Proceedngs of the 8 th IFToMM Internatonal Conference on Rotordynacs Septeber 1-15, 010, KIST, Seoul, Korea NON-SYNCHRONOUS TILTING PAD BEARING CHARACTERISTICS Joach Sched DELTA JS Zurch, Swtzerland Alexander Fedorov Dept. of Physcs and Mechancs, SPbSPU St. Petersburg, Russa Bors S. Grgorev Dept. of Physcs and Mechancs, SPbSPU St. Petersburg, Russa ABSTRACT The characterstcs of flud fl bearngs are speeddependent. Recently t s dscussed ore and ore, that the characterstcs ay also be frequency dependent, especally n case of tltng pad bearngs. In the present paper a odel for tltng pad bearngs ncludng the pad nertas and coordnates for the pad rotatonal angles and ther rotatonal veloctes s created. Dynac bearng coeffcents are calculated ncludng these coordnates by solvng the Reynolds equaton for the perturbed equlbru poston. The perturbaton ncludes the rotor dsplaceents and veloctes as well as the pad tltng angles and angular veloctes. Thus a syste of lnear +Np dfferental equatons wth Np as the nuber of pads s created for the bearng. Fro these equatons transfer functons for the bearng force / rotor dsplaceent relaton can be deterned. These transfer functons clearly yeld frequency dependent stffness and dapng coeffcents, although all coeffcents of the coplete syste wth +Np equatons are only speed dependent. The tltng pad bearng odel wth the +Np equatons s ncluded nto a rotor odel by eans of the rotor dynac progra MADYN 000 [1]. The rotor dynac stablty of the rotor consderng the non-synchronous behavour of the tltng pad bearngs s presented for dfferent bearng geoetres showng cases wth strong and weak frequency dependence. INTRODUCTION Flud fl bearng forces on the rotor are norally odeled by stffness and dapng atrces and soetes wth ass atrces as shown n the followng equaton, whch ncludes the ass ter: The dscusson about the frequency dependence of the rotor dynac coeffcents of tltng pad bearngs s controversal. A good overvew of the research n ths area s (1) gven n the very recent paper of C.R. Carter and D.W. Chlds []. In the sae paper easureents are copared to calculated results. The an concluson s that a bearng odel ncludng stffness, dapng and ass coeffcents yelds good agreeent to easured results wthout consderng any frequency dependence. In contrast to [] the bearng forces n [] are forulated wth stffness and dapng coeffcents only,.e. wthout a ass ter. An adaptaton of the forces to easureents clearly yelds frequency dependent coeffcents. The stffness decreases wth ncreasng frequency, whereas the dapng ncreases. In the odel presented here the bearng forces are forulated n a ore general way than n equaton (1) ncludng coordnates for the tltng of the pads and consderng the pads oent of nerta, whereas n the bearng odels generally used nowadays the coordnates for the pad tltng are elnated. In [4] s shown that ths ples neglectng tangental pad forces whch s true f the pad edately follows any shaft oveent,.e. the pad nerta and the nhbton of the pad tltng by the flud fl dapng are neglected. The resultng stffness and dapng coeffcents for the lateral journal coordnates n [4] are purely speed dependent. The startng pont for the analyss of the bearng forces of the extended odel presented here s the Reynolds equaton, as n all coon odels. The vscosty s assued as constant. In a frst step the Reynolds equaton s solved to receve the statc equlbru of the rotor journal and each pad due to the statc forces. In a second step the equlbru s perturbed. The perturbaton ncludes the rotor dsplaceents and veloctes as well as the pad tltng angles and angular veloctes. NOMENCLATURE d dapng coeffcent e journal excentrcty h ol fl thckness k stffness coeffcent ass coeffcent, preload p t pressure te 1 `
t p x z C p C b F I p L R b R j R p U D G K M X ε pad thckness dsplaceent coordnate axal drecton R p -R j, pad clearance R b -R j, bearng clearance force pad oent of nerta bearng wdth bearng radus rotor journal radus pad radus Ω * R j, crcuferental speed dapng atrx gyroscopc atrx stffness atrx ass atrx vector of coordnatess densonless exctrcty µ ol vscosty ω frequency n [rad/s] Θ crcuferental angle Ψ pad tltng angle Ω rotor speed n [rad/s] Indces -drecton -drecton R ndex for rotor MATHEMATICAL MODEL Bearng force analyss of tltng pad bearngs The general geoetry of one pad of a tltng pad bearng s shown n fgure 1. Usng the followng densonless paraeter wth the reference pressure,,, (1) The Reynolds equaton can be wrtten as follows: h θ p + υ h θ z p h h = + z θ t Further n the paper we wll ot the bar above the varables, workng only wth densonless varables. For the ol fl thckness the followng equaton apples: h ψ 1 ( θ, z) = ε x x ( p ) 0 cosθ cosθ snθ cos θ θ χ sn( θ θ ) wth () () (4) The ol fl thckness can be presented as the su of the thckness n equlbru poston and an addtonal perturbaton ter: wth p (5) (6) h 1 ( θ θ ) ψ χ ( θ θ ) = ε θ sn 0 0 cos cos p 0 p (7) and For a sall perturbaton δh a slar decoposton can be ade for the pressure. (8) x Fg.1 Pad geoetry of a tltng pad bearng x By substtutng (9) nto (), neglectng ters of second and hgher order and usng (7) and (8), we can wrte equatons for p 0 and δp. The structure of the rght hand sde n the equaton for δp allows to wrte the perturbaton pressure ter δp n the followng for: δp + χ ( t, θ, z) = px ( θ, z) x px ( θ z) x p x ( θ z) x +, + 1, + p x ( θ, z) 1 p ( θ, z) δψ + p ( θ, z) δψ ( ) δψ 1δψ (9) x (10) `
Thus we receve a set of equatons for separate coponents of δp fro whch the dynac bearng force can be wrtten n the followng for: (11) (1) The vectors and atrces n (11) and (1) are: X : Vector of the lateral journal postons (coordnates x, x ). Ψ : Vector of the tltng angles of each pad. F : Vector of bearng forces n - and -drecton. I p : Dagonal atrx wth the oents of nertas of each pad : Matrces wth speed dependent dapng and stffness coeffcents. The coeffcents n equatons (11) and (1) are only speed dependent, not frequency dependent. The transfer functons between bearng forces and lateral journal coordnates n equaton (1) can be ganed by assung haronc forces, dsplaceents and tltng angles wth the sae frequency and elnatng the tltng angles. (1) A 4-tltng pad bearng wth 55% offset () A 5-tltng pad bearng wth 55% offset () A 4-tltng pad bearng wth 60% offset The load n all three exaples s orented between the pads. The bearngs are rather lghtly loaded: The specfc bearng load s 5bar. The geoetry of the three bearngs s shown n fgures 1 to. Ol wth a vscosty accordng to ISO VG46 wth an nlet teperature of 40 o C s used. The ean teperature at a speed of 6000rp accordng to a DIN analyss (see [5]) s 55 o C. (1) contan the coplex apltudes n - and - drecton. It can be seen, that equaton (1) s a relaton, whch cannot be sply expressed by constant ass, dapng and stffness coeffcents n the for of equaton (14). Fg.1 4-tltng pad bearng, 55% offset. (14) Cobnng the structure and the tltng pad bearng The equaton of oton for the rotor, whch can also contans the supports and other bearngs than tltng pad bearngs, s as follows: (15) The vector contans all structural coordnates of the rotor and of the bearng supports. The vector F contans the forces on the rotor and the support, aong others the bearng forces of the tltng pad bearngs. Substtutng equaton (11), whch s coupled to equaton (1), nto equaton (15), yelds the equaton of the cobned syste. Equatons (11) and (1) can be rearranged to receve a state space for, whch can be used to odel the tltng pad bearng n a slar way as a agnetc bearng n MADYN 000 [1]. Fg. 5-tltng pad bearng, 55% offset EXAMPLE Bearngs The bearng characterstcs of the extended odel are shown for three dfferent tltng pad bearngs: Fg. 4-tltng pad bearng, 60% offset `
Rotor The effect of the bearngs on a rotor are studed wth the exaple rotor n fgure 4. Fg.4 Exaple rotor The rotor has a axu speed of 6000rp or 100rps. It s rather sl and has a hgh rato of the axu speed to the frst natural frequency (so called flex rato) of about. RESULTS Bearng Characterstcs The frequency dependent stffness and dapng coeffcents for lateral coordnates of the three bearngs at a speed of 6000rp or 100Hz are shown n fgures 5 to 7. The coeffcents are derved fro the transfer functons (equaton (1)) as follows: Fg.6 Stffness and dapng coeffcents at 6000rp, 5-tltng pad bearng 55% offset (16) (17) where,j are ndces for the two drectons,. The synchronous coeffcents n the fgures are arked. They agree exactly wth the classcally calculated purely speed dependent dapng and stffness coeffcents for lateral coordnates accordng to the DIN standard [5]. Fg.7 Stffness and dapng coeffcents at 6000rp, 5-tltng pad bearng 60% offset Fg.5 Stffness and dapng coeffcents at 6000rp, 4-tltng pad bearng 55% offset For the 4-tltng pad bearng wth 55% offset the coeffcents are strongly frequency dependent. The stffness ncreases wth decreasng frequency (.e. n the subsynchronous range) and the dapng decreases. For the dapng of the frst natural ode of a rotor, whch s norally n the sub-synchronous frequency range, ths has an adverse effect. Ths trend s n accordance to the easured results n [] of a 5-tltng pad bearng wth central pvot n load on pad as well as load between pad confguratons. The frequency dependence s lower n case of the 5- tltng pad bearng wth 55% offset. Ths s ost probably due to the shorter pads. Increasng the offset of the 4-tltng pad bearng to 60% practcally elnates the frequency dependence. The stffness as well as dapng coeffcent consderably ncrease copared to 55% offset, whch s an unwanted effect n case of flexble rotors, snce ths norally reduces the syste dapng of the frst natural ode. The odel presented here yelds cross couplng coeffcents. However, they are consderably saller than the 4 `
drect coeffcents, except n case of the dapng of the 4- tltng pad bearng at very low frequences. Neglectng the pad nerta practcally yelds the sae results. Ths eans the frequency dependence of the 4-tltng pad bearng wth 55% offset s not anly caused by the nerta but by the nhbton of the pad tltng by dapng. Natural Modes of the rotor bearng syste at axu speed The frst forward whrlng natural ode of the rotor at 6000rp wth the synchronous coeffcents n fgure 5 has a frequency of 1.6Hz and a dapng rato of.7% (log.dec..%). It s shown n fgure 8. t has the lowest stffness. The 4-tltng pad bearng wth 60% offset yelds a lower frequency, n spte of the hgher stffness, whch can be attrbuted to the hgher dapng coeffcent. The dapng rato s low due to the hgh bearng stffness. - 4-tltng pads 55% offset - 5-tltng pads 55% offset - 4-tltng pads 60% offset Fg.10 Egenvalues of the 1 st ode, extended odel CONCLUSIONS Fg.8 Frst natural ode wth synchronous coeffcents, 4-tltng pad bearngs, 55% offset The egenvalues of the 1 st backward whrlng (-) and forward whrlng (+) bendng ode of the rotor on 4-tltng pad bearngs are shown n fgure 9 for two odels of the tltng pad bearngs: The classc odel wth synchronous purely speed dependent coeffcents (black) and the extended odel (red). It can be clearly seen, that due to the hgher stffness and lower dapng of the bearng n the subsynchronous frequency range the frequences slghtly ncrease and the syste dapng rato consderably decreases. - - Synchronous coeffcents - Extended odel Fg.9 Egenvalues of the 1 st ode, 4-tltng pad bearng, 55% offset The egenvalues of the syste wth the dfferent bearng types calculated wth the extended odel are shown n fgure 10. The 5-tltng pad bearng yelds the best dapng, because - + + Tltng pad bearng characterstcs wth an extended odel ncludng the tltng angle of the pads are calculated. Fro the odel the transfer functon of the bearng force / lateral rotor dsplaceent relaton s derved. The transfer functon can also be expressed as frequency dependent stffness and dapng coeffcents. Usng addtonal ass coeffcents would not yeld frequency ndependent coeffcents. The extended bearng odel s cobned wth the rotor odel n the rotor dynac software MADYN 000. The characterstcs of dfferent tltng pad bearngs wth load on pad arrangeent are calculated at the nonal speed of a correspondng rotor. A 4-tltng pad bearng wth 55% offset yelds strongly frequency dependent stffness and dapng coeffcents; the stffness ncreases and the dapng decreases n the sub-synchronous frequency range. Other researchers have found a slar tendency (see []). A 5- tltng pad bearng wth the sae offset yelds less frequency dependent stffness and dapng coeffcents and a 4-tltng pad bearng wth an offset of 60% practcally has frequency ndependent stffness and dapng coeffcents. The synchronous coeffcents n all cases are slar to classcally calculated coeffcents, where the tltng angles are elnated. Applyng the extended bearng odel to a rather flexble rotor wth a flex rato of about can yeld consderably lower dapng ratos of the 1 st natural ode than usng the classcal odel. For the case of the 4-tltng pad bearng the dapng rato drops to less than one thrd. The extended odel s not yet experentally valdated. However, the followng facts confr t to soe extent: (1) Other researchers have found slar trends. () The synchronous coeffcents are dentcal to classcally calculated coeffcents. () Fro the experence wth soe rotors on tltng pad bearngs wth a sall or no offset can be concluded that 5 `
the dapng s uch lower than an analyss wth classc frequency ndependent coeffcents suggests. Bearngs wth a sall offset are soetes used, because they analytcally yeld a lower stffness and thus a hgher syste dapng rato of the frst bendng ode of flexble rotors. The present results of the extended odel do not confr ths behavor ganed wth classcal odels. REFERENCES 1. Sched, J. et.al., Applcaton of MADYN 000 to Rotordynac Probles n Industral Machnery, Proc. GT007, Montreal, Canada, May (007). Carter, C.R. and Chlds D.W., Measureents Versus Predctons for the Rotordynac Characterstcs of a Fv-Pad Rocker-Pvot Tltng-Pad Bearng n Load-Between-Pad Confguraton, ASME Journal of Engneerng for Gas Turbnes and Power, Vol.11, January (009).. Flack, R.D., Wygant K.D. and Barret L.E., Measured Dynac Perforance of a Tltng Pad Journal Bearng over a Range of Forcng Frequences, Proc. 7th IFTOMM-Conference on Rotor Dynacs, Venna, Austra, Sept. (006). 4. Han, D.C., Statsche und dynasche Egenschaften von Gletlagern be hohen Ufangsgeschwndgketen und be Verkantung. Dssertaton TU Karlsruhe, (1979) 5. DIN 1657 Standard, (1996), Berln: Beuth Verlag. 6 `