Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions
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Summary
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 of the accuracy of approximate and numerical solutions, and for evaluating the role played by various geometric and loading parameters.
Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions offers the first new treatment of closed-form solutions since the works of Leonhard Euler over two centuries ago. It presents simple solutions for vibrating bars, beams, and plates, as well as solutions that can be used to verify finite element solutions. The closed solutions in this book not only have applications that allow for the design of tailored structures, but also transcend mechanical engineering to generalize into other fields of engineering. Also included are polynomial solutions, non-polynomial solutions, and discussions on axial variability of stiffness that offer the possibility of incorporating axial grading into functionally graded materials.
This single-package treatment of inhomogeneous structures presents the tools for optimization in many applications. Mechanical, aerospace, civil, and marine engineers will find this to be the most comprehensive book on the subject. In addition, senior undergraduate and graduate students and professors will find this to be a good supplement to other structural design texts, as it can be easily incorporated into the classroom.
Table of Contents
FOREWORD
PROLOGUE
INTRODUCTION: REVIEW OF DIRECT, SEMI-INVERSE AND INVERSE EIGENVALUE PROBLEMS
Introductory Remarks
Vibration of Uniform Homogeneous Beams
Buckling of Uniform Homogeneous Columns
Some Exact Solutions for the Vibration of Non-Uniform Beams
Exact Solution for Buckling of Non-Uniform Columns
Other Direct Methods (FDM,FEM,DQM)
Eisenberger 's Exact Finite Element Method
Semi-Inverse or Semi-Direct Methods
Inverse Eigenvalue Problems
Connection to the Work by ?Zyczkowski and Gajewski
Connection to Functionally Graded Materials
Scope of the Present Monograph
UNUSUAL CLOSED-FORM SOLUTIONS IN COLUMN BUCKLING
New Closed-Form Solutions for Buckling of a Variable Flexural Rigidity Column
Inverse Buckling Problem for Inhomogeneous Columns
Closed-Form Solution for the Generalized Euler Problem
Some Closed-Form Solutions for the Buckling of Inhomogeneous Columns Under Distributed Variable Loading
UNUSUAL CLOSED-FORM SOLUTIONS FOR ROD VIBRATIONS
Reconstructing the Axial Rigidity of a Longitudinally Vibrating Rod by Its Fundamental Mode Shape
The Natural Frequency of an Inhomogeneous Rod May be Independent of Nodal Parameters
Concluding Remarks
UNUSUAL CLOSED-FORM SOLUTIONS FOR BEAM VIBRATIONS
Apparently First Closed-Form Solutions for Frequencies of Deterministically and/or Stochastically Inhomogeneous Beams (Pinned -Pinned Boundary Conditions)
Apparently First Closed-Form Solutions for Inhomogeneous Beams (Other Boundary Conditions)
Inhomogeneous Beams That May Possess a Prescribed Polynomial Second Mode
Concluding Remarks
BEAMS AND COLUMNS WITH HIGHER-ORDER POLYNOMIAL EIGENFUNCTIONS
Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 1: Buckling
Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 2: Vibration
INFLUENCE OF BOUNDARY CONDITIONS ON EIGENVALUES
The Remarkable Nature of Effect of Boundary Conditions on Closed-Form Solutions for Vibrating Inhomogeneous Bernoulli-Euler Beams
BOUNDARY CONDITIONS INVOLVING GUIDED ENDS
Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Pinned Support
Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Clamped Support
Class of Analytical Closed-Form Polynomial Solutions for Guided-Pinned Inhomogeneous Beams
Class of Analytical Closed-Form Polynomial Solutions for Clamped -Guided Inhomogeneous Beams
VIBRATION OF BEAMS IN THE PRESENCE OF AN AXIAL LOAD
Closed -Form Solutions for Inhomogeneous Vibrating Beams Under Axially Distributed Loading
A Fifth-Order Polynomial That Serves as Both the Buckling and Vibration Modes of an Inhomogeneous Structure
UNEXPECTED RESULTS FOR A BEAM ON AN ELASTIC FOUNDATION OR WITH ELASTIC SUPPORT
Some Unexpected Results in the Vibration of Inhomogeneous Beams on an Elastic Foundation
Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Rotational Spring
Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Translational Spring
NON-POLYNOMIAL EXPRESSIONS FOR THE BEAM'S FLEXURAL RIGIDITY FOR BUCKLING OR VIBRATION
Both the Static Deflection and Vibration Mode of a Uniform Beam Can Serve as Buckling Modes of a Non-Uniform Column
Resurrection of the Method of Successive Approximations to Yield Closed-Form Solutions for Vibrating Inhomogeneous Beams
Additional Closed-Form Solutions for Inhomogeneous Vibrating Beams by the Integral Method
CIRCULAR PLATES
Axisymmetric Vibration of Inhomogeneous Clamped Circular Plates: An Unusual Closed-Form Solution
Axisymmetric Vibration of Inhomogeneous Free Circular Plates: An Unusual, Exact, Closed-Form Solution
Axisymmetric Vibration of Inhomogeneous Pinned Circular Plates: An Unusual, Exact, Closed-Form Solution
EPILOGUE
APPENDICES
REFERENCES
AUTHOR INDEX
SUBJECT INDEX
Editorial Reviews
"This is a most remarkable and thorough review of the efforts that have been made to find closed-form solutions in the vibration and buckling of all manner of elastic rods, beams, columns and plates. The author is particularly, but not exclusively, concerned with variations in the stiffness of structural members. The resulting volume is the culmination of his studies over many years.
"What more can be said about this monumental work, other than to express admiration? The author's solutions to particular problems will be very valuable for testing the validity and accuracy of various numerical techniques. Moreover, the study is of great academic interest, and is clearly a labor of love. The author is to be congratulated on this work, which is bound to be of considerable value to all interested in research in this area."
-- Dr. H.D. Conway, Professor Emeritus, Department of Theoretical and Applied Mechanics, Cornell University
"It is generally believed that closed-form solutions exist for only a relatively few, very simple cases of bars, beams, columns, and plates. This monograph is living proof that there are, in fact, not just a few such solutions. Even in the current age of powerful numerical techniques and high-speed, large-capacity computers, there are a number of important uses for closed-form solutions:
"This book is fantastic. Professor Elishakoff is to be congratulated not only for pulling together a number of solutions from the international literature, but also for contributing a large number of solutions himself. Finally, he has explained in a very interesting fashion the history behind many of the solutions."
-- Dr. Charles W. Bert, Benjamin H. Perkinson Professor Emeritus, Aerospace and Mechanical Engineering, The University of Oklahoma
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