Eigenvalues of Inhomogeneous Structures

Unusual Closed-Form Solutions

Isaac Elishakoff

• Publisher: CRC Press
• Year: 2004
• eBook ISBN: 9781000611007
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

The engineering community generally accepts that there exists only a small set of closed-form solutions for simple cases of bars, beams, columns, and plates. Despite the advances in powerful computing and advanced numerical techniques, closed-form solutions remain important for engineering; these include uses for preliminary design, for evaluation

1

Introduction: Review of Direct, Semi-inverse and Inverse Eigenvalue Problems

This chapter presents a selective review (with emphasis on the word “selective”) of the direct, semi-inverse and inverse eigenvalue problem for structures described by a differential equation with variable coefficients. It gives only a taste of the extensive research that has been conducted since 1759, when Leonhard Euler posed, apparently for the first time, a boundary value problem. Since then numerous studies have been conducted for rods, Bernoulli–Euler beams, Bresse–Timoshenko beams, Kirchhoff–Love and Mindlin–Reissner plates and shells, and structures analyzed via finer, high-order theories. This selective review classifies the solutions as belonging to one of three main classes: (1) direct problems, (2) semi-inverse problems, (3) inverse problems. In addition, some new closed-form solutions are reported that have been obtained via posing an inverse vibration problem. Due to the huge body of literature, the author limits himself to discussing classic theories of structures.

1.1 Introductory Remarks

The vibration and buckling eigenvalue problems in engineering may be roughly categorized as belonging to one of the following three classes: (1) direct or forward problems, (2) inverse or backward problems, (3) semi-inverse or semi-direct problems. Direct problems are associated with the determination of the vibration frequencies and/or the buckling loads of structures with specified configuration; namely, in the vibration context, direct problems call for evaluation of the natural frequencies of rods, beam, plates or shells of specified, uniform or non-uniform cross-section (in the case of rods and beams) or thickness (in the case of plates and shells), for both homogeneous structures, i.e., those with constant elastic properties and mass density, and inhomogeneous ones, in which the elastic properties and/or the material density vary with the coordinates. There are numerous methods of solving direct eigenvalue problems. These methods are conveniently subdivided into the exact and the a...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Foreword
• Prologue
• Chapter 1 Introduction: Review of Direct, Semi-inverse and Inverse Eigenvalue Problems
• Chapter 2 Unusual Closed-Form Solutions in Column Buckling
• Chapter 3 Unusual Closed-Form Solutions for Rod Vibrations
• Chapter 4 Unusual Closed-Form Solutions for Beam Vibrations
• Chapter 5 Beams and Columns with Higher-Order Polynomial Eigenfunctions
• Chapter 6 Influence of Boundary Conditions on Eigenvalues
• Chapter 7 Boundary Conditions Involving Guided Ends
• Chapter 8 Vibration of Beams in the Presence of an Axial Load
• Chapter 9 Unexpected Results for a Beam on an Elastic Foundation or with Elastic Support
• Chapter 10 Non-Polynomial Expressions for the Beam’s Flexural Rigidity for Buckling or Vibration
• Chapter 11 Circular Plates
• Epilogue
• Appendices
• References
• Author Index
• Subject Index