The Second IASTED African Conference on
Modelling and Simulation
~AfricaMS 2008~
SCIENCE AND TECHNOLOGY INNOVATION FOR SUSTAINABLE DEVELOPMENT
September 8 – 10, 2008
Gaborone, Botswana
TUTORIAL SESSION
Bifurcation, Chaos and Stability Theories for High Technology: Modeling, Simulation and Experimental Investigations
Abstract
Following an historical overview of natural sciences development, the significant role of a physical pendulum (and its various modifications), which is a very useful mechanism in the modeling and design of various real processes, is observed.
The nonlinear dynamics of a real plane and periodically forced triple pendulum is investigated experimentally and numerically. The mathematical modeling includes details taking into account some characteristic features (for example real characteristics of joints built by the use of roller bearings) as well as some imperfections (asymmetry of the forcing) of the real system. Parameters of the model are obtained by a combination of the estimation from experimental data and direct measurements of the geometric and physical system parameters. A few versions of the model of resistance in the joints are tested in the identification process. Good agreement between both numerical simulation results and experimental measurements are obtained. Some novel features of the bifurcation and chaotic behavior of the investigated system are also reported, and a novel approach to rolling bearings friction modeling is proposed, among other results.
The introduction of motion limiters (barriers) to the pendulum dynamics allows the study of hybrid pendulum behavior including interaction with the barriers (impacts and/or sliding). In addition, the studied triple inverted pendulum can be used to model a piston-connecting rod-crankshaft system of a mono-cylinder combustion engine showing the well-known six stages of the piston along the cylinder per one engine cycle. The proposed methodology consisting of three fundamental steps including modeling, simulation and experimental investigation may serve as an engineering kit useful for solving a variety of challenging problems occurred in mechanical, civil and mechatronical high technology engineering.
Background knowledge expected of the participants
Participants should possess fundamental knowledge of mathematics, modeling, numerical simulation, and a basic mechatronics background.
Objectives
The objective of the proposed tutorial is to introduce up-to-date achievements to the audience regarding smooth and non-smooth nonlinear dynamical systems governed by ordinary differential and algebraic equations, putting emphasis on modeling of impact and friction phenomena. In addition, it is shown how the introduced theoretical background can be used in the modeling of various lumped mechanical systems and in particular of a four-stroke mono-cylinder combustion engine. The originally constructed mechatronical triple pendulum system (chaos generator) is presented and described from an engineering point of view, and its rich bifurcation and chaotic dynamics is reported including two movies. Finally, modeling and simulation results are compared with those measured via the LabView system being applied to the real experimental rig. In spite of the application-oriented tutorial purpose, a few interesting and not widely known nonlinear phenomena are illustrated and discussed.
This tutorial covers the following fundamental topics:
- modeling of general mechanical systems with rigid limiters of motion;
- numerical schemes for simulation;
- stability and corresponding numerical algorithms;
- modeling of a triple physical pendulum with rigid barriers;
- Lyapunov exponents computation;
- stability of periodic and quasi-periodic orbits;
- smooth and non-smooth bifurcations;
- modeling and analysis of the piston-connecting rod-crankshaft (mono-cylinder combustion) engine;
- experimental rig;
- model parameters identification;
- numerical and experimental analysis.
Time allocations for the major course topics
- Background of Bifurcation, Chaos, Stability, and Applied Numerical Algorithms: 45 min
- Questions/Break: 15 min
- Modeling of the triple physical pendulum and the piston-connecting rod-crankshaft system and simulation results: 45 min
- Questions/Break: 15 min
- Experimental rig presentation and comparison of simulation and measurement results
- Questions/Wrap-up: 15 min
Qualifications of the Instructor
Dr. Jan Awrejcewicz holds an M.S. (1977), Ph.D. (1981), D.Sci. (1990), Prof. (1994), and Full Prof. (1997) degrees from the Mechanical Engineering Faculty of the Technical University of Lódz, Poland. He is a founder and head of the Department of Automatics and Biomechanics, as well as a founder and chairperson of the Ph.D. Schools on Mechanics. He is an author and/or a co-author of 34 monographs, 4 textbooks, 10 conference proceedings, 11 journal special issues, and over 460 purely reviewed papers in international journals, conferences and meetings.
He spent over 10 years abroad conducting research supported by The Fulbright Foundation (University of California, Berkeley), The Alexander von Humboldt Foundation (Technical University of Braunschweig, Germany), The Kosciuszko Foundation (University of Illinois, USA), The Research Centre for Advanced Science and Technology and The Japanese Society for Promotion of Science (Tokyo University, Japan), among others. His research is mainly focused on asymptotic methods, nonlinear dynamics of lumped continuous and discontinuous systems, nonlinear structural mechanics, engineering biomechanics, and recently mechatronics. He is a supervisor of 13 Ph.D. theses, a reviewer of numerous papers and books, a member of a variety of conferences, a member of editorial boards for numerous journals, and a recipient of a few prestigious awards for his scientific activity (see http://www.p.lodz.pl/k16 for more details).
