Eigenvalues of inhomogeneous structures : unusual closed-form solutions

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.

Author Notes

Elishakoff\, Isaac

Foreword	p. xv
Prologue	p. 1
Chapter 1 Introduction: Review of Direct, Semi-inverse and Inverse Eigenvalue Problems	p. 7
1.1 Introductory Remarks	p. 7
1.2 Vibration of Uniform Homogeneous Beams	p. 8
1.3 Buckling of Uniform Homogeneous Columns	p. 10
1.4 Some Exact Solutions for the Vibration of Non-uniform Beams	p. 19
1.4.1 The Governing Differential Equation	p. 21
1.5 Exact Solution for Buckling of Non-uniform Columns	p. 24
1.6 Other Direct Methods (FDM, FEM, DQM)	p. 28
1.7 Eisenberger's Exact Finite Element Method	p. 30
1.8 Semi-inverse or Semi-direct Methods	p. 35
1.9 Inverse Eigenvalue Problems	p. 43
1.10 Connection to the Work by Zyczkowski and Gajewski	p. 50
1.11 Connection to Functionally Graded Materials	p. 52
1.12 Scope of the Present Monograph	p. 53
Chapter 2 Unusual Closed-Form Solutions in Column Buckling	p. 55
2.1 New Closed-Form Solutions for Buckling of a Variable Flexural Rigidity Column	p. 55
2.1.1 Introductory Remarks	p. 55
2.1.2 Formulation of the Problem	p. 56
2.1.3 Uncovered Closed-Form Solutions	p. 57
2.1.4 Concluding Remarks	p. 65
2.2 Inverse Buckling Problem for Inhomogeneous Columns	p. 65
2.2.1 Introductory Remarks	p. 65
2.2.2 Formulation of the Problem	p. 65
2.2.3 Column Pinned at Both Ends	p. 66
2.2.4 Column Clamped at Both Ends	p. 68
2.2.5 Column Clamped at One End and Pinned at the Other	p. 69
2.2.6 Concluding Remarks	p. 70
2.3 Closed-Form Solution for the Generalized Euler Problem	p. 74
2.3.1 Introductory Remarks	p. 74
2.3.2 Formulation of the Problem	p. 76
2.3.3 Column Clamped at Both Ends	p. 79
2.3.4 Column Pinned at One End and Clamped at the Other	p. 79
2.3.5 Column Clamped at One End and Free at the Other	p. 81
2.3.6 Concluding Remarks	p. 83
2.4 Some Closed-Form Solutions for the Buckling of Inhomogeneous Columns under Distributed Variable Loading	p. 84
2.4.1 Introductory Remarks	p. 84
2.4.2 Basic Equations	p. 87
2.4.3 Column Pinned at Both Ends	p. 92
2.4.4 Column Clamped at Both Ends	p. 97
2.4.5 Column that is Pinned at One End and Clamped at the Other	p. 100
2.4.6 Concluding Remarks	p. 105
Chapter 3 Unusual Closed-Form Solutions for Rod Vibrations	p. 107
3.1 Reconstructing the Axial Rigidity of a Longitudinally Vibrating Rod by its Fundamental Mode Shape	p. 107
3.1.1 Introductory Remarks	p. 107
3.1.2 Formulation of the Problem	p. 108
3.1.3 Inhomogeneous Rods with Uniform Density	p. 109
3.1.4 Inhomogeneous Rods with Linearly Varying Density	p. 112
3.1.5 Inhomogeneous Rods with Parabolically Varying Inertial Coefficient	p. 114
3.1.6 Rod with General Variation of Inertial Coefficient (m [greater than sign] 2)	p. 115
3.1.7 Concluding Remarks	p. 118
3.2 The Natural Frequency of an Inhomogeneous Rod may be Independent of Nodal Parameters	p. 120
3.2.1 Introductory Remarks	p. 120
3.2.2 The Nodal Parameters	p. 121
3.2.3 Mode with One Node: Constant Inertial Coefficient	p. 124
3.2.4 Mode with Two Nodes: Constant Density	p. 127
3.2.5 Mode with One Node: Linearly Varying Material Coefficient	p. 129
3.3 Concluding Remarks	p. 131
Chapter 4 Unusual Closed-Form Solutions for Beam Vibrations	p. 135
4.1 Apparently First Closed-Form Solutions for Frequencies of Deterministically and/or Stochastically Inhomogeneous Beams (Pinned-Pinned Boundary Conditions)	p. 135
4.1.1 Introductory Remarks	p. 135
4.1.2 Formulation of the Problem	p. 136
4.1.3 Boundary Conditions	p. 137
4.1.4 Expansion of the Differential Equation	p. 138
4.1.5 Compatibility Conditions	p. 139
4.1.6 Specified Inertial Coefficient Function	p. 140
4.1.7 Specified Flexural Rigidity Function	p. 141
4.1.8 Stochastic Analysis	p. 144
4.1.9 Nature of Imposed Restrictions	p. 151
4.1.10 Concluding Remarks	p. 151
4.2 Apparently First Closed-Form Solutions for Inhomogeneous Beams (Other Boundary Conditions)	p. 152
4.2.1 Introductory Remarks	p. 152
4.2.2 Formulation of the Problem	p. 153
4.2.3 Cantilever Beam	p. 154
4.2.4 Beam that is Clamped at Both Ends	p. 163
4.2.5 Beam Clamped at One End and Pinned at the Other	p. 165
4.2.6 Random Beams with Deterministic Frequencies	p. 168
4.3 Inhomogeneous Beams that may Possess a Prescribed Polynomial Second Mode	p. 175
4.3.1 Introductory Remarks	p. 175
4.3.2 Basic Equation	p. 180
4.3.3 A Beam with Constant Mass Density	p. 182
4.3.4 A Beam with Linearly Varying Mass Density	p. 185
4.3.5 A Beam with Parabolically Varying Mass Density	p. 190
4.4 Concluding Remarks	p. 199
Chapter 5 Beams and Columns with Higher-Order Polynomial Eigenfunctions	p. 203
5.1 Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 1: Buckling	p. 203
5.1.1 Introductory Remarks	p. 203
5.1.2 Choosing a Pre-selected Mode Shape	p. 204
5.1.3 Buckling of the Inhomogeneous Column under an Axial Load	p. 205
5.1.4 Buckling of Columns under an Axially Distributed Load	p. 209
5.1.5 Concluding Remarks	p. 224
5.2 Family of Analytical Polynomial Solutions for Pinned Inhomogeneous Beams. Part 2: Vibration	p. 225
5.2.1 Introductory Comments	p. 225
5.2.2 Formulation of the Problem	p. 226
5.2.3 Basic Equations	p. 227
5.2.4 Constant Inertial Coefficient (m = 0)	p. 228
5.2.5 Linearly Varying Inertial Coefficient (m = 1)	p. 230
5.2.6 Parabolically Varying Inertial Coefficient (m = 2)	p. 231
5.2.7 Cubic Inertial Coefficient (m = 3)	p. 236
5.2.8 Particular Case m = 4	p. 239
5.2.9 Concluding Remarks	p. 242
Chapter 6 Influence of Boundary Conditions on Eigenvalues	p. 249
6.1 The Remarkable Nature of Effect of Boundary Conditions on Closed-Form Solutions for Vibrating Inhomogeneous Bernoulli-Euler Beams	p. 249
6.1.1 Introductory Remarks	p. 249
6.1.2 Construction of Postulated Mode Shapes	p. 250
6.1.3 Formulation of the Problem	p. 251
6.1.4 Closed-Form Solutions for the Clamped-Free Beam	p. 252
6.1.5 Closed-Form Solutions for the Pinned-Clamped Beam	p. 271
6.1.6 Closed-Form Solutions for the Clamped-Clamped Beam	p. 289
6.1.7 Concluding Remarks	p. 308
Chapter 7 Boundary Conditions Involving Guided Ends	p. 309
7.1 Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Pinned Support	p. 309
7.1.1 Introductory Remarks	p. 309
7.1.2 Formulation of the Problem	p. 310
7.1.3 Boundary Conditions	p. 310
7.1.4 Solution of the Differential Equation	p. 311
7.1.5 The Degree of the Material Density is Less than Five	p. 312
7.1.6 General Case: Compatibility Conditions	p. 318
7.1.7 Concluding Comments	p. 322
7.2 Closed-Form Solutions for the Natural Frequency for Inhomogeneous Beams with One Guided Support and One Clamped Support	p. 322
7.2.1 Introductory Remarks	p. 322
7.2.2 Formulation of the Problem	p. 323
7.2.3 Boundary Conditions	p. 323
7.2.4 Solution of the Differential Equation	p. 324
7.2.5 Cases of Uniform and Linear Densities	p. 325
7.2.6 General Case: Compatibility Condition	p. 327
7.2.7 Concluding Remarks	p. 329
7.3 Class of Analytical Closed-Form Polynomial Solutions for Guided-Pinned Inhomogeneous Beams	p. 330
7.3.1 Introductory Remarks	p. 330
7.3.2 Formulation of the Problem	p. 330
7.3.3 Constant Inertial Coefficient (m = 0)	p. 332
7.3.4 Linearly Varying Inertial Coefficient (m = 1)	p. 333
7.3.5 Parabolically Varying Inertial Coefficient (m = 2)	p. 335
7.3.6 Cubically Varying Inertial Coefficient (m = 3)	p. 337
7.3.7 Coefficient Represented by a Quartic Polynomial (m = 4)	p. 338
7.3.8 General Case	p. 340
7.3.9 Particular Cases Characterized by the Inequality n [greater than or equal] m + 2	p. 349
7.3.10 Concluding Remarks	p. 364
7.4 Class of Analytical Closed-Form Polynomial Solutions for Clamped-Guided Inhomogeneous Beams	p. 364
7.4.1 Introductory Remarks	p. 364
7.4.2 Formulation of the Problem	p. 364
7.4.3 General Case	p. 366
7.4.4 Constant Inertial Coefficient (m = 0)	p. 376
7.4.5 Linearly Varying Inertial Coefficient (m = 1)	p. 377
7.4.6 Parabolically Varying Inertial Coefficient (m = 2)	p. 378
7.4.7 Cubically Varying Inertial Coefficient (m = 3)	p. 380
7.4.8 Inertial Coefficient Represented as a Quadratic (m = 4)	p. 385
7.4.9 Concluding Remarks	p. 392
Chapter 8 Vibration of Beams in the Presence of an Axial Load	p. 395
8.1 Closed-Form Solutions for Inhomogeneous Vibrating Beams under Axially Distributed Loading	p. 395
8.1.1 Introductory Comments	p. 395
8.1.2 Basic Equations	p. 397
8.1.3 Column that is Clamped at One End and Free at the Other	p. 398
8.1.4 Column that is Pinned at its Ends	p. 402
8.1.5 Column that is clamped at its ends	p. 407
8.1.6 Column that is Pinned at One End and Clamped at the Other	p. 411
8.1.7 Concluding Remarks	p. 416
8.2 A Fifth-Order Polynomial that Serves as both the Buckling and Vibration Modes of an Inhomogeneous Structure	p. 417
8.2.1 Introductory Comments	p. 417
8.2.2 Formulation of the Problem	p. 419
8.2.3 Basic Equations	p. 421
8.2.4 Closed-Form Solution for the Pinned Beam	p. 422
8.2.5 Closed-Form Solution for the Clamped-Free Beam	p. 431
8.2.6 Closed-Form Solution for the Clamped-Clamped Beam	p. 442
8.2.7 Closed-Form Solution for the Beam that is Pinned at One End and Clamped at the Other	p. 452
8.2.8 Concluding Remarks	p. 460
Chapter 9 Unexpected Results for a Beam on an Elastic Foundation or with Elastic Support	p. 461
9.1 Some Unexpected Results in the Vibration of Inhomogeneous Beams on an Elastic Foundation	p. 461
9.1.1 Introductory Remarks	p. 461
9.1.2 Formulation of the Problem	p. 462
9.1.3 Beam with Uniform Inertial Coefficient, Inhomogeneous Elastic Modulus and Elastic Foundation	p. 463
9.1.4 Beams with Linearly Varying Density, Inhomogeneous Modulus and Elastic Foundations	p. 468
9.1.5 Beams with Varying Inertial Coefficient Represented as an mth Order Polynomial	p. 475
9.1.6 Case of a Beam Pinned at its Ends	p. 480
9.1.7 Beam Clamped at the Left End and Free at the Right End	p. 486
9.1.8 Case of a Clamped-Pinned Beam	p. 491
9.1.9 Case of a Clamped-Clamped Beam	p. 496
9.1.10 Case of a Guided-Pinned Beam	p. 501
9.1.11 Case of a Guided-Clamped Beam	p. 510
9.1.12 Cases Violated in Eq. (9.99)	p. 515
9.1.13 Does the Boobnov-Galerkin Method Corroborate the Unexpected Exact Results?	p. 517
9.1.14 Concluding Remarks	p. 521
9.2 Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Rotational Spring	p. 522
9.2.1 Introductory Remarks	p. 522
9.2.2 Basic Equations	p. 522
9.2.3 Uniform Inertial Coefficient	p. 523
9.2.4 Linear Inertial Coefficient	p. 526
9.3 Closed-Form Solution for the Natural Frequency of an Inhomogeneous Beam with a Translational Spring	p. 528
9.3.1 Introductory Remarks	p. 528
9.3.2 Basic Equations	p. 529
9.3.3 Constant Inertial Coefficient	p. 531
9.3.4 Linear Inertial Coefficient	p. 533
Chapter 10 Non-Polynomial Expressions for the Beam's Flexural Rigidity for Buckling or Vibration	p. 537
10.1 Both the Static Deflection and Vibration Mode of a Uniform Beam Can Serve as Buckling Modes of a Non-uniform Column	p. 537
10.1.1 Introductory Remarks	p. 537
10.1.2 Basic Equations	p. 538
10.1.3 Buckling of Non-uniform Pinned Columns	p. 539
10.1.4 Buckling of a Column under its Own Weight	p. 542
10.1.5 Vibration Mode of a Uniform Beam as a Buckling Mode of a Non-uniform Column	p. 544
10.1.6 Non-uniform Axially Distributed Load	p. 545
10.1.7 Concluding Remarks	p. 547
10.2 Resurrection of the Method of Successive Approximations to Yield Closed-Form Solutions for Vibrating Inhomogeneous Beams	p. 548
10.2.1 Introductory Comments	p. 548
10.2.2 Evaluation of the Example by Birger and Mavliutov	p. 551
10.2.3 Reinterpretation of the Integral Method for Inhomogeneous Beams	p. 553
10.2.4 Uniform Material Density	p. 555
10.2.5 Linearly Varying Density	p. 557
10.2.6 Parabolically Varying Density	p. 559
10.2.7 Can Successive Approximations Serve as Mode Shapes?	p. 563
10.2.8 Concluding Remarks	p. 563
10.3 Additional Closed-Form Solutions for Inhomogeneous Vibrating Beams by the Integral Method	p. 566
10.3.1 Introductory Remarks	p. 566
10.3.2 Pinned-Pinned Beam	p. 567
10.3.3 Guided-Pinned Beam	p. 575
10.3.4 Free-Free Beam	p. 582
10.3.5 Concluding Remarks	p. 590
Chapter 11 Circular Plates	p. 591
11.1 Axisymmetric Vibration of Inhomogeneous Clamped Circular Plates: an Unusual Closed-Form Solution	p. 591
11.1.1 Introductory Remarks	p. 591
11.1.2 Basic Equations	p. 593
11.1.3 Method of Solution	p. 594
11.1.4 Constant Inertial Term (m = 0)	p. 594
11.1.5 Linearly Varying Inertial Term (m = 1)	p. 595
11.1.6 Parabolically Varying Inertial Term (m = 2)	p. 596
11.1.7 Cubic Inertial Term (m = 3)	p. 598
11.1.8 General Inertial Term (m [greater than or equal] 4)	p. 600
11.1.9 Alternative Mode Shapes	p. 601
11.2 Axisymmetric Vibration of Inhomogeneous Free Circular Plates: An Unusual, Exact, Closed-Form Solution	p. 604
11.2.1 Introductory Remarks	p. 604
11.2.2 Formulation of the Problem	p. 605
11.2.3 Basic Equations	p. 605
11.2.4 Concluding Remarks	p. 607
11.3 Axisymmetric Vibration of Inhomogeneous Pinned Circular Plates: An Unusual, Exact, Closed-Form Solution	p. 607
11.3.1 Basic Equations	p. 607
11.3.2 Constant Inertial Term (m = 0)	p. 608
11.3.3 Linearly Varying Inertial Term (m = 1)	p. 609
11.3.4 Parabolically Varying Inertial Term (m = 2)	p. 610
11.3.5 Cubic Inertial Term (m = 3)	p. 612
11.3.6 General Inertial Term (m [greater than or equal] 4)	p. 614
11.3.7 Concluding Remarks	p. 616
Epilogue	p. 617
Appendices	p. 627
References	p. 653
Author Index	p. 711
Subject Index	p. 723

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