Abstract
Aiming at the oil film instability of the sliding bearing at high speeds, a rotor test rig is built to study the non-linear dynamic behaviours caused by the first- and second-mode instability. A lumped mass model (LMM) of the rotor system considering the gyroscopic effect is established, in which the graphite self-lubricating bearing and the sliding bearing are simulated by a spring–damping model and a nonlinear oil film force model based on the assumption of short bearings, respectively. Moreover, a finite element model is also established to verify the validity of the LMM. The researches focus on the effects of two loading conditions (the first- and second-mode imbalance excitation) on the onset of instability and nonlinear responses of the rotor-bearing system by using the amplitude–frequency response, spectrum cascade, vibration waveform, orbit, and Poincaré map. Finally, experiments are carried out on the test rig. Simulation and experiment all show that oil film instability can excite complicated combination frequency components about the rotating frequency and the first-/second-mode whirl/whip frequency.
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Abbreviations
- C :
-
Damping matrix of the global system
- C 1 :
-
Rayleigh damping matrix
- C 2 :
-
Bearing damping matrix
- c :
-
Bearing clearance
- c blx , c bly :
-
Dampings of the left bearing in x and y directions
- D :
-
Bearing diameter
- E :
-
Young’s modulus
- F b :
-
Oil film force vector of the right bearing
- F bx5, F by5 :
-
Oil film forces of the right bearing in x and y directions
- F e :
-
Unbalanced force vector of the rotor system
- f bx5, f by5 :
-
Dimensionless oil film forces of the right bearing in x and y directions
- f n1, f n2 :
-
The first- and second-mode whirl/whip frequencies
- f r :
-
Rotating frequency
- G :
-
Gyroscopic matrix
- g :
-
Acceleration of gravity
- I :
-
Moment of inertia
- J di (i = 1,2,3,4,5):
-
Diametral moment of inertia about any axis perpendicular to the rotor axis
- J pi (i = 1,2,3,4,5):
-
Polar mass moment of inertia about rotor axis
- K :
-
Stiffness matrix of the global system
- k blx k bly :
-
Stiffnesses of the left bearing in x and y directions
- L :
-
Bearing length
- l i :
-
The distance between every two lumped mass points
- M :
-
General mass matrix of the global system
- m i (i = 1,2,3,4,5):
-
Lumped mass
- q :
-
Displacement vector
- \({\tilde{\it{{\bf q}}}}\) :
-
Dimensionless displacement vector
- u 1, u 2 :
-
Eccentricities of disc 1 and disc 2
- x i , y i (i = 1,2,3,4,5):
-
Displacements in x and y directions
- \({\tilde{x}_{i}, \tilde{y}_{i}(i=1, 2, 3, 4, 5)}\) :
-
Dimensionless displacements in x and y directions
- \({\xi_{1}, \xi_{2}}\) :
-
The first and second modal damping ratios
- \({\eta}\) :
-
Lubricant viscosity
- \({\theta_{xi}, \theta_{yi}}\) :
-
Angles of orientation about the x and y axes
- τ :
-
Dimensionless time
- \({\varphi_{1}, \varphi_{2}}\) :
-
Initial phase angle of two discs
- \({\omega}\) :
-
Rotating speed of rotor
- \({\omega_{n1}, \omega_{n2}}\) :
-
The first and second natural frequencies
References
Muszynska A.: Rotordynamics. Taylor & Francis, New York (2005)
Ding Q., Zhang K.P.: Order reduction and nonlinear behaviors of a continuous rotor system. Nonlinear Dyn. 67, 251–262 (2012)
Newkirk B.L., Taylor H.D.: Shaft whipping due to oil action in journal bearings. Gen. Electr. Rev. 28, 559–568 (1925)
Rao J.S.: History of Rotating Machinery Dynamics. Springer, Berlin (2011)
Robertson, D.: XII. Whirling of a journal in a sleeve bearing, The London, Edinburgh, and Dublin. Philos. Mag. J. Sci. 15, 113–130 (1933)
Pinkus, O., Sternlicht, B.: Theory of Hydrodynamic Lubrication. McGraw-Hill, New York (1961)
Morrison, D., Paterson, A.N.: Paper 14: criteria for unstable oil-whirl of flexible rotors. In: Proceedings of the Institution of Mechanical Engineers Conference Proceedings, vol. 179, pp. 45–55, Sage (1964)
Lund J.W.: Stability and damped critical speeds of a lexible rotor in fluid-film bearings. J. Eng. Ind. 96, 509–517 (1974)
Botman M.: Experiments on oil-film dampers for turbo machinery. ASME J. Eng. Power 98, 393–400 (1976)
Ehrich, F.F.: A state of the art survey in rotor dynamics-nonlinear and self-excited vibration phenomena. In: Proceedings of the Second International Symposium on Transport Phenomena, Dynamics, and Design of Rotating Machinery Part II, pp. 3–25, Hemisphere Pub. Co. (1988)
Nikolajsen J.L., Holmes R.: Investigation of squeeze-film isolators for the vibration control of a flexible rotor. ASME J. Mech. Eng. Sci. 21, 247–252 (1979)
Muszynska A.: Whirl and whip—rotor/bearing stability problems. J. Sound Vib. 110, 443–462 (1986)
Muszynska A.: Stability of whirl and whip in rotor/bearing systems. J. Sound Vib. 127, 49–64 (1988)
Childs D., Moes H., van Leeuwen H.: Journal bearing impedance descriptions for rotordynamic applications. J. Lubrif. Technol. 99, 198–210 (1977)
Muszynska A., Bently D.E.: Anti-swirl arrangements prevent rotor/seal instability. ASME J. Vib. Acoust. Stress Reliab. Des. 111, 156–162 (1989)
Mohan S., Hahn E.J.: Design of squeeze film damper supports for rigid rotors. J. Eng. Ind. 96, 976–982 (1974)
Ocvirk, F.W.: Short-Bearing Approximation for Full Journal Bearings. National Advisory Committee for Aeronautics, TN 2808 (1952)
Capone G.: Orbital motions of rigid symmetric rotor supported on journal bearings. La Meccanica Italiana 199, 37–46 (1986)
Capone, G.: Analytical description of fluid-dynamic force field in cylindrical journal bearing. L’Energia Elettrica 3, 105–110 (1991); (in Italian)
Zhang W., Xu X.: Modeling of nonlinear oil-film force acting on a journal with unsteady motion and nonlinear instability analysis under the model. Int. J. Nonlinear Sci. Numer. Simul. 1, 179–186 (2000)
Adiletta G., Guido A.R., Rossi C.: Nonlinear dynamics of a rigid unbalanced rotor in journal bearings. Part I: theoretical analysis. Nonlinear Dyn. 14, 57–87 (1997)
Jing J.P., Meng G., Sun Y. et al.: On the non-linear dynamic behavior of a rotor-bearing system. J. Sound Vib. 274, 1031–1044 (2004)
Jing J.P., Meng G., Sun Y. et al.: On the oil-whipping of a rotor-bearing system by a continuum model. Appl. Math. Model. 29, 461–475 (2005)
de Castro H.F., Cavalca K.L., Nordmann R.: Whirl and whip instabilities in rotor-bearing system considering a nonlinear force model. J. Sound Vib. 317, 273–293 (2008)
Ding Q., Leung A.Y.T.: Numerical and experimental investigations on flexible multi-bearing rotor dynamics. ASME J. Vib. Acoust. 127, 408–415 (2005)
Cheng M., Meng G., Jing J.P.: Numerical study of a rotor-bearing-seal system. Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 221, 779–788 (2007)
Rao T.V.V.L.N., Biswas S., Hirani H. et al.: An analytical approach to evaluate dynamic coefficients and nonlinear transient analysis of a hydrodynamic journal bearing. Tribol. Trans. 43, 109–115 (2000)
Chen C.L., Yau H.T.: Chaos in the imbalance response of a flexible rotor supported by oil film bearings with non-linear suspension. Nonlinear Dyn. 16, 71–90 (1998)
Vance J.M.: Rotordynamics of Turbomachinery. Wiley, New York (1988)
Schweizer B., Sievert M.: Nonlinear oscillations of automotive turbocharger turbines. J. Sound Vib. 321, 955–975 (2009)
Zhao, X.J., He, H., Xu, S.Y.: Influence of the floating-ring bearing parameters on stability of turbocharge rotor-bearing system. In: The 4th International Symposium on Fluid Machinery and Fluid Engineering, pp. 421-425, No. 4ISFMFE.Ch17, Beijing, China (2008)
Gunter E.J., Chen W.J.: DyRoBeS-Dynamics of Rotor Bearing Systems User’s Manual. RODYN Vibration Analysis, Inc., Charlottesville, VA (2000)
Kirk R.G., Alsaeed A.A., Gunter E.J.: Stability analysis of a high-speed automotive turbocharger. Tribol. Trans. 50, 427–434 (2007)
Gunter, E.G., Chen, W.J.: Dynamic analysis of a turbocharger in floating bushing bearings. In: Proceedings of the 3rd International Symposium on Stability Control of Rotating Machinery, Cleveland, OH (2005)
Chen W.J., Gunter E.J.: Introduction to Dynamics of Rotor-Bearing Systems. Trafford Publishing, Victoria, BC (2005)
Tian L., Wang W.J., Peng Z.J.: Effects of bearing outer clearance on the dynamic behaviours of the full floating ring bearing supported turbocharger rotor. Mech. Syst. Signal Process. 31, 155–175 (2012)
Tian L., Wang W.J., Peng Z.J.: Nonlinear effects of unbalance in the rotor-floating ring bearing system of turbochargers. Mech. Syst. Signal Process. 34, 298–320 (2013)
Rao J.S., Raju R.J., Reddy K.V.B.: Experimental investigation on oil whip of flexible rotors. Tribology 3, 100–103 (1970)
Adiletta G., Guido A.R., Rossi C.: Nonlinear dynamics of a rigid unbalanced rotor in journal bearings. Part II: experimental analysis. Nonlinear Dyn. 14, 157–189 (1997)
Fan C.C., Syu J.W., Pan M.C. et al.: Study of start-up vibration response for oil whirl, oil whip and dry whip. Mech. Syst. Signal Process. 25, 3102–3115 (2011)
EI-Shafei A., Tawfick S.H., Raafat M.S. et al.: Some experiments on oil whirl and oil whip. J. Eng. Gas Turbines Power 129, 144–153 (2007)
Sunar M., Al-Shurafa A.M.: The effect of disk location, shaft length and imbalance on fluid induced rotor vibrations. Arab. J. Sci. Eng. 36, 903–918 (2011)
Zhong, Y.E., He, Y.Z., Wang, Z., et al.: Rotor Dynamics. Tsinghua University Press, Beijing (1987); (in Chinese)
Adiletta G., Guido A.R., Rossi C.: Chaotic motions of a rigid rotor in short journal bearings. Nonlinear Dyn. 10, 251–269 (1996)
Friswell, M.I., Penny, J.E.T., Garvey, S.D., et al.: Dynamics of Rotating Machines, pp. 206–208. Cambridge University Press, Cambridge (2010)
Ma H., Shi C.Y., Han Q.K. et al.: Fixed-point rubbing fault characteristic analysis of a rotor system based on contact theory. Mech. Syst. Signal Process. 38, 137–153 (2013)
Bathe K.J., Wilson E.L.: Numerical Methods in Finite Element Analysis. Prentice-Hall, New Jersey (1976)
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Ma, H., Li, H., Niu, H. et al. Numerical and experimental analysis of the first-and second-mode instability in a rotor-bearing system. Arch Appl Mech 84, 519–541 (2014). https://doi.org/10.1007/s00419-013-0815-9
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DOI: https://doi.org/10.1007/s00419-013-0815-9