Chaotic Motion of a Parametrically Excited Dielectric Elastomer

2020 ◽  
Vol 12 (03) ◽  
pp. 2050033
Author(s):  
Hamidreza Heidari ◽  
Amin Alibakhshi ◽  
Habib Ramezannejad Azarboni
Keyword(s):  
Time Integration ◽  
Nonlinear Behavior ◽  
Strain Energy Function ◽  
Nonlinear Ordinary Differential Equation ◽  
Dielectric Elastomer ◽  
Dissipation Function ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Poincaré Sections ◽  
Poincare Sections

In this paper, an effort is made to study the chaotic motions of a dielectric elastomer (DE). The DE is activated by a time-dependent voltage (AC voltage), which is superimposed on a DC voltage. The Gent strain energy function is employed to model the nonlinear behavior of the elastomeric matter. The nonlinear ordinary differential equation (ODE) in terms of the stretch of the elastomer governing the motion of the system is deduced using the Euler–Lagrange method and the Rayleigh dissipation function. This ODE is solved via the use of a time integration-based solver. The bifurcation diagrams of Poincaré sections are generated to identify the chaotic domains. The largest Lyapunov exponents (LLEs) are illustrated for validation of the results obtained by the bifurcation diagrams. Various types of motion for the system are precisely discussed through the depiction of stretch-time responses, phase-plane diagrams, Poincaré sections and power spectral density (PSD) diagrams. The results reveal that the damping coefficient plays an influential role in suppressing the chaos phenomenon. Besides, the initial stretch of the elastomer could affect the chaotic interval of system parameters.

Chaotic Motions in Forced Mixed Rayleigh–Liénard Oscillator with External and Parametric Periodic-Excitations

2018 ◽  
Vol 28 (03) ◽  
pp. 1830005 ◽  
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
A. A. Koukpemedji ◽  
Y. J. F. Kpomahou ◽  
J. B. Chabi Orou
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Melnikov Method ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Poincaré Sections ◽  
Routes To Chaos ◽  
Domain Boundaries ◽  
Bifurcation Structures ◽  
Poincare Sections

This paper addresses the issue of a mixed Rayleigh–Liénard oscillator with external and parametric periodic-excitations. The Melnikov method is utilized to analytically determine the domain boundaries where horseshoe chaos appears. Routes to chaos are investigated through bifurcation structures, Lyapunov exponents, phase portraits and Poincaré sections. The effects of Rayleigh and Liénard parameters are analyzed. Results of analytical investigations are validated and complemented by numerical simulations.

Use of Fractal Dimension in the Characterization of Chaotic Structural Dynamic Sytems

1991 ◽  
Vol 44 (11S) ◽  
pp. S107-S113 ◽  
Author(s):  
E. Hall ◽  
S. Kessler ◽  
S. Hanagud
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Specific Class ◽  
Chaotic Motions ◽  
Structural Dynamic ◽  
Poincaré Sections ◽  
Pde Model ◽  
Forcing Function ◽  
Poincare Sections

The purpose of this paper is to investigate the use of fractal dimensions in the characterization of chaotic systems in structural dynamics. The investigation focuses on the example of a simply-supported, Euler-Bernoulli beam which when subjected to a transverse forcing function of a particular amplitude responds chaotically. Three different nonlinear models of the system are studied: a complex partial differential equation (PDE) model, a simplified PDE model, and a Galerkin approximation to the simpler PDE model. The responses of each model are examined through zero velocity Poincare´ sections. To characterize and compare the chaotic trajectories, the box counting fractal dimension of the Poincare´ sections are computed. The results demonstrate that the fractal dimension is a spatial invariant along the length of the beam for the specific class of forcing function studied, and thus it can be used to characterize chaotic motions. In addition, the three models yield different fractal dimensions for the same forcing which indicates that fractal dimensions can also be used to quantify whether a simplification of a chaotic model accurately predicts the chaotic behavior of the full-blown model. Thus the conclusion of the paper is that fractal dimensions may play an important role in the characterization of chaotic structural dynamic systems.

Study on Synchronization of the Centrifugal Flywheel Governor System

2013 ◽  
Vol 433-435 ◽  
pp. 21-29 ◽  
Author(s):  
Jian Kui Peng ◽  
Jian Ning Yu ◽  
Li Zhang ◽  
Ping Hu
Keyword(s):  
Adaptive Control ◽  
Lyapunov Exponents ◽  
External Disturbance ◽  
Global Bifurcation ◽  
Control Methods ◽  
Chaotic Motions ◽  
Poincaré Sections ◽  
Dynamical Behaviors ◽  
System Synchronization ◽  
Poincare Sections

In this paper, the dynamical behaviors of the centrifugal flywheel governor with external disturbance is studied and it has abundant nonlinear behavior.The influence of system parameter is discussed by Lyapunov exponents spectrum and global bifurcation diagram, which accurately portray the partial dynamic behavior of the centrifugal flywheel governor. The routes to chaos are analyzed using Poincaré sections, which are found to be more complex . Periodic and chaotic motions can be clearly distinguished by Poincaré sections, bifurcation diagrams and Lyapunov exponents. Then, the paper proposes coupledfeedback control and adaptive control methods to achieve the chaotic the centrifugal flywheel governor system synchronization, the numerical simulation was provided in order to show the effectiveness of coupled feedback control and adaptive control methods for the synchronization of the chaotic nonautonomous centrifugal flywheel governor system.

Dynamical Analysis and Chaos Control of a Driven System with One Cubic Nonlinearity: Numerical and Experimental Investigations

2012 ◽  
Vol 486 ◽  
pp. 204-210
Author(s):  
Zhen Wang ◽  
Wei Sun ◽  
Zhou Chao Wei
Keyword(s):  
Dynamical Analysis ◽  
Phase Portraits ◽  
Experimental Investigations ◽  
Bifurcation Diagrams ◽  
Cubic Nonlinearity ◽  
Controlled System ◽  
Poincaré Sections ◽  
Driven System ◽  
Poincare Sections ◽  
Parameter Values

The dynamics of a non-autonomous chaotic system with one cubic nonlinearity is studied through numerical and experimental investigations in this paper. A method for calculating Lyapunov exponents (LEs), Lyapunov dimension (LD) from time series is presented. Furthermore, some complex dynamic behaviors such as periodic, quasi-periodic motion and chaos which occurred in the system are analyzed, and a route to chaos, phase portraits, Poincare sections, bifurcation diagrams are observed. Finally, a first order differential controller for the non-autonomous system is designed. Also some dynamics such as Poincare sections, bifurcation diagrams for specific control parameter values of the controlled system are showed using numerical and experimental simulations.

scholarly journals VALIDATING IDENTIFIED NONLINEAR MODELS WITH CHAOTIC DYNAMICS

1994 ◽  
Vol 04 (01) ◽  
pp. 109-125 ◽  
Author(s):  
LUIS A. AGUIRRE ◽  
S.A. BILLINGS
Keyword(s):  
Lyapunov Exponent ◽  
Correlation Dimension ◽  
Chaotic Dynamics ◽  
Nonlinear Models ◽  
Original System ◽  
Largest Lyapunov Exponent ◽  
Bifurcation Diagrams ◽  
Poincaré Sections ◽  
The Largest Lyapunov Exponent ◽  
Poincare Sections

This paper investigates the effectiveness of several criteria for validating models which exhibit chaotic dynamics. Embedded trajectories, Poincaré sections, bifurcation diagrams, the largest Lyapunov exponent and correlation dimension are considered. The Duffing-Ueda equation and four identified models are used as examples. The results show that models with similar invariants such as Poincaré sections, the largest Lyapunov exponent and correlation dimension may have very different bifurcation behaviours. This suggests that the requirement that an identified model should reproduce the bifurcation pattern of the original system is a very exacting criterion which is well suited for validation purposes.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

A New Method to Find the Forced Response of Nonlinear Systems with Dry Friction

2021 ◽  
Author(s):  
Gregory L. Altamirano ◽  
Meng-Hsuan Tien ◽  
Kiran D'Souza
Keyword(s):  
Dry Friction ◽  
Coulomb Friction ◽  
Nonlinear Response ◽  
Piecewise Linear ◽  
Time Integration ◽  
Nonlinear Behavior ◽  
Forced Response ◽  
New Method ◽  
Analysis Tools ◽  
Compatibility Constraint

Abstract Coulomb friction has an influence on the behavior of numerous mechanical systems. Coulomb friction systems or dry friction systems are nonlinear in nature. This nonlinear behavior requires complex and time demanding analysis tools to capture the dynamics of these systems. Recently, efforts have been made to develop efficient analysis tools able to approximate the forced response of systems with dry friction. The objective of this paper is to introduce a methodology that assists in these efforts. In this method, the piecewise-linear nonlinear response is separated into individual linear responses that are coupled together through compatibility constraint equations. The new method is demonstrated on a number of systems of varying complexity. The results obtained by the new method are validated through the comparison with results obtained by time integration. The computational savings of the new method is also discussed.

Nonlinear Dynamic Analysis on Planetary Gears-Rotor System in Geared Turbofan Engines

2019 ◽  
Vol 29 (06) ◽  
pp. 1950076 ◽  
Author(s):  
Lanlan Hou ◽  
Shuqian Cao
Keyword(s):  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Nonlinear Behavior ◽  
Rotor System ◽  
Chaotic Motions ◽  
Planetary Gears ◽  
Gear Noise ◽  
Turbofan Engines ◽  
Vibration Responses ◽  
Nonlinear Analytical Model

Rotor fatigue and gear noise triggered by nonlinear vibration are the key concerns in Geared Turbofan (GTF) engine which features a new configuration by introducing planetary gears into low-pressure compressor. A nonlinear analytical model of the GTF planetary gears-rotor system is developed, where the torsional effect of rotor and pivotal parameters from gears are incorporated. The nonlinear behavior of the model can be obtained by focusing on the relative torsional vibration responses between gear and rotor. The torsional nonlinear responses are illustrated with bifurcation diagrams, the largest Lyapunov exponents (LLE), Poincaré maps, phase diagrams and spectrum waterfall. Numerical results reveal that the gears-rotor system exhibits abundant torsional nonlinear behaviors, including multiperiodic, quasi-periodic, and chaotic motions. Furthermore, the roads to chaos via quasi-periodicity, period-doubling scenario, mutation and intermittence are demonstrated. The ring gear stiffness at a low value can propel the system into chaos. The damping may complicate the motion, i.e. the system may enter chaos with increasing damping. These results provide an understanding of undesirable torsional dynamic motion for the GTF engine rotor system and therefore serve as a useful reference for engineers in designing and controlling such system.

Shooting With Deflation Algorithm-Based Nonlinear Response and Neimark-Sacker Bifurcation and Chaos in Floating Ring Bearing Systems

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Sitae Kim ◽  
Alan B. Palazzolo
Keyword(s):  
Nonlinear Response ◽  
Nonlinear Behavior ◽  
Rotor System ◽  
Operating Conditions ◽  
Fluid Film ◽  
Stable Limit ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Response Characteristics ◽  
Computation Efficiency

The double-sided fluid film force on the inner and outer ring surfaces of a floating ring bearing (FRB) creates strong nonlinear response characteristics such as coexistence of multiple orbits, Hopf bifurcation, Neimark-Sacker (N-S) bifurcation, and chaos in operations. An improved autonomous shooting with deflation algorithm is applied to a rigid rotor supported by FRBs for numerically analyzing its nonlinear behavior. The method enhances computation efficiency by avoiding previously found solutions in the numerical-based search. The solution manifold for phase state and period is obtained using arc-length continuation. It was determined that the FRB-rotor system has multiple response states near Hopf and N-S bifurcation points, and the bifurcation scenario depends on the ratio of floating ring length and diameter (L/D). Since multiple responses coexist under the same operating conditions, simulation of jumps between two stable limit cycles from potential disturbance such as sudden base excitation is demonstrated. In addition, this paper investigates chaotic motions in the FRB-rotor system, utilizing four different approaches, strange attractor, Lyapunov exponent, frequency spectrum, and bifurcation diagram. A numerical case study for quenching the large amplitude motion by adding unbalance force is provided and the result shows synchronization, i.e., subsynchronous frequency components are suppressed. In this research, the fluid film forces on the FRB are determined by applying the finite element method while prior work has utilized a short bearing approximation. Simulation response comparisons between the short bearing and finite bearing models are discussed.

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