Modern Trends in Structural and Solid Mechanics 2

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Группа авторов. Modern Trends in Structural and Solid Mechanics 2

Table of Contents

List of Illustrations

List of Tables

Guide

Pages

Modern Trends in Structural and Solid Mechanics 2. Vibrations

Preface. Short Bibliographical Presentation of Prof. Isaac Elishakoff

Books by Elishakoff

Books edited or co-edited by Elishakoff

1. Bolotin’s Dynamic Edge Effect Method Revisited (Review)

1.1. Introduction

1.2. Toy problem: natural beam oscillations

1.3. Linear problems solved

1.4. Generalization for the nonlinear case

1.5. DEEM and variational approaches

1.6. Quasi-separation of variables and normal modes of nonlinear oscillations of continuous systems

1.7. Short-wave (high-frequency) asymptotics. Possible generalizations of DEEM

1.8. Conclusion: DEEM, highly recommended

1.9. Acknowledgments

1.10. Appendix

1.11. References

2. On the Principles to Derive Plate Theories. 2.1. Introduction

2.2. Some historical remarks

2.3. Possibilities to formulate plate theories

2.3.1. Theories based on hypotheses

2.3.2. Reduction of the governing equations by mathematical techniques

2.3.3. Direct approach

2.3.4. Consistent approach

2.4. Shear correction

2.5. Conclusion

2.6. References

3. A Softening–Hardening Nanomechanics Theory for the Static and Dynamic Analyses of Nanorods and Nanobeams: Doublet Mechanics. 3.1. Introduction

3.2. Doublet mechanics formulation

3.3. Governing equations. 3.3.1. Static equilibrium equations of a nanorod with periodic microand nanostructures

3.3.2. Equations of motion of a nanorod with periodic micro- and nanostructures

3.3.3. Static equilibrium equations of a nanobeam with periodic micro- and nanostructures

3.3.4. Equations of motion of a nanobeam with periodic micro- and nanostructures

3.4. Analytical solutions. 3.4.1. Axial deformation of nanorods with periodic nanostructures

3.4.2. Vibration analysis of nanorods with periodic nanostructures

3.4.3. Axial wave propagation in nanorods with periodic nanostructures

3.4.4. Flexural deformation of nanobeams with periodic nanostructures

3.4.5. Buckling analysis of nanobeams with periodic nanostructures

3.4.6. Vibration analysis of nanobeams with periodic nanostructures

3.4.7. Flexural wave propagation in nanobeams with periodic nanostructures

3.5. Numerical results

3.6. Conclusion

3.7. References

4. Free Vibration of Micro-Beams and Frameworks Using the Dynamic Stiffness Method and Modified Couple Stress Theory

4.1. Introduction

4.2. Formulation of the potential and kinetic energies

4.3. Derivation of the governing differential equations

4.4. Development of the dynamic stiffness matrix

4.4.1. Axial stiffnesses

4.4.2. Bending stiffnesses

4.4.3. Combination of axial and bending stiffnesses

4.4.4. Transformation matrix

4.5. Application of the Wittrick–Williams algorithm

4.6. Numerical results and discussion

4.7. Conclusion

4.8. Acknowledgments

4.9. References

5. On the Geometric Nonlinearities in the Dynamics of a Planar Timoshenko Beam

5.1. Introduction

5.2. The geometrically exact planar Timoshenko beam

5.3. The asymptotic solution

5.4. The importance of nonlinear terms. 5.4.1. An initial case

5.4.2. The effect of the slenderness

5.4.3. The effect of the end spring

5.4.4. The effect of the resonance order

5.5. Simplified models

5.5.1. Neglecting axial inertia

5.5.2. One-field equation

5.5.3. The Euler–Bernoulli nonlinear beam

5.6. Conclusion

5.7. References

6. Statics, Dynamics, Buckling and Aeroelastic Stability of Planar Cellular Beams. 6.1. Introduction

6.2. Continuous models of planar cellular structures

6.2.1. Timoshenko beam

6.2.2. Shear beam

6.2.3. Elastic constant identification

6.3. The grid beam

6.3.1. Rigid transverse model

6.3.2. Flexible transverse model

6.3.3. Comparison among models

6.4. Buckling

6.4.1. Formulation

6.4.2. Critical loads

6.5. Dynamics

6.5.1. Timoshenko beam

6.5.2. Shear beam and discrete spring–mass model

6.6. Aeroelastic stability

6.6.1. Modeling a base-isolated tower

6.6.2. Critical wind velocity

6.7. References

7. Collapse Limit of Structures under Impulsive Loading via Double Impulse Input Transformation. 7.1. Introduction

7.2. Collapse limit corresponding to the critical timing of second impulse

7.3. Classification of collapse patterns in non-critical case

7.4. Analysis of collapse limit using energy balance law

7.4.1. Collapse Pattern 1’

7.4.2. Collapse Pattern 2’

7.4.3. Collapse Pattern 3’

7.4.4. Collapse Pattern 4’

7.5. Verification of proposed collapse limit via time-history response analysis

7.6. Conclusion

7.7. References

8. Nonlinear Dynamics and Phenomena in Oscillators with Hysteresis

8.1. Introduction

8.2. Hysteresis model and SDOF response to harmonic excitation

8.3. 2DOF hysteretic systems

8.3.1. Equations of motion

8.3.2. Modal characteristics

8.4. Nonlinear modal interactions in 2DOF hysteretic systems. 8.4.1. Top-hysteresis configuration (TC)

8.4.2. Base-hysteresis configuration (BC)

8.5. Conclusion

8.6. Acknowledgments

8.7. Appendix: Mechanical characteristics of SDOF and 2DOF systems

8.8. References

9. Bridging Waves on a Membrane: An Approach to Preserving Wave Patterns

9.1. Introduction

9.2. Problem statement

9.3. Homogenized bridge

9.4. Internal reflections

9.5. Discrete bridge

9.6. Net bridge

9.7. Concluding remarks

9.8. Acknowledgments

9.9. References

10. Dynamic Soil Stiffness of Foundations Supported by Layered Half-Space

10.1. Introduction

10.2. Generation of dynamic soil stiffness. 10.2.1. Dynamic stiffness matrix under point loads

10.2.2. Formulation of the flexibility function

10.2.2.1. Flexibility function of the out-of-plane motion

10.2.2.2. Flexibility function of the in-plane motion

10.2.3. Formulation of Green’s influence function

10.2.3.1. Green’s influence function under a vertical load

10.2.3.2. Green’s influence function under a horizontal load

10.2.3.3. Green’s influence function under three-dimensional loads

10.2.4. Total dynamic soil stiffness by the boundary element method

10.2.4.1. Dynamic soil stiffness for a rigid foundation

10.2.4.2. Dynamic soil stiffness for a flexible foundation

10.3. Numerical examples of the generation of dynamic soil stiffness

10.3.1. A rigid square foundation supported by a layer on half-space

10.3.2. A rigid circular foundation supported by a layer on half-space

10.3.3. A rigid circular foundation supported by half-space and a layer on half-space

10.4. Numerical examples of the generation of FRS

10.5. Conclusion

10.6. References

List of Authors

Index. A, B, C

D, E, F

G, H, I

L, M, N

P, R, S

T, V, W

Summary of Volume 1

Summary of Volume 3

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DEEM can be considered a special case of short-wave (high-frequency) asymptotics. The corresponding algorithms are known as the method of geometric optics, the ray method, the semi-classical approximation, the WKBJ (Wentzel–Kramers–Brillouin–Jeffreys) approach, the method of edge waves, the Keller–Rubinow method, etc. (Keller and Rubinow 1960; Maslov and Fedoryuk 1981; Babich et al. 1985; Babich and Buldyrev 1991; Chen et al. 1991, 1992; Chen and Zhou 1993; Bauer et al. 2015). They were independently developed in various fields of mathematics, mechanics and physics.

Note an interesting fact: Ufimtsev proposed the asymptotic method of edge waves (Ufimtsev 1962, 2003, 2014). According to Rich and Janos (1994) and Mitzner (2003), this theory played a critical role in the design of American stealth aircrafts F-117 and B-2. It is a fascinating example of the direct application of asymptotic formulas in engineering practice!

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