Parameter Identification of an End Sealed SFD Part II: Improved Predictions of Added Mass Coefficients for Grooved SFDs and Oil Seals.

Transcription

1 Texas A&M University Mehanial Engineering Department Turbomahinery laboratory Parameter dentifiation of an End Sealed SFD Part : mproved Preditions of Added Mass Coeffiients for Grooved SFDs and Oil Seals. Researh Progress Report to the Turbomahinery Researh Consortium TRC-SFD--07 Luis San Andrés Mast-Childs Professor Prinipal nvestigator Adolfo Delgado Researh Assistant May 007. TRC Projet 3513/1519C3

2 Exeutive Summary Added mass oeffiients are not adequately predited for SFD onfigurations inluding grooves or other reesses. This defiieny also extends to grooved floating ring oil seals operating under laminar flow onditions. n oil seals, the grooves are intended to redue the ross-oupled stiffness oeffiients and improve the seal stability harateristis. However, experimental results demonstrate that inner grooves do not redue the fore oeffiients as muh as aepted theory predits. Furthermore, preditive odes for this type of seal do not aount for fluid inertia effets, while experiments render large added mass oeffiients. A bulk-flow model is proposed to analye multiple-groove SFD or oil seal onfigurations. A perturbation analysis yields eroth and first order flow equations defined at eah individual flow region (land and grooves) of onstant learane ( ). At the groove regions, an effetive groove depth ( d ) and learane ( = d + ) are defined based on qualitative observations of the laminar flow pattern through annular avities. This depth may differ from the atual physial groove depth. Boundary onditions between adjaent flow regions impose ontinuity of the pressure and flow fields. The boundary onditions at the inlet and exit planes are a funtion of the geometri onfiguration. ntegration of the resulting dynami pressure fields on the journal surfae yields the fore oeffiients (stiffness, damping and inertia). The preditive model indiates that the damping and stiffness oeffiients derease rapidly with inreasing effetive groove depths ( d ). Damping is roughly proportional to 1/ 3. On the other hand, the added mass oeffiient, whih is proportional to 1/, is less sensitive to hanges in the effetive groove depth. Furthermore, preditions show that the added mass oeffiient inreases as the effetive groove depth inreases until it reahes a maximum value and starts deaying. Comparisons of the predited and experimental fore oeffiients on a grooved oil seal and a SFD show exellent orrelation over a narrow range of effetive groove depths. Speifially, for a short and shallow mid-land groove in the test oil seal, preditions of added mass, ross-oupled stiffness, and damping oeffiients orrelate well with experimental data when using a fration (1/ or less) of the atual groove depth. Most importantly, urrent preditions, as well as the test

3 results, indiate that the inner groove in the oil seal does not isolate the adjaent film lands. For SFD with a side feeding groove, predition of damping and added mass oeffiients also orrelate well with test data for a narrow range of effetive inlet groove depths. Future work will inlude further validation of the model with available experimental data and CFD analysis to determine the effetive groove depths from the path of the streamline dividing the thru-flow and reirulation regions at the grooves. 3

4 Table of Contents Exeutive Summary... ntrodution... 7 Literature Review Grooved SFDs Grooved oil seals Analysis Bulk flow formulation Boundary onditions Single film land- lassial solution model... 0 V Model validation with experimental data... 1 V.1 Grooved Oil Seal... 1 V.1.1 Geometry and model desription... V.1. Results and omparisons to experimental data... 3 V. Sealed SFD... 8 V..1 Geometry and model desription... 8 V.. Results and omparisons to experimental data V Conlusions V Referenes List of Tables Table 1 Oil seal onfiguration, operating onditions and fluid properties Table Test onditions for dynami load tests (CCO). Lubriated SFD List of Figures Figure 1 Shemati view of streamlines in axially symmetri grooved annular avity (ΔP= P s -P d ) Figure Shemati view of grooved annular avity divided into flow regions Figure 3 View of rotating and whirling journal and oordinate system for bulk-flow analysis

5 Figure 4 Shemati view of a simple SFD and boundary onditions... 0 Figure 5 Configuration of parallel oil seals tested in [33].... Figure 6 Partial view of test grooved oil seal geometry [33] and flow regions for preditions... Figure 7 Diret damping and added mass oeffiients versus effetive inlet groove-to-seal learane ratio. Solid lines represent preditions and dotted lines enlose the range of experimental values from [33] for a smooth seal (no mid-land groove) Figure 8 Diret damping and added mass oeffiients versus effetive entral groove groove-to-learane ratio. Solid lines represent preditions for three different effetive mid-land seal groove depths ( d =5, 10, 15 ). Dotted lines enlose the range of experimental values from [33] for two seal depths ( d =, +15 )... 5 Figure 9 Diret damping and added mass oeffiients versus effetive mid-land groove depth. Solid lines represent preditions for two different effetive entral groove depths ( d =6, 11 ). Dotted lines enlose the range of experimental values from [33] for two seal depths ( d =0, 15 )... 7 g Figure 10 Cross-oupled stiffness oeffiients versus rotor speed. Solid lines represent preditions for smooth seal and a effetive mid-land groove depth with g ( d =0,6 ). Dotted lines represent experimental values from [33] for a smooth seal and a grooved seal (mid-land groove depth =16) Figure 11 Sealed-end SFD assembly ut view Figure 1 Test squeee film damper geometry and flow regions used for preditions [14] Figure 13 Predited SFD damping and added mass oeffiients versus effetive inlet groove depths. Solid lines represent preditions for two different inlet effetive groove depths ( = 1,7 ). Dotted lines represent range of experimental values from [14]

6 omenlature C g d g d Diret damping oeffiient [.s/m] Clearane [m] Groove learane (d g + ) [m] Effetive groove learane (d + ) [m] Groove depth [m] Effetive groove depth [m] (f,g) X,Y Dynami pressure funtions [Pa] h Film thikness [m] L Axial length [m] k xy,k yx Cross-oupled stiffness oeffiient [/m] k x, Shear flow fators M Added mass oeffiient [Kg] m Mass flow rates [kg/s] x, umber of flow regions P Pressure [Pa] P X,P Y First-order pressure field [Pa] i maginary number ( 1 ) R Journal radius [m] Re * Modified squeee film Reynolds number s Zeroth order pressure axial gradient [Pa/m] t Time [s] V x, V Bulk flow veloities [m/s] X,Y,Z nertial oordinate system [m] x, Cirumferential and axial oordinates [m] Δe Displaement amplitude [m] μ Absolute visosity [Pa.s] Ω Rotor rotational speed [rad/s] ω Rotor whirling frequeny [rad/s] ρ Oil density [kg/m 3 ] θ Angular oordinate [deg] Subsripts 0 Zeroth order solution exp. Derived from experiments g groove Last annular avity setion model Derived from preditions -th annular avity setion 6

7 ntrodution SFDs were originally analyed as journal bearings using the lassial Reynolds equation negleting both the temporal and onvetive fluid inertia terms. Kuma [1], in early 1967, demonstrates that fluid inertia effets should be onsidered in squeee film dampers. n 1975, Reinhart and Lund [], in a lassial paper, derived the fore oeffiients for journal bearings inluding fluid inertia effets. The authors indiate that the added fluid mass term is relatively small for onventional journal bearings but it may be signifiant in long bearings or squeee film dampers. A year later, Brennen [3] presents an analysis to determine the fluid fores for the flow in an annulus surrounding a long whirling ylinder under the assumption of laminar flow. The results evidene the importane of the fluid added mass effet on the dynami response of the test system. n 1983, Tihy [4] uses an order of magnitude analysis to demonstrate that fluid inertia fores are indeed omparable to visous fores in squeee film dampers for Reynolds numbers equal to 10. n the early 80 s extensive work was onduted by several researhers [3-9] to investigate the influene of fluid inertia effets on the dynami response of SFDs. These analytial efforts to study fluid inertia effets have been partially suessful. Preditions of fluid inertia oeffiients present adequate orrelation with experimental data only for the simplest geometries tested [6]. n the ase of onfigurations with feeding grooves and reesses (mainly shallow ones), added mass oeffiient are largely underpredited by about 50% [6]. As a onsequene, analytial efforts onentrated on studying the influene of irumferential grooves in the fored response of SFDs. Up to date, the added mass terms on SFDs remain to be properly predited for most ommon geometries, as evidened in reent experimental work [14]. Suh defiieny also extends to grooved annular oils seals (floating rig seals) operating in the laminar flow regime [15]. The following review introdues some of the most relevant analytial work on fluid inertia effets on grooved SFD and oil seals, as well as some of the most reent experimental investigations evidening large disrepanies between added mass preditions and test results. 7

8 Literature Review This setion presents a brief review of researh onduted to quantify the effet of fluid inertia on the dynami response of grooved SFDs and annular oil seals, as well as experimental work exposing the disrepany between preditions of added mass oeffiients and test results. The publiations in eah subsetion are presented in hronologial order..1 Grooved SFDs San Andrés [16] analyes the fored response of a squeee film damper with a entral groove. The analysis, based on the short length bearing model and for small entered journal motions, inludes the dynami interation of the flow at the interfae between the squeee film lands and a entral (mid-plane) feeding groove. Analytial results show that a SFD with a shallow groove behaves at low frequenies as a single land damper. Dynami fore oeffiients are determined to be frequeny dependent. Preditions show that the ombined ation of fluid inertia and groove volume-liquid ompressibility affets the fore oeffiients for dynami exitation at large frequenies. Arau and San Andrés [17] use bulk flow equations, as in Ref. [16], but inlude irumferential flow (i.e. no short length bearing assumption) and neglet fluid ompressibility at the groove. The bulk flow equations for small entered journal orbits are amenable to be solved exatly. Preditions of tangential and radial fores at the groove and the SFD lands are ompared to experimental results. nterestingly enough, reorded dynami pressure levels at shallow grooves ( g / < 10) and the film land are of the same order of magnitude. Comparisons with experimental data indiate that the preditions reprodue well the fores at the groove, but underestimate the radial fore and overestimate the tangential (damping) fore at the damper film land. Arau and San Andrés [18] present tests revealing the importane of a irumferential feeding groove and reirulation annuli on the fored response of a test damper. The test results show that the radial fores in a groove with depth-to-learane ratios of 5 to 10 are similar or larger to those at the SFD land (depending on the journal whirl orbit amplitude). For unavitated lubriant onditions, the tangential (damping) fores at the groove are smaller than at the damper film land but still of omparable magnitude. 8

9 Zhang and Roberts [19] present an analysis to predit SFD fore oeffiients inluding the effets of the entral groove and oil supply system. The authors make a balane of the flow and pressure starting at the upstream supply pipe, passing through a reirulation groove, and into the damper film land. Preditions are presented in terms of damping and inertia fore oeffiients, whih are ompared with experimental values obtained by Ellis et al. [0] and Zhang et al. [1]. The omparisons show that the derived formulas marginally improve the predition of damping and inertia fore oeffiients. The analysis fouses more on modeling the inlet pipe line than the flow transition between the groove and the damper film land. An expression for the pressure drop at the groove-film land interfae is obtained by using the inertialess axial veloity profile and integrating the axial momentum and ontinuity equations aross the film thikness. Qinghang et al. [] analye the effet of irumferential grooves on the fored response of SFDs. For the groove, the authors used the same model as that in Ref. [16]. For the squeee film land the authors use a similar approah to that in Ref. [3]. The solution of the pressure field and fluid fores along the SFD follow form the ontinuity equation and equating the pressures at the interfae groove-film land. onetheless, the experimental results show that the radial (inertial) fore is underpredited by a fator of three. Della Pietra and Adilleta [1,13] present a omprehensive review of the state of the art on SFDs up to 00. The review disusses the analytial and experimental investigations on fluid inertia effets on grooved SFDs, and urges for better models to predit fore oeffiients on SFDs with ommon geometries (i.e. inluding irumferential grooves and reesses). Lund et al. [4] present analytial preditions and experimental results of dynami fore oeffiients of a SFD with a entral groove and sealed at both ends with O-rings. An equivalent first-order Reynolds equation is derived and solved using a perturbation analysis for small amplitude entered journal orbits. A bulk flow model is used to desribe the onditions in the entral groove. The experiments onsist of unidiretional dynami load exitations on SFDs with various groove depths and widths. The results show that the damping oeffiient fairly agrees with analytial preditions for inreasing groove volumes (i.e. depth and width). On the other hand, the fluid inertia fore oeffiient tends to be overpredited (up to 70 %) as the groove volume inreases. 9

10 Kim and Lee [5] present experiments and preditions of the fore oeffiients in a sealed SFD with feed and disharge grooves. The theoretial development follows from the bulk flow equations with the pressure and veloity fields as originally given by Mulahy [5] and San Andrés [6]. The dynami film pressure at the groove is obtained as in Ref. [16] where the fluid at the groove is regarded as slightly ompressible. The tests and analysis inludes one-stage and two-stage seal onfigurations. For the two-stage seal, preditions of the inertia oeffiient orrelate well experimental data, while the test damping oeffiients are underestimated. Reently, in 006, San Andres and Delgado [14] experimentally identify fore oeffiients on an end sealed SFD desribing irular entered orbits. The SFD is fed from a side plenum, and inorporates irumferential grooves at the inlet and outlet of the damper. The experimental work aims to evaluate the effetiveness of the end seal in preventing air ingestion and extrat the fore oeffiient from single frequeny exitation tests. The identified damping oeffiients are presented as a funtion of the amplitude of motion and orrelate well with preditive formula found in the literature [6]. On the other hand, the added mass oeffiients are largely underpredited. nterestingly enough, the authors state that more realisti preditions, orrelating well with experimental estimations, are obtained using twie the length of the damper film land. This report provides a detailed analysis to support the improved orrelations.. Grooved oil seals The literature in annular liquid seals is extensive but mainly inludes seals operating under turbulent flow regimes (i.e. Re 000). For the urrent analysis, the ase of interest is grooved oil seals operating under laminar flow regimes. Oil seals are known to indue instability problems in ompressors [7, 8]. A general pratie is to groove the oil seals to redue the ross-oupled stiffness oeffiients and to improve their stability harateristis. This literature review inludes analytial and experimental work onduted on grooved oil seals under laminar flow operation. n partiular, the review emphasies the orrelation between preditions and experiments for added mass oeffiients (i.e. fluid inertia effets). Semanate and San Andrés [15] present an analysis to predit fore oeffiients of a grooved oil seal operating under either laminar or turbulent flow regimes. The analysis inludes the solution of the isovisous, bulk flow equations disregarding fluid inertia 10

11 terms. The analysis aounts for inertial and visous loss effets at the seal entrane but neglets pressure variations aross the irumferential grooves. Preditions indiate that the grooved oil seal ross-oupled stiffness and diret damping oeffiients are signifiantly smaller that those of smooth seals. Suh results indiate that the grooves effetively separate the seal lands and thus redue the seal fore oeffiients. For example, for a seal with a single mid-land groove the ode predits that the ross-oupled stiffness and damping oeffiients would be ¼ of those identified for the smooth seal. As mentioned above, fluid inertia effets in the lands are onsidered negligible in the noted referene. Baheti and Kirk [7] obtain the pressure and temperature distributions in a grooved oil seal from a finite element solution of the Reynolds and energy equations. The study inludes square and irular grooves with and without steps. The groove depths are approximately 6 times the seal learane and the lengths are nearly 1/6 of the seal land length. Preditions indiate that the grooves effetively isolate the seal lands as the stiffness and damping fore oeffiients are redued by approximately 60 % for the grooved onfigurations. Childs et al. [8] identify experimentally the rotordynami fore oeffiients and leakage harateristis of a smooth and a grooved oil seals operating in the laminar flow regime. The grooved seal inludes three small round irumferential grooves. This researh work is intended to provide experiments to validate preditions and to assess the effetiveness of the grooves in reduing ross-oupled stiffness effets. The experimentally derived fore oeffiients for the smooth seal are ompared to preditions based on Ref. [9], and the grooved seal fore oeffiients are ompared to preditions from Semante and San Andrés [30]. The experimental fore oeffiients for the smooth seal present reasonable agreement with the preditions based on [9] exept for the added mass terms whih are underpredited by a fator of ~seven. The authors attribute the disrepany to the effets of inlet groove and the exit hamber. However, pressure measurements at both the inlet groove an exit avity show no pressure osillations. On the other hand, preditions based on [30] do not represent well the experimental results from the grooved seal. The measurements do not reflet the large redution in rossoupled stiffnesses as predited in [7, 30]. The urrent analysis hereby presented demonstrates that the exit (disharge) groove does not have an effet on the added mass 11

12 term, and that regardless of the groove depth, the groove will not isolate adjaent film lands. Childs et al. [31] present more experiments on a grooved annular seal to determine the influene of the groove depth on the seal fored response and leakage performane. n this ase, the test seal inludes an internal single square groove. The groove depth-to learane ratios tested are 5, 10 and 15. The tests inlude entered and off-entered journal operation up to 70 % of the radial learane; rotor speeds from 4000 to rpm, and three oil supply pressures (4, 45, 70 bar). The Reynolds numbers for eah ase are well below 000 (i.e. laminar flow ondition). The results from the dynami load tests indiate that the rotordynami oeffiients derease with inreasing groove depths exept for the added mass terms. Comparisons of experimental data with preditions based on Ref. [30] show that the seal fore oeffiients are underpredited, whih indiates that the groove does not effetively separate the seal lands (i.e. K xy (1 land) ¼ K xy ( lands), C xx (1 land) ¼ C xx ( lands)). n addition, the test added mass oeffiients are onsiderably large, up to 30 kg for entered journal operation. Preditions of added mass oeffiients from Ref. [] yield.8 kg (i.e. 10 times smaller than the experimental value). Analysis As presented in the literature review, there is a need for a model that properly predits inertia fore oeffiients on grooved SFDs and oil annular seals. The bulk flow model presently advaned is supported by qualitative observations of the laminar flow pattern through annular avities with grooves. Figure 1 depits a representation of the streamlines for a pressure driven flow on a symmetri annular avity with a supply groove and a mid-land groove. The analysis onsiders two major assumptions for determining the appropriate boundary onditions to obtain the fluid fores for small journal dynami motions. The first assumption onsiders an effetive groove learane ( = d + ) defined by the streamline dividing the reirulation region from the rest of the thru-flow. Thus, the dividing streamline is assumed to at as a physial boundary defining an effetive groove depth (d ), obviously different from the physial groove depth. n terms of damping oeffiients, even for small effetive groove depths (i.e. d =5), the damping ontribution from the groove remains signifiantly lower than that related to 1

13 the ungrooved portion of the annular avity. Reall that damping oeffiients are proportional to 1/ 3 [6]. On the other hand, the added mass oeffiient, whih is proportional to 1/ [6], presents the same order of magnitude at the grooves than at the lands (i.e. ungrooved portion) when based on a effetive groove depth (d ). The seond important assumption relates to the pressure distribution at the deep entral groove. nitially, it was thought that there is no development of dynami pressures at the entral grooves in SFDs. However, experimental results [6,17,18] show that the dynami pressures generated in a groove are of signifiant magnitude when ompared to those generated at the film lands, in partiular for motions about the journal entered ondition. The generation of the dynami pressure at the groove is related to the geometrial symmetry of the system and the large loal squeee film Reynolds number (ρω g /μ). For journal dynami displaements the axial flow through the oil inlet plane must be ero, and thus the dynami pressure field is not null at the groove. A perturbation analysis of the bulk flow equations for entered small amplitude orbits implementing the aforementioned hypotheses follows. mid-land groove oil supply, P s feed plenum groove P d P d y P s - P d >0 P d :disharge pressure Figure 1 Shemati view of streamlines in axially symmetri grooved annular avity (ΔP= P s -P d )..1 Bulk flow formulation The analysis determines the fluid fores developed in multi-groove annular avities enlosing a rotating journal whirling with small amplitudes about a entered position. The model applies to SFDs with multiple onstant learane setions, and to smooth and grooved oil seals operating on the laminar flow regime. Speifially, the formulation is 13

14 detailed for the ase of annular avities with axially symmetri groove arrangements, inluding a entral feeding groove as shown in Figure. The multiple groove annular avity is divided into individual flow regions with uniform learane. n ase of a groove, the depth is expressed in terms of an effetive groove depth (d ), whih differs from the atual physial groove depth. The following derivation applies to eah individual flow region with onstant learane, and with the oordinate system set at the entrane of the orresponding grooved or ungrooved region. Figure 3 depits the journal and the oordinate system used in the analysis for small journal motions about a entered position. Oil supply =1,...Ν Ν V n n+1 1 V Figure Shemati view of grooved annular avity divided into flow regions. x= θr Δe= journal displaement ω= whirling frequeny Ω= rotational speed h ω Ω R Δe Y X Figure 3 View of rotating and whirling journal and oordinate system for bulk-flow analysis. 14

15 Within eah individual flow region the bulk-flow mass flow rates in the irumferential (x) and axial () diretions are: m x = ρ h Vx ; m = ρ hv =,,... (1) where h is the film thikness, ( V, V x ) are bulk-flow veloities for eah flow region, and ρ is the lubriant density. The bulk-flow ontinuity and moment transport equations without fluid advetion terms are linear and given as [3]: ( m x ) + ( m ) + ( ρ h ) 0 = () x t ( m x ) P μ ΩR h = kx Vx + x h t (3) ( m ) P V h = k μ + h t ; =,,... (4) with k x =k =1 for laminar flow and μ is the lubriant visosity. Equations (3) and (4) are written as: m x ( m x ) ρ ρ ρ = + k μ x k μ t 3 h P h hωr x x ( m ) 3 ρh P ρh m = ; =,,... k μ k μ t ; (5) Differentiating with respet to x, and with respet to, adding both mx m equations and disregarding seond order terms yields a Reynolds-like equation for the film pressure of an inompressible fluid [33] 15

16 P P x x t x t =,, h + h = 1 μ h + 6μRΩ h + ρh h ( ) ( ) ( ) ( ) (6) The journal desribes dynami motions of small amplitude (Δe X, Δe Y )<< and frequeny ω about its entered position,. The film thikness (h ) equals { } iωt h = + e Δe os ( θ ) + Δe sin ( θ ) ; Δe, Δe < <, i = 1; =,,... (7) X Y X Y where = + d at a grooved region, of effetive groove depth d. The pressure is expressed as a superposition of a eroth order field (P o ) and a first order (dynami) fields ( P, P ) X Y { X X Y Y } iωt P = P 0 + e Δe P + Δ e P ; =,,... (8) Substitution of Eqs. (7) and (8), into Eq. (9) gives the ero th order equilibrium pressure equations P P d ( P ) P a s x x d =,,... sine the stati omponent of film thikness does not hange along the axial or = = 0 = + irumferential diretions (9) The first order equations for journal displaements along the X and Y diretions are PX PX μω 6 μω + = 1i 3 { 1 + i Re* } os ( θ ) sin( ) 3 θ x PY PY μω 6 μω + = 1i 3 { 1 + i Re* } sin ( θ ) + os( θ) 3 x ; =,,... (10) respetively, where Re* ρω = is a modified loal squeee film Reynolds number, 1 μ The perturbed pressure field PX is simply 16

17 P ( ) = f ( )os( θ ) + g ( )sin( θ) ; =,,... (11) X X X Substituting Eq. (11) into the first of Eqs. (10) and olleting similar terms leads to d fx f X i 3 d R d gx gx 6μ 3 d R { i * } μω = Re ; Ω = ; =,,... (1) The homogenous solutions of Eqs.(1) are: fx ( ) osh sinh H f s = R + f R gx ( ) osh sinh ;,,... H g s = g R + = R (13) with partiular solutions: { * } μωr 6μΩR fx = 1i 1 + ire ; g = (14) P 3 XP 3 Then, f ( ) = f ( ) + f and g ( ) = g ( ) + g. X XH X P X XH X P A similar proedure is to be followed for evaluation of PY where P = f ( )os( θ ) + g ( )sin( θ ); and from Eq. (10) g = f ; f = g Y Y Y Y X Y X The total dynami fluid film fore generated along the annular avity inludes the ontribution of the fores generated at eah flow region L (i.e. film lands and grooves). The total fore (radial and tangential) is defined as L π R f X ( ) d L π R i ( )os( ) K XX - ω M XX + ω C f XX X θ 0 dx d = L KYX - M YX + i ω C = ω YX = gx ( )sin( ) 1 θ = π R gx ( ) d 0 (15) 17

18 with =1,,, and as the number of flow regions. ote that K XX =K YY, C XX =C YY, K XY =-K YX for entered motions. From Eqs. (5), the volumetri flow rates, per unit length are q x ( q x ) 3 m x h P h h ρ = = + ΩR; ρ k μ x k μ t x x ( q ) 3 m h P h ρ q = = ; =,,... ρ k μ k μ t (16) As with the pressure field, the flow rates are defined as the superposition of ero th and first order flow fields, i.e. { Δ Δ }; X Y { X Y } q = q + e q + e q iωt x x e X x Y x 0 iωt q = q + e Δe q + Δ e q ; =,,... and X Y 0 (17) t { Δ X x Δ X Y xy } q x = iωt e i ω e q + e q { X X Y Y } q = iωt e i ω Δ e q + Δ e q ; =,,... t ; (18) Substitution of Eqs. (7), (8) and (9) into Eq. (18) leads to ero th and first order expressions, i.e. 3 3 P P X ΩR Y R Ω ( 1+ Re* ) = + os ( θ ); x ( 1+ Re* ) = + sin ( θ ) Y q x i q i X 1 μ x 1 μ x (19a) 3 3 PX 3 os( ) PY 3 sin( ) θ θ q ( 1+ ire * ) = s; q ( 1 Re* ) X + i = s Y 1 μ 1 μ 1 μ 1 μ ; =,... (19b) For motion perturbations along the X-diretion, 3 PX 3 os( ) θ q ( 1+ ire* ) = s X = 1 μ 1 μ 3 3 os( θ ) dfx dgx C s os( θ) + sin( θ) ; =,,... 1 μ 1 μ d d (0) 18

19 The various oeffiients ( f, g, sf, sg ) =,,.. are obtained from appropriate boundary onditions at the interfaes groove-film lands, and from speified pressure or flow onditions at the inlet and exit planes of the global flow region. The pressure must be single-valued and the axial flow rates, leaving one flow region and entering the next flow region, must be idential. n the ase of a SFD, the rotational speed (Ω) is set to ero..1.1 Boundary onditions First, the stati pressure field is defined using Eq.(9), provided that the supply (inlet) and disharge pressures (P 1, P n+1 ) are speified. One the stati pressure gradients are identified (i.e. s dp = ), the dynami pressure field ( PX d ) is obtained by applying the following boundary onditions: a) The avity exit disharges to ambient pressure and thus there is no generation of dynami pressure at the exit plate (last flow region, =L ). This ondition is written as P = 0 = f ( L )os( θ ) + g ( L )sin( θ ) X x x (1) = L Separating the sine and osine terms and substituting in Eq. (13) yields L osh L R sinh f s μω ; 1 f F F i R + R = = 3 L 6 osh L R sinh g s μω ; g G G R + = = R 3 () b) At interfaes groove and land, the first-order pressures are idential, i.e. P ; X = PX = 0 L 1 = + + 1= f ( L ) = f (0); g ( L ) = g (0) X X X X (3) ) The axial flow rates must math at the interfae between groove and land, i.e. ( * ) ( * 1 ) X + q 1 + i Re = q 1 + i Re ; = X = = 0 L (4) Substituting in Eq. (13) yields 19

20 R R R 3 3 L L 1 f sinh sf osh + + s f = 3 s 3 s R R R R 3 3 L L 1 g sinh sg osh + + s g = R (5) d) Finally, due to geometrial symmetry, the axial flow rate must be null at 1 =0 (groove middle plane). Thus, from Eq. 5, s g 1 = 0. This last boundary ondition implies a nonero dynami pressure value at the groove middle plane. The ombined set of boundary onditions provides the neessary equations to solve for the dynami pressure funtions and thus obtain the fore oeffiients. First, the method is applied to determine the fore oeffiients of a simple one land SFD to validate the results with the lassial formulas []. Subsequently, the appliation of these equations is illustrated for two ases: a grooved oil seal and an end sealed SFD.. Single film land- lassial solution model Figure 4 presents a shemati view of a single SFD land extending from L L. Both ends are exposed to ambient pressure and there is no stati axial pressure drop. L P=P a 1 y P=P a Figure 4 Shemati view of a simple SFD and boundary onditions. The first order pressure fields must be symmetri about =0; hene the hyperboli sine terms disappear in Eq. (13), i.e. s f =s g =0. Thus ( ) ( ) fx( ) = f osh f ; ( ) osh X X g P X R + = gx R (6) 0

21 At the ends the pressure is ambient (P a ), and thus the dynami pressure is set to ero at both ends, hene L f X P PX = fx = 0; f = ; g = 0 L = D osh ( L D) (7) with f X P defined in Eq.(14). ntegrating the pressure aross the damper land K L / XX ω M XX C XX P L / ( L D ) ( ) tanh - + i ω = π R f ( ) d f π R L 1 = (8) L D yields the fore oeffiients 3 tanh ( L ) μ R L XX 1 1 D C = π 3 ( L D) K XX ( L D) ( L ) 3 ρ R L tanh = 0; M = π 1 XX D (9) whih oinide with the lassial damping and inertia fore oeffiients derived by Reinhart and Lund []. V Model validation with experimental data This setion inludes omparisons of experimental and predited damping, stiffness and mass oeffiients for an oil ring seal and an end sealed SFD. The results are presented in terms of an effetive groove learane ( = d + ), where d is the effetive groove depth and is the film land learane. V.1 Grooved Oil Seal The experimental data for the oil seal is obtained from Graviss [33]. Figure 5 depits the atual onfiguration of the grooved oil seal. otie that the test onfiguration is axially symmetri, with parallel seals and a entral supply groove. 1

22 Oil supply 13 mm 4 mm 4.89 mm 11.4 mm 11.4 mm Seal length Disharge plenum 136* 76* 76* Journal (0-15)* Buffer seal Figure 5 Configuration of parallel oil seals tested in [33]. V.1.1 Geometry and model desription Figure 6 depits the modeled portion of the oil seal and the partition into flow regions. Table 1 lists the physial dimensions, fluid properties and operating onditions of the test seal. The seal is divided into 4 flow regions with the following boundary onditions: Dynami pressure set to ero at exit plane (Eq. 1). Pressures and flow rates equated at intermediate boundaries (Eqs. 3, 4) Axial flow rate at inlet groove (symmetry plane) is set to ero ( s = 0) g 1 nlet groove Mid-land groove Land V (1-0)* (1-16)* Figure 6 Partial view of test grooved oil seal geometry [33] and flow regions for preditions.

23 Table 1 Oil seal onfiguration, operating onditions and fluid properties. Dimensions Journal Diameter 117 mm Seal length 4.89 mm Clearane 85.9 μm ½ nlet groove length 17 mm ½ nlet groove depth ( d ) 135 g Mid-land groove length mm Mid-land groove depth ( d ) (0,5,10,15) (tests) g Parameters Shaft speed 4000 rpm rpm Oil density 850 kg/m 3 Oil visosity 0.0 Pa.s Supply Pressure.4-7 bar V.1. Results and omparisons to experimental data The dynami pressure fields over eah flow region are integrated to obtain the seal fore oeffiients as detailed in Eq. (15). Figure 7 presents the damping and added mass oeffiients versus an effetive inlet groove-to-seal learane ratios for the seal onfiguration without mid-land (internal) groove (i.e. = ). The results show that the damping oeffiient (tangential fore) rapidly onverges to a onstant value (C=88 k.s/m) for 10. For deep inlet grooves, the ontribution of the groove to the overall tangential fore (damping fore) is negligible. On the other hand, the diret added mass oeffiient onverges more slowly towards an asymptoti value. However, the most important observation is that the added mass oeffiient presents good orrelation with the experimental results preisely in the neighborhood of magnitudes for whih the overall damping beomes insensitive to the groove depth. Thus, omparisons of the damping values for dereasing d with that for a deep groove may be used to obtain an effetive groove learane that properly predits the mass oeffiients. Of ourse, this reommendation applies to relatively large and deep (entral) grooves. 3

24 Damping oeffiient [k.s/m] Experimental results Preditions = = Central nlet groove-to-learane ratio ( ) / Mass oeffiient [kg] Preditions = = 136 Experimental results Central nlet groove-to-learane ration ( ) / Figure 7 Diret damping and added mass oeffiients versus effetive inlet groove-to-seal learane ratio. Solid lines represent preditions and dotted lines enlose the range of experimental values from [33] for a smooth seal (no mid-land groove). For the onfiguration with an internal groove dividing the original seal length, Figure 8 presents the predited damping and added mass oeffiients versus effetive inlet groove-to-seal learane ratios. The graph inludes three different mid-land grooves with effetive groove depths ( d =5, 10, 15 ). The damping oeffiient quikly dereases for inreasing mid-land groove depths. On the other hand, the added mass oeffiient inreases as the effetive groove depth inreases up to a peak value (i.e. d = 5). As in the previous figure, the added mass oeffiient orrelates well with preditions for the 4

25 smallest, i.e. 6, that has a minimum impat on the overall tangential fore developed by the seal. Damping oeffiient [k.s/m] C exp ( =) C exp ( =+15) C model ( = ) C model ( = + 5) = = V V = 136 C model ( = + 15) Mass oeffiient [kg] entral nlet groove-to-learane ratio M model ( = + 5) M model ( = ) M model ( = + 15) M exp ( =+15) M exp ( =) ( ) / = = V V = entral nlet groove-to-learane ratio Figure 8 Diret damping and added mass oeffiients versus effetive entral groove groove-to-learane ratio. Solid lines represent preditions for three different effetive mid-land seal groove depths ( d =5, 10, 15 ). Dotted lines enlose the range of experimental values from [33] for two seal depths ( d =, +15 ). g ( ) / Figure 9 depits the diret damping and added mass oeffiients versus mid-land groove to-seal learane ratios, inluding two effetive groove depths for the entral groove. These values are within the range of best orrelation in Figure 8. The damping 5

26 steadily dereases as the mid-land groove effetive depth ( d ) inreases, whereas the mass presents a maximum value at 5. t is obvious that using the atual groove depth in the predition would lead to smaller damping oeffiients. n this ase, the midland groove is relatively shallow and short when ompared to the seal land and the entral groove. For this mid-land groove, it seems that the best effetive groove depth is around 1/3 of the physial depth. Thus, the results show that using a fration of the groove depth (even ½) leads to muh better preditions of added mass oeffiients. Although, the effetive depth should depend on the groove atual length and depth, urrent preditions may be generalied to short and shallow mid-land grooves typially used in grooved oil seals. A more general orrelation will be determined using further test omparisons and CFD alulations to determine the effetive groove depth from the streamline dividing the thru-flow and reirulation region. Figure 10 displays the ross-oupled stiffness oeffiient (K xy ) versus rotor speed. The results inlude experimental data from Ref.[33] for no mid-land groove and with a mid-land groove 15 deep (i.e. =16). The preditions are based on an effetive inlet groove depth ( d ) of 11 and a mid-land effetive groove depth ( d ) of 6 (i.e. half of atual groove sie). The preditions orrelate best for the lowest rotor speeds. More importantly, the redution in ross-oupled stiffness is properly aptured using effetive groove depths ( d, of added mass and damping oeffiients. ) that provide good orrelation between experiments and preditions 6

27 00 Damping oeffiient [k.s/m] C exp ( = +15) C model ( = + 6) C model ( = + 11) V C exp ( =) = = V mid-land groove-to-learane ratio ( ) / = = V Mass oeffiient [kg] 40 0 M model ( = + 11) M model ( = + 6) M exp ( = ) V M exp ( =+15) mid-land groove-to-learane ratio ( ) / Figure 9 Diret damping and added mass oeffiients versus effetive mid-land groove depth. Solid lines represent preditions for two different effetive entral groove depths ( d =6, 11 ). Dotted lines enlose the range of experimental values from [33] for two seal depths ( d =0, 15 ). g 7

28 Cross-ouple stiffness Kxy [M/m] K model ( =11, = ) K model ( =11, =7 ) k xy -k = (exp.) yx k xy -k yx = +15 (exp.) V Rotor speed [RPM] Figure 10 Cross-oupled stiffness oeffiients versus rotor speed. Solid lines represent preditions for smooth seal and a effetive mid-land groove depth with ( d =0,6 ). Dotted lines represent experimental values from [33] for a smooth seal and a grooved seal (mid-land groove depth =16). V. Sealed SFD n this setion the model preditions are ompared to experimental results obtained in a sealed SFD as part an experimental researh sponsored by TRC [14]. V..1 Geometry and model desription Figure 11 shows a ut view of the test SFD. The damper is fed from the top and the oils exists the damper at an end disharge groove and four outlet ports. Eah outlet port inludes a.8mm hole flow restritor. Measurements of the dynami pressure are available at the disharge groove. Figure 1 depits a shemati view of the SFD detailing the groove and land dimensions, and the flow region subdivisions. 8

29 Oil inlet Eddy urrent sensor Oil plenum Housing top plate ominal learane in (0.17 mm) nlet groove Plexiglas bearing Plexiglas bearing Journal Vertial plate Shaft Ring arrier Wave spring Disharge groove Reirulation annulus Bottom plate in (50.8 mm) Figure 11 Sealed-end SFD assembly ut view. 78 y Feed plenum 6.35 mm nlet groove 5.4 mm Film land mm Disharge groove Figure 1 Test squeee film damper geometry and flow regions used for preditions [14]. The boundary onditions are: ull axial flow at the feeding groove inlet (Eq. 1). This ondition implies that the dynami pressure in the plenum does not allow for bakflow at the damper inlet. 9

30 Equal pressure and flow rates at internal flow region interfaes. (Eqs. 3,4) Dynami pressure amplitude set at outlet plane from measurements. Table Test onditions for dynami load tests (CCO). Lubriated SFD nlet pressure (P s ) * 31 kpa Disharge groove pressure (P r )* 8.6 kpa-15.5 kpa Frequeny range 0-70 H ( H step) Lubriant temperature (T) C ( F) Visosity () 3.1 P-.8 P Clearane () μm (4.9-5 mils) Orbit amplitude (e) 1-50 μm (0.5- mils) Flow restritors (hole diameter).8 mm *: Gauge pressure. V.. Results and omparisons to experimental data Figure 13 depits the SFD damping and added mass oeffiients versus effetive disharge groove-to-film learane ratios and two arbitrary effetive inlet groove depths ( d = 11,6 ). The results show that both fore oeffiients are insensitive to an outlet groove depth of d 3. Thus, only the atual physial disharge groove depth is onsidered in the parametri study. Figure 14 depits the damping and added mass oeffiients versus effetive inlet groove-to-film learane ratios. Regarding the inlet groove, the results are similar to those found for the oil seal (Figure 7). Again, the best estimates of mass oeffiients orrespond to the smallest effetive groove depths (i.e. 10) that do not have a signifiant impat (>10%) on the SFD tangential fores (damping oeffiient) estimated using the atual groove depth. 30

31 0 Damping oeffiient [k.s/m] C model ( = 6 + ) C model ( = 11 + ) C exp nlet groove-to-learane ratio ( ) / 30 Mass oeffiient [kg] 0 10 M model ( = 6 + ) M model ( = 11 + ) M exp nlet groove-to-learane ratio Figure 13 Predited SFD damping and added mass oeffiients versus effetive inlet groove depths. Solid lines represent preditions for two different inlet effetive groove depths ( = 1,7 ). Dotted lines represent range of experimental values from [14]. ( ) / 31

32 Damping oeffiient [k.s/m] C exp C model y nlet groove-to-learane ratio ( ) / 30 M model y Mass oeffiient [kg] 0 10 M exp nlet groove-to-learane ratio ( ) Figure 14 Predited SFD added mass oeffiient versus effetive inlet groove depth. Solid lines represent preditions. Dotted lines represent range of experimental values from [14]. / 3

33 V Conlusions An improved bulk-flow model to predit fore oeffiients in multiple groove annular avities is presented. The model divides the multiple groove annular avities into flow regions of onstant learane. At the groove regions, an effetive groove depth ( d ) and learane ( = d + ) are defined based on qualitative observations of the laminar flow pattern through annular avities. Continuity of the pressure and axial flow field is imposed at the flow region interfaes. For a entral groove, onsidering geometrial symmetry, the axial flow at the mid plane is set to ero. This observation is supported by experiments [6,17,18] that evidene the generation of relatively large dynami pressures at grooves in SFDs. The preditive model reprodues the lassial fore oeffiients (from Ref. []) for a single land SFD. Furthermore, the model shows that the damping and stiffness oeffiients derease rapidly with inreasing effetive groove depths as these are proportional to 1/ 3. On the other hand, the added mass oeffiient, whih is proportional to 1/, is less sensitive to hanges in the effetive groove depth. The model results explain unusual experimental data obtained in ring oil seals [33] and an end sealed SFD [14] through a parametri study varying the effetive groove depths (d ). Comparisons of predited and experimental fore oeffiients show exellent orrelation for a narrow range of effetive groove depths. Most importantly, the parametri study shows that the best analytial orrelation for the fore oeffiients is obtained for the minimum effetive groove depths for whih the overall damping is not substantially modified (<10%) from that alulated using the atual physial groove depth. The seal ross-oupled stiffness oeffiients are well predited using an effetive groove depth (d ) that best predits also damping and added mass oeffiients. More importantly, unlike the existing models, this formulation effetively predits (i.e. does not overpredit) the redution of ross-oupled stiffness due to the addition of a mid-land groove. n terms of a mid-land groove in a seal, the parametri study shows good orrelation with the experimental data for effetive depth (d ) values smaller or equal to 50% of the groove depth. This result may be generalied for short and shallow grooves like those 33

34 found in oil seals. However, in order to determine the most appropriate effetive depth for other groove sies it is neessary to ondut more omparisons with experiments and to perform CFD alulations to find the effetive groove depth from the streamline dividing the thru-flow from the reirulation region aross the grooves. Future work will inlude suh omparisons to determine appropriate effetive depths aording to the groove geometry. 34

35 V Referenes [1] Kuma, D. C., 1967, Fluid nertia Effets in Squeee films, Appl. Sientifi Researh, 18, pp [] Reinhardt, F., and Lund, J. W., 1975, The nfluene of Fluid nertia on the Dynami Properties of Journal Bearings, ASME J. Lubr. Tehnol., 97(1), pp [3] Brennen, C., 1976, On the Flow in an Annulus Surrounding a Whirling Cylinder, J. Fluid Meh., 75(1), pp [4] Tihy, J. A., 1983, The Effet of Fluid nertia in Squeee Film Damper Bearings: A Heuristi and Physial Desription, ASME Paper 83-GT [5] Mulahy, T. M., 1980, Fluid Fores on Rods Vibrating in Finite Length Annular Regions, Trans. ASME, 47, pp [6] San Andrés, 1985, Effet of Fluid nertia Effet on Squeee Film Damper Fore Response, Ph.D. Dissertation, Deember, Texas A&M University, College Station, TX. [7] San Andrés, L., and Vane, J., 1986, Effet of Fluid nertia on Squeee-Film Damper Fores for Small-Amplitude Cirular-Centered Motions, ASLE Trans., 30(1), pp [8] Zhang J. X., and Roberts J. B., 1993, Observations on the onlinear Fluid Fores in Short Cylindrial Squeee Film Dampers, J. Tribol., 115(3), pp [9] Zhang J. X., 1997, Fluid nertia Effets on the Performane of Short and Long Squeee Film Dampers Exeution Periodi Vibration, J. Tribol., 119(3), pp [10] Qinghang, T., Wei, L., and Jun, Z., 1997, Fluid Fores in Short Squeee-Film Damper Bearings, Tribol. nt., 30(10), pp [11] Zhang J. X.,1997, Fluid nertia Effets on the Performane of Short and Long Squeee Film Dampers Exeution Periodi Vibration, J. Tribol., 119(3), pp [1] Della Pietra, L., and Adiletta, G., 00, The Squeee Film Damper over Four Deades of nvestigations. Part : Charateristis and Operating Features, Shok Vib. Dig, 34(1), pp [13] Della Pietra, L., and Adiletta, G., 00, The Squeee Film Damper over Four Deades of nvestigations. Part : Rotordynami Analyses with Rigid and Flexible Rotors, Shok Vib. Dig., 34(), pp [14] Delgado, A., and San Andrés, L., 006, dentifiation of Fore Coeffiients in a Squeee Film Damper with a Mehanial Seal, TRC report, TRC-SFD-1-06, May. [15] Semanate, J., and San Andrés, L., 1993, Analysis of Multi-Land High Pressure Oil Seals, STLE Tribol. Trans., 36(4), pp [16] San Andrés, L., 199, Analysis of Short Squeee Film Dampers with a Central Groove, J. Tribol., 114(4), pp [17] Arau, G., and San Andrés, L., 1994, Effet of a Cirumferential Feeding Groove on the Dynami Fore Response of a Short Squeee Film Damper, J. Tribol., 116(), pp [18] Arau, G., and San Andrés L., 1996, Experimental Study on the Effet of a Cirumferential Feeding Groove on the Dynami Fore Response of a Sealed Squeee 35

36 Film Damper, J. Tribol.,118(4), pp [19] Zhang J. X., and Roberts J. B.,1996, Fore Coeffiients for a Centrally Grooved Short Squeee Film Damper, J. Tribol., 118(3), pp [0] Ellis, J., Roberts, J. B., and Hosseini, S. A., 1990, The Complete Determination of Squeee-Film Linear Dynami Coeffiients from Experimental Data, J. Tribol., 11(4), pp [1] Zhang, J., Roberts, J. B., and Ellis, J., 1994, Experimental Behavior of a Short Cylindrial Squeee Film Damper Exeuting Cirular Centered Orbits, J. Tribol., 116(3), pp [] Qinghang, T., Xaohua, L, and Dawei, Z., 1998, Analytial Study of the Effet of a Cirumferential Feeding Groove on the Unbalane Response of a Rigid Rotor in a Squeee film Damper, Tribol. nt., 31(5), pp [3] Tihy J. A. and Bou-Said B., 1991, Hydrodynami Lubriation and Bearing Behavior with mpulsive Loads, ASLE Tribol. Trans., 34(4), pp [4] Lund, J., W., and Myllerup, C., M., Hartmann, H., 003, nertia Effets in Squeee- Film Damper Bearings Generated by Cirumferential Oil Supply Groove, J. Vib. Aoust., 15(4), pp [5] Kim, K. J., and Lee, C. W, 005, Dynami Charateristis of Sealed Squeee Film Damper with a Central Feeding Groove, J. Tribol., 17(1), pp [6] Zeidan, F.Y., San Andrés, L., and Vane, J. M., 1996, Design and Appliation of Squeee Film Dampers in Rotating Mahinery, Pro. 5 th Turbomahinery Symposium, Houston, TX, pp [7] Baheti, S., and Kirk, R., 1995, Finite Element Thermo-Hydrodynami Solution of Floating Ring Seals for High Pressure Compressors Using the Finite-Element Method, STLE Tribol. Trans.,38, pp [8] Childs, D. W. Rodrigue, L. E., Cullotta, V., Al-Ghasem, A., and Graviss, M., 006, Rotordynami-Coeffiients and Stati (Equilibrium Loi and Leakage) Charateristis for Short, Laminar-Flow Annular Seals, J. Tribol., 18(), pp [9] Zirkelbak,., and San Andrés, L., 1996, Bulk-Flow Model for the Transition to Turbulene Regime in Annular Seals, STLE Tribol. Trans., 39(4), pp [30] Semanate, J., and San Andrés, L., 1993, Thermal Analysis of Loked Multi-Ring Oil Seals, Tribol. nt., 7, pp [31] Childs, D. W., Graviss, M., and Rodrigue, L. E., 007, The nfluene of Groove Sie on the Stati and Rotordynami Charateristis of Short, Laminar-Flow Annular Seals, ASME J. Tribol, 19(), [3] San Andrés, L., Modern Hydrodynami Lubriation Theory, Tribology Group, Texas A&M University, (aessed on 05/15/07) [33] Graviss, M., 005, The nfluene of a Central Groove on Stati and Dynami Charateristis of an Annular Liquid Seal with Laminar Flow, M.S. Thesis, Texas A&M Univ., College Station, TX. 36