PARAMETRIC ANALYSIS OF A HYDRODYNAMIC THRUST BEARING WITH ELASTIC SLIDER

Transcription

1 THE ANNALS OF UNIVERSITY DUNĂREA DE JOS OF GALAŢI 101 PARAMETRIC ANALYSIS OF A HYDRODYNAMIC THRUST BEARING WITH ELASTIC SLIDER Andrei MINCULESCU, Traian CICONE POLITEHNICA University of Bucharest, Romania cicone@meca.omtr.pub.ro ABSTRACT The use of temperature and pressure induced deformations to generate hydrodynamic lift-off for flat and parallel surfaces is relatively new and scarcely present in literature. The present paper analyses theoretically the load carrying capacity of a typical thrust bearing with elastic pad in order to define optimal pad parameters. The study is made using the original Thrust Bearing Analysis (TBA) Software based on finite differences solution of flow and thermal equations and finite elements solution for solid structure equations. The results, presented for the equivalent slider, have primary revealed that the load carrying capacity of such a bearing is comparable with a similar tilting-pad bearing. The parametric analysis shows the existence of optimal dimensions for maximum load capacity. KEYWORDS: lubrication, TEHD, parallel slider, thrust bearing, design. 1. INTRODUCTION The use of compliant components to increase the overall performances of fluid film pairs is a modern solution, in continuous development. Successful applications can be mentioned in bearings, seals or viscous couplings. It is well-known that the elastic deformations due to temperature gradients or due to the pressure, although small, can be comparable with typical film thicknesses and hence, can be used to control and/or improve film performances. One should recall that elasticity has been used since 60 as a solution for thrust bearings performance amplifier, through the elastic support of pads [1, ]. In this way, elastic deformations of the pad support provide for bearing improved performance characteristics or the extension of fluid film operating conditions to large scale of values of the operating parameters. It should be noted that bearing pad is rigid and elasticity is provided only by its support. It is also well studied the elastic effect due to temperature and pressure gradients in classical planeinclined thrust bearings []. However, creating lift-off effects in a thrust bearing requires adequate shaping of the pads surfaces by difficult, expensive and relatively precisionless manufacturing processes. The present paper, deals with a new solution to generate load carrying capacity in thrust bearings, based on controlled elasticity of the pad which is nominally flat and parallel to the mating runner. This alternative can substitute the traditional technology for creating the convergent gap by machining. The first studies of lift-off effects for flat and parallel surfaces had been conducted by Fogg, Cameron and Wood [] in They evaluated only the effects generated by temperature variation along the bearing (thermal wedge effect). Later, after the report of Ettles and Cameron [4] the effect of thermal distortion of the solids on pressure distribution was recognized as important and a great amount of work has been dedicated to the study of thermo-elastic effects in typical bearing configurations. However, the improvement of bearing performances through controlled elasticity of bearing members has been lately considered a subject of interest. The study of HD effects generated in thrust bearings by using elastic pads, has been initiated in Poland at the Technical University of Gdansk in the 80 and published in a series of papers starting with 1985 [5-8]. The experimental results reported in [9] demonstrate not only the capacity of these bearings to generate HD effects but also their superiority in respect with similar tilting pad thrust bearings. However, theoretical modeling of these bearings has been presented in an extremely synthetically manner, without any details regarding the assumptions and the mathematical model solved for. Approximately in the same period, an important bearing producer (KMC Inc.) deposed a patent for a

2 10 THE ANNALS OF UNIVERSITY DUNĂREA DE JOS OF GALAŢI flexure-pivot tilt pad for journal and thrust bearings and reported experimental results [10], but related theoretical models published after that are scarce. The elastic pads of KMC have been recently modified and patented for thrust bearings by KALSI Inc.[11]. KALSI solution, named load responsive hydrodynamic bearing is primary intended for highly loaded thrust bearings operating at high speeds and severe environmental conditions (high shocks, vibrations); typical applications are downhole drilling tools, mud hammer drills, etc. Although the experimental results are very promising, the phenomenological control information is poor. Kucinschi et al. has recently published an important theoretical study on TEHD effects of grooved elastic thrust washers [1]. Thrust washer operation is very similar phenomenologically to that of elastic pad thrust bearings, i.e. fluid film formation is due to thermo elastic deformations. The idea is to use the deformations that occur in a properly shaped stator in order to obtain a convergent gap. A D finite element model is used to study the influence of temperature and pressure induced elastic deformations on the performances of a radially grooved thrust washer. In conclusion, elastic-pad thrust bearings have been in use for more than 10 years, but theoretical approaches have been sparse in academic literature. Their operation is based on thermal and pressure induced pad distortion. As a consequence, the present paper aims to provide an original contribution in this direction that is complex modeling of TEHD phenomena. The main goal of the present paper is to analyze theoretically an elastic pad thrust bearing in order to define optimal values for the most important design parameters. The analysis is performed on a typical thrust bearing using an original Thrust Bearing Analysis (TBA) software presented in detail in [1].. THE MODEL For the sake of simplicity, a single pad with a simplified geometry was considered. Its main dimensions are illustrated schematically in figure 1. As can be seen, the pad consists of two distinct zones: the elastic zone (inlet zone), elastically deformed under the effect of pressure distribution and thermal gradients and the so-called flat zone (outlet zone), corresponding to the rigid part of the pad (pillar). Film thickness is the sum of two components: minimum film thickness h min corresponding to the undeformed zone of the pad and the pressure and thermal distortion of the pad, h e. A Cartesian system of coordinates is used as in most of numerical analyses of thrust bearings [] so that a so called parallel slider is to be analyzed. ` h max U L e F L g e Runner h min Fig. 1. Schematic of the pad..1. Governing Equations The analysis given below is based on the following assumptions: (1) The fluid is Newtonian, incompressible, in laminar flow without slip at solid boundaries. () -D steady state, generalized Reynolds equation with neglected inertia effects is considered: h p h p h + = 6U (1) x µ x z µ z x () A quasi--d energy equation in the fluid film accounting for conduction and convection. A parabolic temperature variation on the film thickness is assumed. This assumption is often used in order to simplify the numerical procedure [1-14]. Uh h p T T U h p c ρ v = k f x µ µ x y h 1µ x () (4) Viscosity-temperature relationship, based on Reynolds formula, (in order to agree with previous calculated tilting-pad bearing): ( Tm T0 µ µ ) 0 e β = () h where: T m( x ) = T( x, y ) dy (4) 0 (5) -D heat conduction equation in the solid parts of the bearing. (6) -D deformation analysis of the solid structure of the pad, induced by pressure and temperature gradients; the runner is assumed perfectly rigid. The original governing equations can be referred to [1]... Boundary Conditions Three groups of boundary conditions have been imposed (thermal, fluid and solid). For fluid flow, atmospheric (ambient) pressure at pad edges is assumed. g h 0

3 THE ANNALS OF UNIVERSITY DUNĂREA DE JOS OF GALAŢI 10 For the temperature film calculation, the following thermal boundary conditions are assumed: (a) The runner is adiabatic, so that the heat is rejected partly by the lubricant and partly through the pad, to the environmental atmosphere by free convection. (b) Heat flux continuity at the solid/lubricant interface. Structurally, the pad is fixed on the bottom leg (pillar) and free to distort in any direction... Numerical Procedure A numerical solution to the Reynolds equation is obtained, followed by an energy balance calculation, and a structural elastic deformation analysis. The Thrust Bearing Analysis (TBA) software [1] couples a finite element analysis (FEA) software program to solve for pad deflections with a finite difference (FD) solution for pressure and temperature distribution in the film as well as heat transfer in the pad. The analysis steps are coupled and solved in an iterative manner, as outlined in the following: (a) Estimate the initial fluid film thickness based upon the geometry and the operating conditions. (b) Estimate the initial temperature distribution in the gap. (c) Solve for modified Reynolds equation (to include viscosity variation) and calculate the pressure distribution on the pad, the power loss and the flow rate, using the assumed temperature distribution and corresponding fluid viscosity. A Gauss-Seidel iterative method with overrelaxation is used. (d) Solve for energy equation in the film and heat transfer equation in the pad to obtain temperature distribution in the film and in the pad. (e) Calculate -D pad structural deformations due to fluid pressure and temperature. (f) Update fluid film thickness as a result of pad deformation Repeat steps (c) through (f) until an equilibrium (evaluated by changes in pad deflection) is reached..4. Mesh Evaluation The number of elements through the pad thickness as well as the number of elements on the pad surface was varied in order to determine the mesh resolution for a mesh independent solution. FEA meshes were created with 1, and 4 elements for each millimeter of pad thickness. There was no significant difference (from an engineering point of view) between predicted results with each of the three meshes. The results are summarized in figure where the relative errors in film thickness are presented. These relative errors are evaluated taking as reference the densest FEA mesh (4 elements/mm). One can see that for the mesh with elements/mm the errors are less than 0.5% except at the boundary between elastic and flat regions, where the discontinuity is better treated using the densest mesh. This mesh density was considered accurate enough and the optimum solution for the compromise between the accuracy of the results and CPU time for the solver. Relative Error [%].0% 1.5% 1.0% 0.5% elem./mm Reference: 4 elem./mm 1 elem./mm 0.0% X [mm] Fig.. Film thickness error with various mesh densities. Fig.. Screen capture of meshed pad ( elem./mm).. RESULTS AND DISCUSSION A typical thrust bearing of r i =5.5mm inner radius, r o =8.5mm outer radius and 10 pads has been considered for analysis. The main dimensions of the equivalent pad and operating conditions are summarized respectively, in Table 1 and Table. The same thrust bearing, with tilting pads, has been calculated using standard design methodology []. A F=10000N thrust load and the same oil feeding characteristics have been assumed. The results are summarized in Table. The analysis was made for a direct lubrication problem, i.e. given the minimum film thickness, load capacity, F, flow rate, Q, and power loss, P f, are calculated for various dimensional and operating parameters. This choice allows for shorter iterative procedure and gives important benefits in terms of run-time.

4 104 THE ANNALS OF UNIVERSITY DUNĂREA DE JOS OF GALAŢI Table 1. Pad characteristics. Length of the pad L 4 mm Width of the pad B 0 mm Film thickness at exit h 0 0 µm Velocity U 1.5 m/s Convection coefficient H c 115 W/m K Pad thermal conductivity h 45 W/m K Pad thermal expansion coefficient α /K Reference temperature for pad thermal deformation T α 0 o C Shaft temperature T s 50 o C Table. Oil characteristics. Viscosity at reference temperature µ 0.06Pa s Viscosity-temperature coefficient β 0.0 Reference temperature for viscosity T 0 40 o C Oil density ρ 860 kg/m 000 Oil specific heat coefficient C v J/kg K Oil thermal conductivity k f 0.1 W/m K Temperature of oil at inlet T in 45 o C Temperature of ambient oil T amb 40 o C Table. Tilting pad performance characteristics. Minimum film thickness h µm Maximum oil temperature T max 54 o C Flow rate Q 0.1 cm /s Power loss P f 97W Several series of numerical simulations have been done. In a first series a g=8mm total thickness pad has been considered and two important dimensions defining pad elasticity have been analyzed: the length of the elastic zone of the pad, in terms of length ratio λ. the thickness of the elastic zone of the pad, in terms of pad thickness for the elastic zone, g e. Figure 4 shows pad load capacity assuming the outlet film thickness, h 0 =0µm. One can remark that for a relatively wide range of values for these dimensional parameters, the load capacity is greater (up to 0%) than that of the similar tilting pad, operating with the same minimum film thickness. Of great importance is the existence of optimum dimensional values providing for maximum load capacity. These values are inter-related as shown in figure 4. For the studied case (h 0 =0µm) the optimum values are in the range λ opt = and g e =4 5 mm. Figure 5 presents an extended analysis of load capacity and film thickness ratio for various pad thicknesses in the elastic zone and a typical value of elastic zone length, L e =6mm (λ=0.765). One should note that the for optimum load capacity increasing the load requires corresponding increase of pad thickness (g e ) in order to maintain pad deformation within optimal range. This means that film thickness ratio must be kept around an optimum which, for the studied case is about H=.8 as shown in figure 5. In fact, from figure 5 one can remark that optimum film thickness ratio is practically independent of the load. As a consequence, a tentative dimensioning of such an elastic pad in order to operate around an optimum for a given load can be obtained using only the deflection analysis and a rough estimate of the mean fluid film temperature. Load, F [N] g e =5mm g e =4mm g e =4.5mm g e =mm Length ratio, λ g e =.5mm Fig. 4. Load capacity for various dimensions of the elastic zone. Figure 6 presents film thickness variation along the pad (the geometry of the deformed pad)for two values of elastic zone thickness around the optimum for load capacity (see fig. 4) and for two lengths of the elastic zone (L e =6mm respectively L e =8mm). One can remark that pad deflection is slightly nonlinear and can be accurately approximated by a parabola or even by a straight line which is an important conclusion for quasi-analytical models to be developed in the future. A slight necking of the film thickness near the edge between flat and elastic zones (transition between elastic and rigid pad) is to be noted. However this necking (about 1µm) is at least one order of magnitude less than the initial minimum film thickness, h 0, and, therefore, can be neglected. A second series of numerical simulations have been performed for a more rigid design of the pad, defined by an increased thickness of the flat zone (pillar thickness), g=1mm. A plot of film thickness and mid-plan temperature variation along the pad is given in figure 7 for two dimensions of the rigid zone of the pad (pillar thickness), g=8 mm (dashed line) respectively g=1mm (solid line). It can be noticed that the stiffer is the pillar, the greater is the necking (but still small with respect to film thickness) and the

5 THE ANNALS OF UNIVERSITY DUNĂREA DE JOS OF GALAŢI 105 greater is the film thickness ratio. However, the load capacity is practically identical and consequently one can conclude that the thickness of the pillar does not influence considerably pad performance characteristics. This can be better seen in figure 8 where film thickness ratio, H, and load capacity, F, are plotted versus elastic pad thickness, g e, for the same two reference cases of pillar thickness, g. Because of the increased necking, film thickness ratio in the case of rigid pillar is greater, optimal values being around H=.. 4. CONCLUSIONS An original analysis of the performance characteristics for an elastic slider with direct application to thrust bearings has been performed. The numerical application was made for a typical thrust bearing pad which was originally a tilting pad. 000 g=8mm L e =6mm (λ= 0.765) Load H 4 hmin [µm] h min g=8mm - F=117N g=1mm - F=1179N x [mm] L e =6mm (λ= 0.765) Fig. 7. Film thickness and temperature variation along the pad. 100 T g=1mm g=8mm T [C] Load, F [N] h min =0 µ m g e h min =0 µ m h min =5 µ m.5.5 Fig. 5. Load carrying capacity and film thickness ratio for various pad thicknesses. h min [µm] g e = 4.5mm L e = 8mm (λ= 0.8 ) g e = 5mm g e = 4.5mm g e = 4mm L e = 6mm (λ= ) x [mm] Fig. 6. Film thickness for various pad dimensions. H F [N] 1150 L e =6mm (λ= 0.765) g e [mm] Fig. 8. Load capacity and film thickness ratio. The comparison shows that load carrying capacity of parallel surface elastic pad is comparable, and in some cases even greater with respect to that of the standard tilting-pad. The parametric study performed demonstrates the existence of optimal pad dimensions in terms of maximizing load carrying capacity. The numerical simulation of pad deformation shows that, in a simplified quasianalytical approach of the same subject, a parabolic variation of film thickness gives a very good approximation. Moreover, even a linear film thickness variation along the elastic zone could be a solution for simplified models. The present analysis revealed also that the thickness of the elastic zone of the pad is a key design parameter for improved bearing performance. In the future a new iterative algorithm for solving the inverse lubrication problem will be developed. This will allow for more accurate comparisons between elastic pads and tilting pads. H

6 106 THE ANNALS OF UNIVERSITY DUNĂREA DE JOS OF GALAŢI AKNOWLEDGEMENT The authors gratefully acknowledge Michel Fillon, professor at University of Poitiers (Laboratoire de Mécanique des Solides), for having contributed with important works and suggestions to this paper. REFERENCES 1. Tieu A.K., 1975, An investigation of Diaphragm Type Thrust Bearings, ASME J. of Lubr. Technology, 97,, pp Constantinescu V.N. et al., 1986, Sliding Bearings, Allerton Press.. Dowson D., 1998, History of Tribology, sprof. Engng. Publ. 4. Ettles C., Cameron A., 1965, Thermal and elastic distortions in thrust-bearing, Int. Mech. Eng., Lubrication and Wear Convention, pp Olszewski O., 1985, An experimental investigation of a hydrodynamic bearing with an elastic thrust plate, Proc. of 4 th Europeean Tribology Conf., vol. I, Ecully, France 6. Dabrowski L., Wasilczuk M., 1994, Method of evaluating shape of the oil film in the bearing with an elastic thrust plate, Engng. Aero-Hidro- Elasticity Conf., Pilsen, Slovakia 7. Dabrowski L., Olszewski O., Wasilczuk M., 199, An experimental investigation of a hydrodynamic bearing with an elastic thrust plate, Proc. EuroTrib Conf., Budapesta 8. Dabrowski L., Wasilczuk M., 1995, On the accuracy of theoretical models of hydrodynamic thrust bearing, Proceed. of the 1 st Int. Conf BalkanTrib, Thessaloniki. 9. Wasilczuk M., 00, Comparison of an optimal-profile hydrodynamic thrust bearing with a typical tilting-pad thrust bearing, Lubrication Science, 15,, pp de Choudhury P., Hill M.R., Paquette D.J., 199, A Flexible Pad Bearing System for a High Speed Centrifugal Compressor, Proc of 1 st Rotating Machinery Symp. 11. Kalsi M.S. et al., 00, Load-Responsive Hydrodynamic Bearing US Patent No , B1 (Oct. 8, 00). 1. Kucinschi B., DeWitt K., Pascovici M., 004, Thermoelasto-hydrodynamic (TEHD) analysis of a grooved thrust washer, ASME J. of Tribology, 16,, pp Minculescu A., 005, Steady-State Thermo-Elasto- Hydrodynamic Analysis of an Elastic Pad Thrust Bearing - Diploma Thesis, POLITEHNICA Univ. of Bucharest. 14. Knight J.D., Barret L.E., 1988, Analysis of tilting-pad journal bearings with heat transfer effects, ASME J. of Lubrication Techn., vol. 110, pp NOTATIONS B width of the pad, [mm] c v specific heat of the oil, [J/kg K] E Young s modulus, [MPa] F total load on one pad, [N] g total pad thickness, [mm] g e pad thickness in the elastic zone, [mm] h film thickness, [µm] H film thickness ratio, = h max hmin [ ] H c convection coefficient, [W/m K] k thermal conductivity, [W/m K] L total pad length, [m] L e length of the elastic zone of the pad, [m] p pressure, [MPa] P f power loss, [W] Q total flow rate, [m /s] r i inner radius, [m] r o outer radius, [m] T temperature, [ o C] T amb ambient temperature, [ o C] T s shaft constant temperature, [ o C] U velocity, [m/s] x,y,z coordinates Greek symbols α thermal expansion coefficient, [K -1 ] β viscosity-temperature coefficient, [K -1 ] λ pad length parameter, = L e L [ ] µ viscosity, [Pa s] ρ density, [kg/m ] υ Poisson s coefficient, [ ] ω angular velocity, [rad/s] Subscripts in inlet out outlet m mean min minimum value max maximum value e corresponding to elastic zone