The base of the cone is the formula. Cone concept. Methods for obtaining tapered surfaces on a lathe

Which emanates from one point (the top of the cone) and which pass through a flat surface.

It happens that a cone is a part of a body that has a limited volume and which is obtained by combining each segment that connects the vertex and points of a flat surface. The latter, in this case, is base of the cone, and the cone is called resting on a given base.

When the base of the cone is a polygon, this is already pyramid .

Circular cone is a body consisting of a circle (the base of the cone), a point that does not lie in the plane of this circle (the top of the cone and all the segments that connect the top of the cone with the base points).

The segments that connect the top of the cone and the points of the circumference of the base are called generators of the cone... The cone surface consists of a base and a lateral surface.

The lateral surface area is correct n-gonal pyramid inscribed in a cone:

S n \u003d ½P n l n,

where P n is the perimeter of the base of the pyramid, and l n - apothem.

By the same principle: for the lateral surface area of \u200b\u200ba truncated cone with base radii R 1, R 2 and generating l we get the following formula:

S \u003d (R 1 + R 2) l.

Straight and oblique circular cones with equal base and height. These bodies have the same volume:

Cone properties.

When the area of \u200b\u200bthe base has a limit, then the volume of the cone also has a limit and is equal to the third part of the product of the height and the area of \u200b\u200bthe base.

where S - base area, H - height.

Thus, each cone that rests on this base and has a vertex that is located on a plane parallel to the base has an equal volume, since their heights are the same.

The center of gravity of each cone with a limit volume is one quarter of the height from the base.
The solid angle at the apex of a right circular cone can be expressed by the following formula:

where α - cone opening angle.

The lateral surface area of \u200b\u200bsuch a cone, formula:

and the total surface area (that is, the sum of the lateral and base areas), the formula is:

S \u003d πR (l + R),

where R - base radius, l- generatrix length.

Volume of a circular cone, formula:

For a truncated cone (not just straight or circular) volume, the formula is:

where S 1 and S 2 - the area of \u200b\u200bthe upper and lower bases,

h and H - the distance from the plane of the upper and lower base to the top.

The intersection of a plane with a right circular cone is one of the conic sections.

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Lesson objectives:

Educational: introduce the concept of a cone, its elements; consider the construction of a straight cone; consider finding the full surface of the cone; to form the ability to solve problems of finding the elements of the cone.
Developing: develop competent mathematical speech, logical thinking.
Educational: to foster cognitive activity, culture of communication, culture of dialogue.

Lesson form:lesson in the formation of new knowledge and skills.

Form of educational activity:collective form of work.

Methods used in the lesson:explanatory and illustrative, productive.

Didactic material:notebook, textbook, pen, pencil, ruler, board, chalk and crayons, projector and presentation “Cone. Basic concepts. Cone surface area ".

Lesson plan:

Organizational moment (1 min).
Preparatory stage (motivation) (5 min).
Learning new material (15 min).
Solving problems to find the elements of the cone (15 min).
Summing up the lesson (2 min).
Homework assignment (2 min).

DURING THE CLASSES

1. Organizational moment

Purpose: prepare for the assimilation of new material.

2. Preparatory stage

Form: oral work.

Purpose: acquaintance with the new body of revolution.

The cone is translated from the Greek “konos” as “pine cone”.

There are cone-shaped bodies. They can be seen in various objects, from ordinary ice cream to technology, as well as in children's toys (pyramid, firecracker, etc.), in nature (spruce, mountains, volcanoes, tornadoes).

(Slides 1-7 are used)

Teacher activity	Student activities
3. Explanation of the new material Purpose: to introduce new concepts and properties of the cone.
1. A cone can be obtained by rotating a right-angled triangle around one of its legs. (Slide 8) Now let's look at how the cone is constructed. First, draw a circle with center O and line OP, perpendicular to the plane of this circle. We connect each point of the circle with a segment to the point P (the teacher builds a cone in stages). The surface formed by these segments is called conical surface, and the segments themselves - generatrix of the conical surface.	A cone is built in notebooks.
(dictates the definition) (Slide 9) A body bounded by a conical surface and a circle with a boundary L is called cone.	Write down the definition.
The conical surface is called lateral surface of the coneand the circle is base of the cone... The line OP passing through the center of the base and the vertex is called axis of the cone... The axis of the cone is perpendicular to the base plane. The OP segment is called cone height... Point P is called the top of the cone, and the generators of the conical surface - generators of the cone.	On the drawing, the elements of the cone are signed.
What are the two generators of the cone and compare them?	PA and PB, they are equal.
Why are generators equal?	The projections of the inclined are equal as the radii of the circle, which means that the generators themselves are equal.
Write down in a notebook: the properties of the cone:	(Slide 10)
1. All generators of the cone are equal. What are the angles of inclination of the generatrices to the base? Compare them. Why, prove it?	Angles: PCO, PDO. They are equal. Since triangle PAB is isosceles.
2. The angles of inclination of the generatrices to the base are equal. What are the angles between the axis and the generatrices? What about these angles?	SRO and DPO They are equal.
3. The angles between the axis and the generatrices are equal. What are the angles between the axis and the base? What are these angles equal to?	POC and POD. 90 about
4. The angles between the axis and the base are straight. We will only consider a straight cone.
2. Consider the section of a cone by different planes. What is the cutting plane through the axis of the cone?	Triangle.
Which triangle is this?	He is isosceles.
Why?	Its two sides are generators, and they are equal.
What is the base of this triangle?	The diameter of the base of the cone.
This section is called axial. (Slide 11) Draw in notebooks and sign this section. What is the cut plane perpendicular to the cone's axis OP?	A circle.
Where is the center of this circle?	On the axis of the cone.
This section is called a circular section. (Handle 12) Draw in notebooks and sign this section. There are other types of cone sections that are not axial and not parallel to the base of the cone. Let's consider them with examples. (Slide 13)	They draw in notebooks.
3. Now we derive the formula for the full surface of the cone. (Slide 14) For this, the lateral surface of the cone, like the lateral surface of the cylinder, can be turned onto a plane by cutting it along one of the generatrices.
What is the sweep of the lateral surface of the cone? (draws on the board)	Circular sector.
What is the radius of this sector?	Generator of the cone.
And the arc length of the sector?	Circumference.
The area of \u200b\u200bits sweep is taken as the area of \u200b\u200bthe lateral surface of the cone. (Slide 15)	, where is the degree measure of the arc.
What is the area of \u200b\u200bthe circular sector?
So, what is the area of \u200b\u200bthe lateral surface of the cone? Let us express in terms of and. (Slide 16) What is the arc length?
On the other hand, this same arc is the circumference of the base of the cone. What is it equal to?
Substituting into the formula for the lateral surface of the cone, we obtain,. The total surface area of \u200b\u200ba cone is the sum of the lateral and base areas. . Write these formulas down.	Write down:, .h	(Slide 21) L \u003d 5

6. Homework.P.55, 56, No. 548 (b), 549 (b). (Slide 22)

Which emanates from one point (the top of the cone) and which pass through a flat surface.

When the base of the cone is a polygon, this is already pyramid .

The segments that connect the top of the cone and the points of the circumference of the base are called generators of the cone... The cone surface consists of a base and a lateral surface.

The lateral surface area is correct n-gonal pyramid inscribed in a cone:

S n \u003d ½P n l n,

where P n is the perimeter of the base of the pyramid, and l n - apothem.

By the same principle: for the lateral surface area of \u200b\u200ba truncated cone with base radii R 1, R 2 and generating l we get the following formula:

S \u003d (R 1 + R 2) l.

Straight and oblique circular cones with equal base and height. These bodies have the same volume:

Cone properties.

When the area of \u200b\u200bthe base has a limit, then the volume of the cone also has a limit and is equal to the third part of the product of the height and the area of \u200b\u200bthe base.

where S - base area, H - height.

Thus, each cone that rests on this base and has a vertex that is located on a plane parallel to the base has an equal volume, since their heights are the same.

The center of gravity of each cone with a limit volume is one quarter of the height from the base.
The solid angle at the apex of a right circular cone can be expressed by the following formula:

where α - cone opening angle.

The lateral surface area of \u200b\u200bsuch a cone, formula:

and the total surface area (that is, the sum of the lateral and base areas), the formula is:

S \u003d πR (l + R),

where R - base radius, l- generatrix length.

Volume of a circular cone, formula:

For a truncated cone (not just straight or circular) volume, the formula is:

where S 1 and S 2 - the area of \u200b\u200bthe upper and lower bases,

h and H - the distance from the plane of the upper and lower base to the top.

The intersection of a plane with a right circular cone is one of the conic sections.

Definition. Cone apex is the point (K) from which the rays emanate.

Definition. Base of the cone is the plane formed by the intersection of a flat surface and all rays emanating from the top of the cone. A cone can have stems such as a circle, ellipse, hyperbola, and parabola.

Definition. Generatrix of the cone (L) is any line segment that connects the top of the cone with the boundary of the base of the cone. The generator is a segment of the ray emerging from the top of the cone.

Formula. Length of generatrix (L) a straight circular cone through the radius R and the height H (through the Pythagorean theorem):

Definition. Guide A cone is a curve that describes the contour of the base of the cone.

Definition. Side surface a cone is a collection of all generators of a cone. That is, the surface that is formed by the movement of the generatrix along the guide of the cone.

Definition. Surface The cone consists of a lateral surface and a base of the cone.

Definition. Height cone (H) is a line segment that extends from the top of the cone and is perpendicular to its base.

Definition. Axis cone (a) is a straight line passing through the top of the cone and the center of the base of the cone.

Definition. Taper (C) cone is the ratio of the diameter of the base of the cone to its height. In the case of a truncated cone, this is the ratio of the difference between the diameters of the cross sections D and d of the truncated cone to the distance between them: where R is the base radius, and H is the height of the cone.

In mechanical engineering, along with cylindrical ones, parts with conical surfaces in the form of outer cones or in the form of conical holes are widely used. For example, the center of a lathe has two outer cones, one of which is used to install and fix it in the tapered bore of the spindle; an external cone for installation and fastening also have a drill, countersink, reamer, etc. The adapter sleeve for fastening drills with a tapered shank has an external cone and a tapered hole

1. The concept of a cone and its elements

Cone elements. If you rotate the right-angled triangle ABC around the leg AB (Fig. 202, a), then the AVG body is formed, called full cone... The AB line is called the axis or cone height, line AB - generatrix of the cone... Point A is the top of the cone.

When the BV leg rotates around the AB axis, a circle surface is formed, called base of the cone.

The angle of VAG between the lateral sides AB and AG is called taper angle and is denoted by 2α. The half of this angle formed by the side of the AG and the AB axis is called taper angle and is denoted by α. Angles are expressed in degrees, minutes and seconds.

If we cut off its upper part from a full cone with a plane parallel to its base (Fig. 202, b), we get a body called truncated cone... It has two bases, upper and lower. The distance OO 1 along the axis between the bases is called truncated cone height... Since in mechanical engineering for the most part it is necessary to deal with parts of the cones, that is, truncated cones, they are usually simply called cones; in what follows we will call all conical surfaces cones.

The relationship between the elements of the cone. The drawing usually indicates three main dimensions of the cone: the larger diameter D, the smaller one - d and the height of the cone l (Fig. 203).

Sometimes only one of the diameters of the cone is indicated in the drawing, for example, the larger D, the height of the cone l and the so-called taper. Taper is the ratio of the difference between the diameters of the cone to its length. We denote the taper by the letter K, then

If the cone has dimensions: D \u003d 80 mm, d \u003d 70 mm and l \u003d 100 mm, then according to the formula (10):

This means that over a length of 10 mm, the diameter of the cone decreases by 1 mm, or for every millimeter of the length of the cone, the difference between its diameters changes by

Sometimes in the drawing, instead of the angle of the cone, it is indicated cone slope... The slope of the cone shows to what extent the generatrix of the cone deviates from its axis.
The slope of the cone is determined by the formula

where tg α is the slope of the cone;

l - cone height in mm.

Using formula (11), you can use trigonometric tables to determine the angle a of the slope of the cone.

Example 6. Given D \u003d 80 mm; d \u003d 70mm; l \u003d 100 mm. According to the formula (11) we have According to the table of tangents we find the value closest to tan α \u003d 0.05, i.e., tan α \u003d 0.049, which corresponds to the slope angle of the cone α \u003d 2 ° 50 ". Consequently, the angle of the cone 2α \u003d 2 · 2 ° 50 "\u003d 5 ° 40".

The taper slope and taper are usually expressed in simple fractions, for example: 1: 10; 1: 50, or a decimal fraction, for example, 0.1; 0.05; 0.02, etc.

2. Methods for obtaining tapered surfaces on a lathe

On a lathe, conical surfaces are processed in one of the following ways:
a) by turning the upper part of the caliper;
b) lateral displacement of the tailstock body;
c) using a tapered ruler;
d) using a wide incisor.

3. Processing of tapered surfaces by turning the upper part of the caliper

When making on a lathe short outer and inner conical surfaces with a large slope angle, you need to turn the upper part of the support relative to the machine axis at an angle α of the slope of the cone (see Fig. 204). With this method of work, the feed can only be made by hand, by rotating the handle of the lead screw of the upper part of the caliper, and only in the most modern lathes there is a mechanical feed of the upper part of the caliper.

To install the upper part of the support 1 at the required angle, you can use the graduations marked on the flange 2 of the rotary part of the support (Fig. 204). If the angle α of the taper of the cone is given according to the drawing, then the upper part of the support is rotated together with its rotary part by the required number of divisions denoting degrees. The number of divisions is counted relative to the marks marked on the bottom of the caliper.

If the angle α is not given in the drawing, but the larger and smaller diameters of the cone and the length of its conical part are indicated, then the value of the angle of rotation of the support is determined by the formula (11)

Example 7. Given the diameters of the cone D \u003d 80 mm, d \u003d 66 mm, the length of the cone l \u003d 112 mm. We have: According to the table of tangents we find approximately: a \u003d 3 ° 35 ". Therefore, the upper part of the support must be turned by 3 ° 35".

The method of turning the conical surfaces by turning the upper part of the support has the following disadvantages: it usually allows the use of only manual feed, which affects labor productivity and cleanliness of the treated surface; allows turning relatively short tapered surfaces limited by the stroke length of the upper part of the caliper.

4. Processing of tapered surfaces by the method of lateral displacement of the tailstock body

To obtain a conical surface on a lathe, it is necessary to move the tip of the cutter not parallel, but at a certain angle to the center axis when rotating the workpiece. This angle must be equal to the angle α of the taper of the cone. The easiest way to get the angle between the centerline and feed direction is to offset the centerline by moving the trailing center laterally. By shifting the rear center towards the cutter (towards itself), as a result of turning, a cone is obtained, in which the larger base is directed towards the headstock; when the rear center is displaced in the opposite direction, that is, from the cutter (away from you), the larger base of the cone will be on the side of the tailstock (Fig. 205).

The displacement of the tailstock body is determined by the formula

where S is the displacement of the tailstock body from the headstock spindle axis in mm;
D is the diameter of the large base of the cone in mm;
d is the diameter of the small base of the cone in mm;
L is the length of the entire part or the distance between centers in mm;
l is the length of the tapered part of the part in mm.

Example 8. Determine the offset of the tailstock center for turning a truncated cone, if D \u003d 100 mm, d \u003d 80 mm, L \u003d 300 mm and l \u003d 200mm. By formula (12) we find:

The displacement of the tailstock body is made using divisions 1 (Figure 206), marked on the end of the base plate, and at risk 2 at the end of the tailstock body.

If there are no divisions at the end of the plate, then the tailstock housing is shifted using a measuring ruler, as shown in Fig. 207.

The advantage of machining tapered surfaces by offsetting the tailstock body is that long taper lengths can be turned in this way and can be turned with power feed.

Disadvantages of this method: inability to bore tapered holes; loss of time to rearrange the tailstock; the ability to handle only gentle cones; misalignment of the centers in the center holes, which leads to quick and uneven wear of the centers and center holes and causes rejects when the part is re-installed in the same center holes.

Uneven wear of the center holes can be avoided by using a special ball center instead of the usual one (Fig. 208). Such centers are used primarily for the processing of precise tapers.

5. Processing of tapered surfaces using a tapered ruler

For processing tapered surfaces with a slope angle of up to 10-12 °, modern lathes usually have a special device called a tapered ruler. The cone processing scheme using a tapered ruler is shown in Fig. 209.

A plate 11 is attached to the machine bed, on which a tapered ruler 9 is installed. The ruler can be rotated around the pin 8 at the required angle a to the axis of the workpiece. To fix the ruler in the required position, there are two bolts 4 and 10. A slider 7 freely slides along the ruler, which connects to the lower transverse part 12 of the support using a rod 5 and a clamp 6. So that this part of the support can freely slide along the guides, it is disconnected from the carriage 3 by unscrewing the cross screw or disconnecting the nut from the caliper.

If you tell the carriage a longitudinal feed, then the slider 7, captured by the rod 5, will begin to move along the ruler 9. Since the slider is fastened to the cross slide of the slide, they, together with the cutter, will move parallel to the ruler 9. Due to this, the cutter will machine a tapered surface with a slope angle equal to the angle α of rotation of the tapered ruler.

After each pass, the cutter is set to the cutting depth using the handle 1 of the upper part 2 of the support. This part of the caliper must be rotated 90 ° relative to the normal position, i.e. as shown in fig. 209.

If the diameters of the bases of the cone D and d and its length l are given, then the angle of rotation of the ruler can be found by formula (11).

Having calculated the value of tg α, it is easy to determine the value of the angle α from the table of tangents.
The use of a tapered ruler has several advantages:
1) adjusting the ruler is convenient and quick;
2) when switching to the processing of cones, it is not required to disrupt the normal adjustment of the machine, that is, it is not necessary to displace the body of the tailstock; the centers of the machine remain in the normal position, that is, on one axis, due to which the center holes in the parts and the centers of the machine are not triggered;
3) using a tapered ruler, you can not only grind the outer tapered surfaces, but also bore tapered holes;
4) it is possible to work with a longitudinal self-propelled gun, which increases labor productivity and improves the quality of processing.

The disadvantage of a tapered rule is the need to disconnect the slide slide from the cross feed screw. This drawback is eliminated in the design of some lathes, in which the screw is not rigidly connected to its handwheel and toothed wheels of the transverse self-propelled.

6. Processing of tapered surfaces with a wide cutter

The processing of tapered surfaces (external and internal) with a small length of the cone can be done with a wide cutter with an angle in the plan corresponding to the angle α of the slope of the cone (Fig. 210). The cutter feed can be longitudinal and transverse.

However, the use of a wide cutter on conventional machines is only possible with a cone length not exceeding about 20 mm. It is possible to use wider cutters only on particularly rigid machines and parts, if this does not cause vibration of the cutter and the workpiece.

7. Boring and reaming of tapered holes

Tapered hole machining is one of the most difficult turning jobs; it is much more difficult than machining the outer tapers.

The processing of tapered holes on lathes in most cases is carried out by boring with a cutter with a turn of the upper part of the support and less often using a tapered ruler. All calculations associated with turning the upper part of the caliper or tapered ruler are performed in the same way as when turning the outer tapered surfaces.

If the hole must be in solid material, then first a cylindrical hole is drilled, which is then bored with a taper cutter or processed with conical countersinks and reamers.

To speed up boring or reaming, you should first drill a hole with a drill, diameter d, which is 1-2 mm less than the diameter of the small base of the cone (Fig. 211, a). After that, the hole is drilled with one (Fig. 211, b) or two (Fig. 211, c) drills to obtain steps.

After finishing boring of the cone, it is deployed with a conical sweep of the appropriate taper. For tapers with a small taper, it is more profitable to process tapered holes directly after drilling with a set of special reamers, as shown in Fig. 212.

8. Cutting conditions when machining holes with conical reamers

Tapered reamers work in more severe conditions than cylindrical reamers: while cylindrical reamers remove a small allowance with small cutting edges, tapered reamers cut the entire length of their cutting edges located on the generatrix of the cone. Therefore, when working with conical reamers, feed rates and cutting speeds are used less than when working with cylindrical reamers.

When machining holes with conical reamers, the feed is done manually by rotating the tailstock handwheel. Make sure that the tailstock quill moves evenly.

Feeds when unrolling steel 0.1-0.2 mm / rev, while unrolling cast iron 0.2-0.4 mm / rev.

Cutting speed when reaming tapered holes with reamers from high speed steel 6-10 m / min.

Cooling should be used to facilitate the operation of the conical reamers and to obtain a clean and smooth surface. When processing steel and cast iron, an emulsion or sulfofresol is used.

9. Measuring tapered surfaces

The surfaces of the cones are checked with templates and gauges; measurement and at the same time check of the angles of the cone is carried out by protractors. In fig. 213 shows a method for checking a cone using a template.

The outer and inner angles of various parts can be measured with a universal goniometer (Fig. 214). It consists of a base 1, on which the main scale is marked on the arc 130. A ruler 5 is rigidly attached to the base 1. Sector 4, which carries the vernier 3, moves along the base arc. the ability to move along the edge of sector 4.

By means of various combinations in the installation of the measuring parts of the protractor, it is possible to measure angles from 0 to 320 °. The reading value for the vernier is 2 ". The reading obtained when measuring the angles is made according to the scale and the vernier (Fig. 215) as follows: the zero stroke of the vernier indicates the number of degrees, and the stroke of the vernier, which coincides with the stroke of the base scale, indicates the number of minutes. 215 with the stroke of the base scale coincides with the 11th stroke of the vernier, which means 2 "X 11 \u003d 22". Therefore, the angle in this case is equal to 76 ° 22 ".

In fig. 216 shows combinations of measuring parts of a universal protractor, which allow measurement of various angles from 0 to 320 °.

For a more accurate check of the cones in mass production, special gauges are used. In fig. 217, a shows a tapered bushing gauge for checking the outer cones, and in Fig. 217, b-taper gauge-plug for testing tapered holes.

On the gauges, ledges 1 and 2 are made at the ends or risks 3 are applied, which serve to determine the accuracy of the surfaces to be checked.

On. fig. 218 shows an example of checking a tapered bore with a plug gauge.

To check the hole, a gauge (see Fig. 218), having a ledge 1 at a certain distance from the end 2 and two risks 3, is inserted with light pressure into the hole and check if the gauge is swinging in the hole. No wobble indicates that the taper angle is correct. After making sure that the angle of the cone is correct, they begin to check its size. To do this, observe to what place the caliber will enter the checked part. If the end of the taper of the part coincides with the left end of the step 1 or with one of the notches 3 or is between the risks, then the dimensions of the taper are correct. But it may happen that the gauge enters the part so deeply that both risks 3 enter the hole or both ends of the ledge 1 come out of it. This shows that the hole diameter is larger than the specified one. If, on the contrary, both risks are outside the hole or none of the ends of the ledge comes out of it, then the hole diameter is less than required.

The following method is used to accurately check the taper. On the measured surface of the part or gauge, two or three lines are drawn with chalk or a pencil along the generatrix of the cone, then the gauge is inserted or put on the part and turned by a part of a turn. If the lines are erased unevenly, this means that the taper of the part is not machined accurately and needs to be corrected. Erasing lines at the ends of the gauge indicates incorrect taper; the erasure of the lines in the middle of the gauge shows that the taper has a slight concavity, which is usually caused by the inaccurate position of the tip of the cutter in the center height. Instead of chalk lines, you can apply a thin layer of special paint (blue) to the entire conical surface of a part or caliber. This method provides greater measurement accuracy.

10. Defect in the processing of tapered surfaces and measures to prevent it

When processing tapered surfaces, in addition to the mentioned types of scrap for cylindrical surfaces, the following types of scrap are additionally possible:
1) incorrect taper;
2) deviations in the size of the cone;
3) deviations in the dimensions of the diameters of the bases with the correct taper;
4) non-straightness of the generatrix of the conical surface.

1. Incorrect taper is mainly due to inaccurate displacement of the tailstock body, inaccurate rotation of the upper part of the caliper, incorrect installation of tapered rule, improper sharpening or installation of a wide cutter. Therefore, by accurately setting the tailstock housing, the upper part of the caliper or the tapered ruler before starting processing, defects can be prevented. We can correct this type of marriage only if the error in the entire length of the cone is directed into the body of the part, i.e., all diameters of the sleeve are smaller, and that of the conical rod are larger than required.

2. The wrong size of the cone at the correct angle, ie, the wrong size of the diameters along the entire length of the cone, is obtained if not enough or too much material has been removed. Defects can be prevented only by carefully setting the cutting depth along the dial on finishing passes. We will fix the marriage if not enough material has been removed.

3. It can happen that with the correct taper and exact dimensions of one end of the cone, the diameter of the other end is incorrect. The only reason is non-compliance with the required length of the entire tapered section of the part. We will fix the marriage if the part is too long. To avoid this type of scrap, it is necessary to carefully check its length before processing the cone.

4. The non-straightness of the generatrix of the cone being machined is obtained when the cutter is installed above (Fig. 219, b) or below (Fig. 219, c) of the center (in these figures, for greater clarity, the distortions of the generatrix of the cone are shown in a highly exaggerated form). Thus, this type of marriage is also the result of the turner's careless work.

test questions 1. What methods can be used to process tapered surfaces on lathes?
2. When is it recommended to turn the upper part of the caliper?
3. How is the angle of rotation of the upper part of the taper turning caliper calculated?
4. How is the correct rotation of the upper part of the caliper checked?
5. How to check the tailstock housing offset? How to calculate the offset amount?
6. What are the main elements of a tapered ruler? How to adjust a tapered rule for a given part?
7. Set the following angles on the universal goniometer: 50 ° 25 "; 45 ° 50"; 75 ° 35 ".
8. What instruments are used to measure tapered surfaces?
9. Why are ledges or marks made on tapered gauges and how to use them?
10. List the types of rejects when processing tapered surfaces. How to avoid them?