Developments of a truncated cylinder and a cone.

To construct a development of a truncated cylinder, draw a truncated cylinder in two projections (front view and top view), then divide the circle into an equal number of parts, for example 12 (Fig. 243). On the right side of the first projection, draw a straight line AB, equal to the straightened length of the circle, and divide it into the same number of equal parts, i.e. 12. From the division points 1, 2, 3, etc. on the line AB, reconstruct perpendiculars, and from points 1, 2, 3, etc., lying on the circle, draw straight lines parallel to the axial line until they intersect with the inclined section line.

Rice. 243. Constructing a development of a truncated cylinder

Now, on each perpendicular, segments are laid with a compass upwards from line AB, equal in height to the segments indicated on the front view projection by the numbers of the corresponding points. For clarity, two such segments are marked with curly braces. The resulting points on the perpendiculars are connected by a smooth curve.

The construction of the development of the lateral surface of the cone is shown in Fig. 244, a. A full-size lateral projection of the cone is drawn according to the given dimensions of diameter and height. Using a compass, measure the length of the generatrix of the cone, designated by the letter R. Using a compass with a set radius, draw an arc around the center O, which is the extreme point of an arbitrarily drawn straight line OA.

From point A, along an arc, plot (with a compass in small segments) the length of the unfolded circle, equal to πD. The resulting extreme point B is connected to the center O of the arc. The figure AOB will be a development of the lateral surface of the cone.

The development of the lateral surface of a truncated cone is constructed as shown in Fig. 244, b. The profile of the truncated cone is drawn according to the height and diameters of the upper and lower bases of the truncated cone in full size. The generatrices of the cone continue until they intersect at point O. This point is the center, from which arcs are drawn equal to the lengths of the circles of the base and top of the truncated cone. To do this, divide the base of the cone into seven parts. Each such part, i.e. 1/7 of the diameter D, is laid out along a large arc 22 times and from the resulting point B a straight line is drawn to the center of the arc O. After connecting point O with points A and B, a development of the lateral surface of the truncated cone is obtained.

Cylinder (straight circular cylinder) is a body consisting of two circles (the bases of a cylinder), combined by parallel translation, and all the segments connecting the corresponding points of these circles during parallel translation. The segments connecting the corresponding points of the base circles are called generators of the cylinder.

Here's another definition:

Cylinder- a body that is limited by a cylindrical surface with a closed guide and two parallel planes intersecting the generatrices of this surface.

Cylindrical surface- a surface that is formed by the movement of a straight line along a certain curve. The straight line is called the generatrix of the cylindrical surface, and the curved line is called the guide of the cylindrical surface.

Lateral surface of the cylinder- part of a cylindrical surface that is limited by parallel planes.

Cylinder bases- parts of parallel planes cut off by the side surface of the cylinder.

Fig.1 mini

The cylinder is called direct(Cm. Fig.1), if its generators are perpendicular to the planes of the bases. Otherwise the cylinder is called inclined.

Circular cylinder- a cylinder whose bases are circles.

Right circular cylinder (just a cylinder) is a body obtained by rotating a rectangle around one of its sides. Cm. Fig.1.

Cylinder radius is the radius of its base.

Generator of the cylinder- generatrix of a cylindrical surface.

Cylinder height is called the distance between the planes of the bases. Cylinder axis called a straight line passing through the centers of the bases. The section of a cylinder by a plane passing through the axis of the cylinder is called axial section.

The axis of the cylinder is parallel to its generatrix and is the axis of symmetry of the cylinder.

A plane passing through the generatrix of a straight cylinder and perpendicular to the axial section drawn through this generatrix is ​​called tangent plane of the cylinder. Cm. Fig.2.

Development of the lateral surface of the cylinder- a rectangle with sides equal to the height of the cylinder and the circumference of the base.

Cylinder side surface area- development area of ​​the lateral surface. $$S_(side)=2\pi\cdot rh$$ , where h is the height of the cylinder, and r– radius of the base.

Total surface area of ​​a cylinder- area, which is equal to the sum of the areas of the two bases of the cylinder and its side surface, i.e. is expressed by the formula: $$S_(full)=2\pi\cdot r^2 + 2\pi\cdot rh = 2\pi\cdot r(r+h)$$ , where h is the height of the cylinder, and r– radius of the base.

Volume of any cylinder equal to the product of the area of ​​the base and the height: $$V = S\cdot h$$ Volume of a round cylinder: $$V=\pi r^2 \cdot h$$ , where ( r- base radius).

A prism is a special type of cylinder (the generators are parallel to the side ribs; the guide is a polygon lying at the base). On the other hand, an arbitrary cylinder can be considered as a degenerate (“smoothed”) prism with a very large number of very narrow faces. In practice, a cylinder is indistinguishable from such a prism. All properties of the prism are preserved in the cylinder.

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Cone development. Constructing a cone scan.

Calculation of cone development.

Let's take the vertical and horizontal projections of the cone (Fig. 1, a). The vertical projection of the cone will have the form of a triangle, the base of which is equal to the diameter of the circle, and the sides are equal to the generatrix of the cone. The horizontal projection of the cone will be represented by a circle. If the height of the cone H is given, then the length of the generatrix is ​​determined by the formula:

i.e., like the hypotenuse of a right triangle.

Wrap the cardboard around the surface of the cone. By unfolding the cardboard again into one plane (Fig. 1, b), we obtain a sector whose radius is equal to the length of the generatrix of the cone, and the length of the arc is equal to the circumference of the base of the cone. A complete development of the side surface of the cone is performed as follows.

Rice. 1. Cone development:

a - projection; b - scan.

Cone sweep angle.

Taking the generatrix of the cone as the radius (Fig. 1, b), an arc is drawn on the metal, on which a segment of the arc is then laid KM , equal to the circumference of the base of the cone 2 π r. Arc length in 2 π r corresponds to the angle α , the value of which is determined by the formula:

r is the radius of the circle of the base of the cone;

l is the length of the cone generatrix.

The construction of the sweep comes down to the following. Not part of the arc is deposited along the length of the previously drawn arc KM , which is practically impossible, and the chord connecting the ends of this arc and corresponding to the angle α . The magnitude of the chord for a given angle is found in the reference book or indicated on the drawing.

Found points KM connect to the center of the circle. The circular sector obtained as a result of the construction will be the unfolded lateral surface of the cone.

The walls of which would be perfectly smooth is not achieved in every case, even if high-quality drills are used. In addition, the diameter of the hole may differ from the required one by several tenths of millimeters. In order for the gaps to be perfect, manual reaming is needed. These are metal-cutting tools specifically designed for finishing holes after drilling and countersinking operations. Let's look at what this tool is, how it works, why it is needed and how to use it.

Characteristic

A reamer is a cutting tool for making a hole with this device; you can increase its diameter, as well as significantly improve surface cleanliness and dimensional accuracy. Reamers are used for both finishing and pre-processing. There is a standard by which manual scanning is regulated - GOST 7722-77. Hand tools are considered to be tools designed for processing holes with a diameter in the range from 3 to 60 mm (step - 1 mm).

Using these tools, you can obtain dimensions whose accuracy will correspond to the second and third class. As for the surface cleanliness, it can be from Rz 10 to Rz 6.3. It is impossible to achieve such cleanliness by drilling.

The principle of operation of sweeps

Using a tool for processing holes, you can achieve high precision and surface quality - this has already been mentioned above. Manual sweep works on small scales. It is possible to correct holes with such precision because the tool is equipped with several cutting edges. Thus, a manual reamer - depending on the type - can have from 4 to 14 cutting edges. It is due to this that the smallest bites are removed.

The tool works as follows. The reamer needs to be inserted into the hole, then, if it is manual, put on a special wrench and rotate the tool with it. The device will work not only with rotational movements, but also with simultaneous movement down or up the axis. The tool is capable of removing thin layers of metal - from a few tenths to hundredths of a millimeter.

Not only traditional cylindrical holes, but also conical ones can be processed in this way. For this, a conical reamer is used. There are several types of this cutting tool. In this article we will look at each of these types.

What does the scan look like?

And the device looks like this: This is a cylindrical or conical rod, which has longitudinal grooves on the working part. The other part is smooth and can be equipped at the end with a square or conical shank.

The working side of the tool is represented by several departments. The front part is conical and short. Then comes the cutting part itself, then the guiding part and, finally, the rear working part.

This is what the scan looks like. The tool, despite such a large number of working parts, directly cuts metal only with the receiving or working part. The short back side is called the gauge side. Grooves are formed between the cutting teeth. They are designed to remove chips during tool operation. The cutting edges are located along the entire circumference of the rod.

Classification

As you know, reamers are designed for finishing holes. Directly depending on the technological requirements, these tools are used to produce holes in different tolerance ranges - from the fourth class to the first. The accuracy of its operation depends on the design, as well as on the quality of the tool. Different manual reamers are used for different holes - let's look at the main types.

As for the characteristics of the tool, more than one factor plays a role here:

• Allowance amounts for deployment.
• Tool sharpening level.
• Cutting edge geometry, as well as many other factors.

Reamers are distinguished by the type of hole for which they are intended. The shape of the cutting teeth and the material being processed are also important.

In operation, to perform the main part of metalworking operations, the following are used: cylindrical reamers, adjustable tools, conical ones. Along with manual ones, there are also machine ones. These tools can be of different types. There are cylindrical, conical, with replaceable teeth, and with carbide cutting inserts.

Includes a large group of tools - for conical pins, for processing conical threads, for Morse taper, for metric cones. Cylindrical fine-grained tools are used especially widely in plumbing.

Cylindrical

This reamer is designed for machining cylindrical holes.

Manual reaming can be used either with a wrench or with an electric drill at low speeds. This tool can be made in one piece or with the ability to adjust the working diameter.

Conical

This tool is designed to work with conical holes.

They can also be used for traditional cylindrical holes.

Rough, intermediate, finishing

If you need to expand the size of the hole within serious limits, then you cannot do without a set of tools of different cleanliness. A conical reamer, like all others, is divided into rough, intermediate, and finishing.

The first tool is distinguished by teeth located along the entire line in steps. This tool works as follows. Narrow chips are cut using the cutting edge of each stage. Moreover, if the hole was cylindrical, then after such processing it turns into a stepped conical one.

An intermediate metal reamer can cut chips that are much thinner. The cutting part is distinguished by special channels for chip separation. Finishing tools cut metal using the entire working surface. Thus, a cylindrical or conical hole of the required size is formed. As you can see, the principle of operation is quite simple.

Adjustable

Modern cutting tools of this type can be of various designs. You can find expandable and sliding models on the market. Both types work on the same principle - when moving up or down, the diameter of the hole can decrease or increase. The two types of adjustable reamers differ in how they are tightened, as well as in the range of sizes.

So, in the expanding structure there is an upper and lower nut. The size can be changed in the range from 0.25 to 3 millimeters. In sliding reamers, the diameter changes by tightening the screw. The latter forces a special ball in the body to move, which unclenches the cutting parts. The adjustable sliding reamer is considered more accurate, and the diameter can be increased as much as possible from 0.15 to 0.5 millimeters.

As for the last type, the tool is structurally similar to all other reamers. It is a housing made of inexpensive steel and inserted cutting parts. Knives are often made in the form of thin plates. The material used is tool steel. The plates are removable, sharpenable and replaceable.

This metal reaming makes it possible to change the diameter of the hole by tenths and hundredths of a millimeter. Unlike solid ones, they are more economical. In case of wear, the knives can be easily replaced.

What you need to know about

The process of boring a hole is best performed using two classes of tools - rough reaming and finishing. The former are often made from old and worn materials. Before reaming the hole, its end part is ground. This is done so that the reamer can work effectively with each of its teeth. This is also true for cast iron parts. If you neglect such pre-processing, there is a risk of dulling the scan.

When working with the scan, it is better not to rush too much. The feed should be carried out evenly. The slower the tool is fed into the hole, the better the final result. The deployment process does not involve working at high speeds, as is the case with a drill. Experienced mechanics recommend putting away the electric drill and using a wrench instead. In this case, control over the process will be much higher.

Curved surfaces that can be completely aligned with a plane, without stretching or compression, without tears or folds, are called developable. These surfaces include only ruled surfaces and only those in which adjacent generatrices intersect each other or are parallel. This property is possessed by torsi (surfaces formed by straight lines tangent to a directing spatial curve), conical and cylindrical surfaces. The remaining ruled surfaces, as well as all non-ruled surfaces, are not expandable.

Construction of a complete development of a right circular truncated cylinder of revolution

(Fig. 10.41).

To construct a development of a cylinder, it is enough to imagine it as a prism with a large number of faces (in fact, 12-16 such faces are enough), evenly dividing the circumference of the base of the cylinder into an equal number of parts.

If there is any line on the surface of the cylinder, then this line can be transferred to the development of the cylinder along the points belonging to the corresponding generators of this surface.

Constructing a scan of the full surface of a right circular cone (Fig. 10.42).

To construct a development of a right circular cone, it is enough to imagine its surface as a regular pyramid with a large number of faces and then construct its development by finding the actual size of one of the faces, which is an isosceles triangle, along its side and base. The construction of the development of the cone can be seen from the drawing, where the base of the “face” S01 is equal to the chord 0 ` 1 `. The development of the lateral surface of the cone, in this case, contains 12 such “faces”.

The development of the lateral surface will be found more accurately if we determine the angle j 0 at point S on the development using the formula:

j 0 =R/l 360 0, where R is the radius of the base of the cone, and l is the length of the generatrix of the cone.

The points of a certain ABCDE curve belonging to the lateral surface of the cone can be found by the belonging of these points to the corresponding generators of the conical surface. To do this, it is enough to use a rotation method, as shown in the example of point C belonging to the generatrix S2, to find the segments S``B`` 0 =SB, S``D`` 0 =SD and S``E`` 0 =SE .. Place the found segments along the corresponding generators on the development of the cone and draw a line ABCDE through them. To obtain a complete development of the cone surface, it must be supplemented with the base of the cone, tangent at the corresponding point of development of the lateral surface.

Development of the lateral surface of an inclined cone be like the development of an inclined pyramid with a large number of faces, each of which is found on three sides - two lateral “edges” and a “base” (Fig. 10.43).