Vibration of Axially Loaded Structures 1st Edition by Lawrence Virgin 0521880424 9780521880428

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ISBN 10: 0521880424

ISBN 13: 9780521880428

Author: Lawrence N. Virgin

 This 2007 book concerns the vibration and the stability of slender structural components. The loss of stability of structures is an important aspect of structural mechanics and is presented here in terms of dynamic behavior. A variety of structural components are analyzed with a view to predicting their response to various (primarily axial) loading conditions. A number of different techniques are presented, with experimental verification from the laboratory. Practical applications are widespread, ranging from cables to space structures. The book presents methods by which the combined effects of vibration and buckling on various structures can be assessed. Vibrations and buckling are usually treated separately, but in this book their influence on each other is examined together, with examples when a combined approach is necessary. The avoidance of instability is the primary goal of this material.

Vibration of Axially Loaded Structures 1st Table of contents:

1 Context: The Point of Departure

2 Elements of Classical Mechanics

2.1 Introduction

2.2 Newton’s Second Law

2.3 Energy and Work

2.4 Virtual Work and D’Alembert’s Principle

2.5 Hamilton’s Principle and Lagrange’s Equations

2.5.1 Constraints

2.5.2 Conservation Laws

2.6 Nonconservative Forces and Energy Dissipation

2.6.1 Damping

2.6.2 Time-Dependent Forces

2.7 Strain Energy

References

3 Dynamics in the Vicinity of Equilibrium

3.1 The Linear Oscillator

3.2 Oscillator with a Slow Sweep of Frequency

3.3 Dynamics and Stability

3.3.1 Stability Concepts

3.4 Bifurcations

3.4.1 The Saddle-Node Bifurcation

3.4.2 Bifurcations from a Trivial Equilibrium

3.4.3 Initial Imperfections

3.4.4 Bifurcations of Maps

3.5 A Simple Demonstration Model

3.6 Experiments

References

4 Higher-Order Systems

4.1 Introduction

4.2 Multiple-Degree-of-Freedom Systems

4.2.1 The Algebraic Eigenvalue Problem

4.2.2 Normal Modes

4.2.3 Equilibrium, Linearization, and Stability

4.2.4 Routh–Hurwitz Criterion

4.2.5 Lyapunov Functions

4.2.6 Rayleigh’s Quotient

4.3 Distributed Systems

4.3.1 The Differential Eigenvalue Problem

4.3.2 Solution Methods

Modal Analysis and Truncation

Rayleigh–Ritz

Weighted Residuals–Galerkin

The Finite-Element Method

4.3.3 Context Revisited

References

5 Discrete-Link Models

5.1 Introduction

5.2 An Inverted Pendulum

5.2.1 Static Behavior

5.2.2 Geometric Imperfections

5.2.3 Dynamic Behavior

5.2.4 A Note on Inertia

5.3 A Discrete-Strut Model

5.4 An Asymmetric Model

5.5 A Three-Bar Model

5.6 A Snap-Through Model

5.7 Augusti’s Model

5.8 Multiple Loads

5.9 Load-Dependent Supports

5.10 Path Following and Continuation

References

6 Strings, Cables, and Membranes

6.1 Introduction

6.2 The Stretched String

6.2.1 The Wave Equation

6.2.2 Traveling-Wave Solution

6.2.3 Energy Considerations and Rayleigh’s Principle

6.3 A Suspended Cable

6.3.1 The Hanging Chain

6.4 A Rectangular Membrane

References

7 Continuous Struts

7.1 Introduction

7.2 Basic Formulation

7.2.1 The Response

7.2.2 The Temporal Solution

7.2.3 The Spatial Solution

7.3 Rayleigh’s Quotient

7.4 Rayleigh–Ritz Analysis

7.5 A Galerkin Approach

7.6 Higher Modes

7.7 Rotating Beams

7.8 A Strut with a Tangential Load

7.9 Self-Weight

7.9.1 A Hanging Beam

7.9.2 Experiments

7.10 Thermal Loading

7.11 Other Effects

References

8 Other Column-Type Structures

8.1 A Beam on an Elastic Foundation

8.2 Elastically Restrained Supports

8.3 Beams with Variable Cross Section

8.4 Modal Coupling

8.5 Flexural–Torsional Buckling and Vibration

8.6 Type of Loading

8.7 A Continuous Arch

References

9 Frames

9.1 A Beam with General Boundary Conditions

9.2 The Stiffness Method

9.3 A Self-Strained Frame Example

9.4 Modal Analysis

9.5 Large-Deflection Analysis

9.6 A Tubular Structure

References

10 Plates

10.1 Introduction

10.1.1 Brief Review of the Classical Theory

10.1.2 Strain Energy

10.1.3 Boundary and Initial Conditions

10.1.4 The Simplest Case

10.1.5 Initial Imperfections

10.2 The Ritz and Finite-Element Approaches

10.3 A Fully Clamped Plate

10.4 Moderately Large Deflections

10.5 Postbuckling

Experimental Description

10.6 Mode Jumping

10.6.1 Introduction

10.6.2 The Analytic Approach

Equilibrium Paths

Free Vibration

10.6.3 Finite-Element Transient Results

10.7 Cylindrical Shells

References

11 Nondestructive Testing

11.1 Introduction

11.1.1 The Southwell Plot

11.1.2 Examples

11.2 Some Background

11.3 Snap-Through Revisited

11.4 Range of Prediction

11.5 A Box Column

11.6 Plates and Shells

References

12 Highly Deformed Structures

12.1 Introduction to the Elastica

12.2 The Governing Equations

12.3 Case Study A: Self-Weight Loading Revisited

12.3.1 Numerical Results

12.3.2 Experiments

12.4 Case Study B: A Heavy Beam

12.4.1 Numerical Results

12.4.2 Experiments

12.5 Case Study C: A Pinched Loop

12.6 Case Study D: A Beam Loaded by a Cable

12.7 The Softening Loop Revisited

References

13 Suddenly Applied Loads

13.1 Load Classification

13.2 Back to Link Models

13.3 Dynamic Buckling of a Plate

13.4 A Type of Escaping Motion

13.5 Impulsive Loading

13.5.1 Equilibrium Behavior

13.5.2 Behavior under Sudden Loading

13.6 Snap-Through of a Curved Panel

References

14 Harmonic Loading: Parametric Excitation

14.1 An Oscillating End Load

14.2 The Variational Equation

14.3 Mathieu’s Equation

14.4 Pulsating Axial Loads on Shells

14.4.1 A Curved Panel

14.4.2 A Cylindrical Shell

References

15 Harmonic Loading: Transverse Excitation

15.1 Introduction: Resonance Effects

15.1.1 A Single-Mode Approximation

15.1.2 Beyond Buckling

15.2 The Poincar Section

15.3 Continuous Systems

15.4 An Application to Vibration Isolation

15.4.1 Postbuckling of a Strut Revisited

15.4.2 Experimental Verification

15.4.3 The Forced Response

15.5 Forced Excitation of the Thermally Buckled Plate

References

16 Nonlinear Vibration

PART I: FREE VIBRATION

16.1 Introduction

16.2 Abstract Models

16.3 A Mass Between Stretched Springs

16.4 Nonlinear Vibration of Strings

16.5 Nonlinear Vibration of Beams

16.6 Nonlinear Vibration of a Plate

16.7 Nonlinear Vibration in Cylindrical Shells

PART II: FORCED VIBRATION

16.8 Nonlinear Forced Vibration of Strings

16.9 Nonlinear Forced Vibration of Beams

16.10 Persistent Snap-Through Behavior in a Plate

16.11 A Panel in Supersonic Flow

16.12 Chaotic Behavior

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