Chaos and Three-Dimensional Horseshoes in Slowly Varying Oscillators

1988 ◽  
Vol 55 (4) ◽  
pp. 959-968 ◽  
Author(s):  
Stephen Wiggins ◽  
Steven W. Shaw
Keyword(s):  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Equation Of Motion ◽  
Necessary Condition ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Unstable Manifolds ◽  
Additional Equation

We present general results pertaining to chaotic motions in a class of systems termed slowly varying oscillators which consist of weakly perturbed single-degree-of-freedom systems in which parameters vary slowly in time according to an additional equation of motion. Our results include an analytical method for detecting transversal intersections of stable and unstable manifolds (typically a necessary condition for chaotic motions to exist) and a detailed description of the chaotic dynamics that occur when this situation exists.

Bifurcations and Chaotic Motions of a Class of Mechanical System With Parametric Excitations

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen ◽  
Yushu Chen
Keyword(s):  
Mechanical System ◽  
Parametric Excitation ◽  
Chaotic Dynamics ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Critical Curves ◽  
Unstable Manifolds

Bifurcations and chaotic motions of a class of mechanical system subjected to a superharmonic parametric excitation or a nonlinear periodic parametric excitation are studied, respectively, in this paper. Chaos arising from the transverse intersections of the stable and unstable manifolds of the homoclinic and heteroclincic orbits is analyzed by Melnikov's method. The critical curves separating the chaotic and nonchaotic regions are plotted. Chaotic dynamics are compared for these systems with a periodic parametric excitation or a superharmonic parametric excitation, or a nonlinear periodic parametric excitation. Especially, some new dynamical phenomena are presented for the system with a nonlinear periodic parametric excitation.

Chaos in a Weakly Nonlinear Oscillator With Parametric and External Resonances

1991 ◽  
Vol 58 (1) ◽  
pp. 244-250 ◽  
Author(s):  
K. Yagasaki
Keyword(s):  
Invariant Torus ◽  
Chaotic Dynamics ◽  
Averaging Method ◽  
Double Resonance ◽  
External Excitation ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
External Forces

This paper describes a study of the chaotic dynamics of a weakly nonlinear single degree-of-freedom system subjected to combined parametric and external excitation. We consider a case of double resonance in which primary resonances, with respect to parametric and external forces, exist simultaneously. By using the averaging method and Melnikov’s technique, it is shown that chaos may occur in certain parameter regions. These chaotic motions result from the existence of orbits homoclinic to a normally hyperbolic invariant torus which corresponds to a hyperbolic periodic orbit in the averaged system. The mechanism and structure of chaos in this situation are also described. Furthermore, the existence of steady-state chaos is demonstrated by numerical simulation.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

1992 ◽  
Vol 59 (3) ◽  
pp. 693-695 ◽  
Author(s):  
Pi-Cheng Tung
Keyword(s):  
Dynamic Response ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Excited System

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

scholarly journals Development of n-DoF Preloaded Structures for Impact Mitigation in Cobots

2018 ◽  
Vol 10 (5) ◽  
Author(s):  
S. Seriani ◽  
P. Gallina ◽  
L. Scalera ◽  
V. Lughi
Keyword(s):  
Three Dimensional ◽  
Impact Loads ◽  
Single Degree Of Freedom ◽  
Impact Mitigation ◽  
Single Degree ◽  
Future Developments ◽  
Core Issue ◽  
Comprehensive Framework ◽  
Peculiar Behavior ◽  
Good Agreement

A core issue in collaborative robotics is that of impact mitigation, especially when collisions happen with operators. Passively compliant structures can be used as the frame of the cobot, although, usually, they are implemented by means of a single-degree-of-freedom (DoF). However, n-DoF preloaded structures offer a number of advantages in terms of flexibility in designing their behavior. In this work, we propose a comprehensive framework for classifying n-DoF preloaded structures, including one-, two-, and three-dimensional arrays. Furthermore, we investigate the implications of the peculiar behavior of these structures—which present sharp stiff-to-compliant transitions at design-determined load thresholds—on impact mitigation. To this regard, an analytical n-DoF dynamic model was developed and numerically implemented. A prototype of a 10DoF structure was tested under static and impact loads, showing a very good agreement with the model. Future developments will see the application of n-DoF preloaded structures to impact-mitigation on cobots and in the field of mobile robots, as well as to the field of novel architected materials.

The Dynamic Response of a System With Preloaded Compliance

1988 ◽  
Vol 110 (3) ◽  
pp. 278-283 ◽  
Author(s):  
S. W. Shaw ◽  
P. C. Tung
Keyword(s):  
Dynamic Response ◽  
Piecewise Linear ◽  
Harmonic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Linear Features ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stable Periodic ◽  
Model Equations

We consider the dynamic response of a single degree of freedom system with preloaded, or “setup,” springs. This is a simple model for systems where preload is used to suppress vibrations. The springs are taken to be linear and harmonic excitation is applied; damping is assumed to be of linear viscous type. Using the piecewise linear features of the model equations we determine the amplitude and stability of the periodic responses and carry out a bifurcation analysis for these motions. Some parameter regions which contain no simple stable periodic motions are shown to possess chaotic motions.

Strange Attractors in a Three-Dimensional Autonomous Polynomial Equation

2014 ◽  
Vol 24 (09) ◽  
pp. 1450111 ◽  
Author(s):  
Fengjuan Chen ◽  
Jibin Li
Keyword(s):  
Three Dimensional ◽  
Small Neighborhood ◽  
Polynomial Equation ◽  
Strange Attractors ◽  
Stable And Unstable Manifolds ◽  
Small Perturbations ◽  
Unstable Manifolds ◽  
Comprehensive Study

We studied a three-dimensional autonomous polynomial equation. This equation is with a parameter, which we denote as μ. At μ = 0, the system has two saddles, the stable and unstable manifolds of which coincide. We present a comprehensive study on the dynamics of the system for small μ ≠ 0 in a small neighborhood of the unperturbed stable and unstable manifolds, where one of the heteroclinic connections of the two saddles are broken by small perturbations and strange attractors are created.

Estimation of Ship Roll Parameters in Random Waves

1992 ◽  
Vol 114 (2) ◽  
pp. 114-121 ◽  
Author(s):  
J. B. Roberts ◽  
J. F. Dunne ◽  
A. Debonos
Keyword(s):  
Markov Property ◽  
Simulated Data ◽  
Wide Band ◽  
Equation Of Motion ◽  
Roll Motion ◽  
Random Waves ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Ship Roll ◽  
Stochastic Input

The problem of estimating the parameters in an equation of roll motion from roll measurements only, taken in an irregular sea, is discussed. A single degree of freedom equation of motion is assumed, with a wide-band stochastic input and with a linear-in-the-parameters representation of both the damping and restoration terms. A method based on the Markov property of the energy envelope process, associated with the roll motion, is developed which enables all the relevant parameters to be estimated. The method is validated by applying it to some simulated data, for which the true parameters are known.

Bifurcation and Chaos in Friction-Induced Vibration

2002 ◽  
Author(s):  
Yong Li ◽  
Z. C. Feng
Keyword(s):  
Friction Model ◽  
Degree Of Freedom ◽  
Stable Limit ◽  
Mechanical Oscillator ◽  
Chaotic Motions ◽  
Sliding Speed ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Friction Models ◽  
Friction Induced Vibration

Friction-induced vibration is a phenomenon that has received extensive study by the dynamics community. This is because of the important industrial relevance and the evere-volving development of new friction models. In this paper, we report the result of bifurcation study of a single-degree-of-freedom mechanical oscillator sliding over a surface. The friction model we use is that developed by Canudas de Wit et al, a model that is receiving increasing acceptance from the mechanics community. Using this model, we find a stable limit cycle at intermediate sliding speed for a single-degree-of-freedom mechanical oscillator. Moreover, the mechanical oscillator can exhibit chaotic motions. For certain parameters, numerical simulation suggests the existence of a Silnikov homoclinic orbit. This is not expected in a single-degree-of-freedom system. The occurrence of chaos becomes possible because the friction model contains one internal variable. This demonstrates a unique characteristic of the friction model. Unlike most friction models, the present model is capable of simultaneously modeling self-excitation and predicting stick-slip at very low sliding speed as well.

Phase Space Interpretation and Stability Analysis of a Self Excited, Friction Damped Turbine Airfoil

1997 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre
Keyword(s):  
Stability Analysis ◽  
Phase Space ◽  
Asymptotic Methods ◽  
Three Dimensional ◽  
Space Representation ◽  
Free Response ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Dimensional Phase Space ◽  
Phase Space Representation

Abstract The free response motion of a self excited, friction damped, single-degree of freedom, turbine airfoil model is determined utilizing both exact and asymptotic methods. A three-dimensional phase space representation is used to examine the system’s global stability, and to further intuitive understanding of the system dynamics. Conclusions are reached regarding the validity and application of stability predictions through comparison of approximate and exact solutions.

scholarly journals Subsumed Homoclinic Connections and Infinitely Many Coexisting Attractors in Piecewise-Linear Maps

2017 ◽  
Vol 27 (02) ◽  
pp. 1730010 ◽  
Author(s):  
David J. W. Simpson ◽  
Christopher P. Tuffley
Keyword(s):  
Periodic Solutions ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Stable And Unstable Manifolds ◽  
Homoclinic Connections ◽  
Linear Maps ◽  
Continuous Maps ◽  
Piecewise Linear Maps ◽  
Stable Periodic ◽  
Unstable Manifolds

We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for [Formula: see text]-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solutions and components of the stable and unstable manifolds which are available because the maps are piecewise-linear. We also describe a practical approach for finding this phenomenon in the parameter space of a map and illustrate the results with the three-dimensional border-collision normal form.

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