Elastic and Thermoelastic Problems in Nonlinear Dynamics of Structural Members: Applications of the Bubnov-Galerkin and Finite Difference Methods

Preface
Contents
1 Introduction
2 Coupled Thermoelasticity and Transonic Gas Flow
	2.1 Coupled Linear Thermoelasticity of Shallow Shells
		2.1.1 Fundamental Assumptions
		2.1.2 Differential Equations
		2.1.3 Boundary and Initial Conditions
		2.1.4 An Abstract Coupled Problem
		2.1.5 Existence and Uniqueness of Solutions of Thermoelasticity Problems
	2.2 Cylindrical Panel Within Transonic Gas Flow
		2.2.1 Statement and Solution of the Problem
			2.2.1.1 Equations of Motion and Boundary Conditions of a Panel
			2.2.1.2 Equations of Transonic Ideal-Gas Motion
			2.2.1.3 Equations and Boundary Conditions of Panel–Flow Interaction
			2.2.1.4 Analysis of Panel Oscillations
			2.2.1.5 Method of Solution of the Problem
			2.2.1.6 Interaction Between a Flow and a Panel
		2.2.2 Stable Vibrating Panel Within a Transonic Flow
			2.2.2.1 Shock Wave Motion
			2.2.2.2 Interaction of a Flow and a Panel
		2.2.3 Stability Loss of Panel Within Transonic Flow
			2.2.3.1 Stability Criteria
			2.2.3.2 Analysis of Stress–Strain State
3 Estimation of the Errors of the Bubnov–Galerkin Method
	3.1 An Abstract Coupled Problem
	3.2 Coupled Thermoelastic Problem Within the Kirchhoff–Love Model
	3.3 Case of a Simply Supported Plate Within the Kirchhoff Model
	3.4 Coupled Problem of Thermoelasticity Within a Timoshenko-Type Model
4 Numerical Investigations of the Errors of the Bubnov–Galerkin Method
	4.1 Vibration of a Transversely Loaded Plate
	4.2 Vibration of a Plate with an Imperfection in the Form of a Deflection
	4.3 Vibration of a Plate with a Given Variable Deflection Change
5 Coupled Nonlinear Thermoelastic Problems
	5.1 Fundamental Relations and Assumptions
	5.2 Differential Equations
	5.3 Boundary and Initial Conditions
	5.4 On the Existence and Uniqueness of a Solution and on the Convergence of the Bubnov–Galerkin Method
6 Theory with Physical Nonlinearities and Coupling
	6.1 Fundamental Assumptions and Relations
	6.2 Variational Equations of Physically Nonlinear Coupled Problems
	6.3 Equations in Terms of Displacements
7 Nonlinear Problems of Hybrid-Form Equations
	7.1 Method of Solution for Nonlinear Coupled Problems
	7.2 Relaxation Method
	7.3 Numerical Investigations and Reliability of the Results Obtained
	7.4 Vibration of Isolated Shell Subjected to Impulse
	7.5 Dynamic Stability of Shells Under Thermal Shock
	7.6 Influence of Coupling and Rotational Inertia on Stability
	7.7 Numerical Tests
	7.8 Influence of Damping  and Excitation Amplitude A
	7.9 Spatial–Temporal Symmetric Chaos
	7.10 Dissipative Nonsymmetric Oscillations
	7.11 Solitary Waves
8 Dynamics of Thin Elasto-Plastic Shells
	8.1 Fundamental Relations
	8.2 Method of Solution
	8.3 Oscillations and Stability of Elasto-Plastic Shells
9 Mathematical Model of Cylindrical/Spherical Shell Vibrations
	9.1 Fundamental Relations and Assumptions
	9.2 The Bubnov-Galerkin Method
		9.2.1 Closed Cylindrical Shell
		9.2.2 Cylindrical Panel
	9.3 Reliability of the Obtained Results
	9.4 On the Set Up Method in Theory of Flexible Shallow Shells
	9.5 Dynamic Stability Loss of the Shells Under the Step-Type Function of Infinite Length
10 Scenarios of Transition from Periodic to Chaotic Shell Vibrations
	10.1 Novel Models of Scenarios of Transition from Periodic to Chaotic Orbits
	10.2 Sharkovsky's Periodicity Exhibited by PDEs Governing Dynamics of Flexible Shells
	10.3 On the Space-Temporal Chaos
11 Mathematical Models of Chaotic Vibrations of Closed Cylindrical Shells with Circle Cross Section
	11.1 On the Convergence of the Bubnov-Galerkin (BG) Method in the Vase of Chaotic Vibrations of Closed Cylindrical Shells
	11.2 Chaotic Vibrations of Closed Cylindrical Shells Versus Their Geometric Parameters and Area of the External Load Action
12 Chaotic Dynamics of Flexible Closed Cylindrical Nano-Shells Under Local Load
	12.1 Statement of the Problem
	12.2 Algorithm of the Bubnov–Galerkin Method
	12.3 Numerical Experiment
13 Contact Interaction of Two Rectangular Plates Made from Different Materials with an Account of Physical Non-Linearity
	13.1 Statement of the Problem
	13.2 Reduction of PDEs to ODEs
		13.2.1 The Method of Kantorovich-Vlasov (MKV)
		13.2.2 Method of Vaindiner (MV)
		13.2.3 Method of Variational Iteration (MVI)
		13.2.4 Method of Arganovskiy-Baglay-Smirnov (MABS)
		13.2.5 Combined Method (MC)
		13.2.6 Matching of the Methods of Kantorovich-Vlasov and Arganovskiy-Baglay-Smirnov (MKV+MABS)
		13.2.7 Matching of the Methods of Vaindiner and the Arganovskiy-Baglay-Smirnov (MV+MABS)
		13.2.8 Matching of the Methods of Vaindiner with the Method of Variational Iterations (MV+MVI)
		13.2.9 Numerical Example
	13.3 Mathematical Background
		13.3.1 Theorems on Convergence of MVI
		13.3.2 Convergence Theorem
	13.4 Contact Interaction of Two Square Plates
		13.4.1 Computational Examples
	13.5 Dynamics of a Contact Interaction
14 Chaotic Vibrations of Flexible Shallow Axially Symmetric Shells vs. Different Boundary Conditions
	14.1 Problem Statement and the Method of Solution
	14.2 Quantification of True Chaotic Vibrations
	14.3 Modes of Vibrations (Simple Support)
	14.4 Modes of Vibrations (Rigid Clamping)
	14.5 Investigation of Occurrence of Ribs (Simple Nonmovable Shell Support)
	14.6 Shell Vibration Modes (Movable Clamping)
15 Chaotic Vibrations of Two Euler-Bernoulli Beams With a Small Clearance
	15.1 Mathematical Model
	15.2 Principal Component Analysis (PCA)
	15.3 Numerical Experiment
	15.4 Application of the Principal Component Analysis
16 Unsolved Problems in Nonlinear Dynamics of Thin Structural Members
References