83 Figure 5.11 Fitness value considering friction variable as objective function 86 Figure 5.12 Fitness value considering flow coefficient as objective function 86. Figure 5.13 Fitness value considering load as objective function 87. minimization of a multi-objective function 88 Figure 5.16 Almost optimal configuration for minimization of friction variable 89. 109 Figure 7.1 Comparison of flow coefficient for two bearing configurations 112 Figure 7.2 Comparison of friction coefficient for two bearing configurations of non-dimensional configurations 7.32 Figure comparison capacity for two bearing configurations 113.
List of Tables
Notation
WX vertical component (in X direction) of the resultant load WZ vertical component (in Z direction) of the resultant load.
Abstract
Introduction
- State of the Art
- Literature Review
- Journal bearing and two axial groove bearing
- Multi-lobe bearings
- Optimization of Journal bearing
- Scope of the present work
- Organization of the Thesis
It was observed that a small variation of the torque stress parameter ‘l’ had a major influence on the dynamic properties, i.e. the stiffness and damping coefficients. The load orientation is one of the factors affecting the stability of a three-lobe bearing.
Basic Equations, geometry of Lobe Bearings and optimization technique
Introduction
Reynolds equation in the hydrodynamic theory
Linear Perturbation Method
Boundary conditions
Non-Dimensional load Capacity
Solution Scheme
Design Parameters
Dynamic Coefficients
Since the journal performs small harmonic oscillations about its stationary position; the dynamic bearing capacity is expressed as a spring and a viscous damping force. The stiffness and damping coefficients presented in equation 2.16 and where WXt1 and WXt2 are total vertical load due to disturbance of steady state position. Likewise.
Stability Analysis
The steady state equilibrium position of the log is x0,z0 and∆xand ∆dice the perturbed sum from the steady state position at time 't'. The value of the mass parameter at the instability threshold is known as the critical mass parameter (Mcrit) and the corresponding spin ratio is known as the critical spin ratio (λcrit).
Geometry of bearings
- Geometry of two-groove bearing
- Geometry of three-lobe bearing
2.3 presents the geometry and coordinate system used for the analysis of the double wing bearing. These relationships, presented in Equations 2.31 through 2.36, are obtained from Figure 2.6 by following a similar procedure used for the double-wing bearing.
Optimization Techniques
- Real coded Genetic algorithm Computational procedure
Formulation of Multi-Objective function
Summary
Introduction
Estimation of Steady state and dynamic characteristics
Analysis of steady state and dynamic characteristics .1 Groove size for better performance
- Bearing Performance with different Groove locations
Genetic Algorithm (GA) has been used to find the optimal solution as outlined in Chapter 2. The optimal configurations have been obtained for eccentricity ratios from 0.1 to 0.9 in this case. The optimal value of fitness function obtained corresponding to minimization of friction variable has been tabulated for both GA and SQP in Table 3.5.
The optimum groove locations for minimum friction variable, non-dimensional load capacity, flow coefficient, mass parameter and multi-objective function (section 2.11) at different. The optimum results obtained for friction variable, flow coefficient, non-dimensional load capacity and mass parameter are shown in Figures 3.10 to 3.14. It was observed that the optimum configurations correspond to higher eccentricity ratio except for the multi-objective function.
Since the summation of differences is minimum for configuration 1, configuration 1 is therefore considered "close to the optimal configuration". A similar procedure has also been followed for other objective functions to arrive at "near optimal configurations". The various near-optimum configurations obtained are shown in Table 3.10 and also in Fig.
Summary
Analysis of two-lobe bearings
- Introduction
- Estimation of Steady state and dynamic characteristics
- Analysis of steady state and dynamic characteristics .1 Groove size for better performance
- Bearing Performance with different Groove locations
- Determination of optimum location of groove
- Determination of near to the optimum location of groove
- Summary
The optimum groove locations for minimum friction variable, non-dimensional load capacity, flow coefficient and mass parameter at different ε are shown in Fig. The optimal groove locations for the multi-objective function (section 2.11) are shown in Figs. The optimal results obtained for friction variable, flow coefficient, non-dimensional load capacity and mass parameter are shown in Figures 4.10 to 4.14.
It is observed that the optimum groove locations are different for different loading conditions (eccentricity ratio) as well as for different objective functions in the case of two-lobed bearings. The same was observed for double-axis grooved bearings and therefore, "near optimal configurations" were identified. Therefore, it is decided to identify "near optimal configurations" for double lobe bearings for different objective functions following the same procedure described in Chapter 2.
Optimal groove locations for various objective functions, viz. maximizing the dimensionless load capacity, flow coefficient and mass parameter, and minimizing the friction variable were obtained using MatLab's Genetic Algorithm (GA) toolbox. It is observed that the optimum locations of the grooves correspond to a significant improvement in the performance of the two-piece bearing. It was found that the optimal slot locations are not only different for different objective functions, but also different for different eccentricity ratios, depending on the loading conditions.
Analysis of three lobe bearings
Introduction
Estimation of Steady state and dynamic characteristics
Analysis of steady state and dynamic characteristics .1 Groove size for better performance
- Bearing Performance with different Groove locations
- Determination of optimum location of groove
- Determination of near to the optimum location of groove
Therefore, it is clear that there must be a certain configuration that corresponds to the optimal performance of a three-lobe bearing. The optimal configurations in this case were obtained for an eccentricity ratio ranging from 0.05 to 0.441. A comparison of the optimal non-dimensional friction variable, flow coefficient, non-dimensional load capacity and mass parameter with groove location 1200 apart has been made as shown in Tables 5.5 and 5.6.
It is very clear from the comparison that there is a significant improvement in the optimum value of friction variable, flow coefficient, non-dimensional load capacity and mass parameter value than that of three lobe bearing with grooves located 1200 apart. - to the other. When the eccentricity ratio (ε), the starting angle of the first groove (θ1), the starting angle of the second groove (θ2) and the starting angle of the third groove ( . θ3) are variables and it acts as a chromosome as in the case -II, the optimal configuration for the same is shown in table 5.7. The optimal results obtained for the friction variable, flow coefficient, non-dimensional load capacity, mass parameter and for the multi-objective function are shown in Figures 5.10 to 5.14.
It was observed that the optimum groove locations are different for different load conditions (eccentricity ratio) as well as for different objective functions in the case of three-lobe bearing. It is observed that the optimum groove locations correspond to significant performance improvement of three-lobe carrier. Therefore, the identification of locations of grooves for three-lobe carriers was obtained in such a way that the performance characteristics are close to the optimum for any loading condition (eccentricity ratio) for different objective functions.
Analysis of four lobe bearings
Introduction
Estimation of Steady state and dynamic characteristics
Analysis of steady state and dynamic characteristics .1 Groove size for better performance
- Bearing Performance with different Groove locations
- Determination of optimum location of groove
Therefore, it is obvious that there must be a certain configuration which corresponds to the optimal performance of four lobe bearings. The optimal groove locations for minimum friction variable, nondimensional load capacity, nondimensional flow coefficient, mass parameter, and combined objective function at different e are shown in Fig. It is very clear from the comparison that there is a significant improvement in the optimum value of friction variable, flow coefficient, non-dimensional load capacity and mass parameter value than that of four lobe bearings with grooves spaced 900 apart.
When the eccentricity ratio (ε), the initial angle of the first groove (θ1), the initial angle of the second groove (θ2), the initial angle of the third groove (θ3) and the initial angle of the fourth groove (θ4) are variables and act as a chromosome as in Example –II, the optimum configuration for the same is presented in Table 6.7. The optimal results obtained for the friction variable, flow coefficient, dimensionless bearing capacity and mass parameter and for multi-objective function minimization are shown in Figures 6.10 to 6.14. It was found that the optimal locations of the grooves are different for different load conditions (eccentricity ratio) as well as for different target functions in the case of two- and three-pad bearings.
It was therefore decided to follow the same procedure as described in Chapter 2 to identify 'near optimal configurations' of four-pad bearings for different target functions. It is observed that the optimal groove locations correspond to a significant improvement in the performance of the four-pad bearings. bearing. Therefore, the identification of groove locations for the four-blade bearing was achieved so that the performance characteristics are close to optimal for any load condition (eccentricity ratio) for various objective functions.
Concluding Remarks
- Introduction
- Concluding Remarks
- Scope for Future Works
- Summary
For three-blade bearings, the second and third groove locations are sensitive to the type of target function, while the first groove is more or less the same. In the case of four lobes, the locations of the second, third, and fourth grooves are sensitive to the type of objective function, while the locations of the first groove are more or less the same. In the optimal position, the performance characteristics are improved, e.g. friction variable, flow coefficient, dimensionless load capacity and mass parameter values.
The magnitude of the flow coefficient is much higher at optimal groove position for bilobe bearings compared to dual-groove, trilobe, and quadrilobe bearings. The magnitude of the non-dimensional friction variable is much lower for bilobe bearings compared to three-lobe and four-lobe bearings. From a practical point of view, it is easier to machine a two-axial groove or a two-lobe bearing when the grooves are in a horizontal plane.
An attempt has been made in this thesis to study and analyze four commonly used bearing configurations, namely two axial groove bearings, two-lobe bearings, three-lobe bearings and four-lobe bearings. An attempt was made to find out the optimum groove locations of the bearing configurations depending on maximization of flow coefficient, non-dimensional load, mass parameter and minimization of friction variable using Genetic Algorithm. Also a comparison of various performance characteristics of two-lobe, three-lobe and four-lobe bearings was made with the current practice of groove locations of these bearings.
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24] Sinhasan R., Goyal K.C, “Transient response of two lobe journal bearing with non Newtonian lubricant,” 1995, Tribology International, 28(4), pp. 37] Pettinato B en Flack R.D.,” Toetsresultate vir 'n hoogs voorafgelaaide daar lobjoernaal Bearing- Effect of load orientation on static and dynamic characteristics,” 2001, Journal of the Society of Tribologists and Lubrication Engineers, pp. 40] Bhushan G., “Effect of load orientation on the stabiliteit of a three-lobe bearing wat rigiede en buigsame rotors ondersteun,” 2011, World Academy of Science, Engineering and Technology, 57, pp.195-198.
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List of Publications
Vitae
