MECHANICS OF MACHINES AND MECHANISMS - MODELS AND MATHEMATICAL METHODS
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).
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.