Faculty of Mechanical Engineering University of Niš International Federation of Nonlinear Analysts (IFNA).

WCNA 2004 MINISYMPOSIUM INTEGRITY OF DYNAMICAL SYSTEMS (Theory, Applications and Experiments) Http://www.masfak.ni.ac.yu/masfak/topic.php?lang=SR&id=525.

Booklet of Abstracts

Editor: Katica (Stevanoviæ) HEDRIH

Fourth World Congress of Nonlinear Analysts WCNA-2004, June 30-July 7, 2004. Http://kermani.math.fit.edu/ Hyatt Regency Orlando (Near Walt Disney World Resort) NIŠ-Orlando, 2004.

ISBN 86-80587-29-X.

Fourth World Congress of Nonlinear Analysts WCNA-2004 June 30 through July 7, 2004 Hyatt Grand Cypress Resort

Orlando, Florida USA

Lakshmikantham V., Ph. D. Professor Head, Dept. of Mathematical Sciences President, International Federation of Nonlinear Analysts IFNA Organizer/WCNA, Chairman, GOC Florida Institute of Technology, Dept. of Mathematical Sciences Melbourne, FL 32901 email: [email protected] or [email protected]

Donn Miller-Kermani, Ph. D.

Director, International Federation of Nonlinear Analysts (IFNA) Florida Institute of Technology

Dept. of Mathematical Sciences Melbourne, FL 32901

(321) 674-7412 (phone/fax) [email protected]

http://www.fit.edu/org/ifna/ (Information regarding WCNA2004) http://grandcypress.hyatt.com/groupbooking/ifna/

Hyatt Grand Cypress Resort

Fourth World Congress of Nonlinear Analysts WCNA-2004, June 30-July 7, 2004. Members of the Minisymposium Session Chairmen The fourth World Congress (WCNA--2004) of Nonlinear Analysts will be held at the Hyatt Regency Orlando (Near Walt Disney World Resort) under Chernousko L. Felix, the auspices of the International Federation of Nonlinear Analysts (IFNA). Academician of Russian Academy of Sciences, Russia Institute for Problems in Mechanics of The vision of IFNA and WCNA is: the Russian Academy of Sciences • To promote, encourage, and influence more cooperation, pr. Vernadskogo 101-1 understanding, and collaboration in the world community of nonlinear Moscow 119526 Russia analysts from various diverse disciplines; Tel.: +7 (095) 434 02 07 Fax: +7 (095) 938 20 48 • To bring together various disciplines that attempt to understand 117334, MOSKOW, nonlinear phenomena and solve nonlinear problems; and ul. Akad.Zalinskogo 38-8-2 • To help minimize the ever widening gap between the developed and e-mail: "Felix L. Chernousko" [email protected] developing countries by providing scientific and technical research assistance in various forms. Giuseppe Rega, Ph.D. Professor It is with this spirit, the International Federation of Nonlinear Analysts DISG, University of Rome “La Sapienza”, via A. Gramsci 53, 00197, Roma, Italy was established in 1992, as a transdisciplinary world society. Dipartimento di Ingegneria Strutturale e Geotecnica IFNA sponsors the World Congress of Nonlinear Analysts periodically Università di Roma “La Sapienza”, Italy Via A. Gramsci 53 once in four years. 00197 Roma, Italy tel. +39-06-49919195 fax: +39-06-49919192 or +39-06-3221449 cell. 349-5951890 e-mail: [email protected]

Ji-Huan He, Ph.D., Editor of Int. Journal Nonlinear Sciences and Numerical Simulation College of Science, Donghua University 1882 Yan’an Xilu Road, Shanghai 200051 People’s Republic of China Past WCNAs have been held in Tampa, Florida, Athens, Greece, and Catania, Email: [email protected] Italy and have been attended by more than 1600 participants from 90 countries. The WCNA-2004 will provide a perfect opportunity for collaboration, Hiroshi Yabuno, Ph. D. , contacts as well as relaxation. Don"t miss the chance to participate in this Institute of Engineering Mechanics and Systems University of Tsukuba important scientific event!! Remember IFNA membership entitles you to a e-mail:[email protected] reduced registration fee for WCNA-2004 in addition to receiving free copies of the journal, Nonlinear Studies (published quarterly). If you are interested in receiving further Katica (Stevanović) Hedrih information concerning the WCNA-2004, please send the following member of the GOC-WCNA 2004. information via email Organizer of the Session Integrity of dynamical systems [email protected] or by regular mail to: (Theory, Applications and Experiments) WCNA-2004 Address: Katica (Stevanović) HEDRIH, Ph.D. Professor, Dip. Mech. Eng. academician of UHEAS and ANS, member EuroMech, GAMM, Tensor Society, Int. ASME Florida Institute of Technology Faculty of Mechanical Engineering University of Niš Department of Mathematical Sciences Yu-18 000 - Niš, ul. Vojvode Tankosića 3/22 Melbourne, FL 32901 USA. Continuous updated information concerning Telefax: 381 18 41 663 Mobile 063 8 75 75 99 e-mail: [email protected] - e-mail (houm): [email protected] WCNA-2004 will be distributed through email and the web at http://kermani.math.fit.edu

5 6 My sincere thanks are also due to the Ministry of Science and Life Environmental Protection of Republic of Serbia for financial support to the organization of the Minysimposium INTEGRITY OF DYNAMICAL SYSTEMS WCNA 2004. PREFACE Finally, I would like to thank the Centre for Nonlinear Dynamics and Active Structures

at the Faculty of Mechanical Engineering University of Niš, who completed the Booklet of The Booklet of the Abstracts contain 16 abstracts of Invited Lectures accepted Abstracts in a short time period. for presentation at the MINISYMPOSIUM-INTEGRITY OF DYNAMICAL Nonlinear Mechanics is a subject of great importance in the development of SYSTEMS (Theory, Applications and Experiments), science and technology. The aim of the Symposium is to provide a forum to exhibit Http://www.masfak.ni.ac.yu/masfak/topic.php?lang=SR&id=525, to be held in Oralando at Hyatt the progress in this field during the past years and a place to further the interaction Regency Orlando (Near Walt Disney World Resort). The Minysymposium is included in the of modern mathematics and modern mechanics, as well as modern engineering Program of the Fourth World Congress of Nonlinear Analysts - WCNA-2004, sciences. The Symposium is an extraordinary opportunity for scholars to meet and June 30-July 7, 2004. Http://kermani.math.fit.edu/. discuss recent advances in Nonlinear Mechanics and their Applications to the issue The fourth World Congress (WCNA--2004) of Nonlinear Analysts will be held at the of Integrity of Dynamical Systems. The participants cover a wide range of Hyatt Regency Orlando (Near Walt Disney World Resort) under the auspices of the expertise, from pure theoreticians to people primarily oriented toward applications. International Federation of Nonlinear Analysts (IFNA). It is a great honour to held this Symposium under the auspice of the IFNA as an Invited session by GOC WCNA 2004. A significant achievement of the Symposium is expected to be extensive in-depth At the beginning there is a question: What is the integrity of a dynamical discussion on a variety of topics ranging from highly theoretical questions to system? One or more answers,as well as corresponding definitions of the used practical engineering applications. notions, are needed. Further questions also appear in correlations with the first The main conclusion that can be drawn from a total of 16 abstracts of lectures basic question. presented in this Booklet is that nonlinear mechanics keeps on deserving a great Firstly, let us make some remarks as regards the word source integrity. The scientific interst as far as both methodology and applicability are concerned. word integrity originates from the Latin word integritas and has the following We can write: "So what is new in nonlinear dynamics and Mechanics today? meanings: inviolable, untouched state, completeness, intactness, on the whole, The initial scope of applications in solid mechanics has broadened to cover material purity of spirit, chastity, innocence. processing, inelasticity and fracture mechanics. In rigid body dynamics, more Also the Latin word integrum has the following meaning: untouched totality complex systems such as vehicles, robotics and controlled machines have (entirety). The Latin expression in integrum has the following meaning: return in come into the framework of nonlinear dynamics. On the mathematical side of the previous state, or in the previous position. nonlinear dynamics, it is now recognized that spatio-temporal problems, hysteretic We have enough reason to feel ourselves gratified with the content of the invited lectures and time delay problems are new frontiers in this field. Also the term ‘complexity’ of this Minisymposium INTEGRITY OF DYNAMICAL SYSTEMS, which witness how the has been added to the lexicon of the chaos theory to describe the dynamics of many Minysimposium has received active support and extensive attention within academic and interacting sub-systems which can exhibit self-organization and evolution. engineering communities. This shows that the subject of Integrity of Dynamical Systems is Complexity analysis has gained a foothold in biological and some social sciences both a basic issue of interest in applied mathematics and a final aim in nonlinear mechanics, as well as in fluid and chemical physics. It remains to be seen what impact it will with a prosperous vitality and a remarkable progress in recent years. have in applied mechanics and engineering. Vitality and attraction of a research field originate from its social demand. We can point Here, I would like to express my heartfelt thanks to all members of the out that twenty years ago, with the rapid development of economy, science and technology scientific Committee, and also to the participants, for their engagement in people have been facing with the challenge of many nonlinear problems. This situation provides ample opportunities for addressing the topic of the Integrity of Dynamical Systems. organizing the Symposium, including the preparation of manuscripts which will be As a matter of fact, I have been sincerely pleased with my perusing of the content of published in the WCNA IFNA post congress publications (journal). Minisymposium invited lectures. Finally, on behalf of my colleagues in Serbia and Montenegro, I would like to I enjoyed pleasant honor and cooperation with the members of the chairmen session wish all the Invited participants to the Symposium a warm welcome and thanks. professors Chernousko L. Felix, GIUSEPPE REGA, Ji-Huan He and Hiroshi Yabuno. We are deeply convinced that the overall similarity in the addressed topics and the slight differences Katica (Stevanović) Hedrih, in the visual angle of the invited lectures will promote the academic exchanges among all Organizer of the Minisimposium Integrity of Dynamical Systems, participants, with beneficial effects on both the theoretical and the application point of view. Faculty of Mechanical Engineering University of Niš June 2004. I would like to express my heartfelt gratitude to all of them. 7 8

Preliminary Program Invited Lectures - 45-minute of Variational Approach to Nonlinear Problems and a Review Minisymposium on Mathematical Model of Electrospinning Ji-Huan He and Hong-Mei Liu College of Science, Donghua University 1882 Yan’an Xilu Road, Shanghai 200051 Integrity of dynamical systems People’s Republic of China Email: [email protected] (Theory, Applications and Experiments). http://my.fit.edu/~dkermani/organizers.html Stabilization Control and Motion Control for Some Mechanical Systems by Using Bifurcation Control dr Hiroshi Yabuno Institute of Engineering Machanics and Systems University of Tsukuba, e-mail:[email protected] List of the Invited Lectures Stability analysis of a multi-body mechanical system with Opening Lecture - One Hour rigid unilateral constraint and its application Jan Awrejcewicz and Grzegorz Kudra Technical University of Lodz, Department of Automatics and Biomechanics Identifying, Evaluating, and Controlling Dynamical 90-924 Lodz, 1/15 Stefanowskiego, Poland e-mail: [email protected], [email protected]

Integrity Measures in Nonlinear Mechanical Oscillators http://www.p.lodz.pl/k16/

GIUSEPPE REGA ([email protected]) DISG, University of Rome “La Sapienza”, via A. Gramsci 53, 00197, Roma, Italy Dynamical Version of Saint-Venant’s Principle – Historical STEFANO LENCI ([email protected]) Account and Recent Results ISTC, Polytechnic University of Marche, via Brecce Bianche, 60131, Ancona, Italy Prof dr B. Karp Department of Mechanical Engineering, Ben Gurion University of the Negev, Invited Lecture - One Hour P.O. Bax 653, Beer-Sheva 84105, Israel e-mail: [email protected] Ellipsoidal State Estimation for Dynamical Systems Chernousko L. Felix, Integrity of Dynamical Systems Academician of Russian Academy of Sciences, Russia Katica (Stevanović) Hedrih Institute for Problems in Mechanics of member of the GOC-WCNA 2004, and Member of the Session Chairmen. the Russian Academy of Sciences Organizer of the session Integrity of dynamical systems (Theory, Applications and Experiments) pr. Vernadskogo 101-1 Address: Katica (Stevanović) HEDRIH, Ph.D. Professor, Dip. Mech. Eng. Moscow 119526 Russia academician of UHEAS and ANS, member EuroMech, GAMM, Tensor Society, Int. ASME Tel.: +7 (095) 434 02 07, Fax: +7 (095) 938 20 48 Faculty od Mechanical Engineering University of Niš 117334, MOSKOW, Yu-18 000 - Niš, ul. Vojvode Tankosića 3/22 ul. Akad.Zalinskogo 38-8-2 Telefax: 381 18 41 663 Mobile 063 8 75 75 99 e-mail: "Felix L. Chernousko" [email protected] e-mail: [email protected] - e-mail(houm): [email protected]

9 10

Invited Lectures - 30-minute

An Asymptotic Method for Solving Stochastic Theoretical Study on the Friction Resistance of a Flat Plate by Micro-bubbles Integrodifferential Equations Zhi-Yong Liang Ph. D. Professor Svetlana Janković, Center of Physics of Fibrous Materials, College of Science, Donghua University, and Miljana Jovanović, Ph. D. Ass. Professor P.O. Box 471, 1882 Yan’an Xilu Road, Shanghai 200051, China University of Niš, Faculty of Science and Mathematics, Serbia and Montenegro e-mail: [email protected] Visegradska 33, 18 000 Niš email: [email protected], [email protected] Nonlinear Viscoelasticity of Polymer Melts in Pulasatile Flow Loads in Coronary Bypass Electromagnetic Vibration Force Field dr Ana Pejović-Milić, Juan Zhang Department of Mathematics, Physics and Computer Sci., Ryerson University, Canada, e-mail: Department of Applied Mathematics, Donghua University, Shanghai 200051, China [email protected] e-mail: [email protected] Stanislav Pejović, Department of Civil Engineering University of Toronto, Canada, e-mail: [email protected] Jin-Ping Qu Bryan Karney, National Engineering Research Center of Novel Equipment for Polymer Processing, Department of Civil Engineering University of Toronto, Canada South China University of Technology, Guangzhou 510640, China

On Nonlinear Perturbed Stochastic Hereditary Differential On Lyapunov Stability of Perturbed Linear Discrete – Equations Delay Systems with Multiple Delays Miljana Jovanović, Ph. D. Ass. Professor and Svetlana Janković, Ph. D. Professor University of Niš, Faculty of Science and Mathematics, Serbia and Montenegro Visegradska 33, 18 000 Niš S. B. Stojanovic*, D. Lj. Debeljkovic** email: [email protected], [email protected] * University of Nis, Faculty of Technology, Bulevar Oslobodjenja 124, 16000 Leskovac, Serbia and Monte Negro **University of Belgrade, Faculty of Mechanical Engineering Analitycal and Numerical Solution of Non-Stationary State Department of Control Eng., 11000 Belgrade, Serbia and Monte Negro of Stress in a Thin Viscoelastoic Plate Ing. Vitezslav Adamek, On Asymptotic Stability of Particular Class of Linear West Bohemia University in Pilsen, Univerzitni 22, 306 14 Pilsen, (tel.: +420377632335, e-mail: [email protected]), Continuous Singular Time Delay Systems Doc. Ing. Vladislav Las, CSc., D. Lj. Debeljkovic* S. B. Stojanovic**, West Bohemia University in Pilsen, Univerzitni 22, 306 14 Pilsen (tel.: +420377632335, e-mail: M. B. Jovanoviæ***, S. A. Milinkoviæ*** [email protected]), *University of Belgrade, Faculty of Mechanical Eng, Ing. Frantisek Vales, CSc., Dept. of Control Eng., 11000 Belgrade, Serbia and Montenegro Institute of Thermomechanics AS CR, Veleslavinova 11, 301 14 Pilsen, Czech Republic. ** University of Nis, Faculty of Technology, Bulevar Oslobodjenja 124, 16000 Leskovac, Serbia and Montenegro **University of Belgrade, Faculty of Metallurgy and Technology, The Cauchy Problem for Fuzzy fractional Differential 11000 Belgrade, Serbia and Montenegro [email protected] Equations Cheng Yu, Liangjian Hu, Guozhu Gao Department of Applied Mathematics, Donghua University, Shanghai, 200051, Peoples Republic of China Email: [email protected]

11 12 Identifying, Evaluating, and Controlling The first point consists in reconsidering the definition of “safe basin”, Dynamical Integrity Measures which is not generally accepted. For example, it differs (i) if one is interested in in Nonlinear Mechanical Oscillators transient or steady dynamics, (ii) if fractality of the basin can be accepted or must be disregarded, and so on. Its effect on subsequent studies is sometimes noteworthy. GIUSEPPE REGA The second step is the investigation of basins evolution. If “safe basins” ([email protected]) coincide with classical basins of attraction, this can be rigorously pursued through DISG, University of Rome “La Sapienza”, via A. Gramsci 53, 00197, Roma, Italy the detection of invariant manifolds – which are the basin boundaries – and the analysis of their evolution, in particular by studying the global bifurcations. In STEFANO LENCI other cases, while providing only approximate information, analysis of inset ([email protected]) modifications remains as one of the most important tools for in-depth investigation. ISTC, Polytechnic University of Marche, via Brecce Bianche, 60131, Ancona, Italy Erosion profiles for a varying system parameter, e.g., the excitation amplitude, are constructed for several systems and compared with each other. It is noted that, at least in the class of investigated mechanical systems, they are sharp in In past years researchers devoted a lot of attention to the investigation of the neighborhood of resonance frequencies, and dull elsewhere. It is further the nonlinear behaviour of several mechanical systems, which are usually classified observed that, while the erosion is always triggered by homo/heteroclinic according to their characters: hardening vs softening, smooth vs non-smooth, bifurcations of appropriate – commonly hilltop – saddles, it actually proceeds with single-d.o.f. vs multi-d.o.f., and so on. A lot of works deal with the application of different topological mechanisms. They are always extremely complex, and it is the abstract dynamical systems theory to practical oscillators, and concern conjectured that they may involve a cascade of secondary heteroclinic bifurcations attractors identification, their evolution through local and global bifurcations, linking the inset of the dangerous attractor with the outset of other saddles construction of bifurcation diagrams and of attractor-manifold-basin phase embedded in the dynamics. portraits, the discovery of new bifurcations and phenomena (especially in non- The erosion ends with the onset of out-of-well phenomena, namely, with smooth systems), and so on. The final aim is that of enlightening on, and possibly the occurrence of either scattered attractors, or escape, or overturning, etc., explaining in detail, system complex responses. according to the specific features of the considered mechanical oscillator. To a These topics are nowadays well established, although a lot of work still different extent, those phenomena may be linked to the safe basin erosion, and the remains to be done, and more recent advances concern the investigation of the work ends with a discussion of this point and of possible dynamical systems based integrity of the dynamical system. In this respect, an important role is played by the countermeasures for preserving integrity. phenomenon of erosion of the “safe basin” due to the penetration of tongues from [1] Thompson J.M.T., 1989, “Chaotic Behavior Triggering the Escape from a Potential “dangerous” attractors. Well,” Proceedings of the Royal Society of London A, 421, 195-225. Starting from works of Thompson [1], it has been realized that attractors must be paralleled by uncorrupted basins for safe practical applications. Indeed, eroded basins constitute a critical state for the structure corresponding to its incipient failure, which can be instantaneous – and dramatic – in the presence of perturbations or uncertainties. On the other hand, using the structure only when erosion is prevented may be too much conservative, because it can actually survive safely well above the relevant threshold if the erosion is not sharp. These considerations call for a detailed investigation of the safe basin erosion and of its possible control, which constitutes the subject of this work, along with the attempt to overview on similitudes and differences amongst various mechanical systems.

13 14 Ellipsoidal State Estimation for Dynamical Systems

Chernousko L. Felix, Academician of Russian Academy of Sciences, Russia Institute for Problems in Mechanics of the Russian Academy of Sciences pr. Vernadskogo 101-1, Moscow 119526 Russia Tel.: +7 (095) 434 02 07, Fax: +7 (095) 938 20 48 e-mail: "Felix L. Chernousko" [email protected]

16 15 Stabilization Control and Motion Control for 3. Stabilization by an actuator (active nonlinear feedback control) [5]: Some Mechanical Systems by Using Rail vehicles experience the problem of hunting above a critical forward Bifurcation Control speed. It has been clari.ed that the hunting motion can occur below the critical speed obtained by the linear theory, depending on the magnitude of disturbance, in the case when the nonlinear characteristics of the Hiroshi Yabuno bifurcation are subcritical [6]. In this study, the subcritical Hopf bifurcation is [email protected] transformed into a supercritical one by bifurcation control [7] and the 1-1-1, Ten-no-dai, Tsukuba, 305-8573, JAPAN stabilization is theoretically and experimentally carried out independent of the magnitude of disturbance for a wheelset model below the critical speed. Abstract In the latter part, we consider an application of a nonlinear As mechanical systems become faster, lighter, more .exible and phenomenon for the motion control of underactuated manipulator [8]. We more precise, complex phenomena are often encountered that cannot be propose a new method without feedback control, by using the dynamic explained by linear theory. In the .rst half of this lecture, some mechanical stabilization phenomenon which stabilizes inverted pendulum without systems are introduced and the following nonlinear phenomena occurring in feedback. The bifurcation in the phenomenon is theoretically characterized the systems are analyzed theoretically and experimentally: and the appropriate actuation is designed for the perturbation of the 1. The 1/3-order subharmonic resonance in a magnetically levitated bifurcation. body 2. Parametric resonance in a vertically excited cantilever beam References 3. Hunting motion in a railway vehicle wheels et model. [1] Yabuno, H., Endo, Y., and Aoshima, N., 1999, “Stabilization of 1/3-Order Then, by taking into account the physical mechanisms causing Subharmonic Resonance Using an Autoparametric Vibration Absorber,” ASME Journal of Vibration and Acoustics, Vol. 121, pp. 309-315. these phenomena, bifurcation control methods are proposed to stabilize [2] Nayfeh, A. H. and Balachandran, B., Modal Interactions in dynamical and each resonance. structural systems, 1989, ASME Applied Mechanics Review, Vol. 42, No.11, pp. 175- The features of the control strategies are as follows: 201.

1. Stabilization by a 1:2 auto-parametric vibration absorber (passive [3] Yabuno, H., Kawazoe, J., and Aoshima, N., 2004, “Suppression of Parametric Resonance of a Cantilever Beam by a Pendulum-Type Vibration Absorber,” ASME [1]: As seen in the detail review article on nonlinear modal nonlinear control) Journal of Vibration and Acoustics, in press. interactions by Nayfeh and Balachandran [2], there are many researches [4] Haxton, R. S. and Barr, A. D. S., 1972, “The Autoparametric Vibration Absorber,” which deal with autoparametric vibration absorbers for the case when the ASME Journal of Engineering for Industry, Vol. 94, pp. 119-125. main systems are in primary resonance. However, few studies have been [5] Yabuno, H., Okamoto, T., and Aoshima, N., 2002, “E.ect of Lateral Linear focused on the development of the stabilization method for subharmonic Sti.ness on Nonlinear Characteristics of Hunting Motion of a Railway Wheelset”, resonance. In this study, we discuss the e.ect of an autoparametric MECCANICA, Vol. 37, pp. 555-568. vibration absorber on the 1/3-order subharmonic resonance both [6] Xu, G., Troger, H., and Steindl, A., 1990, “Global Analysis of the Loss of Stability theoretically and experimentally. of a Special Railway Body” In W. Schiehlen, (ed), Nonlinear Dynamics in Engineering Systems pp. 345–352, Springer-Verlag (Berlin). 2. Stabilization by a pendulum-type vibration absorber (passive linear [7] Yabuno, H., 1997, “Bifurcation Control of Parametrically Excited Du.ng System control)[3]: Most of studies on pendulum-type vibration absorber (for by a Combined Linear-Plus-Nonlinear Feedback Control”, Nonlinear Dynamics, 12, example, [4]) are concerned with the saturation of the steady state pp. 263–274. amplitude of the main system. In contrast, in this study, we theoretically and [8] Yabuno, H., Goto, K., and Aoshima, N., 2004, “Swing-up and Stabilization of an experimentally show that the pendulum-type vibration absorber suppresses Underactuated Manipulator without State Feedback of a Free Joint,” IEEE Transactions on Robotics and Automation, in press. the steady state amplitude of the beam to be 0, under the parametric excitation. The avoidance is carried out by the shift of the unstable region, i.e., the shift of the trivial bifurcation points, due to the static friction at the pivot of the pendulum, but the autoparametric interaction is not needed. 17 18

Variational Approach to Nonlinear problems By the transformations: x = et and v = x−3u , and making the functional (2) and a review on mathematical model of stationary, we obtain the following equation:

Electrospinning v′′(t) − 7v′(t) +12v = v3/ 2 . (3) Eq.(3) is much easier to solve than the original one. Various new perturbation Ji-Huan He and Hong-Mei Liu methods can be applied here. We hereby apply the homotopy perturbation method: College of Science, Donghua University 1882 Yan’an Xilu Road, Shanghai 200051 v′′(t) − 7v′(t) +12v = pv3/ 2 , p=1. (4) People’s Republic of China, Email: [email protected] The solution is assumed to be expanded in a power series in p Though computer science is growing very fast, and numerical simulation is 2 v = v0 + pv1 + p v2 + ⋅⋅⋅ (5) applied everywhere, non-numerical issues will still play a large role. In this paper, Another example is the KdV equation: we review most well-known nonlinear equations arising in physics, chemistry, u + uu + u = 0 , (6) biology, mechanics, and others, in view of variational theory. Some a new t x xxx variational methods are suggested which can locate multi-branched solutions By introducingΦ Φ a potential Φ defined as u =Φ x , we obtain the Lagrangian: simultaneously, and determine the bifurcation point with ease．We consider the 1 1Φ 1 L = + 3 + Φ 2 . In order to simplify its Euler-Lagrange equation, we electrospinning process as an example to illustrate how to establish governing 2 x t 6 x 2 xx equations and boundary conditions in an energy view, how to predict diameter of the produced nanofibers and how to study the instability (bifurcation) of the make the following transformation: Φ =12(log F)x , which is Hopf-Cole transformation. The Euler-Lagrange equation leads to the linear equation: process. φ Π Furthermore, Hofe-Cole transformation, Backlund transformation, and Fxt + Fxxxx = 0 , which is easyΠ to solve. Darboux transformation are absorbed in the variational methods. We To study the soliton wave in theΠ ion acoustic plasma, by the semi-inverse systematically give a variational approach to Riccati equation, Bernoulli equation, method we obtain the following variational principle: Abel equation, Lambert equation, Thomas-Fermi equation, Logistic equation, φ t2 x2 ⎧ ∂ ∂ 1 2 1 ∂φ 2 φ ⎫ Liouville equation, Monge-Ampere equation, Burgers equation, Boussiesq J (n,v, , ) = ∫ dt∫ ⎨n + nv − nv + n − ( ) − e ⎬dx . (7) t1 x1 ⎩ ∂t ∂x 2 2 ∂x ⎭ equations, Schroedinger equation, Ginzburge-Laudau equation. To illustrate, we consider the Thomas-Fermi equation where φ ,n, and v are respectively the electric potential, ion density and ion ux′′()= x−12// u 32, u()01= , u()∞=0 . (1) velocity. Electrospinning is a method of producing superfine fibers with diameters ranging The energy(in atomic units) for a neutral atom of atomic number Z is from 10 µm down to 10 nm by forcing a molten polymer or a solution through a 2 / 3 6 ⎛ 4 ⎞ E = ⎜ ⎟ Z 7 / 3u′(0) . spinnerette by electric field. A rigorous thermo-electro-hydrodynamics description 7 ⎝ 3π ⎠ of electrospinning is needed for better understanding of the process, which is So it is important to determine a highly accurate value for the initial slope of the obtained by minimizing an energy functional. From the functional, the relationship potential u′()0 . The variational formulation can be easily obtained: between radius r of jet and the axial distance z from nozzle is easily obtained, and it −1/ 2 ∞ ⎧1 2 2 −1/ 2 5/ 2 ⎫ is in an allometric scaling: r ~ z . By further simplification of the function, we J (u) = ∫ ⎨ u′ + x u ⎬dx . (2) 0 ⎩2 5 ⎭ obtain a Bratu equation as stationary condition: 19 20 ∂2v Stability analysis of a multi-body system with + λev = 0 , (8) ∂z2 rigid unilateral constraint and its application λ There are two solutions to Eq.(8) for values of 0 < < λc , and no solutions for λ λ > λ . In case = λ there is only one solution. This can well explain the JAN AWREJCEWICZ c c GRZEGORZ KUDRA bifurcation phenomenon in the process. Department of Automatics and Biomechanics, Technical University of Lodz 1/15 Stefanowski St., 90-924 Lodz, Poland, [email protected] , [email protected]

The work includes modeling of three joined with viscous damping flat physical pendulums, with arbitrarily situated barriers imposed on the position of the system, where each of a body can be externally excited. The system is governed by the set of ordinary differential equations together with the set of algebraic inequalities representing rigid limiters of motion. In the system the impacts are possible (modelled by the use of generalized Newton’s law based on the restitution coefficient rule) as well as the sliding states with permanent activity of some obstacles. The system is simulated by numerical integration of smooth ordinary differential equations (ODE’s) or differential-algebraic equations (DAE’s) (in the case of sliding) between two successive events (impact, start or end of sliding) with detection of these events. Then the linear stability is numerically investigated using the special rules for handling with perturbed solution in points of discontinuity (‘saltation matrices’) based on the theory of Aizerman and Gantmakher. This approach is used then to seek and follow branches constructed through determination of stability and bifurcation analysis of periodic solutions (Floquet multipliers), and to calculate the Lyapunov exponents of attractors in the investigated piece-wise smooth (PWS) system. In the next step, we show some examples of analysis of a special case of the investigated system: three identical coupled rods, where the first rod harmonically excited. We show coexisting stable and unstable periodic solutions, quasi-periodic, chaotic and hyper-chaotic attractors. Some bifurcations of periodic solutions with impacts are shown, i.e. classical (for example the Neimark-Sacker bifurcation leading to quasi-periodicity) as well as non-classical (grazing bifurcations). In the end we show the possible application of the developed model: the piston-connecting rod-crankshaft system of a mono-cylinder four-stroke combustion engine, modelled as an inverted triple pendulum. The obtained self-excited system can be only treated as a first step in more advanced modelling of real processes (since some very important technological details are neglected), but after taking account of some technological details, the proposed model can be useful in impact dynamics analysis of the piston in a cylinder barrel.

21 22 Dynamical Version of Saint-Venant’s a dynamic excitation. This is achieved by introducing a notion of “dynamically equivalent system of loads” analogous to the “statically equipollent system of Principle – Historical Account and Recent loads” introduced by Saint-Venant himself. Following these lines, a definition of Results DSVP within the theory of Linear Elasticity, sound with experimental observations, is proposed. The criteria for its validity and the ways of its application are detailed. B. Karp Except for a purely theoretical interest in that principle, there are several potential applications that can be benefited by acknowledging that principle. It can Department of Mechanical Engineering, Ben Gurion University of the Negev, be employed to enhance our understanding of some experimental devices (either P.O. Bax 653, Beer-Sheva 84105, Israel for interpretation or for construction) such as Split Hopkinson Pressure Bar (Meng and Li, 2003), acoustic microscope (Peterson et. Al., 1998) and delay lines in Extended Abstract which stress history is channeled through waveguides. It is also relevant to vibration characterization (Duva and Simmonds, 1992), elastic constants Saint-Venant’s principle (SVP) is one of cornerstones in the justification characterization (Peterson, 1999) and source recognition (Ziv, 2002). of the use of analytic solutions in engineering practice. The essence of that The work presented refers explicitly to linear or quasi-linear problems in principle (suggested as an assumption) is the assertion that spatial distribution of mechanics of a continuum. The need for extension of the ideas presented to non- end loads is immaterial to the actual distribution of the stresses (strains) far from linear problems and other wave phenomena is naturally follows. the loaded end. The possible existence of a dynamic version of the quasi-static Saint- References Venant’s principle (DSVP) has been raised 5 decades ago (Boley, 1955). Since Boley B.A., 1955, Application of Saint-Venant’s principle in dynamical problems, J. Appl. Mech. then several papers are dedicated to the question of validity of DSVP. These are (Trans. ASME), 22, 204-206. reviewed in the first part of a present communication. One general observation is an absence of any convergence toward an acceptable idea of whether there is a Ciarletta M. and Chirita S., 1999, Thermodynamic restrictions, free energies and Saint-Venant’s dynamic counterpart to the classical SVP with the same generality and validity. principle in the linear theory of viscoelastic materials with voids, Int. J. Solids and Srtuct., Majority of these papers are bouncing between concluding that no valid dynamic 36, 1949-64. version of that principle exists (e.g. Novozhilov and Slepian, 1965) to existence of Duva J.M. and Simmonds J.G., 1992, The influence of two-dimensional end effects on the natural a principle under some restrictive conditions (e.g. Boely, 1955) and finally frequencies of cantilevered beams weak in shear, J. Appl. Mech. (Trans. ASME), 59, 230- appealing to viscosity as a dissipation mechanism enforcing the principle work - 232. decay of dynamic end effects (e.g. Ciarletta and Chirita, 1999). Karp B. and Durban D., 1997, Towards a dynamic version of Saint-Venant's principle, In On the other hand, there are experimental observations (or simulations of Modern practice in stress and vibration analysis, Gilchrist M.D. Ed., A.A. experiments) regarded by several researchers as an evidence for existence and Balkema, ISBN 90 5410 896 7. 3rd international conference, Dublin, Ireland, 3- validity of DSVP in conservative systems (e.g. Ziv, 2002). Nevertheless, no 5 September 1997. coherent version of a Saint-Venant’s principle for dynamic problem for non- Meng H. and Li Q.M., 2003, An SHPB set-up with reduced time-shift and pressure bar length, Int. dissipative systems has been proposed or alternately, those proposed, are proved to J. Impact Eng., 28, 677-696. be not valid. By “dynamical system” we refer here to wave phenomena (including Novozhilov V.V., Slepian L.I., 1965, On Saint-Venant’s principle in the dynamics of beams, vibration) in long bars/plates termed frequently as “waveguides”. PMM, 29, 261-81. In the present paper both analytical and experimental works related, Peterson M.L., 1999, Prediction of longitudinal disturbances in multimode cylindrical waveguide, directly and indirectly, to DSVP are reviewed. Emphasis is given to the apparent Experimental Mech., 39, 36-42. gap between the theoretical approach and the experimental insights. Subsequently, few recently published papers (e.g. Karp and Durban, 1997) and not yet published Peterson M.L., Srinath S., Murphy J., 1998, A waveguide-based acoustic microscope, Ultrasonics, results, bridging over the apparent gap, are presented. 36, 855-863. The main idea in bridging the gap lays in abandoning the attempts to Ziv M., 2002, Source signature and elastic waves in half-space under a sustainable line- extend the classical SVP to dynamic problems utilizing the notion of decay of self- concentrated impulsive normal force, Int. J. Numer. Anal. Meth. Geomech., 26, 373-406. equilibrated end loads. Instead, the original version of SVP is generalized to question the sensitivity of the far field in a waveguide to the spatial distribution of 23 24 The Integrity of Dynamical Systems The Latin word integrare has the following meaning: reconstruct, renew, restore, supplement, add something to the whole as it’s essential part. Also this word has the meaning necessary belong to the whole.

By using the listed meanings of the Latin words integritas, integrum, in integrum and integrare and others with the same word root, we can conclude some Katica (Stevanović) HEDRIH basic properties of the meaning of the notion the integrity of dynamical system. In Faculty od Mechanical Engineering University of Niš, Mathematical Institute SANU the engineering practice we have the notion and words: the integrity of structures Yu-18 000 - Niš, ul. Vojvode Tankosića 3/22, Serbia and Montenegro, Yugoslavia, Telefax: and also the integrity of construction. Then we can use these engineering 381 18 41 663 / Mobile 063 8 75 75 99 experimental and theoretical knowledge and experiences for definitions of the e-mail: [email protected] * e-mail(houm): [email protected] integrity of dynamical systems as a generalized definition and we choose the generalized parameters for examination of the basic properties, content and

Abstract concept of notion of the integrity of dynamical systems. The main research problem for the study of the integrity of dynamical systems is what are the measures of This paper opens a discussion on what are the basic properties, contents and concept of dynamical systems by which we can conclude that the system possesses the integrity of dynamical systems. The main research problem for the study of the integrity of integrity or loses the integrity. dynamical systems is to recover measures of integrity of dynamical system structures and For that reason, at the beginning, we choose one basic system with a basic dynamical system processes by which we can conclude that system dynamics is stable, structure and basic kinematic parameters, with basic equilibrium states and whether the system possesses integrity or whether that system loses integrity. dynamical properties in corresponding free or forced dynamic regimes. For that By using two examples of free and forced transversal vibrations of the elastically reason we choose kinetic parameters, and boundary conditions, and also stability connected double-plate systems and corresponding system with a discontinuity in elastical conditions of the equilibrium positions and the stability conditions of dynamical connection, we show some basic properties of the integrity of basic dynamical system. The integrity of corresponding dynamical processes in the behaviour of the whole system and its processes. By this choice of the basic dynamical system we define the notion of subsystems or in component processes and in response of the whole system with respect to the integrity of this dynamical system. Then we study the modification of this corresponding system with discontinuity have been studied by using methods of Bernoulli’s system due to a loss of the system structure integrity. As a source of the loss of particular integral and Lagrange’s method of constants variation. It is shown that one- dynamical system integrity we can introduce into system the discontinuity of the frequency regimes changes into multi-frequency regimes induced by discontinuity in the system structure by changing kinetic parameters of the dynamical system which is system, which represents the loss of integrity of the system and marks the appearance of the the source of the dynamical processes change or of stability loss. loss of integrity of corresponding processes. 2. The basic system with integrity and the corresponding system with the loss of integrity – Governing equations of the transversal vibrations of elastically 1. Preface connected double-plate systems and corresponding with discontinuity in elastic layer. At the beginning there is a question: What is the integrity of dynamical system? And one or more answers and also corresponding definitions of the used By Hedrih (Stevanović) (2004a), the analytical and numerical analysis of notions are needed. Also more questions appear and these questions are in free and forced transversal vibrations of an elastically connected double plate correlations with first basic question. system is presented. The analytical solutions of the system coupled partial Firstly, some sentences in relations of the word source integrity. The word differential equations of corresponding dynamical free and forced processes are integrity was derived from (originated from) the Latin word integritas and has the obtained by using the method of Bernoulli’s particular integral and Lagrange’s following meanings: inviolable, untouched state, completeness, intactness, on the method of variation constants. It is shown that two-frequency regime for free and whole, purity of spirit, chastity, innocence. three-frequency regime for forced vibrations respectively induced by initial Also the Latin word integrum has the following meaning: untouched conditions and one-frequency external excitation correspond to one mode totality (entirety). The Latin word in integrum has the following meaning: return in vibrations. The analytical solutions show us that elastic connection between plates the previous state, or in the previous position. coursed appearance of two-frequency regime of time function correspond to one

25 26 Transversal vibration beam on the elastic Winckler’s foundation under the eigen amplitude function of one mode, and also that time functions of different action of multi-frequency forces with frequencies in the form of first frequency vibration modes are uncoupled. It is shown for every shape of vibrations. It is resonant range of the beam are studied by Hedrih paper (1973), and some results of proved that in one of the mn -family vibration modes n,m = 1,2,3,4,....∞ of both an investigation of multi-frequency vibrations in single-frequency regime in elastically connected plates two possibilities for appearance of the resonance nonlinear systems with many degrees of freedom and with slowly changing dynamical states are present, and also for appearance of the dynamical absorption. parameters are presented by Stevanović and Rašković article from 1974. The By using MathCad program the corresponding visualizations of the characteristic application of the Krilov-Bogolyubov-Mitropolskiy asymptotic method for the forms of the plate middle surfaces through time are presented. study of the elastic bodies nonlinear oscillations and energetic analysis of the By Hedrih (Stevanović) (2004b), the analytical and numerical analysis of elastic bodies oscillatory motions gave new results in the thesis by Stevanović in free and forced transversal vibrations of an elastically connected double plate 1975. One-frequency transversal oscillations of thin rectangular plate with systems with discontinuity in the elastic layer Winkler type is presented. The nonlinear constitutive material stress-strain relations and nonlinear transversal analytical solutions of the system coupled partial differential equations of vibrations of plate with special analysis of influence of weak nonlinear Boundary corresponding dynamical free and forced vibration processes are obtained by using Conditions are the contents of the articles by Hedrih (1979, 1981 a, b) and then method of Bernoulli’s particular integral and Lagrange’s method of variation asymptotic solution of the nonlinear equations of the thin elastic shell with positive constants. It is shown that to one vibration mode correspond infinite or finite multi- Gauss’ curvature in two-frequency regime is pointed out by article Hedrih (1983). frequency regime for free and also multi-frequency regime for forced vibrations The two-frequency oscillations of the thin elastic shells with finite deformations induced by initial conditions and one-frequency or corresponding number multi- and interactions between harmonics is studied by Hedrih and Mitić (1983) and frequency regime depending on external excitation. It is shown for every shape of multi frequency forced vibrations of thin elastic shells with a positive Gauss's vibrations. Analytical solutions show us that discontinuity caused appearance of curvature and finite displacements by Hedrih in 1984, and also on the mutual multi-frequency regime of time function corresponds to one eigen amplitude influence between modes in nonlinear Systems with small parameter applied to the function of one mode, and also that time functions of different vibrations basic multi-frequency plate oscillations are study by Hedrih and all (1984). mode are coupled. From final expression we can separate the new generalized Multi-frequency forced vibrations of thin elastic shells with a positive eigen amplitude functions with corresponding time eigen functions of the free one Gauss' curvature and finite deformations by H1985, and initial deformation frequency and forced corresponding multi-frequency regime of vibrations. influence of the shell middle surface on the phase-frequency characteristics of the These two examples are good for explanation the basic idea of integrity of nonlinear stationary forced shell vibrations and numerical analysis of the four- dynamical systems. frequency vibrations of thin elastic shells with Gauss' positive curvature and finite deformations were studied by Hedrih and Mitić (1985a, b) and also initial 3. Nonlinear phenomena as a measure of the loss of integrity of dynamical displacement deformation influence of the thin elastic shell middle surface to the processes. resonant jumps. The classical monograph by Andronov, Witt and Haykin (1933, 1981), By using results from the listed papers, which contain dates of existence contains some classical examples of nonlinear systems with one degree of vibration and appearance and disappearance of resonant jumps in amplitude frequency freedom and their corresponding phase portraits as the visualizations of variety of curves and phenomena of mutual influence between oscillatory modes in nonlinear nonlinear dynamics. In the monograph of Iooss and Joseph (1980), the properties of dynamics of plates and shells forced vibrations, we can analyze the integrity of a variety of kinds of bifurcations are pointed out. In the monographs by oscillatory processes of the body with nonlinear properties. Then, we can talk of the Mitropolyskiy, Yu. A (1976) the asymptotic theory, the foundation of the methods loss of integrity of dynamical processes under the external forces with frequency in and asymptotic approximations of solutions of the oscillations problem of nonlinear the resonant frequency range. The external excitation with critical values of the continuous systems are presented. external force frequency, initializes resonant jumps as a loss of stability, and also as By using the asymptotic method of Krilov-Bogolyubov-Mitropolskiy a loss of integrity of dynamical oscillatory processes. In the case when external (1955) and (1964), one frequency and multi-frequency nonlinear oscillations excitation is multi-frequency, then we can conclude that between nonlinear modes regimes of deformable bodies are studied. Specially, one and two-frequency appear mutual influence and results show that a plate oscillates with amplitudes and stationary and nonstationary regime of nonlinear transversal free and forced phases which contain many possibilities of appearance of resonant jumps from vibrations of beams, plates and shells are studied by Hedrih’s articles (1972a, b, c). lower to upper amplitude or phase and also from upper to lower amplitudes and

27 28 phases. In these cases resonant jumps are the loss of integrity processes, which are phenomenology and phenomenological mapping between physical disparate approximately almost sure independent for some values of frequencies far from systems and processes (see Petrovic’s Phenomenological Mapping (1933) and resonant range frequencies, and with important mutual influence of the values of (1911)). By every connection between systems we introduce new possibilities for the frequencies in the resonant ranges containing values for appearance of resonant building new contents of integrity or the loss of structural integrity or the integrity jumps. of processes. Then, we talk about different kind of integrities as relative notions, depending on our aims. Then, the integrity of dynamical system is a relative notion 4. Measures of the integrity of dynamical systems which contains many possibilities of integrity definition with respect to stability: equilibrium positions stability, structural stability, time stability, dynamical system By using the listed examples it is evident that integrity of dynamical stability. Having in mind that science and experience know many different kinds of system can contain many different contents. Then definitions of the notion of stability and unstability, and also of control of dynamical system processes, we can integrity of dynamical systems are not unique. Also, the measures of the integrity conclude that the integrity notion of dynamical system is important in the of dynamical system depend on the corresponding definition, and also corresponds generalized theory of stability, control and design of system with respect to the to the programming processes in the dynamical systems. Then we can talk of theory, experiments and practice of nature and human activity in environmental internal integrity of the system under external excitations and also of integrity of action and protection. the dynamical processes when the system structure changes under constructive And for ending, we can conclude that direct research to the dynamical aims, or under the destruction of basic elements of the system. system integrity is pertinent to different branches of science. Than some new For the measures of dynamical system integrity is possible to take one of scientific and professional society in the world have the aim to continuously the set of kinetic parameters the changes of which in the critical values range are develop a knowledge base relating to all aspects of structural integrity of dynamical sources of the bifurcation processes or of the appearance of nonprogrammed systems with the objective of improving the safety and performance of engineering processes in dynamical system. structures, components, systems and their associated materials. Also, we can learn of the research of dynamic stability of the sustainable development of processes in 5. Concluding Remarks different kinds of systems with pure disparate properties, from biological to engineering, from chemical to social systems. By using two examples of free and forced transversal vibrations of the By Kozlov (1998) a mathematical explanation of the analogies between elastically connected double-plate systems and corresponding system with hydrodynamics, geometric optics, and mechanics is developed. It turns out that discontinuity in elastically connection we show some basic properties of integrity of studying families of trajectories of Hamiltonian systems, in fact, reduces to basic dynamical system. The integrity of corresponding dynamical processes in the studying multidimensional hydrodynamics of an ideal fluid. behaviour of the whole system and its subsystems or in component processes and in For developing intellectual control of dynamical system, it is first necessary to response of all system in correlations to corresponding system with discontinuity recover measures of integrity of dynamical system structures and dynamical have been studied by using methods of Bernoulli’s particular integral and system processes, and analogies between these measures in disparate kinds of Lagrange’s method of constants variation. It is shown that one-frequency regimes dynamical systems. changes into multi-frequency regimes induced by discontinuity in the system, which is the loss of integrity of the system and are marks of the appearance of loss of Acknowledgment integrity of corresponding independent processes. Parts of this research were supported by the Ministry of Sciences, Technologies and Also we can conclude that every discontinued changes of basic kinetic Development of Republic Serbia trough Math parameters of system entail loss of integrity of basic system or basic processes in No. 1616 Real Problems on Mechanics and Faculty of Mechanical Engineering University basic systems. Also, by using some constraints, the connection between two of Niš Grant No. 1828 Dynamics and Control of Active Structures. systems with separate independent integrity is the process of the loss of these separate structural integrity and builds a new content of structural integrity of 8. References common coupled subsystems. We can see appearance of a new task as a control as ematical Institute SANU Belgrade Grants Andronov, A.A., Vitt, A.A., Haykin, S.E., (1981), Teoriya kolebaniy, Nauka, well as optimal control of integrity of complex dynamical systems or of new Moskva., pp. 568. processes in these dynamical systems. For that reason it is important to develop a Gerard Iooss and Daniel Joseph: Elementary Stability and Bifurcation Theory, theory of integrity of dynamical systems and corresponding theory of mathematical Springer Verlag, 1980 (Rusian Edition Moscow, 1983, pp. 300).

29 30 Janković, V. S., Potić, V. P. and Hedrih (Stevanović), K., Parcijalne diferencijalne jednačine i integralne jednačine sa primenama u inženjerstvu (Partial differential equations and integro-differential equations with examples in engineering), Katica (Stevanovi}) Hedrih, One-frequency Proper Nonlinear Vibrations of Thin Univerzitet u Nišu, 1999, str. 347. (in Serbian). Plate, Theoretical and Applied Mechanics, No. 4, 1978., pp. 51-65. Kozlov, V.V., (1998), Dynamical Systems, Springer, p. 184. Katica (Stevanović) Hedrih, Nonlinear Transversal Vibrations of Plate with Mitropolskiy Yu.A. and Moseenkov B.I.,: Asimptotičeskie rešenia uravneniy v Special Analysis of Influence of Weak Nonlinear Boundary Conditions, Abstracts XVI th časnih proizvodnih, Kiev 1976. Mid West Mechanics Conference, Manhattan, Kansas, 1979. Petrović Mihailo, Fenomenološko preslikavanje, Srpska kraljevska akademija, Katica (Stevanovi}) Hedrih: Nonlinear Diferential Equations of the Surface of Štamparija Planeta, Beograd, 1933. Thin Plate, Asbstracts, Section 3., GAMM, Mittelungen, Heft 1, Marz 1981. p.69. Petrović Mihailo, Elementi matematičke fenomenologije, Srpska kraljevska Katica (Stevanović) Hedrih: Asimptotska reenja jednačina nelinearnih oscilacija akademija, Državna štamparija Kraljevine Srbije, Beograd, 1911. plitke ljuske sa pozitivnom Gauss-ovom krivinom u dvofrekventnom režimu (Asimptotic Rašković, D., (1985), Teorija elastičnosti (Theory of Elasticity), Nau~na knjiga, solution of the nonlinear equations of the thin elastic shell with positive Gauss’ curvature in 1985, 414. two-frequency regimeš), Zbornik Simpozijum'83 Savremeni problemi nelinearne dinamike, Rašković, D., (1965), Teorija oscilacija, (Theory of Oscillations), Nau~na knjiga, Društvo za mehaniku Srbije, Arandjelovac 1983. II. 5. str. 183-196. 1965, 503.(in Serbian). Katica (Stevanović) Hedrih: Multi frequency Forced Vibrations of Thin Elastic Hedrih (Stevanović), K., (2004a), Transversal vibrations of elastically connected Shells with a Positive Gauss's Curvature and Finite Displacements, Kurzfassungen double-plate systems, (to appear) Abstracts der wissensschaflichen Jaahrestaagung GAMM, Regensburg 1984. p.I-12. Hedrih (Stevanović), K., (2004b), Transversal vibrations of elastically connected Programm p.36. double-plate systems with discontinuity- First part: Analytical approach, (to appear) Hedrih (Stevanović), K., (1985), Multifrequency forced vibrations of thin elastic Hedrih (Stevanović), K., Discrete Continuum Method, Recent Advances in shells with a positive Gauss' curvature and finite deformations, Theoretical and Applied Analytical Dynamics Control, Stability and Differential Geometry, Proceedings Mechanics, 11, Beograd 1985, pp. 51-58. Mathematical institute SANU, p. 151, 2002, pp.30-57. .ISBN 86-80593-32-X. Hedrih (Stevanović) Hedrih, (1974), Nestacionarne oscilacije pravougaone tanke ploče, 12th Yugoslav Congress of Rational and Applied Mechanics, Ohrid, Vol.4, pp. 51-65, 1974. Katica Stevanović (Hedrih), Nestacionarne oscilacije pravougaone tanke ploče, (Nonstationary Vibrations of Thin Rectangular Plate), Zbornik radova XII jugoslovenskog kongresa teorijske i primenjene mehanike, Ohrid, 1974, C2. str. C2.12.1-11. Katica Stevanovi} (Hedrih), Popere~nie kolebaniÔ balki, le`aÈeyna uprugom osnovanii nahodÔÈeysÔ pod vozdeystviem vozmuÈeysÔ sili s neskolÃkimi garmonikami ~astotami bliskimi pervoy sobstvennoy (Transversal Vibration Beam on the Elastic Foundation uned the Action of the Multi-Frequencies Forces with Frequencies form First Frequency Resonant Range of the Beam), Matemati~eskaÔ fizika, vip.13, Kiev, 1973, str. 161-181. Katica Stevanovi} (Hedrih) and Danilo Ra{kovi}, Isledovanie "vi{e~astotnih kolebaniy" v odno~astotnom re`ime v nelineynih sistemah so mnogimi stepenÔi svobodı i medlenno menÔÓÈimsÔ parametrami (Investigation of Multi-frequencies Vibrations in single-frequency regime in Nonlinear Systems with Many Degrees of the Freedom and with Slowchanging Parameters), Journal Nonlinear Vibrations Problems – Zagadnenia dragan neliniowicz" No. 15., 1974. Warsaw, pp. 201-202. Katica Stevanović (Hedrih), Primena energijske interpretacije asimptotske metode na izučavanje nelinearnih oscilacija elastičnih tela - Energijska analiza oscilatornih kretanja elastičnih tela ( Application of the Asymptotic Method for study of the Elastic Bodies Nonlinear Oscillations – Energetic Analysis of the Elastic Bodies Oscillatory Motions), Doktorska disertacija- Doctoral Ph. D Dissertation), Mašinski fakultet, Niš, 1975., str. 331., Biblioteka MF Niš. Mentori (Supervision) Prof. dr. Ing. Math Danilo P. Rašković (Yugoslavia) i akademik Yuriy Alekseevich Mitropolskiy (Ukraine).

31 32 An Asymptotic Method for Solving Stochastic these papers, we consider discrete perturbations and give suffcient conditions under which the obtained sequence of equations can be treated as a special iterative Integrodifferential Equations procedure of the presented iterative method, so that the sequence of iterations tends with probability one to the solution of the unperturbed equation. Note that this iterative method could be appropriately extended to stochastic differential equations including martingales and martingale measures Svetlana Janković and Miljana Jovanović instead of the Brownian motion process. Faculty of Science and Mathematics, Department of Mathematics, University of Ni¡s Vi¡segradska 33, 18000 Ni¡s, SCG References: E-mail: [email protected], [email protected] [1]. Sv. Janković, Iterative procedure for solving stochastic di_erential equations, Mathematica Balkanica, 1 (1987), 64–71. Non-linear differential equations subjected to deterministic and random excitations have been extensively studied both theoretically and experimentally [2]. Sv. Janković, M. Jovanović, Generalized stochastic perturbation – depending over a long period of time. In mechanics, and much more in engineering practice, di_erential equations, Stochastic Anal. and Appl., 20(6) (2002), 1281–1307. some important roles are played by non–linear random deferential equations, which represent mathematical models of elastic systems motion under random excitations [3]. M. Jovanović, Sv. Janković On perturbed nonlinear Ito type stochastic integrodi_erential equations, J. of Math. Anal. and Appl., 269 (2002), 301– of a Gaussian white noise type. Bearing in mind that a white noise is, at least, a 316. tolerable abstraction and is never a completely faithful representation of a physical noise source, mathematically described as a Gaussian stationary wide-band random [4]. M.G. Murge, B.G. Pachpatte, On generalized Ito type stochastic integral process of small intensity and correlation time with expectation equal to zero, equations, Yokohama Math. J., 34 (1986), 23–33. which is represented as a formal derivative of a Brownian motion process, all such phenomena are essentially based on stochastic deferential or integrodifferential [5]. A.V. Skorokhod, Asymptotic Methods in the Theory of Stochastic Di_erential equations of the Ito type. Since a large number of such stochastic equations are not Equations, Naukova Dumka, Kiev, 1987 (In Russian). effectively solvable, it is convenient to find their approximative solutions, which is the object of the present paper. * Supported by Grant No 1834 of MNTRS through Math. Institute SANU. By following some ideas from paper [1], we consider an analytic iterative method for solving a general stochastic integrodifferential equation of the Ito type, earlier studied by many authors, by Murge and Pachpatte [4], for example. The generality of this equation is in the sense that it includes, as its special cases, many other stochastic differential and integrodifferential equations. We give conditions under which the sequence of iterations converges with probability one to the solution of the original equation. It is well-known that several phenomena in mechanics, physics and engineering depend on small parametric perturbations which, together with Gaussian white noise excitations, mathematically determine perturbed stochastic differential equations of the Ito type, whose coefficients are additionally perturbed and depend on small perturbations. In papers [3] and [4], under some special types of perturbations, their solutions are compared with the solutions of the corresponding unperturbed equations. The rate of the closeness between the solutions of the perturbed and unperturbed equations is measured in the sense of the (2m)-th moment norm, on finite intervals or on intervals whose length tends to infinity as small perturbations tend to zero. By using the results of

33 34 PULASATILE FLOW LOADS IN CORONARY BYPASS

Ana Pejoviæ-Milić Dept. of Math, Physics and Computer Sci., Ryerson University, Canada Stanislav Pejović Dept. of Civil Engineering, University of Toronto, Canada Bryan Karney Dept. of Civil Engineering, University of Toronto, Canada

The fluid dynamics of a localized human bypass loop (Figure 1) is influenced by a number of factors and some of them, such as wave reflections from junctions and narrowing of coronary vessels can cause a partial cut-off within the loop under some singular conditions (Duan at el., 1995) or increase the pressure amplitude (Pejoviæ-Miliæ at el., 2003), thus influence the bypass’s efficacy. Once the mechanical stiffness of the coronary bypass was included into the hemodynamic of the loop, results illustrate that a stiffer bypass with a diameter smaller than that of the coronary artery should be used as an implant to moderate adverse effects in the patient (Alderson et al., 2001). Beside these factors, the bypass length, the exact position of the narrowing in the coronary vessel and its stiffness, the complexity of the vascular network along with the aging of vessels are expected to influence the local hemodynamic function of the bypass loop. The pressure pulsation causes pulsatile flow loads within the human blood vessel system, and thus induces oscillatory motion in the enclosed blood. Stability, mode shape, and the maximum and minimum pressures are all influenced by the system characteristics mentioned above, and all these aspects should be analysed over the zone of typical excitations. In this work a complex mathematical method developed for technical hydraulic systems such as distribution networks, pumps, hydroelectric plants and aeronautical systems, is used to study a pulsatile flow in time domain as well as in steady oscillatory condition, which is excited by the human heart. The equations of motion and continuity

35 36 This analytical approach is applied to a coronary system of healthy vessels, which can be excited by an inside or outside More specifically, it is observed that the oscillatory behaviour force. A wide domain of the frequency range from 0 to 150 Hz was is highly dependant on the length and location of narrowing inside therefore analyzed in frequency domain (Fig. 2), and once significant the coronary artery, as in the case of the coronary arterio stenosis, and frequencies were calculated they were visualized in the time domain on the length, diameter and elasticity of a bypass loop. It is found that (Fig. 3). even in the absence of a bypass graft, wave reflections from the narrowed coronary artery affect the pressure and flow both in the diseased vessel and in the aorta, and thus significantly change the response of even such a simplified system as discussed here.

References:

Alderson H., Zamir M., 2001, Smaller, stiffer coronary bypass can moderate or reverse the adverse effects of wave reflections, Journal of Biomechanics 34 (2001), pp 1455–1462.

Duan B., Zamir M., 1995, Mechanics of Wave Reflections in a Coronary Bypass Loop Model: the Possibility of Partial Flow Cut-off, Journal of Biomechanics 28 (1995), pp 567- 574.

Pejoviæ-Mililæ A., Pejoviæ S. and Karney B., Extreme Theoretical Pressure Oscillations in Coronary Bypass, Proceedings of the International Conference on CSHS03, Belgrade, 2003 (in press).

The computer programs’ subroutines are added to the already extensively tested technical software to simulate boundary conditions typical for blood vessels such as arteries without reflection, backward excitation of the hearth through the coronary artery and some external excitations.

The aim of hydraulic analysis carried out here, by using design and parameters typical in the human coronary circulation, is to evaluate conditions when the bypass functions safely, avoiding possible undesirable effects to the patient. 37 38

On Nonlinear Perturbed Stochastic Hereditary Nonlinear Viscoelasticity of Polymer Melts in Differential Equations Electromagnetic Vibration Force Field

Miljana Jovanović and Svetlana Janković _ Juan Zhang ∗ Faculty of Science and Mathematics, Department of Applied Mathematics , Donghua University, Shanghai 200051, China Department of Mathematics, University of Ni¡s [email protected] Vi¡segradska 33, 18000 Ni¡s, SCG E-mail: [email protected], [email protected] Jin-Ping Qu National Engineering Research Center of Novel Equipment for Polymer Processing, South It is well-known that hereditary phenomena are useful tool for studying China University of Technology, Guangzhou 510640, China real phenomena in continuum mechanics of materials with memories, as a version of the theory of ”fading memory” spaces. Mathematical models in studies of viscoelasticity represent deterministic functional hereditary differential equations, Abstract researched by many authors in many papers and monographs. Especially, Mizel and Trutzer [3] incorporated the effect of the Gaussian white noise on hereditary Recently, the electromagnetic dynamic moulding method and the apparatus phenomena, as a random excitation of the deterministic case, so that mathematical have been introduced successfully into the polymer extrusion and injection models were reproduced by stochastic hereditary differential and integrodifferential processing. One of the key technologies is that an electromagnetic field can cause equations of the Ito type. the mechanical vibration field effectively. That means an electromagnetic winding The present paper deals with the asymptotic behavior of the solution of the affects the processes of polymers plasticization, extrusion or injection and pressure perturbed stochastic hereditary integrodifferential equation of the Ito type, indirectly. At the same time when the screw and the metal moving body are investigated earlier in [1] and [2]. Its coeffcients are linear with respect to the simultaneously rotating pulsatorily, the screw , at the time,makes axial pulsative Lebesgue and Ito integrals and depend on small perturbations. The solution of this displaments, so that the processes of plasticization, extrusion or injection and equation is compared in the (2m)-th moment sense with the solution of the pressure maintaining are all carried out in a periodic oscillating state. At present, appropriate unperturbed equation, on finite intervals or on intervals whose length the electromagnetic dynamic apparatus for plastics has acquired great economic tends to infinity as small perturbations tend to zero. and social benefit progressively. And there has been a great deal of research By following these theoretical considerations, we investigate the validity activity to investigate the behavior of polymers in electromagnetic vibration force of the previous concept and results, by applying them to an example. In fact, for a field. It is well known that polymer melts have viscoelastic behavior, so how to given rate of the closeness between the solutions of the perturbed and unperturbed measure their viscoelastic behavior is worthy and essential to study. equations, we give some numerical estimations of time-intervals, with respect to For conveniently, we use the capillary dynamic rheometer, which has been the small perturbations and deterministic parameters. developed successfully in National Engineering Research Center of Novel Equipment for Polymer Processing in South China University of Technology , to References: study the viscoelastic behavior of the polymer melts. The results show the inherent

[1] Sv. Janković, M. Jovanović, Perturbed stochastic hereditary di_erential equations, nonlinear character of the viscoelastic behavior of polymer melts in Stochastic Anall. and Appl., 20 (3) (2002), 567–589. electromagnetic vibration force field.

[2] Sv. Janković, M. Jovanović, On perturbed stochastic hereditary di_erential equations, Computers & Math. with Appl., 42 (2001), 871–881.

[3] V.J. Mizel, V. Trutzer, On stochastic hereditary equations: existence and ∗ Corresponding author. E-mail: [email protected]; Fax: 86-21-62193606 asymptotic stability, J. of Integral Equations, 7 (1984), 1–72.

• Supported by Grant No 1834 of MNTRS through Math. Institute SANU. 39 40 ANALYTICAL AND NUMERICAL SOLUTION OF The material is assumed incompressible, so the bulk modulus is NON-STATIONARY STATE OF STRESS IN A THIN considered constant and the relaxation function purely describes shape changes. VISCOELASTIC PLATE Naturally, there exist more complicated discrete or continuous material models that describe viscoelastic properties more generally and accurately, but their usage Vitezslav Adamek and Vladislav Las, significantly increases the complexity of the analytical procedure. West Bohemia University in Pilsen, Univerzitni After the introduction of integral constitutive equations into motion 22, 306 14 Pilsen (tel.: +420377632335, e-mail: [email protected]) equations, the system of two dependent partial integro-di_erential equations of the

Frantisek Vales, second order for the unknown functions of deflection and deflection angle is Institute of Thermomechanics AS CR, obtained. Considering the geometry of problem solved and the character of applied Veleslavinova 11, 301 14 Pilsen, Czech Republic. loading, it is advantageous to solve this system by the method of integral transformations. After the Laplace transform application in time domain and ABSTRACT assuming that initial conditions are zero, the original system of equations can be This contribution deals with the non-stationary phenomena in viscoelastic rewritten in the form of the Bessel’s equations. It is well known that it is suitable to solids. The analytical and the numerical solution of a non-stationary state of stress assume the solution of this system in the form of the Hankel’s integrals. If these in a thin homogenous isotropic viscoelastic plate are presented. The geometry of relations are used, two algebraic equations for the deflection and deflection angle problem solved consists of a thin infinite plate with constant thickness. Its upper transforms are obtained. It is easy to solve this equation system and to find complex face is loaded by transverse pressure loading that is non-zero only in the circle area transforms of unknown functions. At this stage of solution process, the first with finite diameter. The amplitude is constant in the whole circle area and changes important results are gained. By using the singularity analysis of complex according to the Heaviside function in time, such that the loading has an impact transforms, the dispersion and attenuation behaviour of the plate are explored. character and it evokes non-stationary stress waves. Considering the axial Owing to the corrections used, two dispersion and two attenuation curves exist. symmetry of applied loading and the infinity of geometry, the problem is solved as These curves correspond to two modes of waves propagated in the plate. After this an axisymmetric one and the cylindrical coordinate system is used. Consequently, analysis, Hankel and Laplace inverse transformations are used for the finding of the unknown functions of plate deflection and deflection angle depend only on time required functions of deflection and deflection angle. Because of the complicated and on one spatial coordinate. forms of the transforms, it is impossible to express unknown functions explicitly, The first section of this work concerns the analytical solution of the so the inverse transformations are performed numerically. problem. Since the analytical solution resulted from exact theory of 3D continuum The second section of this work is focused on the numerical solution of is feasible only in special simplified cases, the theory of thin plates is used for the problem presented. The problem is solved as 3D and axisymmetric one in FE derivation of motion equations. For being in better accord with the exact solution, software MSC.Marc. The spatial discretisation is realized by 8-node elements some corrections are used. In addition to the transverse displacement of an element, (bricks) and 4-node axisymmetric elements with linear base functions. The element its rotational momentum and the effect of shear forces on the plate deflection are size is chosen according to the Fourier spectrum of applied loading. The implicit taken into account. These corrections are analogous to the Timoshenko’s correction Newmarc method is used for the numerical integration in time domain. Its constant in the case of thin beams. The deflection angle is then expressed as the sum of the time step is chosen as the maximum stable time step in the case of the explicit deflection angle caused by bending moments and the angle that corresponds to the method of central di_erences. shear strain of an element. In view of this fact, the geometric relations resulted In the last section of this contribution, numerical and analytical results are from the theory of small deformations are introduced into the motion equations. presented and their analysis and comparison are performed. Additionally, the Afterwards, the constitutive equations in cylindrical coordinates are accuracy of the analytical method used and its suitability for the solution of the derived. The standard linear viscoelastic solid was chosen for the representing of problem presented are discussed. material properties of the plate. This model is one of the simplest discrete models that are suitable for the simulation of nonstationary phenomena, for the phase Acknowledgment: This research was supported by the Research Project MSM velocity of high frequencies does not tend to infinity. 235200003 and by the grant of GA AS CR No A2076001.

41 42 The Cauchy Problem for Fuzzy Fractional Theoretical Study on the Friction Resistance of a Differential Equations Flat Plate by

Micro-bubbles

Cheng Yu, Guozhu Gao and Liangjian Hu, Zhi-Yong Liang Department of Applied Mathematics, Donghua University, Center of Physics of Fibrous Materials, College of Science, Donghua University, Shanghai, 200051, Peoples Republic of China P.O. Box 471, 1882 Yan’an Xilu Road, Shanghai 200051, China Email: [email protected]

Abstract

A lot of experiments have been done to improve the peculiarity of a turbulent boundary layer by micro-bubbles. Researchers usually make micro- bubbles by injecting air into water. Experiments show that the friction resistance of a plate (or a haul) can be reduced by micro-bubbles. The amount of resistance reduction increases with the amount of bubbles when the coming velocity is fixed. The resistant reduction increases with the coming velocity increases when the concentration of bubbles is fixed. But the resistance reduces with the coming velocity increases when the amount of injecting bubbles is fixed. The maximum coefficient of resistant reduction reaches more than 80%. However, the accurate mechanism of resistant reduction by bubbles is not very clear because of the complication of two-phase turbulent flow. This paper tries to do some research work on this problem with spectrum method by numerical simulation. The numerical results show that the present theory is consistent with the previous experiment in quality.

Key words: Micro-bubbles, Flat plate, Friction resistance, and Numerical simulation.

43 44 ON THE ASYMPTOTIC STABILITY OF LINEAR DISCRETE TIME DELAY SYSTEMS The paper of Trinh, Aldeen (1995) presents some new sufficient conditions for robust and D-stability of discrete – delay perturbed systems. It has been shown that these results are less conservative than those reported in literature, S. B. Stojanovic*, D. Lj. Debeljkovic** particularly to Mori et. Al (1982). * University of Nis, Faculty of Technology, This paper extends some of the basic results in the area of Lyapunov Bulevar Oslobodjenja 124, 16000 Leskovac, Serbia and Monte Negro (asymptotic) to linear, discrete, time invariant time-delay systems. These results are **University of Belgrade, Faculty of Mechanical Eng, given in the form of only sufficient conditions, in the form of time delayed Dept. of Control Eng., 11000 Belgrade, Serbia and Monte Negro independent criteria, and represent other generalisation of some previous ones or completely new results. The problem of investigation of time delay systems has been exploited In the latter case it is easy to show that, in the most cases, these results are over many years. Time delay is very often encountered in various technical less conservative then those in existing literature. The first group represents systems, such as electric, pneumatic and hydraulic networks, chemical processes, generalization of some previous results of Mori et. Al (1992) and Trinith et. al long transmission lines, etc. (1995) which are concerned with the cases with only one delay. The existence of pure time lag, regardless if it is present in the control Another group is dealing with suitable decomposition of matrices and or/and the state, may cause undesirable system transient response, or even gives a main contribution of paper and are in the same time less restrictive than instability. Consequently, the problem of stability analysis for this class of systems others given in recent literature. has been one of the main interests for many researchers. In general, the introduction A numerical example has been working out to show the applicability of of time delay factors makes the analysis much more complicated. results derived. In the existing stability criteria, mainly two ways of approach have been To the best knowledge of the authors, such results have not yet been adopted. reported. Namely, one direction is to contrive the stability condition which does not \ include the information on the delay, and the other is the method which takes it into References account. The former case is often called the delay - independent criteria and generally provides simple algebraic conditions. Numerous reports have been published on this matter, with particular [1] Lee T. and Radovic U., (1987), “General decentralized stabilization of large – scale linear continuous and discrete time-delay systems”, Int. J. Control 46 (6) emphasis on the application of Lyapunov’s second method or on using the concept 2127—2140. of the matrix measure. The results concerning Lyapunov stability, for non delay time systems, are [2] Mori, T., Fukuma N. and Kuwahara M., (1982), “Delay - Independent Stability well documented in a number of known references, thus for the sake of brevity are Criteria for Discrete - Delay Systems”, IEEE Trans. Automat. Contr., AC-27 (4) 946-966. omitted here. Discussing the problem of investigation of linear discrete time delay [3] Trinh H. and Aldeen M., (1995a), “D-Stability Analysis of Discrete-Delay Perturbed systems and their Lyapunov stability it should be pointed out that there are not to Systems”, Int. J. Control 61 (2) 493-505. many results dealing with this problem. Several sufficient conditions for asymptotic stability of linear discrete – delay systems were presented in the paper of Mori et al. (1982). Since these conditions are independent of delay and possess simple forms, they provide useful tools to check system stability at the first stage. The study of stabilization problem for general decentralized large - scale linear continuous and discrete time delay systems using local feedback controllers were presented by Lee, Radovic (1987). The local feedback controls were assumed to be memoryless. In that sense, the sufficient stabilization conditions were established. 45 46 ON ASYMPTOTIC STABILITY OF PARTICULAR Generally, the singular differential control systems with time delay can be written CLASS OF LINEAR CONTINUOUS SINGULAR as:

TIME DELAY SYSTEMS

D. Lj. Debeljkovic* S. B. Stojanovic**, M. B. Jovanoviæ***, S. A. Milinkoviæ*** *University of Belgrade, Faculty of Mechanical Eng, Dept. of Control Eng., 11000 Belgrade, Serbia and Montenegro ** University of Nis, Faculty of Technology, Bulevar Oslobodjenja 124, 16000 Leskovac, Serbia and Montenegro ***University of Belgrade, Faculty of Metallurgy and Technology, 11000 Belgrade, Serbia and Montenegro [email protected] When the general time delay systems are considered, in the existing stability criteria, mainly two ways of approach have been adopted. It should be noticed that in some systems we must consider their character Namely, one direction is to contrive the stability condition which does not of dynamic and static state at the same time. Singular systems are those the include the information on the delay, and the other is the method which takes it into dynamics of which are governed by a mixture of algebraic and differential account. equations. Recently many scholars have paid much attention to singular systems The former case is often called the delay - independent criteria and and have obtained many good consequences. The complex nature of singular generally provides simple algebraic conditions. systems causes many difficulties in the analytical and numerical treatment of such In that sense the question of their stability deserves great attention. systems, particularly when there is a need for their control. Our aim is to derive new results concerning asymptotic stability of a The problem of investigation of time delay systems has been exploited particular class of linear continuous singular time delay systems. over many years. Time delay is very often encountered in various technical This paper extends one of the basic results in the area of Lyapunov systems, such as electric, pneumatic and hydraulic networks, chemical processes, (asymptotic) to linear, continuous singular time invariant time-delay systems. This long transmission lines, etc. The existence of pure time lag, regardless if it is result is given in the form of only sufficient conditions and is related to the present in the control or/and the state, may cause undesirable system transient following class of systems: response, or even instability. Consequently, the problem of stability analysis for this class of systems has been one of the main interests for many researchers. In general, the introduction of time delay factors makes the analysis much more .To the best knowledge of the authors, such result have not yet been reported. complicated. We must emphasize that there are a lot of systems that have the phenomena of time delay and singular simultaneously, we call such systems as the singular differential systems with time delay. These systems have many special characters. If we want to describe them more exactly, to design them more accurately and to control them more effectively, we must paid tremendous endeavor to investigate them, but that is obviously very difficult work. In recent references authors had discussed such systems and got some consequences. But in the study of such systems, there are still many problems to be considered.

47 48

APPENDIX I

Poslanica prvog predsednika Akademije nauka dr Josifa Pan~i}a

Serbian Academy of Sciences and Arts “(1) da se naša akademija u svima svojim radovima rukovodi samo istinom i strogo naučnom objektivnošću i da se nikad ne povodi za kojekakvim verovanjima i strujama, koje Društvo srpske slovesnosti (1842) vreme ili socijalne prilike iznesu. Sve nauke potrebuju pomoći od drugih nauka, ali ni jedna The Serbian Scientific Society (1842) nauka nema prava da što drugoj propisuje, da joj pravac ili smisao menja. “... to create a scientific language and to use it to widen, popularize scientific knowledge...” (2)… Osobiti činim pažljive g.g. akademičare da se nikada ne upuštaju u propisivanje under the influence of Vuk Karadzic, in 1845 the Society changes its initial mission into: kojekakvih nomenklatura i tehničkih termina …koliko je polje nauke prostrano, toliko je i “... to study the language of its people, hitory, customs, ancient relics, land and all it posseses...” svaki akademičar slobodan da objekte svog saznanja po volji, sposobnosti i spremi svojoj potraži onde gde se za to oseća sposobnim. Ali za prvi bi početak dobro bilo, da se November, 1, 1886 in Niš, People’s Parliement voted The Royal-Serbian Academy Foundation Act ograničimo na one nauke koje se srpstva i južnog slovenstva, a naročito zemalja (Kraljevsko-srpska akademija). Balkanskog poluostrva tiču. Tu je naša prošlost zakopana, a tu mislim da leži i naša budućnost, u koju ja čvrsto i nepokolebljivo verujem, jer joj tu sve uslove za što bolji uslov 1887. posvednevno gledam …Vaš, Vama davno i svagda odani dr J. Pančić.” Academy of Natural Sciences (Akademija prirodnih nauka) dr Josif Pančić Dimitrije Nešić The epistle of the first president of the academy of sciences Josif Pancic, Ph. D. Ljubomir Klerić “1) That our academy in all of its works be led only by the truth and strict scientific objectivity and that it never allows it slef to be taken and led by prejudice and beliefs Academy of Social Sciences (Akademija nauka društvenih) brought about by the times or social conditions. All sciences need help from other sciences, Čedomir Mijatović but no science has the right to prescribe things to any other, to direct it or change its Milan Dj. Milićević meaning. Ljubomir Kovačević 2)....I call upon academicians that they never get involved in prescribing certain Pantelija Panta Srećković nomenclatures and technical terms .... as much as the areas of science are wide, that much is Academy of Philosophy Sciences (Akademija nauka filozofskih) every academician free to search for the objects of his cognition according to his will, ability Stojan Novaković and qualification wherever he feels able to find them. But for the beginning it would be Milan Kujindžić good, that we limit ourselves to those sciences that would be of benefit to Serbs and Southern Svetislav Vulović Slavs and especially to the Balkan peninsula. There, our past is buried, and there, I believe, Svetomir Nikolajević lies our future, in which I strongly and determinedly believe, because there I see the potentials for better future... Yours to You for long time and always faithfull J. Pancic, Ph. Academy of Arts (Akademija umetnosti) D.” Ljubomir Nenadović Mihajlo Valtrović Matija Ban Promene naziva Davorin Janko The Royal-Serbian Academy - Kraljevsko-srpska akademija (1886) The Serbian Royal Academy - Srpska kraljevska akademija (1892) The First President of the Royal-serbian Academy Serbian Academy of Sciences - Srpska akademija nauka (1947) Serbian Academy of Sciences and Arts - at present- Srpska akademija nauka i umetnosti danas 1887-1888 Dr Josif Pančić (1814-1888)

“…da nauku obradjuje i unapredjuje; do postavlja i održava zdrave osnove naučnom sudu; da Presidents obelodanjuje i izaziva istraživanja naučna u prirodi, društvu i istorijskim spomenicima; da potpomaže odomaćivanja i razvitak viših umetnosti; i da udruženom snagom za napredak prosvete izvršuje sve za The First President of the Royal-serbian Academy što je poptrebna snaga nedovoljna;..” 1887-1888 Dr Josif Pančić (1814-1888) “… to refine and advance science; to set and maintain healthy basis of scientific judgement; to uncover * Serbian Scientific Terminology, and stimulate scientific research in nature, society and historical monuments; to help the domestication * Biology and Natural Sciences and development of higher arts; and to, with joint strength for the advancement of education, execute * Founder of Zoo-Cabinet, Botany-Cabinet and Mineralogy-Cabinet, and * Botanicae Garden in everything for which the available strength is insufficient…” Belgrade

51 52

1888-1889 Čedomilj Mijatović (1842-1932) * Writter of Pariotic esseys, Politician, ecconomist, diplomat 1937-1960 dr Aleksandar Belić (1876-1960) * History of medieval Serbian economy * the most famous Serbian linguist, dialectology, history of language * Introduced the Dinar as the first Serbian Convertible currency * works: “On the nature of language and language development”. 1881 signed the famous secret Convention with Austria-Hungary “ The dictionary of Serbo-Croatian literate and folks language” as the Ministter of Foreign Affairs 1892-1895 Dimitrije Neišić (1836-1904) 1960-1965 dr Ilija Djurišić (1898-1965) * Professor of Mathematics at Great School * a medic, physiologist, the dean of the school of veterinary medicine * Introdused metric measures in Serbia * physiology of nerves and muscules, physiology of heart and blood vessels, experimental hematology,... 1896-1899 Milan Dj. Milićevć (1831-1908) * Writer, historian, Folclorist, translator, education specialist 1965-1971 Velibor Gligorić (1899-1977) * serbian literary critic 1899 Jovan Ristić (1831-1899) * Royal executive, professional politician and respected statesman 1971- 1981 dr Pavle Savić (1909-1994) “Extremes are never the roads of freedom. Temperativness is the road to freedom , and a shield which * physical chemistry and nuclear physics guards it.” * The assistent of Irena Zolio Kiri on research of the problems of nuclear fission, organized the work of the institute Boris Kidric in Vinca.

1899-1900 i 1903-1906 Dr Sima Lozanić (1847-1935) * chemist and the rector of the Great school. 1981-1994 dr Dušan Kanazir ( 1921) * The first honorary Ph.D. of the Belgrade University * a medic, biochemistry, molecular biology, radiobiology, biochemistry on nucleic acids, mechanisms of * Works in organic and nonorganic chemistry, analytical chemistry, agricultural chemistry, ionizing radiation influence on cells electrosynthesis... 1994-1998 dr Aleksandar Despić (1927) 1900-1903 general Jovan Mišković (1844-1908) * technologist, fundamental and applied electrochemistry, chemical engineering * Defense minister and the Serbian army headquarters commander * military geography, history, tactics * founder of the Museum of scince and technology in Belgrade

1999- 2003 Dejan Medaković (1922) 1906- 1915 Stojan Novakovi} (1842-1915) * art hystorian * politican, diplomat, philologist, historian of literacy, one of the greatest minds of the Serbian people * stood for the ideas that the unity of all souther Slavs is necessary 2003- dr Nikola Hajdin (1923) * Applied and Technical Mechanics 1915-1921 dr Jovan M. Žujović (1856-1936) * Bridges Constructions * a famous geologist, Great school professor * Civil Engineering Projekts * founded the Geology institute, the serbian geographical society and the journal “ Geographical annals of the Balkan peninsula” * the supporter of the united state of southern Slavs and a republican.

1921-1927 dr Jovan Cvijić (1865-1927) * the most important Serbian intellectual of that time * the research of karsts, glaciation, hydrography, physical geography, antropogeography, etnopsychology, geopolitics * Hypotheseis on the genesis of the Alpian arc and on the formation of DJerdap canyon.

1928-1931 Slobodan Jovanović (1869-1958) * historian, lawyer, politician * the president of the royal goverment in exile

1931-1937 dr Bogdan Gavrilović (1863-1946) * a famous mathemtician, algebra, geometry and theoretical mathematics, rector of the University * works: Analythical geometry, and linear algebra

53 54

APPENDIX II

57 SCIENTIFIC COMMITTEE SOME REMARKS F. L. Chernousko, Institute for Problems in Mechanics, Russian Academy of Sciences E. Kreuzer, Arbeitsbereich Meerestechnik II-Mechanik, Technische Universität Hamburg-Harburg, ON THE IUTAM SYMPOSIUM ON CHAOTIC DYNAMICS AND F. C. Moon, Sibley School of Mechanical and Aerospace Engineering, Cornell University G. Rega (Chairman), Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza CONTROL OF SYSTEMS AND PROCESSES IN MECHANICS, G. Stepan, Department of Applied Mechanics, Technical University of Budapest.. 8-13 June 2003, Rome, Italy J. M. T. Thompson, Centre for Nonlinear Dynamics, Civil Engineering Building, University College London H. Troger, Institut für Mechanik, Technische Universität Wien D. H. van Campen (IUTAM Representative), Faculty of Mechanical Engineering, Eindhoven University of The IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Technology F. Vestroni (Co-chairman), Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Mechanics was organized by the Department of Structural and Geotechnical Engineering of Sapienza the University of Roma "La Sapienza", Rome, Italy, within the program of the 2003 Symposia of the International Union of Theoretical and Applied Mechanics (IUTAM). Chairman of Key Lectures A smart controller for commercial and military cranes Organizing Committee was Professor Giuseppe Rega and Co-Chairman Professor Fabricio Nayfeh A.H., Masoud Z.N., Nayfeh N.A Vestroni, and in preface of the booklet of Abstract they wrote the following: Virginia Polytechnic Institute and State University, Virginia, USA

"The IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Chaotic clock models: a paradigm for noise in machines Moon F. C. Cornell University, New York, USA Mechanics continues the tradition of earlier IUTAM Symposia in the field of chaotic Control and synchronization of homoclinic chaos and its implication for neurodynamics dynamics in mechanics (London 1993, Cornell, 1997), a research area which engages a Arecchi F.T. University of Firenze and Istituto Nazionale di Ottica Applicata, Italy large and active community of scientists. Nowadays, the matter of control of chaotic Spatial motion of CVT-chains dynamics is becoming increasingly important, too. Accordingly, this new Symposium in the Pfeiffer F., Sedlmayr M. Technical University of Munich, Germany series is aimed at diving deep both into theory and applications to mechanics of nonlinear Some Lectures and, chaotic dynamics, and into their control". Active damping of spatio-temporal dynamics of drill-strings Kreuzer E., Struck H. Technical University of Hamburg, Germany The interest of the mechanics community towards nonlinear/chaotic dynamics of Bifurcation and chaos in mechanical applications: a dynamical system approach to their engineering systems has exploded in the last decade. Successful IUTAM Symposium control Lenci S., Università di Ancona, Italy, Rega G., Università di Roma "La Sapienza", Italy (London 1993, Eindhoven 1996, Cornell, 1997) gave first evidence: (i) of the need to Control of limit cycle oscillations complement analytical/computational treatments with geometrical approaches and Schiehlen W., Guse N. , University of Stuttgart, Germany. Controlled motions of multibody systems along a plane experimental analyses; (ii) of the limitations inherent to the archetypal single/two-degree-of- Chernousko F.L., Russian Academy of Sciences, Russia. freedom systems from the point of view of technical applications characterized by rather Delay equations with fluctuating delay : application to regenerative chatter high-dimensionality; (iii) of the interaction between dynamics and control. Namachchivaya N. S., Universiy of Illinois at Urbana-Champaign, Illinois, USA Van Roessel H.J., University of Alberta, Canada It can bee concluded from the list of all invited participants, that the lectures the newest Nonlinear phenomena in microelectromechanical resonators result in the field of nonlinear continuous and high-dimensional systems, numerical and Balachandran B., Li H., University of Maryland, Maryland, USA Optimal control of retrieval of tethered subsatellite geometrical methods in chaotic dynamics, qualitative and quantitative analysis of chaotic Steindl A., Troger H., Technical University of Wien, Austria. dynamics, Stability and bifurcation analysis, phenomena and criteria for spatio-temporal chaotic oscillations, interaction between chaotic and stochastic phenomena, control of Contribution from Serbia: oscillations and chaos, experimental investigation of chaotic dynamics, Applications Fascinating Nonlinear Dynamics of a Heavy Material Particle(s) Along Circle(s) with in engineering mechanics, interaction problems. Coupled Rotations and Optimal Control by Hedrih (Stevanović) K., University of Niš, Serbia, Yugoslavia, From the scope of the symposium organizer wrote: "It is clear how difficult and involved the passage from simple models to actual engineering systems can be. The International Union of Theoretical and Applied Mechanics (IUTAM) was formed in Nevertheless, there is need to explore practical applications of nonlinear and 1946 with the object of creating a link between persons and national or international chaotic dynamics, and to draw the technical world's attention to the potential of organizations engaged in scientific work (theoretical or applied) in solid and fluid mechanics these new phenomena and to their implication in the design and operating or in related sciences. It does so mainly by organizing international meetings to deal with scientific problems. An International Congress on Theoretical and Applied Mechanics (next conditions advanced engineering systems. This implies considering more realistic ICTAM - 21st INTERNATIONAL CONGRESS OF THEORETICAL AND APPLIED models, obtaining meaningful hints from experimental investigations, MECHANICS 15-21 August, 2004, Warsaw, Poland, http://ictam04.ippt.gov.pl), including developing/generalizing techniques for nonlinear analysis, exploring complex mini-symposia and pre-nominated sessions, is held every four years with unrestricted behaviours, and analysisng needs and features for control of the observed dynamics". participation. Additionally a number of specialized Symposia with invited participants are held every year. These IUTAM Symposia are sometimes held in cooperation with other Unions in the International Council for Science or with the 17 organizations affiliated to IUTAM.

58 59

Participants at the break of Participants of The IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes The IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in in Mechanics, 8-13 June 2003, Rome, Department of Structural and Geotechnical Mechanics, 8-13 June 2003, Rome, Department of Structural and Geotechnical Engineering Engineering of the University of Roma "la Sapienza", Rome, Italy of the University of Roma "la Sapienza", Rome, Italy In the middle N. S. Namashivaya, D. H. von Campen and F. Vestroni

J. M. T. Thompson, V. Beletski, F. Moon, F. Chernousko and H. Troger at The IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in Mechanics, 8-13 June 2003, Rome, Department of Structural and Geotechnical Engineering of the University of Roma "la Sapienza", Rome, Italy Participants of

IUTAM SYMPOSIUM Non-linearity and Chaos in Engineering Dynamics D Materials characterization and structural behaviour under shock and impact. UCL 1993 London UK E Biomechanics. 60 61 F Non-convex and non-smooth mechanics: Defects, Damage and fracture, Contact mechanics; Phase Forth International Conference on transitions and hysteresis; static and dynamical buckling analysis. G Multi-scale modeling of material. Nonlinear Mechanics (ICNM-IV) H Nonlinear coupling of electric, magnetic, thermal field and deformation. L Fluid Flow, vortex dynamics and nonlinear waves. August 14-17, 2002, Shanghai, P.R. China M Turbulence and nonlinear stability. and N Nonlinear dynamics: solitons, bifurcation, chaos and complexity. IUTAM SYMPOSIUM ON DUALITY-COMPLEMENTARITY- P Interface mechanics and free boundary problems. SYMMETRY IN NONLINEAR MECHANICS Q Dissipative systems. R Experimental mechanics and experimental simulation techniques. S Miscellaneous. The Fourth International Conference on Nonlinear Mechanics and IUTAM Symposium Conference Sessions and Minisimposia: on Duality-Complementarity-Symmetry in Nonlinear Mechanics was held in the period Session A1. A2 and A3. Solid mechanics I (Constitutive Models, Finite deformation, Elasticity and August 14-17, 2002 in the new Campus of the Shanghai University, China. In this respect the Plasticity). th 4 ICNM - IV continues the tradition of the three previous International Conferences on Session B1. B2 and B3. Solid Mechanics II . (Nonlinear Theory of Plates and Shells). Nonlinear Mechanics ICNM's held in Shanghai (1985), in Beijing (1993) and in Shanghai Session C1. C2. and C3. Solid Mechanics III . (Structure Mechanics) (1998). Session D1. D2. D3. and D4. Fluid Mechanics. Conference Chairmen were: Prof. Dr Wei-zang Chien, Academmician of Chinese Session E1. E2. E3. and E4. Nonlinear Dynamics . Academy of Sciences, President of Shanghai University, P.R. China and Prof. Dr Session F1. F2. F3. and F4. Mathematical Methods in Mechanics. R.W.Ogden, George Sinclair Professor, Glasgow University, UK. Chairman of the Local Minisimposium 1. (M1): Piezoelectric materials. Organizing Committee was Prof. Dr Zhe-wei Zhou. Minisimposium 2. (M2): Waves and instability in solids. IUTAM Symposium on Complementary-Dual Variational Principles in Nonlinear Minisimposium 3. (M3): Vortex Dynamics. Mechanics (Duality-Complementarity=Symmetry in Nonlinear Mechanics – IUTAM - SDCS), also was held during the Conference ICNM. From Serbia: ICNM –IV and IUTAM Symposium sponsored by Chinese Society of Theoretical and 1. Transversal Creep Vibrations of a Fractional derivative Order Constitutive Relation Applied Mechanics, Sciences and Technology Commission of Shanghai Municipality, Beam by Katica (Stevanović) Hedrih, in Session C1.Solid Mechanics III . (Structure Shanghai University and by IUTAM Symposia Committee, National Natural Science Mechanics) . 2. On Rheonomic Systems with Equivalent Holonomic Scleronomic Conservative System by Foundation of China, National Science Foundation of United State of America, K. C. Wong' Education Foundation , Hong Kong, Shanghai Institute of Applied Mathematics and Katica (Stevanović) Hedrih, in Session E2. Nonlinear Dynamics Mechanics, Shanghai Center for Nonlinear Sciences, Editorial Board of Journal Applied 3. Stress Distribution around the Tip of the Dynamically Propagating Interfacial Crack by Methematics and Mechanics. R. Nikolić and J. Veljković, in Session II3. It is not so often that a series of international conferences have been convening Publications of the Conference: successfully in the country with increasing frequencies and increasing number of Fourth International Conference on Nonlinear Mechanics (ICNM – IV) – Final Program participants. The influence and leading role of Prof. dr Chien Wei-zang have played a crucial and List of Paticipants, Shanghai, 2002., p. 53. part in the success of the series of conferences.

IUTAM Symposium on Complementary-Dual Variational Principles in Nonlinear Prof. Chien published his famous papers on the intrinsic theory of plates and shells at his Mechanics (Duality-Complementarity=Symmetry in Nonlinear Mechanics – IUTAM - age of thirty and completed the work on the plate-shell theory based on non-Kirchhoff SDCS) – Final Program and List of Paticipants, Shanghai, 2002., p. 9. assumptions when he was near 90 years old. He has been plowing and weeding in the field of nonlinear mechanics for almost 60 years. It is greater importance that he cultivated and Proceedings of the Fourth International Conference on Nonlinear Mechanics (ICNM – trained a large amount of younger talents in the area of mechanics, founded the related IV) Edited by Editor-in -Chief Chien Wei zang, and Vice-chief Editors: Cheng Chang jun, institutions, advocated various activities of academic exchange and promoted the Dai Shi-qiang , Guo Xing-ming, DONG LI-yun, Shanghai university Press, Shanghai, 2002., development of nonlinear mechanics in China and in the world. Professor Chien has his 90th p. 1324 +5 . birthday in last, conference year.

Fourth International Conference on Nonlinear Mechanics (ICNM – IV) - CONFERENCE TOPICS A Nonlinear continuum mecghanics. B Modern mathematical methods and numerical simulations in nonlinear mechanics. C Optimization and Control for smart, composite systems. The Fourth International Conference on Nonlinear Mechanics and IUTAM Symposium on Duality-Complementarity-Symmetry in Nonlinear Mechanics, Shanghai 2002. 62 Chairmen: Wei-zang Chien and R. W. Ogden 63

D. H. von Campen, IUTAM Representative at The Fourth International Conference on Nonlinear Mechanics and IUTAM Symposium on Duality-Complementarity-Symmetry in Nonlinear Mechanics

Opening ceremony of Ji Huan He, Editor-and-Chief of Int. Jour. of Nonlinear Science and Applications and D. H. von Campen. IUTAM Representative, and Editor-in Chief of Scientific Jour. Facta Universitatis at The Fourth International Conference on Nonlinear Mechanics and IUTAM Symposium on Duality-Complementarity-Symmetry in Nonlinear Mechanics, S Opening Plenary Lecture was: Statistical Theory of Oscillations by V.V. Kozlov Shanghai 2002 65 64 ON THE 4TH EUROMECH NONLINEAR OSCILLATIONS CONFERENCE, 19-23 August 2002, Moscow, Russia

The 4th EuroMech Nonlinear Oscillations Conference was organized by the Institute for Problems in Mechanics, Russian Academy of Sciences, within the program of the 2002 conference of the European Mechanics Society - EUROMECH. Chairman of Organizing Committee was Professor Klimov A.A., Moscow, Russia and Chairman of the European Nonlinear Oscillations Conference Committee Professor E. Kreuzer, Hamburg, Germany. In From Serbia: preface of the Conference Programme they wrote the following: On Rheonomic Systems in the Field with Turbulent Dampling with Equivalent Holonomic "European Mechanics Society – EUROMECH is an international nongovernmental Nonconservative System. by Katica (Stevanović) Hedrih nonprofit scientific organization. The objective of the Society is to engage in all activities Analysis of Nonlinear Vibrations of a Rotor with a Transverse Crack by V. Marjanović and R. Nikolić. intended to promote in Europe the development of mechanics as a branch of science and engineering. Mechanics deals with motion, flow and deformation of matter, be it fluid or The interest of the mechanics community towards nonlinear/chaotic oscillations of solid, under the action of applied forces, and with any associated phenomena". engineering systems has exploded in the last three decades. Successful ICNO and ENOC The society is governed by the Council whose membrs are being elected according to rules set in Conferences gave first evidence: (i) of the need to complement analytical/computational Statutes. The European Mechanics Council has organized more than 400 Euromech Colloquia on treatments with geometrical approaches and experimental analyses; (ii) of the limitations inherent specialized topics in mechanics. The Nonlinear Oscillations Conference Committee (ENOC) was to the archetypal single/two-degree-of-freedom and with many degree of freedom oscillations established to organize Euromech Conferences on Nonlinear Oscillations. The conference was intended systems from the point of view of technical applications characterized by rather high- to bring together scientists working in different fields of dynamics, to exchange ideas and discuss dimensionality; (iii) of the interaction between oscillations and control of oscillations. current research trends and developments with emphasis on the interdisciplinary character of nonlinear It can bee concluded from the list of all participants, that the lectures the newest result oscillations. in the field of nonlinear continuous and high-dimensional oscillations systems, numerical and In this respect the 4th ENOC continues the tradition of the twelve International Conferences on Nonlinear Oscillations (ICNO's) and the three previous ENOC's held in Hamburg (1993), Prague (1996) geometrical methods in oscillations and chaotic oscillations dynamics, qualitative and quantitative and Copenhagen (1999). analysis of nonlinear oscillations, Stability and bifurcation analysis, phenomena and criteria for spatio-temporal nonlinear and chaotic oscillations, interaction between chaotic and stochastic 4th Euromech Nonlinear Oscillations Conference - CONFERENCE TOPICS phenomena, control of oscillations and chaos, experimental investigation of chaotic dynamics, A Nonlinear dynamics of continuous and discontinuous systems. applications in engineering mechanics, interaction problems. B Qualitative and quantitative analysis of nonlinear dynamic systems. C Analysis of bifurcations and chaos. D Numerical and geometrical methods in nonlinear dynamics. E Spatio-temporal chaotic oscillations. F Nonlinear random vibrations. G Phenomena and criteria of chaotic oscillations. H Computer aided symbolic methods in dynamics L Control of oscillations and chaos. M Nonlinear optics. N Experimental methods in nonlinear dynamics. P Applications in mechanical engineering, electrical engineering, physics, biology, chemistry and other sciences. SCIENTIFIC PROGRAMME The scientific programme involved invited and contributed presentations: Section I. GENERAL THEORY OF OSCILLATIONS. Section II. CONTROL OF COMPLEX SYSTEMS. Section III. NONLINEAR OSCILLATIONS IN CONTINUOUS MEDIA. Section IV. APPLIED PROBLEMS OF NONLINEAR OSCILLATION. Section V. STOCHASTIC AND CHAOTIC OSCILLATIONS. Participants of The 4th EuroMech Nonlinear Oscillations Conference, organized by the Institute for Problems in Mechanics, Russian Academy of Sciences, 2002 - European Mechanics Society – EUROMECH 66

Some of the international scientific meetings on Nonlinear Mechanics and Sciences in the last three decades in the world

The First International Conference on Nonlinear Oscillations - Paris 1932.

EUROMECH, International Conferences European Nonlinear Oscillations Conference ENOC Hamburg, Germany, 1993, Prague, Czech, 1997, Lyngby, Copenhagen 1999, Moscow, Russia, 2002. previous

ICNO International Conference on Nonlinear Oscillations Participants of ICNO Kiev 1961, Varsow, Berlin, Prague, Marsel, ... Kiev 1969, International Conference on Nonlinear Oscillations Kracow 1972,...... Varna 1982,...... Kiev 1969. Chairman ICNO academician Yuri Alekseevich Mitropolskiy

IUTAM Symposim Nonlinearity and Chaos in Engineering Dynamics, University College London 1993, England, Great Brittany. Chairman Professor J.M.T. Thompson

IUTAM Symposim New Applications of Nonlinear and CHAOTIC DYNAMICS IN MECHANICS, Ithaca, NY, USA, 1997. Chairman Professor Francis C. Moon

IUTAM Symposium Chaotic Dynamics and Control of Systems and Processes in Mechanics Roma, Italy, 2003. Chairmen Professors: Giuseppe Rega and Fabricio Vestroni

Participants of ICNO International Conference on Nonlinear Oscillations Varna 1982. 67 68 Review of the international scientific conference: THE SECOND INTERNATIONAL CONFERENCE CONTROL OF OSCILLATIONS AND CHAOS COC 2000 July 5-7,2000, Saint-Petersburg, Russia

Second International Conference "Control of Oscillations and Chaos" - COC 2000, July 5-7,2000, Saint-Petersburg, Russia was held from 5 to 7 June 2000 at House of Scientists (dvorcovaya Naberehnaya 26, St. Petersburg, phone 315-5959, e-mail: [email protected]). Organizers of the conference were: Institute of Mechanical Engineering of Russian Academy of Science, State University of Saint Petersburg, Society of Information Systems and Control of Saint Petersburg and Saint Petersburg Educational and Research Center: Problems of machines and buildings, mechanics and process control. The sponsors and cosponsors of the conference were: The Russian Academy of Sciences (RAS), Russian Fundamental Research Fund, The IEEE Circuits and System Society and The International Union of Theoretical and Applied Mechanics. YURIY ALEKSEEVICH MITROPOLSKIY Academician of Ukrainian National Academy of Sciences and The president of the scientific committee was one of the worlds leading scientists in Academician of Russian Academy of Sciences the field of robotics, RAS academician Feliks Chernousko, while the president of the Chairman of the Scientific Committees ICNO organizing board was professor A. L. Fradkov from the Saint Petersburg Institute for Problems of Mechanical Engineering (61 Bolshoy ave V.O., 1999178, St. Peterburg Russia, http://www.ipme.ru/coc2000.html). Scientific board was formed of thirty scientists from the whole world (Russia, USA, Sweden, Netherlands, Italy, Switzerland, Austria, Poland, Germany, Ukraine, Japan and China) in the field of control and robotics. Many visitors came to the conference and during the conference 7 plenary and 9 semiplenary invited lectures were presented, as well as 116 short scientific contributed lectures classified into 11 sessions; 51 contributed lectures were presented at the poster presentations and total number of authors and co-authors from 24 countries was around 360. The following plenary and semi plenary invited lectures were presented: I.I. Blekhman (Institute for Problems of Mechanical Engineering, RAS, St. Petersburg, Russia) and K.A. Lurie (Worchester Polytechnic Inst. USA): Creating Dynamical Materials as a Problem of Control (Oppening Plenary Lecture). F. L. Chernousko (Institute for Problems in Mechanics, RAS, Moscow, Russia): Snake-like Motions of Multibody Systems over a Rought Plane. M. Dimentberg, D. Iourtchenko (Worchester Polytechnic Inst. USA) and A. Bratus' (Moscow State University, Russia): Optimal Bounded Control of Random Vibration and Hybrid Solutions to dynamic Programming Equations. A.S. Dmitriev, A. I . Panas and S. O. Starkov (Institute of Radio Engineering and Electronics, RAS, Moscow, Russia): Multiple Access Communication Based on Control of Special Chaotic Trajectories. R. J. P. de Figueiredo (University of California, Irwine, USA): A reproducing Participanst of ICNO Kernel Hilbert Space (RKHS) Approach to the Optimal Modeling, Identification, and Design International Conference on Nonlinear Oscillations of Nonlinear Adaptive Systems (IEEE Distinguished Lecture) – (Closing Plenary Lecture) . Cracow 1990.

69 70 K.S. Hedrih (University of Nis, Yugoslavia): Nonlinear Dynamics of a Gyrorotor, Extremely interesting semi plenary lecture of 40 min duration was presented by and Sensitive Dependence on the Initial Conditions of the Forced Vibrations of a Heavy the president of the conference scientific board, academician L. Chernousko (Institute for Gyrorotor. Problems in Mechanics, RAS, Moscow, Russia). In his lecture Snake-like Motions of E. Kreuzer, M. Wendt (Technical University Hamburg, Germany) Multibody Systems over a Rought Plane he presented how motion of a being (snake on a Nonlinear Dynamics of Ship Oscilations. rough surface) was used to design control algorithms and numerical simulations for robot V. F. Lazutkin (St. Petersburg State University, Russia): Exponentially Small motion. At this point I would remind readers that the academician F. L. Chernousko Separatrix Spliting. contributed an invited plenary lecture at Yugoslav Congress of Theoretical and Applied E. Mosekilde (Technical University of Denmark, Lyngby, Denmark): Chaotic Mechanics held in 1995 at the Faculty of Mechanical Engineering in Niš. Synchronization in Living Systems. The plenary lecture presented by IEEE president, professor R. J.P. de Figueiredo P. C. Miller (Technical University of Wupertal, Germany): Nonlinearity (University of California, Irwine, USA) should also be mentioned as distinguished lecture at Estimation and Compensation by Linear Observers: Theory and Applications. the closing of the conference. The title of his contribution is A reproducing Kernel Hilbert F. Pfeiffer, Th. Rossmann, K. Loffler (Technical University of Muenchen, Space (RKHS) Approach to the Optimal Modelling, Identification, and Design of Nonlinear Germany): Control of a Tube Crawling Machine. Adaptive Systems. W. Schiehlen (University of Stuttgart, Germany): Motion Control of Vehicles in Among the authors of the contributed lectures there were authots who presented a Convoy. number of papers as coauthors, so there are multiple papers of those authors in the A. N. Sharkovsky, S. A. Berezovsky (Institute of Mathematics, Kiev, Ukraine): proceedings. Transitions of "Correct-Incorrect" Numerical Calculations for Solutions of Some Problems One of these authors is prof. J. M. Baltezar (Departamento de Estatistica, "Phase Transitions" in Computer tubulence. Matematica aplicada e Computacional, UNESP-Rio Claro, Brazil) from Brazil who has 4 F. E. Udwadia (University of Southern California, Los Angeles, USA): New coauthored papers published in this proceedings. The idea about the content of these papers developements in Dynamics of Constrained Systems with Applications to Tracking Control. can be obtained from the following titles: Nonlinear Oscillations of a Porlat Frame Structure N. Van Dao (Vietnam National University, Hanoi, Vietnam): Delay Control in excited by a nonideal motor (coauthors: Reyollando M.L., and Brasil R.F.), Bifurcational Nonlinear Oscillating Systems. Phenomena in a Nonideal System (coauthors: D. Belato, H. I. Weber and J. M. Rosario), N. van de Wouw (Eindhoven University of technology, The Netherlands), H. Nonlinear Stieltjes Equations and Proximity of Solutions (coauthors: L. Barbanti and J.C. Nijmeijer (University of Twente, Enschede, The Netherlands) and D. H. van Campen Prandini) and A Note of the Use of Melnikov's Chaos Prediction Criterion: Results of a (Eindhoven University of technology, The Netherlands): Statistical Bilinearization in Friction-Driven Self-Excited Vibrating System (coauthors: B.R. Pontes and V.A. Oliveira). Stochastic Nonlinear Dynamics. One of these authors is also F. L. Chernousko (Institute for Problems in O. M. Belotserkovsky, Yu. D. Shevelov (Institute for Computer Aided Design, Mechanics, RAS, Moscow, Russia) with the paper named: Control Algorithm for a Linear RAS, Moscow, Russia) F. A. Maksimov (Instrument Design Bureau, Tula, Russia): A Flow Dynamic system subjected to Complex Constraints; (the first coauthor A. S. Smyshlyaev). about Delta Wings at Various Reynold Numbers. The abstract of this paper is: A linear dynamical controlled system is considered The first plenary lecture at the opening of the conference was presented by with complex constraints imposed on the control variable and state coordinates. A control academician I.I. Blekhman (Institute for Problems of Mechanical Engineering, RAS, St. algorithm is constructed which satisfies all constraints and drives the system to a prescribed Petersburg, Russia) and his co-author K.A. Lurie (Worchester Polytechnic Inst. USA): terminal state in a finite time. It is shown that this algorithm can be used for a feedbak Creating Dynamical Materials as a Problem of Control. This lecture is an important control of the system subjected to perturbations. The operation of this algorithm is illustrated contribution to knowledge and creation about dynamic active materials. It was also one of by an example of the control process of an electrically driven system. the invited lectures at Yugoslav Congress on Theoretical and Applied Mechanics 1997 in The participants were first delivered Preliminary Program and Call for Vrnjačka Banja. Participation, page 28, and at the registration desk following publications were delivered: It should also be mentioned that a group of five leading scientists in the field of Final Program - Control of Oscillations and Chaos (COC 2000), page 40 and three books of mechanics, robotics and control from Germany presented invited lectures. They are: F. Proceedings - 2000 2nd Second International Conference, Control of Oscillations and Chaos Pfeiffer (Technical University of Munich, Germany), W. Schiehlen (University of Stuttgart, COC 2000, July 5-7,2000, Saint-Petersburg, Russia-Proceedings, Edited by F.L. Chernousko Germany), E. Kreuzer (Technical University Hamburg, Germany) and P. C. Miller and A. I. Fradkov, page 584. (Technical University of Wupertal, Germany). They presented top realizations of scientific Information on Second International Conference Control of Oscillations and Chaos results in the field of control through review of concrete models and prototypes. As at ENOC COC 2000, Saint-Petersburg, Russia can be found at web site: 1999 in Kopenhagen, professor F. Pfeiffer showed a certain interest for conditions under http://www.ipmme.ru/coc2000.html. which the research work in Yugoslavia is undertaken. On 22 pages at the beginning of every volume a review of the conference program The plenary lecture of E. Mosekilde (Technical University of Denmark, Lyngby, is given with the sections marked. The names of those sections are: Denmark): Chaotic Synchronization in Living Systems was also interesting, who, regrettably did not print his paper in these proceedings. Professor E. Mosekilde, also, showed interest for cooperation with Yugoslav scientists. 71 72 Control Oscillations, Control of Chaos, Control of Mechanical Systems, Mathematics of patented and which can be produced. Of course the professional results are with the leading Stability and Instability, Identification and Reconstruction of Parameters, Synchronization, and united areas, which deserve respect and which put their authors to the top of the Nonlinear Dynamics and Chaos in Physical Systems, Nonlinear Dynamics and Chaos in scientific creativity. Mechanical Systems, Nonlinear Dynamics and Chaos in Electronics, Communications and As I concluded in the presentations of scientific conferences in the past decade, I Power Systems, Nonlinear Dynamics and Chaos in Biomedical, Ecological and Neural can repeat here, the scientists from western countries show final results of research in the Systems, Methods of System Analysis. form of patented models, but with little explanations on fundamental results, in the form " I show that I projected and constructed a model which works", while the scientists from the Eastern and former and present socialist countries like Russia, Ukraine, China present theoretical, analytical results of the highest level that can be imagined at this moment by the modern civilization of the human mind. But ideas and knowledge can not be protected by a patent nor can they yield profit. Profit is made by those who materialize the ideas in the form of products for the real market. Benefits of participating in the scientific conference: If I would ask a question: What is the benefit of participating in this scientific conference? The answer could develop in two directions. 1. One of the answers is personal and it results in participant's advanced training through getting new knowledge in the fields of nonlinear dynamics, oscillation and chaos control, knowledge about new control prototypes of mechanical, building constructions, electro energetic and economic systems and inspiration for new research and meeting colleagues who perform research in the same fields. 2. Founding Yugoslav center for nonlinear dynamics with three divisions: A. division for mathematical phenomenology of nonlinear dynamics, B. laboratory division for experimental research of nonlinear dynamics and C. division for models with basic workshops in military industry and big state companies of machinery building and electro industry. Such center would gather all competent researchers in the field of nonlinear dynamics from different disparate fields in engineering, medicine, bio-agricultural science, economics social sciences, which meet the models of nonlinear dynamic processes and phenomena. This center would also create conditions for more rational financing of scientific researches and integration of scientific knowledge from disparate fields of science. Such

center would also enable to create a basic new university textbooks and monographs of new Scientific organizer of the international scientific conference: concepts and new minimal optimal essence of fundamental sciences which students of the THE SECOND INTERNATIONAL CONFERENCE CONTROL OF universities should learn and know as future professionals on which the country should count OSCILLATIONS AND CHAOS COC 2000 when opening perspective of survival and planned standard of life and work of people in July 5-7, 2000, Saint-Petersburg, Russia harmony with nature in this region. Such center could perform researches on strategic project Scientists: Feliks Chernousko, R. J. P. de Figueiredo, W. Schiehlen and A. L. Fradkov. "Internet in Serbia". *** 3. Acquisition of fundamental science monographs from different disparate The evaluation of the conference: scientific fields, as well as scientific journals in the fields of nonlinear dynamics and chaos If I would have to evaluate the program of this international scientific conference, published in the last decade, ever since different blockades were imposed on Yugoslavia; then, if I would compare it with the others I participated in so far, and which were in the area that material should be placed in one of national libraries or in the library at the center for of theoretical and applied mechanics, nonlinear mechanics and nonlinear oscillations, I nonlinear mechanics in order to be available to all interested researchers. would have to say that this one differs in topics and in the way the papers were presented, 4. If this is not done fast I fear that people who understand those phenomena and because of its interdisciplinary character and because it contains lectures ranging from purely can transmit their knowledge to young generations will disappear from this region, as well as theoretical, mathematical accomplishments, and accomplishments in the areas of "literacy" in these fields. And that "literacy" to read latest scientific achievements is very fundamental scientific areas of engineering such as: mechanics, fluid mechanics, electronics, important for future scientific research. biomechanics, various computer methods, to those which presented only the realizations of 5. For that reason providing conditions for work of one permanent "international the well-known scientific ideas, through engineering contributions, but finalized to models, seminar-type school of science" in the field of nonlinear dynamics and control is also

73 74 necessary, with pretension to form, besides Center for nonlinear dynamics, International International Symposium Study Research Congress and Communication Center for gathering scientists from the whole world and from different disparate fields of theory and applications. on Analysis, Manifolds and Mechanics I am aware that we can not develop and finance all fields of science, but we can M. C. Chaki Centre for Mathematics and Mathematical form a center for international communications through scientific school for seminars and Sciences through international meetings, which would provide latest original scientific knowledge and Calcutta University, February 5-7, 2003 from the whole world at first with minimal investment and later even for free, because the In honour of Professor M.C. Chaki’s 90th Birth center would finance itself from the participants' fees from the whole world if it acquired Anniversary reputation of prestige and useful meeting. It would provide meeting at our regions of information courses from the latest scientific achievements. And our region also has M. C. Chaki Centre for Mathematics and Mathematical Sciences and geographic advantage for realization of such an idea. Calcutta University, India, in collaboration with Indian Association for Possible practical solutions for basic knowledge in nonlinear dynamics and control the Cultivation of Science, Organizing Committee with Prof Dr U.C. or nonlinear dynamics systems: De, University of Kalyani, and Prof. Dr Manjusha Majumder, Calcutta Creating technology and products based on nonlinear dynamics, processes and University, as well as International Advisory Committee with Dr S. P. phenomena and their control on the basis of fast and slow oscillations and separation and Mukherjee and Dr B. K. Datta organized International Symposium on Analysis, Manifolds and joining of certain dynamics. Design of active constructions and active dynamic materials Mechanics, in Calcutta February 5-7, 2003, in honour of Professor M.C. Chaki’s 90th Birth An- under control. Designing new kinds of dynamic active materials, which should be of interest niversary. for military industry. That is the reason why technology projects should be supported, This biennial venue gathered over 120. authors and co-authors and scientists-participants from through which elaborate schemes and technological documentation and models for Italy, India, Japan, Korea, Turkey, Germany, Romania, Bangladesh and Yugoslavia. application and production would be created. Constructions for oscillation control, which From Scientific Programme: could be finalized as product series of vibration machines for mining, civil engineering, • Paying Respects to Professor Chaki – professors: B. K. Lahiri (University of Kalyani), process technology in food production and agriculture. R. K. Poddar (Calcutta University) • Reading of Written Citation – Dr P.K. Majumder K. S. H. • V. de Sabbata (Italy): Hamiltonian Formulation of Gravitational Theory and Spin Fluctuations; • T. Kawaguchi (Japan): On the Application of Finsler Geometry to Engineering Dynamical Systems: • K. (Stevanovic) Hedrih (Yugoslavia): Vector Method and Phase Plane Method, Approach for the Analysis of Kinetic Parameters and Homoclinic Orbits of Coupled Rotor Dynamics • F. Ozen and S.Altay (Turkey): The Bianchi Identities in a Reimiannian Manifold Admitting a Special type of semi-symmetric Metric Connection; • M. K. Chakrabortty (India): A Theory of Partial Consistency: A Logical Approach. • N. C. Chattopdhyoy and A. Chakraborti (India) Study of Thermal stress generated in and Elastic Half-Space in the context of Generalized Thermoelasticity Theory; • Sarbari Ray-Guha and M.C. Chaki (India): On Conformally Flat Perfect Fluid Pseudo Ricci Symmetric Space-time; • U.C.De (India): On weakly Symmetric Structures on a Reimannian Manifold; • S. Maeda (Japan): Circles and Hypersurfaces in space forms; • T. Adachi (Japan) A comparison Theorem Associated with KäMagnetic Field. • R. Karmakar (Bangladesh): Semiclassical approximation calculations for K-shell Ionization of Atoms by Heavy charged particles; • B. Chaki (India): A Role of the Cosmological Constant for a Classification of Dirac Particles in Quasi Einstein Manifold; Financial support/donation for the Symposium by students and admirers of Prof. Dr M. C. Chaki.

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NONLINEAR ANALYSIS AND IT'S APPLICATIONS M. M. Khrustalev (Russia): Method of Lyapunov-Lagrange Functions in Optimal Control International Congress, September 1-5, 1998, Moscow, Problem for Distributed Parameter Systems. Russia, pp. 190. Dan Kannan (USA): Stability of Some Reducible Singular Markovian Systems. Publisher: Russian Academy of Sciences (RAS), Stability and Nonlinear Dynamics Research B. B. Beletsky (Russia): Applied Problems of the Dynamical Billiards. Center of Mechanical Engineering Research Institute (MERI), Academy of Nonlinear V. V. Kozlov (Russia): First Lyapunov Method for Nonlinear Systems. Sciences (ANS), International Federation of Nonlinear Analysis (IFNA), UNESCO. V. Lakshmikhantham (USA): Advances in Stability Theory of Lyapunov: Old and New. V. E. Pannin, K. V. Frolov (Russia): Physical Mesomechanics of Structural Instabilities in Solid under Loading. This publication contains the 154 abstracts from the Program of International V. Vujičić (Yugoslavia): A new approach to the fundamental concept of classical Congress Nonlinear Analysis and it's Applications was held at Moscow in period September mechanics. September 1-5, 1998. General Chairs of the International Congress were Professors: V. Here we will only quote a selection of the abstracts of the short communications-papers: Lakshmikhantham, IFNA, Florida Institute of Technology, Melbourne (USA) and V. M. S. B. Agase (India): On Constrained Practical Stability. Matrossov, Academy of Nonlinear Sciences (Russia). F. F. Alekseev (Russia): Synthesis and Analysis of the Combined Systems on the Base of The contents of this publication include following two parts: Plenary Papers (p.5) Nonlinear Analysis. and Sectional papers and short communications (p.53). Ya. Goltser (Israel): Invariant Tori and Totally Stable Resonance Modes in Nonlinear Second part Sectional papers and short communications is divided into following Oscillations Theory. Sections: N. I. Matrossova (Russia): On Stability of Sets for Differential Equations in Banach Spaces Stability and Nonlinear Oscillations; by VLF Method with Comparison System in Critical Case. Nonlinear Control Theory; G. I. Melnikov (Russia): Sequences of Nonlinear Differential Inequalities in Estimations of Nonlinear Mechanics; Stability of Motions. Nonlinear Differential Equations and Analysis; M. M. Khapaev (Russia): Stability and Evolution in the Planetary Problem of Three Bodies. Nonlinear Modeling in Natural and Technical Sciences. Norimichi Hirano (Japan): Existence of Subharmonic Periodic Solutions for Second Order Abstracts of papers are allocated to corresponding sections. Ordinary Differential Equations with a Singularity. Here we will only quote a selection of the abstracts of the plenary lectures- papers: A. G. Butkovskiy (Russia): Mathematical Structures in the Theory of Control and Systems. V. A. Trenogin, A. M. Ter-Krikorov, N. A. Sidorov (Russia): Nonlinear Analysis: Inversion S. M. Meerkov (USA): Vibrational Control: Theory and Applications. of the Nonlinear Operators, Extension by Parameters, Bifurcations, Group Invariance and H. H. Dai (Hong Kong): Existence and Analytical Description of Kink Waves in it's Applications. Incompressible Hyperelastic Rods. J. Batt (Germany): Recent Results in the Mathematical Investigation of Two Nonlinear V. F. Krotov (Russia): On Genesis of Mass from Field. Systems of partial Differential Equations Describing the Evolution of Gravitational and K. (Stevanović) Hedrih (Yugoslavia): Axoids in the Nonlinear Dynamics of the Heavy Charged Matter: The Vlasov-Poisson- and the Vlasov-Maxwell-System. Rotors with many Axes of the Rotation (Gyro-Rotors). V. M. Matrosov, E. I. Somov, G. P. Anshakov, V. A. Raevskiy (Russia); Nonlinear E. Galperin (Canada): ETA-Equivalent Solutions of Nonlinear PDE-Problems. Dynamics of the Notis-Stable Systems of the Control Motion of the Space Equipment. Jean-Pierre Caubet (France): Differential Equations with or without impulses: Asymptotic S. Leela (USA): Stability criteria for Fuzzy Differential Equations. Behavior of Solutions and Ergodism. Leon O. Chua ( USA): Edge of Chos: A Mathematical Characterization. S. Pilipović, M. Stojanović (Yugoslavia): G-Quasiasymptotic Expansion to Nonlinear V. B. Bajic (South Africa): The Subsidiary Parametric Functions and Generic Concepts in Generalized Solutions. Qualitative Analysis of System Behavior. S. Heikkila (Finland): On Operator and Differential Equations in Ordered Spaces. T. A. Burton (Carbondale): Periodic Solutions of the Functional Differential Equations. J. S. Florio (Mexico): On the Reduction Theory of Stability in Dynamical and K. V. Frolov (Russia): Human Factor of the Technologic Safety as a Nonlinear Dynamical Semidynamical Systems. Problems.

P. K. C. Wang (China): Bifurcation Problems in Nonlinear Microdistributed Systems.

Nobuaki Obata (Japan): Nonlinear Extension of Quantum Stohastic Analysis of Differential

Equations.

Xilin Fu, Liqin Zhang (China): Oscillation Properties of Nonlinear Impulsive Partial

Differential Equations.

V. V. Rumyantsev (Russia): On General Equations of Analytical Dynamics.

V. V. Bolotin (Russia), A. N. Kounadis (Greece), J. B. Roberts (England): Stability and

Postcritical Vibrations of Nonconservative Deformable Mechanical Systems.

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A PRESENTATION OF SCIENTIFIC CONTENTS OF K.F. Chernih: Nonlinear singular theory of stability. THE INTERNATIONAL CONFERENCE This lecture pointed to the popular areas of mechanics and physics of deformable "NONLINEAR SCIENCES ON THE BORDER OF MILLENIUMS" bodies through: the brittle fracture mechanics, dislocation and disinclination in crystals and dedicated to the 275th Annibersary of the Russian Academy of Sciences the concentrated forces and moments, through comparation of approaches in linear and Saint-Petersburg, June 22-24, 1999. nonlinear setting of the problem. It points to the small number of papers in the area of nonlinear approach to this problems group and mainly through physical and geometrical In the period from June 22nd to 24th, at the University of Saint-Petersburg and at sources of non-linearity. The lecturer pointed to his own results and to the results of his the Institute for precise mechanics and optics - Technical University of Saint-Petersburg, an associates in the area of nonlinear singular theory of elasticity. international scientific conference called "Nonlinear sciences on the border of millenniums" V.A.Yakibovih: The square criteria of absolute stability of nonlinear systems. was held. It was dedicated to the 275th anniversary of the Russian Academy of Science. The M.B. Ignyatev: Vagueness and the phenomenon of adaptation maximum in complex organizers of this conference were The Russian Academy of Science, the Academy of nonlinear systems. Nonlinear sciences, Saint-Petersburg's state-owned Institute of Precise mechanics and optics The lecture takes into consideration the evolution of the problem of vagueness in - the Technical University. science and technology since the appearance of the probability theory, through the The scientific and program committee was constituted of well-known scientists Schredinger's vagueness principle and till the latest worked-out systems with vagueness in from around the world. The participants were also from around the world. linguistics, mechanics, biology, chemistry, physics, numerical systems, and in the artificial The scientific part of the conference was held plenary and by sections, with a intelligence systems. The research of nonlinear systems with vagueness are only starting and certain number of invited lectures. promise many new discoveries. In the year 1963 the phenomenon of adaptation maximum in In the plenary part of the program lectures had an overview character, and in the systems which develop through degrees of freedom increase was discovered. By using the sections they were specialized. phenomenon of adaptation maximum we can explain many anti-entropic processes both in The conference had the following sections: nature and in the society. In the process of interaction with a changeable environment, the Nonlinear mechanics and applications, appearance of random coefficients in the system of equivalent equations allows the system to Nonlinear differential equations and analysis, conduct changes which are of such nature that the more vague coefficients there are the Nonlinear control systems theory, greater are the adaptation possibilities of the system. Considerations about the possibility of Nonlinear methods and models in natural, technical and humanitarian sciences, controlling such systems - the systems which make possible the keeping of such systems Stability and nonlinear oscillations. within the area of adaptation maximum in the current of changes through the removal of old Among others there were the following plenary lectures: boundaries and their integration into a collective system, are presented. V.M. Matrosov: The development of the stability theory at the Russian Academy of The presentation pointed out that stable development is possible only in the zone of Science. adaptation maximum and that is possible for biological, social-economical, physical- A review of papers in the area of the motion stability theory development at the chemical systems and systems of self-sustained reactions. In order to increase the Russian Academy of Science published since the foundation of RAS 275 years ago is given. harmlessness of system they must be kept within the area of the adaptation maximum. On The classic works of Euler (18th century) on the matter of equilibrium stability of elastic the basis of the system with vagueness a unified nonlinear theory of matter has being built, rods and A.M. Ljapunov in the area of setting of method of nonlinear theory of motion which explains the behavior of gases, liquids, rigid bodies, plasma and living structures. A stability and its applications to the problems of mechanics (19th century) are described. After unified theory of matter is being created on the basis of analysis of the transfer from one that the ideas, generalized and modified function methods of Ljapunov in the 19th century point of rest to another. are presented, and the paper also pointed out to the tasks of their further development in the Kolesnikov A.A: A Synergetic approach to nonlinear control systems theory. 21st century and that applies both to the basic methods of nonlinear dynamical analysis of A new synergetic concept in the control theory, based on the fundamental characteristic of different systems, including the logical-dynamical ones, and method of analysis and self-organization of natural disipative systems is proposed in this lecture. Invariance, self- synthesis of complex systems which allow hybrid defining and also many others. He organization, nonlinearity, optimization and synthesis appear as basic terms of the synergetic considered the basic applications of the Ljapunov's function methods in application to control theory developed in the lecture which determine its essence, innovations and problems of mechanics, technology, physics, economy, in the tasks of stability preservation, contents. controlling and projecting and similar, and especially for aero-kinetic, electromechanical, At the basis of the synergetic approach lie two fundamental principles of nature - energy and other systems. they are, first- the invariance principle and second - the principle of contraction (shrinking) - Important problems of science, techniques of global development, no-spherical expansion of phase volume in dissipate dynamical systems of random nature. Based on the control, large-scale systems theories, intellectual support to the accepting of solutions, synergetic concept an essentially new invariant-group approach to the analytical construction dynamical theories of games and many others for the research of which, the Ljapunov's of nonlinear multi-measure dynamical objects control systems with complex function method can be applied, are defined in the conclusion. interconnections, based on the idea of introduction of attracting invariant multiplicities –

79 80 attractors, on which the natural (energetic, mechanical, heat etc.) properties of object The development of aviation technology is in connection with constant battle for consigned in the best way. Such attractors (synergy) form internal dynamic connections, as a the increase of speed and the reduction of weight of flying objects, and in connection with it result of what coherent collective motion in the phase space of the system appears. That the reduction of relative thickness of wings and the increase in the resistance of material permits the realization of a purposive (directed) self-organization of the collective state in which all led to the appearance of flexible and thin constructions, the elastic movements of dynamic systems of different nature. In a developed synergetic approach the laws of control which during flight can not be considered small. Taking into account the elastic movements which take into consideration the internal interactions of concrete physical (chemical, of wings in the process of calculating their load caused the development of linear problems biological) phenomena and processes are synthesized. That approach enabled a breakthrough of aero-elasticity. Big movements also demand that the geometric non-linearities of the in solving the fundamental applied problems of setting up physical (chemical, biological, studied system be taken into consideration. That is especially applicable on compositional social-economical) theories of control as problems of searching for general objective laws of constructions such as the wing with wing fans. control processes. The introduced language of invariants, as a primary element of synergetic As noticed by author, the forces in the hanging supports in the middle area cause control theory, permits the attachment of natural-mathematical unity to the theory and the the appearance of untouchable types of equilibrium, the crossing to which represents the establishment of a direct connection to preservation laws i.e. to basic prepositions of natural catastrophic changes of shape of resistance (wings). The catastrophic crossings into streams properties of objects which correspond to nature. can result statically or represent an oscillatory process by themselves- the oscillations of As the first, the new synergetic approach permits a breakthrough in the area of catastrophic change of shape. synthesis of continuos, discrete, selectively-invariant, multi-criterial, terminal and adaptiv ... The scientific importance of the authors contribution is in the fact, that it brings nonlinear dynamical objects of different physical nature control systems with complex essential changes in the concept of composed bodies - like rods which are composed jointly interconnections. Such approach found a direct application to solving of complex problems in a stream of gas, interaction and it can be formulated in the following way: The of control of nonlinear technical objects (flying machines, turbo-generators, robots, electric phenomenon which was theoretically and experimentally unknown earlier - the catastrophic power suppliers, technological aggregates etc.), as well as in control tasks in ecology, change oscillations of composed bodies in a stream of gas, consist in the fact that bodies of biotechnology etc. prolonged shape - the type of plain rods, which have a big discrepancy between maximum O.V. Vasiljev and V.A. Srochko: On numerical methods of optimal control and minimum bending rigidities of their transversal cuts, composed jointly by the long sides problems solving. of the plain of max rigidity and which have small rigidly immobilized in one of the This lecture considers the types of classes of problems of optimal programmed transversal cuts, angles between those plains, when simultaneously bent by the load of a control in normal dynamical systems. A prehistory of development and advancement of moving stream catastrophically cross into a new equilibrium state in the area of bending numerical solving methods which are connected to the maximum principle and gradient movements and curling of rods, which, when the transversal suts attack angles are small, approximations is given. The lecture also gives an overview of modern approaches to the return to starting shape, forming that way an oscillatory process. development of numerical methods which posses the properties of improving through (the S.A. Zagezda, N.G. Filipov M.P. Yushkov: A combined task of dynamics and use of) effectiveness characteristics. nonlinear non-holonom connections of high order. H. Miyagi (Okinawa): Application of fuzzy relations to nonlinear uncertain By the use of the introduced tangential space in multiplicity's of all the possible systems. positions of a free mechanical system, the equations of Lagrange of the second order are Fuzzy relations are well known as the tools for the description, representation and written in the form of vector equation, which has the shape of the Newton's second law. By symbolic manipulation of knowledge or system uncertainty. Computational algorithms and the use of constraint equations up to the second order the involved tangential spaces fall systematic methodology of fuzzy relational equations and inequalities are engaged in their apart on a direct sum of two subspaces, in one of which the component of the system practical utilization in nonlinear knowledge-based systems, information retrieval, medical acceleration vector is completely determined by constraint equations as a function of time, and psychological diagnosis and some business applications. Solutions of fuzzy relation generalized coordinates and generalized velocities. equations presented by Sanchez have been widely applied to the fuzzy inferences, system On the basis of introduced subspaces a definition of ideal connection is given. That identifications, diagnosis problems etc. In company with relation equation, another important definition is also concerned with linear connections of a high order. As such connections we formula is relation inequality which is also used in the area of fuzzy reasoning and fuzzy refer to movement program, given in the form of a differential equations system. A system problem diagnostics which is supported by the upper or the lower boundary of normal of differential equations is constituted in relation to the required generalized coordinated and searching for solutions by the use of fuzzy relation inequality. Whatsoever, such a method the required control forces, which secure the fulfillment of the given program connections depends on the solution algorithm which constitutes of several rules and that complicates (non-holonom connections of high order). everything. When the program is given in the form of differential equation, nonlinear in In this lecture, an effective way of solving fuzzy relation equations and inequalities relation to higher derivatives, that equation is reduced to a linear form through differentiation by introducing fuzzy operators is presented. Operator definitions are made capable of the in time. In the process the concoction order is increased by one. simpler solving procedures and a method suitable for application is considered. By using the Among the section lectures the following were especially interesting: operators we get a whole set of solutions if we solve the new linear matrix equations. I.V. Matrosov: Modeling of global security and stability of world development with taking V.A. Pavlov: The phenomena of catastrophic change of shape of flying objects. into consideration the bio-mass dynamics, control of BWP distribution, and scientific and technological progress. 81 82 A.V. Bushmanov: The Analysis of Tension-deformational State of Biomechanical Systems. In the lecture author described a system he proposed himself for the analysis of A.A. Tikhonov: On Nonlinear Oscillations of Gravity-oriented Rigid Bodies. global security and stable development. For evaluating global stability the following F. Wegmann and F. Pfeiffer (Germany): The Textile Threads Dynamics. indicators of the security of world development have been used: the prolongation of life, the Katica (Stevanović) Hedrih (Yugoslavia): Differential Equations of Two Material Particles spending of nutritional and industrial products, average social spending per capita in the Dynamics Constrained with a Hereditary Element world, pollution level and the bio-mass of plant inhabited part of land which remained at the and a 20 minute long, improvised speech On Some Research Directions and the Integration end of the 21st century as a natural resource. If the higher values (in the case of pollution of Scientific Knowledge in the are of Nonlinear dynamical Systems. lower) are the border of permissibility, then the global system of world developments in the In the concluding part of the closing session of this international scientific state of security, but if one of the inequalities is disturbed we talk about the disturbance of conference a scientific discussion developed on perspectives of development of sciences that the security conditions. System is set up on the basis of the model of D.Medous World 3.91 deal with nonlinear dynamical processes, a especially connected to nonlinear processes and written in program language C++. stability in social systems, in the course of which the perspective of civilization on Earth as a Additionally the following things are taken into consideration in the system: nonlinear system was analyzed - "noosphere" with all the properties of intellect and other - The control of the total world bruto product - time constants which take into consideration resources of that systems dynamics. I was given the floor and in my speech I pointed to the the time of making certain branches are introduced into control equations. dangers of monopoly control and the system of deciding "by the power of wealth" and about - Dynamics of change of the plant inhabited land bio-mass dependent on the pollution level. the need for the limitation of those powers by the "force of the power of conciseness, moral - In the economic specter the science progress equation is introduced (by using the results of and ethics", as well as that we by our heroic resistance gave the UN again the chance to S.V. Dubrovski), make decisions, which must not be missed by the more powerful in numbers and militarily - The artificial cleaning of air stronger. - The conquest of new types and the regeneration of non-regenerative resources. The president of the scientific committee academician Matrosov held a scientific - The political tension in the world in dependence on the remaining non-regenerative speech on perspectives scientific research on global level in the area of nonlinear sciences resources, quantity of nutritional products per capita, pollution and population density. from the aspect of modeling of social processes, Earth's resources, and global nonlinear - Model contains around 300 mathematical dependence (differential equations and algebra dynamical, intelligent system on Earth - "noosphere", and which will also be in the content relations) some from the published literature, and some worked out by workers of his speech at the, at that time - future, World Conference on Science in the Third By the use of the worked out system a number of scenarios of global development has been Millennium, held in Budapest one week after the end of our conference (from June 29th to studied including both the known (depleted resource crises, ecological and demographically) July 2nd 1999), and in the work of which he participated as a scientist - a member of the and the qualitatively new. Author has been specially noticed that the scenario of Meduzov D. Russian state delegation. "A stable society" can lead to a global collapse, if the listed additional factors are included. Within the boundaries of the proposed models a scenario called "A stable development" was found, and that scenarios property is that it is not destroyed when small disturbances of the system parameters appear. The basic thing in that scenario is the use (20 to 40%) of the bruto world product for control. The work was carried out on a Pentium II computer, with C++ Builder. Harold SZU (University of George Washington: Non-surveying learning sensors in nonlinear sciences. Starting from biological systems and the basis of sensory learning, the author makes conclusions which he uses to make artificial systems which learn by using sensor and on the basis of informational data and introduces Ljapunov's function for the study of such systems. The lecture caused a very vivid discussion and many doubts about the reexamination of the statements and introduced prepositions of the presented theory. Al.A. Kolesnikov: Directed self-organization and controlling of objects with chaotic attractors. V.V. Baranov: The equilibrium method in problems of dynamic solution accepting in vagueness. Participants of THE INTERNATIONAL CONFERENCE C.N. Vasiljev: From classic problems of regulation to intelligent control. "NONLINEAR SCIENCES ON THE BORDER OF MILLENIUMS" P.E. Tovstik and V.Ju. Anisimov: On one model of crack growth. dedicated to the 275th Annibersary of the Russian Academy of Sciences V.A. Shamina: On asymptotic method of setting up nonlinear models of continual Saint-Petersburg, June 22-24, 1999. environments mechanics. Members of Academz of Nonlinear Sciences. A.O. Bochkarev: MGE of Geometrically Nonlinear Problems of Fracture Mechanics. 83 84 Moreover, he is the author or co-author of 230 articles in magazines and scientific proceedings and of The 6th International Conference and Dynamical other 76 publications. Systems, Theory and Applications His scientific works have been published in 35 different scientific magazines and proceedings of 90 conferences. Science has opened for him door to many travels abroad. He has been JUBILEE OF PROFESSOR JAN AWREJCEWICZ taken part in conferences and scientific visits in over 40 countries. His scientific work covers mechanics, mathematics and applied physics, including Jacek Nowakowski biomechan-ics and automatics. His scientific achievements have been noticed abroad where he spent Technical University of Łodz, Department of Automatics and Biomechanics (K-16) more than 7 years in different scientific centres. Professor Jan Awrejcewicz can fluently speak English, ul. Stefanowskiego 1/15, 90-924 Lodz, POLAND Rus-sian and German. Phone: +48(42) 631-23-51 Fax: +48(42) 631-22-25, E-mail: [email protected] Among his international achievements one of the most important are obtaining grants in com-petitions for scientific research from all major world foundations, such as: 1. Alexander von Humboldt Foundation (Technische Universitaet Carolo zu Wilhelmina, Braunschweig, The 6th International Conference – Dynamical Systems, Theory and Applications was held 2 years); in Lodz from 10th to 12th of December 2001. This biennial venue gathered over 50 participants from all 2. Japanese Society for the Promotion of Science (The Tokyo University, Hongo, Bukyo-ku, 1 year); continents of the world: both Americas, Africa, Asia, New Zealand and of course Europe. 3. Research Center for Advanced Science and Technology (The Tokyo University, Komaba, Meguro-ku, The following plenary lectures were performed by invited scientists: Mitsubishi Endowed Chair, 9 months); 1. Trigger of Coupled Singularities by Katica (Stevanovic) Hedrih from Nis, Yugoslavia, 4. Region Rhones Alpes (Lyon; 6 months in 1995 and 3 months in 2001/2002); 2. On the Development of Negative Pressure in Oil Film and the Characteristics of Journal Bearing by 5. The Kosciuszko Foundation Award, New York (Mechanical and Industrial Engineering, Uni-versity Tsuneo Someya from To-kyo, Japan, of Illinois in Urbana-Champaign, 3 months); 3. An Overview of Motion Localization and Energy Pumping in Coupled Nonlinear Oscillators by 6. The Fulbright Senior Grant Award (The University of California, Berkeley, 1 year); Alexander Vakakis from Illinois, USA, As well important as these (from scientific point of view) were also half-year scientific train- 4. Recent Results on Vibrating Problems with Limited Power Supply by Jose M. Balthazar from Rio ings performed in 1986 in the Institute of Fundamental Technological Research of the Polish Acad-emy Claro, Brasil, of Sciences and in Stefan Banach's Mathematical Centre in Warsaw. 5. Order Equations Method in Non-Stationary Gas Dynamics by Rem Barantsev from St. Peters-burg, After split of the Institute of Applied Mechanics Professor Jan Awrejcewicz became head of Russia. Division of Dynamics of Machines and Control and then, in 1994-1998 he is the head of Division of One of the important events of the Conference was the Jubilee session of Professor Jan Automatics and Biomechanics in the Institute of Turbomachinery. Since 1998 he has been head of the Awre-jcewicz, head of the Department of Automatics and Biomechanics. Below I present short Department of Automatics and Biomechanics (K-16). This department has been practically newly biography of the main animator of this Conference. created and it is the greatest success of Professor. Currently the department leads to major research Jan Awrejcewicz was born on 26th September 1952 in Telesze in Wilno district. In 1957 he directions and didactical work which are dynamics, control and biomechanics. These re-search crossed border of 'new Poland' in a carriage with his parents and other repatriates. He graduated eight directions are currently one of the most fashionable in the world. Professor Jan Awrejcewicz was the years of primary and four years of secondary school in Wlocławek. Due to his learning results he was initiator of creation of postgraduate studies, which have been conducted under his guidance since 1995. awarded with a gold medal as the best secondary school's graduate. In 1996-1998 he was the head of Scientific-researching Group for cooperation between Technical In 1971 he began his studies at the Mechanical Department of the Technical University in University of Lodz and the Institute of Fundamental Technological Research of the Pol-ish Academy of Lodz and in 1975 he parallelly commenced philosophical studies at the Department of Philosophy and Sciences. He has been member of two commissions of Scientific Research Com-mittee for a few years. History of the University of Lodz. Just after finishing studies of Applied Mechanics he obtained the best Moreover, he is a member of many national and international organizations, including their result in exam to the Dynamics and Control of Machines Ph. D. Course, which was administered by the choices' – Mechanics' Committee of Polish Academy of Sciences (since 1993) and Nonlinear Vi- late Pro-fessor Zdzislaw Parszewski. In 1978 he finished philosophical studies with a very good grade. brations Committee of IFToMM (since 1995). For many years he has been the head of the Science In 1978/1979 he entered Military School for Officers of Reserve in Olesnica and then a helicopter Commission of the Mechanical Department of the Technical University in Lodz. regiment in Leznica Wielka (near Leczyca) and he became lieutenant of air forces. Professor Jan Awrejcewicz has organized 25 conferences (mainly international) or special On 11 March 1981 he obtained Ph. D. at the Mechanical Department (MD) of the Technical ses-sions within these conferences. He is member of editor committees of 5 scientific journals and was a University in Lodz. His Ph. D. thesis covered problems of self-excited vibrations caused by friction in member of scientific committees of 25 different national and international conferences. assemblies with many degrees of freedom and was promoted by Professor Zdzislaw Parszewski. What's more, he has performed 35 lectures (45-60 minutes) at many famous foreign Afterwards, he worked under guidance of Professor Janusz Szosland for half a year and universities (Hannover, Braunschweig, Munich, Stuttgart, Berlin, Lyon, Glasgow, Tokyo, Kyoto, starting from 1 October 1981 thanks to personal engagement and help from Professor Jerzy Leyko he Senday, Sap-poro, Fukuoka, Melbourne, Beijing, Taejon, Manila, Singapore, Toronto, Hamilton-New commenced work as researcher in the Institute of Applied Mechanics at the Mechanical Depart-ment of Zealand, Vienna, Caracas, Rio de Janeiro, Illinois, Urbana-Champaign, Berkeley). the Technical University of Lodz. He spent first two years on self-education and gaining didactical He also performed seven plenary lectures at national and international conferences. experience. His first publications came out in 1983 (more detailed informations are available at web site Professor Jan Awrejcewicz has promoted 6 Ph. Ds. Beside many distinctions and www.p.lodz.pl/K16). international prizes he has been awarded the Golden Cross of Merit from President of Poland In 1990 he defended his habilitation thesis at the Mechanical Department of the Technical Aleksander Kwaśniewski in 1996. In 1998 he received the Ministry of Education prize for his scientific Uni-versity of Lodz. In 1994 he was distinguished with professor title by president Lech Walesa and publica-tions and in 2001 he was awarded with the Cavalier's Cross of the Poland Restitution Medal. then, in 1997 Ministry of Education of Poland awarded him with the title of ordinary professor. The Scientific Committee of the 6th Conference decided to continue to arrange such a meeting of Professor Jan Awrejcewicz is the author or co-author of 18 monographs, 2 scripts and editor scientists working on nonlinear dynamics and control and the next conference will be organized in of 2 sets of articles (published by Akademie Verlag and Springer Verlag), 9 edited volumes and sci- December 2003. Readers interested in receiving current informations concerning this venue I ask to visit entific proceedings, 5 special editions of foreign journals and author or co-author of 6 other books. the web page of the future conference: http://www.p.lodz.pl/UDYN03. 85 86 Facta Universitatis, Niš, Vol.3, No 12, 2002 pp. 485-486 THE X-TH CONFERENCE ON MECHANICAL VIBRATIONS WITH INTERNATIONAL PARTICIPATION Timisoara, Romania May 23-24, 2002

Pantele Chelu and Liviu Bereteu Department of Mechanics and Vibrations, "Politehnica" University of Timisoara, Bvd. Mihai Viteazul 1, 1900 Timisoara, Romania

The origin of the Conference goes back to 1975, when Professor Gheorghe SILAS has initiated it. The conference is organized every three years in Timisoara at the Mechanical Faculty of "Politehnica" University. This is the only periodical conference organized in Romania in the vibrations field. The main objective of the Conference was to bring together researches of several disciplines to share their latest findings in analytical, computational and experimental methods and to provide a better understanding of various dynamic aspects in the area of mechanical vibrations and biomechanics.

Participants of The 6th International Conference – Dynamical Systems, Theory and Applications, Łodz, Poland, 2001

Participants of The X-th Conference on Mechanical Vibrations with International Participation Timisoara, Romania May 23-24, 2002.

The Conference brought together 185 researchers from both, academia and industry, with a substantial number of international participants. A number of 125 papers were presented during this year’s conference. The conference covered the following five sessions: 1. Plenary session; 2. Vibration and vibro-impacts session; In a break of the conference 3. Identification of mechanical systems and diagnosis of mechanical structures session;

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4. Vibration of mechanical structures and technical acoustics session; 5. Modeling of biomechanical structures session. In the Plenary session, the following four lectures were presented: Radu VOINEA (Romania) – "On some analogies"; Zarko SPASIC (Yugoslavia) – "Computer based vibration monitoring and its role in predictive maintenance" (co-authors Miroslav PILIPOVIC (Yugoslavia) and Peter B. PETROVIC (Yugoslavia)); Gheorghe MIHALAS (Romania), "Modeling and simulation of biological processes – actual tendencies"; Liviu BRINDEU (Romania) – "The personality of professor Gheorghe SILAS in science and technique". During the Vibration and vibro-impacts session, 36 papers were presented covering a large variety of problems associated with: vibration by impact, tests of materials and equipment at vibro-impacts, oscillations and mathematical modeling of nonlinear systems, methods for studying nonlinear systems, stability of motion, modeling of the vibro-cutting process, study of self sustaining vibrations in the working process, chaotic behavior of mechanical systems, etc. The Identification of mechanical systems and diagnosis of mechanical structures session, consisted of 34 papers covering various topics like vibration state diagnosis, dynamic high strain behavior of concrete and mortar, vibro-diagnosis and mathematical modelling of dynamic systems, modal analysis in bridge vibration monitoring, multiple time scale method in galloping´s study, analysis of dynamic behavior of geaers with random manufacturing errors. In the Vibration of mechanical structures and technical acoustics session, 32 papers were presented. The session was devoted to dynamic response of some structures as bars, beams, plates, active control of mechanical structures, dual extremum principles in dynamics of linear thermoelasticity of laminated plates, researches concerning the acoustic screens for motorways, transient motions of a railway set, phonic pollution generated by the road traffic. During the Modelling of biomechanical structures session, 22 papers were presented covering a wide range of topics like the relationship between a mechanical vibration generator and the human body receptor, the applicatuion of Hertzian contact theory to explain the evolution of artrosis process, modelling and control of an artificial muscle, TH biomechanical modelling principles of osteo-articular system, modelling and dynamic study THE 6 INTERNATIONAL CONFERENCE OF TENSOR of human mandible, higher-order spectral analysis used for phonocardiogram processing. SOCIETY ON DIFFERENTIAL GEOMETRY Many papers having a practical character were presented during the conference. & ITS APPLICATIONS The authors of the conference papers are to be congratulated for their service to the of the research community. The proccedings of the conference were published in three volumes AND and a CD-room. MATHEMATICAL FOUNDATIONS The participants at "The X-th Conference on Mechanical Vibration with OF INFORMATION SCIENCES & ITS APPLICATIONS International Participation, Timisoara, Romania, May 23-24, 2002" were unanimous in recommending the continuation of the conference in the future. There was also the decision On occasion of the Anniversary of Akisugu KAWAGUCHI's 100 years birth, of the participants that in the future this conference should have an International Scientific who is the Founder of Tensor Society August 5-9, 2002 Committee. Several personalities participating at the conference put the bases of the future co-operation between "Politehnica" University of Timisoara and the Universities of Belgrade and Nis (Yugoslavia) and also the University of G?dollo from Hungary. The conference closed with visit to the Sag's Abbey situated at 12 km from Timisoara, where the participants served lunch. By all considerations, the conference was a success.

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APPENDIX III

SYMPOSIUMS ON The invited plenary lectures were given by the following Yugoslav scientists: Veljko A. Vujičić, Vladan Djordjević, Ljubomir Grujić, Božidar Vujanović, and other.. NONLINEAR MECHANICS Prof. dr Slobododan Laković, the Dean of the Faculty of Mechanical Engineering IN SERBIA University of Niš and a member of the Organizing Committee, has significantly contributed to the success of this symposium. The Third Yugoslav Symposium on Nonlinear Mechanics was held in the form of Dear participants of the Sixth Symposium on Nonlinear Mechanics, a Minisymposium, as a part of the XXII Yugoslav Conference on Mechanics in Niš in 1995. The Faculty of Mechanical Engineering in Niš, with co-organization by the Faculty of Civil This symposium, entitled Sixth International Symposium on Nonlinear Engineering organized this Congress as well. Mechanics Nonlinear Sciences and Applications Niš 2003 (The 6th ISNM NSA The Chairman of the Organizing Committee was Prof. Katica (Stevanović) Hedrih; NIŠ'2003), as was the case with the previous one, entitled ‘Nonlinear Sciences at the Prof. dr Zoran Boricic, the Dean of the Faculty of Mechanical Engineering, was very Threshold of the Third Millenium’, is organized with the wish to unite in a single hospitable. Academician Vladan Djordjević, the President of YUSM has contributed to the symposium, on the basis of the Mathematical Phenomenology of Mihajlo Petrovic Alas, successful organization of this Congress and the Minisymposium. The Chairman of the quite disparate sciences with the aim of integrating the knowledge of the participants of our Scientific Committee was Prof. dr Veljko Vujičić. Professors Yu. A. Mitropolskiy, V.V. symposium. This is a serious and long-term task of science. Rumyantsev, Felix Chernousko, Anatoliy Martinyuk, Valentina Filchakova, Dan Stamatiu, ... How did it all start? It started here, at the Faculty of Mechanical Engineering, were guest at this symposium. University of Niš. The Fourth Symposium on Nonlinear Mechanics was held in 1997, again in the form of a Minisymposium, as a part of the XIII Yugoslav Congress on Theoretical and Prof. dr Danilo P. Rasković (Ph.D. in mechanical engineering and BS in Applied Mechanics. This Congress, held in the Congress Center – Zvezda in Vrnjacka Banja, mathematics), the first head of the Chair for Mechanics and Automatics, while teaching was organized by the Yugoslav Society for Mechanics. The organization of this Symposium mechanics at the Department for Mechanical Engineering at the newly founded Faculty of was helped by the Institute for Mathematics – SANU and the Faculty of Mechanical Technical sciences, directed his youngest and most talented students towards studying Engineering in Niš and in Belgrade. nonlinear oscillations and nonlinear mechanics. Thereafter, he initiated a cooperation with The Chairman of the Scientific Committee was the academician Nikola Hajdin, the academician Jury Aleksejevitch Mitropolsky and a leading school of nonlinear and the Chairman of the Organizing Committee was Prof. Katica Hedrih. Professors mechanics, asymptotic methods and nonlinear oscillations at the Institute for Mathematics in Anthony Kounadis, Guiseppe Rega, Anton Baltov, ... were guest at this symposium. Kiev. That is how a centre for nonlinear mechanics was established at the Chair for The Fifth Symposium on Nonlinear Mechanics- Nonlinear Sciences at the Mechanics, Faculty of Mechanical Engineering in Niš. They were supported by the Institute Threshold of the Third Millenium was organized with the wish for it to become a tradition for Mathematics – SANU from Belgrade and by the scientist from all the Universities in and to gather the connoisseurs of nonlinear phenomenology from disparate sciences and Yugoslavia, especially Universities in Belgrade and Novi Sad. The organization of dynamic systems and for it to become renown all over the world. symposiums on nonlinear sciences followed. The Chairmen of the Scientific Committee were academicians Jury A. The first symposium on nonlinear mechanics - Nonlinear Dynamics, organized by Mitropolsky, V. M. Matrosov and V. Vujičić, and the Chairmen of the Organizing YUSM and Serbian Society for Mechanics, was held in Arandjelovac. Prof. Jury Committee was Prof. Katica Hedrih and Prof. Z. Boričić. Academicians N. Hajdin, V.V. Aleksejevitch Mitropolskiy gave the invited plenary lecture. All the members of the Chair Rumyantsev and M. Prvanović and Professors D.S. Sophianopoulos, G.T.Michaltos, Ji for Mechanics and the Chair for Hydraulic Engineering of the Faculty of Mechanical Huan He, I. Finogenko, P.S. Krasil’nikov ..... were guests at this symposium. Engineering University of Niš took part in this and other symposiums; they were co- The year of the Fifth Symposium was the year of the 10 th Jubilante issue of the organizers as well. University Journal – Facta Universitatis, new Series – Mechanics, Automatic Control and The Second Yugoslav Symposium on Nonlinear Mechanics entitled: “The First Robotics. These symposiums and the Journal Facta Universitatis are a permanent Yugoslav Conference on Nonlinear Deterministic and Stochastic Processes in characteristic of the University of Niš, Faculty of Mechanical Engineering in Niš and Dynamical Systems with Applications YCNP Niš'91”, organized by the Faculty of scientific achievements of Yugoslav and Serbian scientists in international relations. Mechanical Engineering was held in Niš. The Chairman of the Scientific Committee was We would like to wish all participants of this Symposium a warm welcome to our Prof. dr Veljko Vujičić, while the Chairman of the Organizing Committee was Prof. Katica country and our Faculty. (Stevanović) Hedrih. Proceedings of Abstracts was printed; the papers and invited lectures We would like to welcome you, hoping that creative efforts and achievements will which were approved were printed in the first and the following issues of the University prevail over preemption. Journal – Facta Universitatis, new Series – Mechanics, Automatic Control and Robotics. This Series was the third Series, which was being published, besides the Series Mathematics, October 2000. Katica (Stevanović) Hedrih Informatics and Electronics, and Energetics. William Nash, Kazuyuki Yagasaki ..., gave the invited lectures, at this international symposium. 93 94

Invited Lecturer: Professor Willam Nash from MTI Masachusets at Yugoslav Conference on Deterministic and Stochstic Processes in Dynamical Systems with Applications Nis 1991

THE FIFTH YUGOSLAV SYMPOSIUM ON NONLINEAR MECHANICS - Serbian Symposium on Nonlinear Mechanics, Ara|elovac, 1984; Invited Lecturer Academician RAN and NANU Yu. A. Mitropolskiy (KIEV) in Niš NONLINEAR SCIENCES AT THE THRESHOLD OF THE THIRD MILLENNIUM with Serbian Scientists (YUSNM NIŠ '2000)

held in Niš, Yugoslavia at October 2-5, 2000. at Faculty of Mechanical Engineering, and it dedicated to the 40th Anniversary of the Faculty of Mechanical Engineering and Faculty of Civil Engineering and Architecture, as well as to the 35th Anniversary of the University of Niš. The Symposium is organized under the patronage of the Departmentof Technical Sciences, Serbian Academy of Sciences and Arts.

Academician RAS V. Rumyantsev (Moscow, Russia) in Niš (YUSNM NIŠ '2000-October 2-5, 2000.) at Faculty of Mechanical Engineering.

95 96 From Sacientific Technical Review (2994)

6th international symposium on nonlinear mechanics — nonlinear sciences and applications

Marinko Ugrčić, PhD (Eng) Zoran Drašković, PhD (Mech)

THE 6th International Symposium On Nonlinear Mechanics — Nonlinear Sciences And Applications was held from 24th to 29th August, 2003 in Niš and was organized by the University of Niš, Faculty of Mechanical Engineering in co-operation with the Mathematical Institute of the Serbian Academy of Sciences and Arts SANU, Belgrade. This symposium is a sequel to the previous symposia on Nonlinear Mechanics in Serbia. Nonlinear Mechanics is a subject of great importance in the development of science and technology. The aim of the Symposium is to provide a forum to exhibit the progress in this field during the past three years and a place to promote the interaction of modern mathematics and modern mechanics, as well as modern engineering sciences. Professor W. Schihlen from the University of Stuttgart, vice president of IUTAM, wrote: "So what is new in Nonlinear Dynamics and Mechanics today?" The initial scope of applications in solid mechanics has broadened to cover material processing, inelasticity and fracture mechanics. In rigid body dynamics, more complex systems such as vehicles, robotics and controlled machines have come into the purview (field) of nonlinear dynamics. On the mathematical side of nonlinear dynamics, it is now recognized that spatio-temporal problems, hysteretic and time delay problems are the new frontiers in this field. Also the term "complexity" has been added to the lexicon of the chaos theory to describe the Participants of dynamics of many interacting sub-systems which can exhibit self-organization and The Fifth Yugoslav Symposium on Nonlinear Mechanics -NONLINEAR SCIENCES AT evolution. Complexity analysis has gained a foothold in biological and some social sciences THE THRESHOLD OF THE THIRD MILLENNIUM as well as in fluid and chemical physics. It remains to be seen what impact it will have on (YUSNM NIŠ '2000) applied mechanics and engineering. from Russia, Greece, China, Bulgaria, Rumania, Ukraine and Yugoslavia. mr D. Jovanović (Niš,YU), The authors from abroad (Italia, Germany, Poland, Japan, USA, Romania, Russia, China, Professor P. Krasil'nikov (Moscow, Russia), Professor Ji Huan He mr D. Jovanović (Niš,YU), Professor India, ...) and Serbia & Montenegro (Belgrade, Niš, Novi Sad, Kragujevac, Vranje, ... ) have P. Krasil'nikov (Moscow, Russia), Professor Ji Huan He(Shanghai, China), Academician UHEAS K. submitted nearly hundred papers printed in the Booklet of abstracts. Hedrih (Niš, YU), Academician SASA M. Prvanović (Belgrade, YU), Professor G. Michaltsos (Athens, All papers were classified into three subjects: memorial lectures (3 papers), invited Greece), Z. Vosika (Belgrade YU), Professor D. Sophianopoulos (Athens, Greece), Academician SASA plenary lectures (more than 20 papers) and contributed lectures (more than 70 papers) and N. Hajdin (Belgrade, YU), Ass. Professor P. Rajkovi ć (Niš, YU), Professor G. T. Konstantakopoulos were communicated within the memory sessions, key plenary sessions and plenary sessions, (Athens, Greece), Academician RAS V. Rumyantsev (Moscow, Russia), Professor D. Miki či ć (Belgrade, respectively. YU), Academician ANS V.A. Vujičić,.... We believe that, among all the papers communicated during the Symposium, the following

ones deserve the special attention of the readers from the Military Technical Institute. Memorial lectures

– Milutin Milanković (1879-1958) (V. Đorđević, the Serbian Academy of Sciences and Arts SANU, Belgrade) Milutin Milanković ranks among those great scientists of the world who have marked the 20th century. The chief idea underlying his longstanding work was that climatic variations on the Earth result from regular changes in Celestial Mechanics, which in turn cause cyclic changes in the intensity of insolation. – Mihailo Petrović Alas (1868-1943) (K. (Stevanovic) Hedrih, Faculty of Mechanical Engineering, Niš) 97 98 In essence this symposium is organized with the wish to unite quite disparate sciences in a single place, on the basis of the Mathematical Phenomenology of M. Petrović, with the aim The motion of the dynamical impact damper is studied using numerical simulation. The of integrating the knowledge of the participants. Phenomenological Mapping by M. Petrović regions of existence and stability of different regimes of the system response on the and his Mathematical Phenomenology and Mathematical Analogy can be considered as a harmonic excitation are evaluated. The boundaries of the regions are specified as grazing, continuation of the ideas of Poincare’s mapping and M. Petrović as one of the researchers in period doubling, saddle-node and Hopf bifurcations. Periodic, quasiperiodic and chaotic the row leading to modern researchers who contributed to different kinds of mapping in the impact motions are explained by time series, phase trajectories, bifurcation diagrams and research of non-linear dynamics and dynamical systems. Poincarč maps. – Akitsugu Kawaguchi (1902-1984), Tatomir Anđelić (1903-1993) and Danilo Rašković – On modelling problems in mechanics (theoretical and applied aspects) (Ly.K. (1910-1985) (K. (Ste-vanovic) Hedrih, Faculty of Mechanical Engineering, Niš) Kuzmina, Kazan Aviation Institute, Russia) This lecture is dedicated to three Professors whom Professor K. Hedrih met along her The research subject are complex large-scale systems, for which the original way of Tensor Calculus applications in Mechanics. Professor A. Kawaguchi was the founder mathematical model, an adequate real object, is extremely complex. The principal tasks are and the first president of the Japanese (today international) scientific Tensor Society (since to elaborate universal methods of: modelling, constructing correct simplified models, 1938). Academician T. Anđelić introduced Tensor Calculus into the studies of Mathematics rigorous substantiating of these reduced models in dynamics, estimating errors and and Mechanics at the Faculty of Natural Sciences and Mathematics, University of Belgrade admissible parameters domains by using these reduced models. The suggested method is (1946); the successes of Analytical Mechanics and Continuum Mechanics of "Belgrade combining the stability theory and the perturbations theory methods based on the postulates school of Mechanics" are primarily based on its solid mathematical foundation, in majority of stability and singularity. It allows to work out the effective manners of rigorous analysis on Functional and Tensor Analysis and Differential Geometry. Professor D. Raškovic with general methodology of constructing simplified models and their analysing, with introduced Tensor Calculus into the studies of Mechanics and Engineering at the Technical dividing the original problem to separate particular ones, with decomposing the original Sciences High Schools in Serbia; he brought the high mathematical level of Tensor Calculus model and its dynamic characteristics, with building the shortened models of hierarchy, and closer to students of technical sciences and mechanical engineers. with revealing essential variables and freedom degrees. – Regular, chaotic and hyperchaotic vibrations of nonlinear systems with self, Invited plenary lectures parametric and external exci-tations (J. Warminski, Department of Applied Mechanics, – The State of the art in modelling, analysis, design and engineering applications of Technical University of Lublin, Poland) smart structures (U. Gabbert, Institute of Mechanics, Otto-von-Guericke University of Self-, parametrically and externally excited systems are well known and thoroughly Magdeburg, Germany) investigated in the literature. The analysis of mechanical systems in which self- and A brief introduction into the main ideas and concepts of smart structures is given. An parametric excitation takes place at the same time is carried out in some monographs; in overview about recent research such a case, interactions between two different types of vibrations appear. Analytical, activities in this field is presented. Some of the research results of the author’s group are numerical and analogue investigations show that near parametric resonances of the discussed in more detail, such as the modelling and numerical analysis of piezoelectric smart synchronization phenomenon occur. structures including control. Finally, a look into the future is ventured as a very important – A new concept unifying Lyapunov and orbital stabilities (S.L.-S. Leelamma, State task to adjust research directions. University, New York) – A Generic optimal control of homo/heteroclinic bifurcations and ensuing global Following the idea of J.L. Massera that the distance between the trajectories can be dynamics in different mechanical systems (G. Rega, Dipartimento di Ingegneria measured maintaining different time scales or "clocks" with which time is measured along Strutturale e Geotecnica, Universitŕ di Roma “La Sapienza”, S. Lenci, Istituto di Scienza each motion, a new concept of stability that can unify Lyapunov stability and orbital stability e Tecnica delle Costruzioni, Universitŕ Politecnica, Ancona, Italy) is defined. Also, using the idea of two measures (one to measure the changes in initial values The control method, proposed in recent years for controlling nonlinear dynamics and and the other to measure the changes in the solutions), one can achieve unification of a large chaos, has been applied to various single-d.o.f. systems representing either exact mechanical number of stability notions (various refinements) that are already in the literature. This idea models or reduced dynamics of continuous systems. These results are now reconsidered in a of using different "clocks" (to measure time) while studying the stability of perturbed unitary context with the aim of better understanding the control procedure and highlighting systems relative to a given nonlinear system is well received by applied mathematicians and generic properties. Trying to find out generic features of the method, attention is focused on engineers. the underlying mathematical problem of optimization, which constitutes a method keystone. – The Numerical integration of differential equations for one dynamic system with dry It is shown that the one-side control and the global control of the considered symmetric friction coupling (L. Bereteu, “POLITEHNICA” University of Timişoara, România) oscillators are system-independent, thus strongly confirming the generality of the control An algorithm for numerical integration of the differential equations of motion (based on a method. so-called forth-order Runge-Kutta method) for one dynamic system with dry-friction coupling is derived. Some characteristics, which appear in the integration of these equations, – A More detailed view on the dynamics of the impact damper (F. Peterka, Institute of are given and the results of dynamic response simulations are presented. In order to prevent a Thermomechanics, Academy of Sciences, Prague, Czech Republic). 99 100 major failure of the wind turbine structure by vibratory effects, a solution consists of a cross-section of the diffuser is approximated by a sixth-order polynomial. The system of vibration absorber, placed inside the tower structure (the dry friction forces give the main governing equations is reduced to three ordinary differential equations and solved damping). numerically. The obtained results show that the performance, position of the separation point – Lyapunov and non-Lyapunov stability theory: linear autonomous and non-autonomous and other flow characteristics of diffusers depend on the angle and the Reynolds number. singular systems (D. Lj. De-beljković, Faculty of Mechanical Engineering, Belgrade) – Fascinating nonlinear dynamics of heavy material particles along circles with coupled Singular systems are those in which the dynamics is governed by a combination of rotations (K. (Stevano-vic) Hedrih, Faculty of Mechanical Engineering, of Niš) algebraic and differential equations. These systems (also known as descriptor, semi-state or Some research results of fascinating nonlinear dynamics of heavy material particles along generalized systems) arise naturally as linear approximations of linear and non-linear system the circles with coupled rotations with many different properties of nonlinear dynamics are models in many applications. The complex nature of singular systems causes many presented. The paper considers the class of nonlinear systems with coupled rotation motions difficulties in their analytical and numerical studies, particularly when there is a need for into a system with two degrees of mobility, but with one degree of freedom of motion their control. In that sense the question of their stability deserves a great attention. A defined by one generalized coordinate and one degree of mobility defined by a rheonomic particular class of these systems operates in free as well as in forced regimes. A brief survey coordinate linearly depending on time. The atlas of the phase portraits families and constant of the results concerning their stability in the sense of Lyapunov and finite and practical energies curves of the equivalent systems with respect to the original rheonomic system with stability is presented. quazi-parameter nonlinear vibrations is presented. – The Regular and chaotic behavior exhibited by coupled oscillators with friction (J. – The Controller design for a funnel-shaped smart shell structure (T. Nestorović Awrejcewicz, Technical Uni-versity of Lodz, Poland, L. Dzyubak, Kharkov Poly-technic Trajkov, Faculty of Mechanical Engineering, Niš, U. Gabbert, H. Köppe, Institute for University, Ukraine) Mechanics, Otto-von-Guericke University of Magdeburg, Germany) Using a new approach – applicable to any dynamical system governed by ordinary differential equations and especially suitable for the estimation of regular and chaotic The controller design which enables vibration suppression of a funnel-shaped shell-type motions – the nonlinear behavior of an autonomous two degree-of-freedom mechanical structure (which is a part of a complex medical device – magnetic resonance tomograph) is system with friction is investigated. The domains of a chaotic motion are obtained in various considered. The funnel-shaped structure is modelled using the finite element approach. The sections of a three-dimensional driving parameter space. Chaotic and regular motions are control is achieved with piezoelectric actuators attached to the surface of the funnel (the detected and classified as stick-slip or slip-slip ones. dynamics of the piezoelectric active elements is taken into account in the procedure of the

finite element modeling). The set of linearized constitutive equations of mechanical and Contributed lectures electrical fields is used as a starting point in the finite element modeling procedure. The – A Compressible channel flow over a permeable wall (V. Đorđević, the Serbian Academy experimental verification of the frequency response of the funnel was performed. of Sciences and Arts, Belgrade) The problem of 2-D compressible gas flow through a channel with one permeable wall – Modelling of laminate composites with embedded piezoelectric actuators and sensors (which makes the part of the contour of a porous body) is treated as a problem of strong (U. Gabbert, Institute for Mechanics, Otto-von-Guericke University of Magdeburg, interaction between the channel flow and the flow through the porous body. Simplified Germany, D. Marinković, Faculty of Mechanical Engineering, Niš) equations governing both flows are solved. Exact expressions for the frictions coefficient The Behaviour improvement of structures made of laminate composites can be achieved and the relative increase of the mass flow rate are derived, and their agreement with the by embedding components made of smart material between layers. Over the last few years existing experiments is better than with empirically defined boundary conditions. the use of piezoelectric materials as actuators and sensors in vibration control has been successfully demonstrated. The aim of this paper is to present a model of a plate structure – The Determination of tooth root stress concentration factor at heavy loaded gear made of such a material. The equations governing the mechanical behaviour of a plate are in drives (V. Nikolić, Faculty of Mechanical Engineering, Kragujevac) accordance with the first order plate theory. Based on constitutive equations of piezoelectric A calculation procedure of the strain and stress state of a gear tooth is provided by the material and certain assumptions, governing equations of piezoelectric actuating loads and finite elements method. The basic relations of the finite elements method with a short sensor output are given. The problem of the finite element formulation of the piezolaminated description of the developed computer program are presented. The three dimensional curved plate is pointed out. isoparametric finite element is applicated. The formula for the gear tooth root concentracion factor is derived as well. – Discrete Fourier transform in the problem of the wave packet dynamics (V.A. Zharov, Moscow Institute of Physics and Technology, Russia) – The Calculation of the separation point for the turbulent flow in plane diffusers (M. Vujičić, University of Serb Sarajevo, C. Crnojević, Faculty of Mechanical Engineering, A method of the wave packet dynamic description is suggested for the Tollmien- Belgrade) Schlichting waves in the boundary layer flow of incompressible fluids. It is based The equations of the turbulent boundary layer in the integral form (adjusted for the on combining the one mode spectral wave components equation with the wave internal flow) are used for calculation and the turbulent viscosity model (based on the packet envelope equation. First of all, it gives rise to solving the system of mixing length) is used for closing the system of equations. The velocity profile in each equations for a wave packet envelope that contains the integral equation with the

101 102 singular kernel which transforms into an algebraic function in the wave number – Transformations between surfaces with animations (V. Veličković, Faculty of Science space. This approach is applied when the splitting of nonlinear equations into linear and Mathematics, Niš) and nonlinear parts is used at each time step. The linear part can be solved with the help of The application of the owned software and its exten-sions, the graphical presentations of the wave packet spectral component equation and then the field is transformed from the surfaces of different classes, the visualization of mappings between them and the animations wave number space to the physical space. In the physical space the system of ordinary of transformations of surfaces are given. differential equations is solved by the subsequent inverse Fourier transformation into the wave number space. MTI presentation The following three papers were communicated by the authors from the MTI (Military – Large dynamic structures in turbulent boundary layer (review of experimental works) Technical Institute): (Yu.I. Khlopkov, V.A. Zharov, S.L. Gorelov, Moscow Institute of Physics and Technology, Russia) – An Efficient computation method in fatigue life estimation of damaged structural The data describing vorticity generation and the moment transfer in the turbulent components (K. Maksimović, MTI, Belgrade, V. Nikolić, Faculty of Mechanical boundary layer that has been accumulated for a long time prove convincingly the existence Engineering, Kragujevac, S. Maksimović, MTI, Belgrade) of large scale vortical structures (coherent structures) that can be governed by autonomous This paper focuses on developing an efficient computational method in fatigue life dynamical equations. These structures are responsible for many turbulent boundary layer estimations of damaged structural components. The aim is to examine the strength behavior properties. The experimental results allow to give a definition of the structure and describe of an important constructional element, the lug, when fatigue cracked, and to propose a some of its details. Such results represent a powerful base for theoretical work. stress intensity solution considering various lug geometries. To obtain an efficient algorithm in the crack growth analysis an analytic model for the stress intensity factor of the damaged – The effect of nonlinear excitation of asynchronous electric motors on the work of lug is derived, while numerical modeling, based on the crack finite elements, was used to driving mechanisms of cranes (Z. Marinković, S. Marković, D. Marinković, Goran determine the stress intensity factor solution for an aluminum lug with a through-the-thickness Petrović, Faculty of Mechanical Engineering, Niš) crack. Two laboratory electrically-driven mechanisms for the analysis of the work of driving mechanisms in the period of acceleration are discussed. One case uses a cage, and the other a – The Determination of the critical jet velocity during the penetration into the slide-disc three-phase asynchronous electric motor with non-linear starting characteristics. homogenous steel obstacle (M. Ugrčić, Military Technical Institute, Belgrade) On the basis of non-linear dynamic models, the motion of these mechanisms in the period of The procedure of the experimental determination of the critical jet velocity during the acceleration has been analytically solved and simulated by the program package penetration into the homogenous obstacle is given. The procedure is based on the use of the MATLAB/Simulink. The results of these simulations are in good agreement with logic analyzer and special captures to record the discrete data of the curve - the penetration experimental records, primarily due to the non-linear modeling of curves electric motor length depending on time. A polynomial form of that mathematically fitted functional curve ignition. gives the possibility to determine the critical jet velocity. Also, the results of the experimental determination of the critical velocity of the copper shaped charge jet during the – Crack propagation in discrete model of material (D.B. Jovanović, M.B. Jovanović, penetration into the homogenous steel obstacle are shown. Faculty of Mechanical Engineering, Niš) The theory of fracture mechanics has two main approaches to the problem of crack – On a procedure of obtaining the equations of motion of the material point over the propagation: continuum mechanics and atomic approach. Interactions of different physical smooth surface (Z. Draško-vić, Military Technical Institute, Belgrade) phenomena involved in the initiation and propagation of cracks as well as in the process of In the search for a procedure – more simple but in essence a formal one –for the fracture and damage, have directed the research towards analyzing processes at the atomic derivation of equations of motion of the material point over the smooth surface the presence (and molecular) level. The atomic approach considers cracks inside a discrete model of of a formal step in the usual procedures for obtaining these equations was pointed out. This material (atomic lattice). Two intrinsic interatomic force functions are used to represent the formality needs one more consistent derivation to be performed from the subspace (a non- mechanical interaction between neighboring atoms (discrete masses) in the lattice. The Euclidean one) point of view. released potential energy, as a result of crack propagation through the lattice by braking interatomic bonds, is presented. Special events During the Symposium a visit to the major industrial plants / firms in the region of Niš – A software for visualization and animation in mathematics and physics (E. ("MIN", "EI", "Ja-strebac") was organized for the participants from abroad. Malkowsky, Faculty of Science and Mathematics, Niš) Also, an unforgettable sightseeing tour was realized – its key-points were: the Scull Tower A short survey of the basic concepts and principles of the owned software for geometry (world’s unique monument dating from 1809), Mediana (famous for the archaelogical remains and differential geometry and its extensions is given. Some applications of this software to dating from the Roman period), Ravanica and Manasija (famous Serbian monasteries, the the visualization and animation of certain topics in mathematics and physics (such as the second one especially due to the "Resava school" handwritings) and the Cave of Resava (the illustration of geometric principles in the definition of curves, the study of properties of master-piece of nature). maps, the graphical representation of some minimal surfaces and of potential surfaces and their Gaussian and mean curvature, the growth of crystals, the study of weak topologies by a collection of functions) are presented. 104 103 Organization The organization of the Symposium, as well as of the above mentioned social events, was perfectly realized, mainly due to the permanent care and enormous efforts of Professor Katica (Stevanović) Hedrih and her tireless collaborators from the Faculty of Mechanical Engineering in Niš.

Conclusion The Symposium provided an extraordinary opportunity for the participants to meet and discuss recent advances in Nonlinear Mechanics. The participants represented a wide range of expertise, from pure theoreticians to people primarily oriented towards applications. Significant achievements of the Symposium were very extensive discussions taking place over the whole range from highly theoretical questions to practical engineering applications. The main conclusion that can be drawn from the lectures presented at the Symposium is that Nonlinear Mechanics as a subject has gained in extent as far as both methodology and applicability are concerned.

Received: 17.9.2003

Participants of the 6th International Symposium on Nonlinear Mechanics — Nonlinear Sciences and Applications 6th INM NSA NIŠ 2003.

Participants of the 6th International Symposium on Nonlinear Mechanics — Nonlinear Sciences and Applications 6th INM NSA NIŠ 2003.

105 106

Chair of Mechanics

dr Ing. Dipl. Math. Danilo P. Rašković (1910-1985) The First Head of Chair of Mechanics and Automatic at Faculty of Mechanical Engineering in Niš

Seminar of Theoretical and Applied Mechanics of Chair of Mechanics at Faculty of Mechanical Enigineering University of Nis

Participants of Seminar (1994) with academician of Serbian Academy of Science and Arts Miomir Vukobratovic

Participants of Jubilante Seminar 1994

Participants of Seminar 1994 dedicated to the 60the anniversary of Professor Dusan Stokic

109 110

Congres on Theoretical

and Appled Mechanics Nis’1995

Colleagues form Faculty of Mechanical Engineering University of Podgorica, Montenegro at Chairmans of the Yugoslav Congress of Yugoslav Congres on Theoretical and Theoretical and Applied Mechanics Nis 1995 Appled Mechanics Nis’1995 corresponding memeber of Serbian Academy of Science and Arts Bozidar Vujanovic, academician Serbian Academy of Science and Arts Vladan Djordjevic and dean of Mechanical Engineering Faculty at Nis Zoran Boricic

Participants of Yugoslav Congress of At Congress pererive: corresponding memeber of Theoretical and Applied Mechanics Nis 1995 Serbian Academy of Science and Arts Bozidar Vujanovic and professors Dusan Stokic and Katica (Stevanovic) Hedrih 111 112

6th International Symposium on Nonlinear Mechanics IUTAM ROMA 2003 — Nonlinear Sciences and Applications 6th INM NSA NIŠ 2003.

ICTAM Haifa 1992.

113 114

Katica (Stevanović) Hedrih and Tamara Nestorović the rulers, but it had remained a part of the Serbian State, constantly fighting a battle for its freedom and independence. In the middle ages Niš played an important role in the state. In the historical documents dating from that period, Niš is mentioned as a NIŠ - The Past and the Present Serbian metropolis. It developed from the very foundation of the Serbian State, especially during the reign of Emperor Dušan and King Uroš, but its tremendous š During history and nowadays Ni has been an important place from economic and political prosperity came with the reign of the Serbian duke Lazar geographic, strategic, cultural or any other point of view. It has always been an object Hrebeljanović. of interest of people coming to this region either as travellers passing through for the Niš was occupied by the Turks in 1386 and they ruled for the next five first time or those who are familiar with the town and its sightseeing, cultural, human th centuries. On December the 29 1877 Niš was liberated and this date represents the and other values. Natural beauties of the town surroundings have been attractive even beginning of a new period during which Niš started developing as a free and modern for the first inhabitants of this district who founded the first settlements. town. In 1878 the first secondary school was opened and the next year the National It is assumed that Library was established. As a developing cultural and business center of the region, the town area was inhabited Niš marked the opening of other cultural and business institutions - the National even in the early Stone and Theatre, the first literary journal, banks and various industrial plants. The traffic links Bronze Ages. Many local were also developing, so in 1884 the railway through Niš was built connecting it in archaeological remains more proper way with Belgrade and the rest of Europe. found in the wider area Because of its significant and advantageous geographic position, the town surrounding of the town has often been called “the gateway between the East and the West” or “the Balkan prove this assumption. One crossroads”. It is situated at the very crossing point of several strategic traffic routes of the most famous of international importance. The route along the Morava valley links Middle and archaeological places West Europe with Greece and the Aegean sea, as well as with the Near East along the around Niš is Mediana, South Morava and the Vardar valleys. The Nišava route, stretching along the Nišava famous for the and Moravica valleys, links Europe with Turkey, Asia and the Near East. The third, archaeological remainders trans-Balkan route, connects East Europe, the Vlach lowland and the Morava district dating from different ages, with the Adriatic Sea via the Nišava valley, the Toplica district and Kosovo. Owing especially from the Roman to its advantageous position, the town was always attractive for the invaders. period. Before the World War I Niš became the seat of the Serbian Government. At According to that time the government was situated in the building which is presently occupied by historical data, the the University. It might be interesting to mention that the ultimatum that Austria inhabitants of this area directed to Serbian Government before the beginning of the World War I was during the pre-Roman addressed to the building of the present University. period were Celtic and In the period between the two wars working class grew into an important Illyrian tribes. It is political factor. During the World War II Niš was an important center of the supposed that the town got resistance movement. Some places in the town still remind of tremendous victims but the name by the Celts who also the heroism and patriotism of the people. Such places are Concentration camp called it Naissus, which “Crveni Krst” or the Bubanj hill, where 12,000 patriots were executed. Many national means “the town of heroes from the World War II originated from Niš. Niš was liberated on October 14, nymphs”. They believed that the nymphs lived in the Nišava river, which they called 1944. the magic Nissa meaning “the river of nymphs”.

In the next period the town was an object of many destructions during the Hun invasions in the fifth century AD. Later, the Byzantine Emperor Justinian had the 116 town reconstructed. One of the most famous Roman emperors’ name from this period is connected with Niš It is Contstantinus, who is believed to be born in Niš. This After the World War II the period of the town development in many period was also important as the beginning of the Slavic migrations to the Balkan different ways began. Through the years the town grew up into a modern social, Peninsula. The Slavs continued their settlement, so in the first decades of the seventh economic, educational, medical, cultural and sport center of the Nišava District. century Niš was developing as a Byzantine town mostly populated by the Slavs. Today it represents one of the most important industrial and commercial centers of Serbia and Montenegro (Yugoslavia), with several leading firms in electronic, 115 mechanical, tobacco and textile industries. For the first time Niš was taken over by the Serb monarch Stefan Nemanja Niš is also an important educational center of the region. There are 33 in 1183 after the death of the Byzantine Emperor Comnenus. At first as a part of the elementary and 21 secondary schools with approximately 50,000 pupils and about Serbian State of Raška under Stefan Nemanja and then influenced by the time and 14,000 students at the University. As a university center the town involves: Faculty of winds of changes, the town had its ups and downs during the history. It kept changing the Civil Engineering and Architecture, Faculty of Economics, Faculty of Electronic groups and amateur ensembles and clubs as well as other cultural and educational Engineering, Faculty of Law, Faculty of Mechanical Engineering, Faculty of institutions. Medicine, Faculty of Occupational Safety, Faculty of Philosophy, Faculty of The town is also known for its historical monuments such as the great Fortress, which Technology, Faculty of Natural Sciences which comprises departments of is the symbol and a recognition sign of Niš. Another great monument is the Scull mathematics, physics, chemistry, and Faculty of Teaching. This year the University of Tower, a world unique monument from the Turk period originally made of 959 Serb Niš celebrates the 38th anniversary. The first academic classes in Niš commenced in human sculls as a terrifying sign and a punishment for the heroic resistance of the 1960 with the introduction of undergraduate programs in law/economics, medicine Serbs in the battle on the hill Čegar in 1908. The Čegar monument commemorates and engineering. In 1965 by decision of the People’s Assembly of Serbia the mentioned heroic battle fought against the Turks and conducted by the great Serbian University of Niš was incorporated as a degree-granting institution offering courses at hero Stevan Sindjelić. Archaeological emplacement Mediana is famous for its floor both the undergraduate and postgraduate level. So today in the year of 2003 the mosaics. University of Niš is celebrating its 38th anniversary, while the Faculty of Mechanical Engineering as well as the faculties founded in 1960 are celebrating its 43th A series of cultural anniversary. events of Yugoslav and The University today is characterized with a great number of students, international character take place in highly skilled and well-experienced academic staff and researchers in different fields Niš. Some of the best known are: as well as modern research equipment maintained by the faculty institutes. The the Festival of acting achievements University has 82 specialized research laboratories in the areas basic to the mission of in the Yugoslav film art, the the faculties which operate them. Some of them are more generalized in order to Festival of amateur choirs, the Art provide for joint research undertakings. In addition, there are numerous laboratories Colony Sićevo - the oldest operated by the clinics and institutes of the Faculty of Medicine. Moreover, the Yugoslav art colony, the Literary University operates two research centers of interdisciplinary and multidisciplinary Colony, Jazz Festival, etc. A orientation. The University also has Integrated Computer-Based University popular tourist attraction located Information System JUNIS which among other activities provides university some 10km from Niš is spa Niška computer network. Banja, which is one of the best Publishing activities of the University are also very rich. The University of known health resources in Niš publishes two journals: journal for social theory and practice Theme and Yugoslavia famous for its hot scientific journal Facta Universitatis, comprising several series which cover almost all radioactive water. major fields of study taught at the University. Papers are reviewed and published in The inhabitants of Niš are English, German, French or Russian. Most of the faculties publish their own scientific great lovers of sport and their town journals within their specific scientific orientation. offers them considerable opportunities to enjoy it, either recreationally or actively, The University Library “Nikola Tesla” founded in 1967 today represents a satisfying the taste of those who love football, handball, boxing, basketball, volleyball wide ranging scientific library offering a comprehensive range of facilities and tennis, swimming, diving, riding, etc. services tailored to fulfil a crucial role in supporting the learning, teaching and research activities of the University. Besides the University Library each faculty also has a library providing services that answer specialized needs of its staff and students. 118 117

As a medical center Niš includes thirty clinics and institutes which function as the educational basis for the Medical Faculty, as well as many other institutions supplying the population of this region with top health service. Niš also has a rich inheritance and tradition and a wide variety of cultural and artistic events and institutions which make this town an attractive spot of the Balkans. As it was mentioned the town is celebrating the 38th anniversary of the University and it has already celebrated the more than hundredth anniversaries of its first secondary school, Public Theatre, Public Library and literary journal. The town has been the dwelling place of many historical persons, writers and poets, actors... Modern cultural trends of the town are also influenced by the Public Museum Symphonic Orchestra, Puppet Theatre, University library “Nikola Tesla”, journal “Gradina”, several publishing houses, the Gallery of Modern Arts, numerous choral

Faculty of Mechanical Engineering University of Niš University of Niš

Students of the University of Niš with Professor dr Ing dipl Math Danilo P. Rašković first Head of Chair of Mechanics at Facultz of Mechanical Engineering in Niš 119 120

Faculty of Mechanical Engineering Fortress

Bus Station Hotel Ambasador and Sity Center Title: WCNA 2004 – MINISYMPOSIUM - INTEGRITY OF DYNAMICAL

SYSTEMS (Theory, Applications and Experiments) - Booklet of Abstracts Fourth World Congress of Nonlinear Analysts -WCNA-2004, June 30-July 7, 2004. Hyatt Regency Orlando (Near Walt Disney World Resort). Http://www.masfak.ni.ac.yu/masfak/topic.php?lang=SR&id=525. Http://kermani.math.fit.edu/

Editor-in-Chief: Katica (Stevanović) HEDRIH, Organizer of the Minisymposium Session Integrity of Dynamical Systems

First Edition Publishers: FACULTY OF MECHANICAL ENGINEERING 18000 - NIŠ, A. Medvedeva 14, Tel.: (+381 18) 524-915 Zoran BORIČIĆ, Dean of the Faculty of Mechanical Engineering Niš International Federation of Nonlinear Analysts (IFNA)

Support: Ministry of Sciences and Life Environmental Protection Republic of Serbia and Projects: 1616-Real Problems on Mechanics and 1828 Dynamics and Control of Active Structures

Foreign Communications: Vladimir HEDRIH

Cover design and computer support: Katica (Stevanović) HEDRIH and Tamara Stevanović

ISBN 86-80587-29-X. Number of Copies: 50 copies

Printed by: Sven Štamparija Niš, Stojana Novakovića 10, 18 000 Niš, Phone: +381 18 48142

CIP - Katalogizacija u publikaciji Nar odna bi bl i ot eka Sr bi je, Beograd

531.01 (048) : 530.182 (048) : 53:005.745 (100)

MINISYMPOSIUM INTEGRITY OF DYNAMICAL SYSTEMS ( 2004 ; Niš)

Booklet of Abstracts MINISYMPOSIUM INTEGRITY OF DYNAMICAL SYSTEMS (Theory, Applications and Experiments) Niš 2004 Http://www.masfak.ni.ac.yu/masfak/topic.php?lang=SR&id=525. [ u okviru ] Fourth World Congress of / Nonlinear Analysts - WCNA-2004, June 30-July 7, 2004. Orlando; Http://kermani.math.fit.edu/, Hyatt Regency Orlando (Near Walt Disney World Resort), NIŠ-Orlando, 2004. Editor Katica R. (Stevanovi}) Hedrih. – Niš : Faculty of Mechanical Engineering University of Niš and International Federation of Nonlinear Analysts (IFNA)., 2004. (Niš : Sven). – 118 str. : Ilustr. ; 21 cm Tiraž 50. – Bibliografija uz pojedine apstrakte.

ISBN 86-80587-29-X.

1. Stevanovi} Hedrih, Katica 2. World Congress of Nonlinear Analysts – WCNA (4 ; 2004 ; Orlando) a) Nelinearna mehanika – Bibliografije, referativne; b) Nelinearne teorije – Bibliografije, referativne; COBISS.SR – ID 115496204