On the vibrations of shafts rotating on lubricated bearings

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A contribution is given here to the analysis of vibrations of shafts rotating on lubricated bearings. From the mathematical point of view this involves the study of the classical partial differential equations of the vibrations of shafts accompanied by non-linear boundary conditions, which reflect the complex response of lubricating films. From the mechanical point of view the major difficulty rests in the precise specification of this response: here complete (non-cavitating) and stable-laminar films of lubricant are postulated.


  1. (1)

    A review of work on self-excited vibrations can be found in the paperJournal Bearing Instability: a Review byB. L. Newkirk, published in the Proceedings of the Conference on Lubrication and Wear of the Inst. Mech. Eng., London, 1957. More recently the problem has been studied byE. Frederiksen inNotes on Oil Whirl of Flexible Rotors Supported by Cylindrical Journal Bearings, Ingenioren (International Ed.),1 (1957), 29–36; byJ. H. Halton inElliptical Whirl of Flooded Journal Bearings, Proc. Cambridge Phil. Soc.,54 (1958), 119–127; byFinn Ørbeck inTheory of Oil Whip for Vertical Rotors Supported by Plain Journal Bearings, Trans. A. S. M. E.,80 (1958), 1497–1502; and byYukio Hori inA Theory of Oil Whip, J. Appl. Mech.,26 E (1959) 189–198.A. C. Hagg andG. O. Sankey have studied the problem of forced vibrations inSome Dynamic Properties of Oil Film Journal Bearings with Reference to the Unbalance Vibration of Rotors, J. Appl. Mech.,23 (1956) 302–306.

  2. (2)

    For instance, self-excited oscillations may be avoided when bearings of proper geometry are introduced. See the review paper byB. L. Newkirk quoted in footnote (1) and earlier papers by the same author.

  3. (3)

    See, for instance;J. A. Cole andC. J. Hughes,Oil Flow and Film Extent in Complete Journal Bearings, Proc. I. Mech. E.,170 (1956), 499–510;J. A. Cole andC. J. Hughes,Visual Study of Film Extent in Dynamically Loaded Complete Journal Bearings, Proc. Conference on Lubrication and Wear of the I. Mech. E., London, 1957;M. V. Özdas,The Behaviour of Lubricating Film in Journal Bearings, IXe Congrès International de Mécanique Appliquée, Actes. Tome IV, 259–271, Bruxelles, 1956.

  4. (4)

    See for instance:D. F. Wilcock,Turbulence in High Speed Journal Bearings, Trans. A. S. M. E.,72 (1950), 825–833.

  5. (5)

    O. Reynolds,On the Theory of Lubrication and its Application to Mr. Beauchamp Tower’s Experiments, Phil. Trans. Roy. Soc. AI and II,177 (1886), 157. No direct link is usually claimed between Reynolds equation and the equations of Navier-Stokes; the former is usually established with direct considerations on a « film-flow ». Our developments generalize a procedure ofG. H. Wannier,A Contribution to the Hydrodynamics of Lubrication, Quart. Appl. Math.,8 (1950), 1–32. Note added in proof: See also a paper by H. G. Elrod in the January 1960 issue of the Q. Appl. Math.

  6. (6)

    On the same lines as in the steady-state case:L. N. Tao,General Solution of Reynolds Equation for a Journal Bearing of Finite Width, Q. Appl. Math.,17 (1959), 129–136.

  7. (7)

    A. Sommerfeld,Zur hydrodynamischen Theorie der Schmiermittelreibung, Z. Math. Phys.,50 (1904), 97.D. Robertson,Whirling of a Journal in a Sleeve Bearing, Phil. Mag., (7)24 (1933), 113–130.

  8. (8)

    F. W. Ocvirk, Techn. Note No. 2808, N. A. C. A., Washington, 1952.

  9. (9)

    Studied extensively byM. Muskat andF. Morgan,Studies in Lubrication. I. The Theory of Thick Film Lubrication of a Complete Journal Bearing of Finite Length. J. Appl. Phys.,9 (1938), 393.

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    To the reader who has in mind results of experimental investigations the statement may appear strange, and such as to cast doubts upon the theory. In fact the statement must be takenliterally, in the sense that the equations (3.4), (3.5), (3.21) have no simplestrictly periodic solutions; it has already been observed in the study of other non-linear systems that strictly periodic motions may not exist, whereas motions exist which are « very nearly » periodic. The distinction is important only from a mathematical point of view. See, for instance:F. Colombo,Sopra il fenomeno dell’azione asincrona, Rend. Sem. Mat. Padova,24 (1955), 353–395; the phenomena mentioned in this paper have other points in common with those studied here. The remark made above explains also the limitations of an approach to the dynamics of journal bearings suggested ayJ. H. Halton, loc. cit. footnote (1).

  11. (13)

    The remark may lead to an explanation of some of the discrepancies in the published results of experimental work, the argument being that, when a very flexible shaft approaches perfect mechical conditions, oil film instability is not excited when the speed 2ω(r) 1 is reached and speeds in excess of 2ω(f) 1 must be attained before its onset. On the other hand if disturbances of sufficient magnitude occur at speeds just above twice the first « restrained » critical ω(r) 1 oil whirl will set in immediately. See for instance the reduced range of stability in « bumping runs » compared with « quiet runs » as reported byB. L. Newkirk andJ. F. Lewis,Oil Film Whirl — An Investigation of Disturbances due to Oil Films in Journal Bearings, Trans. A. S. M. E.,78 (1956), 21–27 (particularly Tables 1 and 2). The remark would explain also the « inertia effect » in resonant whirl « characterized by a resistance to whip when the shaft is stable and unwillingness to pull out of whip once whip has developed » (O. Pinkus,Experimental Investigation of Resonant Whip, Trans. A. S. M. E.,78 (1956), 975–983).

  12. (13)

    We may also note that many years agoA. C. Hagg suggested « that rotors whose mass is contained between bearings may have a higher stable speed then overhung rotors, even though the lowest natural frequencies are the same ». (A. C. Hagg,The Influence of Oil Film Journal Bearings on the Stability of Rotating Machines, J. Appl. Mech., Trans. A. S. M. E.,68 (1946), A211-A220).

  13. (16)

    During the final revision of the manuscript of this paper, three memoirs have appeared on this and related topics:R. E. D. Bishop,The Vibration of Rotating Shafts, Journ. Mech. Eng. Science,1 (1959), 50–65;R. E. D. Bishop andG. M. L. Gladwell,The Vibration and Balancing of an Unbalanced Flexible Rotor, ibid., 66–77;G. M. L. Gladwell andR. E. D. Bishop,The Receptances of Uniform and Non-uniform Rotating Shafts, ibid., 78–91.

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To Antonio Signorini on his 70th birthday.

The Author is with the Englis Electric Co., Ltd, at the Nelson Research Laboratories.

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Capriz, G. On the vibrations of shafts rotating on lubricated bearings. Annali di Matematica 50, 223–248 (1960).

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  • Boundary Condition
  • Differential Equation
  • Partial Differential Equation
  • Major Difficulty
  • Mathematical Point