CN103394768A - Slotting cutter of inverse-tapered spline straight gear and designing method of slotting cutter - Google Patents

Abstract

The invention discloses a slotting cutter of an inverse-tapered spline straight gear and a designing method of the slotting cutter. Basic parameters of the slotting cutter comprise a tooth number Zo, a front angle gamma, a tooth crest rear angle alpha e, a reference circle diameter dfo and a reference circle pressure angle alpha fo, wherein the reference circle diameter dfo is equal to Zom, and m represents the modulus of the inverse-tapered spline straight gear; and the reference circle pressure angle alpha fo meets the formula. According to the slotting cutter of the inverse-tapered spline straight gear, an adopted inclined surface, namely a V-V plane, of a larger end of the inverse-tapered spline straight gear is designed according to the principle of meshing the bevel gear and the slotting cutter, the slotting cutter has higher tooth crest intensity, the problem of low separation and processing efficiency of double gears in the prior art is thoroughly solved, and the geometric accuracy of an inserted tooth profile of the inverse-tapered spline straight gear is guaranteed. The invention further provides the designing method of the inverse-tapered spline straight gear. With the adoption of the designing method, the high-accuracy and high-strength slotting cutter of the inverse-tapered spline straight gear can be obtained, and the accuracy of the inserted tooth profile of the inverse-tapered spline straight gear is improved.

Description

Back taper spline spur gear pinion cutter and method for designing thereof

Technical field

The present invention relates to a kind of cutter, relate in particular to a kind of pinion cutter and method for designing thereof of processing back taper spline spur gear.

Background technology

Back taper spline spur gear generally all is used on the doubly-linked gear of transmission gear shift (or gear shift) of motor vehicle, the main internal spline that plays gear coaxial with another (or concentric) gear when rotating is meshed, it has good guide effect in engagement is rotated, so adopted widely in mechanical rotation, its structure as shown in Figure 1.Processing in the past will obtain the back taper spline spur gear profile of tooth of high accuracy (7 grades of gears), all to adopt this doubly-linked gear to separate processing (operation is gear hobbing, roll flute, boring, car hole, grinding out, assembling), although so the processing profile accuracy is guaranteed, time-consuming.

, for the problems referred to above, need a kind of pinion cutter that can process back taper spline spur gear of design, to solve the time-consuming problem of above-mentioned separation processing.

Summary of the invention

In view of this, the invention provides a kind of back taper spline spur gear pinion cutter, but adopt this pinion cutter time processing when processing back taper spline spur gear, and can guarantee the machining accuracy of back taper spline spur gear profile of tooth, and then solved the time-consuming problem of separation processing doubly-linked gear in the past.In addition, the present invention also provides the method for designing of back taper spline spur gear pinion cutter, and the pinion cutter that designs by the method has the advantages such as high strength, high accuracy.

The present invention solves the problems of the technologies described above by following technological means:

One of purpose of the present invention is to provide a kind of back taper spline spur gear pinion cutter, and its basic parameter comprises tooth number Z _o, anterior angle γ, tooth top relief angle α _e, reference diameter d _foWith pressure angle of graduated circle α _fo, described reference diameter d _fo=Z _oM, in following formula, m is back taper spline spur gear modulus; Described pressure angle of graduated circle α _foMeet formula ${\mathrm{\α}}_{\mathrm{fo}}=\arctan \left[\frac{\sin \mathrm{\ψ}\tan {\mathrm{\β}}_f+\cos \mathrm{\ψ}\tan {\mathrm{\α}}_f}{1-\tan r\tan {\mathrm{\α}}_e}\right],$ In following formula, ψ is that angle between the pinion cutter direction of motion and back taper spline spur gear axial line is namely inserted established angle processed, β _fFor the angle, reference circle rear flank of back taper spline spur gear, α _fPressure angle of graduated circle for back taper spline spur gear.

Further, described slotting established angle processed

D in formula _iFor the root diameter of back taper spline spur gear, d _IxiaoFor back taper spline spur gear small end root diameter, B is the thickness of back taper spline spur gear.

Further, described back taper spline spur gear small end root diameter d _Ixiao=2A _Xiao-d _eo, A in formula _XiaoFor the centre-to-centre spacing between back taper spline spur gear small end face and pinion cutter, d _eoFor the pinion cutter outside diameter circle.

Further, the centre-to-centre spacing between described back taper spline spur gear small end face and pinion cutter

α in following formula _XiaoFor the angle of engagement between back taper spline spur gear small end face and pinion cutter, Z is the number of teeth of back taper spline spur gear; Described meshingangleα _XiaoMeet formula ${\mathrm{inv\α}}_{\mathrm{xiao}}=\frac{2\tan {\mathrm{\α}}_f({\mathrm{\ξ}}_{\mathrm{xiao}}+{\mathrm{\ξ}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\α}}_f,$ ξ in formula _XiaoFor back taper spline spur gear small end modification coefficient, ξ _oFor the pinion cutter modification coefficient; Described back taper spline spur gear small end modification coefficient ξ _XiaoMeet formula

S in formula _FxiaoFor back taper spline spur gear small end graduated arc transverse tooth thickness; Described S _FxiaoMeet formula S _Fxiao=S _f-2Btan β _f, S in formula _fFor back taper spline spur gear circular tooth thickness.

Further, described pinion cutter outside diameter circle d _eo=2A _o-d _i, A in following formula _oFor the centre-to-centre spacing between back taper spline spur gear large end face and pinion cutter; Centre-to-centre spacing between described back taper spline spur gear large end face and pinion cutter α in following formula _oFor the angle of engagement between back taper spline spur gear large end face and pinion cutter; Described meshingangleα _oMeet formula ${\mathrm{inv\α}}_o=\frac{2\tan {\mathrm{\α}}_f(\mathrm{\ξ}+{\mathrm{\ξ}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\α}}_f,$ In formula, ξ is back taper spline spur gear modification coefficient.

Further, described back taper spline spur gear modification coefficient

Further, also comprise for grinding pinion cutter toothed surfaces desired parameters d _boAnd β _bo, described d _bo, for the base circle diameter (BCD) of pinion cutter toothed surfaces, meet formula d _bo=d _foCos α _foDescribed β _bo, for the Base spiral angle of pinion cutter toothed surfaces, meet formula β _bo=arctan (sin α _foTan α _e).

Two of purpose of the present invention is to provide a kind of method for designing of above-mentioned back taper spline spur gear pinion cutter, and the major parameter of the large end end face of described back taper spline spur gear comprises modulus m, pressure angle of graduated circle α _f, reference circle rear flank angle β _f, outside diameter circle d _e, reference diameter d _f, root diameter d _iWith circular tooth thickness S _f, described method for designing comprises the following steps:

S1) obtain slotting established angle ψ processed according to back taper spline spur gear and pinion cutter basic parameter;

S2) be the equivalent parameter of bevel gear on the V-V plane with the parameter conversion of the large end end face of back taper spline spur gear;

S3) in conjunction with pinion cutter modification coefficient ξ _o, tooth number Z _o, anterior angle γ and tooth top relief angle α _eDeng basic parameter, according to step S2) the equivalent parameter of back taper spline spur gear on the v-v plane that draw, principle with bevel gear and pinion cutter engagement is obtained remaining geometric parameter of pinion cutter, and the pinion cutter tooth profile parameter that draws according to above-mentioned steps can process high-precision back taper spline spur gear pinion cutter.

Further, inserting established angle ψ processed step S1) tries to achieve by the following method:

S11) determine meshingangleα between back taper spline spur gear large end face and pinion cutter _o, this parameter alpha _oCan pass through formula ${\mathrm{inv\α}}_o=\frac{2\tan {\mathrm{\α}}_f(\mathrm{\ξ}+{\mathrm{\ξ}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\α}}_f$ Calculating is tried to achieve;

S12) according to step S11) in the angle of engagement parameter alpha that obtains _oDetermine the centre-to-centre spacing A between back taper spline spur gear large end face and pinion cutter _o, this centre-to-centre spacing parameter A _oCan pass through formula Calculating is tried to achieve;

S13) according to step S12) in the centre-to-centre spacing parameter A that draws _oDetermine pinion cutter outside diameter circle d _eo, described pinion cutter outside diameter circle d _eoCan pass through formula d _eo=2A _o-d _iCalculating is tried to achieve;

S14) ask back taper spline spur gear small end face parameter according to theory of plane engagement;

S141) at first determine back taper spline spur gear small end graduated arc transverse tooth thickness S _Fxiao, described S _FxiaoMeet formula S _Fxiao=S _f-2Btan β _f

S142) according to step S141) in the back taper spline spur gear small end graduated arc transverse tooth thickness S that determines _FxiaoDetermine back taper spline spur gear small end modification coefficient ξ _Xiao, described ξ _XiaoMeet formula

S143) by step S142) in the back taper spline spur gear small end modification coefficient ξ that tries to achieve _XiaoIn conjunction with pinion cutter modification coefficient ξ _oCan determine the meshingangleα between back taper spline spur gear small end face and pinion cutter _Xiao, described meshingangleα _XiaoCan pass through formula ${\mathrm{inv\α}}_{\mathrm{xiao}}=\frac{2\tan {\mathrm{\α}}_f({\mathrm{\ξ}}_{\mathrm{xiao}}+{\mathrm{\ξ}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\α}}_f$ Obtain;

S144) by step S143) definite meshingangleα _XiaoDetermine the centre-to-centre spacing A between back taper spline spur gear small end face and pinion cutter _Xiao, described centre-to-centre spacing A _XiaoCan pass through formula

Obtain;

S145) by step S144) the back taper spline spur gear small end face that obtains and the centre-to-centre spacing A between pinion cutter _XiaoDetermine back taper spline spur gear small end root diameter d _Ixiao, described d _IxiaoCan pass through formula d _Ixiao=2A _Xiao-d _eoObtain;

S15) by step S14) the back taper spline spur gear small end face parameter that draws finally determines to insert established angle ψ processed, and described ψ can pass through formula

Try to achieve, in formula, B is the thickness of back taper spline spur gear.

Further, step S2) on the V-V plane equivalent parameter of bevel gear comprise Equivalent modulus m _v, virtual number of teeth Z _v, equivalent outside diameter circle d _ev, equivalent circle-dividing diameter d _fv, equivalent root diameter d _iv, equivalent base circle diameter (BCD) d _bv, equivalent pressure angle of graduated circle α _fv, described Equivalent modulus m _vEquate described virtual number of teeth Z with the modulus m of back taper spline spur gear _vMeet formula

Described equivalent outside diameter circle d _evMeet formula $d_{\mathrm{ev}}=\frac{d_e}{\cos \mathrm{\ψ}}+[2B-(d_e-d_i)\tan \mathrm{\ψ}]\sin \mathrm{\ψ},$ Described equivalent circle-dividing diameter d _fvMeet formula d _fv=mZ _v, described equivalent root diameter d _ivMeet formula

Described equivalent pressure angle of graduated circle α _fvMeet formula tan α _fv=sin ψ tan β _f+ cos ψ tan α _f, described equivalent base circle diameter (BCD) d _bvMeet formula d _bv=mZ _vCos α _fv

On described V-V plane, the equivalent parameter of bevel gear also comprises equivalent graduated arc transverse tooth thickness S _fv, described S _fvDetermine by following steps:

A) ask the distance of back taper spline spur gear large end face to S-S plane and V-V plane point of intersection f:

Y _f=0.5 (d _f-d _i) tan ψ, Y _fFor the distance of back taper spline spur gear large end face to S-S plane and V-V plane point of intersection f;

S _fs＝S _f-2Y _ftanβ _f

X _fsS-S plane reference circle d _fChordal tooth thickness, namely the fs point is to the distance at back taper spline spur gear center;

Δ X=0.5d _f-X _fs, Δ X is S-S plane reference circle d _fChordal height, namely the fs point is to the distance of f;

B) ask at f _vThe centre-to-centre spacing X of spot projection _fv:

X _fv＝X _fs+ΔX?sinψsinψ

C) ask the height Y of back taper spline spur gear large end face and fv intersection point _fv:

Y _fv＝Y _f-ΔXsinψcosψ

D) ask the fv point circular tooth thickness S on Sv-Sv plane _fsvDiameter d _sv:

S _fsv＝S _f-2Y _fv?tanβ _f

Following formula solves with Newton iteration method:

$f\left(d_{\mathrm{sv}}\right)=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\α}}_f-\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{{2X}_{\mathrm{fv}}}\right)}^2-1}\right)-\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}\right)$

$f^{\′}\left(d_{\mathrm{sv}}\right)=\frac{{2X}_{\mathrm{fv}}}{d_{\mathrm{sv}}\sqrt{{d_{\mathrm{sv}}}^2-{\left({2X}_{\mathrm{fv}}\right)}^2}}-\frac{\sqrt{{d_{\mathrm{sv}}}^2-{d_b}^2}}{d_{\mathrm{sv}}{d}_b}$

$d_{\mathrm{sv}}=d_{\mathrm{sv}}-\frac{f\left(d_{\mathrm{sv}}\right)}{f^{\′}\left(d_{\mathrm{sv}}\right)}$

$\frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\α}}_f-\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}\right)$

$X_{\mathrm{fsv}}={0.5d}_{\mathrm{sv}}\cos \frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}$

$S_{\mathrm{fsv}}=d_{\mathrm{sv}}\sin \frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}$

$d_{\mathrm{fv}}=2\sqrt{{\left(\frac{X_{\mathrm{fsv}}}{d_{\mathrm{sv}}}\right)}^2-{\left(\frac{S_{\mathrm{sv}}}{2}\right)}^2}={\mathrm{mZ}}_v$

Final: $S_{\mathrm{fv}}=d_{\mathrm{fv}}\arctan (1/\sqrt{{\left(\frac{S_{\mathrm{fsv}}}{d_{\mathrm{fv}}}\right)}^2-1})$

The present invention has following beneficial effect:

1) back taper spline spur gear pinion cutter of the present invention, its inclined-plane that adopts the large end of back taper spline spur gear are that on the V-V plane, the principle design with bevel gear and pinion cutter engagement forms, its tooth top relief angle α _eCan equally with common pinion cutter select 6 °, compare the tooth top relief angle and adopt α _eThe pinion cutter of=ψ+6 ° has higher tooth top intensity, and pinion cutter of the present invention can be fit to the function of general gear shapping machine and be not limited to the rapidoprint of workpiece;

2) back taper spline spur gear pinion cutter of the present invention, its parameter pressure angle of graduated circle α _foMeet derivation formula ${\mathrm{\α}}_{\mathrm{fo}}=\arctan \left[\frac{\sin \mathrm{\ψ}\tan {\mathrm{\β}}_f+\cos \mathrm{\ψ}\tan {\mathrm{\α}}_f}{1-\tan r\tan {\mathrm{\α}}_e}\right],$ Make it have higher machining accuracy, not only thoroughly solved doubly-linked gear in the past and separated the low problem of working (machining) efficiency, and guaranteed the geometric accuracy of the back taper spline spur gear profile of tooth of slotting system;

3) method for designing of back taper spline spur gear pinion cutter provided by the invention, adopting the back taper large end of spline spur gear inclined-plane is to design the profile of tooth of pinion cutter on the V-V plane with the principle of bevel gear and pinion cutter engagement, adopt this method for designing can access high accuracy, high-intensity back taper spline spur gear pinion cutter, thus solved simple employing with the pinion cutter of the large transverse plane theory of engagement of back taper spline spur gear design in the deficiency aspect profile accuracy and tooth top intensity.

Description of drawings

The invention will be further described below in conjunction with drawings and Examples.

Fig. 1 back taper spline spur gear structure chart;

Fig. 2 is generalized section shown in K-K in Fig. 1;

Fig. 3 is that the tooth top relief angle is the pinion cutter structural representation of ψ+6 °;

Fig. 4 is that the tooth top relief angle is the slotting controlling angle schematic diagram of the pinion cutter of 6 °;

Fig. 5 is that the tooth top relief angle is the slotting controlling angle schematic diagram of another kind of the pinion cutter of 6 °;

Fig. 6 is equivalent size parameter schematic diagram on back taper spline spur gear V-V plane;

Fig. 7 is back taper spline spur gear geometrical expansion Fig. 1;

Fig. 8 is back taper spline spur gear geometrical expansion Fig. 2;

Fig. 9 is back taper spline spur gear geometrical expansion Fig. 3;

Figure 10 is back taper spline spur gear V-V planar tooth coordinate analysis figure;

Figure 11 is that back taper spline spur gear V-V plane divides circular thickness to solve figure;

Figure 12 is for plugging in back taper spline spur gear engagement figure processed.

The specific embodiment

The present invention is described in detail below with reference to drawings and Examples:

1 one kinds of back taper spline spur gear pinion cutters of embodiment

The back taper spline spur gear that has a following parameters take processing is as example: tooth number Z=39, modulus m=1.8, pressure angle of graduated circle α _f=20 °, reference circle rear flank angle β _f=4 °, outside diameter circle d _e=71.45, root diameter d _i=67, across number of teeth K=5, base tangent length W _K=24.88.

Back taper spline spur gear pinion cutter described in the present embodiment, its basic parameter is tooth number Z _o=56, anterior angle γ=5 °, tooth top relief angle α _e=6 °, reference diameter d _fo=Z _oM=100.8, circular tooth thickness S _fo=2.696, when ψ meets formula

The time, the basic parameter pressure angle of graduated circle α of described pinion cutter _fo, meet following formula:

Be to solve all the other parameters of described back taper spline spur gear pinion cutter on the V-V plane with the principle of bevel gear and pinion cutter engagement according to the basic parameter of above-mentioned back taper spline spur gear pinion cutter by the large end of back taper spline spur gear inclined-plane, as follows: base circle diameter (BCD) d _bo=d _foCos α _fo=94.412592337, angle, basic circle rear flank (Base spiral angle) β _bo=arctan (sin α _foTan α _e)=0.036802708rad=2 ° 07 '.

While using back taper spline spur gear pinion cutter of the present invention to plug in back taper spline spur gear processed, the gear shaper workbench established angle is necessary for the established angle ψ slotting processed of design, to guarantee the back taper spline spur gear precision of processing.But adopt back taper spline spur gear pinion cutter of the present invention time processing when processing back taper spline spur gear, and can guarantee the machining accuracy of back taper spline spur gear profile of tooth, and then solved the time-consuming problem of separation processing doubly-linked gear in the past.

The method for designing of 2 one kinds of back taper spline spur gear pinion cutters of embodiment

The invention provides a kind of method for designing of back taper spline spur gear pinion cutter 2, the major parameter of the large end face 11 of described back taper spline spur gear 1 comprises modulus m, pressure angle of graduated circle α _f, reference circle rear flank angle β _f, outside diameter circle d _e, reference diameter d _f, root diameter d _iWith circular tooth thickness S _f, described method for designing comprises the following steps:

S1) obtain and insert established angle ψ processed according to the basic parameter of back taper spline spur gear 1 and pinion cutter 2, as Figure 1-3, known:

Back taper spline spur gear infolute function: inv α _f=tan α _f-α _f

Back taper spline spur gear reference diameter: d _f=Zm

Back taper spline spur gear base circle diameter (BCD): d _b=Zmcos α _f

The large end modification coefficient of back taper spline spur gear:

The pinion cutter number of teeth: Z _o

Pinion cutter modification coefficient: ξ _o

Its solution procedure is as follows:

S11) determine meshingangleα between back taper spline spur gear large end face 11 and pinion cutter 2 _o, this parameter alpha _oCan pass through formula ${\mathrm{inv\α}}_o=\frac{2\tan {\mathrm{\α}}_f(\mathrm{\ξ}+{\mathrm{\ξ}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\α}}_f$ Calculating is tried to achieve;

S12) according to step S11) in the angle of engagement parameter alpha that obtains _oDetermine the centre-to-centre spacing A between back taper spline spur gear large end face 11 and pinion cutter 2 _o, this centre-to-centre spacing parameter A _oCan pass through formula

Calculating is tried to achieve;

S13) according to step S12) in the centre-to-centre spacing parameter A that draws _oDetermine pinion cutter outside diameter circle d _eo, described pinion cutter outside diameter circle d _eoCan pass through formula d _eo=2A _o-d _iCalculating is tried to achieve;

S14) ask back taper spline spur gear small end face parameter according to theory of plane engagement;

S141) at first determine back taper spline spur gear small end graduated arc transverse tooth thickness S _Fxiao, described S _FxiaoMeet formula S _Fxiao=S _f-2Btan β _f

S142) according to step S141) in the back taper spline spur gear small end graduated arc transverse tooth thickness S that determines _FxiaoDetermine back taper spline spur gear small end modification coefficient ξ _Xiao, described ξ _XiaoMeet formula

S143) by step S142) in the back taper spline spur gear small end modification coefficient ξ that tries to achieve _XiaoIn conjunction with pinion cutter modification coefficient ξ _oCan determine the meshingangleα between back taper spline spur gear small end face and pinion cutter _Xiao, described meshingangleα _XiaoCan pass through formula ${\mathrm{inv\α}}_{\mathrm{xiao}}=\frac{2\tan {\mathrm{\α}}_f({\mathrm{\ξ}}_{\mathrm{xiao}}+{\mathrm{\ξ}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\α}}_f$ Obtain;

S144) by step S143) definite meshingangleα _XiaoDetermine the centre-to-centre spacing A between back taper spline spur gear small end face and pinion cutter _Xiao, described centre-to-centre spacing A _XiaoCan pass through formula

Obtain;

S145) by step S144) the back taper spline spur gear small end face that obtains and the centre-to-centre spacing A between pinion cutter _XiaoDetermine back taper spline spur gear small end root diameter d _Ixiao, described d _IxiaoCan pass through formula d _Ixiao=2A _Xiao-d _eoObtain;

S15) by step S14) the back taper spline spur gear small end face parameter that draws finally determines to insert established angle ψ processed, and described ψ can pass through formula

Try to achieve, in formula, B is the thickness of back taper spline spur gear.

After trying to achieve slotting established angle processed, other parameter can design by common pinion cutter design the pinion cutter that can process back taper spline spur gear profile of tooth, and just this kind pinion cutter relief angle is α _e=ψ+6 °, as shown in pinion cutter in Fig. 32, because this relief angle is larger, cause tooth top intensity poor, so this kind pinion cutter generally all is used for processing softer material, as plastic gear, Lv Hejin gear, copper alloy gear etc., and there is no the oblique-feeding organisation of working on common gear shapping machine,, so it is less that this kind pinion cutter uses, be difficult for adopting during the processing steel part.

When the established angle ψ slotting processed that adopts above-mentioned steps to try to achieve designs pinion cutter with the large end face theory of plane engagement, pinion cutter relief angle α the same as common pinion cutter _e=6 °, as shown in pinion cutter in Fig. 42, while using this pinion cutter to plug in back taper spline spur gear processed, the pinion cutter direction of motion and ψ angle of back taper spline spur gear shaft core line inclination, gear shaper workbench can be done the adjustment at ψ angle, as shown in Figure 3.Although the pinion cutter tooth top intensity of this scheme design is enough, the back taper spline spur gear profile accuracy of processing is poor, to less demanding back taper spline spur gear or feasible.For obtaining high strength, high-precision back taper spline spur gear pinion cutter, the inventor provides a kind of brand-new method for designing in following step, and its particular content is as follows:

S2) be the equivalent parameter of bevel gear 3 on the V-V plane with the parameter conversion of the large end end face of back taper spline spur gear, as shown in Figure 4, described equivalent parameter comprises:

Equivalent modulus m _v, its numerical value equates with the modulus m of back taper spline spur gear;

Virtual number of teeth Z _v, it meets formula

Equivalent outside diameter circle d _ev, meet formula $d_{\mathrm{ev}}=\frac{d_e}{\cos \mathrm{\ψ}}+[2B-(d_e-d_i)\tan \mathrm{\ψ}]\sin \mathrm{\ψ};$

Equivalent circle-dividing diameter d _fv, meet formula

Equivalent root diameter d _iv, meet formula

Equivalent pressure angle of graduated circle α _fv, described equivalent pressure angle of graduated circle α _fvCan be tried to achieve by the geometrical expansion relational expression, as shown in Figure 5, wherein:

$\sin \mathrm{\ψ}=\frac{\mathrm{BO}}{\mathrm{AB}}$

${\tan \mathrm{\β}}_f=\frac{\mathrm{BG}}{\mathrm{BO}}$

$\cos \mathrm{\ψ}=\frac{\mathrm{AO}}{\mathrm{AB}}$

${\tan \mathrm{\α}}_f=\frac{\mathrm{EO}}{\mathrm{AO}}$

${\tan \mathrm{\α}}_{\mathrm{fv}}=\frac{\mathrm{BD}}{\mathrm{AB}}=\frac{\mathrm{BG}+\mathrm{DG}}{\mathrm{AB}}=\frac{\mathrm{BG}+\mathrm{EO}}{\mathrm{AB}}=\frac{\mathrm{BG}}{\mathrm{AB}}+\frac{\mathrm{EO}}{\mathrm{AB}}=\frac{\mathrm{BO}}{\mathrm{AB}}{\tan \mathrm{\β}}_f+\frac{\mathrm{AO}}{\mathrm{AB}}{\tan \mathrm{\α}}_f$

$=\sin \mathrm{\ψ}{\tan \mathrm{\β}}_f+\cos \mathrm{\ψ}{\tan \mathrm{\α}}_f$

Can push away to obtain α by following formula _fvMeet formula α _fv=arctan (sin ψ tan β _f+ cos ψ tan α _f);

Equivalent base circle diameter (BCD) d _bv, meet formula d _bv=mZ _vCos α _fv

S3) in conjunction with pinion cutter modification coefficient ξ _o, tooth number Z _o, anterior angle γ and tooth top relief angle α _eDeng basic parameter, according to step S2) the equivalent parameter of back taper spline spur gear on the v-v plane that draw, obtain remaining geometric parameter of pinion cutter with bevel gear 3 and the principle of pinion cutter 2 engagements, the pinion cutter tooth profile parameter that draws according to above-mentioned steps can process high-precision back taper spline spur gear pinion cutter.

Step S3) principle with bevel gear and pinion cutter engagement described in is obtained remaining geometric parameter of pinion cutter, and its concrete grammar is as follows:

The basic parameter of known pinion cutter, modification coefficient ξ o, tooth number Z o, anterior angle γ, tooth top relief angle α e,, according to the bevel gear theory of engagement, as shown in Figure 8, try to achieve:

${\mathrm{inv\α}}_{\mathrm{olv}}=\frac{2\tan {\mathrm{\α}}_{\mathrm{fv}}({\mathrm{\ξ}}_v+{\mathrm{\ξ}}_o)}{Z_v+Z_o}+{\mathrm{inv\α}}_{\mathrm{fv}},$ α in formula _olvThe angle of engagement on the V-V plane for slotting tool and back taper spline spur gear;

A _olvThe centre-to-centre spacing on the V-V plane for pinion cutter and back taper spline spur gear;

d _jvFor back taper spline spur gear pitch diameter;

d _joFor plugging in back taper spline spur gear pinion cutter pitch diameter;

d _eo=2A _olv-d _iv, d _eoFor the pinion cutter outside diameter circle;

d _io=2A _olv-d _ev, d _ioFor the pinion cutter root diameter;

${\mathrm{\α}}_{\mathrm{fo}}=\arctan \left[\frac{\tan {\mathrm{\α}}_{\mathrm{fv}}}{1-\tan r\tan {\mathrm{\α}}_e}\right],$ α _fPressure angle of graduated circle for back taper spline spur gear;

d _bo=mZ _oCos α _fo, d _boBase circle diameter (BCD) for the pinion cutter toothed surfaces;

Measure profile of tooth apart from pinion cutter front end face 2.5mm place:

${\mathrm{\&rho;}}_{\mathrm{eo}}=0.5\sqrt{{(d_{\mathrm{eo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{{\mathrm{<figure-callout\; id="2"\; label="d\; bo"\; filenames="HDA0000368729720000021.png,HDA0000368729720000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{bo}}}^2}$

$d_{\mathrm{qo}}=2\sqrt{{(A_{\mathrm{olv}}\sin {\mathrm{\&alpha;}}_{\mathrm{olv}}-0.5\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="d\; ev"\; filenames="HDA0000368729720000021.png,HDA0000368729720000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{ev}}}^2+{d_{\mathrm{bv}}}^2})}^2+{\left(0.5d_{\mathrm{bo}}\right)}^2}$

${\mathrm{\&rho;}}_{\mathrm{io}}=0.5\sqrt{{(d_{\mathrm{qo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{{\mathrm{<figure-callout\; id="2"\; label="d\; bo"\; filenames="HDA0000368729720000021.png,HDA0000368729720000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{bo}}}^2}$

The pitch circle of back taper spline spur gear can be projected to (d on the XOZ plane according to above profile of tooth coordinate Calculation _j=d _jvCos ψ), then make space back taper spline spur gear profile of tooth model with pinion cutter pitch circle engagement and become figure, also provable to design back taper spline spur gear pinion cutter with the bevel gear theory of engagement be rational.

Embodiment 3 proofs adopt the correctness of V-V plane equivalent parameter designing pinion cutter

Compare and whether overlap with the profile of tooth of the equivalent calculation of parameter on V-V plane by back taper spline spur gear large end face calculation of parameter V-V planar tooth again, namely whether proof is correct with V-V plane equivalent parameter designing pinion cutter, followingly with the spatial coordinates calculation formula, proves:

1. use back taper spline spur gear large end face calculation of parameter V-V planar tooth

According to the coordinate of back taper spline spur gear large end face calculation of parameter V-V planar tooth in space, as shown in Figure 6, wherein:

d _a＝d _e+[2B-(d _e-d _i)tanψ]sinψcosψ

(get and count: N=1,2,3 ... n)

d _n＝d _a-NΔd

Y _n＝(d _n-d _i)tanψ

S _fn＝S _f-2Y _ntanβ _f

$\frac{S_n}{d_n}=\frac{S_{\mathrm{fn}}}{d_f}+\mathrm{inv}{\mathrm{\&alpha;}}_f-\sqrt{{\left(\frac{d_n}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_n}{d_b}\right)}^2-1}\right),$ D in formula _nFor d in the V-V plane _iv≤ d _n≤ d _evGet spot diameter, S _nFor d in the V-V plane _iv≤ d _n≤ d _evGet spot diameter, ask for its d _nCircular tooth thickness, finally ask the profile of tooth coordinate on its V-V plane, and the profile of tooth coordinate of asking for equivalent draws two profile of tooth figure with 3d space and compares use;

$X_n=0.5d_n\cos \frac{S_n}{d_n}$

ΔX _n＝0.5d _n-X _n

X _v＝X _n+ΔX _nsinψsinψ

Y _v＝Y _n-ΔX _nsinψcosψ

S _fsv＝S _f-2Y _vtanψ

Following formula solves with Newton iteration method:

$f\left(d_{\mathrm{vn}}\right)=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\&alpha;}}_f-\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{vn}}}{{2X}_v}\right)}^2-1}\right)-\sqrt{{\left(\frac{d_{\mathrm{vn}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{vn}}}{d_b}\right)}^2-1}\right)$

$f^{\&prime;}\left(d_{\mathrm{vn}}\right)=\frac{{2X}_v}{d_{\mathrm{vn}}\sqrt{{d_{\mathrm{vn}}}^2-{\left({2X}_v\right)}^2}}-\frac{\sqrt{{d_{\mathrm{vn}}}^2-{d_b}^2}}{d_{\mathrm{vn}}{d}_b}$

$d_{\mathrm{vn}}=d_{\mathrm{vn}}-\frac{f\left(d_{\mathrm{vn}}\right)}{f^{\&prime;}\left(d_{\mathrm{vn}}\right)}$

$\frac{S_{\mathrm{vn}}}{d_{\mathrm{vn}}}=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\&alpha;}}_f-\sqrt{{\left(\frac{d_{\mathrm{vn}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{vn}}}{d_b}\right)}^2-1}\right)$

F on the V-V plane _vPoint coordinates:

$X_{\mathrm{vn}}={0.5d}_{\mathrm{vn}}\cos \frac{S_{\mathrm{vn}}}{d_{\mathrm{vn}}}$

$Y_{\mathrm{vn}}=\frac{d_a}{2\sin \mathrm{\&psi;}}-X_{\mathrm{vn}}\tan \mathrm{\&psi;}$

$Z_{\mathrm{vn}}={0.5d}_{\mathrm{vn}}\sin \frac{S_{\mathrm{vn}}}{d_{\mathrm{vn}}}$

2. use the equivalent calculation of parameter profile of tooth coordinate on back taper spline spur gear V-V plane, obtain equivalent circle-dividing diameter d _fvEquivalent divide circular thickness S _fv, as shown in Figure 7:

Ask the distance of back taper spline spur gear large end face to S-S plane and V-V Plane intersects:

Y _f＝0.5(d _f-d _i)tanψ

S _fs＝S _f-2Y _ftanβ _f

$X_{\mathrm{fs}}=0.5d_f\cos \frac{S_{\mathrm{fs}}}{d_f}$

ΔX＝0.5d _f-X _fs

Ask at f _vThe centre-to-centre spacing X of spot projection _fv:

X _fv＝X _fs+ΔX?sinψsinψ

Ask the fv point height Y of back taper spline spur gear large end face and Sv-Sv Plane intersects _fv:

Y _fv＝Y _f-ΔX?sinψcosψ

Ask the fv point circular tooth thickness S on Sv-Sv plane _fsvDiameter d _sv:

S _fsv＝S _f-2Y _fvtanβ _f

Following formula solves with Newton iteration method:

$f\left(d_{\mathrm{sv}}\right)=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\&alpha;}}_f-\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{{2X}_{\mathrm{fv}}}\right)}^2-1}\right)-\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}\right)$

$f^{\&prime;}\left(d_{\mathrm{sv}}\right)=\frac{{2X}_{\mathrm{fv}}}{d_{\mathrm{sv}}\sqrt{{d_{\mathrm{sv}}}^2-{\left({2X}_{\mathrm{fv}}\right)}^2}}-\frac{\sqrt{{d_{\mathrm{sv}}}^2-{d_b}^2}}{d_{\mathrm{sv}}{d}_b}$

$d_{\mathrm{sv}}=d_{\mathrm{sv}}-\frac{f\left(d_{\mathrm{sv}}\right)}{f^{\&prime;}\left(d_{\mathrm{sv}}\right)}$

$\frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\&alpha;}}_f-\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}\right)$

$X_{\mathrm{fsv}}={0.5d}_{\mathrm{sv}}\cos \frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}$

$S_{\mathrm{fsv}}=d_{\mathrm{sv}}\sin \frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}$

$d_{\mathrm{fv}}=2\sqrt{{\left(\frac{X_{\mathrm{fsv}}}{d_{\mathrm{sv}}}\right)}^2-{\left(\frac{S_{\mathrm{sv}}}{2}\right)}^2}={\mathrm{mZ}}_v$

$S_{\mathrm{fv}}=d_{\mathrm{fv}}\arctan (1/\sqrt{{\left(\frac{S_{\mathrm{fsv}}}{d_{\mathrm{fv}}}\right)}^2-1})$

${\mathrm{\&xi;}}_v=\frac{S_{\mathrm{fv}}-0.5\mathrm{\&pi;m}}{2m\tan {\mathrm{\&alpha;}}_{\mathrm{fv}}}$

Calculate V-V plane equivalent tooth shape coordinate:

(get and count: N=1,2,3 ... n)

d _n＝d _ev-NΔd

$\frac{S_n}{d_n}=\frac{S_{\mathrm{fv}}}{d_{\mathrm{fv}}}+{\mathrm{inv\&alpha;}}_{\mathrm{fv}}-\sqrt{{\left(\frac{d_n}{d_{\mathrm{bv}}}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_n}{d_{\mathrm{bv}}}\right)}^2-1}\right)$

$X_n={0.5d}_n\cos \frac{S_n}{d_n}$

$Y_n=\frac{d_a}{2\sin \mathrm{\&psi;}}-X_n\tan \mathrm{\&psi;}$

$Z_n={0.5d}_n\cos \frac{S_n}{d_n}$

With 1., 2. space profile of tooth coordinate draws with the SPL of AutoCAD, then with the space tooth curve that 1. 3d dynamic view can be found out, 2. calculate, overlap, and in calculating formula 2.: d _fv=mZ _v, adopting V-V plane equivalent parameter when this has also proved design back taper spline spur gear pinion cutter is rational with bevel gear and the pinion cutter theory of engagement.

Embodiment 4 back taper spline spur gear pinion cutter design examples

Design back taper spline spur gear pinion cutter, at first will calculate and insert oblique angle processed ψ value, then according to embodiment 3 2. described in the method equivalent that solves back taper spline spur gear divide circular thickness Sfv, obtain the pinion cutter geometric parameter with the V-V theory of plane engagement finally.

Take the back taper spline spur gear with following parameters described in processing and implementation example 1 as example: tooth number Z=39, modulus m=1.8, pressure angle of graduated circle α _f=20 °, reference circle rear flank angle β _f=4 °, outside diameter circle d _e=71.45, root diameter d _i=67, across number of teeth K=5, base tangent length W _K=24.88.

The basic parameter of described back taper spline spur gear pinion cutter is tooth number Z _o=56, anterior angle γ=5 °, tooth top relief angle α _e=6 °, reference diameter d _fo=Z _oM=100.8, circular tooth thickness S _fo=2.696, design and try to achieve all the other geometric parameters of pinion cutter.

Separate: by design back taper spline spur gear and pinion cutter basic parameter, tried to achieve:

(1). with large end face theory of plane engagement design pinion cutter (first determining to insert oblique angle ψ processed)

invα _f＝tanα _f+α _f＝0.01490438387rad

$S_f=m[\frac{W_k}{m\cos {\mathrm{\&alpha;}}_f}-(K-1)\mathrm{\&pi;}-\mathrm{Zinv}{\mathrm{\&alpha;}}_f]=2.810988126\mathrm{mm}$

$\mathrm{\&xi;}=\frac{S_f-0.5\mathrm{\&pi;m}}{2m\tan {\mathrm{\&alpha;}}_f}=-0.01255082972$

${\mathrm{\&xi;}}_o=\frac{S_{\mathrm{fo}}-0.5\mathrm{\&pi;m}}{2m\tan {\mathrm{\&alpha;}}_f}=-0.1$

${\mathrm{inv\&alpha;}}_{\mathrm{ol}}=\frac{2\tan {\mathrm{\&alpha;}}_f(\mathrm{\&xi;}+{\mathrm{\&xi;}}_o)}{Z+Z_o}+\mathrm{inv}{\mathrm{\&alpha;}}_f=0.01404195962\mathrm{rad}$

α _ol＝0.34241944396rad

$A_{\mathrm{ol}}=\frac{m\cos {\mathrm{\&alpha;}}_f(Z+Z_o)}{2\cos {\mathrm{\&alpha;}}_{\mathrm{ol}}}=85.2955475088\mathrm{mm}$

d _eo＝2A _ol-d _i＝103.5910950177mm

When pinion cutter and the engagement of back taper spline spur gear small end, the parameter of back taper spline spur gear is:

S _fxiao＝S _f-2Btanβ _f＝1.901939571mm

${\mathrm{\&xi;}}_{\mathrm{xiao}}=\frac{S_{\mathrm{fxiao}}-0.5\mathrm{\&pi;m}}{2m\tan {\mathrm{\&alpha;}}_f}=-0.706325935\mathrm{mm}$

${\mathrm{inv\&alpha;}}_{\mathrm{xiao}}=\frac{2\tan {\mathrm{\&alpha;}}_f({\mathrm{\&xi;}}_{\mathrm{xiao}}+{\mathrm{\&xi;}}_o)}{Z+\mathrm{Zo}}+{\mathrm{inv\&alpha;}}_f=0.008725886\mathrm{rad}$

α _xiao＝0.293471786rad

$A_{\mathrm{xiao}}=\frac{m\cos {\mathrm{\&alpha;}}_f(Z+\mathrm{Zo})}{2\cos {\mathrm{\&alpha;}}_{\mathrm{xiao}}}=83.932212481\mathrm{mm}$

d _ixiao＝2A _xiao-d _eo＝64.273329944mm

(2). ask the parameter of back taper spline spur gear on the V-V plane:

$d_{\mathrm{ev}}=\frac{d_e}{\cos \mathrm{\&psi;}}+[2B-(d_e-d_i)\tan \mathrm{\&psi;}]\sin \mathrm{\&psi;}=75.48172162\mathrm{mm}$

$d_{\mathrm{iv}}=\frac{d_i}{\cos \mathrm{\&psi;}}=68.45788673\mathrm{mm}$

α _fv＝arctan(sinψtanβ _f+cosψtanα _f)＝0.35488423rad＝20°20′00″

invα _fv＝tanα _fv+α _fv＝0.015689228rad

$Z_v=\frac{Z}{\cos \mathrm{\&psi;}}=39.84862063$

d _fv＝Z _vm＝71.727517139mm

d _bv＝d _fvcosα _fv＝67.257940489mm

(3). divide circular thickness S according to solving the equivalent of back taper spline spur gear on the V-V plane in 2. stating _fv:

Y _fs＝0.5(d _f-d _i)tanψ＝0.335590161mm

S _fs＝S _f-2Y _fs?tanβ _f＝2.764054626mm

$X_{\mathrm{fs}}=0.5d_f\cos \frac{S_{\mathrm{fs}}}{d_f}=35.072795545\mathrm{mm}$

ΔX＝0.5d _f-X _fs＝0.027204455mm

X _fsv＝X _fs+ΔXsinψsinψ＝35.073941905mm

Y _fsv＝Y _fs-ΔXsinψcosψ＝0.330124636mm

S _fsv＝S _f-2Y _fsvtanβ _f＝2.764818999mm

d _sv(initial value)=d _f

$f\left(d_{\mathrm{sv}}\right)=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\&alpha;}}_f-\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{{2X}_{\mathrm{fv}}}\right)}^2-1}\right)-\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}\right)$ $={9.87550378410}^{-10}$

$f^{\&prime;}\left(d_{\mathrm{sv}}\right)=\frac{{2X}_{\mathrm{fv}}}{d_{\mathrm{sv}}\sqrt{{d_{\mathrm{sv}}}^2-{\left({2X}_{\mathrm{fv}}\right)}^2}}-\frac{\sqrt{{d_{\mathrm{sv}}}^2-{d_b}^2}}{d_{\mathrm{sv}}{d}_b}=0.36678376167$

$d_{\mathrm{sv}}=d_{\mathrm{sv}}-\frac{f\left(d_{\mathrm{sv}}\right)}{f^{\&prime;}\left(d_{\mathrm{sv}}\right)}=70.202291747\mathrm{mm}$

$\frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}=\frac{S_{\mathrm{fsv}}}{d_f}+{\mathrm{inv\&alpha;}}_f-\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}+\arctan \left(\sqrt{{\left(\frac{d_{\mathrm{sv}}}{d_b}\right)}^2-1}\right)=0.039373002\mathrm{rad}$

$X_{\mathrm{sv}}={0.5d}_{\mathrm{sv}}\cos \frac{S_{\mathrm{sv}}}{d_{\mathrm{sv}}}=35.073941905\mathrm{mm}$

$d_{\mathrm{fv}}=2\sqrt{{\left(\frac{X_{\mathrm{fsv}}}{d_{\mathrm{sv}}}\right)}^2-{\left(\frac{S_{\mathrm{sv}}}{2}\right)}^2}=71.72751706\mathrm{mm}$

d _fv＝mZ _v＝71.72751714mm

This circle-dividing diameter and equivalent circle-dividing diameter differ: Δ d _fv=8.28375 * 10 ^-8

${\mathrm{\&xi;}}_v=\frac{S_{\mathrm{fv}}-0.5\mathrm{\&pi;m}}{2m\tan {\mathrm{\&alpha;}}_{\mathrm{fv}}}=-0.047515286$

${\mathrm{inv\&alpha;}}_{\mathrm{olv}}=\frac{2\tan {\mathrm{\&alpha;}}_{\mathrm{fv}}({\mathrm{\&xi;}}_v+{\mathrm{\&xi;}}_o)}{Z_v+Z_o}+{\mathrm{inv\&alpha;}}_{\mathrm{fv}}=0.011454857\mathrm{rad}$

$A_{\mathrm{olv}}=\frac{m\cos {\mathrm{\&alpha;}}_{\mathrm{fv}}(Z_v+Z_o)}{2\cos {\mathrm{\&alpha;}}_{\mathrm{olv}}}=85.995152703\mathrm{mm}$

d _eo＝2A _olv-d _iv＝103.532418677mm

d _io＝2A _olv-d _ev＝95.788583781mm

d _fo＝mZ _o＝100.8

β _bo＝arctan(sinα _fo?tanα _e)＝0.036802708rad＝2°07′

d _bo＝Z _omcosα _fo＝94.412592337

Measure profile of tooth apart from pinion cutter front end face 2.5mm place:

${\mathrm{\&rho;}}_{\mathrm{eo}}=0.5\sqrt{{(d_{\mathrm{eo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{{\mathrm{<figure-callout\; id="2"\; label="d\; bo"\; filenames="HDA0000368729720000021.png,HDA0000368729720000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{bo}}}^2}=20.595408109\mathrm{mm}\&ap;25.59\mathrm{mm}$

$d_{\mathrm{qo}}=2\sqrt{{(A_{\mathrm{olv}}\sin {\mathrm{\&alpha;}}_{\mathrm{olv}}-0.5\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="d\; ev"\; filenames="HDA0000368729720000021.png,HDA0000368729720000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{ev}}}^2+{d_{\mathrm{bv}}}^2})}^2+{\left(0.5d_{\mathrm{bo}}\right)}^2}=97.023443807\mathrm{mm}$

${\mathrm{\&rho;}}_{\mathrm{io}}=0.5\sqrt{{(d_{\mathrm{qo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{{\mathrm{<figure-callout\; id="2"\; label="d\; bo"\; filenames="HDA0000368729720000021.png,HDA0000368729720000022.png"\; state="\{\{state\}\}">d</figure-callout>}}_{\mathrm{bo}}}^2}=11.178227234\mathrm{mm}\&ap;11.17\mathrm{mm}$

Explanation is finally, above embodiment is only unrestricted in order to technical scheme of the present invention to be described, although with reference to preferred embodiment, the present invention is had been described in detail, those of ordinary skill in the art is to be understood that, can modify or be equal to replacement technical scheme of the present invention, and not breaking away from aim and the scope of technical solution of the present invention, it all should be encompassed in the middle of claim scope of the present invention.

Claims (10)

1. back taper spline spur gear pinion cutter, its basic parameter is tooth number Z _o, anterior angle γ, tooth top relief angle α _e, reference diameter d _foWith pressure angle of graduated circle α _fo, it is characterized in that: described reference diameter d _fo=Z _oM, in following formula, m is back taper spline spur gear modulus; Described pressure angle of graduated circle α _foMeet formula

In following formula, ψ is that angle between the pinion cutter direction of motion and back taper spline spur gear axial line is namely inserted established angle processed, β _fFor the angle, reference circle rear flank of back taper spline spur gear, α _fPressure angle of graduated circle for back taper spline spur gear.

2. back taper spline spur gear pinion cutter according to claim 1, is characterized in that: described slotting established angle processed

D in formula _iFor the root diameter of back taper spline spur gear, d _IxiaoFor back taper spline spur gear small end root diameter, B is the thickness of back taper spline spur gear.

3. back taper spline spur gear pinion cutter according to claim 2, is characterized in that: described back taper spline spur gear small end root diameter d _Ixiao=2A _Xiao-d _eo, A in formula _XiaoFor the centre-to-centre spacing between back taper spline spur gear small end face and pinion cutter, d _eoFor the pinion cutter outside diameter circle.

4. back taper spline spur gear pinion cutter according to claim 3, is characterized in that: the centre-to-centre spacing between described back taper spline spur gear small end face and pinion cutter

α in following formula _XiaoFor the angle of engagement between back taper spline spur gear small end face and pinion cutter, Z is the number of teeth of back taper spline spur gear; Described meshingangleα _XiaoMeet formula

ξ in formula _XiaoFor back taper spline spur gear small end modification coefficient, ξ _oFor the pinion cutter modification coefficient; Described back taper spline spur gear small end modification coefficient ξ _XiaoMeet formula

S in formula _FxiaoFor back taper spline spur gear small end graduated arc transverse tooth thickness; Described S _FsiaoMeet formula S _Fxiao=S _f-2Btan β _f, S in formula _fFor back taper spline spur gear circular tooth thickness.

5. back taper spline spur gear pinion cutter according to claim 4, is characterized in that: described pinion cutter outside diameter circle d _eo=2A _o-d _i, A in following formula _oFor the centre-to-centre spacing between back taper spline spur gear large end face and pinion cutter; Centre-to-centre spacing between described back taper spline spur gear large end face and pinion cutter

α in following formula _oFor the angle of engagement between back taper spline spur gear large end face and pinion cutter; Described meshingangleα _oMeet formula In formula, ξ is back taper spline spur gear modification coefficient.

6. back taper spline spur gear pinion cutter according to claim 5, is characterized in that: described back taper spline spur gear modification coefficient

7. the back taper spline spur gear pinion cutter described according to claim 6, is characterized in that: also comprise for grinding pinion cutter toothed surfaces desired parameters d _boAnd β _bo, described d _bo, for the base circle diameter (BCD) of pinion cutter toothed surfaces, meet formula d _bo=d _foCos α _foDescribed β _bo, for the Base spiral angle of pinion cutter toothed surfaces, meet formula β _bo=arctan (sin α _foTan α _e).

8. method for designing as back taper spline spur gear pinion cutter as described in any one in claim 1-7, the major parameter of the large end face of described back taper spline spur gear comprises modulus m, pressure angle of graduated circle α _f, reference circle rear flank angle β _f, outside diameter circle d _e, reference diameter d _f, root diameter d _iWith circular tooth thickness S _f, it is characterized in that, comprise the following steps:

S1) obtain slotting established angle ψ processed according to back taper spline spur gear and pinion cutter basic parameter;

S2) be the equivalent parameter of bevel gear on the V-V plane with the parameter conversion of back taper spline spur gear large end face;

S3) in conjunction with pinion cutter modification coefficient ξ _o, tooth number Z _o, anterior angle γ and tooth top relief angle α _eDeng basic parameter, according to step S2) the equivalent parameter of back taper spline spur gear on the v-v plane that draw, principle with bevel gear and pinion cutter engagement is obtained remaining geometric parameter of pinion cutter, and the pinion cutter tooth profile parameter that draws according to above-mentioned steps can process high accuracy, high-intensity back taper spline spur gear pinion cutter.

9. method for designing according to claim 8, is characterized in that, step S1) described in insert established angle ψ processed and try to achieve by the following method:

S11) determine meshingangleα between back taper spline spur gear large end face and pinion cutter _o, this parameter alpha _oCan pass through formula

Calculating is tried to achieve;

S12) according to step S11) in the angle of engagement parameter alpha that obtains _oDetermine the centre-to-centre spacing A between back taper spline spur gear large end face and pinion cutter _o, this centre-to-centre spacing parameter A _oCan pass through formula

Calculating is tried to achieve;

S13) according to step S12) in the centre-to-centre spacing parameter A that draws _oDetermine pinion cutter outside diameter circle d _eo, described pinion cutter outside diameter circle d _eoCan pass through formula d _eo=2A _o-d _iCalculating is tried to achieve;

S14) ask back taper spline spur gear small end face parameter according to theory of plane engagement;

S141) at first determine back taper spline spur gear small end graduated arc transverse tooth thickness S _Fxiao, described S _FxiaoMeet formula S _Fxiao=S _f-2Btan β _f

S142) according to step S141) in the back taper spline spur gear small end graduated arc transverse tooth thickness S that determines _FxiaoDetermine back taper spline spur gear small end modification coefficient ξ _Xiao, described ξ _XiaoMeet formula

S143) by step S142) in the back taper spline spur gear small end modification coefficient ξ that tries to achieve _XiaoIn conjunction with pinion cutter modification coefficient ξ _oCan determine the meshingangleα between back taper spline spur gear small end face and pinion cutter _Xiao, described meshingangleα _XiaoCan pass through formula

Obtain;

S144) by step S143) definite meshingangleα _XiaoDetermine the centre-to-centre spacing A between back taper spline spur gear small end face and pinion cutter _Xiao, described centre-to-centre spacing A _XiaoCan pass through formula

Obtain;

S145) by step S144) the back taper spline spur gear small end face that obtains and the centre-to-centre spacing A between pinion cutter _XiaoDetermine back taper spline spur gear small end root diameter d _Ixiao, described d _IxiaoCan pass through formula d _Ixiao=2A _Xiao-d _eoObtain;

S15) by step S14) the back taper spline spur gear small end face parameter that draws finally determines to insert established angle ψ processed, and described ψ can pass through formula

Try to achieve, in formula, B is the thickness of back taper spline spur gear.

10. method for designing according to claim 9, is characterized in that, step S2) described on the V-V plane equivalent parameter of bevel gear comprise Equivalent modulus m _v, virtual number of teeth Z _v, equivalent outside diameter circle d _ev, equivalent circle-dividing diameter d _fv, equivalent root diameter d _iv, equivalent base circle diameter (BCD) d _bv, equivalent pressure angle of graduated circle α _fv, described Equivalent modulus m _vEquate described virtual number of teeth Z with the modulus m of back taper spline spur gear _vMeet formula

Described equivalent outside diameter circle d _evMeet formula

Described equivalent circle-dividing diameter d _fvMeet formula d _fv=mZ _v, described equivalent root diameter d _ivMeet formula

Described equivalent pressure angle of graduated circle α _fvMeet formula tan α _fv=sin ψ tan β _f+ cos ψ tan α _f, described equivalent base circle diameter (BCD) d _bvMeet formula d _bv=mZ _vCos α _fv

On described V-V plane, the equivalent parameter of bevel gear also comprises equivalent graduated arc transverse tooth thickness S _fv, described S _fvDetermine by following steps:

A) ask the distance of back taper spline spur gear large end face to S-S plane and V-V plane point of intersection f:

Y _f=0.5 (d _f-d _i) tan ψ, Y _fFor the distance of back taper spline spur gear large end face to S-S plane and V-V plane point of intersection f;

S _fs＝S _f-2Y _ftanβ _f

X _fsS-S plane reference circle d _fChordal tooth thickness, namely the fs point is to the distance at back taper spline spur gear center;

Δ X=0.5d _f-X _fs, Δ X is S-S plane reference circle d _fChordal height, namely the fs point is to the distance of f;

B) ask at f _vThe centre-to-centre spacing X of spot projection _fv:

X _fv＝X _fs+ΔX?sinψsinψ

C) ask the height Y of back taper spline spur gear large end face and fv intersection point _fv:

Y _fv＝Y _f-ΔXsinψcosψ

D) ask the fv point circular tooth thickness S on Sv-Sv plane _fsvDiameter d _sv:

S _fsv＝S _f-2Y _fv?tanβ _f

Following formula solves with Newton iteration method:

Final:

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