Normal modes, uncoupling, and stability for a class of nonlinear oscillators

Pecelli, G.; Thomas, Edward S.

doi:10.1090/qam/548988

Cited by 15 publications

(15 citation statements)

References 19 publications

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“…This equation includes a curvature of the modal line as a system parameter. Pecelli and Tomas [97,98] considered variational equations for rectilinear NNMs in the form of the Lame equations. Boundaries of the instability regions are treated.…”

Section: Stability and Bifurcations Of Nnmsmentioning

confidence: 99%

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Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

Mikhlin

Аврамов

2010

Applied Mechanics Reviews

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Two principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer-Rosenberg and Shaw-Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described.

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Section: Stability and Bifurcations Of Nnmsmentioning

confidence: 99%

“…The nonlinear stability was considered for the two-DOF conservative systems in Refs. [97,98] using the Siegel and Moser approach [118]. The analysis is made for the system having restoring forces with linear and cubic terms with respect to general coordinates.…”

Section: Stability Of Nnms In High Approximations (Nonlinear Stability)mentioning

confidence: 99%

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

Mikhlin

Аврамов

2010

Applied Mechanics Reviews

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“…The linear stability results are not dependent on initial conditions, this is a property of linear variational equations. But it is known [20,21] that additional nonlinear instability regions (obtained if we take into account nonlinear terms) have a smaller dimension in parameter space than instability regions obtained by the linearized stability analysis. Examples verify that the stability analysis is independent of initial variations if the initial variations are small.…”

Section: Lyapunov Stability Definition and Its Computer Realizationmentioning

confidence: 99%

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

Mikhlin

Shmatko

Manucharyan

2004

Computers & Structures

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This paper deals with forced vibrations of two-DOF systems with more than one equilibrium positions. Such systems may be obtained by digitization of elastic post-buckling systems. A vibration mode, which is periodic at small force amplitudes and becomes chaotic as the force amplitudes are slowly increased, is selected. It is possible to formulate and solve the problem of stability of a periodic or chaotic vibration mode in a space with greater dimension using the classical Lyapunov stability definition and some calculating procedures. Instability of phase trajectories is used as a criterion of the chaotic behavior in dynamical systems. Trajectories with very close initial values are compared. Use of the Lyapunov stability definition shows mutual stability/instability of the trajectories. Calculations permit to observe an appearance and enlargement of the chaotic behavior regions. Specific results are obtained for the nonautonomous Duffing equation and pendulum system.

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“…A detailed treatment has been given by van der Burgh (1988van der Burgh ( , 1974, including higher order approximations (van der Burgh, 1975). Pecelli and Thomas (1979) study a friction-less spring system and in particular the stability of certain in-and out-phase periodic solutions. Figure 5.…”

Section: 51mentioning

confidence: 99%

Asymptotic analysis of hamiltonian systems

Verhulst

1983

Lecture Notes in Mathematics

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Normal modes, uncoupling, and stability for a class of nonlinear oscillators

Cited by 15 publications

References 19 publications

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

Asymptotic analysis of hamiltonian systems

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