Two-dimensional map for impact oscillator with drift

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Abstract

An impact oscillator with drift is considered. The model accounts for viscoelastic impacts and is capable of mimicking the dynamics of progressive motion, which is important in many applications. To simplify the analysis of this system, a transformation decoupling the original coordinates is introduced. As a result, the bounded oscillations are separated from the drift motion. To study the bounded dynamics, a two-dimensional analytical map is developed and analyzed. In general, the dynamic state of the system is fully described by four variables: time tau, relative displacement p and velocity y of the mass, and relative displacement q of the slider top. However, this number can be reduced to two if the beginning of the progression phase is being monitored. The lower and upper bounds of the map domain are approximated. A graphical method of iteration of the two-dimensional map, similar to the cobweb method used in the one-dimensional case, is proposed. The results of numerical iterations of this two-dimensional map are presented, and a comparison is given between bifurcation diagrams calculated for this map and for the original system of differential equations.

Original languageEnglish
Pages (from-to)036201-036210
Number of pages10
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume70
Issue number3
DOIs
Publication statusPublished - Sep 2004

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oscillators
iteration
Iteration
Graphical Methods
Motion
Bifurcation Diagram
System of Differential Equations
Decoupling
Progression
chutes
progressions
decoupling
Upper and Lower Bounds
Simplify
differential equations
diagrams
Oscillation
oscillations
Model

Keywords

  • Multiple friction oscillators
  • Border-collision bifurcations
  • Smooth dynamical-systems
  • Grazing bifurcations
  • Nonlinear dynamics
  • Rate prediction
  • Model

Cite this

@article{5569b21883404263bfb726bb0a138e59,
title = "Two-dimensional map for impact oscillator with drift",
abstract = "An impact oscillator with drift is considered. The model accounts for viscoelastic impacts and is capable of mimicking the dynamics of progressive motion, which is important in many applications. To simplify the analysis of this system, a transformation decoupling the original coordinates is introduced. As a result, the bounded oscillations are separated from the drift motion. To study the bounded dynamics, a two-dimensional analytical map is developed and analyzed. In general, the dynamic state of the system is fully described by four variables: time tau, relative displacement p and velocity y of the mass, and relative displacement q of the slider top. However, this number can be reduced to two if the beginning of the progression phase is being monitored. The lower and upper bounds of the map domain are approximated. A graphical method of iteration of the two-dimensional map, similar to the cobweb method used in the one-dimensional case, is proposed. The results of numerical iterations of this two-dimensional map are presented, and a comparison is given between bifurcation diagrams calculated for this map and for the original system of differential equations.",
keywords = "Multiple friction oscillators, Border-collision bifurcations, Smooth dynamical-systems, Grazing bifurcations, Nonlinear dynamics, Rate prediction, Model",
author = "Pavlovskaia, {Ekaterina Evgenievna} and Marian Wiercigroch and Celso Grebogi",
year = "2004",
month = "9",
doi = "10.1103/PhysRevE.70.036201",
language = "English",
volume = "70",
pages = "036201--036210",
journal = "Physical Review. E, Statistical, Nonlinear and Soft Matter Physics",
issn = "1539-3755",
publisher = "AMER PHYSICAL SOC",
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TY - JOUR

T1 - Two-dimensional map for impact oscillator with drift

AU - Pavlovskaia, Ekaterina Evgenievna

AU - Wiercigroch, Marian

AU - Grebogi, Celso

PY - 2004/9

Y1 - 2004/9

N2 - An impact oscillator with drift is considered. The model accounts for viscoelastic impacts and is capable of mimicking the dynamics of progressive motion, which is important in many applications. To simplify the analysis of this system, a transformation decoupling the original coordinates is introduced. As a result, the bounded oscillations are separated from the drift motion. To study the bounded dynamics, a two-dimensional analytical map is developed and analyzed. In general, the dynamic state of the system is fully described by four variables: time tau, relative displacement p and velocity y of the mass, and relative displacement q of the slider top. However, this number can be reduced to two if the beginning of the progression phase is being monitored. The lower and upper bounds of the map domain are approximated. A graphical method of iteration of the two-dimensional map, similar to the cobweb method used in the one-dimensional case, is proposed. The results of numerical iterations of this two-dimensional map are presented, and a comparison is given between bifurcation diagrams calculated for this map and for the original system of differential equations.

AB - An impact oscillator with drift is considered. The model accounts for viscoelastic impacts and is capable of mimicking the dynamics of progressive motion, which is important in many applications. To simplify the analysis of this system, a transformation decoupling the original coordinates is introduced. As a result, the bounded oscillations are separated from the drift motion. To study the bounded dynamics, a two-dimensional analytical map is developed and analyzed. In general, the dynamic state of the system is fully described by four variables: time tau, relative displacement p and velocity y of the mass, and relative displacement q of the slider top. However, this number can be reduced to two if the beginning of the progression phase is being monitored. The lower and upper bounds of the map domain are approximated. A graphical method of iteration of the two-dimensional map, similar to the cobweb method used in the one-dimensional case, is proposed. The results of numerical iterations of this two-dimensional map are presented, and a comparison is given between bifurcation diagrams calculated for this map and for the original system of differential equations.

KW - Multiple friction oscillators

KW - Border-collision bifurcations

KW - Smooth dynamical-systems

KW - Grazing bifurcations

KW - Nonlinear dynamics

KW - Rate prediction

KW - Model

U2 - 10.1103/PhysRevE.70.036201

DO - 10.1103/PhysRevE.70.036201

M3 - Article

VL - 70

SP - 36201

EP - 36210

JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics

SN - 1539-3755

IS - 3

ER -