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Parametric Excitation
Mechanical systems with a finite number of degrees of freedom (DOF) are studied, whose parameters (mass, damping, stiffness) are varied in time with an assigned periodic law. This form of solicitation is called parametricexcitation. The equations of motion are derived for linear systems with one or two DOF, as well as for discretized continuous systems. The fundamental Floquet theorem is introduced for a generic periodic system, linear and continuous in time, and the deriving conditions for stability are discussed in terms of characteristic exponents as well as characteristic multipliers. These results are reinterpreted in the light of an alternative discrete-time vision, supplied by the Poincaré map, leading to the definition of a transfer matrix, which relates the states of the system at equispaced instants. The study is particularized to single DOF systems, described by the Hill and Mathieu equations, both undamped and damped. These equations highlight two different bifurcation mechanisms, called of divergence and flip type. The bifurcations occur on several transition curve of the amplitude and frequency bifurcation parameter plane, which separate stable from unstable states. The asymptotic construction of the transition curves is illustrated, both for the Mathieu equation and for a physical system (namely, the Bolotin beam reduced to a single DOF). Successively, a nonlinear system with a single DOF is analyzed, for which a family of periodic solutions, bifurcating from the transition curves, is found. Finally, linear systems with multiple DOF are considered, which reveal the existence of a new mechanism of bifurcation, called of Neimark-Sacker type. This involves two modes of the system, which are in combination resonance with the frequency of the parametric excitation, and gives rise to further regions of instability on the parameter plane. The case of a two DOF system in principal combination resonance is developed in detail, by making use of a perturbation method.
Parametric Excitation
Mechanical systems with a finite number of degrees of freedom (DOF) are studied, whose parameters (mass, damping, stiffness) are varied in time with an assigned periodic law. This form of solicitation is called parametricexcitation. The equations of motion are derived for linear systems with one or two DOF, as well as for discretized continuous systems. The fundamental Floquet theorem is introduced for a generic periodic system, linear and continuous in time, and the deriving conditions for stability are discussed in terms of characteristic exponents as well as characteristic multipliers. These results are reinterpreted in the light of an alternative discrete-time vision, supplied by the Poincaré map, leading to the definition of a transfer matrix, which relates the states of the system at equispaced instants. The study is particularized to single DOF systems, described by the Hill and Mathieu equations, both undamped and damped. These equations highlight two different bifurcation mechanisms, called of divergence and flip type. The bifurcations occur on several transition curve of the amplitude and frequency bifurcation parameter plane, which separate stable from unstable states. The asymptotic construction of the transition curves is illustrated, both for the Mathieu equation and for a physical system (namely, the Bolotin beam reduced to a single DOF). Successively, a nonlinear system with a single DOF is analyzed, for which a family of periodic solutions, bifurcating from the transition curves, is found. Finally, linear systems with multiple DOF are considered, which reveal the existence of a new mechanism of bifurcation, called of Neimark-Sacker type. This involves two modes of the system, which are in combination resonance with the frequency of the parametric excitation, and gives rise to further regions of instability on the parameter plane. The case of a two DOF system in principal combination resonance is developed in detail, by making use of a perturbation method.
Parametric Excitation
Luongo, Angelo (Autor:in) / Ferretti, Manuel (Autor:in) / Di Nino, Simona (Autor:in)
Stability and Bifurcation of Structures ; Kapitel: 13 ; 493-551
17.02.2023
59 pages
Aufsatz/Kapitel (Buch)
Elektronische Ressource
Englisch
Floquet theorem , Characteristic exponents , Characteristic multipliers , Poincaré map , Mathieu equation , Divergence and flip bifurcations , Transition curves , Strutt diagram , Bolotin beam , Nonlinear system parametrically excited , Neimark-Sacker bifurcation , Combination resonance Engineering , Mechanical Statics and Structures , Solid Mechanics , Mechanical Engineering , Structural Materials , Solid Construction , Building Construction and Design
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