Research Papers

A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory

[+] Author and Article Information
S. H. Mirtalaie

Department of Mechanical Engineering,
Shahid Bahonar University of Kerman,
Kerman 76169-133, Iran
e-mail: h_mirtalaie@yahoo.com

M. A. Hajabasi

Department of Mechanical Engineering,
Shahid Bahonar University of Kerman,
Kerman 76169-133, Iran
e-mail: hajabasi@yahoo.com

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received September 22, 2015; final manuscript received December 3, 2015; published online January 22, 2016. Assoc. Editor: Izhak Bucher.

J. Vib. Acoust 138(2), 021012 (Jan 22, 2016) (13 pages) Paper No: VIB-15-1401; doi: 10.1115/1.4032327 History: Received September 22, 2015; Revised December 03, 2015

The linear lateral free vibration analysis of the rotor is performed based on a new insight on the Timoshenko beam theory. Rotary inertia, gyroscopic effects, and shear deformations are included, but the torsion is neglected and a new dynamic model is presented. It is shown that if the total rotation angle of the beam cross section is considered as one of the degrees-of-freedom of the Timoshenko rotor, as is common in the literature, some terms are missing in the modeling of the global dynamics of the system. The total deflection of the beam cross section is divided into two steps, first the Euler angles relations are employed to establish the curved geometry of the beam due to the elastic deformation of the beam centerline and then the shear deformations was superposed on it. As a result of this methodology and the mutual interaction of shear and Euler angles some variable coefficient terms appeared in the kinetic energy of the system which makes the problem be classified as the parametrically excited systems. A linear coupled variable coefficient system of differential equations is derived while the variable coefficient terms have been missing in all previous studies in the literature. The free vibration behavior of parametrically excited system is investigated by perturbation method and compared with the common Rayleigh, Timoshenko, and higher-order shear deformable spinning beam models in the rotordynamics. The effects of rotating speed and slenderness ratio are studied on the forward and backward natural frequencies and the critical speeds of the system are examined. The study demonstrates that the shear and Euler angles interaction affects the high-frequency free vibrations behavior of the spinning beam especially for higher slenderness ratio and rotating speeds of the rotor.

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Rao, J. S. , 1991, Rotor Dynamics, Wiley, New York.
Muszynska, A. , 2005, Rotordynamics, CRC Press, Boca Raton.
Yamamoto, T. , and Ishida, Y. , 2001, Linear and Nonlinear Rotordynamics: A Modern Treatment With Applications, Wiley, Germany.
Ehrich, F. F. , 1999, Handbook of Rotor Dynamics, Krieger, Malabar, FL.
Lalanne, M. , and Ferraris, G. , 1998, Rotordynamics Prediction in Engineering, 2nd ed., Wiley, Germany.
Genta, G. , 2005, Dynamics of Rotating Systems, Springer, New York.
Adams, L. M. , and Adama, J. R. , 2001, Rotating Machinery Vibrations From Analysis to Troubleshooting, Dekker, New York.
Lee, C. W. , 1993, Vibration Analysis of Rotors, Kluwer Academic Publishers, The Netherlands.
Casch, R. , Markert, R. , and Pfutzner, H. , 1979, “ Acceleration of Unbalanced Flexible Rotors Through the Critical Speeds,” J. Sound Vib., 63(3), pp. 393–409. [CrossRef]
Bernasconi, O. , 1987, “ Bisynchronous Torsional Vibration in Rotating Shafts,” ASME J. Appl. Mech., 54(4), pp. 893–897. [CrossRef]
Wang, Y. , 1997, “ Prediction of Periodic Response of Rotor Dynamic Systems With Nonlinear Supports,” ASME J. Vib. Acoust., 119(3), pp. 346–353. [CrossRef]
Lee, C. W. , Katz, R. , Ulsoy, A. G. , and Scott, R. A. , 1988, “ Modal Analysis of a Distributed Parameter Rotating Shaft,” J. Sound Vib., 122(1), pp. 119–130. [CrossRef]
Gladwell, G. M. L. , and Bishop, R. E. D. , 1959, “ The Vibration of Rotating Shafts Supported in Flexible Bearings,” J. Mech. Eng. Sci., 1(3), pp. 195–206. [CrossRef]
Bishop, R. E. D. , 1959, “ The Vibration of Rotating Shafts,” J. Mech. Eng. Sci., 1(1), pp. 50–65. [CrossRef]
Bishop, R. E. D. , and Parkinson, A. G. , 1965, “ Second Order Vibration of Flexible Shafts,” Philos. Trans. R. Soc. A, 259(1095), pp. 1–31. [CrossRef]
Pai, P. F. , Qian, X. , and Du, X. , 2013, “ Modeling and Dynamic Characteristics of Spinning Rayleigh Beams,” Int. J. Mech. Sci., 68, pp. 291–303. [CrossRef]
Eshleman, R. L. , and Eubanks, R. A. , 1969, “ On the Critical Speeds of a Continuous Rotor,” J. Eng. Ind., 91(4), pp. 1180–1188. [CrossRef]
Meirovitch, L. , and Silverberg, L. M. , 1985, “ Control of Non-Self-Adjoint Distributed-Parameter Systems,” J. Optim. Theory Appl., 47(1), pp. 77–90. [CrossRef]
Ziegler, H. , 1977, Principles of Structural Stability, 2nd ed., Birkhauser Verlag, Basel, Switzerland.
Badlani, M. , Kleinhenz, W. , and Hsiao, C. C. , 1978, “ The Effect of Rotary Inertia and Shear Deformation on the Parametric Stability of Unsymmetric Shafts,” Mech. Mach. Theory, 13(5), pp. 543–553. [CrossRef]
Kim, W. , Argento, A. , and Scott, R. A. , 2001, “ Forced Vibration and Dynamic Stability of a Rotating Tapered Composite Timoshenko Shaft: Bending Motions in End-Milling Operations,” J. Sound Vib., 246(4), pp. 583–600. [CrossRef]
Nayfeh, A. H. , Pai, P. F. , 2004, Linear and Nonlinear Structural Mechanics, Wiley Interscience, New York.
Bower, A. F. , 2009, Applied Mechanics of Solids, CRC Press, Boca Raton.
Nayfeh, A. H. , 1973, Perturbation Methods, Wiley-Interscience, New York.
Nayfeh, A. H. , 1981, Introduction to Perturbation Techniques, Wiley-Interscience, New York.
Boukhalfa, A. , 2014, “ Dynamic Analysis of a Spinning Functionally Graded Material Shaft by the ρ-Version of the Finite Element Method,” Lat. Am. J. Solids Struct., 11(11), pp. 2018–2038. [CrossRef]
Vance, J. M. , 1988, Rotordynamics of Turbomachinery, Wiley, Germany.


Grahic Jump Location
Fig. 3

Lateral displacements and shear rotation angles of a shaft element

Grahic Jump Location
Fig. 2

Euler angle rotations

Grahic Jump Location
Fig. 1

Rotor configuration and the coordinate system

Grahic Jump Location
Fig. 4

Time response of the rotor at ξ=0.3



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