CN103206515A - Loaded tooth surface contact analysis method direct at epicycloid bevel gear errors - Google Patents

Abstract

The invention relates to a loaded tooth surface contact analysis method direct at epicycloid bevel gear errors, and belongs to the field of theory of non linear vibration. Gear pair elastic deformation, support system torsion, gear processing errors and machine tool setting errors are taken into consideration to derive a novel error loaded contact analysis method. When analyzing one loaded tooth surfaced contact, real situation can be closer. The loaded tooth surface contact analysis method is usually used for setting parameters and machine tool parameters via V-H-J (V is the longitudinal set parameter of a gear pair, and the upward direction of a small gear is active; H is a coordinate set parameter of the gear pair, and the direction facing towards the big end of the small gear is active; J is the adjusting parameter along a coordinate direction, and the direction, distant from a big gear, of the small gear is active) within a real contact area under the conditions of loading and delivering errors and simulated load distribution and processing, so that impact of the errors is compensated to improve delivering performance and correct size and position of the contact area.

Description

A kind of method that loads Tooth Contact Analysis at the overlikon spiral bevel gear error

Technical field

The present invention relates to a kind of method at overlikon spiral bevel gear error loading Tooth Contact Analysis, belong to gear nonlinear dynamics field.

Background technique

Crin Gen Beierge bevel gear (Klingelnberg Spiral Bevel Gear) is as one of two canine tooth systems of helical bevel gear, have characteristics such as stable drive, bearing capacity height, hard surface skiving technology, thereby being specially adapted to high-power and high pulling torque heavy load transmission field, is the core transmission component in the key areas such as heavy high-grade, digitally controlled machine tools, car transmissions, Aero-Space equipment.Steadily reliable in transmission process in order to guarantee gear, reduce vibration and noise, must carry out dynamic analysis to gear train assembly.Effectively setting up gear engagement mathematical model is the basis of dynamic analysis.

Domestic and international many scholars have carried out more extensive and deep research to gear teeth face contact analysis dynamics, but majority all is the Tooth Contact Analysis of considering only to comprise the lathe step-up error, and concentrate on straight toothed spur gear, and the loading flank of tooth Contact Analysis and Study of considering transmission error, shaft distortion, bearing deformation, tooth distortion, flank of tooth machining error and lathe step-up error at crin Gen Beierge bevel gear seldom.Truth can be pressed close to more when loading Tooth Contact Analysis to one when considering these sum of errors amount of deformation, and the size and location of transmitting performance and revising area of contact can be improved.

Summary of the invention

The purpose of this invention is to provide a kind of method that loads Tooth Contact Analysis at the overlikon spiral bevel gear error, this method can apply in the loading Tooth Contact Analysis of crin Gen Beierge bevel gear error, when loading Tooth Contact Analysis to one, can press close to truth more, this method can be improved the transmission performance of gear and revise the area of contact size simultaneously, also provides basis and foundation for overlikon spiral bevel gear vibration control, noise reduction and defects detection.

The present invention adopts following technological means to realize:

A kind of method at overlikon spiral bevel gear error loading Tooth Contact Analysis is characterized in that this method may further comprise the steps:

1) distortion and transmission error

1.1) transmission error under loading environment

Transmission error under the loading environment is made up of three parts: position deviation error delta E _a, the transmission error Δ E that the gear resiliently deformable causes _bThe transmission error Δ E that causes with the bending deflection of gear _c, therefore, comprehensive transmission error Δ E is

ΔE=ΔE _a+ΔE _b+ΔE _c (1)

1.2) load distribution and compatibility of deformation rule

Suppose to have the tooth of n to mesh, can obtain the load distribution relation

$\left\{\begin{array}{c}T={\mathrm{\&Sigma;}}_{i-1}^n{T}_i={\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">T</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">T</figure-callout>}}_2+...+T_n\\ T_i=0.5d_i\&CenterDot;F_i\&CenterDot;{\cos \mathrm{\&alpha;}}_i\end{array}\right.---\left(2\right)$

In the formula, T is resultant couple, T _iBe the moment of the gear pair of i, d _iBe the gear diameter of point of contact, F _iBe the normal force of point of contact, α _iLoading direction and tangential angle;

The right transmission deviation of each engaging tooth is identical, therefore, releases the gear transmission formula

$\left\{\begin{array}{c}\mathrm{\&Delta;}{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">E</figure-callout>}}_1=\mathrm{\&Delta;}{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">E</figure-callout>}}_2=...\mathrm{\&Delta;}{E}_n\\ \mathrm{\&Delta;}{E}_i=\mathrm{\&Delta;}{E}_{\mathrm{ia}}=\mathrm{\&Delta;}{E}_{\mathrm{ib}}=\mathrm{\&Delta;}{E}_{\mathrm{ic}}\end{array}\right.---\left(3\right)$

In the formula, Δ E _iIt is the transmission error of the gear pair of i;

1.3) axle distortion

The axle of gear reverses and is

The bending deflection of gear is f _Xr, f _YrAnd f _Zr, the axle relevant parameter of small gear is

f _Xl, f _Yl, and f _Zl

The supporting structure of overlikon spiral bevel gear can be reduced to an overhang or a support beam, uses Fi and Ti can calculate bending deflection and torsional deflection, and can calculate the displacement of point of contact

${\stackrel{\&RightArrow;}{f}}_{\mathrm{ai}}=(f_{\mathrm{xr}},f_{\mathrm{yr}},f_{\mathrm{zr}})---\left(4\right)$

1.4) bearing deformation

Consider supporting structure, Fi produces one

Power because the effect of axial force, the axial displacement of small gear is f _z, because effect and the bearing radial internal clearance of tangential force and radial force, be f along the displacement of the axial bearing centre of y _y, because effect and the bearing axial internal clearance of axial force and radial force, be f along the displacement of the axial bearing centre of x _x, therefore, the flank of tooth displacement that is produced by bearing deformation is

$\left\{\begin{array}{c}{\stackrel{\&RightArrow;}{f}}_{\mathrm{bi}}=(f_x,f_y,f_z)\\ f_x=\mathrm{sign}\left(R_{\mathrm{zi}}\right)({\mathrm{\&delta;}}_z+u_a)\\ f_y=\mathrm{sign}\left(R_{\mathrm{yi}}\right)({\mathrm{\&delta;}}_y+u_r/2)\cos \mathrm{\&epsiv;}\\ f_z=\mathrm{sign}\left(R_{\mathrm{xi}}\right)({\mathrm{\&delta;}}_x+u_r/2)\sin \mathrm{\&epsiv;}\end{array}\right.---\left(5\right)$

Formula (5), sign (R) are sign functions; u _a, u _rBe respectively radial internal clearance and the axial internal clearance of bearing; ε is radially opposite force and the axially angle of opposite force; δ _x, δ _y, δ _yBe axial deformation and the radial deformation that is caused by axial force and radial force effect;

1.5) the tooth distortion

1.5.1) be out of shape at the tooth at contact point of gear surface place

According to the Hertz formula, the tooth distortion at the contact point of gear surface place can expand into

${\mathrm{\&delta;}}_c={\mathrm{\&lambda;}}^3\sqrt{\frac{9}{128}{\mathrm{AF}}^2{(\frac{1-{{\mathrm{\&mu;}}_1}^2}{{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">E</figure-callout>}}_1}+\frac{1-{{\mathrm{\&mu;}}_2}^2}{E_2})}^2}---\left(6\right)$

In the formula, λ is a coefficient in the elastomechanics; E ₁, E ₂, μ ₁, μ ₂Be respectively Young's modulus and Poisson's ratio; A is the relative mean value of curvature; F is contact force;

1.5.2) tooth bending deflection

Calculate the bending deflection of tooth with the overhang method of Westinghouse, at the bending deflection value δ of contact point of gear surface _wFor

${\mathrm{\&delta;}}_w=\frac{{\mathrm{FL}}^3}{3\mathrm{EI}}{\{1+1.3(t/L)+[0.25+0.75(1-\mathrm{\&mu;})]\&CenterDot;{(t/L)}^2+0.35{(t/L)}^3\}}^{-1}---\left(7\right)$

In the formula, I is the quadrature moment of inertia in load cross section; T is the normal tooth thickness in load cross section; L is the tooth depth of load(ing) point; E is elasticity modulus of materials; μ is the material Poisson's ratio;

2) flank of tooth machining error

Consider tooth thickness error and tangential resultant error, the actual effect error is

$\left\{\begin{array}{c}{\mathrm{\&delta;}}_{\mathrm{jgr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\&CenterDot;\mathrm{\&Delta;}{F}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{u}}_r}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\\ {\mathrm{\&delta;}}_{\mathrm{kgr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\&CenterDot;{\mathrm{\&Delta;F}}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{k}}_{\mathrm{gr}}}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\\ {\mathrm{\&delta;}}_{\mathrm{igr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\&CenterDot;{\mathrm{\&Delta;E}}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{i}}_{\mathrm{gr}}}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\end{array}\right.---\left(8\right)$

In the formula,

It is the tooth thickness error of gear;

Be respectively axially, radially with tangential direction on unit an amount of; R _rThe radially radius of tooth; Δ F ' _IrIt is the tangential resultant error of gear; r ₁It is the Pitch radius of gear; k ₁Be conversion coefficient, coefficient k ₁Be to use apportionment ratio, the relative compensating effect of tooth thickness error and tangential resultant error are determined, k ₁Definition be associated with apportionment ratio and the total number of teeth in engagement of the actual tangential resultant error that meets normal distribution law;

3) lathe step-up error

The composition that lathe arranges parameter is: the error delta τ of cutter spacing setting angle τ; By cutterhead head center to carriage center M _dThe lathe that produces of distance apart from error delta M _dWith the error delta r in cutterhead head radius r;

3.1) cutter spacing sets angle τ and helixangle _mBetween relation

τ=2sin ^-1(M _d/ 2k), wherein k cutter range setting can be obtained setting the lathe of angular error generation apart from error delta M because of cutter spacing by this relation _d

$\mathrm{\&Delta;}{M}_d=k\cos \frac{\mathrm{\&tau;}}{2}\mathrm{\&Delta;\&tau;}---\left(9\right)$

Cutter spacing sets angular error

$\mathrm{\&Delta;\&tau;}={\mathrm{\&tau;}}^{\&prime;}=-\frac{R_m(r{Z}_p\cos v+{\mathrm{<figure-callout\; id="0"\; label="R\; m\; Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">R</figure-callout>}}_m{Z}_{}{}_0\sin {\mathrm{\&beta;}}_m)\cos ({\mathrm{\&beta;}}_m-v)}{k{M}_d{Z}_p\cos v}---\left(10\right)$

In the formula, cos β _m=M _dZ _p/ (2R), v=sin ^-1(0.5M _dZ ₀/ r) and R be pitch cone radius, R=R _mAnd β=β _m, lathe is apart from M _d, helixangle _m, cutterhead head radius r and pitch cone radius R; Cutterhead head number is Z ₀, the crown gear number is Z _p

3.2) cutterhead head radius r and helixangle _mBetween relation

Be generally used for adjusting the error delta r of the blade pad of eccentric cutter and the combined error definition cutting end radius that slip is counted, this error can change the Δ β in the helix angle _m, while R _mAnd M _dRemain unchanged,

$\mathrm{\&Delta;\&tau;}={\mathrm{\&tau;}}^{\&prime;}=\frac{r{R}_m(r{Z}_p\cos v+{\mathrm{<figure-callout\; id="0"\; label="R\; m\; Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">R</figure-callout>}}_m{Z}_{}{}_0\sin {\mathrm{\&beta;}}_m)\cos ({\mathrm{\&beta;}}_m-v)}{r^2{Z}_p\cos v-r{Z}_p{R}_m\sin ({\mathrm{\&beta;}}_m-v)\cos v-{\mathrm{<figure-callout\; id="0"\; label="R\; m\; Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">R</figure-callout>}}_m{Z}_{}{}_0\cos ({\mathrm{\&beta;}}_m-v)\cos {\mathrm{\&beta;}}_m}---\left(11\right)$

Following formula (11) has provided cutterhead head radius r and helixangle _mBetween relation;

3.3) relation between lathe step-up error and the area of contact position

According to tooth surface equation and Tooth Contact Analysis equation, by the importing lathe parameter error relevant with them is set and sets up a new meshing condition equation

$\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})---\left(12\right)$

In the formula, the three dimensional space distance vector of conjugation point of contact $\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-$ $\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;}),$ $\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})$ Be to be respectively the left and right vector that revolves the conjugation point of contact in the processing engagement on the member flank of tooth; Δ r is the error of cutterhead head radius.

4) flank of tooth contact error analysis

The gear teeth face contact analysis is based on the rigidity of gear pair and support system, when considering the distortion of support system, the relative position of mesh tooth face will change in the space, this means the support system distortion and establishing error by the lathe that the lathe value of setting produces is equivalent in magnitude, according to formula (4), (5), (12) can obtain

In the formula, α is the blade pressure angle; R _rAnd R _lIt is respectively the radius of gearwheel and pinion point of contact; δ _r, δ _lIt is respectively the generation angle of gearwheel and small gear; V be gear pair parameter vertically is set, and the upward direction of small gear is initiatively; H is that the coordinate axes of gear pair arranges parameter, and is initiatively towards the big extreme direction of small gear; J is the adjustment parameter along change in coordinate axis direction, and small gear is initiatively away from the direction of gearwheel.

Beneficial effect:

Characteristics of the present invention are only to have considered the lathe step-up error in the traditional loading Tooth Contact Analysis method of gear, and this method is at overlikon spiral bevel gear and consider the gear pair resiliently deformable, support system is reversed, Errors in Gear Processing and lathe step-up error have proposed a kind of new error and have loaded the contact analysis method.Can press close to truth more when being added in Tooth Contact Analysis by this method to one.The inventive method can apply to crin Gen Beierge bevel gear be used for to solve loading, transmission error, load distribute and machining simulation under real contact area by V-H-J parameter is set and lathe arranges parameter, the influence of compensating error improves the size and location of transmitting performance and revising area of contact.Summary of the invention comprises three parts.In first portion, traditional Tooth Contact Analysis formula is proposed; In the second portion, derive each sum of errors amount of deformation formula (transmission error, shaft distortion, bearing deformation, tooth distortion, flank of tooth machining error and lathe step-up error); In the third part, each sum of errors amount of deformation is introduced the loading Tooth Contact Analysis formula of deriving and making new advances in the Tooth Contact Analysis formula, proved the method that the present invention proposes by example at last.

By following description and accompanying drawings, the present invention can be more clear, and description of drawings is used for explaining the inventive method and embodiment.

Description of drawings

Fig. 1 summary flow chart of the present invention;

Fig. 2 coordinate system figure that rolls;

Fig. 3 gear tooth thickness error schematic diagram;

Fig. 4 lathe is apart from schematic diagram;

Fig. 5 cutter spacing sets angle τ and helixangle _mBetween relation principle figure;

Fig. 6 cutterhead head radius r and helixangle _mBetween relation principle figure;

The routing routing diagram of Fig. 7 error load Tooth Contact Analysis;

Fig. 8 is at the following transmission error curve of loading environment.

Embodiment

Shown in Fig. 1-8, a kind of method at overlikon spiral bevel gear error loading Tooth Contact Analysis of the present invention, its concrete steps are as follows:

1 Tooth Contact Analysis

The three dimensional space distance vector of conjugation point of contact

$\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}=-H\stackrel{\&RightArrow;}{p_l}+J\stackrel{\&RightArrow;}{p_r}+V\stackrel{\&RightArrow;}{e_l}---\left(1\right)$

In the formula, V be gear pair parameter vertically is set, and the upward direction of small gear is initiatively.H is that the coordinate axes of gear pair arranges parameter, and is initiatively towards the big extreme direction of small gear.J is the adjustment parameter along change in coordinate axis direction, and small gear is initiatively away from the direction of gearwheel, With

It is the unit vector in the three dimensional space.

$\left\{\begin{array}{c}H=\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}\&CenterDot;(\stackrel{\&RightArrow;}{e_l}\&times;\stackrel{\&RightArrow;}{p_r})\\ J=\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}\&CenterDot;({\stackrel{\&RightArrow;}{e}}_l\&times;\stackrel{\&RightArrow;}{p_l})\\ V=-\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}\&CenterDot;\stackrel{\&RightArrow;}{e_l}\end{array}\right.---\left(2\right)$

Formula (1) and formula (2) have provided the Tooth Contact Analysis formula.

2 distortion and transmission errors

The distortion of system has comprised the resiliently deformable of gear, the distortion of gear shaft and the resiliently deformable of spring bearing.

2.1 the transmission error under loading environment

Transmission error under the loading environment is made up of three parts: position deviation error delta E _a, the transmission error Δ E that the gear resiliently deformable causes _bThe transmission error Δ E that causes with the bending deflection of gear _cTherefore, comprehensive transmission error Δ E is

ΔE=ΔE _a+ΔE _b+ΔE _c (3)

2.2 load distributes and the compatibility of deformation rule

Suppose to have the tooth of n to mesh, can obtain the load distribution relation

$\left\{\begin{array}{c}T={\mathrm{\&Sigma;}}_{i=1}^n{T}_i={\mathrm{<figure-callout\; id="1"\; label="T"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">T</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="T"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">T</figure-callout>}}_2+...+T_n\\ T_i={0.5d}_i\&CenterDot;F_i\&CenterDot;{\cos \mathrm{\&alpha;}}_i\end{array}\right.---\left(4\right)$

In the formula, T is resultant couple, T _iBe the moment of the gear pair of i, d _iBe the gear diameter of point of contact, F _iBe the normal force of point of contact, α _iLoading direction and tangential angle.

The right transmission deviation of each engaging tooth is identical.Therefore, release the gear transmission formula

$\left\{\begin{array}{c}\mathrm{\&Delta;}{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">E</figure-callout>}}_1=\mathrm{\&Delta;}{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">E</figure-callout>}}_2=...=\mathrm{\&Delta;}{E}_n\\ \mathrm{\&Delta;}{E}_i=\mathrm{\&Delta;}{E}_{\mathrm{ia}}=\mathrm{\&Delta;}{E}_{\mathrm{ib}}=\mathrm{\&Delta;}{E}_{\mathrm{ic}}\end{array}\right.---\left(5\right)$

In the formula, Δ E _iIt is the transmission error of the gear pair of i.

2.3 the distortion of axle

Fig. 2 has shown at spur roller gear system of coordinates O _r(Ol) point of intersection of system axis: O _BrBe the center of gear compound graduation circle,

Be that point arrives point of contact p,

Point is to the big end of gear, and the axle of gear reverses and is

The bending deflection of gear is f _Xr, f _YrAnd f _ZrThe reversing with the bending deflection parameter of axle of small gear is similar to gearwheel, and relevant parameter is

f _Xl, f _Yl, and f _Zl

The supporting structure of overlikon spiral bevel gear can be reduced to an overhang or a support beam.Use F _iAnd T _iCan calculate bending deflection and torsional deflection.Next step can calculate the displacement of point of contact

${\stackrel{\&RightArrow;}{f}}_{\mathrm{ai}}=(f_{\mathrm{xr}},f_{\mathrm{yr}},f_{\mathrm{zr}})---\left(6\right)$

2.4 bearing deformation

Consider supporting structure, F _iProduce one

Power.Because the effect of axial force, the axial displacement of small gear is f _zBecause effect and the bearing radial internal clearance of tangential force and radial force, be f along the displacement of the axial bearing centre of y _yBecause effect and the bearing axial internal clearance of axial force and radial force, be f along the displacement of the axial bearing centre of x _xTherefore, the flank of tooth displacement that is produced by bearing deformation is

$\left\{\begin{array}{c}{\stackrel{\&RightArrow;}{f}}_{\mathrm{bi}}=(f_x,f_y,f_z)\\ f_x=\mathrm{sign}\left(R_{\mathrm{zi}}\right)({\mathrm{\&delta;}}_z+u_a)\\ f_y=\mathrm{sign}\left(R_{\mathrm{yi}}\right)({\mathrm{\&delta;}}_y+u_r/2)\cos \mathrm{\&epsiv;}\\ f_z=\mathrm{sign}\left(R_{\mathrm{xi}}\right)({\mathrm{\&delta;}}_x+u_r/2)\sin \mathrm{\&epsiv;}\end{array}\right.---\left(7\right)$

Formula (7), sign (R) are sign functions; u _a, u _rBe respectively radial internal clearance and the axial internal clearance of bearing; ε is radially opposite force and the axially angle of opposite force; δ _x, δ _y, δ _yBe axial deformation and the radial deformation that is caused by axial force and radial force effect.

2.5 tooth distortion

1) tooth at the contact point of gear surface place is out of shape

According to the Hertz formula, the tooth distortion at the contact point of gear surface place can expand into

${\mathrm{\&delta;}}_c={\mathrm{\&lambda;}}^3\sqrt{\frac{9}{128}{\mathrm{AF}}^2{(\frac{1-{{\mathrm{\&mu;}}_1}^2}{{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">E</figure-callout>}}_1}+\frac{1-{{\mathrm{\&mu;}}_2}^2}{E_2})}^2}---\left(8\right)$

In the formula, λ is a coefficient in the elastomechanics; E ₁, E ₂, μ ₁, μ ₂Be respectively Young's modulus and Poisson's ratio; A is the relative mean value of curvature; F is contact force.

2) tooth bending deflection

Calculate the bending deflection of tooth with the overhang method of Westinghouse.Bending deflection value δ in contact point of gear surface _wFor

${\mathrm{\&delta;}}_w=\frac{{\mathrm{FL}}^3}{3\mathrm{EI}}{\{1+1.3(t/L)+[0.25+0.75(1-\mathrm{\&mu;})]\&CenterDot;{(t/L)}^2+0.35{(t/L)}^3\}}^{-1}---\left(9\right)$

In the formula, I is the quadrature moment of inertia in load cross section; T is the normal tooth thickness in load cross section; L is the tooth depth of load(ing) point; E is elasticity modulus of materials; μ is the material Poisson's ratio.

3 flank of tooth machining errors

In gear coordinate system (Fig. 3), can obtain the transverse tooth thickness in a method phase cross section by the vector conversion of contact points.Transverse tooth thickness

For

${\stackrel{\&RightArrow;}{S}}_r={\stackrel{\&RightArrow;}{R}}_{\mathrm{br}}-{\stackrel{\&RightArrow;}{R}}_{\mathrm{br}}^{\&prime;}---\left(10\right)$

In the formula,

With

Radial vector for gear.Represent that with vector symbol tooth thickness error is

${\stackrel{\&RightArrow;}{T}}_{\mathrm{<figure-callout\; id="0"\; label="gr"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">gr</figure-callout>}0}={\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\frac{{\stackrel{\&RightArrow;}{S}}_r}{\left|{\stackrel{\&RightArrow;}{S}}_r\right|}---\left(11\right)$

In the formula, It is the tooth thickness error of gear.From the gear tooth thickness error of the rotational coordinates system of Fig. 6 .1, can obtain

${\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}=M{(-{\mathrm{\&delta;}}_m)}_j\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}0}---\left(12\right)$

In the formula, δ _mIt is working pressure angle; Tooth thickness error

Axially, radially with tangential direction on influence the gap is set, also can be at V, H produces the variable of adjusted value on the J direction.Therefore, can be the error of gap variable as an actual effect

$\left\{\begin{array}{c}{\mathrm{\&delta;}}_{\mathrm{jgr}}^{\&prime;}={\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{u}}_r\\ {\mathrm{\&delta;}}_{\mathrm{kgr}}^{\&prime;}={\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{k}}_{\mathrm{gr}}\\ {\mathrm{\&delta;}}_{\mathrm{igr}}^{\&prime;}={\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{i}}_{\mathrm{gr}}\end{array}\right.---\left(13\right)$

In the formula,

Be respectively axially, radially with tangential direction on unit an amount of.Consider tooth thickness error and tangential resultant error, the actual effect error is

$\left\{\begin{array}{c}{\mathrm{\&delta;}}_{\mathrm{jgr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\&CenterDot;\mathrm{\&Delta;}{F}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{u}}_r}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\\ {\mathrm{\&delta;}}_{\mathrm{kgr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\&CenterDot;{\mathrm{\&Delta;F}}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{k}}_{\mathrm{gr}}}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\\ {\mathrm{\&delta;}}_{\mathrm{igr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">k</figure-callout>}}_1\&CenterDot;{\mathrm{\&Delta;E}}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{i}}_{\mathrm{gr}}}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\end{array}\right.---\left(14\right)$

In the formula, Δ F ' _IrIt is the tangential resultant error of gear; r ₁It is the Pitch radius of gear; k ₁It is conversion coefficient.Coefficient k ₁Be to use apportionment ratio, the relative compensating effect of tooth thickness error and tangential resultant error are determined.k ₁Definition be associated with apportionment ratio and the total number of teeth in engagement of actual tangential resultant error (meeting normal distribution law usually).

4 lathe step-up errors

The composition that lathe arranges parameter error is: the error delta τ of cutter spacing setting angle τ; By cutterhead head center to carriage center M _dThe lathe that produces of distance apart from error delta M _dWith the error delta r in cutterhead head radius r.

4.1 cutter M _dAnd concern β between the helix angle _m

Fig. 4 can obtain lathe apart from M _d, helixangle _m, cutterhead head radius r, and the relation between the pitch cone radius R.

${\mathrm{<figure-callout\; id="2"\; label="M\; d"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">M</figure-callout>}}_d^{}={\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">R</figure-callout>}}^2+r^2-2R\&CenterDot;r\sin ({\mathrm{\&beta;}}_m-v)---\left(15\right)$

$\sin v=\frac{{\mathrm{<figure-callout\; id="0"\; label="M\; d\; Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">M</figure-callout>}}_d{Z}_{}{}_0}{2r}=\frac{{\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0\cos {\mathrm{\&beta;}}_m}{Z_{p}r}---\left(16\right)$

In the formula, cos β _m=M _dZ _p/ (2R), v=sin ^-1(0.5M _dZ ₀/ r) and R be pitch cone radius.Solved M _dDifferential, R=R on a M _mAnd β=β _m, obtain

M _dM _d′=R _mR′+rr′-R′rsin(β _m-v)-R _mr′sin(β _m-v)-R _mr(1-v′)cos(β _m-v) (17)

$v^{\&prime;}=\frac{Z_0[r(R^{\&prime;}\cos {\mathrm{\&beta;}}_m-R_m\sin {\mathrm{\&beta;}}_m)-R_m{r}^{\&prime;}\cos {\mathrm{\&beta;}}_m]}{{\mathrm{<figure-callout\; id="2"\; label="Z\; p\; r"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">Z</figure-callout>}}_p{r}^{}{}^2\cos v}---\left(18\right)$

4.2 cutter spacing sets angle τ and helixangle _mBetween relation

τ=2sin ^-1(M _d/ 2k), wherein k is the cutter range setting, can be obtained setting the lathe of angular error generation apart from error delta M because of cutter spacing by this relation _d

$\mathrm{\&Delta;}{M}_d=k\cos \frac{\mathrm{\&tau;}}{2}\mathrm{\&Delta;\&tau;}---\left(19\right)$

Lathe is Δ M apart from error _dThe time, Fig. 5 has shown that the helix angle of tooth trace changes.But, R _mWith r be constant, therefore

M _dM _d′=-R _mr(1-v′)cos(β _m-v) (20)

$v^{\&prime;}=\frac{-Z_0{R}_m\sin {\mathrm{\&beta;}}_m}{Z_{p}r\cos v}{\mathrm{\&beta;}}^{\&prime;}---\left(21\right)$

Three formula (19) (20) (21) according to the front can calculate cutter spacing and set angular error

$\mathrm{\&Delta;\&tau;}={\mathrm{\&tau;}}^{\&prime;}=-\frac{R_m(r{Z}_p\cos v+R_m{\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0\sin {\mathrm{\&beta;}}_m)\cos ({\mathrm{\&beta;}}_m-v)}{k{M}_d{Z}_p\cos v}---\left(22\right)$

In the formula, cutterhead head number is Z ₀, the crown gear number is Z _pAccording to top formula (22), the relation between central point and the helix angle sets angular error Δ τ definition by cutter spacing.

4.3 cutterhead head radius r and helixangle _mBetween relation

Be generally used for adjusting the error delta r of the blade pad of eccentric cutter and the combined error definition cutting end radius that slip is counted.This error can change the Δ β in the helix angle _m, as shown in Figure 6, while R _mAnd M _dRemain unchanged.

Got by formula (17) and formula (18)

$v^{\&prime;}=-\frac{Z_0{R}_m\left(r\sin {\mathrm{\&beta;}}_m+r^{\&prime;}\cos {\mathrm{\&beta;}}_m\right)}{{\mathrm{<figure-callout\; id="2"\; label="Z\; p\; r"\; filenames="HDA00003071680400042.png"\; state="\{\{state\}\}">Z</figure-callout>}}_p{r}^{}{}^2\cos v}---\left(23\right)$

r′[r-R _msin(β _m-v)]=R _mr(1-v′)cos(β _m-v) (24)

Provide simultaneously

$\mathrm{\&Delta;\&tau;}={\mathrm{\&tau;}}^{\&prime;}=\frac{r{R}_m(r{Z}_p\cos v+R_m{\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0\sin {\mathrm{\&beta;}}_m)\cos ({\mathrm{\&beta;}}_m-v)}{r^2{Z}_p\cos v-r{Z}_p{R}_m\sin ({\mathrm{\&beta;}}_m-v)\cos v-{\mathrm{<figure-callout\; id="0"\; label="R\; m\; Z"\; filenames="HDA00003071680400023.png"\; state="\{\{state\}\}">R</figure-callout>}}_m{Z}_{}{}_0\cos ({\mathrm{\&beta;}}_m-v)\cos {\mathrm{\&beta;}}_m}---\left(25\right)$

Following formula (25) has provided cutterhead head radius Δ r and helixangle _mBetween relation.

4.4 the relation between lathe step-up error and the area of contact position

According to tooth surface equation and Tooth Contact Analysis equation, this paper arranges the parameter error relevant with them by the lathe that imports and sets up a new meshing condition equation

$\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})---\left(26\right)$

In the formula, the three dimensional space distance vector of conjugation point of contact $\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;}),$

It is respectively the left and right vector that revolves the conjugation point of contact in the processing engagement on the member flank of tooth; Δ r is the error of cutterhead head radius.

5 flank of tooth contact error analysis (V-H-J Equivalent Calculation formula under the load effect)

The gear teeth face contact analysis is based on the rigidity of gear pair and support system.When considering the distortion of support system, the relative position of mesh tooth face will change in the space.This means the support system distortion and establishing error by the lathe that the lathe value of setting produces is equivalent in magnitude.According to formula (6), (7), (26) can obtain

In the formula, α is the blade pressure angle; R _rAnd R _lIt is respectively the radius of gearwheel and pinion point of contact; δ _r, δ _lIt is respectively the generation angle of gearwheel and small gear.

The example of 6 error load Tooth Contact Analysis

Herein, we analyze a gear pair that processes with spiral AMK852 milling machine.Gear pair has method phase modulus 3.510,35 ° of helix angles, 20 ° of pressure angles, facewidth 20.00mm.Small gear and gearwheel are respectively left-handed and dextrorotation.Small gear has 21 teeth, and gearwheel has 26 teeth.We calculate a series of parameter according to the routing routing diagram of Fig. 7 in test.

Fig. 8 has provided the transmission error curve under loading environment, and line I, II and III have represented the transmission error of engagement beginning, transmission error in the engagement, the transmission error that engagement finishes respectively.The transmission error curve of the transmission error curve of Jia Zaiing and loading is not included on the comprehensive transmission error curve.From simulation result, it is more stable along with the increase of load and drive ratio to find to transmit performance.

Sum up by above instance analysis: this method is reasonably, effectively to the transmission characteristics that improves overlikon spiral bevel gear.The inventive method can apply to crin Gen Beierge bevel gear be used for to solve loading, transmission error, load distribute and machining simulation under real contact area by V-H-J parameter is set and lathe arranges parameter, the influence of compensating error improves the size and location of transmitting performance and revising area of contact.This method can be pressed close to truth more when loading Tooth Contact Analysis to one.

Claims (1)

1. one kind loads the method for Tooth Contact Analysis at the overlikon spiral bevel gear error, it is characterized in that this method may further comprise the steps:

1) distortion and transmission error

1.1) transmission error under loading environment

Transmission error under the loading environment is made up of three parts: position deviation error delta E _a, the transmission error Δ E that the gear resiliently deformable causes _bThe transmission error Δ E that causes with the bending deflection of gear _c, therefore, comprehensive transmission error Δ E is

ΔE=ΔE _a+ΔE _b+ΔE _c (1)

1.2) load distribution and compatibility of deformation rule

Suppose to have the tooth of n to mesh, can obtain the load distribution relation

$\left\{\begin{array}{c}T={\mathrm{\&Sigma;}}_{i-1}^n{T}_i=T_1+T_2+...+T_n\\ T_i=0.5d_i\&CenterDot;F_i\&CenterDot;{\cos \mathrm{\&alpha;}}_i\end{array}\right.---\left(2\right)$

In the formula, T is resultant couple, T _iBe the moment of the gear pair of i, d _iBe the gear diameter of point of contact, F _iBe the normal force of point of contact, α _iLoading direction and tangential angle;

The right transmission deviation of each engaging tooth is identical, therefore, releases the gear transmission formula

$\left\{\begin{array}{c}\mathrm{\&Delta;}{E}_1=\mathrm{\&Delta;}{E}_2=...\mathrm{\&Delta;}{E}_n\\ \mathrm{\&Delta;}{E}_i=\mathrm{\&Delta;}{E}_{\mathrm{ia}}=\mathrm{\&Delta;}{E}_{\mathrm{ib}}=\mathrm{\&Delta;}{E}_{\mathrm{ic}}\end{array}\right.---\left(3\right)$

In the formula, Δ E _iIt is the transmission error of the gear pair of i;

1.3) axle distortion

The axle of gear reverses and is

The bending deflection of gear is f _Xr, f _YrAnd f _Zr, the axle relevant parameter of small gear is

f _Xl, f _Yl, and f _Zl

The supporting structure of overlikon spiral bevel gear can be reduced to an overhang or a support beam, uses Fi and Ti can calculate bending deflection and torsional deflection, and can calculate the displacement of point of contact

${\stackrel{\&RightArrow;}{f}}_{\mathrm{ai}}=(f_{\mathrm{xr}},f_{\mathrm{yr}},f_{\mathrm{zr}})---\left(4\right)$

1.4) bearing deformation

Consider supporting structure, Fi produces one

Power because the effect of axial force, the axial displacement of small gear is f _z, because effect and the bearing radial internal clearance of tangential force and radial force, be f along the displacement of the axial bearing centre of y _y, because effect and the bearing axial internal clearance of axial force and radial force, be f along the displacement of the axial bearing centre of x _x, therefore, the flank of tooth displacement that is produced by bearing deformation is

$\left\{\begin{array}{c}{\stackrel{\&RightArrow;}{f}}_{\mathrm{bi}}=(f_x,f_y,f_z)\\ f_x=\mathrm{sign}\left(R_{\mathrm{zi}}\right)({\mathrm{\&delta;}}_z+u_a)\\ f_y=\mathrm{sign}\left(R_{\mathrm{yi}}\right)({\mathrm{\&delta;}}_y+u_r/2)\cos \mathrm{\&epsiv;}\\ f_z=\mathrm{sign}\left(R_{\mathrm{xi}}\right)({\mathrm{\&delta;}}_x+u_r/2)\sin \mathrm{\&epsiv;}\end{array}\right.---\left(5\right)$

Formula (5), sign (R) are sign functions; u _a, u _rBe respectively radial internal clearance and the axial internal clearance of bearing; ε is radially opposite force and the axially angle of opposite force; δ _x, δ _y, δ _yBe axial deformation and the radial deformation that is caused by axial force and radial force effect;

1.5) the tooth distortion

1.5.1) be out of shape at the tooth at contact point of gear surface place

According to the Hertz formula, the tooth distortion at the contact point of gear surface place can expand into

${\mathrm{\&delta;}}_c={\mathrm{\&lambda;}}^3\sqrt{\frac{9}{128}{\mathrm{AF}}^2{(\frac{1-{{\mathrm{\&mu;}}_1}^2}{E_1}+\frac{1-{{\mathrm{\&mu;}}_2}^2}{E_2})}^2}---\left(6\right)$

In the formula, λ is a coefficient in the elastomechanics; E ₁, E ₂, μ ₁, μ ₂Be respectively Young's modulus and Poisson's ratio; A is the relative mean value of curvature; F is contact force;

1.5.2) tooth bending deflection

Calculate the bending deflection of tooth with the overhang method of Westinghouse, at the bending deflection value δ of contact point of gear surface _wFor

${\mathrm{\&delta;}}_w=\frac{{\mathrm{FL}}^3}{3\mathrm{EI}}{\{1+1.3(t/L)+[0.25+0.75(1-\mathrm{\&mu;})]\&CenterDot;{(t/L)}^2+0.35{(t/L)}^3\}}^{-1}---\left(7\right)$

In the formula, I is the quadrature moment of inertia in load cross section; T is the normal tooth thickness in load cross section; L is the tooth depth of load(ing) point; E is elasticity modulus of materials; μ is the material Poisson's ratio;

2) flank of tooth machining error

Consider tooth thickness error and tangential resultant error, the actual effect error is

$\left\{\begin{array}{c}{\mathrm{\&delta;}}_{\mathrm{jgr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+k_1\&CenterDot;\mathrm{\&Delta;}{F}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{u}}_r}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\\ {\mathrm{\&delta;}}_{\mathrm{kgr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+k_1\&CenterDot;{\mathrm{\&Delta;F}}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{k}}_{\mathrm{gr}}}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\\ {\mathrm{\&delta;}}_{\mathrm{igr}}^{\&prime;}=\frac{({\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}+k_1\&CenterDot;{\mathrm{\&Delta;E}}_{\mathrm{ir}}^{\&prime;}\&CenterDot;R_r/r_1)\&CenterDot;{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}\&CenterDot;{\stackrel{\&RightArrow;}{i}}_{\mathrm{gr}}}{{\stackrel{\&RightArrow;}{T}}_{\mathrm{gr}}}\end{array}\right.---\left(8\right)$

In the formula,

It is the tooth thickness error of gear;

Be respectively axially, radially with tangential direction on unit an amount of; R _rThe radially radius of tooth; Δ F ' _IrIt is the tangential resultant error of gear; r ₁It is the Pitch radius of gear; k ₁Be conversion coefficient, coefficient k ₁Be to use apportionment ratio, the relative compensating effect of tooth thickness error and tangential resultant error are determined, k ₁Definition be associated with apportionment ratio and the total number of teeth in engagement of the actual tangential resultant error that meets normal distribution law;

3) lathe step-up error

The composition that lathe arranges parameter is: the error delta τ of cutter spacing setting angle τ; By cutterhead head center to carriage center M _dThe lathe that produces of distance apart from error delta M _dWith the error delta r in cutterhead head radius r;

3.1) cutter spacing sets angle τ and helixangle _mBetween relation

τ=2sin ^-1(M _d/ 2k), wherein k is the cutter range setting, can be obtained setting the lathe of angular error generation apart from error delta M because of cutter spacing by this relation _d

$\mathrm{\&Delta;}{M}_d=k\cos \frac{\mathrm{\&tau;}}{2}\mathrm{\&Delta;\&tau;}---\left(9\right)$

Cutter spacing sets angular error

$\mathrm{\&Delta;\&tau;}={\mathrm{\&tau;}}^{\&prime;}=-\frac{R_m(r{Z}_p\cos v+R_m{Z}_0\sin {\mathrm{\&beta;}}_m)\cos ({\mathrm{\&beta;}}_m-v)}{k{M}_d{Z}_p\cos v}---\left(10\right)$

In the formula, cos β _m=M _dZ _p/ (2R), v=sin ^-1(0.5M _dZ ₀/ r) and R be pitch cone radius, R=R _mAnd β=β _m, lathe is apart from M _d, helixangle _m, cutterhead head radius r and pitch cone radius radius R; Cutterhead head number is Z ₀, the crown gear number is Z _p

3.2) cutterhead head radius r and helixangle _mBetween relation

Be generally used for adjusting the error delta r of the blade pad of eccentric cutter and the combined error definition cutting end radius that slip is counted, this error can change the Δ β in the helix angle _m, while R _mAnd M _dRemain unchanged,

$\mathrm{\&Delta;\&tau;}={\mathrm{\&tau;}}^{\&prime;}=\frac{r{R}_m(r{Z}_p\cos v+R_m{Z}_0\sin {\mathrm{\&beta;}}_m)\cos ({\mathrm{\&beta;}}_m-v)}{r^2{Z}_p\cos v-r{Z}_p{R}_m\sin ({\mathrm{\&beta;}}_m-v)\cos v-R_m{Z}_0\cos ({\mathrm{\&beta;}}_m-v)\cos {\mathrm{\&beta;}}_m}---\left(11\right)$

Following formula (11) has provided cutterhead head radius Δ r and helixangle _mBetween relation;

3.3) relation between lathe step-up error and the area of contact position

According to tooth surface equation and Tooth Contact Analysis equation, by the importing lathe parameter error relevant with them is set and sets up a new meshing condition equation

$\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})---\left(12\right)$

In the formula, the three dimensional space distance vector of conjugation point of contact $\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-$ $\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;}),$ $\mathrm{\&Delta;}\stackrel{\&RightArrow;}{R_d}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})=\stackrel{\&RightArrow;}{R_{\mathrm{br}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})-\stackrel{\&RightArrow;}{R_{\mathrm{bl}}^p}(\mathrm{\&Delta;r},\mathrm{\&Delta;\&tau;})$ It is respectively the left and right vector that revolves the conjugation point of contact in the processing engagement on the member flank of tooth; Δ r is the error of cutterhead head radius.

4) flank of tooth contact error analysis

The gear teeth face contact analysis is based on the rigidity of gear pair and support system, when considering the distortion of support system, the relative position of mesh tooth face will change in the space, this means the support system distortion and establishing error by the lathe that the lathe value of setting produces is equivalent in magnitude, according to formula (4), (5), (12) can obtain

In the formula, α is the blade pressure angle; R _rAnd R _lIt is respectively the radius of gearwheel and pinion point of contact; δ _r, δ _lIt is respectively the generation angle of gearwheel and small gear; V be gear pair parameter vertically is set, and the upward direction of small gear is initiatively; H is that the coordinate axes of gear pair arranges parameter, and is initiatively towards the big extreme direction of small gear; J is the adjustment parameter along change in coordinate axis direction, and small gear is initiatively away from the direction of gearwheel.

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