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Stability analysis of a substructured model of the rotating beam. (English) Zbl 1170.74345

Summary: 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.

MSC:

74H55 Stability of dynamical problems in solid mechanics
74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)

Software:

HomCont; AUTO2000
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Full Text: DOI

References:

[1] Schilhans, M.J.: Bending frequency of a rotating cantilever beam. J. Appl. Mech. 25, 28–30 (1958) · Zbl 0079.39903
[2] 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) · doi:10.2514/3.20195
[3] 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) · Zbl 1235.74193 · doi:10.1016/0022-460X(87)90410-X
[4] Wu, S.C., Haug, E.J.: Geometric non-linear substructuring for dynamics of flexible mechanical systems. J. Numer. Methods Eng. 26, 2211–2226 (1988) · Zbl 0661.73059 · doi:10.1002/nme.1620261006
[5] 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) · doi:10.1006/jsvi.1998.1563
[6] 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) · Zbl 1060.70011 · doi:10.1023/A:1015567829908
[7] 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)
[8] 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)
[9] 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) · Zbl 1336.74038
[10] 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) · doi:10.1006/jsvi.2000.3416
[11] 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) · doi:10.1016/j.jsv.2003.12.044
[12] Shabana, A.A.: Dynamics of Multibody Systems, 2nd edn. Cambridge University Press, New York (1998) · Zbl 0932.70002
[13] Goriely, A., Tabor, M.: Nonlinear dynamics of filaments I. Dynamical instabilities. Phys. D 105, 20–44 (1997) · Zbl 0962.74513 · doi:10.1016/S0167-2789(96)00290-4
[14] Antman, S.S.: Nonlinear Problems of Elasticity. Springer, Berlin (1995) · Zbl 0820.73002
[15] 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) · Zbl 1149.70316 · doi:10.1007/s00332-005-0700-y
[16] Valverde, J., van der Heijden, G.: Stability and bifurcation analysis of a spinning space tether. J. Nonlinear Sci. (2008, submitted)
[17] 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)
[18] Strogatz, S.H.: Nonlinear Dynamics and Chaos. Perseus Books, Cambridge (1994)
[19] 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)
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