scholarly journals Armbruster – Guckenheimer – Kim Hamiltonian System in $\bs 1$:$\bs 1$ Resonance

2021 ◽  
Vol 17 (1) ◽  
pp. 59-76
Author(s):  
M. Alvarez-Ramírez ◽  
◽  
A. García ◽  
J. Vidarte ◽  
◽  
...  
Keyword(s):  
Hamiltonian System ◽  
Periodic Solutions ◽  
Degree Of Freedom ◽  
Galactic Potential ◽  
Stable Periodic ◽  
Two Parameters ◽  
Two Degree Of Freedom

This article deals with the autonomous two-degree-of-freedom Hamiltonian system with Armbruster – Guckenheimer – Kim galactic potential in 1:1 resonance depending on two parameters. We detect periodic solutions and KAM 2-tori arising from linearly stable periodic solutions not found in earlier papers. These are established by using reduction, normalization, averaging and KAM techniques.

On the Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems

1959 ◽  
Vol 26 (3) ◽  
pp. 377-385
Author(s):  
R. M. Rosenberg ◽  
C. P. Atkinson
Keyword(s):  
Free Vibrations ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Natural Modes ◽  
Theoretical Predictions ◽  
Stability Chart ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

Abstract The natural modes of free vibrations of a symmetrical two-degree-of-freedom system are analyzed theoretically and experimentally. This system has two natural modes, one in-phase and the other out-of-phase. In contradistinction to the comparable single-degree-of-freedom system where the free vibrations are always orbitally stable, the natural modes of the symmetrical two-degree-of-freedom system are frequently unstable. The stability properties depend on two parameters and are easily deduced from a stability chart. For sufficiently small amplitudes both modes are, in general, stable. When the coupling spring is linear, both modes are always stable at all amplitudes. For other conditions, either mode may become unstable at certain amplitudes. In particular, if there is a single value of frequency and amplitude at which the system can vibrate in either mode, the out-of-phase mode experiences a change of stability. The experimental investigation has generally confirmed the theoretical predictions.

scholarly journals Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zihan Wang ◽  
Jieqiong Xu ◽  
Shuai Wu ◽  
Quan Yuan
Keyword(s):  
Control Strategy ◽  
Stability Criterion ◽  
Control Method ◽  
Degree Of Freedom ◽  
Local Analysis ◽  
Grazing Bifurcation ◽  
Vibroimpact System ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

scholarly journals Dynamics and Stability of a Two Degree of Freedom Oscillator With an Elastic Stop

2005 ◽  
Vol 1 (1) ◽  
pp. 94-102 ◽  
Author(s):  
Madeleine Pascal
Keyword(s):  
Periodic Solutions ◽  
Analytical Form ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Periodic Motions ◽  
External Excitation ◽  
Time Duration ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Two Degree Of Freedom

A two degree of freedom oscillator with a colliding component is considered. The aim of the study is to investigate the dynamic behavior of the system when the stiffness obstacle changes to a finite value to an infinite one. Several cases are considered. First, in the case of rigid impact and without external excitation, a family of periodic solutions are found in analytical form. In the case of soft impact, with a finite time duration of the shock, and no external excitation, the existence of periodic solutions, with an arbitrary value of the period, is proved. Periodic motions are also obtained when the system is submitted to harmonic excitation, in both cases of rigid or soft impact. The stability of these periodic motions is investigated for these four cases.

Impulsive Steering Between Coexisting Stable Periodic Solutions With an Application to Vibrating Plates

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Daniël W. M. Veldman ◽  
Rob H. B. Fey ◽  
Hans Zwart
Keyword(s):  
Periodic Solutions ◽  
Degree Of Freedom ◽  
Limited Information ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Nonlinear Mechanical ◽  
Vibrating Plates ◽  
Stable Periodic ◽  
Nonlinear Mechanical Systems ◽  
Hardening Nonlinearity

Single-degree-of-freedom (single-DOF) nonlinear mechanical systems under periodic excitation may possess multiple coexisting stable periodic solutions. Depending on the application, one of these stable periodic solutions is desired. In energy-harvesting applications, the large-amplitude periodic solutions are preferred, and in vibration reduction problems, the small-amplitude periodic solutions are desired. We propose a method to design an impulsive force that will bring the system from an undesired to a desired stable periodic solution, which requires only limited information about the applied force. We illustrate our method for a single-degree-of-freedom model of a rectangular plate with geometric nonlinearity, which takes the form of a monostable forced Duffing equation with hardening nonlinearity.

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