Abstract
The state-space of a mechanical system, made of positions and velocities, is introduced. In this space, the definition of stability of equilibrium is formulated, according to Lyapunov. Then, families of systems, dependent on a parameter, are considered and the general definition of bifurcation is given. Confining the attention to bifurcations from an equilibrium point, bifurcations are classified into static and dynamic. On the ground of purely phenomenological considerations, the critical and postcritical behaviors of the Euler beam are illustrated. With a similar approach, the different types of static bifurcations, manifested by conservative systems, without or with imperfections, are browsed. The dependence of the shape of the critical mode on bifurcation parameters is discussed; then, the possible occurrence of multiple modes and their potentially dangerous effect is commented. Finally, dynamic bifurcations are sketched, exhibited by nonconservative systems, such as elastic structures subject to forces depending on position, velocity, or time.
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Notes
- 1.
The theorem was originally formulated for discrete systems and then generalized by Koiter [7] to continuous systems.
- 2.
An exception concerns elastoplastic systems, which will be examined in the static field only.
- 3.
As an example, the single degree of freedom system \(\ddot {x}+c\left (\mu \right )\dot {x}+x=0\), with \(c\left (\mu \right )=0\) when μ ≤ 0 and \(c\left (\mu \right )=1\) when μ > 0, admits in \(\left (x,\dot {x}\right )=\left (0,0\right )\): (i) a center point, when μ ≤ 0, and (ii) a stable focus, when μ > 0. Therefore μ = 0 is a bifurcation point at which stability is preserved.
- 4.
The basin of attraction of an asymptotically stable equilibrium point is the locus of all the initial conditions, in the state-space, which bring the system to come back to the equilibrium point. For asymptotic stability, dam** must be added to an elastic structure.
- 5.
The jump is usually accompanied by a “snap,” as occurring for “click-clack” of metal capsules.
- 6.
Or on a geometric manifold of dimension m − 2 belonging to a m-parameter space.
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Luongo, A., Ferretti, M., Di Nino, S. (2023). Phenomenological Aspects of Bifurcation of Structures. In: Stability and Bifurcation of Structures. Springer, Cham. https://doi.org/10.1007/978-3-031-27572-2_2
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DOI: https://doi.org/10.1007/978-3-031-27572-2_2
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