Solution of the problem of the form and tension of a filament in its three-dimensional balloon during twisting

Abstract

Systems of differential equations in cartesian and cylindrical coordinates are obtained to describe a fl exible fi lament rotating in a resisting medium. The concept of the unit aerodynamic drag coeffi cient of the fi lament is introduced in addition to the assumptions used in previous studies. The systems of differential equations that were obtained are solved in analytical form using the assumptions that were made. A value is found for the new concept of unit aerodynamic drag coeffi cient, and ways are described to study it and determine its value for different fi laments in practical applications. Complete mathematical expressions are obtained to calculate parameters that are of practical importance in the twisting of a fi lament -its tension, the maximum radius (diameter) of the torsion balloon, and the conditions under which multiple balloons are formed. It is established that the problem of the form and tension of a fi lament during its twisting cannot be solved in fi nal form without allowance for the resistance of the medium.The theoretical framework that determines the main operational and design parameters of the working spindle of doubling machines is the spatial position of a fi lament that freely rotates about the vertical axis of the spindle and is unwound from or wound onto a package. This position is usually referred to as the balloon of the fi lament, and the process itself is referred to as ballooning. The most signifi cant progress in the study of this topic was made by Prof. A. P. Minakov, who was the fi rst to obtain a system of differential equations that describe the form and tension of a fi lament positioned spatially in an balloon and rotating in a resisting medium while moving along its own axis [1,2]. This and subsequent treatments of the problem of the location of a fi lament in space with allowance for the resistance of the surrounding medium were solved only with the problem in its most basic formulation, i.e. a system of differential equations was obtained in cartesian coordinates. It was shown that in special cases, without allowance for the resistance of air, the system leads to particular solutions [3][4][5]. However, the problem of calculating the form and tension of a fi lament in two and three dimensions has not been completely solved in analytical form. Such a solution is obtained below with certain assumptions.