JPS6222729B2 - - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION Recently, research on the drum-shaped worm gear, which is expected to have a high load capacity, has progressed, and its superior performance has been gradually demonstrated. However, the problem lies in the method of manufacturing this gear. Normally, machining the tooth surface of a worm is not very difficult, including grinding, but a major problem remains: how to precisely machine the tooth surface of a wheel. In this case, it would be very expensive to manufacture an hourglass-shaped worm gear hob with an irregular pressure angle and variable lead thread surface as the cutting surface, and the Lorentz gear hob used in conventional Hindley-type hourglass-shaped worm gears would be very expensive. The method cannot also be used with other hourglass worm gears. This invention aims to provide a simple wheel cutting method that can be commonly applied to various types of hourglass-shaped worm gears, and will be described in detail below in comparison with known techniques. The conventionally known Lorentz gear cutting method is a simple wheel cutting method for Hindley type hourglass-shaped worm gears. This machining method focuses on the fact that the tooth surface of the worm is a linear surface created by a straight cutting edge, and uses the generatrix located at the end of the worm tooth surface as the cutting edge of the flying tool for wheel cutting. It's a method. Therefore, it cannot be applied to the cutting of wheels of hourglass-shaped worm gears, which have been recently developed in various fields and whose worm tooth surfaces are ground by flat or conical surfaces. Furthermore, even when limited to Hindley type hourglass worm gears, it could not be applied to the purpose of modifying the tooth profile of the wheel as intended by the designer. When cutting a drum-shaped worm wheel, this invention replaces the theoretical tooth surface of the wheel with a linear surface that is sufficiently close to it, and then cuts this linear surface using a machining tool with a single straight cutting blade. It's something I'm trying to release. At this time, the tool and workpiece are made to perform rotational motion at a constant rotation ratio, and the following properties of the linear surface given to the wheel are given as intended by the designer, and the linear cutting edge, tool axis, and workpiece axis are It is possible to determine the relative position of 1. Determine one generatrix of the line woven surface at any position on the work blank. 2. Setting the radius of curvature of the normal cut in the direction perpendicular to this generatrix at any two points on this generatrix to an arbitrary value. 3. The direction of the normal line on the generatrix must be set in a constant and arbitrary direction. The relationships between these various properties, the cutting edge straight line, and each axis are as shown below, and all the specifications necessary for machining can be determined based on these relational expressions. In Figure 1, _A2 is the work axis and A is the generatrix to be determined. The relationship between A ₂ and A is determined by the designer and is known. The axis of the tool with this generatrix as the cutting edge straight line is A _c . There is one common perpendicular line Co between A ₂ and A _c , and the instantaneous axis of relative motion of the rotational motion of the workpiece and tool around A ₂ and A _c (see Note 1) A _p is perpendicular to Co. are doing. Now, let the angular velocity of the rotational motion of the workpiece around A ₂ be ω ₂ and that of the tool around A _c be ω _c , and the ratio of the two, that is, the rotation ratio, be i=ω _c /ω ₂ . Let h be the conversion pitch (see Note 2) of the relative screw motion around the instantaneous axis A _p . The angle between A ₂ and A _p is Γ
_2. Let Γ _c be the angle formed by A _p and A _c . At this time A _2
The case where the angle from A _p to A _c is clockwise is defined as the case where Γ ₂ and Γ _c are positive. Further, the distance between the axes of A ₂ and A _p and that of A _p and A _c are respectively Γ ₂ and Γ _c . To understand the local shape of a linear surface created by a straight cutting edge on a workpiece, it is sufficient to consider the shape equivalent to a circle of curvature in the case of a planar figure. It is to consider this line-woven surface locally as a line-woven screw surface (a line-woven surface is given such a name to any surface created by straight lines. However, what is industrially important is those lines. The question is whether a woven surface can be easily and accurately created.In that sense, a threaded surface can be said to relatively satisfy the above requirements.Although not strictly speaking, this linear woven surface can be approximated by a threaded surface. Moreover, it would be easier to create this thread surface with a straight line, so we decided to approximate it with a linear thread surface.) This axis is called the curvature axis (see Note 3). The axis of curvature at the moment when the straight cutting edge is at position A is A _H
shall be. The generatrix A, the instantaneous axis A _p and the axis of curvature A _H share one common perpendicular. Let this be C. The angle from A to A _H is Φ with the clockwise direction being positive.
Similarly, let the angle from ρ, A to A _p be Φr. A
Let the distance between the axes of and A _p be Φr . Further, the angle formed by two common perpendicular lines Co and C is defined as Θ, with the clockwise direction from C to Co being positive. Also, let the distance between both straight lines be Θ . In this case, due to the properties of the instantaneous axis (see Note 4), the following formula holds true. isinΓ _c = −sinΓ ₂ …(1) i Γ _c cosΓ _c = − Γ ₂ cosΓ ₂ …(2) h=−Γ ₂ tanΓ _c …(3) All normals on the generatrix of interest are In other words, the condition that the wire faces in a certain direction is the condition that the wire weave surface becomes a developable surface in the vicinity of this generatrix. Linear surface theory (Note 5)
Therefore, this condition is h+ Φr cotΦr=0...(4) If the curvature radius of the normal cut in the direction perpendicular to the generatrix of any two points A and B on this generatrix is R _A and R _B (the 3
), tanΦρ=R _B −R _A /AB ...(5) From the Euler Savary equation (Note 6) in space, tanΦρ=(icosΓ _c − cosΓ ₂ ) sin ² Φr/(icos Γ _c − cosΓ ₂ ) sin Φr cos
Φr+sinΓ ₂ cosΘ...(6) Let the unit vector on the work axis A ₂ be a ₂ and that on the generatrix A be a. a ₂ and a are known vectors. If the tooth surface is a developable surface and the normal (Note 7) to a certain generatrix is c=n×a...(7) where c is the common perpendicular C of the generatrix, instantaneous axis, and curvature axis.
It is a unit vector on . Equation (7) shows that this common perpendicular line is on the tangential plane of this developable surface. This common perpendicular passes through the recursion point of the developable surface. If that point is P, then =R _A /tanΦρ (8) The common perpendicular C is uniquely determined by equations (7) and (8).
Based on this and equations (1) to (6), it is possible to arbitrarily give i and determine the tool axis A _c (Note 8). As described above, the generatrix of the linear weave surface given as the wheel tooth surface, the radius of curvature of the normal cut in the direction perpendicular to the generatrix at any two points on the generatrix, and the normal line on the generatrix are all set in a constant direction. If you specify
The arrangement of the tool axis can be determined for any rotation ratio between the workpiece and the tool. Therefore, by creating a flying tool whose generatrix is a straight cutting edge and processing it into a workpiece at the determined relative position, a linear weave surface meeting the above conditions can be obtained. As described above, the present invention has the great advantage that, while the Lorentz gear cutting method is effective only for Hindley worm gears, it can be applied to approximate cutting of wheels of any type of hourglass-shaped worm gears. In addition, since the Lorentz gear cutting method uses a straight line on the worm tooth surface as the cutting edge of the flying tool, it had the disadvantage that the approximation accuracy deteriorates as the reduction ratio becomes smaller. Since the gear cutting method according to the invention directly deals with the wheel tooth surface,
It doesn't have its drawbacks. Furthermore, since there is a considerable degree of freedom in the line weave surface to be provided, there is also the advantage that the wheel tooth surface can be modified and cut as desired. In the above description, the method was explained as an approximation cutting method for a drum-shaped wheel, but it is also applicable to wheel cutting of a cylindrical worm gear, approximate cutting of a ring gear tooth surface of a hypoid gear, or approximate cutting of a gear tooth surface of a spiroid gear. It is possible. (Note 1) Momentary axes are two rigid bodies. In this case, when the workpiece axis A ₂ and the tool axis A _c are in relative motion, at a certain moment one rigid body, the workpiece axis A ₂ , is the other rigid body, the tool. It can be considered that a screw movement (a combined rotation and translation movement) is being performed around a certain axis A _p with respect to the axis A _c . Conversely, the motion of A _c with respect to A ₂ can be viewed in the same way. Such an axis A _p is called an instantaneous axis. On a plane, for example, in Fig. 4, the rigid body A,
In the case of B and C, points such as Pac, Pab, and Pbc are the instantaneous centers. This instantaneous axis can be found from the following three equations. That is, isinΓ _c = -sinΓ ₂ ...(1) i Γ _c cosΓ _c = - Γ ₂ cosΓ ₂ ...(2) h = - Γ ₂ tanΓ _c ...(3) Given the three equations, A ₂ axes If the relative position of the and A _c axis is determined as Γ ₂ + Γ _c = Γ ... (a) Γ ₂ + Γ _c = Γ ... (b), then the five unknowns Γ _c , Γ _c , Γ ₂ , Γ
₂ , h are all determined using 5 equations. (Note 2) What is the conversion pitch h?As explained above regarding the instantaneous axis, the relative motion of two rigid bodies can be considered to be a screw movement about the instantaneous axis. In this screw movement, the ratio of rotation ω to translation ω is called conversion pitch h. For example, the conversion pitch of the screw motion of the rigid body A _c in the instantaneous axis A _p with respect to the rigid body A ₂ is determined by the above equation (3). (Note 3) The entire wheel tooth surface is considered to be a line-woven surface whose generatrix is the first contact line that appears at the center of the wheel tooth surface. In reality, as shown in FIG. 12, the first and second tooth surfaces are smoothly connected, so it is necessary to get as close to the actual tooth surface as possible. Therefore, it is necessary to know the local shape of the linear surface, which can be well explained by the Euler-Savary equation in space. In addition, the straight cutting edge A, the instantaneous axis A _p , and the curvature axis A _H of the linear surface created by the cutting edge A share one common perpendicular line. From that (6)
The expression is derived. (Note 4) How to derive equations (1), (2), and (3) will be explained using quantities called binary vectors and binary angles, but then the explanation of binary vectors and binary angles becomes I'll omit it since it will be necessary. For reference, please refer to Masao Naruse's ``Study of Gears'' edited by Yokendo (P77-115). (Note 5) Regarding the theory of linear surfaces; In general, the trajectory drawn by a directed straight line A, which is a function of a certain parameter t, as t changes is called a linear surface.
The two adjacent straight lines on the line weave surface are marked A and A as shown in Figure 5.
+dA, the angle between the two straight lines is dΦ, and the distance is dΦ
Then, P=dΦ/dΦ (c) The quantity P is called the distribution coefficient of the linear surface, and P represents the properties of the curved surface in the vicinity of A. Common perpendicular line of two straight lines A _H
Considering, A _H becomes the axis of curvature of the linear surface (same as the center of curvature of the plane). If the angle between the two straight lines B and B+dB is dδ, and the distance is dδ , then with K and K as certain variables, h _H = dδ/dδ = P+K・K/1+K ² ...(d) That is, the axis of curvature A _H is It is also the instantaneous screw movement axis of A. In particular, the wire weave surface when P=constant, K=constant, and K =constant is called the wire weave screw surface (Figure 6). At this time, if the angle between the instantaneous axis A _H and the straight line A is ε, and the distance is ε , then K=cot ε ε = (PK-K)/1+K ² , and from this relationship, P=h _H + ε cotε ...(e ) becomes. For example, when P = 0, K = 0, K = 0, h _H = 0, δ = 0, δ = 0. Furthermore, when P=0, K≠0, and K =0, h _H =0, δ=cot ^-1 K δ =0, resulting in a conical line weave surface as shown in FIG. Also, P≠0, K≠0, K ≠0 However, when P+K K = 0, δ=cot ^-1 K, δ = PK-K/1+K ² , h _H =0, and the rotating monoplane hyperboloid in Fig. 9 becomes. The above is an overview of the theory of linear surfaces. Here, in order for the linear weave surface to become a developable surface (a surface that can be developed into a flat surface, in the above example, Figures 7 and 8 are developable surfaces, and Figure 9 is not a developable surface), it is necessary to Adjacent bus lines A must intersect one after another. Therefore, h _H =dδ/dδ=0. In other words, equation (e) for the developable surface is h _H + ε cotε=0...(f) Now, since the axis of curvature A _H of the line-woven surface is also the instantaneous screw axis, equations (d) to (f) is being guided. Naturally, the instantaneous axis A _p
A similar relationship holds true between and A. That is, for a developable surface, h+ Φr cotΦr=0...(4) This is what equation (4) means. (Note 6) Euler-Savary's formula in space Euler-Savary's formula is useful for giving the curvature of a trochoid curve in a planar figure. That is, in Fig. 11, when points O and A' are fixed and rolled while touching each other at point P, point C on circle A is the center of curvature M of the trajectory (trochoid) x-x drawn with respect to circle O.
We show how to find . =r =P ∠A'PC=θ a; radius of circle A'R; radius of circle O, then 1/P-r+1/rcosθ=1/R+1/a M: Trochoid Center of curvature of x-x Space In a figure, a rigid body around the axes A ₂ and A _c
The instantaneous axes of A ₂ and A _c are A _p and the curvature axis of the line weave surface is A _H. At this time, the Euler Savary equation in space is used to find the position of the A _H axis and the size of the radius of curvature. That is, when expressed using binary angles (see Note 4), sinΦρcosΘ/sin(Φρ−Φr) sinΦ
r = cotΓ ₂ + cotΓ _c = -ω+εω/ω ₂ sinΓ ₂ ...(g) Here, Φρ is the binary angle between the axis of curvature of the weave surface A _H and the generating line A of the weave surface Θ; A _H and The common perpendicular line of A is C, and the work axis A ₂
When the common perpendicular line between C and tool axis A _c is _C0 , C and Co
Binary angle formed by Φr; When the instantaneous axis is A _p , the binary angle between A _p and A, where Φρ=φρ+ε φρ , Θ=θ+ε
θ , Φr=φr+ε φr , Γ ₂ =γ ₂ +ε γ ₂ ,
Γ _c = γc + ε γc This equation is the Eulerian sabery equation in space. Equation (6) is obtained by comparing the real parts of the above equations. tan Φρ=(icos Γ _c − cos _{Γ 2} ) sin ² Φr/(icos Γ _c − cos Γ ₂ ) sin Φr cos
Φr+sinΓ ₂ cosΘ...(6) Also, Φρ can be found by comparing the imaginary parts. (Note 7) The normal line along the generatrix line is the normal line to the tooth surface of the worm wheel on a certain generatrix line A on the tooth surface of a theoretical drum-shaped worm wheel. Therefore, this normal n is theoretically determined and known. (Note 8) About finding A _c if C and i are given (see Figures 12 and 13). coordinate origin A ₂
If the center of the blank on the axis is A ₂ ⊥Co, then A ₂ is Z
Can be selected as the axis. Also, the common perpendicular line between the theoretical worm wheel axis _A2 and the theoretical worm axis is defined as the y-axis. First, n and a are the theoretical wormwheel tooth surface (first
Figure 2) is determined by selecting one bus line A on the top. Therefore, C is determined from equation (7). The Tomoki curve is also known, and the P point is determined. If we determine the radius of curvature RA of the theoretical worm wheel at that point A,
is known theoretically. Therefore, by equation (8), Φρ
is determined. Once A and C are determined, A _p is given as A _p (Φr, Φr ). Therefore, Co is A ₂ and A _p (Φr,
Co(Φr, Φr ) is found as the common perpendicular line of Φr ). Also, Γ ₂ and Γ ₂ are also Γ ₂ (Φr, Φ
r), Γ ₂ (Φr, Φr ). Once Co and C are determined, Θ and Θ are respectively Θ(Φ
r, Φr ) and Θ(Φr, Φr ). R _A , R
_B , is known on A. From equations (5) and (6), Φr can be found as Φr=Φr( Φr , _Γc , _Γ2 ). Also, Γ ₂ ( Φr , Γ _c ), Γ ₂ ( Φr , Γ _c ) From equation (4), h=h ( Φr , Γ _c , Γ ₂ ) From equation (3), Γ _c = Γ _c ( Φr , Γ ₂ ) Also, by substituting this Γ _c into _equations Γ ₂ − _{Γ 2} ( Φr ) Γ ₂ ( Φr ) (1) and ( ₂ ), we _get , Γ ₂ )=cos Γ _c ( Φr , Γ ₂ )=
−Γ ₂ ( Φr ) cosΓ ₂ ( Φr ) From these two equations, Φ
r, Γ ₂ are determined, and Γ ₂ and others are determined one after another.

[Brief explanation of the drawing]

FIG. 1 shows the arrangement of each axis at the moment when the cutting edge coincides with the generatrix. FIG. 2 shows the angular relationship between each axis. FIG. 3 is an explanation of the positional relationship between the radii of curvature R _A and R B of the normal cut in the direction perpendicular to the generatrix of two arbitrary points A and _B.
Figure 4 is an explanatory diagram corresponding to (Note 1). Figures 5 to 10 are explanatory diagrams corresponding to (Note 5). Figure 11 is an explanatory diagram corresponding to (Note 6). Figures 12 and 13 are drawings corresponding to (Note 8). In the diagram; A ₂ ... workpiece axis, A ... generatrix to be determined, A _c ... tool axis, Co ... A ₂ and A
Common perpendicular line between _c , C...generating line, A...common perpendicular line between instantaneous axis A _p and curvature axis A _H , A _p ...moment of relative motion of rotational motion of workpiece and tool around A ₂ and A _c shaft,
ω ₂ ...The rotational motion angular velocity of the workpiece around A ₂ ,
ω _c ...Rotational angular velocity of the tool around A _c , i=ω
_c / ω ₂ , h... conversion pitch of relative screw motion around the instantaneous axis, Γ ₂ ... angle between A ₂ and A _p , Γ _c ...
Angle formed by A _p and A _c , Γ ₂ ... Instantaneous distance between A ₂ and A ₀ , Γ _c ... Center distance between A _p and A _c , A _H ... Moment when the straight cutting edge is at position A axis of curvature, Φρ...Angle from A to A _H (clockwise direction is positive), Φ
r...Angle from A to A _p , Θ...Angle between Co and C (positive clockwise from C to Co), Θ ...Common perpendicular
Distance between Co and C, R _A , R _B ... Radius of curvature of the normal cut in the direction perpendicular to the generatrix of any two points A and B on the generatrix, a ₂ ... Unit vector on A ₂ , a ...Unit vector on the generatrix A, n...Normal on the generatrix determined by the designer, c...Unit vector on the common perpendicular to the generatrix, instantaneous axis, and curvature axis.

Claims (1)

[Scope of Claims] 1. Replace the tooth surface of the theoretical drum-shaped worm wheel with a line weave surface that is sufficiently similar to the tooth surface, and cut this line weave surface with a machining tool having a single straight cutting blade to remove the line An approximate cutting method for a drum-shaped worm wheel characterized in that the woven surface has the following properties (1) to (4). (1) One generatrix of the wire weave surface is placed at any position on the wheel blank. (2) Set the radius of curvature of the normal cut in the direction perpendicular to the generatrix at any two points on the generatrix to an arbitrary value. (3) The directions of all the normals of the tooth surface along the generatrix are fixed, and the directions are chosen arbitrarily, and at this time, this line-woven surface has the property of a developable surface in the vicinity of the generatrix. thing. (4) The straight cutting edge of the dancing tool coincides with the above generatrix line at a certain moment. In addition, the tool and workpiece were rotated at a constant rotational ratio, this rotational ratio was arbitrarily determined, and the arrangement of the tool axis, workpiece axis, and linear cutting blade was determined so as to satisfy all of the conditions (1) to (3) above. thing.