FolJowing the formulation of the laws of mechanics by Newton, Lagrange sought to clarify and emphasize their geometrical character. Poincare and Liapunov successfuIJy developed analytical mechanics further along these lines. In this approach, one represents the evolution of all possible states (positions and momenta) by the flow in phase space, or more efficiently, by mappings on manifolds with a symplectic geometry, and tries to understand qualitative features of this problem, rather than solving it explicitly. One important outcome of this line of inquiry is the discovery that vastly different physical systems can actually be abstracted to a few universal forms, like Mandelbrot's fractal and Smale's horse-shoe map, even though the underlying processes are not completely understood. This, of course, implies that much of the observed diversity is only apparent and arises from different ways of looking at the same system. Thus, modern nonlinear dynamics 1 is very much akin to classical thermodynamics in that the ideas and results appear to be applicable to vastly different physical systems. Chaos theory, which occupies a central place in modem nonlinear dynamics, refers to a deterministic development with chaotic outcome. Computers have contributed considerably to progress in chaos theory via impressive complex graphics. However, this approach lacks organization and therefore does not afford complete insight into the underlying complex dynamical behavior. This dynamical behavior mandates concepts and methods from such areas of mathematics and physics as nonlinear differential equations, bifurcation theory, Hamiltonian dynamics, number theory, topology, fractals, and others.

The nonlinear normal modes of a parametrically excited cantilever beam are constructed by directly applying the method of multiple scales to the governing integral-partial differential equation and associated boundary conditions. The effect of the inertia and curvature nonlin earities and the parametric excitation on the spatial distribution of the deflection is examined. The results are compared with those obtained by using a single-mode discretization. In the absence of linear viscous and quadratic damping, it is shown that there are nonlinear normal modes, as defined by Rosenberg, even in the presence of a principal parametric excitation. Furthermore, the nonlinear mode shape obtained with the direct approach is compared with that obtained with the discretization approach for some values of the excitation frequency. In the single-mode discretization, the spatial distribution of the deflection is assumed a priori to be given by the linear mode shape centsn, which is parametrically excited, as Equation (41). Thus, the mode shape is not influenced by the nonlinear curvature and nonlinear damping. On the other hand, in the direct approach, the mode shape is not assumed a priori; the nonlinear effects modify the linear mode shape centsn. Therefore, in the case of large-amplitude oscillations, the single-mode discretization may yield inaccurate mode shapes. References 1. Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. v., Pilipchuk, V. N., and Zevin A. A., Nonnal Modes and Localization in Nonlinear Systems, Wiley, New York, 1

Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach describes the control behavior of mechanical objects, such as wave equations, plates, and shells. It shows how the differential geometric approach is used when the coefficients of partial differential equations (PDEs) are variable in space (waves/plates), when the PDEs themselves are defined on curved surfaces (shells), and when the systems have quasilinear principal parts. To make the book self-contained, the author starts with the necessary background on Riemannian geometry. He then describes differential geometric energy methods that are generalizations of the classical energy methods of the 1980s. He illustrates how a basic computational technique can enable multiplier schemes for controls and provide mathematical models for shells in the form of free coordinates. The author also examines the quasilinearity of models for nonlinear materials, the dependence of controllability/stabilization on variable coefficients and equilibria, and the use of curvature theory to check assumptions. With numerous examples and exercises throughout, this book presents a complete and up-to-date account of many important advances in the modeling and control of vibrational and structural dynamics.

The control of vibrating systems is a significant issue in the design of aircraft, spacecraft, bridges and high-rise buildings. This 2001 book discusses the control of vibrating systems, integrating structural dynamics, vibration analysis, modern control and system identification. Integrating these subjects is an important feature in that engineers will need only one book, rather than several texts or courses, to solve vibration control problems. The book begins with a review of basic mathematics needed to understand subsequent material. Chapters then cover more recent and valuable developments in aerospace control and identification theory, including virtual passive control, observer and state-space identification, and data-based controller synthesis. Many practical issues and applications are addressed, with examples showing how various methods are applied to real systems. Some methods show the close integration of system identification and control theory from the state-space perspective, rather than from the traditional input-output model perspective of adaptive control. This text will be useful for advanced undergraduate and beginning graduate students in aerospace, mechanical and civil engineering, as well as for practising engineers.

This is the first ever book that provides a comprehensive coverage of automotive control systems. The presentation of dynamic models in the text is also unique. The dynamic models are tractable while retaining the level of richness that is necessary for control system design. Much of the mateiral in the book is not available in any other text.

Written by the world 's leading researchers on various topics of linear, nonlinear, and stochastic mechanical vibrations, this work gives an authoritative overview of the classic yet still very modern subject of mechanical vibrations. It examines the most important contributions to the field made in the past decade, offering a critical and comprehensive portrait of the subject from various complementary perspectives.

The subject of random vibrations of elastic systems has gained, over the past decades, great importance, specifically due to its relevance to technical problems in hydro- and aero-mechanics. Such problems involve aircraft, rockets and oil-drilling platforms; elastic vibrations of structures caused by acoustic radiation of a jet stream and by seismic disturbances must also be included. Appli cations of the theory of random vibrations are indeed numerous and the development of this theory poses a challenge to mathematicians, mechanicists and engineers. Therefore, a book on random vibrations by a leading authority such as Dr. V.V. Bolotin must be very welcome to anybody working in this field. It is not surprising that efforts were soon made to have the book translated into English. With pleasure I acknowledge the co-operation of the very competent translater, I Shenkman; of Mrs. C. Jones, who typeJ the first draft; and of Th. Brunsting, P. Keskikiikonen and R. Piche, who read it and suggested where required, corrections and changes. I express my gratitude to Martinus Nijhoff Publishers BV for entrust ing me with the task of editing the English translation, and to F.J. van Drunen, publishers of N. Nijhoff Publishers BV, who so kindly supported my endeavours. Special acknowledgement is due to Mrs. L. Strouth, Solid Mechanics Division, University of Waterloo, for her competent and efficient preparation of the final manuscript."

As robots are becoming more and more sophisticated the interest in robot dynamics is increasing. Within this field, contact problems are among the most interesting, since contacts are present in almost any robot task and introduce serious complexity to system dynamics, strongly influencing robot behavior. The book formulates dynamic models of robot interaction with different kinds of environment, from pure geometrical constraints to complex dynamic environments. It provides a number of examples. Dynamic modeling is the primary interest of the book but control issues are treated as well. Because dynamics and contact control tasks are strongly related the authors also provide a brief description of relevant control issues.
The book will be of interest to engineers working in research and development in robotics and automation and to both graduate and postgraduate students. The work will also be valuable to readers involved in manufacturing, robotics, automation, computer and control engineering.

The last two decades have witnessed an enormous growth with regard to ap plications of information theoretic framework in areas of physical, biological, engineering and even social sciences. In particular, growth has been spectac ular in the field of information technology, soft computing, nonlinear systems and molecular biology. Claude Shannon in 1948 laid the foundation of the field of information theory in the context of communication theory. It is in deed remarkable that his framework is as relevant today as was when he 1 proposed it. Shannon died on Feb 24, 2001. Arun Netravali observes "As if assuming that inexpensive, high-speed processing would come to pass, Shan non figured out the upper limits on communication rates. First in telephone channels, then in optical communications, and now in wireless, Shannon has had the utmost value in defining the engineering limits we face." Shannon introduced the concept of entropy. The notable feature of the entropy frame work is that it enables quantification of uncertainty present in a system. In many realistic situations one is confronted only with partial or incomplete information in the form of moment, or bounds on these values etc.; and it is then required to construct a probabilistic model from this partial information. In such situations, the principle of maximum entropy provides a rational ba sis for constructing a probabilistic model. It is thus necessary and important to keep track of advances in the applications of maximum entropy principle to ever expanding areas of knowledge."

This volume contains 44 papers presented at the Third Contact Mechanics International Symposium (CMIS 2001) held in Praia da Consola9ao, Peniche (portugal), June 17-21,2001. This Symposium was the direct continuation of the first two CMIS held in Lausanne (1992) and in Carry-Le-Rouet (1994). Other related meetings, in what concerns scientific topics and participants, took place in the nineties at La Grande Motte (1990), Vadstena (1996), Ferrara (1997), Munich (1998) and Grenoble (1999). The Symposium aimed at gathering researchers with interests in a wide range of topics in theoretical, computational and experimental contact mechanics. The call for papers mentioned topics in tribology, mathematical formulations and analysis, numerical methods in non-smooth mechanics, impact problems, instabilities and technological problems. The total number of participants was 102, from Universities and Research Institutes of 19 countries. The Scientific Committee reviewed 102 submitted abstracts, and the final program consisted of 6 main lectures, 43 oral communications and 36 poster presentations (see Appendix A). The papers in this book correspond to almost all the main lectures and oral communications, and they are assembled in 5 chapters: * Dynamics and Impact * Instabilities, Oscillations and Waves * Contact Models, Results and Applications * Mathematical Analysis * Numerical Methods. We thank all the authors for their valuable contributions to this volume. We are indebted to the members of the Scientific Committee for their help in refereeing the submitted abstracts and manuscripts. We also thank the Series editor, Prof. Graham Gladwell, for his assistance in the revision process.

Optical Microscanners and Microspectrometers using Thermal Bimorph Actuators shows how to design and fabricate optical microsystems using innovative technologies and and original architectures. A barcode scanner, laser projection mirror and a microspectrometer are explained in detail, starting from the system conception, discussing simulations, choice of cleanroom technologies, design, fabrication, device test, packaging all the way to the system assembly.

An advanced microscanning device capable of one- and two-dimensional scanning can be integrated in a compact barcode scanning system composed of a laser diode and adapted optics. The original design of the microscanner combines efficiently the miniaturized thermal mechanical actuator and the reflecting mirror, providing a one-dimensional scanning or an unique combination of two movements, depending on the geometry. The simplicity of the device makes it a competitive component.

The authors rethink the design of a miniaturized optical device and find a compact solution for a microspectrometer, based on a tunable filter and a single pixel detector. A porous silicon technology combines efficiently the optical filter function with a thermal mechanical actuator on chip. The methodology for design and process calibration are discussed in detail. The device is the core component of an infrared gas spectrometer.

0.1 The partial differential equation (1) (Au)(x) = L aa(x)(Dau)(x) = f(x) m lal9 is called elliptic on a set G, provided that the principal symbol a2m(X, ) = L aa(x) a lal=2m of the operator A is invertible on G X (~n \ 0); A is called elliptic on G, too. This definition works for systems of equations, for classical pseudo differential operators ("pdo), and for operators on a manifold n. Let us recall some facts concerning elliptic operators. 1 If n is closed, then for any s E ~ , is Fredholm and the following a priori estimate holds (2) 1 2 Introduction If m > 0 and A : C=(O; C') -+ L (0; C') is formally self - adjoint 2 with respect to a smooth positive density, then the closure Ao of A is a self - adjoint operator with discrete spectrum and for the distribu- tion functions of the positive and negative eigenvalues (counted with multiplicity) of Ao one has the following Weyl formula: as t -+ 00, (3) n 2m = t / II N+-(1,a2m(x,e))dxde T*O\O (on the right hand side, N+-(t,a2m(x,e))are the distribution functions of the matrix a2m(X,e) : C' -+ CU).

This book explores two important aspects of the optimal control of oscillatory systems: the initiation of optimal oscillatory regimes and control possibilities for random disturbances. The main content of the book is based upon assertions of the optimal control theory and the disturbance theory. All theoretical propositions are illustrated by examples with exact mechanical context. An appendix covers the necessary mathematical prerequisites.

At the end of the twentieth century, nonlinear dynamics turned out to be one of the most challenging and stimulating ideas. Notions like bifurcations, attractors, chaos, fractals, etc. have proved to be useful in explaining the world around us, be it natural or artificial. However, much of our everyday understanding is still based on linearity, i. e. on the additivity and the proportionality. The larger the excitation, the larger the response-this seems to be carved in a stone tablet. The real world is not always reacting this way and the additivity is simply lost. The most convenient way to describe such a phenomenon is to use a mathematical term-nonlinearity. The importance of this notion, i. e. the importance of being nonlinear is nowadays more and more accepted not only by the scientific community but also globally. The recent success of nonlinear dynamics is heavily biased towards temporal characterization widely using nonlinear ordinary differential equations. Nonlinear spatio-temporal processes, i. e. nonlinear waves are seemingly much more complicated because they are described by nonlinear partial differential equations. The richness of the world may lead in this case to coherent structures like solitons, kinks, breathers, etc. which have been studied in detail. Their chaotic counterparts, however, are not so explicitly analysed yet. The wavebearing physical systems cover a wide range of phenomena involving physics, solid mechanics, hydrodynamics, biological structures, chemistry, etc.

We are honoured to present this collection of selected papers from the International Conference on Mixing and Crystallization, held at the Tioman Island, Malaysia in April, 1998. We are grateful to the editorial board comprising five eminent researchers in the field of mixing and crystallization for their thoughtful review and suggestions. In order to make this book as current as possible some of the papers have been thoroughly revised, which caused some delay in bringing out this edited version. We received necessary support from the Institute of Post Graduate Studies and Research, the University of Malaya and the Special Research Centre for Multiphase Processes, and the University of Newcastle, Australia in organizing this conference. We are indebted to the Institute of Chemical Engineers, United Kingdom, and the Institution of Engineers, Malaysia for their sponsorship. We would like to thank K.C. Lim, Dr. C. Ramakanth and Ms. Zubaidah for their help at the various stages of editing. We would also like to express our gratitude to Professor Mohd. Ali Hashim and Dr. Nafis Ahmed for their help and encouragement. Finally, I would like to thank Kluwer Academic Publishers for publishing this book. Bhaskar Sen Gupta Shaliza Ibrahim University of Malaya, Kuala Lumpur xi CFD MODELLING OF HYDRODYNAMIC CONDITIONS WITHIN THE WAKE OF MIXING IMPELLER BLADES 1 G.D. RIGByl., G. LANE . AND G.M. EVANSl.

This book collects invited lectures presented and discussed on the AMAS & ECCOMAS Workshop/Thematic Conference SMART'o3. The SMART'o3 Conference on Smart Materials and Structures was held in a 19th century palace in Jadwisin near Warsaw, 2-5 September 2003, Poland .It was organized by the Advanced Materials and Structures (AMAS) Centre of Excellence at the Institute of Fundamental Technological Research (IFTR) in Warsaw, ECCOMAS - European Community on Computational Methods in Applied Sciences and SMART-TECH Centre at IFTR. The idea of the workshop was to bring together and consolidate the community of Smart Materials and Structures in Europe. The workshop was attended by 66 participants from n European countries (Austria, Belgium, Finland, France, Germany, Italy, Poland, Portugal, Spain, U.K., Ukraine), 1 participant from Israel and 1 participant from the USA. The workshop program was grouped into the following major topics: 4 sessions on Structural Control (18 presentations), 3 sessions on Vibration Controland Dynamics (14 presentations), 2 sessions on Damage Identification (10 presentations), 2 sessions on Smart Materials (9 presentations). Each session was composed of an invited lecture and some contributed papers. Every paper scheduled in the program was presented, so altogether 51 presentations were given. No sessions were run in parallel. The workshop was attended not only by researchers but also by people closely related to the industry. There were interesting discussions on scientific merits of the presented papers as well as on future development of the field and its possible industrial applications.

T his book presents a t.hooretical framewerk and control methodology for a class of complcx dyna.mical systenis characterized by high state space dimension, multiple inpu t.s anrl out puts. significant nonlinearity, parametric uncertainty and unmodellod dyuarni cs. The book start.s wit.h an inl.rod uct.orv Chapter 1 where the peculiari- ties of control problcrns Ior complex systems are discussed and motivating examples from different fiolds of seience and technology are given. Chapter 2 prcscnts SO Il I(' rcsults of nonlinear control theory which assist in reading subsequent chaptors. The main notions and concepts of stability theory are int roduced. and problems of nonlinear transformation of sys- tem coordinates an' discussod. On this basis, we consider different design techniques and approaches t 0 linearization. stabilization and passification of nonlinear dynamical SySt('IIIS. Chapter 3 gives an cx posit.ion of the Speed-Gradient method and its ap- plications to nonlinear aud adaptive control. Convergence and robustness properties are exam iued. I~ roblcms of rcgulat ion, tracking, partial stabiliza- tion and control of 11amiItonia.n systerns are considered .

In determining the response of offshore structures, it is of utmost importance to determine, in the most correct manner, all factors which contribute to the total force acting on these structures. Applying the Morison formula (Morison et. al. , 1950) to calculate forces on offshore slender structures, uncertainties related to the understanding of the wave climate, the hydrodynamic force coefficients and the kinematics of ocean waves represent the most important contributions to the uncertainties in the prediction of the total forces on these structures (Haver and Gudmestad, 1992). Traditional calculation of forces on offshore structures involves the use of regular waves with the following non-linearities inco1porated use of regular wave theories inco1porating higher order terms use of Morison equation having a nonlinear drag term inclusion of the effect of the free surface by integrating all contributions to total forces and moments from the sea floor to the free surface of the waves In order to describe the sea more realistically, the ocean surface is to be described as an irregular sea surface represented by its energy spectrum. The associated decomposition of the sea surface is given as a linear sum of linear waves. The total force is found by integrating the contribution from all components in the wave spectrum to the free surface. The kinematics of each component must therefore be determined.

This monograph presents an updated source of information on the state of the art in advanced control of articulated and mobile robots. It includes relevant selected problems dealing with enhanced actuation, motion planning and control functions for articulated robots, as well as of sensory and autonomous decision capabilities for mobile robots. The basic idea behind the book is to provide a larger community of robotic researchers and developers with a reliable source of information and innovative applications in the field of control of cooperating and mobile robots. This book is the outcome of the research project MISTRAL (Methodologies and Integration of Subsystems and Technologies for Anthropic Robotics and Locomotion) funded in 2001-2002 by the Italian Ministry for Education, University and Research. The thorough discussion, rigorous treatment, and wide span of the presented work reveal the significant advances in the theoretical foundation and technology basis of the robotics field worldwide.

Acoustical imaging has become an indispensable tool in a variety of fields. Since its introduction, the applications have grown and cover a variety of techniques, producing significant results in fields as disparate as medicine and seismology. Cutting-edge trends continue to be discussed worldwide.

This book contains the proceedings of the 27th International Symposium on Acoustical Imaging (AI27), which took place in Saarbrucken, Germany, from March 24th to March 27th 2003. The Symposium belongs to a conference series in existence since 1968. AI27 comprised sessions on:

• Medical Imaging,
• Non-Destructive Testing,
• Seismic Imaging,
• Physics and Mathematics of Acoustical Imaging,
• Acoustic Microscopy.

During two well-attended workshops the applications of quantitative acoustical imaging in biology and medical applications, and in near-field imaging of materials, were discussed. Based on its cross-disciplinary aspects, the authors of the papers of AI27 present experiments, theory and construction of new instruments.

Audience: This volume will be of interest to engineers and researchers of all levels in the field, in industry or academia, and for those newcomers who want to get acquainted with the state-of-the-art in acoustical imaging. "

The survival of the Aeronautical Industries of Europe in the highly competitive World Aviation Market is strongly dependent on such factors as time-to-market of a new or derivative aircraft and on its manufacturing costs but also on the achievement of a competitive technological advantage by which an increased market share can be gained. Recognizing this, cooperative research is continuously encouraged and co-financed by the European Union in order to strengthen the scientific and technological base of the Aeronautical Industries thus providing - among others - the technological edge needed for survival. Corresponding targets of research within Area 3, Technologies for Transport Means, and here in particular Area 3A, Aeronautics Technologies, of the Industrial and Materials Technologies Program ( Brite -EuRam III, 1994 -1998) have been identified to be aircraft efficiency, cost effectiveness and environmental impact. Concerning aircraft efficiency - relevant to the present research - a reduction in aircraft drag of 10%, a reduction in aircraft fuel consumption of 30%, and a reduction in airframe, engine and system weight of 20% are envisaged. Meeting these objectives has, of course, also a strong positive impact on the environment.

Undeservedly little attention is paid in the vast literature on the theories of vibration and plasticity to the problem of steady-state vibrations in elastoplastic bodies. This problem, however, is of considerable interest and has many important applications. The problem of low-cyclic fatigue of metals, which is now in a well de veloped state is one such application. The investigations within this area are actually directed to collecting experimental facts about repeated cyclic loadings, cf. 47J. Theoretical investigations within this area usually con sider the hysteretic loops and the construction of models of plasticity theory which are applicable to the analysis of repeated loadings and the study of the simplest dynamic problems. Another area of application of the theory of the vibration of elastoplas tic bodies is the applied theory of amplitude-dependent internal damping. Another name for this theory is the theory of energy dissipation in vibrat ing bodies. In accordance with the point of view of Davidenkov "internal damping" in many metals, alloys and structural materials under consider able stress presents exactly the effect of micro plastic deformations. There fore, it may be described by the methods of plasticity theory. This point of view is no doubt fruitful for the theory of energy dissipation in vibrating bodies, as it allows one to write down the constitutive equations appropri ate both for vibrational analysis of three-dimensional stress states and an investigation of nonharmonic deformation. These problems are known to be important for the theory of internal damping."

Asymptotic methods of nonlinear mechanics developed by N. M. Krylov and N. N. Bogoliubov originated new trend in perturbation theory. They pene- trated deep into various applied branches (theoretical physics, mechanics, ap- plied astronomy, dynamics of space flights, and others) and laid the founda- tion for lrumerous generalizations and for the creation of various modifications of thesem. E!f,hods. A great number of approaches and techniques exist and many differen. t classes of mathematical objects have been considered (ordinary differential equations, partial differential equations, delay diffe,'ential equations and others). The stat. e of studying related problems was described in mono- graphs and original papers of Krylov N. M. , Bogoliubov N. N. [1], [2], Bogoli- ubov N. N [1J, Bogoliubov N. N. , Mitropolsky Yu. A. [1], Bogoliubov N. N. , Mitropol- sky Yu. A. , Samoilenko A. M. [1], Akulenko L. D. [1], van den Broek B. [1], van den Broek B. , Verhulst F. [1], Chernousko F. L. , Akulenko L. D. and Sokolov B. N. [1], Eckhause W. [l], Filatov A. N. [2], Filatov A. N. , Shershkov V. V. [1], Gi- acaglia G. E. O. [1], Grassman J. [1], Grebennikov E. A. [1], Grebennikov E. A. , Mitropolsky Yu. A. [1], Grebennikov E. A. , Ryabov Yu. A. [1], Hale J . K. [I]' Ha- paev N. N. [1], Landa P. S. [1), Lomov S. A. [1], Lopatin A. K. [22]-[24], Lykova O. B.

Results of experimental research on aerodynamic and acoustic control of subsonic turbulent jets by acoustic excitation are presented. It was demonstrated that these control methods, originated by authors, not only can intensify mixing (by acoustic irradiation at low frequency), but also notably ease it (at high-frequency irradiation). This research monograph presents the updated results of the authors supplemented by other investigations conducted in USA, Germany and Great Britain. The methods for the numerical simulation of subsonic turbulent jets under acoustic excitation are described in detail, and examples are reviewed of practical applications, including reduction of turbojet engine noise and acoustic control of self-sustained oscillations in wind tunnels.