Nonlinear Dynamics and Chaos of an Unbalanced Flexible Rotor Supported by Deep Groove Ball Bearings with Radial Internal Clearance

  • T. C. GuptaEmail author
  • K. Gupta
  • D. K. Sehgal
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 1011)


The flexible rotor modeling has an advantage over rigid rotor modeling, by its suitability to model rotors of complex shapes with multiple discs and bearings. In this paper, the non-linear dynamic behavior of a horizontal unbalanced flexible rotor supported on deep groove ball bearings is theoretically studied in detail for instability and chaos. A generalized Timoshenko beam FE formulation, which can be used for both flexible and rigid rotor systems with equal effectiveness, is developed. The shape functions are derived from the exact solutions of governing differential equations. The formulation is validated for mass, stiffness and, gyroscopic matrices and for nonlinear dynamic response. The steady state quasi-periodic solution is obtained by the non-autonomous shooting method, which also gives the monodromy matrix. The eigenvalues of monodromy matrix give information about stability and nature of bifurcation. The maximum value of Lyapunov exponent is used to decide upon the chaotic nature of the dynamic response. Different sets of clearance values, unbalance excitation force, and shaft flexibility are investigated for the presence of instability and chaos. Finally the range of parameters is established for the same.


Flexible rotor Ball bearing Shooting method Lyapunov exponents Quasi-periodic response Instability Chaos 


  1. 1.
    Perret, H.: Elastiche spielschwingungen konstant belaster walzlger. Werkstatt und Betrieb 3, 354–358 (1950)Google Scholar
  2. 2.
    Sunnersjo, C.S.: Varying compliance vibrations of rolling bearings. J. Sound Vib. 58(3), 363–373 (1978)CrossRefGoogle Scholar
  3. 3.
    Fukata, S., Gad, E.H., Kondou, T.A., Tamura, H.: On the radial vibrations of ball bearings (computer simulation). Bull. JSME 28, 899–904 (1985)Google Scholar
  4. 4.
    Mevel, B., Guyader, J.L.: Routes to chaos in ball bearings. J. Sound Vib. 162, 471–487 (1993)zbMATHCrossRefGoogle Scholar
  5. 5.
    Tiwari, M., Gupta, K., Prakash, O.: Effect of radial internal clearance of a ball bearing on the dynamics of a balanced horizontal rotor. J. Sound Vib. 238(5), 723–756 (2000)CrossRefGoogle Scholar
  6. 6.
    Tiwari, M., Gupta, K., Prakash, O.: Dynamic response of an unbalanced rotor supported on ball bearings. J. Sound Vib. 238(5), 757–779 (2000)CrossRefGoogle Scholar
  7. 7.
    Tiwari, M., Gupta, K., Prakash, O.: Experimental study of a rotor supported by deep groove ball bearing. Int. J. Rotat. Mach. 8(4), 243–258 (2002)CrossRefGoogle Scholar
  8. 8.
    Chang-qing, B., Qingyu, X.: Dynamic model of ball bearings with internal clearance and waviness. J. Sound Vib. 294(1–2), 23–48 (2006)CrossRefGoogle Scholar
  9. 9.
    Harsha, S.P.: Nonlinear dynamic analysis of a high-speed rotor supported by rolling element bearings. Nonlinear Dyn. 1, 65–100 (2006)CrossRefGoogle Scholar
  10. 10.
    Dong-Soo, L., Dong-Hoo, L.: A dynamic analysis of a flexible rotor in ball bearing with nonlinear stiffness characteristics. Int. J. Rotat. Mach. 3(2), 73–80 (1997)CrossRefGoogle Scholar
  11. 11.
    El-Saeidy, F.M.A.: Finite element modeling of rotor-shaft-rolling bearing systems with consideration of bearing nonlinearities. J. Vib. Control 4, 541–602 (1998)CrossRefGoogle Scholar
  12. 12.
    Reddy, J.N.: On locking-free shear deformable beam beam finite elements. Comput. Methods in Appl. Mech. Eng. 149, 113–132 (1997)zbMATHCrossRefGoogle Scholar
  13. 13.
    Nayfeh, A.H., Balachandram, B.: Applied Non Linear Dynamics: Analytical, Computational and Experimental Methods. John Willey & Sons, New York (1995)Google Scholar
  14. 14.
    Ku, D.M.: Finite element analysis of natural whirl speeds for rotor-bearing systems with internal damping. Mech.l Syst. Signal Process. 12(5), 599–610 (1998)CrossRefGoogle Scholar
  15. 15.
    Hashish, E., Sankar, T.S.: Finite element and modal analysis of rotor-bearing system under stochastic loading conditions. Trans. ASME, J. Vib. Acoust. Stress Reliab. Design 106, 80–89 (1984)Google Scholar
  16. 16.
    Harris, T.A.: Roller Bearing Analysis. John Wiley and Sons, New York (1984)Google Scholar
  17. 17.
    Kramer, E.: Dynamics of Rotors and Foundations. Springer-Verlag, New York (1993)Google Scholar
  18. 18.
    Bathe, K.J.: Finite Element Procedures. Prentice Hall of India, New-Delhi (1996)Google Scholar
  19. 19.
    Parker, T., Chua, L.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (1989)zbMATHGoogle Scholar
  20. 20.
    Wolf, A., Swift, J., Swinney, H., Vastano, J.: Determining lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Gupta, T.C., Gupta, K., Sehgal, D.K.: Nonlinear vibration analysis of an unbalanced flexible rotor supported by ball bearings with radial internal clearance. ASME Turbo-Expo-2008, Paper No. GT-2008-51204 (2008)Google Scholar
  22. 22.
    Sang-Kyu, C., Noah, S.T.: Response and stability analysis of piecewise linear oscillations under multi-forcing frequencies. Nonlinear Dyn. 3, 105–121 (1992)CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNational Institute of TechnologyJaipurIndia
  2. 2.Department of Mechanical EngineeringIndian Institute of TechnologyDelhiIndia
  3. 3.Department of Applied MechanicsIndian Institute of TechnologyDelhiIndia

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