Семинар за рачунарство и примењену математику, 16. октобар 2018.

• 15. Октобар, 2018
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Наредни састанак Семинара биће одржан у уторак, 16. октобра 2018. у сали 301ф Математичког института САНУ са почетком у 14:15.

Предавач: Катица (Стевановић) Хедрих, Математички институт САНУ, Београд

Наслов предавања: ROLLING HEAVY BALL OVER THE SPHERE AND ANALYSIS OF DYNAMICS OF VIBRO-IMPACT SYSTEMS WITH ROLLING BALLS OVER THE SPHERE

Апстракт:
The ball as a rigid body has six degrees of freedom of motion, but when rolling around the immobile sphere then has three degrees of freedom of movement. The limitations come from the assumption that it rolls around the sphere, so there is a link that the center of the ball is always in the sphere of radius equal to the sum or difference between radius of the sphere and ball, depending whether the ball is rolling inside or outside of the sphere. The other two constraints come from the assumption that rolling without slipping the graze can therefore determine the relation between the angular velocity of rolling about two orthogonal axes tangent to the sphere at the point of contact between the ball and the sphere. Constraints are geometrical and stationary, and system is holonomic and scleronomic.

We propose that system is in gravitational field and that rolling is activated by gravitational force and initial kinetic and potential energy given to ball at initial moment. For mathematical description of the rolling of heavy rigid homogeneous ball over the sphere inside as well as outside of sphere surface, spherical coordinates are used: angle in circular and angle in meridional directions, and angle of ball self rotation about radial direction. Nonlinear differential equations are derived. Angle coordinate in circular direction is cyclic coordinate, and an integral, for circular-cyclic coordinate is derived. Integral constant depends of initial condition and is determined. The main nonlinear differential equation is expressed by angle meridional coordinate and corresponding first integral is derived. The equation of first integral is equation of phase trajectory and by use of this equation and corresponding set of initial conditions phase trajectory portraits are graphically presented. An elliptic integral is derived. By usage of new Hedrih’s results in theory of collision between two rolling bodies geometry, kinematics and dynamics of successive collisions of two rolling balls over the surface of sphere is analyzed and the methodology for investigate vibro-impact nonlinear dynamics of vibro-impact system with rolling bodies over the sphere surface is presented.

A mathematical analogy between differential equations and phase trajectory portraits of dynamics of rolling heavy ball and material heavy mass particle moving along sphere surface is identified. (This lecture is dedicated to 150 years of birthday of Mihailo Petrovic, founding father of Serbian mathematics)