Analytical and Numerical Investigations of Stable Periodic Solutions of the Impacting Oscillator With a Moving Base and Two Fenders

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Barbara Blazejczyk-Okolewska ◽  
Krzysztof Czolczynski ◽  
Andrzej Okolewski
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Analytical Method ◽  
Equations Of Motion ◽  
Moving Frame ◽  
Stable Periodic Solution ◽  
Bifurcation Diagrams ◽  
Numerical Investigations ◽  
Moving Base ◽  
Stable Periodic

A vibrating system with impacts, which can be applied to model the cantilever beam with a mass at its end and two-sided impacts against a harmonically moving frame, is investigated. The objective of this study is to determine in which regions of parameters characterizing system, the motion of the oscillator is periodic and stable. An analytical method to obtain stable periodic solutions to the equations of motion on the basis of Peterka's approach is presented. The results of analytical investigations have been compared to the results of numerical simulations. The ranges of stable periodic solutions determined analytically and numerically with bifurcation diagrams of spectra of Lyapunov exponents show a very good conformity. The locations of stable periodic solution regions of the system with a movable frame and two-sided impacts differ substantially from the locations of stable periodic solution regions for the system: (i) with a movable frame and one-sided impacts and (ii) with an immovable frame and two-sided impacts.

ON THE EXISTENCE OF A STABLE PERIODIC SOLUTION OF TWO IMPACTING OSCILLATORS WITH DAMPING

2004 ◽  
Vol 14 (11) ◽  
pp. 3931-3947 ◽  
Author(s):  
KRZYSZTOF CZOLCZYNSKI ◽  
TOMASZ KAPITANIAK
Keyword(s):  
Periodic Solution ◽  
Periodic Motion ◽  
Equations Of Motion ◽  
Analytical Determination ◽  
Stable Periodic Solution ◽  
Numerical Investigations ◽  
Stable Periodic ◽  
Existence Of Periodic Solutions ◽  
The Stability

A system that consists of two impacting oscillators with damping has been considered in this paper. In the first part, a method of analytical determination of the existence of periodic solutions to the equations of motion and a method of analysis of the stability of these solutions are presented. The results of the computations carried out by these methods have been illustrated with a few examples. In the second part of the paper, the results of some numerical investigations are presented. The goal of these studies is to determine, in which regions of parameters characterizing the system, the periodic motion with one impact per period exists and is stable.

ON THE EXISTENCE OF A STABLE PERIODIC SOLUTION OF AN IMPACTING OSCILLATOR WITH TWO FENDERS

2004 ◽  
Vol 14 (09) ◽  
pp. 3115-3134 ◽  
Author(s):  
KRZYSZTOF CZOLCZYNSKI ◽  
TOMASZ KAPITANIAK
Keyword(s):  
Periodic Solution ◽  
Equations Of Motion ◽  
Analytical Determination ◽  
Stable Periodic Solution ◽  
Numerical Investigations ◽  
Stable Periodic ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Damped Oscillator

A system that consists of a damped oscillator impacting two immovable fenders has been considered in this paper. In the first part a method of analytical determination of the existence of periodic solutions to the equations of motion and a method of analysis of the stability of these solutions have been presented. The results of the computations carried out by means of these methods have been illustrated by a few examples. In the second part of the paper, the results of some numerical investigations have been presented. The goal of these studies was to determine, in which regions of parameters characterizing the system, the motion of the oscillator is periodic and stable.

scholarly journals Periodic Solutions of a System of Delay Differential Equations for a Small Delay

2002 ◽  
Vol 7 (2) ◽  
pp. 295
Author(s):  
Adu A.M. Wasike ◽  
Wandera Ogana
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Original System ◽  
System Of Equations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Stable Periodic ◽  
Delay Differential

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system.  This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.   

scholarly journals Twist Periodic Solutions in the Relativistic Driven Harmonic Oscillator

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Daniel Núñez ◽  
Andrés Rivera
Keyword(s):  
Periodic Solution ◽  
Harmonic Oscillator ◽  
Periodic Solutions ◽  
Relativistic Effects ◽  
Stable Periodic Solution ◽  
One Dimensional ◽  
Stable Periodic ◽  
The One ◽  
Hill’S Equations ◽  
Twist Theorem

We study the one-dimensional forced harmonic oscillator with relativistic effects. Under some conditions of the parameters, the existence of a unique stable periodic solution is proved which is of twist type. The results depend on a Twist Theorem for nonlinear Hill’s equations which is established and proved here.

scholarly journals Permanence and Periodic Solution of Predator-Prey System with Holling Type Functional Response and Impulses

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Weibing Wang ◽  
Jianhua Shen ◽  
Juan J. Nieto
Keyword(s):  
Periodic Solution ◽  
Functional Response ◽  
Impulsive Effect ◽  
Two Dimensional ◽  
Stable Periodic Solution ◽  
Predator Prey ◽  
Prey System ◽  
Stable Periodic ◽  
Holling Type Functional Response

We considered a nonautonomous two dimensional predator-prey system with impulsive effect. Conditions for the permanence of the system and for the existence of a unique stable periodic solution are obtained.

ON THE INFLUENCE OF THE RESONANT FREQUENCIES RATIO ON A STABLE PERIODIC SOLUTION OF TWO IMPACTING OSCILLATORS

2006 ◽  
Vol 16 (12) ◽  
pp. 3707-3715 ◽  
Author(s):  
KRZYSZTOF CZOLCZYNSKI ◽  
TOMASZ KAPITANIAK
Keyword(s):  
Periodic Solution ◽  
Periodic Motion ◽  
Stable Periodic Solution ◽  
Resonant Frequencies ◽  
Stable Periodic

A system that consists of two impacting oscillators with damping has been considered in this paper. The goal of the studies was to determine the relation between the values of resonant frequencies of the oscillators and the existence of their stable periodic motion. This paper indicates various origins of the periodicity of motion and offers a some advice to the designers of systems with impacts. Especially, the results of the considerations point out some potentially dangerous consequences of the improper value of the resonant frequencies ratio.

scholarly journals Global asymptotic stability in a periodic Lotka-Volterra system

1985 ◽  
Vol 27 (1) ◽  
pp. 66-72 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Volterra System ◽  
Periodic Coefficients ◽  
Globally Asymptotically Stable ◽  
Stable Periodic

AbstractA set of easily verifiable sufficient conditions are obtained for the existence of a globally asymptotically stable periodic solution in a Lotka-Volterra system with periodic coefficients.

scholarly journals Chemical reaction dynamics: analysis of the brusselator model

2010 ◽  
Vol 51 ◽  
Author(s):  
Liana Stonkienė ◽  
Donatas Švitra
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Chemical Reaction ◽  
Nonlinear Differential Equations ◽  
Reaction Dynamics ◽  
Dynamics Analysis ◽  
Stable Periodic Solution ◽  
Brusselator Model ◽  
Chemical Reaction Dynamics ◽  
Stable Periodic

It is observed the differential equations system. The stable periodic solution of the nonlinear differential equations system is constructed, which is based on the theory of bifurcations.

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