reached gr, whereas gr (1 - m p ) × arctan( m p ) + 2 ×p - au z (28) gt = 2 ×p æ æ × çj + a uç z è (1 - m ) è m (29) The slip of links in the joint occurs for the following range of the angle of rotation: j gr £ j £ (30) t = arctan (µ p ) (31) As the value of the drum rotation angle is rising so is decreasing the values of the force in the preceding vertical link S V and of the reaction between the front torus of the link and the bottom of the seat N while the value of the reaction between the rear torus of the horizontal link and the tooth flank F is rising. The values of forces up to the rotation angle N 0 at which the value of reaction N falls to zero have been determined from the above dependencies. The third range (P3 in Fig. 1) starts from this moment lasting from the time the reaction N reaches a zero value until the front torus contacts the next horizontal link with the bottom of the next seat: j N0 £ j £ (32) The system of forces acting on the horizontal link changes for the angle > N 0 and the force T occurs on the tooth flank necessary for maintaining the balance of the horizontal link preventing the slip of the rear torus of the horizontal link towards the seat bottom at the tooth flank (P3 in Fig. 1). Equations for the balance of the horizontal link are then assuming the following form: S × cos(j - n ) - S × cos(n ) - F × sin(b ) + T × cos(b ) = 0 H V (33) (34) (35) - S × sin(j - n ) - S × sin (n ) + F × cos( b ) + T × sin(b ) = 0 d d ù V é n H d - S × ê( p + d ) × sin(n - e ) + × sin(g p )ú + S × × sin(g t ) - T × = 0 2 2 2 ë û The forces S V, F and T as the function of the run-on force S H can be determined from the equation notated with the relationships (33), (34) and (35): T S SV SH SV SH × cos( b - n ) - cos ( b -n + j) (36) T SH × cos( b - n ) - cos ( b -n + j) (37) F S cos(j - n ) sin (b ) SV S cos(n ) sin( b ) T S tan(b ) (38) If the reaction N does not reach the zero value in the second range until the moment the front torus of the next horizontal link contacts the bottom of the next seat, i.e. for: j N0 ³ (39) then the third range of loading the sprocket drum elements does not occur. 3. Computer tests A longwall scraper conveyor with the length of L = 350 m equipped with a single main drive and a single auxiliary drive was subjected to computer tests. The driving systems with asynchronous motors with the power of 400 kW each were equipped with sprocket drums with the number of teeth of z = 8 and the pitch angle of 45°. The drums were co-working with two central chains sized 34 ×126 with the joint module of m p = 0,85. The upper conveyor branch was loaded with the mined coal with the intensity of 300 kg/m. The conditions of non-slackened chains and permanently slackened chains were simulated during the tests. A non-slackened chain condition refers to the longwall conveyor's such dynamic condition where no interlink slackening exists in the chain (Dolipski, 1997). This means that the static and dynamic elastic elongation of the chain was compensated by initial tension. In the condition of constant slackening, interlink slackening occurs constantly in the run off point from the drive sprocket drum. In the case where the condition of constant chain slackening occurred in the tested longwall conveyor, interlink slackening was accumulated in the run off point from the sprocket drum in the auxiliary drive (Fig. 3). For the steady-state motion of the investigated conveyor (in the condition of constant chains` slackening), the maximum dynamic load value in the chain in the place it runs on the sprocket drum in the main drive was S11A H ,Umax = 416,2 kN. The sprocket drum in the main = 195,5 kNm (Fig. 3). drive was loaded with the maximum moment of K,Umax MA In the case where the elongation of the chain pitch was p/p = 1%, the maximum value of the force of reaction in the contact point of the rear torus of the horizontal link with the sprocket drum tooth flank in the main drive F11 A was 388,5 kN (Fig. 4). On the other hand, the maximum value of the force of reaction between the front torus of the horizontal link and the bottom of the sprocket drum seat in the main drive for the conveyor's steady-state motion N11 A the tooth flank towards the bottom seat (where N = 0) T11 A was 124,7 kN and the maximum value of the force preventing the slip of the rear torus of the horizontal link at was 17,2 kN. The value of the chains' initial tension was increased in the next stage of the computer tests (in practise this is done when the longwall conveyor is at standstill). As a result, the elastic elongations of the chain were completely compensated and the condition of non-slackened chains occurred in the conveyor. A higher value of the set initial tension of the chains, in the steady-state motion of the conveyor, increased a dynamic load in the chain in the place where it runs on the sprocket drum in the main drive to S11A was K,Umax MA H ,Umax = 532,6 kN. The maximum load of the drive drum = 195,5 kNm (Fig. 5). & AW B &W S t11A S11 tB Fig. 3. Start-up and steady-state motion of the longwall scraper conveyor in the condition of constant slackening of chains: a) rotational velocity of sprocket drums, b) dynamic loads in the chain, c) dynamic loads of sprocket drums The examples of curves for dynamic loads for the sprocket drum teeth and seats in the main drive in the steady-state motion of the investigated longwall conveyor are presented in Fig. 6. In the case where the elongation of the link chain pitch was p/p = 1%, the maximum values of the loads were: N11 A = 160,2 kN, F11 A = 493,2 kN and T11 A = 22,7 kN. E & d H S11A t Fig. 4. Dynamic loads of teeth and bottoms of sprocket drum seats in the main drive in the steady-state motion of the conveyor in the condition of constant slackening of chains for the chain pitch elongation of p/p = 1% The abrasive wear of the links joints, being the main cause of the higher pitch of the chain links, is decisive for the position of links in the sprocket drum seats, causing the rear torus of the horizontal link to be settled higher and higher on the tooth flank. This shortens the time from the moment the front torus of the horizontal link contacts the seat bottom until the rear torus contacts this link with the tooth flank and extends the time from the moment this reaction reaches a zero value, in the contact point of the front torus of the horizontal link with the seat bottom, until the front torus of the next horizontal link contacts the bottom of the next seat. An increase in pitch p/p within the examined range is affecting adversely the maximum value of the force T (it occurs when N = 0 and is perpendicular to the reaction F) existing in the steady-state motion of the conveyor. In the condition of a non-slackened chain, its value grew from 22,7 kN to 98,7 kN, and between 17,2 kN to 80,0 kN (Fig. 7) in the condition of constant slackening. It should be emphasised that the increased value of the required friction force at the tooth flank increases the probability of link slip at the tooth flank causing the faster abrasive wear of the teeth and effecting adversely a sprocket wheel's efficiency. A relationship between the maximum value of the force loading the tooth flank in the contact point of the rear link torus and the maximum value of the force loading the seat bottom in the contact point with the front torus of the link (F/N) is growing nonlinearily as the elongation of the links pitch is growing from 3,1 for pitch growth by 1% to 11,5 for pitch growth by 4%. As the pitch of the links is growing so is growing the relationship between the maximum value of the required friction force preventing the slip of the rear torus at the tooth flank and the & A W &W B d tH A S11 H tB S11 Fig. 5. Start-up and steady-state motion of the longwall scraper conveyor in the condition of non-slackening of chains: a) rotational velocity of sprocket drums, b) dynamic loads in the chain, c) dynamic loads of sprocket drums maximum value of the force loading the bottom of the seat in the contact point with the front torus of the link (T/N) from 0,1 for pitch growth by 1% to 2,4 for pitch growth by 4% (Fig. 8). Growth in the relationships of the forces is progressing almost identically for the condition of constant chain slackening and non-slackening. E & d H S11A t Fig. 6. Dynamic loads of teeth and bottoms of sprocket drum seats in the main drive in the steady-state motion of the conveyor in the condition of constant non-slackening of chains for the chain pitch elongation of p/p = 1% Fig. 7. Impact of the chain pitch elongation p/p on the maximum value of the friction force T preventing the slip of the rear torus of the horizontal link on the tooth flank towards the seat bottom in the steady-state motion of the longwall conveyor dE dE &E &E d E & E Fig. 8. Impact of the chain pitch elongation p/p on the relationship between the maximum values of the forces F/N and T/N 4. Summary An expanded physical and mathematical model of the longwall scraper conveyor allows to determine dynamic loads not only of sprocket drums but also their teeth and seats during start-up and in the steady-state motion. A characteristic feature of a scraper conveyor is the occurrence of high dynamic loads not only during start-up but also for the entire duration of steady-state motion. The following conclusions have been drawn on the basis of the computer tests of the steady-state motion of the longwall scraper conveyor equipped with sprocket drums with the number of teeth z = 8, loaded with the mined rock at its entire length: 1. The elongation of the chain pitch, in practise caused mainly by the abrasive wear of the joints links, is causing the rear torus of the horizontal link to settle higher and higher at the tooth flank (higher values of angles and u ). This shortens the time from the moment the front torus of the horizontal link contacts the bottom of the seat until the moment the rear torus of this link contacts the tooth flank. The consequence of this fact is a smaller maximum value of the load on the seat bottom in the contact point with the front torus of the link and a larger maximum value of the required friction force preventing the slip of the rear torus at the tooth flank both, in the condition of constant slackening as well as in the condition of an non-slackened chain. 2. The relationship between the maximum value of the force loading the tooth flank in the contact point with the rear torus of the link and the maximum value of the force loading the seat bottom in the contact point with the front torus of the link and the relationship between the maximum value of the required friction force preventing the slip of the rear torus at the tooth flank to the maximum value loading the seat bottom in the contact point with the front torus of the link are rising in a non-linear manner as the elongation of the links pitch is growing. Growth in the relationships is progressing almost identically for the condition of constant chain slackening and chain non-slackening. 6. If the chain pitch is increased between 1% to 4%, the maximum value of the friction force preventing the slip of the rear torus of the horizontal link at the tooth flank towards the seat bottom is growing over four times. If the value of the expanded friction force induced by the pressing force of the rear torus of the horizontal link on the tooth flank is at least equal to the value of the considered friction force, then the system of forces is balanced. If, however, the friction force coming from the pressure of the rear torus on the tooth flank is smaller than the value of this friction force, then the slip of the rear torus on the tooth flank towards the seat bottom occurs. For this reason, the high values of the considered friction force in the contact place of the rear torus of the horizontal link with the tooth flank are disadvantageous, as they increase a possibility of link slip occurring on the tooth flank causing the greater abrasive wear of the tooth flank and deteriorating the durability of the sprocket drum and compromising chain meshing efficiency. The assignment carried out under the development project No. N R09 0026 06/2009 financed by the Ministry of Science and Higher Education under decision No. 0481/R/ T02/2009/06

Journal

Archives of Mining Sciencesde Gruyter

Published: Dec 1, 2012

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