Nonlinear Response of Flexible Robotic Manipulators Performing Repetitive Tasks

1989 ◽  
Vol 111 (3) ◽  
pp. 470-479 ◽  
Author(s):  
D. A. Streit ◽  
C. M. Krousgrill ◽  
A. K. Bajaj
Keyword(s):  
Nonlinear Response ◽  
Period Doubling ◽  
Flexible Manipulator ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Periodic Motions ◽  
Dynamical Equations ◽  
Chaotic Motions ◽  
Compliant Motion ◽  
Repetitive Tasks

The dynamics of a flexible manipulator is described by two distinct types of variables, one describing the nominal motion and the other describing the compliant motion. For a manipulator programmed to perform repetitive tasks, the dynamical equations governing the compliant motion are parametrically excited. Nonlinear dynamics of a two-degree-of-freedom model is investigated in parameter regions where the nominal motion is predicted by the Floquet theory to be unstable. Multiple time scales technique is used to study the nonlinear response, and it is shown that the compliant coordinates can execute small but finite amplitude periodic motions. In one particular case, the amplitude of these periodic motions is shown to bifurcate to a periodic solution which subsequently undergoes period-doubling bifurcations leading to chaotic motions.

Switching Bifurcation and Chaos in an Extended Fermi-Acceleration Oscillator

2008 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Local Stability ◽  
Period Doubling ◽  
Time History ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Numerical Predictions ◽  
Global View ◽  
Acceleration Model

The Fermi acceleration oscillator is extensively used to interpret many physical and mechanical phenomena. To understand dynamic behaviors of a particle (or a bouncing ball) in such a Fermi oscillator, a generalized Fermi acceleration model is developed. This model consists of a particle moving vertically between a fixed wall and the piston in a vibrating oscillator. The motion switching bifurcation of a particle in such a generalized Fermi oscillator is investigated through the theory of discontinuous dynamical systems. The analytical conditions for the motion switching are developed for numerical predictions. Thus, periodic motions in the Fermi-acceleration oscillator are given and the corresponding local stability and bifurcation are presented. Periodic and chaotic motions in such an oscillator are presented via the displacement time-history. From switching bifurcation and period-doubling bifurcation, parameter maps of periodic and chaotic motions will be developed for a global view of motions in the Fermi acceleration oscillator. To illustrate motion switching phenomena, the acceleration responses of the particle and base in the Fermi oscillator are presented. Poincare mapping sections are also used to illustrate chaos, and energy dissipation in chaotic motions can be evaluated.

DOUBLE-HOPF BIFURCATION IN AN OSCILLATOR WITH EXTERNAL FORCING AND TIME-DELAYED FEEDBACK CONTROL

2006 ◽  
Vol 16 (12) ◽  
pp. 3523-3537 ◽  
Author(s):  
ZHEN CHEN ◽  
PEI YU
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Approximate Solutions ◽  
Delayed Feedback Control ◽  
Critical Conditions ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Feedback Controls ◽  
Chaotic Motions ◽  
Double Hopf Bifurcation

In this paper, an oscillator with time delayed velocity feedback controls is studied in detail. Particular attention is given to internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which double-Hopf bifurcation occurs. Then center manifold theory is applied to obtain an ODE system described on a four-dimensional center manifold. Further, the technique of multiple-time scales is employed to find the approximate solutions of periodic and quasi-periodic motions. Finally, numerical simulation results are presented to not only validate the analytical predictions, but also show chaotic motions for some certain parameter values.

ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250093 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
JIANZHE HUANG
Keyword(s):  
Dynamical Systems ◽  
Analytical Solutions ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Damping ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Dynamical

In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.

The Dynamics of an Impact Print Hammer

1988 ◽  
Vol 110 (2) ◽  
pp. 193-200 ◽  
Author(s):  
P. C. Tung ◽  
S. W. Shaw
Keyword(s):  
Period Doubling ◽  
Cycle Period ◽  
Flexible Beam ◽  
Qualitative Behavior ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Linear Spring ◽  
The Stability ◽  
Energy Type ◽  
The Impact

A mathematical model is developed to describe the characteristic behavior of an impact print hammer of the stored energy type. The armature of the impact print hammer is represented by a rigid mass held against a backstop by a preloaded linear spring with negative stiffness which characterizes the net effect of a permanent magnet and a prestressed flexible beam acting on the armature. Periodic sine pulses are adopted to represent currents which release the armature to strike the ribbon and paper which is represented by a linear spring and a linear viscous dashpot. A coefficient of restitution is employed to characterize the instantaneous behavior of impact and rebound at the backstop. In this paper, periodic motions with n impacts against the backstop per forcing cycle, period doubling bifurcations, and chaotic motions are found. The stability of the periodic motions is investigated as is the influence of various parameters on the performance of the impact print hammer. With this simple model we can predict much of the qualitative behavior of the actual physical system.

Stability and Bifurcation for the Equispaced, Periodic Motion of a Horizontal Impact Damper

2001 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Periodic Motion ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Impact Damper ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Good Agreement

Abstract Stability and bifurcation conditions for the asymmetric, periodic motion of a horizontal impact damper under a periodic excitation are developed through four mappings for two switch-planes relative to discontinuities. Period-doubling bifurcation for equispaced motion does not occur, but the asymmetric period-1 motions change to the asymmetric, period-2 ones through a period doubling bifurcation. A numerical prediction for equispaced to chaotic motions is completed. The numerical and analytical predictions of the periodic motion are in very good agreement. The asymmetric, periodic motions are also simulated.

Model Reduction for Nonlinear Elastic Structures With Inertial Quadratic Nonlinearities

2009 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Model Reduction ◽  
Time Scales ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Nonlinear Response ◽  
Normal Modes ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Parametric Resonances ◽  
Quadratic Nonlinearities

In order to achieve accurate and high fidelity nonlinear response predictions, discrete models usually obtained through Galerkin approximation utilizing linear normal modes of the structure, need to retain a large number of degrees of freedom. This is specially the case if the structural response has the possibility of modal interactions. Then, a possible approach suggested in the literature to decrease the required degrees of freedom while retaining same accuracy is to use nonlinear normal modes of the structure to perform further model reduction. In this work, we discuss model reduction for nonlinear structural systems under harmonic excitations. The analysis needs to carefully consider the possibility of external resonances, parametric resonances, combination parametric resonances (the parametric excitation frequency being near the sum or difference of frequencies of two modes), and internal resonances. A master-slave separation of degrees of freedom is used, and a nonlinear relation between the slave coordinates and the master coordinates is constructed based on the multiple time scales approximation. More specifically, three cases are considered: external resonance of a mode without any internal resonance, and subharmonic as well as superharmonic excitation for systems with 1:2 internal resonance. The steady state periodic responses determined by the method of multiple time scales are compared to exact solutions of the discrete model computed by the bifurcation analysis and parameter continuation software AUTO. It is seen that for systems with essential inertial quadratic nonlinearities, the technique based on nonlinear model reduction through multiple time scales approximation over-predict the softening nonlinear response.

SOME REMARKS ON THE MULTIPLE TIME SCALES OF DIFFUSION IN MINERALS AND MELTS

2018 ◽  
Author(s):  
Yan Liang ◽  
◽  
Daniele J. Cherniak ◽  
Chenguang Sun
Keyword(s):  
Time Scales ◽  
Multiple Time Scales ◽  
Multiple Time

scholarly journals Odor response adaptation in Drosophila—a continuous individualization process

2021 ◽  
Vol 383 (1) ◽  
pp. 143-148
Author(s):  
Shadi Jafari ◽  
Mattias Alenius
Keyword(s):  
Sensory Neurons ◽  
Olfactory System ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Olfactory Perception ◽  
Adaptation Mechanisms ◽  
Metabolic States ◽  
The Central Nervous System ◽  
Olfactory Adaptation ◽  
Central Adaptation

AbstractOlfactory perception is very individualized in humans and also in Drosophila. The process that individualize olfaction is adaptation that across multiple time scales and mechanisms shape perception and olfactory-guided behaviors. Olfactory adaptation occurs both in the central nervous system and in the periphery. Central adaptation occurs at the level of the circuits that process olfactory inputs from the periphery where it can integrate inputs from other senses, metabolic states, and stress. We will here focus on the periphery and how the fast, slow, and persistent (lifelong) adaptation mechanisms in the olfactory sensory neurons individualize the Drosophila olfactory system.

scholarly journals Are We Doing ‘Systems’ Research? An Assessment of Methods for Climate Change Adaptation to Hydrohazards in a Complex World

2019 ◽  
Vol 11 (4) ◽  
pp. 1163 ◽  
Author(s):  
Melissa Bedinger ◽  
Lindsay Beevers ◽  
Lila Collet ◽  
Annie Visser
Keyword(s):  
Climate Change ◽  
Human Nature ◽  
Climate Change Adaptation ◽  
Time Scales ◽  
Spatial Scales ◽  
Local Context ◽  
Multiple Time Scales ◽  
Future Research ◽  
Multiple Time ◽  
Systems Research

Climate change is a product of the Anthropocene, and the human–nature system in which we live. Effective climate change adaptation requires that we acknowledge this complexity. Theoretical literature on sustainability transitions has highlighted this and called for deeper acknowledgment of systems complexity in our research practices. Are we heeding these calls for ‘systems’ research? We used hydrohazards (floods and droughts) as an example research area to explore this question. We first distilled existing challenges for complex human–nature systems into six central concepts: Uncertainty, multiple spatial scales, multiple time scales, multimethod approaches, human–nature dimensions, and interactions. We then performed a systematic assessment of 737 articles to examine patterns in what methods are used and how these cover the complexity concepts. In general, results showed that many papers do not reference any of the complexity concepts, and no existing approach addresses all six. We used the detailed results to guide advancement from theoretical calls for action to specific next steps. Future research priorities include the development of methods for consideration of multiple hazards; for the study of interactions, particularly in linking the short- to medium-term time scales; to reduce data-intensivity; and to better integrate bottom–up and top–down approaches in a way that connects local context with higher-level decision-making. Overall this paper serves to build a shared conceptualisation of human–nature system complexity, map current practice, and navigate a complexity-smart trajectory for future research.

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