Abstract
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.
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Perret, H.: Elastiche spielschwingungen konstant belaster walzlger. Werkstatt und Betrieb 3, 354–358 (1950)
Sunnersjo, C.S.: Varying compliance vibrations of rolling bearings. J. Sound Vib. 58(3), 363–373 (1978)
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)
Mevel, B., Guyader, J.L.: Routes to chaos in ball bearings. J. Sound Vib. 162, 471–487 (1993)
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)
Tiwari, M., Gupta, K., Prakash, O.: Dynamic response of an unbalanced rotor supported on ball bearings. J. Sound Vib. 238(5), 757–779 (2000)
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)
Chang-qing, B., Qingyu, X.: Dynamic model of ball bearings with internal clearance and waviness. J. Sound Vib. 294(1–2), 23–48 (2006)
Harsha, S.P.: Nonlinear dynamic analysis of a high-speed rotor supported by rolling element bearings. Nonlinear Dyn. 1, 65–100 (2006)
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)
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)
Reddy, J.N.: On locking-free shear deformable beam beam finite elements. Comput. Methods in Appl. Mech. Eng. 149, 113–132 (1997)
Nayfeh, A.H., Balachandram, B.: Applied Non Linear Dynamics: Analytical, Computational and Experimental Methods. John Willey & Sons, New York (1995)
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)
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)
Harris, T.A.: Roller Bearing Analysis. John Wiley and Sons, New York (1984)
Kramer, E.: Dynamics of Rotors and Foundations. Springer-Verlag, New York (1993)
Bathe, K.J.: Finite Element Procedures. Prentice Hall of India, New-Delhi (1996)
Parker, T., Chua, L.: Practical Numerical Algorithms for Chaotic Systems. Springer, Berlin (1989)
Wolf, A., Swift, J., Swinney, H., Vastano, J.: Determining lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)
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)
Sang-Kyu, C., Noah, S.T.: Response and stability analysis of piecewise linear oscillations under multi-forcing frequencies. Nonlinear Dyn. 3, 105–121 (1992)
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Gupta, T.C., Gupta, K., Sehgal, D.K. (2011). Nonlinear Dynamics and Chaos of an Unbalanced Flexible Rotor Supported by Deep Groove Ball Bearings with Radial Internal Clearance. In: Gupta, K. (eds) IUTAM Symposium on Emerging Trends in Rotor Dynamics. IUTAM Bookseries, vol 1011. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0020-8_28
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DOI: https://doi.org/10.1007/978-94-007-0020-8_28
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