EP0335255A2 - Method for grinding a polygon cone on an NC grinding machine - Google Patents

Abstract

A method is used to grind a cone (14a), the radial cross-sectional area of which shows the shape of an n-polygon profile with a predetermined eccentricity and a predetermined arithmetic mean D _m, the diameter of the circumference and the incircle of the polygon profile. A grinding wheel (40a) rotatable about a first axis (42a) engages along a common surface line (41a) at an intersection at a sharp angle (λ) around a second axis (43a). The grinding wheel (40) carries out an oscillating movement, in which the surface line (41a) in the plane spanned by the axes (42a, 43a) with the n-fold frequency of the rotary movement of the workpiece blank (46a) and with a double eccentricity Amplitude is shifted in parallel.

gehorcht, wobei D_m2 der arithmetische Mittelwert der Durchmesser des Polygon-Profils am spitzen Ende des Kegels (14a) und β_max ein vorgegebener Winkel von vorzugsweise 14° ist.

In order to be able to grind a polygon cone (14a) with a technically producible grinding wheel with a straight surface line (41a), the radial cross-sectional areas of which are approximated as close as possible to the constant thickness profile according to DIN 32 711 from the point of view of strength, the amplitude is determined with an eccentricity e 'set that the relationship

obeys, where D _{m2 is} the arithmetic mean of the diameter of the polygon profile at the tip end of the cone (14a) and β _{max is} a predetermined angle of preferably 14 °.

Description

The invention relates to a method for grinding a cone, the radial cross-sectional area of which has the shape of an n-polygon profile with a predetermined eccentricity and a predetermined arithmetic mean of the diameter of the circumference and the incircle of the polygon profile, in which one a first axis rotatable grinding wheel engages on a workpiece blank rotatable about a second axis along a common surface line, the axes intersecting at an acute angle, at which the grinding wheel also carries out an oscillating movement in which the surface line is in the plane spanned by the axes with the n-fold frequency of the rotary movement of the workpiece blank and with an amplitude corresponding to twice the eccentricity.

Such a method is known from the company publication "Fortuna Polygon System - Working documents on shaft-hub connections".

As is well known, polygon connections of shafts and hubs are a type of positive driver connection for the transmission of torques. In addition to cylindrical polygons, conical polygons are also used. It is known to additionally apply a small axial clamping force to a tapered polygon connection between a shaft and a hub, in order to achieve a positive connection in addition to the positive locking of the polygon surfaces. In this way, connections can be made which have a considerably shorter axial length than circular conical connections with the same transmissible torque.

The polygon shapes used in known polygon connections are standardized according to DIN 32 711 and DIN 32 712. The polygon profiles with the designation P3G according to DIN 31 711 are so-called constant-thickness profiles because the diameters running through the axis of the profile are all of the same length. This is from This is particularly advantageous because it enables torque transmission with extremely even voltage distribution.

Polygonal profiles in cylindrical or conical shape are conventionally produced using kinematic polygon grinding machines, in which the grinding wheel lies in a common surface line on the workpiece blank. The grinding wheel is oscillated by means of an eccentric and a push rod in two axes which are perpendicular to one another and to the grinding wheel axis. The grinding wheel axis thus describes an outer surface of an elliptical cylinder. The polygon profile is then generated on its surface by synchronously rotating the workpiece blank. By adjusting the main axes of the ellipse and the basic feed of the grinding wheel in the direction of the workpiece blank, a polygon profile with freely selectable eccentricity and freely selectable mean diameter can be ground in this way, whereby the arithmetic mean of the diameter of the incircle and understands the circumference of the polygon profile.

While cylindrical polygon profiles can be ground as uniform thicknesses in accordance with DIN 32 711 in the manner described, difficulties have been encountered when grinding conical polygons.

It has been shown that when grinding a conical polygon connection there are considerable difficulties with regard to the accuracy of fit of the shaft and the hub and that a uniform thickness cannot be achieved for all radial cross sections along the axis of the polygon cone.

For this reason, tapered polygon connections have not been able to establish themselves in practice because, due to irregularities in shape, local stress peaks occur at the contact surfaces of the shaft and hub, which can lead to rapid wear and misalignment of the connection.

The invention is therefore based on the object of developing a method of the type mentioned in such a way that a tapered polygon connection is provided which meets the highest demands on fit and in which the stress distribution is as uniform as possible, so that the advantages of Polygon connections, in particular the advantage of self-centering, are retained even in long-term operation with large torque values.

mit den Randbedingungen:
e′ < e

ρ₂ =

- 8e′
wobei D_m2 der arithmetische Mittelwert der Durchmesser und ρ₂ der praktisch minimal mögliche Krümmungsradius des Polygon-­Profils am spitzen Ende des Kegels und β_max ein vorgegebener Winkel ist.This object is achieved in that the amplitude is set with an eccentricity value that the relationship

with the boundary conditions:
e ′ <e

ρ₂ =

- 8e ′
where D _{m2 is} the arithmetic mean of the diameters and ρ₂ is the practically minimum possible radius of curvature of the polygon profile at the tip end of the cone and β _{max is} a predetermined angle.

The object underlying the invention is completely achieved in this way. It has been shown that, for fundamental mathematical reasons, it is not possible to use a grinding wheel which has a straight surface line, ie. a cylindrical or conical grinding wheel to grind a polygon cone, the radial cross-sectional areas of which have a uniform thickness shape at each position of the longitudinal axis. The dimensioning of the eccentricity value in the above-mentioned manner has the advantage that a real grindable polygonal cone shape is obtained, which nevertheless gives almost the same values with regard to the stress distribution as a theoretically definable constant-thickness polygonal cone. The differences from this theoretical optimum are practically no longer measurable with the dimensioning of the eccentricity value described above.

In this way, it is possible to open up new areas of application for polygon cone connections, because torque transmission connections are now available that are easy to manufacture and also meet long-term requirements, with which the theoretically undisputed advantages of polygon connections are now also available for conical connections Practice can be implemented. This opens up completely new possibilities, particularly in the field of tool holder construction, because compared to conventional tool holders with circular cones, a significant reduction in the axial overall length is possible.

In a preferred embodiment of the invention, the angle β _{max is} 14 °.

It has been shown that this angle represents an optimal value at which the torque transmission on the one hand and the voltage distribution on the other hand occur to almost the same extent as is the case with a theoretically defined constant-thickness polygon cone. The angle β stands for the intersection angle of the tangent in a point of the polygon profile with a tangent to a circle through the point, the center of the circle coinciding with the center of the polygon. The angle β is thus, for example in the case of a triangular polygon profile, at the three points of smallest curvature and at the three points of greatest curvature, i.e. every 60 degrees over the circumference zero. Between these zero values, the tangent intersection angle β assumes a maximum, and it has been shown that the described optimal conditions are present when the angle maximum is set to 14 °.

In a particularly preferred embodiment of the method according to the invention, the grinding wheel is adjusted along a X-axis running at right angles to the surface line in predetermined axial steps and the workpiece blank about the second axis with predetermined angle steps by means of a numerical control device.

This measure has the advantage that the method according to the invention can be carried out by means of an NC-controlled grinding machine, in which the kinematics of conventional polygon grinding machines with an eccentric and push rod an NC coordinate control is replaced. This enables precise manufacturing for internal or external grinding of external or internal polygon cones with standard grinding machines, such as those used for cam grinding.

In a further preferred embodiment of the invention, the grinding wheel is adjusted in predetermined axial steps along the X axis as a superimposition of a continuous radial infeed movement and a path movement in accordance with the polygon profile.

This measure has the advantage that large oversizes can be carried along a spiral path by the superposition of the so-called delivery operation and rail operation.

In a further preferred embodiment of the invention, the workpiece blank is adjusted with predetermined angular steps about the second axis at an angular velocity which is set depending on the angle of rotation after the second derivative of the eccentricity value according to the angle of rotation.

This measure has the advantage that the so-called related chip removal volume Q 'can be set optimally, so that local overloads or overheating can be safely avoided.

In a further preferred embodiment of the invention, the movement of the grinding wheel in the X-axis is based on the relationship:
X = √ (Rs + r₁ + s) ² + (s ′) ² - (R _s + r₁)
With:
R _s = radius of the grinding wheel
r₁ = inner radius of the polygon profile at the tip end of the cone;
s = profile survey value
= e (1 - cos (n Θ))
with Θ = angle of rotation around the second axis
s ′ = ds / dΘ

These measures have the advantage that a simple path control of the grinding wheel is possible with a conventionally numerically controlled grinding machine.

mit:
D₁ = der größere Durchmesser des Polygon-Profils am stumpfen Ende des Kegels;
D_m1 = der arithmetische Mittelwert der Durchmesser des Polygon-Profils am stumpfen Ende des Kegels;
L = axiale Länge des Kegels.In a further preferred embodiment of this embodiment, the grinding wheel is pivoted about an axis perpendicular to its axis of rotation, and preferably according to the relationship:

With:
D₁ = the larger diameter of the polygon profile at the blunt end of the cone;
D _m1 = the arithmetic mean of the diameter of the polygon profile at the blunt end of the cone;
L = axial length of the cone.

These measures have the advantage that a fluctuation in the angle of inclination of the surface lines of the polygon cone is compensated for.

Further advantages result from the description and the attached drawing.

It goes without saying that the features mentioned above and those yet to be explained below can be used not only in the combination specified in each case, but also in other combinations or on their own without departing from the scope of the present invention.

• Fig. 1 eine perspektivische Ansicht eines Polygon-Kegels als Teil eines Werkzeughalters mit zugehöriger Spindelaufnahme;
• Fig. 2 ein ebenes Polygon-Profil mit den zugehörigen Parametern;
• Fig. 3 zwei Koordinatensysteme zur Erläuterung der Defi­nition von Gleichdick-Polygonen;
• Fig. 4 eine perspektivische Darstellung zur Erläuterung der Unterschiede beim Herstellen von Polygon-­Zylindern und Polygon-Kegeln;
• Fig. 5 ein Diagramm zur Erläuterung eines Tangenten-­Schnittwinkels;
• Fig. 6 eine grafische Darstellung eines Verlaufes des Tangenten-Schnittwinkels gemäß Fig. 5 über dem Umfangswinkel eines Polygon-Profils;
• Fig. 7 eine perspektivische Darstellung eines nach dem erfindungsgemäßen Verfahren definierten Polygon-­Kegels;
• Fig. 8 bis 11 vier Prinzipdarstellungen zur Erläuterung von Geometriefehlern nach dem Stand der Technik bzw. zur Vermeidung von Geometriefehlern bei Polygon-­Kegeln nach der Erfindung;
• Fig. 12 eine perspektivische Darstellung, ähnlich Fig. 4, zur Erläuterung einer ersten Variante eines erfin­dungsgemäßen Verfahrens zum Außenschleifen;
• Fig. 13 eine Darstellung, ähnlich Fig. 12, jedoch zum Innenschleifen;
• Fig. 14 eine äußerst schematisierte Draufsicht auf eine numerisch gesteuerte Schleifmaschine, mit der die in den Fig. 12 und 13 illustrierten Verfahren durch­geführt werden können;
• Fig. 15 eine Darstellung, ähnlich Fig. 12, für eine andere Variante eines erfindungsgemäßen Verfahrens zum Außenschleifen;
• Fig. 16 eine Darstellung, ähnlich Fig. 13, jedoch für die in Fig. 15 dargestellte Verfahrensvariante, jedoch beim Innenschleifen;
• Fig. 17 eine Darstellung, ähnlich Fig. 14, für eine nume­risch gesteuerte Schleifmaschine, mit der das in den Fig. 15 und 16 illustrierte Verfahren durchge­führt werden kann.
• Fig. 18 eine schematisierte Darstellung zur Erläuterung der Verhältnisse beim Eingriff einer Schleifscheibe in ein Werkstück;
• Fig. 19 und 20 Diagramme zur Erläuterung der Geometrie- bzw. Bearbeitungsparameter bei der Konfiguration gemäß Fig. 18;
• Fig. 21 eine perspektivische Darstellung zur Erläuterung von Schleifscheibenparametern, wie sie zur Durch­führung des Verfahrens gemäß den Fig. 18 bis 20 eingestellt werden.

Embodiments of the invention are shown in the drawing and are explained in more detail in the following description. Show it:

• Figure 1 is a perspective view of a polygon cone as part of a tool holder with associated spindle holder.
• 2 shows a flat polygon profile with the associated parameters;
• 3 shows two coordinate systems for explaining the definition of uniform thickness polygons;
• 4 shows a perspective illustration to explain the differences in the production of polygon cylinders and polygon cones;
• 5 is a diagram for explaining a tangent intersection angle;
• FIG. 6 shows a graphic representation of a profile of the tangent intersection angle according to FIG. 5 over the circumferential angle of a polygon profile;
• 7 shows a perspective illustration of a polygon cone defined by the method according to the invention;
• 8 to 11 show four schematic diagrams to explain geometry errors according to the prior art or to avoid geometry errors in polygon cones according to the invention;
• FIG. 12 shows a perspective illustration, similar to FIG. 4, to explain a first variant of a method for external grinding according to the invention;
• 13 shows a representation similar to FIG. 12, but for internal grinding;
• FIG. 14 is an extremely schematic plan view of a numerically controlled grinding machine with which the methods illustrated in FIGS. 12 and 13 can be carried out;
• 15 shows a representation, similar to FIG. 12, for another variant of a method for external grinding according to the invention;
• FIG. 16 shows an illustration, similar to FIG. 13, but for the method variant shown in FIG. 15, but for internal grinding;
• FIG. 17 shows an illustration, similar to FIG. 14, for a numerically controlled grinding machine with which the method illustrated in FIGS. 15 and 16 can be carried out.
• 18 shows a schematic illustration to explain the conditions when a grinding wheel engages in a workpiece;
• 19 and 20 are diagrams for explaining the geometry or machining parameters in the configuration according to FIG. 18;
• 21 is a perspective view for explaining grinding wheel parameters as they are set to carry out the method according to FIGS. 18 to 20.

In Fig. 1, 10 designates a tool holder, which can carry a tool, for example a drill 11 or a milling cutter or another tool, at its lower end. The tool sits with the end facing away from the machining end in a usually cylindrical flange 12. Above the flange 12 there is a positive and / or non-positive driver connection, which in the example shown in FIG. 1 consists of a polygon cylinder 13 and there is a polygon cone 14 placed thereon. A complementary spindle holder 15 of a machine tool has a correspondingly shaped polygon inner cylinder 16 and a polygon inner cone 17. The dimensions of the pairings 13/16 and 14/17 are chosen so that the surfaces of the cylinders or cones lie close together when the tool holder 10 is inserted into the spindle holder 15.

An arrow 18 indicates the direction in which the tool holder 10 can be inserted into or removed from the spindle receptacle 15. In addition, the arrow 18 symbolizes an axial tensile force that can be exerted on the tool holder 10 clamped in the spindle holder 15. For this purpose, the polygon cone 14 can be provided in its upper end face with a bore 19 which widens inwards and into which an arrangement of the pull rod and collet of the machine tool can engage in order to pull the tool holder 10 used upwards into the spindle receptacle 15 how this is known per se.

FIG. 2 shows a radial section through a polygon cone, which shows a circumferential polygon profile 30. The polygon profile 30 is delimited by a radius 31 and an incircle 31. The radius 31 has a diameter D and the incircle 32 has a diameter d. The effective diameter D _{m of} the polygon profile 30 is equal to the arithmetic mean of the diameters D and d of the circumference 31 and incircle 32. The difference between the diameters D and d is 2e, e being the so-called eccentricity of the polygon profile 30.

If the polygon profile 30 is a uniform thickness according to DIN 32 711, this means that each diameter D _m running through the center M of the polygon profile 30 has the same length, regardless of the angle α that the diameter D _m assumes an imaginary y or x axis.

As is well known, polygon profiles are epicycloids, which are created by rolling a rolling circle on a fixed circle and looking at the locus of a point that is located within the rolling circle at a distance from its periphery.

If the ratio of the diameter of the rolling circle to the fixed circle is an integer, closed epicycloid curves are created. It can be read in the relevant literature that the equal-thickness polygon profiles of interest arise when the distance between the fixed circle and the rolling circle becomes infinitely large and the epicycloid that is then formed is mapped into finite dimensions.

cos γ - 2e cos 2γ - e cos 4γ

y₁ =

sinγ + 2e sin2γ + e sin4γ3 shows a coordinate system yx in which points P of a polygon of interest in the present context are to be represented. From the literature it can be seen that you can use a rotating auxiliary coordinate system y₁-x₁, whose origin O₁ lies on an ellipse 35 and whose inclination to the yx coordinate system with the origin O corresponds to an angle γ. The ellipse 35 has a main axis, the half length of which corresponds to the product of the number of corners n and the eccentricity e of the polygon profile, while the half length of the minor axis is equal to the eccentricity e. Around a point P of the polygon profile, the polar coordinates of which are in the y / x coordinate system Have elevation angle eine to construct in the plane of FIG. 3, one chooses a parameter representation according to the angle γ, which obeys the following relationships:

x₁ =

cos γ - 2e cos 2γ - e cos 4γ

y₁ =

sinγ + 2e sin2γ + e sin4γ

To define the auxiliary coordinate system y₁-x₁, first draw an auxiliary line through the origin O of the coordinate system y-x, at an angle 3γ to the x-axis. The intersection of this auxiliary line with the ellipse 35 gives the origin O₁ of the auxiliary coordinate system y₁-x₁. With the help of the equations given above, the coordinates x 1 and y 1 are now determined in the auxiliary coordinate system and the position of the point P is thus reached.

Appropriate conversions can also be used to derive polar coordinates for the point P in the y-x coordinate system from the formulas given above.

The polygon profile obtained in this way has a uniform thickness in accordance with DIN 32 711.

4 shows a grinding wheel 40 which can be cylindrical or conical in shape. In both cases, the grinding wheel 40 has a straight surface line 41. The cutting disk 40 can be rotated about a grinding wheel axis 42, which is not drawn to scale in FIG. 4 is. The surface line 41 can run parallel to the grinding wheel axis 42 if a cylindrical grinding wheel 40 is used, but it can also run inclined to the grinding wheel axis 42 if the grinding wheel 40 has a conical shape.

Let us first consider the case where a first workpiece blank 45 can be rotated about a workpiece axis 43 which runs parallel to the surface line 41. If the rotational movement of the first workpiece blank 45 is synchronized with an oscillating stroke movement H _s (φ) of the grinding wheel 40, a polygon cylinder is obtained if the frequency of the stroke movement is, for example, just three times the rotational frequency of the first workpiece blank 45 and the amplitude of the stroke movement corresponds to the amount 2e. The axis of the lifting movement H _s (φ) of the grinding wheel 40 is referred to in the technical language of grinding machine technology as the X axis.

If one now inclines the workpiece axis 43 by an angle λ into a position 43 ', one obtains a second workpiece blank 46, which has the shape of a polygon cone, under otherwise unchanged conditions.

In Fig. 4, the workpiece axes are marked with Z and Z ', and the length of the polygon cone is designated with L. In the technical language of grinding machine technology, the workpiece axis 43 is referred to as the "C axis". The workpiece blank 45 rotates about the C-axis at an angular velocity dΘ / dt, so that an angular increment ΔΘ is measured in the X-axis with an incremental stroke movement H _s (φ) of the grinding wheel 40.

Since the only straight surface lines in cylindrical or conical outer surfaces are the outer lines, the axes 42 and 43 or 42 and 43 'must lie in one plane, so they must not be skewed. In the case of the inclined workpiece axis 43 ', the axes 42 and 43' intersect outside of the representation of Fig. 4. Also the surface line 41 lies in the plane spanned by the axes 42 and 43 or 42 and 43 '.

The oscillating movement of the closing disk 40 takes place in such a way that the surface line 41 is deflected in a direction perpendicular to its extension with a predetermined amplitude and a predetermined frequency. If a polygon cylinder (workpiece blank 45) is ground, the amplitude of the deflection is just twice the eccentricity e, while in the case of the inclined workpiece axis 43 'this amount can only be multiplied by the factor cos λ.

It can now be demonstrated by analytical observation that a polygon cone ground according to FIG. 4 is designed in such a way that its radial cross-sectional areas over the axial length are not equal thickness profiles. In the case of a constant thickness profile, the relative eccentricity, ie the quotient of the eccentricity e and the mean diameter D _{m, must be} constant over the axial length. In the case of polygon cones which are ground with a straight surface line 41 of a grinding wheel 40, however, this is not the case because the relative eccentricity changes continuously over the axial length.

This means that in the case of a polygon cone ground according to FIG. 4, at best one of the radial cross-sectional areas has an even thickness profile, while all other radial cross-sectional areas deviate from this ideal shape. In other words, in a workshop dimensioning of a polygon cone with an upper and a lower end face, each of which has the shape of a constant thickness profile, production according to FIG. 4 is not possible, because the "surface lines" of such Polygon cones would not be straight lines that could be ground with a surface line 41 of a grinding wheel 40 as the line of action of the machining tool. Rather, it can be shown by analytical observation that with such a workshop-sized polygon cone, both the lower end face and the upper end face, which are each designed as a constant thickness profile, have their own cone, the tips of which on the axis of the polygon - cone does not collapse.

On the basis of the following considerations, an approximate solution is therefore to be shown with which non-ideal polygon cones can be generated, which come as close as possible to the ideal polygon cone with a continuous uniform thickness profile over the axial length, but on the other hand ground using a straight surface line 41 can be.

5 shows a section of the illustration in FIG. 2, where a section of the polygon profile 30 is entered in a quadrant of the yx coordinate system. Any point on the polygon profile 30 is denoted by P.

A diameter D _m is now drawn in, which passes through the point P and by definition through the center M of the polygon profile 30, which is also the origin of the yx coordinate system. With the resulting radius MP, a circle K around M is now drawn in FIG. 5. The tangent to the circle K in the point P is also drawn in with T _k in FIG. 5.

The associated radius of curvature ρ and the tangent T _P perpendicular to the radius of curvature ρ of the polygon profile 30 at point P are now also drawn at point P.

Since the radius of curvature ρ generally does not pass through the center M of the polygon profile 30, a tangent intersection angle β results between the tangents T _K and T _P

If we now consider the point P 1 of the polygon profile 30, which is also on the y-axis, it follows by simple consideration that the minimum radius of curvature ρ _min present in the point P 1 passes through the center M. The tangent intersection angle β is zero in points P₁. The same applies to a point P₂, which is offset from the point P₁ by 60 ° on the polygon profile 30. At point P₂, the polygon profile 30 is known to have its largest radius of curvature ρ _max , which also goes through the center M. At point P₂, the tangent intersection angle β is therefore zero.

These relationships can be seen from the diagram in FIG. 6, where the tangent intersection angle β is plotted over the angle α which the diameter D _m includes with the y-axis. With α = 0 ° and α = 60 ° as well as with all other multiples The tangent intersection angle β is 60 °, as can clearly be seen from the course 50. Between 0 ° and 60 ° there is a pronounced maximum 51, which corresponds to a maximum tangent intersection angle β _max .

It can theoretically be shown that this maximum tangent intersection angle β _{max is} a measure of the above-described compromise between a technically producible and a constant thickness polygon cone that is optimal in terms of strength.

gehorcht. Für ein dreieckiges Polygon-Profil und den bereis erläuterten optimalen maximalen Tangentenschnittwinkel β_max von 14° ergibt dies:
e′ = 0,04156 D_m2.It has been found through theoretical considerations that the triangular polygon cones of interest here, with a radial cross-sectional area which is as close as possible to the polygon profile P3G according to DIN 32 711, can be obtained in a technically feasible manner if the maximum tangent intersection angle β _max increases about 14 °. Furthermore, it can be shown that in this case a modified eccentricity e ′ must be applied, which corresponds to the relationship:

obey. For a triangular polygon profile and the optimal maximum tangent intersection angle β _max of 14 ° already explained, this results in:
e ′ = 0.04156 D _m2 .

This modified eccentricity e 'is to be set constant over the axial length L of the polygon cone.

The conditions must: e ′ <e

ρ₂ = - 8e ′
ρ₂> 2.5 mm
be observed, where ρ₂ is practically the minimum possible radius of curvature ρmin of the P3 polygon profile at the tip of the cone, because with polygon profiles of the type of interest here, a polygon outer profile and an associated polygon inner profile must always be ground and a practical one Limiting the dimensions means that polygon inner profiles cannot be ground with any thin grinding wheels. A practical limit is typically a grinding wheel diameter of 5 millimeters, so that as a boundary condition in the method according to the invention it is important to ensure that no polygon profiles are determined which have minimal radii of curvature of less than 2.5 millimeters.

7 shows a practical calculation case with all parameters of interest here.

A triangular polygon cone 14 should be produced, the lower, larger end face of which is a polygon profile 30/1 and the upper, smaller end face of which is a polygon profile 30/2.

In the given example, a polygon shaft cone 40 g 6 / 1.4 x 30 with a polygon hub cone 40 H 7 / 1.4 x 30 with a taper of 1:10 and an inclination angle of λ = 2.8624 ° should be ground will.

For illustration purposes, a geometry analysis according to the state of the art will be presented first. The values from the above-mentioned standard for the polygon profile initially result:
D _m1 = 40 mm
e = 1.4 mm

β_max′ = 12,79°
β_max = 11,86°From this, the geometric sizes of the cone are calculated according to the formulas given above as follows:
D₁ = D _m1 + 2 · e = 40 + 2 · 1.4 = 42.8 mm
d 1 = D _m1 - 2e = 40 - 21.4 = 37.2 mm
D = D _{m1 m2} - 2 · L tan λ = 40 - 2 X 30 tan (2.8624 °) = 37.0 mm
D₂ = D _m2 + 2 · e = 37.0 + 2 · 1.4 = 39.8 mm
d₂ = D _m2 - 2e = 37.0 - 2 1.4 = 34.2 mm

β _max ′ = 12.79 °
β _max = 11.86 °

From the geometry analysis of the polygon cone it can be seen that the tangent contact angle of the small cone diameter is larger than the tangent contact angle of the large cone diameter. This means that the cone described is not homogeneous in the sense of the cone definition.

8, it can be seen that the cone by definition consists of a guide curve D, which lies in the directrix plane ΠΔ perpendicular to the Z axis at a distance L from the coordinate origin 0, and each generatrix G meets at the apex Z.

From this point of view it can be specified that the inclination of the grinding wheel with the angle λ does not result in a geometric polygon cone, since no apex is made possible as a cone tip.

bzw. als Differenz der axialen Längen der Kegel
Δ L = l′ - l = 428,0 - 400,0 = 28,0 mm
Δ L = l - l˝ = 400,0 - 372,0 = 28,0 mmIf one calculates the respective axial lengths of the cones according to FIG. 9, the result is:

or as the difference in the axial lengths of the cones
Δ L = l ′ - l = 428.0 - 400.0 = 28.0 mm
Δ L = l - l˝ = 400.0 - 372.0 = 28.0 mm

As can be seen from Fig. 9, the cone generating the polygon flank has the apex in Z and the polygon tip in Z₂. In the transition from the flank to the tip, a cone is generated which corresponds to the angle of inclination λ. All other generators have an offset vertex, which corresponds to the distance ± Δ L. It can be seen that the polygon cone described has a different geometry within the lateral surface in the flanks, tips and in the transition region.

The generated polygon profiles in the directrix plane ΠΔ, as an orthogonal section to the axis of rotation OZ within the generatrix G, are not the same thickness. A polygon cone in the geometrical sense can only be created if the condition with regard to the tangent contact angle tan β _max ′ = tan β _{max is} fulfilled. This means that the relative eccentricity E _i = e _i / D _mi must remain constant along the cone generators.

If we now consider the boundary conditions given above for the grinding wheel diameter D _si for internal grinding, the following results:
D _si = 0.8 (d₂ - 16e) = 0.8 (34.2 - 16.1.4) = 9.44 mm

Daraus berechnen sich die Durchmesser D₂, d₂ am spitzen Ende des Polygon-Kegels wie folgt:
D₂ = D_m2 + 2 e′ = 37,0 + 2 · 1,295 = 39,59 mm
d₂ = D_m2 - 2 e′ = 37,0 - 2 · 1,295 = 34,41 mm

If one now transfers the above considerations with the shaft / hub connection unchanged to the method according to the invention, the modified eccentricity e ′ must first be determined from the variables calculated above:

From this, the diameters D₂, d₂ at the tip end of the polygon cone are calculated as follows:
D₂ = D _m2 + 2 e ′ = 37.0 + 2 · 1.295 = 39.59 mm
d₂ = D _m2 - 2 e ′ = 37.0 - 2 · 1.295 = 34.41 mm

From the above consideration it can be seen that the angles of inclination of the cone vary at the locations of the largest diameter D, the smallest diameter d and the mean diameter D _m , namely in the range:
Δλ = λ ′ - λ = 3.0624 - 2.8624 = 0.2 °
Δλ = λ - λ˝ = 2.8624 - 2.6623 = 0.2 °

In a preferred embodiment of the invention, this variation in the angle of inclination of the surface lines of the polygon cone can be compensated for by correspondingly pivoting the grinding wheel or the workpiece.

The geometric condition for the tangent contact angle is fulfilled. The tangent contact angle of the small cone diameter is equal to the tangent contact angle of the large cone diameter.

This means that a constant change in the angle of inclination λ can produce a geometric polygon cone as a function of the workpiece rotation, as illustrated in FIG. 10.

As a check, the lengths of the respective cones are determined again, and this results, as also shown in FIG. 11:

As you can see, there is now a common vertex Z because all lengths l, l ′ and l˝ are of equal length.

In the geometrical relationships shown in FIG. 11, a common apex Z thus arises, which corresponds to the cone tip. The generated polygon profiles in the directrix plane ΠΔ in an orthogonal section to the axis of rotation OZ within the generatrix D are of equal thickness. The analytical condition for the relative eccentricity E _i = e _i / D _mi is fulfilled because the eccentric value is not a constant parameter when considering the geometry of the polygon cone.

In practice, the polygon cones can be manufactured in different ways.

FIG. 12 illustrates a first manufacturing method, in which the same reference numerals are used again, but with the addition of an "a".

The polygon cone 14a is produced from a workpiece blank 46a by adjusting the workpiece axis 43a by the angle λ with respect to the axis 42a of the grinding wheel 40a. A surface line 41a then results. In this variant of the method, the grinding wheel 40a therefore remains in its unchanged position, while the workpiece blank 46a was pivoted with its axis 43a by the angle λ.

While FIG. A shows the process of external grinding, the complementary process of internal grinding is illustrated in FIG. 13 with the addition "b" at the reference numerals.

A small roller-shaped grinding wheel 40b rotates about the axis 42b, which is unchanged in its direction, while a workpiece blank 46b is positioned with its axis 43b at an angle λ with respect to the grinding wheel axis 42b. In this way, a polygon inner cone 17b is obtained.

FIG. 14 illustrates the grinding machine required for the methods according to FIGS. 12 and 13, which is designated as a whole by 60.

A workpiece chuck 61 holds the workpiece blank 46a or 46b clamped, a tailstock 62 possibly also being used as a counter bearing. A grinding spindle 63 can be moved in the known manner in the direction of the so-called x-axis at 90 ° to the axis of the workpiece chuck 61 or of the tailstock 62, the latter axis being adjustable as a so-called C-axis in defined angular steps, by means of an im Workpiece chuck 61 contained drive.

The workpiece chuck 61 can be rotated together with the entire workpiece holder about a vertical axis 64, the so-called U-axis. In this way, the workpiece blank 46a or 46b with its axis 43a or 43b can be moved according to the angle λ Fig. 12 and 13 compared to the unchanged orientation 42a and 42b of the grinding wheel 40a and 40b.

The grinding wheel 40a or 40b now performs an oscillating movement in the direction of the x-axis during the rotation of the workpiece blank 46a or 46b, with a frequency that is three times as large as that in the described application of the triangular polygon cone 14 Rotation frequency of the workpiece blank 46a or 46b in the workpiece chuck 61. The amplitude of the oscillating movement Δ X in this case is 2e 'cosλ.

FIG. 15 shows a further variant of the external grinding method, the letter "c" being added to the reference numerals.

In this variant of the method, the axis 43c of the workpiece blank 46c is left unchanged, while the axis 42c of the grinding wheel 40c is set at an angle λ with respect to its starting position. The same applies to the internal grinding, which is shown in Fig. 16, wherein the letter "d" has been added to the reference numerals.

It also applies here that the axis 42d of the grinding wheel 40d was adjusted by the angle λ, while the axis 43d of the workpiece blank 46d remained unchanged.

While a polygon cone 14c is obtained in the former case, a polygon inner cone 17d is again formed in the latter case.

In FIG. 16, an arrow 70 also illustrates that the grinding wheel 40d, just like the grinding wheel 40b according to FIG. 13 or also like the grinding wheels 40a and 40c according to FIGS. 12 and 15, closes if very long surface lines 41a or 41c grinding, can still be moved in the direction of their axis, should this be necessary in individual cases.

FIG. 17 again shows a schematic illustration of the grinding machine 60, which is now set to carry out the method according to FIGS. 15 and 16.

The workpiece chuck 61 remains in the original position in FIG. 17, while the grinding spindle 63 can be rotated about a vertical axis 65, the so-called B axis.

In the case of FIG. 17 as well, the statements made above apply to the frequency and the amplitude of the oscillating movement of the grinding wheel 40c or 40d in the direction of the X axis of FIG. 17.

A numerical control device 66 of the type known per se is used to set all of these method parameters. In the numerical control device 66 there is a computing stage which, from the values D _m1 , D _m2 , e ′, D₁, D₂, d₁, d₂ and the predetermined values L and λ determined above, and the path points of the polygon profiles 30/1 and 30 / 2 determined, preferably in polar coordinates, so that the oscillating movement of the grinding wheels 40a to 40d can be adjusted according to path points at which the desired polygon cones 14a, 14c or 17b, 17d ultimately result.

The oscillating movement of the grinding wheels 40a to 40d per se is referred to as a so-called "web operation", to which a feed movement as a so-called "feed operation" must of course be superimposed to remove an oversize.

While grinding non-rotationally symmetrical workpieces, for example for grinding cams of a camshaft, it has already been proposed to separate the infeed mode and the rail mode in time by first immersing the grinding wheel in the infeed mode to the depth of the finished surface and then in pure web operation grinds the cam, in the present case of grinding polygon cones, however, an embodiment is preferred in which the web operation and the delivery operation are superimposed. In practice, this means that the grinding wheel also works towards the finished surface on a spirally wound path to remove large oversizes, the grinding wheels 40a to 40d always executing a superimposed movement, which are components of both pure rail operation and pure feed operation contains.

Either a constant angular velocity or an angular velocity varying over the circumference of the workpiece blank can be set to rotate the workpiece blanks.

Varying the angular velocity over the circumference, if the first derivative of the eccentricity value e 'after the angle of rotation Θ is taken into account, the case can be obtained that the so-called related chip removal volume Q' is constant during the entire grinding process. This leads to a uniform thermal load on the grinding wheel and thus to a high surface quality.

18 to 21 will now be used to explain how the grinding machine can be set in the numerical example described above with reference to FIGS. 7 to 11.

For example, a polygonal cone with the same thickness along the generatrix for a given cone taper on a CNC grinding machine using three axes in rail operation when using a controller to determine the X, C and B coordinates and in conjunction with a profiled grinding wheel (conical shape).

The contour equation in the parameter form is used to describe the polygon curve when grinding in path mode between the X and C axes.

In Fig. 18, the contour parameters s are drawn in as the profile elevation value and s' as the height offset value. The height offset value s' is the first derivative of the contour parameter s after the angle of rotation Θ. Depending on the angle of rotation Θ, the values s and s ′ can be represented in a Cartesian coordinate system using the following equations:
s = e (1 - cos (n · Θ))
s ′ = n · e · sin (n · Θ)

In Fig. 18 is p'p˝ the X-way as an oscillating movement of the X-axis with a height offset PP˝ on the tangent T in the contact point P, corresponding to a grinding wheel radius R _s or R _s '. Then for the traverse path X of the grinding wheel:
X = √ (R _s + r₁ + s) ² + (s ′) ² - (R _s + r₁)

If one designates α as the manufacturing angle of rotation, this results in:

The polygon profile K for the eccentric value e, which corresponds to the large cone diameter, and the polygon profile k, which determines the small cone diameter at an eccentric value e ', must be produced at the same time during the surface creation in order to produce an equally thick polygon cone to edit.

In this regard, is adjusted during the machining of the B-axis in railway operation to the XC-axes with an angle ± Δλ, so that the conically shaped grinding wheel at an angle δ enables a contact point P₁ on the polygon profile k with a height offset value H ' that deviates from the height offset value H of the polygon profile K.

The dressing angle δ is described according to the following parameter equations:
H = R _s · sin α
H ′ = H - (e - e ′) · n

Applying the above considerations to the application example already numbered above, the following results for a grinding wheel diameter D _s of 600 millimeters:
s = e · (1 - cos 3Θ)
s ′ = 3 · e · sin 3Θ

Fig. 19 shows in a diagram the relationships for s, s' and Δλ in the following points: Θ = 0 ° s = 0 mm s ′ = 0 mm Δλ = -0.2 ° Θ = 30 ° s = 1.4 mm s ′ = 4.2 mm Δλ = 0 ° Θ = 60 ° s = 2.8 mm s ′ = 0 mm Δλ = + 0.2 ° Θ = 90 ° s = 1.4 mm s ′ = -4.2 mm Δλ = 0 ° Θ = 120 ° s = 0 mm s ′ = 0 mm Δλ = -0.2 °

If you transfer these geometry values to a path operation for the grinding machine when the C, X and B axes are moving, the following results when considering the formula:
X = √ (300 + 18.6 + s) ² + (s ′) ² - (300 + 18.6)
using the values given in the table above, the following table of values for axes C, X and B: C = 0 ° X = 0.0000 mm B = -0.2 ° C = 30 ° X = 1.4276 mm B = 0 ° C = 60 ° X = 2.8000 mm B = + 0.2 ° C = 90 ° X = 1.4276 mm B = 0 ° c = 120 ° X = 0.0000 mm B = -0.2 °

The values mentioned above in the table are plotted in FIG. 20.

Finally, the required grinding wheel profile for the example numbered above is given. With the values:
Θ = 30 °
s = 1.4 mm
s ′ = 4.2 mm
R _s = 300 mm
r₁ = 18.6 mm
L = 30 mm
results from the formula

The values:
H = R _s · sin α = 300 · sin 0.752 ° = 3.937 mm
H ′ = H - (e - e ′) · n = 3.937 - (1.4 - 1.295) · 3 = 3.622 mm

These variables are shown again in FIG. 21 for illustration.

Claims (8)

1. A method for grinding a cone (14; 17), the radial cross-sectional area of which has the shape of an n-polygon profile (30) with a predetermined eccentricity (e) and a predetermined arithmetic mean (D _m ) of the diameter (D, d) of the Has the circumference (31) and the incircle (32) of the polygon profile (30), in which a grinding wheel (40) rotatable about a first axis (42) along a common surface line (41) on one around a second axis (43) rotatable workpiece blank (46), the axes (42, 43) intersecting at an acute angle (), at which the grinding wheel (40) also carries out an oscillating movement, in which the surface line (41) in the direction of the axes ( 42, 43) spanned plane with the n-fold frequency of the rotary movement of the workpiece blank (46) and with an amplitude corresponding to twice the eccentricity (2e cos), characterized in that the amplitude is set with an eccentricity value (e ′), of the relationship:

with the boundary conditions:
e ′ <e

ρ ₂ =

- 8e ′
obeys, where D _{m2 is} the arithmetic mean of the diameter (D₂, d₂) and ρ ₂ the practically minimum possible radius of curvature of the polygon profile (30/2) at the tip end of the cone (14/17) and β _max a predetermined angle. 2. The method according to claim 1, characterized in that the angle β _{max is} 14 °. 3. The method according to one or both of claims 1 or 2, characterized in that the grinding wheel (40) along an X-axis running at right angles to the surface line (41) in predetermined axial steps and the workpiece blank (46) about the second axis (43) can be adjusted with predetermined angular steps by means of a numerical control device (66). 4. The method according to claim 3, characterized in that the adjustment of the grinding wheel (40) is carried out in predetermined axial steps along the X-axis as a superimposition of a continuous radial feed movement and a path movement according to the polygon profile (30). 5. The method according to claim 3 or 4, characterized in that the adjustment of the workpiece blank (46) with predetermined angular steps about the second axis (43) at an angular velocity (dΘ / dt), which depends on the angle of rotation after the second derivative of the eccentricity (e ') Is set according to the angle of rotation (Θ). 6. The method according to one or more of claims 3 to 5, characterized in that the movement of the grinding wheel (40) in the X-axis according to the relationship:
X = √ (R _s + r₁ + s) ² + (s ′) ² - (R _s + r₁)
is set with:
R _s = radius of the grinding wheel (40)
r₁ = inner radius of the polygon profile (30/2) at the tip end of the cone (14/17);
s = profile survey value
= e (1 - cos (n · Θ))
with Θ = angle of rotation about the second axis (43)
s ′ = ds / dΘ 7. The method according to claim 6, characterized in that the grinding wheel (40) is pivoted about an axis perpendicular to its axis of rotation (B). 8. The method according to claim 7, characterized in that the pivot angle Δλ according to the relationship:

is set with
D₁ = the larger diameter of the polygon profile (30/1) at the blunt end of the cone (14/17);
D _m1 = the arithmetic mean of the diameters (D₁, d₁) of the polygon profile (30/1) at the blunt end of the cone (14/17);
L = axial length of the cone (14/17).

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