The computation of according to Chebyshev

 

In the "About cutting clothes" [1] P. L. Chebyshev proved the possibility of construction of development tapered slim shells of cloth for different surfaces. Thus he proceeded from the fact that the warp and weft of the fabric to its original flat shape intersect orthogonally and that when dressing the surface of the fabric, change the angles between the filaments, length of filaments remains the same. Chebyshev also assumed that when changing the rectangular cells of the tissue in parallelograms fabric resists the stretching it only along the warp and weft.In line with this, the fabric clinging to the surface, warp and weft are stretched by forces acting along their length. But in order for a stretched thread on the surface were in equilibrium, they must be geodesics. It is a known equilibrium condition of the filaments on the surface in the General case can be strongly made by only one warp yarn and one weft thread, as all threads of the fabric does not coincide with the geodesic lines on prosverlivayut surfaces.Hence, to ensure equilibrium of the threads of the fabric on the surface, Chebyshev came to need the location of two intersecting threads of the fabric orthogonal to the geodesics. He took them for original coordinate axes on the surface (x, y), considering the coordinates of the length of the warp and weft of the fabric (Fig. IV-4). Threads of the fabric Tchebyshev considered as the coordinate lines, forming on the surface of the curved chebyshevskii network whose elements are infinitesimal parallelograms.The length of the diagonal of the elementary parallelogram ds = √(dу2 + dх2 + 2cosφdydx) defines the distance between two adjacent points on the surface (see Fig. IV-4). Based on this equation, Chebyshev solved the problem of dressing the surface of the fabric in the form of the following formulas:

x = S + 1/6(k2₀u2 + k₀k₂u3 + 1/4k2₂u⁴)S3 + 1/8(k₀k₁u2 + 1/2k₁k₂u3)S⁴ + . . .

y = u - (1/2k₀u + 1/4k₂u2)S2 + 1/6k₁uS3 + . . ., (1)

where x and y rectangular coordinates defining the shape of the scanner shell fabric on the plane; S is the shortest distance from the axis Oy to the points of suture line (border) of the shell surface (see Fig. IV-4); u is the ordinate of the point of intersection of the line of shortest distance (S) the axis Oy; k₀, k₁, k₂ are the coefficients of the expansion in a power series of the Gaussian curvature of the surface at a given point (x, y).

To verify the formula, Chebyshev determined the shape of the sweep sheath to a ball consisting of two parts, and found a simple method of constructing arcs of circles of radii r and r (Fig. IV-5).

These radii are determined by formulas: r = 0,65 Rш; R = 2,46 Rш, if AB = sun; OA = OS = 0,5 πRш; S = 1,42 Rш where Rш - the radius of the ball.

The cutting sheath is produced by positioning the warp and weft of the fabric on the axes Oh, Oh.

Experimental verification of the shells of different fabrics (canvas, side canvas, wool suiting fabrics), suggests the possibility of manufacture for the Chebyshev expansions a tight-fitting sheath of fabric for the surface of a sphere consisting of two parts.

The solution of the problem of dressing the surfaces with a cloth after Chebyshev engaged in other mathematics.* However, their results cannot be directly used in the design of clothes due to the fact that they solved the problems of differential geometry, is the study of methods for the determination of shell fabric. Along with this, their research shows that a strict solution of the problem of dressing surfaces leads to considerable mathematical complications. Therefore, to solve this problem for computation of shell fabrics should approximate methods.Application of these methods greatly simplifies the solution of the problem of dressing the surface and provides a high accuracy of the computation of items of clothing, as their formation occurs at a slight angle of misalignment of the threads of the fabric (15 - 18°). In this section we described approximate methods of computation of shells of tissue, developed at the Department of garment manufacturing of MTILP. The application of these methods to the computation of items of clothing covered in a separate paragraph.

The coordinates S and u that appear in the Chebyshev formula are orthogonal geodesic lines; they can be identified by direct measurement of the surface using a geodesic polygon. But despite this, the Chebyshev formula (1) was not used in the design of clothing, as it was not developed methods of their use for computation of various surfaces. These formulas represent the power series that define the coordinates of the scan (x, y). Therefore, for approximate calculations of formula (1) can be simplified.

Limited to the first two terms of the series in formulas (1) and noticing that the first one of the second member may be represented in the form of a square of the sum have x = S + 1/6(k₀u + 1/2k₂u2)2S3; y = u - 1/2(k₀u + 1/2k₂u2)2S2. (2)

Excluding from these formulas the expression k₀u + 1/2k₂u2, we obtain an approximate equation of sweep shells of cloth for different surfaces. x = S + 2/3((u - y)2 : S). (3)

For computation of based on this equation it is necessary to define on the surface of the additional survey line (see Fig. IV-4). This line is the shortest distance between the points of the suture line of the shell (D) and the ox axis (E), located at a shortest distance S from the axis Oy.In a small corner of the warp threads of the fabric in the shell additional survey line of u and the corresponding ordinate of the approximately equal, since they are placed on the surface at a small distance from each other and connect the same point on the seam of the shell (D) and the ox axis (E), located at a shortest distance S from the axis Oy.In a small corner of the warp threads of the fabric in the shell additional survey line of u and the corresponding ordinate of the approximately equal, since they are placed on the surface at a small distance from each other and connect the same point on the seam of the shell (D) with the axis ox. Hence, substituting in equation (3) u = u, we find approximate formulas for the computation of shells of tissue with a small warp threads on the surface x = S + 2/3((u - u2) : S); y = u.4)

the computation of membranes with a large warp threads of the fabric on the surface (more than 20' - 30') is produced also by the formula (3) in determining u at more accurate formulas [5]

y = Rаrс sin( u/u sin u/R) (5) or u = u - u3/6R2, (6) where R = uS/(u - u).

When calculating scan for concave surfaces (i.e., when u < u) in the formula (5) take the positive R, in the formula (6) to the minus sign in the right side change to a plus. Experimental verification of the formulas (4) showed quite satisfactory results in the manufacture of tightly-fitting shells, fabric, for 1/3 of the surface of the balloon and parts of the surface of the dummy. When determining from the formulas (5, 6) is sufficiently accurate results will be obtained by calculating the scans for 1/2 of the surfaces of the sphere and ellipsoid of revolution, and sweep from one side to the pseudosphere [5].Based on the formulas of the Chebyshev formulas (3, 4, 5, 6) are used to determine the sweep of clothing items that provide the desired form of the primary sample models and reduce the fabric consumption for products.

The specified form of sample models is ensured by the fact that the Chebyshev formulas obtained on the basis of the equilibrium conditions of the threads of the fabric on the surface. Economical same amount of fabric on products is obtained due to the fact that the scanner shell fabric, constructed according to specified formulas, have the minimum area.

Be aware that the minimum area of the scan depends on the position of the reference axes of coordinates on the surface in terms of lines of stitches of the shell. For example, if the source axis, the component parts of the shell of the hemisphere do not coincide with their axes of symmetry, the area of the sweep is larger than the area of scanner shell in which the reference axes are the axes of symmetry [6].

Another example: the area of sweep of the shelves outerwear is greater when the axis Aw is parallel to the Board compared to the area of sweep in which the same axis is set in the middle of the convex parts on the surface. Consequently, the location of the source of the coordinate axes in the orthogonal geodesics provides the relative (conditional) minimum area of the scan that has the minimum value in the predetermined coordinate axes.This complies with the terms of the computation of detailing in the design of which is the axis of reference coordinate set, considering the operational, technological and other requirements.

 

Determination of the deformation of the fabric in the shell

 

When computation of shells of tissue is necessary to determine not only the coordinates of the scan and the angles between them on the surface, but the deformation of the fabric by lines of stitches, to have the appropriate parameters for the manufacture of shells. When dressing cloth prosverlivayut surfaces by changing the angle between the threads can happen the elongation (stretching) or compression (fit) fabric at the seam due to changes in the lengths of the diagonals of the cells of the tissue, when threads deviate from the right angle.The magnitude of this deformation (Δl) on any part of the seam is determined by the difference between the length of this section l₁ in the shell on the surface and l₀ in the scan plane (Fig. IV-6, a); Δl = l₁ - l₀.7)

, the Strain at the tensile fabric has a positive value, and when boarding - negative.

A sprain occurs when the threads of the fabric a and B around the seam line of the angle φ is greater than 90' (Fig. IV-6, b), and landing, when the angle φ is less than 90' (Fig. IV-6). Fit or stretch of the fabric is determined on the basis of measurements of the same plot line of the seam in the shell on the surface (l₁) and the scan plane (l₀). In determining the form, scan using an auxiliary grid (the canvas) the deformation of the tissue can determine without sweep sheath on the basis of length measurements of small sections of the seam line (l₁) and the corresponding threads of the mesh (a, B).The expectation was that a small portion of the length of the seam line, curvilinear triangle on the surface can be considered approximately flat and straight. In this case, the sides a and B of a right triangle in the scan (see Fig. IV-6, a) and oblique triangle in a shell (see Fig. IV-6 a, b, C) have the same length. As a result, the length of a small section of the seam line in the scan l₀ = √(a2 + b2).Substituting this value l₀ in equation (7), we find the formula to determine the deformation of a small portion of the seam line in the shell by measuring the length of this phase and the corresponding threads of the support grid on the surface: Δl = l₁ - √(a2 + b2).8) the length of the small section of the seam line of the oblique-angled triangle (see Fig. IV-6 a, b, C) l₁ = √(a2 + b2 - 2abcosφ). (9) Substituting this value l₁ in the previous equation (8), we obtain Δl = √(a2 + b2 - 2abcosφ) - √(a2 + b2), or by replacing the angle y between the threads of his deviation from the direct (skew angle thread angle) φ = ±φ₀, we find Δl = √(a2 + b2 ± 2absinφ₀) - √(a2 + b2). (10) It follows that the larger the absolute value of the skew angle of the threads (φ₀), the more strength (at +φ₀) or fit (in -φ₀) fabric in the seam. And depending on the angle of the seam line to the thread, the amount of deformation of the fabric at the seam line will be different.

Denote by α the angle of the seam to thread b in the plane (see Fig. IV-6, a), and using ε = Δl/l₀ is the relative deformation (stretching or landing) of the fabric in the seam.

Of a right triangle are a = l₀sinα; S = l₀соsα.

Substituting this into equation (10) and dividing it left and right to l₀, after conversion and replacement in the left part of Δl/l₀ on ε, we find ε = √(1 ± sin2αsinφ₀ - 1). (11) as in various clothing items φ₀ usually does not exceed 15', according to the well-known formula √(a2 ± x) ≈ a ± x/2a approximately (with an error not exceeding 1%), we obtain ε = ±1/2sin2αsinφ₀. (12) As you know, the sine increases with the angle change from 0 to 90'. As a result, from formula (12) follows, that at one and the same angle of the bias threads (φ₀) the deformation of a tissue at the suture line (ε) increases when changing α from 0 to 45', with α = 0 ε = 0; at α = 45' e = ±1/2sinφ₀. Taking the largest value α = 45', φ₀ = 15', we find that the maximum deformation of the fabric in clothing is ε = ±1/2sin15' = ± 0,13 or 13%. Such deformation is, in some cases, small areas of seam. In most cases, α < 45' and φ₀ < 15', and therefore deformation of the fabric in clothing much less.So, even if φ₀ = 15' and α = 15', then ε = ±6,5%. If for the same α = 15' φ₀ = 10', then ε =4%. Finally, if α = 5', φ₀ = 5 - 10°, ε = 0,6 x 1,5%.

All this suggests that in the design of clothing items you need to pay attention to the position of the lines of stitches in relation to the threads of the fabric, especially when large deviations of the angle between the yarns from 90'. With the decrease in the slope of the line of the weld to the threads of the fabric in the scan (α) is a reduction of this angle in the shell, located on the surface (γ, see Fig. IV-6, b). Therefore, the identification and the reduction of the slope of the line of the weld to the threads of the fabric can be produced when determining the location of seams on the surface, i.e. to the calculation and construction of development.Thus, it is necessary to focus on the position of the seams relative to the original axes, which may produce a reduction of the angles of the suture line to the threads of the fabric.

Determining the position of the seams and the source of the coordinate axes, it is necessary to strive to ensure that when significant misalignment of the threads of the seam as little as possible deviated from the threads of the fabric. This gives you the opportunity to provide a predetermined three-dimensional shape with minimal deformation of the fabric by lines of stitches. Along with this it is necessary to consider the aesthetic and technical requirements for the product. Form lines of stitches should be beautiful and not have significant deviations from the samples.

Must remain appropriate location of the fabric pattern on the details, especially in prominent places of the product put on the person. It is also impossible to violate the technical conditions of the cutting in relation to the location of the warp and weft of cloth on the details.

The corresponding location of the reference axes of coordinates in the computation of is the fact that these axes on the surface wear of the body must be orthogonal geodesic lines, and the scan - to coincide with the direction of the warp and weft of the fabric, defined by the technical conditions of the cutting. Orthogonal geodesic lines on the surface, as is known, are defined using the geodetic polygon. To the original coordinate axes coincide with the direction of threads of the fabric when cutting, you need to check the specified axes on the surface.To do this, define the coordinates of the scan points, which sets the direction of the threads of the fabric in detail according to the technical conditions of cutting.

For example, according to the specifications of the cutting threads of the fabric in the back of the jacket should be parallel with the middle line cut from waist to hem. To check the proper positioning of the specified axes, it is necessary to determine the abscissa of the sweep of the back, i.e. the distance from the axis Oy to the seam line at the waist area and hem. These coordinates (distances) in the appropriate location of the coordinate axes must be equal. If this condition is not met, it is necessary to clarify the location of the coordinate axes on the surface.

The determination of the coordinates when checking the reference axes is: unfolding surfaces by measuring the shortest distance geodesic polygon, and prosverlivayut surfaces - design method according to (4).

 

The computation of flat shells

 

The position of the reference axes of the flat shells, which, when bending double folded fabrics are placed on a plane (the collar, the lower portions of the sleeves, trousers, etc.), normally installed directly in the technical specifications cutting, measuring distances from one axis of the coordinate points defining a direction of threads of the fabric to the details. These items belong to the flat shells prosverlivayut surfaces. When laying them on the plane of the fabric is folded in half, forming the edges of the curve of the fold line by changing the angle between the warp and weft.So, for example, is formed in a curve the front edge of the sleeve upper garment, bend, stand collar, rear halves of the pants, etc.

Curve, fold the fabric in flat shells formed in three ways:

1. By varying the angle between the yarns of the fabric in the upper and lower portions of the parts for aligning the fold line with the direction of the warp threads or the weft (Fig. IV-7, a);

2. By varying the angle between the yarns of the fabric in only one part of the details, for example the top (Fig. IV-7, b);

3. By varying the angle between the yarns of the fabric in the upper and lower portions without overlapping the fold line with the direction of the threads (Fig. IV-7).

Flat shell to calculate their developments can be defined as a flat pattern or drawing details are an indication of the lines of stitches and the directions of threads of the fabric according to the specifications of the cutting.

Scan flat shell obtained by the first method, you determine a graphical method. To do this, first ask for drawing shell, the axis of the Ow on the fold line of the fabric and perpendicular to it axis ox (Fig. IV-8). On the axis OU are asked a series of points from which the conduct of the abscissa is parallel to Oh. The abscissa h, h, ..., h, h and intervals Δu between them along the axis OU will get a scan of the shell in the rectangular coordinate axes (Fig. IV-8, b).

To determine the sweep of a flat shell, obtained by the second method, a set of rectangular axes ox, OU (Fig. IV-9, a). In this case the fabric when bending on a curve is compressed at the bend line due to the angle change between the threads along this line. In a small area between the fold line and the axis Oh of the filament of the x-axis are arranged on straight oblique lines. As can be seen from Fig. IV-9, b, of the shell to the right of the axis Oy, in which the strands do not have a warp that is already unfolding.Scan the rest of the shell, you can define a graphic by saving a small pieces of thread a₁, a₂, ... h, h ... on the continuation of the abscissa h, h ... the first part of the shell (see Fig. IV-9 b). Reamer shell, obtained by the third method, you can define graphically (Fig. IV-10). After drawing in the drawing the shell of the original coordinate axes XY are first in rectangular coordinates scan the same part for the x-h, h ...and for the intervals Δu between them on the axis Oy (Fig. IV-10 b).More on the continuation of the abscissa of the first part of the shell lay abscissa h, h ... a₁, a₂, ... of the second part and find the scan. Take into account that the lines a₁, a₂, ... abscissa between the fold line and the axis Oh of the shell are, as in the second method approximately at right angles to the fold line (see Fig. IV-10).

From the obtained scans, it follows that one and the same flat shell at different ways of its formation has a significantly different shape of the scan (see Fig. IV-8, 9, 10). This is due to the different arrangement of threads of the fabric in the shell (see Fig. IV-7). Filament of the x-axis are arranged on straight, and the axis Oy - curved lines.

Curved the threads of the fabric in flat membranes leads to significant complications in the determination of the coordinate scan calculation and graphical methods. Therefore, the form of scans of flat membranes, it is recommended to define a support grid. In this case, the angles between the threads (network angles φ) and deformation fabric defined the same as in the bulk shells. Network determine the angles between the threads of the net with a protractor (angles φ₁, φ₂ ... Fig. IV-11).Deformation of the fabric by lines of stitches shell find by formulas (7 and 8) based on linear dimensions of the mesh or directly on measurements of the same plots of the suture line in the shell and flatten.

In all major ways of forming flat skins thread axis OU have the same slope along the straight filaments of the x-axis. Consequently, the angles φ and angles of the warp threads (φ₀ = 90' - φ) vary along the thread axis OU, along the threads of the x-axis they have a constant value.

The skew angle φ₀ is determined by the angle between polychordal AD the fold line of the fabric and a tangent AB to the curved filament (see Fig. IV-11). At the beginning of the coordinate axes, the skew angle φ₀ is equal to 0, at the end of the fold line of the fabric it has the highest value, which is determined by the formula tg(φ₀/2) = k(2f - f₁)/2a (13) or approximately (φ₀ < 30' with an accuracy of 2%) φ₀ = k(2f - f₁)/a, (14) where K - coefficient taking into account the curvature of the fold line of the fabric: for a straight bend line K = 1,0; for the curve is approximately 2.0; f - deflection curve WITH and polychord as the fold line of the fabric (Fig. IV-11 a) or the projection on the axis ox and the normal to the axis of the straight line fold of the fabric (Fig. IV-11, b); f₁ - deflection IN the axis Oy for the third method of forming the fold line.

For the first method f₁ = f φ₀ = kf/a. (14a)

For the second method f₁ = 0 φ₀ = 2kf/a (14b)

From the formula (14) it follows that the skew angle of the threads is two times less in bending of the fabric in a straight line (K = 1) than when bending it at the curve (K ≈ 2).

Therefore, in the shells with curved fold line, the skew angle can be reduced by increasing the curvature of the crease line near the x-axis and straightening it in the opposite end of the tangential curved portion (Fig. IV-11). Therefore, the skew angle of the thread changes depending on the shape of the crease line of the fabric, and the results of calculations by formulas (14) for K = 2 are approximate.

From the formula (14) and (14 b), it follows that the skew angle of φ₀ threads in the second method two times more than the first. The warp yarns in the second method can be reduced if technical conditions of cutting the axis ox at the bottom of the shell is located at the end of the crease line (Fig. IV-12 a), and in its upper part can be set with its axis And in the middle of the crease line (Fig. IV-12 b). This shows that the arc AB which defines the skew angle of the threads φ₀ while the secondary axis And two times less compared to the arc AC, which determines the skew angle φ₀ at the core of the x-axis.This is due to the rotation of the thread the OS on the angle φ₀/2, which reduces the skew angle at the other end of the fold line, i.e. at the point O (see Fig. IV-12 b). In accordance with additional axis And will be two times less than the maximum angle of misalignment of the threads in the shell, obtained by the second method. It will have approximately the same magnitude as in the first method of forming the fold line of the fabric when in this method, the axis ox is located at the end of the fold line of the shell.

From the formula (14; 14a; 14b) it follows that the third method of forming the fold line of the flat of shells gives the skew angle of the threads of the fabric more than the first and less than the second method. In this method, the skew angle is smaller in the lower part of the shell than at the top.

The third method of forming the curve of the fold line of the fabric, as well as the second method, allows the reduction of the skew angle of the filaments by moving the axes of the upper part of the shell to the middle crease line (Fig. IV-12).

With the decrease of skew angle in all cases examined, decreases the deformation of tissue along the line of its bend, the magnitude of which ∆l along the axis Oy (or the normal to the ox axis, see Fig. 11) is determined by the formula ∆l = ftgφ₀ (15) or approximately ∆l = kf(2f - f₁)/a. (16)

According to the formula (16) determines the amount of displacement of the auxiliary grid at the ends of the fold lines of the shell in the direction of the x-axis when defining scans of flat shells by using a grid. The degree of deformation of the lines of stitches, as indicated above, is determined by formulas (7 and 8) based on measurements of the minor grid. Note that in all the ways of the formation of a flat shells, you can change the amount of deformation of the fabric at the seam line due to bending of the filaments of the x-axis in those portions of the shell, where the curved thread axis OU are not associated with the axis ox (not pass through it, Fig. IV-13).Thus it is possible to reduce to zero the deformation fabric in the seam. The question of the degree of curvature of the thread is solved using an auxiliary mesh with the curvature of its threads.

The above-obtained formulas (14, 16) to determine the skew angle of the threads and the deformation of the fabric, as well as the possibilities of reducing this angle and the deformation of the fabric are also valid in the case when the curve of the bend line of the shell has a convex shape. If the bend line has a convex-concave shape, break it into several sections, the ends of which are fitted at the points of inflection. Examples of the application of the above methods of calculation of flat membranes in the design of items of clothing (sleeves, collars, details Trouser) are described below.

 

The computation of given tissue thickness

 

When presenting methods for the computation of the various shells was not taken into account the thickness of the fabric. Meanwhile, for dressing curved surfaces the thick tissues of the size of the scans may not be entirely accurate. They may be somewhat less when the calculation is carried out by measuring surface wear and more, when the measurements produced by the surface of the shell. To determine the exact size of the scan measurement of the surface necessary to produce in the middle of the fabric thickness of the shell.However, it is necessary to dress the surface of the auxiliary shell material, the thickness of which is less than half the thickness of the fabric main shell. Thus it is possible to establish correction factors for the computation of standard parts found in clothing and other products. But since these data are not yet available and cannot be set for all cases, you must consider settlement methods of definition of coordinates of the scan taking into account the thickness of the fabric.These coordinates can be determined by the previously derived formulas (4, 5, 6), and substituting in them the length of the geodesic lines of the surface, established taking into account the thickness of the fabric; or by specification of coordinates of the scan, obtained on the basis of the surface measurement using an auxiliary grid. Due to the fact that this calculation is made by the same formulae, it is sufficient to consider only one of these two ways.

To determine the coordinates of the scan taking into account the thickness of the fabric in the second method, measured on the surface using calipers or calipers the distance between the points of intersection of the axes of coordinates and lines of stitches threads of the net, combined with the surface. In other words, measured on the surface of the chord subtending the individual abscissa and ordinate on the threads of the mesh. After removing the mesh from the surface determines the coordinates of the scan on those same threads of the net.So get the length of the chords, are located in straight lines, and the length of the corresponding spatial curves, which define the coordinates of the scanner. To switch to the flat curve lines, highlight on the surface coated of a thick fabric, a narrow strip with the warp yarn or weft in the middle, having a length corresponding to any ordinate or the abscissa of the shell. Secure the ends of the strips in the plane so that the distance between them was equal to the chord of the spatial curve of the line that has the thread strip on the surface.Curving narrow strip, it obviously can be positioned at various flat curved lines and, in particular, on the arc of a circle (Fig. IV-14).

Denote by via the Rc - the radius of the arc of the middle line of the strip; x and x or u and y - the length of the strips in the outer and middle lines; δ - the thickness of the fabric; and the chord of the arc; α - the Central angle of the circular arc.

From Fig. IV-14 have a = Rcsin α/2, x = αRc, hence, eliminating Rc, we get AA = 2хѕіп α/2.

Expanding the range of sin α/2 = α/2 - α3/(8 x 31) + α⁵/(32 x 51) ... and putting two of the first member of the series in the previous equation, we find an approximation (error less α⁴/4000 with α < 1) the formula (6) α = ±√(24(x - a)/a). (17) of the same figure have x = α(Rc + δ/2), but as Rc = x/α, x = x - α δ/2.

Replacing α by the formula (17), we find formulas for the computation of given thickness of fabric: x = x ± δ/2√(24(x - a)/x); y = y ± δ/2√(24(y - a)/x) (18)

the minus Sign in front of δ take if the computation of the dimension of the outer surface of the shell, and plus - measurement of surface wear of the body.

To account for the thickness of the fabric in the computation of geodetic lines (in formulas 3, 4, 5), it is necessary to substitute in these formulas refined values of the geodesic u, s, u by the formulas similar to the above (18).

Based on the calculation according to the formula ∆h = πδ is solved a question on necessity of definition of coordinates of the scan of individual parts taking into account the thickness of the fabric. For example, when the tissue thickness of 1 mm obtained ∆h = 3,14 mm. in the distribution of this quantity for 4 - 6 parts (shelves, backs, etc.) for each item, on average, less than 1 mm, for products from the tissues is of no practical value. Coordinate scans of parts must be carried out on thick tissues with a thickness of 2 - 3 mm, where ∆h = 6 - 9 mm.

 

the computation of multilayer shells

 

The thickness of the fabric also take into account in the calculation of multilayer shells. In cases where these shells are large, the computation of the surface produced by the model made on a smaller scale. The presence of certain dependence of the coordinates of the expansions from the geodesic lines of a surface (4, 5, 6) allows the computation of shells by models based on geometric similarity.Transition rates from scans obtained on the model surface to its natural size is determined by the ratio of the sizes of a natural sweep to the size of the sweep model to the original coordinate axes ox and OU.

These dimensions determine the surface measurement of natural and model for orthogonal geodesic lines of the original coordinate axes. Coordinates natural sweep is determined by the following formulas obtained on the basis of geometric similarity xn = λx,xm, yn = λy -ym, (19) λx = Ln.n/a PoE.m or λy = uo n/uo m, (20) where xn, yn coordinates natural sweep; xm, ym are the coordinates of the scanner models that are determined by the formulas (4, 5, 6) or by means of auxiliary grid; PoE.n, IO.n and PoE.n, IO.m - orthogonal geodesic lines of the original coordinate axes of the natural surface and the model; λx is the transition rate in x-axis, λy - axis OU.

In the layered shells of the coordinates of the scan of the model xm, ym the natural surface xn, yn increases with the number of layers of fabric. Arguing in the same way as in the derivation of formula (18), the coordinates of the natural expansions of the individual layers of each part of a multilayered shell can be represented in the form of arcs of circles of a certain initial radius R increases with increasing shell layers (Fig. IV-15). When the same thickness of each layer and a constant Central angle α of the arcs of the circle xn, xn the difference between the abscissa at the transition from one layer to another will obviously be constant: ∆x = HN - HN = HN - h = ... (21) Hence, to determine the abscissa and ordinate of each layer fabrics have HN = HN + ∆x, xn = HN + ∆x ... (22) UN = UN + ∆y, an = UN + ∆u, ...

the First coordinate natural sweep in, UN are determined by formulas (19, 20).

The coordinates ∆x, ∆from the transition from one layer to another are defined the same as in the single-layer shell, with the thickness of the tissue: ∆x = h/2√(24(HN - a)/HN); ∆y = h/2√(24(an - a)/UN). (23)

Increments ∆x and ∆y can also be determined experimentally by manufacturing a multilayer shell on the surface of the model. In this case, in the manufacturing process of the shell surface every few layers of tissue determine the coordinates of the scan; subtract from the coordinates of the following layers of the shell the coordinates of the previous layers; find the corresponding increments ∆x, ∆y. Adding these increments to the corresponding natural coordinates of the scan, get scan individual layers of genuine shell.

When designing multilayer shells developable surfaces (cylinder, cone) the increments ∆x, ∆u accept the same for all coordinates of the scanner this part of the shell. On prosverlivayut surfaces to separate coordinate scan each part of a multilayered shell separately it is necessary to determine their increments ∆x, ∆y. Some multi-layered shell have different thickness layers of fabric, they consist of main parts, covering the entire surface, and additional parts, creating a thickening of the membranes (Fig. IV-16).The computation of such shells produced in the manufacturing process of the shell on the surface of the model. The definition of natural nets the first of the main and supplementary parts of the shell is performed by formulas (19, 20, 23), taking the coordinatesxm, ym, and the increments ∆x, ∆y in scan of the shell model. After the formation of the uneven layer shell the coordinates of the scan (natural and model) changes due to the fact that the Central angles α₁, α₂ arcs of a circle to the transformed coordinates of the scan differed from the original angle α (see Fig. IV-16)**. However, the chords of these arcs preserve the original length (a and λxa). In addition, it is possible to determine the deflection of these arcs (f₁ and f₂). Due to this, the natural coordinates of the scanner can be determined by approximate formula arc length of circle:

(24)

The same equation for the coordinates of the scanner models have

 

in addition, accurate formulas,

 

where F is the thickness of the layers additional parts of the sheath which is determined by measurement on the model surface.

Solving the last three equations together, we find

(24)

similarly we find for the determination of y

(25) (25, a)

the Calculation of these formulas, as in the previous work on the measurement of the outer surface of the layer is determined by the scan. Moreover, the formulas do not include the thickness of the first layer (h/2), in order to have a small allowance in the scan in case of inaccuracy of the calculation.

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* E. H. Friedland. About the dressing of the surfaces with a cloth, proceedings of MTILP, sat 24, 1963.

** In this case, as in the previous, spatial curves the coordinates represent the to find the method in the form of flat curves arcs of a circle of a certain radius R.

 

Literature

 

1. Chebyshev, P. L. collected works, vol V, publishing house of as USSR, 1951.

5. Suboticki A.V., Lermontov, D. B. Determination of coordinates of the scan shells of fabrics for different surfaces, Izvestiya vuzov, Technology of light industry, 1966, № 4.

6. Suboticki A. V., Melikov, E. H. the Design of shells made of woven materials, journal of applied physics, Technology of light industry, 1961, No. 2.