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Bifurcation and buckling analysis of a unilaterally confined self-rotating cantilever beam. (English) Zbl 1200.74063
Summary: A nonlinear dynamic model of a simple non-holonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton’s Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon (i.e. catastrophe phenomenon) is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is non-zero. The results obtained by two methods are consistent.

MSC:
74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] Godoy, L.A., Mirasso, A.E.: On the elastic stability of static non-holonomic systems. Int. J. of Solids and Structures 40, 3439–3462 (2003) · Zbl 1038.70004
[2] Chateau, X., Nguyen, Q.S.: Buckling of elastic structures in unilateral contact with or without friction. Eur. J. Mech. A/Solids 10(1), 71–89 (1991) · Zbl 0735.73043
[3] Domokos, G., Holmes, P., Royce, B.: Constrained Euler buckling. J. Nonlinear Science, 7(3), 281–314 (1997) · Zbl 0876.34047
[4] Chai, H.: The post-buckling behavior of a bi-laterally constrained column. J. Mech. Phys. Solids 46(7), 1155–1181 (1998) · Zbl 1063.74510
[5] Holmes, P., Domokos, G., Schmitt, J., Szeberenyi, I.: Constrained Euler buckling: an interplay of compution and analysis. Comput. Meth. Appl. Mech. Eng. 17(3–4), 175–207 (1999) · Zbl 0949.74023
[6] Guo, Y.T., Ren, W.M.: Some advances in confined buckling. Adv. Mech. 34(1), 41–52 (2004) (in chinese)
[7] Kane, T.R., Ryan, R.R., Banerjee, A.K.: Dynamics of a cantilever beam attached to a moving base. J. of Guidance, Control and Dynamics, 10(2), 139–151 (1987)
[8] Bloch, A.M.: Stability analysis of a rotating flexible system. Acta Applicandae Mathematicae 15, 211–234 (1989) · Zbl 0682.70021
[9] Haering, W.J., Ryan, R.R.: New formulation for flexible beams undergoing large overall motions. J. of Guidance, Control and Dynamics, 17(1), 76–83 (1994) · Zbl 0787.73037
[10] Lee, S.Y., Kuo, Y.H.: Bending frequency of a rotating beam with an elastically restrained root. J. Appl. Mech. 58, 209–214 (1991)
[11] Xiao, S.F., Chen, B.: Modeling and stability investigation of a rigid-flexible coupling system. Acta Mechanica Sinica 29(4), 439–447 (1997) (in chinese)
[12] Xiao, S.F., Chen, B.: Modeling and bifurcation analysis of internal cantilever beam system on a steadily rotating ring. Science in China, Series A 41(5), 527–533 (1998) · Zbl 0908.73035
[13] Xiao, S.F., Chen, B.: Modeling and bifurcation analysis of the centre rigid-body mounted on an external Timoshenko beam. Appl. Math. Mech. 20(12), 1389–1393 (1999) · Zbl 0949.70505
[14] Xiao, S.F., Chen, B.: Global bifurcation analysis of a cantilever beam vertically fixed in centrifugal field. Acta Mechanica Sinica, 32(5), 559–565 (2000) (in chinese)
[15] Xiao, S.F., Du Q., Chen, B.: Modal test and analysis of cantilever beam with tip mass. Acta Mechanica Sinica 18, 407–413 (2002)
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