Whirling Analysis of Axial-Loaded Multi-Step Timoshenko Rotor Carrying Concentrated Masses

Subject Areas : Engineering

K Torabi ¹ , H Afshari ² , H Najafi ³

1 - Faculty of Mechanical Engineering, University of Isfahan, Isfahan, Iran
2 - Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
3 - Department of Solid Mechanics, Faculty of Mechanical Engineering, Politecnico di Milano, Milan, Italy

Received: 2016-11-12 Accepted : 2017-01-14 Published : 2017-03-30

Keywords: Axial load, Concentrated mass, Whirling analysis, Timoshenko rotor, Multi-step, Rotational inertia, Transfer matrix method (TMM),

Abstract :

In this paper, exact solution for two-plane transverse vibration analysis of axial-loaded multi-step Timoshenko rotor carrying concentrated masses is presented. Each attached element is considered to have both translational and rotational inertia. Forward and backward frequencies and corresponding modes are obtained using transfer matrix method (TMM). The effect of the angular velocity of spin, value of the translational and rotational inertia, position of the attached elements and applied axial force on the natural frequencies are investigated for various boundary conditions.

References:

[1] Chen Y., 1963, On the vibration of beams or rods carrying a concentrated mass, Journal of Applied Mechanics 30: 310-311.
[2] Laura P., Maurizi M.J., Pombo J.L., 1975, A note on the dynamics analysis of an elastically restrained-free beam with a mass at the free end, Journal of Sound and Vibration 41: 397-405.
[3] Rossit C.A., Laura P., 2001, Transverse vibrations of a cantilever beam with a spring mass system attached on the free end, Ocean Engineering 28: 933-939.
[4] Rao G.V., Saheb K.M., Janardhan G.R., 2006, Fundamental frequency for large amplitude vibrations of uniform Timoshenko beams with central point concentrated mass using coupled displacement field method, Journal of Sound and Vibration 298: 221-232.
[5] Rossit C.A., Laura P., 2001, Transverse normal modes of vibration of a cantilever Timoshenko beam with a mass elastically mounted at the free end, Journal of the Acoustical Society of America 110: 2837-2840.
[6] Laura P., Filipich C.P., Cortinez V.H., 1987, Vibrations of beams and plates carrying concentrated masses, Journal of Sound and Vibration 117: 459-465.
[7] Rossi R.E., Laura P., 1990, Vibrations of a Timoshenko beam clamped at one end and carrying a finite mass at the other, Applied Acoustics 30: 293-301.
[8] Maiz S., Bambill D., Rossit C., Laura P., 2007, Transverse vibration of Bernoulli–Euler beams carrying point masses and taking into account their rotary inertia, Journal of Sound and Vibration 303: 895-908.
[9] Lin H.Y., 2009, On the natural frequencies and mode shapes of a multi-span Timoshenko beam carrying a number of various concentrated elements, Journal of Sound and Vibration 319: 593-605.
[10] Gutierrez R.H., Laura P., Rossi R.E., 1991, Vibrations of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and carrying a finite mass at the other, Ocean Engineering 18: 129-145.
[11] Nelson H.D., 1980, A finite rotating shaft element using Timoshenko beam theory, Journal of Mechanical Design 102: 793-803.
[12] Edney S.L., Fox C.H.J., Williams E.J., 1990, Tapered Timoshenko finite elements for rotor dynamics analysis, Journal of Sound and Vibration 137: 463-481.
[13] Zu J.W.Z., Han R.P.S., 1992, Natural frequencies and normal modes of a spinning Timoshenko beam with general boundary conditions, Journal of Applied Mechanics 59: 197-204.
[14] Jun O.S., Kim J.O., 1999, Free bending vibration of a multi-step rotor, Journal of Sound and Vibration 224: 625-642.
[15] Banerjee J.R., Su H., 2006, Dynamic stiffness formulation and free vibration of a spinning composite beam, Computers & Structures 84: 1208-1214.
[16] Hosseini S.A.A., Khadem S.E., 2009, Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia, Mechanism and Machine Theory 44: 272-288.
[17] Hosseini S.A.A., Zamanian M., Shams Sh., Shooshtari A., 2014, Vibration analysis of geometrically nonlinear spinning beams, Mechanism and Machine Theory 78: 15-35.
[18] Afshari H., Irani M., Torabi K., 2014, Free whirling analysis of multi-step Timoshenko rotor with multiple bearing using DQEM, Modares Mechanical Engineering 14: 109-120.
[19] Wu J.S., Chen C.T., 2007, A lumped-mass TMM for free vibration analysis of a multi-step Timoshenko beam carrying eccentric lumped masses with rotary inertias, Journal of Sound and Vibration 301: 878-897.
[20] Wu J.S., Chen C.T., 2008, A continuous-mass TMM for free vibration analysis of a non-uniform beam with various boundary conditions and carrying multiple concentrated elements, Journal of Sound and Vibration 311: 1420-1430.
[21] Wu J.S., Chang B.H., 2013, Free vibration of axial-loaded multi-step Timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method, European Journal of Mechanics - A/Solids 38: 20-37.
[22] Khaji N., Shafiei M., Jalalpour M., 2009, Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions, International Journal of Mechanical Sciences 51: 667-681.
[23] Torabi K., Afshari H., Najafi H., 2013, Exact solution for free vibration analysis of multi-step Bernoulli-Euler and Timoshenko beams carrying multiple attached masses and taking into account their rotary inertia, Journal of Solid Mechanics 5: 336-349.
[24] Genta G., 2005, Dynamics of Rotating Systems, Springer, New York.
[25] Hutchinson J.R., 2001, Shear coefficients for Timoshenko beam theory, Journal of Applied Mechanics 68: 87-92.