Asymptotic methods in the theory of plates with mixed boundary conditions

Title:

Publication Information:

New York : Wiley, 2014

Physical Description:

xi, 272 p. ; 25 cm.

ISBN:

9781118725191

Subject Term:

Plates (Engineering) -- Mathematical models

Asymptotic expansions

Added Author:

Andrianov, Igor V.

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Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010325330	TA660.P6 A62 2014	Open Access Book	Book	Searching... Unknown

Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions comprehensively covers the theoretical background of asymptotic approaches and their use in solving mechanical engineering-oriented problems of structural members, primarily plates (statics and dynamics)
with mixed boundary conditions.

The first part of this book introduces the theory and application of asymptotic methods and includes a series of approaches that have been omitted or not rigorously treated in the existing literature. These lesser known approaches include the method of summation and construction of the asymptotically equivalent functions, methods of small and large delta, and the homotopy perturbations method.

The second part of the book contains original results devoted to the solution of the mixed problems of the theory of plates, including statics, dynamics and stability of the studied objects. In addition, the applicability of the approaches presented to other related linear or nonlinear problems is addressed.

Key features:

* Includes analytical solving of mixed boundary value problems

* Introduces modern asymptotic and summation procedures

* Presents asymptotic approaches for nonlinear dynamics of rods, beams and plates

* Covers statics, dynamics and stability of plates with mixed boundary conditions

* Explains links between the Adomian and homotopy perturbation approaches

Asymptotic Methods in the Theory of Plates with Mixed Boundary Conditions is a comprehensive reference for researchers and practitioners working in the field of Mechanics of Solids and Mechanical Engineering, and is also a valuable resource for graduate and postgraduate students from Civil and Mechanical Engineering.

Author Notes

Igor V. Andrianov obtained his Master (1971) and PhD (1975) degrees from the Dnepropetrovsk State University (Ukraine). He obtained the Doctor of Sciences degree in Mechanics of Solids from the Moscow State Institute of Electronics and Mathematics (Russia) in 1990. During 1974-1977, he was a Research Scientist at the Dnipropetrovs'k State University, during 1977-1990, an Associated Professor, and during 1990-1997, a Full Professor of Mathematics in the Dnepropetrovsk Civil Engineering Institute. Currently he is a Research Scientist in the RWTH Aachen University in Germany. Prof. Andrianov is the author or co-author of 12 books and over 250 papers in refereed journals and conference proceedings. He has presented papers at numerous International Conferences, and he has supervised 25 PhD Theses. His research interests are in Mechanics of Solids, Nonlinear Dynamics, and Asymptotic Methods.

Jan Awrejcewicz graduated from Lodz University of Technology in 1977 (Mechanics) and from the University of Lodz in 1978 (Philosophy). He obtained his PhD (Habilitation) in 1981 (1990), and become a Full Professor in 1997. He has authored and/or co-authored 17 monographs in English; 2 textbooks; 12 edited conference proceedings; 275 journal papers; 340 conference papers; 18 chapters in books. He has served as an editor of 9 books, and as a Guest-Editor of 15 journal special issues. His research includes Nonlinear Mechanics, Mechatronics and Control, and Biomechanics. He is a recipient of the Humboldt Research Award.

Vladyslav V. Danishevs'kyy obtained his Masters (1996), Ph.D. (1999) degrees, and Doctor of Sciences degree in Structural Mechanics (2008) from the Prydniprovska State Academy of Civil Engineering and Architecture, Ukraine. He is a Professor at this State Academy. He has authored 2 monographs and over 70 refereed papers. Among his awards are the Soros Post-Graduate Student's Award (1997), Prize of the National Academy of Sciences of Ukraine for the best academic achievement among young scientists (2000), Alexander von Humboldt Foundation Research Fellowship (2001), NATO Research Fellowship (2003), NATO Reintegration Grant (2005), and institutional academic co-operation grant of the Alexander von Humboldt Foundation (2007). He conducted research at the Institute of General Mechanics in the RWTH Aachen University, Germany (2001-2002) and at the Group of Physics of Materials in the University of Rouen, France (2003-2004). His research interests are in asymptotic methods, nonlinear dynamics, and heterogeneous materials and structures.

Andrey O. Ivankov, PhD is the author of more than author of more than 30 research publications. His main areas of research include ODE, PDE, Mixed BVPs and Padé approximants.

Preface	p. ix
List of Abbreviations	p. xiii
1 Asymptotic Approaches	p. 1
1.1 Asymptotic Series and Approximations	p. 1
1.1.1 Asymptotic Series	p. 1
1.1.2 Asymptotic Symbols and Nomenclatures	p. 5
1.2 Some Nonstandard Perturbation Procedures	p. 8
1.2.1 Choice of Small Parameters	p. 8
1.2.2 Homotopy Perturbation Method	p. 10
1.2.3 Method of Small Delta	p. 13
1.2.4 Method of Large Delta	p. 17
1.2.5 Application of Distributions	p. 19
1.3 Summation of Asymptotic Series	p. 21
1.3.1 Analysis of Power Series	p. 21
1.3.2 Padé Approximants and Continued Fractions	p. 24
1.4 Some Applications of PA	p. 29
1.4.1 Accelerating Convergence of Iterative Processes	p. 29
1.4.2 Removing Singularities and Reducing the Gibbs-Wilbraham Effect	p. 31
1.4.3 Localized Solutions	p. 32
1.4.4 Hennite-Padé Approximations and Bifurcation Problem	p. 34
1.4.5 Estimates of Effective Characteristics of Composite Materials	p. 34
1.4.6 Continualization	p. 35
1.4.7 Rational Interpolation	p. 36
1.4.8 Some Other Applications	p. 37
1.5 Matching of Limiting Asymptotic Expansions	p. 38
1.5.1 Method of Asymptotically Equivalent Functions for Inversion of Laplace Transform	p. 38
1.5.2 Two-Point PA	p. 41
1.5.3 Other Methods of AEFs Construction	p. 43
1.5.4 Example: Schrödinger Equation	p. 45
1.5.5 Example: AEFs in the Theory of Composites	p. 46
1.6 Dynamical Edge Effect Method	p. 49
1.6.1 Linear Vibrations of a Rod	p. 49
1.6.2 Nonlinear Vibrations of a Rod	p. 51
1.6.3 Nonlinear Vibrations of a Rectangular Plate	p. 54
1.6.4 Matching of Asymptotic and Variational Approaches	p. 58
1.6.5 On the Normal Forms of Nonlinear Vibrations of Continuous Systems	p. 60
1.7 Continualization	p. 61
1.7.1 Discrete and Continuum Models in Mechanics	p. 61
1.7.2 Chain of Elastically Coupled Masses	p. 62
1.7.3 Classical Continuum Approximation	p. 64
1.7.4 "Splashes"	p. 65
1.7.5 Envelope Continualization	p. 66
1.7.6 Improvement Continuum Approximations	p. 68
1.7.7 Forced Oscillations	p. 69
1.8 Averaging and Homogenization	p. 71
1.8.1 Averaging via Multiscale Method	p. 71
1.8.2 Frozing in Viscoelastic Problems	p. 74
1.8.3 The WKB Method	p. 75
1.8.4 Method of Kuzmak-Wliitham (Nonlinear WKB Method)	p. 11
1.8.5 Differential Equations with Quickly Changing Coefficients	p. 79
1.8.6 Differential Equation with Periodically Discontinuous Coefficients	p. 84
1.8.7 Periodically Perforated Domain	p. 88
1.8.8 Waves in Periodically Nonhomogenous Media	p. 92
References	p. 95
2 Computational Methods for Plates and Beams with Mixed Boundary Conditions	p. 105
2.1 Introduction	p. 105
2.1.1 Computational Methods of Plates with Mixed Boundary Conditions	p. 105
2.1.2 Method of 'Boundary Conditions Perturbation	p. 107
2.2 Natural Vibrations of Beams and Plates	p. 109
2.2.1 Natural Vibrations of a Clamped Beam	p. 109
2.2.2 Natural Vibration of a Beam with Free Ends	p. 114
2.2.3 Natural Vibrations of a Clamped Rectangular Plate	p. 118
2.2.4 Natural Vibrations of the Orthotropic Plate with Free Edges Lying on an Elastic Foundation	p. 123
2.2.5 Natural Vibrations of the Plate with Mixed Boundary Conditions "Clamping-Simple Support"	p. 128
2.2.6 Comparison of Theoretical and Experimental Results	p. 133
2.2.7 Natural Vibrations of a Partially Clamped Plate	p. 135
2.2.8 Natural Vibrations of a Plate with Mixed Boundary Conditions "Simple Support-Moving Clamping"	p. 140
2.3 Nonlinear Vibrations of Rods, Beams and Plates	p. 144
2.3.1 Vibrations of the Rod Embedded in a Nonlinear Elastic Medium	p. 144
2.3.2 Vibrations of the Beam Lying on a Nonlinear Elastic Foundation	p. 153
2.3.3 Vibrations of the Membrane on a Nonlinear Elastic Foundation	p. 155
2.3.4 Vibrations of the Plate on a Nonlinear Elastic Foundation	p. 158
2.4 SSS of Beams and Plates	p. 160
2.4.1 SSS of Beams with Clamped Ends	p. 160
2.4.2 SSS of the Beam with Free Edges	p. 163
2.4.3 SSS of Clamped Plate	p. 166
2.4.4 SSS of a Plate with Free Edges	p. 170
2.4.5 SSS of the Plate with Mixed Boundary Conditions "Clamping-Simple Support"	p. 172
2.4.6 SSS of a Plate with Mixed Boundary Conditions "Free Edge-Moving Clamping"	p. 180
2.5 Forced Vibrations of Beams and Plates	p. 184
2.5.1 Forced Vibrations of a Clamped Beam	p. 184
2.5.2 Forced Vibrations of Beam with Free Edges	p. 189
2.5.3 Forced Vibrations of a Clamped Plate	p. 190
2.5.4 Forced Vibrations of Plates with Free Edges	p. 194
2.5.5 Forced Vibrations of Plate with Mixed Boundary Conditions "Clamping-Simple Support"	p. 197
2.5.6 Forced Vibrations of Plate with Mixed Boundary Conditions "Free Edge - Moving Clamping"	p. 202
2.6 Stability of Beams and Plates	p. 207
2.6.1 Stability of a Clamped Beam	p. 207
2.6.2 Stability of a Clamped Rectangular Plate	p. 209
2.6.3 Stability of Rectangular Plate with Mixed Boundary Conditions "Clamping-Simple Support	p. 211
2.6.4 Comparison of Theoretical and Experimental Results	p. 219
2.7 Some Related Problems	p. 221
2.7.1 Dynamics of Nonhomogeneous Structures	p. 221
2.7.2 Method of Ishlinskii-Leibenzon	p. 224
2.7.3 Vibrations of a String Attached to a Spring-Mass-Dashpot System	p. 230
2.7.4 Vibrations of a String with Nonlinear BCs	p. 233
2.7.5 Boundary Conditions and First Order Approximation Theory	p. 238
2.8 Links between the Adomian and Homotopy Perturbation Approaches	p. 240
2.9 Conclusions	p. 263
References	p. 264
Index	p. 269

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