Nonlinear Dynamics, Chaos, and Mechanics

1990 ◽  
Vol 43 (5S) ◽  
pp. S23-S39 ◽  
Author(s):  
Philip Holmes
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Chaos Theory ◽  
Nonlinear Differential Equations ◽  
Analytical Approaches

Nonlinear Dynamics or “Chaos Theory” is an ill-defined but energetic and rapidly developing subject which cuts across the boundaries of traditional disciplines. In this review, I describe a small part of it: some of the analytical approaches to nonlinear differential equations which have been developed in the last ten to fifteen years. I illustrate them with applications in solid and fluid mechanics.

scholarly journals Deterministic Chaos Theory: Basic Concepts

2016 ◽  
Vol 39 (1) ◽  
Author(s):  
Mauro Cattani ◽  
Iberê Luiz Caldas ◽  
Silvio Luiz de Souza ◽  
Kelly Cristiane Iarosz
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Chaos Theory ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Deterministic Chaos ◽  
Chaotic Motions ◽  
And Mathematics ◽  
Irregular Behavior ◽  
Mathematics And Physics

This article was written to students of mathematics, physics and engineering. In general, the word chaos may refer to any state of confusion or disorder and it may also refer to mythology or philosophy. In science and mathematics it is understood as irregular behavior sensitive to initial conditions. In this article we analyze the deterministic chaos theory, a branch of mathematics and physics that deals with dynamical systems (nonlinear differential equations or mappings) with very peculiar properties. Fundamental concepts of the deterministic chaos theory are briefly analyzed and some illustrative examples of conservative and dissipative chaotic motions are introduced. Complementarily, we studied in details the chaotic motion of some dynamical systems described by differential equations and mappings. Relations between chaotic, stochastic and turbulent phenomena are also commented.

scholarly journals Principal Component Analysis in the Nonlinear Dynamics of Beams: Purification of the Signal from Noise Induced by the Nonlinearity of Beam Vibrations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
A. V. Krysko ◽  
Jan Awrejcewicz ◽  
Irina V. Papkova ◽  
Olga Szymanowska ◽  
V. A. Krysko
Keyword(s):  
Nonlinear Dynamics ◽  
Principal Component Analysis ◽  
Differential Equations ◽  
Geometric Nonlinearity ◽  
Nonlinear Differential Equations ◽  
Principal Component ◽  
Component Analysis ◽  
Harmonic Load ◽  
Linear And Nonlinear ◽  
The Impact

The paper discusses the impact of the von Kármán type geometric nonlinearity introduced to a mathematical model of beam vibrations on the amplitude-frequency characteristics of the signal for the proposed mathematical models of beam vibrations. An attempt is made to separate vibrations of continuous mechanical systems subjected to a harmonic load from noise induced by the nonlinearity of the system by employing the principal component analysis (PCA). Straight beams lying on Winkler foundations are analysed. Differential equations are obtained based on the Bernoulli-Euler, Timoshenko, and Sheremetev-Pelekh-Levinson-Reddy hypotheses. Solutions to linear and nonlinear differential equations are found using the principal component analysis (PCA).

On the Nonlinear Kinematic Oscillations of Railway Wheelsets

2016 ◽  
Vol 11 (5) ◽  
Author(s):  
Mate Antali ◽  
Gabor Stepan
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Vibration Amplitude ◽  
Nonlinear Differential Equations ◽  
Analytical Formula ◽  
Local Geometry ◽  
Nonlinearity Factor

In this paper, nonlinear dynamics of a railway wheelset is investigated during kinematic oscillations. Based on the nonlinear differential equations, the notion of nonlinearity factor is introduced, which expresses the effect of the vibration amplitude on the frequency of the oscillations. The analytical formula of this nonlinearity factor is derived from the local geometry of the rail and wheel profiles. The results are compared to the ones obtained from the rolling radius difference (RRD) function.

Research on Fluid Mechanics with Slip Flow of Viscoelastic Fluid in the Micro Channel

2013 ◽  
Vol 387 ◽  
pp. 51-54
Author(s):  
Zheng Liu ◽  
Jing Zhu ◽  
Lian Cun Zheng
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Applied Mathematics ◽  
Nonlinear Differential Equations ◽  
Slip Flow ◽  
Fluid Motion ◽  
Physical Factors ◽  
Stagnation Region ◽  
Analytical Approximations

Stagnation flow, an import research branch of fluid mechanics, describing the fluid motion near the stagnation region, exists on all solid bodies moving in a fluid. And stagnation point boundary layer flow problems described by partial differential equations have attracted many scholars attention nowadays. These problems have become difficult and hot in the study of applied mathematics, mechanics and materials engineering. This paper has transformed the governing boundary layer equations into a system of nonlinear differential equations through the similarity transformation, and the analytical approximations of solutions are derived by homotopy analysis method (HAM). In addition, the effects of physical factors (such as the slip parameter, Magnetic field parameter and Reynolds number) on the flow are examed and discussed graphically. They have a great impact on the speed.

Robust Adaptive Neural Estimation of Restoring Forces in Nonlinear Structures

2001 ◽  
Vol 68 (6) ◽  
pp. 880-893 ◽  
Author(s):  
E. B. Kosmatopoulos ◽  
A. W. Smyth ◽  
S. F. Masri ◽  
A. G. Chassiakos
Keyword(s):  
Nonlinear Dynamics ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Adaptive Methods ◽  
Reinforced Concrete Structure ◽  
Response Measurement ◽  
New Approach ◽  
Model Based ◽  
Restoring Forces ◽  
On Line

The availability of methods for on-line estimation and identification of structures is crucial for the monitoring and active control of time-varying nonlinear structural systems. Adaptive estimation approaches that have recently appeared in the literature for on-line estimation and identification of hysteretic systems under arbitrary dynamic environments are in general model based. In these approaches, it is assumed that the unknown restoring forces are modeled by nonlinear differential equations (which can represent general nonlinear characteristics, including hysteretic phenomena). The adaptive methods estimate the parameters of the nonlinear differential equations on line. Adaptation of the parameters is done by comparing the prediction of the assumed model to the response measurement, and using the prediction error to change the system parameters. In this paper, a new methodology is presented which is not model based. The new approach solves the problem of estimating/identifying the restoring forces without assuming any model of the restoring forces dynamics, and without postulating any structure on the form of the underlying nonlinear dynamics. The new approach uses the Volterra/Wiener neural networks (VWNN) which are capable of learning input/output nonlinear dynamics, in combination with adaptive filtering and estimation techniques. Simulations and experimental results from a steel structure and from a reinforced-concrete structure illustrate the power and efficiency of the proposed method.

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