


ARTICLE 

Year : 2011  Volume
: 1
 Issue : 2  Page : 9499 

Effect of L/D Ratio on the Performance of an Inverted Threelobe Pressure Dam Bearing
NK Batra^{1}, Gian Bhushan^{2}, NP Mehta^{1}
^{1} Maharishi Markendeshwar Engineering College, Ambala (Haryana), India ^{2} National Institute of Technology, Kurukshetra (Haryana), India
Date of Web Publication  24Oct2011 
Correspondence Address: N K Batra Maharishi Markendeshwar Engineering College, Ambala (Haryana) India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09768580.86641
Abstract   
An inverted threelobe pressure dam bearing which is produced by cutting a pressure dams on the upper lobe and two relieftracks on the lower two lobes of an ordinary inverted threelobe bearing is found to be more stable than an inverted threelobe bearing. In this paper an inverted threelobe pressure dam bearing supporting rigid and flexible rotors is analytically investigated to determine its performance when L/D ratio is varied in the range 0.81.5. The static and dynamic characteristics are studied at various L/D ratios. The results show that the stability of an inverted threelobe pressure dam bearing increases with decrease in L/D ratios both for rigid as well as flexible rotors. Keywords: Finite element method, L/D ratio, inverted threelobe pressure dam bearing
How to cite this article: Batra N K, Bhushan G, Mehta N P. Effect of L/D Ratio on the Performance of an Inverted Threelobe Pressure Dam Bearing. J Eng Technol 2011;1:949 
How to cite this URL: Batra N K, Bhushan G, Mehta N P. Effect of L/D Ratio on the Performance of an Inverted Threelobe Pressure Dam Bearing. J Eng Technol [serial online] 2011 [cited 2019 Nov 14];1:949. Available from: http://www.onlinejet.net/text.asp?2011/1/2/94/86641 
1. Introduction   
The present era in the industry is to run the turbomachines at high speeds. The ordinary circular bearings, which are the most common type of bearings, are found to be unstable at high speeds. It is found that the stability of ordinary journal bearing increases through the use of multilobes and the incorporation of pressure dams. The analysis of multilobe bearing by analytical means was first published by Pinkus ^{[1]} , followed by Lund and Thomson ^{[2]} and Malik et al. ^{[3]} . Analytical dynamic analysis ^{[4],[5]} has shown that cylindrical pressure dam bearings are found to be very stable. Also, an experimental stability analysis of such types of bearings ^{[6],[7]} showed that the analytical stability analysis reflects the general trends in the experimental data. The study of noncylindrical pressure dam bearings such as finiteelliptical, halfelliptical, offset halves, threelobe and fourlobe pressure dam bearings have proved that by incorporation of a pressure dam, the performance of bearings is improved ^{[8],[9],[10],[11],[12],[13],[14]} . L/D ratio is one of the important ^{[6],[7],[8],[9]} parameters that affect the stability of a bearing. The effect of L/D ratio on the stability of circular bearings was discussed by Lund ^{[10]} , Badgley et al. ^{[11]} and Hori ^{[12]} . The effect of L/D ratio on the performance of twolobe, threelobe and four lobe pressure dam bearings was studied by Mehta ^{[13]} and Rattan ^{[14]} and Gian ^{[15]} , respectively. The present study is undertaken to investigate the effect of L/D ratio on the performance of an inverted threelobe pressure dam bearing supporting rigid and flexible rotors.
2. Bearing Geometary   
[Figure 1] shows the geometry of an inverted threelobe pressure dam bearing. A rectangular dam or step of depth S _{d} and width L _{d} is cut circumferentially in lobe 1 of the bearing. Circumferential relief tracks or grooves of certain depth and width L _{t} are also cut centrally in lobes 2 and 3 of the bearing. Lobe 1 with pressure dam and lobes 2 and 3 with relief tracks are shown in [Figure 2]. The relief tracks are assumed to be so deep that their hydrodynamics effects can be neglected.
For a concentric position of the rotor, there are two reference clearances of the bearing: A major clearance c given by a circle circumscribed by the lobe radius and a minor clearance c _{m} given by an inscribed circle. Thus, the center of each lobe is shifted by a distance ep= cc _{m} , known as the ellipticity of the bearing. The various eccentricities and ellipticities are nondimensionalized by dividing the major clearance c.
If l _{1} and l _{2} are the circumferential lengths of the bearing before and after the dam, then
The various eccentricity ratios and attitude angles of the lobes of an inverted threelobe pressure dam bearing are given by:
3. Analysis   
The Reynolds equation for the laminar flow is:
The above equation is nondimensionalized by making the following substitutions:
The nondimensionalized equation thus obtained is
The various assumptions made in deriving the Reynolds equation are that the fluid is Newtonian, no slip occurs at the bearing surface, inertia terms are neglected, oil viscosity is constant and curvature is negligible. The Reynolds equation is analyzed for a pressure profile using the finite element method. The solution of this equation is obtained by minimizing the following variation integral ^{[16]} over the individual elements:
where = dimensionless film pressure in the e ^{th} element.
The Reynolds equation is an elliptical partial differential equation and hence must be solved as a boundary value problem. According to MaCallion et al., ^{[17]} for a bearing having oil supplied at zero pressure, the largest possible extent of positive pressure region is given by the boundary conditions that both pressure and pressure gradient are zero at the breakdown and build up boundaries of oilfilm. However, it has been shown ^{[18]} that even by setting the negative hydrodynamics pressure to zero as they occur in any iteration step, the results tend to satisfy the abovementioned boundary conditions in the limit. The latter approach has been followed in the present analysis. Stiffness and damping coefficients are determined separately for each lobe and then added. The values of these stiffness and damping coefficients, shaft flexibility, and dimensionless speed are then used to evaluate the coefficients of the characteristic equation ^{[19]} , which is a polynomial of the 6 ^{th} order for flexible rotors as given below (for a rigid rotor, F = 0).
This characteristic equation is given as:
where
For a rigid rotor, the value of F (dimensionless flexibility) is taken as 0. The system is considered as stable if the real part of all roots is negative. For a particular bearing geometry and eccentricity ratio, the values of dimensionless speed are increased until the system becomes unstable. The maximum value of speed for which the bearing is stable is then adopted as the dimensionless threshold speed.
The present analysis has been done for the bearing with the following parameters
The ellipticity ratio (δ) = 0.5 is selected for the present study. The value of L/D ratio is varied from 0.8 to 1.5 and the bearing is investigated for its static and dynamic characteristics.
4. Result and Discussion   
The effects of L/D ratio on the static characteristics of an inverted threelobe pressure dam bearing are shown in [Figure 3], [Figure 4], [Figure 5], [Figure 6] and [Figure 7]. The values of L/D ratios considered for this purpose are 0.8, 1.0 and 1.5. It is observed from [Figure 3] and [Figure 4] that with the increase in L/D ratio, eccentricity ratio decreases, whereas the attitude angle increases for a particular value of Sommerfeld number. The minimum film thickness is observed to increase with an increase in L/D ratio when considered for a particular value of Sommerfeld number [Figure 5]. [Figure 6] and [Figure 7] show the effect of L/D ratio on oilflow and friction coefficients. There is a considerable fall in the oilflow coefficient, whereas there is no significant change in the friction coefficient with the increase in L/D ratio when considered for a particular value of Sommerfeld number. The effect of L/D ratio on the stability of an inverted threelobe bearing supporting a rigid rotor is shown in [Figure 8]. The plots show that both the zone of infinite stability and the minimum threshold speed increase with decrease in L/D ratio. The zone of infinite stability increases from 0.26 to 1.02 and the minimum threshold speed increases from 9.3 to 18.65 when L/D ratio decreases from 1.5 to 0.8. These effects are due to reduction in loadcarrying capacity of the bearing.  Figure 8: Effect of L/D ratio on the stability of inverted threelobe pressure dam bearing supporting a rigid rotor (F=0)
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[Figure 9] and [Figure 10] show the effects of L/D ratio on the stability of an inverted threelobe bearing supporting flexible rotors. The results are found to be similar to that of the bearing supporting a rigid rotor. It is also observed from these plots of the stability that for a particular L/D ratio, the minimum threshold speed is reduced with the increase in flexibility of the rotor while there is no change in the zone of infinite stability.  Figure 9: Effect of L/D ratio on the stability of inverted threelobe pressure dam bearing supporting a flexible rotor (F=0.5)
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 Figure 10: Effect of L/D ratio on the stability of inverted threelobe pressure dam bearing supporting a flexible rotor (F=4.0)
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5. Conclusions   
 The eccentricity ratio decreases while attitude increases with an increase in L/D ratio.
 The minimum oilfilm thickness increases with an increase in L/D ratio.
 The oilflow coefficient decreases with an increase in L/D ratio.
 The friction coefficient remains almost unchanged with an increase in L/D ratio.
 Both the minimum threshold speed and the zone of infinite stability increase with decrease in L/D ratio for a fourlobe bearing supporting rigid rotor as well as flexible rotor. Therefore, the stability of an inverted threelobe pressure dam bearing increases with decrease in L/D ratio.
 For a particular L/D ratio, the minimum threshold speed reduces with increase in the rotor flexibility while the zone of infinite stability remains unchanged.
6. Notations
References   
1.  O. Pinkus, "Analysis and Characteristics of ThreeLobe Bearings", ASME Journal of Basic Engineering, Vol. 81, pp. 19, 1959. 
2.  J. W. Lund, and K. K. Thomson, "A Calculation Method and Data for the Dynamic Coefficients of Oil Lubricated Journal Bearings", Proceedings of the ASME Design and Engineering Conference, Minneapolis, pp. 1, 1978. 
3.  M. Malik, M. Chandra and R. Sinhasan, "Design Data for OffsetHalves Journal Bearings in Laminar and Turbulent Flow Regimes", ASLE Trans., pp. 133140, 1982. 
4.  J. C. Nicholas and P. E. Allaire, "Analysis of Step Journal Bearings Finite Length Stability", ASLE Trans., 1980, pp. 197207. [PUBMED] 
5.  J. C. Nicholas, L. E. Barrett, and M. E. Leader, "Experimental  Theoretical Comparison of Instability Onset Speeds for a Three Mass Rotor Supported by Step Journal Bearing", Trans. ASME, Journal of Mechanical Design, pp. 344351, 1980. [PUBMED] 
6.  R. D. Flack, M. E. Leader, and E. J. Gunter, "An Experimental Investigation on the Response of a Flexible Rotor Mounted in Pressure Dam Bearings", Trans. ASME, Journal of Mechanical Design, pp. 842850, 1980. [PUBMED] 
7.  N. P. Mehta, A. Singh, and B. K. Gupta, "Dynamic Analysis of Finite HalfElliptical Pressure Dam Bearings with Rotor Flexibility Effects", ASLE Trans., Vol. 29, no. 1, pp. 6166, 1986. 
8.  N. P. Mehta, and A. Singh, "Stability of Finite OrthogonallyDisplaced Pressure Dam Bearings", ASME Journal of Tribology, Vol. 109, no. 4, pp. 718720, 1987. 
9.  N. P. Mehta, and S. S. Rattan, "Performance of ThreeLobe Pressure Dam Bearings", Wear, pp. 181185, 1993. 
10.  J. W. Lund, "SelfExcited Stationary Whirl Orbits of a journal in Sleeve Bearings", Ph.D. Thesis, Renesselaer Polytechnic Institute,Troy, N.Y., 1966. 
11.  R. H. Badgley, and J. F. Booker, "Turborotor Instability: Effect of Initial Transients on Plane Motion", Trans. ASME, Journal of Lubrication Technology, pp. 625633, 1969. 
12.  Y. Hori, "A. Theory of Oil Whip", Trans. ASME, Series E, pp.189, 1959. 
13.  N. P. Mehta, A. Singh, and B. K. Gupta, Stability of Finite Elliptical Pressure Dam Bearings with Rotor Flexibility Effects, ASLE Trans., Vol. 29, no. 4, pp. 548557, 1986. 
14.  S. S. Rattan, "Stability analysis of pressure dam bearings using finite element method", Ph.D. Thesis, Kurukshetra University, 1995. 
15.  G. Bhushan, S. S. Rattan, and N. P. Mehta, "Effect of Load Orientation on the Stability of a Threelobe Pressure Dam Bearing with Rigid and Flexible Rotors", pp. 10, Vol. 1, DOI:10.4103/09768580.74532. 
16.  G. F. Booker, and K. H. Hubner, "Application of Finite Elements to Lubrication: An Engineering Approach", Trans ASME, Journal of Lubrication Technology, pp. 313323, 1972. 
17.  H. MaCallion, A. J. Smalley, T. Lioyd and R. Harsnell, "A Comparison of performance for Steadily Loaded Journal Bearings", Proc. Lub. and Wear Conference (Institute of Mechanical Engineers, London, no 12, pp. 180, 1986. 
18.  J. F. Booker and P. K. Goenka, "Spherical Bearings: Static and Dynamic Analysis via the Finite Element Method", Trans. ASME J. Lub. Tech. (1980). 
19.  E. J. Hahn, "The Excitability of Flexibility Rotors in Short Sleeve Bearings", Trans. ASME Journal of. Lubrication Technology, pp. 196 205, 1975. 
Authors   
N. K. Batra is presently serving as a faculty member in Mechanical Engineering Department of M.M. Engineering College, Mullana. His areas of interest include Tribology and Fluid Mechanics. He is a Life Member of the Indian Society of Technical Education (ISTE) and Member of Society of Automotive engineers. He has 15 publications in international and Indian journals and conferences.
Dr. Gian Bhushan is presently serving as Associate Professor in the Department of Mechanical Engineering at National Institute of Technology Kurukshetra Haryana INDIA. He received his Ph.D degree in 2004 from Kurukshetra University, Kurukshetra. He is supervising 5 Ph D research scholars and has supervised 17 M Tech dissertations. He has more than twenty five research papers in International/National journals and conferences to his credit. He received The Sir Rajendra Nath Mookerjee Memorial Prize for the best paper in the Journal of Institute of Engineers (Mech. Division) for the year 2002. He is a life member of the Tribology Society of India. His research interest areas include Tribology, Fluid Machines & CAE.
Dr. N. P. Mehta is at present Director, Technical Institutions, Mullana and Pro ViceChancellor, M.M. University, Mullana. Prior to joining M.M.University, Dr. Mehta was Principal, Regional Engineering College, Kurukshetra and later on Founder Director (VC), National Institute of Technology (Deemed to be University), Kurukshetra for about seven years. He has forty four years experience in teaching/ research. He graduated in Mechanical Engineering from Punjab University, Chandigarh in 1966 and did his Master's Degree in Mechanical Engineering from Kurukshetra University, Kurukshetra in 1972. He was awarded Ph.D by Allahabad University, Allahabad in 1982. He has more than 100 research publications to his credit in National & International Journals/ Conferences. He won Sir Rajendra Nath Mukerjee Memorial Prize and Gold Medal  2003 for the Best Paper in Mechanical Engineering. His Excellency, the Governor of Haryana, presented him "The Outstanding Engineer Award," on 36th Engineer's Day, Sept. 15, 2003. Seven research scholars got Ph.D under his guidance and four are presently registered for the Ph.D Degree.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6], [Figure 7], [Figure 8], [Figure 9], [Figure 10]
