This paper shows a theoretical vibration analysis regarding excitation due to elliptical shaft journals in sleeve bearings of electrical motors, based on a simplified rotordynamic model. It is shown that elliptical shaft journals lead to kinematic constraints regarding the movement of the shaft journals on the oil film of the sleeve bearings and therefore to an excitation of the rotordynamic system. The solution of the linear differential equation system leads to the mathematical description of the movement of the rotor mass, the shaft journals, and the sleeve bearing housings. Additionally the relative movements between the shaft journals and the bearing housings are deduced, as well as the bearing housing vibration velocities. The presented simplified rotordynamic model can also be applied to rotating machines, other than electrical machines. In this case, only the electromagnetic spring value cm has to be put to zero. 1. Introduction Many different kinds of excitation exist in rotating machinery—for example, mechanical unbalance and misalignment of the coupling—which may cause vibrations [1–6]. Besides these typical excitations also specific excitations associated with the type of the rotating machine occur. This paper focuses on vibrations of electrical machines; therefore also electromagnetic forces have to be considered, which may cause vibrations. These electromagnetic forces mainly occur if an eccentricity of the air gap in the electrical machine exists. The eccentricity can be divided into static eccentricity—that means the smallest air gap remains at constant position—and dynamic eccentricity, which means that the smallest air gap changes its position by the rotation of the rotor. Static eccentricity is, for example, caused by production tolerances regarding concentricity and fitting tolerance between stator housing, end shields, bearing housing, and so forth. Dynamic eccentricity is caused if the rotor is bent or if the rotor core is eccentrically positioned on the rotor shaft due to, for example, tolerances in the punching process of the rotor sheets. In this case a rotating magnetic force occurs, which is called UMP (unbalanced magnetic pull), which has been investigated in many publications as in Dorrell [7], Smith and Dorrell [8], Schuisky [9], Holopainen [10], Arkkio et al. [11], Belmans et al. [12], Stoll [13], and Werner [14, 15]. Most of the vibrations in rotating machinery occur with the frequency of the rotor rotation [1–5], for example, mechanical unbalance or UMP in electrical machines. Due to misalignment of the coupling between
References
[1]
M. I. Friswell, J. E. T. Penny, S. D. Garvey, and A. W. Lees, Dynamics of Rotating Machines, Cambridge University Press, Cambridge, UK, 2010.
[2]
L. Maurice and J. R. Adams, Rotating Machinery Vibration, CRC Press, Taylor & Francis Group, Boca Raton, Fla, USA, 2010.
[3]
J. M. Vance, F. J. Zeidan, and B. Murphy, Machinery Vibration and Rotordynamics, John Wiley & Sons, Hoboken, NJ, USA, 2010.
[4]
J. S. Rao, Rotor Dynamics, John Wiley & Sons, New York, NY, USA, 1996.
[5]
R. Gasch, R. Nordmann, and H. Pfützner, Rotordynamik, Springer, Berlin, Germany, 2002.
[6]
W. Kliem, C. Pommer, and J. Stoustrup, “Stability of rotor systems: a complex modelling approach,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 49, no. 4, pp. 644–655, 1998.
[7]
D. G. Dorrell, “Experimental behaviour of unbalanced magnetic pull in 3-phase induction motors with eccentric rotors and the relationship with tooth saturation,” IEEE Transactions on Energy Conversion, vol. 14, no. 3, pp. 304–309, 1999.
[8]
A. C. Smith and D. G. Dorrell, “Calculation and measurement of unbalanced magnetic pull in cage induction motors with eccentric rotors. I. Analytical model,” IEE Proceedings: Electric Power Applications, vol. 143, no. 3, pp. 193–201, 1996.
[9]
W. Schuisky, “Magnetic pull in electrical machines due to the eccentricity of the rotor,” Electricity Research Association Transaction, vol. 295, pp. 391–399, 1972.
[10]
T. P. Holopainen, Electromechanical interaction in rotor dynamics of cage induction motors, Ph.D. thesis, VTT Technical Research Centre of Finland, Helsinki University of Technology, Finland, 2004.
[11]
A. Arkkio, M. Antila, K. Pokki, A. Simon, and E. Lantto, “Electromagnetic force on a whirling cage rotor,” IEE Proceedings: Electric Power Applications, vol. 147, no. 5, pp. 353–360, 2000.
[12]
R. Belmans, A. Vandenput, and W. Geysen, “Calculation of the flux density and the unbalanced pull in two pole induction machines,” Archiv für Elektrotechnik, vol. 70, no. 3, pp. 151–161, 1987.
[13]
R. L. Stoll, “Simple computational model for calculating the unbalanced magnetic pull on a two-pole turbogenerator rotor due to eccentricity,” IEE Proceedings: Electric Power Applications, vol. 144, no. 4, pp. 263–270, 1997.
[14]
U. Werner, Rotordynamische analyze von asynchronmaschinen mit magnetischen unsymmetrien, dissertation, Technical University of Darmstadt, Shaker, Aachen, Germany, 2006.
[15]
U. Werner, “Mathematical analysis of rotor shaft displacements in asynchronous machines; a critical speed or just a rotation of the orbit axis?” ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik, vol. 89, no. 7, pp. 514–535, 2009.
[16]
IEC 60034-14, Rotating electrical machines—part 14: mechanical vibration of certain machines with shaft heights 56 mm and higher—measurement, evaluation and limits of vibration severity, International Electrotechnical Commissio, 2007.
[17]
ANSI /API 541, Form-wound-Squirrel-Cage Induction Motors-500 Horse power and Larger, API, 2004.
[18]
O. Reynolds, “On the theory of lubrication,” Philosophical Transaction of the Royal Society, vol. 177, pp. 157–234, 1886.
[19]
A. A. Gnanadoss and M. R. Osborne, “The numerical solution of reynolds' equation for a journal bearing,” Quarterly Journal of Mechanics and Applied Mathematics, vol. 17, no. 2, pp. 241–246, 1964.
[20]
J. Glienicke, Feder- und d？mpfungskonstanten von gleitlagern für turbomaschinen und deren einfluss auf das schwingungsverhalten eines einfachen rotors, dissertation, Technische Hochschule Karlsruhe, 1966.
[21]
J. Lund and K. Thomsen, “A calculation method and data for the dynamics of oil lubricated journal bearings,” in Topics in Fluid Film Bearing and Rotor Bearing System Design and Optimization, pp. 1–28, ASME, New York, NY, USA, 1978.
[22]
A. Tondl, Some Problems of Rotor Dynamics, Chapman & Hall, London, UK, 1965.
