Method for manufacturing an eccentric transition between pipes. Methodological recommendations for the course of descriptive geometry Development eccentric transition calculation guide

The main dimensions of a round cone transition (Fig. 129) are: D-diameter of the lower base; d-diameter of the upper base; h - the height of the transition and the opening angle of the transition, which is formed from the intersection of the side faces of the side view of the transition as they continue.

Rice. 129. Development of full and truncated cones

The opening angle in transitions is assumed to be 25-35°, unless there are special instructions in the drawings.

At an opening angle of 25-35°, the transition height is approximately 2 (D-d).

Transitions from round to circular cross-section can have accessible and inaccessible vertices. In the first case, the lateral edges of the lateral type of transition intersect within the sheet when they are continued, in the second case - beyond its boundaries.

The production of a transition from a round to a round section begins with the construction of a development and cutting of individual elements of the transition.

Let's consider techniques for constructing a scan of conical transitions, which are a truncated cone.

A complete cone is the body shown in Fig. 129,a, with base diameter D and top diameter O.

If you roll a cone on a plane around the vertices O, you will get a trace, which will be the development of the cone. The length of the arc constituting the trace of the circle of the base of the cone with diameter D is equal to D, and the radius of size R is equal to the length of the side generatrix of the cone 1.

Unfolding a forward transition with an accessible vertex. If we cut the cone parallel to the base, we will get a truncated cone (Fig. 129, b).

To draw the development of a truncated cone, we construct its side view (ABVG in Fig. 129, c) according to the diameter of the lower base D = 320 mm, the upper base d = 145 mm and the height h = 270 mm given for this example.

To construct a scan, we continue lines AG and BV until they intersect at point O (Fig. 129, c). If the construction is done correctly, then point O must be located on the center line.

We place a compass at point O and draw two arcs: one through point A and the other through point D; from an arbitrary point B 1 on the lower arc we plot the circumference of the base of the cone, which is determined by multiplying the diameter D by 3.14. Points B 1 and H are connected to vertex O. Figure D 1 B 1 HH 1 will be the development of a truncated cone. To the resulting development we add allowances for folds, as shown in the figure.

The above method of constructing the development of a truncated cone is possible provided that the lateral generatrices AG and BV, when extended, intersect at an accessible distance from the base of the cone, i.e., at an accessible vertex of the cone.

Development of a direct transition with an inaccessible vertex. If the diameter of the upper circle of the cone differs little in size from the diameter of the lower circle, then straight lines AG and BV will not intersect within the picture. In such cases, approximate constructions are used to draw the development.

One of the simplest methods for approximate construction of the sweep of a transition with a small taper is the method of L.A. Laptop.

For example, let us construct a transition scan with a height h = 750 mm, a diameter of the lower base D = 570 mm, and a diameter of the upper base d = 450 mm. To determine the height of the development I, we draw a side view of the transition according to the given dimensions, as shown in Fig. 130, a. The length I of the lateral generatrix of the lateral view of the transition will be the height of the scan. The construction of the sweep of this transition according to the method of L. A. Lapshov (Fig. 130, b) is carried out as follows.

Rice. 130. Development of a circular cross-section transition according to the method of L. A. Lapshov

First, we determine the approximate dimensions of the development, so that when drawing the development, it is possible to correctly position it on the sheets of roofing steel in order to reduce waste and save materials. To do this, we calculate the width of the transition sweep at the lower and upper bases.

The width of the development at the lower base is 3.14 x D = 3.14 x 570 = 1,790 mm, the width of the development at the upper base is 3.14 x d = 3.14 x 450 = 1,413 mm.

Since the width of the development is greater than the length of the sheet (1,420 mm), and the height is greater than the width of the sheet (710 mm), the picture for the transition along the length and width will be composed of a sheet with extensions.

The total width of the picture with allowances for folds (single closing fold 10 mm wide and intermediate double fold 13 mm wide) will be equal to 1,790 + 25 + 43 = 1,858 mm.

To construct a scan in the picture, we draw the O-O axis at a distance of approximately 930 mm from the edge (1,858:2). At a distance of 20 mm from the bottom edge of the sheet, we set aside the height of the scan, the size of which we take from the side view, and find points A and B, as shown in Fig. 130, b. Points A and B will be the extreme points of the transition sweep axis. From point B to the left on a line perpendicular to it, draw a segment equal to 0.2 (D - d), find point B and connect it. straight line with point A. In our example, this segment is equal to 0.2 (570 - 450) = 24 mm. This value is a correction for the accuracy of the marking and is determined practically. From points A and B we draw perpendicular lines to the left and put the values ​​3.14 on them. x d / 8 and 3.14 x D / 8, i.e. 1/8 of the sweep, we get points 3, 3 1, which we connect with a straight line three more times to the left along 1/8 of the transition sweep and get the left half. transition sweep.

We construct the curves forming the upper and lower sweep arcs using a square and a ruler, as shown in Fig. 130, b.

To the resulting curves we add the width of the flange to the flanges and cut the cutting line with scissors

Then we bend the cut part of the material to the right side of the development according to the template (shaded in the figure) and cut off the excess material. To the resulting development we add an allowance for the longitudinal closing fold.

Development of an oblique transition of circular cross-section. An oblique transition is one in which the centers of the upper and lower bases lie on different axes in one or two planes. The distance between these axes is called the center offset.

Oblique transitions of circular cross-section are used to connect a round fan intake opening with round-section air ducts if their centers lie on different axes.

The development of an oblique transition of a circular cross-section, the surface of which is the lateral surface of a truncated cone, is performed by dividing the entire surface of the oblique transition into auxiliary triangles.

Let us need to construct a development of an oblique transition with a height of H = 400 mm; diameter of the lower base D = 600 mm; diameter of the upper base d = 280 mm; displacement of centers in one plane / = 300 mm.

We build a side view of the oblique transition (Fig. 131, a). To do this, set aside the line AB = 600 mm. From the center of this line - the lower base of the cone - we draw the O 1 -O 1 axis and plot the height H = 400 mm on it. From the top point of height H, draw a horizontal line and mark the offset size on it to the left - 300 mm, find the center O - the upper base. From center O we lay off 140 mm to the left and to the right - half the diameter of the upper base - and find the extreme points B and D. We connect points A and B, B and D with straight lines and get a side view of the oblique transition AVGB.

Rice. 131. Development of an oblique transition of a circular cross-section with displacement of the centers of the upper and lower bases in the same plane

To construct a development of half of the transition, we divide its surface into a number of auxiliary triangles.

To do this, we divide the large and small semicircles, each into 6 equal parts, and the division points of the small semicircle are designated by numbers 1", 3", 5", 7", 9", 11" and 13", and the division points of the large semicircle by numbers 1 ", 3", 5", 7", 9", 11" and 13",

Connecting points 1"-1", 1"-3", 3"-3", 3"-5", etc., we get lines 1 1, 2 1, 3 1, 4 1, 5 1, 6 1 , 7 1, 8 1, 9 1, 10 1, 11 1, 12 1 and 13 1, which divide the side surface of half of the transition into auxiliary triangles, on three sides of which there are 1"-1", 1"-3" And 3"-1", etc. - you can construct a development of these triangles.

In these triangles, the only true dimensions on the plan are sides 1"-3", 3"-5", 1"-3", 3"-5", etc.

The sides of the triangles, indicated on the plan by lines under the numbers 1 1, 2 1, 3 1, 4 1, etc., are not true quantities, and therefore are depicted on the plan in abbreviated form (projections).

The true values ​​of these sides will be the hypotenuses of a right triangle, in which one leg is equal to the transition height H, and the other leg is equal to the dimensions of the lines 1 1, 2 1, 3 1, 4 1, 5 1, etc. (Fig. 131, e).

To determine the true values ​​of these lines, we build a series of right-angled triangles with legs a-b equal to H, and legs b - 1 1, b - 2 1, b - 3 1, b - 4 1, etc., equal to lines 1 1, 2 1, 3 1, 4 1, etc. In these triangles (Fig. 131, c) we find the lengths of the hypotenuses 1, 2, 3, 4, etc.

In order not to obscure the construction, the dimensions of lines with odd numbers 1 1, 3 1, 5 1, etc. are placed on one side of leg b-a, and with even numbers 2 1, 4 1, etc. - on the other side leg b-a.

We construct the development of half of the oblique transition as follows (Fig. 131, d).

We draw the center line O-O and on it we lay a line 1"-1", equal to hypotenuse 1. From point 1" with a radius equal to 1"-3", we draw a notch with a compass, and from point 1" with a radius equal to hypotenuse 2, we draw Compass another notch and find point 3". Triangle 1" 1" 3" will be the first triangle of the scan. In the same way, a second triangle is attached to it along sides 1"-3" and hypotenuse 3. The remaining triangles are constructed in the same way. The resulting points 1", 3", 5", etc., as well as points 1", 3", 5", etc., are connected by smooth curves, as shown in the figure.

To the resulting contour of the development of half of the oblique transition, allowances for folds and flanges are added.

Using this pattern, the second symmetrical half of the pattern is cut out.

Development of an oblique transition with displacement of the centers of the upper and lower bases in two planes. Suppose we need to construct a scan of an oblique transition having a center offset in the horizontal plane e = 300 mm and a center offset in the vertical plane e 1 = 150 mm; diameter of the lower base D = 700 mm; diameter of the upper base d = 400 mm; height H = 400 mm.

We build a side view, as described above (Fig. 132, a).

Rice. 132. Side view and plan of an oblique transition of a circular cross-section with offset centers of the upper and lower bases in two planes

To build a plan (Fig. 132, b) we proceed as follows.

We build a rectangle with a horizontal side equal to 300 mm (displacement e) and a vertical side equal to 150 mm (displacement e 1). We place the horizontal side of the rectangle between the axes of the upper and lower bases, as shown in Fig. 132, b.

The centers of the upper and lower bases of the oblique transition with an offset in two planes will be located at the vertices of the opposite corners of the rectangle along the diagonal. We draw the O-O axis on this diagonal and build a plan for half of the oblique transition on it. Dividing the plan into separate triangles and constructing a development is performed in the same way as for an oblique transition with an offset in one plane.

After making the transitions, flanges are placed on them, as indicated above.

The invention relates to metal forming and can be used in the manufacture of eccentric transitions between large-diameter pipes in the production of heat exchangers. A straight cone blank is obtained, from which a truncated eccentric cone blank is formed with bases of small and large diameters and a conical surface, one of the lines of which is perpendicular to the bases. The formation of a truncated eccentric cone blank is carried out by trimming the ends of a straight cone blank. An eccentric transition is obtained by flanging large and small diameters using a punch and a matrix. Moreover, for flanging a small diameter, the workpiece of a truncated eccentric cone is placed vertically with the small diameter upwards, the matrix is ​​placed around the small diameter with its inner surface touching the outer surface of the workpiece at at least four points, the punch is advanced inside the small diameter of the workpiece parallel to the line on the conical surface perpendicular to the bases. For large-diameter flanging, a punch is used instead of a matrix and, accordingly, a matrix is ​​used instead of a punch. Technological capabilities are expanding. 7 ill.

Drawings for RF patent 2492016

The invention relates to metal forming and can be used in the manufacture of eccentric transitions between large-diameter pipes in the production of heat exchangers.

There is a known method for producing pipes in cold pipe rolling mills, according to which a pre-prepared initial hollow billet is fed along the rolling axis by a certain amount (feed amount) into the deformation zone and is compressed by rotating rolls with a variable radius of the stream while simultaneously moving the rolling stand (direct movement of the stand) in direction of billet supply (Technology and equipment for pipe production; textbook for universities / V.Ya. Osadchiy, A.S. Vavilin, V.G. Zimovets, A.P. Kolikov. - M.: Intermetingzhiniring, 2007. - p. 448-452). In the final (extreme) position of the stand, the strands of the rolls form a caliber, the size of which ensures the free passage of the workpiece through it (the idle section of the longitudinal development of the strand profile). At this moment, the workpiece with the mandrel is rotated around its axis at a given angle (turned), after which the rolling stand moves in the opposite direction to its original position (reverse stroke of the stand) with simultaneous deformation of the workpiece section that was previously compressed during the forward stroke of the stand. Next, the workpiece is bent again and the above-described cycle of processing the workpiece on the mandrel is repeated many times until a finished pipe is obtained.

The described method of rolling pipes involves metal deformation using interchangeable tools and equipment in the form of gauges, gears and racks, made up of pairs of absolutely identical parts, which creates symmetry of the deformation process relative to the horizontal plane. In this process, the mandrel is self-aligned in the radial direction relative to the inner diameter of the pipe, which does not significantly reduce the value of the eccentric component of the wall difference and reduces the accuracy of cold-deformed pipes obtained by this method. In addition, high rolling forces, which require an increase in the mass of the deforming equipment and cause large elastic deformations of the stand, also lead to a decrease in the accuracy of the finished pipes, including those made of difficult-to-deform steels and alloys.

The closest to the proposed method is the method of manufacturing pipes with an eccentric transition by relative displacement of sections of a tubular blank with a conical transition, according to which a cylindrical section with a smaller diameter is rigidly fixed, and an internal support is created on a cylindrical section with a larger diameter, then it and the conical transition are sequentially bent relative to a cylindrical section with a smaller diameter (USSR Author's Certificate No. 806210, published 02/23/1981 - prototype).

The known method can only be applied to pipes of small diameter and does not allow the production of transitions of large diameter, i.e. diameter more than 1 m.

The problem is solved by the fact that in the method of manufacturing an eccentric transition, including obtaining a blank of a straight cone, forming from it a blank of a truncated eccentric cone with bases of small and large diameters and a conical surface, one of the lines of which is perpendicular to the bases, according to the invention, the formation of a blank of a truncated eccentric cone is carried out by cutting the ends of the workpiece of a straight cone, which is placed with a large diameter downwards, tilted until one line is taken on its conical surface in a vertical position, from the top point of which a horizontal line is drawn, along which the upper part of the workpiece of a straight cone is cut off, and its lower part is cut off along a horizontal line, drawn from the top point of the large base, raised when tilted, the eccentric transition is obtained by flanging large small diameters using a punch, and for flanging a small diameter, the workpiece of a truncated eccentric cone is placed vertically with the small diameter upward, the matrix is ​​placed around the small diameter with its inner surface touching the outer surface of the workpiece at no less than four points, the punch is advanced inside the small diameter of the workpiece parallel to a line on the conical surface perpendicular to the bases, and for flanging of a large diameter, a punch is used instead of a matrix and, accordingly, a matrix is ​​used instead of a punch.

The essence of the invention

In the field of mechanical engineering, and more precisely in the field of manufacturing heat exchangers, today there is the task of manufacturing eccentric transitions between large-diameter pipes with flanged ends. This task, as a rule, is either not performed or is performed using workaround technologies that damage the structure of the metal. Existing equipment is not adapted to solve these particular problems, and enterprises that have this equipment are often still forced to resort to workaround technologies when fulfilling orders.

The proposed invention solves the problem of manufacturing an eccentric transition of large diameter.

A general view of the eccentric transition is shown in Fig. 1. In Fig.1, d is the small diameter of the transition, D is the larger diameter of the transition, I is the length of the cylindrical part of the transition of small diameter, L is the length of the cylindrical part of the transition of a larger diameter, S is the thickness of the transition wall, H is the length of the transition.

When manufacturing an eccentric transition, a reamer is made, from which a blank for the future transition is subsequently produced by sharp welding and shaping.

The workpiece is bent using a 3-roll machine. Figure 2 shows a diagram of the bending of the transition blank on a three-roll machine: 1 - transition blank, 2 - ends of the blank, 3 - rolls. After bending, the workpiece is butt welded at the ends 3. A straight cone of the workpiece is obtained. Next, begin trimming the ends of the workpiece. For ease of marking when cutting the ends of a straight cone, self-aligning construction laser levels are used to mark the vertical and horizontal planes simultaneously. Figure 3 shows the marking diagram: 4 - self-aligning construction laser levels, 5 - vertical planes, 6 - horizontal planes, 7 - cone generatrix. A straight cone 8 is placed with a large diameter at the bottom. The straight cone is tilted so that one line on the surface of the cone 8 takes a vertical position in the vertical plane. From the top point “A” of the vertical line, draw a horizontal line 9. Along this line, cut off the upper part of the cone 8. From the top point “B” on the larger diameter of the cone 8, which turns out to be raised when tilted, draw a horizontal line 10. Along this line, cut off the lower part cone 8. An eccentric cone 11 is obtained. Thus, an eccentric cone blank is made from a simple truncated cone, taking into account allowances by cutting off part of a straight cone. As a result, we obtain a flattened workpiece of the eccentric transition.

Prepare the matrix and punch for each end of the transition based on the thickness of the matrix and punch of at least 5 sizes of the cylindrical part of the transition I and L. For a small diameter of the transition, the preparation diagram is shown in Fig. 4 and 5, for a larger diameter of the transition - in Fig. 6 and 7.

In Figs 4 and 5 the following are indicated: 11 - eccentric cone, 12 - matrix, 13 - stops, 14 - zones for removing weld reinforcement, 15 - punch, 16 - cup, 17 - press. The width of the matrix 12 and the punch 15 is assigned to at least 3 corresponding thicknesses of the transition wall S. The matrix 12 and the punch 15 are equipped with devices for carrying out lifting operations (not shown). For a small transition diameter d, the diameter of the punch 15 is selected as nominal with a tolerance of + for the tolerance for changes in the thickness of the transition metal, the diameter of the matrix 12 is calculated from the diameter of the punch, +2 wall thickness S, +2 tolerance for wall thickness, +1.5 mm. For a larger transition diameter D, the main one is the matrix, and the derivative is the punch (the matrix diameter is chosen at nominal value with a tolerance in - for the tolerance for changes in wall thickness, the punch diameter is calculated from the matrix diameter, - 2 wall thickness, - 2 tolerances for wall thickness, - 1, 5 mm). The roughness of the working surfaces of the punch and matrix is ​​at least grade 11.

Figures 6 and 7 indicate: 11 - eccentric cone, 14 - zones for removing weld reinforcement, 17 - press, 18 - matrix, 19 - stops, 20 - punch, 21 - cup.

Prepare the equipment.

On the punch 15 for a small diameter and, accordingly, for a matrix with a larger diameter 18, after the radius of curvature of the inlet part there is a section with a slope of 20°±1°, the cylindrical part of the punch 15 or matrix 18 is at least half of their thickness. For a small diameter transition for the punch 15, a glass 16 is used for attachment to the press 17 with the possibility of removing the punch from the glass. The height of the cup 16 is calculated from the condition of the transition length H with tolerances + 3 punch thickness. For matrix 12, at least 3 stands are prepared with a height of 3 matrix thicknesses.

For a larger transition diameter, a glass 21 is prepared for the matrix and a stand for the punch 20 similarly to the above (Fig. 6).

A small diameter transition is flanged. To do this, the weld reinforcements in the stamping zone 14 are first removed (Fig. 4, Fig. 6). A matrix 12 is placed on a vertically mounted eccentric cone 11 with a small diameter upwards and installed in the working position, i.e. position of matrix 12 during stamping. The oval of small diameter is expanded in the area of ​​the smaller axis, and the oval is adjusted to the circle. For expansion, a hydraulic set is used, for example, for straightening bodies with a maximum force of at least 3 tons and a set of extensions. Ensure that the inner surface of the matrix 12 and the outer surface of the eccentric cone 11 touch at least 4 points “B” (for flanging of a larger diameter “D”). Check the largest gap between the matrix 12 and the eccentric cone 11. This affects the size of the gap between the lower plane of the matrix and the welded stops. From below, under the matrix, 4 stops are welded to the eccentric cone diametrically in 2 perpendicular planes with a gap equal to the thickness of the transition wall + half the maximum gap (Fig. 4). For stamping, a press with a maximum force of at least 100 tons and a span height capable of placing a pre-assembled structure under the working cylinder is used. Under the working cylinder, a structure is assembled from a matrix 12 on supports and an eccentric cone 11 with a glass 16 inserted into it with an installed punch 15. The glass 16 is attached to the platform of the working cylinder of the press 17. Lubricant (graphite or a mixture) is applied to the punch 15 and the inner surface of the eccentric cone 11 graphite powder and industrial oil or a mixture of talc and liquid soap). Turn on the press 17, move the punch 15 inside the eccentric cone 11 parallel to the vertical plane 5. Upon completion of stamping, remove the punch 15 from the glass 16 and the eccentric cone 11 from the glass 16. Flanging of the small diameter is completed.

For flanging of a larger diameter, preparatory operations are performed similar to those for flanging of a small diameter, with a difference in the operations for the punch and the matrix (instead of the matrix - a punch and, accordingly, instead of a punch - a matrix) (Fig. 6). Subsequently, flanging of a larger diameter is performed (Fig. 7).

Example of concrete execution

A transition with a flange of 1100-1600×12 mm is made. The flange size is 40 mm at both ends. According to Fig.1 d=1100 mm, D=1600 mm, I=L=40 mm, S=12 mm, r=R=20 mm, H=1500 mm.

The operations are performed according to Figs. 1-7. An eccentric transition with high surface quality is obtained.

Application of the proposed method will make it possible to perform an eccentric transition of a larger diameter.

CLAIM

A method for manufacturing an eccentric transition for connecting large-diameter pipes, including obtaining a straight cone blank, forming from it a truncated eccentric cone blank with bases of small and large diameters and a conical surface, one of the lines of which is perpendicular to the bases, characterized in that the formation of a truncated eccentric cone blank is carried out by cutting the ends of the straight cone workpiece, which is placed with a large diameter downwards, tilted until one line is taken on its conical surface in a vertical position, the upper part of the straight cone workpiece is cut off along a horizontal line drawn from the top point of the vertical line of the conical surface, and its lower part is cut off along the horizontal line drawn from the top point of the large base, raised when tilted, flanging of large and small diameters is carried out using a punch and a matrix, and for flanging a small diameter, a truncated eccentric cone blank is placed vertically with the small diameter upward, the matrix is ​​placed around the small diameter touching its inner surface the outer surface of the workpiece at at least four points, the punch is advanced inside the small diameter of the workpiece parallel to a line on the conical surface perpendicular to the bases, and for flanging a large diameter, a punch is used instead of a matrix and, accordingly, instead of a punch, a matrix is ​​used.

We often encounter surface developments in everyday life, in production and in construction. To make a case for a book (Fig. 169), sew a cover for a suitcase, a tire for a volleyball, etc., you must be able to construct developments of the surfaces of a prism, ball and other geometric bodies. A development is a figure obtained by combining the surface of a given body with a plane. For some bodies, scans can be accurate, for others they can be approximate. All polyhedra (prisms, pyramids, etc.), cylindrical and conical surfaces, and some others have precise developments. Approximate developments have a ball, a torus and other surfaces of revolution with a curved generatrix. We will call the first group of surfaces developable, the second - non-developable.

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When constructing developments of polyhedra, you will have to find the actual size of the edges and faces of these polyhedra using rotation or changing projection planes. When constructing approximate developments for non-developable surfaces, it will be necessary to replace sections of the latter with developable surfaces close in shape to them.

To construct a scan of the lateral surface of the prism (Fig. 170), it is assumed that the scan plane coincides with the face AADD of the prism; other faces of the prism are aligned with the same plane, as shown in the figure. The face ССВВ is preliminarily combined with the face ААВВ. Fold lines in accordance with GOST 2.303-68 are drawn with thin solid lines with a thickness of s/3-s/4. Points on the scan are usually denoted by the same letters as on the complex drawing, but with index 0 (zero). When constructing a development of a straight prism according to a complex drawing (Fig. 171, a), the height of the faces is taken from the frontal projection, and the width from the horizontal one. It is customary to build a scan so that the front side of the surface is facing the observer (Fig. 171, b). This condition is important to observe because some materials (leather, fabrics) have two sides: front and back. The bases of the ABCD prism are attached to one of the faces of the side surface.

If point 1 is specified on the surface of the prism, then it is transferred to the development using two segments marked on the complex drawing with one and two strokes, the first segment C1l1 is laid to the right of point C0, and the second segment is laid vertically (to point l0).

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Similarly, a development of the surface of the cylinder of rotation is constructed (Fig. 172). Divide the surface of the cylinder into a certain number of equal parts, for example 12, and unfold the inscribed surface of a regular dodecagonal prism. The sweep length with this construction turns out to be slightly less than the actual sweep length. If significant accuracy is required, then a graphic-analytical method is used. The diameter d of the circumference of the base of the cylinder (Fig. 173, a) is multiplied by the number π = 3.14; the resulting size is used as the development length (Fig. 173, b), and the height (width) is taken directly from the drawing. The bases of the cylinder are attached to the development of the side surface.

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If point A is given on the surface of the cylinder, for example, between the 1st and 2nd generatrices, then its place on the development is found using two segments: a chord marked with a thick line (to the right of point l1), and a segment equal to the distance of point A from the upper base of the cylinder , marked in the drawing with two strokes.

It is much more difficult to construct the development of a pyramid (Fig. 174, a). Its edges SA and SC are straight lines in general position and are projected onto both projection planes by distortion. Before constructing the development, it is necessary to find the actual value of each edge. The size of the edge SB is found by constructing its third projection, since this edge is parallel to the plane P3. The ribs SA and SC are rotated around a horizontally projecting axis passing through the vertex S so that they become parallel to the frontal plane of projections P (the actual value of the rib SB can be found in the same way).

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After such a rotation, their frontal projections S 2 A 2 and S 2 C 2 will be equal to the actual size of the ribs SA and SC. The sides of the base of the pyramid, like horizontal straight lines, are projected onto the projection plane P 1 without distortion. Having three sides of each face and using the serif method, it is easy to construct a development (Fig. 174, b). Construction begins from the front face; a segment A 0 C 0 = A 1 C 1 is laid out on a horizontal straight line, the first notch is made with a radius A 0 S 0 - A 2 S 2 the second - with a radius C 0 S 0 = = G 2 S 2 ; at the intersection of the serifs, point S„ is obtained. Accept the order side A 0 S 0 ; from point A 0 make a notch with radius A 0 B 0 =A 1 B 1 from point S 0 make a notch with radius S 0 B 0 =S 3 B 3 ; at the intersection of the serifs, point B 0 is obtained. Similarly, the face S 0 B 0 C 0 is attached to the side S 0 G 0 . Finally, the base triangle A 0 G 0 S 0 is attached to side A 0 C 0 . The lengths of the sides of this triangle can be taken directly from the development, as shown in the drawing.

The development of a cone of rotation is constructed in the same way as the development of a pyramid. Divide the circumference of the base into equal parts, for example into 12 parts (Fig. 175, a), and imagine that a regular dodecagonal pyramid is inscribed in the cone. The first three faces are shown in the drawing. The surface of the cone is cut along the generatrix S6. As is known from geometry, the development of a cone is represented by a sector of a circle whose radius is equal to the length of the cone generatrix l. All generatrices of a circular cone are equal, therefore the actual length of the generatrix l is equal to the frontal projection of the left (or right) generatrix. From the point S 0 (Fig. 175, b) a segment of 5000 = l is laid vertically. An arc of a circle is drawn with this radius. From the point O 0, the segments Ol 0 = O 1 l 1, 1 0 2 0 = 1 1 2 1, etc. are laid off. By setting aside six segments, we get point 60, which is connected to the vertex S0. The left part of the scan is constructed in the same way; The base of the cone is attached below.

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If you need to put point B on the scan, then draw the generatrix SB through it (in our case S 2), apply this generatrix to the scan (S 0 2 0); rotating the generatrix with point B to the right until it aligns with the generatrix S 3 (S 2 5 2), find the actual distance S 2 B 2 and set it aside from the point S 0. The found segments are marked on the drawings with three strokes.

If it is not necessary to plot points on the cone scan, then it can be constructed faster and more accurately, since it is known that the scan sector angle is a=360°R/l, the radius of the base circle, and l is the length of the cone generatrix.

You will need

• Pencil Ruler square compass protractor Formulas for calculating angles using arc length and radius Formulas for calculating sides of geometric figures

Instructions

On a sheet of paper, build the base of the desired geometric body. If you are given a parallelepiped or, measure the length and width of the base and draw a rectangle on a piece of paper with the appropriate parameters. To construct a development a or a cylinder, you need the radius of the base circle. If it is not specified in the condition, measure and calculate the radius.

Consider a parallelepiped. You will see that all its faces are located at an angle to the base, but the parameters of these faces are different. Measure the height of the geometric body and, using a square, draw two perpendiculars to the length of the base. Plot the height of the parallelepiped on them. Connect the ends of the resulting segments with a straight line. Do the same on the opposite side of the original one.

From the intersection points of the sides of the original rectangle, draw perpendiculars to its width. Plot the height of the parallelepiped on these straight lines and connect the resulting points with a straight line. Do the same on the other side.

From the outer edge of any of the new rectangles, the length of which coincides with the length of the base, construct the top face of the parallelepiped. To do this, draw perpendiculars from the intersection points of the length and width lines located on the outside. Set aside the width of the base on them and connect the points with a straight line.

To construct a development of a cone through the center of the base circle, draw a radius through any point on the circle and continue it. Measure the distance from the base to the top of the cone. Set aside this distance from the intersection point of the radius and the circle. Mark the vertex point of the side surface. Using the radius of the side surface and the length of the arc, which is equal to the circumference of the base, calculate the sweep angle and set it aside from the straight line already drawn through the top of the base. Using a compass, connect the previously found intersection point of the radius and the circle with this new point. The cone scan is ready.

To construct the development of a pyramid, measure the heights of its sides. To do this, find the middle of each side of the base and measure the length of the perpendicular drawn from the top of the pyramid to this point. Having drawn the base of the pyramid on a sheet of paper, find the midpoints of the sides and draw perpendiculars to these points. Connect the resulting points with the intersection points of the sides of the pyramid.

The development of a cylinder consists of two circles and a rectangle located between them, the length of which is equal to the length of the circle, and the height is the height of the cylinder.

The development of the surface of a cone is a flat figure obtained by combining the side surface and base of the cone with a certain plane.

Options for constructing a sweep:

Development of a right circular cone

The development of the lateral surface of a right circular cone is a circular sector, the radius of which is equal to the length of the generatrix of the conical surface l, and the central angle φ is determined by the formula φ=360*R/l, where R is the radius of the circle of the base of the cone.

In a number of problems of descriptive geometry, the preferred solution is to approximate (replace) a cone with a pyramid inscribed in it and construct an approximate development, on which it is convenient to draw lines lying on the conical surface.

Construction algorithm

1. We fit a polygonal pyramid into a conical surface. The more lateral faces an inscribed pyramid has, the more accurate the correspondence between the actual and approximate development.
2. We construct the development of the lateral surface of the pyramid using the triangle method. We connect the points belonging to the base of the cone with a smooth curve.

Example

In the figure below, a regular hexagonal pyramid SABCDEF is inscribed in a right circular cone, and the approximate development of its lateral surface consists of six isosceles triangles - the faces of the pyramid.

Consider the triangle S 0 A 0 B 0 . The lengths of its sides S 0 A 0 and S 0 B 0 are equal to the generatrix l of the conical surface. The value A 0 B 0 corresponds to the length A’B’. To construct a triangle S 0 A 0 B 0 in an arbitrary place in the drawing, lay off the segment S 0 A 0 =l, after which from points S 0 and A 0 we draw circles with radius S 0 B 0 =l and A 0 B 0 = A'B' respectively. We connect the intersection point of circles B 0 with points A 0 and S 0.

We construct the faces S 0 B 0 C 0 , S 0 C 0 D 0 , S 0 D 0 E 0 , S 0 E 0 F 0 , S 0 F 0 A 0 of the pyramid SABCDEF similarly to the triangle S 0 A 0 B 0 .

Points A, B, C, D, E and F, lying at the base of the cone, are connected by a smooth curve - an arc of a circle, the radius of which is equal to l.

Inclined cone development

Let's consider the procedure for constructing a scan of the lateral surface of an inclined cone using the approximation (approximation) method.

Algorithm

1. We inscribe the hexagon 123456 into the circle of the base of the cone. We connect points 1, 2, 3, 4, 5 and 6 with the vertex S. The pyramid S123456, constructed in this way, with a certain degree of approximation is a replacement for the conical surface and is used as such in further constructions.
2. We determine the natural values ​​of the edges of the pyramid using the method of rotation around the projecting line: in the example, the i axis is used, perpendicular to the horizontal projection plane and passing through the vertex S.
   Thus, as a result of the rotation of edge S5, its new horizontal projection S’5’ 1 takes a position in which it is parallel to the frontal plane π 2. Accordingly, S’’5’’ 1 is the actual size of S5.
3. We construct a scan of the lateral surface of the pyramid S123456, consisting of six triangles: S 0 1 0 6 0 , S 0 6 0 5 0 , S 0 5 0 4 0 , S 0 4 0 3 0 , S 0 3 0 2 0 , S 0 2 0 1 0 . The construction of each triangle is carried out on three sides. For example, △S 0 1 0 6 0 has length S 0 1 0 =S’’1’’ 0 , S 0 6 0 =S’’6’’ 1 , 1 0 6 0 =1’6’.

The degree to which the approximate development corresponds to the actual one depends on the number of faces of the inscribed pyramid. The number of faces is chosen based on the ease of reading the drawing, the requirements for its accuracy, the presence of characteristic points and lines that need to be transferred to the development.

Transferring a line from the surface of a cone to a development

Line n lying on the surface of the cone is formed as a result of its intersection with a certain plane (figure below). Let's consider the algorithm for constructing line n on a scan.

Algorithm

1. We find the projections of points A, B and C at which line n intersects the edges of the pyramid S123456 inscribed in the cone.
2. We determine the natural size of the segments SA, SB, SC by rotating around the projecting straight line. In the example under consideration, SA=S’’A’’, SB=S’’B’’ 1 , SC=S’’C’’ 1 .
3. We find the position of points A 0 , B 0 , C 0 on the corresponding edges of the pyramid, plotting on the scan the segments S 0 A 0 =S''A'', S 0 B 0 =S''B'' 1, S 0 C 0 =S''C'' 1 .
4. We connect points A 0 , B 0 , C 0 with a smooth line.

Development of a truncated cone

The method described below for constructing the development of a right circular truncated cone is based on the principle of similarity.