SYNTHESIS OF A FLUID JOURNAL BEARING USING A GENETIC ALGORITHM

SYNTHESIS OF A FLUID JOURNAL BEARING USING A GENETIC ALGORITHM A. MANFREDINI and P. VIGNI Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione (DIMNP) - University of Pisa Via Diotisalvi, - 51 Pisa, ITALY; e-mail: p.vigni@ing.unipi.it ABSTRACT The synthesis of a hydrodynamic radial journal bearing is presented. The fluid-dynamic behaviour of the bearing is analysed both assuming a temperature-dependent viscosity of the lubricant film and applying an empirical procedure suitable to provide a better correlation with the performance of fluid film bearings tested under laboratory conditions. A genetic algorithm has been employed to select the optimum bearing parameters corresponding to several load-frequency combinations and to applications where the load and frequency may change from time to time within given bounds. Keywords: lubrication, journal bearings, thermal-hydrodynamics, synthesis, genetic algorithm 1 INTRODUCTION Fluid film bearings are a common means of supporting rotating shafts subjected to high radial loads in a high variety of rotating machinery such as pumps, turbines and compressors. A closed form of the Reynolds equation is possible only for bearing whose length is very large compared to their diameter or in the case of very small length-to-diameter ratios [1]. Use of high-speed computers allowed for a numerical solution of the Reynolds equation for any given geometry and boundary conditions. The most widely known and used results has been obtained by Raimondi and Boyd [] and are based on the assumption of an isoviscous film independent of pressure and temperature variations. Over the last few years many experimental investigations on this subject have shown that the classic theory is not able to take into account for the load-carrying capacity and the temperature rise in the fluid film. Fogg [] introduced the concept of the thermal wedge in order to explain the additional load-carrying capacity due to the thermal expansion of the lubricating fluid, while Cameron [] suggested that a hydrodynamic pressure could be created in the film between two rotating disks as a consequence of the variation of viscosity across the film thickness. Experiments of Cole [5] have indicated the presence of strong temperature gradients in a journal bearing subjected to high speeds, both across the film and along the plane of relative motion. The thermal effects and the thermal-hydrodynamic phenomena in the lubricating film have also been analysed in a great deal of researches, for a large number of film situations and bearing geometric configurations (see, for example, the references [, 7, ]). Even though both the circumferential and axial pressure distributions are correctly evaluated by these theories, the dependence of the peak pressure on the rotational speed of the journal bearing is not appropriately predicted. The pressure peak experimentally results approximately proportional to the square root of the journal frequency, instead of the almost linear proportionality obtained with the analytical calculations. A better agreement with the experimental data has been obtained by Wang and Seireg [9] who assumed that the velocity variation and heat generation in the fluid film take place in a central zone with the same length and width as the bearing, but with a significant smaller thickness than the fluid film thickness. Even though the existence of a thin shear zone with high velocity gradients has been revealed in some experimental investigations [1, 11, 1], no explanation has still be found on this particular rheological behaviour of the lubricating film. A simpler approach to the problem by Seireg [1] is based on the calculation of a modified Sommerfeld number to be used in a standard isoviscous analysis. This number is calculated on the basis of experimental findings and no theoretical confirmation is developed for the proposed method. On the other side, the optimisation of the performances of a hydrodynamic bearing in the past has relied heavily on empirical rules, due to the complexity of the interaction between the different parameters governing the behaviour of such bearings. In this work the approximate synthesis of hydrodynamic journal bearings is studied using an optimisation method based on genetic algorithms (GAs) [1], [1]. Unlike conventional methods, which usually requires the function of interest to be well behaved, GAs tolerate noisy and discontinuous function evaluations. Due to their stochastic nature, they are able to search the entire solution space with more chance of finding the global optimum than conventional methods. They also do not suffer by getting stuck on a relative optimum and so failing to reach the absolute optimum. The base analysis (viscous case) relates to the optimisation of a journal bearing taking into account for the variation of the fluid viscosity with the average bearing temperature. The second calculation (thermalhydrodynamic case) employs the empirical procedure suggested by Seireg to improve the agreement between the calculated behaviour of a hydrodynamic bearing and its experimental performance. The algorithm has finally been extended in order to select the optimum bearings operating within a determined range of imposed loads and frequencies. OPTIMISATION PROBLEM A full journal bearing rotating at a constant frequency and supporting a given load is considered. The most significant variables selected for the synthesis of the bearing are the diameter D, the length L and the radial clearance C (geometry of the system), the oil viscosity µ, the applied load W and rotational frequency N. In all the optimisation cases, the quantities D, N, W are input data for the bearing design, as in most of practical situa-

tions. The diameter D is assumed equal to 5 cm. The design parameters for the calculations are therefore the ratio L/D, the clearance C and the oil inlet temperature T i. The constraints on the design are shown in Table 1 and represent the bounds on the oil thickness h, the oil pressure P and inlet temperature, the bearing length and clearance. Parameter Min Max Min. film thickness (µm) 1.7 Length-diameter ratio (-).5 1. Clearance (µm) 5. 5. Oil maximum pressure (MPa) Oil inlet temperature (K).. Table 1. Design and optimisation constraints A constraint on bearing stability according to the work of Lund and Saibel [15] has also been imposed. Finally, an objective function - the penalty function - is defined describing the goal of the optimisation. It is designed to find the journal bearing capable to support the given load, with the least oil temperature rise in the bearing lubrication film and the minimum oil flowrate. The penalty function is a weighted sum between two conflicting objectives, i.e. the dimensionless oil flowrate and temperature rise in the bearing, and provides a very significant criterion about the efficiency of the optimisation algorithm. The mathematical modelling of the hydrodynamic bearing performances is based on the numerical solutions of the Reynolds equation obtained by Raimondi & Boyd for an isoviscous lubricating film. These results are presented as charts where the most important characteristics of a bearing, such as the temperature rise, minimum oil film thickness, maximum oil film pressure, oil mass flowrate, coefficient of friction and so on, are reported as a function of the ratio L/D and the dimensionless Sommerfeld number ( S= ). D µ NLD C W These curves have been fitted with appropriate relationships in order to allow a fast computational analysis of a bearing. The synthesis is performed employing a GA, which works with a population of individuals, i.e. hydrodynamic journal bearings with their specific design parameters. These parameters are coded into a real vector, which constitutes the genetic code of the bearing. The GA can begin with a completely random set of parameters for the whole first population. Each bearing is analysed to evaluate its objective function, which is used to determine the fitness of each member of the population in its environment. The GAs select the individuals for the next population from the current one, based on their fitness. For this selection a roulettewheel model is used where the bearing with higher objective function have a higher likelihood of being selected. The newly selected individuals are arranged in pairs and a new member of the population is created simulating natural crossover. Uniform and average crossover are available as alternative operators for the creation of a new population. The resulting real parameters are than randomly mutated at a given mutation rate. This rate is several order of magnitudes higher than that observed in living organisms. Increased mutation rate was found to improve the convergence speed of the algorithm, even though too high values of this variable lead to an essentially random process. Each vector obtained through this manipulation represents a bearing of the new generation. This process is repeated until a satisfactory convergence level is reached. GAs are generally robust and relatively easy to apply once the objectives have been clearly defined. They are ideal in those cases where the GA requires no real understanding of the analysis of a system, but receives just a single number from each result. The evaluations of individuals already analysed (steady-state reproduction with replacement) is avoided. Elitism was implemented as an option in this GA, even though it seldom produce an improvement in the convergence rate. Both absolute fitness with scaling based on the inverse of penalty function and relative fitness based on differences between the values of the penalty function have been coded in the program. DISCUSSION OF THE RESULTS The base calculation has been performed taking into account for the dependence of the oil dynamic viscosity on the average film temperature, through the relationship: ( ) b ( T+θ µ T = µ ) e which well fits experimental data. The two constants b and θ depend on the oil type. The simultaneous dependence of the oil viscosity on the average film temperature and of the oil temperature rise on the Sommerfeld number (which, in turn, depends on the average oil viscosity) required an iterative optimisation procedure. The optimal ratio radial clearances-radius values (Figure 1) are higher for high loads and low frequencies to satisfy the minimum film thickness requirement. At intermediate frequencies a new increase of the radial clearance is required to counterbalance the sudden drop of the values of the optimal length-to-diameter ratio (Figure ). Even though a longer bearing should allow for a lower penalty function, the decrease of this variable is basically caused by the stability requirement. Figure shows that the optimal oil inlet temperature is relatively low at lower frequencies and higher loads, in order to assure sufficiently high values of the oil viscosity to maintain the required film thickness. At higher values of the frequency the optimal oil inlet temperature results almost independent on the bearing load. The ratio between the maximum and the mean oil pressure (P = W/LD) is shown Figure. It decreases as the external load increases, except for the lower load, and shows a weak dependence on the rotational frequency. The oil volumetric flowrate (Figure 5) and the corresponding oil temperature rise (Figure ) both increase with the frequency, but show a weak dependence on the bearing load. The predicted frictional losses (Figure 7) show a nearly quadratic increase with the rotational frequency while the dependence on the external load is very weak. In the second calculation an empirical procedure is adopted, based on the existence of a particular value of the frequency, for which the maximum pressure calculated by the viscous theory is equal to the experimental one. By this particular value of the frequency, a modified Sommerfeld number can be calculated and used instead of the classical one in a standard analysis, leading to results closer to experimental findings. The optimum

radial clearance (Figure ) is now much lower than in the previous calculations and is fundamentally determined by the minimum film thickness requirement, reaching the lower imposed constraint as the rotational frequency increases. The stability criterion predicts the possibility of a bearing instability only for the lower value of the external load. As a consequence, the optimum length-to diameter ratio (Figure 9) is much higher than in the previous calculation, reaching the upper constraint for the higher loads. Except for the lower applied load and for low values of frequencies, the optimum oil inlet temperature (Figure 1) results almost independent both on the external load and the rotational frequency. The ratio between the maximum and the mean pressure in the bearing (Figure 11) shows a similar dependence on the external load and rotational frequency as in the first calculation, even though the related values are now sensibly lower. The required oil volumetric flowrate (Figure 1) shows a very similar behaviour with respect to the previous calculation, with a bit lower values. The corresponding lubricant temperature rise (Figure 1) is instead sensibly lower with respect to the values predicted by the classical theory and seems to depend only on the applied load, while a very weak dependence on the rotational frequency is detected. The frictional losses (Figure 1) are now much lower with respect to the previous case and show an almost linear dependence on the rotational frequency of the bearing. In both calculations the minimum film thickness resulted very close to the minimum value imposed as a design constraint. The penalty functions obtained for each optimal bearing are finally presented in Figures 15 and 1. The above synthesis procedure has been extended so that the optimum parameters are searched for a hydrodynamic bearing operating within a range of frequencies and loads. The region under consideration has been divided in five sub-regions, each presenting a x load-frequency grid. For each sub-region the penalty function has been evaluated by simply summing the values obtained at each node of the grid with a particular configuration of the bearing. The results obtained employing the thermal-hydrodynamic calculation are shown in Table. These parameters should be applied to those cases where the load and frequency may change from time to time within the given bounds. CONCLUSIONS The design and operation of a hydrodynamic journal bearing were optimised using a GA. Results were compared to those obtained using a standard search technique. The comparison confirmed that the GAs are suitable to solve synthesis and optimisation problems relating to fluid journal bearings. Nay, GA s allowed to find better results concerning to the penalty function. Consequently, others synthesis calculations, considering more refined descriptions of the bearing behaviour, were performed with the same procedure. This study demonstrates the GA s ability to perform efficiently on a realworld optimisation problem providing better or comparable results relating to other search methods. 5 REFERENCES [1] Seireg, A. A.: Friction and Lubrication in Mechanical Design. New York Dekker 199 [] Raimondi, A. A. and Boyd, J.: A solution for the Finite Journal Bearing and Its Application to Analysis and Design: I-III. ASLE Trans., Vol. 1, 195, pp.159-9. [] Fogg, A.: Film Lubrication of Parallel Thrust Surfaces. Proc. Inst. Mech. Eng., Vol. 155, 195, pp. 9-7. [] Cameron, A.: Hydrodynamic Lubrication of Rotating Discs in Pure Sliding. New Type of Oil Film Formation. J. Inst. Petrol., Vol. 7, pp. 71. [5] Cole, J. A.: An Experimental Investigation of Temperature Effects in Journal Bearings. Proc. Conf. Of Lubrication and Wear, paper, 1957, pp. 111. [] Seireg, A., and Ezzat, H.: Thermohydrodynamic Phenomena in Fluid Film Lubrication. ASME J. Lubr. Technol., Vol.95, 197. [7] Rohde S. M., and Oh, K. P.: A Thermoelastohydrodynamic Analysis of a Finite Slider Bearing. ASME J. Lubr. Technol., 1975, pp 5-. [] Boncompain, R., Fillon, M., and Frene, J.: Analysis of Thermal Effects in Hydrodynamic Bearings. ASME J. Tribol., Vol.1, 19, pp. 19-. [9] Wang, N. Z., and Seireg, A.: Empirical Prediction of Shear Layer Thickness in Lubricating Films. Journal Tribol., Vol. 117, 1995, pp. -9. [1] Batchelor, J. K.: Note on a Class of Solutions of Navier-Stokes Equations Representing Steady Rotationally-Symmetric Flow. Q. J. Mech. Appl. Maths., Vol., 1951, pp. 9-1. [11] Szeri, A. Z., Schneider, S. J., Labbe, F., and Kaufmann, H. N.: Flow between Rotating Disks, Part 1: Basic Flow. Journal of Fluid Mechanics, Vol. 1, 19, pp. 1-11. [1] Sirivat, A., Rajagopal, K. R., and Szeri, A. Z.: An Experimental Investigation of the Flow of Non- Newtonian Fluids between Rotating Disks. Journal of Fluid Mechanics, Vol. 1, 19, pp. -5. [1] Davis, L.: Handbook of Genetic Algorithms. Van Nostrand Reinhold, New York, USA, 1991 [1] Goldberg, D. E.: Genetic Algorithms in Search, Optimisation and Machine Learning. Addison Wesley Publ. Co. Inc., Reading(MA),USA, 199 [15] Lund, J. W., and Seibel, A.: Stability and Damped Critical Speeds of a Flexible Rotor in Fluid Film Bearings. ASME Trans. J. Eng. Indust., paper n. 7, 197. Reg. N (Hz) W (N) µ (kpa s) C/D(*1 ) L/D T max (K) h /D(*1 ) Q (l/s) 1.1.1 Loss 1 1- -7. 1.75 1...5.1 1 1-7-9..5 1. 9.7.5. 1-1 -9.9 1. 1. 1.9.5.7 1 1- -7.7 1..5.9.5.7 1 5 1-7-9. 1..99 11.5.5.511 Table : Optimum bearing parameters and corresponding operative characteristics

5 Radial clearance to radius ratio * 1 7 5 W =9 N W = N Radial clearance to radius ratio*1 1 W =9 N Figure 1: Optimum clearance Figure : Optimum clearance.55 1.1.5 W = 9 N 1 Length/Diameter ratio.5..5..5 W = N Length/Diameter ratio.9..7..5.. W = 9 N. Figure : Optimum length/diameter ratio 5. Figure 9: Optimum length/diameter ratio 5 Oil inlet temperature (K) 9 Oil inlet temperature (K) 9 Figure : Optimum oil inlet temperatures Figure 1: Optimum oil inlet temperatures 9 Normalised pressure (Pmax/P) 7 1 15 9 Normalised pressure (Pmax/P) 1 15 1 9 FREQUENCY Hz Figure : Normalised pressure in optimum bearings FREQUENCY Hz Figure 11: Normalised pressure in optimum bearings

.1. Volumetric oil flowrate (l/s)... Volumetric oil flowrate (l/s).5.....1 Figure 5: Oil mass flowrate for optimum bearings Figure 1: Oil mass flowrate for optimum bearings 1 1 1 Temperature rise (K) 1 1 1 W =111 N Temperature rise (K) 1 Figure : Temperature rise in optimum bearings Figure 1: Temperature rise in optimum bearings 1 5 W = N Friction loss (W) W =111 N Friction loss (W) 15 1 5 Figure 7: Frictional losses in optimum bearings Figure 1: Frictional losses in optimum bearings Penalty function (-) Penalty function (-) 1 1 1 Figure 15: Penalty function for optimum bearing Figure 1: Penalty function for optimum bearings