This paper presents the various arrangements of grooving location of twogroove oil journal bearing for optimum performance. An attempt has been made to find out the effect of different configurations of two groove oil journal bearing by changing groove locations. Various groove angles that have been considered are 10°, 20°, and 30°. The Reynolds equation is solved numerically in a finite difference grid satisfying the appropriate boundary conditions. Determination of optimum performance is based on maximization of nondimensional load, flow coefficient, and mass parameter and minimization of friction variable using genetic algorithm. The results using genetic algorithm are compared with sequential quadratic programming (SQP). The two grooved bearings in general have grooves placed at diametrically opposite directions. However, the optimum groove locations, arrived at in the present work, are not diametrically opposite.
Journal bearings are used extensively in rotating machines because of their low wear and good damping characteristics. Fluidfilm journal bearings are available to support a rotating shaft in a turbo machinery system. A full circular journal bearing has a much simple configuration but exhibits instability at higher rotational speeds. It is relatively less expensive compared to the multilobe bearings. It is well known that whirl instability occurs at high speed in oil journal bearing. Present day bearings, at over increasing speeds and loads, confront the engineer with many new problems. Excessive power losses reduce the efficiency of the engine, and high bearing temperature poses a danger to material of the bearing as well as the lubricant. Instability arising mainly in the form of oil whip may ruin not only the bearing but the machine itself. New bearing designs are sought to meet the new requirements. A journal bearing fed by two axial grooves has a wide practical application due to its good load carrying capacity and ability to operate when reversal of shaft rotation occurs [
David et al. [
It has been observed that GAs have been successfully applied for optimizing bearing performance. However, the performance of twogroove journal bearing has not been optimized pertaining to location of groove positions with multiple objectives. In view of this, an attempt has been made in this paper to obtain an optimum configuration of the two grooves positions around the circumference of the hydrodynamic journal bearing for maximum oil flow, minimum friction loss, maximum load bearing capacity, and maximum critical speed visàvis mass parameter, a function of speed.
The oil flow rate depends on several factors, such as the viscosity of the lubricant, the geometry (length, diameter, and radial clearance) of the bearing, operating eccentricity, the inlet oil pressure, the arrangement of feeding sources, and groove location of the bearing. The pressure developed in the film due to journal motion also contributes to the flow. An adequate oil flow takes away frictional heat and does not allow rapid rise in temperature.
The calculation of friction loss within a bearing oil film is an integral part of the design of the bearing. The friction loss appears as heat, raises the temperature of the lubricant and lowers its viscosity, which is a key parameter of the bearing analysis. Therefore, the accurate prediction of friction loss is desired. The friction force is calculated by integrating shear stress over the journal surface. It is desired to keep the friction loss at minimum.
The load carrying capacity of the bearing within a bearing is developed due to pressure developed in the film. For a more accurate analysis, careful consideration of film extent needs to be included. This is expected to influence hydrodynamic leakage significantly and load carrying ability under some circumstances. If the feeding groove (in which pressure is zero) falls in the load carrying film, this part of the bearing makes no contribution to the loadcarrying ability. Thus the location of the groove plays a role in determining the load carrying ability of the bearing.
Plain circular bearing is mostly replaced by some other bearings, as plain bearing does not suit the stability requirements of highspeed machines and precision machine tools. Grooved circular bearings and multilobe bearings with two lobes, three lobes, and four lobes are commonly used. The critical mass parameter (a measure of stability) is a function of speed. The higher the critical speed is, the higher the stability limit is. The larger the eccentricity ratio is, the more stable the shaft is. If the eccentricity ratio is larger than 0.8, in particular, the shaft is always stable. In engineering analysis it is essential to know the critical speed at which oil whirl occurs and avoid it during operation. It has been found that severe whirl occurs when the shaft speed is approximately twice the bearing critical frequency.
To facilitate the optimum bearing design in the present paper, the nondimensional values of flow coefficient, load, and mass parameters along with friction variables for different configurations in groups are estimated. The optimum performance is determined on the basis of maximization of flow, load, mass parameter, and minimization of friction variable.
The Reynolds equation in two dimensions for an incompressible fluid is the governing equation. It can be written in a dimensionless form as
The pressure and film thickness can be expressed for small amplitude of vibration as
The nondimensional steady state load components as well as the nondimensional steady state load are given by
The flow coefficient in the dimensionless form can be written as
The nondimensional linearised equations of journal motion can be written as [
Now,
It has been found that the location of the groove has an influence on flow (
The problem is framed with four objectives. The variables used in the problem are in caseI starting angle of first groove (
Variable bounds for the bearing problem.
Case  Variable  Lower bound  Upper bound 

I  Starting angle of first groove  0°  180° 
Starting angle of second groove  170°  350°  
 
II 

0.1  0.9 
Starting angle of first groove  0°  180°  
Starting angle of second groove  170°  350° 
In this problem three variables called genes will form a chromosome. A set of chromosome is called population. With uniform probability distribution all chromosomes in the population are initialized. The population of each generation will have feasible design variables (chromosome) in terms of their allowable ranges but may be infeasible otherwise. The main steps involved in the genetic algorithm are discussed below and shown in flow chart (Figure
Flow chart for realcoded genetic Algorithm.
Realoded GA comprises of mainly six steps as follows.
There are mainly four userdefined parameters in the program, population size, maximum number of generation, cross over probability, and mutation probability. The best value of population size is 50. It is found that the program is converging very fast with these values. Cross over probability and mutation probability are more sensitive parameters for this program.
Second stage of program is to initialize the population size. So, 50 chromosomes are initialized using random probability for each variable span.
The selection operator involves randomly choosing members of the population to enter a mating pool. The operator is carefully formulated to ensure that better members of the population (with higher fitness) have a greater probability of being selected for mating, but that worse members of the population still have a small probability of being selected. Having some probability of choosing worse members is important to ensure that the search process is global and does not simply converge to the nearest local optimum. Selection is one of the important aspects of the GA process, and there are several ways for the selection.
Recombination is carried out through crossover and mutation operation in GA. The crossover operator is a method for sharing information between chromosomes. It ensures that the probability of reaching any point in the search space is never zero. The crossover operator is the main search operator in the GA. The search power of a crossover operator is defined as a measure of how flexible the operator is to create an arbitrary point in the search space. Crossover is useful in problems where building block exchange is necessary. It has been found that GAs may work well with large crossover probability and with a small mutation probability. A single point crossover preserves the structure of the parent string to the maximum. From a set of crossover operator, linear, blended crossover, and simulated binary crossover operators, it is found that, from trial run, the simulated binary crossover gives better convergence in limited time.
From biological view, mutation is any change of DNA material that can be reproduced. From computer science view, mutation is a genetic operator that follows crossover operator. It usually acts on only one individual chosen based on a probability or fitness function. One or more genetic components of the individual are scanned. And this component is modified based on some userdefinable probability or condition. Without mutation, offspring chromosomes would be limited to only the genes available within the initial population. Mutation should be able to introduce new genetic material as well as modify the existing one. With these new gene values, the genetic algorithm may be able to arrive at a better solution than was previously possible. Mutation operator prevents premature convergence to local optima by randomly sampling new points in the search space. There are many types of mutation, and these types depend on the representation itself. Random mutation finds a better suitability with the existing problem.
Elite preservation forms a new population from the initial population and mutated one. This operator is responsible for convergence of the fitness by allowing better value to pass to the next generation.
The groove position located around the circumference is grouped as follows.
GroupI: HzHz configuration: grooves are placed in a horizontal position 180° apart, that is, diametrically opposite to each other (Figure
GroupII: UpUp configuration: both grooves are placed (5° to 80°) above the horizontal position (Figure
GroupIII: UpHz configuration: the left groove is (5° to 80°) above the horizontal position, and the other groove is in horizontal position (Figure
GroupIV: DnDn configuration: both grooves are placed (5° to 80°) below the horizontal position (Figure
GroupV: HzUp configuration: the left groove is in horizontal position, and the other groove is (5° to 80°) above the horizontal position (Figure
GroupVI: DnUp configuration: one of the grooves is (5° to 80°) below the horizontal position (the left one), and the other groove (the right one) is (5° to 80°) above the horizontal position; groove position is varied at 5° interval (Figure
GroupVII: DnHz configuration: the left groove is (5° to 80°) below the horizontal position, and the other groove is in horizontal position (Figure
GroupVIII: HzDn configuration: the left groove is in horizontal position, and the other groove is (5° to 80°) below the horizontal position; groove position is varied at 5° interval (Figure
GroupIX: UpDn configuration: the left groove is (5° to 80°) above horizontal position, and the other groove is (5° to 80°) below the horizontal position; groove position is varied at 5° interval (Figure
HzHz configuration of twogroove oil journal bearing.
UpUp configuration of twogroove oil journal bearing.
UpHz configuration of twogroove oil journal bearing.
DnDn configuration of twogroove oil journal bearing.
HzUp configuration of twogroove oil journal bearing.
DnUp configuration of twogroove oil journal bearing.
DnHz configuration of twogroove oil journal bearing.
HzDn configuration of twogroove oil journal bearing.
UpDn configuration of twogroove oil journal bearing.
To ascertain the size of the groove for better performance, a comparison of nondimensional load is made for different groove angles as shown in Table
Comparison of nondimensional load values using 10°, 20°, and 30° groove angles.

 

10° groove  20° groove  30° groove  
0.200  0.077  0.074  0.0715 
0.400  0.1865  0.181  0.175 
0.600  0.406  0.399  0.389 
0.800  1.135  1.123  1.107 
A code has been developed to calculate the steady state and dynamic characteristics for given values of
The optimum value of fitness function obtained corresponding to minimization of friction variable has been tabulated for both GA and SQP in Table
Comparison of GA and SQP results.

Objective function value (minimum friction variable)  

GA results  SQP results  
0.100  25.841  25.841 
0.200  12.575  12.575 
0.300  7.991  7.991 
0.400  5.603  5.603 
0.500  4.050  4.050 
0.600  3.023  3.023 
0.700  2.146  2.146 
0.800  1.501  1.501 
0.900  0.358  0.358 
Similarly maximum load, maximum flow, and maximum mass parameter values are also found to match both methods. It has been observed as stated above that the results using both methods are found to be the same. However, GA has been used in this work as GA, being a heuristic search and optimization technique inspired by natural evolution, has been successfully applied to a wide range of realworld problems of significant complexity [
Initially a single objective function has been taken up. The generic algorithm convergence rate to true optima depends on the probability of crossover and mutation, on one hand, and the maximum generation, on the other hand. In order to preserve a few very good strings and reject lowfitness strings, a high crossover probability is preferred. The mutation operator helps to retain the diversity in the population but disrupts the progress towards a converged population and interferes with beneficial action of the selection and crossover. Therefore, a low probability, 0.001–0.1, is preferred. The genetic algorithm updates its population on every generation, with a guarantee of better or equivalent fitness strings. For wellbehaved functions, 30–40 generations are sufficient. For steep and irregular functions, 50–100 generations are preferred [
The optimum groove locations for minimum nondimensional friction variable, nondimensional load, nondimensional flow, and mass parameter at different
Variation of friction variable at optimum grooving location for different
Variation of optimum nondimensional load at different eccentricity ratios.
Variation of flow at optimum grooving location for different eccentricity ratios.
Variation of mass parameter at optimum grooving location for different eccentricity ratios.
From the results shown in Figures
Similarly by combining all the objective functions at a time the optimum configurations obtained is tabulated (Table
The optimum configurations combining all the objective functions at a time.




0.100  0.346  336.325 
0.200  0.141  196.512 
0.300  1.238  208.919 
0.400  0.469  240.390 
0.500  0.281  241.176 
0.600  0.382  231.287 
0.700  0.785  222.092 
0.800  0.476  343.712 
It has been observed from the tabulated results in Table
If the three variables, namely, eccentricity ratio (
Variable bounds for the bearing problem.
Variable  Lower bound  Upper bound 


0.100  0.900 
Starting angle of first groove ( 
0°  180° 
Starting angle of second groove ( 
170°  350° 
Fitness value considering friction variable as objective function.
Fitness value considering flow as objective function.
Fitness value considering load as objective function.
Fitness value considering mass parameter as objective function.
Again by combining all the objective functions at a time the fitness value plot has been obtained as shown in Figure
Fitness value combining all the objective function.
The optimum locations for each objective function including that of multiobjective function have been shown in Table
Optimum location considering different objectives.
Optimum location for objectives 




Minimum friction variable  0.900  0  232.906 
Maximum flow  0.899  5.660  301.960 
Maximum load carrying capacity  0.899  0.626  308.230 
Maximum mass parameter  0.811  0.890  308.230 
Optimization of the combined objectives  0.268  3.670  349.990 
After carefully looking at the results presented above, it appears that one may get near optimal results by placing a single groove and eliminating the second groove entirely. In view of this, an attempt has been made to find the optimum groove location for a singlegrove bearing and compared with twogroove cases for each of the objective functions as presented in Table
Comparison of optimum locations of grooves for twogroove and singlegroove bearings.
Comparison  Objective function 




For two groove  Minimum friction variable  0.900  0  232.906 
For single groove  0.100  180  —  
For two groove  Maximum flow  0.899  5.66  301.96 
For single groove  0.100  180  —  
For two groove  Maximum load  0.899  0.626  308.23 
For single groove  0.657  349.038  —  
For two groove  Maximum mass parameter  0.811  0.890  308.23 
For single groove  0.704  223.435  —  
For two groove  Optimization of all the combined objectives  0.268  3.67  349.99 
For single groove  0.657  349.038  — 
A dimensional example has been shown below to demonstrate how to convert the nondimensional parameters to dimensional parameters.
Let
So,
Taking minimum film thickness as
Hence,
Conversion of nondimensional results to dimensional ones.
Eccentricity ratio  Objective function  Optimum groove locations  Present nondimensional result  Dimensional values 

0.5  Maximization of flow 

0.8329 (optimum flow variable)  Flow, 
Minimization of friction 

4.050 (optimum friction variable)  Coefficient of Friction, 
From the results presented here, it can be inferred that the second groove location is sensitive to the type of objective function whereas the first groove is more or less the same for any objective function. The practice and the notion of convenience of keeping groove positions 180° apart need to be thoroughly looked into as the present results show that optimum groove locations are not 180° apart for any of the objective functions considered in the present work. Experimental verification of the present result may lead to a new approach of production of bearings with optimum groove locations; however, it is beyond the scope of the present work and hopefully experimentalists have a problem in hand.
For the purpose of validation of results the steady state characteristics of twogroove oil journal bearing having 20° groove angles placed in horizontal position for
Comparison of present results with [




0.103  1.453 [1.470]  75.860 [75.990] 
0.150  0.980 [0.991]  70.462 [71.580] 
0.224  0.629 [0.635]  63.4598 [63.540] 
0.352  0.352 [0.358]  56.100 [55.410] 
0.460  0.232 [0.235]  49.925 [49.270] 
0.559  0.157 [0.159]  45.1075 [44.330] 
0.650  0.106 [0.108]  40.120 [39.720] 
0.734  0.070 [0.071]  35.432 [35.160] 
0.773  0.0562 [0.056]  33.160 [32.860] 
0.811  0.043 [0.044]  30.614 [—] 
0.883  0.023 [0.024]  25.142 [25.020] 
Radial clearance (m)
Diameter of the journal (m)
Length of the bearing (m)
Bearing radius (m)
Eccentricity (m)
Eccentricity ratio
Coefficient of absolute viscosity of the lubricant (Pas)
Coefficient of friction, friction variable
Speed of the journal in r.p.s
Bearing attitude angle
Film thickness (m)
Nondimensional film thickness
Position of starting of the groove
Position of end of the groove
Sliding speed
Steady state pressure (Pa)
Nondimensional steady state pressure
Load carrying capacity (N)
Nondimensional load carrying capacity
Vertical direction
Horizontal direction
Vertical component (in
Vertical component (in
Load per unit bearing area
Sommerfeld number
Nondimensional flow coefficient,
Perturbed pressures
Perturbed eccentricity ratio and attitude angle around the steady state value
Stiffness coefficients (N/m)
Nondimensional stiffness coefficients
Damping coefficient (N·s/m)
Nondimensionaldamping co efficient
Time (s)
Journal rotational speed (rad/s), frequency of journal vibration
Nondimensional time,
Whirl ratio
Rotor mass (kg), mass parameter,
Steady state value.