Chepizhnyi A. Geometric method of identifying trajectory kinematic properties of a point in compound motion

The thesis considers a new approach to identifying kinematic properties of a point’s compound motion. As a rule, two coordinate systems – moving and fixed – are applied. The moving coordinate system conducts relative motion with regard to the fixed system, and the point, which moves in the moving system, conducts relative motion with regard to it. The sum of two motions produces absolute motion with regard to the fixed coordinate system. Transportation and relative motions are usually given as dependencies in time functions. The work views Frenet-Serret frame of the directing curve as a moving coordinate system. The arc length of the directing curve is an independent parameter in this case. Transportation motion of a trihedron is now defined, as its position on the curve depends on the current value of an arc coordinate, curvature and torsion of the directing curve, which is the transportation trajectory of the trihedral motion. Such approach enables the usage of Frenet formulas. In turn, it requires working out a vector equation of absolute motion of a point, which is then gradually differentiated applying Frenet formulas to receive absolute velocity and acceleration. A special case highlighted in this work is when the point conducts relative motion in the trihedron osculating plane. As a result, velocity and acceleration expressions in projections to cover the unit axes of the trihedron trajectory (Frenet formulas) are acquired. These expressions include such equations as relative motion of a point in osculating plane and their derivatives, velocity of the trihedron motion along the directing curve, curvature and torsion of the curve and their derivatives. The work also considers certain examples of applying the results in direct and inverse problems of compound motion of a point.They are applied, respectively, in direct problems in kinematic analysis of the points of the driven unit in planar mechanisms, and in inverse problems in identifying acceleration of chemical fertilizing particles along different blades of dispersing units in track.