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Dynamic Bifurcations Induced by Follower Forces
Follower forces acting on elastic beams are considered, and their nonconservative character is discussed. The linearized bifurcation analysis of a double pendulum, known as Ziegler column, is performed. In absence of damping, the mechanism of flutterFluttercirculatory system (or circulatory, or reversible, Hopf bifurcation) is illustrated, caused by the collision of two pairs of purely imaginary eigenvalues. The effect of added damping is then discussed, changing flutter into a generic Hopfbifurcation, in which only one pair of purely imaginary eigenvalues are involved. Damping, although dissipative, is generally found to exert a destabilizing effect, since it reduces the critical load; furthermore, the damped critical load is not in continuity with that of the undamped system, when damping tends to zero. All these surprising phenomena constitute the so-called Ziegler paradox./COMP: Set “Ziegler paradox” to Roman. By still making reference to the double pendulum, a nonlinear analysis is developed, aimed at describing the amplitude of the limit cycle, as a function of the bifurcation parameter. To achieve this result, the Lindstedt-Poincaré perturbation method is used, which allows to determine the periodic solutions of a weakly nonlinear system. The analytical results are then compared with those deriving from numerical integrations. Finally, a viscoelastic fixed-free beam, loaded at the tip by a compression follower force, known as the Beck beam, is studied. The linear analysis carried out for the Ziegler discrete system is repeated for the continuous Beck system. The role of the internal and external damping acting on the beam is discussed with regard to the Ziegler paradox.
Dynamic Bifurcations Induced by Follower Forces
Follower forces acting on elastic beams are considered, and their nonconservative character is discussed. The linearized bifurcation analysis of a double pendulum, known as Ziegler column, is performed. In absence of damping, the mechanism of flutterFluttercirculatory system (or circulatory, or reversible, Hopf bifurcation) is illustrated, caused by the collision of two pairs of purely imaginary eigenvalues. The effect of added damping is then discussed, changing flutter into a generic Hopfbifurcation, in which only one pair of purely imaginary eigenvalues are involved. Damping, although dissipative, is generally found to exert a destabilizing effect, since it reduces the critical load; furthermore, the damped critical load is not in continuity with that of the undamped system, when damping tends to zero. All these surprising phenomena constitute the so-called Ziegler paradox./COMP: Set “Ziegler paradox” to Roman. By still making reference to the double pendulum, a nonlinear analysis is developed, aimed at describing the amplitude of the limit cycle, as a function of the bifurcation parameter. To achieve this result, the Lindstedt-Poincaré perturbation method is used, which allows to determine the periodic solutions of a weakly nonlinear system. The analytical results are then compared with those deriving from numerical integrations. Finally, a viscoelastic fixed-free beam, loaded at the tip by a compression follower force, known as the Beck beam, is studied. The linear analysis carried out for the Ziegler discrete system is repeated for the continuous Beck system. The role of the internal and external damping acting on the beam is discussed with regard to the Ziegler paradox.
Dynamic Bifurcations Induced by Follower Forces
Luongo, Angelo (author) / Ferretti, Manuel (author) / Di Nino, Simona (author)
Stability and Bifurcation of Structures ; Chapter: 11 ; 403-438
2023-02-17
36 pages
Article/Chapter (Book)
Electronic Resource
English
Follower forces , Flutter , Ziegler column , Viscoelastic Beck beam , Ziegler paradox , Destabilizing effect of damping , Whitney umbrella , Limit cycle of the Ziegler column , Lindstedt-Poincaré perturbation method , Internal <italic>vs</italic> external damping Engineering , Mechanical Statics and Structures , Solid Mechanics , Mechanical Engineering , Structural Materials , Solid Construction , Building Construction and Design
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