Methods for processing conical surfaces. Methods for turning a cone, processing shaped surfaces Turning conical internal and external surfaces
Conical surfaces can be processed in several ways: with a wide cutter, with the upper slide of the caliper rotated, with the tailstock body offset, using a copying-cone ruler and using special copying devices.
Processing cones with a wide cutter. Conical surfaces 20-25 mm long are processed with a wide cutter (Fig. 151, a). To obtain the required angle, use an installation template, which is applied to the workpiece, and to its inclined work surface the cutter is brought in. Then the template is removed and the cutter is brought to the workpiece (Fig. 151.6). Machining cones with the upper slide of the caliper turned (Fig. 152, a, b). The rotating plate of the upper part of the caliper can be rotated relative to the transverse slide of the caliper in both directions; To do this you need to release the screw
152 PROCESSING OF CONICAL SURFACES (CONES) WITH THE UPPER SLIDE OF THE CALIPER ROTATED:
Key screws securing the PLATE. The rotation angle is controlled with an accuracy of one degree using the divisions of the rotary plate.
Advantages of the method: the ability to process cones with any slope angle; ease of setting up the machine. Disadvantages of the method: the impossibility of processing long conical surfaces, since the processing length is limited by the stroke length of the upper support (for example, the 1KG2 machine has a stroke length of 180 mm); Grinding is carried out by manual feed, which reduces productivity and deteriorates the quality of processing.
When processing with the upper part of the support rotated, the feed can be mechanized using a device with a flexible shaft (Fig. 153). The flexible shaft 2 receives rotation from the lead screw or from the machine lead roller through bevel or spiral gears.
(ІК620М, 163, etc.) with a mechanism for transmitting rotation to the screw of the upper part of the caliper. On such a machine, regardless of the angle of rotation of the upper support. you can get automatic feeding.
If the outer conical surface of the shaft and the inner conical surface of the bushing must mate, then the taper of the mating surfaces must be the same. To ensure the same taper, processing of such surfaces is carried out without readjusting the position of the upper part of the caliper (Fig. 154 a, b). In this case, to process a conical hole, a boring cutter with a head bent to the right from the rod is used, and the spindle is rotated in reverse.
The rotary plate of the upper part of the caliper is adjusted to the required rotation angle using an indicator using a pre-fabricated reference part. The indicator is fixed in the tool holder, and the tip of the indicator is set exactly in the center and brought to the conical surface of the standard near the smaller section, while the indicator arrow is set to “zero”; then the caliper is moved so that the indicator pin touches the workpiece and the needle is at zero the entire time. The position of the caliper is fixed with clamping nuts.
Machining of conical surfaces by shifting the tailstock. Long outer tapered surfaces are machined by displacing the tailstock housing. The workpiece is installed in the centers. The tailstock body is shifted in the transverse direction using a screw so that the workpiece becomes “skewed.” When turned on
Feeding the support carriage, the cutter, moving parallel to the spindle axis, will grind the conical surface.
The amount of displacement H of the tailstock body is determined from the LAN triangle (Fig. 155a):
H = L sin a. From trigonometry it is known that for small angles (up to 10°) the sine is almost equal to the tangent of the angle. For example, for an angle of 7°, the sine is 0.120 and the tangent is 0.123.
The method of shifting the tailstock is used, as a rule, to process workpieces with small slope angles, so we can assume that sina = tga. Then
Ig. g D-d L D-d
I = L tg a ~ L ------------- = ----- MM.
The tailstock can be shifted by ±15 mm.
Example. Determine the amount of displacement of the tailstock for turning the workpiece shown in Fig. 155.6, if L=600 mm /=500 mm D=80 mm; d=60 mm.
I= 600----===600 ■ _______ =12mm.
The amount of displacement of the tailstock body relative to the plate is controlled by the divisions at the end of the plate or using the cross-feed dial. To do this, a bar is attached to the tool holder, which is brought to the tailstock quill, and the position of the dial is fixed. Then the cross slide is moved back to the calculated value along the limb, and then the tailstock is shifted until it comes into contact with the bar.
Setting up the machine for turning cones by displacing the tailstock can be done using a reference part. To do this, the reference part is fixed in the centers and the tailstock is shifted, using an indicator to control the parallelism of the generatrix of the reference part to the feed direction. For the same purpose you can use
1 55 PROCESSING OF OUTER CONICAL SURFACES (CONES) BY DISPLACEMENT OF THE TAILSTOCK:
Use a cutter and a strip of paper: the cutter is in contact with the conical surface along a smaller and then a larger diameter so that a strip of paper is pulled between the cutter and this surface with some resistance (Fig. 156).
According to the law of conservation of energy, the energy spent on the cutting process cannot disappear: it turns into another form - into thermal energy. Cutting heat occurs in the cutting zone. During the cutting process more...
A feature of modern technological progress is automation based on advances in electronic technology, hydraulics and pneumatics. The main areas of automation are the use of tracking (copying) devices, automation of machine control and control of parts. Automatic control …
8.1. Processing methods When processing shafts, there are often transitions between processed surfaces that have conical shape. If the length of the cone does not exceed 50 mm, then it is processed with a wide cutter (8.2). In this case, the cutting edge of the cutter must be set in plan relative to the axis of centers at an angle corresponding to the angle of inclination of the cone on the workpiece. The cutter is given a transverse feed or longitudinal direction. To reduce the distortion of the generatrix of the conical surface and the deviation of the angle of inclination of the cone, the cutting edge of the cutter is installed along the axis of rotation of the part.
It should be taken into account that when processing a cone with a cutter with a cutting edge longer than 10-15 mm, vibrations may occur. The level of vibration increases with increasing length of the workpiece and with decreasing its diameter, as well as with decreasing angle of inclination of the cone, with the approach of the cone to the middle of the part and with an increase in the overhang of the cutter and when it is not firmly secured. Vibrations cause marks and deteriorate the quality of the treated surface. When processing hard parts with a wide cutter, vibrations may not occur, but the cutter may shift under the influence of the radial component of the cutting force, which can lead to a violation of the cutter’s adjustment to the required angle of inclination. The offset of the cutter also depends on the processing mode and feed direction.
Conical surfaces with large slopes can be processed with the upper slide of the support with the tool holder (8.3) rotated at an angle a equal to the angle of inclination of the cone being processed. The cutter is fed manually (using the handle of the upper slide), which is a disadvantage of this method, since uneven feeding leads to an increase in the roughness of the machined surface. This method is used to process conical surfaces, the length of which is commensurate with the stroke length of the upper slide.
Long conical surfaces with an inclination angle сс = 84-10° can be processed by shifting the rear center (8.4), the value of which d = = L sin а. At small angles sin a«tg a, and h = L(D-d)/2l. If L = /, then /i = (D - -d)/2. The amount of tailstock displacement is determined by the scale marked on the end of the base plate on the flywheel side and the mark on the end of the tailstock housing. The division value on the scale is 1 mm. If there is no scale on the base plate, the amount of tailstock displacement is measured using a ruler attached to the base plate. The amount of tailstock displacement is controlled using a stop (8.5, a) or an indicator (8.5, b). The back side of the cutter can be used as a stop. The stop or indicator is brought to the tailstock quill, their initial position is fixed along the dial of the cross-feed handle or along the indicator arrow. Tailstock shifted by an amount greater than h (see 8.4), and the stop or indicator is moved (with the cross feed handle) by an amount h from the original position. Then the tailstock is shifted towards the stop or indicator, checking its position by the indicator arrow or by how tightly a strip of paper is clamped between the stop and the pi-zero. The position of the tailstock can be determined from the finished part or sample, which is installed at the centers of the machine.
Then the indicator is installed in the tool holder, brought to the part until it touches the tailstock and moved (with a support) along the forming part. The tailstock is shifted until the deviation of the indicator needle is minimal along the length of the generatrix of the conical surface, after which the tailstock is secured. The same taper of parts in a batch processed by this method is ensured when minimal deviations blanks in length and center holes in size (depth). Since the displacement of the centers of the machine causes wear of the center holes of the fogs, the conical surfaces are pre-processed and then corrected center holes, perform final finishing processing. To reduce the breakdown of center holes and wear of centers, it is advisable to use centers with rounded tops.
Conical surfaces with a = 0-j-12° are processed using copying devices. A plate / (8.6, a) with a tracing ruler 2 is attached to the machine bed, along which a slider 5 moves, connected to the support 6 of the machine by a rod 7 using a clamp 8. To freely move the support in the transverse direction, it is necessary to disconnect the cross-feed screw. When the caliper 6 moves longitudinally, the cutter receives two movements: longitudinal from the caliper and transverse from the tracing ruler 2. The angle of rotation of the ruler relative to the axis 3 is determined by the divisions on the plate /. The ruler is secured with bolts 4. The cutter is fed to the cutting depth using the handle for moving the upper slide of the caliper.
Processing of external and end conical surfaces 9 (8.6, b) is produced using a copier 10, which is installed in the tailstock quill or in the turret head of the machine. A device 11 with a follower roller 12 and a pointed cutter through passage is fixed in the tool holder of the transverse support. When the caliper moves transversely, the follower finger, in accordance with the profile of the follower 10, receives longitudinal movement by a certain amount, which is transmitted to the cutter. The outer conical surfaces are processed with passing cutters, and the inner ones with boring cutters.
To obtain a conical hole in solid material(8.7, a-d) the workpiece is pre-processed (drilled, countersinked, bored), and then finally processed (reamed, bored). Deployment is performed sequentially as a set conical reamers(8.8, a-c). A hole with a diameter 0.5-1.0 mm smaller than the diameter of the reamer guide cone is first drilled into the workpiece. Then the hole is processed sequentially with three reamers: the cutting edges of the rough reamer (the first) have the shape of ledges; the second, semi-finish reamer removes the irregularities left by the rough reamer; the third, finishing reamer has continuous cutting edges along the entire length and calibrates the hole.
High precision conical holes are pre-processed with a conical countersink and then with a conical reamer. To reduce metal removal with a countersink, the hole is sometimes machined stepwise with drills different diameters. 8.2. Processing of center holes In parts such as shafts, it is often necessary to make center holes, which are used for further processing of the part and for restoring it during operation.
The center holes of the shaft must be on the same axis and have the same dimensions at both ends of the shaft, regardless of the diameters of the end journals of the shaft. Failure to comply with these requirements reduces the processing accuracy and increases the wear of centers and center holes.
The most common are center holes with a cone angle of 60° (8.9, a; Table 8.1). Sometimes when processing large, heavy workpieces, this angle is increased to 75 or 90°. The top of the working part of the center should not rest against the workpiece, therefore the center holes always have a cylindrical recess of small diameter d at the top. To protect the center holes from damage during repeated installation of the workpiece, center holes with a safety chamfer with an angle of 120° are provided in the centers (8.9, b).
Figure 8.10 shows how the rear center of the machine wears out when the center hole in the workpiece is made incorrectly. If the center holes a are misaligned and the centers b are misaligned (8.11), the workpiece is mounted with a skew, which causes significant shape errors outer surface details.
Center holes in workpieces are processed different ways. The workpiece is secured in a self-centering chuck, and a drill chuck with a centering tool is inserted into the tailstock quill.
Center holes with a diameter of 1.5-5 mm are processed with combined center drills without a safety chamfer (8.12, d) and with a safety chamfer (8.12, d). Center holes of other sizes are processed separately, first with a cylindrical drill (8.12, a), and then with a single-tooth (8.12, b) or multi-tooth (8.12, e) countersink. Center holes are processed with a rotating workpiece and manual feeding of the centering tool. The end of the workpiece is pre-cut with a cutter. Required size The center hole is determined by the recess of the centering tool, using the tailstock flywheel dial or the quill scale (stop). To ensure the alignment of the center holes, the workpiece is pre-marked, and during alignment it is supported by a steady rest. The center holes are marked using a marking square (8.13). The intersection of several marks determines the position of the center hole at the end of the shaft. After marking, the center hole is marked.
Measuring the taper of outer conical surfaces can be done with a template or a universal protractor. For more precise measurements For cones, bushing gauges are used. Using a bushing gauge, not only the angle of the cone is checked, but also its diameters (8.14). 8.14 is applied to the treated surface of the cone. Bushing gauge for checking outer cones (a) and an example of its use (b) 2-3 marks with a pencil, then put the bushing gauge on the part being measured, pressing lightly along the axis and turning it. With a correctly executed cone, all risks are erased, and the end conical part located between marks A and B of the bushing gauge.
When measuring conical holes, a plug gauge is used. The correct processing of a conical hole is determined in the same way as when measuring external cones by the mutual fit of the surfaces of the part and the plug gauge.
In mechanical engineering, along with cylindrical ones, parts with conical surfaces in the form of external 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 serves to install and secure it in the conical hole of the spindle; outer cone for installation and fastening they also have a drill, countersink, reamer, etc. The adapter sleeve for fastening drills with a conical shank has an outer cone and a conical hole
1. The concept of a cone and its elements
Elements of a cone. If you rotate the right triangle ABC around the leg AB (Fig. 202, a), then a body ABG is formed, called full cone. Line AB is called the axis or cone height, line AB - generatrix of the cone. Point A is the top of the cone.
When the leg BV rotates around the axis AB, a circle surface is formed, called base of the cone.
The angle VAG between the lateral sides AB and AG is called cone angle and is denoted by 2α. Half of this angle formed by the lateral side AG and the axis AB is called cone angle and is denoted by α. Angles are expressed in degrees, minutes and seconds.
If you cut it off from a full cone top part plane parallel to its base (Fig. 202, b), then 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 having to deal with parts of cones, i.e. truncated cones, they are usually simply called cones; From now on we will call all conical surfaces cones.
The connection between the elements of the cone. The drawing usually indicates three main dimensions of the cone: the larger diameter D, the smaller diameter d and the height of the cone l (Fig. 203). 
Sometimes the drawing indicates only one of the cone diameters, for example, the larger D, the cone height l and the so-called taper. Taper is the ratio of the difference between the diameters of a cone and its length. Let us denote the taper by the letter K, then
If the cone has dimensions: D = 80 mm, d = 70 mm and l = 100 mm, then according to 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 the extent to which the generatrix of the cone deviates from its axis.
The slope of the cone is determined by the formula
where tan α is the slope of the cone;
l is the height of the cone in mm.
Using formula (11), you can use trigonometric tables determine the angle a of the cone.
Example 6. Given D = 80 mm; d=70mm; l= 100 mm. Using formula (11), we have From the table of tangents we find the value closest to tan α = 0.05, i.e. tan α = 0.049, which corresponds to the cone slope angle α = 2°50". Therefore, the cone angle 2α = 2 ·2°50" = 5°40".
Cone slope and taper are usually expressed as a simple fraction, for example: 1:10; 1:50, or decimal, for example, 0.1; 0.05; 0.02, etc.
2. Methods for producing conical surfaces on a lathe
On lathe processing of conical surfaces is carried out in one of the following ways:
a) turning the upper part of the caliper;
b) transverse displacement of the tailstock body;
c) using a cone ruler;
d) using a wide cutter.
3. Machining conical surfaces by turning the upper part of the caliper
When producing short external and internal conical surfaces on a lathe with high angle slope, you need to rotate the upper part of the support relative to the axis of the machine at an angle α of the slope of the cone (see Fig. 204). With this method of operation, feeding can only be done by hand, rotating the handle of the lead screw of the upper part of the support, and only the most modern lathes have a mechanical feed of the upper part of the support.
To set the upper part of the caliper 1 to the required angle, you can use the divisions marked on the flange 2 of the rotating part of the caliper (Fig. 204). If the slope angle α of the cone is specified according to the drawing, then the upper part of the caliper is rotated together with its rotating part by the required number of divisions indicating degrees. The number of divisions is counted relative to the mark 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 caliper rotation angle is determined by formula (11)
Example 7. The given cone diameters are D = 80 mm, d = 66 mm, cone length l = 112 mm. We have: 
Using the table of tangents we find approximately: a = 3°35". Therefore, the upper part of the caliper must be rotated 3°35".
The method of turning conical surfaces by turning the upper part of the caliper has the following disadvantages: it usually allows the use of only manual feed, which affects labor productivity and the cleanliness of the treated surface; allows you to grind relatively short conical surfaces limited by the stroke length of the upper part of the caliper.
4. Processing of conical surfaces using the method of transverse displacement of the tailstock body
To obtain a conical surface on a lathe, when rotating the workpiece, it is necessary to move the tip of the cutter not parallel, but at a certain angle to the axis of the centers. This angle must be equal to the slope angle α of the cone. The simplest way to obtain the angle between the center axis and the feed direction is to shift the center line by moving the back center in the transverse direction. By shifting the rear center towards the cutter (toward itself) as a result of grinding, a cone is obtained, the larger base of which is directed towards the headstock; when the rear center shifts to the opposite side, i.e., away 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 axis of the headstock spindle 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 conical part of the part in mm.
Example 8. Determine the displacement of the center of the tailstock for turning a truncated cone if D = 100 mm, d = 80 mm, L = 300 mm and l = 200 mm. Using formula (12) we find:
The tailstock housing is shifted using divisions 1 (Fig. 206) marked on the end of the base plate, and mark 2 on the end of the tailstock housing.
If there are no divisions at the end of the plate, then move the tailstock body using a measuring ruler, as shown in Fig. 207.
The advantage of machining conical surfaces by displacing the tailstock body is that this method can be used to turn long cones and grind with mechanical feed.
Disadvantages of this method: inability to bore conical holes; loss of time for rearranging the tailstock; the ability to process only shallow cones; misalignment of the centers in the center holes, which leads to rapid and uneven wear of the centers and center holes and causes defects during the secondary installation of the part in the same center holes.
Uneven wear of the center holes can be avoided if a special ball center is used instead of the usual one (Fig. 208). Such centers are mainly used when processing precision cones.
5. Machining conical surfaces using a conical ruler
To process conical surfaces with a slope angle of up to 10-12°, modern lathes usually have a special device called a cone ruler. The scheme for processing a cone using a cone ruler is shown in Fig. 209.
A plate 11 is attached to the machine bed, on which a conical ruler 9 is mounted. The ruler can be rotated around pin 8 at the required angle a to the axis of the workpiece. To secure the ruler in the required position, two bolts 4 and 10 are used. A slider 7 slides freely along the ruler, connecting to the lower transverse part 12 of the caliper using a rod 5 and a clamp 6. So that this part of the caliper can slide freely along the guides, it is disconnected from the carriage 3 by unscrewing the cross screw or disconnecting its nut from the caliper.
If you give 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 attached to the transverse slide of the caliper, they, together with the cutter, will move parallel to the ruler 9. Thanks to this, the cutter will process a conical surface with an inclination angle , equal to the angleα rotation of the cone ruler.
After each pass, the cutter is set to the cutting depth using the handle 1 of the upper part 2 of the caliper. 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 using formula (11).
Having calculated the value of tan α, it is easy to determine the value of angle α using the table of tangents.
The use of a cone ruler has a number of advantages:
1) setting up the ruler is convenient and quick;
2) when switching to processing cones, there is no need to disrupt the normal setup of the machine, i.e., there is no need to move the tailstock body; the centers of the machine remain in the normal position, i.e. on the same axis, due to which the center holes in the part and the centers of the machine do not work;
3) using a conical ruler, you can not only grind the outer conical surfaces, but also bore conical holes;
4) it is possible to work with a longitudinal self-propelled machine, which increases labor productivity and improves the quality of processing.
The disadvantage of a tapered ruler is the need to disconnect the caliper 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 the gear wheels of the transverse self-propelled machine.
6. Machining conical surfaces with a wide cutter
Machining of conical surfaces (external and internal) with a short cone length can be done with a wide cutter with a plan angle corresponding to the slope angle α of the cone (Fig. 210). The cutter feed can be longitudinal or transverse.
However, the use of a wide cutter on conventional machines is only possible with a cone length not exceeding approximately 20 mm. Wider cutters can only be used 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
Machining tapered holes is one of the most difficult turning jobs; it is much more difficult than processing external cones.
The machining of conical holes on lathes is in most cases carried out by boring with a cutter with turning 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 the tapered ruler are performed in the same way as when turning the outer conical surfaces.
If the hole must be in solid material, then first drill cylindrical hole, which is then bored into a cone with a 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 this, the hole is drilled with one (Fig. 211, b) or two (Fig. 211, c) drills to obtain steps.
After finishing boring the cone, it is reamed using a conical reamer of the appropriate taper. For cones with a small taper, it is more profitable to process the conical holes immediately after drilling with a set of special reamers, as shown in Fig. 212.
8. Cutting modes when processing holes with conical reamers
Conical reamers work under more difficult conditions than cylindrical reamers: while cylindrical reamers remove a small amount of stock with small cutting edges, conical reamers cut their entire length cutting edges, located on the generatrix of the cone. Therefore, when working with conical reamers, feeds and cutting speeds are used less than when working with cylindrical reamers.
When processing holes with conical reamers, the feed is done manually by rotating the tailstock handwheel. It is necessary to ensure that the tailstock quill moves evenly.
Feed when reaming steel is 0.1-0.2 mm/rev, when reaming cast iron 0.2-0.4 mm/rev.
Cutting speed when reaming conical holes using reamers from high speed steel 6-10 m/min.
To facilitate the operation of conical reamers and obtain a clean and smooth surface cooling should be used. When processing steel and cast iron, an emulsion or sulfofresol is used.
9. Measuring conical surfaces
The surfaces of the cones are checked with templates and gauges; measuring and simultaneously checking the angles of the cone is carried out using protractors. In Fig. 213 shows a method for checking a cone using a template.
External and internal corners 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 an arc 130. A ruler 5 is rigidly attached to the base 1. Sector 4 moves along the arc of the base, carrying a vernier 3. A square 2 can be attached to the sector 4 by means of a holder 7, in which, in turn, a removable ruler 5 is fixed. The square 2 and the removable ruler 5 have ability to move along the edge of sector 4.
Through 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 on the vernier is 2". The reading obtained when measuring angles is made using the scale and vernier (Fig. 215) as follows: the zero stroke of the vernier shows the number of degrees, and the vernier stroke, coinciding with the stroke of the base scale, shows the number of minutes. In Fig. . 215 the 11th stroke of the vernier coincides with the stroke of the base scale, which means 2 "X 11 = 22". Therefore, the angle in this case is 76°22".
In Fig. 216 shows combinations of measuring parts of a universal protractor, allowing the measurement of various angles from 0 to 320°.
For more accurate testing of cones in mass production, special gauges are used. In Fig. 217, and shows a conical bushing gauge for checking outer cones, and in Fig. 217, b-conical plug gauge for checking conical holes.
On the gauges, ledges 1 and 2 are made at the ends or marks 3 are applied, which serve to determine the accuracy of the surfaces being checked.
On the. rice. 218 provides an example of checking a conical hole with a plug gauge.
To check the hole, a gauge (see Fig. 218), which has a ledge 1 at a certain distance from the end 2 and two marks 3, is inserted with light pressure into the hole and checked to see if the gauge is swinging in the hole. No wobble indicates that the cone angle is correct. Once you are sure that the angle of the cone is correct, proceed to check its size. To do this, observe to what point the gauge will enter the part being tested. If the end of the part's cone coincides with the left end of ledge 1 or with one of the marks 3 or is between the marks, then the dimensions of the cone are correct. But it may happen that the gauge enters the part so deeply that both marks 3 enter the hole or both ends of the ledge 1 come out of it. This indicates that the hole diameter is larger than specified. If, on the contrary, both risks are outside the hole or none of the ends of the ledge come out of it, then the diameter of the hole is less than the required one.
To accurately check the taper, use the following method. On the surface of the part or gauge to be measured, draw two or three lines with chalk or a pencil along the generatrix of the cone, then insert or put the gauge on the part and turn it part of the turn. If the lines are erased unevenly, this means that the cone of the part is not processed accurately and needs to be corrected. The erasing of lines at the ends of the gauge indicates an incorrect taper; the erasing of the lines in the middle part of the caliber shows that the taper has a slight concavity, which is usually caused by the inaccurate location of the tip of the cutter along the height of the centers. Instead of chalk lines, you can apply a thin layer of special paint (blue) to the entire conical surface of the part or gauge. This method gives greater measurement accuracy.
10. Defects in the processing of conical surfaces and measures to prevent them
When processing conical surfaces, in addition to the mentioned types of defects for cylindrical surfaces, additionally possible the following types marriage:
1) incorrect taper;
2) deviations in the dimensions of the cone;
3) deviations in 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 tapered ruler, improper sharpening or installation of a wide cutter. Therefore, by accurately positioning the tailstock housing, the upper part of the caliper or the cone ruler before starting processing, defects can be prevented. This type of defect can be corrected only if the error along the entire length of the cone is directed into the body of the part, that is, all the diameters of the sleeve are smaller, and those of the conical rod are larger than required.
2. Wrong size cone with the correct angle, i.e., incorrect diameters along the entire length of the cone, occurs if not enough or too much material is removed. Defects can be prevented only by carefully setting the cutting depth along the dial on finishing passes. We will correct the defect if not enough material was filmed.
3. It may turn out that with the correct taper and exact dimensions of one end of the cone, the diameter of the second end is incorrect. The only reason is failure to comply with the required length of the entire conical section of the part. We will correct the defect if the part is too long. To avoid this type of defect, it is necessary to carefully check its length before processing the cone.
4. Non-straightness of the generatrix of the cone being processed is obtained when the cutter is installed above (Fig. 219, b) or below (Fig. 219, c) the center (in these figures, for greater clarity, the distortions of the generatrix of the cone are shown in a greatly exaggerated form). Thus, this type of defect is the result of the inattentive work of the turner.
Control questions 1. In what ways can conical surfaces be processed on lathes?
2. In what cases is it recommended to rotate the upper part of the caliper?
3. How is the angle of rotation of the upper part of the support for turning a cone calculated?
4. How do you check that the top of the caliper is rotated correctly?
5. How to check the displacement of the tailstock housing?. How to calculate the amount of displacement?
6. What are the main elements of a cone ruler? How to set up a tapered ruler for this part?
7. Set the following angles on the universal protractor: 50°25"; 45°50"; 75°35".
8. What tools are used to measure conical surfaces?
9. Why are there ledges or risks on conical gauges and how to use them?
10. List the types of defects when processing conical surfaces. How to avoid them?
Conical surfaces include surfaces formed by the movement of a rectilinear generatrix l along a curved guide T. The peculiarity of the formation of a conical surface is that
Rice. 95
Rice. 96
in this case, one point of the generatrix is always motionless. This point is the vertex of the conical surface (Fig. 95, A). The determinant of a conical surface includes the vertex S and guide T, wherein l"~S; l"^ T.
Cylindrical surfaces are those formed by a straight generatrix / moving along a curved guide T parallel to the given direction S(Fig. 95, b). A cylindrical surface can be considered as a special case of a conical surface with a vertex at infinity S.
The determinant of a cylindrical surface consists of a guide T and directions S forming l, while l" || S; l" ^ t.
If the generators of a cylindrical surface are perpendicular to the projection plane, then such a surface is called projecting. In Fig. 95, V a horizontally projecting cylindrical surface is shown.
On cylindrical and conical surfaces, given points are constructed using generatrices passing through them. Lines on surfaces, such as a line A in Fig. 95, V or horizontal h in Fig. 95, a, b, are constructed using individual points belonging to these lines.
Surfaces of revolution
Surfaces of revolution include surfaces formed by rotating line l around straight line i, which represents the axis of rotation. They can be linear, such as a cone or cylinder of revolution, and non-linear or curved, such as a sphere. The determinant of the surface of revolution includes the generatrix l and the axis i.
During rotation, each point of the generatrix describes a circle, the plane of which is perpendicular to the axis of rotation. Such circles of the surface of revolution are called parallels. The largest of the parallels is called equator. Equator determines the horizontal outline of the surface if i _|_ P 1 . In this case, the parallels are the horizontals of this surface.
Curves of a surface of revolution resulting from the intersection of the surface by planes passing through the axis of rotation are called meridians. All meridians of one surface are congruent. The frontal meridian is called the main meridian; it determines the frontal outline of the surface of revolution. The profile meridian determines the profile outline of the surface of rotation.
It is most convenient to construct a point on curved surfaces of revolution using surface parallels. In Fig. 103 point M built on parallel h4.
Surfaces of revolution have found the most wide application in technology. They limit the surfaces of most engineering parts.
A conical surface of revolution is formed by rotating a straight line i around the straight line intersecting with it - axis i (Fig. 104, a). Dot M on the surface constructed using the generatrix l and parallel h. This surface is also called a cone of revolution or a right circular cone.
A cylindrical surface of revolution is formed by rotating a straight line l around an axis i parallel to it (Fig. 104, b). This surface is also called a cylinder or a right circular cylinder.
A sphere is formed by rotating a circle around its diameter (Fig. 104, c). Point A on the surface of the sphere belongs to the main
Rice. 103
Rice. 104
meridian f, dot IN- equator h, a point M built on an auxiliary parallel h".
A torus is formed by rotating a circle or its arc around an axis lying in the plane of the circle. If the axis is located within the resulting circle, then such a torus is called closed (Fig. 105, a). If the axis of rotation is outside the circle, then such a torus is called open (Fig. 105, b). An open torus is also called a ring.
Surfaces of revolution can also be formed by other second-order curves. Ellipsoid of revolution (Fig. 106, A) formed by rotating an ellipse around one of its axes; paraboloid of revolution (Fig. 106, b) - by rotating the parabola around its axis; A one-sheet hyperboloid of revolution (Fig. 106, c) is formed by rotating a hyperbola around an imaginary axis, and a two-sheet (Fig. 106, d) is formed by rotating a hyperbola around a real axis.
IN general case surfaces are depicted as not limited in the direction of propagation of the generating lines (see Fig. 97, 98). To solve specific problems and obtain geometric shapes, they are limited to cut planes. For example, to obtain a circular cylinder, it is necessary to limit a section of the cylindrical surface to the cutting planes (see Fig. 104, b). As a result, we get its upper and lower bases. If the cutting planes are perpendicular to the axis of rotation, the cylinder will be straight; if not, the cylinder will be inclined.
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Rice. 106
To obtain a circular cone (see Fig. 104, a), it is necessary to cut along the top and beyond. If the cutting plane of the base of the cylinder is perpendicular to the axis of rotation, the cone will be straight; if not, it will be inclined. If both cutting planes do not pass through the vertex, the cone will be truncated.
Using the cut plane, you can get a prism and a pyramid. For example, a hexagonal pyramid will be straight if all its edges have the same slope to the cutting plane. In other cases it will be slanted. If it is completed With using cutting planes and none of them passes through the vertex - the pyramid is truncated.
A prism (see Fig. 101) can be obtained by limiting a section of the prismatic surface to two cutting planes. If the cutting plane is perpendicular to the edges of, for example, an octagonal prism, it is straight; if not perpendicular, it is inclined.
By choosing the appropriate position of the cutting planes, you can obtain various shapes geometric shapes depending on the conditions of the problem being solved.
Question 22
A paraboloid is a type of second-order surface. A paraboloid can be characterized as an open non-central (that is, without a center of symmetry) second-order surface.
Canonical equations of a paraboloid in Cartesian coordinates:
2z=x 2 /p+y 2 /q
If p and q are of the same sign, then the paraboloid is called elliptical.
If different sign, then the paraboloid is called hyperbolic.
if one of the coefficients is zero, then the paraboloid is called a parabolic cylinder.
Elliptical paraboloid
2z=x 2 /p+y 2 /q
Elliptic paraboloid if p=q
2z=x 2 /p+y 2 /q
Hyperbolic paraboloid
2z=x 2 /p-y 2 /q
Parabolic cylinder 2z=x 2 /p (or 2z=y 2 /q)
Question23
A real linear space is called Euclidean , if it defines an operation scalar multiplication : any two vectors x and y are associated with a real number ( denoted by (x,y) ), and this accordingly satisfies the following conditions, whatever may be vectors x,y and z and number C:
2. (x+y , z)=(x , z)+(y , z)
3. (Cx, y)= C(x, y)
4. (x, x)>0 if x≠0
The simplest corollaries from the above axioms:
1. (x, Cy)=(Cy, x)=C(y, x) therefore always (X, Cy)=C(x, y)
2. (x, y+z)=(x, y)+ (x, z)
3. ()= (x i , y)
(
)= (x , y k)
Methods for processing conical surfaces. Machining of conical surfaces on lathes is carried out in the following ways: by turning the upper slide of the caliper, by transversely moving the tailstock body, using a cone ruler, or with a special wide cutter.
By turning the upper slide of the caliper, grind short conical surfaces with different slope angles a. The upper slide of the caliper is set to the value of the slope angle according to the divisions marked around the circumference of the support flange of the caliper. If V In the drawing of the part, the slope angle is not indicated, then it is determined by the formula: and the table of tangents.
Feeding with this method of operation is carried out manually by rotating the handle of the screw of the upper caliper slide. The longitudinal and transverse slides must be locked at this time.
Conical surfaces with a small cone angle for a relatively long workpiece length process With using a transverse displacement of the tailstock housing. With this processing method, the cutter is moved by a longitudinal feed in the same way as when turning cylindrical surfaces. The conical surface is formed as a result of the displacement of the rear center of the workpiece. When the rear center is shifted away from you, the diameter D the large base of the cone is formed at the right end of the workpiece, and when shifted “towards itself” - at the left. The amount of lateral displacement of the tailstock housing b determined by the formula: where L- distance between centers (length of the entire workpiece), l- length of the conical part. At L = l(cone along the entire length of the workpiece). If K or a are known, then , or
Rear housing offset money are made using the divisions marked on the end of the base plate and the mark on the end of the tailstock housing. If there are no divisions at the end of the plate, then the tailstock body is shifted using a measuring ruler.
Machining of conical surfaces using a tapered ruler is carried out with the simultaneous implementation of longitudinal and transverse feeds of the cutter. The longitudinal feed is carried out, as usual, from the roller, and the transverse feed is carried out by means of a cone ruler. A plate is attached to the machine bed , on which the conical ruler is installed . The ruler can be rotated around the finger at the required angle a° to the axis of the workpiece. The position of the ruler is fixed with bolts . The slider sliding along the ruler is connected to the lower transverse part of the support by means of a clamp rod . In order for this part of the caliper to slide freely along its guides, it is disconnected from the carriage , by removing or disconnecting the cross feed screw. If the carriage is now given a longitudinal feed, the rod will move the slider along the conical ruler. Since the slider is connected to the transverse slide of the caliper, they, together with the cutter, will move parallel to the cone ruler. Thus, the cutter will process a conical surface with a slope angle equal to the angle of rotation of the conical ruler.
The depth of cut is set using the handle of the upper slide of the caliper, which must be rotated at an angle of 90° relative to its normal position.
Cutting tools and cutting modes for all considered methods of processing cones are similar to those for turning cylindrical surfaces.
Conical surfaces with short cone length can be machined special wide cutter with a plan angle corresponding to the inclination angle of the cone. The feed of the cutter can be longitudinal or transverse.
