Wave gear device having tapered flexible external gear

With a wave gear device, the gears employ homothetic curve tooth profiles AD, BE. Furthermore, a transposition is applied to the external teeth along the tooth trace such that the movement loci M2, M3 of the external teeth in a section perpendicular to the axis, from the aperture end to the inner end, share the movement locus M1 of the aperture end and bottom portion thereof, and a continuous meshing of the teeth in the tooth trace direction is achieved. Furthermore, the tooth bottom rim thickness of the aperture end of the external teeth is optimized using a modified Goodman diagram, and a tooth bottom rim thickness which takes into account the relationship between the tooth profile and the transmitted torque from the aperture end to the inner end is employed for the flexible external gear.

TECHNICAL FIELD

The present invention relates to a wave gear device having a tapered flexible external gear furnished with a tooth profile capable of continuous meshing over a wide range in the tooth trace direction. In more detail, the present invention relates to a wave gear device having a tapered flexible external gear, whereby maximization of transmission load torque is possible through optimization of the rim thickness established in the flexible external gear.

BACKGROUND ART

Since the invention of the wave gear device by C. W. Musser (Patent Document 1) up to the present day, inventions for devices of various types have been conceived by its originator, as well as by numerous researchers, including the present inventor. Even limiting the discussion to inventions relating to the tooth profile, numerous different types have been proposed. For example, in Patent Document 2, the present inventor proposed using an involute tooth profile as a basic tooth profile; and in Patent Documents 3 and 4 proposed a tooth profile design method employing a procedure for approximating, by means of a rack, meshing of the teeth of a rigid internal gear and a flexible external gear of a wave gear device, to derive an addendum tooth profile that affords contact of the two gears over a wide area.

Typically, a wave gear device has a ring-shaped rigid internal gear, a flexible external gear disposed coaxially to the inside thereof, and a wave generator fitting inside thereof. The flexible external gear is provided with a flexible cylindrical barrel portion, a diaphragm radially extending from the rear end of this cylindrical barrel portion, and external teeth formed on the outside peripheral face section of the cylindrical barrel portion at the front end opening side thereof. The flexible external gear is flexed into ellipsoidal shape by the wave generator, and meshes with the rigid internal gear at both ends in the major axis direction of the ellipse.

The external teeth of the flexible external gear flexed into ellipsoidal shape have an increasing amount of flexure, proportional to the distance from the diaphragm, from the diaphragm side towards the front end opening along the tooth trace direction thereof. Moreover, sections of the tooth portion of the flexible external gear undergo repeated flexure in radial directions in association with rotation of the wave generator. However, to date, sufficient consideration has not been given to a rational method for establishing a tooth profile in a manner that takes into consideration such flexural action (coning) of the flexible external gear by the wave generator.

In Patent Document 5, the present inventor proposed a wave gear device provided with a tooth profile by which continuous meshing is possible, with consideration given to coning of the teeth. In the device proposed in Patent Document 5, an arbitrary axis-perpendicular cross section location in the tooth trace direction of the flexible external gear is selected as a principal cross section, and at a major axis position in an ellipsoidal rim neutral line of the flexible external gear in the principal cross section, an amount of flexure 2 κmn (where κ is the flexural coefficient, m is the module, and n is a positive integer) with respect to a rim neutral circle prior to flexure is established in such a way as to flex to a standard-deflection state of 2 mn (κ=1).

Additionally, using rack meshing to approximate meshing of the flexible external gear and the rigid internal gear, in axis-perpendicular cross sections at locations including the principal cross section in the tooth trace direction of the flexible external gear, movement loci of the teeth of the flexible external gear with respect to the teeth of the rigid internal gear in association with rotation of the wave generator are derived; a first homothetic curve BC is derived by scaling down, by a ratio λ (λ<1) while employing a point B as the homothetic center, a curve segment extending from a point A of an apical portion to the point B in the next bottom portion in a movement locus of a standard deflection obtained in the principal cross section, and this first homothetic curve BC is adopted as the basic tooth profile for the addendum of the rigid internal gear.

Furthermore, a second homothetic curve is derived by scaling, by a ratio (1−λ)/λ while employing an end point C of the first homothetic curve BC as the homothetic center, of a curve obtained by 180 degree rotation of the first homothetic curve BC about a center at the point C, and this second homothetic curve is adopted as the basic tooth profile for the addendum of the flexible external gear.

In addition to this, a transposition is applied to tooth profile sections to both sides, in the tooth trace direction, from the principal cross section in the tooth profile of the flexible external gear, doing so in such a way that both negative deflecting-side movement loci obtained in axis-perpendicular cross sections flexed to a negative deflection state (flexural coefficient λ<1) to the diaphragm side from the principal cross section in the external teeth of the flexible external gear, and positive deflection-side movement loci obtained in axis-perpendicular cross sections flexed to a positive deflection state (flexural coefficient λ>1) to the opening side from the principal cross section, describe curves that contact the bottom part of the standard-deflecting movement locus in the principal cross section. The resultant flexible external gear is a tapered flexible gear having an addendum circle of progressively smaller diameter from the opening side towards the diaphragm side in the tooth trace direction.

With a wave gear device in which such a tooth profile has been formed, centering on continuous meshing of tooth profile over a wide range in the principal cross section, effective meshing can be achieved in a tooth trace range extending to the opening end from the principal cross section, and in a tooth trace range extending to the diaphragm side from the principal cross section. Therefore, greater torque can be transmitted, as compared with a conventional wave gear device in which meshing takes place over a narrow tooth trace range.

PRIOR ART DOCUMENTS

Patent Documents

DISCLOSURE OF THE INVENTION

Problems to be Solved by the Invention

Prior inventions relating to tooth profiles of wave gear devices were made in independently in a manner unrelated to the rim thickness of the flexible external gear. Specifically, no consideration whatsoever was given to the relationship between tooth profile, and root rim thickness of the flexible external gear which is related to transmission load torque.

Even when a tooth profile that takes coning of the flexible external gear into consideration, making continuous meshing possible, is established such that increased transmission load torque becomes possible, the result is that the transmission load torque of the flexible external gear cannot be increased, unless there is provided root rim thickness such that transmission of the increased transmission load torque is possible. In cases of a shifted tooth profile in which addendum modifications involving different amounts are applied along the tooth trace direction with consideration to coning of the external teeth, it is necessary to establish an appropriate root rim thickness according to the tooth profile (amount of addendum modification) at each location in the tooth trace direction, in order to make possible an increase in the transmission load torque.

With the foregoing in view, it is an object of the present invention to make possible increase in transmission load torque, through optimized establishment of rim thickness of a flexible external gear for a wave gear device having a tapered flexible external gear capable of continuous meshing over a wide range in the tooth trace direction.

Means Used to Solve the Problems

In order to achieve the aforementioned object, in the wave gear device of the present invention, the tooth profiles of both gears and the external tooth rim thickness are established according to the procedure of (1) to (6) below.

(1) Designating an axis-perpendicular cross section at an opening end location in the tooth trace direction of the external teeth of a flexible external gear, as a standard-deflecting principal cross section having a flexural coefficient κ=1, and deriving respective homothetic curve tooth profiles to be employed for specifying addendum tooth profiles of the teeth of both gears, from movement loci of the external teeth of the flexible external gear with respect to the internal teeth of the rigid internal gear in the principal cross section in question.

(2) Employing, as the tooth profile of the principal cross section at the opening end location of the external teeth of the flexible external gear, a composite tooth profile specified by the addendum profile specified in the aforedescribed manner, a linear tooth profile connected thereto, and an appropriate dedendum tooth profile that avoids interference.

(3) Employing, as the tooth profile for the internal teeth of the rigid internal gear, a composite tooth profile specified by the addendum profile specified in the aforedescribed manner, a linear tooth profile connected thereto, and an appropriate dedendum tooth profile that avoids interference.

(4) In consideration of coning of the flexible external gear, adopting as the tooth profile in cross sections other than the opening end in the tooth trace direction of the external teeth of the flexible external gear, a shifted tooth profile in which a tooth profile shifting is applied to the composite tooth profile adopted as the tooth profile of the principal cross section at the opening end location. In other words, a tooth profile shifting is applied to the tooth profile of the principal cross section, in such a way that relative movement loci with respect to the internal teeth of the rigid internal gear obtained in cross sections in the tooth trace direction of the external teeth of the flexible external gear share the bottom portion thereof with the movement locus of the principal cross section of the opening end location of the external teeth, whereby continuous meshing of both gears over a wide range in the tooth trace direction is achieved.

(5) A modified Goodman diagram is employed to establish optimal root rim thickness at the opening end location in the tooth trace direction of the external teeth of the flexible external gear.

(6) In consideration of the tooth profile and the transmitted torque, a modified Goodman diagram is employed to establish (on the basis of the optimal root rim thickness at the opening end location) the root rim thickness at locations other than the opening end in the tooth trace direction of the external teeth.

Effect of the Invention

According to the present invention, there is provided a wave gear device capable of continuous meshing, provided with a tapered flexible external gear with which continuous meshing is achieved over a wide range in the principal cross section of an opening end location of the external teeth, as well as achieving meshing over a wide range in the tooth trace direction, wherein the root rim thickness of the flexible external gear, which had been given no consideration whatsoever in the prior art, can be established at optimal thickness at each location in the tooth trace direction, so as to be commensurate with the transmission load torque. Therefore, according to the present invention, it is possible to greatly improve the transmission load torque of a flexible external gear of a wave gear device, as compared with the prior art.

MODE FOR CARRYING OUT THE INVENTION

A wave gear device in which the present invention is applied will be described below, making reference to the drawings.

(Constitution of Wave Gear Device)

FIG. 1is a front view of a wave gear device to which the present invention is directed. The cross sectional views inFIG. 2show, in axis-including cross section, a condition in which the opening of the flexible external gear thereof is flexed into ellipsoidal shape, whereinFIG. 2(a)shows a state prior to deformation,FIG. 2(b)shows a cross section including the major axis of an ellipsoidal curve subsequent to deformation, andFIG. 2(c)shows a cross section including the minor axis of an ellipsoidal curve subsequent to deformation, respectively. InFIG. 2(a) to (c), the solid lines indicate a flexible external gear of cup shape, and the broken lines show a flexible external gear of silk hat shape.

As shown in these drawings, the wave gear device1has a ring-shaped rigid internal gear2, a flexible external gear3disposed to the inside thereof, and a wave generator4of ellipsoidal contours fitting inside thereof. The rigid internal gear2and the flexible external gear3are both spur gears of module m. The difference in number of teeth between the two gears is 2n (n is a positive integer), with the rigid internal gear2having the greater number. The flexible external gear3is flexed into ellipsoidal shape by the wave generator4of ellipsoidal contours, and meshes with the rigid internal gear2in sections at either end of the ellipsoidal shape in the major axis L1direction. As the wave generator4rotates, the locations at which the two gears2,3mesh move in a circumferential direction, generating between the two gears2,3relative rotation according to the difference in number of teeth between the two gears. The flexible external gear3is provided with a flexible cylindrical barrel part31, a diaphragm32continuing on from the rear end31bthereof and spreading out in a radial direction, a boss33continuing on from the diaphragm32, and external teeth34formed on an outside peripheral surface section at an opening31aside of the cylindrical barrel part31.

Due to the wave generator4of ellipsoidal contours fitting within an inside peripheral surface section of the external tooth formation section of the cylindrical barrel part31, the cylindrical barrel part31experiences a progressively increasing amount of flexure towards the outside or towards the inside in a radial direction, towards the opening end31afrom a rear end31bon the diaphragm side. As shown inFIG. 2(b), in a cross section that includes the major axis L1of the ellipsoidal shape, the amount of flexure towards the outside progressively increases in proportion to the distance from the rear end31bto the opening end31a; and as shown inFIG. 2(c), in a cross section that includes the minor axis L2of the ellipsoidal shape, the amount of flexure towards the inside progressively increases in proportion to the distance from the rear end31bto the opening end31a. Consequently, the external teeth34formed on the outside peripheral surface section at the opening end31aside likewise experience varying amounts of flexure in axis-perpendicular cross sections in the tooth trace direction thereof. Specifically, the amount of flexure progressively increases, in a manner proportional to the distance from the rear end31b, from the location of the inner end34bon the diaphragm side towards the location of the opening end34aon the opening side in the tooth trace direction of the external teeth34.

(Tooth Profile Shape of Both Gears)

FIG. 3is a descriptive diagram showing an example of the tooth profiles of both gears2,3; andFIG. 4is a descriptive diagram showing a tooth profile contour shape in the tooth trace direction of the flexible external gear3. The tooth profile shape of the external teeth34shown inFIG. 3is that at the location of the opening end34a(principal cross section) thereof, and the tooth profile shape in a section extending from the opening end34ato the inner end34bof the external teeth34is a shifted tooth profile shape obtained by applying minus tooth profile shifting to the tooth profile shape shown inFIG. 3, in the manner discussed below. As a result, as shown inFIG. 4, the flexible external gear3is a tapered flexible external gear in which the diameter of the addendum circle becomes progressively smaller from the opening end34atowards the inner end34balong the tooth trace direction. In contrast to this, the tooth profile shape of the inner teeth24is unchanging along the entire tooth trace direction, and is established to have the tooth profile shape shown inFIG. 3.

As shown inFIG. 3, the tooth profile shape at the opening end34a(principal cross section) of the external teeth34is defined by an external tooth addendum tooth profile section41of convex curving shape, an external tooth linear tooth profile section42continuous therewith, an external tooth dedendum tooth profile section43of concave curving shape continuous therewith, and an external tooth root section44continuous therewith. The tooth profile shape of the internal teeth24is defined by an internal tooth addendum tooth profile section51of convex curving shape, an internal tooth linear tooth profile section52continuous therewith, an internal tooth dedendum tooth profile section53of concave curving shape continuous therewith, and an internal tooth root section54continuous therewith.

(Method of Forming Tooth Profiles of Both Gears)

Next, the method of forming the tooth profiles of the external teeth34and the internal teeth24will be described making reference toFIG. 3,FIG. 5, andFIG. 6.

(Movement Loci of Teeth Through Rack Approximation)

FIG. 5is a diagram showing movement loci of the external teeth34of the flexible external gear3with respect to the internal teeth24of the rigid internal gear2, obtained through rack approximation of the relative motion of the two gears2,3. In the drawing, the x axis is the translation direction of the rack, and the y axis shows a direction perpendicular thereto. Here, in an axis-perpendicular cross section at any location in the tooth trace direction of the external teeth34of the flexible external gear3, the amount of flexure by the external tooth34in question at the major axis location L1in an ellipsoidal rim neutral line with respect to the rim neutral line prior to flexure of the external tooth34in question into ellipsoidal shape, is 2 κmn, where κ is the flexural coefficient. Movement loci of the external teeth34of the flexible external gear3are given by equation 1.
x=0.5 mn(θ−κ sin θ)
y=κmn cos θ

Assuming, for simplicity of description, that m=1 and n=1 (the tooth count differential is 2), the movement locus is as described by the following equation.
x=0.5(θ−κ sin θ)
y=κ cos θ

The origin of the y axis inFIG. 5is the average position of amplitude of the movement loci. Of the movement loci, the standard-deflecting movement locus M1is one obtained in a benchmark, standard deflecting flexural state in which the flexural coefficient κ=1, while the negative deflecting movement loci M2, M3are ones obtained in a negative deflecting flexural state in which the flexural coefficient κ<1. In the present invention, the principal cross section serving as the foundation for formation of the tooth profile of the two gears2,3is an axis-perpendicular cross section at the location of the opening end34ain the tooth trace direction of the external teeth34of the flexible external gear3. The negative deflecting movement locus M2is a locus obtained in an axis-perpendicular cross section at a medial location in the tooth trace direction of the external teeth34, and the negative deflecting movement locus M3is a locus obtained at the location of the inner end34bin the tooth trace direction of the external teeth34. InFIG. 5, the movement locus M2is one obtained at flexural coefficient κ=0.85, and the movement locus M3is one obtained at flexural coefficient κ=0.7.

(Method of Forming Tooth Profile in Principal Cross Section)

FIG. 6is a descriptive diagram showing a utilization range of the standard deflecting movement locus M1, employed in forming the tooth profiles of the external teeth34and the inner teeth24. In the drawing, the parameter θ of the standard deflecting movement locus M1of the principal cross section (the cross section of the opening end34a) has a range from π (point B: the bottom portion of the movement locus) to 0 (point A: the apical portion of the movement locus), and with point B as the homothetic center, the standard deflecting movement locus M1undergoes homothetic transformation by a ratio λ (λ<1), to obtain a first homothetic curve BC.FIG. 6shows a case in which λ=0.6. The first homothetic curve BC is adopted as the tooth profile curve employed for defining the addendum tooth profile of the rigid internal gear2.

The first homothetic curve BC is then rotated by 180 degrees about the end point C of the first homothetic curve BC, to obtain a curve B′C. A second homothetic curve AC is obtained through transformation of this curve B′C at a ratio (1−λ)/λ at a homothetic center at the end point C. The second homothetic curve AC is adopted as the tooth profile curve employed for defining the addendum tooth profile of the flexible external gear3.

The tooth profile curves for defining these addendum tooth profiles are expressed by the following equations.

Basic equation for addendum tooth profile of rigid internal gear:
xCa=0.5{(1−λ)π+λ(θ−sin θ)}
yCa=λ(1+cos θ)} (0≦θ≦π)

Basic equation for addendum tooth profile of flexible external gear:
xFa=0.5(1−λ)(π−θ+sin θ)}
yFa=(1−λ)(1+cos θ)} (0≦θ≦π)
(Tooth Profile Shape of Principal Cross Section of External Teeth)

The tooth profile curve AC for defining the addendum tooth profile, derived in the aforedescribed manner, is employed in forming an external tooth tooth profile in the principal cross section (an axis-perpendicular cross section of the opening end34a) of the external teeth34in the following manner. To describe with reference toFIG. 3andFIG. 6, a straight line L is drawn to intersect, at a pressure angle α, the tooth profile curve AC for defining the addendum tooth profile of the flexible external gear3, and a curve segment AD lying between the end point A of the tooth profile curve AC and an intersection point D with the straight line L is derived. Adopting this curve segment AD as the tooth profile curve defining a normal addendum tooth profile, the tooth profile curve in question is employed to form the external tooth addendum tooth profile section41. The external tooth linear tooth profile section42is defined by a line segment of the straight line L extending from the intersection point D. Further, the external tooth deddendum tooth profile section43is defined by a predetermined convex curve connecting between the external tooth linear tooth profile section42and the external tooth root section44which is defined by a predetermined external tooth root curve, doing so in such a way as to ensure predetermined radial clearance of the external tooth linear tooth profile section42with respect to the internal teeth24.

(Tooth Profile Shape of Internal Teeth)

Likewise, the tooth profile curve BC employed for defining the addendum tooth profile is employed to form the tooth profile of the internal teeth24. To describe with reference toFIG. 3andFIG. 6, a straight line L is drawn to intersect, at a pressure angle α, the tooth profile curve BC for defining the addendum tooth profile of the rigid internal gear2, and a curve segment BE lying between the end point B of the tooth profile curve BC and an intersection point E with the straight line L is derived. Adopting this curve segment BE as the tooth profile curve defining a normal addendum tooth profile, the tooth profile curve in question is employed to form the internal tooth addendum tooth profile section51. The internal tooth linear tooth profile section52is defined by a line segment of the straight line L extending from the intersection point E. Further, the internal tooth deddendum tooth profile section53is defined by a predetermined convex curve connecting the internal tooth linear tooth profile section52and the internal tooth root section54which is defined by a predetermined external tooth root curve, doing so in such a way as to ensure predetermined radial clearance of the internal tooth linear tooth profile section52with respect to the external teeth34.

The tooth profile sections43,44,53,54of the deddendums of the two gears do not participate in meshing. Consequently, these dedendum tooth profile sections43,44,53,54can be designed freely, provided that there is no interference with the respective corresponding addendum tooth profile sections51,52,41,42.

In this way, tooth profile shapes are formed at locations of principal cross sections (axis-perpendicular cross sections of the opening end34aof the external teeth34) in both of the gears2,3shown inFIG. 3. In the present example, the pressure angle of the linear tooth profile α is 9 degrees. From the standpoint of machining of the gears, it is preferable to avoid sections in which the pressure angle of the addendum tooth profile is close to zero, and to connect the linear tooth profiles to the deddendum tooth profiles from points of a pressure angle of close to 6 degrees to 10 degrees.

(Tooth Profile Shape at Locations Other than Principal Cross Section in External Teeth)

With regard to meshing of the tooth profiles of the principal cross section established in the aforedescribed manner, during intermeshing of the addendum tooth profiles of the two gears2,3, when the flexible external gear3moves along the standard deflecting movement locus M1shown inFIG. 5with respect to the rigid internal gear2, the addendum tooth profiles come into continuous contact due to the nature of the homothetic curve. In contrast to this, in axis-perpendicular cross sections lying towards the diaphragm side from the principal cross section in the external teeth34, the deflection coefficient κ<1. As shown inFIG. 5, the negative deflecting movement loci M2, M3interfere with the non-deflecting movement locus M1, and for as long as this persists, continuous intermeshing of the addendum tooth profiles, such as that taking place in the case of the principal cross section, cannot be sustained.

Accordingly, a shifted tooth profile in which a tooth profile shifting is applied to the tooth profile of the principal cross section (the axis-perpendicular cross section of the opening end34a) is adopted as the external tooth tooth profile of axis-perpendicular cross sections in a section extending from the opening end34ato the inner end34bin the external teeth34. Specifically, the shifted tooth profile shapes are obtained by applying minus tooth profile shifting to the external-tooth tooth profile of the opening end34a, doing in such a way that movement loci obtained through rack approximation of the external teeth34with respect to the internal teeth24in axis-perpendicular cross sections from the opening end34ato the inner end34bcontact the bottom portion B of the movement locus M1obtained at the opening end34aconstituting the principal cross section location. In so doing, proper meshing in at least localized fashion can be ensured on all cross sections in the tooth trace direction of the external teeth34.

To discuss in more specific terms, in each of axis-perpendicular cross sections taken towards the location of the inner end34bon the diaphragm side from the opening end34aof the external teeth34, an amount of addendum modification mnh is established according to the flexural coefficient κ at each of the axis-perpendicular cross section locations, doing so in such a way that the movement locus in each axis-perpendicular cross section contacts the bottom part B of the movement locus M1at the opening end34a. In a case in which m=1 and n=1, the amount of addendum modification (tooth profile shifting) is h, and assumes a negative value represented by the following expression.
h=κ−1

Through application of tooth profile shifting in this manner, the root rim thickness t in each axis-perpendicular cross section in the tooth trace direction of the external teeth34is
t=κt1

Here, t1: root rim thickness in principal cross section (axis-perpendicular cross section at the opening end).

(Method of Establishing Root Rim Thickness of External Teeth and Amount of Addendum Modification of Teeth)

FIG. 7shows the procedure of the present invention in which a so-called modified Goodman diagram is employed to determine the root rim thickness of a flexible external gear, and amounts of addendum modification for the teeth. In the wave gear device1, where σbis the tensile stress associated with bending of the root rim surface on the major axis in association with deformation of the flexible external gear3to ellipsoidal shape, σbis defined by the following expression, taken from a basic formula of material mechanics.
σb=3Et/(RD)

Here, E: Young's modulust: root rim thicknessR: reduction ratioD: diameter of rim neutral circle prior to deformation

Additionally, where σnis the tensile stress arising on the major axis due to output torque T, the surface area of the root receiving the load is DL, and therefore σnis defined by the following expression.
σn=T/(DLt)

Here, L: tooth width of flexible external gear

Consequently, the stress arising on the major axis of the flexible external gear3is the sum of σband σn, and the stress arising at the root rim surface on the minor axis of the flexible external gear3is compressive stress −σb. Thus, stress amplitude of the flexible external gear3arising due to rotation of the wave generator4is:
((σb+σn)−(−σb))/2=σb+σn/2

and average stress is:
((σb+σn)+(−σb))/2=σn/2,

On the same plane, a straight line is drawn connecting a point A (the vertical coordinate of which is σA) at which the substantial fatigue limit of alternating stress of the steel constituting the material of the flexible external gear3is plotted on the vertical axis, and a point B (the horizontal coordinate of which is σB) at which the center of yield stress and tensile strength of the steel is plotted on the horizontal axis, to create a so-called modified Goodman diagram. The triangular area bounded by this straight line, the horizontal axis, and the vertical axis is the permissible range of points produced by plotting the average stress of the root rim surface of the flexible external gear3on the horizontal axis, and the stress amplitude thereof on the vertical axis.

Here, in a case in which an ellipsoidal rim neutral curve of the flexible external gear3is given, a point P is derived by plotting, on the vertical axis, of the stress amplitude (σb+σn/2) appearing at the root rim surface in the principal cross section (the axis-perpendicular cross section at the opening end34a) arising due to rotation of the wave generator4, and plotting, on the horizontal axis, of the average stress σn/2. In the first instance, it is necessary for this point P to be included within the aforedescribed triangular area.

At this time, the transmission load torque T transmitted by the flexible external gear3is proportional to the product of the root rim thickness t and the tensile stress σn. The root rim thickness t is proportional to the tensile stress σbin association with bending. Consequently, the torque T of the flexible external gear is proportional to the product of the tensile stress σband the tensile stress σn. Here, σbis represented by a line segment PQ, where Q designates the intersection point of a straight line parallel to the vertical axis and passing through point P, and a straight line forming a 45 degree angle to the horizontal axis and passing through the origin. From the above, the torque T is proportional to the area of an oblong shape bounded by straight lines parallel to the horizontal axis and passing respectively through point P and point Q to the vertical axis, and the line segment PQ.

Consequently, the point at which the torque transmitted by the flexible external gear3of given specifications reaches maximum is the midpoint M of a line segment AC, where C is the intersection point of the modified Goodman diagram and a straight line passing through the origin and forming a 45 angle to the horizontal axis, and the root rim thickness corresponding to the midpoint M is the optimal value. Consequently, in this case, from the geometric relationships in the diagram:
σb=σA/2
σn=σAσB/(σA+σB)

The optimal value tmof root rim thickness t1in the principal cross section (location of the opening end34a) of the external teeth is given by the following expression.
t1=tm=σARD/(6E)

As stated above, when minus tooth profile shifting is applied to the external teeth34, and the root rim thickness thereof is designated as the root rim thickness t1at the location of the opening end34ain the tooth trace direction of the external teeth34, the root rim thickness at locations other than the opening end34ais κt1. Consequently, when the root rim thickness t1at the opening end34ais set to the optimal rim thickness tmin the aforedescribed manner, the root rim thickness t in each of axis-perpendicular cross sections taken from the opening end34ato the inner end34bof the external teeth34is set to κtm.

In other words, the rim thickness of the flexible external gear is determined in such a fashion as to take progressively smaller values, in such a way that points corresponding to root rim thickness of axis-perpendicular cross sections lying in the tooth trace direction from the opening end34ato the inner end34bof the external teeth34are plotted to the right side of the midpoint M in the modified Goodman diagram. During this process, it is necessary for the coordinate points representing stress amplitude and average stress on the modified Goodman diagram to lie within the permissible range mentioned earlier.

In the present invention, as shown below, the condition in question is met, and the coordinate points of the modified Goodman diagram representing the stress state of the flexible external gear3lie in the triangular area constituting the permissible range in the diagram in question.

Specifically, with regard to the root rim thickness of axis-perpendicular cross sections from the opening end34ato the inner end34b, when the root rim thickness of the opening end34aof the flexible external gear has been assigned the optimal value tm, in order to sustain meshing of the tooth profiles along the tooth trace, the root rim thickness is set to κtm, which is equivalent to applying tooth profile shifting of a coefficient 1−κ (κ<1) to the teeth, doing so in such a way that the bottom portions of the movement loci of the external teeth34of the flexible external gear3to the internal teeth24of the rigid internal gear2in each of the axis-perpendicular cross sections are made congruent. At this time, according to the decrease in the rim thickness, the tensile stress of the rim at any location increases in the manner σnm/κ, with respect to the tensile stress σnmof the opening end34adue to torque.

Meanwhile, bending stress σbarising on the major axis in any cross section of the flexible external gear is proportional to the product of the rim thickness κtmand the amount of flexure w. Where the value of bending stress with respect to tmis designated as σbm,
σb=κ2σbm=κ2σA/2.
From the equation for a straight line, the vertical coordinate corresponding to the horizontal coordinate of average stress σnm/κ/2 on the modified Goodman diagram is:
−(σA/σB)/σnm/2/κ+σA
Here, employing the relationship σbm=σA/2, in the axis-perpendicular cross section of the opening end34a, from the relationship:
(σB−σnm/2)(σA/σB)=σbm+σnm/2=σA/2+σnm/2
the following result is obtained.
σnm=σAσB/(σA+σB)

Consequently, the vertical coordinate of a modified Goodman straight line corresponding to the average stress σnm/κ/2 of a cross section of coordinate κ is given by the following expression.

In contrast to this, the stress amplitude of a cross section of coordinate κ is:

The difference of the two is:

σA⁡(1-κ22-12⁢⁢κ)[Expression⁢⁢3]
and this value is positive with respect to the range of actual values of κ (in the present example, from 1 to 0.7), thereby showing that the coordinate values with respect to the rim thickness κt lie within the permissible range.
(State of Meshing of Teeth)

FIG. 8is a graph showing movement loci of external teeth on axis-perpendicular cross sections at the opening end34a(principal cross section), and at a medial location and at the inner end34bin the tooth trace direction, in the external teeth34for which the tooth profile has been established in the aforedescribed manner. The movement loci M2a, M3aof the shifted tooth profiles at the medial location and at the inner end34bcontact the movement locus M1at the opening end34ain the bottom portion B, the shapes of the loci being mutually homothetic, with the exception of portions of the apical portions. This shows that, with the tooth profile of the present invention, it is possible to obtain a state of meshing over the entirety of the tooth trace, with the exception of portions of the apical portions.

Next,FIG. 9AtoFIG. 9Care descriptive diagrams showing, through rack approximation, the condition of meshing of external teeth and internal teeth for which tooth profiles have been established in the aforedescribed manner.FIG. 9Ashows that obtained at the opening end location of the external teeth,FIG. 9Bthat at a medial location in the tooth trace direction of the external teeth, andFIG. 9Cthat at the inner end location of the external teeth. The movement loci at each location in the tooth trace direction of the external teeth have good congruence in sections leading to the bottom portions thereof, whereby it may be appreciated that a state of meshing of the external teeth and the internal teeth is obtained over the entirety of the tooth trace.