ὅδε οἶκος, ὦ ἑταῖρε, μνημεῖον ἐστιν ζῴων τῶν σοφῶν ἀνδρῶν, καὶ τῶν ἔργων αὐτῶν

**Mechanics Colloquim **

** PROGRAM**

MATEMATIČKI INSTITUT SANU

ODELJENJE ZA MEHANIKU

Pozivamo Vas da učestvujete u radu sednica Odeljenja i to:

__SREDA, 1. decembar 2010. u 18 sati:__

Lecture No 1136

* Mr Snezana Milicev and Prof. dr Nevena Stevanovic, University of Belgrade, Faculty of Mechanical Engineering *

** A non-isothermal Couette slip gas flow **

Abstract: Micro-Couette flow as a shear-driven flow is encountered often in Micro-Electro-Mechanical-Systems (MEMS), such as microcomb mechanisms, microbearings and micromotors. Moreover, the most frequently conditions in these systems correspond to the slip flow regime (Kn<0.1), so the results in that region are very useful. As the thermal effects are very often present in MEMS, many authors have been investigated the heat transfer in Couette flow. The microscopic approach is used mostly, either by solving kinetic Boltzmann equation or by direct simulation Monte-Carlo DSMC.

In this paper micro-Couette slip gas flow for moderately high and low Reynolds numbers is analyzed by macroscopic approach analytically and numerically. The gas flow between parallel plates that is preserved in relative motion is considered in two cases: the temperatures of the walls are equal and constant, i.e. the isothermal walls, and the temperatures of the walls are different, but constant. The Knudsen number is Kn?0.1, which corresponds to the slip flow and continuum. The flow is defined by continuity, Navier-Stokes and energy continuum equations, along with the velocity slip and the temperature jump first order boundary conditions. The gas flow is subsonic and the ratio Ma2/Re is taken to be of the order of a small parameter. An analytical solution for velocity and temperature is obtained by developing a perturbation scheme. The first approximation corresponds to the continuum flow conditions, while the others represent the contribution of the rarefaction effect. Moreover, the same system of continuum governing equations with corresponding first order boundary conditions is solved numerically, by Runge-Kutta method, in order to verify presented analytical solutions. The influences of the viscosity and conductivity dependence on temperature, the dissipation and the rarefaction on the velocity and temperature profiles are explored. It is shown, in the case of isothermal walls, that temperature influence on viscosity and conductivity is insignificant on the velocity and temperature fields. Then, the exact analytical solution for constant viscosity and conductivity is found. It is shown that, although very simple, it is complete substitution to the exact numerical solution for the isothermal walls case. The results for the velocity and temperature fields are also compared with some numerical and analytical results of other authors and good agreement is achieved.

__SREDA, 8. decembar 2010. u 18 sati:__

Lecture No 1137

* Jelena Dimitrijevic, dipl. mas. ing., Military Technical Institute of Serbian Army, Department for aircraft Vojislav Devic, dipl. mas. ing., Military Technical Institute of Serbian Army, Department for aircraft mr Predrag Kovacevic, dipl. mas. ing, Military Technical Institute of Serbian Army, Department for aircraft *

Abstract: During designing and development of an aircraft prototype it is required to be confirm that the aircraft is free from flutter in the range of the designed speed. Wing, ailerons, stabilizer, or whole aircraft flutter is a phenomenon of self-induced undamped oscillations begins of coupling aerodynamic, elastic and inertial forces at high flight speeds. Flutter is very dangerous because it can cause aircraft construction crash. This presentation presents Computational and Experimental Modal Analysis procedures that are applied during designing and examination of the LASTA prototype aircraft.

Computational Modal Analysis is done by using Finite Element Method. Main assemblies of the aircraft wing, fuselage and stabilizers are modeled and analyzed separate. While modeling main assemblies we took care of the real construction elastic and inertial characteristics representation. This presentation gives the results of Modal Analysis of the LASTA aircraft. Critical flutter speeds calculation is done according to these results.

After the prototype manufacturing, Experimental Modal Analysis is required to obtain data that are more precise for further critical flutter speed calculation. Aircraft preparation, testing method, excitation position, data acquisition and processing are presented in the presentation. Obtained results modal frequencies, damping and mode shapes are presented, too. According to these results final critical flutter speed calculation is done. In addition, the aircraft finite element model will be verified and fit (according to these results) for further use in the analysis of the aeroelastic phenomena and aircraft dynamic response.

References [1] FELIPPA,C.A.: Introduction to Finite Element Methods, Department of Aerospace Engineering Sciences, September 2006. [2] BATHE,K.J., WILSON,E.L.: Numerical Methods in Finite Element Analysis, Prentice-hall, inc., Englewood Cliffs, New Jersey, 1976. [3] WILSON,E.L.: Numerical Methods in Offshore Engineering, John Wiley & Sons, 1987. [4] WILKINSON,J.H.: The Algebric Eigenvalue Problem, Clarendon press, Oxford, 1965. [5] OJALVO,I.U.: Proper Use of Lanczos Vectors for Large Eigenvalue Problems, Computers & Structures, vol. 20, No. 1-3, 1985. [6] HUGHES,T.J.R.: The Finite Element Method, Static and Dynamic Finite Element Analysis, Prentice Hall, 1987. [7] FEDERAL AVIATION ADMINISTRATION: Advisory Circular, Means of Compliance With Section 23.629-1B, Flutter, 2004 [8] HUTIN,C.: Modal Analysis Using Appropriated Excitation Techniques, Data Physics SA, Voisins le Bretonneux, France, 2000. [9] WELARANTA,S.: Advanced Data Acquisition and Signal Analysis Packages Can Enable Effective Ground Vibration Testing, Aerospace Testing International, May 2003. [10] PEETERS,B., HENDRICX,W., DEBILLE,J., CLIMENT,H.: Modern Solutions for Ground Vibration Testing of Large Aircraft, LMS International, Leuven, Belgium and EADS CASA, Getafe, Spain, January 2009. [11] KOVACEVIC,P., DIMITRIJEVIC,J., DEJANOVIC, N., BANOVIC,N.: Ispitivanje vibracija na zemlji aviona Lasta P1, Vojnotehnicki institut VS, maj 2010. [12] KOVACEVIC,P., DIMITRIJEVIC,J.: Obrada podataka sa ispitivanja vibracija na zemlji aviona Lasta P1, Vojnotehnicki institut VS, maj 2010.

__SREDA, 15. decembar 2010. u 18 sati:__

Lecture No 1138

* Prof. dr Ivana Kovacic, Department of Mechanics, Faculty of Technical Sciences, Novi Sad, Serbia *

** ON GEORG DUFFING, SOME OF THE EQUATIONS NAMED AFTER HIM AND ASSOCIATED PHENOMENA **

Abstract: The classical Duffing Equation is one of the most important equations in the theory of nonlinear dynamics, which models a harmonically excited single-degree-of-freedom oscillator with a linear-plus-cubic restoring force. This equation is named after Georg Duffing, a German engineer, who published a comprehensive book Forced oscillations with variable natural frequency and their technical significance in 1918. Since then there has been a tremendous amount of work done on this equation and its various forms, including the development of different methods and the use of these methods to investigate the rich dynamic behaviour and associated phenomena of different and disparate îscillatory systems modelled by them. The first objective of this lecture is to give a historical background to Duffings work and life, as not much is known and published about him. The story behind this is very interesting, because Georg Duffing was not an academic; he was an engineer, who carried out academic work in his spare time. The second objective is to compare and illustrate some of the characteristic nonlinear phenomena, including multi-valued amplitudefrequency response, hysteresis and the occurrence of jumps in the classical Duffing Oscillator and the one that contains pure cubic nonlinearity and is also under the action of a constant force. The lecture is also to announce the appearance of the book The Duffing Equation: Nonlinear Oscillators and their Behaviour

http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470715499.html

which brings together the results of a wealth of disseminated research literature on the Duffing Equation.

__SREDA, 22. decembar 2010. u 18 sati: __

Lecture No 1139

* Mr Ljljana Veljovic, Faculty of Mechanical Engineering University of Kragujevac *

Abstract: Brief review of realized models of gyro-rotors, applied gyr control and stabilizations and motion of ships, aircrafts, vehicles, and torpedo. With development of micro and nano- technology it feels a need for new class of gyroscopic systems. An analysis of principle of gyro-system work is presented as well as analysis of component motions. Most of realized gyro0stabilizators are realized on the basis of coupled rotations, with resultant rotation around fixed point. Gyroscopic moments are analyzed. (see References [1-2], [9-10], [15-16]). For a model of gyro-rotor, by use vector method, based on the mass moment vectors for an axis and pole, introduced by Hedrih (Stevanovic) K. [4-8],, vector expressions for linear momentum and angular momentum of a heavy rigid body rotation around two axes without intersection are derived, as well as their derivatives. These vector expressions are used for obtaining expressions of the kinetic parameters of nonlinear dynamics of considered system dynamics or vibrations. The following vector expressions: for kinetic pressures to bearings on the self rotation axis and to the axis of precession rotation, as well as their components are pointed out and analyzed. For the special case that heavy rigid disk is eccentrically and skew positioned on the self rotation axis which rotate in the horizontal plane around vertical axis with constant angular velocity on a distance, we derived: the nonlinear differential equation of the system dynamics in the gravitational field and corresponding equations of the phase trajectory as well as kinetic pressure components on self rotation bearing and vector rotator. Series of graphical presentation are presented.

Reference:

[1] Avramov K., Borysluk O., Bifurication of Elastic Rotors in Journal Bearings, The Third International Conference Nonlinear Dynamics 2010, pp. 21-26 [2 Áóëüŕęîâ Á. Â., Ďđčęëŕäíŕ˙ ňĺîđč˙ ăčđîńęîďîâ, Čçäŕňĺëńňâî Ěîńęîâńęîăî óíčâĺđńčňĺňŕ, 1976, 400 [3] S.K. Kim, D. M. Tilbury: Mathematical Modeling and Experimental Identification of a Model Helicopter, in Journal of Guidance, Control and Dynamics, August 31, 2000 [4] Hedrih (Stevanovic), K., The Vector Method of the Heavy Rotor Kinetic Parameter Analysis and Nonlinear Dynamics , Monograph, University of Nis, 2001, pp. 252., YU ISBN 86-7181- 046-1. [5] Hedrih (Stevanovic), K., (1992), On some interpretations of the rigid bodies kinetic parameters, XVIIIth ICTAM HAIFA, Apstracts, pp. 73-74. [6] Hedrih (Stevanovic), K. (1998), Vectors of the Body Mass Moments, Monograph paper, Topics from Mathematics and Mechanics, Mathematical institute SANU, Belgrade, Zbornik radova 8(16), 1998, pp. 45-104. published in 1999 .(in English), (Zentralblatt Review). [7] Hedrih (Stevanovic), K., (1993), Same vectorial interpretations of the kinetic parameters of solid material lines, ZAMM. Angew.Math. Mech. 73(1993) 4-5, T153-T156. [8] Hedrih (Stevanovic), K.: (1993), The mass moment vectors at n-dimensional coordinate system, Tensor, Japan, Vol 54 (1993), pp. 83-87. [9] S.K. Kim, D. M. Tilbury: Mathematical Modeling and Experimental Identification of an Unmanned Helicopter Robot with Flybar Dynamics, in Journal of Robotic Systems 21 (3), 95-116 (2004), 2004 Wiley Periodicals, Inc. Published online in Wiley Inter Science (www.interscience.wiley.com), DOI: 10. 1002/rob.20002 [10] Stephen C. Spry, Anouck R. Girard: (2008), Gyroscopic Stabilization of Unstable Vehicles: configurations, dynamics and control, in Vehicle System Dynamics, Volume 46, Issue S1 2008, pp.247-260, DOI: 10.1080/00423110801935863 [11] Katica (Stevanovic) Hedrih and Ljiljana Veljovic, (2008), Nonlinear dynamics of the heavy gyro-rotor with two skew rotating axes, Journal of Physics: Conference Series, 96 (2008) 012221 DOI:10.1088/1742-6596/96/1/012221, IOP Publishing http://www.iop.org/EJ/main/-list=current/ [12] Hedrih (Stevanovic) K., A Trigger of Coupled Singularities, MECCANICA, Vol.38, No. 6, 2003., pp. 623-642. , International Journal of the Italian Association of Theoretical and Applied Mechanics, CODEN MECC B9, ISSN 025-6455, Kluwer Academic Publishers [13] Hedrih (Stevanovic K., (2008), The optimal control in nonlinear mechanical systems with trigger of the coupled singularities, in the book: Advances in Mechanics: Dynamics and Control: Proceedings of the 14th International Workshop on Dynamics and Control / [ed. by F.L. Chernousko, G.V. Kostin, V.V. Saurin] : A.Yu. Ishlinsky Institute for Problems in Mechanics RAS. Moscow: Nauka, pp. 174-182, ISBN 978-5-02-036667-1. [14] Hedrih (Stevanovic) K, Veljovic Lj., (2010), The Kinetic Pressure of the Gzrorotor Eigen Shaft Bearings and Rotators, The Third International Conference Nonlinear Dynamics 2010, pp. 78-83 [15] Yu. G. Martynenko, I. V. Merkuryev, V. V. Podalkov: Control of Nonlinear Vibrations of Vibrating Ring Micro gyroscope, in Mechanic of Solids, 2008, Vol. 43, No. 3, pp. 379-390, Allerton Press, Inc., 2008, ISSN 0025-6544 [16] Strogatz, Steven H. (1994). Nonlinear Systems and Chaos, Perseus publishing

__SREDA, 29. decembar 2010. u 18 sati: __

Lecture No 1140

* Lecture by Professor Djordje Musicki is canceled and will be scheduled in the end of March 2011 *

Prof dr Katica Stevanovic Hedrih

Mechanics of Materials (period 2006-2010)

Abstract: Numerous research results in the fields of nonlinear mechanics, gyro-rotor dynamics, fractional order system dynamics, vibro-impact system dynamics, energy interactions in the hybrid system dynamics, biomechanics and fracture mechanics were obtained. Series of papers was published in international Journals and presented at World and international scientific meetings. . A complete list of publications can be found at the web site of the Mathematical Institute of the SASA at the address http://www.mi.sanu.ac.rs/projects/projects.htm. Extensive international cooperation was realized with scientists from Ukraine, Portugal, Russia, Poland, Greece, China, Japan, India, Turkey, Brazil, Germany, Romania and Macedonia. Three Mini-symposia was organized in 2007 (in Greece, Russia and China), two mini-symposia was organized in 2009, and one included in the program of European Solid Mechanics Conference Lisbon 2009. One doctoral dissertation and one magistar of science thesis were submitted and defended and three doctoral dissertations are in final forms. Selected list of the Keywords of the research results are: Partial and partial fractional differential equations, nonlinear phenomena, constitutive relation of stress-strain state, hybrid systems, delay, optimal control, stability, rigid body, deformable body, analytical methods, theorems, applications, resonant jumps, bifurcations, plate, belt, kinetics, energy transfer, multi-pendulum, chains, DNA helix, stochastic, eigen chains, eigen main chain normal coordinates, energy interaction, Lyapunov exponents, subsystems, interactions, trigonometric method, vector method, mass moment vectors, kinetic pressures and impacts, rotors, rotators, homoclinic orbits, discrete continuum method, transversal vibrations.

__SREDA, 12. januar 2011. u 18 sati: __

Lecture No 1141

* Prof dr Milutin Marjanov, Matemtical Institute SANU Belgrade *

Abstract: What is the period of the Earth's rotation around its axis? As a rule, answer to this question is that that one turn around the axis of the Earth requires, approximately, 24 hours and that this period is cold the mean solar day. Either, more precisely, some four minutes shorter stellar day. In fact, neither of these days stands for the Earth's full spin period. Earth turns around its axis considerably more slowly. The only frame of reference in which its intrinsic rotation is clearly defined, while its orbital motions are excluded, has to be the one related with the Moon. The full spin period of our planet is about 50 minutes longer lunar day the time between one lunar zenith to the next. The term tidal day is also in use in oceanography

Predavanja ce se odrzavati sredom sa pocetkom u 18.00 casova, u sali 301 F na trecem spratu zgrade Matematickog instituta SANU, Knez Mihailova 36/III, (zgrada preko puta glavne zgrade SANU).

Poziv naucnicima i istrazivacima da prijave svoja predavanja:

Prijava potencijalnog predavaca treba da sadrzi apstrakt predavanja do jedne stranice na srpskom jeziku cirilicom i prevod na engleski jezik, kao i CV obima do dve stranice. Prijavu poslati na adresu upravnika Odelenja za mehaniku u vidu Word DOC na adresu: khedrih@eunet.rs

Start of each lecture is at each Wednesday at 18,00 h in room 301 F at Mathematical Institute SANU, street Knez Mihailova 36/III.

Announcement and Invitation:

All scientists and researchers in area of Mechanics are invited to contribute to the Program of Mechanics Colloquium of Mathematical Institute of Serbian Academy of Sciences and Arts. One page Abstract of proposed Lecture with short CV is necessary to submit in world doc to Head of Department of Mechanics (address: khedrih@eunet.rs), one month before first day in the next month.

Sekretar Odeljenja

mr Srdjan V. Jovic

sekretar Odelenja za mehaniku

Matematickog instituta SANU, Beograd

e-mail: jovic003@yahoo.com

mr Srdjan V. Jovic

sekretar Odelenja za mehaniku

Matematickog instituta SANU, Beograd

e-mail: jovic003@yahoo.com

Upravnik Odeljenja

Prof. Dr. Katica R. (Stevanovic) Hedrih

Prof. Dr. Katica R. (Stevanovic) Hedrih