CN103394768B - Back taper spline spur gear pinion cutter and method for designing thereof - Google Patents

Abstract

The invention discloses a kind of back taper spline spur gear pinion cutter and method for designing thereof, 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 above formula, m is back taper spline spur gear modulus; Described pressure angle of graduated circle α _fomeet formula <maths num=" 0001 " > </maths> back taper spline of the present invention spur gear pinion cutter, it adopts the principle design that the inclined-plane of the large end of back taper spline spur gear and V-V plane engage with pinion cutter with bevel gear to form, it has higher tooth top intensity, not only the thorough low doubly-linked gear in the past that solves is separated the low problem of working (machining) efficiency, and ensure that the geometric accuracy of plugging in the back taper spline spur gear profile of tooth made; Present invention also offers a kind of method for designing of back taper spline spur gear pinion cutter, adopt this method for designing can obtain the back taper spline spur gear pinion cutter of high accuracy, high strength, and the back taper spline spur gear profile accuracy of inserting system also increases.

Description

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

Technical field

The present invention relates to a kind of cutter, particularly relate 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 is generally all used on the doubly-linked gear of transmission gear shift (or gear shift) of motor vehicle, the main internal spline playing gear coaxial with another (or concentric) gear when rotating is meshed, it has good guide effect in engagement rotation, 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 adopt this doubly-linked gear to be separated processing (operation is gear hobbing, roll flute, boring, Che Kong, grinding out, assembling), although processing profile accuracy like this is guaranteed, time-consuming.

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

Summary of the invention

In view of this, the invention provides a kind of back taper spline spur gear pinion cutter, adopt this pinion cutter can time processing when processing back taper spline spur gear, and the machining accuracy of back taper spline spur gear profile of tooth can be ensured, and then solve and be separated the time-consuming problem of processing doubly-linked gear in the past.In addition, present invention also offers the method for designing of back taper spline spur gear pinion cutter, the pinion cutter designed by the method has the advantage such as high strength, high accuracy.

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

An object 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 above formula, m is back taper spline spur gear modulus; Described pressure angle of graduated circle α _fomeet formula ${\mathrm{\&alpha;}}_{\mathrm{fo}}=\arctan \left[\frac{\sin \mathrm{\&psi;}\tan {\mathrm{\&beta;}}_f+\cos \mathrm{\&psi;}\tan {\mathrm{\&alpha;}}_f}{1-\tan r\tan {\mathrm{\&alpha;}}_e}\right],$ In above formula, ψ is the angle i.e. slotting established angle processed between the pinion cutter direction of motion and back taper spline spur gear axial line, β _ffor the angle, reference circle rear flank of back taper spline spur gear, α _ffor the pressure angle of graduated circle of 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 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 above 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 ${\mathrm{inv\&alpha;}}_{\mathrm{xiao}}=\frac{2\tan {\mathrm{\&alpha;}}_f({\mathrm{\&xi;}}_{\mathrm{xiao}}+{\mathrm{\&xi;}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\&alpha;}}_f,$ ξ in formula _xiaofor back taper spline spur gear small end modification coefficient, ξ _ofor 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 above 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 above formula _ofor the angle of engagement between back taper spline spur gear large end face and pinion cutter; Described meshingangleα _omeet formula ${\mathrm{inv\&alpha;}}_o=\frac{2\tan {\mathrm{\&alpha;}}_f(\mathrm{\&xi;}+{\mathrm{\&xi;}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\&alpha;}}_f,$ 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 _bofor the base circle diameter (BCD) of pinion cutter toothed surfaces, meet formula d _bo=d _focos α _fo; Described β _bofor the Base spiral angle of pinion cutter toothed surfaces, meet formula β _bo=arctan (sin α _fotan α _e).

Two of object 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) slotting established angle ψ processed is obtained according to back taper spline spur gear and pinion cutter basic parameter;

S2) by the parameter conversion of large for back taper spline spur gear end end face be the equivalent parameters of bevel gear in V-V plane;

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) equivalent parameters of back taper spline spur gear in v-v plane that draw, the principle engaged with pinion cutter with bevel gear obtains remaining geometric parameter of pinion cutter, and the pinion cutter tooth profile parameter drawn according to above-mentioned steps can process high-precision back taper spline spur gear pinion cutter.

Further, step S1) described in insert established angle ψ processed try to achieve by the following method:

S11) meshingangleα between back taper spline spur gear large end face and pinion cutter is determined _o, this parameter alpha _oby formula ${\mathrm{inv\&alpha;}}_o=\frac{2\tan {\mathrm{\&alpha;}}_f(\mathrm{\&xi;}+{\mathrm{\&xi;}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\&alpha;}}_f$ 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 _oby 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 _eoby formula d _eo=2A _o-d _icalculating is tried to achieve;

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

S141) back taper spline spur gear small end graduated arc transverse tooth thickness S is first determined _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 _othe meshingangleα between back taper spline spur gear small end face and pinion cutter can be determined _xiao, described meshingangleα _xiaoby formula ${\mathrm{inv\&alpha;}}_{\mathrm{xiao}}=\frac{2\tan {\mathrm{\&alpha;}}_f({\mathrm{\&xi;}}_{\mathrm{xiao}}+{\mathrm{\&xi;}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\&alpha;}}_f$ Obtain;

S144) by step S143) meshingangleα determined _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 _xiaoby formula obtain;

S145) by step S144) centre-to-centre spacing A between the back taper spline spur gear small end face that obtains and pinion cutter _xiaodetermine back taper spline spur gear small end root diameter d _ixiao, described d _ixiaoby 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 is by formula try to achieve, in formula, B is the thickness of back taper spline spur gear.

Further, step S2) described in V-V plane the equivalent parameters 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 _vequal with the modulus m of back taper spline spur gear, described virtual number of teeth Z _vmeet formula described equivalent outside diameter circle d _evmeet formula $d_{\mathrm{ev}}=\frac{d_e}{\cos \mathrm{\&psi;}}+[2B-(d_e-d_i)\tan \mathrm{\&psi;}]\sin \mathrm{\&psi;},$ 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;

In described V-V plane, the equivalent parameters of bevel gear also comprises equivalent graduated arc transverse tooth thickness S _fv, described S _fvdetermined by following steps:

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

Y _f=0.5 (d _f-d _i) tan ψ, Y _ffor back taper spline spur gear large end face is to the distance of 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 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 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+ΔXsinψsinψ

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

Y _fv＝Y _f-ΔXsinψcosψ

D) the fv point circular tooth thickness S of Sv-Sv plane is asked _fsvdiameter d _sv:

S _fsv＝S _f-2Y _fvtanβ _f

Following formula Newton iteration method solves:

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

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

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

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

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

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

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

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, it adopts the principle design that the inclined-plane of the large end of back taper spline spur gear and V-V plane engage with pinion cutter with bevel gear to form, its tooth top relief angle α _eequally with common pinion cutter can select 6 °, compare 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 applicable 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 ${\mathrm{\&alpha;}}_{\mathrm{fo}}=\arctan \left[\frac{\sin \mathrm{\&psi;}\tan {\mathrm{\&beta;}}_f+\cos \mathrm{\&psi;}\tan {\mathrm{\&alpha;}}_f}{1-\tan r\tan {\mathrm{\&alpha;}}_e}\right],$ Make it have higher machining accuracy, not only thoroughly solve doubly-linked gear in the past and be separated the low problem of working (machining) efficiency, and ensure that the geometric accuracy of plugging in the back taper spline spur gear profile of tooth made;

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

Accompanying drawing explanation

Below in conjunction with drawings and Examples, the invention will be further described.

Fig. 1 back taper spline spur gear structure chart;

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

The pinion cutter structural representation of Fig. 3 to be tooth top relief angle be ψ+6 °;

The slotting controlling angle schematic diagram of Fig. 4 to be tooth top relief angle be pinion cutter of 6 °;

The another kind of Fig. 5 to be tooth top relief angle be pinion cutter of 6 ° inserts controlling angle schematic diagram;

Fig. 6 is equivalent size parameter schematic diagram in 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 processed engagement figure.

Detailed description of the invention

Below with reference to drawings and Examples, the present invention is described in detail:

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

There is the back taper spline spur gear of following parameters: tooth number Z=39, modulus m=1.8, pressure angle of graduated circle α for processing _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 time, the basic parameter pressure angle of graduated circle α of described pinion cutter _fo, meet following formula:

All the other parameters of described back taper spline spur gear pinion cutter can be solved by the principle that back taper spline spur gear large end inclined-plane and V-V plane engage with pinion cutter with bevel gear according to the basic parameter of above-mentioned back taper spline spur gear pinion cutter, 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 '.

When using back taper spline spur gear pinion cutter of the present invention to plug in back taper spline spur gear processed, gear shaper workbench established angle is necessary for the established angle ψ slotting processed of design, to ensure the back taper spline spur gear precision of processing.Adopt back taper spline spur gear pinion cutter of the present invention can time processing when processing back taper spline spur gear, and the machining accuracy of back taper spline spur gear profile of tooth can be ensured, and then solve and be separated the time-consuming problem of processing doubly-linked gear in the past.

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

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) slotting established angle ψ processed is obtained 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) meshingangleα between back taper spline spur gear large end face 11 and pinion cutter 2 is determined _o, this parameter alpha _oby formula ${\mathrm{inv\&alpha;}}_o=\frac{2\tan {\mathrm{\&alpha;}}_f(\mathrm{\&xi;}+{\mathrm{\&xi;}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\&alpha;}}_f$ 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 _oby 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 _eoby formula d _eo=2A _o-d _icalculating is tried to achieve;

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

S141) back taper spline spur gear small end graduated arc transverse tooth thickness S is first determined _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 _othe meshingangleα between back taper spline spur gear small end face and pinion cutter can be determined _xiao, described meshingangleα _xiaoby formula ${\mathrm{inv\&alpha;}}_{\mathrm{xiao}}=\frac{2\tan {\mathrm{\&alpha;}}_f({\mathrm{\&xi;}}_{\mathrm{xiao}}+{\mathrm{\&xi;}}_o)}{Z+\mathrm{Zo}}+\mathrm{inv}{\mathrm{\&alpha;}}_f$ Obtain;

S144) by step S143) meshingangleα determined _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 _xiaoby formula obtain;

S145) by step S144) centre-to-centre spacing A between the back taper spline spur gear small end face that obtains and pinion cutter _xiaodetermine back taper spline spur gear small end root diameter d _ixiao, described d _ixiaoby 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 is by formula try to achieve, in formula, B is the thickness of back taper spline spur gear.

Try to achieve after inserting established angle processed, other parameter can design by common pinion cutter design the pinion cutter can processing back taper spline spur gear profile of tooth, and just this kind of 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 of pinion cutter is generally all used for processing softer material, as plastic gear, Lv Hejin gear, copper alloy gear etc., and on common gear shapping machine, there is no oblique-feeding organisation of working, so this kind of pinion cutter uses less, not easily adopt during processing steel part.

When the established angle ψ slotting processed adopting above-mentioned steps to try to achieve is with large end face theory of plane engagement design pinion cutter, pinion cutter relief angle α the same as common pinion cutter _e=6 °, as shown in pinion cutter in Fig. 42, when using this pinion cutter to plug in back taper spline spur gear processed, the pinion cutter direction of motion and back taper spline spur gear shaft core line tilt a ψ angle, and gear shaper workbench can do the adjustment at ψ angle, as shown in Figure 3.Although the pinion cutter tooth top intensity of program 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 present inventor provides a kind of brand-new method for designing in a subsequent step, and its particular content is as follows:

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

Equivalent modulus m _v, its numerical value is equal 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 $d_{\mathrm{ev}}=\frac{d_e}{\cos \mathrm{\&psi;}}+[2B-(d_e-d_i)\tan \mathrm{\&psi;}]\sin \mathrm{\&psi;};$

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 geometrical expansion relational expression, as shown in Figure 5, wherein:

$\sin \mathrm{\&psi;}=\frac{\mathrm{BO}}{\mathrm{AB}}$

${\tan \mathrm{\&beta;}}_f=\frac{\mathrm{BG}}{\mathrm{BO}}$

$\cos \mathrm{\&psi;}=\frac{\mathrm{AO}}{\mathrm{AB}}$

${\tan \mathrm{\&alpha;}}_f=\frac{\mathrm{EO}}{\mathrm{AO}}$

${\tan \mathrm{\&alpha;}}_{\mathrm{fv}}=\frac{\mathrm{BD}}{\mathrm{AB}}=\frac{\mathrm{BG}+\mathrm{DG}}{\mathrm{AB}}=\frac{\mathrm{BG}+\mathrm{EO}}{\mathrm{AB}}=\frac{\mathrm{BG}}{\mathrm{AB}}+\frac{\mathrm{EO}}{\mathrm{AB}}=\frac{\mathrm{BO}}{\mathrm{AB}}{\tan \mathrm{\&beta;}}_f+\frac{\mathrm{AO}}{\mathrm{AB}}{\tan \mathrm{\&alpha;}}_f$

$=\sin \mathrm{\&psi;}{\tan \mathrm{\&beta;}}_f+\cos \mathrm{\&psi;}{\tan \mathrm{\&alpha;}}_f$

α is can be derived from by above 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) equivalent parameters of back taper spline spur gear in v-v plane that draw, the principle engaged with pinion cutter 2 with bevel gear 3 obtains remaining geometric parameter of pinion cutter, and the pinion cutter tooth profile parameter drawn according to above-mentioned steps can process high-precision back taper spline spur gear pinion cutter.

Step S3) described in the principle engaged with pinion cutter with bevel gear obtain remaining geometric parameter of pinion cutter, 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, tries to achieve:

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

a _olvthe centre-to-centre spacing in V-V plane for pinion cutter and back taper spline spur gear;

d _jvfor back taper spline spur gear pitch diameter;

d _jofor plug in back taper spline spur gear pinion cutter pitch diameter;

D _eo=2A _olv-d _iv, d _eofor pinion cutter outside diameter circle;

D _io=2A _olv-d _ev, d _iofor pinion cutter root diameter;

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

D _bo=mZ _ocos α _fo, d _bofor the base circle diameter (BCD) of pinion cutter toothed surfaces;

Profile of tooth is measured apart from pinion cutter front end face 2.5mm place:

${\mathrm{\&rho;}}_{\mathrm{eo}}=0.5\sqrt{{(d_{\mathrm{eo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{d_{\mathrm{bo}}}^2}$

$d_{\mathrm{qo}}=2\sqrt{{(A_{\mathrm{olv}}\sin {\mathrm{\&alpha;}}_{\mathrm{olv}}-0.5\sqrt{{d_{\mathrm{ev}}}^2+{d_{\mathrm{bv}}}^2})}^2+{\left(0.5d_{\mathrm{bo}}\right)}^2}$

${\mathrm{\&rho;}}_{\mathrm{io}}=0.5\sqrt{{(d_{\mathrm{qo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{d_{\mathrm{bo}}}^2}$

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

Embodiment 3 proves the correctness adopting V-V plane equivalent parameters design pinion cutter

Calculate V-V planar tooth by back taper spline spur gear large end face parameter again to compare with the profile of tooth that the equivalent parameters of V-V plane calculates and whether overlap, whether namely prove with V-V plane equivalent parameters design pinion cutter correct, following spatial coordinates calculation formula proves:

1. V-V planar tooth is calculated by back taper spline spur gear large end face parameter

The coordinate of V-V planar tooth in space is calculated according to back taper spline spur gear large end face parameter, 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 V-V plane _iv≤ d _n≤ d _evget spot diameter, S _nfor d in V-V plane _iv≤ d _n≤ d _evget spot diameter, ask for its d _ncircular tooth thickness, finally asks the profile of tooth coordinate of its V-V plane, and the profile of tooth coordinate 3d space asked for equivalent draws two profile of tooth figure 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 Newton iteration method solves:

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

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

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

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

F in 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. calculate profile of tooth coordinate with the equivalent parameters of back taper spline spur gear V-V plane, equivalent circle-dividing diameter d be obtained _fvequivalent divide circular thickness S _fv, as shown in Figure 7:

Ask back taper spline spur gear large end face to the distance of 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+ΔXsinψsinψ

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-ΔXsinψcosψ

Ask the fv point circular tooth thickness S of Sv-Sv plane _fsvdiameter d _sv:

S _fsv＝S _f-2Y _fvtanβ _f

Following formula Newton iteration method solves:

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

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

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

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

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

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

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

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

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

Calculate V-V plane equivalent tooth shape coordinate:

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

d _n＝d _ev-NΔd

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

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

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

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

The SPL of space profile of tooth coordinate AutoCAD 1., is 2. drawn, then with 3d dynamic view can find out 1., the space tooth curve that 2. calculates overlaps, and in calculating formula 2.: d _fv=mZ _v, when this also demonstrates design back taper spline spur gear pinion cutter, adopt V-V plane equivalent parameters to be rational with bevel gear and the pinion cutter theory of engagement.

Embodiment 4 back taper spline spur gear pinion cutter design example

Design back taper spline spur gear pinion cutter, 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, finally obtain pinion cutter geometric parameter by V-V theory of plane engagement.

The back taper spline spur gear with following parameters for described in process embodiment 1: 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: tried to achieve by design back taper spline spur gear and pinion cutter basic parameter:

(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 engages with 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 back taper spline spur gear in the parameter of 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 back taper spline spur gear in 2. stating at the equivalent of V-V plane _fv:

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

S _fs＝S _f-2Y _fstanβ _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 differs with equivalent circle-dividing diameter: 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 _fotanα _e)＝0.036802708rad＝2°07′

d _bo＝Z _omcosα _fo＝94.412592337

Profile of tooth is measured apart from pinion cutter front end face 2.5mm place:

${\mathrm{\&rho;}}_{\mathrm{eo}}=0.5\sqrt{{(d_{\mathrm{eo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{d_{\mathrm{bo}}}^2}=20.595408109\mathrm{mm}\&ap;25.59\mathrm{mm}$

$d_{\mathrm{qo}}=2\sqrt{{(A_{\mathrm{olv}}\sin {\mathrm{\&alpha;}}_{\mathrm{olv}}-0.5\sqrt{{d_{\mathrm{ev}}}^2+{d_{\mathrm{bv}}}^2})}^2+{\left(0.5d_{\mathrm{bo}}\right)}^2}=97.023443807\mathrm{mm}$

${\mathrm{\&rho;}}_{\mathrm{io}}=0.5\sqrt{{(d_{\mathrm{qo}}-5\tan {\mathrm{\&alpha;}}_e)}^2-{d_{\mathrm{bo}}}^2}=11.178227234\mathrm{mm}\&ap;11.17\mathrm{mm}$

What finally illustrate is, above embodiment is only in order to illustrate technical scheme of the present invention and unrestricted, although with reference to preferred embodiment to invention has been detailed description, those of ordinary skill in the art is to be understood that, can modify to technical scheme of the present invention or equivalent replacement, and not departing from aim and the scope of technical solution of the present invention, it all should be encompassed in the middle of right of the present invention.

Claims (10)

1. a 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 above formula, m is back taper spline spur gear modulus; Described pressure angle of graduated circle α _fomeet formula in above formula, ψ is the angle i.e. slotting established angle processed between the pinion cutter direction of motion and back taper spline spur gear axial line, β _ffor the angle, reference circle rear flank of back taper spline spur gear, α _ffor the pressure angle of graduated circle of 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 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 above 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 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.

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 above 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 above 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. according to the back taper spline spur gear pinion cutter described in claim 6, it is characterized in that: also comprise for grinding pinion cutter toothed surfaces desired parameters d _boand β _bo, described d _bofor the base circle diameter (BCD) of pinion cutter toothed surfaces, meet formula d _bo=d _focos α _fo; Described β _bofor the Base spiral angle of pinion cutter toothed surfaces, meet formula β _bo=arctan (sin α _fotan α _e).

8. the method for designing of back taper spline spur gear pinion cutter according to any one of 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) slotting established angle ψ processed is obtained according to back taper spline spur gear and pinion cutter basic parameter;

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

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) equivalent parameters of back taper spline spur gear in v-v plane that draw, the principle engaged with pinion cutter with bevel gear obtains remaining geometric parameter of pinion cutter, and the pinion cutter tooth profile parameter drawn according to above-mentioned steps can process the back taper spline spur gear pinion cutter of high accuracy, high strength.

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

S11) meshingangleα between back taper spline spur gear large end face and pinion cutter is determined _o, this parameter alpha _oby formula ${\mathrm{inv\&alpha;}}_o=\frac{2{\mathrm{tan\&alpha;}}_f(\mathrm{\&xi;}+{\mathrm{\&xi;}}_o)}{Z+Zo}+{\mathrm{inv\&alpha;}}_f$ 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 _oby 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 _eoby formula d _eo=2A _o-d _icalculating is tried to achieve;

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

S141) back taper spline spur gear small end graduated arc transverse tooth thickness S is first determined _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 _othe meshingangleα between back taper spline spur gear small end face and pinion cutter can be determined _xiao, described meshingangleα _xiaoby formula ${\mathrm{inv\&alpha;}}_{xiao}=\frac{2{\mathrm{tan\&alpha;}}_f({\mathrm{\&xi;}}_{xiao}+{\mathrm{\&xi;}}_o)}{Z+Zo}+{\mathrm{inv\&alpha;}}_f$ Obtain;

S144) by step S143) meshingangleα determined _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 _xiaoby formula obtain;

S145) by step S144) centre-to-centre spacing A between the back taper spline spur gear small end face that obtains and pinion cutter _xiaodetermine back taper spline spur gear small end root diameter d _ixiao, described d _ixiaoby 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 is by 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 in V-V plane the equivalent parameters 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 _vequal with the modulus m of back taper spline spur gear, described virtual number of teeth Z _vmeet formula $Z_v=\frac{Z}{cos\mathrm{\&psi;}},$ Described equivalent outside diameter circle d _evmeet formula $d_{ev}=\frac{d_e}{cos\mathrm{\&psi;}}+\&lsqb;2B-(d_e-d_i)tan\mathrm{\&psi;}\&rsqb;\sin \mathrm{\&psi;},$ 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;

In described V-V plane, the equivalent parameters of bevel gear also comprises equivalent graduated arc transverse tooth thickness S _fv, described S _fvdetermined by following steps:

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

Y _f=0.5 (d _f-d _i) tan ψ, Y _ffor back taper spline spur gear large end face is to the distance of 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 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 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+ΔXsinψsinψ

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

Y _fv＝Y _f-ΔXsinψcosψ

D) the fv point circular tooth thickness S of Sv-Sv plane is asked _fsvdiameter d _sv:

S _fsv＝S _f-2Y _fvtanβ _f

Following formula Newton iteration method solves:

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

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

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

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

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

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

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

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