Relation Between the Single Orths of a Point on the Axis of the Deformed Thread in the Case of Euler and Lagrange Coordinates

Abstract

The values of the position vectors and unit tangents have been determined by research, defining the positions of points on the axis of the deformed thread using Eulerian and Lagrangian coordinates. The position vector defines the point's position on the line relative to the moving coordinate system. It determines the point's position on the line relative to the stationary coordinate system. Consequently, when the thread is displaced, the point moves to a new position. The new position vector defines the point's position on the line. The displacement vector of the point on the thread axis characterizes the shift of points due to deformation at the thread intersection. This has made it possible to use these theoretical relation at the initial stage of designing the technological process to derive equations for determining thread tension in the fabric and knit forming zone, reduce thread breakage rates, and enhance the quality of the finished products