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.

scholarly journals The Stick-Slip Vibration and Bifurcation of a Vibro-Impact System with Dry Friction

2014 ◽  
Vol 8 (1) ◽  
pp. 308-313 ◽  
Author(s):  
Quanfu Gao ◽  
Xingxiao Cao
Keyword(s):  
Periodic Motion ◽  
Dry Friction ◽  
Low Frequency ◽  
Period Doubling ◽  
Periodic Motions ◽  
Vibratory System ◽  
Stick Slip ◽  
The Impact ◽  
Two Degree Of Freedom ◽  
Impact Motion

In this paper, the periodic motion, bifurcation and chatter of two-degree-of-freedom vibratory system with dry friction and clearance were studied. Slip-stick motion and the impact of system motions were analyzed and numerical simulations were also carried out. The results showed that the system possesses rich dynamics characterized by periodic motion, stick-slip-impact motion, quasi-periodic motion and chaotic attractors, and the routs from periodic motions to chaos observed via Hof bifurcation or period-doubling bifurcation. Furthermore, it was found that there exists the chatter phenomena induced by dry friction in low frequency, and the windows of chaotic motion are broadened in the area of higher excitation frequencies as the dry friction increases.

Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-Of-Freedom Vibro-Impact System

2000 ◽  
Vol 68 (4) ◽  
pp. 670-674 ◽  
Author(s):  
G. L. Wen and ◽  
J. H. Xie
Keyword(s):  
Finite Number ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Period Doubling Bifurcation ◽  
Periodic Response ◽  
Impact System ◽  
Route To Chaos ◽  
Two Degree Of Freedom ◽  
Impact Motion ◽  
Period Doubling Bifurcations

A nontypical route to chaos of a two-degree-of-freedom vibro-impact system is investigated. That is, the period-doubling bifurcations, and then the system turns out to the stable quasi-periodic response while the period 4-4 impact motion fails to be stable. Finally, the system converts into chaos through phrase locking of the corresponding four Hopf circles or through a finite number of times of torus-doubling.

Nonlinear Normal Modes and Localization in Elastic Vibro-Impact Systems With Multiple Constraints

2007 ◽  
Author(s):  
David Wagg
Keyword(s):  
Mathematical Formulation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Lumped Mass ◽  
Localized Modes ◽  
Impact Oscillator ◽  
Mass System ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In this paper we consider the dynamics of compliant mechanical systems subject to combined vibration and impact forcing. Two specific systems are considered; a two degree of freedom impact oscillator and a clamped-clamped beam. Both systems are subject to multiple motion limiting constraints. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom lumped mass system are considered. We then consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, which is a form of periodic localization. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. A numerical example of a sticking orbit for this system is shown and we discuss identifying a nonlinear normal modal basis for the system. This is achieved by extending the normal modal basis to include localized modes. Finally preliminary experimental results from a clamped-clamped vibroimpacting beam are considered and a simplified model discussed which uses an extended modal basis including localized modes.

Discrete modular serpentine robotic tail: design, analysis and experimentation

Robotica ◽  
2018 ◽  
Vol 36 (7) ◽  
pp. 994-1018 ◽  
Author(s):  
Wael Saab ◽  
William S. Rone ◽  
Pinhas Ben-Tzvi
Keyword(s):  
Dynamic Models ◽  
Design Analysis ◽  
Degree Of Freedom ◽  
Legged Robots ◽  
Design Parameters ◽  
Inertial Loading ◽  
Forward Kinematic ◽  
The Impact ◽  
Accuracy And Repeatability ◽  
Two Degree Of Freedom

SUMMARYThis paper presents the design, analysis and experimentation of a Discrete Modular Serpentine Tail (DMST). The mechanism is envisioned for use as a robotic tail integrated onto mobile legged robots to provide a means, separate from the legs, to aid stabilization and maneuvering for both static and dynamic applications. The DMST is a modular two-degree-of-freedom (DOF) articulated, under-actuated mechanism, inspired by continuum and serpentine robotic structures. It is constructed from rigid links with cylindrical contoured grooves that act as pulleys to route and maintain equal displacements in antagonistic cable pairs that are connected to a multi-diameter pulley. Spatial tail curvatures are produced by adding a roll-DOF to rotate the bending plane of the planar tail curvatures. Kinematic and dynamic models of the cable-driven mechanism are developed to analyze the impact of trajectory and design parameters on the loading profiles transferred through the tail base. Experiments using a prototype are performed to validate the forward kinematic and dynamic models, determine the mechanism's accuracy and repeatability, and measure the mechanism's ability to generate inertial loading.

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 Stability Improvement of the Periodic Vibro-Impact Processes

1999 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Krzysztof Tomczak
Keyword(s):  
Periodic Orbits ◽  
Periodic Motion ◽  
Feedback Loop ◽  
Degree Of Freedom ◽  
Loop Control ◽  
Analytical Stability ◽  
Impact Processes ◽  
Delay Loop ◽  
Sufficient Stability ◽  
Two Degree Of Freedom

Abstract In this paper an attention is focused on stabilisation improvement of periodic orbits of a one- and two-degree-of-freedom nonautonomous vibro-impact systems. This approach, among others, includes two problems. 1. A possibility of a sufficient stability improvement of the considered vibro-impact periodic motion using a feedback loop control is presented. 2. An original analytical averaging technique applied to a one-degree-of-freedom system is demonstrated. It enables to predict the efficient delay loop coefficients in order to achieve the desired stabilisation. In addition, an efficient delay loop control applied to two-degree-of-freedom system is proposed and illustrated.

scholarly journals A Solution Procedure Combining Analytical and Numerical Approaches to Investigate a Two-Degree-of-Freedom Vibro-Impact Oscillator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1374
Author(s):  
Nicolae Herisanu ◽  
Vasile Marinca
Keyword(s):  
Analytical Solutions ◽  
Initial Conditions ◽  
Solution Procedure ◽  
Degree Of Freedom ◽  
Impact Oscillator ◽  
New Approach ◽  
Analytical Technique ◽  
The Stability ◽  
Two Degree Of Freedom ◽  
Numerical Approaches

In this paper, a new approach is proposed to analyze the behavior of a nonlinear two-degree-of-freedom vibro-impact oscillator subject to a harmonic perturbing force, based on a combination of analytical and numerical approaches. The nonlinear governing equations are analytically solved by means of a new analytical technique, namely the Optimal Auxiliary Functions Method (OAFM), which provided highly accurate explicit analytical solutions. Benefiting from these results, the application of Schur principle made it possible to analyze the stability conditions for the considered system. Various types of possible motions were emphasized, taking into account possible initial conditions and different parameters, and the explicit analytical solutions were found to be very useful to analyze the kinetic energy loss, the contact force, and the stability of periodic motions.

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