Mechanical systems may be considered as an example of nonlinear science, since the nonlinear effects can arise from a number of sources, such as geometric nonlinearities, nonlinear body forces, constitutive relations, kinematics or boundary conditions. This book presents the general methods of investigation of chaotic behaviour such as Lyapunov exponents, Melnikov method and Poincare maps. These methods are then applied to nonlinear mechanical systems where the chaotic oscillations are present. Throughout the book the emphasis is on the equal importance of both mathematical preciseness as well as mechanical systems applications. The book is intended to be of interest to mathematicians who are interested in applications as well as for mechanical engineers with an interest in the theory of oscillations.
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Höhe: 240 mm
Breite: 157 mm
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ISBN-13
978-0-471-93524-7 (9780471935247)
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Schweitzer Klassifikation
Part 1 Introduction: chaotic and stochastic processes; Poincare map. Part 2 Attractor: definitions; typical attractors of n < 5 dimensional dissipative systems; example of attractors of 5-dimensional system; reconstruction of attractor from time series. Part 3 Determination of chaotic behaviour: definition of Lyapunov exponents; calculation of Lyapunov exponents for explicity known equations of motion; estimation of Lyapunov exponents from time series; autoregressive method for estimation of Lyapuniv exponents; dimension of attractor; Melnikov method; weak damping; power spectrums of near homoclinic orbit; large damping; almost periodic excitation; general remarks on excitation. Part 4 Routes to chaos: period doubling; intermitent transition; break of torus. Part 5 Duffing's equation: nonlinear stiffness; harmonic balance method and periodic solutions; Ueda's Japanese attractor; the equation of a buckled beam; mechanical examples; Poincare map; approximate criteria for aperiodic behaviour; necessary and sufficient condition for chaos; systems with time delay; quasiperiodic excitation; necessary condition for chaos; combined bifurcation; numerical examples; map on the interval and Duffing's oscillator; analytical condition for chaos by period doubling route. Part 6 Oscillators with limit cycles - Van der Pol's equation: mechanical example of a system with a limit cycle; existence of limit cycles; routes to chaos; transient states of chaotic attractor. Part 7 Pendulum: dynamics of pendulum; necessary condition for chaotic behaviour; symmetry breaking and chaotic behaviour; the map of a circle and a pendulum. Part 8 Other chaotic oscillators: chaotic behaviour of rotor system; Freud's pendulum; chaos generated by the cutting process; oscillations of shells and arches; oscillators with dry fraction; piece-wise linear oscillations; impact oscillators; coupled anharmonic oscillators; stability of the system forced by chaotic force. Part 9 Strange nonchaotic attractors: definition; main properties and example; quasiperiodic excitation; periodic excitation. Part 10 Fractial basin boundaries: definition; main properties; examples of fractal basin boundaries; spatiotemporal dynamics. Appendices: stability by the first approximation and Hurwitz criterion; stable manifold theorem for fixed point; averaging; Hopf bifurcation.
