scholarly journals Codimension-Two Grazing Bifurcations in Three-Degree-of-Freedom Impact Oscillator with Symmetrical Constraints

2015 ◽  
Vol 2015 ◽  
pp. 1-15
Author(s):  
Qunhong Li ◽  
Pu Chen ◽  
Jieqiong Xu
Keyword(s):  
Periodic Motion ◽  
Mapping Method ◽  
Degree Of Freedom ◽  
Poincaré Mapping ◽  
Impact Oscillator ◽  
Codimension Two ◽  
Poincare Mapping ◽  
Discontinuity Mapping ◽  
Three Degree Of Freedom ◽  
Grazing Bifurcations

This paper investigates the codimension-two grazing bifurcations of a three-degree-of-freedom vibroimpact system with symmetrical rigid stops since little research can be found on this important issue. The criterion for existence of double grazing periodic motion is presented. Using the classical discontinuity mapping method, the Poincaré mapping of double grazing periodic motion is obtained. Based on it, the sufficient condition of codimension-two bifurcation of double grazing periodic motion is formulated, which is simplified further using the Jacobian matrix of smooth Poincaré mapping. At the end, the existence regions of different types of periodic-impact motions in the vicinity of the codimension-two grazing bifurcation point are displayed numerically by unfolding diagram and phase diagrams.

Degenerate grazing bifurcations in a three-degree-of-freedom impact oscillator

2019 ◽  
Vol 97 (1) ◽  
pp. 525-539 ◽  
Author(s):  
Shan Yin ◽  
Jinchen Ji ◽  
Shuning Deng ◽  
Guilin Wen
Keyword(s):  
Degree Of Freedom ◽  
Impact Oscillator ◽  
Three Degree Of Freedom ◽  
Grazing Bifurcations

scholarly journals An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Yu Wang ◽  
Heng Cao ◽  
JinLin Jiang
Keyword(s):  
Domain Of Attraction ◽  
Mapping Method ◽  
Simple Cell ◽  
Iteration Algorithm ◽  
Poincaré Mapping ◽  
Cell Mapping ◽  
Improved Method ◽  
Stable Domain ◽  
Simple Cell Mapping ◽  
Poincare Mapping

An indicator of a passive biped walker’s global stability is its domain of attraction, which is usually estimated by the simple cell mapping method. It needs to calculate a large number of cells’ Poincare mapping result in the estimating process. However, the Poincare mapping is usually computationally expensive and time-consuming due to the complex dynamical equation of the passive biped walker. How to estimate the domain of attraction efficiently and reliably is a problem to be solved. Based on the simple cell mapping method, an improved method is proposed to solve it. The proposed method uses the multiple iteration algorithm to calculate a stable domain of attraction and effectively decreases the total number of Poincare mappings. Through the simulation of the simplest passive biped walker, the improved method can obtain the same domain of attraction as that calculated using the simple cell mapping method and reduce calculation time significantly. Furthermore, this improved method not only proposes a way of rapid estimating the domain of attraction, but also provides a feasible tool for selecting the domain of interest and its discretization level.

scholarly journals Dynamical analysis of a single degree-of-freedom impact oscillator with impulse excitation

2017 ◽  
Vol 9 (7) ◽  
pp. 168781401771661 ◽  
Author(s):  
Jun Wang ◽  
Yongjun Shen ◽  
Shaopu Yang
Keyword(s):  
Periodic Motion ◽  
Dynamical Behavior ◽  
Degree Of Freedom ◽  
Dynamical Response ◽  
Restitution Coefficient ◽  
Impact Oscillator ◽  
Vibration Period ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Impulse Excitation

In this article, the dynamical behavior of a single degree-of-freedom impact oscillator with impulse excitation is studied, where the mass impacts at one stop and is shocked with impulse excitation at the other stop. The existing and stability conditions for periodic motion of the oscillator are established. The effects of system parameters on dynamical response are discussed under different initial velocities. It is found that smaller shock gap than impact gap could make the periodic motion more stable. The decrease in natural frequency would consume less impact energy, make the vibration frequency smaller, and reduce the vibration efficiency. Finally, the dynamical properties are further analyzed under a special case, that is, the shock gap approaches zero. It could be seen that the larger shock coefficient and impact restitution coefficient would make vibration period smaller. Based on the stability condition, there are an upper limit for the product of shock coefficient and impact restitution coefficient, so that a lower limit of corresponding vibration period exists.

Analytical determination of bifurcations in an impact oscillator

1994 ◽  
Vol 347 (1683) ◽  
pp. 353-364 ◽  
Keyword(s):  
Analytical Methods ◽  
Coefficient Of Restitution ◽  
Analytical Determination ◽  
Impact Oscillator ◽  
Grazing Bifurcation ◽  
Codimension Two ◽  
Dimension One ◽  
Codimension Two Bifurcation ◽  
Grazing Bifurcations

It is well known that some solutions for a sinusoidally driven oscillator with linear stiffness and impacts at rigid stops modelled with a coefficient of restitution impact law can be located analytically. Recently, new co-dimension one bifurcations called grazing bifurcations have been found in such systems. Here we present analytical results which show how the type of grazing bifurcation changes with parameter, and that when the type of grazing bifurcation changes a codimension two bifurcation occurs. The simplest grazing bifurcations involve orbits of period-1, but we show that the same analytical methods can be used to locate some subharmonics and their bifurcations.

scholarly journals Dynamic Behaviors of a Two-Degree-of-Freedom Impact Oscillator with Two-Sided Constraints

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Songtao Li ◽  
Qunhong Li ◽  
Zhongchuan Meng
Keyword(s):  
Bifurcation Point ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Impact Oscillator ◽  
Grazing Bifurcation ◽  
Constraint System ◽  
Codimension Two ◽  
Codimension One ◽  
Discontinuity Mapping ◽  
Elastic Constraints

The dynamic model of a vibroimpact system subjected to harmonic excitation with symmetric elastic constraints is investigated with analytical and numerical methods. The codimension-one bifurcation diagrams with respect to frequency of the excitation are obtained by means of the continuation technique, and the different types of bifurcations are detected, such as grazing bifurcation, saddle-node bifurcation, and period-doubling bifurcation, which predicts the complexity of the system considered. Based on the grazing phenomenon obtained, the zero-time-discontinuity mapping is extended from the single constraint system presented in the literature to the two-sided elastic constraint system discussed in this paper. The Poincare mapping of double grazing periodic motion is derived, and this compound mapping is applied to obtain the existence conditions of codimension-two grazing bifurcation point of the system. According to the deduced theoretical result, the grazing curve and the codimension-two grazing bifurcation points are validated by numerical simulation. Finally, various types of periodic-impact motions near the codimension-two grazing bifurcation point are illustrated through the unfolding diagram and phase diagrams.

Neimark-Sacker Bifurcations Near Degenerate Grazing Point in a Two Degree-of-Freedom Impact Oscillator

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
Shan Yin ◽  
Jinchen Ji ◽  
Shuning Deng ◽  
Guilin Wen
Keyword(s):  
Periodic Motion ◽  
Small Neighborhood ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Grazing Impact ◽  
Mapping Technique ◽  
Impact Oscillator ◽  
Parameter Continuation ◽  
The Impact ◽  
Two Degree Of Freedom

Saddle-node or period-doubling bifurcations of the near-grazing impact periodic motions have been extensively studied in the impact oscillators, but the near-grazing Neimark-Sacker bifurcations have not been discussed yet. For the first time, this paper uncovers the novel dynamic behavior of Neimark-Sacker bifurcations, which can appear in a small neighborhood of the degenerate grazing point in a two degree-of-freedom impact oscillator. The higher order discontinuity mapping technique is used to determine the degenerate grazing point. Then, shooting method is applied to obtain the one-parameter continuation of the elementary impact periodic motion near degenerate grazing point and the peculiar phenomena of Neimark-Sacker bifurcations are revealed consequently. A two-parameter continuation is presented to illustrate the relationship between the observed Neimark-Sacker bifurcations and degenerate grazing point. New features that differ from the reported situations in literature can be found. Finally, the observed Neimark-Sacker bifurcation is verified by checking the existence and stability conditions in line with the generic theory of Neimark-Sacker bifurcation. The unstable bifurcating quasi-periodic motion is numerically demonstrated on the Poincaré section.

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