CN104462638A - Design method of high-order modified Pascal spiral curve non-circular gear pair - Google Patents

Abstract

The invention discloses a design method of a high-order modified Pascal spiral curve non-circular gear pair. The study on a Pascal spiral curve gear is only limited within the situation of the periodic transmission ratio symmetric change, and the application range of the Pascal spiral curve gear can be expanded through the deep study on a high-order modified spiral curve gear and a non-circular gear conjugated with the high-order modified spiral curve gear. The design method specifically comprises the following steps that firstly, a pitch curve equation of the high-order modified Pascal spiral curve non-circular gear pair is established, and a center distance is calculated through the numerical method; secondly, the concavity and convexity of a pitch curve are verified, the maximum modulus under the situation that undercutting is not carried out when a gear is machined through the gear shaping method is obtained, the pressure angle change range is calculated, the maximum pressure angle value is verified, and the contact ratio when the high-order modified Pascal spiral curve non-circular gear pair is meshed is calculated. According to the design method, a whole set of thorough design theoretical basis is provided for the high-order modified Pascal spiral curve non-circular gear pair in actual application, and the using and popularizing of a high-order modified Pascal spiral curve non-circular gear are promoted.

Description

The method for designing of high-order denatured Pascal curve noncircular gear pair

Technical field

The present invention relates to a kind of non-circular gear pair designing method, be specifically related to a kind of method for designing of high-order denatured Pascal curve noncircular gear pair.

Background technology

Non-circular gear mechanism, cam mechanism and linkage assembly can both realize non-at the uniform velocity transmission, but non-circular gear has advantages such as transmission efficiency is high, stable movement, reliable operation, be widely used in a lot of fields.

In the research of the non-circular gear of Pascal curve type gear and conjugation thereof, Ren Tingzhi etc. give the equation of pitch curve and the computing method of centre distance, pitch curve length, basic radius of circle and pressure angle, and go out tooth profile equation by pitch curve equation inference, the approximate formula of high-order Pascal curve gear expression formula and the centre distance thereof of also having derived further, but these researchs are all only confined to the situation of ratio of gear periodic symmetry change, do not comprise the method for designing of the non-circular gear of all Pascal curve type gears and conjugation thereof.To the further investigation of the non-circular gear of high-order denatured snail linear gears and conjugation thereof, will the range of application of expansion Pascal curve non-circular gear.In actual applications, Pascal curve gear is applied on rice transplanter transplanting mechanism, is also applied in blade differential pump design.

Summary of the invention

The object of the invention is for the deficiencies in the prior art, a kind of method for designing of high-order denatured Pascal curve noncircular gear pair is proposed, for high-order denatured Pascal curve non-circular gear provides a whole set of perfect design theory basis in actual applications, all Pascal curve non-circular gear transmission mechanisms can be applied to, facilitate promoting the use of of high-order denatured Pascal curve non-circular gear.First this method for designing sets up the pitch curve equation of high-order denatured Pascal curve noncircular gear pair, and utilizes numerical calculations centre distance; Then verify pitch curve concavity and convexity, solve gear shaping method machining gears not root cut the maximum modulus in situation, calculating pressure angle variation range, verification maximum pressure angle value, calculates the registration during engagement of high-order denatured Pascal curve noncircular gear pair.

For solving the problems of the technologies described above, technical scheme of the present invention is:

Concrete steps of the present invention are as follows:

Step one, set up the mathematical model of driving wheel in high-order denatured Pascal curve noncircular gear pair.

The pitch curve of driving wheel is by the n of mechanical periodicity ₁bar pitch curve line segment forms, and every bar pitch curve line segment comprises asymmetrical first denaturation curve section r ₁₁with the second denaturation curve section r ₁₂; The angular displacement of driving wheel in first period of change, the radius vector of corresponding driving wheel is:

In formula, b is the generation circular diameter of single order Pascal curve non-circular gear, and l is length; m ₁₁, m ₁₂be respectively the first denaturation curve section r ₁₁with the second denaturation curve section r ₁₂sex change coefficient, and be positive number, and meet relational expression:

$\frac{\mathrm{\π}}{n_1{m}_{11}}+\frac{\mathrm{\π}}{n_1{m}_{12}}=\frac{2\mathrm{\π}}{n_1}---\left(2\right)$

The numerical evaluation of step 2, high-order denatured Pascal curve noncircular gear pair centre distance a.Constructed fuction is as follows:

In formula, n ₂for the exponent number of engaged wheel 2; Utilize the unimodal interval of advance and retreat method determination centre distance a, and utilize Fibonacci method to determine accurate centre distance.

The pitch curve equation of step 3, calculating engaged wheel is as follows:

In formula, r ₂for the angular displacement of engaged wheel corresponding radius vector.

The concavity and convexity of step 4, differentiation driving wheel and engaged wheel.

The curvature radius calculation formula of driving wheel is as follows:

The curvature radius calculation formula of engaged wheel is as follows:

In formula, the ratio of gear of driving wheel and engaged wheel the first order derivative of ratio of gear the second derivative of ratio of gear

Driving wheel is corresponding curvature radius calculation formula:

Engaged wheel is corresponding curvature radius calculation formula:

First denaturation curve section r ₁₁in, the radius vector of driving wheel asks single order, second derivative obtains:

Second denaturation curve section r ₁₂in, the radius vector of driving wheel asks single order, second derivative obtains:

In formula (6), radius-of-curvature molecule perseverance be just, order formula (1), (8) and (9) are substituted into, must be or time, the radius-of-curvature of driving wheel pitch curve obtains extreme value k ₁₁=(1+n ₁ ²m ₁₁ ²) b ²+ (2+n ₁ ²m ₁₁ ²) bl+l ²or k ₁₂=(1+n ₁ ²m ₁₁ ²) b ²-(2+n ₁ ²m ₁₁ ²) bl+l ².As (1+n ₁ ²m ₁₁ ²) b ²-(2+n ₁ ²m ₁₁ ²) bl+l ²during > 0, driving wheel first denaturation curve section r ₁₁curvature ρ ₁> 0, first denaturation curve section r ₁₁without indent; Order push away to obtain driving wheel first denaturation curve section r ₁₁condition without indent is:

$\left[(1-({n_1}^2{m_{11}}^2+1)\mathrm{\ϵ})\right](1-\mathrm{\ϵ})>0---\left(12\right)$

Namely $\mathrm{\&epsiv;}<\frac{1}{{n_1}^2{m_{11}}^2+1}.$

In like manner, formula (1), (10) and (11) are substituted into k ₁, release driving wheel second denaturation curve section r ₁₂condition without indent is:

$\mathrm{\&epsiv;}<\frac{1}{{n_1}^2{m_{12}}^2+1}---\left(\text{13}\right)$

So the pitch curve of driving wheel without the condition of indent is:

$\left\{\begin{array}{c}\mathrm{\&epsiv;}\&le;\frac{1}{{n_1}^2{m_{11}}^2+1}\\ \mathrm{\&epsiv;}\&le;\frac{1}{{n_1}^2{m_{12}}^2+1}\end{array}\right.---\left(14\right)$

In formula (7), radius-of-curvature molecule perseverance be just, order formula (1), (8) and (9) are substituted into, must be or time, with the first denaturation curve section r of driving wheel ₁₁the radius-of-curvature of the engaged wheel pitch curve be meshed obtains extreme value k ₂₁=a ((1+n ₁ ²m ₁₁ ²) b ²+ (2+n ₁ ²m ₁₁ ²) bl-abn ₁ ²m ₁₁ ²+ l ²) or k ₂₂=a ((1+n ₁ ²m ₁₁ ²) b ²+ abn ₁ ²m ₁₁ ²-(n ₁ ²m ₁₁ ²+ 2) bl+l ²).

As (1+n ₁ ²m ₁₁ ²) b ²+ b (2l+n ₁ ²m ₁₁ ²l-an ₁ ²m ₁₁ ²)+l ²during > 0, order push away with the first denaturation curve section r of driving wheel ₁₁the engaged wheel pitch curve be meshed without the condition of indent is:

(1+n ₁ ²m ₁₁ ²)ε ²+(2+n ₁ ²m ₁₁ ²)ε-γεn ₁ ²m ₁₁ ²+1＞0 (15)

In like manner, formula (1), (10) and (11) are substituted into k ₂, release the second denaturation curve section r with driving wheel ₁₂the engaged wheel pitch curve be meshed without the condition of indent is:

(1+n ₁ ²m ₁₂ ²)ε ²+(2+n ₁ ²m ₁₂ ²)ε-γεn ₁ ²m ₁₂ ²+1＞0 (16)

So engaged wheel pitch curve without the condition of indent is:

$\left\{\begin{array}{c}(1+{n_1}^2{m_{11}}^2){\mathrm{\&epsiv;}}^2+(2+{n_1}^2{m_{11}}^2)\mathrm{\&epsiv;}-\mathrm{\&gamma;\&epsiv;}{n_1}^2{m_{11}}^2+1>0\\ (1+{n_1}^2{m_{12}}^2){\mathrm{\&epsiv;}}^2+(2+{n_1}^2{m_{12}}^2)\mathrm{\&epsiv;}-\mathrm{\&gamma;\&epsiv;}{n_1}^2{m_{12}}^2+1>0\end{array}\right.---\left(17\right)$

Step 5, solve gear shaping method machining gears not root cut the maximum modulus in situation, computing formula is as follows:

$m_{\max }\&le;\frac{{\mathrm{\&rho;}}_{\min }{\sin }^2{\mathrm{\&alpha;}}_0}{h_a^{*}}---\left(18\right)$

By the radius-of-curvature that formula (6) and (7) are tried to achieve, get minimum value ρ _min, substitute into formula (18), calculate gear not root cut the maximum modulus m in situation _max.Wherein, α ₀for the normal pressure angle of pinion cutter, for pinion cutter addendum coefficient.

Step 6, pressure angle variation range when solving the non-circular gear auxiliary driving of high-order denatured Pascal curve, and verify maximum pressure angle.

When the left side flank profil of driving wheel is active side, pressure angle α ₁₂computing formula is as follows:

When the right side flank profil of driving wheel is active side, pressure angle α ₁₂computing formula is as follows:

In formula (19) and (20), μ ₁for the angle of pitch curve between tangent line positive dirction and radius vector.

Registration when step 7, the noncircular gear pair engagement of calculating high-order denatured Pascal curve, and verify minimum registration.Registration calculating formula:

${\mathrm{\&epsiv;}}_{\mathrm{\&alpha;}}=\frac{u_1+u_2}{\mathrm{\&pi;}m\cos {\mathrm{\&alpha;}}_0}---\left(21\right)$

In formula, $u_1=\sqrt{{({\mathrm{\&rho;}}_1+h_{{\mathrm{\&alpha;}}_1})}^2-{\left({\mathrm{\&rho;}}_1\cos {\mathrm{\&alpha;}}_0\right)}^2}-{\mathrm{\&rho;}}_1\sin {\mathrm{\&alpha;}}_0,u_2=\sqrt{{({\mathrm{\&rho;}}_2+h_{{\mathrm{\&alpha;}}_2})}^2-{\left({\mathrm{\&rho;}}_2\cos {\mathrm{\&alpha;}}_0\right)}^2}-{\mathrm{\&rho;}}_2\sin {\mathrm{\&alpha;}}_0;$ for the ad. of driving wheel, for the ad. of engaged wheel; M is the modulus of driving wheel.

The beneficial effect that the present invention has:

1, the present invention is that high-order denatured Pascal curve non-circular gear provides a whole set of perfect design theory basis in actual applications, all Pascal curve non-circular gear transmission mechanisms can be applied to, facilitate promoting the use of of high-order denatured Pascal curve non-circular gear.

2, the present invention can obtain periodically variable ratio of gear, and in each cycle, ratio of gear change is asymmetric, therefore can be applicable to the occasion that the change of special ratio of gear requires.

3, the present invention adopts the exact value of numerical solution computing center distance, and be easy to programming realization, solving precision is high, convenient and swift.

Accompanying drawing explanation

Fig. 1 is the pitch curve mesh schematic representation of single order Pascal curve noncircular gear pair;

Fig. 2 is the pitch curve mesh schematic representation of high-order denatured Pascal curve noncircular gear pair;

Fig. 3 is that in six blade differential pumps, the first impeller three rank sex change Pascal curve drives wheel set and the second impeller three rank sex change Pascal curve to drive the pitch curve mesh schematic representation of wheel set;

Fig. 4 is that in six blade differential pumps, the first impeller three rank sex change Pascal curve drives wheel set and the second impeller three rank sex change Pascal curve to drive the transmission ratio curve figure of wheel set;

Fig. 5 is that the leakage fluid dram aperture of six blade differential pumps becomes the schematic diagram starting greatly discharge opeing;

Fig. 6 is the schematic diagram that the leakage fluid dram aperture of six blade differential pumps starts to diminish;

Fig. 7 is the schematic diagram that the leakage fluid dram aperture of six blade differential pumps is about to close.

Embodiment

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

The method for designing of high-order denatured Pascal curve noncircular gear pair, concrete steps are as follows:

Step one, as shown in Figure 1, set up the mathematical model of driving wheel 1 in high-order denatured Pascal curve gear pair.The pitch curve equation of single order Pascal curve non-circular gear is wherein, for there is circular diameter in b, l is length, for the angular displacement that radius vector r is corresponding.As shown in Figure 2, in high-order denatured Pascal curve gear pair, the pitch curve of driving wheel 1 is by the n of mechanical periodicity ₁bar pitch curve line segment forms, and every bar pitch curve line segment comprises asymmetrical first denaturation curve section r ₁₁with the second denaturation curve section r ₁₂; The angular displacement of driving wheel 1 radius vector corresponding in first period of change is the (angular displacement of driving wheel 1 consistent at radius vector expression formula corresponding to all the other periods of change and first period of change):

In formula, m ₁₁, m ₁₂be respectively the first denaturation curve section r ₁₁with the second denaturation curve section r ₁₂sex change coefficient, and be positive number, and meet relational expression:

$\frac{\mathrm{\&pi;}}{n_1{m}_{11}}+\frac{\mathrm{\&pi;}}{n_1{m}_{12}}=\frac{2\mathrm{\&pi;}}{n_1}---\left(2\right)$

The numerical evaluation of step 2, high-order denatured Pascal curve gear pair centre distance a.Closure condition according to non-circular gear pitch curve obtains following relational expression:

In formula, n ₂for the exponent number of engaged wheel 2, i is the ratio of gear of driving wheel and engaged wheel,

Constructed fuction is as follows:

Utilize the unimodal interval of advance and retreat method determination centre distance a, and utilize Fibonacci method to determine accurate centre distance.

The pitch curve equation of step 3, calculating engaged wheel 2 is as follows:

In formula, r ₂for the angular displacement of engaged wheel corresponding radius vector.

The concavity and convexity of step 4, differentiation driving wheel 1 and engaged wheel 2; Concavity and convexity affects the stationarity of the intensity of non-circular gear, the convenience of processing and transmission.

According to infinitesimal geometry knowledge, the curvature radius calculation formula of driving wheel is as follows:

The curvature radius calculation formula of engaged wheel is as follows:

In formula, the first order derivative of ratio of gear the second derivative of ratio of gear

Driving wheel 1 is corresponding curvature radius calculation formula:

Engaged wheel 2 is corresponding curvature radius calculation formula:

First denaturation curve section r ₁₁in, the radius vector of driving wheel asks single order, second derivative obtains:

Second denaturation curve section r ₁₂in, the radius vector of driving wheel asks single order, second derivative obtains:

If radius-of-curvature be positive, then the pitch curve of driving wheel is in angular displacement place is evagination; Otherwise pitch curve is in angular displacement place is indent.In formula (8), radius-of-curvature molecule perseverance be just, as long as ensure radius-of-curvature denominator just can ensure what driving wheel 1 was evagination.Formula (1), (10) and (11) are substituted into k ₁, can be or time, driving wheel pitch curve radius-of-curvature obtains extreme value, and its value is k ₁₁=(1+n ₁ ²m ₁₁ ²) b ²+ (2+n ₁ ²m ₁₁ ²) bl+l ²or k ₁₂=(1+n ₁ ²m ₁₁ ²) b ²-(2+n ₁ ²m ₁₁ ²) bl+l ².Therefore, as (1+n ₁ ²m ₁₁ ²) b ²-(2+n ₁ ²m ₁₁ ²) bl+l ²during > 0, driving wheel first denaturation curve section r ₁₁curvature ρ ₁> 0, first denaturation curve section r ₁₁without indent; Order push away to obtain driving wheel first denaturation curve section r ₁₁condition without indent is:

$\left[(1-({n_1}^2{m_{11}}^2+1)\mathrm{\&epsiv;})\right](1-\mathrm{\&epsiv;})>0---\left(14\right)$

Namely $\mathrm{\&epsiv;}<\frac{1}{{n_1}^2{m_{11}}^2+1}.$

In like manner, formula (1), (12) and (13) are substituted into k ₁, release driving wheel second denaturation curve section r ₁₂condition without indent is:

$\mathrm{\&epsiv;}<\frac{1}{{n_1}^2{m_{12}}^2+1}---\left(\text{15}\right)$

So the pitch curve of driving wheel without the condition of indent is:

$\left\{\begin{array}{c}\mathrm{\&epsiv;}\&le;\frac{1}{{n_1}^2{m_{11}}^2+1}\\ \mathrm{\&epsiv;}\&le;\frac{1}{{n_1}^2{m_{12}}^2+1}\end{array}\right.---\left(16\right)$

If radius-of-curvature be positive, then the pitch curve of engaged wheel is in angular displacement place is evagination; Otherwise pitch curve is in angular displacement place is indent.In formula (9), radius-of-curvature molecule perseverance be just, as long as ensure denominator the pitch curve just can protecting engaged wheel is evagination.Formula (1), (10) and (11) are substituted into k ₂, can be or time, the radius-of-curvature of the engaged wheel pitch curve be meshed with the first denaturation curve section r11 of driving wheel obtains extreme value k ₂₁=a ((1+n ₁ ²m ₁₁ ²) b ²+ (2+n ₁ ²m ₁₁ ²) bl-abn ₁ ²m ₁₁ ²+ l ²) or k ₂₂=a ((1+n ₁ ²m ₁₁ ²) b ²+ abn ₁ ²m ₁₁ ²-(n ₁ ²m ₁₁ ²+ 2) bl+l ²).

Therefore, as (1+n ₁ ²m ₁₁ ²) b ²+ b (2l+n ₁ ²m ₁₁ ²l-an ₁ ²m ₁₁ ²)+l ²during > 0, order push away with the first denaturation curve section r of driving wheel ₁₁the engaged wheel pitch curve be meshed without the condition of indent is:

(1+n ₁ ²m ₁₁ ²)ε ²+(2+n ₁ ²m ₁₁ ²)ε-γεn ₁ ²m ₁₁ ²+1＞0 (17)

In like manner, formula (1), (12) and (13) are substituted into k ₂, release the second denaturation curve section r with driving wheel ₁₂the engaged wheel pitch curve be meshed without the condition of indent is:

(1+n ₁ ²m ₁₂ ²)ε ²+(2+n ₁ ²m ₁₂ ²)ε-γεn ₁ ²m ₁₂ ²+1＞0 (18)

So engaged wheel pitch curve without the condition of indent is:

$\left\{\begin{array}{c}(1+{n_1}^2{m_{11}}^2){\mathrm{\&epsiv;}}^2+(2+{n_1}^2{m_{11}}^2)\mathrm{\&epsiv;}-\mathrm{\&gamma;\&epsiv;}{n_1}^2{m_{11}}^2+1>0\\ (1+{n_1}^2{m_{12}}^2){\mathrm{\&epsiv;}}^2+(2+{n_1}^2{m_{12}}^2)\mathrm{\&epsiv;}-\mathrm{\&gamma;\&epsiv;}{n_1}^2{m_{12}}^2+1>0\end{array}\right.---\left(19\right)$

Step 5, solve gear shaping method machining gears not root cut the maximum modulus in situation, computing formula is as follows:

$m_{\max }\&le;\frac{{\mathrm{\&rho;}}_{\min }{\sin }^2{\mathrm{\&alpha;}}_0}{h_a^{*}}---\left(20\right)$

By the radius-of-curvature that formula (8) and (9) are tried to achieve, get minimum value ρ _min, substitute into formula (20), calculate gear not root cut the maximum modulus m in situation _max.Wherein, α ₀for the normal pressure angle of pinion cutter, for pinion cutter addendum coefficient.

Step 6, pressure angle variation range when solving the non-circular gear auxiliary driving of high-order denatured Pascal curve, and verify maximum pressure angle.

When the left side flank profil of driving wheel 1 is active side, pressure angle α ₁₂computing formula is as follows:

When the right side flank profil of driving wheel 1 is active side, pressure angle α ₁₂computing formula is as follows:

In formula (21) and (22), μ ₁for the angle of pitch curve between tangent line positive dirction and radius vector.Pressure angle α when obtaining driving wheel 1 transmission by formula (21), (22) ₁₂maximal value, require α ₁₂maximal value be no more than 65 °.

Registration when step 7, the noncircular gear pair engagement of calculating high-order denatured Pascal curve, and verify minimum registration.Registration during non-circular gear engagement is effective length of action and the ratio of rolling circle tooth pitch, registration ε _αfor:

${\mathrm{\&epsiv;}}_{\mathrm{\&alpha;}}=\frac{u_1+u_2}{\mathrm{\&pi;}m\cos {\mathrm{\&alpha;}}_0}---\left(23\right)$

In formula, $u_1=\sqrt{{({\mathrm{\&rho;}}_1+h_{{\mathrm{\&alpha;}}_1})}^2-{\left({\mathrm{\&rho;}}_1\cos {\mathrm{\&alpha;}}_0\right)}^2}-{\mathrm{\&rho;}}_1\sin {\mathrm{\&alpha;}}_0,u_2=\sqrt{{({\mathrm{\&rho;}}_2+h_{{\mathrm{\&alpha;}}_2})}^2-{\left({\mathrm{\&rho;}}_2\cos {\mathrm{\&alpha;}}_0\right)}^2}-{\mathrm{\&rho;}}_2\sin {\mathrm{\&alpha;}}_0;$ for the ad. of driving wheel, for the ad. of engaged wheel; M is the modulus of driving wheel.

The method for designing of this high-order denatured Pascal curve noncircular gear pair, for high-order denatured Pascal curve non-circular gear provides a whole set of perfect design theory basis in actual applications, facilitates promoting the use of of high-order denatured Pascal curve non-circular gear.Below just for three rank sex change Pascal curve non-circular gear driving mechanisms of six blade differential pumps, illustrate and design high-order denatured Pascal curve non-circular gear according to engineering requirements.

Six blade differential pumps needed to complete three same movement rules within 2 π cycles, therefore can by three rank Pascal curve gear drive, for obtaining better performance, according to six blade differential pump structure and the characteristics of motion, need when adjacent two panels blade openings is little (as best shown in figures 5 and 7) to make the absolute motion speedup of the differential motion of adjacent two blades and every sheet blade, be conducive to opening chamber and reducing liquid resistance; As shown in Figure 6, time adjacent two panels blade openings is maximum, the differential of adjacent two blades is maximum, needs to make the absolute motion of every sheet blade slack-off, improves flow.On high-order basis, introduce sex change coefficient, this designing requirement can be realized, therefore adopt three rank sex change Pascal curve non-circular gears to drive.As shown in Figure 3, first impeller three rank sex change Pascal curve active drive wheel 3 is identical with the pitch curve parameter that the second impeller three rank sex change Pascal curve active drive takes turns 5, and phase 90 ° is installed, the driven driving wheel of the first impeller three rank sex change Pascal curve 4 is identical with the pitch curve parameter of the driven driving wheel of the second impeller three rank sex change Pascal curve 6.

In the design parameter of the first impeller three rank sex change Pascal curve active drive wheel 3, first impeller three rank sex change Pascal curve driven driving wheel 4, second impeller three rank sex change Pascal curve active drive wheel 5 and the driven driving wheel 6 of the second impeller three rank sex change Pascal curve, there is circular diameter b=2mm, length l=24mm; First denaturation curve section r of the first impeller three rank sex change Pascal curve active drive wheel 3 and the second impeller three rank sex change Pascal curve active drive wheel 5 ₁₁sex change Coefficient m ₁₁=0.95.

1, the first impeller three rank sex change Pascal curve active drive wheel 3 and the second impeller three rank sex change Pascal curve active drive wheel 5 exist first period of change in corresponding radius vector be:

2, the pitch curve of the driven driving wheel of the first impeller three rank sex change Pascal curve 4 and the driven driving wheel 6 of the second impeller three rank sex change Pascal curve can be obtained according to formula (5).The ratio of gear i of the first impeller three rank sex change Pascal curve active drive wheel 3 and the driven driving wheel 4 of the first impeller three rank sex change Pascal curve ₃₄, the second impeller three rank sex change Pascal curve active drive wheel 5 and ratio of gear i of the driven driving wheel 6 of the second impeller three rank sex change Pascal curve ₅₆curve as shown in Figure 4.

3, the first impeller three rank sex change Pascal curve drives wheel set and the second impeller three rank sex change Pascal curve to drive the centre distance initial value of wheel set to be a ₀=40mm, adopts numerical evaluation mode, and by the unimodal interval of advance and retreat method determination centre distance a, recycling Fibonacci method calculates accurate centre distance a=48.1657mm.

4, the actual practicality determining non-circular gear whether easy to process, therefore requires that pitch curve is convex.Judge formula (16) and (19) according to concavity and convexity, can show that the pitch curve of the first impeller three rank sex change Pascal curve active drive wheel 3, first impeller three rank sex change Pascal curve driven driving wheel 4, second impeller three rank sex change Pascal curve active drive wheel 5 and the driven driving wheel of the second impeller three rank sex change Pascal curve 6 is all convex.

5, maximum pressure angle calculates:

Get α ₀=20 °, pressure angle when the first impeller three rank sex change Pascal curve active drive wheel 3 engages with the driven driving wheel 4 of the first impeller three rank sex change Pascal curve is calculated respectively by formula (21), (22), and the pressure angle of the second impeller three rank sex change Pascal curve active drive wheel 5 when engaging with the driven driving wheel 6 of the second impeller three rank sex change Pascal curve, thus the maximum pressure angle of trying to achieve when the first impeller three rank sex change Pascal curve drives wheel set and the second impeller three rank sex change Pascal curve to drive wheel set to engage is 33.46 °.

6, root does not cut maximum modulus calculating:

Get obtain the m of the first impeller three rank sex change Pascal curve active drive wheel 3, first impeller three rank sex change Pascal curve driven driving wheel 4, second impeller three rank sex change Pascal curve active drive wheel 5 and the driven driving wheel 6 of the second impeller three rank sex change Pascal curve _max=1.72mm.

7, minimum registration calculates:

Get m=1.5,

Can obtain, minimum registration when the first impeller three rank sex change Pascal curve drives wheel set and the second impeller three rank sex change Pascal curve to drive wheel set to engage

Claims (1)

1. the method for designing of high-order denatured Pascal curve noncircular gear pair, is characterized in that: the concrete steps of this method for designing are as follows:

Step one, set up the mathematical model of driving wheel in high-order denatured Pascal curve noncircular gear pair;

The pitch curve of driving wheel is by the n of mechanical periodicity ₁bar pitch curve line segment forms, and every bar pitch curve line segment comprises asymmetrical first denaturation curve section r ₁₁with the second denaturation curve section r ₁₂; The angular displacement of driving wheel in first period of change, the radius vector of corresponding driving wheel is:

In formula, b is the generation circular diameter of single order Pascal curve non-circular gear, and l is length; m ₁₁, m ₁₂be respectively the first denaturation curve section r ₁₁with the second denaturation curve section r ₁₂sex change coefficient, and be positive number, and meet relational expression:

$\frac{\mathrm{\&pi;}}{n_1{m}_{11}}+\frac{\mathrm{\&pi;}}{n_1{m}_{12}}=\frac{2\mathrm{\&pi;}}{n_1}---\left(2\right)$

The numerical evaluation of step 2, high-order denatured Pascal curve noncircular gear pair centre distance a; Constructed fuction is as follows:

In formula, n ₂for the exponent number of engaged wheel 2; Utilize the unimodal interval of advance and retreat method determination centre distance a, and utilize Fibonacci method to determine accurate centre distance;

The pitch curve equation of step 3, calculating engaged wheel is as follows:

In formula, r ₂for the angular displacement of engaged wheel corresponding radius vector;

The concavity and convexity of step 4, differentiation driving wheel and engaged wheel;

The curvature radius calculation formula of driving wheel is as follows:

The curvature radius calculation formula of engaged wheel is as follows:

In formula, the ratio of gear of driving wheel and engaged wheel the first order derivative of ratio of gear the second derivative of ratio of gear

Driving wheel is corresponding curvature radius calculation formula:

Engaged wheel is corresponding curvature radius calculation formula:

First denaturation curve section r ₁₁in, the radius vector of driving wheel asks single order, second derivative obtains:

Second denaturation curve section r ₁₂in, the radius vector of driving wheel asks single order, second derivative obtains:

In formula (6), radius-of-curvature molecule perseverance be just, order formula (1), (8) and (9) are substituted into, must be or time, the radius-of-curvature of driving wheel pitch curve obtains extreme value k ₁₁=(1+n ₁ ²m ₁₁ ²) b ²+ (2+n ₁ ²m ₁₁ ²) bl+l ²or k ₁₂=(1+n ₁ ²m ₁₁ ²) b ²-(2+n ₁ ²m ₁₁ ²) bl+l ²; As (1+n ₁ ²m ₁₁ ²) b ²-(2+n ₁ ²m ₁₁ ²) bl+l ²during > 0, driving wheel first denaturation curve section r ₁₁curvature ρ ₁> 0, first denaturation curve section r ₁₁without indent; Order push away to obtain driving wheel first denaturation curve section r ₁₁condition without indent is:

[(1-(n ₁ ²m ₁₁ ²+1)ε)](1-ε)＞0 (12)

Namely $\mathrm{\&epsiv;}<\frac{1}{{n_1}^2{m_{11}}^2+1};$

In like manner, formula (1), (10) and (11) are substituted into k ₁, release driving wheel second denaturation curve section r ₁₂condition without indent is:

$\mathrm{\&epsiv;}<\frac{1}{{n_1}^2{m_{12}}^2+1}---\left(13\right)$

So the pitch curve of driving wheel without the condition of indent is:

$\left\{\begin{array}{c}\mathrm{\&epsiv;}\&le;\frac{1}{{n_1}^2{m_{11}}^2+1}\\ \mathrm{\&epsiv;}\&le;\frac{1}{{n_1}^2{m_{12}}^2+1}\end{array}\right.---\left(14\right)$

In formula (7), radius-of-curvature molecule perseverance be just, order formula (1), (8) and (9) are substituted into, must be or time, with the first denaturation curve section r of driving wheel ₁₁the radius-of-curvature of the engaged wheel pitch curve be meshed obtains extreme value k ₂₁=a ((1+n ₁ ²m ₁₁ ²) b ²+ (2+n ₁ ²m ₁₁ ²) bl-abn ₁ ²m ₁₁ ²+ l ²) or k ₂₂=a ((1+n ₁ ²m ₁₁ ²) b ²+ abn ₁ ²m ₁₁ ²-(n ₁ ²m ₁₁ ²+ 2) bl+l ²);

As (1+n ₁ ²m ₁₁ ²) b ²+ b (2l+n ₁ ²m ₁₁ ²l-an ₁ ²m ₁₁ ²)+l ²during > 0, order push away with the first denaturation curve section r of driving wheel ₁₁the engaged wheel pitch curve be meshed without the condition of indent is:

(1+n ₁ ²m ₁₁ ²)ε ²+(2+n ₁ ²m ₁₁ ²)ε-γεn ₁ ²m ₁₁ ²+1＞0 (15)

In like manner, formula (1), (10) and (11) are substituted into k ₂, release the second denaturation curve section r with driving wheel ₁₂the engaged wheel pitch curve be meshed without the condition of indent is:

(1+n ₁ ²m ₁₂ ²)ε ²+(2+n ₁ ²m ₁₂ ²)ε-γεn ₁ ²m ₁₂ ²+1＞0 (16)

So engaged wheel pitch curve without the condition of indent is:

$\left\{\begin{array}{c}(1+{n_1}^2{m_{11}}^2){\mathrm{\&epsiv;}}^2+(2+{n_1}^2{m_{11}}^2)\mathrm{\&epsiv;}-{{\mathrm{\&gamma;\&epsiv;n}}_1}^2{m_{11}}^2+1>0\\ (1+{n_1}^2{m_{12}}^2){\mathrm{\&epsiv;}}^2+(2+{n_1}^2{m_{12}}^2)\mathrm{\&epsiv;}-{\mathrm{\&gamma;}{\mathrm{\&epsiv;n}}_1}^2{m_{12}}^2+1>0\end{array}\right.---\left(17\right)$

Step 5, solve gear shaping method machining gears not root cut the maximum modulus in situation, computing formula is as follows:

$m_{\max }\&le;\frac{{\mathrm{\&rho;}}_{\min }{\sin }^2{\mathrm{\&alpha;}}_0}{h_a^{*}}---\left(18\right)$

By the radius-of-curvature that formula (6) and (7) are tried to achieve, get minimum value ρ _min, substitute into formula (18), calculate gear not root cut the maximum modulus m in situation _max; Wherein, α ₀for the normal pressure angle of pinion cutter, for pinion cutter addendum coefficient;

Step 6, pressure angle variation range when solving the non-circular gear auxiliary driving of high-order denatured Pascal curve, and verify maximum pressure angle;

When the left side flank profil of driving wheel is active side, pressure angle α ₁₂computing formula is as follows:

When the right side flank profil of driving wheel is active side, pressure angle α ₁₂computing formula is as follows:

In formula (19) and (20), μ ₁for the angle of pitch curve between tangent line positive dirction and radius vector;

Registration when step 7, the noncircular gear pair engagement of calculating high-order denatured Pascal curve, and verify minimum registration; Registration calculating formula:

${\mathrm{\&epsiv;}}_{\mathrm{\&alpha;}}=\frac{u_1+u_2}{\mathrm{\&pi;}m\cos {\mathrm{\&alpha;}}_0}---\left(21\right)$

In formula, $u_1=\sqrt{{({\mathrm{\&rho;}}_1+h_{{\mathrm{\&alpha;}}_1})}^2-{\left({\mathrm{\&rho;}}_1\cos {\mathrm{\&alpha;}}_0\right)}^2}-{\mathrm{\&rho;}}_1\sin {\mathrm{\&alpha;}}_0,u_2=\sqrt{{({\mathrm{\&rho;}}_2+h_{{\mathrm{\&alpha;}}_2})}^2-{\left({\mathrm{\&rho;}}_2\cos {\mathrm{\&alpha;}}_0\right)}^2}-{\mathrm{\&rho;}}_2\sin {\mathrm{\&alpha;}}_0;h_{{\mathrm{\&alpha;}}_1}$ For the ad. of driving wheel, for the ad. of engaged wheel; M is the modulus of driving wheel.