Arch. Min. ci., ol. 57 (2012), No 4,. 1101 1119 Electronic version (in color) of this aer is available: htt://mining.archives.l DOI 10.2478/v10267-012-0073-7 MARIAN DOLIPKI*, ERYK REMIORZ*, PIOTR OBOTA* DETERMINATION OF DYNAMIC LOAD OF PROCKET DRUM TEET AND EAT BY MEAN OF A MATEMATICAL MODEL OF TE LONGWALL CONEYOR WYZNACZENIE OBCIĄŻEŃ DYNAMICZNYC ZĘBÓW I GNIAZD BĘBNA ŁAŃCUCOWEGO ZA POMOCĄ MODELU MATEMATYCZNEGO PRZENOŚNIKA ŚCIANOWEGO craer conveyors are one of the key machines forming art of mechanised longwall systems. They are currently the only means of transorting 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 shae corresonding to a geometric olygon. 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 relationshis between the srocket drum and the links are decisive for the osition of the chain links in the seats. The abrasive wear of the chain arts and of the drive drum arts occurring due to conveyor oeration is increasing the chain itch and decreasing the wheel itch. The osition of a link in the seats changes as a result along with the load on the srocket drum teeth and seats. rocket 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 resents a hysical model and a mathematical model of a longwall conveyor created for the urose of determination of dynamic loads of the srocket drum teeth and seats. The results of comuter simulations are also resented (dynamic loads: in chains, dynamic loads of srocket drums and dynamic loads of srocket drums teeth and seats) carried out using the created mathematical model for a 350 m long face conveyor. Keywords: scraer conveyor, srocket drum, mathematical and hysical model, dynamic loads, teeth, seats Koncentracja rodukcji węgla kamiennego wymusza otrzebę rowadzenia intensywnych badań maszyn górniczych w asekcie zwiększenia ich niezawodności i żywotności. Jedną z odstawowych maszyn wchodzących w skład ścianowych komleksów zmechanizowanych są rzenośniki zgrzebłowe. Przenośniki zgrzebłowe ścianowe są obecnie jedynymi środkami odstawy urobku z wyrobisk ścianowych w koalniach węgla kamiennego. W czasie swojego rozwoju wyosażane były w różne tyy łańcuchów ociągowych, z których najleszym okazał się łańcuch ogniwowy. Przenośniki ścianowe mogą być wyosażone w jeden łańcuch, dwa łańcuchy skrajne, trzy łańcuchy lub dwa łańcuchy środkowe, rzy czym ostatnie rozwiązanie stosowane jest najczęściej. * INTITUTE OF MINING MECANIATION, FACULTY OF MINING AND GEOLOGY, ILEIAN UNIERITY OF TEC- NOLOGY, AKADEMICKA 2, 44-100 GLIWICE, POLAND
1102 iła uciągu wywołana naędem rzekazywana jest łańcuchowi orzez koła naędowe, które mają ostać geometryczną wieloboku i wyosażone są najczęściej w 5 8 zębów (gniazd). Ogniwa oziome nabiegające na bęben układają się w gniazdach koła naędowego i wchodzą w zazębienie z segmentami zębów. O ołożeniu ogniw łańcucha w gniazdach decydują relacje geometryczne omiędzy bębnem łańcuchowym a ogniwami. Zużycie ścierne elementów łańcucha i bębna naędowego nastęujące na skutek eksloatacji rzenośnika owoduje zwiększenie odziałki łańcucha i zmniejszenie odziałki koła. W efekcie zmienia się zarówno ołożenie ogniw w gniazdach jak i obciążenie zębów i gniazd bębna łańcuchowego. Obecnie najsłabszym elementem w rzenośnikach ścianowych są bębny łańcuchowe. Zachodzi zatem ilna otrzeba oznania obciążeń dynamicznych zębów i gniazd tych bębnów. Dla otrzeb wyznaczania obciążeń dynamicznych zębów i gniazd bębna łańcuchowego został rozbudowany model fizyczny i matematyczny rzenośnika ścianowego o elementy zazębienia łańcuchowego. Dyskretny model fizyczny i matematyczny rzenośnika ścianowego zbudowano wcześniej i wielokrotnie zweryfikowano go doświadczalnie. Po rozbudowaniu o elementy zazębienia łańcuchowego model fizyczny rzyjmuje ostać jak na rysunku 1. Ruch w tym rozbudowanym modelu fizycznym oisuje układ nieliniowych równań różniczkowych zwyczajnych drugiego rzędu (wzory 1, 2 i 3). Podczas wsółdziałania bębna łańcuchowego o wymiarach normowych z łańcuchem o wydłużonej odziałce nabiegające ogniwo oziome nie styka się z dnem gniazda na całej swej długości. Ten wariant zazębienia charakteryzuje się tym, że ogniwa oziome łańcucha znajdujące się na kole gniazdowym o liczbie zębów z są nachylone względem den gniazd od kątem ε tak, że ich torusy rzednie stykają się dnami gniazd a torusy tylne stykają się z bokami roboczymi segmentów zębów koła o kącie ochylenia względem dna gniazda β. W celu jednoznacznego oisu ołożenia ogniw łańcucha w gniazdach koła wyznaczono arametry ε, u i α u (rys. 2). Przy analizowaniu wsółdziałania bębna łańcuchowego z łańcuchem ogniwowym uwzględniono zjawisko ruchliwości ogniw w rzegubach odczas wzajemnego rzechylania ogniw, którego nastęstwem jest rzemieszczanie się unktu styku ogniw. Przechylaniu ogniwa oziomego względem ogniwa ionowego towarzyszy toczenie się ogniwa oziomego względem ogniwa ionowego w wyniku anującego w rzegubie tarcia lub oślizg ogniw w rzegubie w zależności od wartości modułu rzegubu m i wartości wsółczynnika tarcia w rzegubie µ. Podczas toczenia ogniwa oziomego w rzegubie nastęuje rzemieszczanie się unktu styku ogniw w rzegubie, zaś odczas oślizgu ogniwa oziomego ołożenie unktu styku w rzegubie ogniwa ionowego ozostaje bez zmian. Ze względu na owtarzalność ołożenia ogniw w gniazdach koła łańcuchowego o liczbie zębów z odczas ich nabiegania nastęuje cykliczne obciążanie kolejnych den gniazd, flanek zębów i ogniw łańcucha siłami w czasie obrotu bębna łańcuchowego o kąt odziałowy φ = 2π/z. Podczas analizy obciążenia elementów bębna łańcuchowego rzyjęto zmienność kąta obrotu bębna od chwili zetknięcia się torusa rzedniego nabiegającego ogniwa oziomego z dnem gniazda (φ = 0) do chwili zetknięcia się torusa rzedniego kolejnego ogniwa oziomego z dnem nastęnego gniazda (φ = 2π/z). W zakresie obrotu bębna łańcuchowego o kąt odziałowy wyróżniono trzy rzedziały charakteryzujące się odmiennym sosobem obciążenia elementów bębna łańcuchowego (P1, P2 i P3 na rys. 1). Wzory od (4) do (39) oisują obciążenia dna gniazda i flanki zęba bębna łańcuchowego w tych rzedziałach. Utworzony model matematyczny ozwolił na komuterowe wyznaczenie obciążeń dynamicznych łańcuchów, bębnów naędowych oraz zębów i gniazd bębnów łańcuchowych w rzenośniku ścianowym o długości 350 m (rys. 3 8). W czasie badań symulowano stan nieluzowania łańcuchów i stan stałego luzowania. Na odstawie rzerowadzonych badań komuterowych ruchu ustalonego ścianowego rzenośnika zgrzebłowego, wyosażonego w bębny łańcuchowe o liczbie zębów z = 8, obciążonego urobkiem węglowym na całej długości sformułowano nastęujące wnioski: 1. Wydłużenie odziałki łańcucha, w raktyce sowodowane głównie zużyciem ściernym rzegubów ogniw, owoduje osiadanie torusa tylnego ogniwa oziomego coraz wyżej na flance zęba (wzrost wartości kątów ε oraz α u ). Prowadzi to do skracania czasu od chwili zetknięcia się torusa rzedniego ogniwa oziomego z dnem gniazda do chwili zetknięcia się torusa tylnego tego ogniwa z flanką zęba. Konsekwencją tego jest zmniejszanie się wartości maksymalnej obciążenia dna gniazda w unkcie styku z torusem rzednim ogniwa oraz wzrost maksymalnej wartości wymaganej siły tarcia zaobiegającej oślizgowi torusa tylnego o flance zęba zarówno w stanie stałego luzowania jak i w stanie nieluzowania łańcucha. 2. tosunek maksymalnej wartości siły obciążającej flankę zęba w unkcie styku z torusem tylnym ogniwa do maksymalnej wartości siły obciążającej dno gniazda w unkcie styku z torusem rzednim
1103 ogniwa oraz stosunek maksymalnej wartości wymaganej siły tarcia zaobiegającej oślizgowi torusa tylnego o flance zęba do maksymalnej wartości siły obciążającej dno gniazda w unkcie styku z torusem rzednim ogniwa rosną nieliniowo ze wzrostem wydłużenia odziałki ogniw. Wzrosty te rzebiegają niemal identycznie dla stanu stałego luzowania i stanu nieluzowania łańcucha. 3. Zwiększenie odziałki łańcucha od 1% do 4% sowodowało onad czterokrotny wzrost wartości maksymalnej siły tarcia zaobiegającej oślizgowi torusa tylnego ogniwa oziomego o flance zęba w stronę dna gniazda. Jeżeli wartość siły tarcia rozwiniętego wywołanego siłą nacisku torusa tylnego ogniwa oziomego na flankę zęba jest co najmniej równa wartości rozatrywanej siły tarcia to układ sił jest w równowadze. Jeśli natomiast siła tarcia ochodząca od nacisku torusa tylnego na flankę zęba jest mniejsza od wartości tej siły tarcia to nastęuje oślizg torusa tylnego o flance zęba w stronę dna gniazda. Z tego względu duże wartości rozważanej siły tarcia w miejscu styku torusa tylnego ogniwa oziomego z flanką zęba są niekorzystne, gdyż zwiększają możliwość wystąienia oślizgu ogniwa o flance zęba co owoduje zwiększenie zużycia ściernego flanki zęba obniżając trwałość bębna łańcuchowego i owodując zmniejszenie srawności zazębienia łańcuchowego. łowa kluczowe: rzenośnik ścianowy, bęben łańcuchowy, model fizyczny i matematyczny, obciążenia dynamiczne zębów, obciążenia dynamiczne gniazd 1. Introduction The concentration of hard coal roduction necessitates the intensive tests of mining machines in the context of imroving their reliability and life (Doliski, 1997; oseinie et al., 2011; Krauze et al., 2009; Krauze, 2004; Langosch & Ruel, 2008). craer conveyors are one of the key machines forming art of mechanised longwall systems. Longwall scraer conveyors are currently the only means of transorting the mined rock from longwalls in hard coal mines. The conveyors, throughout their develoment, have been equied with the different tyes of drive chains, with a link chain turning out to be the best. Its essential advantages include simle roduction technology, the links can be rotated relatively in the horizontal and vertical lane, high tensile strength, easy installation and the simle and fast linking of the broken sections using connecting links. A mining lain link chain consists of alternate horizontal (active) and vertical (assive) links. The chains are made of a steel rod with the d diameter and are shaed 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 shae corresonding to a geometric olygon. 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 relationshis between the srocket drum and the links are decisive for the osition of the chain links in the seats. The abrasive wear of the chain arts and of the drive drum arts occurring due to conveyor oeration is increasing the chain itch and decreasing the wheel itch. As a result the osition of links in the seats changes along with the load on the srocket drum teeth and seats. Investigations into srocket drums have been concentrated to date on the meshing geometry and on the calculation of static loads (Doliski, 1997; trümfel, 1989; Uhr, 1993), and the investigations aimed at determining dynamic loads in longwall conveyors were concentrated on chains and coulings (Doliski, 1997, 2001; Kallrath & Brychta, 1986; Wölfe & Flöte, 2000; Ziegler et al., 2007). rocket 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.
1104 2. Discrete hysical and mathematical model igh-caacity longwall scraer 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 scraer chain by means of a srocket drum equied with two seat wheels. A hysical and mathematical longwall conveyor model has been extended with the chain meshing elements for the urose of dynamic loads determination of the srocket drum teeth and seats. A discrete hysical and mathematical longwall conveyor model was built earlier (Doliski, 1997) and verified exerimentally many times. The hysical model, after exanding it with the chain meshing elements, assumes its form as in Fig. 1. Motion in this exanded hysical model is described with the following system of ordinary nonlinear second-order differential equations: m11 q11 Z11 [] k1b( q11 B R0B) h1b( q 11 B R0B) 11 Z12[] k11( q12q11) h11( q 12q 11) 11W1 10... m1 iq 1i Z1 i [] k1 ( i1) ( q1 i q1 ( i1) ) h1 ( i1) ( q 1iq 1( i1) ) 1i Z1 ( i1) [] k1i( q1 ( i1) q1i) h1i ( q 1( i1) q 1i) 1i W1i 0... m1jq 1j Z1j[] k1( j1) ( q1j q1( j1) ) h1( j1) ( q 1j q 1 ( j 1) ) 1j Z [] k ( R q ) h ( R q ) W IA A Z11A [] 11A R11 A Z12A [] 12 A R12 A Z211 [] k21 A( q211 AR01A) h21a( q 211 AR01A) 21A R21 A Z221 [] k22 A( q221 AR02A) h22a ( q 221 AR02 A) 22 A R22 A ka1 ( A1 A) ha1 ( A1 A) Z11 A 11A R11A Z12A12AR12 A Z21A [][ 21A] R21AZ22A[ ] [ 22 A] R22 A I k h k h Z R 1A 1A A 0A 1j 1A A 0A 1j 1j 1j 0 ( ) ( ) ( ) ( ) A1 A1 A1 A1 A A1 A1 A A2 A1 A2 A2 A1 A2 11A 11A 11A Z12 A12 A R12AZ21A[][ 21A] R21 AZ22A[][ 22A] R22A0... I k h Z R Z R ( ) ( ) A4 A4 A4 A4 A3 A4 A4 A3 11A 11A 11 A 12A 12 A 12A [][ ] [][ ] Z R Z R M 21A 21A 21A 22A 22A 22A A
1105 m21 q 11 Z21 [] k2a( q21 A R0A) h2a( q 21 AR0A) 21 Z22[ ] k21( q22q21) h21( q 22 q 21) 21W2 10... m2 i q 2iZ2i[] k2 ( i1) ( q2 iq2( i1) ) h2 ( i1) ( q 2iq 2( i1) ) 2i Z k ( q q ) h ( q ( ) q ) W... 2( i1) [] 2i 2( i1) 2i 2i 2 i1 2i 2i 2i 0 m2 jq 2jZ2j[] k2 ( j1) ( q2 j q2 ( j1) ) h2 ( j1) ( q 2jq 2( j1) ) 2j Z2 B[] k2b( BR0 Bq2j) h2 B ( BR0 B q 2j) 2j W2 j 0 IB B Z21B[] 21B R21B Z22B [] 22B R22B Z111 [] k11 B( q111 BR01B) h11 B ( q 111 BR01B) 11 B R11 B Z121 [] k12 B( q121 BR02 B) h12b( q 121 BR02 B) 12B R12B k h Z R Z R ( ) ( ) B1 B1 B B1 B1 B 21B 21B 21B 22B 22B 22B Z [][ ] R Z [][ ] R 11B 11 B 11B 12B 12B 12B I k h k h Z R ( ) ( ) ( ) ( ) [][ ] [][ ] B1 B1 B1 B1 B B1 B1 B B2 B1 B2 B2 B1 B2 11B 11B 11B Z R Z R Z R 12B 12B 12B 21B 21B 21B 22B 22B 22B 0... I k h Z R Z R ( ) ( ) B4 B4 B4 B4 B3 B4 B4 B3 21B 21B 21B 22B 22B 22B [][ ] [][ ] Z R Z R M 11B 11B 11B 12B 12B 12B B (1) whereas: ( ) ( ) 1A k1a A R0A q1j h1a A R0A q1j 1A ( ) ( ) 2B k2b B R0B q2j h2b BR0B q2j 2B i = 2, 3,..., j 1 (2) (3) κ = 1, 2 where: h a substitute daming coefficient of additional longitudinal chain damers, h A1, h B1 a substitute daming coefficient of srocket drum in the main drive (index A) and auxiliary drive (index B),
1106 h A2, h B2 a substitute daming coefficient of reducer in the main drive (index A) and auxiliary drive (index B), h A3, h B3 a substitute daming coefficient of couling in the main drive (index A) and auxiliary drive (index B), h A4, h B4 a substitute daming coefficient of the driving motor in the main drive (index A) and auxiliary drive (index B), [ ] eaviside s function (square bracket means the content of the eaviside s function argument), I A, I B moment of inertia of srocket drum in the main drive (index A) and auxiliary drive (index B), I A1, I B1 moment of inertia of reducer reduced onto the srocket drum shaft in the main and auxiliary drive, I A2, I B2 moment of inertia of the outut member of the couling reduced onto the srocket drum shaft in the main and auxiliary drive, I A3, I B3 moment of inertia of the inut member of the couling reduced onto the srocket drum shaft in the main and auxiliary drive, I A4, I B4 moment of inertia of the motor rotor reduced onto the srocket drum shaft in the main and auxiliary drive, k secific rigidity of the chain s flexible bond, k A1, k B1 secific torsional rigidity of the drive drum reduced onto the srocket drum shaft in the main drive (A) and auxiliary drive (B), k A2, k B2 secific torsional rigidity of the reducer reduced onto the srocket drum shaft in the main drive (A) and auxiliary drive (B), k A3, k B3 secific torsional rigidity of the couling reduced onto the srocket drum shaft in the main drive (A) and auxiliary drive (B), k A4, k B4 secific torsional rigidity of the asynchronous motor rotor reduced onto the srocket drum shaft in the main drive (A) and auxiliary drive (B), m substitute vibrating mass, M A, M B driving torque of the asynchronous motor in the main and auxiliary drive reduced onto the srocket drum shaft, q translation coordinates, R 11A, R 12A functions of radiuses of the chain No. 1 and No. 2 running on the srocket drum of the main drive, R 21B, R 22B functions of radiuses of the chain No. 1 and No. 2 running on the srocket drum of the auxiliary drive, R 21A, R 22A functions of radiuses of the chain No. 1 and No. 2 running off the srocket drum of the main drive, R 11B, R 12B functions of radiuses of the chain No. 1 and No. 2 running off the srocket drum of the auxiliary drive, R 01A, R 02A itch radiuses of srocket wheels of the drive A, R 01B, R 02B itch radiuses of srocket wheels of the drive B, static load in the chain (being the function of resistance of motion, residual initial chain tension and ower distribution factor), W force of external friction in the lace of arranging the substitute vibrating mass,
1107 Z the chain break coefficient (Z = 0 for broken chain, Z = 1 if chain breakage does not occur), φ rotation coordinates. When a srocket drum with standard dimensions is co-working with a chain with the elongated itch, 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 ε 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 β. The following arameters are determined in order to clearly describe the location of the chain links in the wheel seats (Fig. 2) (Doliski et al., 2010): the links' inclination angle relative to the bottoms of the wheel seats ε; 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 olygon u; the rotation angle of the vertical link relative to the receding horizontal link in the middle of the joint with the horizontal link rear torus α u. The longer relative elongation of the itch the greater values are achieved by the arameters describing the location of links in the srocket wheel seats (ε, u and α u ). The mobility effect of links in joints, when the links are tilting mutually, has been considered when analysing the co-working of the srocket drum with the link chain, the result of which is the dislacement of the contact oint of the links. The tilting of the horizontal link relative to the vertical link is accomanied 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 sli of links in the joint deending on the value of the joint module m and the value of the friction coefficient in the joint µ. When the horizontal link is rolling, the contact oint of the links is dislacing in the joint, and the osition of the contact oint in the vertical link joint remains unchanged during the sli of the horizontal link. Due to the reeatability of the links location in the srocket 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 srocket drum is rotated by the itch angle of φ = 2π/z. It was assumed when analysing the loading of the srocket 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 (φ = 0) until the front torus of another horizontal link is contacting the bottom of the next seat (φ = 2π/z). Three ranges, characterised by the different loading of the srocket drum arts, are differentiated for srocket drum rotation by the itch 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 φ changing within the range of: 2 u 0 (4) z The horizontal link that is meshed is loaded with the run-on force (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 sli of the front torus on the seat bottom
1108 Fig. 1. Physical model of a longwall scraer conveyor with the srocket drums marked is accomanied by the friction force, erendicular to the reaction, deendent uon the value of reaction and friction coefficient on the seat bottom µ g and the force transmitted from the horizontal link on the receding vertical link (P1 in Fig. 1). The tilting of the horizontal link relative
1109 Fig. 2. Location of chain links in the srocket wheel seats to the receding vertical link is accomanied by link rolling or sli in the joint resulting in the dislacement of the contact oint of the links by the angle γ. The mobility of links in the joint of the horizontal link and of the receding 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 λ and this is accomanied by the rolling or sli of the link in the joint causing the dislacement of the contact oint of the links by the angle γ t. The equations of balances of forces acting on the horizontal link are then assuming the following form: cos( ) cos( ) N 0 (5) g sin( ) sin( ) N (6) d g d d t sin( ) N ( d) sin( ) sin( ) 0 2 2 2 (7) where: 2 n u (8) z The forces and N in the function of the run-on force can be determined from the equations recorded with the relationshis (5) and (6): cos( ) sin( ) g cos( ) sin( ) g (9)
1110 N sin( ) g cos( ) sin( ) (10) and the deviation angle λ 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: whereas: gr 0 (φ λ) φ gr (11) (1 m ) arctan( ) (12) m m ( ) (1 m ) (13) The sli of links in the joint occur for the following range of the angle of rotation: whereas: gr 2 u (14) z γ = arctan(μ ) (15) imilarly, the mobility range of the deviation angle of the vertical link in the axis of the horizontal link λ should consider the rolling or sli of links in the joint. Links are rolling for the following range: whereas: gr 0 λ λ gr (16) (1 m ) arctan( ) (17) m t m (1 m ) (18) meanwhile the sli of links in the joint occur for the following angle of rotation: whereas: λ > λ gr (19) γ t = arctan(µ ) (20)
1111 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 φ changing within the following range: 2 u N0 z (21) A horizontal link is loaded with the run-on force, 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 oint of the rear torus of the horizontal link with the tooth flank and with the force transmitted from the horizontal link onto the receding vertical link (P2 in fig. 1). The equations of balances of forces acting on the horizontal link are then assuming the following form: cos( ) cos( ) Fsin( ) 0 (22) sin( ) sin( ) Fcos( ) N 0 (23) n d t d n ( d) sin( ) sin( ) sin( ) F( d) cos( ) 0 2 2 (24) The forces, F and N in the function of the run-on force can be determined from the equation notated with the relationshis (22), (23) and (24): n 2 ( ) sin( ) sin( t n d d ) sin( ) 2( d) cos( ) cos( ) n dsin( ) sin( ) 2 ( d) cos( ) cos ( ) (25) F cos( ) cos( ) (26) sin( ) sin( ) N F sin( ) sin( ) cos( ) (27) The variability range of the vertical link rotation angle relative to the receding 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 sli 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 λ and lasts until the limit angle is reached φ gr, whereas ( 1 m ) gr 2 u arctan( ) (28) m z t m 2 u ( 1 m ) z (29)
1112 The sli of links in the joint occurs for the following range of the angle of rotation: whereas: gr 2 (30) z γ t = arctan(µ ) (31) As the value of the drum rotation angle φ is rising so is decreasing the values of the force in the receding vertical link 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 u to the rotation angle φ N0 at which the value of reaction N falls to zero have been determined from the above deendencies. 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: (32) z N0 2 The system of forces acting on the horizontal link changes for the angle φ > φ N0 and the force T occurs on the tooth flank necessary for maintaining the balance of the horizontal link reventing the sli 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: cos( ) cos( ) F sin( ) Tcos( ) 0 (33) sin( ) sin( ) F cos( ) Tsin( ) 0 (34) n d d t d ( d) sin( ) sin( ) sin( ) T 0 2 2 2 (35) The forces, F and T as the function of the run-on force can be determined from the equation notated with the relationshis (33), (34) and (35): T T cos( ) cos ( ) (36) cos( ) cos ( ) (37) F cos( ) cos( ) T 1 (38) sin ( ) sin( ) tan( )
1113 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: N0 2 (39) z then the third range of loading the srocket drum elements does not occur. 3. Comuter tests A longwall scraer conveyor with the length of L = 350 m equied with a single main drive and a single auxiliary drive was subjected to comuter tests. The driving systems with asynchronous motors with the ower of 400 kw each were equied with srocket drums with the number of teeth of z = 8 and the itch angle of 45. The drums were co-working with two central chains sized 34 126 with the joint module of m = 0,85. The uer conveyor branch was loaded with the mined coal with the intensity of 300 kg/m. The conditions of non-slackened chains and ermanently 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 (Doliski, 1997). This means that the static and dynamic elastic elongation of the chain was comensated by initial tension. In the condition of constant slackening, interlink slackening occurs constantly in the run off oint from the drive srocket 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 oint from the srocket 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 lace it runs on the srocket drum in the main drive was,umax 11 A = 416,2 kn. The srocket drum in the main drive was loaded with the maximum moment of M K,Umax A = 195,5 knm (Fig. 3). In the case where the elongation of the chain itch was Δ/ = 1%, the maximum value of the force of reaction in the contact oint of the rear torus of the horizontal link with the srocket U drum tooth flank in the main drive F max 11 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 U srocket drum seat in the main drive for the conveyor s steady-state motion N max 11 A was 124,7 kn and the maximum value of the force reventing the sli of the rear torus of the horizontal link at U the tooth flank towards the bottom seat (where N = 0) T max 11 A was 17,2 kn. The value of the chains initial tension was increased in the next stage of the comuter tests (in ractise this is done when the longwall conveyor is at standstill). As a result, the elastic elongations of the chain were comletely comensated 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 lace where it runs on the srocket drum in the main drive to,umax 11 A = 532,6 kn. The maximum load of the drive drum was M K,Umax A = 195,5 knm (Fig. 5).
1114 ϕ A ϕ B 11A 11B Fig. 3. tart-u and steady-state motion of the longwall scraer conveyor in the condition of constant slackening of chains: a) rotational velocity of srocket drums, b) dynamic loads in the chain, c) dynamic loads of srocket drums The examles of curves for dynamic loads for the srocket drum teeth and seats in the main drive in the steady-state motion of the investigated longwall conveyor are resented in Fig. 6. In the case where the elongation of the link chain itch was Δ/ = 1%, the maximum values of U the loads were: N max U 11 A = 160,2 kn, F max U 11 A = 493,2 kn and T max 11 A = 22,7 kn.
1115 11A Fig. 4. Dynamic loads of teeth and bottoms of srocket 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 itch elongation of Δ/ = 1% The abrasive wear of the links joints, being the main cause of the higher itch of the chain links, is decisive for the osition of links in the srocket 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 oint 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 itch Δ/ within the examined range is affecting adversely the maximum value of the force T (it occurs when N = 0 and is erendicular 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 emhasised that the increased value of the required friction force at the tooth flank increases the robability of link sli at the tooth flank causing the faster abrasive wear of the teeth and effecting adversely a srocket wheel s efficiency. A relationshi between the maximum value of the force loading the tooth flank in the contact oint of the rear link torus and the maximum value of the force loading the seat bottom in the contact oint with the front torus of the link (F/N) is growing nonlinearily as the elongation of the links itch is growing from 3,1 for itch growth by 1% to 11,5 for itch growth by 4%. As the itch of the links is growing so is growing the relationshi between the maximum value of the required friction force reventing the sli of the rear torus at the tooth flank and the
1116 ϕ A ϕ B 11A 11B Fig. 5. tart-u and steady-state motion of the longwall scraer conveyor in the condition of non-slackening of chains: a) rotational velocity of srocket drums, b) dynamic loads in the chain, c) dynamic loads of srocket drums maximum value of the force loading the bottom of the seat in the contact oint with the front torus of the link (T/N) from 0,1 for itch growth by 1% to 2,4 for itch growth by 4% (Fig. 8). Growth in the relationshis of the forces is rogressing almost identically for the condition of constant chain slackening and non-slackening.
1117 11A Fig. 6. Dynamic loads of teeth and bottoms of srocket 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 itch elongation of Δ/ = 1% Fig. 7. Imact of the chain itch elongation Δ/ on the maximum value of the friction force T reventing the sli 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
1118 Fig. 8. Imact of the chain itch elongation Δ/ on the relationshi between the maximum values of the forces F/N and T/N 4. ummary An exanded hysical and mathematical model of the longwall scraer conveyor allows to determine dynamic loads not only of srocket drums but also their teeth and seats during start-u and in the steady-state motion. A characteristic feature of a scraer conveyor is the occurrence of high dynamic loads not only during start-u but also for the entire duration of steady-state motion. The following conclusions have been drawn on the basis of the comuter tests of the steady-state motion of the longwall scraer conveyor equied with srocket drums with the number of teeth z = 8, loaded with the mined rock at its entire length: 1. The elongation of the chain itch, in ractise 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 ε and α 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 oint with the front torus of the link and a larger maximum value of the required friction force reventing the sli 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 relationshi between the maximum value of the force loading the tooth flank in the contact oint with the rear torus of the link and the maximum value of the force loading the seat bottom in the contact oint with the front torus of the link and the relationshi between the maximum value of the required friction force reventing the sli of the rear torus at the tooth flank to the maximum value loading the seat bottom in the contact oint
1119 with the front torus of the link are rising in a non-linear manner as the elongation of the links itch is growing. Growth in the relationshis is rogressing almost identically for the condition of constant chain slackening and chain non-slackening. 6. If the chain itch is increased between 1% to 4%, the maximum value of the friction force reventing the sli 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 exanded friction force induced by the ressing 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 ressure of the rear torus on the tooth flank is smaller than the value of this friction force, then the sli 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 lace of the rear torus of the horizontal link with the tooth flank are disadvantageous, as they increase a ossibility of link sli occurring on the tooth flank causing the greater abrasive wear of the tooth flank and deteriorating the durability of the srocket drum and comromising chain meshing efficiency. The assignment carried out under the develoment roject No. N R09 0026 06/2009 financed by the Ministry of cience and igher Education under decision No. 0481/R/ T02/2009/06 References Doliski M., 1997. Dynamika rzenośników łańcuchowych. Podręcznik akademicki. Wyd. Pol. Śl., Gliwice. Doliski M., 2001. Przeciążenie dynamiczne srzęgieł odatnych odczas rozruchu rzenośnika ścianowego z silnikami dwubiegowymi. Mechanizacja i Automatyzacja Górnictwa, nr 3. Doliski M., Remiorz E., obota P., Osadnik J., 2010. Wływ zwiększenia odziałki łańcucha na ołożenie jego ogniw w gniazdach bębnów łańcuchowych. Wiadomości Górnicze, nr 9. oseinie..; Ataei M., Khalokakaie R., 2011. Reliability modeling of water system of longwall shearer machine. Arch. Min. ci., ol. 56, No 2. Kallrath E., Brychta P., 1986. trömungskulungen für schweranlaufende trebförderer. Glückauf, nr 12. Krauze K., Kotwica K., Rączka W., 2009. Laboratory and underground tests of cutting heads with disc cutters. Arch. Min. ci., ol. 54, No 2. Krauze K., 2004. election of combined cutter-loader arameters for unidirectional and bidirectional longwall mining systems and the investment costs involved in mining oerations. Arch. Min. ci., ol. 49, No 2. Langosch U., Ruel U., 2008. tate of the art dimensioning of shield suort to otimize longwall roof control. Arch. Min. ci., ol. 53, No 3. trümfel., 1989. Zum Antrieb von Ketten mit großer Teilung. dhf, nr 11 Uhr M., 1993. Untersuchungen zur erbesserung des Zusammenwirkens von Kette und Kettenrad. Glückauf, nr 6 Wölfe M., Flöte K., 2000. Causes of vibration and stress loadings of chain-oerated face equiment resulting from vibration. orträge anläßlich des zweiten internationalen Kolloquiums ochleistungs-trebbetriebe. RWT Aachen, 13-14 Juni. Ziegler M., Ketting M., cholten J., 2007. Dynamische Beansruchung von obel- und trebförderanlagen. Glückauf, Nr. 11. Received: 20 February 2012