Chaotic Vibrations

A revision of a professional text on the phenomena of chaotic vibrations in fluids and solids. Major changes reflect the latest developments in this fast-moving topic, the introduction of problems to every chapter, additional mathematics and applications, more coverage of fractals, numerous computer and physical experiments. Contains eight pages of 4-color pictures.

Covering the whole spectrum of vibration theory and itsapplications in both civil and mechanical engineering, Mechanicaland Structural Vibrations provides the most comprehensive treatmentof the subject currently available. Based on the author s manyyears of experience in both academe and industry, it is designed tofunction equally well as both a day-to-day working resource forpracticing engineers and a superior upper-level undergraduate orgraduate-level text. Features a quick-reference format that, Mechanical and StructuralVibrations gives engineers instant access to the specific theory orapplication they need. Saves valuable time ordinarily spent wadingthrough unrelated or extraneous material. And, while they arethoroughly integrated throughout the text, applications to bothcivil and mechanical engineering are organized into sections thatpermit the reader to reference only the material germane to his orher field. Students and teachers will appreciate the book's practical,real-world approach to the subject, its emphasis on simplicity andaccuracy of analytical techniques, and its straightforward,step-by-step delineation of all numerical methods used incalculating the dynamics and vibrations problems, as well as thenumerous examples with which the author illustrates those methods.They will also appreciate the many chapter-end practice problems(solutions appear in appendices) designed to help them rapidlydevelop mastery of all concepts and methods covered. Readers will find many versatile new concepts and analyticaltechniques not covered in other texts, including nonlinearanalysis, inelastic response of structural and mechanicalcomponents of uniform and variable stiffness, the "dynamic hinge,""dynamically equivalent systems," and other breakthrough tools andtechniques developed by the author and his collaborators. Mechanical and Structural Vibrations is both an excellent text forcourses in structural dynamics, dynamic systems, and engineeringvibration and a valuable tool of the trade for practicing engineersworking in a broad range of industries, from electronic packagingto aerospace. Timely, comprehensive, practical--a superior student text and anindispensable working resource for busy engineers Mechanical and Structural Vibrations is the first text to cover theentire spectrum of vibration theory and its applications in bothcivil and mechanical engineering. Written by an author with over aquarter century of experience as a teacher and practicing engineer,it is designed to function equally well as a working professionalresource and an upper-level undergraduate or graduate-level textfor courses in structural dynamics, dynamic systems, andengineering vibrations. Mechanical and Structural Vibrations: * Takes a practical, application-oriented approach to the subject * Features a quick-reference format that gives busy professionalsinstant access to the information needed for the task at hand * Walks readers, step-by-step, through the numerical methods usedin calculating the dynamics and vibration problems * Introduces many cutting-edge concepts and analytical tools notcovered in other texts * Is packed with real-world examples covering everything from thestresses and strains on buildings during an earthquake to thoseaffecting a space craft during lift-off * Contains chapter-end problems--and solutions--that help studentsrapidly develop mastery of all important concepts and methodscovered * Is extremely well-illustrated and includes more than 300diagrams, tables, charts, illustrations, and more

An ideal text for students that ties together classical and modern topics of advanced vibration analysis in an interesting and lucid manner. It provides students with a background in elementary vibrations with the tools necessary for understanding and analyzing more complex dynamical phenomena that can be encountered in engineering and scientific practice. It progresses steadily from linear vibration theory over various levels of nonlinearity to bifurcation analysis, global dynamics and chaotic vibrations. It trains the student to analyze simple models, recognize nonlinear phenomena and work with advanced tools such as perturbation analysis and bifurcation analysis. Explaining theory in terms of relevant examples from real systems, this book is user-friendly and meets the increasing interest in non-linear dynamics in mechanical/structural engineering and applied mathematics and physics. This edition includes a new chapter on the useful effects of fast vibrations and many new exercise problems.

This volume introduces new approaches to modeling strongly nonlinear behaviour of structural mechanical units: beams, plates and shells or composite systems. The text draws on bifurcation theory and chaos, emphasizing control and stability of objects and systems.

This book consists of lecture notes for a semester-long introductory graduate course on dynamical systems and chaos taught by the authors at Texas A&M University and Zhongshan University, China. There are ten chapters in the main body of the book, covering an elementary theory of chaotic maps in finite-dimensional spaces. The topics include one-dimensional dynamical systems (interval maps), bifurcations, general topological, symbolic dynamical systems, fractals and a class of infinite-dimensional dynamical systems which are induced by interval maps, plus rapid fluctuations of chaotic maps as a new viewpoint developed by the authors in recent years. Two appendices are also provided in order to ease the transitions for the readership from discrete-time dynamical systems to continuous-time dynamical systems, governed by ordinary and partial differential equations. Table of Contents: Simple Interval Maps and Their Iterations / Total Variations of Iterates of Maps / Ordering among Periods: The Sharkovski Theorem / Bifurcation Theorems for Maps / Homoclinicity. Lyapunoff Exponents / Symbolic Dynamics, Conjugacy and Shift Invariant Sets / The Smale Horseshoe / Fractals / Rapid Fluctuations of Chaotic Maps on RN / Infinite-dimensional Systems Induced by Continuous-Time Difference Equations

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:BeamsPlatesPanelsShellsIntroduction to Fractal Dynamics:Cantor Set and Cantor DustKoch Snowflake1D MapsSharkovsky's TheoremJulia SetMandelbrot's SetIntroduction to Chaos and Wavelets:Routes to ChaosQuantifying Chaotic DynamicsSimple Chaotic Models:IntroductionAutonomous SystemsNon-Autonomous SystemsDiscrete and Continuous Dissipative Systems:IntroductionLinear FrictionNonlinear FrictionHysteretic FrictionImpact DampingDamping in Continuous 1D SystemsEuler-Bernoulli Beams:IntroductionPlanar BeamsLinear Planar Beams and Stationary Temperature FieldsCurvilinear Planar Beams and Stationary Temperature and Electrical FieldsBeams with Elasto-Plastic DeformationsMulti-Layer BeamsTimoshenko and Sheremetev-Pelekh Beams:The Timoshenko BeamsThe Sheremetev-Pelekh BeamsConcluding RemarksPanels:Infinite Length PanelsCylindrical Panels of Infinite LengthPlates and Shells:Plates with Initial ImperfectionsFlexible Axially-Symmetric Shells Readership: Post-graduate and doctoral students, applied mathematicians, physicists, mechanical and civil engineers. Key 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 physicistsCovers 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 scalesContains 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