EP1884686B1  Conical involute gear and gear pair  Google Patents
Conical involute gear and gear pair Download PDFInfo
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 EP1884686B1 EP1884686B1 EP20070113429 EP07113429A EP1884686B1 EP 1884686 B1 EP1884686 B1 EP 1884686B1 EP 20070113429 EP20070113429 EP 20070113429 EP 07113429 A EP07113429 A EP 07113429A EP 1884686 B1 EP1884686 B1 EP 1884686B1
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 F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
 F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
 F16H—GEARING
 F16H55/00—Elements with teeth or friction surfaces for conveying motion; Worms, pulleys or sheaves for gearing mechanisms
 F16H55/02—Toothed members; Worms
 F16H55/08—Profiling
 F16H55/0806—Involute profile
 F16H55/0813—Intersectingshaft arrangement of the toothed members

 Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSSSECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSSREFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
 Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
 Y10T—TECHNICAL SUBJECTS COVERED BY FORMER US CLASSIFICATION
 Y10T74/00—Machine element or mechanism
 Y10T74/19—Gearing
 Y10T74/19949—Teeth
 Y10T74/19958—Bevel

 Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSSSECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSSREFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
 Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
 Y10T—TECHNICAL SUBJECTS COVERED BY FORMER US CLASSIFICATION
 Y10T74/00—Machine element or mechanism
 Y10T74/19—Gearing
 Y10T74/19949—Teeth
 Y10T74/19963—Spur

 Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSSSECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSSREFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
 Y10—TECHNICAL SUBJECTS COVERED BY FORMER USPC
 Y10T—TECHNICAL SUBJECTS COVERED BY FORMER US CLASSIFICATION
 Y10T74/00—Machine element or mechanism
 Y10T74/19—Gearing
 Y10T74/19949—Teeth
 Y10T74/19963—Spur
 Y10T74/19972—Spur form
Description
 The present invention relates to a conical gear having an involute tooth profile and a gear pair in which the conical gears are engaged.
 In general, a bevel gear is used for engaging a plurality of gears which respectively rotate around a plurality of intersecting axes. In order to smoothly transmit rotation by the bevel gear, there are frequently used spiral bevel gears 103 and 104 having circular arc shaped teeth 101 and 102 and formed in a pseudospiral shape, as shown in
Fig. 10(A) .  In the manufacture of the bevel gears 103 and 104 mentioned above, the teeth 101 and 102 are machined one by one by a milling tool 107 in which a plurality of cutting chips 106 are arranged in an outer circumferential portion of a disc 105, as shown in
Fig. 10(B) . Accordingly, the working time is elongated and the working efficiency is lowered. Further, since a gear generating process is not employed, it is hard to form the involute tooth profile. Accordingly, the involute tooth profile may be formed by using a formed cutting blade or grinding stone in place of the milling tool 107. However, in the teeth 101 and 102 of both of the bevel gears 103 and 104, since the diameter of an inside circular arc is different from the diameter of an outside circular arc, a precise engaging position between the bevel gears 103 and 104 exists only at one position in an axial direction of the bevel gears 103 and 104 even if the involute tooth profile is obtained. Accordingly, in order to precisely engage the bevel gears 103 and 104, a high assembling precision is required.  In addition, as shown in
Fig. 10(C) , since one tooth surface of both tooth surfaces of each of the teeth 101 and 102 is convex, and the other tooth surface is concave, a concavoconvex relationship between a drive tooth surface and a driven tooth surface is counterchanged by reversing rotating directions of both bevel gears 103 and 104. Accordingly, if the rotating direction of the gear is different, a great difference is generated in a transmission efficiency. In the case of the gear having no involute tooth profile, the transmission efficiency is further lowered.  Further, as is apparent from
Fig. 10(A) , in the case where a crossing angle (δ1 + δ2) between the axes of the bevel gears 103 and 104 is small, a distance (hereinafter, referred to as a condistance) R between large diameter ends of both the bevel gears 103 and 104 and a conical center thereof is elongated. Each of the bevel gears 103 and 104 is supported on a rotating table at a time of being machined. Further, each of the teeth 101 and 102 is cut by the milling tool 107 while rotating the table in increments of one pitch of each of the teeth 101 and 102. Accordingly, the longer the condistance R becomes, the larger the radius of rotation of the table must be. As a result, a wide working space is necessary.  On the other hand,
JP 694101 Fig. 11 , one gear 311 in two gears constituting a gear pair is conical, and the other gear 312 is cylindrical. Further, as described in the specification of this publication, in order to engage the gears 311 and 321 in a wide range, an addendum modification coefficient of the conical gear 311 is changed in such a manner as to draw a nonlinear gentle curve in a face width direction of the gear 311. Accordingly, it is possible to bring the tooth surfaces of both the gears 311 and 321 into contact with each other approximately over a whole face width.  However, in each of the gears 311 and 321 disclosed in
JP 694101A Fig. 1B of the present application andFig. 4 ofJP No. 694101A JP 694101A  Further,
JP 694101A  Conical gears having an involute tooth profile and a gear pair in which the conical gears are engaged are known from the generic
DE 24 46 172 A1 ,US 4 703 670 A andJP 6 094 101 A  It is the object of the present invention to provide a conical involute gear that precisely engages, is easily manufactured, and achieves a wide conical angle, even in the case of helical teeth, and to provide a gear pair constituted by the conical involute gears.
 The object of the invention is achieved with a conical involute gear according to claim 1 and a gear pair according to claim 7.
 Further advantageous developments of the invention are subjectmatter of the dependent claims.


Fig. 1(A) is an explanatory view showing a concept of the present invention; 
Fig. 1(B) is an explanatory view showing a concept of a conventional structure; 
Fig. 2 is a perspective view partially showing a conical involute gear in accordance with the present embodiment; 
Fig. 3 is an explanatory view showing an addendum modification state and a spiral angle of a tooth; 
Fig. 4(A) is a diagrammatic front elevational view showing an engagement state of the teeth; 
Fig. 4(B) is a diagrammatic front elevational view showing the engagement state of the teeth; 
Fig. 5 is a perspective view partially showing a contact locus of the teeth; 
Fig. 6 is an explanatory view showing an engagement state of a conical involute gear; 
Fig. 7 is an explanatory view showing dimensions and angles of various portions of the conical involute gear; 
Fig. 8 is an explanatory view showing a gear generating process by a gear hobbing machine; 
Fig. 9 is an explanatory view showing another gear generating process by the gear hobbing machine; 
Fig. 10(A) is an explanatory view showing an engagement state of a conventional bevel gear; 
Fig. 10(B) is an explanatory view showing machining by a milling cutter; 
Fig. 10(C) is an explanatory enlarged view showing a tooth of a bevel gear; and 
Fig. 11 is an explanatory view showing a gear pair having a conventional structure.  A description will be given of one embodiment of the present invention with reference to
Fig. 1(A) andFigs. 2 to 9 .  As shown in
Figs. 2, 3 and6 , a gear pair 31 is constituted by a smalldiameter conical involute gear (hereinafter, referred to as a smalldiameter conical gear) 32, and a largediameter conical involute gear (hereinafter, referred to as a largediameter conical gear) 33. The gears 32 and 33 are engaged with each other, and have predetermined conical angles δ1 and δ2, respectively. The conical gear 32 has a set of teeth 34, of the number of teeth z_{1}, and the conical gear 33 has a set of teeth 35, of the number of teeth z_{2}. The teeth 34 and 35 have an involute tooth profile and are constituted by helical teeth having a spiral angle β.  An addendum modification coefficient of each of the teeth 34 and 35 is nonlinearly changed in each of a face width direction. In other words, as shown in
Figs. 3 ,6 and7 , the addendum modification coefficient (an addendum modification amount) of each of the teeth 34 and 35 is zero in an arbitrary intermediate portion in the face width direction thereof, is negative in a smalldiameter side from the center portion, and is positive in a largediameter side from the center portion. In this embodiment, a position at which the addendum modification coefficient is zero is defined as a reference pitch point P.Fig. 3 shows a tooth 34A existing at the position at which the addendum modification coefficient is zero, a tooth 34B profile shifted to the negative side, and a tooth 34C profile shifted to the positive side, respectively.  In this case, values of the conical angles δ1 and δ2 can be optionally set in correspondence to a specification of each of the gears 32 and 33. In this embodiment, these values are set in such a manner as to satisfy a relationship of numerical expression 1 in the present embodiment. In the expression 1, δ1 denotes the conical angle of the smalldiameter conical gear 32, 52 denotes the conical angle of the largediameter conical gear 33, z_{1} denotes the number of teeth of the smalldiameter conical gear 32, and z_{2} denotes the number of teeth of the largediameter conical gear 33.
$$\begin{array}{l}\phantom{\rule{1em}{0ex}}\\ \begin{array}{l}\mathrm{\Sigma}={\delta}_{1}+{\delta}_{2}\\ {\mathrm{tan\delta}}_{1}=\frac{\mathrm{sin\; \Sigma}}{\frac{{z}_{2}}{{z}_{1}}+\mathrm{cos\; \Sigma}}\\ {\mathrm{tan\delta}}_{2}=\frac{\mathrm{sin\; \Sigma}}{\frac{{z}_{2}}{{z}_{1}}+\mathrm{cos\; \Sigma}}\end{array}\end{array}$$  In this embodiment, the smalldiameter conical gear 32 and the largediameter conical gear 33 constructing the gear pair 31 are defined as an aggregation of infinitude of imaginary cylindrical gears 321 and 331 in which a face width is zero, as shown in
Fig. 1A andFigs. 6 and7 . These imaginary cylindrical gears 321 and 331 respectively have a center axis uu and a center axis vv which are in parallel to a line gg (hereinafter, the line gg is referred to as a conical line) extending along a conical surface passing through the reference pitch point P of the gears 32 and 33. Accordingly, in the same manner as the imaginary cylindrical gears 321 and 331, there exist infinitude of center axes uu and center axes vv. Further, on one line ww orthogonal to the conical line gg at the reference pitch point P, the imaginary cylindrical gears 321 and 331 are defined as a helical spur gear (hereinafter, the helical spur gear in which the addendum modification amount is zero is set to standard imaginary cylindrical gears 321A and 331A) in which the addendum modification amount is zero. The positions of the standard imaginary cylindrical gears 321A and 331A are optionally set. As shown inFig. 3 , the imaginary cylindrical gears 321 and 331 in a smaller diameter side than the standard imaginary cylindrical gears 321A and 331A are profile shifted to the negative side, and the imaginary cylindrical gears 321 and 331 in a larger diameter side than the standard imaginary cylindrical gears 321A and 331A are profile shifted to the positive side.  Parameters of the imaginary cylindrical gears 321 and 331 are set in accordance with a numerical expression 2. Reference symbol m_{v} in the expression denotes a module of the imaginary cylindrical gears 321 and 331, reference symbol β denotes a spiral angle of the teeth of the imaginary cylindrical gears 321 and 331, that is, the conical gears 32 and 33, reference symbol m_{n} denotes a module of the imaginary cylindrical gears 321 and 331 in a vertical cross section of a tooth which is set such that a cross sectional involute tooth profile can be obtained or which is perpendicular to the spiral angle β in the imaginary cylindrical gears 321 and 331, reference symbol α_{v} denotes a pressure angle of the imaginary cylindrical gears 321 and 331, reference symbol α_{n} denotes a pressure angle of the imaginary cylindrical gears 321 and 331 in the vertical cross section of the tooth, reference symbol z denotes the number of teeth of the conical gears 32 and 33, reference symbol z_{v} denotes the number of teeth of the imaginary cylindrical gears 321 and 331, and reference symbol 5 denotes a conical angle of the conical gears 32 and 33, respectively.
$$\begin{array}{l}\phantom{\rule{1em}{0ex}}\\ \phantom{\rule{7em}{0ex}}\begin{array}{ll}\begin{array}{c}\mathrm{Module}\end{array}& {\mathrm{m}}_{\mathrm{v}}={\mathrm{m}}_{\mathrm{n}}\mathrm{/}\mathrm{cos\beta}\\ \mathrm{Pressure\; angle}& {\mathrm{\alpha}}_{\mathrm{v}}\mathrm{=}{\mathrm{tan}}^{\mathrm{}\mathrm{1}}\left({\mathrm{tan\alpha}}_{\mathrm{n}}\mathrm{/}\mathrm{cos\beta}\right)\\ \mathrm{Number\; of\; teeth}& {\mathrm{Z}}_{\mathrm{v}}\mathrm{=}\mathrm{Z}/\mathrm{cos\delta}\end{array}\end{array}$$ mentioned above are engaged with each other without being in biased contact or being in point contact with each other, as shown inFig. 5 , it is possible to obtain a continuous engagement over a whole region in an axial direction (in the face width direction) of the gear pair 31. Respective data of the conical gears 32 and 33 in this case can be set in accordance with the following expressions.Fig. 5 expresses an engagement locus 35a of the teeth. Accordingly, if all the teeth of the gear pair 31 shows a linear engagement locus shown inFig. 5 , it is possible to bring the teeth 34 and 35 of the gears 32 and 33 into surface contact with each other in a wide range.  In
Fig. 7 , a center distance a between the imaginary cylindrical gears 321 and 331 engaging with each other is expressed by numerical expression 3. Reference symbol r_{1'} in the expression denotes a diameter of a pitch circle of the imaginary cylindrical gear 321, reference symbol r_{2'} denotes a diameter of a pitch circle of the imaginary cylindrical gear 331, reference symbol R denotes a condistance of the conical gears 32 and 33, and reference symbol y denotes a distance from a largediameter end of the gears 321 and 331 on the conical line gg to the imaginary cylindrical gears 321 and 331, respectively.$$\begin{array}{c}\phantom{\rule{1em}{0ex}}\\ \mathrm{a}\mathrm{=}{\mathrm{r}}_{\mathrm{1}}\mathrm{\u02b9}\mathrm{+}{\mathrm{r}}_{\mathrm{2}}\mathrm{\u02b9}\mathrm{=}\left(\mathrm{R}\mathrm{}\mathrm{y}\right)\left({\mathrm{tan\delta}}_{\mathrm{1}}\mathrm{/}{\mathrm{cos\delta}}_{\mathrm{2}}\right)\\ {\mathrm{r}}_{\mathrm{1}}\mathrm{\u02b9}\mathrm{=}\left(\mathrm{R}\mathrm{}\mathrm{y}\right){\mathrm{tan\delta}}_{\mathrm{1}}\phantom{\rule{6em}{0ex}}{\mathrm{r}}_{\mathrm{2}}\mathrm{\u02b9}\mathrm{=}\left(\mathrm{R}\mathrm{}\mathrm{y}\right){\mathrm{tan\delta}}_{\mathrm{2}}\end{array}$$  A center distance a_{0} of the standard imaginary cylindrical gears 321A and 331A is shown in numerical expression 4. Reference symbol y_{0} in the expression denotes a distance from the large diameter end of the gears 321 and 331 to the standard imaginary cylindrical gears 321A and 331A, reference symbol z_{v1} denotes the number of teeth of the smalldiameter standard imaginary cylindrical gear 321A, and reference symbol z_{v2} denotes the number of teeth of the largediameter standard imaginary cylindrical gear 331A, respectively.
$$\begin{array}{l}\phantom{\rule{1em}{0ex}}\\ {a}_{0}={r}_{1}+{r}_{2}=\left(\mathit{R}{y}_{0}\right)\left(\mathrm{tan}{\delta}_{1}+\mathrm{tan}{\delta}_{2}\right)\\ \begin{array}{c}{r}_{1}=\left(\mathit{R}{y}_{0}\right){\mathrm{tan\delta}}_{1}=\frac{{m}_{v}{z}_{v1}}{2}\\ {r}_{2}=\left(\mathit{R}{y}_{0}\right){\mathrm{tan\delta}}_{2}=\frac{{m}_{v}{z}_{v2}}{2}\end{array}\Rightarrow \begin{array}{c}{y}_{0}=\frac{R\mathrm{tan}{\delta}_{1}{m}_{v}{z}_{v1}/2}{\mathrm{tan}{\delta}_{1}}\\ {y}_{0}=\frac{R\mathrm{tan}{\delta}_{2}{m}_{v}{z}_{v2}/2}{\mathrm{tan}{\delta}_{2}}\end{array}\end{array}$$  Next, an engagement pressure angle α_{v'} in the rightangled surface of the conical line of the imaginary cylindrical gears 321 and 331 is shown in numerical expression 5. The rightangled surface of the conical line corresponds to the surface perpendicular to the width direction of the conical involute gear. Reference symbol α_{v} in the expression denotes a pressure angle of the standard imaginary cylindrical gears 321A and 331A, reference symbol x_{1} denotes an addendum modification coefficient of the smalldiameter imaginary cylindrical gear 321, reference symbol x_{2} denotes an addendum modification coefficient of the largediameter imaginary cylindrical gear 331, respectively. Accordingly, if the imaginary cylindrical gears 321 and 331 satisfy the expression (1) in the numerical expression 5, all the imaginary cylindrical gears 321 and 331 can achieve a normal engagement with no gap, as shown in
Figs. 4(A) and 4(B) . As a result, it is possible to obtain a continuous engagement in all the regions in a tooth trace direction of the gear pair 31.$$\begin{array}{c}\phantom{\rule{1em}{0ex}}\\ \mathrm{cos}\phantom{\rule{4em}{0ex}}{\alpha}_{v}\u02b9=\frac{{a}_{0}}{a}\mathrm{cos}{\alpha}_{v}\end{array}$$ $$\mathit{inv}{\alpha}_{v}\u02b9=\frac{2\left({x}_{1}+{x}_{2}\right)}{{z}_{v1}+{z}_{v2}}\mathrm{tan}{\alpha}_{v}+\mathit{inv}{\alpha}_{v}\u02b9$$
in which
invα = tanα  α  The radii of the imaginary cylindrical gears 321 and 331 are expressed as numerical expression 6.
$$\begin{array}{c}\phantom{\rule{1em}{0ex}}\\ {\begin{array}{c}{r}_{1}\end{array}}^{\u02b9}={r}_{1}\frac{\mathrm{cos}{\alpha}_{v}}{\mathrm{cos}{\alpha}_{v}\u02b9}\phantom{\rule{4em}{0ex}}{r}_{2}\u02b9={r}_{2}\frac{\mathrm{cos}{\alpha}_{v}}{\mathrm{cos}{\alpha}_{v}\u02b9}\end{array}$$  The relationship between the addendum modification coefficients x_{1} and x_{2} of the imaginary cylindrical gears 321 and 331 is shown in numerical expression 7.
$$\begin{array}{c}\phantom{\rule{1em}{0ex}}\\ {x}_{1}+{x}_{2}=\frac{\left({z}_{v1}+{z}_{v2}\right)\left(\mathit{inv}{\alpha}_{v}\u02b9\mathit{inv}{\alpha}_{v}\right)}{2\mathrm{tan}\phantom{\rule{1em}{0ex}}{\alpha}_{v}}\end{array}$$  In order to suitably engage the imaginary cylindrical gears 321 and 331 with each other, it is necessary to bring the thickness of the tooth into line with the width of the tooth space on the pitch circle in which the imaginary cylindrical gears 321 and 331 are engaged with each other, as shown in
Figs. 4(A) and 4(B) . In other words, if the thickness of the tooth and the width of the tooth space are equal on the pitch circle in which the imaginary cylindrical gears 321 and 331 are engaged, it is possible to suitably engage the gears with each other. In this case, a relationship of numerical expression 8 is satisfied. Reference symbol p' in the expression denotes a tooth pitch on the engagement pitch circle after being profile shifted, and reference symbols p_{1'} and p_{2'} denote tooth pitches of the imaginary cylindrical gears 321 and 331 on the engagement pitch circle after being profile shifted, respectively. Reference symbols s_{1'} and s_{2'} denote tooth thicknesses of the imaginary cylindrical gears 321 and 331 on the engagement pitch circle after being profile shifted, respectively. Reference symbols e_{1'} and e_{2'} denote widths of the tooth spaces of the imaginary cylindrical gears 321 and 331 on the engagement pitch circle after being profile shifted, respectively. Reference symbol s denotes a tooth thickness on the engagement pitch circle of the standard imaginary cylindrical gears 321A and 331A, and reference symbol x denotes an addendum modification coefficient of the standard imaginary cylindrical gears 321A and 331A, respectively.$$\begin{array}{l}\phantom{\rule{1em}{0ex}}\\ \phantom{\rule{8em}{0ex}}\begin{array}{l}\mathit{p\u02b9}={p}_{1}\u02b9=\frac{2\pi {r}_{1}\u02b9}{{z}_{v1}}{p}_{2}\u02b9=\frac{2\pi {r}_{2}\u02b9}{{z}_{v2}}\\ \mathit{p\u02b9}={s}_{1}\u02b9+{e}_{1}\u02b9={s}_{2}\u02b9+{e}_{2}\u02b9\\ {s}_{1}\u02b9={e}_{t}\u02b9={s}_{1}\u02b9={e}_{2}\u02b9\\ \mathit{s\u02b9}=s\frac{\mathit{r\u02b9}}{r}+2\mathit{r\u02b9}\left(\mathit{inv}\uef22{\alpha}_{v}\mathit{inv}\uef22{\alpha}_{v}\u02b9\right)\\ s={m}_{v}\left(\frac{\mathit{\pi}}{2}+2\times \mathrm{tan}{\alpha}_{v}\right)\end{array}\end{array}$$  The addendum modification coefficients x_{1} and x_{2} of the imaginary cylindrical gears 321 and 331 are determined on the basis of numerical expression 9.
$$\begin{array}{c}{x}_{1}=\frac{{z}_{v1}\left(\mathit{inv}{\alpha}_{v}\u02b9\mathit{inv}{\alpha}_{v}\right)}{2\mathrm{tan}{\alpha}_{v}}={z}_{v1}f\left(\mathit{inv}{\alpha}_{v}\u02b9\right)\\ {x}_{2}=\frac{{z}_{v2}\left(\mathit{inv}{\alpha}_{v}\u02b9\mathit{inv}{\alpha}_{v}\right)}{2\mathrm{tan}{\alpha}_{v}}={z}_{v2}f\left(\mathit{inv}{\alpha}_{v}\u02b9\right)\end{array}$$  The addendum modification coefficients x_{1} and x_{2} are functions of the pressure angles α_{v} and α_{v'}, which are variables. Accordingly, the addendum modification coefficients x_{1} and x_{2} are nonlinear functions. An allocation of the addendum modification coefficients x_{1} and x_{2} is not only related to the conical angles δ1 and δ2 of the imaginary cylindrical gears 321 and 331, but also related to the number of teeth of the imaginary cylindrical gears 321 and 331, as shown in numerical expression 10. Accordingly, the conical involute gears 32 and 33 can obtain a nonlinear addendum modification coefficient, and can accordingly obtain an engagement of being in line contact with each other.
$$\begin{array}{c}\phantom{\rule{1em}{0ex}}\\ \frac{{x}_{1}}{{x}_{2}}=\frac{{z}_{v1}}{{z}_{v2}}=\frac{{z}_{1}\mathrm{cos}{\delta}_{2}}{{z}_{2}\mathrm{cos}{\delta}_{1}}\end{array}$$  As mentioned above, in accordance with this embodiment, a concept of the imaginary cylindrical gears 321 and 331 is introduced, and the conical gears 32 and 33 are respectively defined as the aggregate of the imaginary cylindrical gears 321 and 331. Further, the structure is made such as to precisely engage the imaginary cylindrical gears 321 and 331 with each other by changing the addendum modification coefficient by using the number of teeth taking the conical angles of the conical gears 32 and 33 into consideration in the respective imaginary cylindrical gears 321 and 331 (refer to the numerical expression 10). Accordingly, it is possible to achieve a smooth engagement between the conical gears 32 and 33, under a condition having the same module, pressure angle and number of teeth, and being different only in the addendum modification coefficient.
 Further, in each of the cross sections perpendicular to the face width, the pressure angle and the engagement angle come into line in the case where the profile shift does not exist. However, the pressure angle and the engagement angle are different in the case where the profile shift exists. Accordingly, in each of the surfaces perpendicular to the face width, it is necessary that a total of the addendum modification coefficients of the gear pair satisfy a predetermined relationship as shown in the numerical expression 7 and the numerical expression 9, and the pitches of the gears on the engagement pitch circle come into line as shown in the numerical expression 8. In other words, if these relationships are satisfied in each plane perpendicular to the face width, it is possible to obtain an engagement position at one point as shown in
Figs. 4(A) and 4(B) . Accordingly, it is possible to obtain the conical involute gear which is simultaneously engaged in all the regions of the face width and comes into surface contact. Further, even if the spiral angle β exists in the conical gears 32 and 33, the module and the pressure angle are set in correspondence with the spiral angle β as is apparent from the numerical expression 2. Accordingly, it is possible to bring the conical gears 32 and 33 into contact with each other in a wide area, and it is possible to further smoothly engage with each other.  Each of the conical gears 32 and 33 can be formed by gear cutting by using a gear hobbing machine shown in
Figs. 8 and 9 . As shown inFig. 8 , a position of a hob spindle Ha is fixed in a state in which an axis C of each of the conical gears 32 and 33 (the workpieces) is inclined. Further, the gear cutting work of each of the conical gears 32 and 33 is executed such that the addendum modification coefficient is changed to be nonlinear by moving the work in a direction of arrow Q shown inFig. 8 while rotating the workpiece axis C as well as moving the hob H along the tooth trace at a time of executing the gear cutting.  Further, as shown in
Fig. 9 , the workpiece axis C may be operated as follows without inclining the workpiece axis C. In other words, the workpiece axis C may be moved in a direction of arrow Q shown inFig. 9 in such a manner that the gear corresponding to the conical angles δ1 and δ2 of the conical gears 32 and 33 can be generated in correspondence to the movement in the gear trace direction of the hob H. In this case, the other movement than the movement mentioned above in the direction of arrow Q is executed to the work axis C, in such a manner that a change amount of the addendum modification coefficient is generated in addition to the generation of the tooth.  Further, in place of moving the workpiece axis C in the direction of arrow Q, a hob axis Ha may be moved in the direction of arrow Q. As mentioned above, it is possible to easily execute the gear generating tooth cutting with respect to each of the conical gears 32 and 33, only by controlling the movement of the workpiece axis C or the hob axis Ha in the direction of arrow Q, while using the gear hobbing machine in the same manner as the generation of the normal involute gear. Further, since the modules, the numbers of teeth, and the pressure angles of the respective conical gears 32 and 33 are equal, it is possible to execute the gear generating process using the same cutter.
 The embodiment mentioned above has the following advantages.
 (1) The conical gears 32 and 33 respectively correspond to the aggregates of the imaginary cylindrical gears 321 and 331. Accordingly, it is possible to change the addendum modification coefficient of each of the imaginary cylindrical gears 321 and 331 by using the number of teeth of each of the imaginary cylindrical gears 321 and 331. Accordingly, it is possible to bring the conical gears 32 and 33 into surface contact with each other, and it is possible to achieve a suitable engagement between the conical gears 32 and 33.
 (2) Since the spiral angle β of the conical gears 32 and 33 is reflected on the change of the addendum modification coefficient, it is possible to obtain a precise engagement in a wide area on the basis of the involute tooth profile, regardless whether the conical gears 32 and 33 are constituted by straight teeth or helical teeth.
 (3) Since it is possible to properly allocate the addendum modification coefficient to each of the conical gears 32 and 33, it is possible to engage the conical gears 32 and 33 with no gap in all the regions in the tooth trace direction of the conical gears 32 and 33.
 (4) Each of the conical gears 32 and 33 can be easily generated and machined by using the gear hobbing machine. Further, since the modules, the numbers of teeth, and the pressure angles are equal in the conical gears 32 and 33 constructing the gear pair 31, it is possible to generate and machine teeth by the same cutter. Accordingly, it is possible to simplify an initial setup at a time of machining, and it is possible to further easily manufacture each of the conical gears 32 and 33.
 (5) It is possible to set the standard imaginary cylindrical gears 321A and 331A in which the addendum modification amount is zero, in an arbitrary intermediate portion in the face width direction, and it is possible to respectively set the imaginary cylindrical gears 321 and 331 that are profile shifted to a positive side, and the imaginary cylindrical gears 321 and 331 that are profile shifted to a negative side, on both sides of the imaginary cylindrical gears 321A, 331A having no addendum modification. Accordingly, it is possible to form the conical gears 32 and 33 having a large conical angle.
Claims (10)
 A conical involute gear (32, 33) having an involute tooth profile, being characterized in that the conical involute gear (32, 33) is defined as an aggregate of a plurality of imaginary cylindrical gears (321, 331) having a face width of zero, and each of the imaginary cylindrical gears (321, 331) has a center axis (uu, vv) which is in parallel to a conical surface of the conical involute gear (32, 33) passing through a reference pitch point (P).
 The conical involute gear (32, 33) according to claim 1, being characterized in that each of the imaginary cylindrical gears (321, 331) has the same module.
 The conical involute gear (32, 33) according to claim 1 or 2, being characterized in that an addendum modification coefficient of each of the imaginary cylindrical gears (321, 331) changes nonlinearly in a face width direction of the conical involute gear (32, 33).
 The conical involute gear (32, 33) according to any one claims 1 to 3, characterized in that the imaginary cylindrical gear (321, 331) in which the addendum modification amount is zero is a standard imaginary cylindrical gear (321A, 331A), the standard imaginary cylindrical gear (321A, 331A) is arranged between a largediameterend and a sqnalldiameter end of the conical involute gear (31, 32), the imaginary cylindrical gears (321, 331) between the smaller diameter end and the standard imaginary cylindrical gear (321A, 331A) are negatively profiled shifted, and the imaginary cylindrical gears (321, 331) between the larger diameter end and the standard imaginary cylindrical gear (321A, 331A) are positively profiled shifted.
 The conical involute gear (32, 33) according to any one of claims 1 to 4, characterized in that in the case of setting the number of teeth of the conical involute gear (32, 33) to z, setting a conical angle of the conical involute gear (32, 33) to δ, and setting the number of teeth of the imaginary cylindrical gear (321, 331) to z_{v}, a relational expression z_{v} = z/cosδ is satisfied.
 The conical involute gear (32, 33) according to any one of claims 1 to 5, characterized in that the imaginary cylindrical gear (321, 331) has helical teeth.
 A gear pair (31) in which at least one of two involute gears engaging with each other is a conical involute gear (32, 33), being characterized in that the conical involute gear (32, 33) is the conical involute gear (32, 33) according to any one claims 1 to 6.
 The gear pair (31) according to claim 7, characterized in that in the case of setting a pressure angle of the imaginary cylindrical gears (321, 331) to α_{v}, setting a pressure angle of the imaginary cylindrical gear (321, 331) in a vertical cross section of a tooth to α_{n}, setting a module of the imaginary cylindrical gear (321, 331) to m_{v}, setting a spiral angle to β, and setting a module of the imaginary cylindrical gear (321, 331) in a vertical cross section of a tooth to m_{n}, relational expressions m_{v} = m_{n} /cosβ, and α_{v} = tan ^{1} (tanα_{n}/cosβ) are satisfied.
 The gear pair (31) according to claim 7 or 8, characterized in that in the case of setting the imaginary cylindrical gear (321, 331) in which the addendum modification amount is zero to a standard imaginary cylindrical gear (321A, 331A), setting a pressure angle of the standard imaginary cylindrical gear (321A, 331A) to α_{v}, setting an engagement pressure angle in a surface perpendicular to a conical line of the imaginary cylindrical gear (321, 331) to α_{v}', setting an addendum modification coefficient of a smalldiameter imaginary cylindrical gear (321) to x_{1}, setting an addendum modification coefficient of a largediameter imaginary cylindrical gear (331) to x_{2}, the number of teeth of the smalldiameter imaginary cylindrical gear (321) to z_{v1}, and setting the number of teeth of the largediameter imaginary cylindrical gear (331) to z_{v2}, a relational expression inv α_{v}' = [2(x_{1} + x_{2})/z_{v1} + z_{v2}] tanα_{v} + invα_{v}, in which invα = tanα  α, is satisfied.
 A gear pair (31) according to any one of claims 7 to 9, characterized in that in the case of setting an addendum modification coefficient of a smalldiameter imaginary cylindrical gear (321) to x_{1}, setting an addendum modification coefficient of a largediameter imaginary cylindrical gear (331) to x_{2}, the number of teeth of the smalldiameter imaginary cylindrical gear (321) to z_{v1}, setting the number of teeth of the largediameter imaginary cylindrical gear (331) to z_{v2}, setting the number of teeth of a smalldiameter conical involute gear (32) to z_{1}, setting the number of teeth of a largediameter conical involute gear (33) to z_{2}, a conical angle of a smalldiameter conical gear (32) to δ1, and setting a conical angle of a largediameter conical gear (33) to δ2, a relational expression (x_{1}/x_{2}) = z_{v1}/z_{v2} = z_{1}cosδ2/ z_{2}costδ1 is satisfied.
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