TOOTHING OF A GEARWHEEL

Abstract

The invention includes a toothing of a gearwheel having a plurality of teeth, the tooth flanks of which have a main region and a tooth root region. The tooth root region extends from a root circle (FP) as far as a main circle (d.sub.H), as considered in the end section or normal section through the axis of rotation of the gearwheel. As considered in each case in the end section or normal section, the tooth flanks in the tooth root region are designed as a Bézier curve from a relevant diameter (d.sub.r) in the direction of the tooth root. The Bézier curve merges in each case at a main point (P.sub.0, P.sub.3) in the relevant diameter (d.sub.r) in a continuous tangent into the tooth profile of the main region.

Claims

1. A toothing of a gearwheel having a plurality of teeth, said toothing comprising tooth flanks including a main region and a tooth root region; whereby, when considered in a cross section: said tooth root region extends from a root circle as far as a main circle (d.sub.H) through the axis of rotation of the gearwheel; said tooth flanks in said tooth root region are designed as a Bézier curve starting from a relevant diameter (d.sub.r) in the direction toward the root circle; and said Bézier curve merges in each case at a main point (P.sub.0, P.sub.3) in said relevant diameter (d.sub.r) in a continuous tangent into the tooth profile of said main region.

2. The toothing according to claim 1, wherein the relevant diameter (d.sub.r) coincides with the main circle (d.sub.H).

3. The toothing according to claim 1, wherein the Bézier curve includes at least two control points (P.sub.1, P.sub.2), each of which are positioned in the tooth root region on a tangent (t.sub.1, t.sub.2) at a main point (P.sub.0, P.sub.3).

4. The toothing according to claim 1, wherein the Bézier curve is a Bézier curve of the third or higher degree, in particular a cubic Bézier curve including precisely two control points (P.sub.1, P.sub.2), each of which is positioned on a tangent (t.sub.1, t.sub.2) proceeding through a main point (P.sub.0, P.sub.3), whereby the distance (k) of each control point (P.sub.1, P.sub.2) relative to its main point (P.sub.0, P.sub.3) on a tangent (t.sub.1, t.sub.2) is calculated as follows:
k=(0.25+0.1×f)×1
with:
0<k≦1 and 0≦f≦3, whereby 1 represents the distance of the one main point (P0, P3) from the intersection (S) of the tangents (t.sub.1, t.sub.2).

5. The toothing according to claim 4, wherein the factor f is preferably between 0.5 and 1.5.

6. The toothing according to claim 1, wherein the tooth flanks of adjacent teeth located in the face section or normal section are designed symmetrical relative to each other, whereby the axis of symmetry (y) intersects at the root point in the root circle; and whereby the tangents (t.sub.1, t.sub.2) proceed symmetrical to the axis of symmetry (y) and, in the case of external toothing, intersect radially inside the relevant diameter (d.sub.r) in an intersection (S).

7. The toothing according to claim 1, wherein the tooth flanks of adjacent teeth located in the face section or normal section are always designed asymmetrical relative to each other.

8. The toothing according to claim 1, wherein the Bézier curve includes a control polygon, whereby the entire control polygon that connects main points (P.sub.0,P.sub.3) as well as the at least two control points (P.sub.1, P.sub.2) with each other is positioned inside the area spanned by tangents (t.sub.1, t.sub.2) and the Bézier curve.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0028] The above-mentioned and other features and advantages of this invention, and the manner of attaining them, will become more apparent and the invention will be better understood by reference to the following description of embodiments of the invention taken in conjunction with the accompanying drawings, wherein:

[0029] FIG. 1 illustrates the parameters on an involute externally toothed gearwheel having spur gear toothing, shown in face section in two embodiments;

[0030] FIG. 2 illustrates the arrangement of the tooth root region on a gearwheel having symmetric toothing according to FIG. 1 consistent with the invention;

[0031] FIG. 3 illustrates the arrangement of the tooth root region on a gear having asymmetric toothing according to FIG. 1 consistent with the invention;

[0032] FIGS. 4a and 4b show two additional embodiments in a further development of the illustration in FIG. 1; and

[0033] FIG. 5 is a detailed illustration of toothing as illustrated in FIG. 1 or in FIG. 4b.

[0034] Corresponding reference characters indicate corresponding parts throughout the several views. The exemplifications set out herein illustrates embodiments of the invention, and such exemplifications are not to be construed as limiting the scope of the invention in any manner.

DETAILED DESCRIPTION OF THE INVENTION

[0035] FIG. 1 illustrates the parameters on a tooth gap 1 for two embodiments in a partial face section, vertical to the non-illustrated axis of rotation of an externally toothed gearwheel—viewed in direction of the axis of rotation. The reference values are hereby the coordinates x, y, whereby the y-axis is at the same time the symmetrical axis of tooth gap 1. The x-axis or more precisely the origin of the illustrated x-y coordinate system shall thereby proceed through the non-illustrated rotational axis of the gearwheel. The illustration of FIG. 1 thereby shows on the left of the symmetry axis one design form whereby the inventive toothing has no protuberance. The illustration on the right of the symmetry axis in contrast shows a design form whereby a protuberance 6 is provided. The latter is exaggerated and not to scale.

[0036] The sections of the two teeth 2 indicated in the two illustrations are thereby restricted in their tip region 3 by a tip circle that is not illustrated. The tip circle may be consistent with the outside diameter of tip region 3. Tooth profile 4 that herein is selected as an example is an involute tooth flank shape that each time is used up to a diameter d.sub.N of the so-called useful circle of the non-illustrated tooth flank of the tooth of a mating gearwheel or respectively gear element meshing with this gearwheel. In regard to both embodiments illustrated in FIG. 1, the section between the tip circle in the region of tips 3 of teeth 2 and a useful circle d.sub.N is referred to in the following as “useful region”. Moreover, reference is also to be made to the diameter up to which a tooth of a mating gearwheel or respective tooth element meshing with this gearwheel engages into the tooth gap. This diameter is typically described as a free circle diameter d.sub.FR. The region between useful circle d.sub.N and the deepest point—that is the radially innermost point of tooth gap 1 in which the so-called root circle d.sub.f is positioned, which in the case of the herein illustrated external toothing always adjoins in direction of the gearwheel center-is identified in the following text as the tooth root region of tooth gap 1. For the case that an internal toothing is provided, the region between useful circle d.sub.N and the radially outermost point of tooth gap 1 adjoins free circle diameter d.sub.FR in the direction away from the gearwheel center.

[0037] The intersection of symmetry axis y with root circle d.sub.F is thereby root point FP of tooth gap 1.

[0038] The parameters described herein thus far are usual and common parameters on all gearwheels and will be relied on in the subsequent detailed description of the inventive arrangement of the tooth root region that herein is illustrated in the inventive manner.

[0039] In addition, additional parameters are significant for the herein described embodiments of the involute tooth profile 4. Thus, the so-called base circle d.sub.b is drawn in FIG. 1 which is relevant for the design of tooth profile 4 of the involute toothing. Moreover, a brief reference should be made to module m that is commonly used in toothing and that results from the pitch diameter which is not illustrated here, divided by the number of teeth; or respectively, division p divided by factor π. Moreover, both embodiments of FIG. 1 show a main circle d.sub.H which together with the tip circle defines a main region of the tooth profile. In the current example the main region directly adjoins the tooth root region. In the left side of the illustration in FIG. 1, the main region is consistent with the useful region. Main circle d.sub.H and useful circle d.sub.N therefore coincide with each other. In contrast, in the right side of the illustration, part of the main region in addition to the ancillary region is a protuberance region in which protuberance 6 is located. In this case the protuberance region proceeds between useful circle d.sub.N and a protuberance circle d.sub.P that is located radially internally opposite same. Accordingly, in the case of internal toothing, protuberance circle d.sub.P would be positional radially outside useful circle d.sub.N. In the current example, the main region is therefore limited in the radial direction toward the inside by main circle d.sub.H which is consistent with protuberance circle d.sub.P. Protuberance circle d.sub.P can thereby pass through the radial innermost end of the protuberance profile which, for example, may be part of an arc. If internal toothing is provided the main region would therefore be limited similarly in the radial direction toward the outside by respective main circle d.sub.H, so that protuberance circle d.sub.P can pass accordingly through the radial outermost end of the protuberance profile.

[0040] Moreover, the diameter or respectively the radius can be recognized in FIG. 1 which is relevant for the invention and which is to be designated as relevant diameter d.sub.r. In terms of the current invention, relevant diameter d.sub.r—if it is imposed on main circle d.sub.H—is consistent with the so-called form circle of conventional toothing. In the left side of the illustration in FIG. 1, relevant diameter d.sub.r is selected from the arithmetic mean between useful circle diameter d.sub.N and free circle diameter d.sub.FR, so that a certain safety distance is created between relevant diameter d.sub.r and useful circle diameter d.sub.N. This ensures that a tooth (not illustrated here) of a mating gear element intermeshing with the gearwheel in each case runs off the calculated form of tooth flank 4—in this case the involute—and does not engage in a load bearing manner on the inventively designed shape of the flank in the tooth root region. In the right side of the illustration in FIG. 1, relevant diameter d.sub.r is selected smaller in the current example than protuberance circle d.sub.P. This also contributes to ensure a certain safety distance, in particular to protuberance 6. In the case of internal toothing the protuberance circle would be selected to be accordingly greater than the relevant diameter. This may be the case, but is not obligatory.

[0041] One alternative in the selection of the relevant diameter d.sub.r according to FIG. 1 is illustrated in FIGS. 4a and 4b, in each case in a partial face section. Corresponding elements are identified with corresponding reference identifications. In FIG. 4a it can be seen that relevant diameter d.sub.r coincides with main circle d.sub.H and at the same time is consistent with useful circle d.sub.N. In this case no protuberance is provided analog to the left side of the illustration in FIG. 1. Such a protuberance 6 is however illustrated greatly exaggerated and not to scale in FIG. 4b. In the current example it is located between useful circle d.sub.N and relevant diameter d.sub.r which, in this case coinciding with main circle d.sub.H and at the same time with protuberance circle d.sub.P.

[0042] In FIG. 2 the shape of the tooth root is illustrated in one inventive arrangement of an example of external toothing, without protuberance. The illustrated toothing could also be implemented as internal toothing, with or without protuberances. The elements already referred to in FIG. 1 are identified also in FIG. 2 with the same references. Of the diameters discussed in FIG. 1 only relevant diameter d.sub.r is now shown in FIG. 2. As already mentioned—due to safety and tolerance based reasons—tooth profile 4 of the useful region in the current design example merges in the region of relevant diameter d.sub.r in a continuous tangent into the inventive arrangement of the tooth root form in the tooth root region. At points P.sub.0 and P.sub.3 which are also referred to as main points where diameter d.sub.r intersects tooth profile 4, this transition occurs from involute tooth profile 4 into a Bézier curve 5.

[0043] According to the embodiment in FIG. 2 where the plane of symmetry of the illustrated and adjacent tooth flanks 4 proceeds perpendicular to the drawing plane through the y-axis, the tooth root region is also symmetrical to the plane of symmetry that proceeds through the y-axis.

[0044] Tangents t.sub.1 and t.sub.2 intersect at main points P.sub.0 and P.sub.3 at intersection point S on axis of symmetry y. In the current example control points P.sub.1 and P.sub.2 are located on tangents t.sub.1, t.sub.2. Near main points P.sub.0 and P.sub.3 additional control points Q.sub.0 and Q.sub.2 are positioned on tangents t.sub.1 and t.sub.2. One additional control point Q.sub.1 is moreover provided. Control points Q.sub.0, Q.sub.1 and Q.sub.2 respectively form the end points of the illustrated vertical dash-dot line. Control point Q.sub.1 is thereby positioned on a double dash-dot line that connects control points P.sub.1 and P.sub.2. The respective main and control points in FIG. 2 are connected with each other and are identified as a control polygon. The latter is thereby position inside the area spanned by tangent t.sub.1 and t.sub.2 as well as Bézier curve 5.

[0045] Distance k of control points P1 and P2 from the corresponding main points P0 and P3 along the respective tangent t1 and t2 is thereby selected so that it is according to the following relationship:

k=(0.25+0.1×f)×1

with:

0<k≦1 and 0≦f≦3,

whereby 1 indicates the distance of the respective main point P.sub.0, P.sub.3 from intersection S of tangent t.sub.1, t.sub.2.

[0046] For construction of the Bézier curve—as it is illustrated in the remaining figures—one can proceed as previously discussed also as indicated in the remaining figures.

[0047] A comparative arrangement for an asymmetric external toothing can be seen in FIG. 3. Such an asymmetry is also conceivable for internal toothing. In each case a protuberance could also be provided, even though not illustrated. In the case of symmetric toothing, the y-axis that is illustrated herein as a dash-dot line would represent the symmetry axis analog to the y-axis in FIG. 2. Also, in the case of the asymmetric illustrated toothing, tooth profile 4 that in this case is arranged differently on the two sides of tooth gap 1 merges at the respective main points P.sub.0, P.sub.3 tangentially into Bézier curve 5. Control points P.sub.1, P.sub.2 are again positioned on these tangents t.sub.1, t.sub.2 which virtually continue the tangential transition in the main points P.sub.0, P.sub.3, in this case in a downward direction. An intersection S of tangents t.sub.1, t.sub.2 also occurs hereby in most cases; however, it is not located on the symmetry axis or respectively the y-axis, as can be seen from the illustration in FIG. 3, even though the intersection in this case is no longer within the illustration.

[0048] FIG. 5 is a detailed view of the toothing shown for example in the right side of the illustration in FIG. 1 or FIG. 4b. A continuous tangent transition between the protuberance profile and protuberance 6 in the main region and Bézier curve 5 in the tooth root region is seen in FIG. 5 in an example of an external toothing. The transition occurs herein in main point P.sub.0 through which relevant diameter d.sub.r proceeds. The transition is hereby located at a distance of one undercut F.sub.S from tooth flank 2, measured parallel to the tooth flank. The same also applies to an accordingly designed internal toothing.

[0049] In principle, and independent of a specific embodiment illustrated in the drawings, Bézier curve 5 can be mathematically represented by means of the Bernstein polynomial:

[00001]$\stackrel{.fwdarw.}{X}\left(t\right)=\underset{i=0}{\overset{n}{.Math.}}.Math.\left(\begin{array}{c}n\\ i\end{array}\right).Math.t^i.Math.{\left(1-t\right)}^{n-i}.Math.{\stackrel{.fwdarw.}{P}}_i$

{right arrow over (P)}.sub.i are hereby the directional vectors to the support points (main- and control points)

[00002]${\stackrel{.fwdarw.}{P}}_0=\left(\begin{array}{c}{P}_{0.Math.x}\\ P_{0.Math.y}\\ 0\end{array}\right).Math..Math.{\stackrel{.fwdarw.}{P}}_1=\left(\begin{array}{c}{P}_{1.Math.x}\\ P_{1.Math.y}\\ 0\end{array}\right).Math..Math.{\stackrel{.fwdarw.}{P}}_2=\left(\begin{array}{c}{P}_{2.Math.x}\\ P_{2.Math.y}\\ 0\end{array}\right).Math..Math.{\stackrel{.fwdarw.}{P}}_3=\left(\begin{array}{c}{P}_{3.Math.x}\\ P_{3.Math.y}\\ 0\end{array}\right)$

[0050] The following applies for cubic Bézier curves:

[00003]$\stackrel{.fwdarw.}{X}\left(t\right)=\underset{i=0}{\overset{3}{.Math.}}.Math.\left(\begin{array}{c}3\\ 1\end{array}\right).Math.t^i.Math.{\left(1-t\right)}^{3-i}.Math.{\stackrel{.fwdarw.}{P}}_i={\left(1-t\right)}^3.Math.{\stackrel{.fwdarw.}{P}}_0+3.Math.{t\left(1-t\right)}^2.Math.{\stackrel{.fwdarw.}{P}}_1+3.Math.t^2\left(1-t\right).Math.{\stackrel{.fwdarw.}{P}}_2+t^3.Math.{\stackrel{.fwdarw.}{P}}_3.Math..Math.\stackrel{.fwdarw.}{X}\left(t\right)=\left(-{\stackrel{.fwdarw.}{P}}_0+3.Math.{\stackrel{.fwdarw.}{P}}_1-3.Math.{\stackrel{.fwdarw.}{P}}_2+{\stackrel{.fwdarw.}{P}}_3\right).Math.t^3+\left(3.Math.{\stackrel{.fwdarw.}{P}}_0-6.Math.{\stackrel{.fwdarw.}{P}}_1+3.Math.{\stackrel{.fwdarw.}{P}}_2\right).Math.t^2+\left(-3.Math.{\stackrel{.fwdarw.}{P}}_0+3.Math.{\stackrel{.fwdarw.}{P}}_1\right).Math.t+{\stackrel{.fwdarw.}{P}}_0$

[0051] With the introduction of the vectorial factors the following applies:

{right arrow over (D)}=−{right arrow over (P)}.sub.0+3.Math.{right arrow over (P)}.sub.1−3.Math.{right arrow over (P)}.sub.2+{right arrow over (P)}.sub.3

{right arrow over (C)}=3.Math.{right arrow over (P)}.sub.0−6.Math.{right arrow over (P)}.sub.1+3.Math.{right arrow over (P)}.sub.2

{right arrow over (B)}=−3.Math.{right arrow over (P)}.sub.0+3.Math.{right arrow over (P)}.sub.1

{right arrow over (A)}={right arrow over (P)}.sub.0

[0052] Resulting thus in the parameter shape of the Bézier curve:

{right arrow over (X)}(t)={right arrow over (D)}.Math.t.sup.3+{right arrow over (C)}.Math.t.sup.2+{right arrow over (B)}.Math.t+{right arrow over (A)}

[0053] If all points {right arrow over (X)} for t∈[0;1] are calculated, then the Bézier curve results between {right arrow over (P)}.sub.0 and {right arrow over (P)}.sub.3 with control points {right arrow over (P)}.sub.1 and {right arrow over (P)}.sub.2.

[0054] One specific example for a symmetric toothing of a gearwheel pair by way of values that are selected from the aforementioned value ranges is explained below. The selected identifications and formula symbols are those that are commonly used and recognized.

[0055] A gearwheel toothed according to one embodiment and its mating gearwheel can for example have the following parameters: [0056] Gearwheel: 1 [0057] Module: 4 [0058] Number of teeth: 50 [0059] Pressure angle: 20° [0060] Addendum modification coefficient: 0.2 [0061] Mating gearwheel: 2 [0062] Module: 4 [0063] Number of teeth: 35 [0064] Pressure angle: 20° [0065] Addendum modification coefficient: 0.321 [0066] Distance from axis: 172 mm [0067] Comparison: Toothing according to DIN 867 [0068] Hobbed fillet contour without protuberance [0069] tip height factor h.sub.aP0=1.389 [0070] tip rounding factor ρ.sub.aP0=0.25 [0071] Tool-machining allowance, in other words finishing: 0.0 [0072] (* indicates: module dependent) [0073] Useful circle diameter: d.sub.n1=195.617 mm [0074] Root circle diameter: d.sub.f1=189.79 mm [0075] Transition diameter involute Bézier curve; this is consistent with the relevant diameter d.sub.r: d.sub.ü1=193.48 mm

[0076] From this data the transition point from the useful region to the tooth root region of the toothing can be easily determined with the coordination system origination in the center of the gearwheel with the tooth gap center on the y-axis.

[0077] Pressure angle at the transition diameter: a.sub.ü=14.796°

[0078] Only the control point is still missing for determining the Bézier curve. On the transition diameter the transition point—here identified as main point P.sub.0 or P.sub.3—is defined on the left as well as on the right flank (in the case of symmetric toothing symmetrical to the y-axis). A tangent t.sub.1, t.sub.2 is applied to the involute at both main points. The intersection of tangent t.sub.1, t.sub.2 applied to the left flank with that to the right flank results in intersection S.

[0079] The two control points P.sub.1 and P.sub.2 can now theoretically be positioned on each point of the straight P.sub.oS or P.sub.3S whereby the variable k∈[0; 1] is defined.

[0080] After determining the points, a cubic equation is developed in vector style with the Bernstein polynomial:

{right arrow over (X)}(t)={right arrow over (D)}.Math.t.sup.3+{right arrow over (C)}.Math.t.sup.2+B.Math.t+{right arrow over (A)}

[0081] With:

[00004]$D=\left(\begin{array}{c}-3.578\\ 0\end{array}\right)$$C=\left(\begin{array}{c}5.368\\ 6.639\end{array}\right)$$B=\left(\begin{array}{c}1.754\\ -6.639\end{array}\right)$$A=\left(\begin{array}{c}-1.771\\ 96.724\end{array}\right)$

[0082] This now creates the Bézier curve.

[0083] As a result, an improvement of approximately 35% compared to conventional toothing is achieved with this calculation, in addition to a calculation saving of up to 25 calculations compared with the toothing mentioned at the beginning with tangent function and arc.

[0084] The production of such gearwheels can occur for example by means of milling or grinding machines that are freely movable in several axes and are freely programmable, or by way of suitable hobbing cutters that are derived from the inventive tooth root form.

[0085] While this invention has been described with respect to at least one embodiment, the present invention can be further modified within the spirit and scope of this disclosure. This application is therefore intended to cover any variations, uses, or adaptations of the invention using its general principles. Further, this application is intended to cover such departures from the present disclosure as come within known or customary practice in the art to which this invention pertains and which fall within the limits of the appended claims.