Abstract
One of the most well-known situations in which nonlinear effects must be taken into account to obtain realistic results is the rotating beam problem. This problem has been extensively studied in the literature and has even become a benchmark problem for the validation of nonlinear formulations. Among other approaches, the substructuring technique was proven to be a valid strategy to account for this problem. Later, the similarities between the absolute nodal coordinate formulation and the substructuring technique were demonstrated. At the same time, it was found the existence of a critical angular velocity, beyond which the system becomes unstable that was dependent on the number of substructures. Since the dependence of the critical velocity was not so far clear, this paper tries to shed some light on it. Moreover, previous studies were focused on a constant angular velocity analysis where the effects of Coriolis forces were neglected. In this paper, the influence of the Coriolis force term is not neglected. The influence of the reference conditions of the element frame are also investigated in this paper.
Similar content being viewed by others
References
Schilhans, M.J.: Bending frequency of a rotating cantilever beam. J. Appl. Mech. 25, 28–30 (1958)
Kane, T.R., Ryan, R.R., Banerjee, A.K.: Dynamics of a cantilever beam attached to a moving base. AIAA J. Guid. Control Dyn. 10(2), 139–151 (1987)
Simo, J.C., Vu-Quoc, L.: The role of non-linear theories in transient dynamic analysis of flexible structures. J. Sound Vib. 119(3), 487–508 (1987)
Wu, S.C., Haug, E.J.: Geometric non-linear substructuring for dynamics of flexible mechanical systems. J. Numer. Methods Eng. 26, 2211–2226 (1988)
Escalona, J.L., Hussien, H.A., Shabana, A.A.: Application of the absolute nodal coordinate formulation to multibody system dynamics. J. Sound Vib. 214(5), 833–951 (1998)
Berzeri, M., Shabana, A.A.: Study of the centrifugal stiffening effect using the finite element absolute nodal coordinate formulation. Multibody Syst. Dyn. 7, 357–387 (2002)
García-Vallejo, D., Sugiyama, H., Shabana, A.A.: Finite element analysis of the geometric stiffening effect: a correction in the floating frame of reference formulation. Proc. Inst. Mech. Eng., Proc. Part K, J. Multi-Body Dyn. 219, 187–202 (2005)
García-Vallejo, D., Sugiyama, H., Shabana, A.A.: Finite element analysis of the geometric stiffening effect: non-linear elasticity. Proc. Inst. Mech. Eng., Proc. Part K, J. Multi-Body Dyn. 219, 203–211 (2005)
Maqueda, L.G., Bauchau, O.A., Shabana, A.A.: Effect of the centrifugal forces on the finite element eigenvalue solution of rotating blades: a comparative study. In: Proceedings of the 2007 ECCOMAS Thematic Conference on Multibody Dynamics, Milan, Italy, June 25–28 (2007)
Omar, M., Shabana, A.A.: A two-dimensional shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243(3), 565–573 (2001)
Dufva, K., Sopanen, J., Mikkola, A.: A two-dimensional shear deformable beam element based on the absolute nodal coordinate formulation. J. Sound Vib. 280, 719–738 (2005)
Shabana, A.A.: Dynamics of Multibody Systems, 2nd edn. Cambridge University Press, New York (1998)
Goriely, A., Tabor, M.: Nonlinear dynamics of filaments I. Dynamical instabilities. Phys. D 105, 20–44 (1997)
Antman, S.S.: Nonlinear Problems of Elasticity. Springer, Berlin (1995)
Valverde, J., Escalona, J.L., Domínguez, J., Champneys, A.R.: Stability and bifurcation analysis of a spinning space tether. J. Nonlinear Sci. 16(5), 507–542 (2006)
Valverde, J., van der Heijden, G.: Stability and bifurcation analysis of a spinning space tether. J. Nonlinear Sci. (2008, submitted)
Valverde, J., van der Heijden, G.: Stability of a whirling conducting rod in the presence of a magnetic field. Application to the problem of space tethers. In: Proceedings of the ASME DETC and CIE Conference, Long Beach, CA (2005)
Strogatz, S.H.: Nonlinear Dynamics and Chaos. Perseus Books, Cambridge (1994)
Doedel, E.J., Paffenroth, R.C., Champneys, A.R., Fairgrieve, T., Kuznetsov, Y.A., Oldeman, B.E., Sandstede, B., Wang, X.: AUTO2000: Continuation and bifurcation software for ordinary differential equations (with HomCont), Reference Manual, Concordia University, Canada (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Valverde, J., García-Vallejo, D. Stability analysis of a substructured model of the rotating beam. Nonlinear Dyn 55, 355–372 (2009). https://doi.org/10.1007/s11071-008-9369-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-008-9369-8