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General information about cones
The conical surface is characterized by the following parameters (Fig. 4.31): smaller d and larger D diameters and the distance l between the planes in which the circles with diameters D and d are located. The angle a is called the angle of inclination of the cone, and the angle 2α is called the angle of the cone.
The ratio K = (D - d) / l is called taper and is usually denoted with a division sign (for example, 1:20 or 1:50), and in some cases, a decimal fraction (for example, 0.05 or 0.02).
The ratio Y = (D - d) / (2l) = tanα is called the slope.
Tapered Surface Processing Methods
When machining shafts, there are often conical transitions between surfaces. If the length of the cone does not exceed 50 mm, then it can be machined by plunge-cutting wide incisor... The angle of inclination of the cutting edge of the cutter in the plan must correspond to the angle of inclination of the cone on the machined part. A lateral feed motion is imparted to the cutter.
To reduce the distortion of the generatrix of the conical surface and reduce the deviation of the angle of inclination of the cone, it is necessary to set the cutting edge of the cutter along the axis of rotation of the workpiece.
It should be borne in mind that when machining a taper with a cutter with cutting edge with a length of more than 15 mm, vibrations can occur, the level of which is the higher, the longer the length of the workpiece, the smaller its diameter, the smaller the angle of inclination of the cone, the closer the cone is to the middle of the workpiece, the greater the overhang of the cutter and the lower the strength of its fastening. As a result of vibrations, traces appear on the treated surface and its quality deteriorates. When machining rigid parts with a wide cutter, vibrations may be absent, but at the same time, the cutter may shift under the action of the radial component of the cutting force, which leads to a violation of the cutter adjustment to the required angle of inclination. (The cutter offset depends on the machining mode and feed direction.)
Conical surfaces with large slopes can be machined by turning the upper slide of the support with a tool holder (Fig.4.32) at an angle α, equal to the angle tilt of the cone to be machined. The cutter is fed manually (by the handle of moving the upper slide), which is a disadvantage of this method, since the unevenness of the manual feed leads to an increase in the roughness of the machined surface. In this way, conical surfaces are processed, the length of which is commensurate with the stroke length of the upper slide.
A conical surface of great length with an angle α = 8 ... 10 ° can be processed when the tailstock is displaced (Fig.4.33)
At small angles sinα ≈ tgα
h≈L (D-d) / (2l),
where L is the distance between the centers; D - larger diameter; d - smaller diameter; l is the distance between the planes.
If L = l, then h = (D-d) / 2.
The displacement of the tailstock is determined by the scale marked on the end of the base plate from the flywheel side, and the risk at the end of the tailstock housing. The scale division is usually 1 mm. In the absence of a scale on the base plate, the displacement of the tailstock is counted along a ruler attached to the base plate.
To ensure the same taper of a batch of parts processed in this way, it is necessary that the dimensions of the workpieces and their center holes had minor deviations. Since the offset of the machine centers causes wear on the center holes of the workpieces, it is recommended that the tapered surfaces be pre-machined, then the center holes are corrected and then the final finishing is performed. To reduce the breakdown of center holes and wear of centers, it is advisable to perform the latter with rounded tops.
Processing of tapered surfaces with the use of copiers is quite common. A plate 7 is attached to the machine bed (Fig. 4.34, a) with a guide ruler 6, along which slider 4 moves, connected to the support 1 of the machine by a rod 2 using a clamp 5. For free movement of the support in the transverse direction, it is necessary to disconnect the screw of the transverse movement of the feed. When the caliper 1 moves longitudinally, the cutter receives two movements: longitudinal from the caliper and transverse from the guide ruler 6. The transverse movement depends on the angle of rotation of the guide ruler 6 about the rotation axis 5. The angle of rotation of the ruler is determined by the divisions on the plate 7, fixing the ruler with bolts 8. The movement of the cutter feed to the cutting depth is carried out by the handle for moving the upper slide of the support. The outer conical surfaces are machined with straight cutters.
Methods for processing internal conical surfaces
The processing of the inner conical surface 4 of the workpiece (Figure 4.34, b) is carried out according to the copier 2 installed in the quill of the tailstock or in the turret of the machine. In the tool holder of the transverse caliper, a device 1 with a copying roller 3 and a pointed through cutter is installed. With the transverse movement of the caliper, the tracing roller 3 in accordance with the profile of the tracer 2 receives a longitudinal movement, which is transmitted through the device 1 to the cutter. The inner conical surfaces are machined with boring bits.
To receive tapered bore in a solid material, the workpiece is first pretreated (drilled, bored), and then finally (unrolled). Deployment is performed sequentially in a set conical reamers... Diameter pre drilled hole 0.5 ... 1 mm less than the lead-in diameter of the reamer.
If a tapered hole is required high precision, then it is processed with a conical countersink before deployment, for which a hole with a diameter of 0.5 mm less than the diameter of the cone is drilled in the solid material, and then a countersink is used. To reduce the allowance for countersinking, stepped drills of different diameters are sometimes used.
Center hole machining
In parts such as shafts, center holes are often made, which are used for subsequent turning and grinding of the part and for restoring it during operation. Based on this, the alignment is carried out especially carefully.
The center holes of the shaft must be on the same axis and have the same tapered holes at both ends, regardless of the diameters of the end journals of the shaft. If these requirements are not met, the machining accuracy decreases and the wear of centers and center holes increases.
Center hole designs are shown in Fig. 4.35. Most widespread have center holes with a taper angle of 60 °. Sometimes in heavy shafts, this angle is increased to 75 or 90 °. In order for the top of the center not to rest against the workpiece, cylindrical recesses with a diameter d are made in the center holes.
To protect against damage, reusable center holes are made with a safety chamfer at an angle of 120 ° (Figure 4.35, b).
For machining center holes in small workpieces, use different methods... The workpiece is fixed in a self-centering chuck, and a drill chuck with a centering tool is inserted into the tailstock quill. Center holes large sizes processed first with a cylindrical drill (Figure 4.36, a), and then with a single-tooth (Figure 4.36, b) or multi-tooth (Figure 4.36, c) countersink. Center holes with a diameter of 1.5 ... 5 mm are processed with combined drills without a safety chamfer (Fig. 4.36, d) and with a safety chamfer (Fig. 4.36, d).
Center holes are machined with a rotating workpiece; the movement of the centering tool feed is carried out manually (from the flywheel of the tailstock). The end in which the center hole is processed is pre-cut with a cutter.
The required center hole size is determined by the recess of the centering tool, using the tailstock flywheel dial or quill scale. To ensure the alignment of the center holes, the part is preliminarily marked, and long parts are supported with a steady rest during alignment.
The center holes are marked with a square.
After marking, the center hole is nibbled. If the diameter of the shaft journal does not exceed 40 mm, then it is possible to punch the center hole without preliminary marking using the device shown in Fig. 4.37. The housing 1 of the device is installed with the left hand on the end of the shaft 3 and the center of the hole is marked with a hammer blow on the center punch 2.
If, during operation, the tapered surfaces of the center holes were damaged or unevenly worn out, then they can be corrected with a cutter. In this case, the upper carriage of the support is rotated through the taper angle.
Tapered surface inspection
The taper of the outer surfaces is measured with a template or universal goniometer... For more accurate measurements, sleeve gauges are used (Fig. 4.38), with which they check not only the angle of the cone, but also its diameters. On the processed surface of the cone, two or three risks are applied with a pencil, then a sleeve gauge is put on the measured cone, slightly pressing on it and turning it along the axis. With a correctly executed cone, all risks are erased, and the end tapered part located between marks A and B.
When measuring tapered holes, a plug gauge is used. The correctness of the machining of a tapered hole is determined (as in the case of measuring the outer tapers) by the mutual adhesion of the surfaces of the part and the plug gauge. If thin layer paint applied to the cork gauge will be erased at a small diameter, then the cone angle in the part is large, and if at large diameter- the angle is small.
The processing of conical surfaces on lathes is performed different ways: by turning the upper part of the caliper; displacement of the tailstock housing; by turning the tapered ruler; wide incisor. The application of this or that method depends on the length of the conical surface and the angle of inclination of the cone.
The processing of the outer cone by turning the upper slide of the support is advisable in cases where it is necessary to obtain a large angle of inclination of the cone with a relatively small length of it. The greatest length of the generatrix of the cone should be slightly less than the stroke of the carriage of the upper support. The processing of the outer taper by the method of displacement of the tailstock body is convenient for obtaining long, shallow tapers with a small slope angle (3 ... 5). To do this, the tailstock body is shifted in the transverse direction from the center line of the machine along the guides of the headstock base. The workpiece to be processed is fixed between the centers of the machine in a drive chuck with a clamp. Tapering with a tapered (copying) ruler attached to the back of the bed lathe on a slab, used to obtain a gentle cone of considerable length. The workpiece is mounted in centers or in a three-jaw self-centering chuck. The cutter, fixed in the tool holder of the machine support, receives simultaneous movement in the longitudinal and transverse directions, as a result of which it processes tapered surface blanks.
The processing of the outer cone with a wide cutter is used when it is necessary to obtain a short cone (l<25 мм) с большим углом уклона. Широкий проходной резец, режущая кромка которого длинней образующей конуса, устанавливают в резце держатель так, чтобы главная режущая кромка резца составляла с осью заготовки угол а, равный углу уклона конуса. Обработку можно вести как с продольной, так и с поперечной подачей. На чертежах деталей часто не указывают размеры, необходимые для обработки конус и их необходимо подсчитывать. Для подсчета неизвестных элементов конусов и их размеров (в мм) можно пользоваться следующими формулами
a) taper K = (D - d) / l = 2tg
b) the angle of inclination of the cone tg = (D - d) / (2l) = K / 2
c) slope i = K / 2 = (D - d) / (2l) = tg
d) larger diameter of the cone D = Kl + d = 2ltg
e) smaller diameter of the cone d = D - K1 = D - 2ltg
f) the length of the cone l = (D - d) K = (D - d) / 2tg
The processing of inner tapered surfaces on lathes is also carried out in various ways: with a wide cutter, turning the upper part (slide) of the support, turning the tapered (copying) ruler. Internal conical surfaces up to 15 mm long are processed with a wide cutter, the main cutting edge of which is set at the required angle to the axis of the cone, carrying out a longitudinal or transverse feed. This method is used when the slope angle of the cone is large, and no high requirements are imposed on the accuracy of the slope angle of the cone and surface roughness. Internal cones longer than 15 mm at any angle of inclination are processed by turning the upper slide of the support using manual feed.
The end of the workpiece to be machined should protrude from the chuck no more than 2.0 - 2.5 of the workpiece diameter. The main cutting edge of the cutter is set at the desired taper angle using a template or goniometer. You can grind the cone with transverse and longitudinal feed.
When the taper of the workpiece protrudes from the chuck for more than 20 mm or the length of the cutting edge of the cutter is more than 15 mm, vibrations occur, which make it impossible to machine the taper. Therefore, this method is used to a limited extent.
Remember! The length of the taper to be cut with wide incisors should not exceed 20 mm.
Questions
- When is a cone cut with wide incisors?
- What is the disadvantage of machining cones with wide incisors?
- Why should the workpiece cone not protrude more than 20 mm from the chuck?
To grind short outer and inner conical surfaces with a taper slope angle α = 20 ° on a lathe, you need to turn the upper part of the support relative to the machine axis at an angle α.
 |
With this method, the feed can be done by hand by rotating the screw handle 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.
If the angle a is given, then the upper part of the caliper is rotated using the divisions drawn, usually in degrees, on the disc of the rotary part of the caliper. The minutes have to be set by eye. Thus, in order to rotate the upper part of the caliper by 3 ° 30 ', you need to put the zero stroke approximately between 3 and 4 °.
Disadvantages of turning tapered surfaces with a turn of the upper part of the caliper:
- labor productivity decreases and the cleanliness of the treated surface deteriorates;
- the resulting tapered surfaces are relatively short, limited by the stroke length of the upper part of the caliper.
Questions
- How should the upper part of the caliper be installed if the taper slope angle a is specified according to the drawing with an accuracy of 1 °?
- How to install the upper part of the caliper if the angle is specified with an accuracy of 30 '(up to 30 minutes)?
- List the disadvantages of turning tapered surfaces with the top of the caliper turning.
Exercises
- Set up the machine for turning the tapered surface at an angle of 10 °, 15 °, 5 °, 8 ° 30 ′, 4 ° 50 ′.
- Make a center punch as shown below.
Technological map for the manufacture of a center punch
|
Blank | Forging | ||||||||||
| Material | Steel U7 | |||||||||||
| P / p No. | Processing sequence | Instruments | Equipment and fixtures | |||||||||
| worker | marking and measuring | |||||||||||
| 1 | Cut the workpiece with an allowance | Bench saw | Vernier caliper, measuring ruler | Locksmith vices | ||||||||
| 2 | Trim the butt to the length with a centering allowance | Scoring cutter | Calipers | Lathe, three-jaw chuck | ||||||||
| 3 | Center on one side | Center drill | Calipers | Lathe, drill chuck | ||||||||
| 4 | Roll the cylinder at length L— (l 1 + l 2) |
|
Knurling | Calipers | Three-jaw lathe chuck, center | |||||||
| 5 | Grind the cone at length l 1 at an angle α, grind the sharpened at an angle of 60 ° | Straight cutter bent | Calipers | |||||||||
| 6 | Trim end with centering to length l | Straight cutter bent | Calipers | Three-jaw lathe chuck | ||||||||
| 7 | Turn the taper of the striker at length l 2 | Straight cutter bent | Calipers | Three-jaw lathe chuck | ||||||||
| 8 | Turn rounding of the striker | Straight cutter bent | Radius template | Three-jaw lathe chuck | ||||||||
"Locksmithing", I. G. Spiridonov,
G.P.Bufetov, V.G. Kopelevich
Tapered holes with a large apex angle are processed as follows: the workpiece is fixed in the chuck of the headstock and to reduce the allowance for boring, the hole is processed with drills of different diameters. First, the workpiece is processed with a smaller diameter drill, then with a medium diameter drill and, finally, with a large diameter drill. The sequence of drilling a part for a taper. Tapered holes are usually bored by turning the upper part ...
When processing conical surfaces, the following types of rejects are possible: incorrect taper, deviations in the dimensions of the cone, deviations in the dimensions of the diameters of the bases with correct taper, non-straightness of the generatrix of the conical surface. Incorrect taper is mainly due to inaccurately positioned cutter, inaccurate rotation of the upper part of the caliper. By checking the installation of the tailstock body, the upper part of the caliper before starting to work, you can prevent this kind of ...
In the sixth and seventh grades, you were introduced to the various jobs performed on a lathe (for example, external cylindrical turning, cutting off parts, drilling). Many workpieces processed on lathes can have an outer or inner tapered surface. Tapered parts are widely used in mechanical engineering (eg drill spindle, drill shanks, lathe centers, tailstock quill hole)….
Conical surfaces include surfaces formed by displacement of a straight generatrix l along a curved guide T. A feature of the formation of a conical surface is that
Rice. 95
Rice. 96
in this case, one point of the generator is always motionless. This point is the apex of the conical surface (Fig. 95, a). The conical surface qualifier includes the vertex S and guide T, wherein l"~ S; l"^ T.
Cylindrical surfaces include surfaces formed by a straight generatrix / moving along a curved guide T parallel to a given direction S(fig. 95, b). A cylindrical surface can be viewed as a special case of a conical surface with a vertex at infinity S.
The identifier of a cylindrical surface consists of a guide T and directions S forming l, while l "|| S; l "^ t.
If the generatrices 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, the given points are plotted using generators passing through them. Lines on surfaces, such as a line a in fig. 95, v or horizontally 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 the rotation of the line l around the line i, which is 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.
Each point of the generator during rotation 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 horizontal lines of this surface.
Curved surfaces 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 defines the frontal outline of the surface of revolution. The profile meridian defines the profile outline of the surface of revolution.
It is most convenient to plot a point on curved surfaces of revolution using surface parallels. In fig. 103 point M built on the parallel h 4.
Surfaces of revolution have found the widest application in technology. They constrain the surfaces of most mechanical engineering parts.
The conical surface of revolution is formed by rotating a straight line i around the straight line intersecting with it - the i-axis (Fig. 104, a). Point M on the surface is constructed using the generator l and the parallel h. This surface is also called a cone of revolution or a right circular cone.
The cylindrical surface of revolution is formed by the rotation of the straight line l around the parallel axis i (Fig. 104, b). This surface is also called a cylinder or a straight circular cylinder.
The sphere is formed by rotating a circle around its diameter (Fig. 104, c). Point A on the surface of the sphere belongs to the principal
Rice. 103
Rice. 104
meridian f, point V- equator h, but 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 formed 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 be formed by other curves of the second order. Ellipsoid of revolution (Fig. 106, a) formed by rotating an ellipse around one of its axes; paraboloid of revolution (Fig. 106, b) - by the rotation of the parabola around its axis; a one-sheet hyperboloid of rotation (Fig. 106, c) is formed by the rotation of the hyperbola around the imaginary axis, and a two-sheet hyperboloid (Fig. 106, d) - by the rotation of the hyperbola around the real axis.
In the general case, surfaces are depicted as not limited in the direction of propagation of generating lines (see Fig. 97, 98). To solve specific problems and obtain geometric shapes, they are limited to trimming planes. For example, to get a circular cylinder, it is necessary to limit the area of the cylindrical surface with the trim planes (see Fig. 104, b). As a result, we get its upper and lower bases. If the trimming planes are perpendicular to the axis of rotation, the cylinder will be straight, if not, the cylinder will be inclined.
Rice. 105
Rice. 106
To obtain a circular cone (see Fig. 104, a), it is necessary to cut along the top and outside it. If the trim 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 trim planes do not pass through the vertex, the cone will be truncated.
Using the trim plane, you can get a prism and a pyramid. For example, a hexagonal pyramid will be straight if all of its edges have the same slope to the trim plane. In other cases, it will be sloped. If it is done with using trim planes and none of them passes through the top - the pyramid is truncated.
A prism (see Fig. 101) can be obtained by limiting the area of the prismatic surface to two trim planes. If the plane of the cut is perpendicular to the edges, for example, an octagonal prism, it is straight, if not perpendicular, it is inclined.
By choosing the appropriate position of the trim planes, you can get various shapes of geometric shapes, depending on the conditions of the problem being solved.
Question 22
Paraboloid is a type of surface of the second order. A paraboloid can be characterized as an open off-center (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 of opposite sign, then the paraboloid is called hyperbolic.
if one of the coefficients is zero, then the paraboloid is called a parabolic cylinder.
Elliptical parabolic
2z = x 2 / p + y 2 / q
Elliptic parabolic if p = q
2z = x 2 / p + y 2 / q
Hyperbolic parabolic
2z = x 2 / p-y 2 / q
Parabolic cylinder 2z = x 2 / p (or 2z = y 2 / q)
Q23
Real linear space is called Euclidean if an operation is defined in it 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 the vectors x, y and z and the number C are:
2. (x + y, z) = (x, z) + (y, z)
3. (Cx, y) = C (x, y)
4. (x, x)> 0 if x ≠ 0
The simplest consequences of the above axioms:
1. (x, Cy) = (Cy, x) = C (y, x) hence always (X, Cy) = C (x, y)
2. (x, y + z) = (x, y) + (x, z)
3. () = (x i, y)
(
) = (x, y k)
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 an 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 the VAG between the lateral sides AB and AG is called taper angle and is denoted by 2α. Half of this angle, formed by the lateral side of the AH and the AB axis, is called taper slope and is denoted by α. Angles are expressed in degrees, minutes and seconds.
If we cut off its upper part from a full cone by 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 = 80 mm, d = 70 mm and l = 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 taper 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 = 80 mm; d = 70mm; l = 100 mm. According to the formula (11) we have. According to the table of tangents, we find the value closest to tan α = 0.05, i.e., tan α = 0.049, which corresponds to the slope angle of the cone α = 2 ° 50 ". Consequently, the angle of the cone 2α = 2 · 2 ° 50 "= 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 short outer and inner conical surfaces with a large slope angle on a lathe, 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 markings 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 lower part 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 = 80 mm, d = 66 mm, the length of the cone l = 112 mm. We have: 
According to the table of tangents, we find approximately: a = 3 ° 35 ". Therefore, the upper part of the caliper 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 the 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 transverse 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 should be equal to the angle α of the slope of the cone. The easiest way to get the angle between the center axis and the feed direction is to offset the center line 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 = 100 mm, d = 80 mm, L = 300 mm and l = 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 body 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 process only gentle cones; misalignment of the centers in the center holes, which leads to rapid 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 transverse screw or disconnecting its 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 caliper, they, together with the cutter, will move parallel to the ruler 9. Due to this, the cutter will process 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) adjustment of 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 tailstock body; the centers of the machine remain in the normal position, that is, on the same 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 the tapered ruler 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 conical 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 caliper 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 is to be in solid material, then first a cylindrical hole is drilled, which is then bored with a taper cutter or processed with tapered 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 corresponding taper. For cones with a small taper, it is more profitable to process the tapered holes immediately 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 tapered reamers, the feed is made 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 made of 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 verification of the angles of the cone is carried out with goniometers. 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, carrying 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 = 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-tapered 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. rice. 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. The absence of 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. Erasure of lines at the ends of the gauge indicates incorrect taper; the erasure of the lines in the middle of the gauge shows that the cone has a slight concavity, which is usually caused by inaccurate positioning of the tip of the cutter in 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 a part or caliber. This method provides greater measurement accuracy.
10. Defect in the processing of conical 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 a tapered ruler, improper sharpening or installation of a wide cutter. Therefore, by accurately setting the tailstock body, 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 may 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-observance of the required length of the entire conical 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) 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 the result of the turner's careless work.
Control questions 1. What methods can be used to process tapered surfaces on lathes?
2. When is it recommended to rotate 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 displacement of the tailstock body? .How to calculate the amount of displacement?
6. What are the main elements of a tapered ruler? How to adjust a tapered ruler 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 can you avoid them?
