Chaos in Structural Mechanics (Understanding Complex Systems)

This monograph is devoted to construction of novel theoretical approaches of m- eling non-homogeneous structural members as well as to development of new and economically ef?cient (simultaneously keeping the required high engineering ac- racy)computationalalgorithmsofnonlineardynamics(statics)ofstronglynonlinear behavior of either purely continuous mechanical objects (beams, plates, shells) or hybrid continuous/lumped interacting mechanical systems. In general, the results presented in this monograph cannot be found in the - isting literature even with the published papers of the authors and their coauthors. We take a challenging and originally developed approach based on the integrated mathematical-numerical treatment of various continuous and lumped/continuous mechanical structural members, putting emphasis on mathematical and physical modeling as well as on the carefully prepared and applied novel numerical - gorithms used to solve the derived nonlinear partial differential equations (PDEs) mainly via Bubnov-Galerkin type approaches. The presented material draws on the ?elds of bifurcation, chaos, control, and s- bility of the objects governed by strongly nonlinear PDEs and ordinary differential equations (ODEs), and may have a positive impact on interdisciplinary ?elds of n- linear mechanics, physics, and applied mathematics. We show, for the ?rst time in a book, the complexity and fascinating nonlinear behavior of continual mechanical objects, which cannot be found in widely reported bifurcational and chaotic dyn- ics of lumped mechanical systems, i. e., those governed by nonlinear ODEs. Introduction Theory of Non-homogeneous Shells Preliminary Remarks Fundamental Relations and Assumptions Non-homogeneity of a Shell Variational Equations Equations of Motion Boundary and Initial Conditions Non-dimensional Form of Equations Variable Parameters of Stiffness Flexural Stiffness Coefficient of a Shell Element Generalized Functions Static Instability of Rectangular Plates Fundamental Concepts of the Theory of Elastic Stability Two Fundamental Forms of the Energetic Criterion of Bifurcational Stability Loss Bubnov-Galerkin Methods Devoted to Shell Stability Investigations Subdomains Method Colocation Method Least-Squares Method Method of Moments Galerkin Method A Comparison of the Weighting Error Methods Relations to Other Methods Theoretical Properties Computational Advantages of Galerkin Methods Summary Bubnov-Galerkin Method of High-Order Approximations and the Numerical Algorithm Shells with Additions of Other Materials Static Stability of a Shell Central Square Element of Non-homogeneity Central Cross Addition of Non-homogeneity ``Perforation''-Type Non-homogeneity Vibrations of Rectangular Shells Linear and Weakly Nonlinear Vibrations of Mechanical Systems Natural Vibrations of Non-homogeneous Shells The Solution Method Description of Results Free Nonlinear Vibrations of Plates and Shells The Solution Method Spectral Analysis of Solutions Method Convergence Spectral Analysis of Free Vibrations Dynamic Loss of Stability of Rectangular Shells Types of Dynamic Buckling Perfect Constructions The Concept of Finite-time Stability Mathematical Models of Vibrating and Dynamic Systems Synchronization, Chaos, and Quasi-Periodicity Static Bifurcations and Catastrophe Theory ``Wrinkle-Type'' Catastrophe or a Limit Point A ``Fold-Type'' Catastrophe or Symmetric Bifurcation Dynamic Bifurcations Criteria for Practical Computations Stability Loss of Homogeneous Shells under Transverse Loads Feasibility of the Obtained Results Buckling Load and Parameter kx=ky of a Homogeneous Shell Stability Loss of Heterogeneous Shells Under Transverse Load Relation Between Buckling Load and the Surfaceof an Extra Element Relation Between the Buckling Load and Stiffness Coefficient of an Extra Element Relation Between Buckling Load and the Numberof Reinforcement Elements Situated Along One Sideof a Shell Relation Between Buckling Load and the Width of a Rib (Cross-Type Heterogeneity, Fig. 2.8b) Stability of a Closed Cylindrical Shell Subjected to an Axially Non-symmetrical Load Equations of Motion The Influence of Imperfection on the Stability of Shells The Load Resulting from a Wind-Type Flow The Problem of Statics Dynamics Composite Shells Equations Static Stability of Composite Shells Three-Layered Shell Dynamic Stability Interaction of Elastic Shells and a Moving Body Vibration of Construction and Moving Lumped Body (One-Sided Constraint Case) Moving Load Equations Non-dimensional Form of Lumped Body Equations Boundary and Initial Problem for a Shell Shell Rise Shell Vibrations with Two-Sided Moving LumpedBody Constraints Shell Subjected to Transversal Rigid Body Impact Shells with Constant Velocity Moving Load Shell and Load Moving with Constant Acceleration Shell and Load Moving with ConstantNegative Acceleration Conclusions Chaotic Vibrations of Sectorial Shells Introduction Statement of the Problem Static Problems and Reliability of Results Convergence of a Finite Difference Method Investigation of Chaotic Vibrations of Spherical Sector-Type Shells Boundary Conditions The Influence of Sector Angle Vibrations of Sector-Type Shells Versus Sloping Parameter Transitions from Harmonic to Chaotic Vibrations Control of Chaotic Vibrations of Flexible Spherical Sector-Type Shells Scenarios of Transition from Harmonic to Chaotic Motion Historical Background Landau-Hopf Scenario (LH) Scenario by Ruelle, Takens, and Newhouse Scenario by Feigenbaum Scenario by Pomeau-Manneville Synchronization of Frequencies Dynamics of Closed Flexible Cylindrical Shells Introduction Fundamental Equations Bubnov-Galerkin Method and Fourier Representation Static Problems of Closed Cylindrical Shell Theory Dynamics of Closed Cylindrical Shells Convergence of the Fourier Representation for a Non-stationary Problem Vibrations of Closed Cylindrical Shells Subjected to Transversal Sinusoidal Load Dependence of Vibration Character on Width of the Pressure Zone Dependence of Vibration Character on the Linear Shell Dimension Scenarios of Shell Vibration Transition into Chaos Versus l Feigenbaum Scenario The Ruelle-Takens-Feigenbaum Scenarios Conclusions Controlling Time-Spatial Chaos of Cylindrical Shells Introduction Mathematical Model Bubnov-Galerkin Method and Fourier Transformation Control of Chaos Conclusions Chaotic Vibrations of Flexible Rectangular Shells Fundamental Equations Bubnov-Galerkin Method with Higher Approximations Method of Finite Differences Comparison of Results Obtained Conclusions Determination of Three-layered Nonlinear Uncoupled Beam Dynamics with Constraints Introduction Fundamental Relations Formulation of the Problem and Computational Algorithm Structurally Nonlinear Problems Structurally and Physically Nonlinear Problems Special Case Conclusions Bifurcation and Chaos of Sandwich Beams Introduction Problem Formulation and Computational Algorithm Numerical Results All Three Beams are Linearly Elastic All Three Beams are Nonlinearly Elastic Conclusions Nonlinear Vibrations of the Euler-Bernoulli Beam Introduction Problem Formulation Finite Differences Method Influence of Damping Coefficients on the Frequency Characteristics Power Spectra Waves Generated by a Longitudinal Impact Conclusions Bibliography Index

this Volume Introduces And Reviews Novel Theoretical Approaches To Modeling Strongly Nonlinear Behaviour Of Either Individual Or Interacting Structural Mechanical Units Such As Beams, Plates And Shells Or Composite Systems Thereof.

the Approach Draws Upon The Well-established Fields Of Bifurcation Theory And Chaos And Emphasizes The Notion Of Control And Stability Of Objects And Systems The Evolution Of Which Is Governed By Nonlinear Ordinary And Partial Differential Equations. Computational Methods, In Particular The Bubnov-galerkin Method, Are Thus Described In Detail.

"This volume introduces and reviews novel theoretical approaches to modeling strongly nonlinear behaviour of either individual or interacting structural mechanical units such as beams, plates and shells or composite systems thereof." "The approach draws upon the well-established fields of bifurcation theory and chaos and emphasizes the notion of control and stability of objects and systems the evolution of which is governed by nonlinear ordinary and partial differential equations. Computational methods, in particular the Bubnov-Galerkin method, are thus described in detail"--Jacket