Bifurcation and Chaos in Nonsmooth Mechanical Systems.

By: Awrejcewicz, JanContributor(s): Lamarque, Claude-HenriSeries: World Scientific Series on Nonlinear Science Series A SerPublisher: Singapore : World Scientific Publishing Co Pte Ltd, 2003Copyright date: ©2003Description: 1 online resource (564 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789812564801Subject(s): Bifurcation theory.;Chaotic behavior in systems.;Differential equations, NonlinearGenre/Form: Electronic books. Additional physical formats: Print version:: Bifurcation and Chaos in Nonsmooth Mechanical SystemsDDC classification: 515 LOC classification: QA380.A9 2003Online resources: Click to View
Contents:
Intro -- Preface -- Contents -- Chapter 1 Introduction to Discontinuous ODEs -- 1.1 Introduction -- 1.2 Filippov's Theory -- 1.3 Aizerman's Theory -- 1.4 Examples -- 1.4.1 Example 1 -- 1.4.2 Example 2 -- 1.5 Boundary Value Problem -- Chapter 2 Mathematical Background for Multivalued Formulations -- 2.1 Origin of Nonlinearities -- 2.2 Smooth and Nonsmooth Nonlinearities -- 2.3 Examples and Dynamical Equilibria -- 2.3.1 Models with friction -- 2.3.2 Another simple example with impact -- 2.4 Existence and Uniqueness -- 2.4.1 The frame of maximal monotone operators -- 2.4.2 Ill-posed problems -- 2.5 Stochastic Frame -- Chapter 3 Numerical Schemes and Analytical Methods -- 3.1 Numerical Schemes -- 3.1.1 Deterministic cases -- 3.1.2 Stochastic case -- 3.2 Analytical Methods -- 3.2.1 Simple case -- 3.2.2 Second case -- Chapter 4 Properties of Numerical Schemes -- 4.1 Dynamics of Systems with Friction or Elastoplastic Terms -- 4.2 Systems with Impacts -- 4.2.1 Introduction -- 4.2.2 Dynamical system and resolution methods -- 4.2.3 Sticking motion and accumulation of impacts -- 4.2.4 Behaviour of the numerical methods -- 4.2.5 Numerical results -- 4.2.6 Computing times -- 4.3 Conclusion -- Chapter 5 Bifurcations of a Particular van der Pol-Duffing Oscillator -- 5.1 The Analysed System and the Averaged Equations -- 5.2 "0" Type Bifurcations -- 5.3 Complex Bifurcations -- 5.4 Observations of Strange Attractors Using Numerical Simulations -- Chapter 6 Stick-Slip Oscillator with Two Degrees of Freedom -- 6.1 Introduction -- 6.2 Disc - Flexible Arm Oscillator -- 6.2.1 Equations of motion and phase flow -- 6.2.2 Trivial solutions - analytical investigations -- 6.2.3 Discussion of the analytical results -- 6.2.4 Stability of equilibria. Numerical investigations -- 6.2.5 The integration of the equations of motion.
6.2.6 Calculations of periodic orbits and their stability -- 6.2.7 Evolution of periodic orbits -- 6.2.8 Observations of chaos -- 6.3 Two Horizontally Situated Masses -- 6.3.1 An overview of the methods of analysis -- 6.3.2 Numerical analysis and results -- 6.3.3 Concluding remarks -- Chapter 7 Piecewise Linear Approximations -- 7.1 Introduction -- 7.2 Exact and Approximated Models -- 7.2.1 Exact model -- 7.2.2 Approximated models -- 7.3 Approximation and Global Dynamic Behavior -- 7.4 Numerical Results -- 7.4.1 Numerical method -- 7.4.2 Periodic solutions -- 7.4.3 Basins of attraction -- 7.5 Conclusion -- Chapter 8 Chua's Circuit with Discontinuities -- 8.1 Introduction -- 8.2 Mechanical Realizations of Chua's Circuit -- 8.2.1 Introduction -- 8.2.2 Mechanical models of Chua's circuit -- 8.2.3 Concluding remarks -- 8.3 Generalized Double Scroll Chua's Circuit -- 8.3.1 Introduction -- 8.3.2 Mechanical point of view -- 8.3.3 Existence and uniqueness of solutions -- 8.3.4 Analytical calculation of the solution -- 8.3.5 Numerical results -- 8.3.6 Conclusion -- Chapter 9 Mechanical System with Impacts and Modal Approaches -- 9.1 Introduction -- 9.2 Single Degree of Freedom System -- 9.2.1 Analytical solution -- 9.2.2 Periodic solutions -- 9.2.3 Modal superposition -- 9.3 Two Degrees of Freedom Systems -- 9.3.1 Weak coupling -- 9.3.2 Strong coupling -- 9.3.4 Two colliding rigid bodies -- 9.4 Conclusion -- Chapter 10 One DOF Mechanical System with Friction -- 10.1 Introduction -- 10.2 Modelling the Pendulum with Friction -- 10.2.1 Existence and uniqueness -- 10.2.2 Numerical scheme -- 10.2.3 Numerical estimation of order -- 10.3 Numerical Results -- 10.3.1 Oscillations of the free pendulum -- 10.3.2 Global behavior -- 10.3.3 Lyapunov exponents -- 10.4 The Melnikov Analysis -- 10.5 Conclusion.
Chapter 11 Modelling the Dynamical Behaviour of Elasto-Plastic Systems -- 11.1 Rheological Systems with "Friction -- 11.1.1 Introduction -- 11.1.2 Existence and uniqueness results -- 11.1.3 Numerical simulations -- 11.1.4 Conclusion -- Chapter 12 A Mechanical System with 7 DOF -- 12.1 Mathematical Model -- 12.1.1 Final equations with finite k3 -- 12.1.2 Introduction of new coordinates -- 12.1.3 Numerical scheme -- 12.2 Numerical Results and Comments for Finite k3 -- Chapter 13 Stability of Singular Periodic Motions in Single Degree of Freedom Vibro-Impact Oscillators and Grazing… -- 13.1 Introduction -- 13.2 Mechanical System and Change of Coordinates -- 13.3 Local Expansion of the Poincaré Map -- 13.3.1 General case (ß1 # 0) -- 13.3.2 Particular case (ß1 = 0) -- 13.4 Stability of the Nondifferentiable Fixed Point -- 13.4.1 General case (ß1 # 0) -- 13.4.2 Particular case (ß1 = 0) -- 13.5 Applications -- 13.5.1 Linear harmonic oscillator -- 13.5.2 Forced damped pendulum -- 13.6 Conclusion -- Chapter 14 Triple Pendulum with Impacts -- 14.1 Introduction -- 14.2 Investigated Pendulum and Governing Equations (Without Impacts) -- 14.3 Introduction of the Obstacles -- 14.4 Calculation of the Fundamental Solution Matrices for Dynamical Systems with Impacts -- 14.5 Simplification of the System -- 14.6 The Method Used for Integration of the System and its Accuracy -- 14.7 Numerical Examples -- 14.8 Concluding Remarks -- Chapter 15 Analytical Prediction of Stick-Slip Chaos -- 15.1 Introduction -- 15.2 The Melnikov's Method -- 15.3 Analyzed System -- 15.4 Analytical Results -- 15.4.1 Numerical results -- Chapter 16 Thermoelasticity, Wear and Stick-Slip Movements of a Rotating Shaft with a Rigid Bush -- 16.1 Introduction -- 16.1.1 Statement of the problem -- 16.1.2 Solution of the problem -- 16.1.3 Steady-state solution analysis.
16.1.4 Analysis of steady-state solution in the presence of wear (kz # 0) -- 16.1.5 Numerical analysis of the transient solution -- Chapter 17 Control for Discrete Models of Buildings Including Elastoplastic Terms -- 17.1 Introduction -- 17.2 Reminder about Prandtl Rheological Model -- 17.3 The Studied Models with n DOF -- 17.4 Existence and Uniqueness Results -- 17.4.1 Reminder about maximal monotone graphs and ß -- 17.4.2 Mathematical study of a differential system -- 17.5 Numerical Scheme -- 17.6 Control Procedure -- 17.7 Algorithm of Control -- 17.7.1 Improvement of the control -- 17.7.2 Riccati equation -- 17.8 Numerical Results for a System with 3 DOF -- 17.8.1 System with 3 DOF under stochastic loading -- 17.9 Extension to Nonlinear Cases -- 17.9.1 System with 3 DOF under stochastic loading -- 17.10 Conclusion -- Bibliography -- Index.
Summary: This book presents the theoretical frame for studying lumped nonsmoothdynamical systems: the mathematical methods are recalled, and adaptednumerical methods are introduced (differential inclusions, maximalmonotone operators, Filippov theory, Aizerman theory, etc.
Holdings
Item type Current library Call number Status Date due Barcode Item holds
Ebrary Ebrary Afghanistan
Available EBKAF0008559
Ebrary Ebrary Algeria
Available
Ebrary Ebrary Cyprus
Available
Ebrary Ebrary Egypt
Available
Ebrary Ebrary Libya
Available
Ebrary Ebrary Morocco
Available
Ebrary Ebrary Nepal
Available EBKNP0008559
Ebrary Ebrary Sudan

Access a wide range of magazines and books using Pressreader and Ebook central.

Enjoy your reading, British Council Sudan.

Available
Ebrary Ebrary Tunisia
Available
Total holds: 0

Intro -- Preface -- Contents -- Chapter 1 Introduction to Discontinuous ODEs -- 1.1 Introduction -- 1.2 Filippov's Theory -- 1.3 Aizerman's Theory -- 1.4 Examples -- 1.4.1 Example 1 -- 1.4.2 Example 2 -- 1.5 Boundary Value Problem -- Chapter 2 Mathematical Background for Multivalued Formulations -- 2.1 Origin of Nonlinearities -- 2.2 Smooth and Nonsmooth Nonlinearities -- 2.3 Examples and Dynamical Equilibria -- 2.3.1 Models with friction -- 2.3.2 Another simple example with impact -- 2.4 Existence and Uniqueness -- 2.4.1 The frame of maximal monotone operators -- 2.4.2 Ill-posed problems -- 2.5 Stochastic Frame -- Chapter 3 Numerical Schemes and Analytical Methods -- 3.1 Numerical Schemes -- 3.1.1 Deterministic cases -- 3.1.2 Stochastic case -- 3.2 Analytical Methods -- 3.2.1 Simple case -- 3.2.2 Second case -- Chapter 4 Properties of Numerical Schemes -- 4.1 Dynamics of Systems with Friction or Elastoplastic Terms -- 4.2 Systems with Impacts -- 4.2.1 Introduction -- 4.2.2 Dynamical system and resolution methods -- 4.2.3 Sticking motion and accumulation of impacts -- 4.2.4 Behaviour of the numerical methods -- 4.2.5 Numerical results -- 4.2.6 Computing times -- 4.3 Conclusion -- Chapter 5 Bifurcations of a Particular van der Pol-Duffing Oscillator -- 5.1 The Analysed System and the Averaged Equations -- 5.2 "0" Type Bifurcations -- 5.3 Complex Bifurcations -- 5.4 Observations of Strange Attractors Using Numerical Simulations -- Chapter 6 Stick-Slip Oscillator with Two Degrees of Freedom -- 6.1 Introduction -- 6.2 Disc - Flexible Arm Oscillator -- 6.2.1 Equations of motion and phase flow -- 6.2.2 Trivial solutions - analytical investigations -- 6.2.3 Discussion of the analytical results -- 6.2.4 Stability of equilibria. Numerical investigations -- 6.2.5 The integration of the equations of motion.

6.2.6 Calculations of periodic orbits and their stability -- 6.2.7 Evolution of periodic orbits -- 6.2.8 Observations of chaos -- 6.3 Two Horizontally Situated Masses -- 6.3.1 An overview of the methods of analysis -- 6.3.2 Numerical analysis and results -- 6.3.3 Concluding remarks -- Chapter 7 Piecewise Linear Approximations -- 7.1 Introduction -- 7.2 Exact and Approximated Models -- 7.2.1 Exact model -- 7.2.2 Approximated models -- 7.3 Approximation and Global Dynamic Behavior -- 7.4 Numerical Results -- 7.4.1 Numerical method -- 7.4.2 Periodic solutions -- 7.4.3 Basins of attraction -- 7.5 Conclusion -- Chapter 8 Chua's Circuit with Discontinuities -- 8.1 Introduction -- 8.2 Mechanical Realizations of Chua's Circuit -- 8.2.1 Introduction -- 8.2.2 Mechanical models of Chua's circuit -- 8.2.3 Concluding remarks -- 8.3 Generalized Double Scroll Chua's Circuit -- 8.3.1 Introduction -- 8.3.2 Mechanical point of view -- 8.3.3 Existence and uniqueness of solutions -- 8.3.4 Analytical calculation of the solution -- 8.3.5 Numerical results -- 8.3.6 Conclusion -- Chapter 9 Mechanical System with Impacts and Modal Approaches -- 9.1 Introduction -- 9.2 Single Degree of Freedom System -- 9.2.1 Analytical solution -- 9.2.2 Periodic solutions -- 9.2.3 Modal superposition -- 9.3 Two Degrees of Freedom Systems -- 9.3.1 Weak coupling -- 9.3.2 Strong coupling -- 9.3.4 Two colliding rigid bodies -- 9.4 Conclusion -- Chapter 10 One DOF Mechanical System with Friction -- 10.1 Introduction -- 10.2 Modelling the Pendulum with Friction -- 10.2.1 Existence and uniqueness -- 10.2.2 Numerical scheme -- 10.2.3 Numerical estimation of order -- 10.3 Numerical Results -- 10.3.1 Oscillations of the free pendulum -- 10.3.2 Global behavior -- 10.3.3 Lyapunov exponents -- 10.4 The Melnikov Analysis -- 10.5 Conclusion.

Chapter 11 Modelling the Dynamical Behaviour of Elasto-Plastic Systems -- 11.1 Rheological Systems with "Friction -- 11.1.1 Introduction -- 11.1.2 Existence and uniqueness results -- 11.1.3 Numerical simulations -- 11.1.4 Conclusion -- Chapter 12 A Mechanical System with 7 DOF -- 12.1 Mathematical Model -- 12.1.1 Final equations with finite k3 -- 12.1.2 Introduction of new coordinates -- 12.1.3 Numerical scheme -- 12.2 Numerical Results and Comments for Finite k3 -- Chapter 13 Stability of Singular Periodic Motions in Single Degree of Freedom Vibro-Impact Oscillators and Grazing… -- 13.1 Introduction -- 13.2 Mechanical System and Change of Coordinates -- 13.3 Local Expansion of the Poincaré Map -- 13.3.1 General case (ß1 # 0) -- 13.3.2 Particular case (ß1 = 0) -- 13.4 Stability of the Nondifferentiable Fixed Point -- 13.4.1 General case (ß1 # 0) -- 13.4.2 Particular case (ß1 = 0) -- 13.5 Applications -- 13.5.1 Linear harmonic oscillator -- 13.5.2 Forced damped pendulum -- 13.6 Conclusion -- Chapter 14 Triple Pendulum with Impacts -- 14.1 Introduction -- 14.2 Investigated Pendulum and Governing Equations (Without Impacts) -- 14.3 Introduction of the Obstacles -- 14.4 Calculation of the Fundamental Solution Matrices for Dynamical Systems with Impacts -- 14.5 Simplification of the System -- 14.6 The Method Used for Integration of the System and its Accuracy -- 14.7 Numerical Examples -- 14.8 Concluding Remarks -- Chapter 15 Analytical Prediction of Stick-Slip Chaos -- 15.1 Introduction -- 15.2 The Melnikov's Method -- 15.3 Analyzed System -- 15.4 Analytical Results -- 15.4.1 Numerical results -- Chapter 16 Thermoelasticity, Wear and Stick-Slip Movements of a Rotating Shaft with a Rigid Bush -- 16.1 Introduction -- 16.1.1 Statement of the problem -- 16.1.2 Solution of the problem -- 16.1.3 Steady-state solution analysis.

16.1.4 Analysis of steady-state solution in the presence of wear (kz # 0) -- 16.1.5 Numerical analysis of the transient solution -- Chapter 17 Control for Discrete Models of Buildings Including Elastoplastic Terms -- 17.1 Introduction -- 17.2 Reminder about Prandtl Rheological Model -- 17.3 The Studied Models with n DOF -- 17.4 Existence and Uniqueness Results -- 17.4.1 Reminder about maximal monotone graphs and ß -- 17.4.2 Mathematical study of a differential system -- 17.5 Numerical Scheme -- 17.6 Control Procedure -- 17.7 Algorithm of Control -- 17.7.1 Improvement of the control -- 17.7.2 Riccati equation -- 17.8 Numerical Results for a System with 3 DOF -- 17.8.1 System with 3 DOF under stochastic loading -- 17.9 Extension to Nonlinear Cases -- 17.9.1 System with 3 DOF under stochastic loading -- 17.10 Conclusion -- Bibliography -- Index.

This book presents the theoretical frame for studying lumped nonsmoothdynamical systems: the mathematical methods are recalled, and adaptednumerical methods are introduced (differential inclusions, maximalmonotone operators, Filippov theory, Aizerman theory, etc.

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2019. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

There are no comments on this title.

to post a comment.