Preface

Nonlinearity, bifurcation and chaos are common features exhibited by both natural and technological processes. Therefore, in this book theoretical and applied studies are addressed with emphasis on engineering dynamical systems.

This book includes chapters of both theoretical and applied studies of nonlinearity, bifurcation and chaos not published yet.

**Chapter 1** by I. Andrianow et al. presents a novel approach to the problem of inhomogeneous compression of shells with a zero Gaussian curvature. The authors provide a meromorphic continuation of the polynomial asymptotics of a solution of differential equations within the nonlinear shell theory. They apply Padé approximation of two variable functions (2D PAs) and then confirm the obtained results by experiments with stainless steel specimens using a holographic interferometry.

Digital pulse-width modulation (digital-PWM) controllers yield a novel alternative to control power converters. These controllers have many advantages such as the programmability, high reliability and easy implementation of the advanced control algorithms. They can be designed in the measured variables in order to guarantee the necessary computing time of the signal control. However, the performance of PWM controllers is affected by delays. Investigations of the incidence of delay in digital-PWM controllers based on two novel techniques, i.e. the so called Zero Average Dyamics (ZAD) and Fixed-Point Inducting Control (FPIC) are presented in **chapter 2**  by F. Angulo and G. Olivar.

In **chapter 3** J. M. Balthazar et al. present research results of the behavior of Nano- and Micro-Electro-Mechanical Systems (NEMS and MEMS). The main interest of this chapter is focused on the relationship between structure and nanoscale materials properties and the molecular dynamics of the atomic force microscopy. In particular, micro-cantilever beams and interaction of their vibration modes are analyzed. In addition, various aspects of Atomic Force Microscopy (AFM) and modeling and dynamic behavior of Electro-Mechanical System (EMS) are presented with respect to their control by taking into account the nonlinear and chaotic dynamic behavior.

### XII Preface

A. O. Belyakov and A. P. Seyranian in **chapter 4** study the dynamics of a pendulum of variable length, rotations of a pendulum with elliptically moving pivot and twirling of a hula-hoop. They assumed that the quasi-linearity of the analyzed systems permits the derivation of higher order approximations by the averaging method. To prove correctness of the method they compared all approximate solutions with results obtained by numerical simulation.

Preface XI

**Jan Awrejcewicz**

**Peter Hagedorn**

Poland

Germany

The Lodz University of Technology

Dynamics and Vibrations Group, FNB, Mechanical Engineering, TU Darmstadt

**Chapter 10** by N. A. Magnitskii is devoted to the theory of development of complexity in nonlinear systems through subharmonic and harmonic cascades of bifurcations of stable limit cycles or stable two-dimensional or many-dimensional invariant tori. It is illustrated and shown that this universal theory of dynamical chaos is applicable to all kinds of nonlinear differential equations including dissipative and conservative, nonautonomous and autonomous dynamical systems governed by ordinary and partial

In **chapter 11** B.M. Podlevskyi presents numerical examples for the numerical algorithms of finding the eigenvalue curves and the bifurcation points. The methods allow finding branching lines of solutions of nonlinear integral equations, obtained as a result of solution of a synthesis problem, where also problems associated with the

Heterogenous agent models are present in various fields of economic analysis, such as marker models, exchange rate models, monetary policy models, overlapping generations models, asset price models and models of socio-economic behavior. In **chapter 12** M. Verbič discusses the role of memory in an asset pricing model with heterogeneous beliefs. A particular emphasis is paid to how memory in the fitness measure affects the stability of evolutionary adaptive systems and the survival of

We hope that the presented book will be useful to academic researchers, engineers as

differential equations, as well as differential equations with delay.

linearization procedure of the integral operator are rigorously solved.

technical trading.

well as post-graduate students.

Jet flows or multi-jet flows are frequently observed to adhere around nearby solid boundaries. This general class of phenomena, which may be found in both liquid and gaseous jets, is known as the Conada effect. In **chapter 5** A. Dumitrache et al. present a number of approaches to study both external and internal flows with emphasis on the Conada effect.

N. M. Evstigneev and N. A. Magnitskii in **chapter 6** study the laminar-turbulent transition processes for some particular initial-boundary value problems. Using the 3D Navier-Stokes equations for incompressible fluids, they consider two problems: an advection problem for flow over a backward facing step and a free convection problem as the Oberbeck-Boussinesq approximation of the Rayleigh-Benard convection. Then, applying the previously confirmed and validated numerical methods, nonlinear dynamics phenomena within the phase space and its three and four dimensional subspaces are studied.

U. Fuellekrug in **chapter 7** illustrates the application of the non-linear identification by an analytical example and from measured data of a large aircraft. The applied analysis of technical structures is validated by computational models, which is usually done assuming a linear elasto-mechanical structures. However, in order to improve the accuracy of simulations, the author considers equations of structures taking into account the non-linearities and then compares the obtained results with the experimental dynamic identification and modal analysis.

T. Kopecki in **chapter 8** aims on research of the problem of obtaining credible results of nonlinear finite elements method analyses of thin-walled load-bearing structures subjected to post-critical loads. This chapter is highly recommended for designers working in the aircraft industry, since the post-critical loads decide about a structure's deformation state being the effect of a rapid (dangerous) change of the structure's shape occurring when the critical load levels are exceeded.

It is well known that thin-walled constructions are used in sports, industry, automotive, as well as the aerospace and civil engineering. As most structures exhaust carrying capacity not by exceeding the allowable stress but rather by the stability loss, not only critical load but also the postbuckling behavior of thin-walled structures subjected to static and dynamic load should be rigorously studied. **Chapter 9** written by T. Kubiak, devoted to buckling and postbuckling behavior of thin-walled ortotrophic structures subjected to static and dynamic loads, addresses this problem, and the results obtained there can directly be applied by engineers and designers.

**Chapter 10** by N. A. Magnitskii is devoted to the theory of development of complexity in nonlinear systems through subharmonic and harmonic cascades of bifurcations of stable limit cycles or stable two-dimensional or many-dimensional invariant tori. It is illustrated and shown that this universal theory of dynamical chaos is applicable to all kinds of nonlinear differential equations including dissipative and conservative, nonautonomous and autonomous dynamical systems governed by ordinary and partial differential equations, as well as differential equations with delay.

X Preface

obtained by numerical simulation.

four dimensional subspaces are studied.

experimental dynamic identification and modal analysis.

shape occurring when the critical load levels are exceeded.

Conada effect.

A. O. Belyakov and A. P. Seyranian in **chapter 4** study the dynamics of a pendulum of variable length, rotations of a pendulum with elliptically moving pivot and twirling of a hula-hoop. They assumed that the quasi-linearity of the analyzed systems permits the derivation of higher order approximations by the averaging method. To prove correctness of the method they compared all approximate solutions with results

Jet flows or multi-jet flows are frequently observed to adhere around nearby solid boundaries. This general class of phenomena, which may be found in both liquid and gaseous jets, is known as the Conada effect. In **chapter 5** A. Dumitrache et al. present a number of approaches to study both external and internal flows with emphasis on the

N. M. Evstigneev and N. A. Magnitskii in **chapter 6** study the laminar-turbulent transition processes for some particular initial-boundary value problems. Using the 3D Navier-Stokes equations for incompressible fluids, they consider two problems: an advection problem for flow over a backward facing step and a free convection problem as the Oberbeck-Boussinesq approximation of the Rayleigh-Benard convection. Then, applying the previously confirmed and validated numerical methods, nonlinear dynamics phenomena within the phase space and its three and

U. Fuellekrug in **chapter 7** illustrates the application of the non-linear identification by an analytical example and from measured data of a large aircraft. The applied analysis of technical structures is validated by computational models, which is usually done assuming a linear elasto-mechanical structures. However, in order to improve the accuracy of simulations, the author considers equations of structures taking into account the non-linearities and then compares the obtained results with the

T. Kopecki in **chapter 8** aims on research of the problem of obtaining credible results of nonlinear finite elements method analyses of thin-walled load-bearing structures subjected to post-critical loads. This chapter is highly recommended for designers working in the aircraft industry, since the post-critical loads decide about a structure's deformation state being the effect of a rapid (dangerous) change of the structure's

It is well known that thin-walled constructions are used in sports, industry, automotive, as well as the aerospace and civil engineering. As most structures exhaust carrying capacity not by exceeding the allowable stress but rather by the stability loss, not only critical load but also the postbuckling behavior of thin-walled structures subjected to static and dynamic load should be rigorously studied. **Chapter 9** written by T. Kubiak, devoted to buckling and postbuckling behavior of thin-walled ortotrophic structures subjected to static and dynamic loads, addresses this problem, and the results obtained there can directly be applied by engineers and designers.

In **chapter 11** B.M. Podlevskyi presents numerical examples for the numerical algorithms of finding the eigenvalue curves and the bifurcation points. The methods allow finding branching lines of solutions of nonlinear integral equations, obtained as a result of solution of a synthesis problem, where also problems associated with the linearization procedure of the integral operator are rigorously solved.

Heterogenous agent models are present in various fields of economic analysis, such as marker models, exchange rate models, monetary policy models, overlapping generations models, asset price models and models of socio-economic behavior. In **chapter 12** M. Verbič discusses the role of memory in an asset pricing model with heterogeneous beliefs. A particular emphasis is paid to how memory in the fitness measure affects the stability of evolutionary adaptive systems and the survival of technical trading.

We hope that the presented book will be useful to academic researchers, engineers as well as post-graduate students.

> **Jan Awrejcewicz** The Lodz University of Technology Poland

**Peter Hagedorn** Dynamics and Vibrations Group, FNB, Mechanical Engineering, TU Darmstadt Germany

**Chapter 1** 

© 2012 Olevs'kyy et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Applications of 2D Padé Approximants in** 

**and Experimental Justification** 

Additional information is available at the end of the chapter

approximation of two variable functions (2D PAs).

http://dx.doi.org/10.5772/48822

**1. Introduction** 

Igor Andrianov, Jan Awrejcewicz and Victor Olevs'kyy

**Nonlinear Shell Theory: Stability Calculation** 

The widest class of shells used in the civil and mechanical engineering is the class of shells with developable principal surface. The stress-strain state of shell structures under loads, which corresponds to buckling, is inhomogeneous, significantly bended, and nonlinear. Permanent interest of researchers in the problem of inhomogeneous compression of shells of zero Gaussian curvature has not led so far to a correct solution. Therefore, there is a need for the development and application of new methods that allow considering the problem in a

The approximate analytic integration of nonlinear differential equations of the theory of flexible elastic shells in most practical cases is based on the method of continuation of solution on the artificially introduced parameter. They can be satisfactorily applied only with an effective method of summation. The most natural analytical continuation method is that using Padé approximants (PAs). PAs effectively solves the problem of analytical continuation of power series, and this is a basis of their successful application in the study of applied problems. Currently, the method of PAs is one of the most promising non-linear methods of summation of power series, and the localization of its singular points. Recently, the method of PAs for single-variable functions has been successfully extended to the

A method that provides polynomial asymptotics of the exact solution of the general form and its meromorphic continuation based on 2D Padé approximants is proposed in this work. Several examples of displacements, stability and vibration calculations for inhomogeneous loaded shells with developable principal surface are presented. The accuracy of 2D PAs theoretical results are confirmed by experiments with stainless steel

and reproduction in any medium, provided the original work is properly cited.

complex setting, the most appropriate to study real behavior of structures.
