Hydrodynamic Lubrication 2011 Part 7 pptx

112 5 Stability of a Rotating Shaft — Oil Whip Fig. 5.38a-c. Stability charts for a shaft in porous bearings [36]. a Φ = 10 −2 , b Φ = 10 −4 , c Φ = 0 (nonporous bush) such as an earthquake. This was confirmed experimentally also by simulating an artificial miniearthquake with a hammer by hitting the pedestal which carries the rotating shaft (Adams [63]). Further, if an imbalance and the bearing load of a rotor are changed variously while it is running in the hysteresis domain, it has been shown that all kinds of rotor responses can occur, e.g., periodic, quasiperiodic, and chaotic [63]. These examples are shown in Fig. 5.39. Which one of these three actually occurs and when the tran- sition from one state to another occurs is very sensitively related to the operating conditions of the rotor. Thus, it has been reported that chaotic phenomena are useful 5.10 Prevention of Oil Whip 113 Fig. 5.39. Chaos in rotor-bearing system [63] as safe diagnostic tools in assessing risks associated with the stable limit cycle within the hysteresis loop. [63]. 5.10 Prevention of Oil Whip It is very important to prevent oil whip in rotating machines. Some common methods are listed below: 1. Raise the critical speed of the shaft. The stable region in the stability chart is thereby expanded. Further, even if the shaft becomes unstable, if the rotating speed is lower than twice the critical speed, the shaft will not start a violent oil whip. The stable region can also be expanded by lowering the (length/diameter) ratio of the bearing. 2. Increase the eccentricity ratio of the journal (to reduce the floating height of the journal). The operation point of the shaft moves thereby into the stable region of the stability chart. The shaft is stable whenever the eccentricity ratio is larger than 0.8. To increase the eccentricity ratio, lowering the coefficient of viscousity of the oil, reducing the bearing length, increasing the bearing pressure, and in- creasing the bearing clearance are useful. While the above-mentioned methods are effective in circular bearings, the following methods using special bearings are also possible. 3. Use a noncircular bearing such as a two circular arc bearing or a three circular arc bearing. In these bearings, the radius of curvature of the metal surface is larger than that of a circular bearing, and hence the effective eccentricity ratio of the journal is large and so stability is high. 4. Use floating bush bearings. In this case, the bearing has two oil films, one in- side and one outside the floating bush. Stability is generally improved by using a floating bush bearing, but the stability chart is complicated and sometimes sta- bility can decrease. It is recommended that stability be examined for each case. 5. Use tilting pad bearings. In this case, the pad can tilt freely, and hence the cou- pling terms of the oil film coefficients are zero. Therefore, stability is essentially 114 5 Stability of a Rotating Shaft — Oil Whip high. The structure of the bearing is complicated, however, and hence the struc- tural strength of the bearing can be low. It is suitable for low bearing pressures and high shaft speeds. In summary, a circular bearing is adequate when bearing the pressure is high and the shaft speed is low. A tilting pad bearing is recommended when the bearing pressure is low and the shaft speed is high. A noncircular bearing or a floating bush bearing is used in intermediate cases. Many papers have been published on the sta- bility of two circular arc and three circular arc bearings [40] [44], and also on the stability of tilting pad bearings [21] [27] [35] [46]. References 1. A. L. Kimball, “Internal Friction Theory of Shaft Whirling”, General Electric Review, April, 1924, pp. 244 -251. 2. B. L. Newkirk and H. D. Taylor, “Shaft Whipping Due to Oil Action in Journal Bearings”, General Electric Review, August, 1925, pp. 559 - 568. 3. D. Robertson, “Whirling of a Journal in a Sleeve Bearing”, Philosophical Magazine, Series 7, Vol. 15, January 1933, pp. 113 - 130. 4. H. Poritsky, “A Contribution to the Theory of Oil Whip”, Trans. ASME, Vol. 75, 1953, pp. 1153 - 1161. 5. T. Okazaki, Personal Communication, 1954. 6. T. Okazaki and Y. Hori, “The Theory of Oil-Whip in Jouranl Bearings” (in Japanese), Trans. JSME, Vol. 21, No. 102, February 1955, pp. 125 - 130. Vol. 22, No. 114, Febru- ary 1956, p. 90. 7. Y. Hori, “A Theory of Oil-Whip in Journal Bearings,” Proc. of the 5th Japan National Congress for Applied Mechanics, September 7 - 9, Tokyo, 1955, pp. 395 - 398. 8. M. Hamanaka, Personal Communication, 1956. 9. B. L. Newkirk and J. F. Lewis, “Oil Film Whirl - An Investigation of Disturbances Due to Oil Films in Journal Bearings”, Trans. ASME, Vol. 78, 1956, pp. 21 - 27. 10. B. L. Newkirk, “Varieties of Shaft Disturbances Due to Fluid Films in Journal Bearings”, Trans. ASME, Vol. 78, 1956, pp. 985 - 988. 11. O. Pinkus, “Experimental Investigation of Resonant Whip”, Trans. ASME , Vol. 78, 1956, pp. 975 - 983. 12. V.T. Morgan and A. Cameron, “Mechanism of Lubrication in Porous Bearings”, Proceed- ings of Conference on Lubrication and Wear, IMechE, London, 1957, pp. 151 - 157. 13. T. Yamamoto, “On Critical Speeds of a Shaft Supported by a Ball Bearing”, Trans. ASME, Series E, J. of Applied Mechanics, Vol. 26, No. 2, June 1959, pp. 199 - 204. 14. Y. Hori, “Oil Whip” (in Japanese), Journal of JSME, Vol. 61, No. 478, November 1958, pp. 1348 - 1356. 15. Y. Hori, “A Theory of Oil Whip”, Trans. ASME, Series E, J. of Applied Mechanics, Vol. 26, No. 2, June 1959, pp. 189 - 198. 16. Y. Hori, “Study on Oil Whip” (in Japanese), Dissertation, University of Tokyo, December 1959. 17. T. Someya, “Stabilit ¨ at einer in zylindrischen Gleitlagern laufenden, unwuchtfreien Welle”, Ingenieur-Archiv, 33. Band, 2. Heft, 1963, pp. 85 - 108. References 115 18. H. Gotoda, “Research on the Vibration of a Rotating Shaft Supported by Journal Bearings (in Japanese), Report of the Research Subcommittee on Radial Gasturbines, No. 4, JSME, July 15, 1963, pp. 114 - 206, Appendix pp. 207 - 244. 19. T. Someya, “Stability of a Balanced Shaft Running in Cylindrical Journal Bearings”, Proc. Lubrication and Wear Second Convention, Eastbourne, Sponsored by IMechE, London, May 28-30, 1964, Paper 21, pp. 3 - 21. 20. H. Gotoda, “On the Vibration of a Rotating Shaft Supported by Journal Bearings (First Report, Unbalance Vibration)” (in Japanese), Trans. JSME, Vol. 30, No. 215, July 1964, pp. 887 - 900. 21. J.W. Lund, “Spring and Damping Coefficients for the Tilting Pad Journal Bearing”, ASLE Trans., Vol. 7, 1964, pp. 342 - 352. 22. M. Funakawa and A. Tatara, “Stability Criterion of an Elastic Rotor in Journal Bearings” (in Japanese), Trans. JSME, Vol. 30, No. 218, October 1964, pp. 1238 - 1244. 23. T. Someya, “Schwingungs- und Stabilit ¨ atsverhalten einer in zylindrischen Gleitlagern laufenden Welle mit Unwucht”, VDI - Forschungsheft 510, 1965. pp. 1 - 36. 24. E. Nakagawa and H. Aoki, “Approximate Solution for Elastic and Damping Properties of Oil Film in Journal Bearings” (in Japanese), Trans. JSME, Vol. 31, No. 229, Septem- ber 1965, pp. 1398 - 1408. 25. A. Cameron, “Principles of Lubrication”, Longman, London, 1966 26. F.K. Orcutt and E.B. Arwas, “The Steady State and Dynamic Characteristics of a Full Cir- cular Bearing and a Partial Arc Bearing in the Laminar and Turbulent Flow Regimes”, Trans. ASME, J. Lub. Tech., Vol. 89, July 1967, pp. 143 - 152. 27. F.K. Orcutt, “The Steady-State and Dynamic Characteristics of the Tilting-Pad Journal Bearing in Laminar and Turbulent Flow Regimes”, Trans. ASME, J. Lub. Tech., Vol. 89, July 1967, pp. 392 - 404. 28. K. Ono and A. Tamura, “On the Vibrations of a Horizontal Shaft Supported on Oil- Lubricated Journal Bearings” (in Japanese), Trans. JSME, Vol. 34, No. 258, February 1968, pp. 285 - 297. 29. T. Someya, “Dynamic Problems of Journal Bearing - Case Where It is Lubricated with Non-Compressive Liquid -” (in Japanesse), Journal of JSME, Vol 72, No. 610, Novwember 1969, pp. 1513 - 1523. 30. A. Tatara, “An Experimental Study on the Stabilizing Effect of Floating Bush Journal Bearings” (in Japanese), Journal of JSME, Vol. 72, No. 610, Novwember 1969, pp. 1564 - 1569. 31. M. Harada and H. Aoki, “The Dynamic Characteristics of Fully Circular Journal Bear- ings in the Turbulent Region” (in Japanese), Journal of Japan Society of Lubrication Engineers, Vol. 16, No. 6, June 1971, pp. 429 - 436. 32. K. Shiraki, “Troubleshooting of Vibration Problems in the Field” (in Japanese), Journal of JSME, Vol. 75, No. 639, April 1972, pp. 507 - 524. 33. M. Tanaka and Y. Hori, “Stability Characteristics of Floating Bush Bearings”, Trans. ASME, Series F, Vol. 94, No. 3, July 1972, pp. 248 - 259. 34. G.L. Falkenhagen, E.J. Gunter and F.T. Schuller, “Stability abd Transient Motion of a Vertical Three-Lobe Bearing System”, J. Engineering for Industry, Trans. ASME, Vol. 94, 1972, pp. 665 - 667. 35. S. Iida, “A Study on the Vibration Characteristics of Tilting Pad Journal Bearings” (in Japanese), Trans. JSME, Vol. 40, No. 331, March 1974, pp. 875 - 884. 36. Y. Hori and K. Okoshi, “Stability of a Rotating Shaft Supported by Porous Bearings”, Proc. the JSLE-ASLE International Lubrication Conference, Tokyo, June 9 - 11, 1975, pp. 333 - 340. 116 5 Stability of a Rotating Shaft — Oil Whip 37. A.G. Holmes, C.M.McC. Ettles and I.W. Mayes, “The Dynamics of Multi-Rotor Systems Supported on Oil Film Bearings”, Trans. ASME, Series L, Vol. 100, No. 1, 1978, pp. 156 - 164. 38. K. Kikuchi, M. Takagi and S. Kobayashi, “Effect of Alignment on Vibration of Multi- Bearing Rotor System (First Report, Vibration Characteristics of Three-Bearing, One-Rotor System) (in Japanese), Trans. JSME, Vol. 45, No. 400, December 1979, pp. 1349 - 1356. 39. R. Uematsu and Y. Hori, “Influence of Misalignment of Support Journal Bearings on Stability of A Multi-Rotor System”, Tribology International, Vol. 13, No. 5, Oct. 1980, pp. 249 - 252. 40. P.E. Allaire and R.D. Flack, “Journal Bearing Design for High Speed Turbomachinery”, in Bearing Design - Historical Aspects, Present Technology and Future Problems, ASME Publication, Edited by W.J. Anderson, 1980, New York. 41. D.F. Li, K.C. Choy and P.E. Allaire, “Stability and Transient Characteristics of Four Mul- tilobe Journal Bearing Configurations”, Journal of Luburication Technology, Trans. ASME, Vol. 102, 1980, pp. 291 - 299. 42. T. Nasuda and Y. Hori, “Influence of Misalignment of Support Journal Bearings on Sta- bility of Multi-Rotor Systems”, Proc. IFToMM Int. Conf. on Rotordynamic Problem in Power Plants, Rome, September 28 - October 1, 1982, pp. 389 - 395. 43. M. Akkok and C.M.M. Ettles, “Journal Bearing Response to Excitation and Behaviour in the Unstable Region”, ASLE Trans., Vol. 27, 1984, pp. 341 - 351. 44. H. Hashimoto, S. Wada and H. Tsunoda, “Performance Characteristics of Elliptical Jour- nal Bearings in Turbulent Flow Regime”, Bulletin, Japan Society of Mechanical En- gineers, 27 - 232, 1984, pp. 2265 - 2271. 45. M. Tanaka, K. Fukuda and Y. Hori, “Friction of Porous Metal Bearing” (in Japanese), Journal of the Faculty of Engineering, University of Tokyo, A, Vol. 23, 1985, pp. 16 - 17. 46. H. Hashimoto, S. Wada and T. Marukawa, “Performance Characteristics of Large Scale Tilting-Pad Journal Bearings”, Bulletin, Japan Society of Mechanical Engineers,28- 242, 1985, pp. 1761 - 1768. 47. M. Malik and Y. Hori, “An Approximate Nonlinear Transient Analysis of Journal Bearing Response in Unstable Region of Linearized System”, Proc. IFToMM-JSME Interna- tional Conference on Rotordynamics, Tokyo, September 14 - 17, 1986, pp. 217 - 220. 48. Y. Hori, T. Kato, R. Wu and M. Tanaka, “Dynamic Characteristics of a Rotor of an Electric Motor for a Downhole Pump: A Vertical Flexible Rotor in a Three Lobe Stator”, Proc. International Conference on Mechanical Dynamics, Shengyang, China, August 3 - 6, 1987, pp. 516 - 519. 49. T. Someya, Editor, “Journal Bearing Databook”, Springer Verlag, Berlin Heidelberg, 1988 50. J. Mitsui, “Method of Calculation for Bearing Characteristics”, in “Journal Bearing Data- book”, T. Someya, Editor, Springer Verlag, Berlin Heidelberg, 1988, pp. 231 - 240. 51. Y. Hori, “Anti-Earthquake Considerations in Rotordynamics”, Proc. 4th IMechE Interna- tional Conference on Vibrations in Rotating Machinery, Edinburgh, September 1988, pp. 1 - 8. 52. T. Kato and Y. Hori, “Application of the Matrix Form of the Reynolds Equation to Dy- namic Journal Bearings” (in Japanese), Trans. JSME, C, Vol. 54, No. 500, April 1988, pp. 935 - 942. 53. T. Kato and Y. Hori, “Fast Method for Calculating Dynamic Coefficients of Finite Width Journal Bearings with Quasi-Reynolds Boundary Condition”, Trans. ASME. J. of Tri- bology, Vol. 110, No. 3, July 1988, pp. 387 - 393. References 117 54. T. Kato and Y. Hori, “On the Cross Terms of the Damping Coefficients of Finite Width Journal Bearings” (in Japanese), Trans. JSME, C, Vol. 54, No. 505, September 1988, pp. 2214 - 2217. 55. Y. Hori and T. Kato, “Seismic Effect on the Stability of a Rotor Supported by Oil Film Bearings” (in Japanese), Trans. JSME, C, Vol. 55, No. 511, March 1989, pp. 614 - 617. 56. T. Kato and Y. Hori, “Theoretical Condition for the C xy = C yx Relation in Fluid Film Jour- nal Bearings”, Trans. ASME, J. Tribology, Vol. 111, No. 3, July 1989, pp. 426 - 429. 57. Y. Hori and T. Kato, “Earthquake-Induced Instability of a Rotor Supported by Oil Film Bearings”, Trans. ASME, J. Vibration and Acoustics, Vol. 112, April 1990, pp. 160 - 165. 58. T. Kato, K. Koguchi and Y. Hori, “Seismic Response of a Multirotor System Supported by Oil Film Bearings” (in Japanese), Trans. JSME, C, Vol. 57, No. 544, December 1991, pp. 3761 - 3768. 59. T. Kato, H. Matsuoka and Y. Hori, “Seismic Response of a Multirotor System Supported by Oil Film Bearing (Effect of Misalignment)” (in Japanese), Trans. JSME,C,Vol. 58, No. 549, May 1992, pp. 1572 - 1578. 60. T. Kato, K. Koguchi and Y. Hori, “Seismic Response of a Linearly Stable Multirotor Sys- tem”, Proc. IMechE International Conference on Vibrations in Rotating Machinery, Bath, June. 1992, pp. 7 - 11. 61. T. Kato, H. Matsuoka and Y. Hori, “Seismic Response of a Linearly Stable, Misaligned Multirotor System”, Tribology Transaction, S.T.L.E., Vol. 36, 1993-2, pp. 311 - 315. 62. C.E. Koeneke, M. Tanaka and H. Motoi, “Axial Oil Film Rupture in High Speed Bearings Due to the Effect of the Centrifugal Force”, Trans. ASME, J. Tribology, Vol. 117, No. 3, July 1995, pp. 394 - 398. 63. Maurice L. Adams, Michael L. Adams and J-S Guo, “Simulations and experiments of the non-linear hysteresis loop for rotor-bearing instability”, Proc. IMechE International Conference on Vibrations in Rotating Machinery, September 9 - 12, Oxford, 1996, pp. 309 - 319. 64. K. Hatakenaka, M. Tanaka and K. Suzuki, “A Modified Reynolds Equation with Cen- trifugal Force being Considered and Its Application to Floating Bush Bearing” (in Japanese), Trans. JSME, C, Vol. 65, No. 636, August 1999, pp. 3395 - 3400. 65. K. Hatakenaka, M. Tanaka and K. Suzuki, “A Study on the Dynamic Characteristics of High Speed Journal Bearing with Axial Oil Film Rupture and Its Application o Sta- bility Analysis of Floating Bush Bearing” (in Japanese), Trans. JSME, C, Vol. 65, No. 640, December 1999, pp. 4840 - 4845. 66. Y. Hori, Compilation of Manufacturers’ Data from Various Sorces (2003). 6 Foil Bearings A bearing surface is usually made so rigid that it will not deform under journal load or fluid film pressure. In some bearings, however, the bearing surface is made of foil or tape (metal foil or high polymer film, for example) that is sufficiently flexible. This kind of bearing is called a foil bearing. Figure 6.1a shows its fundamental form, where the angle β is called the wrap angle. Figure 6.1b, a combination of three basic units, is an example of a practical form of the foil bearing. Foil bearings were first studied by H. Blok and J. J. Van Rossum [1]. Fig. 6.1a,b. Foil bearings. a simple form, b combined form In the case of a foil bearing, the foil deforms due to the pressure in the fluid film. If the foil deforms, the bearing clearance will naturally change and, in response to it, the pressure generated will also change. Thus, the foil shape (clearance shape) and the fluid film pressure interact closely with each other. Figure 6.2 shows the distribution of pressure and clearance for a foil bearing Fig. 6.1a developed on a straight line. In a foil bearing, particularly when the wrap angle 120 6 Foil Bearings is large, it is known that the pressure and the clearance are almost constant over a quite wide range of the lubricating domain. Constancy of the pressure over a wide range of a foil bearing means that there is no particular force causing the shaft to whirl, and hence a shaft in a foil bearing has excellent stability [7]. Since the foil is not very strong mechanically, it can be said that a foil bearing is suitable to support a shaft with a low bearing load and low stability. A rotating shaft in an instrument used under zero-gravity conditions is a good example. Fig. 6.2. Pressure and clearance of a foil bearing, showing a maximum in the pressure and a minimum in the clearance Further, it is known that near the exit of the lubricating domain of a foil bearing, the pressure and the bearing clearance change as shown in Fig. 6.2, with a maximum and a minimum. A sharp increase in pressure is called a pressure spike. The relationship between the magnetic head and the magnetic tape of a magnetic tape storage device for a computer is similar to that between the shaft and the foil of a foil bearing. The fact that the film thickness has a minimum near the exit of the lubricating domain is particularly important in this case. The smaller the clearance between the magnetic tape and the read/write element is, the higher the recording density can be. Therefore, a read/write element is installed at the minimum clearance position in magnetic tape storage devices. In this case, the surrounding air is auto- matically drawn into the space between the tape and the magnetic head and forms a fluid film. In connection with magnetic tape storage, much research has been carried out into foil bearings [3]-[12]. In this section, a finite element method for a fluid film lubrication problem [13] is applied to a foil bearing, and the theoretical results are compared with experiments (Hori et al. [14] [15]). The profile of a lubricating surface of a magnetic head is often complicated, and the finite element method is suitable to the solution of such a problem. 6.1 Basic Equations 121 6.1 Basic Equations For a foil bearing, since the clearance distribution (foil shape) and the pressure dis- tribution are mutually related, its analysis is mathematically a solution of the si- multaneous equations of fluid film pressure and foil deformation. For simplicity, the following assumptions will be made: 1. Reynolds’ equation is applicable to the fluid film. 2. Compressibility of the fluid can be disregarded. 3. The foil deforms easily. 4. Tension in the foil is constant regardless of time and location. 5. Flow and pressure in the fluid are uniform in the width direction of the foil tape. Some notes should be added to these assumptions. In item (1), the extent of the domain in which Reynolds’ equation can be applied is not clearly defined because the air in a very large space enters a gradually decreasing space and finally into a very thin clearance and then flows out again to the surroundings. Item (2) is valid when the pressure is low, but in some cases compressibility cannot be ignored. Item (3) does not hold in some cases where rigidity of the foil cannot be disregarded. Item (4) means that the viscosity of the fluid and the mass of the foil are very small. Item (5) means that the dimension of the lubricating domain in the flow direction is sufficiently small compared with that in the width direction of the tape. On the above assumptions, the system will be described by the following simul- taneous equations: d dx  h 3 6µ dp dx  = U dh dx (6.1) p = T  1 R − d 2 h dx 2  (6.2) Equation 6.1 is Reynolds’ equation for the fluid film where p is the fluid film pres- sure, h is the film thickness, µ is viscosity of the fluid, and x is the coordinate in the direction of foil movement. Equation 6.2, which is basically an equation of the balance of the fluid film pressure p and the foil tension T , shows the relation be- tween pressure distribution p and film thickness distribution h in the fluid film on the assumption that the foil tension T is constant. R is the radius of the shaft. The following boundary conditions are assumed (see Fig. 6.3): 1. At a point x 1 (the entrance of the lubricating domain), which is located upstream far enough but not too far from the entrance z 1 of the contact domain of the circular shaft and the foil at rest, it is assumed that pressure p is equal to the ambient pressure (i.e., zero) and that the clearance h is equal to h 1 when the foil is not moving. 2. At a point x 2 (the exit of the lubricating domain), which is located downstream far enough but not too far from the exit z 2 of the contact domain, as for the case of the entrance, it is assumed that pressure p is equal to the ambient pressure and that the clearance h is equal to h 2 when the foil is not moving. 122 6 Foil Bearings Fig. 6.3. Boundary conditions [15] The locations of x 1 , and x 2 are such that the film pressure generated can still be disregarded and the clearance is not so large that Reynolds’ equation can still be used. It is difficult to determine the positions of x 1 and x 2 exactly, but it is expected that a little deviation from the exact position does not greatly affect the calculated results of pressure and clearance. Thus, the boundary conditions are as follows:  p = 0 and h = h 1 at x = x 1 p = 0 and h = h 2 at x = x 2 (6.3) 6.2 Finite Element Solution of the Basic Equations In solving the basic equations, Eqs. 6.1 and 6.2 under the boundary conditions Eq. 6.3, a finite element method is used [13]. It can be conveniently applied to a lubri- cating surface of complicated shape. 6.2.1 Reynolds’ Equation First, consider the following integral concerning the pressure distribution p(x) over the interval (x 1 , x 2 ): J{p} =  x 2 x 1 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ h 3 12µ  dp dx  2 − hU dp dx ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ dx (6.4) J{p} is a function of the function p(x) and is generally called a functional. When an arbitrary small change δp(x) is given to the function p(x), the first variation δJ{p} of J{p} will be as follows: δJ{p} = −  x 2 x 1  d dx  h 3 6µ dp dx  − U dh dx  δpdx (6.5) Equating this to zero yields the following stationary condition of the functional J{p}: [...]... angle 6.3.1 Single Cylinder Heads Figure 6 .7 shows the nondimensional pressure distribution p and the nondimensional film thickness distribution h of a single cylinder head (cf Fig 6.5a,ba ) for small, intermediate, and large wrap angles β Definitions of p and h are given in the figure together with that of β Toward the end of the transition from Fig 6.7a to Fig 6.7b, the region of constant film thickness... is clear in the case of a single cylinder head, is not clearly seen except for the pressure valley just behind the exit In the flat part, the pressure is lower and the film thickness is larger than those in the cylindrical part 6.3 Characteristics of Foil Bearings 129 Fig 6.7a–c Pressure and clearance distribution in a single cylinder head [15] a β = 2.48, b β = 5.08, c β = 9.53 130 6 Foil Bearings ∗... 100 nm has been achieved, i.e., of the order of the mean free path of an air molecule λ = 70 nm Therefore, the air cannot be regarded as a continuum but shows 6.4 Additional Topics 131 Fig 6.9 Dependencies of the positions of hmin , pmin , and pmax on β [15] the particulate nature of the molecules The effect of the particulate nature appears as slip between the solid surface and the air (slip flow) and... Equating this to zero gives the following stationary condition: x2 δJ{h} = − x1 p 1 d2 h − + 2 T R dx δh dx = 0 (6. 27) which is equivalent to the equation of balance for the foil, Eq 6.2, because δh is an arbitrary quantity Therefore, the film thickness h is obtained by solving Eq 6. 27 The mathematical procedure hereafter is the same as that of the previous section If the film thickness in element hi... xi+1 K pi = i=1 N V= xi i=1 N h3 T i R Ri dx 12µ i xi+1 Vi = hi URi dx i=1 (6.15) (6.16) xi i=1 Next, we equate the first variation of the functional of Eq 6.14 to zero: N δJ{p} = i=1 ∂J δpi = 0 ∂pi (6. 17) Since δpi is an arbitrary variable and K p is symmetrical, the following relation is obtained: ∂J ∂pi = 2K p P − V = 0 (6.18) 1 V 2 (6.19) or Kp P = This is a matrix representation of the simultaneous... h are given in the figure together with that of β Toward the end of the transition from Fig 6.7a to Fig 6.7b, the region of constant film thickness and that of constant pressure begin to appear In Fig 6.7c, wide domains of constant film thickness and constant pressure are clearly seen; the minimum film thickness appears near the exit and, corresponding to this, 128 6 Foil Bearings the maximum pressure (pressure... the exit are well known as the exit effects of a foil bearing In the case of a magnetic tape memory storage device, the read/write element is installed near the point of minimum film thickness in Fig 6.7c, as stated before It is known that the recording density goes up in almost inverse proportion to the size of the clearance between the read/write element and the recording surface Figure 6.8 shows the... dependence of the nondimensional constant film thickness ∗ h (the film thickness h∗ at the point where the pressure gradient becomes zero for the first time is defined as the constant film thickness, cf Fig 6.7c), the minimum film thickness hmin , the maximum pressure pmax , and the minimum pressure pmin on the nondimensional wrap angle β As seen in the figure, these values are almost constant for β > 5 The constant... p and the film thickness ¯ ¯ h in the case of a double cylinder head of wrap angle β = 2.48 The solid line in the figure corresponds to the case l = 1.93, where l is the nondimensional length of the flat part connecting the two cylinders and the dashed line is the case l = 0 The latter is equivalent to a single cylinder head For l = 1.93, two maxima of pressure and two minima of film thickness are seen,... domain into N elements as shown in Fig 6.4, and assume that the pressure in the ith element can be approximated by the following linear formula: pi (x) = xi+1 − x x − xi , xi+1 − xi xi+1 − xi pi pi+1 (6 .7) where [ ] shows a matrix This expression shows that the pressure at the ends (nodes) of the ith element are equal to pi and pi+1 , respectively, and the pressure changes linearly between them This can . , Vol. 78 , 1956, pp. 975 - 983. 12. V.T. Morgan and A. Cameron, “Mechanism of Lubrication in Porous Bearings”, Proceed- ings of Conference on Lubrication and Wear, IMechE, London, 19 57, pp. 151. ASME, Vol. 94, 1 972 , pp. 665 - 6 67. 35. S. Iida, “A Study on the Vibration Characteristics of Tilting Pad Journal Bearings” (in Japanese), Trans. JSME, Vol. 40, No. 331, March 1 974 , pp. 875 - 884. 36 Society of Lubrication Engineers, Vol. 16, No. 6, June 1 971 , pp. 429 - 436. 32. K. Shiraki, “Troubleshooting of Vibration Problems in the Field” (in Japanese), Journal of JSME, Vol. 75 , No. 639,