Eigenvalues of inhomogenous structures : unusual closed-form solutions /

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Bibliographic Details
Author / Creator:	Elishakoff, Isaac.
Imprint:	Boca Raton : CRC Press, c2005.
Description:	xv, 729 p. : ill. ; 25 cm.
Language:	English
Subject:	Structural dynamics -- Mathematical models. Buckling (Mechanics) -- Mathematical models. Eigenvalues. Buckling (Mechanics) -- Mathematical models. Eigenvalues. Structural dynamics -- Mathematical models.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/5546308

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ISBN:	0849328926 (alk. paper)
Notes:	Includes bibliographical references (p. 653-710) and indexes.

Table of Contents:

Foreword
Prologue
Chapter 1. Introduction: Review of Direct, Semi-inverse and Inverse Eigenvalue Problems
1.1. Introductory Remarks
1.2. Vibration of Uniform Homogeneous Beams
1.3. Buckling of Uniform Homogeneous Columns
1.4. Some Exact Solutions for the Vibration of Non-uniform Beams
1.4.1. The Governing Differential Equation
1.5. Exact Solution for Buckling of Non-uniform Columns
1.6. Other Direct Methods (FDM, FEM, DQM)
1.7. Eisenberger's Exact Finite Element Method
1.8. Semi-inverse or Semi-direct Methods
1.9. Inverse Eigenvalue Problems
1.10. Connection to the Work by Zyczkowski and Gajewski
1.11. Connection to Functionally Graded Materials
1.12. Scope of the Present Monograph
Chapter 2. Unusual Closed-Form Solutions in Column Buckling
2.1. New Closed-Form Solutions for Buckling of a Variable Flexural Rigidity Column
2.1.1. Introductory Remarks
2.1.2. Formulation of the Problem
2.1.3. Uncovered Closed-Form Solutions
2.1.4. Concluding Remarks
2.2. Inverse Buckling Problem for Inhomogeneous Columns
2.2.1. Introductory Remarks
2.2.2. Formulation of the Problem
2.2.3. Column Pinned at Both Ends
2.2.4. Column Clamped at Both Ends
2.2.5. Column Clamped at One End and Pinned at the Other
2.2.6. Concluding Remarks
2.3. Closed-Form Solution for the Generalized Euler Problem
2.3.1. Introductory Remarks
2.3.2. Formulation of the Problem
2.3.3. Column Clamped at Both Ends
2.3.4. Column Pinned at One End and Clamped at the Other
2.3.5. Column Clamped at One End and Free at the Other
2.3.6. Concluding Remarks
2.4. Some Closed-Form Solutions for the Buckling of Inhomogeneous Columns under Distributed Variable Loading
2.4.1. Introductory Remarks
2.4.2. Basic Equations
2.4.3. Column Pinned at Both Ends
2.4.4. Column Clamped at Both Ends
2.4.5. Column that is Pinned at One End and Clamped at the Other
2.4.6. Concluding Remarks
Chapter 3. Unusual Closed-Form Solutions for Rod Vibrations
3.1. Reconstructing the Axial Rigidity of a Longitudinally Vibrating Rod by its Fundamental Mode Shape
3.1.1. Introductory Remarks
3.1.2. Formulation of the Problem
3.1.3. Inhomogeneous Rods with Uniform Density
3.1.4. Inhomogeneous Rods with Linearly Varying Density
3.1.5. Inhomogeneous Rods with Parabolically Varying Inertial Coefficient
3.1.6. Rod with General Variation of Inertial Coefficient (m [greater than sign] 2)
3.1.7. Concluding Remarks
3.2. The Natural Frequency of an Inhomogeneous Rod may be Independent of Nodal Parameters
3.2.1. Introductory Remarks
3.2.2. The Nodal Parameters
3.2.3. Mode with One Node: Constant Inertial Coefficient
3.2.4. Mode with Two Nodes: Constant Density
3.2.5. Mode with One Node: Linearly Varying Material Coefficient
3.3. Concluding Remarks
Chapter 4. Unusual Closed-Form Solutions for Beam Vibrations
4.1. Apparently First Closed-Form Solutions for Frequencies of Deterministically and/or Stochastically Inhomogeneous Beams (Pinned-Pinned Boundary Conditions)
4.1.1. Introductory Remarks
4.1.2. Formulation of the Problem
4.1.3. Boundary Conditions
4.1.4. Expansion of the Differential Equation
4.1.5. Compatibility Conditions
4.1.6. Specified Inertial Coefficient Function
4.1.7. Specified Flexural Rigidity Function
4.1.8. Stochastic Analysis
4.1.9. Nature of Imposed Restrictions
4.1.10. Concluding Remarks
4.2. Apparently First Closed-Form Solutions for Inhomogeneous Beams (Other Boundary Conditions)
4.2.1. Introductory Remarks
4.2.2. Formulation of the Problem
4.2.3. Cantilever Beam
4.2.4. Beam that is Clamped at Both Ends
4.2.5. Beam Clamped at One End and Pinned at the Other
4.2.6. Random Beams with Deterministic Frequencies
4.3. Inhomogeneous Beams that may Possess a Prescribed Polynomial Second Mode
4.3.1. Introductory Remarks
4.3.2. Basic Equation
4.3.3. A Beam with Constant Mass Density
4.3.4. A Beam with Linearly Varying Mass Density
4.3.5. A Beam with Parabolically Varying Mass Density
4.4. Concluding Remarks
Chapter 5. Beams and Columns with Higher-Order Polynomial Eigenfunctions
5.1. Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 1: Buckling
5.1.1. Introductory Remarks
5.1.2. Choosing a Pre-selected Mode Shape
5.1.3. Buckling of the Inhomogeneous Column under an Axial Load
5.1.4. Buckling of Columns under an Axially Distributed Load
5.1.5. Concluding Remarks
5.2. Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 2: Vibration
5.2.1. Introductory Comments
5.2.2. Formulation of the Problem
5.2.3. Basic Equations
5.2.4. Constant Inertial Coefficient (m = 0)
5.2.5. Linearly Varying Inertial Coefficient (m = 1)
5.2.6. Parabolically Varying Inertial Coefficient (m = 2)
5.2.7. Cubic Inertial Coefficient (m = 3)
5.2.8. Particular Case m = 4
5.2.9. Concluding Remarks
Chapter 6. Influence of Boundary Conditions on Eigenvalues
6.1. The Remarkable Nature of Effect of Boundary Conditions on Closed-Form Solutions for Vibrating Inhomogeneous Bernoulli-Euler Beams
6.1.1. Introductory Remarks
6.1.2. Construction of Postulated Mode Shapes
6.1.3. Formulation of the Problem
6.1.4. Closed-Form Solutions for the Clamped-Free Beam
6.1.5. Closed-Form Solutions for the Pinned-Clamped Beam
6.1.6. Closed-Form Solutions for the Clamped-Clamped Beam
6.1.7. Concluding Remarks
Chapter 7. Boundary Conditions Involving Guided Ends
7.1. Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Pinned Support
7.1.1. Introductory Remarks
7.1.2. Formulation of the Problem
7.1.3. Boundary Conditions
7.1.4. Solution of the Differential Equation
7.1.5. The Degree of the Material Density is Less than Five
7.1.6. General Case: Compatibility Conditions
7.1.7. Concluding Comments
7.2. Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Clamped Support
7.2.1. Introductory Remarks
7.2.2. Formulation of the Problem
7.2.3. Boundary Conditions
7.2.4. Solution of the Differential Equation
7.2.5. Cases of Uniform and Linear Densities
7.2.6. General Case: Compatibility Condition
7.2.7. Concluding Remarks
7.3. Class of Analytical Closed-Form Polynomial Solutions for Guided-Pinned Inhomogeneous Beams
7.3.1. Introductory Remarks
7.3.2. Formulation of the Problem
7.3.3. Constant Inertial Coefficient (m = 0)
7.3.4. Linearly Varying Inertial Coefficient (m = 1)
7.3.5. Parabolically Varying Inertial Coefficient (m = 2)
7.3.6. Cubically Varying Inertial Coefficient (m = 3)
7.3.7. Coefficient Represented by a Quartic Polynomial (m = 4)
7.3.8. General Case
7.3.9. Particular Cases Characterized by the Inequality n [greater than or equal] m + 2
7.3.10. Concluding Remarks
7.4. Class of Analytical Closed-Form Polynomial Solutions for Clamped-Guided Inhomogeneous Beams
7.4.1. Introductory Remarks
7.4.2. Formulation of the Problem
7.4.3. General Case
7.4.4. Constant Inertial Coefficient (m = 0)
7.4.5. Linearly Varying Inertial Coefficient (m = 1)
7.4.6. Parabolically Varying Inertial Coefficient (m = 2)
7.4.7. Cubically Varying Inertial Coefficient (m = 3)
7.4.8. Inertial Coefficient Represented as a Quadratic (m = 4)
7.4.9. Concluding Remarks
Chapter 8. Vibration of Beams in the Presence of an Axial Load
8.1. Closed-Form Solutions for Inhomogeneous Vibrating Beams under Axially Distributed Loading
8.1.1. Introductory Comments
8.1.2. Basic Equations
8.1.3. Column that is Clamped at One End and Free at the Other
8.1.4. Column that is Pinned at its Ends
8.1.5. Column that is clamped at its ends
8.1.6. Column that is Pinned at One End and Clamped at the Other
8.1.7. Concluding Remarks
8.2. A Fifth-Order Polynomial that Serves as both the Buckling and Vibration Modes of an Inhomogeneous Structure
8.2.1. Introductory Comments
8.2.2. Formulation of the Problem
8.2.3. Basic Equations
8.2.4. Closed-Form Solution for the Pinned Beam
8.2.5. Closed-Form Solution for the Clamped-Free Beam
8.2.6. Closed-Form Solution for the Clamped-Clamped Beam
8.2.7. Closed-Form Solution for the Beam that is Pinned at One End and Clamped at the Other
8.2.8. Concluding Remarks
Chapter 9. Unexpected Results for a Beam on an Elastic Foundation or with Elastic Support
9.1. Some Unexpected Results in the Vibration of Inhomogeneous Beams on an Elastic Foundation
9.1.1. Introductory Remarks
9.1.2. Formulation of the Problem
9.1.3. Beam with Uniform Inertial Coefficient, Inhomogeneous Elastic Modulus and Elastic Foundation
9.1.4. Beams with Linearly Varying Density, Inhomogeneous Modulus and Elastic Foundations
9.1.5. Beams with Varying Inertial Coefficient Represented as an mth Order Polynomial
9.1.6. Case of a Beam Pinned at its Ends
9.1.7. Beam Clamped at the Left End and Free at the Right End
9.1.8. Case of a Clamped-Pinned Beam
9.1.9. Case of a Clamped-Clamped Beam
9.1.10. Case of a Guided-Pinned Beam
9.1.11. Case of a Guided-Clamped Beam
9.1.12. Cases Violated in Eq. (9.99)
9.1.13. Does the Boobnov-Galerkin Method Corroborate the Unexpected Exact Results?
9.1.14. Concluding Remarks
9.2. Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Rotational Spring
9.2.1. Introductory Remarks
9.2.2. Basic Equations
9.2.3. Uniform Inertial Coefficient
9.2.4. Linear Inertial Coefficient
9.3. Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Translational Spring
9.3.1. Introductory Remarks
9.3.2. Basic Equations
9.3.3. Constant Inertial Coefficient
9.3.4. Linear Inertial Coefficient
Chapter 10. Non-Polynomial Expressions for the Beam's Flexural Rigidity for Buckling or Vibration
10.1. Both the Static Deflection and Vibration Mode of a Uniform Beam Can Serve as Buckling Modes of a Non-uniform Column
10.1.1. Introductory Remarks
10.1.2. Basic Equations
10.1.3. Buckling of Non-uniform Pinned Columns
10.1.4. Buckling of a Column under its Own Weight
10.1.5. Vibration Mode of a Uniform Beam as a Buckling Mode of a Non-uniform Column
10.1.6. Non-uniform Axially Distributed Load
10.1.7. Concluding Remarks
10.2. Resurrection of the Method of Successive Approximations to Yield Closed-Form Solutions for Vibrating Inhomogeneous Beams
10.2.1. Introductory Comments
10.2.2. Evaluation of the Example by Birger and Mavliutov
10.2.3. Reinterpretation of the Integral Method for Inhomogeneous Beams
10.2.4. Uniform Material Density
10.2.5. Linearly Varying Density
10.2.6. Parabolically Varying Density
10.2.7. Can Successive Approximations Serve as Mode Shapes?
10.2.8. Concluding Remarks
10.3. Additional Closed-Form Solutions for Inhomogeneous Vibrating Beams by the Integral Method
10.3.1. Introductory Remarks
10.3.2. Pinned-Pinned Beam
10.3.3. Guided-Pinned Beam
10.3.4. Free-Free Beam
10.3.5. Concluding Remarks
Chapter 11. Circular Plates
11.1. Axisymmetric Vibration of Inhomogeneous Clamped Circular Plates: an Unusual Closed-Form Solution
11.1.1. Introductory Remarks
11.1.2. Basic Equations
11.1.3. Method of Solution
11.1.4. Constant Inertial Term (m = 0)
11.1.5. Linearly Varying Inertial Term (m = 1)
11.1.6. Parabolically Varying Inertial Term (m = 2)
11.1.7. Cubic Inertial Term (m = 3)
11.1.8. General Inertial Term (m [greater than or equal] 4)
11.1.9. Alternative Mode Shapes
11.2. Axisymmetric Vibration of Inhomogeneous Free Circular Plates: An Unusual, Exact, Closed-Form Solution
11.2.1. Introductory Remarks
11.2.2. Formulation of the Problem
11.2.3. Basic Equations
11.2.4. Concluding Remarks
11.3. Axisymmetric Vibration of Inhomogeneous Pinned Circular Plates: An Unusual, Exact, Closed-Form Solution
11.3.1. Basic Equations
11.3.2. Constant Inertial Term (m = 0)
11.3.3. Linearly Varying Inertial Term (m = 1)
11.3.4. Parabolically Varying Inertial Term (m = 2)
11.3.5. Cubic Inertial Term (m = 3)
11.3.6. General Inertial Term (m [greater than or equal] 4)
11.3.7. Concluding Remarks
Epilogue
Appendices
References
Author Index
Subject Index

Eigenvalues of inhomogenous structures : unusual closed-form solutions /

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