There are 2 ways to build a cone sweep:
- Divide the base of the cone into 12 parts (we enter a regular polyhedron - a pyramid). You can divide the base of the cone into more or less parts, because. the smaller the chord, the more accurate the construction of the sweep of the cone. Then transfer the chords to the arc of the circular sector.
- Construction of a sweep of the cone, according to the formula that determines the angle of the circular sector.
Since we need to plot the intersection lines of the cone and the cylinder on the development of the cone, we still have to divide the base of the cone into 12 parts and inscribe the pyramid, so we will immediately go along 1 way to build the development of the cone.
Algorithm for constructing a sweep of a cone
- We divide the base of the cone into 12 equal parts (we enter the correct pyramid).
- We are building side surface cone, which is a circular sector. The radius of the circular sector of the cone is equal to the length of the generatrix of the cone, and the length of the arc of the sector is equal to the circumference of the base of the cone. We transfer 12 chords to the arc of the sector, which will determine its length, as well as the angle of the circular sector.
- We attach the base of the cone to any point of the arc of the sector.
- Through the characteristic points of intersection of the cone and the cylinder we draw generators.
- Find the natural size of the generators.
- We build data generators on the development of the cone.
- We connect the characteristic points of intersection of the cone and the cylinder on the sweep.
More details in the video tutorial on descriptive geometry in AutoCAD.
During the construction of the sweep of the cone, we will use the Array in AutoCAD - a Circular array and an array along the path. I recommend watching these AutoCAD video tutorials. The AutoCAD 2D video course at the time of this writing contains the classic way to build a circular array and interactive when building an array along a path.
Sometimes the task arises - to make a protective umbrella for an exhaust or chimney, an exhaust deflector for ventilation, etc. But before you start manufacturing, you need to make a pattern (or scan) for the material. On the Internet there are all sorts of programs for calculating such sweeps. However, the problem is so easy to solve that you will quickly calculate it with a calculator (on a computer) than you will search, download and deal with these programs.
Let's start with a simple option - the development of a simple cone. The easiest way to explain the principle of calculating the pattern is with an example.
Suppose we need to make a cone with a diameter of D cm and a height of H centimeters. It is quite clear that a circle with a cut segment will act as a blank. Two parameters are known - diameter and height. Using the Pythagorean theorem, we calculate the diameter of the workpiece circle (do not confuse it with the radius finished cones). Half diameter (radius) and height form right triangle. That's why:
So, now we know the radius of the workpiece and we can cut out the circle.
Calculate the angle of the sector to be cut out of the circle. We argue as follows: The diameter of the workpiece is 2R, which means that the circumference is Pi * 2 * R - i.e. 6.28*R. We denote it by L. The circle is complete, i.e. 360 degrees. And the circumference of the finished cone is Pi * D. We denote it by Lm. It is, of course, less than the circumference of the workpiece. We need to cut a segment with an arc length equal to the difference between these lengths. Apply the ratio rule. If 360 degrees gives us the full circumference of the workpiece, then the desired angle should give the circumference of the finished cone.
From the ratio formula, we obtain the size of the angle X. And the cut sector is found by subtracting 360 - X.
From a round blank with a radius R, a sector with an angle (360-X) must be cut. Be sure to leave a small strip of overlapping material (if the cone mount will overlap). After connecting the sides of the cut sector, we get a cone of a given size.
For example: We need a chimney hood cone with a height (H) of 100 mm and a diameter (D) of 250 mm. According to the Pythagorean formula, we obtain the radius of the workpiece - 160 mm. And the circumference of the workpiece, respectively, 160 x 6.28 = 1005 mm. At the same time, the circumference of the cone we need is 250 x 3.14 = 785 mm.
Then we get that the ratio of angles will be: 785 / 1005 x 360 = 281 degrees. Accordingly, it is necessary to cut the sector 360 - 281 = 79 degrees.
Calculation of the pattern blank for a truncated cone.
Such a detail is sometimes needed in the manufacture of adapters from one diameter to another or for Volpert-Grigorovich or Khanzhenkov deflectors. They are used to improve draft in a chimney or ventilation pipe.
The task is slightly complicated by the fact that we do not know the height of the entire cone, but only its truncated part. In general, there are three initial numbers: the height of the truncated cone H, the diameter of the lower hole (base) D, and the diameter of the upper hole Dm (at the cross section of the full cone). But we will resort to the same simple mathematical constructions based on the Pythagorean theorem and similarity.
Indeed, it is obvious that the value (D-Dm) / 2 (half the difference in diameters) will relate with the height of the truncated cone H in the same way as the radius of the base to the height of the entire cone, as if it were not truncated. We find the total height (P) from this ratio.
(D – Dm)/ 2H = D/2P
Hence Р = D x H / (D-Dm).
Now knowing the total height of the cone, we can reduce the solution of the problem to the previous one. Calculate the development of the workpiece as if for a full cone, and then “subtract” from it the development of its upper, unnecessary part. And we can calculate directly the radii of the workpiece.
We obtain by the Pythagorean theorem a larger radius of the workpiece - Rz. This Square root from the sum of the squares of the heights P and D/2.
The smaller radius Rm is the square root of the sum of squares (P-H) and Dm/2.
The circumference of our workpiece is 2 x Pi x Rz, or 6.28 x Rz. And the circumference of the base of the cone is Pi x D, or 3.14 x D. The ratio of their lengths will give the ratio of the angles of the sectors, if we assume that the full angle in the workpiece is 360 degrees.
Those. X / 360 = 3.14 x D / 6.28 x Rz
Hence X \u003d 180 x D / Rz (This is the angle that must be left to get the circumference of the base). And you need to cut accordingly 360 - X.
For example: We need to make a truncated cone 250 mm high, base diameter 300 mm, top hole diameter 200 mm.
We find the height of the full cone P: 300 x 250 / (300 - 200) = 600 mm
According to the Pythagorean method, we find the outer radius of the workpiece Rz: The square root of (300/2) ^ 2 + 6002 = 618.5 mm
By the same theorem, we find the smaller radius Rm: The square root of (600 - 250)^2 + (200/2)^2 = 364 mm.
We determine the angle of the sector of our workpiece: 180 x 300 / 618.5 = 87.3 degrees.
On the material we draw an arc with a radius of 618.5 mm, then from the same center - an arc with a radius of 364 mm. The arc angle can have approximately 90-100 degrees of opening. We draw radii with an opening angle of 87.3 degrees. Our preparation is ready. Don't forget to allow for seam edges if they overlap.
Or
Where:
- radius of the circle,
- circle diameter,
- circumference,
- Pi (),
As a rule, the value () up to the second sign (3,14) is used for calculation, but in some cases, this may not be enough.
- Truncated cone with accessible vertex: A cone that can be used to determine the position of the vertex.
- A truncated cone with an inaccessible vertex: A cone, during the construction of which the position of the vertex is difficult to determine, in view of its remoteness.
- Triangulation: a method for constructing unfolded surfaces of non-developing, conical, general view and with a return edge.
- It should be remembered: Regardless of whether the surface under consideration is developable or non-developable, only an approximate development can be plotted graphically. This is due to the fact that in the process of removing and postponing dimensions and performing other graphic operations, errors are inevitable due to the design features of drawing tools, physical abilities eyes and errors from replacing arcs with chords and angles on the surface with flat angles. Approximate developments of curves of non-developing surfaces, in addition to graphical errors, contain errors obtained due to the mismatch of the elements of such surfaces with flat approximating elements. Therefore, in order to obtain a surface from such a development, in addition to bending, it is necessary to partially stretch and compress its individual sections. Approximate scans, when carefully performed, are accurate enough for practical purposes.
The material presented in the article implies that you have an idea about the basics of drawing, know how to divide a circle, find the center of a segment with a compass, remove / transfer dimensions with a compass, use patterns, and the corresponding reference material. Therefore, the explanation of many points in the article is omitted.
Construction of a cylinder sweep
Cylinder
A body of revolution with the simplest unfolding, having the shape of a rectangle, where two parallel sides correspond to the height of the cylinder, and the other two parallel sides correspond to the circumference of the bases of the cylinder.
Truncated cylinder (fish)
truncated cylinder
Preparation:
- To create a sweep, draw a quadrilateral ACDE full size (see drawing).
- Let's draw a perpendicular BD, out of plane AC exactly D, cutting off from the construction the straight part of the cylinder ABDE which can be adjusted as needed.
- From the center of the plane CD(dot O) draw an arc with a radius of half the plane CD, and divide it into 6 parts. From the resulting points O, draw perpendicular lines to the plane CD. From points on a plane CD, draw straight lines perpendicular to the plane BD.
Building:
- Line segment BC transfer, and turn it into a vertical. From a point B, vertical BC, draw a ray perpendicular to the vertical BC.
- Take the size with a compass C-O 1 B, point 1 . We remove the size B1-C1 1 .
- Take the size with a compass O 1 -O 2, and set aside on the beam, from the point 1 , point 2 . We remove the size B2-C2, and set aside the perpendicular from the point 2 .
- Repeat until point is delayed D.
- The resulting verticals, from the point C, vertical BC, to the point D- connect with a curved curve.
- The second half of the sweep is mirrored.
Any cylindrical slices are constructed in a similar way.
Note: Why "Rybina"- if you continue building a sweep, while building half from the point D, and the second in the opposite direction from the vertical BC, then the resulting pattern will look like a fish, or a fish tail.
Construction of a development of a cone
Cone
The reaming of the cone can be done in two ways. (See drawing)
- If the size of the side of the cone is known, from the point O, an arc is drawn with a compass, with a radius equal to the side of the cone. Two points are plotted on the arc ( A 1 And B1 ABOUT.
- A life-size cone is built, from a point O, exactly A, a compass is placed, and an arc is drawn passing through the points A And B. Two points are plotted on the arc ( A 1 And B1), at a distance equal to the circumference and connected to a point ABOUT.
For convenience, half of the circumference can be set aside from, on both sides of the centerline of the cone.
A cone with a displaced apex is constructed in the same way as a truncated cone with displaced bases.
- Construct the circumference of the base of the cone in top view, full size. Divide the circle into 12 or more equal parts, and put them on a straight line one by one.
A cone with a rectangular (polyhedral) base.
Cones with polyhedral base
- If the cone has an even, radial base: ( When constructing a circle in a top view, by setting the compass to the center, and outlining the circle along an arbitrary vertex, all the vertices of the base fit on the arc of the circle.) Construct a cone, by analogy with the development of an ordinary cone (build the base in a circle, from a top view). Draw an arc from a point O. Put a point in an arbitrary part of the arc A 1, and alternately put all the faces of the base on the arc. The end point of the last face will be B1.
- In all other cases, the cone is built according to the principle of triangulation ( see below).
Truncated cone with accessible apex
Frustum
Construct a truncated cone ABCD full size (See drawing).
Parties AD And BC continue until the intersection point appears O. From the point of intersection O, draw arcs, with radius OB And OC.
On the arc OC, set aside the circumference DC. On the arc OB, set aside the circumference AB. Connect the resulting points with segments L1 And L2.
For convenience, half of the circumference can be set aside from, on both sides of the centerline of the cone.
How to plot the circumference of an arc:
- With the help of a thread, the length of which is equal to the circumference.
- With the help of a metal ruler, which should be bent “in an arc”, and put the appropriate risks.
Note: It is not at all necessary that the segments L1 And L2, if they continue, will converge at a point O. To be completely honest, they should converge, but taking into account the corrections for the errors of the tool, material and eye, the intersection point may be slightly lower or higher than the top, which is not a mistake.
Truncated cone with a transition from a circle to a square
Cone with a transition from a circle to a square
Preparation:
Construct a truncated cone ABCD full size (see drawing), build a top view ABB 1 A 1. Divide the circle into equal parts (in the above example, the division of one quarter is shown). points AA 1-AA 4 connect segments with a dot A. Hold Axis O, from the center of which draw a perpendicular O-O 1, height equal height cone.
Below, the primary dimensions are taken from the top view.
Building:
- Remove size AD and build an arbitrary vertical AA0-AA1. Remove size AA0-A, and put an "approximate point" by making a go-ahead with a compass. Remove size A-AA 1, and on the axis O, from the point O O 1 AA 1, to the expected point A. Connect dots with line segments AA 0 -A-AA 1.
- Remove size AA 1-AA 2, from the point AA 1 put a "approximate point", making a go-ahead with a compass. Remove size A-AA 2, and on the axis O, from the point O, postpone the segment, remove the size from the received point to the point O 1. Make a go-ahead with a compass from a point A, to the expected point AA 2. Draw a segment A-AA 2. Repeat until the segment is delayed A-AA 4.
- Remove size A-AA 5, from the point A set a point AA5. Remove size AA 4-AA 5, and on the axis O, from the point O, postpone the segment, remove the size from the received point to the point O 1. Make a go-ahead with a compass from a point AA 4, to the expected point AA5. Draw a segment AA 4-AA 5.
Build the rest of the segments in the same way.
Note: If the cone has an accessible vertex, and SQUARE foundation - then the construction can be carried out according to the principle truncated cone with an accessible vertex, and the base is cones with a rectangular (polyhedral) base. The accuracy will be lower, but the construction is much simpler.
Instead of the word “pattern”, “sweep” is sometimes used, but this term is ambiguous: for example, a reamer is a tool for increasing the diameter of a hole, and in electronic technology there is a concept of a reamer. Therefore, although I am obliged to use the words “cone sweep” so that search engines can find this article using them, I will use the word “pattern”.
Building a pattern for a cone is a simple matter. Let us consider two cases: for a full cone and for a truncated one. On the picture (click to enlarge) sketches of such cones and their patterns are shown. (I note right away that we will only talk about straight cones with a round base. We will consider cones with an oval base and inclined cones in the following articles).
1. Full taper
Designations:
Pattern parameters are calculated by the formulas:
;
;
Where 
.
2. Truncated cone
Designations:
Formulas for calculating pattern parameters: 
;
;
;
Where 
.
Note that these formulas are also suitable for the full cone if we substitute .
Sometimes, when constructing a cone, the value of the angle at its vertex (or at the imaginary vertex, if the cone is truncated) is fundamental. The simplest example is when you need one cone to fit snugly into another. Let's denote this angle with a letter (see picture).
In this case, we can use it instead of one of the three input values: , or . Why "together O", not "together e"? Because three parameters are enough to construct a cone, and the value of the fourth is calculated through the values of the other three. Why exactly three, and not two or four, is a question that is beyond the scope of this article. A mysterious voice tells me that this is somehow connected with the three-dimensionality of the “cone” object. (Compare with the two initial parameters of the two-dimensional circle segment object, from which we calculated all its other parameters in the article.)
Below are the formulas by which the fourth parameter of the cone is determined when three are given.
4. Methods for constructing a pattern
- Calculate the values on the calculator and build a pattern on paper (or immediately on metal) using a compass, ruler and protractor.
- Enter formulas and source data into a spreadsheet (for example, Microsoft Excel). The result obtained is used to build a pattern using graphic editor(for example, CorelDRAW).
- use my program, which will draw on the screen and print out a pattern for a cone with given parameters. This pattern can be saved as a vector file and imported into CorelDRAW.
5. Not parallel bases
As far as truncated cones are concerned, the Cones program still builds patterns for cones that have only parallel bases.
For those who are looking for a way to construct a truncated cone pattern with non-parallel bases, here is a link provided by one of the site visitors:
A truncated cone with non-parallel bases.
It is necessary to build a development of surfaces and transfer the line of intersection of the surfaces to the development. This problem is based on surfaces ( cone and cylinder) with their line of intersection given in previous task 8.
To solve such problems in descriptive geometry, you need to know:
- the order and methods of constructing surface developments;
- mutual correspondence between the surface and its development;
- special cases of constructing sweeps.
Solution procedurehadachi
1.
Note that a sweep is a figure obtained in
as a result of cutting the surface along some generatrix and gradually unbending it until it is completely aligned with the plane. Hence the sweep, direct circular cone- a sector with a radius equal to the length of the generatrix, and a base equal to the circumference of the base of the cone. All sweeps are built only from natural values.
Fig.9.1
- the circumference of the base of the cone, expressed in natural value, is divided into a number of shares: in our case - 10, the accuracy of the sweep construction depends on the number of shares ( fig.9.1.a);
- we postpone the received shares, replacing them with chords, on the length
an arc drawn with a radius equal to the length of the generatrix of the cone l=|Sb|. We connect the beginning and end of the count of the shares with the top of the sector - this will be the development of the side surface of the cone.
Second way:
- we build a sector with a radius equal to the length of the generatrix of the cone.
Note that in both the first and second cases, the extreme right or left generators of the cone l=|Sb| are taken as the radius, because they are expressed in natural size;
- at the top of the sector, we set aside the angle a, determined by the formula:
Fig.9.2
Where r- the value of the radius of the base of the cone;
l is the length of the generatrix of the cone;
360 is a constant value converted to degrees.
To the sweep sector we build the base of the cone of radius r.
2.
According to the conditions of the problem, it is required to move the line of intersection
surfaces of the cone and cylinder on the development. To do this, we use the properties of one-to-one between the surface and its development, in particular, we note that each point on the surface corresponds to a point on the development and to each line on the surface there corresponds a line on the development.
From this follows the sequence of transferring points and lines
from the surface to the development.
Fig.9.3
For reaming a cone. Let us agree that the cut of the surface of the cone is made along the generatrix S’
a’
. Then the points 1, 2, 3,…6
will lie on circles (arcs on the sweep) with radii correspondingly equal to the distances taken along the generatrix S’
A’
from the top S’
to the corresponding cutting plane with points 1’
, 2’, 3’…6’ -|
S1|, |
S2|, |
S3|….|
S6| (Fig.9.1.b).
The position of the points on these arcs is determined by the distance taken from the horizontal projection from the generatrix Sa along the chord to the corresponding point, for example, to the point c, ac=35 mm ( fig.9.1.a). If the distance along the chord and the arc are very different, then to reduce the error, you can divide a larger number of shares and put them on the corresponding sweep arcs. In this way, any points are transferred from the surface to its development. The resulting points will be connected by a smooth curve along the pattern ( fig.9.3).
For cylinder reaming.
The development of a cylinder is a rectangle with a height equal to the height of the generatrix and a length equal to the circumference of the base of the cylinder. Thus, to construct a sweep of a right circular cylinder, it is necessary to construct a rectangle with a height equal to the height of the cylinder, in our case 100mm, and a length equal to the circumference of the base of the cylinder, determined by known formulas: C=2 R=220mm, or by dividing the circumference of the base into a series of shares, as indicated above. We attach the base of the cylinder to the upper and lower parts of the obtained sweep.
Let us agree that the cut is made along the generatrix AA 1 (A’ A’ 1 ; AA1) . Note that the cut should be made along the characteristic (reference) points for a more convenient construction. Given that the length of the sweep is the circumference of the base of the cylinder C, from point A’= A’ 1 section of the frontal projection, we take the distance along the chord (if the distance is large, then it is necessary to divide it into shares) to the point B’ (in our example, 17mm) and put it aside on the scan (along the length of the base of the cylinder) from point A. From the resulting point B we draw a perpendicular (generatrix of the cylinder). Dot 1 should be on this perpendicular) at a distance from the base, taken from the horizontal projection to the point. In our case, the point 1 lies on the axis of symmetry of the sweep at a distance 100/2=50mm (fig.9.4).
Fig.9.4
And so we do to find all other points on the sweep.
We emphasize that the distance along the length of the sweep to determine the position of the points is taken from the frontal projection, and the distance along the height is taken from the horizontal, which corresponds to their natural values. We connect the obtained points with a smooth curve along the pattern ( fig.9.4).
In problem variants, when the intersection line splits into several branches, which corresponds to the complete intersection of surfaces, the methods for constructing (transferring) the intersection line to the development are similar to those described above.
Section: Descriptive geometry /