Abstract
Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton’s principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.
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Chung, J., Yoo, H.H.: Dynamic analysis of a rotating cantilever beam by using the finite element method. J. Sound Vib. 249, 147–164 (2002)
Chung, J., Jung, D., Yoo, H.H.: Stability analysis for the flapwise motion of a cantilever beam with rotary oscillation. J. Sound Vib. 273, 1047–1062 (2004)
Yoo, H.H., Lee, S.H., Shin, S.H.: Flapwise bending vibration analysis of rotating multi-layered composite beams. J. Sound Vib. 286, 745–761 (2005)
Lin, S.M., Leen, S.Y.: Prediction of vibration and instability of rotating damped beams with an elastically restrained root. Int. J. Mech. Sci. 46, 1173–1194 (2004)
Lin, C.Y., Chen, L.W.: Dynamic stability of a rotating beam with a constrained damping layer. J. Sound Vib. 267, 209–225 (2003)
Fung, E.H.K., Yau, D.T.W.: Vibration characteristics of a rotating flexible arm with ACLD treatment. J. Sound Vib. 269, 165–182 (2004)
Hosseini, S.A.A., Khadem, S.E.: Vibration and reliability of a rotating beam with random properties under random excitation. Int. J. Mech. Sci. 49(12), 1377–1388 (2007)
Dasa, S.K., Rayb, P.C., Pohit, G.: Free vibration analysis of a rotating beam with non-linear spring and mass system. J. Sound Vib. 301, 165–188 (2007)
Wei, K., Meng, G., Zhou, S., Liu, J.: Vibration control of variable speed/acceleration rotating beams using smart materials. J. Sound Vib. 298, 1150–1158 (2006)
Gunda, J.B., Ganguli, R.: Stiff-string basis functions for vibration analysis of high speed rotating beams. J. App. Mech. 75, 024502-1–024502-5 (2008)
Valverde, J., García-Vallejo, D.: Stability analysis of a sub-structured model of the rotating beam. Nonlinear Dyn. 55(4), 355–372 (2009)
Yang, J.B., Jiang, L.J., Chen, D.C.H.: Dynamic modelling and control of a rotating Euler–Bernoulli beam. J. Sound Vib. 274, 863–875 (2004)
Lin, S.M.: PD control of a rotating smart beam with an elastic root. J. Sound Vib. 312, 109–124 (2008)
Lee, S.-.Y., Sheu, J.-.J., Lin, S.-.M.: In-plane vibrational analysis of rotating curved beam with elastically restrained root. J. Sound Vib. 315, 1086–1102 (2008)
Abolghasemi, M., Jalali, M.A.: Attractors of a rotating viscoelastic beam. Int. J. Non-Linear Mech. 38, 739–751 (2003)
Bhadbhade, V., Jalili, N., Mahmoodi, S.N.: A novel piezoelectrically actuated flexural/torsional vibrating beam gyroscope. J. Sound Vib. 311, 1305–1324 (2008)
Luo, A.C.J., Han, R.P.S.: Analytical predictions of chaos in a non-linear rod. J. Sound Vib. 227, 523–544 (1999)
Mahmoodi, S.N., Jalili, N., Khadem, S.E.: An experimental investigation of non-linear vibration and frequency response analysis of cantilever viscoelastic beams. J. Sound Vib. 311, 1409–1419 (2008)
Mahmoodi, S.N., Khadem, S.E., Kokabi, M.: Non-linear free vibrations of Kelvin–Voigt visco-elastic beams. Int. J. Mech. Sci. 49, 722–732 (2007)
Esmailzadeh, E., Jalali, M.A.: Non-linear oscillations of viscoelastic rectangular plates. Nonlinear Dyn. 18, 311–319 (1999)
Esmailzadeh, E., Shahani, A.R.: Longitudinal and rotational coupled vibration of viscoelastic bars with tip mass. Int. J. Non-Linear Mech. 34, 111–116 (1999)
Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley-Interscience, New York (1979)
Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley-Interscience, New York (2004)
Younesian, D., Esmailzadeh, E., Sedaghati, R.: Existence of periodic solutions for the generalized form of Mathieu equation. Nonlinear Dyn. 39, 345–353 (2005)
Rao, S.S.: Vibration of Continuous Systems. Wiley, New York (2007)
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Younesian, D., Esmailzadeh, E. Non-linear vibration of variable speed rotating viscoelastic beams. Nonlinear Dyn 60, 193–205 (2010). https://doi.org/10.1007/s11071-009-9589-6
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DOI: https://doi.org/10.1007/s11071-009-9589-6