Mechanics Colloquim

PROGRAM

PROGRAM ZA OKTOBAR 2015.

Pozivamo Vas da učestvujete u radu sednica Odeljenja i to:

Sreda, 7. oktobar 2015. u 18 casova, sala 301f, MI SANU:
Katica R. (Stevanovic) Hedrih, Matematicki institut SANU, Projekat ON174001
CENTRAL COLLISION OF TWO ROLLING BODIES: THEORY AND EXAMPLES OF VIBRO-IMPACT SYSTEM NON-LINER DYNAMICS

Abstract: This chapter is focused to central collision of two rolling rigid and heavy smooth balls and using elements of mathematical phenomenology and phenomenological mapping obtain corresponding post collision and outgoing angular velocities of the balls and applied these results for investigation vibro-impact dynamics of two rolling balls along circle trace. This task is fully solved and obtained results are original and new! Original plans of component impact velocities and angular velocity of each of two different rolling balls in central collision and corresponding outgoing angular velocities are presented. Use Petrovi..s elements of mathematical phenomenology, especially mathematical analogy between kinetic parameters of collision of two bodies in translator motion and collision of two rolling different size balls, new original expressions of two outgoing angular velocities for each of rolling balls after collision are defined. Using this new and original result vibro-impact dynamics of two rolling different heavy balls on the circle trace in vertical plane in period of series collisions is investigated. Use series of the elliptic integrals, new nonlinear equations for obtaining angles of balls positions at positions of collisions are defined. Branches of phase trajectories of the balls in vibro-impact dynamics are theoretically presented. Using previous new and original result, the vibro-impact dynamics of two rolling heavy different disks on the rotating circle trace in vertical plane in period of series of collisions is investigated. Use series of the elliptic integrals, new nonlinear equations for obtaining angles of disks positions at positions of collisions are defined. Phase trajectories of the disks in vibro-impact dynamics are theoretically presented. Two cases of vibro-impact dynamics when phase portraits contain trigger of coupled singularities and homoclinic orbit in the form of number .eight. as well as in the case without that trigger of coupled singularities are discussed. Phase trajectory branches of both rolling disks in period from initial positions to first collision between rolling disks are presented. Keywords: Theory, rolling balls, collision, pre-impact, post-impact, impulse, moment of impulse, impact forces, impact couple, rolling trace, arrival angular velocity, impact angular velocity, outgoing angular velocity, theorems, collision of rolling balls in circle line, phase trajectory, angular velocity discontinuity, collision of rolling disks on rotate circle trace.

References
1. M. Petrovic, Elementi matematicke fenomenologije (Elements of mathematical phenomenology), Srpska kraljevska akademija, Beograd, 1911. str. 89. http://elibrary.matf.bg.ac.rs/handle/123456789/476?locale-attribute=sr
2. M. Petrovic, Fenomenolosko preslikavanje (Phenomenological mapp), Srpska kraljevska akademija, Beograd, 1933. str. 33. http://elibrary.matf.bg.ac.rs/handle/123456789/475
3. Elements of mathematical phenomenology and phenomenological mapping in non-linear dynamics, Edited by Katica R. (Stevanovic) Hedrih, Ivan Kosenko, Pavel Krasilnikov and Pol D. Spanos, Special Issue of International Journal of Non-Linear, Mechanics, Volume 73, Pages 1-128 (July 2015) http://www.sciencedirect.com.proxy.kobson.nb.rs:2048/science/journal/002074 62/73
4. K. R. (Stevanovic) Hedrih, Beseda o Mihailu Petrovicu (Address to Mihailo Petrovic) , Legende Beogradskog Univerziteta Legends about University of Belgrade), Univerzitet u Beogradu, Univerzitetska biblioteka .Svetozar Markovic. u Beogradu, 2005, str. 37.-48.
5. K. R. (Stevanovic) Hedrih, Beseda o Mihailu Petrovi.u i fascinantnoj nelinearnoj dinamici (Address about Mihailo Petrovic and fascinate non-linear dynamics) , Rektorat Univerziteta u Beogradu i Srpska akademija nauka i umetnosti, Maj mesec matematike .- Srpski matematicari , Naucni skup maj 2012, Zavod za izdanje udzbenika, Beograd 2014-2015, pp. (to appear, in press)
6. K. R. Hedrih (Stevanovic), V. Raicevic, S. Jovic, Vibro-impact of a Heavy Mass Particle Moving along a Rough Circle with Two Impact Limiters, Freund Publishing House Ltd., International Journal of Nonlinear Sciences & Numerical Simulation 10(11): 1713-1726, 2009.
7. Hedrih (Stevanovic) K R., Raicevic V. and Jovic S., Phase Trajectory Portrait of the Vibro-impact Forced Dynamics of Two Heavy Mass Particles Motions along Rough Circle, Communications in Nonlinear Science and Numerical Simulations, 2011 16 (12):4745-4755, DOI 10.1016/j.cnsns.2011.05.027.
8. K. R. Hedrih (Stevanovic) (200), Nonlinear Dynamics of a Gyro-rotor, and Sensitive Dependence on initial Conditions of a Heav Gyro-rotor Forced Vibration/Rotation Motion, Semi-Plenary Invited Lecture, Proceedings: COC 2000, Edited by F.L. Chernousko and A.I. Fradkov, IEEE, CSS, IUTAM, SPICS, St. Petersburg, Inst. for Problems of Mech. Eng. of RAS, 2000., Vol. 2 of 3, pp. 259-266.
9. K. R. Hedrih (Stevanovic) (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, 2008. pp. 174-182, ISBN 978-5-02-036667-1.
10. K. R. Hedrih (Stevanovic) (2010), Discontinuity of kinetic parameter properties in nonlinear dynamics of mechanical systems, Keynote Invited Lecture, 9 Congresso Temico de Dinica, Controle e Aplicaesm, June 07-11, 2010. UneSP, Sao Paolo (Serra negra), Brazil, Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications, Serra Negra, 2010, pp. 8-40. SP - ISSN 2178-3667.
11. K. R. Hedrih (Stevanovic) (2012), Energy and Nonlinear Dynamics of Hybrid Systems, Chapter in Book: Edited by A. Luo, Dynamical Systems and Methods, Springer. 2012, Part 1, 29-83, DOI: 10.1007/978-1-4614-0454-5_2


Sreda, 14. oktobar 2015. u 18 casova, Sala 301f, MI SANU
Bozidar Jovanovic, Matematicki institut SANU
DINAMIKA BILIJARA I SIMETRICNE KVADRIKE Rezime: Prikazacemo nove rezultate iz dinamike bilijara definisane simetricnim kvadrikama u pseudo-Euklidskim prostorima. U slucaju kada su trajektorije svetlosnog tipa, sistem se posmatra koristeci okvir kontaktne integrabilnosti. Rad je motivisan istrazivanjima Vladimira Dragovica i Milene Radnovic, a dobijen je u saradnji sa Vladimirom Jovanovicem (Univerzitet u Banja Luci).

Sreda, 21. oktobar 2015. u 18 casova, sala 301f, MI SANU
Djordje Musicki, Fizicki fakultet, Univerzitet u Beogradu i Matematicki institut SANU
PROSIRENJE VUJANOVIC-DJUKICEVE NETERINE TEOREME ZA KONTINUALNE SISTEME, prvi deo

Rezime: Kao sto je poznato, teorema Emmy Noether, primenjena na mehaniku predstavlja jedan algoritam za nalazenje invarijanata, tj. integrala kretanja sistema cestica i prema njoj svakoj transformaciji generalisanih koordinata i vremena koja odrzava dejstvo invarijantno odgovara jedan integral (ili konstanta) kretanja. Docnije je ona uopstavana, ali samo za konzervativne sisteme i neke specijalne slucajeve nekonzervativnih, a opste uopstenje za nekonzervativne sisteme dali su B. Vujanovic i Dj. Djuki. (1975. godine). Oni su to postigli pogodnom transformacijom d.Alambert-Lagrange -ovog principa, cime su istovremeno resili i problem kako naci takve transformacije generalisanih koordinata i vremena kojima odgovara neki integral kretanja.
U ovom saopstenju data je generalizacija ove Vujanovic-Djukiceve Noether-ine teoreme na mehanicke kontinuirane sisteme. Pri tome je za razliku od navedenih autora, ova generalizacija izvrsena na direktan nacin, uopstavanjem uobicajenog dokazaNeterine teoreme, tj. polazeci od totalne varijacije dejstva za kontinuirane sisteme i primenjujuci odgovarajuce opste Lagrange-eve jednacine. U tom cilju formulisana je odgovarajuca totalna varijacija dejstva i na navedeni nacin, a po analogiji sa Vujanovic-Djukic Noether-inom teoremom u analitickoj mehanici, dobijena odgovarajuca generalisana Noether-ina teorema za kontinuriane sisteme. Potom je izvr.ena analiza dobijenih rezultata i pokazano je kakvi integrali kretanja proizlaze iz ove Noetherine teoreme, koji se bitno razlikuju od odgovarajucih integrala kretanja u mehanici cestica, ukljucujuci i dobijanje integrala energije prostornog tipa.

Sreda, 28. oktobar 2015. u 18 casova, sala 301f, MI SANU
Djordje Musicki, Fizicki fakultet, Univerzitet u Beogradu i Matematicki institut SANU
PROSIRENJE VUJANOVIC-DJUKICEVE NETERINE TEOREME ZA KONTINUALNE SISTEME, drugi deo

Rezime: U ovom saopstenju, koje se nadovezuje na prethodno, uvedeni su tzv. pseudokonzervativni sistemi za kontinuirane sisteme, po analogiji sa odgovarajucim u mehanici cestica (Dj. Musicki, 2012), cime je izvrsen jedan drugi, komplementarni prilaz ovoj problematici. Oni su definisani kao takvi nekonzervativni kontinuirani sistemi cije se Lagrange-eve jednacine uvodjenjem nove gustine Lagranzijana mogu svesti na Euler-Lagrange-eve jednacine, i formulisan je uslov da se neki nekonzervativan sistem moze smatrati pseudokonzervativnim. Analizirani su energijskim odnosi ovakvih sistema i pokazano je da pod izvesnim uslovima oni imaju izvesne integrale (ili konstante) energije u sirem smislu, koji uz karakteristicne razlike pokazuju i izvesne slicnosti sa odgovaraju.im integralima energije u mehanici cestica. Pokazano je kako se mogu naci takvi integrali energije, kako pomocu dobijene generalisane Noether-ine teoreme koja odgovara pseudokonzervativnim sistemima, tako i neosredno pomocu jednog sistema parcijalnih diferencijalnih jednacina. Dobijeni rezultati su ilustrovani na jednom primeru: oscilacije zice u viskoznoj sredini. Posto je pokazano da je ovaj sistem pseudokonzervativna, primenjen je odgovarajuci uslov za postojanje integrala kretanja i nadjeno je jedna partikularno resenje ovog uslova, kad integrali kretanja postaju integrali energije. Na osnovu toga nadjen je odgovarajuci integral energije u sirem smislu, koji pokazuje sve karakteristike takvih integrala energije za nekonzervativne sisteme.

Predavanja su namenjena sirokom krugu slusalaca, ukljucujuci studente redovnih i doktorskih studija. Odrzavaju se sredom sa pocetkom u 18 casova u sali 301f na trecem spratu zgrade Matematickog instituta SANU, Knez Mihailova 36.