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Structural Vibration: Exact Solutions PDF

Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates offers an introduction to structural vibration and highlights the importance of the natural frequencies in design. It focuses on free vibrations for analysis and design of structures and machine and presents the exact vibration solutions for strings, membranes, beams, and plates.

This book emphasizes the exact solutions for free transverse vibration of strings, membranes, beams, and plates. It explains the intrinsic, fundamental, and unexpected features of the solutions in terms of known functions as well as solutions determined from exact characteristic equations. The book provides:

  • A single-volume resource for exact solutions of vibration problems in strings, membranes, beams, and plates
  • A reference for checking vibration frequency values and mode shapes of structural problems
  • Governing equations and boundary conditions for vibration of structural elements
  • Analogies of vibration problems

Structural Vibration: Exact Solutions for Strings, Membranes, Beams, and Plates provides practicing engineers, academics, and researchers with a reference for data on a specific structural member as well as a benchmark standard for numerical or approximate analytical methods.

Review – Structural Vibration: Exact Solutions

“The book is arranged in a way from simple to complex, enabling an easy walk into the whole subject. The exact solutions are illustrated not only by the analytical expressions, but also by numerical results in table or figure form. Thus, they can be directly used as benchmarks for comparison with numerical or approximate analytical solutions, or they can be simply taken over in evaluating the dynamic behavior of a practical structure. … The materials presented are very solid. This book presents a collection of exact solutions for vibration of strings, beams (bars), and plates, which are very common both in engineering and natural science. Complicating effects can be involved, however, including non-uniform geometry, non-homogeneous material property, internal support, elastic foundation, etc. The mathematical techniques used to obtain the solutions are very attractive and mathematically strict, and will provide a base for the study of more involved problems. The book should be very useful for researchers, engineers, and students in various engineering areas such as aerospace, civil engineering, mechanical engineering, ocean engineering, chemical engineering, etc. It can also be used as a reference for those who are working in physics, biology, geology, material science, and nanotechnology.”
––W. Q. Chen, Department of Engineering Mechanics, Zhejiang University, Hangzhou, China

“The book provides an excellent comprehensive treatment of analytical vibration analysis of continuous structural systems. It provides the exact solutions for strings, membranes, beams, and plates along with tabulated numerical frequencies which makes this book essential not only for academics but also for practicing engineers and designers. … The authors have done an admirable job in the organization and the flow of the book. Direct explanations tell the story of structural vibrations, moving from basic string model to more complex non-uniform plates. In each chapter, the authors present the current research development based upon the latest research articles.”
––Huseyin Yuce, New York City College of Technology, Brooklyn, USA

“This book is a new reference on a special topic (exact eigensolutions) for certain structural components, which can be quite useful for people in R&D of structural systems, educators of engineering vibrations, and developers of numerical algorithms for structural vibration problems.”
––Bingen Yang, Dept. of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA, U.S.A

Preface

There is a staggering number of research studies on the vibration of structures. Based on a simple search using the Science Citation Index, the numbers of references associated with the following words are 1,000 for “vibration and string,” 2,000 for “vibration and membrane,” 7,000 for “vibration and plate,” and 16,000 for “vibration and beam, bar or rod.” This clearly illustrates the importance of the subject of free and forced vibrations for analysis and design of structures and machines. The free vibration of a structural member eventually ceases due to energy dissipation, either from the material strains or from the resistance of the surrounding fluid.

The frequency of such a system will be lowered by damping. But since damping also causes the amplitude to decay, the resonance with a forced excitation of a strongly damped system will not be as important as the weakly damped system. In this book, we shall consider the undamped system, which models the weakly damped system, and only focus on the exact solutions for free transverse vibration of strings, bars, membranes, and plates because these solutions elucidate the intrinsic, fundamental, and unexpected features of the solutions. They also serve as benchmarks to assess the validity, convergence, and accuracy of numerical methods and approximate analytical methods. We define exact solutions to mean solutions in terms of known functions as well as those solutions determined from exact characteristic equations.

However, this book will not cover longitudinal in-plane/ translational vibrations, shear waves, torsional oscillations, infinite domains (wave propagation), discrete systems (such as linked masses), and frames. The exact solutions for a wide range of differential equations are useful to academics teaching differential equations, as they may draw the practical problems associated with the differential equations.

There are seven chapters in this book. Chapter 1 gives the introduction to structural vibration and the importance of the natural frequencies in design. Chapter 2 presents the vibration solutions for strings. Chapter 3 presents the vibration solutions for membranes. Chapter 4 deals with vibration of bars and beams. Chapter 5 gives the vibration solutions for isotropic plates with uniform thickness. Chapter 6 deals with plates with complicating effects such as the presence of in-plane forces, internal spring support, internal hinge, elastic foundation, and nonuniform thickness distribution. Chapter 7 presents vibration solutions for nonisotropic plates, such as orthotropic, sandwich, laminated, and functionally graded plates. Owing to the vastness of the literature, there may be relevant papers that escaped our search in the Science Citation Index. To these authors, we offer our sincere apology. Such omissions shall be rectified in a future edition.

Finally, we wish to express our thanks to Dr. Tay Zhi Yung and Mr. Ding Zhiwei of the National University of Singapore for checking the manuscript and plotting the vibration mode shapes and also to Dr. Liu Bo of The Solid Mechanics Research Centre, Beihang University, China, for contributing the sections on rectangular isotropic and orthotropic Mindlin plates.

Contents – Structural Vibration: Exact Solutions


Preface………………………………………………………………………………………………………..xi
About the Authors…………………………………………………………………………………….. xiii
Chapter 1 Introduction to Structural Vibration……………………………………………..1
1.1 What is Vibration?……………………………………………………………1
1.2 Brief Historical Review on Vibration of Strings,
Membranes, Beams, and Plates………………………………………….2
1.3 Importance of Vibration Analysis in Structural Design ………..4
1.4 Scope of Book …………………………………………………………………6
References ………………………………………………………………………………..6
Chapter 2 Vibration of Strings……………………………………………………………………9
2.1 Introduction …………………………………………………………………….9
2.2 Assumptions and Governing Equations for Strings………………9
2.3 Boundary Conditions……………………………………………………… 10
2.4 Constant Property String………………………………………………… 11
2.5 Two-Segment Constant Property String……………………………. 12
2.5.1 Different Densities …………………………………………….. 13
2.5.2 A Mass Attached on the Span ……………………………… 16
2.5.3 A Supporting Spring on the Span ………………………… 18
2.6 Transformation for Nonuniform Tension and Density ………… 18
2.7 Constant Tension and Variable Density……………………………..20
2.7.1 Power Law Density Distribution …………………………..20
2.7.2 Exponential Density Distribution………………………….22
2.8 Variable Tension and Constant Density……………………………..25
2.8.1 Vertical String Fixed at Both Ends ……………………….26
2.8.2 Vertical String with Sliding Spring on Top
and a Free Mass at the Bottom……………………………..28
2.9 Free-Hanging Nonuniform String…………………………………….30
2.10 Other Combinations ………………………………………………………. 31
References ……………………………………………………………………………… 31
Chapter 3 Vibration of Membranes…………………………………………………………… 33
3.1 Introduction ………………………………………………………………….. 33
3.2 Assumptions and Governing Equations……………………………. 33
3.3 Constant Uniform Normal Stress and Constant Density ……..34
3.3.1 Rectangular Membrane……………………………………….34
3.3.2 Three Triangular Membranes ……………………………… 35
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3.3.3 Circular and Annular Membranes…………………………38
3.3.4 Circular Sector Membrane and Annular Sector
Membrane………………………………………………………….40
3.4 Two-Piece Constant-Property Membranes………………………… 42
3.4.1 Two-Piece Rectangular Membrane………………………. 42
3.4.2 Two-Piece Circular Membrane …………………………….44
3.5 Nonhomogeneous Membranes………………………………………… 47
3.5.1 Rectangular Membrane with Linear Density
Distribution………………………………………………………..49
3.5.2 Rectangular Membrane with Exponential
Density Distribution …………………………………………… 51
3.5.3 Nonhomogeneous Circular or Annular Membrane…. 52
3.5.3.1 Power Law Density Distribution…………….. 52
3.5.3.2 A Special Annular Membrane………………..58
3.6 Hanging Membranes ………………………………………………………60
3.6.1 Membrane with a Free, Weighted Bottom Edge …….. 61
3.6.2 Vertical Membrane with All Sides Fixed……………….63
3.7 Discussion……………………………………………………………………..66
References ………………………………………………………………………………68
Chapter 4 Vibration of Beams …………………………………………………………………. 71
4.1 Introduction ………………………………………………………………….. 71
4.2 Assumptions and Governing Equations……………………………. 71
4.3 Single-Span Constant-Property Beam……………………………….73
4.3.1 General Solutions……………………………………………….73
4.3.2 Classical Boundary Conditions with Axial Force…… 75
4.3.3 Elastically Supported Ends ………………………………….82
4.3.4 Cantilever Beam with a Mass at One End ……………..83
4.3.5 Free Beam with Two Masses at the Ends……………….84
4.4 Two-Segment Uniform Beam…………………………………………..85
4.4.1 Beam with an Internal Elastic Support ………………….86
4.4.2 Beam with an Internal Attached Mass…………………..89
4.4.3 Beam with an Internal Rotational Spring ………………93
4.4.4 Stepped Beam…………………………………………………….95
4.4.5 Beam with a Partial Elastic Foundation…………………99

Structural Vibration: Exact Solutions
4.5 Nonuniform Beam……………………………………………………….. 109
4.5.1 Bessel-Type Solutions……………………………………….. 110
4.5.1.1 The Beam with Linear Taper……………….. 113
4.5.1.2 Two-Segment Symmetric Beams with
Linear Taper………………………………………. 114
4.5.1.3 Linearly Tapered Cantilever with
an End Mass………………………………………. 116
4.5.1.4 Other Bessel-Type Solutions………………… 122
4.5.2 Power-Type Solutions……………………………………….. 122
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4.5.2.1 Results for m = 6, n = 2………………………. 128
4.5.2.2 Results for m = 8, n = 4………………………. 128
4.5.3 Isospectral Beams and the m = 4, n = 4 Case ……… 130
4.5.4 Exponential-Type Solutions……………………………….. 133
4.6 Discussion…………………………………………………………………… 136
References ……………………………………………………………………………. 137
Chapter 5 Vibration of Isotropic Plates……………………………………………………. 139
5.1 Introduction ………………………………………………………………… 139
5.2 Governing Equations and Boundary Conditions for
Vibrating Thin Plates …………………………………………………… 139
5.3 Exact Vibration Solutions for Thin Plates……………………….. 141
5.3.1 Rectangular Plates with Four Edges Simply
Supported ……………………………………………………….. 141
5.3.2 Rectangular Plates with Two Parallel Sides
Simply Supported…………………………………………….. 142
5.3.3 Rectangular Plates with Clamped but Vertical
Sliding Edges…………………………………………………… 151
5.3.4 Triangular Plates with Simply Supported Edges ….. 155
5.3.5 Circular Plates…………………………………………………. 157
5.3.6 Annular Plates…………………………………………………. 160
5.3.7 Annular Sector Plates……………………………………….. 161
5.4 Governing Equations and Boundary Conditions for
Vibrating Thick Plates………………………………………………….. 172
5.5 Exact Vibration Solutions for Thick Plates ……………………… 184
5.5.1 Polygonal Plates with Simply Supported Edges……. 184
5.5.2 Rectangular Plates……………………………………………. 185
5.5.3 Circular Plates…………………………………………………. 197
5.5.4 Annular Plates………………………………………………….200
5.5.5 Sectorial Plates………………………………………………… 201
5.6 Vibration of Thick Rectangular Plates Based on 3-D
Elasticity Theory ………………………………………………………….209
References ……………………………………………………………………………. 211
Chapter 6 Vibration of Plates with Complicating Effects…………………………… 215
6.1 Introduction ………………………………………………………………… 215
6.2 Plates with In-Plane Forces…………………………………………… 215
6.2.1 Rectangular Plates with In-Plane Forces…………….. 215
6.2.1.1 Analogy with Beam Vibration……………… 217
6.2.1.2 Plates with Free Vertical Edge …………….. 218

Structural Vibration: Exact Solutions
6.2.2 Circular Plates with In-Plane Forces…………………… 221
6.3 Plates with Internal Spring Support ………………………………..224
6.3.1 Rectangular Plates with Line Spring Support……….225
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6.3.1.1 Case 1: All Sides Simply Supported………226
6.3.1.2 Case 2: Both Horizontal Sides Simply
Supported and Both Vertical Sides
Clamped…………………………………………….227
6.3.2 Circular Plates with Concentric Spring Support……227
6.3.2.1 Case 1: Plate Is Simply Supported at
the Edge …………………………………………….229
6.3.2.2 Case 2: Plate Is Clamped at the Edge…….229
6.3.2.3 Case 3: Free Plate with Support……………. 231
6.4 Plates with Internal Rotational Hinge …………………………….. 232
6.4.1 Rectangular Plates with Internal Rotational Hinge ….232
6.4.1.1 Case 1: All Sides Simply Supported……… 233
6.4.1.2 Case 2: Two Parallel Sides Simply
Supported, with a Midline Internal
Rotational Spring Parallel to the Other
Two Clamped Sides ……………………………. 233
6.4.2 Circular Plates with Concentric Internal
Rotational Hinge ……………………………………………… 233
6.4.2.1 Case 1: Plate Is Simply Supported at
the Edge ……………………………………………. 235
6.4.2.2 Case 2: Plate Is Clamped at the Edge……. 235
6.4.2.3 Case 3: Plate Is Free at the Edge …………..236
6.5 Plates with Partial Elastic Foundation……………………………..236
6.5.1 Plates with Full Foundation ………………………………. 237
6.5.2 Rectangular Plates with Partial Foundation………….238
6.5.3 Circular Plates with Partial Foundation……………….238
6.5.3.1 Case 1: Plate Is Simply Supported at
the Edge …………………………………………….240
6.5.3.2 Case 2: Plate Is Clamped at the Edge…….240
6.5.3.3 Case 3: Plate Is Free at the Edge …………..240
6.6 Stepped Plates……………………………………………………………… 241
6.6.1 Stepped Rectangular Plates……………………………….. 241
6.6.1.1 Case 1: Plate Is Simply Supported on
All Sides…………………………………………….244
6.6.1.2 Case 2: Plate Is Simply Supported
on Opposite Sides and Clamped on
Opposite Sides ……………………………………244
6.6.2 Stepped Circular Plates……………………………………..245
6.6.2.1 Case 1: Circular Plate with Simply
Supported Edge…………………………………..247
6.6.2.2 Case 2: Circular Plate with Clamped
Edge ………………………………………………….247
6.6.2.3 Case 3: Circular Plate with Free Edge……247
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Structural Vibration: Exact Solutions
© 2010 Taylor & Francis Group, LLC
6.7 Variable-Thickness Plates………………………………………………249
6.7.1 Case 1: Constant Density with Parabolic Thickness….251
6.7.2 Case 2: Parabolic Sandwich Plate ………………………. 252
6.8 Discussion…………………………………………………………………… 252
References ……………………………………………………………………………. 253
Chapter 7 Vibration of Nonisotropic Plates……………………………………………… 255
7.1 Introduction ………………………………………………………………… 255
7.2 Orthotropic Plates………………………………………………………… 255
7.2.1 Governing Vibration Equation …………………………… 255
7.2.2 Principal Rigidities for Special Orthotropic Plates…258
7.2.2.1 Corrugated Plates………………………………..258
7.2.2.2 Plate Reinforced by Equidistant
Ribs/Stiffeners……………………………………259
7.2.2.3 Steel-Reinforced Concrete Slabs…………..260
7.2.2.4 Multicell Slab with Transverse
Diaphragm ………………………………………… 261
7.2.2.5 Voided Slabs ……………………………………… 261
7.2.3 Simply Supported Rectangular Orthotropic
Plates………………………………………………………… 262
7.2.4 Rectangular Orthotropic Plates with Two
Parallel Sides Simply Supported…………………………262
7.2.4.1 Two Parallel Edges (i.e., y = 0 and
y = b) Simply Supported, with Simply
Supported Edge x = 0 and Free Edge
x = a (designated as SSSF plates)………….264
7.2.4.2 Two Parallel Edges (i.e., y = 0, and
y = b) Simply Supported, with Clamped
Edge x = 0 and Free Edge x = a
(designated as SCSF plates) …………………..264
7.2.4.3 Two Parallel Edges (i.e., y = 0 and
y = b) Simply Supported, with
Clamped Edges x = 0 and x = a
(designated as SCSC plates)………………….265
7.2.4.4 Two Parallel Edges (i.e., y = 0 and
y = b) Simply Supported, with Clamped
Edge x = 0 and Simply Supported Edge
x = a (designated as SCSS plates)…………..265
7.2.4.5 Two Parallel Edges (i.e., y = 0 and
y = b) Simply Supported, with Free
Edges x = 0 and x = a (designated as
SFSF plates)…………………………………………… 265
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7.2.5 Rectangular Orthotropic Thick Plates …………………265
7.2.6 Circular Polar Orthotropic Plates……………………….. 279
7.3 Sandwich Plates……………………………………………………………280
7.4 Laminated Plates…………………………………………………………. 281
7.5 Functionally Graded Plates ……………………………………………286
7.6 Concluding Remarks…………………………………………………….289
References …………………………………………………………………………….290

1.1 WHAT IS VIBRATION? – Structural Vibration: Exact Solutions

Vibration may be regarded as any motion that repeats itself after an interval of time, or one may define vibrations as oscillations of a system about a position of equilibrium (Kelly 2007). Examples of vibratory motion include the swinging of a pendulum, the motion of a plucked guitar string, tidal motion, the chirping of a male cicada by rubbing its wings, the flapping of airplane wings in turbulence, the soothing motion of a massage chair, or the swaying of a slender tall building due to wind or an earthquake.

The key parameters in describing vibration are amplitude, period, and frequency. The amplitude of vibration is the maximum displacement of a vibrating particle or body from its position of equilibrium, and this is related to the applied energy. The period is the time taken for one complete cycle of the motion. The frequency is the number of cycles per unit time or the reciprocal of the period. The angular (or circular) frequency is the product of the frequency and 2π, and hence its unit is radians per unit time. Vibrations may be classified as either free vibration or forced vibration. Free vibration takes place when a system oscillates under the action of forces inherent within the system itself—when externally imposed forces are absent.

A system under free vibration will vibrate at one or more of its natural frequencies, which are dependent on the mass and stiffness distributions as well as the boundary conditions. In contrast, forced vibration occurs when an external periodic force is applied to the system. When the effects of friction can be neglected, the vibrations are referred to as undamped. Realistically, all vibrations are damped to some degree. If a free vibration is only slightly damped, its amplitude gradually decreases until the motion comes to an end after a certain time. If the damping is sufficiently large, vibration is suppressed, and the system then quickly regains its original equilibrium position. A damped forced vibration is maintained so long as the periodic force that causes the vibration is applied.

The amplitude of the vibration is affected by the magnitude of the damping forces. From an energy viewpoint, vibration may be defined as a phenomenon that involves alternating interchange of potential energy and kinetic energy. If the system is damped, then some energy is dissipated in each cycle of the vibration, and the vibratory motion will ultimately come to an end. If a steady motion of vibration is to be maintained, then the energy dissipated due to damping has to be compensated by an external source

About the Author – Structural Vibration: Exact Solutions

C. Y. Wang is a professor in the Department of Mathematics with a joint appointment in mechanical engineering at the Michigan State University, East Lansing, Michigan. He obtained his BS from Taiwan University and PhD from Massachusetts Institute of Technology. Prof. Wang has published about 170 papers in solid mechanics (elastica, torsion, buckling, and vibrations of structural members), 170 papers in fluid mechanics (exact Navier-Stokes solutions, Stokes flow, unsteady viscous flow), and 120 papers in other areas (biological, thermal, electromechanics). Prof. Wang wrote a monograph Perturbation Methods (Taiwan University Press) and is a coauthor of Exact Solutions for Buckling of Structural Members (CRC Press). He has served as a technical editor for Applied Mechanics Reviews.

C. M. Wang is a professor in the Department of Civil and Environmental Engineering and the director of the Engineering Science Programme, Faculty of Engineering, National University of Singapore. He is a chartered structural engineer, a fellow of the Singapore Academy of Engineering, a fellow of the Institution of Engineers Singapore, and a fellow of the Institution of Structural Engineers. His research interests are in the areas of structural stability, vibration, optimization, plated structures, and Mega-Floats. He has published more than 400 scientific publications, co-edited three books, Analysis and Design of Plated Structures: Stability and Dynamics: Volumes 1 and 2 (Woodhead Publishing) and Very Large Floating Structures (Taylor & Francis) and co-authored three books: Vibration of Mindlin Plates (Elsevier), Shear Deformable Beams and Plates: Relationships with Classical Solutions (Elsevier), and Exact Solutions for Buckling of Structural Members (CRC Press). He is the editor-in-chief of the International Journal of Structural Stability and Dynamics and the IES Journal Part A: Civil and Structural Engineering and an editorial board member of Engineering Structures, Advances in Applied Mathematics and Mechanics, Ocean Systems Engineering, and International Journal of Applied Mechanics.

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