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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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On the Stability of a Rotating Blade with Geometric Nonlinearity

Journal of Applied Nonlinear Dynamics 1(3) (2012) 263--286 | DOI:10.5890/JAND.2012.06.002

Fengxia Wang; Albert C.J. Luo

Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, IL62026-1805, USA

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The stability of a rotating, nonlinear blade under a time-varying torque is investigated. The nonlinear blade is presented based on the geometric nonlinearity.Compared to the nonlinear model, the linear model of a rotating blade with effective load is investigated. The two models of rotating blades are reduced to the ordinary differential equations through the Galerkin method, and gyroscopic systems with parametric excitations are obtained. From such gyroscope systems, the stability of the two models is studied by the generalized harmonic balance method, and the approximate, analytical solutions for the two models are obtained. The stability regions in parameter maps are presented for a better understanding of the stability of such parametric, time-varying systems. The analytical and numerical solutions are illustrated. This study provides an efficient and accurate way to determine the stability of gyroscope systems with time-varying excitations.


  1. [1]  Meirovitch, L. (1967), Analytical Methods in Vibrations, McGraw Hill, New York.
  2. [2]  Likins, P.W., Barbera, F.J. and Baddeley, V. (1973), Mathematical Modeling of Spinning Elastic Bodies for Modal Analysis, AIAA journal, 11, 1251-1258.
  3. [3]  Vigneron, F.R. (1975), Comment on "Mathematical Modeling of Spinning Elastic Bodies for Modal Analysis", AIAA Journal, 13, 126-127.
  4. [4]  Sharf, I. (1996), Geometrically Nonlinear Beam Element for Dynamics Simulation of Multibody Systems, International journal for numerical methods in engineering, 39, 763-786.
  5. [5]  Marcelo Areias Trindade and Robens Sampaio (2003), Dynamics of Beams Undergoing Large Rotations Accounting for Arbitrary Axial Deformation, Mecnica Computational, XXII, 2591-2611.
  6. [6]  García-Vallejo, D., Sugiyama, H. and Shabana, A.A. (2005), Finite Element Analysis of the Geometric Stiffening Effect: Non-Linear Elasticity. Proc. Inst. Mech. Eng., Proceedings. Part K, J, Multi-Body Dynamics, 219, 203-211.
  7. [7]  Kane, T.R., Ryan, R.R., and Banerjee, A.K. (1987), Dynamics of Cantilever Beam Attached to a Moving Base, AIAA. Journal of Guidance Control Dynamics, 10 (2), 139-151.
  8. [8]  Simo, J.C. and Vu-Quoc, L. (1987), The Role of Nonlinear Theories in Transient Dynamics Analysis of Flexible Structures, Journal of Sound and Vibration, 119 (3), 487-508.
  9. [9]  Sharf, I. (1995), Simulation of Flexible-Link Manipulators with Inertial and Geometric Nonlinearities, ASME Journal of Dynamic Systems, Measurement, and Control, 117, 74-87.
  10. [10]  Sharf, I. (1999), Nonlinear StrainMeasures, Shape Functions and Beam Elements for Dynamics of Flexible Beams, Multibody System Dynamics, 3, 189-205.
  11. [11]  Crespo da Silva, M.R.M. and Glynn, C. C. (1979), Out-of-plane Vibrations of a Beam Including Nonlinear Inertia and Nonlinear Curvature Effects, International Journal of Non-Linear Mechanics, 13, 261-271.
  12. [12]  Crespo da Silva and M.R.M. (1998), A Reduced-Order Analytical Model for the Nonlinear Dynamics of a Class of Flexible Multi-Beam Structures, International Journal of Solids and Structures, 35 (25), 3299-3315.
  13. [13]  Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillations, Wiley Interscience, New York.
  14. [14]  Anderson T.J., Balachandran, B. and Nayfeh, A.H. (1994), Nonlinear Resonances in a Flexible Cantilever Beam Anderson, Journal of Vibration and Acoustics, Transactions of the ASME, 116 (4), 480-484.
  15. [15]  Bauchau, O.A., Bottasso C.L. and Nikishkov Y.G. (2001), Modeling Rotorcraft Dynamics with Finite Element Multibody Procedures, Mathematical and Computer Modeling, 33, 1113-1137.
  16. [16]  Shabana, A.A. (2003), Dynamics of Multibody Systems, Cambridge University Press, Cambridge.
  17. [17]  Valverde, J. and García-Vallejo D. (2009), Stability Analysis of a Structured Model of The Rotating Beam, Nonlinear dynamics, 55 (4), 355-372.
  18. [18]  Nassar, Y.N. Al and B.O. Al-Bedoor (2003), On the Vibration of a Rotating Blade on a Torsionally Flexible Shaft, Journal of sound and vibration, 259 (5), 1237-1242. ,
  19. [19]  Turhan, Ö. and Bulut, G. (2005), Dynamic Stability of Rotating Blades (Beams) Eccentrically Clamped to a Shaft with Fluctuating Speed, Journal of Sound and Vibration, 280, 945-964.
  20. [20]  Blanch, G.K. and Clemm, D. S. (1969), Mathieu's equation for complex parameters: tables of characteristic values, U.S. Government Print Office.
  21. [21]  Takahashi, K. (1981), An Approach to Investigate the Instability of the Multiple-Degree-Of-Freedom Parametric Dynamic Systems, Journal of Sound and Vibration, 78 (4), 519-529.
  22. [22]  Luo, A.C.J. and Huang, J.Z. (2012), Analytical Dynamics of period-m flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, Article No: 1250093 (29 pages).
  23. [23]  Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP-L&H Scientific, Glen Carbon.
  24. [24]  Hodges, D.H. and Bless, R.R. (1977), On the Axial Vibrations of Rotating Bars, International Journal of Nonlinear Mechanics, 12 (6), 293-296.
  25. [25]  Hodges, D.H. and Bless, R.R. (1994), Axial Instability of Rotating Rods Revised, International Journal of Nonlinear Mechanics, 29 (6), 879-887.
  26. [26]  Meirovitch, L. (1997), Principles and Techniques of Vibrations. McGraw Hill, New York.
  27. [27]  Ider, S.K. and Amirouche, M.L. (1989), Nonlinear Modeling of Flexible Multibody Systems Dynamics Subjected to Variable Constraints, ASME Journal of Applied Mechanics, 56, 444-450.