CN101943245A - Tooth shape optimization design method for linear-circular arc tooth outline internal engaged cylindrical gear pair - Google Patents

Abstract

A method for optimal design of the tooth shape of a linear-arc tooth profile internal meshing cylindrical gear pair. The designed internal gear pair is composed of a linear tooth profile internal ring gear and an arc tooth profile external gear, and the internal gear pair is a cylindrical spur gear. Firstly, the modulus and number of teeth of the gear pair are determined according to the working conditions of the gear pair. After determining the addendum circle, index circle, and dedendum circle of the inner and outer gears of the gear pair, the tooth profile of the inner gear is designed as a simple straight line tooth profile, and then the circular arc The tooth profile replaces the external gear conjugate tooth profile and the meshing transmission error caused by the minimum optimal design method determines the tooth shape parameters of the external gear arc tooth profile, that is, the arc radius of the arc tooth profile and the arc center position. The beneficial effect of the present invention is that the circular arc tooth profile can completely replace the conjugate tooth profile of the external gear, and the internal gear pair can be hardened by the grinding process of the forming method and the grinding process of the circular arc gear respectively. Surface grinding, precision can reach level 6.

Description

The secondary profile of tooth Optimization Design of straight line-circular arc profile internal spur gear

Technical field

The invention belongs to the mechanical transmissioning technology field, the Optimization Design of the secondary profile of tooth of the straight line that is specifically related to use in a kind of machine driven system-circular arc profile internal spur gear.

Background technique

Advantages such as internal gear pair has velocity ratio big, low vibration, low noise, odontotripsis is little, antiscuffing capacity is strong are widely used in planetary train, few tooth difference and zero tooth difference internal gear pair, the pivoting support.Improve the performance of internal gear pair, making greater efforts to promote that it applies is one of main direction of gear research.What employing was more in the present internal gear pair is involute profile, because domestic internal gear grinder bed is less, the corresponding lathe of import costs an arm and a leg, so will realize generally that the hard flank of tooth difficulty of internal gear is very big, and gear can not realize that the hard flank of toothization will have a strong impact on the overall performance and the working life of gear pair.Seek to be easy on a kind of technology realize that the internal gear tooth of efficient roll flute replaces involute profile, significant engineering using value is arranged.It is a viable process that internal gear tooth is designed to straight line, uses this flank profil to adopt its precision of forming method roll flute can reach 6 grades.

The internal gear of internal gear pair adopts straight-side profile, adopting what flank profil with the external gear of its engagement also is the problem that can not ignore, because the bearing capacity of gear and profile accuracy and pitch error are closely related, the external gear flank profil that meshes with straight-side profile is conjugate curve, its difficulty of processing is very big, substantially do not adopt, relevant at present research is, in the patent of application number 200720046096.0 " straight line-circular arc profile internal gear transmission ", what be meshed with straight-side profile is circular arc profile, can use the processing technology of circular arc external gear, but this circular arc profile is by external gear conjugate profiles and external gear teeth tip circle, standard pitch circle, 3 decisions that root circle intersects, substitute the external gear conjugate profiles certain error is arranged, in the patent of application number 200910064116.0 " straight line-involute profile internal spur gear pair ", external gear adopts is involute profile with modulus, and with the straight-side profile of involute profile engagement should be tooth bar, handling its tooth profile error like this is clearly.

Summary of the invention

The technical problem to be solved in the present invention is, after the internal gear of internal gear pair adopts straight-side profile, the external gear profile geometry that finds a kind of design method design and internal gear to mesh, the gear pair tooth profile error of designing is that the meshed transmission gear angular errors is very little, less than 0.005 °, make circular arc profile can substitute the external gear conjugate profiles fully.

The technical solution adopted for the present invention to solve the technical problems is: straight line provided by the present invention-circular arc profile internal gear pair is made up of straight-side profile ring gear and circular arc profile external gear that (Fig. 1, Fig. 2), internal gear pair is a spur gear wheel.At first determine the gear pair modulus and the number of teeth according to gear pair driving torque and velocity ratio, top circle, standard pitch circle, the root circle of external gear have just been decided in the gear pair, internal gear tooth is designed to simple straight-side profile, profile of tooth between its top circle and the root circle is represented with linear equation, determine to make the tooth profile parameter of the external gear circular arc profile of engagement driving error minimum again with Optimization Design, make circular arc profile can substitute the external gear conjugate profiles fully.

The profile of tooth of internal gear is a straight-side profile, gets system of coordinates shown in Figure 3, and its equation is:

$\left.\begin{array}{c}{x}_1=-\frac{1}{2}\mathrm{<figure-callout\; id="2"\; label="m\; z"\; filenames="HSA00000268767500021.png"\; state="\{\{state\}\}">m</figure-callout>}{z}_{}{}_2\sin \frac{\mathrm{\&pi;}}{2z_2}-h\sin \mathrm{\&gamma;}+2\mathrm{ht}\sin \mathrm{\&gamma;}\\ y_1=\frac{1}{2}\mathrm{<figure-callout\; id="2"\; label="m\; z"\; filenames="HSA00000268767500021.png"\; state="\{\{state\}\}">m</figure-callout>}{z}_{}{}_2\cos \frac{\mathrm{\&pi;}}{2z_2}-h\cos \mathrm{\&gamma;}+2\mathrm{ht}\cos \mathrm{\&gamma;}\end{array}\right\}---\left(1\right)$

In the formula: m is a module, z ₂Be the internal gear number of teeth, γ is internal gear profile of tooth half-angle (angle of straight-side profile and Y-axis, 25～32 ° of spans, shown in Figure 3), and h is a whole depth, and t is a continually varying natural number between 0 to 1.

External gear profile of tooth optimization method step is as follows:

1, determines the selection of design variable and parameter

According to the internal gear pair transmission condition, determine velocity ratio i, number of teeth z in advance ₁, z ₂, can calculation of parameter determine module m by power such as contact strength, flexural strengths, and predetermined addendum coefficient h _a ^*, the dedendum of the tooth coefficient h _f ^*After, the ring gear profile of tooth just depends on profile of tooth half-angle γ, the circular arc external gear profile of tooth that is engaged with just depends on radius of arc R and center of arc position, and the position O of this center of arc _C(Fig. 2) can determine, so design variable is got γ and R by formula (2), (3) calculating.

X _CO＝O _CP×cosγ＝(R-R _b+r ₂×sinγ)cosγ (2)

Y _CO＝r ₁-O _CP×sinγ＝r ₁-(R-R _b+r ₂×sinγ)sinγ (3)

In the formula (Fig. 4)

O _CP＝O _CN ₁-PN ₁＝R-R _b+r ₂*sinγ (5)

External gear Pitch radius r ₁For Internal gear Pitch radius r ₂For

X＝[x ₁，x ₂] ^T＝[γ，R] ^T (6)

2, set up objective function

This paper discussion be to guarantee (to determine module earlier) under enough contact strength conditions, the profile of tooth that can make internal gear carry out the correct engagement transmission, and the inside and outside form of gear tooth all can grinding.Internal gear pair can carry out the correct condition of engagement driving continuously be circular pitch equate and the engagement contact ratio greater than 1.The conjugate profiles that substitutes straight-side profile with circular arc profile can cause tooth profile error, and promptly circular pitch is unequal, will produce the engagement driving angular errors.This mathematics model serves as minimum as the optimization aim function with straight line-circular arc profile engagement driving error, and the engagement contact ratio is considered in constraint conditio greater than 1.Straight line-circular arc profile engagement driving error is (Fig. 4):

In the formula: the i-velocity ratio, for

A-central moment,

γ, R in the formula (7) are used x respectively ₁, x ₂Replace to such an extent that objective function is

Monodentate Tooth Profile error is on the pitch circle

3, set up constraint function

(1) engagement contact ratio

The engagement contact ratio is end-acted angle and the ratio of angular pitch

$\mathrm{\&epsiv;}=\frac{{\mathrm{\&alpha;}}_1+{\mathrm{\&alpha;}}_2}{\frac{2\mathrm{\&pi;}}{z_2}}---\left(10\right)$

In the formula: α ₁, α ₂Be respectively the ring gear corner of engagement starting point and terminal point correspondence, have

${\mathrm{\&alpha;}}_1=\arcsin \sqrt{{\sin }^2(\mathrm{\&gamma;}+\frac{\mathrm{\&pi;}}{2z_2})+1-{\left(\frac{r_{a2}}{r_2}\right)}^2}-\mathrm{\&gamma;}---\left(11\right)$

${\mathrm{\&alpha;}}_2=\mathrm{\&gamma;}-\arcsin \frac{R_b*(r_2-r_1)-\sqrt{{{\mathrm{<figure-callout\; id="2"\; label="R\; b"\; filenames="HSA00000268767500021.png"\; state="\{\{state\}\}">R</figure-callout>}}_b}^2*{(r_1-r_2)}^2-({{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000268767500021.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^2-2r_1{r}_2)*({r_1}^2-{r_{a1}}^2+{R_b}^2)}}{{{\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000268767500021.png"\; state="\{\{state\}\}">r</figure-callout>}}_2}^2-2r_1{r}_2}---\left(12\right)$

In the formula: r _A1, r _A2Be respectively outer, internal gear Outside radius, substitution formula (10), constraint function is

g ₁(x)=1-ε≤0 is promptly

(2) restriction of internal gear limit profile of tooth half-angle γ

The profile of tooth half-angle can not be got excessive, is zero otherwise the tooth root garden will be disappeared or make tooth top thick.

$g_2\left(x\right)=\mathrm{\&gamma;}-\arctan \frac{r_1*\sin \frac{\mathrm{\&pi;}}{2z_1}}{r_{a1}-r_1*\cos \frac{\mathrm{\&pi;}}{2z_1}}\&le;0---\left(14\right)$

$g_3\left(x\right)=\mathrm{\&gamma;}-\arctan \frac{r_{a1}*\sin \frac{\mathrm{\&pi;}}{z_1}-r_1*\sin \frac{\mathrm{\&pi;}}{2z_1}}{r_1*\cos \frac{\mathrm{\&pi;}}{2z_1}-r_{a1}*\cos \frac{\mathrm{\&pi;}}{z_1}}\&le;0---\left(15\right)$

(3) restriction of internal gear pair limit of contact point

Straight-side profile should satisfy following formula radially not being unlimited the extension

$g_4\left(x\right)=a+r_{a2}-r_1*\sqrt{1+{\sin }^2(\mathrm{\&gamma;}+\frac{\mathrm{\&pi;}}{2z_2})}\&le;0---\left(16\right)$

For simplifying mathematical optimization models, internal gear pair meshing interference etc. are not listed constraint function here in by the motion simulation check.

4, programming is calculated

According to above-mentioned optimization mathematical model, programme with the MATLAB language, notice parameter in the objective function during computation optimization

Be the corner of internal gear, pair of tooth profile from engaging-in to nibbling out,

From-α ₁Change to α ₂, calculate α with profile of tooth half-angle γ=25 ° of substitution formulas (11), (12) earlier ₁, α ₂Value,

From-α ₁Change to α ₂In the process, the engagement angular errors changes, and makes the maximum value of engagement angular errors be minimum by optimizing cycle calculations, and γ value substitution formula (15), (16) with the optimization result who obtains calculate α again ₁, α ₂Value, search makes maximum value in the angular errors be minimum design variable again, repeats said process, up to satisfying convergence precision.

5, the result is optimized in output, and definite tooth profile parameter

The optimization result of constraint conditio is satisfied in output, parameter is optimized: ring gear profile of tooth half-angle γ and external gear profile of tooth radius of arc R, determine ring gear profile of tooth flank profil by formula (1), determine external gear profile of tooth flank profil by formula (2) (3) and R, show straight line-circular arc profile engagement driving error, and can calculate monodentate Tooth Profile error on the pitch circle by formula (9).

The invention has the beneficial effects as follows: ring gear volume in internal gear pair is big, the manufacture cost height, after adopting simple straight-side profile, can go out the hard flank of tooth by the forming method grinding, be used for the grinding wheel dressing convenience of grinding and be easy to guarantee precision, adopt and the most approaching garden curved tooth exterior feature of conjugate profiles with the external gear of its engagement, can use the method grinding external gear of grinding garden gear with curved teeth, thus the working life of having improved internal gear pair greatly.The circular arc profile that utilizes above-mentioned optimization method to design can substitute external gear conjugate curve flank profil fully, and the gear pair angular errors that causes is not more than 0.005 °, and the gear pair precision can reach 6 grades, and New _ Typed Internal Gears transmission meshing performance is good.

Description of drawings

Fig. 1 is the secondary schematic representation of straight line of the present invention-circular arc profile internal spur gear

Fig. 2 is the local enlarged diagram of the tooth curve of Fig. 1

Fig. 3 is the secondary system of coordinates of straight line of the present invention-circular arc profile internal spur gear

Fig. 4 is the secondary engagement driving error analysis of a straight line of the present invention-circular arc profile internal spur gear schematic representation

Fig. 5 is the angular errors figure of straight line of the present invention-circular arc profile internal spur gear pair in R=31, γ=30 o'clock

Specific implementation method

According to internal gear pair transmission requirement, certain internal gear pair profile of tooth before optimizing profile of tooth setting parameter be: module m is 3, outer, internal gear number of teeth z ₁, z ₂Be respectively 24,60, preliminary election internal gear profile of tooth half-angle γ is 26 °, addendum coefficient

The dedendum of the tooth coefficient Tooth root transition arc radius ρ _f=0.2m.So, the flank profil of internal gear is determined by formula (1) in the internal gear pair.

The design variable initial value is got X ⁰=[25,25] ^TWhen being optimized calculating, notice parameter in the objective function with the MATLAB Programming with Pascal Language

Be the corner of internal gear, pair of tooth profile from engaging-in to nibbling out,

From-α ₁Change to α ₂, calculate α with profile of tooth half-angle γ=25 ° of substitution formulas (11), (12) earlier ₁, α ₂Value,

From-α ₁Change to α ₂In the process, the engagement angular errors changes, and makes the maximum value of engagement angular errors be minimum by optimizing cycle calculations, and γ value substitution formula (11), (12) with the optimization result who obtains calculate α again ₁, α ₂Value, search makes maximum value in the angular errors be minimum design variable again, repeats said process, up to satisfying convergence precision.

Optimizing the result is: work as X=[x ₁, x ₂] ^T=[γ, R] ^T=[30,31] ^TThe time, it is optimum that target reaches.Fig. 5 is

From α ₁=-0.0906 changes to α ₂(with 20 calculation level output results) angular errors figure, wherein hard-over error in=0.0405 whole engagement process:

Monodentate transmission maximum error: f on the pitch circle _p=4.5 μ m occur in firm engagement place, minimal error Just be, the engagement contact ratio is ε=1.2519＞1, and external gear circular arc profile centre coordinate is: [X _CO, Y _CO]=[-25.0932,21.5125].

Result of calculation shows that monodentate Tooth Profile error only is several microns on the pitch circle, and this error in gear transmission is very little, can use in most occasions.As seen the internal gear pair of designing so not only can normal engaged, and has good meshing quality performance, for the sclerosis of the internal gear pair flank of tooth provides may.

Claims (3)

1. the secondary profile of tooth Optimization Design of straight line-circular arc profile internal spur gear is made up of big ring gear and little external gear, and it is characterized in that: the internal gear profile of tooth is a straight-side profile, and the external gear profile of tooth that meshes with internal gear is a circular arc profile.

2. the secondary profile of tooth Optimization Design of straight line according to claim 1-circular arc profile internal spur gear, it is characterized in that: the straight-side profile of internal gear profile of tooth is determined by following equation:

$\left.\begin{array}{c}{x}_1=-\frac{1}{2}m{z}_2\sin \frac{\mathrm{\&pi;}}{2z_2}-h\sin \mathrm{\&gamma;}+2\mathrm{ht}\sin \mathrm{\&gamma;}\\ y_1=\frac{1}{2}m{z}_2\cos \frac{\mathrm{\&pi;}}{2z_2}-h\cos \mathrm{\&gamma;}+2\mathrm{ht}\cos \mathrm{\&gamma;}\end{array}\right\}---\left(1\right)$

In the formula: m is a module, z ₂Be the internal gear number of teeth, γ is an internal gear profile of tooth half-angle, and h is a whole depth, and t is a continually varying natural number between 0 to 1.

3. the secondary profile of tooth Optimization Design of straight line according to claim 1-circular arc profile internal spur gear, it is characterized in that: external gear profile of tooth optimal design step is as follows:

(1) determines the selection of design variable and parameter

Determine module m, velocity ratio i, number of teeth z in advance according to the internal gear pair transmission condition ₁, z ₂, and predetermined addendum coefficient h _a ^*, the dedendum of the tooth coefficient h _f ^*The optimal design variable is:

X＝[x ₁，x ₂] ^T＝[γ，R] ^T (2)

In the formula: γ is a ring gear profile of tooth half-angle, and R is the radius of arc of circular arc external gear profile of tooth.

(2) set up objective function

The engagement driving error that causes with the alternative conjugate curve flank profil of circular arc profile is:

In the formula: the i-velocity ratio, for

A-central moment,

With straight line-circular arc profile engagement driving error serves as minimum as the optimization aim function.γ, R in the formula (3) are used x respectively ₁, x ₂Replace to such an extent that objective function is

(3) set up constraint function

(a) engagement contact ratio

$g_1\left(x\right)=1-\frac{({\mathrm{\&alpha;}}_1+{\mathrm{\&alpha;}}_2)*z_2}{2\mathrm{\&pi;}}\&le;0---\left(5\right)$

In the formula: α ₁, α ₂Be respectively the ring gear corner of engagement starting point and terminal point correspondence, calculate by following formula:

${\mathrm{\&alpha;}}_1=\arcsin \sqrt{{\sin }^2(\mathrm{\&gamma;}+\frac{\mathrm{\&pi;}}{2z_2})+1-{\left(\frac{r_{a2}}{r_2}\right)}^2}-\mathrm{\&gamma;}---\left(6\right)$

${\mathrm{\&alpha;}}_2=\mathrm{\&gamma;}-\arcsin \frac{R_b*(r_2-r_1)-\sqrt{{R_b}^2*{(r_1-r_2)}^2-({r_2}^2-2r_1{r}_2)*({r_1}^2-{r_{a1}}^2+{R_b}^2)}}{{r_2}^2-2r_1{r}_2}---\left(7\right)$

(b) restriction of the maximum profile of tooth half-angle of internal gear γ

$g_2\left(x\right)=\mathrm{\&gamma;}-\arctan \frac{r_1*\sin \frac{\mathrm{\&pi;}}{2z_1}}{r_{a1}-r_1*\cos \frac{\mathrm{\&pi;}}{2z_1}}\&le;0---\left(8\right)$

$g_3\left(x\right)=\mathrm{\&gamma;}-\arctan \frac{r_{a1}*\sin \frac{\mathrm{\&pi;}}{z_1}-r_1*\sin \frac{\mathrm{\&pi;}}{2z_1}}{r_1*\cos \frac{\mathrm{\&pi;}}{2z_1}-r_{a1}*\cos \frac{\mathrm{\&pi;}}{z_1}}\&le;0---\left(9\right)$

(c) restriction of internal gear pair limit of contact point

$g_4\left(x\right)=a+r_{a2}-r_1*\sqrt{1+{\sin }^2(\mathrm{\&gamma;}+\frac{\mathrm{\&pi;}}{2z_2})}\&le;0---\left(10\right)$

(4) programming is calculated

According to above-mentioned optimization mathematical model, programme with the MATLAB language, notice parameter in the objective function during computation optimization

Be the corner of internal gear, pair of tooth profile from engaging-in to nibbling out,

From-α ₁Change to α ₂, calculate α with profile of tooth half-angle γ=25 ° of substitution formulas (6), (7) earlier ₁, α ₂Value, From-α ₁Change to α ₂In the process, the engagement angular errors changes, and makes the maximum value of engagement angular errors be minimum by optimizing cycle calculations, and γ value substitution formula (6), (7) with the optimization result who obtains calculate α again ₁, α ₂Value, search makes maximum value in the angular errors be minimum design variable again, repeats said process, up to satisfying convergence precision.

(5) result is optimized in output, and definite tooth profile parameter

The optimization result of constraint conditio is satisfied in output, and parameter is optimized: ring gear profile of tooth half-angle γ and external gear profile of tooth radius of arc R.Determine ring gear profile of tooth flank profil by formula (1), determine external gear profile of tooth flank profil by formula (11) (12) and R, the display-object functional value is straight line-circular arc profile engagement driving error, and can calculate monodentate Tooth Profile error on the pitch circle by formula (13).

X _CO＝O _CP×cosγ＝(R-R _b+r ₂×sinγ)cosγ (11)

Y _CO＝r ₁-O _CP×sinγ＝r ₁-(R-R _b+r ₂×sinγ)sinγ (12)

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