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

Mechanics Colloquim

PROGRAM

MATEMATIČKI INSTITUT SANU
ODELJENJE ZA MEHANIKU

PROGRAM ZA APRIL 2015.

Pozivamo Vas da učestvujete u radu sednica Odeljenja i to:

Sreda, 1. april 2015. u 18 casova, sala 301f:
Katica R. (Stevanovic) Hedrih, Matematicki institut SANU
ELEMENTS OF GEOMETRY, KINEMATICS AND DYNAMICS OF BILLIARDS

"In connection with the game of billiards .... there are various dynamic tasks, whose solutions contain in this event. I think that people who know Theoretical mechanics, and even students of polytechnics, with interest familiarize themselves with explanations of all the original phenomenon that can be observed from the time of movement billiard balls"
Gaspar-Gistav de Koriolis,
Mathematical theory of billiards game.
G Coriolis (1990). Thrie mathatique des effets du jeu de billard ; suivi des deux celebres memoires publi en 1832 et 1835 dans le Journal de l'ole Polytechnique: Sur le principe des forces vives dans les mouvements relatifs des machines & Sur les uations du mouvement relatif des systes de corps (Originally published by Carilian-Goeury, 1835 ed.). itions Jacques Gabay. ISBN 2-87647-081-0.

Abstract: Displays the elements of the dynamics of billiards, systems whose dynamics are different phenomena observed dynamics of the system. Starting from the geometric basis for switching to the impact theory, which is basically a theory of the dynamics of each ball of billiards. Shown are the plans of translational and angular velocities of rolling of one ball before and after the collision, the two balls collide, as well as three balls in simultaneous collisions. The equations are given for the impulse of movement and kinetic energy before and after the collision in the aforementioned cases.
Then expose the theory of the collision of two mass particles, as well as two balls or impact of mass particle, as well as the balls in the barrier. The output is hypotheses on collision and impact, define the various types of collision and impact. This problem is associated with the dynamics of the system with one side retaining constraints.
Will be talk about the competition and Royal Society. Royal Society Society in London in 1668 announced a competition for the solution of problems of the dynamics of impact and on this competition have submitted their works, by now known scientists Vilis (Wallis, 1616-1703, Mechanica sive de mote-1688) and Hajgens (Huygens . De motu corporum ex percusione). Using the results of the collision submitted by the Royal Society learned Willis and Huygens, and giving their generalizations, Isaac Newton founded the fundamental basics of the theory of impacts. And before Newton and Huygens and Willis, was exploring the dynamics of impacts. Thus, for example, collision problems are dealt with Galileo Galilei, who came to the conclusion that the impact force in relation to the pressure force infinitely large, but it came to the knowledge of the relationship of impact impulse and linear momentum.
It will be shown Karnoova teorema (Lazare Carnot 1753-1824., Principes fondamenteaux de l.uilibre et de movement - 1803), who says that "In a collision, the system inelastic material bodies loss of kinetic energy is equal to the kinetic energy lost speed." The explanations on experimental method for obtaining coefficients of restitution of different types of impacts and collisions will be presented.
Comparing the elements of mathematical phenomenology and identifying qualitative and mathematical analogy between geometry of moving geometric point in the plane with defined constraints propagation ray of light with the refusal of the obstacles and suggests that the trajectory of geometric point and ray of light analogue and can be used as a baseline determination of the trajectory of billiards ball.
But as billiard balls spherical bodies orbit of their dynamics depend on the type of impact limiters in the form of the surface, and angles of impact velocity and outgoing velocities of mass center balls depend on the type of impact: whether the impact is skew or central! Only in the case that the sphere neglected dimensions, so it can be regarded as a geometric point, these angles are equal! In all other cases, the ball gets in the co limitation kinetic energy of translation and rotation, a void to change the angles of the outgoing and incoming velocity, if the balls are not homogeneous and of equal mass.
The conclusion points to the importance of expanding elements of the dynamics of billiards, crossing the dynamics discrete vibro-impacts systems, particularly rolling heavy balls with mutual collisions, when the balls rolling on curved lines that rotate. This dynamics is associated with the dynamics of balls in rolling bearings.

Keywords: Billiards, ball, rolling without slipping, collision, alternation of directional velocity, impact velocity, the uplink speed, trajectory of the center of mass, central and skew collision, the impulse force, kinetic energy, shock and collision, the collision of two balls, collision three balls, rolling balls along rotate curvilinear lines, one side retaining constraints.

Acknowledgment: Parts of this research were supported by Ministry of Sciences of Republic Serbia trough Mathematical Institute SANU Belgrade Grant ON174001:.Dynamics of hybrid systems with complex structures; Mechanics of materials.., and Faculty of Mechanical Engineering, University of Nis.

References

1. G Coriolis (1990). Thrie mathatique des effets du jeu de billard ; suivi des deux celebres memoires publi en 1832 et 1835 dans le Journal de l'ole Polytechnique: Sur le principe des forces vives dans les mouvements relatifs des machines & Sur les uations du mouvement relatif des systes de corps (Originally published by Carilian-Goeury, 1835 ed.). itions Jacques Gabay. ISBN 2-87647-081-0.
2. G Coriolis (1832). "Sur le principe des forces vives dans les mouvements relatifs des machines". J. De l'Ecole royale polytechnique 13: 268-302.
3. G-G Coriolis (1835). "Sur les uations du mouvement relatif des systes de corps". J. De l'Ecole royale polytechnique 15: 144.154.
4. V.V. Koslov i D. V. Trev, BiliardGeneti.eskoe bbedenie dinamiku sistem s udarami, Izdatelstvo Moskovskogo universiteta, 1991. Moskva, str. 192.
5. Persson, A., 1998 How do we understand the Coriolis Force? Bull. Amer. Meteor. Soc. 79, 1373-1385.
6. D.amer, Maks (1957). Concepts of Force. Dover Publications, Inc.. ISBN 0-486-40689-X.
7. Robert Byrne (1990). Byrne's Advanced Technique in Pool and Billiards. Harcourt Trade. p. 49. ISBN 0-15-614971-0.
8. Hedrih (Stevanovi.) K., (2005), Nonlinear Dynamics of a Heavy Material Particle Along Circle which Rotates and Optimal Control, Chaotic Dynamics and Control of Systems and Processes in Mechanics (Eds: G. Rega, and F. Vestroni), p. 37-45. IUTAM Book, in Series Solid Mechanics and Its Applications, Editerd by G.M.L. Gladwell, Springer. 2005, XXVI, 504 p., Hardcover ISBN: 1-4020-3267-6.
9. Hedrih (Stevanovi.) K., (2004), A Trigger of Coupled Singularities, MECCANICA, Vol.39, No. 3, 2004., pp. 295-314. , DOI: 10.1023/B:MECC.0000022994.81090.5f,
10. Hedrih (Stevanovi.), K., (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.
11. Hedrih (Stevanovi. 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, 2008. pp. 174-182, ISBN 978-5-02-036667-1.
12. Hedrih (Stevanovi.) K., (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.
13. Hedrih (Stevanovi.) K., (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
14. Hedrih (Stevanovi.) K R., Rai.evi. V. and Jovi. 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.
15. Hedrih (Stevanovi.) K., Rai.evi. V., Jovi. S., 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.

Sreda, 8. april 2015. u 18 casova, sala 301f:
MODEL ZA ANALIZU OSTECENjA KONSTRKCIJA USLED CIKLICNIH PLASTICNIH DEFORMACIJA

Rezime: Posmatra se najednostavniji nacin uvodjenja ostecenja konstrucija koje nastaje kao posledica zamora u plasticnoj oblasti. Polazeci od Prajzakovog histerezisnog opertaora, napravljen je model za analizu elastoplasticnog ponasanja materijala pri aksijalnom naprezanju i savijanju silama u plasticnoj oblasti. Usled plasticnih deformacija dolazi do pojave zamornih prslina (ostecenja). U izlaganju ce biti pokazano da se i ovaj fenomen moze modelirati uvodjenjem Prajzakovog operatora. Na nekoliko primera resetkastih nosaca pokazane su prednosti ovog nacina modeliranja u odnosu na postojece u literaturi i u komercijalnim programima (SAP, ABAQUS).

Sreda, 22. april 2015. u 18 casova, sala 301f:
Bojan Arbutina, Matematicki fakultet, Univerzitet u Beogradu
EKSPLOZIVNI UDARI SA KOSMICKIM ZRACENjEM . MODIFIKOVANO SEDOVLjEVO RESENjE

Rezime: Udarni talasi javljaju se pri razmatranju raznih astrofizickih fenom ena I objekata, poput supernovih i njihovih ostataka. Na ovom predavanju bic e prikazano Sedovljevo resenje za eksplozivne udare, koje opisuje evoluciju ostataka supernovih u adijabatskoj fazi. Razmotricemo i njegovu modifikaciju u slucaju prisustva kosmickog zracenja, odnosno, dod atne komponente sa funkcijom raspodele cestica u faznom prostoru u obliku st epenog zakona, uz obican gas.

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