Numerical Study of an Impact Oscillator

1995 ◽  
Author(s):  
Kannan Marudachalam ◽  
Faruk H. Bursal
Keyword(s):  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Initial Conditions ◽  
Numerical Study ◽  
Harmonic Excitation ◽  
General Purpose ◽  
Constrained Systems ◽  
Global Dynamics ◽  
Impact Oscillator ◽  
Stick Slip

Abstract Systems with discontinuous dynamics can be found in diverse disciplines. Meshing gears with backlash, impact dampers, relative motion of components that exhibit stick-slip phenomena axe but a few examples from mechanical systems. These form a class of dynamical systems where the nonlinearity is so severe that analysis becomes formidable, especially when global behavior needs to be known. Only recently have researchers attempted to investigate such systems in terms of modern dynamical systems theory. In this work, an impact oscillator with two-sided rigid constraints is used as a paradigm for studying the characteristics of discontinuous dynamical systems. The oscillator has zero stiffness and is subjected to harmonic excitation. The system is linear without impacts. However, the impacts introduce nonlinearity and dissipation (assuming inelastic impacts). A numerical algorithm is developed for studying the global dynamics of the system. A peculiar type of solution in which the trajectories in phase space from a certain set of initial conditions merge in finite time, making the dynamics non-invertible, is investigated. Also, the effect of “grazing,” a behavior common to constrained systems, on the dynamics of the system is studied. Based on the experience gained in studying this system, the need for an efficient general-purpose numerical algorithm for solving discontinuous dynamical systems is motivated. Investigation of stress, vibration, wear, noise, etc. that are associated with impact phenomena can benefit greatly from such an algorithm.

Numerical Study on Mold Filling Process of Casting

2008 ◽  
Vol 575-578 ◽  
pp. 87-92
Author(s):  
Xiao Qiang Pan ◽  
Hong Zhu Sun ◽  
Jun Da Chen ◽  
Yu Ling Zhu
Keyword(s):  
Numerical Simulation ◽  
Initial Conditions ◽  
Numerical Study ◽  
Solidification Process ◽  
Mold Filling ◽  
General Purpose ◽  
Filling Process ◽  
Numerical Heat Transfer ◽  
Multiphase Fluid Flow ◽  
Multiphase Fluid

Techniques of numerical simulation on mold filling process of casting are investigated in this paper. The mathematical model is formed on the ground of some selected theories in computational fluid dynamics (CFD), Numerical Heat Transfer (NHT) and computational methods to interfacial tracking. The discrete solution to the governing equations appeals to Finite Volume Method (FVM) on structured mesh. As for viscous turbulence flow and multiphase fluid flow in mold filling, engineering turbulence model and Volume of Fluid (VOF) method are adopted in the algorithms, respectively. As a debut, the general-purpose CFD software is used to establish the practicable mechanical model for the simulation. By means of numerical simulation, variation and distribution of velocity, temperature, stress and configuration of casting, etc. with respect to time and space in the filling process can be quantitatively analysed in detail, which is helpful for engineers to optimize their design of technics with less time and less cost and is meaningful to provide the subsequent simulation, solidification process of casting, with initial conditions.

scholarly journals COMBINATORIAL CONTROL OF GLOBAL DYNAMICS IN A CHAOTIC DIFFERENTIAL EQUATION

2001 ◽  
Vol 11 (08) ◽  
pp. 2145-2162 ◽  
Author(s):  
ERIK M. BOLLT
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Dynamical Systems ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Discrete Dynamical System ◽  
Global Dynamics ◽  
Chaotic Dynamical System ◽  
Push Forward ◽  
Global Representation

Controlling chaos has been an extremely active area of research in applied dynamical systems, following the introduction of the Ott, Grebogi, Yorke (OGY) technique in 1990 [Ott et al., 1990], but most of this research based on parametric feedback control uses local techniques. Associated with a dynamical system which pushes forward initial conditions in time, transfer operators, including the Frobenius–Perron operator, are associated dynamical systems which push forward ensemble distributions of initial conditions. In [Bollt, 2000a, 2000b; Bollt & Kostelich, 1998], we have shown that such global representations of a discrete dynamical system are useful in controlling certain aspects of a chaotic dynamical system which could only be accessible through such a global representation. Such aspects include invariant measure targeting, as well as orbit targeting. In this paper, we develop techniques to show that our previously discrete time techniques are accessible also to a differential equation. We focus on the Duffing oscillator as an example. We also show that a recent extension of our techniques by Góra and Boyarsky [1999] can be further simplified and represented in a convenient and compact way by using a tensor product.

Loss of Determinacy at Small Scales, with Application to Multiple Timescale and Nonsmooth Dynamics

2021 ◽  
Vol 31 (03) ◽  
pp. 2150041
Author(s):  
S. Webber ◽  
M. R. Jeffrey
Keyword(s):  
Coupled Oscillators ◽  
Blow Up ◽  
Initial Conditions ◽  
Nonsmooth Dynamics ◽  
Global Dynamics ◽  
Time Loss ◽  
Piecewise Smooth Systems ◽  
Stick Slip ◽  
Small Scales ◽  
Multiple Timescale

A singularity is described that creates a forward time loss of determinacy in a two-timescale system, in the limit where the timescale separation is large. We describe how the situation can arise in a dynamical system of two fast variables and three slow variables or parameters, with weakly coupling between the fast variables. A wide set of initial conditions enters the [Formula: see text]-neighborhood of the singularity, and explodes back out of it to fill a large region of phase space, all in finite time. The scenario has particular significance in the application to piecewise-smooth systems, where it arises in the blow up of dynamics at a discontinuity and is followed by abrupt recollapse of solutions to “hide” the loss of determinacy, and yet leave behind a remnant of it in the global dynamics. This constitutes a generalization of a “micro-slip” phenomenon found recently in spring-coupled blocks, whereby coupled oscillators undergo unpredictable stick-slip-stick sequences instigated by a higher codimension form of the singularity. The indeterminacy is localized to brief slips events, but remains evident in the indeterminate sequencing of near-simultaneous slips of multiple blocks.

TWO-DIMENSIONAL DYNAMICAL SYSTEMS WITH PERIODIC COEFFICIENTS

2009 ◽  
Vol 19 (11) ◽  
pp. 3777-3790 ◽  
Author(s):  
BRUNO ROSSETTO ◽  
YING ZHANG
Keyword(s):  
Dynamical Systems ◽  
Phase Plane ◽  
Initial Conditions ◽  
Numerical Study ◽  
Period Doubling ◽  
Periodic Coefficient ◽  
Duffing Equation ◽  
Liapunov Exponent ◽  
Periodic Coefficients ◽  
Autonomous Dynamical Systems

This paper is devoted to the properties conferred on phase space by a vector field of generic second order autonomous dynamical systems with periodic coefficients, called parametric autonomous dynamical systems (PADS). At first, an associated periodical parametric linear equation (APPLE) is defined at every point of the phase plane. The exact value of the Floquet–Liapunov exponent of the APPLE is computed using a fast algorithm, without integration. The role of Floquet–Liapunov exponents is known to establish the stability of periodic solutions. In this work, it is pointed out that, under certain conditions, they bring information on local characteristics of PADS solutions according to their location in the phase plane, such as sensitivity to initial conditions, oscillation frequency, period doubling, parametric resonance, funneling. Then, an invariant manifold of an associated constant coefficients equivalent system (ACCES) is defined. It is shown that this manifold is periodically crossed by solutions of the PADS. This manifold crossing property contributes to the structure of the PADS solutions in phase plane. The implementation of this method is shown on a Van der Pol equation with a periodic coefficient in order to illustrate all kinds of solution patterns near the manifold in the phase plane according to the Floquet–Liapunov exponent local value. The manifold crossing property can be observed in all cases. Then, a parametric Duffing equation is processed. A numerical study shows some chaos routes, their bifurcation diagram and the top Liapunov exponent variations. The ACCES of the Duffing equation does not have any slow manifold. However, the Floquet–Liapunov exponent computation allows to specify the locus in the phase plane where the curvature of the trajectories changes, giving rise to chaos.

scholarly journals Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2471
Author(s):  
Tommaso Bradde ◽  
Samuel Chevalier ◽  
Marco De Stefano ◽  
Stefano Grivet-Talocia ◽  
Luca Daniel
Keyword(s):  
Dynamical Systems ◽  
Power System ◽  
System Dynamics ◽  
Real Time ◽  
Initial Conditions ◽  
Test System ◽  
Initial State ◽  
Power System Dynamics ◽  
Vector Fitting ◽  
Time Modeling

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

Pseudorandom Number Generators Based on Chaotic Dynamical Systems

2001 ◽  
Vol 08 (02) ◽  
pp. 137-146 ◽  
Author(s):  
Janusz Szczepański ◽  
Zbigniew Kotulski
Keyword(s):  
Dynamical Systems ◽  
Communication Systems ◽  
Initial Conditions ◽  
Pseudorandom Number ◽  
New Approach ◽  
Chaotic Dynamical Systems ◽  
Pseudorandom Number Generators ◽  
Transmission Of Information ◽  
Extreme Sensitivity ◽  
Contemporary Technology

Pseudorandom number generators are used in many areas of contemporary technology such as modern communication systems and engineering applications. In recent years a new approach to secure transmission of information based on the application of the theory of chaotic dynamical systems has been developed. In this paper we present a method of generating pseudorandom numbers applying discrete chaotic dynamical systems. The idea of construction of chaotic pseudorandom number generators (CPRNG) intrinsically exploits the property of extreme sensitivity of trajectories to small changes of initial conditions, since the generated bits are associated with trajectories in an appropriate way. To ensure good statistical properties of the CPRBG (which determine its quality) we assume that the dynamical systems used are also ergodic or preferably mixing. Finally, since chaotic systems often appear in realistic physical situations, we suggest a physical model of CPRNG.

Energy Balancing Method to Identify Nonlinear Damping of Bolted-Joint Interface

2016 ◽  
Vol 693 ◽  
pp. 318-323 ◽  
Author(s):  
Xin Liao ◽  
Jian Run Zhang
Keyword(s):  
Dry Friction ◽  
Harmonic Excitation ◽  
Joint Interface ◽  
Damping Factor ◽  
Viscous Damping ◽  
Bolted Joint ◽  
Energy Balancing ◽  
Stick Slip ◽  
Non Linear ◽  
Linear Damping

The interface of bolted joint commonly focuses on the research of non-linear damping and stiffness, which affect structural response. In the article, the non-linear damping model of bolted-joint interface is built, consisting of viscous damping and Coulomb friction. Energy balancing method is developed to identify the dry-friction parameter and viscous damping factor. The corresponding estimation equations are acquired when the input is harmonic excitation. Then, the vibration experiments with different bolted preloads are conducted, from which amplitudes in various input levels are used to work out the interface parameters. Also, the fitting curves of dry-friction parameters are also obtained. Finally, the results illustrate that the most interface of bolted joint in lower excitation levels occurs stick-slip motion, and the feasibility of the identification approach is demonstrated.

CHARACTERIZING LOSS OF MEMORY IN CHAOTIC DYNAMICAL SYSTEMS

1991 ◽  
Vol 05 (14) ◽  
pp. 2323-2345 ◽  
Author(s):  
R.E. AMRITKAR ◽  
P.M. GADE
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Chaotic Dynamical Systems

We discuss different methods of characterizing the loss of memory of initial conditions in chaotic dynamical systems.

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