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

**Seminar
MECHANICS OF MACHINES AND MECHANISMS - MODELS AND MATHEMATICAL METHODS**

**PROGRAM**

**Plan rada Seminara Mehanika mašina i mehanizama - modeli i matematičke metode za OKTOBAR 2019.**

**UTORAK, 22.10.2019. u 17:00, Sala 301f, MI SANU, Kneza Mihaila 36**

*Katica (Stevanović) Hedrih, Mathematical Institute of SASA*

**ROLLING A HEAVY BALL OVER THE COORDINATE SURFACES OF ORTHOGONAL CURVILINEAR COORDINATE SYSTEMS (In Memory of scientists and academician RAS V. V. Rumyantsev and V. M. Matrosov)**

In the last author’s paper rolling heavy ball over the sphere surface is described in curvilinear sphere coordinates using meridian and circular angle coordinates. Rolling of ball is decomposed into two components of the rolling, one along meridians and second along comparators of the spherical curvilinear coordinate lines. Investigation shown that constraints are pure geometrical and stationary, and that system is holonomic and scleronomic. In this lecture, on the basis of previous results, a natural approach for investigation rolling ball over the coordinate surfaces and corresponding parallel surfaces in different orthogonal curvilinear coordinate system is presented. Rolling ball motion is decomposed, into two components of rolling along orthogonal coordinate lines of the curved coordinate surface.

In the series of the author’s published References, the tree basic vectors of tangent space of a kinetic point moving in space, and for different curvilinear coordinate systems (spherical, cylindrical, different generalized elliptical curvilinear coordinate system, cylindrical coordinate system with orthogonal curvilinear coordinates; three dimensional three parabolic coordinate system with orthogonal curvilinear coordinates; elliptical coordinate system with orthogonal curvilinear coordinates; three dimensional oblate spheroidal coordinate system with orthogonal curvilinear coordinates) are determined with corresponding angular velocity and velocities of the basic vector extensions. These results for determination basic vectors of the tangent space of vector position of corresponding surface point, we can use to define in each point of the curved coordinate surface tangent directions to the curved coordinate surface, along which rolling ball moves.
Through the contact point between rolling ball and curved coordinate surface by three basic vectors of tangent space of vector position of this contact point it can be define tangent to the component rolling trace of the ball and also direction of the momentary axis of ball rolling.

We use the Lagrange equations of the second kind for independent two generalized curvilinear coordinates, in the curvilinear orthogonal coordinate system, to derive system of two ordinary nonlinear differential equations describing of a rolling ball dynamics over the coordinate surface of the curvilinear orthogonal coordinate system. Kinetic and potential energies are expressed by these independent two generalized curvilinear coordinates.

It can be conclude, that presented approach for mathematical description and investigation motion of a rolling ball, without slipping, over the surface is possible by use decomposition of the rolling into two components rolling along two rolling motion along orthogonal coordinate elementary arches in each contact point between rolling ball and surface of the rolling.
Lecture is based on the author’s research results presented at Oll-Russian Congress of the Fundamental Problems of Theoretical and Applied Mechanics, Ufa August 20-24, 2019 (XII Всероссийский съезд по фундаментальным проблемам теоретической и прикладной механики, Уфа, 20-24 августа 2019).
Keywords. Rolling ball; coordinate surface of orthogonal curvilinear coordinates; decomposition of the rolling motions; two components of rolling ball motion along orthogonal curvilinear lines of surface; rolling about rolling orthogonal momentary rolling axes. (Project ON174001 Dynamics of hybrid system complex structures).

The lecture will present a brief overview of the development of rail traffic in the world and in Serbia. The special attention will be paid to the problems of the braking of railway vehicles which were in that time has crucial influence on the development of rail traffic. The genius technical solution of the Serbian inventor Dobrivoje S. Božić (1885-1967), as well as its significance and influence on the development of railway traffic from ancient times to nowadays, will be described in detail.

He is the first man in the world who is on an completely new way applied the triple valve in the braking system of railway vehicles. This revolutionary invention contributed to faster, safer and more quality development of railway traffic, and thus the entire business and economy system of the world. For many years, this inventor and scientist was not mentioned, so the domestic and the world's public was deprived of the knowledge about the true value of his work. It is important to note that, by the number of patents, he is in the third place among the Serbs - right after Tesla and Pupin. He has more than 60 registered patents in more than 18 different countries over the world.

In this sense, the aim of this lecture is to bring closer genius technical achievements of Dobrivoje S. Božić in the field of braking of railway vehicles, to scientific-professional public. In addition, the special attention in lecture will be paid to impact of policy on making an important business decisions, which had a significant impact on Božić's life and destiny – to be glorious but unknown.

Seminar Mehanika mašina i mehanizama - modeli i matematičke metode započeo je sa radom u junu 2018.god. Seminar se održava do dva puta mesečno, utorkom u periodu od 17.00 - 19.00 u Matematičkom institutu SANU.

Prof. dr Katica (Stevanović) Hedrih

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dr Ivana Atanasovska

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Milan Cajić

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