Chaos in Structural Mechanics (Understanding Complex Systems)

✍ Scribed by Jan Awrejcewicz, Vadim Anatolevich Krys'ko

• Publisher: Springer
• Year: 2008
• Tongue: English
• Leaves: 424
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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 TheoryWrinkle-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