Combination Resonance of Nonlinear Rotating Balanced Shafts Subjected to Periodic Axial Load

Document Type: Research Paper


Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran


Dynamic behavior of a circular shaft with geometrical nonlinearity and constant spin, subjected to periodic axial load is investigated. The case of parametric combination resonance is studied. Extension of shaft center line is the source of nonlinearity. The shaft has gyroscopic effect and rotary inertia but shear deformation is neglected. The equations of motion are derived by extended Hamilton principle and discretized by Galerkin method. The multiple scales method is applied to the complex form of equation of motion and the system under parametric combination resonance is analyzed. The attention is paid to analyze the effect of various system parameters on the shape of resonance curves and amplitude of system response. Furthermore, the role of external damping on combination resonance of linear and nonlinear systems is discussed. It will be shown that the external damping has different role in linear and nonlinear shaft models. To validate the perturbation results, numerical simulation is used. 


[1] Chen L. W., Ku D. M., 1990, Dynamic stability analysis of a rotating shaft by the finite element method, Journal of Sound and Vibration 143(1): 143-151.
[2] Lee H. P., Tan T. H., Leng G. S. B., 1997, Dynamic stability of spinning timoshenko shafts with a time-dependent spin rate, Journal of Sound and Vibration 199(3): 401-415.
[3] Sheu H.C., Chen L.W., 2000, A lumped mass model for parametric instability analysis of cantilever shaft–disk systems, Journal of Sound and Vibration 234(2): 331-348.
[4] Pei Y.C., 2009, Stability boundaries of a spinning rotor with parametrically excited gyroscopic system, European Journal of Mechanics - A/Solids 28(4): 891-896.
[5] Bartylla D., 2012, Stability investigation of rotors with periodic axial force, Mechanism and Machine Theory 58: 13-19.
[6] Mailybaev A. A., Seyranian A. P., 2013, Instability of a general rotating system with small axial asymmetry and damping, Journal of Sound and Vibration 332(2): 346-360.
[7] Mailybaev A. A., Spelsberg-Korspeter G., 2015, Combined effect of spatially fixed and rotating asymmetries on stability of a rotor, Journal of Sound and Vibration 336: 227-239.
[8] Bolotin V.V., 1964, The Dynamic Stability of Elastic System , Holden-day, Sanfransico, CA.
[9] Yamamoto T., Ishida Y., Aizawa K., 1979, On the subharmonic oscillations of unsymmetrical shafts, Bulletin of JSME 22(164): 164-173.
[10] Yamamoto T., Ishida Y., Ikeda T., 1981, Summed-and-differential harmonic oscillations of an unsymmetrical shaft, Bulletin of JSME 24(187): 183-191.
[11] Yamamoto T., Ishida Y., Ikeda T., Yamada M.,1981, Subharmonic and summed-and-differential harmonic oscillations of an unsymmetrical rotor, Bulletin of JSME 24(187): 192-199.
[12] Yamamoto T., Ishida Y., Ikeda T., Suzuki H., 1982, Super-summed-and-differential harmonic oscillations of an unsymmetrical shaft and an unsymmetrical rotor, Bulletin of JSME 25(200): 257-264.
[13] Yamamoto T., Ishida Y., Ikeda T., Yamamoto M., 1982, Nonlinear forced oscillations of a rotating shaft carrying an unsymmetrical rotor at the major critical speed, Bulletin of JSME 25(210): 1969-1976.
[14] Yamamoto T., Ishida Y., Ikeda T., 1983,Vibrations of a rotating shaft with rotating nonlinear restoring forces at the major critical speed, Transactions of the Japan Society of Mechanical Engineers Series C 49(448): 2133-2140.
[15] Ishida Y., Ikeda T., Yamamoto T., 1986, Effects of nonlinear spring characteristics on the dynamic unstable region of an unsymmetrical rotor, Bulletin of JSME 29(247): 200-207.
[16] Ishida Y., Liu J., Inoue T., Suzuki A., 2008,Vibrations of an asymmetrical shaft with gravity and nonlinear spring characteristics (Isolated Resonances and Internal Resonances), Journal of Vibration and Acoustics 130: 041004.
[17] Shahgholi M., Khadem S. E., 2012, Primary and parametric resonances of asymmetrical rotating shafts with stretching nonlinearity, Mechanism and Machine Theory 51: 131-144.
[18] Shahgholi M., Khadem S.E., Bab S., 2015, Nonlinear vibration analysis of a spinning shaft with multi-disks, Meccanica 50: 2293-2307.
[19] Ghorbanpour Arani A., Haghparast E., Amir S., 2012, Analytical solution for electro-mechanical behavior of piezoelectric rotating shaft reinforced by BNNTs under nonaxisymmetric internal pressure, Journal of Solid Mechanics 4: 339-354.
[20] Hosseini S. A. A., Zamanian M., 2013, Multiple scales solution for free vibrations of a rotating shaft with stretching nonlinearity, Scientia Iranica 20(1): 131-140.
[21] Ishida Y., Nagasaka I., Inoue T., Lee S., 1996, Forced oscillations of a vertical continuous rotor with geometric nonlinearity, Nonlinear Dynamics 11(2): 107-120.
[22] Nayfeh A.H., Pai P.F., 2004, Linear and Nonlinear Structural Mechanics, Wiley, New York.
[23] Nayfeh A. H.,1981, Introduction to Perturbation Techniques, Wiley, New York.
[24] Nayfeh A. H., Mook D. T., 1979, Nonlinear Oscillations, Wiley, New York.
[25] Valeev K. G., 1963, On the danger of combination resonances, Journal of Applied Mathematics and Mechanics 27(6): 1745-1759.