© Strojni{ki vestnik 50(2004)12,598-612 © Journal of Mechanical Engineering 50(2004)12,598-612 ISSN 0039-2480 ISSN 0039-2480 UDK 621.833.3:621.824 UDC 621.833.3:621.824 Kratki znanstveni prispevek (1.03) Short scientific paper (1.03) Dinami~ni model rotorskega sistema z gibkim ~lenom in dvema nesoosnima gredema A Dynamic Model of a Rotor System Consisting of a Flexible Link with Misaligned Shafts Marijonas Bogdevi~ius - Bronislovas Spruogis - Vytautas Turla Zanesljivost in trajnost strojno gnanih sistemov, ki jih uporabljamo v prenosih, sta odvisni od izbire povezovalnih elementov v gonilih. Pomembno je, da pri taksnih sistemih pravilno ocenimo in izravnamo pojav nesoosnosti V pričujočem prispevku predstavljamo dinamični model rotorskega sistema, ki sestoji iz motorja in delovnega stroja z gredema, ki sta povezani z elastično izravnalno gredno vezjo. Enačbe gibanja polovičnih grednih vezi in rotorskih sistemov smo izpeljali na način, ki omogoča določitev kinetične in potencialne energije, kinetičnih in dinamičnih parametrov deformiranega sistema ter lege in oblike gibkega člena, ki povezuje sistem. Izračunali smo tudi krivuljo pospeškov elastičnih sil gibkega člena (gredne vezi). © 2004 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: gonila kompleksna, vezi gredne, nesoosnost, modeli dinamični, metode končnih elementov) The reliability and durability of the machine-drive systems used in transport depend on the selection of the connecting elements used in the drives. In such systems the evaluation and compensation of incoaxiality is an important task. In this paper we discuss the dynamic model of a rotary system consisting of an engine and a working machine with the shafts connected by an elastic compensation coupling. The equations of motion of the half couplings and the rotary systems are derived in such a way that the kinetic and potential energies, the kinetic and dynamic parameters of the deformed system, and the location and shape of the flexible link joining the system can be determined. We have also calculated the hodograph of the elastic forces of the flexible link (coupling). © 2004 Journal of Mechanical Engineering. All rights reserved. (Keywords: complex drive, couplings, misalignment, dynamic models, finite element methods) 0 UVOD Rotorski sistem sestoji iz mnogih, sočasno vrtečih se, členov. Zaradi nesoosnosti grednih vezi, neuravnoteženosti posameznih delov, zunanjih in drugih motenj ter zaradi sprememb v dovedeni energiji se členi vrtijo nepravilno. Ti dejavniki povzročijo povečanje dinamične obremenitve v strojih in mehanizmih, kar poveča vrtilno vibriranje. V pričujoči študiji smo raziskovali, kako lahko zmanjšamo vrtilno vibriranje in sile, ki ga povzročajo. Ugotovili smo, da je eden najbolj učinkovitih načinov zmanjšanja vibracij izboljšanje konstrukcije strojnih sestavnih delov in zamenjava teh elementov z deli, ki so odporni proti vibriranju. Za ta namen lahko uporabimo učinkovite naprave za prenos kroženja in za stabilizacijo, npr.: različne sklopke in dušilnike vibriranja. 0 INTRODUCTION A rotor system consists of many synchronously rotating links. Because of the misalignment of the coupling-link shafts, the non-balanced parts, the external and other disturbances, and the variations in the supplied energy, the links rotate irregularly. These factors result in an increased dynamic load in machines and mechanisms that gives rise to rotary vibration. In this study we have investigated how to reduce these rotary vibrations and the forces that cause them. We found that one of the most effective ways of decreasing vibrations is to improve the construction of the machine’s assembly elements and substitute these elements with parts that are resistant to vibration. For this we can use effective rotary-motion transmission and stabilization devices in the form of various clutches and vibration dampers. 2 jgnnatäüllMliBilrSO | | ^SsFÜWEIK | stran 598 Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model Tu predstavljamo dinamično analizo nelinearne, torzijske, gibke gredne vezi z elastičnimi členi [1]. Z rezultati analize ravnovesnega stanja in prehodnega osciliranja smo določili optimalna razmerja grednih vezi. Mohiuddin in Khulief [2] sta prikazala splošni dinamični model velikega sistema rotorskih ležajev z razpokano gredjo. Model predvideva gredi s klinastimi deli, več diskov in anizotropne ležaje. Upoštevali smo tudi vpliv togosti gibke gredne vezi na amplitudo torzijskega vrtilnega momenta [3]. Trdnost in zanesljivost zobnikov v stroju sta močno odvisni od izbire pravih elementov gredne vezi. V primeru takšnih strojev je zelo pomembno, da pravilno ocenimo izravnavo nesoosnosti gredi. V pričujoči študiji predstavljamo dinamični model rotorskega sistema, ki sestoji iz asinhronega motorja in delovnega stroja, čigar gredi sta povezani s izravnalno sklopko. Sklopka, ki rabi kot gibek člen med motorjem in gredema delovnega stroja, ima lahko različne prostostne stopnje. Glede nesoosnosti pa smo dinamiko povezanih gredi rešili z uporabo metode končnih elementov. Za prikaz geometrijske oblike gredne vezi in člena med polovičnima grednima vezema v primeru nesoosnih gredi uporabimo zapletene elemente. Izpeljemo matematične odvisnosti med globalnimi točkami sklopk v začetnih in deformiranih legah in določimo njihove soodnose. Ugotovimo matematičen odnos med silami, ki delujejo na zapletene elemente sklopke; prav tako izračunamo sile in momente, ki delujejo v polovični gredni vezi in se prenašajo na pripadajoči gredi ter na podporne dele. Razvijemo enačbe gibanja za sistem polovične gredne vezi in rotorja ter jih združimo v matematični model. Dobljena rešitev omogoči, da izračunamo kinetično in potencialno energijo, kinematične in dinamične parametre deformiranih sistemov ter ugotovimo lego in obliko gibke gredne vezi. 1 DINAMIČNI MODEL ROTORSKEGA SISTEMA Analizirali bomo sistem, ki je sestavljen iz motorja in delovnega stroja, čigar gredi sta povezani z gibko gredno vezjo (gibko sklopko - GS). Motor in delovni stroj sta pritrjena na gibke opornike, ki dušijo vibriranje. Predpostavljamo, da so okvir delovnega stroja in diski rotorjev popolnoma togi, kar pomeni, da ima sistem zgoščene parametre. Dinamiko povezanih gredi bomo raziskali z metodo končnih elementov (MKE). Zunanje vzbujanje sistema je posledica radialne in kotne nesoosnosti povezanih gredi motorja in delovnega stroja. Naredili smo naslednje predpostavke: deformacije so majhne, gredi rotorjev sta gibki telesi in bočne sile so zanemarljive. We present a dynamic analysis of a nonlinear torsional flexible coupling with elastic links [1]. The results of analyses of the steady-state and transient-oscillation performances are applied to determine optimum proportions for the couplings. Mohiuddin and Khulief [2] presented a general dynamic model for a large-scale rotor-bearing system with a cracked shaft. The model accommodates shafts with tapered portions, multiple disks and anisotropic bearings. The influence of the stiffness of a flexible coupling on the amplitude of the torsional torque is considered [3]. The strength and the reliability of the gears in machines very much depend on choosing the right coupling elements. For such machines it is very important to evaluate the compensation of the shafts’ misalignment. In this study we present a dynamic model of a rotor system consisting of an asynchronous engine and a working machine, the shafts of which are connected by a compensating clutch. The clutch, which serves as a flexible link between the engine and the shafts of the working machine, can have many degrees of freedom. For the case of misalignment, the dynamics of the connected shafts is solved using the finite-element method. We present the geometry of the coupling as well as the link between its half-couplings using complex elements for the case when the shafts are misaligned. We derive mathematical dependencies between the global points of the clutches in the initial and deformed positions and determine their interrelations. We establish a mathematical relation between the forces influencing the complex elements of the clutch; and the forces and moments acting in the half-coupling, which are transferred to the corresponding shafts and the support parts, are calculated. We develop the equations of motion for the half-coupling and the rotor system and join them in a mathematical model. The solution allows us to calculate the kinetic and potential energies, the kinematics and dynamics parameters of the deformed systems, as well as establish the location and the shape of the flexible coupling. 1 DYNAMIC MODEL OF A ROTOR SYSTEM We will analyse a system composed of an engine and a working machine, whose shafts are connected by a flexible coupling (a flexible clutch – FC). The engine and the working machine are attached to flexible supports that suppress the vibration. We assume that the frame of the working machine and the disks of the rotors are absolutely stiff, i.e., the system possesses concentrated parameters. The dynamics of the connected shafts will be investigated using the finite-element method (FEM). The external excitation of the system is caused by the radial and angular misalignment of the connected engine and working-machine shafts. We have made the following assumptions: the deformations are small, the shafts of the rotors are flexible bodies, and the lateral forces are negligible. | lgfinHi(s)bJ][M]lfi[j;?n 0412 stran 599 I^BSSIfTMlGC Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model Slika 1 prikazuje dinamični model rotorskega sistema, čigar globalni sistem koordinat je OXYZ. Za sistem prve polovične gredne vezi smo izbrali koordinatni sistem OXY1Z, za sistem druge polovične gredne vezi pa smo izbrali koordinatni sistem O2X2YZ2. Vsaka polovična gredna vez ima lahko šest prostostnih stopenj. Poleg tega ima lahko gibka gredna vez (GS) še precej več notranjih prostostnih stopenj (v odvisnosti od strukture povezovalne gredne vezi). Dinamiko modela smo razrešili z uporabo MKE. Vrednosti začetnih deformacij gredi so znane. Analizirali bomo, kako nastane deformacija rotorskega sistema, prav tako bomo ugotovili premike, kote deformacije in njihove odvode ([4] do [10]). Fig. 1 shows the dynamic model of the rotor system, whose global system of coordinates is OXYZ. For the system of the first half-coupling we selected the O1X1Y1Z1 coordinate system, while for the system of the second half-coupling we selected the O2X2Y2Z2, coordinate system. Every half-coupling can have six degrees of freedom. Furthermore, a flexible coupling (FC) can have many more internal degrees of freedom (depending on the structure of the connecting coupling). The dynamics is solved using the FEM. The values of the initial deformations of a shaft are known. We will analyse how the deformation of the rotor system occurs, and we will also find the displacements, the angles of deformation and their derivatives ([4] to [10]). Sl. 1. Dinamični model rotorskega sistema: 1 - rotor motorja; 2 - gibki oporniki; 3 - prva polovična gredna vez; 4 - gibki povezovalni elementi polovične gredne vezi; 5 - druga polovična gredna vez; 6 - rotor delovnega stroja; I - gonilna gred; II - gnana gred Fig. 1. Dynamic model of the rotor system: 1 - engine rotor; 2 - flexible supports; 3 - first half-coupling; 4 - half coupling s flexible connecting elements; 5 - second half coupling; 6 - rotor of the working machine; I - driving shaft; II - driven shaft 1.1 Geometrijska oblika sklopke Analizirali bomo splošen primer sklopke: gonilna gred I in gnana gred II rotorskega sistema sta povezani s polovično gredno vezjo. Števili prve in druge polovične gredne vezi sta i = 1 in i = 2. Števila zapletenih elementov (KE) v gibki gredni vezi so j = 1, 2, ... NZ, pri čemer je NZ skupno število zapletenih 1.1 Geometry of the Clutch We will analyse a general case for the clutch: the driving shaft I and the driven shaft II of the rotor system are connected by a half-coupling. The numbers of the first and second half-couplings are i = 1 and i = 2. The numbers of the complex elements (CE) in the flexible coupling are j = 1, 2, ... NZ, where prva polovična^ gredna vez first half-clutch elastični elementi elastic elements druga polovična gredna vez second half-clutch Sl. 2. Shematični prikaz geometrije gredne vezi, če je NZ = 8 Fig. 2. Schematic view of the coupling geometry when NZ = 8 2 jgnnatäüllMliBilrSO | | ^SSfiflMffiC | stran 600 Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model elementov v gibki gredni vezi (sl. 2). Zahtevni elementi so razvrščeni v krogih s premeroma R11 in R21 (sl. 2). Točke so določene s koordinatnim sistemom OijXijYijZij. Polovični gredni vezi sta povezani z zapletenimi elementi, od katerih vsak sestoji iz treh elementov (sl. 3). NZ is the total number of complex elements in the flexible coupling (Fig.2). The complex elements are arranged in circles with radii R11 and R (Fig. 2). The points are defined in the coordinate system OXYZ. The half-couplings are interlinked by complex ij elements, with each of them consisting of three elements (Fig. 3). Sl. 3. Zahtevni element: 1 - prvi element deluje v vzdolžni smeri; 2 - drugi element deluje v prečni smeri ; 3 - tretji element deluje pravokotno na prečno in vzdolžno smer Fig. 3. Complex element: 1 - the first element acts in the axial direction; 2 - the second element acts in the radial direction; 3 - the third element acts perpendicular to the radial and axial directions 1.2 Sile in momenti sil, ki delujejo na zapletene elemente Koordinate točk O1 in O2 v koordinatnem sistemu OXYZ (sl. 1) so naslednje: 1.2 Forces and moments of the forces acting on the complex elements The coordinates of points O1 and O2 in the coordinate system OXYZ (Fig. 1) are as follows: {X} {X} X0 O1+ {} {X 0 O2 +{U(t)}O (1) (2), kjer so {X}O1, {X}O2 začetne koordinate točk O1 in O2, in sta {U(t)}O1, {U(t)}O2 premika točk O in O2. Razmerje med koordinatnima sistemoma OX1Y1Z1 in O1X1 Y1Z1i lahko izrazimo takole: where {X0}O1, {X0}O2 are the initial coordinates of the points O1 and O2, and {U(t)}O1, {U(t)}O2 are the displacements of the points O1 and O2. The relationship between the coordinate systemsO1X1Y1Z1 and O1iX1iY1iZ1i can be expressed as follows: {^}{} + [o]{ } (3), kjer so {X}1i koordinate točke O1 v koordinatnem sistemu OX YZ1, [G1 ] matrika koordinatne premene in {X1 } koordinate zapletenega elementa i polovične gredne vezi 1. Razmerje med koordinatnima sistemoma O2X2Y2Z2 in O2X2Y2Z2 lahko izrazimo na naslednji where {X1}1i are the coordinates of the point O1i in the O1X1Y1Z1 coordinate system, [G1i] is the matrix of coordinates transformation, and {X1i} are the coordinates of the complex element iof the half-coupling 1. The relationship between the coordinate systems O2X2Y2Z2 and O2iX2iY2iZ2i can be expressed as follows: { X2 } = { X2 } 2i+ [ G2i ]{ X2i } (4), I isfinHi(s)bJ][M]ifln;?n 04 stran 601 I^HSSTTIMIDC Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model kjer so {X}2 koordinate točke O2 v koordinatnem sistemu O2X2Y2Z, [G2] matrika koordinatne premene in {X } koordinate zahtevnega elementa i polovične gredne vezi 2. Razdalja med točkama O1 in O2 je naslednja: where {X2}2i are the coordinates of point O2i in the O2X2Y2Z2 coordinate system, [G2i] is the matrix of coordinates transformation, and {X2 i } are the coordinates of the complex element i of the half-coupling 2. The distance between the points O1i and O2i is as follows: {Li,1 (t)} = {X} 2i -{X} {xo} + { U ( t )} +A2 ]{^} - { X0 } O1- { U ( t )} O1- [ A 1 ]{ X 1 1 1i (5) in časovni odvod vektorja {L 1(t)} je: and the derivative of vector {Li,1(t)} with respect to time is equal to: d dt Li,1 ()} = { U ()}O2 +tA2l{ } -U ()}O1 - ^]{} 1 i (6), kjer sta [A1], [A2] matriki koordinatne premene; where [A1], [A2] are the matrices of coordinates transformation; dt Premik med točkama O in O2i je: d[Ai] = [A i], in/and d { U(t) } = { U(t) } i = 1,2 dt The displacement between the points O1i and O2i is: DL,1(t) = |{Li ,1(t)}|-{Li ,1(t = 0)}| (7). Premik prvega elementa v i-tem zahtevnem elementu je: The displacement of the first element in the i-th complex element is: DUi,1 ( t ) DLi ,1 ( t ) , DLi1 ( t ) <0 0, (8). Hitrost premika prvega elementa v i-tem zahtevnem elementu izrazimo takole: The velocity at which the displacement of the first element in the i-th complex element takes place is expressed as: d DUi,1 t= (^())dd t{ ()} - dd t{ ( 0 )}dd t{ ()} {&1,t ()}T { 1,i ()} Lt Lt (9). Razdalja med točko 1 in točko 2 v i-tem zahtevnem elementu prve polovične gredne vezi in druge polovične gredne vezi je naslednja: {L1,i (t)} The distance between point 1 and point 2 in the i-th complex element of the first half-coupling and the second half-coupling is equal to: {Li,2 (t)}={X} {X} {X 0 O1 = { X0 O2+ { U t O2+ [ A2 ]({^} + [ g2i ]{ x2,2 }) - U t O1 - A1 ]({x} + [ G1i ]{ X1i,1 }) (10), kjer je {*). }T = [a1x, a1y, 0]; {X2i,2}T = [-a2x, -a2y, 0] Časovni odvod je: where {X1i,1}T = [a1x, a1y, 0]; {X2i,2}T = [-a2x, -a2y, 0] (see Fig.3). The derivative with respect to time is as follows: dt ()} = { UtL+[A2]({^} + [ G2i ]{ X2i,2 }) - { U ()}O1 -[A1]({ X1 } 1i+ [ G1i ]{ X1 ,1 }) (11). Premik med točkama 2 in 1 v drugem elementu i-tega zahtevnega elementa je: The displacement between points 2 and 1 in the second element of the i-th complex element is equal to: DLi,2 t= { 2 ()}|- { 2 ( t=0 )}| DUi,2 t 0, DL,2(t)>0 DL,2, DL,2(t)<0 (12). (13). Hitrost premika v drugem elementu zahtevnega i-tega elementa lahko izrazimo takole: The velocity of the displacement in the second element of the i-th complex element can be expressed as: 2 isnnatäieflMliBilrSfl | | ^SsFÜWEIK | stran 602 Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model d {& ()}T { ()} DU1, 2 t= DLi,2 t = i,2 i,2 & () dt( ())L t{Li,2 }L t (14). Razdalja med točko 3 in točko 4 v i-tem zahtevnem elementu prve in druge polovične gredne vezi je: The distance between point 3 and point 4 in the i-th complex element of the first and the second half-coupling is equal to: {Li,3 (t)}={X} {X} ={X0} 1i,3 O2 {()} []({ } [ ]{ }) U t O2 + A2 X2 2i +G2i X2i 4 - {X 0 O1 { ()} []({ } [ ]{}) U t O1 - A1 X1 1i +G1i X1i,3 (15), kjer je {X . }T = [b1x, 0, -a 1z]; {X2i4}T = [-b2x, 0, -a 2z] where {X1i,3}T = [b1x, 0, -a 1z]; {X2i4}T = [-b2x, 0, -a 2z] Časovni odvod vektorja je: The derivative of the vector with respect to time is as follows: dt {Li,3 () } = {U ( t }+[A2]( { X2 }2i + [ G2i ] {X2i,4})-{U () } O1 -[A1]( { X1 }1i + [ G1i ] {X1 ,3}) (16). Premik tretjega elementa i-tega zahtevnega elementa The displacement of the 3rd element of the i-th je naslednji: complex element is as follows: DUi,3 (t) = DLi,3 (t) = {Li,3 (t)} - {Li,3 (t = 0)} (17). Hitrost premika v tretjem elementu i-tega zahtevnega elementa lahko izrazimo takole: The velocity of displacement in the third element of the i-th complex element can be expressed as: d T DUi,3 t ) =jd{ ()} -\{k( =0 )}\y &i,3 ()}T { i,3 ()} Lt Lt dt Li,3 t (18). i,3 i,3 { ()} Sila, ki deluje na prvi element i-tega zahtevnega The force acting on the first element of the i-th elementa, je: kjer je { i,1 ()} i,1 ( i ,1 ()& i,1 ()) F t =F DU t,DU t where complex element is as follows: ()} Li ,1 t (19), {Li,1 (t)} n1 F i,1 ( DU i ,1 ( t ) ,DU i ,1 ( t )) = ( F T ,i,1 +f T,i,2 FN+f T,i,3 FN ) sign ( DU i,2 ( t )) +X( kÄk 'DUk ( t ) +h DU k > ( t ) (20), k=1 where FT,i1 is the frictional force; Fi ,N2,Fi ,N3 are the normal kjer je FT,i,1 sila trenja; Fi ,N2,Fi ,N3 sta normalni sili, ki sta iz drugega in tretjega elemeta preneseni na prvi element; f, f sta koeficienta trenja; k1k, h1 k sta koeficienta togosti oziroma dušenja; n je'število elementov. Sila, ki deluje na drugi element i-tega zahtevnega elementa, je enaka: forces transferred to the first element from the second and third elements; fT,i,2, fT,i,3 are the frictional coefficients; k1i,2i,k, h1i,2i,k are the stiffness and damping coefficients, respectively; and n1 is the number of elements. The force acting on the second element of the i-th complex element is equal to: kjer je { i,2 ()} i,2 ( i,2 ()& i,2 ()) F t =F DU t,DU t where {Li,2 (t)} {Li,2 (t)} (21), n2 F i,2 (ACi2 ( t ) ,DU i2 ( t )) = ( F T i,2 +f T,i,1 FN+f T FN ) sign ( DU i,2 ()) + X( k DUk ( t ) + h • DU 1,2 ( t ) (22), k=1 kjer je FT,i,2 sila trenja; Fi ,N2,Fi ,N3 sta normalni sili, ki sta iz prvega in tretjega elementa preneseni na drugi strukturni element; fT,i,2, fT, sta koeficienta trenja; k, h12k sta koeficienta togosti oziroma dušenja; n je število elementov. Sila, ki deluje na tretji sestavni element, je enaka: where FT,i,2 is the frictional force; Fi ,N2 ,Fi ,N3 are the normal forces transferred to the second structural element from the first and third elements; fT,i,2, fT,i,3 are the frictional coefficients; k12,k, h12,k are the stiffness and damping coefficients, respectively; n2 is the number of elements. The force acting on the third structural element is equal to: I isfinHi(s)bJ][M]ifln;?n 04 stran 603 I^HSSTTIMIDC Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model {}(){{Li,3((t))}} F t =F DU t,DU t i,3 i,3() i,3 i,3() &i,3() Lt n3 F i,3 ( DU.3 ( t ) , DU i3 ( t )) = Xk • DUk 3 ( t ) + h • DUk 3 ( t ) (23) (24), kjer sta *, L34,t koeficienta togosti oziroma dušenja. Moment sile lahko izrazimo takole: where k34,k, h34,k are the stiffness and damping coefficients, respectively. The moment of force can be expressed as follows: {M}={r}x{F}=[B]{F} (25), kjer je [B] antisimetrična matrika: where [B] is the skew-symmetric matrix: [B] 0 -rz ry rz 0 -rx -ry rx 0 in so r, r, r projekcije razdalje. ' Vektor vsote sil, ki delujejo na prvi, drugi in tretji element v vsakem zahtevnem elementu prve polovične gredne vezi glede na točko O1, je: and r, r , r are the projections of radius. Z "The vector of the total force relative to the point 01 of the forces acting on the first, second and third elements in each complex element of the first half-coupling is: NZ 3 {FO1} = XX{Fi,j} (26). i=1 j=1 Vektor celotnega momenta sil, ki delujejo na The vector of the total moment relative to prvi, drugi in tretji element v vsakem zahtevnem the point 01of the forces acting on the first, second elementu prve polovične gredne vezi glede na točko and third elements in each complex element of the O1, je: first half-coupling is: {M NZ 3 XX{ i=1 j=1 x F i=1 j=1 O1 ]ij {Fi, j } (27), kjer so {r02} koordinate točke, v kateri deluje sila {K }; [Bm].. je antisimetrična matrika, ki je dobljena iz vektorja {rrt,}.. Vektor vsote sil, ki delujejo na prvi, drugi in tretji element v vsakem zahtevnem elementu druge polovične gredne vezi glede na točko 02, je: Vektor celotnega momenta sil, ki delujejo na prvi, drugi in tretji element v vsakem zahtevnem elementu druge polovične gredne vezi glede na točko O2, je: where {r2}.. are the coordinates of the point in which the force #.} acts; [B1] is the skew-symmetric matrix that is'generated from the vector {r } . The vector of the total force relative to the point 02 of the forces acting on the first, second and third elements in each complex element of the second half-coupling is: NZ 3 XXF j} (28). i=1 j=1 The vector of the total moment relative to the point 02 of the forces acting on the first, second and third elements in each complex element of the second half-coupling is: {M NZ 3 -XX{ i=1 j=1 x F -XX[B i=1 j=1 O2]ij {Fi,j } (29), kjer so {r} koordinate točke, v kateri deluje sila -{K }; -[B2] 'je antisimetrična matrika, ki je dobljena iz vektorja {r02},. 1.3 Enačbe gibanja rotorja Osnovna shema splošnega rotorskega sistema je prikazana v diagramu 1. Rotorski sistem sestoji iz dveh gredi, na katerih sta pritrjeni dve where {rO2}i,j are the coordinates of the point in which the force {Fi,j} acts; -[BO2]i,j is the skew-symmetric matrix generated from vector {rO2}i,j. 1.3 Equations of Rotor Motion The principal scheme of a general rotor system is presented in Fig.1. The rotor system consists of shafts, on which half-couplings are 2 jgnnatäfcJllMlililrSO | | ^MSSfiflMffiC | stran 604 Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model Sl. 4. Palični končni element Fig. 4 The beam finite element polovični gredni vezi; gredi sta podprti z dvema ali več gibkimi oporniki (dušeni ali nedušeni). Vsaka gred je razdeljena v palične končne elemente z naslednjimi lastnostmi: material elementov je homogen z gostoto r, Youngov modul je E in strižni modul je G; intervalno dušenje gredi zanemarimo; prerez je krožen, uporabljena je Bernoulli-Eulerjeva teorija togega (konzolnega) vpetja. Število prostostnih stopenj v vozliščni točki ješest (sl. 4). Posplošene koordinate paličnih končnih elementov so: mounted, supported by two or more flexible supports (damped or undamped). The shaft is divided into beam finite elements with the following properties: the material of the element is homogeneous, with density r, Young’s modulus E and shear modulus G; the interval damping of the shaft is neglected; the cross-section is circular; the theory of Bernoulli-Euler clam is applied. The number of degrees of freedom at a nodal point is six (Fig. 4). The generalized coordinates in the beam finite elements are: [u1,v1, w1,j1,q ,u2,v2, w2,j2,q J (30), kjer sta qy, qz kotna premika rotorja: where, qy, qz are angular rotor displacements, dw ; dx Geometrične značilnosti paličnega končnega elementa so naslednje: L je dolžina elementa; A je prerez; J je polarni vztrajnostni moment glede na os x X; J, J sta prečna vztrajnostna momenta glede na osi Y, Z; Jp je polarni vztrajnostni moment; J je prečni vztrajnostni moment. Kinetično energijo paličnega končnega elementa lahko izrazimo takole: dv qz= dx The geometrical properties of the beam finite element are as follows: L is the length of the element; A is the cross-sectional area; Jx is the polar moment of inertia with respect to the X axis; Jy, Jz are the transverse moments of inertia with respect to the Y, Z axes; Jp is the polar moment of inertia; JD is the transverse moment of inertia. The kinetic energy of the beam finite element can be expressed as: 1 L T =-^ rA u T u v ¦ v w w dx (31), kjer so u,v,w premiki glede na osi X, Y in Z. Kinetična energija diska je: where u&,v&, w& are displacements with respect to the X, Y and Z axes. The kinetic energy of the disk is: WX \4>] disk ¦ r i J }0y f [Tdisk\ dx (32), kjer je Ldis k- dolžina diska; [Td sk] je matrika disk where Ldisk- is the length of the disk; [Tdisk] is the matrix [Tdisk ]=r Jp 0 -Jpqy 0 JD 0 -Jpqy 0 JD grin^diJjpsflDsijai 04-12 stran 605 |^T§§inR001!C Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model Deformacijsko energijo paličnega elementa lahko izrazimo: The strain energy of the beam element can be expressed as: P (e) J EA (du l dx EJ dv dx2 EJ dx2 GJ dj dx d2v dx2 d2w dx2 dx (33), kjer je P vzdolžna obremenitev. Premiki in kotni premiki rotorja so v približku naslednji: where P is the axial load. The displacements and the angular rotor displacements are approximated as follows: u j v =[N1]{q}, q w qz J [N2]{q} (34). [] N1 N5 0 0 0 0 0 N6 0 0 0 0 0 0 N1 0 0 0 N2 0 N3 0 0 0 N4 0 0 N1 0 -N2 0 0 0 N3 0 -N4 0 N2 [] 0 0 0 N5 0 0 0 0 0 N6 0 0 0 0 dN2 -dx 0 dN2 dx 0 0 0 dN3 dx 0 dN4 dx 0 0 dN1 dx 0 0 0 dN2 dx 0 dN3 dx 0 0 0 dN4 dx kjer so Ni = Ni(x) (i = 1, 2, ..., 6) oblikovne funkcije; N1(x), ..., N4(x) polinomi tretjega reda; in N5(x), N6(x) polinoma prvega reda. Če v obrazcih za energijo (31) in (32), upoštevamo obrazca (34), dobimo naslednja obrazca za kinetično in deformacijsko energijo paličnega končnega elementa z diskom: where Ni = Ni(x) (i = 1, 2, ..., 6) are the shape functions; N1(x), ..., N4(x) are the third-order polynomials; and N5(x), N6(x) are the first-order polynomials. Substituting expressions (34) into the energy expressions (31) and (32) gives us the following kinetic and strain-energy expressions for a beam finite element with a disk: Te= 12{} ([M 1] + [M 2( q )]) { q } 2 kjer je where L 0 L [] N3 dx 0 0 0 d2N1 dx2 0 0 d2N1 [M2 e (q)] = l[N2]T[Tdisk][N2]dx [Ke] = L ([N3]T[D][N3]dx 0 dN (35) (36), (37) (38) (39). dx2 0 0 dN5 dx 0 0 6 dx 0 0 0 d2N2 dx2 0 d2N3 dx2 0 d2N2 0 0 0 d2N3 dx2 0 dx2 0 0 dN6 dx 0 0 0 dN dx d2N4 0 dx2 0 [D] = diag(EA,(EJz + P),(EJy + P),GJp) (40). Enačbe gibanja končnega elementa rotorja dobimo z uporabo Lagrangeve enačbe drugega reda in jih lahko prikažemo z matrično enačbo: The equations of motion of the finite element of the rotor are obtained by using the second-order Lagrange equation, and can be represented by the matrix equation: 2 jgnnatäüllMliBilrSO | | ^SsFÜWEIK | stran 606 Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model ([MW] + [MW(q)]){q} + [M 2W(q)]{q} + [^W]{q} = {FW(,q,q )} (41). 1.4 Enačbe gibanja sklopke Učinke vrteče se polovične gredne vezi lahko izpeljemo iz enačb gibanja togega diska. Enačbo kinetične energije sklopke lahko izrazimo: 1.4 Equations of Motion of the Clutch The effects of a rotating half-coupling can be derived from the equations of motion for a rigid disk. The kinetic-energy expression of the clutch can be presented as: q 1 Iq 3J 1 T1 [] q 1 q 2 Iq 3J + 2 q 8 T2 [] + " q 11 q 12 q&10 Iq 9 J [A][r4][A] q q Iq 9J 1 q 4 ^+ 2^5 Iq 6J [D1 ]T [T3 ][D1] q 4 q 5 Iq 6J + (42), q}r([M3] + [M4(q)]){q} kjer so where [T1] = diag (m1,m1,m1) [T2] = diag (m2,m2,m2) [T3] = diag (Jp1, JD1, JD1) [T4] = diag (Jp2 , JD2 , JD2) [] D1 1 0 -qy1 [A] = 1 0 -qy2 0 1 j1 0 1 j2 0 -j1 1 0 -j2 1 m1 , m sta masi prve oziroma druge polovične gredne vezi; J , J sta masni polarni vztrajnostni moment in masni prečni vztrajnostni moment polovične gredne vezi (i = 1, 2). Enačbe gibanja sklopk dobimo z uporabo Lagrangejeve enačbe drugega reda in jih lahko prikažemo z matrično enačbo: m1, m2 are the mass of the first and the second half-couplings, respectively; Jpi, JDi are mass polar inertia moment and transverse inertia moment of half-coupling (i = 1, 2). The equations of motion of the clutches are obtained by using the second-order Lagrange equation and can be represented by the matrix equation: ([M3cuch] + [M4cuch ( q )]){ q } + [M4cuch ( q )]{ q } = { Fcuch ( q,q )} (43), kjer je {Fc,u,ch (q,q)} nelinearni vektor obremenitve. where {Fcu'ch (q,q)} is the non-linear load vector. 1.5 Sistem enačb asinhronega motorja 1.5 System of Equations of the Asynchronous Engine Sistem enačb asinhronega motorja lahko zapišemo kot ([11], [12] in [14]): The system of equations of the asynchronous engine can be written as ([11], [12] and [14]): { Z } = [A^]{ Z } + { B^( Z,j 1 )} (44), kjer sta [A ] in {B(Z,j 1)} matrični oziroma where [Aasyn] and {B„(Z,j 1)} are the matrix vektorski element, ki sta odvisna od rotorske in statorske induktivnosti ter od števila polarnih dvojic; j 1 je kotna hitrost rotorja asinhronega motorja. Vrtilni moment asinhronega motorja je nelinearna funkcija elementov vektorja {Z}, Masyn(Z). 1.6 Sistem enačb rotorskega sistema Splošni matematični model rotorskega sistema lahko oblikujemo po enačbah (44), (41) in (42): sistemov enačb gibanja asinhronega motorja, rotorjev in sklopk. Splošni sistem enačb lahko and vector elements that depend on rotor and stator inductivities, and the number of pole pairs; j&1 is the angular velocity of the rotor of an asynchronous engine. The torque of an asynchronous engine is a nonlinear function of the elements of the vector {Z}, Masyn(Z). 1.6 System of Equations of the Rotor System A general mathematical model of the rotor system can be constructed from (44), (41) and (42): the systems of equations of motion of an asynchronous engine, the rotors and the clutch, gfin^OtJJIMISCSD 04-12 stran 607 |^BSSIfTMlGC Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model prikažemo takole: respectively. A general system of equations can be represented as follows: [M ( Y )]{ Y } + [C ( Y ) kjer so [0] [0] [0] !Mq(q)] -\a 1 |_ asyn J [0] [0] Kq [M (Y)] [K] [E] je enotna matrika 4 x 4. Poznamo mnogo metod, ki jih lahko uporabimo za numerično časovno integracijo sistema enačb (45). Na splošno lahko te metode klasificiramo kot izrecne ali posredne sheme ([15] in [16]). Izrecne sheme so preproste z računskega vidika, a dolžina njihovega časovnega intervala je odvisna od stabilnosti sistema. Posredne sheme zahtevajo več preračunavanja za posamezničasovni korak, a njihove omejitve dolžine časovnega koraka niso tako stroge. Uporabljali smo posredno shemo, osnovano na trapeznem pravilu. Ob predpostavki, da so vse spremenljivke v enačbi (45) znane za čas tk, smo trapezno pravilo za ta primer razvili takole: + [ K ]{ Y } = { F ( t,Y,Y )} where (45), [E] [0] [0] fCq(q)] { F ( Z,q,q )}\ [E] is the 4x 4 identity matrix. There are many methods that can be used for the numerical time integration of a system of equations (45). Generally speaking, these methods can be classified as either explicit or implicit schemes ([15] and [16]). Explicit schemes are computationally simple but the time-step size is limited by stability considerations. Implicit schemes require more computation per time step, but time-step size limitations are much less stringent. We have used an implicit scheme based on the trapezoidal rule. Assuming all the variables in equation (45) are known at time tk, the trapezoidal rule for this problem is: Dt ({Y& 1), YL = { Yl Dt 2 ({Y& & Y + Y 1) (46) in če združimo obrazca (46), dobimo vektor pospeška za čas tk+1: and combining expressions (46), we obtain the vector of acceleration at time tk+1: PLD4t { Y } {} -{Y }-{Y } (47). Z upoštevanjem formul (46) in (47) v sistemu enačb (45) dobimo sistem nelinearnih algebrskih enačb: Substituting expressions (46) and (47) into the system of equations (45) we obtain a non-linear algebraic system of equations: [F( { Y )] = 0 (48). Dobljeni sistem nelinearnih algebrskih enačb (48) smo rešili z uporabo Newtonove metode ([11] do [14]). 2 NUMERIČNI REZULTATI Preučili smo rotorski sistem z gibko sklopko in neporavnanima grednima osema. Sklopka sestoji iz prosto nameščenih obročev (zahtevni element), čigar osi ležijo pravokotno na os sklopke (sl. 5). Izračunali smo prečne togosti sklopke in izdelali krivulje pospeškov sil za različne vrednosti neporavnanosti grednih osi (e) ter različna števila obročev (NZ) (sl. 6). The obtained system of nonlinear algebraic equations (48) was solved using Newton’s method ([11] to [14]). 2 NUMERICAL RESULTS A rotor system with a flexible clutch and the misalignment of the shaft axes was investigated. The clutch consists of freely located rings (complex element), the axes of which are perpendicular to the axis of the clutch (Fig.5). The radial rigidities of the clutch were calculated and hodographs of the forces for various values of misalignment of the shaft axes (e) and different numbers of rings (NZ) (Fig. 6) were constructed. 2 jgnnatäüllMliBilrSO | | ^SsFÜWEIK | stran 608 Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model a) b) Sl. 5. Gonilna polovična gredna vez z obroči, nameščenimi v režah a - celovit prikaz, b - gonilna polovična sklopka z obroči, nameščenimi v režah Fig.5 Driving half-coupling with rings located in slits a - general view, b - driving half-clutch with rings located in slits a) b) 300,0 150,0 0,0 -150,0 -300,0 -300,0 400,0 200,0 0,0 -200,0 / 3 2 C S ^ r \ \\ Cb s '/ / 1 / —' _,* S S ) 1 /*"-» -600,0 -300,0 0,0 Ry, N 300,0 600,0 Sl. 6. Krivulje pospeškov togosti prečnih sil: sklopka s spremenljivim številom obročev (NZ) in neporavnanostjo grednih osi(e): Fig. 6. Stiffness hodographs of the radial forces: the clutch with a variable number of rings (NZ) and misalignment of the shaft axes(e): a - NZ = 3; b - NZ = 5; c - NZ = 7; 1 - e = 0,5 mm; 2 - e = 1,0 mm; 3 - e = 1,5 mm; r = 0,09m; a1 = a1y = 0,025 m; a1z = 0,01 m; b = b2 = 0,025 m; a2 = a1; a2 = a; a2 = a1; klr = k12i = k 1x 34i = 105 N/m; i = 1 x .. NZ y 1y z gfin^OtJJlMlSCSD 04-12 stran 609 |^BSSITIMIGC Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model Slika 6 kaže, da so radialne sile večje pri večji neporavnanosti grednih osi, kadar imajo povezovalni obroči enake prečne togosti. Poleg tega se sile prečnih togosti povečajo in se oblika krivulje pospeškov togosti prečnih sil približa obliki kroga, kadar se poveča tudi število obročev (NZ), ki povezujejo gonilno in gnano polovično gredno vez. Prečna togost sklopke se periodično spremeni glede na število zahtevnih elementov (obročev) (NZ) in frekvenco vrtenja gredi (w). Frekvenca prečne togosti (W) je enaka: Fig. 6 shows that when the connecting rings are of the same radial rigidities, the radial forces are larger for larger misalignments of the shaft axes. In addition, the forces of the radial rigidities increase and the shape of the stiffness hodograph of the radial forces approaches that of a circle when the number (NZ) of rings connecting the driving and the driven half-couplings is increased. The radial stiffness of the clutch changes periodically according to the number of complex elements (rings) (NZ) and the frequency of the shaft rotation (w). The frequency of radial stiffness (W) is equal to: W-NZ w (49). Kadar se sistem vrti, se lahko pojavijo parametrične vibracije. Da bi zmanjšali pojav parametričnih vibracij, moramo povečati frekvenco prečne togosti sklopke tako, da le-ta ni v fazi s frekvenco vrtenja gredi. Frekvenca prečne togosti sklopke se poveča s povečanim številom elastičnih elementov (NZ), vendar mora biti povečanje NZ zmerno, sicer bo sklopka toga, kar pa ni zaželeno. Lahko povečamo tudi število zahtevnih elementov (NZ), s čimer zmanjšamo njihovo prečno togost. V tem primeru se približamo bandažni sklopki z izboljšanimi značilnostmi vrtilnega gibanja. Ker so takšne sklopke bolj zanesljive, so njihove obratovalne značilnosti primerljive z značilnostmi gumenih (bandažnih) sklopk. [9]. Naš model vključuje rotorski sistem, ki sestoji iz asinhronega motorja (4A100/4SY3), dveh gredi in gredne vezi, narejene iz dveh polovičnih grednih vezi in sedmih zahtevnih elementov (obročev) - sl. 7. Premera in dolžini obeh gredi so d1 = d = 0,040 m in L1 = L2 = 1,0 m; vztrajnostna momenta mas polovičnih grednih vezi sta Ip1 = Ip 2 = 0,624 kgm2; ID1 = ID = 0,312 kgm2; ki = 106 N/m; h = 103 Ns/m; (i = 1, 2, ..., 8). Obremenitveni moment i je M = 10 Nm. Časovna odvisnost premika prve polovične gredne vezi v smeri osi Y je prikazana na sliki 8, odvisnost amplitude premika od frekvence pa je podana na sliki 9. When the system rotates, parametric vibrations can occur. To reduce the occurrence of parametric vibrations, it is necessary to increase the frequency of the clutch radial stiffness in such a way that it is not in phase with the frequency of shaft rotation. But the frequency of the clutch radial stiffness increases with an increasing number of elastic elements (NZ). However, the increase in NZ must be moderate, because, otherwise, we will have a stiff clutch, which is undesirable. One can also increase the number of complex elements (NZ), simultaneously decreasing their radial stiffness. In such a case, we approach a tyre-type clutch with improved characteristics of rotational motion. As such clutches are much more reliable, their operational characteristics are better compared with rubber (tyre-type) clutches [9]. A rotor system consisting of an asynchronous engine (4A100/4SY3), two shafts and a coupling made of two half-couplings and seven complex elements (rings) (Fig.7) is considered. The diameters and the lengths of both shafts are equal tod1 = d2 = 0.040 m and L1 = L2 = 1.0 m, the inertia moments of the half-couplings’ masses are Ip1 = Ip2 = 0.624 kgm2; ID1 = ID2 = 0.312 kgm2; ki = 106 N/m; hi = 103 Ns/m; (i = 1, 2, ..., 8). The load moment Mload = 10 Nm. The dependence of the displacement of the first half-coupling in the direction of the Y axis on time is shown in Fig. 8, while the dependence of the displacement amplitude on frequency is given in Fig.9. Sl. 7. Shema rotorskega sistema Fig. 7. A scheme of the rotor system 2 jgnnatäüllMliBilrSO | | ^SsFÜWEIK | stran 610 Bogdevi~ius M., Spruogis B., Turla V.: Dinami~ni model - A Dynamic Model q (38), m 0,0E + 00 -3 ,0 E -0 6 -6 ,0 E -0 6 -9 ,0 E -0 6 -1 ,2 E -0 5 -1 ,5 E -0 5 Ulli mini im ll 111 1 III llll Allllll ill III flilfin llllllll pNl If w 5|B II ||ll ||i' 1° III 0 ,0 1 ,0 2 ,0 3 ,0 4 ,0 5 ,0 6,0 t, s Sl. 8. Časovna odvisnost premika q38 prve polovične gredne vezi v smeri osi Y Fig. 8. The dependence of the displacement q38 of the first half coupling in the direction of the Y axis on time -5,0 -7,0 -9,0 -11,0 n 0 100 200 300 400 500 f, Hz Sl. 9. Odvisnost amplitude premika q38 od frekvence Fig. 9 The dependence of the displacement q38 amplitude on frequency 1. 2. 3 SKLEPI Naš dinamični model rotorskega sistema z gibko, centrifugalno gredno vezjo opiše prostorska, geometrična, tehnološka in konstrukcijska odstopanja pa tudi dinamične posebnosti. Z modelom lahko ovrednotimo sistem z želenim številom prostostnih stopenj na naslednje načine: - s povezovanjem koordinat začetnega sistema z ustreznimi koordinatami deformiranega sistema, pri čemer uporabljamo matematične odvisnosti; - s prikazom medsebojnih odnosov med začetnimi in deformiranimi točkami v različnih koordinatnih smereh; - s prikazom matematičnega odnosa med silami sestavljenih elementov rotorskega sistema in momenti, pa tudi njihov vpliv na gredi in oporne dele; - z določitvijo kinetične in potencialne energije kinematičnih in dinamičnih parametrov deformiranega sistema, lege in oblike elastičnih grednih vezi, ki povezujeta sistem. 1. 2. 3 CONCLUSIONS Our dynamic model of a rotor system with a flexible, centrifugal coupling describes the spatial, the geometrical, the technological and the constructional deviations as well as the dynamic peculiarities. The model can evaluate the system with a desired number of degrees of freedom in the following ways: - Connecting the initial system’s coordinates with the corresponding coordinates of the deformed system using mathematical dependences; - Showing the interrelations of the initial and deformed points in various directions of the coordinates; - Showing a mathematical relation between the forces of the rotor system’s composite elements and the moments, as well as their influence on the shafts and the supporting parts; - Finding the kinetic and potential energies of the deformed system’s kinematic and dynamic parameters, and the position and the shape of the elastic couplings connecting the system. [1] 4 LITERATURA 4 REFERENCES Bert, C.W., S. Wu (2003) Dynamic analysis of a nonlinear torsional flexible coupling with elastic links. Journal of Mechanical Design. 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Computer Methods in Appied Mechanics and Engineering. Vol. 17, 277-313. Naslova avtorjev: prof. dr. Marijonas Bogdevičius prof. dr. Bronislovas Spruogis Fakulteta za transportno tehniko Tehnična univ Vilnius Gediminas Sauletekio al. 11 LT-2040 Vilnus, Litva marius@ti.vtu.lt doc. dr. Vytautas Turla Fakulteta za mehaniko Tehnična univ. Vilnius Gediminas J. Basanavicius str. 28 LT-2006 Vilnus, Litva pgkatedra@me.vtu.lt Authors’ Addresses: Prof.Dr. Marijonas Bogdevičius ProfDr. Bronislovas Spruogis Faculty of Transport Eng. Vilnius Gediminas Tech. Univ. Sauletekio al. 11 LT-2040 Vilnus, Lithuania marius@ti.vtu.lt doc. dr. Vytautas Turla Mechanical Faculty Vilnius Gediminas Tech. Univ. J. Basanavicius str. 28 LT-2006 Vilnus, Lithuania pgkatedra@me.vtu.lt Prejeto: Received: 28.7.2003 Sprejeto: Accepted: 30.9.2004 Odprto za diskusijo: 1 leto Open for discussion: 1 year 2 jgnnatäüllMliBilrSO | | ^SSfiflMlGC | stran 612