Deterministic Chaos In One Dimensional Continuous Systems

This book focuses on the computational analysis of nonlinear vibrations of structural members (beams, plates, panels, shells), where the studied dynamical problems can be reduced to the consideration of one spatial variable and time. The reduction is carried out based on a formal mathematical approach aimed at reducing the problems with infinite dimension to finite ones. The process also includes a transition from governing nonlinear partial differential equations to a set of finite number of ordinary differential equations. Beginning with an overview of the recent results devoted to the analysis and control of nonlinear dynamics of structural members, placing emphasis on stability, buckling, bifurcation and deterministic chaos, simple chaotic systems are briefly discussed. Next, bifurcation and chaotic dynamics of the Euler–Bernoulli and Timoshenko beams including the geometric and physical nonlinearity as well as the elastic–plastic deformations are illustrated. Despite the employed classical numerical analysis of nonlinear phenomena, the various wavelet transforms and the four Lyapunov exponents are used to detect, monitor and possibly control chaos, hyper-chaos, hyper-hyper-chaos and deep chaos exhibited by rectangular plate-strips and cylindrical panels. The book is intended for post-graduate and doctoral students, applied mathematicians, physicists, teachers and lecturers of universities and companies dealing with a nonlinear dynamical system, as well as theoretically inclined engineers of mechanical and civil engineering.Contents: Bifurcational and Chaotic Dynamics of Simple Structural Members: Beams Plates Panels Shells Introduction to Fractal Dynamics: Cantor Set and Cantor Dust Koch Snowflake 1D Maps Sharkovsky's Theorem Julia Set Mandelbrot's Set Introduction to Chaos and Wavelets: Routes to Chaos Quantifying Chaotic Dynamics Simple Chaotic Models: Introduction Autonomous Systems Non-Autonomous Systems Discrete and Continuous Dissipative Systems: Introduction Linear Friction Nonlinear Friction Hysteretic Friction Impact Damping Damping in Continuous 1D Systems Euler-Bernoulli Beams: Introduction Planar Beams Linear Planar Beams and Stationary Temperature Fields Curvilinear Planar Beams and Stationary Temperature and Electrical Fields Beams with Elasto-Plastic Deformations Multi-Layer Beams Timoshenko and Sheremetev-Pelekh Beams: The Timoshenko Beams The Sheremetev-Pelekh Beams Concluding Remarks Panels: Infinite Length Panels Cylindrical Panels of Infinite Length Plates and Shells: Plates with Initial Imperfections Flexible Axially-Symmetric Shells Readership: Post-graduate and doctoral students, applied mathematicians, physicists, mechanical and civil engineers. Bifurcation;Chaos;Structural Members;PDEs and ODEs;Lyapunov Exponents;WaveletsKey Features: Includes fascinating and rich dynamics exhibited by simple structural members and by the solution properties of the governing 1D non-linear PDEs, suitable for applied mathematicians and physicists Covers a wide variety of the studied PDEs, their validated reduction to ODEs, classical and non-classical methods of analysis, influence of various boundary conditions and control parameters, as well as the illustrative presentation of the obtained results in the form of colour 2D and 3D figures and vibration type charts and scales Contains originally discovered, illustrated and discussed novel and/or modified classical scenarios of transition from regular to chaotic dynamics exhibited by 1D structural members, showing a way to control chaotic and bifurcational dynamics, with directions to study other dynamical systems modeled by chains of nonlinear oscillators