Chaos in nonlinear dynamic systems: Helicopter vibration mechanisms (original) (raw)
Related papers
2007
This paper determines whether chaotic dynamics exists in a flying vibratory system. Acceleration signals were measured at nine different locations or orientations of the flying object during a test flight. Steady-state acceleration data were extracted to reconstruct pseudo phase-space trajectories from which two dynamical indices including the correlation dimension and the maximum Lyapunov exponent are calculated. Although generally the correlation dimension depends on the embedding dimension, it is found that in three out of the nine-channel acceleration signals, the correlation dimension saturates when the embedding dimension reaches a critical value. The phenomenon indicates a possible existence of chaotic motion. The maximum Lyapunov exponents calculated for the same three-channel data are all positive which again implies the possible existence of chaos. To determine whether the experimental time series that demonstrate chaotic characteristics are in fact deterministic (rather t...
Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems
The first part of the paper was aimed at analyzing the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method [1], Rosenstein method [2], Kantz method [3], method based on the modification of a neural network [4, 5], and the synchronization method [6, 7]) for the classical problems governed by difference and differential equations (Hénon map [8], hyper-chaotic Hénon map [9], logistic map [10], Rössler attractor [11], Lorenz attractor [12]) and with the use of both Fourier spectra and Gauss wavelets [13]. It was shown that a modification of the neural network method [4, 5] makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyper-chaos, hyper hyper-chaos and deep chaos [14-16]. Different algorithms for computation of Lyapunov exponents were validated by comparison with the known dynamical systems spectra of the Lyapunov exponents. The ca...
Chaos Control in Mechanical Systems
Shock and Vibration, 2006
Chaos has an intrinsically richness related to its structure and, because of that, there are benefits for a natural system of adopting chaotic regimes with their wide range of potential behaviors. Under this condition, the system may quickly react to some new situation, changing conditions and their response. Therefore, chaos and many regulatory mechanisms control the dynamics of living systems, conferring a great flexibility to the system. Inspired by nature, the idea that chaotic behavior may be controlled by small perturbations of some physical parameter is making this kind of behavior to be desirable in different applications. Mechanical systems constitute a class of system where it is possible to exploit these ideas. Chaos control usually involves two steps. In the first, unstable periodic orbits (UPOs) that are embedded in the chaotic set are identified. After that, a control technique is employed in order to stabilize a desirable orbit. This contribution employs the close-return method to identify UPOs and a semi-continuous control method, which is built up on the OGY method, to stabilize some desirable UPO. As an application to a mechanical system, a nonlinear pendulum is considered and, based on parameters obtained from an experimental setup, analyses are carried out. Signals are generated by numerical integration of the mathematical model and two different situations are treated. Firstly, it is assumed that all state variables are available. After that, the analysis is done from scalar time series and therefore, it is important to evaluate the effect of state space reconstruction. Delay coordinates method and extended state observers are employed with this aim. Results show situations where these techniques may be used to control chaos in mechanical systems.
Some Methods of Analysis of Chaos in Mechanical Systems
American Journal of Mechanical Engineering, 2014
Solution of non-linear dynamic systems is dependent on exact knowledge of the initial conditions. Even a slight deviation of these values can cause substantial change in the overall course of the event. It then appears chaotic. Development of such dynamic system can be represented using abstract phase space through attractors, fractals, etc. A typical example is Lorenz attractor, which is in a three-dimensional view shaped as two intertwined spirals. Rössler attractor is a relatively simple system on which chaos in geometric form can be shown in a time sequence. Among the non-traditional oscillators in non-linear mechanics can be classified Duffing and Van der Pol oscillators. This paper shows an example of a chaotic attractor formed in a non-periodic mode, obtained in an experiment of water dripping from an unclosed valve.
Quantifying chaotic responses of mechanical systems with backlash component
Mechanical Systems and Signal Processing, 2007
This paper presents a detailed numerical and experimental study of a mechanical system comprising a backlash element that shows chaotic behaviour in a certain excitation range. It aims to quantify the chaotic behaviour of the responses and correlate the quantification parameters to the parameters of the (non-linear) system, in particular the backlash size. The motivation for this investigation is to be able ultimately to identify the parameters of non-linear systems without necessarily being able to ensure periodic behaviour. Application of surrogate data tests is utilised to prove the presence of the chaotic behaviour in the response. The Simple Non-linear Noise Reduction is applied to the resulting data to have a better interpretation of the chaotic in the response. r (T. Tjahjowidodo).
ENERGEIA. Sociedad Ibero-Americana de Metodología Económica (ISSN 1666-5732), 2020
Chaos theory refers to the behaviour of certain deterministic nonlinear dynamical systems whose solutions, although globally stable, are locally unstable. These chaotic systems describe aperiodic, irregular, apparently random and erratic trajectories, i.e., deterministic complex dynamics. One of the properties that derive from this local instability and that allow characterizing these deterministic chaotic systems is their high sensitivity to small changes in the initial conditions, which can be measured by using the so-called Lyapunov exponents. The detection of chaotic behaviour in the underlying generating process of a time series has important methodological implications. When chaotic behaviour is detected, then it can be concluded that the irregularity of the series is not necessarily random, but the result of some deterministic dynamic process. Then, even if such process is unknown, it will be possible to improve the predictability of the time series and even to control or stabilize the evolution of the time series. This article provides a summary of the main current concepts and methods for the detection of chaotic behaviour from time series.
Chaos-Genetic Algorithm for the System Identification of a Small Unmanned Helicopter
Journal of Intelligent & Robotic Systems, 2012
This paper focuses on the system identification of a small unmanned helicopter in hover or low-speed flight conditions. A novel genetic algorithm including chaotic optimization operation named chaos-genetic algorithm (CGA) is proposed to identify the linear helicopter model. Based on the input-output data collected from real flight tests, the identification performance of CGA is compared with those calculated by the traditional genetic algorithm (TGA) and the prediction error method (PEM). The accuracy of the identified model is verified by simulation in time domain. Additionally, the small unmanned helicopter is stabilized by a linear quadratic Gaussian (LQG) regulator based on the proposed identified model. In the automatic flight experiments, the achievement of automatic takeoff and landing, hovering performance within a 1.2 m diameter circle and point-to-point horizontal polyline flight also demonstrates the accuracy of the identified model and the effectiveness of the proposed method.
2014
Solution of non-linear dynamic systems is dependent on exact knowledge of the initial conditions. Even a slight deviation of these values can cause substantial change in the overall course of the event. It then appears chaotic. Development of such dynamic system can be represented using abstract phase space through attractors, fractals, etc. A typical example is Lorenz attractor, which is in a three-dimensional view shaped as two intertwined spirals. Rössler attractor is a relatively simple system on which chaos in geometric form can be shown in a time sequence. Among the non-traditional oscillators in non-linear mechanics can be classified Duffing and Van der Pol oscillators. This paper shows an example of a chaotic attractor formed in a non-periodic mode, obtained in an experiment of water dripping from an unclosed valve.
Analytical solutions for periodic motions to chaos in nonlinear systems with/without time-delay
2013
In this paper, the analytical dynamics of periodic flows to chaos in nonlinear dynamical systems is presented from the ideas of Luo (Continuous dynamical systems, Higher Education Press/L&H Scientific, Beijing/Glen Carbon, 2012). The analytical solutions of periodic flows and chaos in autonomous systems are discussed through the generalized harmonic balance method, and the analytical dynamics of periodically forced nonlinear dynamical systems is presented as well. The analytical solutions of periodic motions in free and periodically forced vibration systems are presented. The similar ideas are extended to time-delayed nonlinear systems. The analytical solutions of periodic flows to chaos for time-delayed, nonlinear systems with/without periodic excitations are presented, and time-delayed, nonlinear vibration systems will be also discussed. The analytical solutions of periodic flows and chaos are independent of small parameters, which are different from the traditional perturbation methods. The methodology presented herein will provide the accurate analytical solutions of periodic motions to chaos in dynamical systems with/without time-delay. This approach can handle nonlinear dynamical systems with either single time-delay or multiple time-delays.