[institut] Dopunjen plan rada za oktobar 2015. godine

Sat Oct 10 14:52:35 CEST 2015

Postovane kolege,

saljemo vam dopunjen plan predavanja u okviru Odeljenja za mehaniku
Matematickog
instituta SANU za oktobar 2015. godine.

Molimo vas da, ukoliko ste u
mogucnosti, prilozeni plan odstampate i okacite na oglasnim tablama vasih
institucija.

S postovanjem,
Katarina Kukic
Sekretar Odeljenja za mehaniku

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PLAN RADA ZA OKTOBAR 2015. GODINE

Sreda 7.10.2015. 18 casova, sala 301f

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 Petroviċ’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 matematičke 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 Petroviću 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, 9º Congresso Temático de Dinâmica, Controle e Aplicaçõesm, 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 14.10.2015. 18 casova, sala 301f
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 21.10.2015. 18 casova, sala 301f
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. Djukić
(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 izvršena
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 21.10.2015. 18 casova, sala 301f
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 odgovarajućim
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.

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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.

dr Katarina Kukic
Sekretar Odeljenja za mehaniku
Matematickog instituta SANU

dr Vladimir Dragovic
Upravnik Odeljenja za mehaniku
Matematickog instituta SANU

http://www.mi.sanu.ac.rs/colloquiums/mechcoll.htm
mehanika at mi.sanu.ac.rs

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