ANALYTIKAL SEARCHING OF MOVING AND FIXED AXOIDS OF THE FRENET THRIHEDRAL OF THE DIRECTING CURVE

Abstract

The accompanying Frenet trihedral of a directing curve at movement on it carries out some spatial movement. In theoretical mechanics, the spatial motion of a solid body is described analytically, and at each instant it is considered as the sum of rotational and translational motions. If the accompanying trihedral is regarded as a solid body, its motion is entirely due to the differential characteristics of the directing curve, viz the curvature and torsion at the point of location of the trihedral.

 The spatial motion of a solid body at any given time can be decomposed into many variants of rotational and translational displacement, each of which depends on the choice of the point of the solid body , that is, the pole with respect to which the motion is decomposed. For the points of the body acting as a pole, the vector and the magnitude of the rotational motion are unchanged and for the translational motion they are variable. In a solid at a particular point in time, you can find the pole for which the vectors of rotational and translational motions will coincide in the direction. This direction is the axis of the kinematic screw. Around this axis, at a specific point in time, the body rotates at a certain angular velocity and slides along it at a certain linear velocity. The relation between these velocities is a parameter of the kinematic screw. When moving the body, the axis of the kinematic screw changes its direction and position in the body, that is, it forms а ruled surface. Multitude of the positions of the axes of the kinematic screw can be considered with respect to the fixed coordinate system and with respect to the movable one (in our case, in the Frenet trihedral system). In the first case we get a fixed axoid, and in the second - a moving axoid. When moving a solid body, the moving axoid rolls over a fixed one and simultaneously slides along a common directing line of contact.

The article shows the position of the axis of the kinematic screw in the accompanying Frenet trihedral, the set of which forms a moving axoid, and also, through the directing cosines, the position in a fixed system is found, that is, a fixed axoid is found. According to the developed algorithm, it is possible to construct fixed and moving axoids with a common axis of the kinematic screw for any point of the directing curve. Parametric equations of moving and fixed axoids are given.

Keywords: moving and fixed axoids, kinematic screw, Frene trihedral, directing curve, curvature, torsion.