The present invention relates to a method of designing a hypoid gear.
A design method of a hypoid gear is described in Ernest Wildhaber, Basic Relationship of Hypoid Gears, American Machinist, USA, Feb. 14, 1946, p. 108-111 and in Ernest Wildhaber, Basic Relationship of Hypoid Gears II, American Machinist, USA, Feb. 28, 1946, p. 131-134. In these references, a system of eight equations is set and solved (for cone specifications that contact each other) by setting a spiral angle of a pinion and an equation of a radius of curvature of a tooth trace, in order to solve seven equations with nine variables which are obtained by setting, as design conditions, a shaft angle, an offset, a number of teeth, and a ring gear radius. Because of this, the cone specifications such as the pitch cone angle Γgw depend on the radius of curvature of the tooth trace, and cannot be arbitrarily determined.
In addition, in the theory of gears in the related art, a tooth trace is defined as “an intersection between a tooth surface and a pitch surface”. However, in the theory of the related art, there is no common geometric definition of a pitch surface for all kinds of gears. Therefore, there is no common definition of the tooth trace and of contact ratio of the tooth trace for various gears from cylindrical gears to hypoid gears. In particular, in gears other than the cylindrical gear and a bevel gear, the tooth trace is not clear.
In the related art, the contact ratio mf of tooth trace is defined by the following equation for all gears.
m
f
=F tan ψ0/p
where, p represents the circular pitch, F represents an effective face width, and ψ0 represents a spiral angle.
Table 1 shows an example calculation of a hypoid gear according to the Gleason method. As shown in this example, in the Gleason design method, the tooth trace contact ratios are equal for a drive-side tooth surface and for a coast-side tooth surface. This can be expected because of the calculation of the spiral angle ψ0 as a virtual spiral bevel gear with ψ0=(ψpw+ψgw)/2 (refer to
The present inventors, on the other hand, proposed in Japanese Patent No. 3484879 a method for uniformly describing the tooth surface of a pair of gears. In other word, a method for describing a tooth surface has been shown which can uniformly be used in various situations from a pair of gears having parallel axes, which is the most widely used configuration, to a pair of gears whose axes do not intersect and are not parallel with each other (skew position).
There is a desire to determine the cone specifications independent from the radius of curvature of the tooth trace, and to increase the degree of freedom of the design.
In addition, in a hypoid gear, the contact ratio and the transmission error based on the calculation method of the related art are not necessarily correlated to each other. Of the contact ratios of the related art, the tooth trace contact ratio has the same value between the drive-side and the coast-side, and thus the theoretical basis is brought into question.
An advantage of the present invention is that a hypoid gear design method is provided which uses the uniform describing method of the tooth surface described in JP 3484879, and which has a high degree of freedom of design.
Another advantage of the present invention is that a hypoid gear design method is provided in which a design reference body of revolution (pitch surface) which can be applied to the hypoid gear, the tooth trace, and the tooth trace contact ratio are newly defined using the uniform describing method of the tooth surface described in JP 3484879, and the newly defined tooth trace contact ratio is set as a design index.
According to one aspect of the present invention, there is provided a design method of a hypoid gear wherein an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers vc which is common to rotation axes of the first gear and the second gear, an intersection Cs between the instantaneous axis S and the line of centers vc, and an inclination angle ΓS of the instantaneous axis S with respect to the rotation axis of the second gear are calculated based on a shaft angle Σ, an offset E, and a gear ratio i0 of a hypoid gear, basic coordinate systems C1, C2, and Cs are determined from these variables, and specifications are calculated based on the coordinate systems. In particular, the specifications are calculated by setting a common point of contact of pitch cones of the first gear and the second gear as a design reference point Pw.
When an arbitrary point (design reference point) Pw is set in a static space, six cone specifications which are in contact at the point Pw are represented by coordinates (ucw, vcw, zcw) of the point Pw based on a plane (pitch plane) St defined by a peripheral velocity V1w and a peripheral velocity V2w at the point Pw and a relative velocity Vrsw. Here, the cone specifications refer to reference circle radii R1w and R2w of the first gear and the second gear, spiral angles ψpw and ψgw of the first gear and the second gear, and pitch cone angles γpw and Γgw of the first gear and the second gear. When three of these cone specifications are set, the point Pw is set, and thus the remaining three variables are also set. In other words, in various aspects of the present invention, the specifications of cones which contact each other are determined based merely on the position of the point Pw regardless of the radius of curvature of the tooth trace.
Therefore, it is possible to set a predetermined performance as a design target function, and select the cone specifications which satisfy the target function with a high degree of freedom. Examples of the design target function include, for example, a sliding speed of the tooth surface, strength of the tooth, and the contact ratio. The performance related to the design target function is calculated while the cone specification, for example the pitch cone angle Γgw, is changed, and the cone specification is changed and a suitable value is selected which satisfies the design request.
According to one aspect of the present invention, an contact ratio is employed as the design target function, and there is provided a method of designing a hypoid gear wherein a pitch cone angle Γgcone of one gear is set, an contact ratio is calculated, the pitch cone angle Γgcone is changed so that the contact ratio becomes a predetermined value, a pitch cone angle Γgw is determined, and specifications are calculated based on the determined pitch cone angle Γgw. As described above, the contact ratio calculated by the method of the related art does not have a theoretical basis. In this aspect of the present invention, a newly defined tooth trace and an contact ratio related to the tooth trace are calculated, to determine the pitch cone angle. The tooth surface around a point of contact is approximated by its tangential plane, and a path of contact is made coincident to an intersection of the surface of action (pitch generating line Lpw), and a tooth trace is defined as a curve on a pitch hyperboloid obtained by transforming the path of contact into a coordinate system which rotates with each gear. Based on the tooth trace of this new definition, the original contact ratio of the hypoid gear is calculated and the contact ratio can be used as an index for design. A characteristic of the present invention is in the definition of the pitch cone angle related to the newly defined tooth trace.
According to another aspect of the present invention, it is preferable that, in the hypoid gear design method, the tooth trace contact ratio is assumed to be 2.0 or more in order to achieve constant engagement of two gears with two or more teeth.
When the pitch cone angle Γgw is set to an inclination angle Γs of an instantaneous axis, the contact ratios of the drive-side and the coast-side can be set approximately equal to each other. Therefore, it is preferable for the pitch cone angle to be set near the inclination angle of the instantaneous axis. In addition, it is also preferable to increase one of the contact ratios of the drive-side or coast-side as required. In this process, first, the pitch cone angle is set at the inclination angle of the instantaneous axis and the contact ratio is calculated, and a suitable value is selected by changing the pitch cone angle while observing the contact ratio. It is preferable that a width of the change of the pitch cone angle be in a range of ±5° with respect to the inclination angle Γs of the instantaneous axis. This is because if the change is out of this range, the contact ratio of one of the drive-side and the coast-side will be significantly reduced.
More specifically, according to one aspect of the present invention, a hypoid gear is designed according to the following steps:
(a) setting a shaft angle Σ, an offset E, and a gear ratio i0 of a hypoid gear;
(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i0, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers vc with respect to rotation axes of the first gear and the second gear, an intersection Cs between the instantaneous axis S and the line of centers vc, and an inclination angle Γs of the instantaneous axis S with respect to the rotation axis of the second gear, and determining coordinate systems C1, C2, and Cs for calculation of specifications;
(c) setting three variables including one of a reference circle radius R1w of the first gear and a reference circle radius R2w of the second gear, one of a spiral angle ψpw of the first gear and a spiral angle ψgw of the second gear, and one of a pitch cone angle γpw of the first gear and a pitch cone angle Γgw of the second gear;
(d) calculating the design reference point Pw, which is a common point of contact of pitch cones of the first gear and the second gear, and the three other variables which are not set in the step (c), based on the three variables which are set in the step (c);
(e) setting a contact normal gwD of a drive-side tooth surface of the second gear;
(f) setting a contact normal gwC of a coast-side tooth surface of the second gear; and
(g) calculating specifications of the hypoid gear based on the design reference point Pw, the three variables which are set in the step (c), the contact normal gwD of the drive-side tooth surface of the second gear, and the contact normal gwC of the coast-side tooth surface of the second gear.
According to another aspect of the present invention, a hypoid gear is designed according to the following steps:
(a) setting a shaft angle Σ, an offset E, and a gear ratio i0 of a hypoid gear;
(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i0, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers vc with respect to rotation axes of the first gear and the second gear, an intersection Cs between the instantaneous axis S and the line of centers vc, and an inclination angle Γs of the instantaneous axis S with respect to the rotation axis of the second gear, and determining coordinate systems C1, C2, and Cs for calculation of specifications;
(c) setting three variables including one of a reference circle radius R1w of the first gear and a reference circle radius R2w of the second gear, one of a spiral angle ψpw of the first gear and a spiral angle ψgw of the second gear, and one of a pitch cone angle γpw of the first gear and a pitch cone angle Γgw of the second gear;
(d) calculating the design reference point Pw, which is a common point of contact of pitch cones of the first gear and the second gear, and the three other variables which are not set in the step (c), based on the three variable which are set in the step (c);
(e) calculating a pitch generating line Lpw which passes through the design reference point Pw and which is parallel to the instantaneous axis S;
(f) setting an internal circle radius R2t and an external circle radius R2h of the second gear;
(g) setting a contact normal gwD of a drive-side tooth surface of the second gear;
(h) calculating an intersection P0D between a reference plane SH which is a plane orthogonal to the line of centers vc and passing through the intersection Cs and the contact normal gwD and a radius R20D of the intersection P0D around a gear axis;
(i) calculating an inclination angle φs0D of a surface of action SwD which is a plane defined by the pitch generating line Lpw and the contact normal gwD with respect to the line of centers vc, an inclination angle ψsw0D of the contact normal gwD on the surface of action SwD with respect to the instantaneous axis S, and one pitch PgwD on the contact normal gwD;
(j) setting a provisional second gear pitch cone angle Γgcone, and calculating an contact ratio mfconeD of the drive-side tooth surface based on the internal circle radius R2t and the external circle radius R2h;
(k) setting a contact normal gwC of a coast-side tooth surface of the second gear;
(l) calculating an intersection P0C between the reference plane SH which is a plane orthogonal to the line of centers vc and passing through the intersection Cs and the contact normal gwC and a radius R20c of the intersection P0C around the gear axis;
(m) calculating an inclination angle φs0C of a surface of action SwC which is a plane defined by the pitch generating line Lpw and the contact normal gwC with respect to the line of centers vc, an inclination angle ψsw0C of the contact normal gwC on the surface of action SwC with respect to the instantaneous axis S, and one pitch PgwC on the contact normal gwC;
(n) setting a provisional second gear pitch cone angle Γgcone, and calculating an contact ratio mfconeC of the coast-side tooth surface based on the internal circle radius R2t and the external circle radius R2h;
(o) comparing the contact ratio mfconeD of the drive-side tooth surface and the contact ratio mfconeC of the coast-side tooth surface, and determining whether or not these contact ratios are predetermined values;
(p) when the contact ratios of the drive-side and the coast-side are the predetermined values, replacing the provisional second gear pitch cone angle Γgcone with the second gear pitch cone angle Γgw obtained in the step (c) or in the step (d);
(q) when the contact ratios of the drive-side and the coast-side are not the predetermined values, changing the provisional second gear pitch cone angle Γgcone and re-executing from step (g);
(r) re-determining the design reference point Pw, the other one of the reference circle radius R1w of the first gear and the reference circle radius R2w of the second gear which is not set in the step (c), the other one of the spiral angle ψpw of the first gear and the spiral angle ψgw of the second gear which is not set in the step (c), and the first gear pitch cone angle γpw based on the one of the reference circle radius R1w of the first gear and the reference circle radius R2w of the second gear which is set in the step (c), the one of the spiral angle ψpw of the first gear and the spiral angle ψgw of the second gear which is set in the step (c), and the second gear pitch cone angle Γgw which is replaced in the step (p), and
(s) calculating specifications of the hypoid gear based on the specifications which are set in the step (c), the specifications which are re-determined in the step (r), the contact normal gwD of the drive-side tooth surface of the second gear, and the contact normal gwC of the coast-side tooth surface of the second gear.
According to another aspect of the present invention, a hypoid gear is designed according to the following steps:
(a) setting a shaft angle Σ, an offset E, and a gear ratio i0 of a hypoid gear;
(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i0, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers vc with respect to rotation axes of the first gear and the second gear, an intersection Cs between the instantaneous axis S and the line of centers vc, and an inclination angle Γs of the instantaneous axis S with respect to the rotation axis of the second gear, to determine coordinate systems C1, C2, and Cs for calculation of specifications;
(c) setting three variables including one of a reference circle radius R1w of the first gear and a reference circle radius R2w of the second gear, one of a spiral angle ψpw of the first gear and a spiral angle ψgw of the second gear, and one of a pitch cone angle γpw of the first gear and a pitch cone angle Γgw of the second gear;
(d) calculating the design reference point Pw, which is a common point of contact of pitch cones of the first gear and the second gear, and the three other variables which are not set in the step (c), based on the three variables which are set in the step (c);
(e) calculating a pitch generating line Lpw which passes through the design reference point Pw and which is parallel to the instantaneous axis S;
(f) setting an internal circle radius R2t and an external circle radius R2h of the second gear;
(g) setting a contact normal gwD of a drive-side tooth surface of the second gear;
(h) calculating an intersection P0D between a reference plane SH which is a plane orthogonal to the line of centers vc and passing through the intersection Cs and the contact normal gwD and a radius R20D of the intersection P0D around a gear axis;
(i) calculating an inclination angle φa0D of a surface of action SwD which is a plane defined by the pitch generating line Lpw and the contact normal gwD with respect to the line of centers vc, an inclination angle ψsw0D of the contact normal gwD on the surface of action SwD with respect to the instantaneous axis S, and one pitch PgwD on the contact normal gwD;
(j) setting a provisional second gear pitch cone angle Γgcone, and calculating an contact ratio mfconeD of the drive-side tooth surface based on the internal circle radius R2t and the external circle radius R2h;
(k) setting a contact normal gwC of a coast-side tooth surface of the second gear;
(l) calculating an intersection P0c between the reference plane SH which is a plane orthogonal to the line of centers vc and passing through the intersection Cs and the contact normal gwC and a radius R20c of the intersection P0c around the gear axis;
(m) calculating an inclination angle φs0C of a surface of action SwC which is a plane defined by the pitch generating line Lpw and the contact normal gwC with respect to the line of centers vc, an inclination angle ψsw0C of the contact normal gwC on the surface of action SwC with respect to the instantaneous axis S, and one pitch PgwC on the contact normal gwC;
(n) setting a provisional second gear pitch cone angle Γgcone, and calculating an contact ratio mfconeC of the coast-side tooth fconeC surface based on the internal circle radius R2t and the external circle radius R2h;
(o) comparing the contact ratio mfconeD of the drive-side tooth surface and the contact ratio mfconeC of the coast-side tooth surface, and determining whether or not these contact ratios are predetermined values;
(p) changing, when the contact ratios of the drive-side and the coast-side are not the predetermined values, the provisional second gear pitch cone angle Γgcone and re-executing from step (g);
(q) defining, when the contact ratios of the drive-side and the coast-side are the predetermined values, a virtual cone having the provisional second gear pitch cone angle Γgcone as a cone angle;
(r) calculating a provisional pitch cone angle γpcone of the virtual cone of the first gear based on the determined pitch cone angle Γgcone; and
(s) calculating specifications of the hypoid gear based on the design reference point Pw, the reference circle radius R1w of the first gear and the reference circle radius R2w of the second gear which are set in the step (c) and the step (d), the spiral angle ψpw of the first gear and the spiral angle ψgw of the second gear which are set in the step (c) and the step (d), the cone angle Γgcone of the virtual cone and the cone angle γpcone of the virtual cone which are defined in the step (q) and the step (r), the contact normal gwD of the drive-side tooth surface of the second gear, and the contact normal gwC of the coast-side tooth surface of the second gear.
According to another aspect of the present invention, in a method of designing a hypoid gear, a pitch cone angle of one gear is set equal to an inclination angle of an instantaneous axis, and the specifications are calculated. When the pitch cone angle is set equal to the inclination angle of the instantaneous axis, the contact ratios of the drive-side tooth surface and the coast-side tooth surface become almost equal to each other. Therefore, a method is provided in which the pitch cone angle is set to the inclination angle of the instantaneous axis in a simple method, that is, without reviewing the contact ratios in detail.
More specifically, according to another aspect of the present invention, the hypoid gear is designed according to the following steps:
(a) setting a shaft angle Σ, an offset E, and a gear ratio i0 of a hypoid gear;
(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i0, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers vc with respect to rotational axes of the first gear and the second gear, an intersection Cs between the instantaneous axis S and the line of centers vc, and an inclination angle Γs of the instantaneous axis with respect to the rotational axis of the second gear;
(c) determining the inclination angle Γs of the instantaneous axis as a second gear pitch cone angle Γgw; and
(d) calculating specifications of the hypoid gear based on the determined second gear pitch cone angle Γgw. According to another aspect of the present invention, in a method of designing a hypoid gear, a design reference point Pw is not set as a point of contact between the pitch cones of the first gear and second gear, but is determined based on one of reference circle radii R1w and R2w of the first gear and the second gear, a spiral angle ψrw, and a phase angle βw of the design reference point, and the specifications are calculated.
More specifically, according to another aspect of the present invention, a hypoid gear is designed according to the following steps:
(a) setting a shaft angle Σ, an offset E, and a gear ratio i0 of a hypoid gear;
(b) calculating, based on the shaft angle Σ, the offset E, and the gear ratio i0, an instantaneous axis S which is an axis of a relative angular velocity of a first gear and a second gear, a line of centers vc with respect to rotation axes of the first gear and the second gear, an intersection Cs between the instantaneous axis S and the line of centers vc, and an inclination angle Γs of the instantaneous axis S with respect to the rotation axis of the second gear, to determine coordinate systems C1, C2, and Cs for calculation of specifications;
(c) setting one of a reference circle radius R1w of the first gear and a reference circle radius R2w of the second gear, a spiral angle ψrw, and a phase angle βw of a design reference point Pw, to determine the design reference point;
(d) calculating the design reference point Pw and a reference circle radius which is not set in the step (c) from a condition where the first gear and the second gear share the design reference point Pw, based on the three variables which are set in the step (c);
(e) setting one of a reference cone angle γpw of the first gear and a reference cone angle Γgw of the second gear;
(f) calculating a reference cone angle which is not set in the step (e), based on the shaft angle Σ and the reference cone angle which is set in the step (e);
(g) setting a contact normal gwD of a drive-side tooth surface of the second gear;
(h) setting a contact normal gwC of a coast-side tooth surface of the second gear; and
(i) calculating specifications of the hypoid gear based on the design reference point Pw, the reference circle radii R1w and R2w, and the spiral angle ψrw which are set in the step (c) and the step (d), the reference cone angles γpw and Γgw which are set in the step (e) and the step (f), and the contact normals gwC and gwD which are set in the step (g) and the step (h).
The designing steps of these two aspects of the present invention can be executed by a computer by describing the steps with a predetermined computer program. A unit which receives the gear specifications and variables is connected to the computer and a unit which provides a design result or a calculation result at an intermediate stage is also connected to the computer.
A preferred embodiment of the present invention will now be described with reference to the drawings.
In the following description, a small diameter gear in a pair of hypoid gears is referred to as a pinion, and a large diameter gear is referred to as a ring gear. In addition, in the following, the descriptions may be based on the tooth surface, tooth trace, etc. of the ring gear, but because the pinion and the ring gear are basically equivalent, the description may similarly be based on the pinion.
The plane of action G20 is expressed by vq2c=−Rb2 using the coordinate system Cq2. In the coordinate system C2, the inclination angle of the plane of action G20 to the plane v2c=0 is the angle χ20, and the plane of action G20 is a plane tangent to the base cylinder (radius Rb20).
The relationships between the coordinate systems C2 and Cq2 become as follows because the z2c axis is common.
u
2c
=q
2c cos χ20−vq2 sin χ20
v
2c
=q
2c sin χ20+vq2c cos χ20
Because the plane of action G20 meets vq2c=−Rb20, the following expressions (1), are satisfied if the plane of action G20 is expressed by the radius Rb20 of the base cylinder.
u
2c
=q
2c cos χ20+Rb20 sin χ20
v
2c
=q
2c sin χ20−Rb20 cos χ20
z
2c
=z
2c (1)
If the line of centers g0 is defined to be on the plane of action G20 and also defined such that the line of centers g0 is directed in the direction in which the q2c axis component is positive, an inclination angle of the line of centers g0 from the q2c axis can be expressed by ψb20 (the direction shown in the figure is positive). Accordingly, the inclination angle of the line of centers g0 in the coordinate system. C2 is defined to be expressed in the form of g0 (φ20, ψb20) with the inclination angle φ20 (the complementary angle of the χ20) of the plane of action G20 with respect to the line of centers vc, and ψb2.
As for the gear I, coordinate systems C1 (u1c, v1c, z1c) and Cq1 (g1c, vq1c, z1c), a plane of action G10, a radius Rb1 of the base cylinder, and the inclination angle g0 (φ10, ψb10) of the line of centers g0 can be similarly defined. Because the systems share a common z1c axis, the relationship between the coordinate systems C1 and Cq1 can also be expressed by the following expressions (2).
u
1c
=q
1c cos χ10+Rb10 sin χ10
v
1c
=q
1c sin χ10−Rb10 cos χ10
z
1c
=z
1c (2)
The relationship between the coordinate systems C1 and C2 is expressed by the following expressions (3).
u
1c
=−u
2c cos Σ−z2c sin Σ
v
1c
=v
2c
+E
z
1c
=u
2c sin Σ−z2c cos Σ (3)
sin Ωs/sin(Ωs−Ω)=ω10/ω20; or
sin Γs/sin)Σ−Γs)=ω10/ω20 (4)
wherein Σ=π−Ω (shaft angle) and Γs=π−Ωs. The positive directions are shown in the figure. In other words, the angle Γs is an inclination of the instantaneous axis S with respect to the ring gear axis IIs on the plane SH, and the angle Γs will hereinafter be referred to as an inclination angle of the instantaneous axis.
The location of Cs on the line of centers vc can be obtained as follows.
C
2
C
s
=E tan Γ
s/{tan(Σ−Γs)+tan Γs} (5).
The expression is effective within a range of 0≦Γs≦π, and the location of Cs changes together with Γs, and the location of the point Cs is located above C1 in the case of 0≦Γs≦π/2, and the location of the point Cs is located under C1 in the case of π/2≦Γs≦π.
Because the instantaneous axis S can be determined in a static space in accordance with the aforesaid expressions (4) and (5), the coordinate system CS is defined as shown in
If the points C1 and C2 are expressed to be C1 (0, vcs1, 0) and C2 (0, vcs2, 0) by the use of the coordinate system Cs, vcs1 and vcs2 are expressed by the following expressions (6).
If it is noted that C2 is always located above Cs with respect to the vc axis, the relationships among the coordinate system Cs and the coordinate systems C1 and C2 can be expressed as the following expressions (7) and (8) with the use of vcs1, vcs2, Σ, and Γs.
u
1c
=u
c cos(Σ−Γs)+zc sin(Σ−Γs)
v
1c
=v
c
−v
cs1
z
1c
=−u
c sin(Σ−Γs)+zc cos(Σ−Γs) (7)
u
2c
=−u
c cos Γs+zc sin Γs
v
2c
=v
c
−v
cs2
z
2c
=−u
c sin Γs−zc cos Γs (8)
The relationships among the coordinate system Cs and the coordinate systems C1 and C2 are conceptually shown in
2.1 Relationship Between Relative Velocity and Path of Contact g0
v
rs=ωr×r+Vs (9)
where
ωr=ω10−ω20
ωr=ω20 sin Σ/sin(Σ−Γs)=ω10 sin Σ/sin Γs
Vs=ω10×[C1Ss]−ω20×[C2Cs]
Vs=ω20E sin Γs=ω10E sin(Σ−Γs).
Here, [C1Cs] indicates a vector having C1 as its starting point and Cs as its endpoint, and [C2Cs] indicates a vector having C2 as its starting point and Cs as its end point.
The relative velocity Vrs exists on a tangential plane of the surface of a cylinder having the instantaneous axis S as an axis, and an inclination angle ψ relative to Vs on the tangential plane can be expressed by the following expression (10).
cos ψ=|Vs|/|Vrs| (10)
Because the path of contact g0 is also the line of centers of a tooth surface at the point of contact, g0 is orthogonal to the relative velocity Vrs at the point P. That is,
V
rs
·g
0=0
Consequently, g0 is a directed straight line on a plane N normal to Vrs at the point P. If the line of intersection of the plane N and the plane SH is designated by Hn, Hn is in general a straight line intersecting with the instantaneous axis S, with g0 necessarily passing through the Hn if an infinite intersection point is included. If the intersection point of g0 with the plane SH is designated by P0, then P0 is located on the line of intersection Hn, and g0 and P0 become as follows according to the kinds of pairs of gears.
Because Vs=0, Vrs simply means a peripheral velocity around the instantaneous axis S. Consequently, the plane N includes the S axis. Hence, Hn coincides with S, and the path of contact g0 always passes through the instantaneous axis S. That is, the point P0 is located on the instantaneous axis S. Consequently, for these pairs of gears, the path of contact g0 is an arbitrary directed straight line passing at the arbitrary point P0 on the instantaneous axis.
(2) Case of Gear Other than that Described Above (Σ≠0, π or E≠0)
In the case of a hypoid gear, a crossed helical gear or a wormgear, if the point of contact P is selected at a certain position, the relative velocity Vrs, the plane N, and the straight line Hn, all peculiar to the point P, are determined. The path of contact g0 is a straight line passing at the arbitrary point P0 on Hn, and does not, in general, pass through the instantaneous axis S. Because the point P is arbitrary, g0 is also an arbitrary directed straight line passing at the point P0 on a plane normal to the relative velocity Vrs0 at the intersection point P0 with the plane SH. That is, the aforesaid expression (9) can be expressed as follows.
V
rs
=V
rs0+ωr×[P0P]·g0
Here, [P0P] indicates a vector having P0 as its starting point and the P as its end point. Consequently, if Vrs0·g0=0, Vrs·g0=0, and the arbitrary point P on g0 is a point of contact.
Among pairs of gears having two axes with known positional relationship and the angular velocities, pairs of gears with an identical path of contact g0 have an identical tooth profile corresponding to g0, with the only difference between them being which part of the tooth profile is used effectively. Consequently, in design of a pair of gears, the position at which the path of contact g0 is disposed in a static space determined by the two axes is important. Further, because a design reference point is only a point for defining the path of contact g0 in the static space, the position at which the design reference point is selected on the path of contact g0 does not cause any essential difference. When an arbitrary path of contact g0 is set, the g0 necessarily intersects with a plane SH including the case where the intersection point is located at an infinite point. Thus, the path of contact g0 is determined with the point P0 on the plane SH (on an instantaneous axis in the case of cylindrical gears and bevel gears) as the reference point.
u
c0
=O
s
P
0
v
c0=0
z
c0
=C
s
O
s
For cylindrical gears and bevel gears, uc0=0. Furthermore, the point Os is the intersection point of a plane Ss, passing at the reference point P0 and being normal to the instantaneous axis S, and the instantaneous axis S.
2.3 Definition of Inclination Angle of Path of Contact g0
The relative velocity Vrs0 at the point P0 is concluded as follows with the use of the aforesaid expression (9).
V
rs0=ωr×[uc0]+Vs
where, [uc0] indicates a vector having Os as its starting point and P0 as its end point. If a plane (uc=uc0) being parallel to the instantaneous axis S and being normal to the plane SH at the point P0 is designated by Sp, Vrs0 is located on the plane Sp, and the inclination angle ψ0 of Vrs0 from the plane SH (vc=0) can be expressed by the following expression (11) with the use of the aforesaid expression (10).
Incidentally, ψ0 is assumed to be positive when uc0≧0, and the direction thereof is shown in
If a plane passing at the point P0 and being normal to Vrs0 is designated by S, the plane Sn is a plane inclining to the plane Ss by the ψ0, and the path of contact g0 is an arbitrary directed straight line passing at the point P0 and located on the plane Sn. Consequently, the inclination angle of g0 in the coordinate system Cs can be defined with the inclination angle ψ0 of the plane Sn from the plane Ss (or the vc axis) and the inclination angle n0 from the plane Sp on the plane Sn, and the defined inclination angle is designated by g0 (ψ0, φn0). The positive direction of φn0 is the direction shown in
2.4. Definition of g0 by Coordinate System Cs
For a pair of gears having the set shaft angle Σ thereof, the offset E thereof, and the directions of angular velocities, the path of contact g0 can generally be defined in the coordinate system Cs by means of five independent variables of the design reference point P0 (uc0, vc0, zc0) and the inclination angle g0 (ψ0, φn0). Because the ratio of angular velocity i0 and vc0=0 are set as design conditions in the present embodiment, there are three independent variables of the path of contact g0. That is, the path of contact g0 is determined in a static space by the selections of the independent variables of two of (zc0), φn0, and ψ0 in the case of cylindrical gears because zc0 has no substantial meaning, three of zc0, φn0, and ψ0 in the case of a bevel gear, or three of zc0, φn0, and ψ0 (or uc0) in the case of a hypoid gear, a worm gear, or a crossed helical gear. When the point P0 is set, ψ0 is determined at the same time and only φn0 is a freely selectable variable in the case of the hypoid gear and the worm gear. However, in the case of the cylindrical gears and the bevel gear, because P0 is selected on an instantaneous axis, both of ψ0 and φn0 are freely selectable variables.
When the coordinate system Cs is rotated around the zc axis by βw, to realize a coordinate system Crs (urc, vrc, zc: Crs), the tangential plane Spw can be expressed by urc=rw, and the following relationship is satisfied between urc=rw and the inclination angle ωrw of Vrsw.
where Vs represents a sliding velocity in the direction of the instantaneous axis and ωr represents a relative angular velocity around the instantaneous axis.
The expression (12) shows a relationship between rw of the arbitrary point Pw (rw, βw, zcw: Cs) and the inclination angle ψrw of the relative velocity Vrsw thereof. In other words, when ψrw is set, rw is determined. Because this is true for arbitrary values of βw and zcw, Pw with a constant ψrw defines a cylinder with a radius rw. This cylinder is called the tangential cylinder of the relative velocity.
When rw (or ψrw) and βw are set, Pw is determined on the plane zc=zcw. Because this is true for an arbitrary value of zcw, points Pw having the same rw (or ψrw) and the same βw draw a line element of the cylinder having a radius rw. This line element is called a pitch generating line Lpw. A directed straight line which passes through a point Pw on a plane Snw orthogonal to the relative velocity Vrsw at the arbitrary point Pw on the pitch generating line Lpw satisfies a condition of contact, and thus becomes a contact normal.
The pitch generating line Lpw is uniquely determined by the shaft angle Σ, offset E, gear ratio i0, inclination angle ψrw of relative velocity Vrsw, and rotation angle βw from the coordinate system Cs to the coordinate system Crs. A pair of hyperboloids which are obtained by rotating the pitch generating line Lpw around the two gear axes, respectively, contact each other in a line along Lpw, and because the line Lpw is also a line of intersection between the surfaces of action, the drive-side D and the coast-side C also contact each other along the line Lpw. Therefore, the hyperboloids are suited as revolution bodies for determining the outer shape of the pair of gears. In the present invention, the hyperboloids are set as the design reference revolution bodies, and are called the pitch hyperboloids. The hyperboloids in the related art are revolution bodies in which the instantaneous axis S is rotated around the two gear axes, respectively, but in the present invention, the pitch hyperboloid is a revolution body obtained by rotating a parallel line having a distance rw from the instantaneous axis.
In the cylindrical gear and the bevel gear, Lpw coincides with the instantaneous axis S or zc (rw→0) regardless of ψrw and βw, because of special cases of the pitch generating line Lpw (Vs→0 as Σ→0 or E→0 in the expression (12)). The instantaneous axis S is a line of intersection of the surfaces of action of the cylindrical gear and the bevel gear, and the revolution bodies around the gear axes are the pitch cylinder of the cylindrical gear and the pitch cone of the bevel gear.
For these reasons, the pitch hyperboloids which are the revolution bodies of the pitch generating line Lpw have the common definition of the expression (12) from the viewpoint that the hyperboloid is a “revolution body of line of intersection of surfaces of action” and can be considered to be a design reference revolution body for determining the outer shape of the pair of gears which are common to all pairs of gears.
In the present invention, a curve on the pitch hyperboloid (which is common to all gears) obtained by transforming a path of contact to a coordinate system which rotates with the gear when the tooth surface around the point of contact is approximated with its tangential plane and the path of contact is made coincident with the line of intersection of the surfaces of action (pitch generating line Lpw) is called a tooth trace (curve). In other words, a. tooth profile, among arbitrary tooth profiles on the tooth surface, in which the path of contact coincides with the line of intersection of the surface of action is called a tooth trace. The tooth trace of this new definition coincides with the tooth trace of the related art defined as an intersection between the pitch surface (cone or cylinder) and the tooth surface in the cylindrical gears and the bevel gears and differs in other gears. In the case of the current hypoid gear, the line of intersection between the selected pitch cone and the tooth surface is called a tooth trace.
A total contact ratio m is defined as a ratio of a maximum angular displacement and an angular pitch of a contact line which moves on an effective surface of action (or effective tooth surface) with the rotation of the pair of gears. The total contact ratio m can be expressed as follows in terms of the angular displacement of the gear.
m=(θ2max−θ2min)/(2θ2p)
where θ2max and θ2min represent maximum and minimum gear angular displacements of the contact line and 2θ2p represents a gear angular pitch.
Because it is very difficult to represent the position of the contact line as a function of a rotation angle except for special cases (involute helicoid) and it is also difficult to represent such on the tooth surface (curved surface), in the stage of design, the surface of action has been approximated with a plane in a static space, a path of contact has been set on the surface of action, and an contact ratio has been determined and set as an index along the path of contact.
The contact ratio of the tooth surface is defined in the following manner depending on how the path of contact passing through Pw=P0 is defined on the surface of action Sw=Sw0:
(1) Contact Ratio mz Orthogonal Axis
This is a ratio between a length separated by an effective surface of action (action limit and the tooth surface boundary) of lines of intersection h1z and h2z (P0Pz1sw and P0Pz2sw in
(2) Tooth Trace Contact Ratio mf
This is a ratio between a length of Lpw0 which is parallel to the instantaneous axis separated by the effective surface of action and a pitch in this direction;
(3) Transverse Contact Ratio ms
This is a ratio between a length separated by an effective surface of action of a line of intersection (P0Pssw in
This includes cases where the path of contact is set in a direction of g0 (P0PGswn in
This is a sum of contact ratios in two directions (for example, (2) and (3)) which are normal to each other on the surface of action, and is used as a substitute for the total contact ratio.
In addition, except for points on gw=g0, the pitch (length) would differ depending on the position of the point, and the surface of action and the tooth surface are actually not planes. Therefore, only an approximated value can be calculated for the contact ratio. Ultimately, a total contact ratio determined from the angular displacement must be checked.
In general, a gear design can be considered, in a simple sense, to be an operation, in a static space (coordinate system Cs) determined by setting the shaft angle Σ, offset E, and gear ratio i0, to:
(1) select a pitch generating line and a design reference revolution body (pitch hyperboloid) by setting a design reference point Pw (rw(ψrw), βw, zcw: Crs); and
(2) set a surface of action (tooth surface) having gw by setting an inclination angle (ψrw, φnrw: Crs) of a tooth surface normal gw passing through Pw.
In other words, the gear design method (selection of Pw and gw) comes down to selection of four variables including rw (normally, ψrw is set), βw, zcw (normally, R2w (gear pitch circle radius) is set in place of zcw), and φnrw. A design method for a hypoid gear based on the pitch hyperboloid when Σ, E, and i0 are set will be described below.
3.7 Hypoid Gear (−π/2<βw<π/2)
(1) Various hypoid gears can be realized depending on how βw is selected, even with set values for ψrw (rw) and zcw (R2w).
(a) From the viewpoint of the present invention, the Wildhaber (Gleason) method is one method of determining Pw by determining βw through setting of a constraint condition to “make the radius of curvature of a tooth trace on a plane (
(b) In a preferred embodiment described in section 4.2A below, βw is selected by giving a constraint condition that “a line of intersection between a cone circumscribing at Pw and the surface of action is coincident with the pitch generating line Lpw”. As a result, as will be described below, the tooth trace contact ratios on the drive-side and the coast-side become approximately equal to each other.
(2) Gear radius R2w, βw, and ψrw are set and a design reference point Pw (ucw, vcw, zcw: Cs) is determined on the pitch generating line Lpw. The pitch hyperboloids can be determined by rotating the pitch generating line Lpw around each tooth axis. A method of determining the design reference point will be described in section 4.2B below.
(3) A tooth surface normal gw passing through Pw is set on a plane Snw normal to the relative velocity Vrsw of Pw. The surface of action Sw is determined by gw and the pitch generating line Lpw.
A method of designing a hypoid gear using the pitch hyperboloid will now be described in detail.
When the shaft angle Σ, offset E, and gear ratio i0 are set, the inclination angle Γs of the instantaneous axis, and the origins C1 (0, vcs1, 0: Cs) and C2 (0, vcs2, 0: Cs) of the coordinate systems C1 and C2 are represented by the following expressions.
sin Γs/sin(Σ−Γs)=i0
v
cs2
=E tan Γ
s/{tan(Σ−Γs)+tan Γs}
v
cs1
=v
cs2
−E
The reference point Pw is set in the coordinate system Cs as follows.
P
w(ucw,vcw,zcw:Cs)
If Pw is set as Pw(rw, βw, zcw: Cs) by representing Pw with the cylindrical radius rw of the relative velocity and the angle from the uc axis, the following expressions hold.
u
cw
=r
w cos βw
v
cw
=r
w sin βw
The pitch generating line Lpw is determined as a straight line which passes through the reference point Pw and which is parallel to the instantaneous axis (inclination angle Γs), and the pitch hyperboloids are determined as revolution bodies of the pitch generating line Lpw around the gear axes.
If the relative velocity of Pw is Vrsw, the angle ψrw between Vrsw and the pitch generating line Lpw is, based on expression (12),
tan ψrw=rw sin Σ/{E sin(Σ−Γs)sin Γs}
Here, ψrw is the same anywhere on the same cylinder of the radius rw.
When transformed into coordinate systems C1 and C2, Pw (u1cw, v1cw, z1cw: C1), Pw(u2cw, v2cw, z2cw: C2), and pinion and ring gear reference circle radii R1w and R2w can be expressed with the following expressions.
u
1cw
=u
cw cos(Σ−Γs)+zcw sin(Σ−Γs)
v
1cw
=v
cw
−v
cs1
z
1cw
=−u
cw sin(Σ−Γs)+zcw cos(Σ−Γs)
u
2ce
=−u
cw cos Γs+zcw sin Γs
v
2cw
=v
cw
−v
cs2
z
2cw
=−u
cw sin Γs−zcw cos Γs
R
1w
2
=u
1cw
2
+v
1cw
2
R
2w
2
=u
2cw
2
+v
2cw
2 (13)
A pitch hyperboloid which is a geometric design reference revolution body is difficult to manufacture, and thus in reality, in general, the gear is designed and manufactured by replacing the pitch hyperboloid with a pitch cone which passes through the point of contact Pw. The replacement with the pitch cones is realized in the present embodiment by replacing with cones which contact at the point of contact Pw.
The design reference cone does not need to be in contact at Pw, but currently, this method is generally practiced. When βw is changed, the pitch angle of the cone which contacts at Pw changes in various manners, and therefore another constraint condition is added for selection of the design reference cone (βw). The design method would differ depending on the selection of the constraint condition. One of the constraint conditions is the radius of curvature of the tooth trace in the Wildhaber (Gleason) method which is already described. In the present embodiment, βw is selected with a constraint condition that a line of intersection between the cone which contacts at Pw and the surface of action coincides with the pitch generating line Lpw.
As described, there is no substantial difference caused by where on the path of contact g0 the design reference point is selected. Therefore, a design method of a hypoid gear will be described in which the point of contact Pw is set as the design reference point and circular cones which contact at Pw are set as the pitch cones.
Intersection points between a plane Snw normal to the relative velocity Vrsw of the reference point Pw and the gear axes are set as O1nw and O2nw (
O
1nw(0,0,−E/(tan ε2w sin Σ):C1)
O
2nw(0,0,−E/(tan ε1w sin Σ):C2)
where sin ε1w=v1cw/R1w and sin ε2w=v2cw/R2w.
In addition, O1nwPw and O2nwPw can be expressed with the following expressions.
O
1nw
P
w
={R
1w
2+(−E/(tan ε2w sin Σ)−z1cw)2}1/2
O
2nw
P
w
={R
2w
2+(−E/(tan ε1w sin Σ)−z2cw)2}1/2
Therefore, the cone angles γpw and Γgw of the pinion and ring gear can be determined with the following expressions, taking advantage of the fact that O1nwPw and O2nwPw are back cone elements:
cos γpw=R1w/O1nwPw
cos Γgw=R2w/O2nwPw (14)
The expression (14) sets the pitch cone angles of cones having radii of R1w and R2w and contacting at Pw.
The relative velocity and peripheral velocity are as follows.
V
rsw/ω20={E sin Γs)2+(rw sin Σ/sin(Σ−Γs))2}1/2
V
1w/ω20=i0R1w
V
2w/ω20=R2w
When a plane defined by peripheral velocities V1w and V2w is Stw, the plane Stw is a pitch plane. If an angle formed by V1w and V2w is ψv12w and an angle formed by Vrsw and V1w is ψvrs1w (
cos(ψv12w)=(V1w2+V2w2−Vrsw2)/(2V1w×V2w)
cos(ψvrs1w)=(Vrsw2+V1w2−V2w2)/(2V1w×Vrsw)
If the intersections between the plane St and the pinion and gear axes are O1w and O2w, the spiral angles of the pinion and the ring gear can be determined in the following manner as inclination angles on the plane Stw from PwO1w and PwO2w (
ψpw=π/2−ψvrs1w
ψgw=π/2−ψv12w−ψvrs1w (15)
When a pitch point Pw(rw, βw, zcw: Cs) is set, specifications of the cones contacting at Pw and the inclination angle of the relative velocity Vrsw can be determined based on expressions (13), (14) and (15). Therefore, conversely, the pitch point Pw and the relative velocity Vrsw can be determined by setting three variables (for example, R2w, ψpw, Γgw) from among the cone specifications and the inclination angle of the relative velocity Vrsw. Each of these three variables may be any variable as long as the variable represents Pw, and the variables may be, in addition to those described above, for example, a combination of a ring gear reference radius R2w, a ring gear spiral angle ψgw, and a gear pitch cone angle Γgw, or a combination of the pinion reference radius R1w, the ring gear spiral angle ψpw, and Γgw.
Normally, an addendum aG and an addendum angle αG=aG/O2wPw are determined and the tip cone angle is determined by Γgf=Γs+αG. Alternatively, another value may be arbitrarily chosen for the addendum angle αG.
4.2A.4 Inclination Angle of Normal gW at Reference Point Pw
(1) Expression of Inclination Angle of gW in Coordinate System Cs
An intersection between gw passing through Pw(ucw, vcw, zcw: Cs) and the plane SH (βw=0) is set as P0 (uc0, 0, zc0: Cs) and the inclination angle of gw is represented with reference to the point P0 in the coordinate system Cs, by gw (ψ0, φn0: Cs). The relationship between P0 and Pw is as follows (
u
c0
=u
cw+(vcw/cos ψ0)tan φn0
z
c0
=z
cw
−v
cw tan ψ0 (16)
(2) Expression of Inclination Angle of gw on Pitch Plane Stw and Plane Snw (
When a line of intersection between the plane Snw and the pitch plane Stw is gtw, an inclination angle on the plane Snw from gtw is set as φnw. The inclination angle of gw is represented by gw (ψgw, φnw) using the inclination angle ψgw of Vrsw from PwO2w on the pitch plane Stw and φnw.
(3) Transformation Equation of Contact Normal gw
In the following, transformation equations from gw (ψgw, φnw) to gw (ψ0, φn0: Cs) will be determined.
When gw (φ2w, ψb2w: C2) is transformed from the coordinate system C2 to the coordinate system Cs, gw (ψ0, φn0: Cs) can be represented as follows:
sin φn0=cos ψb2w sin φ2w cos Γs+sin ψb2w sin Γs
tan ψ0=tan φ2w sin Γs−tan ψb2w cos Γs/cos φ2w (19)
With the expressions (17), (18), and (19), gw (ψgw, φnw) can be represented by gw (ψ0, φn0: Cs).
4.2B Reference Point Pw Based on R2w, βw, ψrw
As described above at the beginning of section 4.2A, the pitch cones of the pinion and the gear do not have to contact at the reference point Pw. In this section, a method is described in which the reference point Pw is determined on the coordinate system Cs without the use of the pitch cone, and by setting the gear reference radius R2w, a phase angle βw, and a spiral angle ψrw of the reference point.
The reference point Pw is set in the coordinate system Cs as follows:
P
w(ucw,vcw,zcw:Cs)
When Pw is represented with the circle radius rw of the relative velocity, and an angle from the uc axis βw, in a form of Pw(rw, βw, zcw: Cs),
u
cw
=r
w cos βw
v
cw
=r
w sin βw
In addition, as the phase angle βw of the reference point and the spiral angle ψrw are set based on expression (12) which represents a relationship between a radius rw around the instantaneous axis of the reference point Pw and the inclination angle ψrw of the relative velocity,
r
w
=E tan ψrw×sin(Σ−Γs)sin Γs/sin Σ
ucw and vcw are determined accordingly.
Next, Pw(ucw, vcw, zcw: Cs) is converted to the coordinate system C2 of rotation axis of the second gear. This is already described as expression (13).
u
2cw
=−u
cw cos Γs+zcw sin Γs
v
2cw
=v
cw
−v
cs2
z
2cw
=−u
cw sin Γs−zcw cos Γs (13a)
Here, as described in section 4.1, vcs2=E tan Γs/{tan(Σ−Γs)+tan Γs}. In addition, there is an expression in expression (13) describing:
R
2w
2
=u
2cw
2
+v
2cw
2 (13b)
Thus, by setting the gear reference radius R2w, zcw is determined based on expressions (13a) and (13b), and the coordinate of the reference point Pw in the coordinate system Cs is calculated.
Once the design reference point Pw is determined, the pinion reference circle radius R1w can also be calculated based on expression (13).
Because the pitch generating line Lpw passing at the design reference point Pw is determined, the pitch hyperboloid can be determined. Alternatively, it is also possible to determine a design reference cone in which the gear cone angle Γgw is approximated to be a value around Γs, and the pinion cone angle γpw is approximated by Σ−Γgw. Although the reference cones share the design reference point Pw, the reference cones are not in contact with each other. The tip cone angle can be determined similarly to as in section 4.2A.3.
A contact normal gw is set as gw (ψrw, φnrw; Crs) as shown in
Conversion of the contact normal from the coordinate system Crs to the coordinate system Cs will now be described.
(1) A contact normal gw (ψrw, φnrw; Crs) is set.
(2) When the displacement on the contact normal gw is Lg, the axial direction components of the displacement Lg on the coordinate system Crs are:
L
urs
=−L
g sin φnrw
L
vrs
=L
g cos φnrw·cos ψrw
L
zrs
=L
g cos φnrw·sin ψrw
(3) The axial direction components of the coordinate system Cs are represented with (Lurs, Lvrs, Lzrs) as:
L
uc
=L
urs·cos βw−Lvrs·sin βw
L
vc
=L
urs·sin βw+Lvrs·cos βw
L
zc
=L
zrs
(4) Based on these expressions,
(5) From
(6) From
tan φs0=−Luc/Lvc=(sin φnrw·cos βw+cos φnrw·cos ψrw·sin βw)/(−sin φnrw·sin βw+cos φnrw·cos ψrw·cos βw)
sin ψsw0=Lsc/Lg=cos φnrw·cos ψrw
The simplest practical method is a method in which the design reference point Pw is determined with βw set as βw=0, and reference cones are selected in which the gear cone angle is around Γgw=Γs and the pinion cone angle is around γpw=Σ−Γgw. In this method, because βw=0, the contact normal gw is directly set as gw (ψ0, φn0; Cs).
An contact ratio mf along Lpw and an contact ratio mfcone along a direction of a line of intersection (PwPgcone in
Because the contact normal gw is represented in the coordinate system Cs with gw=g0 (ψ0, φn0: Cs) the point Pw(u2cw, v2cw, z2cw: C2) represented in the coordinate system C2 is converted into the point Pw(q2cw, −Rb2w, z2cw: Cq2) on the coordinate system Cq2 in the following manner:
The inclination angle g0 (φ20, ψb20: C2) of the contact normal gw=g0, the inclination angle φa0 of the surface of action Sw0, and the inclination angle ψsw0 of g0 (=P0PGswn) on Sw0 (
tan φ20=tan φn0 cos(Γs−ψ0)
sin ψb20=sin φn0 sin(Γs−ψ0)
tan φs0=tan φno cos ψ0
tan ψsw0=tan ψs sin φs0
or sin ψsw0=sin φn0 sin ψ0 (20a)
tan φ20=tan φn0 cos Γs/cos ψ0+tan ψ0 sin Γs
sin ψb20=sin φn0=sin Γs−cos φn0 sin ψ0 cos Γs
tan φa0=tan φn0/cos ψ0
tan ψsw0=tan ψ0 cos φa0 (20b)
The derivation of φs0 and φsw0 are detailed in, for example, Papers of Japan Society of Mechanical Engineers, Part C, Vol. 70, No. 692, c2004-4, Third Report of Design Theory of Power Transmission Gears.
In the following, a calculation is described in the case where the path of contact coincides with the contact normal gw=g0. If it is assumed that with every rotation of one pitch Pw moves to Pg, and the tangential plane W translates to Wg, the movement distance PwPg can be represented as follows (
P
w
P
g
=P
gw
=R
b2w(2θ2p)cos ψb20 (21)
where Pgw represents one pitch on g0 and 2θ2p represents an angular pitch of the ring gear.
When the intersection between Lpw and Wg is P1w, one pitch Pfw=P1wPw on the tooth trace Lpw is:
P
fw
=P
gw/sin ψsw0 (22)
The relationship between the internal and external circle radii of the ring gear and the face width of the ring gear is:
R
2t
=R
2h
−F
g/sin Γgw
where R2t and R2h represent internal and external circle radii of the ring gear, respectively, Fg represents a gear face width on the pitch cone element, and Γgw represents a pitch cone angle.
Because the effective length F1wp of the tooth trace is a length of the pitch generating line Lpw which is cut by the internal and external circles of the ring gear:
F
1wp={(R2h2−v2pw2)1/2−(R2t2−v2pw2)1/2}/sin Γs (23)
Therefore, the general equation for the tooth trace contact ratio mf would be:
m
f
=F
1wp
/P
fw (24)
The pitch generating line Lpw coincides with the instantaneous axis (Γs=0), and Pw may be anywhere on Lpw. Normally, Pw is taken at the origin of the coordinate system Cs, and, thus, Pw (ucw, vcw, zcw: Cs) and the contact normal gw=g0 (ψ0, φn0: Cs) can be simplified as follows, based on expressions (20) and (20a):
P
w(0,0,0:Cs),Pw(0,−vcs2,O:C2)
P
0(q2pw=−vcs2 sin χ20,−Rb2w=−vcs2 cos χ20,0:Cq2)
φ20=φs0,ψb20=−ψsw0
tan ψb20=−tan ψsw0=−tan ψ0 sin φ20
In other words, the plane Sw0 and the plane of action G20 coincide with each other. It should be noted, however, that the planes are viewed from opposite directions from each other.
These values can be substituted into expressions (21) and (22) to determine the tooth trace contact ratio mf with the tooth trace direction pitch Pfw and expression (24):
where R2w=Rb2w/sin φ20 represents a radius of a ring gear reference cylinder, p=R2w (2θ2p) represents a circular pitch, and F=F1wp represents the effective face width.
The expression (25) is a calculation equation of the tooth trace contact ratio of the cylindrical gear of the related art, which is determined with only p, F, and ψ0 and which does not depend on φn0. This is a special case, which is only true when Γs=0, and the plane Sw0 and the plane of action G20 coincide with each other.
For the bevel gears and the hypoid gears, the plane Sw0 does not coincide with G20 (Sw0≠G20), and thus the tooth trace contact ratio mf depends on φn0, and would differ between the drive-side and the coast-side. Therefore, the tooth trace contact ratio mf of the bevel gear or the hypoid gear cannot be determined with the currently used expression (25). In order to check the cases where the currently used expression (25) can hold, the following conditions (a), (b), and (c) are assumed:
(a) the gear is a bevel gear; therefore, the pitch generating line Lpw coincides with the instantaneous axis and the design reference point is Pw(0, 0, zcw: Cs);
(b) the gear is a crown gear; therefore, Γs=π/2; and
(c) the path of contact is on the pitch plane; therefore, φn0=0.
The expressions (20), and (20b)-(24) can be transformed to yield:
φ20=ψ0,ψb20=0,φs0=0,ψsw0=ψ0
R
b2w
=R
2w cos φ20=R2w cos ψ0
P
gw
=R
b2w(2θ2p)cos ψb20=R2w(2θ29)cos ψ0
P
fw
=|P
gw/sin ψsw0|=|R2w(2θ2p)/tan ψ0|
m
f
=F
1wp
/P
fw
=F tan ψ0/R2w(2θ2p)=F tan ψ0/p (26)
The expression (26) is identical to expression (25). In other words, the currently used expression (25) holds in bevel gears which satisfy the above-described conditions (a), (b), and (c). Therefore,
(1) strictly, the expression cannot be applied to normal bevel gears having Γs different from π/2 (Γs≠π/2) and φn0 different from 0 (φn0≠0); and
(2) in a hypoid gear (E≠0), the crown gear does not exist and ε2w differs from 0 (ε2w≠0).
For these reasons, the tooth trace contact ratios of general bevel gears and hypoid gears must be determined with the general expression (24), not the expression (26).
4.4 Calculation Method of Contact Ratio mfcone Along Line of Intersection of Gear Pitch Cone and Surface of Action Sw0
The tooth trace contact ratios of the hypoid gear (Gleason method) is calculated based on the expression (26), with an assumption of a virtual spiral bevel gear of ψ0=(ψpw+ψgw)/2 (
P
gcone(ucgcone,vcs2,zcgcone:Cs)
P
gcone(u2cgcone,0,zc2gcone:C2)
where
ucgcone=ucw+(vcw−vcs2)tan Γs0
zcgcone={(vcs2−vcw)/cos φs0} tan ψgcone+zcw
u2cgcone=−ucgcone cos Γs+zcgcone sin Γs
z2cgcone=−ucgcone sin Γs−zcgcone cos Γs
ψgcone represents an inclination angle of Pw Pgcone from P0Pssw on Sw0.
Because Pgcone is a point on a cone surface of a cone angle Γgcone passing through Pw, the following relationship holds.
u
2cgcone
−R
2w=−(z2cgcone−z2cw)tan Γgcone (27)
When a cone angle Γgcone is set, ψgcone can be determined through expression (27). Therefore, one pitch Pcone along PwPgcone is:
P
cone
=P
gw/cos(ψgcone−ψsw0) (28)
The contact length F1wpcone along PwPgcone can be determined in the following manner.
In
u
cws
=u
cw
+v
cw tan φs0
z
cws
=z
cw−(vcw/cos φs0)tan ψgcone
If an arbitrary point on the straight line PwPgcone is set as Q(ucq, vcq, zcq: Cs) (
v
cq={(zcq−zcws)/tan ψgcone} cos φa0
u
cq
=u
cws
−v
cq tan φs0
If the point Q is represented in the coordinate system C2 using expression (13), to result in Q (u2cq, v2cq, z2cq: C2), the radius R2cq of the point Q is:
u
2cq
=−u
cq cos Γs+zcq sin Γs
v
2cq
=v
cq
−v
cs2
R
2cq=√(u2cq2+v2cq2)
If the values of zcq where R2cq=R2h and R2cq=R2t are zcqh and zcqt, the contact length F1wpcone is:
F
1wpcone=(zcqh−zcqt)/sin ψgcone (29)
Therefore, the contact ratio mfcone along PwPgcone is:
m
fcone
=F
1wpcone
/P
cone (30)
The value of mfcone where ψgcone→π/2 (expression (30)) is the tooth trace contact ratio mf (expression (24)).
Table 1 shows specifications of a hypoid gear designed through the Gleason method. The pitch cone is selected such that the radius of curvature of the tooth trace=cutter radius Rc=3.75″. In the following, according to the above-described method, the appropriateness of the present embodiment will be shown with a test result by:
(1) first, designing a hypoid gear having the same pitch cone and the same contact normal as Gleason's and calculating the contact ratio mfcone in the direction of the line of intersection of the pitch cone and the surface of action, and
(2) then, designing a hypoid gear with the same ring gear reference circle radius R2w, the same pinion spiral angle ψpw, and the same inclination angle φnw of the contact normal, in which the tooth trace contact ratio on the drive-side and the coast-side are approximately equal to each other.
When values of a shaft angle Σ=90°, an offset E=28 mm, and a gear ratio i0=47/19 are set, the intersection Cs between the instantaneous axis and the line of centers and the inclination angle Γs of the instantaneous axis are determined in the following manner with respect to the coordinate systems C1 and C2:
Based on Table 1, when values of a ring gear reference circle radius R2w=89.255 mm, a pinion spiral angle ψpw=46.988°, and a ring gear pitch cone angle Γgw=62.784° are set, the system of equations based on expressions (13), (14), and (15) would have a solution:
Therefore, the pitch point Pw is:
The pitch generating line Lpw is determined on the coordinate system Cs as a straight line passing through the reference point Pw and parallel to the instantaneous axis (Γs=67.989°).
In the following calculations, the internal and external circle radii of the ring gear, R2t=73.87 and R2h=105 are set to be constants.
5.2 Contact Ratio mfconeD of Tooth Surface D (Represented with Index of D) with Contact Normal gwD
Based on Table 1, when gwD is set with gwD(ψgw=30.859°, φnwD=15°), gwD can be converted into coordinate systems Cs and C2 with expressions (17), (18), and (19), to yield:
The surface of action SwD can be determined on the coordinate system Cs by the pitch generating line Lpw and gwD. In addition, the intersection P0d between gwD and the plane SH and the radius R20D around the gear axis are, based on expression (16):
The contact ratio mfconeD in the direction of the line of intersection between the pitch cone and the surface of action is determined in the following manner.
The inclination angle φs0D of the surface of action SwD, the inclination angle ψsw0D of gwD on SwD, and one pitch PgwD on gwD are determined, based on expressions (20), (20b), and (21), as:
When gwC(ψgw=30.859°, φnwC=−27.5° is set, similar to the tooth surface D,
The inclination angle φs0C of the surface of action SwC, the inclination angle ψsw0C of gwC on Swc, and one pitch PgwC on gwC are, based on expressions (20), (20b), and (21):
According to the Gleason design method, because Γgw=Γgcone=62.784°, the contact ratio along the line of intersection between the pitch cone and the surface of action are mfcone63D=1.70 and mfcone63C=2.92, which is very disadvantageous for the tooth surface D. This calculation result can be considered to be explaining the test result of
In addition, when the ring gear cone angle Γgcone=Γs=67.989°, ψgcone=−89.99° in both the drive-side and the coast-side. Thus, the line of intersection between the cone surface and the surface of action coincides with the pitch generating line Lpw, the tooth trace contact ratio of the present invention is achieved, and the contact ratio is approximately equal between the drive-side and the coast-side. Because of this, as shown in
αg=Σδt×ag/(ag+ap) (31)
a
g
+a
p
=h
k(action tooth size) (32)
where Σδt represents a sum of the ring gear addendum angle and the ring gear dedendum angle (which changes depending on the tapered tooth depth), αg represents the ring gear addendum angle, ag represents the ring gear addendum, and ap represents the pinion addendum.
The virtual pitch cones Cpv of the ring gear and the pinion defined here do not contact each other, although the cones pass through the reference point Pw1.
The addendum and the addendum angle are defined as shown in
A design method by the virtual pitch cone Cpv will now be additionally described.
As already described, if the surface of action intersects the cone surface having the cone angle of Γgw=67.989° over the entire face width, the above-described tooth trace contact ratio can be realized. In other words, in
In order for the cone surface having the cone angle of 67.989° (approximately 68°) and the surface of action to intersect over the entire face width without a change in the pitch cone Cp1, the ring gear addendum angle αg may be increased so that the tip cone angle Γf is 68°. As shown in
5.4 Hypoid Gear Specifications and Test Results when Γgw is Set Γgw=Γs=67.989°
Table 2 shows hypoid gear specifications when Γgw is set Γgw=Γs=67.989°. Compared to Table 1, identical ring gear reference circle radius R2w=89.255 mm and pinion spiral angle ψpw=46.988° are employed, and the ring gear pitch cone angle is changed from Γgw=62.784° to 67.989°. As a result, Pw and Γgw differ as shown in
With the pressure angles φnwD and φnwC identical to Table 1, if gwD(30.768°, 15°) and gwC(30.768°, −27.5°) are set, the inclination angles would differ, in the static coordinate system Cs, from gwD and gwC of Table 1:
The inclination angles of gwD and gwC on the surface of action, and one pitch are:
The tooth trace contact ratios are calculated in the following manner based on expressions (22), (23), and (24):
5.5 Specifications of Hypoid Gear when βw=0
Table 3 shows specifications of a hypoid gear when βw is set to 0 (βw=0) in the method of determining the design reference point Pw based on R2w, βw, and ψrw described in section 4.2B.
In the above-described design of hypoid gears, the design is aided by a computer aided system (CAD) shown in
The program can be briefly described as follows. First, a design request value of the hypoid gear and values of variables for determining a tooth surface are acquired. A pitch cone angle Γgcone of one gear is provisionally set and used along with the acquired values of the variables, and an contact ratio mfconeD of the drive-side tooth surface and an contact ratio mfconeC of the coast-side tooth surface based on the newly defined tooth trace as described above are calculated. The pitch cone angle Γgcone is changed and the calculation is repeatedly executed so that these contact ratios become predetermined values. When the contact ratios of the tooth surfaces become predetermined values, the pitch cone angle at this point is set as a design value Γgw, and the specifications of the hypoid gear are calculated. The predetermined value of the contact ratio designates a certain range, and values in the range. Desirably, the range of the contact ratio is greater than or equal to 2.0. The range may be changed between the drive-side and the coast-side. The initial value of the pitch cone angle Γgcone to be provisionally set is desirably set to the inclination angle Γs of the instantaneous axis S.
Another program calculates the gear specifications by setting the pitch cone angle Γgw, to the inclination angle Γs of the instantaneous axis from the first place, and does not re-adjust the pitch cone angle according to the contact ratio. Because it is known that the contact ratios of the tooth surfaces become approximately equal to each other when the pitch cone angle Γgw is set to the inclination angle Γs of the instantaneous axis, such a program is sufficient as a simple method.
Number | Date | Country | Kind |
---|---|---|---|
2008-187965 | Jul 2008 | JP | national |
2008-280558 | Oct 2008 | JP | national |
2009-111881 | May 2009 | JP | national |
The present application is a divisional application of U.S. application Ser. No. 13/054,323, filed Feb. 3, 2011, which is a National Stage Entry of PCT/JP2009/063234, filed Jul. 16, 2009, which claims priority to each of JP 2009-111881, filed May 1, 2009, JP 2008-280558, filed Oct. 30, 2008, and JP 2008-187965, filed Jul. 18, 2008. The disclosures of each of the above applications are hereby incorporated by reference in their entireties.
Number | Date | Country | |
---|---|---|---|
Parent | 13054323 | Feb 2011 | US |
Child | 14508422 | US |