Eigenvalues of Inhomogeneous Structures: Unusual Closed-Form Solutions

by Elishakoff; Isaac

• ISBN13: 9780849328923
• ISBN10: 0849328926
• Format: Hardcover
• Copyright: 2004-10-28
• Publisher: CRC Press

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	xv
Prologue	1	(6)
Introduction: Review of Direct, Semi-inverse and Inverse Eigenvalue Problems	7	(48)
Introductory Remarks	7	(1)
Vibration of Uniform Homogeneous Beams	8	(2)
Buckling of Uniform Homogeneous Columns	10	(9)
Some Exact Solutions for the Vibration of Non-uniform Beams	19	(5)
The Governing Differential Equation	21	(3)
Exact Solution for Buckling of Non-uniform Columns	24	(4)
Other Direct Methods (FDM, FEM, DQM)	28	(2)
Eisenberger's Exact Finite Element Method	30	(5)
Semi-inverse or Semi-direct Methods	35	(8)
Inverse Eigenvalue Problems	43	(7)
Connection to the Work by Zyczkowski and Gajewski	50	(2)
Connection to Functionally Graded Materials	52	(1)
Scope of the Present Monograph	53	(2)
Unusual Closed-Form Solutions in Column Buckling	55	(52)
New Closed-Form Solutions for Buckling of a Variable Flexural Rigidity Column	55	(10)
Introductory Remarks	55	(1)
Formulation of the Problem	56	(1)
Uncovered Closed-Form Solutions	57	(8)
Concluding Remarks	65	(1)
Inverse Buckling Problem for Inhomogeneous Columns	65	(9)
Introductory Remarks	65	(1)
Formulation of the Problem	65	(1)
Column Pinned at Both Ends	66	(2)
Column Clamped at Both Ends	68	(1)
Column Clamped at One End and Pinned at the Other	69	(1)
Concluding Remarks	70	(4)
Closed-Form Solution for the Generalized Euler Problem	74	(10)
Introductory Remarks	74	(2)
Formulation of the Problem	76	(3)
Column Clamped at Both Ends	79	(1)
Column Pinned at One End and Clamped at the Other	79	(2)
Column Clamped at One End and Free at the Other	81	(2)
Concluding Remarks	83	(1)
Some Closed-Form Solutions for the Buckling of Inhomogeneous Columns under Distributed Variable Loading	84	(23)
Introductory Remarks	84	(3)
Basic Equations	87	(5)
Column Pinned at Both Ends	92	(5)
Column Clamped at Both Ends	97	(3)
Column that is Pinned at One End and Clamped at the Other	100	(5)
Concluding Remarks	105	(2)
Unusual Closed-Form Solutions for Rod Vibrations	107	(28)
Reconstructing the Axial Rigidity of a Longitudinally Vibrating Rod by its Fundamental Mode Shape	107	(13)
Introductory Remarks	107	(1)
Formulation of the Problem	108	(1)
Inhomogeneous Rods with Uniform Density	109	(3)
Inhomogeneous Rods with Linearly Varying Density	112	(2)
Inhomogeneous Rods with Parabolically Varying Inertial Coefficient	114	(1)
Rod with General Variation of Inertial Coefficient (m > 2)	115	(3)
Concluding Remarks	118	(2)
The Natural Frequency of an Inhomogeneous Rod may be Independent of Nodal Parameters	120	(11)
Introductory Remarks	120	(1)
The Nodal Parameters	121	(3)
Mode with One Node: Constant Inertial Coefficient	124	(3)
Mode with Two Nodes: Constant Density	127	(2)
Mode with One Node: Linearly Varying Material Coefficient	129	(2)
Concluding Remarks	131	(4)
Unusual Closed-Form Solutions for Beam Vibrations	135	(68)
Apparently First Closed-Form Solutions for Frequencies of Deterministically and/or Stochastically Inhomogeneous Beams (Pinned--Pinned Boundary Conditions)	135	(17)
Introductory Remarks	135	(1)
Formulation of the Problem	136	(1)
Boundary Conditions	137	(1)
Expansion of the Differential Equation	138	(1)
Compatibility Conditions	139	(1)
Specified Inertial Coefficient Function	140	(1)
Specified Flexural Rigidity Function	141	(3)
Stochastic Analysis	144	(7)
Nature of Imposed Restrictions	151	(1)
Concluding Remarks	151	(1)
Apparently First Closed-Form Solutions for Inhomogeneous Beams (Other Boundary Conditions)	152	(23)
Introductory Remarks	152	(1)
Formulation of the Problem	153	(1)
Cantilever Beam	154	(9)
Beam that is Clamped at Both Ends	163	(2)
Beam Clamped at One End and Pinned at the Other	165	(3)
Random Beams with Deterministic Frequencies	168	(7)
Inhomogeneous Beams that may Possess a Prescribed Polynomial Second Mode	175	(24)
Introductory Remarks	175	(5)
Basic Equation	180	(2)
A Beam with Constant Mass Density	182	(3)
A Beam with Linearly Varying Mass Density	185	(5)
A Beam with Parabolically Varying Mass Density	190	(9)
Concluding Remarks	199	(4)
Beams and Columns with Higher-Order Polynomial Eigenfunctions	203	(46)
Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 1: Buckling	203	(22)
Introductory Remarks	203	(1)
Choosing a Pre-selected Mode Shape	204	(1)
Buckling of the Inhomogeneous Column under an Axial Load	205	(4)
Buckling of Columns under an Axially Distributed Load	209	(15)
Concluding Remarks	224	(1)
Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 2: Vibration	225	(24)
Introductory Comments	225	(1)
Formulation of the Problem	226	(1)
Basic Equations	227	(1)
Constant Inertial Coefficient (m = 0)	228	(2)
Linearly Varying Inertial Coefficient (m = 1)	230	(1)
Parabolically Varying Inertial Coefficient (m = 2)	231	(5)
Cubic Inertial Coefficient (m = 3)	236	(3)
Particular Case m = 4	239	(3)
Concluding Remarks	242	(7)
Influence of Boundary Conditions on Eigenvalues	249	(60)
The Remarkable Nature of Effect of Boundary Conditions on Closed-Form Solutions for Vibrating Inhomogeneous Bernoulli--Euler Beams	249	(60)
Introductory Remarks	249	(1)
Construction of Postulated Mode Shapes	250	(1)
Formulation of the Problem	251	(1)
Closed-Form Solutions for the Clamped--Free Beam	252	(19)
Closed-Form Solutions for the Pinned--Clamped Beam	271	(18)
Closed-Form Solutions for the Clamped--Clamped Beam	289	(19)
Concluding Remarks	308	(1)
Boundary Conditions Involving Guided Ends	309	(86)
Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Pinned Support	309	(13)
Introductory Remarks	309	(1)
Formulation of the Problem	310	(1)
Boundary Conditions	310	(1)
Solution of the Differential Equation	311	(1)
The Degree of the Material Density is Less than Five	312	(6)
General Case: Compatibility Conditions	318	(4)
Concluding Comments	322	(1)
Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Clamped Support	322	(8)
Introductory Remarks	322	(1)
Formulation of the Problem	323	(1)
Boundary Conditions	323	(1)
Solution of the Differential Equation	324	(1)
Cases of Uniform and Linear Densities	325	(2)
General Case: Compatibility Condition	327	(2)
Concluding Remarks	329	(1)
Class of Analytical Closed-Form Polynomial Solutions for Guided-Pinned Inhomogeneous Beams	330	(34)
Introductory Remarks	330	(1)
Formulation of the Problem	330	(2)
Constant Inertial Coefficient (m = 0)	332	(1)
Linearly Varying Inertial Coefficient (m = 1)	333	(2)
Parabolically Varying Inertial Coefficient (m = 2)	335	(2)
Cubically Varying Inertial Coefficient (m = 3)	337	(1)
Coefficient Represented by a Quartic Polynomial (m = 4)	338	(2)
General Case	340	(9)
Particular Cases Characterized by the Inequality n ≥ m + 2	349	(15)
Concluding Remarks	364	(1)
Class of Analytical Closed-Form Polynomial Solutions for Clamped--Guided Inhomogeneous Beams	364	(31)
Introductory Remarks	364	(1)
Formulation of the Problem	364	(2)
General Case	366	(10)
Constant Inertial Coefficient (m = 0)	376	(1)
Linearly Varying Inertial Coefficient (m = 1)	377	(1)
Parabolically Varying Inertial Coefficient (m = 2)	378	(2)
Cubically Varying Inertial Coefficient (m = 3)	380	(5)
Inertial Coefficient Represented as a Quadratic (m = 4)	385	(7)
Concluding Remarks	392	(3)
Vibration of Beams in the Presence of an Axial Load	395	(66)
Closed-Form Solutions for Inhomogeneous Vibrating Beams under Axially Distributed Loading	395	(22)
Introductory Comments	395	(2)
Basic Equations	397	(1)
Column that is Clamped at One End and Free at the Other	398	(4)
Column that is Pinned at its Ends	402	(5)
Column that is clamped at its ends	407	(4)
Column that is Pinned at One End and Clamped at the Other	411	(5)
Concluding Remarks	416	(1)
A Fifth-Order Polynomial that Serves as both the Buckling and Vibration Modes of an Inhomogeneous Structure	417	(44)
Introductory Comments	417	(2)
Formulation of the Problem	419	(2)
Basic Equations	421	(1)
Closed-Form Solution for the Pinned Beam	422	(9)
Closed-Form Solution for the Clamped--Free Beam	431	(11)
Closed-Form Solution for the Clamped--Clamped Beam	442	(10)
Closed-Form Solution for the Beam that is Pinned at One End and Clamped at the Other	452	(8)
Concluding Remarks	460	(1)
Unexpected Results for a Beam on an Elastic Foundation or with Elastic Support	461	(76)
Some Unexpected Results in the Vibration of Inhomogeneous Beams on an Elastic Foundation	461	(61)
Introductory Remarks	461	(1)
Formulation of the Problem	462	(1)
Beam with Uniform Inertial Coefficient, Inhomogeneous Elastic Modulus and Elastic Foundation	463	(5)
Beams with Linearly Varying Density, Inhomogeneous Modulus and Elastic Foundations	468	(7)
Beams with Varying Inertial Coefficient Represented as an mth Order Polynomial	475	(5)
Case of a Beam Pinned at its Ends	480	(6)
Beam Clamped at the Left End and Free at the Right End	486	(5)
Case of a Clamped--Pinned Beam	491	(5)
Case of a Clamped--Clamped Beam	496	(5)
Case of a Guided--Pinned Beam	501	(9)
Case of a Guided--Clamped Beam	510	(5)
Cases Violated in Eq. (9.99)	515	(2)
Does the Boobnov--Galerkin Method Corroborate the Unexpected Exact Results?	517	(4)
Concluding Remarks	521	(1)
Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Rotational Spring	522	(6)
Introductory Remarks	522	(1)
Basic Equations	522	(1)
Uniform Inertial Coefficient	523	(3)
Linear Inertial Coefficient	526	(2)
Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Translational Spring	528	(9)
Introductory Remarks	528	(1)
Basic Equations	529	(2)
Constant Inertial Coefficient	531	(2)
Linear Inertial Coefficient	533	(4)
Non-Polynomial Expressions for the Beam's Flexural Rigidity for Buckling or Vibration	537	(54)
Both the Static Deflection and Vibration Mode of a Uniform Beam Can Serve as Buckling Modes of a Non-uniform Column	537	(11)
Introductory Remarks	537	(1)
Basic Equations	538	(1)
Buckling of Non-uniform Pinned Columns	539	(3)
Buckling of a Column under its Own Weight	542	(2)
Vibration Mode of a Uniform Beam as a Buckling Mode of a Non-uniform Column	544	(1)
Non-uniform Axially Distributed Load	545	(2)
Concluding Remarks	547	(1)
Resurrection of the Method of Successive Approximations to Yield Closed-Form Solutions for Vibrating Inhomogeneous Beams	548	(18)
Introductory Comments	548	(3)
Evaluation of the Example by Birger and Mavliutov	551	(2)
Reinterpretation of the Integral Method for Inhomogeneous Beams	553	(2)
Uniform Material Density	555	(2)
Linearly Varying Density	557	(2)
Parabolically Varying Density	559	(4)
Can Successive Approximations Serve as Mode Shapes?	563	(1)
Concluding Remarks	563	(3)
Additional Closed-Form Solutions for Inhomogeneous Vibrating Beams by the Integral Method	566	(25)
Introductory Remarks	566	(1)
Pinned--Pinned Beam	567	(8)
Guided--Pinned Beam	575	(7)
Free--Free Beam	582	(8)
Concluding Remarks	590	(1)
Circular Plates	591	(26)
Axisymmetric Vibration of Inhomogeneous Clamped Circular Plates: an Unusual Closed-Form Solution	591	(13)
Introductory Remarks	591	(2)
Basic Equations	593	(1)
Method of Solution	594	(1)
Constant Inertial Term (m = 0)	594	(1)
Linearly Varying Inertial Term (m = 1)	595	(1)
Parabolically Varying Inertial Term (m = 2)	596	(2)
Cubic Inertial Term (m = 3)	598	(2)
General Inertial Term (m ≥ 4)	600	(1)
Alternative Mode Shapes	601	(3)
Axisymmetric Vibration of Inhomogeneous Free Circular Plates: An Unusual, Exact, Closed-Form Solution	604	(3)
Introductory Remarks	604	(1)
Formulation of the Problem	605	(1)
Basic Equations	605	(2)
Concluding Remarks	607	(1)
Axisymmetric Vibration of Inhomogeneous Pinned Circular Plates: An Unusual, Exact, Closed-Form Solution	607	(10)
Basic Equations	607	(1)
Constant Inertial Term (m = 0)	608	(1)
Linearly Varying Inertial Term (m = 1)	609	(1)
Parabolically Varying Inertial Term (m = 2)	610	(2)
Cubic Inertial Term (m = 3)	612	(2)
General Inertial Term (m ≥ 4)	614	(2)
Concluding Remarks	616	(1)
Epilogue	617	(10)
Appendices	627	(26)
References	653	(58)
Author Index	711	(12)
Subject Index	723