CN106372321A - Cycloidal gear tooth curve variable step size discretization method - Google Patents

Abstract

The invention discloses a cycloidal gear tooth curve variable step size discretization method, which comprises the following steps of S1, setting a cycloidal gear tooth curve parameter; S2, deriving out a cycloidal gear tooth curve equation; S3, deriving out a formula of the curvature radius Rho of the practical tooth curve of a cycloidal gear; S4, deriving out an equation of phi relevant to a pressure angle alpha; S5, deriving out an inflexion equation of the cycloidal gear tooth curve; S6, giving the curve linear interpolation initial step size discretization method. The discretization errors and given errors are compared for automatically regulating the closing step size, so that the maximum step size smaller than the closing errors can be obtained; on the premise of meeting the precision requirements, the number of cutting feeding times is reduced; the processing efficiency is improved. The cycloidal gear tooth curve variable step size discretization method provided by the invention has the advantage that the good integral index is realized in the aspects of closing precision, discretization efficiency and calculation speed.

Description

A kind of step Discrete method of cycloid gear tooth curve

Technical field

The present invention relates to a kind of discrete method of cycloidal curve, more particularly, to a kind of variable step of cycloid gear tooth curve Discrete method.

Background technology

Rv decelerator is one of critical component of industrial robot.Robot rv decelerator has small volume, lightweight, biography Dynamic steadily, no impact, noiselessness, kinematic accuracy height, the advantages of gear ratio is big, bearing capacity is high, be widely used in electronics, space flight The industries such as aviation, robot.The complex structure of robot rv decelerator, including pin wheel housing, planetary wheel carrier, three crank axles, pendulum Line wheel and be arranged on three pairs of taper needle bearings between crank axle and planetary wheel carrier, be arranged on eccentric and pendulum on crank axle Three pairs of bearings between line wheel, two pairs of bearings being arranged between pin wheel housing and planetary wheel carrier, the standard of robot motion to be realized Really and meet its service life, the requirement on machining accuracy of these parts and installation accuracy require very high.

Cycloid gear as the strength member of high accuracy rv decelerator, return to decelerator rotating switching by its profile error Journey error and transmission accuracy all have a major impact.Two-dimensional curve is discrete to refer to that finding one group of end to end straightway replaces former song Line, ensures that approximate error is less than specification error simultaneously.The approach method of general curve has equidistant method, unique step method and equal error Method etc..Equidistantly method is high in machining efficiency, but error of fitting is larger；Unique step method is in the little part interpolation efficiency of curvature of curve relatively Low；Equal error method approximation accuracy is high but algorithm is complicated, and calculating speed is slow.

In sum it is necessary to provide that a kind of approximation accuracy is high, discrete efficiency high and the fast cycloid tooth gear teeth of calculating speed Deformation step-length discrete method.

Content of the invention

The purpose of the present invention is for the deficiencies in the prior art, provides that a kind of approximation accuracy is high, discrete efficiency high and calculating The step Discrete method of fireballing cycloid gear tooth curve.

The present invention is achieved by the following technical solutions:

The invention discloses a kind of step Discrete method of cycloid gear tooth curve, comprise the steps:

S1. cycloid gear profile's factories are set: pin tooth is distributed radius of circle as r_z, pin tooth radius be r_z, eccentric throw be a, The cycloid gear number of teeth is z_a, needle pin number be z_b, curtate ratio is k₁, k₁=az_b/r_z；

S2. derive that cycloid gear tooth profile curve equation is as follows:

$\left\{\begin{array}{c}x=r_z\sin \mathrm{\ψ}-{\mathrm{asinz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-\sin \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ y=r_z\cos \mathrm{\ψ}-{\mathrm{acosz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{cosz}}_b\mathrm{\ψ}-\cos \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right.$

Wherein, ψ is the angle that round as a ball center turns over around basic circle center；

S3. derive that the formula of radius of curvature ρ of cycloid gear actual tooth profile curve is as follows:

$\mathrm{\ρ}={\mathrm{\ρ}}_0+r_z=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_a\mathrm{\ψ}-(1+z_b{k}_1^2)}+r_z$

Wherein ρ₀Radius of curvature for cycloid gear theoretic profile curve；

S4. derive that ψ is as follows with regard to the equation of pressure angle α:

$\mathrm{\α}=arccos\frac{k_1{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}$

S5. cycloid gear tooth curve flex point equation is gone out according to equation inference in s2 as follows:

$\frac{d^{2}y}{{\mathrm{dx}}^2}=-\frac{\begin{array}{c}\left({\mathrm{az}}_b\sin \right(z_b\mathrm{\ψ})-r_z\sin (\mathrm{\ψ})+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\sin \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\sin \left(z_b\mathrm{\ψ}\right)+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b^2\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{\mathrm{sinz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\\ +\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\cos \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\cos \left(z_b\mathrm{\ψ}\right)+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b^2\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{\mathrm{cosz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\end{array}}{{\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})-r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})}^3}$

S6. provide the discrete method of curve linear interpolation initial step length, specifically include following steps:

(1) flex point of ψ is determined according to above-mentioned steps s4, s5；

(2) determine initial discrete point step size α₀

Set α₀For initial discrete point step size, η is advance constant, and λ is to retreat coefficient, ε₀For the approximate error allowing, ε ' is Actual approximate error, ε is close approximation error, sets up close approximation error ε equation as follows:

$\mathrm{\ϵ}=\sqrt{\begin{array}{c}{(\frac{a\sin \left(z_b{\mathrm{\α}}_0\right)}{2}-\sin (\frac{z_b{\mathrm{\α}}_0}{2})-r_z\frac{\sin \left({\mathrm{\α}}_0\right)}{2}+r_z\sin (\frac{{\mathrm{\α}}_0}{2})-\frac{r_z}{2}\frac{\sin \left({\mathrm{\α}}_0\right)-k_1\sin \left(z_b{\mathrm{\α}}_0\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0+1}}-r_z\frac{\sin \left(\frac{{\mathrm{\α}}_0}{2}\right)-k_1\sin \left(\frac{z_b{\mathrm{\α}}_0}{2}\right)}{\sqrt{k_1^2-2k_1\cos \frac{z_a{\mathrm{\α}}_0}{2}+1)}})}^2+\\ {(\frac{a}{2}-\frac{r_z}{2}+\frac{r_z}{2}+\frac{a\cos \left(z_b{\mathrm{\α}}_0\right)}{2}-a\cos (\frac{z_b{\mathrm{\α}}_0}{2})-\frac{r_z}{2}\cos ({\mathrm{\α}}_0)+r_z\cos (\frac{{\mathrm{\α}}_0}{2})+\frac{r_z}{2}\frac{\cos \left({\mathrm{\α}}_0\right)-k_1\cos \left(z_b{\mathrm{\α}}_0\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0+1}}-r_z\frac{\cos \left(\frac{{\mathrm{\α}}_0}{2}\right)-k_1\cos \left(\frac{z_b{\mathrm{\α}}_0}{2}\right)}{\sqrt{k_1^2-2k_1\cos \frac{z_a{\mathrm{\α}}_0}{2}+1)}})}^2\end{array}}$

(3) provide the step Discrete method of cycloid gear profile of tooth

Set α₁=α₀+α₀, using (x (α on Equation for Calculating cycloid gear tooth curve in step (2)₀),y(α₀))、(x(2 α₀),y(2α₀)) close approximation error ε between 2 points；

IfThen export α₁Parameter discrete point；IfBy initial discrete point step size α₀It is multiplied by advance system Number η recalculates close approximation error ε；IfBy initial discrete point step size α₀It is multiplied by retrogressing coefficient lambda to recalculate approximately Approximate error ε；Judge α₁Whether arrival curve terminal, does not reach then repetition above step；Otherwise, terminate to calculate close approximation by mistake Difference ε.

Further, in step s2, cycloid gear tooth profile curve equation derivation is as follows:

Assume that basic circle maintains static, take the center o of basic circle_aFor the initial point of x-y rectangular coordinate, when round as a ball r on basic circle r from When one point is rolled into another, its center o is around the center o of basic circle_aThe angle turning over is ψ, and the absolute corner of round as a ball r is θ_a, rolling The relative rotation of circle r is θ_b, then on theoretic profile any point m coordinate (x₀, y₀) it is:

$\left\{\begin{array}{c}{x}_0=r_z\sin \mathrm{\ψ}-a\sin {\mathrm{\θ}}_a\\ y_0=r_z\cos \mathrm{\ψ}-{\mathrm{acos\θ}}_a\end{array}\right.$

When round as a ball r rolls across a tooth around basic circle rAnd round as a ball r is when relatively turning over a full circle, θ_b=2 π, so θ_b= z_aψ；And θ_a=θ_b+ ψ=z_bThe common normal of ψ, m point and the angle of x-axis are γ, then γ meets equation below:

$\left\{\begin{array}{c}cos\mathrm{\γ}=\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-\sin \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ \sin \mathrm{\γ}=\frac{-k_1{\mathrm{cosz}}_b\mathrm{\ψ}+\cos \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right.$

Corresponding k point coordinates (x, y) in practical tooth can be obtained, that is, cycloid gear tooth profile curve equation is as follows:

$\left\{\begin{array}{c}x=x_0+r_z\cos \mathrm{\γ}=r_z\sin \mathrm{\ψ}-{\mathrm{asinz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-\sin \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ y=y_0-r_z\sin \mathrm{\γ}=r_z\cos \mathrm{\ψ}-{\mathrm{acosz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{cosz}}_b\mathrm{\ψ}-\cos \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right..$

Further, in step s3, the formulation process of radius of curvature ρ of cycloid gear actual tooth profile curve is as follows:

According to asking curvature radius formula and cycloid gear tooth profile curve equation in differential, cycloid gear theoretic profile can be tried to achieve Radius of curvature ρ of curve₀, that is,

${\mathrm{\ρ}}_0=\frac{{\[{\left(\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}\right)}^2+{\left(\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}\right)}^2\]}^{\frac{3}{2}}}{\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}\×\frac{d^2{y}_0}{{\mathrm{d\ψ}}^2}-\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}\×\frac{d^2{x}_0}{{\mathrm{d\ψ}}^2}}$

And

$\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}=r_z(cos\mathrm{\ψ}-k_1{\mathrm{cosz}}_b\mathrm{\ψ}),\frac{d^2{x}_0}{{\mathrm{d\ψ}}^2}=r_z(-\sin \mathrm{\ψ}+k_1{z}_b{\mathrm{sinz}}_b\mathrm{\ψ})$

$\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}=r_z(-\sin \mathrm{\ψ}+k_1{\mathrm{sinz}}_b\mathrm{\ψ}),\frac{d^2{y}_0}{{\mathrm{d\ψ}}^2}=r_z(-\cos \mathrm{\ψ}+k_1{z}_b{\mathrm{cosz}}_b\mathrm{\ψ})$

After arrangement:

${\mathrm{\ρ}}_0=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_{\mathrm{\α}}\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_{\mathrm{\α}}\mathrm{\ψ}-(1+z_b{k}_1^2)}$

Because the actual tooth profile curve of cycloid gear is the equidistant curve of theoretical curve, so cycloid gear actual tooth profile curve Radius of curvature ρ be:

$\mathrm{\ρ}={\mathrm{\ρ}}_0+r_z=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_a\mathrm{\ψ}-(1+z_b{k}_1^2)}+r_z.$

Specifically, described pressure angle α is the angle between the direction of power and velocity attitude, and in step s4, ψ is with regard to pressure angle α Equation inference process as follows:

Pressure angle α can be obtained according to the cosine law and sine with regard to the concrete equation of ψ is:

$\mathrm{\α}=\arccos \frac{{\mathrm{az}}_b{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{r_z^2+a^2{z}_b^2-2r_z{\mathrm{az}}_b{\mathrm{cosz}}_a\mathrm{\ψ}}}=\arccos \frac{k_1{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}$

Reverse ψ can obtain:

$a^2{z}_b^2{\cos }^2{z}_a\mathrm{\ψ}-2{\cos }^2{\mathrm{\αr}}_z{\mathrm{az}}_b{\mathrm{cosz}}_a\mathrm{\ψ}+{\cos }^2{\mathrm{\αr}}_z^2-a^2{z}_b^2{\sin }^2\mathrm{\α}=0.$

Preferably, described pressure angle α is a periodic variable in complete cycloid gear tooth curve is formed, and range of variables is [α_min, π-α_min], the variable cycle is 2 π/z_a.

Further, when ψ is in [0, π/z_a] in, radius of curvature ρ of cycloid gear theoretic profile curve₀During for bearing,

$\mathrm{\ψ}=\frac{1}{z_a}arccos\frac{{\cos }^2\mathrm{\α}+sin\mathrm{\α}\sqrt{(k_1+cos\mathrm{\α})(k_1-cos\mathrm{\α})}}{k_1}.$

Further, when ψ is in [0, π/z_a] in, radius of curvature ρ of cycloid gear theoretic profile curve₀For timing,

$\mathrm{\ψ}=\frac{1}{z_a}arccos\frac{{\cos }^2\mathrm{\α}-sin\mathrm{\α}\sqrt{(k_1+cos\mathrm{\α})(k_1-cos\mathrm{\α})}}{k_1}.$

Further, in step s5, cycloid gear tooth curve flex point equation inference process is as follows:

Cycloid gear tooth curve flex point formula is:

$\frac{d^{2}y}{{\mathrm{dx}}^2}=\frac{\frac{d^{2}y}{{\mathrm{d\ψ}}^2}\frac{dx}{d\mathrm{\ψ}}-\frac{dy}{d\mathrm{\ψ}}\frac{d^{2}x}{{\mathrm{d\ψ}}^2}}{{\left(\frac{dx}{d\mathrm{\ψ}}\right)}^3}$

Can obtain

$\frac{d^{2}y}{{\mathrm{dx}}^2}=-\frac{\begin{array}{c}\left({\mathrm{az}}_b\sin \right(z_b\mathrm{\ψ})-r_z\sin (\mathrm{\ψ})+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\sin \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\sin \left(z_b\mathrm{\ψ}\right)+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b^2\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{\mathrm{sinz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\\ +\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\cos \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\cos \left(z_b\mathrm{\ψ}\right)+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b^2\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{\mathrm{cosz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\end{array}}{{\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})-r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})}^3}.$

The invention has the beneficial effects as follows:

(1) present invention is by being compared discretization error and assigned error and adjust automatically approaches step-length, thus obtaining Less than the maximum step-length of approximate error, on the premise of meeting required precision, decrease feed number of times, improve working (machining) efficiency；

(2) the step Discrete method of the cycloid gear tooth curve of the present invention is higher than the approximation accuracy of equidistant method, than The discrete efficiency high of unique step method, faster than the calculating speed of equal error method；

(3) the step Discrete method of the cycloid gear tooth curve of the present invention is in approximation accuracy, discrete efficiency and calculating Speed aspect has good aggregative indicator.

Brief description

In order to be illustrated more clearly that technical scheme, below will be to required in embodiment or description of the prior art Use accompanying drawing be briefly described it should be apparent that, drawings in the following description are only embodiments of the invention, for this For the those of ordinary skill of field, on the premise of not paying creative work, can also be obtained other attached according to these accompanying drawings Figure.

Fig. 1 is cycloid gear tooth curve figure in the embodiment of the present invention 1.

Fig. 2 is that the actual tooth profile curve of cycloid gear in the embodiment of the present invention 1 is divided with the optimum combination of theoretic profile curve Cloth curve chart.

Fig. 3 is the mechanical periodicity curve chart of cycloid gear flank profil pressure angle in the embodiment of the present invention 1.

Fig. 4 is for ψ in the embodiment of the present invention 1 in [0, π/z_a] scope inner cycloidal gear tooth curve linear interpolation point coordinates Figure.

Fig. 5 is for ψ in the embodiment of the present invention 1 in [0, π/z_a] scope inner cycloidal gear tooth curve linear interpolation point approximate Approximate error curve chart.

Specific embodiment

Below in conjunction with accompanying drawing, the technical scheme in the embodiment of the present invention is clearly and completely described it is clear that being retouched The embodiment stated is only a part of embodiment of the present invention, rather than whole embodiments.Based on the embodiment in the present invention, The every other embodiment that those of ordinary skill in the art are obtained on the premise of not making creative work, broadly falls into this The scope of invention protection.

Embodiment 1:

The invention discloses a kind of step Discrete method of cycloid gear tooth curve, comprise the steps:

S1. cycloid gear profile's factories: pin tooth distribution radius of circle r are set_z=76.5mm, pin tooth radius r_z=3mm, Eccentric throw a=1.5mm, cycloid gear number of teeth z_a=39, needle pin number z_b=40, curtate ratio k₁=az_b/r_z=0.784；

S2. derive cycloid gear tooth profile curve equation

As shown in Figure 1 it is assumed that basic circle maintains static, take the center o of basic circle_aFor the initial point of x-y rectangular coordinate, as round as a ball r On basic circle r when c point is rolled into b point, its center o is around the center o of basic circle_aThe angle turning over is ψ, and the absolute corner of round as a ball r is θ_a, the relative rotation of round as a ball r is θ_b, then on theoretic profile any point m coordinate (x₀, y₀) it is:

$\left\{\begin{array}{c}{x}_0=r_{z}sin\mathrm{\ψ}-asin{\mathrm{\θ}}_a\\ y_0=r_z\cos \mathrm{\ψ}-{\mathrm{acos\θ}}_a\end{array}\right.$

When round as a ball r rolls across a tooth around basic circle rAnd round as a ball r is when relatively turning over a full circle, θ_b=2 π, so θ_b= z_aψ；And θ_a=θ_b+ ψ=z_bThe common normal of ψ, m point and the angle of x-axis are γ, then γ meets equation below:

$\left\{\begin{array}{c}cos\mathrm{\γ}=\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-\sin \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ \sin \mathrm{\γ}=\frac{-k_1{\mathrm{cosz}}_b\mathrm{\ψ}+\cos \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right.$

Corresponding k point coordinates (x, y) in practical tooth can be obtained, that is, cycloid gear tooth profile curve equation is as follows:

$\left\{\begin{array}{c}x=r_z\sin \mathrm{\ψ}-{\mathrm{asinz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-\sin \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ y=r_z\cos \mathrm{\ψ}-{\mathrm{acosz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{cosz}}_b\mathrm{\ψ}-\cos \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right.$

Cycloid gear profile's factories substitution cycloid gear tooth profile curve equation is obtained:

$\left\{\begin{array}{c}x=\frac{153sin\left(\mathrm{\ψ}\right)}{2}-\frac{3sin\left(40\mathrm{\ψ}\right)}{2}+\frac{\frac{40sin\left(40\mathrm{\ψ}\right)}{17}-3sin\left(\mathrm{\ψ}\right)}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51}}}\\ y=\frac{153\cos \left(\mathrm{\ψ}\right)}{2}-\frac{3cos\left(40\mathrm{\ψ}\right)}{2}+\frac{\frac{40\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos \left(\mathrm{\ψ}\right)}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51}}}\end{array}\right.$

S3. derive the formula of radius of curvature ρ of cycloid gear actual tooth profile curve

According to asking curvature radius formula and cycloid gear tooth profile curve equation in differential, cycloid gear theoretic profile can be tried to achieve Radius of curvature ρ of curve₀, that is,

${\mathrm{\ρ}}_0=\frac{{\[{\left(\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}\right)}^2+{\left(\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}\right)}^2\]}^{\frac{3}{2}}}{\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}\×\frac{d^2{y}_0}{{\mathrm{d\ψ}}^2}-\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}\×\frac{d^2{x}_0}{{\mathrm{d\ψ}}^2}}$

And

$\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}=r_z(cos\mathrm{\ψ}-k_1{\mathrm{cosz}}_b\mathrm{\ψ}),\frac{d^2{x}_0}{{\mathrm{d\ψ}}^2}=r_z(-\sin \mathrm{\ψ}+k_1{z}_b{\mathrm{sinz}}_b\mathrm{\ψ})$

$\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}=r_z(-\sin \mathrm{\ψ}+k_1{\mathrm{sinz}}_b\mathrm{\ψ}),\frac{d^2{y}_0}{{\mathrm{d\ψ}}^2}=r_z(-\cos \mathrm{\ψ}+k_1{z}_b{\mathrm{cosz}}_b\mathrm{\ψ})$

After arrangement:

${\mathrm{\ρ}}_0=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_{\mathrm{\α}}\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_{\mathrm{\α}}\mathrm{\ψ}-(1+z_b{k}_1^2)}$

As shown in Fig. 2 working as ρ₀Be on the occasion of when, represent cycloid gear tooth curve concave；Work as ρ₀During for negative value, represent pendulum Line gear tooth curve convex, cycloid gear curvature by rotate forward negative during, necessarily have the curvature of any to be equal to 0, that is, Radius of curvature tends to the curvature transfer point (flex point) that infinitely-great point is cycloid gear tooth curve, curvature transfer point two lateral curvature Rate sign changes, thus leading to the normal direction of flank profil to change.

Because the actual tooth profile curve of cycloid gear is the equidistant curve of theoretical curve, so cycloid gear actual tooth profile curve Radius of curvature ρ be:

$\mathrm{\ρ}={\mathrm{\ρ}}_0+r_z=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_a\mathrm{\ψ}-(1+z_b{k}_1^2)}+r_z$

Substitute into cycloid gear profile's factories to obtain:

$\mathrm{\ρ}=\frac{153{(\frac{4201}{2601}-\frac{80cos\left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}}{2(\frac{1640cos\left(39\mathrm{\ψ}\right)}{51}-\frac{66601}{2601})}+3$

S4. derive the equation with regard to pressure angle α for the ψ

Described pressure angle α is the angle between the direction of power and velocity attitude.

As shown in figure 1, pressure angle α can be obtained according to the cosine law and sine with regard to the concrete equation of ψ being:

$\stackrel{\&overbar;}{mp}=\sqrt{r_z^2+r_b^2-2r_z{r}_b{\mathrm{cosz}}_{\mathrm{\α}}\mathrm{\ψ}}$

$\frac{\stackrel{\&overbar;}{mp}}{{\mathrm{sinz}}_a\mathrm{\ψ}}=\frac{r_b}{\cos \mathrm{\α}}$

Then

$\mathrm{\α}=\arccos \frac{{\mathrm{az}}_b{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{r_z^2+a^2{z}_b^2-2r_z{\mathrm{az}}_b{\mathrm{cosz}}_a\mathrm{\ψ}}}=\arccos \frac{k_1{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}$

Reverse ψ can obtain:

$a^2{z}_b^2{\cos }^2{z}_a\mathrm{\ψ}-2{\cos }^2{\mathrm{\αr}}_z{\mathrm{az}}_b{\mathrm{cosz}}_a\mathrm{\ψ}+{\cos }^2{\mathrm{\αr}}_z^2-a^2{z}_b^2{\sin }^2\mathrm{\α}=0;$

As shown in figure 3, described pressure angle α is a periodic variable in complete cycloid gear tooth curve is formed, variable model Enclose for [α_min, π-α_min], the variable cycle is 2 π/z_a；

Because cycloid gear tooth curve has periodically and has symmetry, therefore, the present invention within each cycle Only discuss ψ in [0, π/z_a] in, α is first decremented to α by pi/2 within this range_min, after be incremented to pi/2 during ψ with regard to pressure The equation of angle α, other interval equations can be drawn by the periodicity of function；

(1) when ψ is in [0, π/z_a] in, radius of curvature ρ of cycloid gear theoretic profile curve₀During for bearing,

$\mathrm{\ψ}=\frac{1}{z_a}arccos\frac{{\cos }^2\mathrm{\α}+sin\mathrm{\α}\sqrt{(k_1+cos\mathrm{\α})(k_1-cos\mathrm{\α})}}{k_1}$

Substitute into cycloid gear profile's factories to obtain:

$\mathrm{\ψ}=\frac{1}{39}arccos(\frac{51{\cos }^2\mathrm{\α}}{40}+\frac{51sin\mathrm{\α}}{40}\sqrt{(\frac{40}{51}+cos\mathrm{\α})(\frac{40}{51}-cos\mathrm{\α})})$

(2) when ψ is in [0, π/z_a] in, radius of curvature ρ of cycloid gear theoretic profile curve₀For timing,

$\mathrm{\ψ}=\frac{1}{z_a}arccos\frac{{\cos }^2\mathrm{\α}-sin\mathrm{\α}\sqrt{(k_1+cos\mathrm{\α})(k_1-cos\mathrm{\α})}}{k_1}$

Substitute into cycloid gear profile's factories to obtain:

$\mathrm{\ψ}=\frac{1}{39}arccos(\frac{51{\cos }^2\mathrm{\α}}{40}-\frac{51sin\mathrm{\α}}{40}\sqrt{(\frac{40}{51}+cos\mathrm{\α})(\frac{40}{51}-cos\mathrm{\α})})$

S5. cycloid gear tooth curve flex point equation is gone out according to equation inference in s2

Cycloid gear tooth profile curve equation is continuous and second order can be micro-, and knee of curve formula is:

$\frac{d^{2}y}{{\mathrm{dx}}^2}=\frac{\frac{d^{2}y}{{\mathrm{d\ψ}}^2}\frac{dx}{d\mathrm{\ψ}}-\frac{dy}{d\mathrm{\ψ}}\frac{d^{2}x}{{\mathrm{d\ψ}}^2}}{{\left(\frac{dx}{d\mathrm{\ψ}}\right)}^3}$

Can obtain

$\frac{d^{2}y}{{\mathrm{dx}}^2}=-\frac{\begin{array}{c}\left({\mathrm{az}}_b\sin \right(z_b\mathrm{\ψ})-r_z\sin (\mathrm{\ψ})+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\sin \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\sin \left(z_b\mathrm{\ψ}\right)+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b^2\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{\mathrm{sinz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\\ +\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\cos \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\cos \left(z_b\mathrm{\ψ}\right)+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b^2\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{\mathrm{cosz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\end{array}}{{\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})-r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})}^3}$

Substitute into cycloid gear profile's factories to obtain:

$\frac{d^{2}y}{{\mathrm{dx}}^2}=\frac{\begin{array}{c}(\frac{153\sin \left(\mathrm{\ψ}\right)}{2}-60\sin (40\mathrm{\ψ})+\frac{\frac{1600\sin \left(40\mathrm{\ψ}\right)}{17}-3\sin \left(\mathrm{\ψ}\right)}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51}}}+\frac{520(\frac{40\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos (\mathrm{\ψ}))\sin \left(39\mathrm{\ψ}\right)}{17{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}})\\ \left(\begin{array}{c}(\frac{153\sin \left(\mathrm{\ψ}\right)}{2}-2400\sin (40\mathrm{\ψ})+\frac{\frac{6400\sin \left(40\mathrm{\ψ}\right)}{17}-3\sin \left(\mathrm{\ψ}\right)}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51}}}+\frac{1040(\frac{1600\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos (\mathrm{\ψ}))\sin \left(39\mathrm{\ψ}\right)}{17{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}})\\ +\frac{20280(\frac{40\sin \left(40\mathrm{\ψ}\right)}{17}-3\sin (\mathrm{\ψ}))cos\left(39\mathrm{\ψ}\right)}{17{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}}-\frac{811200(\frac{40\sin \left(40\mathrm{\ψ}\right)}{17}-3\sin (\mathrm{\ψ}))\sin \left(39\mathrm{\ψ}\right){)}^2}{289{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{5}{2}}}\end{array}\right)\\ +\left(60\cos \right(40\mathrm{\ψ})-\frac{153\cos \left(\mathrm{\ψ}\right)}{2}+\frac{\frac{1600\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos \left(\mathrm{\ψ}\right)}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51}}}+\frac{520(\frac{40\sin \left(40\mathrm{\ψ}\right)}{17}-3\sin (\mathrm{\ψ}))\sin \left(39\mathrm{\ψ}\right)}{17{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}})\\ \left(\begin{array}{c}2400\cos \left(40\mathrm{\ψ}\right)-\frac{153\cos \left(\mathrm{\ψ}\right)}{2}+\frac{\frac{1600\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos \left(\mathrm{\ψ}\right)}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51}}}+\frac{20280(\frac{40\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos (\mathrm{\ψ}))\cos \left(39\mathrm{\ψ}\right)}{17{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}})\\ +\frac{1040(\frac{1600\sin \left(40\mathrm{\ψ}\right)}{17}-3\sin (\mathrm{\ψ}))cos\left(39\mathrm{\ψ}\right)}{17{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}}+\frac{811200(\frac{40\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos (\mathrm{\ψ}))\sin \left(39\mathrm{\ψ}\right){)}^2}{289{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{5}{2}}}\end{array}\right)\end{array}}{{\left(60\cos \right(40\mathrm{\ψ})-\frac{153\cos \left(\mathrm{\ψ}\right)}{2}-\frac{\frac{1600\cos \left(40\mathrm{\ψ}\right)}{17}-3\cos \left(\mathrm{\ψ}\right)}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51}}}+\frac{520(\frac{40\sin \left(40\mathrm{\ψ}\right)}{17}-3\sin (\mathrm{\ψ}))\sin \left(39\mathrm{\ψ}\right)}{17{(\frac{4201}{2601}-\frac{80\cos \left(39\mathrm{\ψ}\right)}{51})}^{\frac{3}{2}}})}^3}$

S6. provide the discrete method of curve linear interpolation initial step length, specifically include following steps:

(1) determine ψ in [0, π/z_a] in flex point with π/39000 (radian) unique step search for flex point marginal value, obtain as Lower local tables:

$*\&doublerightarrow;0^{{\stackrel{\·}{0}}_{{03958501}_1{303}_{0}0288}^{067738}05368}025457$

$\&doublerightarrow;{\stackrel{\·}{.}}_{1415930}^{1415930}{1415930}_{{\stackrel{\·}{.}}^{016755161}016916268}016835714-0_{{}^{01707}.0455}03128$

Linear interpolation is as follows:

$\frac{207-\mathrm{\γ}}{-0.00288}=\frac{\mathrm{\γ}-206}{-0.011303}$

$\mathrm{\γ}\≈\frac{\mathrm{\π}}{39000}*206.8\≈0.0167$

Thus show that at flex point, radian is about 0.0167.

(2) determine initial discrete point step size α₀

The present invention is to ψ in [0, π/z_a] in the range of curve carry out discrete, its trocoid gear flank profil discrete point can be by Symmetry obtains；As shown in figure 4, in departure process, setting α₀For initial discrete point step size, η is advance constant, and λ is to retreat system Number, ε₀For the approximate error allowing, ε ' is actual approximate error, and ε is close approximation error, wherein η=1.1, λ=0.9, ε₀= 0.5μm；

If initial coordinate point (x₀,y₀) it is (0.000,72.000), next discrete point (x₁,y₁), by following equations reverse α₀；

Line segment midpoint equation between 2 points is as follows:

$\left\{\begin{array}{c}{x}_l=\frac{1}{2}\left(\right(r_{z}sin{\mathrm{\α}}_0-a\sin z_b{\mathrm{\α}}_0+r_z\frac{k_1{\mathrm{sinz}}_b{\mathrm{\α}}_0-{\mathrm{sin\α}}_0}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0}})+0)\\ y_l=\frac{1}{2}\left(\right(r_z\cos {\mathrm{\α}}_0-a\cos z_b{\mathrm{\α}}_0+r_z\frac{k_1{\mathrm{cosz}}_b{\mathrm{\α}}_0-{\mathrm{cos\α}}_0}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0}})+72)\end{array}\right.$

Mid point of curve equation between 2 points is as follows:

$\left\{\begin{array}{c}{x}_c=r_{z}sin\frac{{\mathrm{\α}}_0}{2}-asin\frac{z_b{\mathrm{\α}}_0}{2}+r_z\frac{k_{1}sin\frac{z_b{\mathrm{\α}}_0}{2}-sin\frac{{\mathrm{\α}}_0}{2}}{\sqrt{1+k_1^2-2k_{1}cos\frac{z_a{\mathrm{\α}}_0}{2}}}\\ y_c=r_z\cos \frac{{\mathrm{\α}}_0}{2}-acos\frac{z_b{\mathrm{\α}}_0}{2}+r_z\frac{k_1\cos \frac{z_b{\mathrm{\α}}_0}{2}-cos\frac{{\mathrm{\α}}_0}{2}}{\sqrt{1+k_1^2-2k_{1}cos\frac{z_a{\mathrm{\α}}_0}{2}}}\end{array}\right.$

Set up close approximation error ε equation as follows:

$\mathrm{\ϵ}=\sqrt{{(x_l-x_c)}^2+{(y_l-y_c)}^2}$

$\mathrm{\ϵ}=\sqrt{\begin{array}{c}{(\frac{a\sin \left(z_b{\mathrm{\α}}_0\right)}{2}-\sin (\frac{z_b{\mathrm{\α}}_0}{2})-r_z\frac{\sin \left({\mathrm{\α}}_0\right)}{2}+r_z\sin (\frac{{\mathrm{\α}}_0}{2})-\frac{r_z}{2}\frac{\sin \left({\mathrm{\α}}_0\right)-k_1\sin \left(z_b{\mathrm{\α}}_0\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0+1}}-r_z\frac{\sin \left(\frac{{\mathrm{\α}}_0}{2}\right)-k_1\sin \left(\frac{z_b{\mathrm{\α}}_0}{2}\right)}{\sqrt{k_1^2-2k_1\cos \frac{z_a{\mathrm{\α}}_0}{2}+1)}})}^2+\\ {(\frac{a}{2}-\frac{r_z}{2}+\frac{r_z}{2}+\frac{a\cos \left(z_b{\mathrm{\α}}_0\right)}{2}-a\cos (\frac{z_b{\mathrm{\α}}_0}{2})-\frac{r_z}{2}\cos ({\mathrm{\α}}_0)+r_z\cos (\frac{{\mathrm{\α}}_0}{2})+\frac{r_z}{2}\frac{\cos \left({\mathrm{\α}}_0\right)-k_1\cos \left(z_b{\mathrm{\α}}_0\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0+1}}-r_z\frac{\cos \left(\frac{{\mathrm{\α}}_0}{2}\right)-k_1\cos \left(\frac{z_b{\mathrm{\α}}_0}{2}\right)}{\sqrt{k_1^2-2k_1\cos \frac{z_a{\mathrm{\α}}_0}{2}+1}})}^2\end{array}}$

Substitute into cycloid gear profile's factories to obtain:

$\mathrm{\ϵ}=\sqrt{\begin{array}{c}{(\frac{3\sin \left(20{\mathrm{\α}}_0\right)}{2}-\frac{153\sin \left(\frac{{\mathrm{\α}}_0}{2}\right)}{2}-\frac{3\sin \left(40{\mathrm{\α}}_0\right)}{4}+\frac{153\sin \left({\mathrm{\α}}_0\right)}{4}+\frac{\frac{\mathrm{40}\sin \left(\mathrm{40}{\mathrm{\α}}_0\right)}{\mathrm{17}}-3\sin \left({\mathrm{\α}}_0\right)}{2\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39{\mathrm{\α}}_0\right)}{51}}}+\frac{3\sin \left(\frac{{\mathrm{\α}}_0}{2}\right)-\frac{\mathrm{40}\sin \left(\mathrm{20}{\mathrm{\α}}_0\right)}{\mathrm{17}}}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(\frac{39{\mathrm{\α}}_0}{2}\right)}{51}}})}^2+\\ {(\frac{3\cos \left(20{\mathrm{\α}}_0\right)}{2}-\frac{153\cos \left(\frac{{\mathrm{\α}}_0}{2}\right)}{2}-\frac{3\cos \left(40{\mathrm{\α}}_0\right)}{4}+\frac{153\cos \left({\mathrm{\α}}_0\right)}{4}+\frac{3\cos \left(\frac{{\mathrm{\α}}_0}{2}\right)-\frac{\mathrm{40}\cos \left(\mathrm{40}{\mathrm{\α}}_0\right)}{\mathrm{17}}}{\sqrt{\frac{4201}{2601}-\frac{80\cos \left(\frac{39{\mathrm{\α}}_0}{2}\right)}{51}}}+\frac{\frac{\mathrm{40}\cos \left(\mathrm{40}{\mathrm{\α}}_0\right)}{\mathrm{17}}-3\cos \left({\mathrm{\α}}_0\right)}{2\sqrt{\frac{4201}{2601}-\frac{80\cos \left(39{\mathrm{\α}}_0\right)}{51}}}+36)}^2\end{array}}$

With 1 × 10^-5(radian) unique step searches for α₀Marginal value, obtains following form:

Linear interpolation is as follows:

$\frac{0.522-0.500}{2.6-{\mathrm{\α}}_0}=\frac{0.500-0.483}{{\mathrm{\α}}_0-2.5}$

Draw α₀≈2.544×10^-4Radian.

(3) provide the step Discrete method of cycloid gear profile of tooth

Set α₁=α₀+α₀, using (x (α on Equation for Calculating cycloid gear tooth curve in step (2)₀),y(α₀))、(x(2 α₀),y(2α₀)) close approximation error ε between 2 points；

IfThen export α₁Parameter discrete point；IfBy initial discrete point step size α₀It is multiplied by advance system Number η recalculates close approximation error ε；IfBy initial discrete point step size α₀It is multiplied by retrogressing coefficient lambda to recalculate approximately Approximate error ε；Judge α₁Whether arrival curve terminal, does not reach then repetition above step；Otherwise, terminate to calculate close approximation by mistake Difference ε.

As shown in Figure 4 and Figure 5, can get cycloid gear tooth curve ψ in [0, π/z_a] in the range of linear interpolation coordinate And error, concrete numerical value see table:

The invention has the beneficial effects as follows:

(1) present invention is by being compared discretization error and assigned error and adjust automatically approaches step-length, thus obtaining Less than the maximum step-length of approximate error, on the premise of meeting required precision, decrease feed number of times, improve working (machining) efficiency；

(2) the step Discrete method of the cycloid gear tooth curve of the present invention is higher than the approximation accuracy of equidistant method, than The discrete efficiency high of unique step method, faster than the calculating speed of equal error method；

(3) the step Discrete method of the cycloid gear tooth curve of the present invention is in approximation accuracy, discrete efficiency and calculating Speed aspect has good aggregative indicator.

The above is the preferred embodiment of the present invention it should be noted that for those skilled in the art For, under the premise without departing from the principles of the invention, some improvements and modifications can also be made, these improvements and modifications are also considered as Protection scope of the present invention.

Claims (8)

1. a kind of step Discrete method of cycloid gear tooth curve is it is characterised in that comprise the steps:

S1. cycloid gear profile's factories are set: pin tooth is distributed radius of circle as r_z, pin tooth radius be r_z, eccentric throw be a, cycloid Number of gear teeth is z_a, needle pin number be z_b, curtate ratio is k₁, k₁=az_b/r_z；

S2. derive that cycloid gear tooth profile curve equation is as follows:

$\left\{\begin{array}{c}x=r_{z}sin\mathrm{\ψ}-a\sin z_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-sin\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ y=r_{z}cos\mathrm{\ψ}-{\mathrm{acosz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{cosz}}_b\mathrm{\ψ}-cos\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right.$

Wherein, ψ is the angle that round as a ball center turns over around basic circle center；

S3. derive that the formula of radius of curvature ρ of cycloid gear actual tooth profile curve is as follows:

$\mathrm{\ρ}={\mathrm{\ρ}}_0+r_z=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_a\mathrm{\ψ}-(1+z_b{k}_1^2)}+r_z$

Wherein ρ₀Radius of curvature for cycloid gear theoretic profile curve；

S4. derive that ψ is as follows with regard to the equation of pressure angle α:

$\mathrm{\α}=arccos\frac{k_1{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}$

S5. cycloid gear tooth curve flex point equation is gone out according to equation inference in s2 as follows:

$\begin{array}{c}\left({\mathrm{az}}_b\sin \right(z_b\mathrm{\ψ})-r_z\sin (\mathrm{\ψ})+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\sin \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\sin \left(z_b\mathrm{\ψ}\right)+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b^2\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{\mathrm{sinz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\\ +\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\end{array}$

$\frac{d^{2}y}{{\mathrm{dx}}^2}=-\frac{\left(\begin{array}{c}{r}_z\cos \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\cos \left(z_b\mathrm{\ψ}\right)+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b^2\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{\mathrm{cosz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)}{{\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})-r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})}^3}$

S6. provide the discrete method of curve linear interpolation initial step length, specifically include following steps:

(1) flex point of ψ is determined according to above-mentioned steps s4, s5；

(2) determine initial discrete point step size α₀

Set α₀For initial discrete point step size, η is advance constant, and λ is to retreat coefficient, ε₀For the approximate error allowing, ε ' is reality Approximate error, ε is close approximation error, sets up close approximation error ε equation as follows:

$\mathrm{\ϵ}=\sqrt{\begin{array}{c}{(\frac{a\sin \left(z_b{\mathrm{\α}}_0\right)}{2}-\sin (\frac{z_b{\mathrm{\α}}_0}{2})-r_z\frac{\sin \left({\mathrm{\α}}_0\right)}{2}+r_z\sin (\frac{{\mathrm{\α}}_0}{2})-\frac{r_z}{2}\frac{\sin \left({\mathrm{\α}}_0\right)-k_1\sin \left(z_b{\mathrm{\α}}_0\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0+1}}-r_z\frac{\sin \left(\frac{{\mathrm{\α}}_0}{2}\right)-k_1\sin \left(\frac{z_b{\mathrm{\α}}_0}{2}\right)}{\sqrt{k_1^2-2k_1\cos \frac{z_a{\mathrm{\α}}_0}{2}+1)}})}^2+\\ {(\frac{a}{2}-\frac{r_z}{2}+\frac{r_z}{2}+\frac{a\cos \left(z_b{\mathrm{\α}}_0\right)}{2}-a\cos (\frac{z_b{\mathrm{\α}}_0}{2})-\frac{r_z}{2}\cos ({\mathrm{\α}}_0)+r_z\cos (\frac{{\mathrm{\α}}_0}{2})+\frac{r_z}{2}\frac{\cos \left({\mathrm{\α}}_0\right)-k_1\cos \left(z_b{\mathrm{\α}}_0\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a{\mathrm{\α}}_0+1}}-r_z\frac{\cos \left(\frac{{\mathrm{\α}}_0}{2}\right)-k_1\cos \left(\frac{z_b{\mathrm{\α}}_0}{2}\right)}{\sqrt{k_1^2-2k_1\cos \frac{z_a{\mathrm{\α}}_0}{2}+1)}})}^2\end{array}}$

(3) provide the step Discrete method of cycloid gear tooth curve

Set α₁=α₀+α₀, using (x (α on Equation for Calculating cycloid gear tooth curve in step (2)₀),y(α₀))、(x(2α₀), y(2α₀)) close approximation error ε between 2 points；

IfThen export α₁Parameter discrete point；

IfBy initial discrete point step size α₀It is multiplied by advance constant η and recalculate close approximation error ε；IfWill be just Begin discrete point step size α₀It is multiplied by retrogressing coefficient lambda and recalculate close approximation error ε；Judge α₁Whether arrival curve terminal, does not reach Then repeat above step；Otherwise, terminate to calculate close approximation error ε.

2. a kind of step Discrete method of cycloid gear tooth curve according to claim 1 is it is characterised in that step In s2, cycloid gear tooth profile curve equation derivation is as follows:

Assume that basic circle maintains static, take the center o of basic circle_aFor the initial point of x-y rectangular coordinate, when round as a ball r on basic circle r from one When point is rolled into another, its center o is around the center o of basic circle_aThe angle turning over is ψ, and the absolute corner of round as a ball r is θ_a, round as a ball r Relative rotation be θ_b, then on theoretic profile any point m coordinate (x₀, y₀) it is:

$\left\{\begin{array}{c}{x}_0=r_{z}sin\mathrm{\ψ}-asin{\mathrm{\θ}}_a\\ y_0=r_z\cos \mathrm{\ψ}-{\mathrm{acos\θ}}_a\end{array}\right.$

When round as a ball r rolls across a tooth around basic circle rAnd round as a ball r is when relatively turning over a full circle, θ_b=2 π, so θ_b=z_aψ. And θ_a=θ_b+ ψ=z_bThe common normal of ψ, m point and the angle of x-axis are γ, then γ meets equation below:

$\left\{\begin{array}{c}cos\mathrm{\γ}=\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-sin\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ sin\mathrm{\γ}=\frac{-k_1{\mathrm{cosz}}_b\mathrm{\ψ}+\cos \mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right.$

Corresponding k point coordinates (x, y) in practical tooth can be obtained, that is, cycloid gear tooth profile curve equation is as follows:

$\left\{\begin{array}{c}x=x_0+r_{z}cos\mathrm{\γ}=r_{z}sin\mathrm{\ψ}-a\sin z_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{sinz}}_b\mathrm{\ψ}-sin\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\\ y=y_0-r_{z}sin\mathrm{\γ}=r_z\cos \mathrm{\ψ}-{\mathrm{acosz}}_b\mathrm{\ψ}+r_z\frac{k_1{\mathrm{cosz}}_b\mathrm{\ψ}-cos\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}\end{array}\right..$

3. a kind of cycloid gear tooth curve according to claim 1 and 2 step Discrete method it is characterised in that In step s3, the formulation process of radius of curvature ρ of cycloid gear actual tooth profile curve is as follows:

According to asking curvature radius formula and cycloid gear tooth profile curve equation in differential, cycloid gear theoretic profile curve can be tried to achieve Radius of curvature ρ₀, that is,

${\mathrm{\ρ}}_0=\frac{{\[{\left(\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}\right)}^2+{\left(\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}\right)}^2\]}^{\frac{3}{2}}}{\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}\×\frac{d^2{y}_0}{{\mathrm{d\ψ}}^2}-\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}\×\frac{d^2{x}_0}{{\mathrm{d\ψ}}^2}}$

And

$\frac{{\mathrm{dx}}_0}{d\mathrm{\ψ}}=r_z(cos\mathrm{\ψ}-k_1{\mathrm{cosz}}_b\mathrm{\ψ}),\frac{d^2{x}_0}{{\mathrm{d\ψ}}^2}=r_z(-sin\mathrm{\ψ}+k_1{z}_b{\mathrm{sinz}}_b\mathrm{\ψ})$

$\frac{{\mathrm{dy}}_0}{d\mathrm{\ψ}}=r_z(-sin\mathrm{\ψ}+k_1{\mathrm{sinz}}_b\mathrm{\ψ}),\frac{d^2{y}_0}{{\mathrm{d\ψ}}^2}=r_z(-cos\mathrm{\ψ}+k_1{z}_b{\mathrm{cosz}}_b\mathrm{\ψ})$

After arrangement:

${\mathrm{\ρ}}_0=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_a\mathrm{\ψ}-(1+z_b{k}_1^2)}$

Because the actual tooth profile curve of cycloid gear is the equidistant curve of theoretical curve, so the song of cycloid gear actual tooth profile curve Rate radius ρ is:

$\mathrm{\ρ}={\mathrm{\ρ}}_0+r_z=\frac{{(1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ})}^{\frac{3}{2}}{r}_z}{k_1(1+z_b){\mathrm{cosz}}_a\mathrm{\ψ}-(1+z_b{k}_1^2)}+r_z.$

4. a kind of step Discrete method of cycloid gear tooth curve according to claim 3 is it is characterised in that described Pressure angle α is the angle between the direction of power and velocity attitude, and in step s4, ψ is as follows with regard to the equation inference process of pressure angle α:

Pressure angle α can be obtained according to the cosine law and sine with regard to the concrete equation of ψ is:

$\mathrm{\α}=arccos\frac{{\mathrm{az}}_b{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{r_z^2+a^2{z}_b^2-2r_z{\mathrm{az}}_b{\mathrm{cosz}}_a\mathrm{\ψ}}}=arccos\frac{k_1{\mathrm{sinz}}_a\mathrm{\ψ}}{\sqrt{1+k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}}}$

Reverse ψ can obtain:

$a^2{z}_b^2{\cos }^2{z}_a\mathrm{\ψ}-2{\cos }^2{\mathrm{\αr}}_z{\mathrm{az}}_b{\mathrm{cosz}}_a\mathrm{\ψ}+{\cos }^2{\mathrm{\αr}}_z^2-a^2{z}_b^2{\sin }^2\mathrm{\α}=0.$

5. a kind of step Discrete method of cycloid gear tooth curve according to claim 4 is it is characterised in that described Pressure angle α is a periodic variable in complete cycloid gear tooth curve is formed, and range of variables is [α_min, π-α_min], variable week Phase is 2 π/z_a.

6. a kind of step Discrete method of cycloid gear tooth curve according to claim 5 is it is characterised in that work as ψ In [0, π/z_a] in, radius of curvature ρ of cycloid gear theoretic profile curve₀During for bearing,

$\mathrm{\ψ}=\frac{1}{z_a}arccos\frac{{\cos }^2\mathrm{\α}+sin\mathrm{\α}\sqrt{(k_1+cos\mathrm{\α})(k_1-cos\mathrm{\α})}}{k_1}.$

7. a kind of step Discrete method of cycloid gear tooth curve according to claim 5 is it is characterised in that work as ψ In [0, π/z_a] in, radius of curvature ρ of cycloid gear theoretic profile curve₀For timing,

$\mathrm{\ψ}=\frac{1}{z_a}arccos\frac{{\cos }^2\mathrm{\α}-sin\mathrm{\α}\sqrt{(k_1+cos\mathrm{\α})(k_1-cos\mathrm{\α})}}{k_1}.$

8. a kind of cycloid gear tooth curve according to claim 1 or 7 step Discrete method it is characterised in that In step s5, cycloid gear tooth curve flex point equation inference process is as follows:

Cycloid gear tooth curve flex point formula is:

$\frac{d^{2}y}{{\mathrm{dx}}^2}=\frac{\frac{d^{2}y}{{\mathrm{d\ψ}}^2}\frac{dx}{d\mathrm{\ψ}}-\frac{dy}{d\mathrm{\ψ}}\frac{d^{2}x}{{\mathrm{d\ψ}}^2}}{{\left(\frac{dx}{d\mathrm{\ψ}}\right)}^3}$

Can obtain

$\begin{array}{c}\left({\mathrm{az}}_b\sin \right(z_b\mathrm{\ψ})-r_z\sin (\mathrm{\ψ})+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\\ \left(\begin{array}{c}{r}_z\sin \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\sin \left(z_b\mathrm{\ψ}\right)+r_z\frac{\sin \left(\mathrm{\ψ}\right)-k_1{z}_b^2\sin \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{\mathrm{sinz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)\\ +\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})\end{array}$

$\frac{d^{2}y}{{\mathrm{dx}}^2}=-\frac{\left(\begin{array}{c}{r}_z\cos \left(\mathrm{\ψ}\right)-{\mathrm{az}}_b^2\cos \left(z_b\mathrm{\ψ}\right)+r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b^2\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}-\frac{k_1{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1\cos \left(z_b\mathrm{\ψ}\right))\cos \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}\\ +\frac{2k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1{z}_b\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}}+\frac{3k_1^2{z}_a^2{r}_z\left(\cos \right(\mathrm{\ψ})-k_1{\mathrm{cosz}}_b\mathrm{\ψ}){\left(\sin \right(z_a\mathrm{\ψ}\left)\right)}^2}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{5}{2}}}\end{array}\right)}{{\left(r_z\cos \right(\mathrm{\ψ})-{\mathrm{az}}_b\cos (z_b\mathrm{\ψ})-r_z\frac{\cos \left(\mathrm{\ψ}\right)-k_1{z}_b\cos \left(z_b\mathrm{\ψ}\right)}{\sqrt{k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1}}+\frac{k_1{z}_a{r}_z\left(\sin \right(\mathrm{\ψ})-k_1\sin \left(z_b\mathrm{\ψ}\right))\sin \left(z_a\mathrm{\ψ}\right)}{{(k_1^2-2k_1{\mathrm{cosz}}_a\mathrm{\ψ}+1)}^{\frac{3}{2}}})}^3}.$