CN102853054B - Curve conjugation based symmetrical cambered surface conjugate curve gears and engagement pair thereof - Google Patents

Abstract

The invention discloses curve conjugation based symmetrical cambered surface conjugate curve gears and an engagement pair thereof. The engagement pair comprises a gear I and a gear II, wherein the gear I and the gear II are engaged with each other in a point contact mode, and a contact curve gamma 1 which is composed of engagement points on a tooth profile cambered surface of the gear I and a contact curve gamma 2 which is composed of engagement points on a tooth profile cambered surface of the gear II are conjugate curves. According to the engagement pair of the curve conjugation based symmetrical cambered surface conjugate curve gears, the contact curve gamma 1 of the gear I and the contact curve gamma 2 of the gear II which form the engagement pair are conjugate curves, and engagement points between the gear I and the gear II move along contact curves; the engagement pair has the advantages of point contact engagement, tooth profiles of point contact have the high contact strength, the contact transmission process is close to pure rolling along the axial direction, and the transmission efficiency is high; and the selection and the determination of the small tooth number and the large modulus can be achieved on the conditions of the same transmission ratio and the same center-to-center distance, the transmission requirements of high speed, heavy load, high power and high efficiency can be met, and the engagement pair has broad application prospects.

Description

Based on symmetrical cambered surface conjugate curve gear and the engagement pair thereof of curve conjugation

Technical field

The invention belongs to gear transmission technology field, be specifically related to a kind of symmetrical cambered surface conjugate curve gear based on curve conjugation and engagement pair thereof.

Background technique

Gear is typical Key basic parts in equipment manufacture, is widely used in production practices.In mechanical transmission, conjugation engagement pair often decides gear-driven performance.Mainly there is the gear-driven form of linear contact lay and point cantact two kinds engagement system in current gears meshing pair: in linear contact lay situation, between the engagement pair flank of tooth, sliding ratio is large in general, and tooth surface abrasion is serious and easily produce heat and power consumption, can reduce transmission efficiency; And the main feature of point cantact gear is exactly roll engagement because it is approximate pure significantly to reduce its sliding ratio, involute gear and circular tooth gear are the representative wherein with outstanding advantages respectively.

Involutes Gears Transmission is widely used due to many advantages such as the separability of its centre distance and manufacture, measurement are convenient, but also there is the shortcomings such as relative sliding velocity is comparatively large, bearing capacity is poor, have impact on transmission performance.In order to overcome the shortcoming of Involutes Gears Transmission, carry out the research to the Helical gear Transmission scheme that the convex-concave flank of tooth is meshed gradually.Circular tooth gear is with the transmission of concavo-convex circular arc profile point cantact form, there is higher contact strength and bearing capacity, transmission process Surface of action easily forms dynamic pressure oil film and rolls close to pure, transmission efficiency is high, and there is good running-in, the features such as long service life, are widely used in the industrial fields such as handling machinery, mine, metallurgy, chemical industry, weaving.But the distance between circular tooth gear contact points and position all have a certain impact to transmission stability, bearing capacity, also exist the shortcoming such as centre distance sensitivity, responsive to axial force, teeth bending strength be low, hinder the raising of bearing capacity and transmission accuracy, be difficult to meet the requirement of some key areas to Circular profile gearing.

In view of this, it is secondary that the present invention is intended to explore a kind of high performance gears meshing, this engagement pair has that sliding ratio between the flank of tooth is little, contact strength is large and bearing capacity is high, without responsive to axial force and the good advantage of running-in characteristic, can meet high speed, heavily loaded, high-power and high efficiency transmission requirement.

Summary of the invention

The technical problem to be solved in the present invention proposes a kind of symmetrical cambered surface conjugate curve gear based on curve conjugation and engagement pair thereof, this gears meshing pair has that sliding ratio between the flank of tooth is little, contact strength large and the advantage that bearing capacity is high, can meet at a high speed, heavily loaded, high-power and high efficiency transmission requirement.

Realize above-mentioned technical purpose, first the present invention discloses a kind of symmetrical cambered surface conjugate curve gear based on curve conjugation, and the tooth curve of this gear is circular arc, and the curvilinear equation of the inter_curve that described tooth curve is made up of contact points is:

$\left\{\begin{array}{c}x=X\left(\mathrm{\θ}\right)\\ y=Y\left(\mathrm{\θ}\right)\\ z=f\left(\mathrm{\θ}\right)\end{array}\right.{\mathrm{\θ}}_1\≤\mathrm{\θ}\≤{\mathrm{\θ}}_2$

In formula, θ is parameter of curve angle; θ ₁, θ ₂for Line of contact span, the parameter of curve angle that namely flank profil engaging-in some place is corresponding is to nibbling out a parameter of curve angle that place is corresponding;

The centre-point curve that the center of circle of flank profil curved surface is formed is the equidistant curve of inter_curve along flank profil curved surface common normal line direction, and the curvilinear equation of centre-point curve is:

$\left\{\begin{array}{c}{x}_h=x\±\mathrm{\ρ}\·\frac{n_x}{\sqrt{n_x^2+n_y^2+n_z^2}}\\ y_h=y\±\mathrm{\ρ}\·\frac{n_y}{\sqrt{n_x^2+n_y^2+n_z^2}}\\ z_h=z\±\mathrm{\ρ}\·\frac{n_z}{\sqrt{n_x^2+n_y^2+n_z^2}}\end{array}\right.$

In formula, ρ is the radius of curvature of curved tooth contour curved surface; n _x, n _y, n _zbe that method vows n decomposition method vector along reference axis direction under gear system of coordinates respectively, n is that inter_curve Γ vows along the method in given wrapping angle direction at contact points place.

Further, the flank profil curved surface of gear be the centre of sphere along the ball race tubulose envelope surface that described centre-point curve moves, its equation is respectively:

In formula, α is ball family parameter, and meets 0≤α≤2 π.

Further, described inter_curve is symmetrical curve axisymmetricly, and described inter_curve seamlessly transits.

Further, described inter_curve is spatially spiral line, and its helix angle is gradually changed to symmetry center by two ends, and the helix angle being positioned at symmetry center place is 0.

The invention also discloses a kind of symmetrical cambered surface conjugate curve gears meshing based on curve conjugation secondary, comprise mutual point cantact meshed gears I and gear II, the inter_curve Γ that described gear I flank profil curved surface is made up of contact points ₁with the inter_curve Γ that described gear II flank profil curved surface is made up of contact points ₂for conjugate curve;

Described inter_curve Γ ₁curvilinear equation be:

$\left\{\begin{array}{c}{x}_1=X\left(\mathrm{\θ}\right)\\ y_1=Y\left(\mathrm{\θ}\right)\\ z_1=f\left(\mathrm{\θ}\right)\end{array}\right.{\mathrm{\θ}}_1\≤\mathrm{\θ}\≤{\mathrm{\θ}}_2$

In formula, θ is parameter of curve angle; θ ₁, θ ₂for Line of contact span, the parameter of curve angle that namely flank profil engaging-in some place is corresponding is to nibbling out a parameter of curve angle that place is corresponding;

By conjugate curve principle, described inter_curve Γ ₂curvilinear equation be:

In formula, i ₂₁for gear ratio; φ ₁, φ ₂be the angle that gear 1,2 turns over respectively, and have relation φ ₂=i ₂₁φ ₁; A is the reference center distance between outer gearing coupling gear; N is that conjugate curve are vowed along the method in given wrapping angle direction at contact points place; υ ⁽¹²⁾it is the speed of related movement vector between contact points place gear.

Further, the tooth curve of described gear I and gear II is circular arc, the centre-point curve Γ ' of the center of circle formation of described gear I flank profil curved surface ₁for inter_curve Γ ₁along the equidistant curve in flank profil curved surface common normal line direction; The centre-point curve Γ ' of the center of circle formation of described gear II flank profil curved surface ₂for inter_curve Γ ₂along the equidistant curve in flank profil curved surface common normal line direction.

Further, the tooth curve of described gear I and gear II is respectively dome arc and concave circular arc;

The centre-point curve Γ ' of described gear I ₁curvilinear equation be:

${{\mathrm{\Γ}}^{\′}}_1:\left\{\begin{array}{c}{x}_{1h}=x_1+{\mathrm{\ρ}}_1\·\frac{n_{x1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\\ y_{1h}=y_1+{\mathrm{\ρ}}_1\·\frac{n_{y1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\\ z_{1h}=z_1+{\mathrm{\ρ}}_1\·\frac{n_{z1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\end{array}\right.$

The centre-point curve Γ ' of described gear II ₂curvilinear equation be:

${{\mathrm{\Γ}}^{\′}}_2:\left\{\begin{array}{c}{x}_{2h}=x_2-{\mathrm{\ρ}}_2\·\frac{n_{x2}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\\ y_{2h}=y_2-{\mathrm{\ρ}}_2\·\frac{n_{y2}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\\ z_{2h}=z_2-{\mathrm{\ρ}}_2\·\frac{n_{z1}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\end{array}\right.$

In formula, ρ ₁for the radius of curvature of the dome curved tooth contour curved surface of gear I; ρ ₂for the radius of curvature of the concave circular arc flank profil curved surface of gear II; n _x1, n _y1, n _z1, n _x2, n _y2, n _z2that method vows n decomposition method vector along reference axis direction under gear system of coordinates respectively.

Further, the flank profil curved surface ∑ of described gear I ₁for the centre of sphere is along described centre-point curve Γ ' ₁the ball race tubulose envelope surface of motion, its equation is:

The flank profil curved surface ∑ of described gear II ₂for the centre of sphere is along described centre-point curve Γ ' ₂the ball race tubulose envelope surface of motion, its equation is:

In formula, α is ball family parameter, and meets 0≤α≤2 π.

Further, described inter_curve Γ ₁for symmetrical curve axisymmetricly, and described inter_curve Γ ₁seamlessly transit; Described inter_curve Γ ₂for with described inter_curve Γ ₁conjugation symmetrical curve axisymmetricly, and described inter_curve Γ ₂seamlessly transit.

Further, described inter_curve Γ ₁for spatially spiral line, its helix angle is gradually changed to symmetry center by two ends, and the helix angle being positioned at symmetry center place is 0.

Beneficial effect of the present invention is:

1, the symmetrical cambered surface conjugate curve gears meshing that the present invention is based on curve conjugation is secondary, the gear I inter_curve Γ of composition engagement pair ₁with gear II inter_curve Γ ₂conjugate curve each other, the contact points between gear I and gear II moves along inter_curve; This engagement pair inherits the feature of point cantact engagement, and the flank profil of point cantact has high contact strength, and Contact Transmission process is rolled close to pure vertically, and transmission efficiency is high; Same velocity ratio, concentricity apart from the little number of teeth can be realized under condition, the selection of large modulus determines, can meet at a high speed, heavily loaded, high-power and high efficiency transmission requirement, have broad application prospects.

2, pass through inter_curve Γ ₁with inter_curve Γ ₂all be set to symmetrically seamlessly transit curve, make the profile of tooth of engagement pair symmetrical, the effect eliminating axial force makes transmission more steady, and the flank of tooth is easy to manufacturing, and engagement pair is had without responsive to axial force and the good advantage of running-in characteristic.

Accompanying drawing explanation

Fig. 1 is the flank profil curved surface schematic diagram of the symmetrical cambered surface conjugate curve gear that the present invention is based on curve conjugation;

Fig. 2 is the gear entity structure schematic diagram of the present embodiment;

Fig. 3 is the gear-profile Surface forming schematic diagram of the present embodiment;

The system of coordinates that Fig. 4 adopts for the symmetrical cambered surface conjugate curve gears meshing pair that the present invention is based on curve conjugation;

Fig. 5 is the normal tooth profile engagement pair schematic diagram of the present embodiment gears meshing pair;

Fig. 6 is the flank profil Surface forming schematic diagram of the present embodiment gears meshing pair.

Embodiment

Below in conjunction with accompanying drawing, the specific embodiment of the present invention is elaborated.

As shown in Figure 1, for the present invention is based on the flank profil curved surface schematic diagram of the symmetrical cambered surface conjugate curve gear of curve conjugation; Fig. 2 is the gear entity structure schematic diagram of the present embodiment.

The present embodiment is based on the symmetrical cambered surface conjugate curve gear of curve conjugation, and the tooth curve 1 of this gear is circular arc, and the curvilinear equation of the inter_curve 2 that described tooth curve 1 is made up of contact points is:

$\left\{\begin{array}{c}x=X\left(\mathrm{\θ}\right)\\ y=Y\left(\mathrm{\θ}\right)\\ z=f\left(\mathrm{\θ}\right)\end{array}\right.{\mathrm{\θ}}_1\≤\mathrm{\θ}\≤{\mathrm{\θ}}_2$

In formula, θ is parameter of curve angle; θ ₁, θ ₂for Line of contact span, the parameter of curve angle that namely flank profil engaging-in some place is corresponding is to nibbling out a parameter of curve angle that place is corresponding.

The centre-point curve that the center of circle of flank profil curved surface 4 is formed is the equidistant curve of inter_curve 2 along flank profil curved surface common normal line direction, and the curvilinear equation of centre-point curve is:

$\left\{\begin{array}{c}{x}_h=x\±\mathrm{\ρ}\·\frac{n_x}{\sqrt{n_x^2+n_y^2+n_z^2}}\\ y_h=y\±\mathrm{\ρ}\·\frac{n_y}{\sqrt{n_x^2+n_y^2+n_z^2}}\\ z_h=z\±\mathrm{\ρ}\·\frac{n_z}{\sqrt{n_x^2+n_y^2+n_z^2}}\end{array}\right.$

In formula, ρ is the radius of curvature of curved tooth contour curved surface; n _x, n _y, n _zbe that method vows n decomposition method vector along reference axis direction under gear system of coordinates respectively, n is that inter_curve Γ vows along the method in given wrapping angle direction at contact points place.

As shown in Figure 3, the flank profil curved surface 4 of gear be the centre of sphere along the ball race tubulose envelope surface that described centre-point curve moves, its equation is respectively:

In formula, α is ball family parameter, and meets 0≤α≤2 π.

Preferably, inter_curve 2 is symmetrical curve axisymmetricly, and inter_curve 2 seamlessly transits, and as shown in Figure 1, inter_curve 2 connects in symmetrical central area 3 place smooth transition, ensures stationarity during gears meshing.

Preferably, inter_curve 2 is spatially spiral line, and its helix angle is gradually changed to symmetry center by two ends, and the helix angle being positioned at symmetry center place is 0, ensures the smooth transition of inter_curve 2 at symmetrical central area 3 place.

The present embodiment can be dome arc based on the tooth curve 1 of the symmetrical cambered surface conjugate curve gear of curve conjugation, also can be concave circular arc.

Below in conjunction with the embodiment of the engagement pair of the above-mentioned symmetrical cambered surface conjugate curve gear composition based on curve conjugation of employing.

As shown in Figure 4, the secondary system of coordinates adopted of the symmetrical cambered surface conjugate curve gears meshing for the present invention is based on curve conjugation; Fig. 5 is the normal tooth profile engagement pair schematic diagram of the present embodiment gears meshing pair.

The present embodiment is secondary based on the symmetrical cambered surface conjugate curve gears meshing of curve conjugation, comprises mutual point cantact meshed gears I and gear II, the inter_curve Γ that gear I flank profil curved surface is made up of contact points ₁with the inter_curve Γ that gear II flank profil curved surface is made up of contact points ₂for conjugate curve;

Inter_curve Γ ₁curvilinear equation be:

$\left\{\begin{array}{c}{x}_1=X\left(\mathrm{\θ}\right)\\ y_1=Y\left(\mathrm{\θ}\right)\\ z_1=f\left(\mathrm{\θ}\right)\end{array}\right.{\mathrm{\θ}}_1\≤\mathrm{\θ}\≤{\mathrm{\θ}}_2$

In formula, θ is parameter of curve angle; θ ₁, θ ₂for Line of contact span, the parameter of curve angle that namely flank profil engaging-in some place is corresponding is to nibbling out a parameter of curve angle that place is corresponding;

By conjugate curve principle, described inter_curve Γ ₂curvilinear equation be:

In formula, i ₂₁for gear ratio; φ ₁, φ ₂be the angle that gear 1,2 turns over respectively, and have relation φ ₂=i ₂₁φ ₁; A is the reference center distance between outer gearing coupling gear; N is that conjugate curve are vowed along the method in given wrapping angle direction at contact points place; υ ⁽¹²⁾it is the speed of related movement vector between contact points place gear.

The gear I of the present embodiment and the tooth curve of gear II are circular arc, the centre-point curve Γ ' of the center of circle formation of gear I flank profil curved surface ₁for inter_curve Γ ₁along the equidistant curve in flank profil curved surface common normal line direction.The centre-point curve Γ ' of the center of circle formation of gear II flank profil curved surface ₂for inter_curve Γ ₂along the equidistant curve in flank profil curved surface common normal line direction.The tooth curve of gear I and gear II can be all adopt dome arc, also can be to adopt dome arc and concave circular arc respectively, and the gear I of the present embodiment and the tooth curve of gear II are respectively dome arc and concave circular arc.

According to inter_curve Γ ₁the centre-point curve Γ ' of gear I can be obtained ₁curvilinear equation be:

${{\mathrm{\Γ}}^{\′}}_1:\left\{\begin{array}{c}{x}_{1h}=x_1+{\mathrm{\ρ}}_1\·\frac{n_{x1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\\ y_{1h}=y_1+{\mathrm{\ρ}}_1\·\frac{n_{y1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\\ z_{1h}=z_1+{\mathrm{\ρ}}_1\·\frac{n_{z1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\end{array}\right.$

According to inter_curve Γ ₂the centre-point curve Γ ' of gear II can be obtained ₂curvilinear equation be:

${{\mathrm{\Γ}}^{\′}}_2:\left\{\begin{array}{c}{x}_{2h}=x_2-{\mathrm{\ρ}}_2\·\frac{n_{x2}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\\ y_{2h}=y_2-{\mathrm{\ρ}}_2\·\frac{n_{y2}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\\ z_{2h}=z_2-{\mathrm{\ρ}}_2\·\frac{n_{z1}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\end{array}\right.$

In formula, ρ ₁for the radius of curvature of the dome curved tooth contour curved surface of gear I; ρ ₂for the radius of curvature of the concave circular arc flank profil curved surface of gear II; n _x1, n _y1, n _z1, n _x2, n _y2, n _z2that method vows n decomposition method vector along reference axis direction under gear system of coordinates respectively.

As shown in Figure 6, the flank profil curved surface ∑ of gear I ₁for the centre of sphere is along described centre-point curve Γ ' ₁the ball race tubulose envelope surface of motion, its equation is:

The flank profil curved surface ∑ of gear II ₂for the centre of sphere is along described centre-point curve Γ ' ₂the ball race tubulose envelope surface of motion, its equation is:

In formula, α is ball family parameter, and meets 0≤α≤2 π.

The present embodiment is secondary based on the symmetrical cambered surface conjugate curve gears meshing of curve conjugation, the gear I inter_curve Γ of composition engagement pair ₁with gear II inter_curve Γ ₂conjugate curve each other, the contact points between gear I and gear II moves along inter_curve; This engagement pair inherits the feature of point cantact engagement, and the flank profil of point cantact has high contact strength, and Contact Transmission process is rolled close to pure vertically, and transmission efficiency is high; Same velocity ratio, concentricity apart from the little number of teeth can be realized under condition, the selection of large modulus determines, can meet at a high speed, heavily loaded, high-power and high efficiency transmission requirement, have broad application prospects.

Preferably, inter_curve Γ ₁for symmetrical curve axisymmetricly, and described inter_curve Γ ₁seamlessly transit; Described inter_curve Γ ₂for with described inter_curve Γ ₁conjugation symmetrical curve axisymmetricly, and described inter_curve Γ ₂seamlessly transit.By by inter_curve Γ ₁with inter_curve Γ ₂all be set to symmetrically seamlessly transit curve, make the profile of tooth of engagement pair symmetrical, the effect eliminating axial force makes transmission more steady, and the flank of tooth is easy to manufacturing, and engagement pair is had without responsive to axial force and the good advantage of running-in characteristic.

As shown in Figure 4, system of coordinates S (O-x, y, z), S _p(O _p-x _p, y _p, z _p) be two system of coordinates fixed in space, z-axis overlaps with the rotational axis of gear I, z _paxle overlaps with the rotational axis of gear II, and two axial lines is parallel.X-axis and x _paxle overlaps, and their direction is exactly the beeline direction of diaxon, and OO _pequal beeline i.e. centre distance a.System of coordinates S ₁(O ₁-x ₁, y ₁, z ₁) be connected with gear I, system of coordinates S ₂(O ₂-x ₂, y ₂, z ₂) be connected with gear II, when initial position they respectively with S, S _poverlap.Gear 1 is with uniform angular velocity ω ⁽¹⁾rotate around z-axis, gear 2 is with uniform angular velocity ω ⁽²⁾around z _paxle rotates, regulation ω ⁽¹⁾forward is identical with z-axis forward, ω ⁽²⁾forward and z _paxle forward is contrary.From initial position through after a period of time, S ₁and S ₂move to position shown in figure.Gear I turns over φ around z-axis ₁angle, gear II is around z _paxle turns over φ ₂angle.

As shown in Figure 4, inter_curve Γ ₁with inter_curve Γ ₂in the engagement of any point of contact P place.Inter_curve Γ ₁with inter_curve Γ ₂be respectively a smoothed curve on the secondary middle gear I of gears meshing and gear II flank profil curved surface, and inter_curve Γ ₂according to given inter_curve Γ under certain movement condition ₁the curve of the conjugation with it obtained, the two forming curves conjugation engages.

The inter_curve Γ of setting the present embodiment ₁for spatially spiral line, its helix angle is gradually changed to symmetry center by two ends, and the helix angle being positioned at symmetry center place is 0, as shown in Figure 4, if inter_curve Γ ₁curvilinear equation be:

$\left\{\begin{array}{c}{x}_1=R\cos \mathrm{\θ}\\ y_1=R\sin \mathrm{\θ}\\ z_1=p\left(\mathrm{\θ}\right)\·\mathrm{\θ}\end{array}\right.$

In formula, R is cylndrical surface, helical curve place radius; P (θ) is the linear representation about helical curve parameter θ, and can be expressed as p (θ)=k θ, k is herein linear dimensions.

Under the system of coordinates shown in Fig. 4, push away to obtain inter_curve Γ on gear II according to conjugate curve principles ₂equation be:

$\left\{\begin{array}{c}{x}_2=R\cos [(i_{21}+1){\mathrm{\φ}}_1+\mathrm{\θ}]-a\cos \left(i_{21}{\mathrm{\φ}}_1\right)\\ y_2=R\sin [(i_{21}+1){\mathrm{\φ}}_1+\mathrm{\θ}]-a\sin \left(i_{21}{\mathrm{\φ}}_1\right)\\ z_2=k{\mathrm{\θ}}^2\\ E\cos {\mathrm{\φ}}_1-F\sin {\mathrm{\φ}}_1=M\end{array}\right.$

In formula,

$E=-i_{21}a[u\frac{-R^3\sin \mathrm{\θ}-4k^2\mathrm{\θR}(\cos \mathrm{\θ}+\mathrm{\θ}\sin \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^2}+v\frac{2\mathrm{kR}(\sin \mathrm{\θ}-\mathrm{\θ}\cos \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^{3/2}}]$

$F=i_{21}a[u\frac{-R^3\cos \mathrm{\θ}+4k^2\mathrm{\θR}(\sin \mathrm{\θ}-\mathrm{\θ}\cos \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^2}+v\frac{2\mathrm{kR}(\cos \mathrm{\θ}+\mathrm{\θ}\sin \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^{3/2}}]$

$M=(1+i_{21})[u\frac{4k^2\mathrm{\θR}}{{(R^2+4k^2{\mathrm{\θ}}^2)}^2}+v\frac{2{\mathrm{kR}}^2\mathrm{\θ}}{{(R^2+4k^2{\mathrm{\θ}}^2)}^{3/2}}]$

As shown in Figure 6, the flank profil curved surface ∑ of gear I gear I ₁for the centre of sphere is along described centre-point curve Γ ' ₁the ball race tubulose envelope surface of motion, the flank profil curved surface ∑ of gear II ₂for the centre of sphere is along described centre-point curve Γ ' ₂the ball race tubulose envelope surface of motion.The centre-point curve Γ ' of the center of circle formation of gear I flank profil curved surface ₁for inter_curve Γ ₁along the equidistant curve in flank profil curved surface common normal line direction.The centre-point curve Γ ' of the center of circle formation of gear II flank profil curved surface ₂for inter_curve Γ ₂along the equidistant curve in flank profil curved surface common normal line direction.The double wedge of its middle gear I is wide equidistant along common normal line forward, and distance is ρ ₁; The recessed flank profil of gear II is oppositely equidistant along common normal line, and distance is ρ ₂, as shown in Figure 5, have ρ ₂> ρ ₁.In ball race spheroid the radius of a ball centered by distance between curve and inter_curve.

Therefore, the equation of the flank profil curved surface of gear I is:

The equation of the flank profil curved surface of gear II is:

In formula,

$\left\{\begin{array}{c}{n}_{x1}=u\frac{-R^3\cos \mathrm{\θ}+4k^2\mathrm{\θR}(\sin \mathrm{\θ}-\mathrm{\θ}\cos \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^2}+v\frac{2\mathrm{kR}(\cos \mathrm{\θ}+\mathrm{\θ}\sin \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^{3/2}}\\ n_{y1}=u\frac{-R^3\sin \mathrm{\θ}-4k^2\mathrm{\θR}(\cos \mathrm{\θ}+\mathrm{\θ}\sin \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^2}+v\frac{2\mathrm{kR}(\sin \mathrm{\θ}-\mathrm{\θ}\cos \mathrm{\θ})}{{(R^2+4k^2{\mathrm{\θ}}^2)}^{3/2}}\\ n_{z1}=u\frac{2{\mathrm{kR}}^2}{{(R^2+4k^2{\mathrm{\θ}}^2)}^2}+v\frac{R^2}{{(R^2+4k^2{\mathrm{\θ}}^2)}^{3/2}}\end{array}\right.$

$\left\{\begin{array}{c}{n}_{x2}=n_{x1}\cos \left[(1+i_{21}){\mathrm{\φ}}_1\right]-n_{y1}\sin \left[(1+i_{21}){\mathrm{\φ}}_1\right]\\ n_{y2}=n_{x1}\sin \left[(1+i_{21}){\mathrm{\φ}}_1\right]+n_{y1}\cos \left[(1+i_{21}){\mathrm{\φ}}_1\right]\\ n_{z2}=n_{z1}\end{array}\right.$

As shown in Figure 5, on the cross section that the flank profil curved surface of gear I and the flank profil curved surface of gear II engage at any time, the tooth curve 5 of gear I contacts at a certain point of contact place with the tooth curve 6 of gear II, wherein, the tooth curve 5 of gear I is dome arc, and its radius of curvature is ρ ₁; The tooth curve 6 of gear II is concave circular arc, and its radius of curvature is ρ ₂.K is the contact points of any time two gear, and the centre of curvature of tooth curve 5 and the centre of curvature of tooth curve 6 are in the common normal line crossing K point.Rotate with certain angular velocity along with gears meshing is secondary, at a time contact points moves a distance along respective inter_curve, and the flank profil of gear I and gear II moves a distance vertically simultaneously and enters the participation engagement of next contact points place.The radius of arc of the tooth curve 6 of gear II is greater than the radius of arc of the tooth curve 5 of gear I, and according to reality engagement situation, tooth curve 6 pairs of tooth curves 5 form local and contain, and make it have higher engagement driving intensity.

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

Claims (10)

1. based on a symmetrical cambered surface conjugate curve gear for curve conjugation, it is characterized in that: the tooth curve of this gear is circular arc, and the curvilinear equation of the inter_curve that described tooth curve is made up of contact points is:

$\left\{\begin{array}{c}x=X\left(\mathrm{\θ}\right)\\ y=Y\left(\mathrm{\θ}\right)\\ z=f\left(\mathrm{\θ}\right)\end{array}\right.{\mathrm{\θ}}_1\≤\mathrm{\θ}\≤{\mathrm{\θ}}_2$

In formula, θ is parameter of curve angle; θ ₁, θ ₂for Line of contact span, the parameter of curve angle that namely flank profil engaging-in some place is corresponding is to nibbling out a parameter of curve angle that place is corresponding;

The centre-point curve that the center of circle of flank profil curved surface is formed is the equidistant curve of inter_curve along flank profil curved surface common normal line direction, and the curvilinear equation of centre-point curve is:

$\left\{\begin{array}{c}{x}_h=x\±\mathrm{\ρ}\·\frac{n_x}{\sqrt{n_x^2+n_y^2+n_z^2}}\\ y_h=y\±\mathrm{\ρ}\·\frac{n_y}{\sqrt{n_x^2+n_y^2+n_z^2}}\\ z_h=z\±\mathrm{\ρ}\·\frac{n_z}{\sqrt{n_x^2+n_y^2+n_z^2}}\end{array}\right.$

In formula, ρ is the radius of curvature of curved tooth contour curved surface; n _x, n _y, n _zbe that method vows n decomposition method vector along reference axis direction under gear system of coordinates respectively, n is that inter_curve Γ vows along the method in given wrapping angle direction at contact points place.

2., according to claim 1 based on the symmetrical cambered surface conjugate curve gear of curve conjugation, it is characterized in that: the flank profil curved surface of gear be the centre of sphere along the ball race tubulose envelope surface that described centre-point curve moves, its equation is respectively:

In formula, α is ball family parameter, and meets 0≤α≤2 π.

3. according to claim 1 or 2 based on the symmetrical cambered surface conjugate curve gear of curve conjugation, it is characterized in that: described inter_curve is symmetrical curve axisymmetricly, and described inter_curve seamlessly transits.

4., according to claim 3 based on the symmetrical cambered surface conjugate curve gear of curve conjugation, it is characterized in that: described inter_curve is spatially spiral line, and its helix angle is gradually changed to symmetry center by two ends, and the helix angle being positioned at symmetry center place is 0.

5. the symmetrical cambered surface conjugate curve gears meshing based on curve conjugation is secondary, it is characterized in that: comprise mutual point cantact meshed gears I and gear II, the inter_curve Γ that described gear I flank profil curved surface is made up of contact points ₁with the inter_curve Γ that described gear II flank profil curved surface is made up of contact points ₂for conjugate curve;

Described inter_curve Γ ₁curvilinear equation be:

$\left\{\begin{array}{c}{x}_1=X\left(\mathrm{\θ}\right)\\ y_1=Y\left(\mathrm{\θ}\right)\\ z_1=f\left(\mathrm{\θ}\right)\end{array}\right.{\mathrm{\θ}}_1\≤\mathrm{\θ}\≤{\mathrm{\θ}}_2$

In formula, θ is parameter of curve angle; θ ₁, θ ₂for Line of contact span, the parameter of curve angle that namely flank profil engaging-in some place is corresponding is to nibbling out a parameter of curve angle that place is corresponding;

By conjugate curve principle, described inter_curve Γ ₂curvilinear equation be:

In formula, i ₂₁for gear ratio; φ ₁, φ ₂be the angle that gear 1,2 turns over respectively, and have relation φ ₂=i ₂₁φ ₁; A is the reference center distance between outer gearing coupling gear; N is that conjugate curve are vowed along the method in given wrapping angle direction at contact points place; υ ⁽¹²⁾it is the speed of related movement vector between contact points place gear.

6. the symmetrical cambered surface conjugate curve gears meshing according to claim 5 based on curve conjugation is secondary, it is characterized in that: the tooth curve of described gear I and gear II is circular arc, the centre-point curve Γ ' of the center of circle formation of described gear I flank profil curved surface ₁for inter_curve Γ ₁along the equidistant curve in flank profil curved surface common normal line direction; The centre-point curve Γ ' of the center of circle formation of described gear II flank profil curved surface ₂for inter_curve Γ ₂along the equidistant curve in flank profil curved surface common normal line direction.

7. the symmetrical cambered surface conjugate curve gears meshing according to claim 6 based on curve conjugation is secondary, it is characterized in that: the tooth curve of described gear I and gear II is respectively dome arc and concave circular arc;

The centre-point curve Γ ' of described gear I ₁curvilinear equation be:

${{\mathrm{\Γ}}^{\′}}_1:\left\{\begin{array}{c}{x}_{1h}=x_1+{\mathrm{\ρ}}_1\·\frac{n_{x1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\\ y_{1h}=y_1+{\mathrm{\ρ}}_1\·\frac{n_{y1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\\ z_{1h}=z_1+{\mathrm{\ρ}}_1\·\frac{n_{z1}}{\sqrt{n_{x1}^2+n_{y1}^2+n_{z1}^2}}\end{array}\right.$

The centre-point curve Γ ' of described gear II ₂curvilinear equation be:

${{\mathrm{\Γ}}^{\′}}_2:\left\{\begin{array}{c}{x}_{2h}=x_2-{\mathrm{\ρ}}_2\·\frac{n_{x2}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\\ y_{2h}=y_2-{\mathrm{\ρ}}_2\·\frac{n_{y2}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\\ z_{2h}=z_2-{\mathrm{\ρ}}_2\·\frac{n_{z1}}{\sqrt{n_{x2}^2+n_{y2}^2+n_{z2}^2}}\end{array}\right.$

In formula, ρ ₁for the radius of curvature of the dome curved tooth contour curved surface of gear I; ρ ₂for the radius of curvature of the concave circular arc flank profil curved surface of gear II; n _x1, n _y1, n _z1, n _x2, n _y2, n _z2that method vows n decomposition method vector along reference axis direction under gear system of coordinates respectively.

8. the symmetrical cambered surface conjugate curve gears meshing according to claim 7 based on curve conjugation is secondary, it is characterized in that: the flank profil curved surface ∑ of described gear I ₁for the centre of sphere is along described centre-point curve Γ ' ₁the ball race tubulose envelope surface of motion, its equation is:

The flank profil curved surface ∑ of described gear II ₂for the centre of sphere is along described centre-point curve Γ ' ₂the ball race tubulose envelope surface of motion, its equation is:

In formula, α is ball family parameter, and meets 0≤α≤2 π.

9. the symmetrical cambered surface conjugate curve gears meshing based on curve conjugation according to any one of claim 5-8 is secondary, it is characterized in that: described inter_curve Γ ₁for symmetrical curve axisymmetricly, and described inter_curve Γ ₁seamlessly transit; Described inter_curve Γ ₂for with described inter_curve Γ ₁conjugation symmetrical curve axisymmetricly, and described inter_curve Γ ₂seamlessly transit.

10. the symmetrical cambered surface conjugate curve gears meshing according to claim 9 based on curve conjugation is secondary, it is characterized in that: described inter_curve Γ ₁for spatially spiral line, its helix angle is gradually changed to symmetry center by two ends, and the helix angle being positioned at symmetry center place is 0.