A method is proposed for determining the thread length in the knit loop using the law of normal probability distribution. An equation was obtained for determining the thread length in the loop for calculation with an integral method in the assumption that the shape of the loop is described by a Gaussian curve. The proposed method can be used to obtain sufficiently high results, since the deviation is no greater than 4%.It is first necessary to have a mathematical description of the structure of the designed knit which is distinguished by the variety and complexity of the weave, for implementing automated design of knits.Ensuring sufficient accuracy of the model means taking into consideration all important properties and links deriving from secondary unimportant properties and links in idealizing a real object. Solving this problem is not only a function of the character of the object but also of the problem posed. In representing a real object with sufficient accuracy, the mathematical model must at the same time be as simple as possible, since working with a complicated model is not only difficult but can also be almost impossible to do. The contradictory nature of these requirements frequently requires giving up accuracy in the interest of simplicity, but such a compromise is only acceptable if the model still reflects the important properties of the real object. The development of methods of simplifying real objects and systems to construct maximally simple mathematical models is a central problem in any sector. In modeling real objects, it is useful to be oriented toward mathematical models of standard form which are provided by the corresponding apparatus.We calculated the loop thread length of one of the basic elements in constructing knit fabrics. A relatively large number of equations has been proposed for solving this problem. Some of them are based on purely empirical dependences and other are based on approximating the results to the geometric shape of the loop.We propose describing the loop shape as a Gaussian curve. A comparison of the Gaussian curve with the shape of the loop photographed in the electron microscope ( Fig. 1) explains our hypothesis on the use of this curve to determine the knit loop length, i.e., ( )

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A comparatively simple repeating element whose relative deformation corresponds to the deformation of the entire sample can be distinguished in many materials made of chemical (including rigid-chain) fibres. A simple program based on the MATLAB mathematics package that allows calculating the relative deformation of each element and consequently also the deformation of the sample is described. The result of calculating the deformation of a repeating element consisting of two circles of different radius twisted to opposite sides is reported as an example of running the program.Fibres and thread made from chemical, including rigid-chain, polymers are used for manufacturing fabrics, knits, and other textile materials for different applications. Since such fibres and thread stretch very little on deformation of the sample, deformation primarily takes place due to bending of the fibres and thread. In most textile materials except for nonwovens, a comparatively simple repeating element whose relative deformation corresponds to the relative deformation of the entire sample can be distinguished. A photomicrograph of one type of knit is shown in Fig. 1a and a photomicrograph of a column separated (prepared) from this knit is shown in Fig. 1b. A comparison of photomicrographs 1a and b shows that the shape and size of the loops changes little when the fibre is free from interacting with the other fibres. This indicates that the fibre used in manufacturing the knit is plastically deformed, as otherwise it would be in the form of a straight line. After calculating the relative deformation of such a repeating element, we obtain the relative deformation of the entire sample.It is convenient to use the two-dimensional nonlinear theory of elasticity for these calculations. We will assume that only concentrated, and not distributed, forces act on the examined element (henceforth the elastic line).Consider a two-dimensional elastic line obtained as a result of the fact that a fibre of arbitrary initial curvature (for example, plastically deformed) is loaded with some finite number of concentrated forces and external moments. We will use the approach proposed by Popov.*The elastic line can always be divided into segments in the general case so that concentrated forces F 0 and F 1 and external bending moments M 0 , M 1 were applied only on ends 0 and 1 (Fig. 2) of the examined line segment (they are taken with consideration of the effect of the cut parts of the fibre on segment 0-1). If there are no distributed forces on segment 0-1, then from the condition of equilibrium of the segment, we have F 0 + F 1 = 0 or F 0 = -F 1 . We stipulate the direction of force F 0 applied at the initial point of segment 0-1 of the elastic line as the basic direction.Let us introduce angle δ read counterclockwise from direction F 0 to axis X (or clockwise from axis X to the direction of force F 0 ) (i.e., the slope of axis X to force vector F 0 ). We will consider the initial curvature of the rod (fibre) (the curvature is a quantity inverse to the radi...

677.023 L. A. Kudryavin, and M. V. ShablyginThis article examines certain aspects of the unwinding of thin filaments of chemical fibers from smalldiameter bobbins. The aspects discussed are related either to the penetration of turns in lower-lying layers of the winding by turns in higher layers or to the "sticking" of turns (if the filament is coated) deep in the winding. Both of these phenomena lead to rupture of the filament as it is being unwound from the bobbin. A method is described for calculating the optimum parameters of a flanged cylindrical bobbin in order to prevent filament rupture due to penetration and sticking of the turns. The optimum parameters that are calculated are for filaments of specified lengths.Thin filaments being unwound from cylindrical bobbins will rupture if their length on the bobbin exceeds a certain value L. This phenomenon is usually related to the fact that the upper layers of the filament press on the lower layers and this pressure causes the former to penetrate the latter. During the unwinding operation, additional force is necessary to remove the upper layers from the lower layers that they penetrated. If a filament is not strong enough, it will rupture. This limits the length of the filament that can be wound on the feed bobbin and complicates the production process.We are proposing a theoretical method that makes it possible to determine the parameters of a cylindrical bobbin in such a way as to prevent the rupture of a filament of a prescribed length as it is unwound from the bobbin. To do this, first we calculate the change in the pressure between the layers of the microwire as the radius of a layer ρ decreases (Fig. 1). We introduce the following notation: r -the internal winding radius; R -the external radius; ρ -the radius of the layer being unwound; B -the distance between the flanges of the bobbin.We will assume that during the winding operation the plane of any turn of the microwire is nearly perpendicular to the axis of the bobbin. This assumption is valid in the case when the diameter of the microwire is small compared to the radius of the bobbin and the outermost turns are wound almost flush with the flanges of the bobbin. The subsequent turns are then wound adjacent to one another. These are the conditions that usually exist when a thin filament is wound on a bobbin. Then the length of one turn of the radius ρ will be equal to 2πρ.We theoretically dissect the bobbin with a plane that passes through its axis. This section is shown in Fig. 1a. We then use the letter n to represent the number of turns that intersect a unit area of this plane. The value of n determines the winding density. Let us examine a layer of very small thickness dρ in the winding (Fig. 1b). Since the layer is very thin, all of the turns that end up in this layer will have nearly the same radius ρ and, thus, the same length 2πρ. The area of the cross section of this layer which coincides with the plane passing through the axis of the bobbin (Fig. 1a) is equal to Bdρ, while the number of turns ...

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