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Stability and Bifurcation Linear Analysis
Some mathematical tools, needed to perform stability and bifurcation analysis, are introduced. After a brief introduction to general dynamical systems, the discussion is focused on mechanical systems. The nonlinear equations of motion of a general system are introduced and then expanded in series. Structural and geometric matrices, related to internal and external forces, respectively, are identified. The linear stability analysis of an equilibrium point, based on detection of the eigenvalues of the linearized equation of motion, is conducted. Discussion is then focused on conservative and circulatory systems and on the effects of damping on their stability. The general concepts are exemplified with reference to the planar mathematical pendulum, for which both the dynamic criterion and the energy criterion of stability are used. Moving on to consider parameter-dependent systems, the notions of equilibrium path, and trivial or non-trivialfundamental paths, are introduced; moreover, the techniques for determining bifurcation points occurring along them, are illustrated. The mechanisms leading to bifurcations of conservative and circulatory systems, without or with damping, are discussed.
Stability and Bifurcation Linear Analysis
Some mathematical tools, needed to perform stability and bifurcation analysis, are introduced. After a brief introduction to general dynamical systems, the discussion is focused on mechanical systems. The nonlinear equations of motion of a general system are introduced and then expanded in series. Structural and geometric matrices, related to internal and external forces, respectively, are identified. The linear stability analysis of an equilibrium point, based on detection of the eigenvalues of the linearized equation of motion, is conducted. Discussion is then focused on conservative and circulatory systems and on the effects of damping on their stability. The general concepts are exemplified with reference to the planar mathematical pendulum, for which both the dynamic criterion and the energy criterion of stability are used. Moving on to consider parameter-dependent systems, the notions of equilibrium path, and trivial or non-trivialfundamental paths, are introduced; moreover, the techniques for determining bifurcation points occurring along them, are illustrated. The mechanisms leading to bifurcations of conservative and circulatory systems, without or with damping, are discussed.
Stability and Bifurcation Linear Analysis
Luongo, Angelo (author) / Ferretti, Manuel (author) / Di Nino, Simona (author)
Stability and Bifurcation of Structures ; Chapter: 3 ; 35-67
2023-02-17
33 pages
Article/Chapter (Book)
Electronic Resource
English
Dynamical systems , Phase diagrams , Linear stability analysis , Structural and geometric matrices , Eigenvalue analysis , Marginal stability , Conservative , Circulatory and damped systems , Equilibrium paths , Bifurcations from trivial or non-trivial paths , Bifurcation mechanisms Engineering , Mechanical Statics and Structures , Solid Mechanics , Mechanical Engineering , Structural Materials , Solid Construction , Building Construction and Design
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