Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft

The nonlinear axial-lateral-torsional free vibration of the rotating shaft is analyzed by employing the Rayleigh beam theory. The effects of lateral, axial and torsional deformations, gyroscopic forces and rotary inertia are taken into account, but the shear deformations are neglected. In the new developed dynamic model, the nonlinearities are originated from the stretching of beam centerline, nonlinear curvature and twist and inertial terms which leads to the coupling between the axial, lateral and torsional deformations. The deformed configuration of the cross section of the beam is represented by the axial and lateral deformations, also the geometry of the beam in the deformed configuration is represented by Euler angles. A system of coupled nonlinear differential equations is obtained which is examined by the method of multiple scales and the nonlinear natural frequencies are determined. The accuracy of the solutions is inspected by comparing the free vibration response of the system with the numerical integration of the governing equations. The effect of the spin speed and radius-to-length ratio of the rotating shaft on the free vibrational behavior of the system is inspected. The study demonstrates the effect of axial-lateral-torsional coupling on the nonlinear free vibrations of the rotating shaft.

[1]  H. Nijmeijer,et al.  Friction-induced limit cycling in flexible rotor systems: An experimental drill-string set-up , 2006 .

[2]  B. O. Al‐Bedoor.  Modeling the coupled torsional and lateral vibrations of unbalanced rotors , 2001 .

[3]  A. Nayfeh,et al.  Linear and Nonlinear Structural Mechanics , 2002 .

[4]  K. Koser,et al.  TORSIONAL VIBRATIONS OF THE DRIVE SHAFTS OF MECHANISMS , 1997 .

[5]  S. H. Mirtalaie,et al.  A New Methodology for Modeling and Free Vibrations Analysis of Rotating Shaft Based on the Timoshenko Beam Theory , 2016 .

[6]  S.A.A. Hosseini,et al.  Multiple scales solution for free vibrations of a rotating shaft with stretching nonlinearity , 2013 .

[7]  Dishan Huang,et al.  Characteristics of torsional vibrations of a shaft with unbalance , 2007 .

[8]  Qing Hua Qin,et al.  Coupled torsional-flexural vibration of shaft systems in mechanical engineering—I. Finite element model , 1996 .

[9]  Anoop Chawla,et al.  Coupled bending, longitudinal and torsional vibrations of a cracked rotor , 2004 .

[10]  B. O. Al-Bedoor.  TRANSIENT TORSIONAL AND LATERAL VIBRATIONS OF UNBALANCED ROTORS WITH ROTOR-TO-STATOR RUBBING , 2000 .

[11]  H. D. Nelson,et al.  The Dynamics of Rotor-Bearing Systems Using Finite Elements , 1976 .

[12]  Jong-Shyong Wu,et al.  Computer method for torsion-and-flexure-coupled forced vibration of shafting system with damping , 1995 .

[13]  Vimal Singh,et al.  Perturbation methods , 1991 .

[14]  A. Bower.  Applied Mechanics of Solids , 2009 .

[15]  Tejas H. Patel,et al.  Vibration response of misaligned rotors , 2009 .

[16]  An-Chen Lee,et al.  A modified transfer matrix method for the coupling lateral and torsional vibrations of symmetric rotor-bearing systems , 2006 .

[17]  S. E. Khadem,et al.  FREE VIBRATIONS ANALYSIS OF A ROTATING SHAFT WITH NONLINEARITIES IN CURVATURE AND INERTIA , 2009 .

[18]  Agnes Muszynska,et al.  Whirl and whip—Rotor/bearing stability problems , 1986 .

[19]  Yehia A. Khulief,et al.  COUPLED BENDING TORSIONAL VIBRATION OF ROTORS USING FINITE ELEMENT , 1999 .

[20]  N. Verichev.  Chaotic torsional vibration of imbalanced shaft driven by a limited power supply , 2012 .

[21]  An Sung Lee,et al.  Prediction of maximum unbalance responses of a gear-coupled two-shaft rotor-bearing system , 2005 .

[22]  Fulei Chu,et al.  External and internal coupling effects of rotor's bending and torsional vibrations under unbalances , 2007 .

[23]  J. Łuczko,et al.  A geometrically non-linear model of rotating shafts with internal resonance and self-excited vibration , 2002 .

[24]  Guilhem Michon,et al.  Modeling and analysis of nonlinear rotordynamics due to higher order deformations in bending , 2011 .

[25]  Qing Hua Qin,et al.  Coupled torsional-flexural vibration of shaft systems in mechanical engineering—II. FE-TM impedance coupling method , 1996 .

[26]  M. Li,et al.  Coupled axial-lateral-torsional dynamics of a rotor-bearing system geared by spur bevel gears , 2002 .

[27]  R. Eshleman,et al.  On the Critical Speeds of a Continuous Rotor , 1969 .

[28]  Haiyan Hu,et al.  Dynamic analysis of a spiral bevel-geared rotor-bearing system , 2003 .

[29]  F. F. Ehrich,et al.  Observations of Nonlinear Phenomena in Rotordynamics , 2008 .

[30]  Andrew D. Dimarogonas,et al.  Coupled longitudinal and bending vibrations of a rotating shaft with an open crack , 1987 .

[31]  H. M. Khanlo,et al.  Chaotic vibration analysis of rotating, flexible, continuous shaft-disk system with a rub-impact between the disk and the stator , 2011 .

[32]  Giancarlo Genta,et al.  Dynamics of Rotating Systems , 2005 .

[33]  A. Nayfeh.  Introduction To Perturbation Techniques , 1981 .

[34]  Y. Ishida,et al.  Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications , 2002 .

[35]  J. S. Rao,et al.  Theoretical analysis of lateral response due to torsional excitation of geared rotors , 1998 .

[36]  J. S. Rao,et al.  History of Rotating Machinery Dynamics , 2011 .

[37]  Tejas H. Patel,et al.  Coupled bending-torsional vibration analysis of rotor with rub and crack , 2009 .

[38]  Xiaomin Zhao,et al.  Nonlinear lateral-torsional coupled motion of a rotor contacting a viscoelastically suspended stator , 2012 .

[39]  An-Chen Lee,et al.  A modified transfer matrix method for the coupled lateral and torsional vibrations of asymmetric rotor-bearing systems , 2008 .