CN1752486A - Geometric design method for cross shaft straight-teeth conical gear pair with changeable transmission ratio - Google Patents

Abstract

The present invention relates to a variable transmission ratio concurrent axis coniflex gear pair geometrical designing method. Based on spherical engagement theory research said method can implement parameter design of given transmission ratio relationship and concurrent axis coniflex gear pair of interaxial included angle. Tooth profile calculation and solid model creation, and utilizes given transmission relationship and interaxial included angle to define parameter equation of a pair of pitch cone faces and utilizes the kinematics relationship of spherical engagement and CAD software to form non-cone gear solid model. Said non-cone gear pair can simplify existent variable transmission ratio transmission mechanism and can be substituted for combination of several pairs of gear pairs, so that it has extensive universality.

Description

The secondary geometric design method of variable ratio concurrent aces straight bevel gear

Affiliated technical field

The present invention relates to the secondary geometric design method of a kind of variable ratio concurrent aces straight bevel gear, belong to the special gear auxiliary driving in the gear pair transmission field.

Background technique

At the gear pair transmission field, the gear of pair of meshing extensively adopts the stable drive ratio transmission, then belongs to special gear auxiliary driving type for the variable ratio situation.For flat-teeth wheel set (two gear shafts are parallel) variable ratio transmission, the design method and the processing method of comparative maturity arranged, as Li Fusheng: noncircular gear, China Machine Press, 1977, and be applied in a lot of fields; But for space gear pair (two gear shafts intersect or staggered) variable ratio transmission, people study lessly and design method very immature.

The kinematic relation of two gear shaft variable ratios is called the nonlinear motion relation again, and a pair of pitch curve shape of this type gear pair has one at least for non-circular, so this type gear transmission is called the noncircular gear transmission again.Along with the development of Machine Design and manufacturing technology, the particularly development of CAD/CAM technology and extraordinary manufacturing technology provides new technical support for the design and the manufacturing of variable ratio space gear pair.For the cone gear pair, its design realizes by the virtual gear of back cone generate approx, finishes on the gear generator of special use and make the principle that mainly is based on itself and circular rack rolling movement.For variable ratio bevel gear pair mechanism, still there are not practical research, analysis and design and calculation method so far, become the big obstacle that this transmission is generally applied.

Present existing research and application mainly are in automobile differential, as, Wang Xiaochun, Wu Xutang, Peng Wei: a kind of drive ratio-changing differential mechanism, publication number: CN 1043981A, open day: the design philosophy that provides a kind of non-cone gear in the 1990-7-18 document: it was based on cone gear, realized that by the modification to the cone gear flank profil velocity ratio is the change by a small margin or the fluctuation in cycle with the circular pitch.The problem that this design philosophy exists at present has 2 points: the one, and the excursion of velocity ratio is too little, is not enough to overcome the influence of skidding; The 2nd, the period of change of velocity ratio is too short, easily produces pulsation and impacts.Its reason is that the velocity ratio of existing variable ratio differential mechanism is is a period of change with gear circular pitch, and its cycle is too short, and amplitude of variation is too little.For breaking through above-mentioned restriction, promptly develop a kind of brand-new bevel gear pair, in the whole circle of planet wheel interval, its velocity ratio Changing Pattern is set, can enlarge its numerical value change scope like this, thereby improve the locking coefficient of differential mechanism, reduce pulsation and impact simultaneously, reach the cross-country requirement of automobile.Jia Jumin, Gao Bo, Qiao Yongwei: the research of offroad vehicle variable ratio differential mechanism, automotive engineering, a kind of research method of non-cone gear has been proposed among the 2003-11-9, this method utilizes differential geometric theory with the approximate generate plane of sphere pitch curve pitch curve, then on the basis of plane pitch curve, carry out the research of the related content such as profile of tooth, contact ratio, undercut, pressure angle of non-Bevel Gear Drive by theory of plane engagement.But the degree of approximation of this research method is not assessed in the document, thus its levels of precision and the design, manufacture view application prospect may have a greatly reduced quality.

Summary of the invention

Technology of the present invention is dealt with problems: overcome the deficiencies in the prior art, provide a kind of variable ratio concurrent aces bevel gear pair accurate geometric design method, realize the geometric programming problem of variable ratio transmission to solve the concurrent aces bevel gear pair, this method has realized the universal design of variable ratio bevel gear pair, not only simplify drive mechanism to a great extent, and improve a lot at aspects such as levels of precisions.

Technical solution of the present invention is: the secondary geometric design method of variable ratio concurrent aces straight bevel gear is characterized in that comprising the following steps:

(1) velocity ratio according to the practice of engineering design bevel gear pair concerns i ₁₂=f (θ ₁) and between centers angle ₁+ ₂= ₀, realize that then the pitch cone parametric equation of this non-linear concurrent aces drive connection is:

The initiative taper pitch cone:

From the mantle pitch cone:

(2) design gear teeth parameter and make undercut judgement and strength check according to above-mentioned bevel gear pair pitch cone equation, concrete grammar is:

The modular angle α of definition bevel gear _mFor an one tooth pitch cone launch fan-shaped on distribute the number of degrees of pairing central angle divided by π.The modular angle of given non-cone gear is α _m, addendum is α _a, height is α at the bottom of the tooth _f, definition α _a=f α _m(f is an addendum coefficient).

The osculating circle conical surface of the definition conical surface is: at (the θ of the conical surface of shape such as = (θ), θ+Δ θ) in the interval, a circular cone is tangent with it, gets Δ θ → 0, the conical surface that then obtains is its osculating circle conical surface at the θ place, and the pitch cone angle of this conical surface is ρ=ρ (θ).

For shape of describing in (1) such as the conical surface parametric equation of = (θ), have:

Get ρ _Min=min{| ρ |, (θ ∈ (0,2 π)) },

When undercut not taking place, maximum tooth is risen α _aShould satisfy:

${\mathrm{\α}}_a=\arcsin \left\{\sin \right[\arcsin \left(\frac{\sin {\mathrm{\ρ}}_{\min }}{\cos {\mathrm{\α}}_n}\right)-\arcsin \left(\sin {\mathrm{\ρ}}_{\min }\cos {\mathrm{\α}}_n\right)\left]\cos {\mathrm{\α}}_n\right\}$

As above defined α _a=f α _m, the minimum number of teeth z that undercut does not then take place should satisfy:

Be similar to everywhere stressed of non-cone gear according to the stressed method of cone gear, judge minimum modulus angle and maximum number of teeth z according to limit case _Max(noncircular gear generally is not used in the high-speed overload condition, and stressing conditions can be very not abominable).

Pitch cone angle of spread formula is:

The central angle that pitch cone launches in the pitch cone span is β _c, number of teeth z also need satisfy relation so:

β _c＝πα _mz

Can handle tooth profile parameter with reference to the spur bevel gear standard for choosing of addendum coefficient, tooth top crack coefficient with strength check, comprehensive above condition determines that suitable gear parameter chooses, if each condition of this step can not satisfy, can relax the undercut condition (when overlap coefficient is enough big) or the design of set-up procedure (1).

(3) according to step (1), (2), the right tooth curve of design bevel gear, as follows:

Addendum cone and tooth base cone for bevel gear are set out by = (θ), according to addendum α _a, high α at the bottom of the tooth _f, determine the parametric equation of addendum cone γ=γ (θ) and tooth base cone β=β (θ).

Definition is that the segmental arc that is positioned on the circle that R is a radius on the sphere of radius is an orthodrome with R.As shown in Figure 3, the tangent line azimythal angle δ of definition spherical curve is: cross the tangent line BQ that B makes orthodrome AB, cross the tangent line BP that B makes spherical curve, then the angle of BP and BQ is δ.The representation of δ is:

As Fig. 6 a, 6b, then the representation of addendum cone and tooth base cone is:

cosγ＝coscosα _a-sinsinα _asinδ

cosβ＝coscosα _f+sinsinα _fsinδ

When $\mathrm{\&delta;}<\frac{\mathrm{\&pi;}}{2}$ The time, get: θ _γ=θ-Δ θ _γ, θ _β=θ+Δ θ _β

When $\mathrm{\&delta;}>\frac{\mathrm{\&pi;}}{2}$ The time, get: θ _γ=θ+Δ θ _γ, θ _β=θ-Δ θ _β

For profile of tooth, by number of teeth z and modulus angle α _mDetermine the distributing position of each tooth on pitch cone, note corresponding to contact points the profile of tooth phase difference of half tooth pitch of principal and subordinate wheel.

For Plane flank of tooth cone gear also claim circular rack, as shown in Figure 4, tooth-shape angle is α _n, relation is then arranged:

tanτ＝tanα _nsecγ

sinα＝sinγcosα _n

From above-mentioned conclusion, the parametric equation of the bevel-gear spherical flank profil of can deriving.As Fig. 5 a to shown in Fig. 5 d, for the conical surface of shape such as = (θ), a ₀, a ' ₀Be the starting point of a tooth, when moving to a place, the central angle of the arc correspondence that is unfolded is:

The angle parameter of several necessity is:

τ＝arctan(tanα _nsecγ)

α＝arcsin(sinγcosα _n)

, can get the profile of tooth angle of distribution and be in the position of pitch curve relation by profile of tooth:

$\left\{\begin{array}{c}{\mathrm{\&lambda;}}_{\mathrm{lu}}=\mathrm{\&pi;}-\mathrm{\&delta;}+\mathrm{\&tau;},{\mathrm{\&lambda;}}_{\mathrm{ld}}=\mathrm{\&delta;}-\mathrm{\&tau;}\\ {\mathrm{\&lambda;}}_{\mathrm{rd}}=\mathrm{\&pi;}-\mathrm{\&delta;}-\mathrm{\&tau;},{\mathrm{\&lambda;}}_{\mathrm{ru}}=\mathrm{\&delta;}+\mathrm{\&tau;}\end{array}\right.$

Wherein: l represents the left side profile of tooth, and r represents the right side profile of tooth, and u represents the profile of tooth that pitch cone is above, and d represents the profile of tooth that pitch cone is following.

Then the parametric equation of the flank of tooth is:

Wherein, lu, rd gets "+", ld, ru gets "-".

Can calculate the complete profile of tooth of bevel gear by the addendum cone of above-mentioned bevel gear and tooth base cone equation and profile of tooth equation.Consider the complexity of above-mentioned formula, need adopt the method for numerical calculation during application, obtain the coordinate of series of points on the tooth curve.

(4), provide the following method that realizes the 3D modeling with cad tools by the analysis of above steps:

Utilize CAD software development (as the VBA developing instrument of AutoCAD2000), adopt generating cutting this bevel gear modeling.

The cone gear that with the pitch cone angle is is as generating cutting tools, and is special, when getting

Circular rack the time, can only model becoming pitch cone is the non-cone gear of convex, this can judge concavity by osculating circle conical surface pitch cone angle in the step (2), and decision does not interfere the maximum pitch cone angle that allows of the cone gear that needs.

Get this bevel pinion shaft of initial position and overlap with the x axle, geared surface is in the face of the negative semiaxis of x, and the summit is positioned at coordinate round dot O.

With θ=0 is reference, for last 1 the P| θ=θ of pitch cone = (θ) ₀, the position of following definite cutter:

A. cutter rotation (around the rotation of x axle), angle is

B. cutter teeth face tilt (around y axle rotation), angle is

C. cutter rotates (around the rotation of z axle) around Gear axis, and angle is θ ₀

D. ask

And get δ in δ ∈ (0, π) in the interval, cutter is around OP axle rotation, and angle is:

With this position wheel blank is done poor Boolean calculation.When the P point changed continuously, the circulation said process can obtain a designed non-cone gear model, used the same method to design another non-cone gear model of pairing with it.Utilize these models to do three dimensional viewing, can offer forming methods such as rapid prototyping or powder metallurgy processing equally with processing entity model or part to non-cone gear shape and engagement situation thereof.

The present invention's advantage compared with prior art: realized the universal design of variable ratio bevel gear pair, can simplify drive mechanism to a great extent, for example be used for substituting the transmission combination of noncircular gear and cone gear, substitute cam mechanism or linkage mechanism etc., realize high-precision, compact structure function transmission.Be used on the existing differential mechanism, also can better improve the performance of differential mechanism.

Description of drawings

Fig. 1 is the foundation of equation of cone in the system of coordinates;

Fig. 2 is the correlation of conical surface engagement;

Fig. 3 is the definition at the δ angle, tangent line azimythal angle of spherical curve;

Fig. 4 is Cone gear profile of tooth feature;

Fig. 5 is the bevel gear profile of tooth feature of = (θ), wherein Fig. 5 a is the above left side gear profile of tooth of pitch cone, Fig. 5 b is the following left side gear profile of tooth of pitch cone, and Fig. 5 c is the following right side gear profile of tooth of pitch cone, and Fig. 5 d is the above right side gear profile of tooth of pitch cone;

Fig. 6 is the feature of the bevel gear tooth top conical surface and the tooth base cone face of = (θ), and wherein Fig. 6 a is $\mathrm{\&delta;}<\frac{\mathrm{\&pi;}}{2}$ The time feature, Fig. 6 b is $\mathrm{\&delta;}>\frac{\mathrm{\&pi;}}{2}$ The time feature;

The limit case that Fig. 7 judges for the bevel gear undercut;

Fig. 8 is tooth base cone, pitch cone and the addendum cone intersection on a certain sphere of non-cone gear;

Fig. 9 is the tooth curve of non-cone gear on a certain sphere;

Figure 10 is used for the bevel gear model to become

Z=40 circular rack cutter model, it can only the model one-tenth joint z conical surface be the bevel gear of convex;

Figure 11 is used for the bevel gear model to become

Z=50 cone gear cutter model, it can be processed the minimum pitch cone angle of the pitch cone concave portions osculating circle conical surface and is

Bevel gear;

The non-cone gear model of Figure 12 for adopting Figure 10 circular rack model to become processing to obtain;

Figure 13 becomes to process the gear structure figure of the non-cone gear differential mechanism that obtains for adopting Figure 11 cone gear model, and its differential gear is 3 rank, and the part pitch cone is a spill, and planet wheel is the protruding pitch cones in 2 rank.

Embodiment

As Fig. 1, the equation of describing a conical surface is: = (θ), and wherein is the azimythal angle, θ is a round angle.As Fig. 2, as two conical surface ₁= ₁(θ ₁) and ₂= ₂(θ ₂) do nonslipping rolling and satisfy angle between rotating shaft ₁+ ₂= ₀Be definite value.

Embodiment 1: the parameter designing of a pair of non-cone gear, solid modelling and rapid prototyping processing with realize the between centers angle be 60 ° and the non-cone gear that satisfies following drive connection to being example: f (θ ₁)=1.1558+0.5795cos (2 θ ₁)

By step (1), the equation of the pitch cone of master, driven wheel can be expressed as:

By step (2), the selection number of teeth is z=21, and addendum coefficient f=1, backlash coefficient are c=0.2, and modular angle is ${\mathrm{\&alpha;}}_m=\frac{\mathrm{\&pi;}}{20},$ Whole Pitch surfaces is a convex.

Make numerical value by step (3) and calculate, can obtain pitch cone, addendum cone and tooth base cone curve (as Fig. 8) and the profile of tooth distribution (as Fig. 9) of this non-cone gear on certain sphere.

By step (4), employing realizes solid modelling by the software of AutoCAD2000 secondary development tool VBA establishment, because pitch cone is a convex, the cutter model of selecting for use as shown in figure 10, obtain non-cone gear model (provide one of them here, the corresponding contact points of driving and driven form of gear tooth differs 1/2 tooth pitch) as shown in figure 12.

Utilize rapid prototyping to process gear mesh, and design frame and motor driving part, this can realize correct engagement and transmission to gear.

Embodiment 2: the parameter designing of non-cone gear differential mechanism and mockup generate

Here adopt non-cone gear to design the differential gear and the planetary pinion of differential mechanism.Transmission ratio function pass with n rank cone gear is reference, considers the characteristics such as symmetry properties of differential mechanism, and differential gear is 3 rank bevel gears, and planetary pinion is 2 rank bevel gears (oval gear).The design velocity ratio concern shape as:

$f\left({\mathrm{\&theta;}}_1\right)=\frac{2}{3}\frac{(1+2e\cos \left(3{\mathrm{\&theta;}}_1\right)+e^2)}{1-e^2}$

Obvious the two between centers angle is

Choosing modular angle earlier is $\mathrm{\&alpha;}=\frac{\mathrm{\&pi;}}{50},$ Coefficient e is about 0.2, adopts numerical method to approach to obtain the gear pair parameter to be: e=0.2041, the number of teeth is differential gear 36 teeth, planetary pinion 24 teeth.Can judge that by step (2) differential gear partly is spill, so adopt as Figure 11 pitch cone angle is Model can avoid cutter interference as cutter, the transmission ratio function that obtains at last is:

f(θ ₁)＝0.7246+0.2840cos(3θ ₁)

The theoretical locking coefficient of this differential mechanism is so:

$K=\frac{0.7246+0.2840}{0.7246-0.2840}=2.289$

Equally, can find the solution the tooth curve of this gear mesh by step (3).

Realize the solid modelling of this gear mesh by step (4), note the difference of gear teeth starting point.Figure 13 is the gear structure figure of this differential mechanism.

Claims (6)

1, the secondary geometric design method of variable ratio concurrent aces straight bevel gear, it is characterized in that: geometry designs is a starting point with the sphere theory of engagement, and design procedure is:

(1) given concurrent aces angle concerns ₁+ ₂= ₀Concern i with velocity ratio ₁₂=f (θ ₁), calculate unique pitch cone equation that satisfies this condition;

(2) by undercut decision condition and existing strength check condition, selected suitable modular angle and tooth profile parameter;

(3) by above-mentioned selected parameter, selected engagement starting point is calculated the gear pair tooth curve;

(4) adopt generating cutting and numerical computation method to generate non-cone gear mockup mockup.

2, the secondary geometric design method of variable ratio concurrent aces straight bevel gear according to claim 1, it is characterized in that: the pitch cone parametric equation of realizing in the described step (1) is:

The initiative taper pitch cone:

From the mantle pitch cone:

3, the secondary geometric design method of variable ratio concurrent aces straight bevel gear according to claim 1, it is characterized in that: the undercut decision method of design is in the described step (2):

When undercut not taking place, maximum tooth is risen α _aShould satisfy:

${\mathrm{\&alpha;}}_a=\arcsin \left\{\sin \right[\arcsin \left(\frac{\sin {\mathrm{\&rho;}}_{\min }}{\cos {\mathrm{\&alpha;}}_n}\right)-\arcsin \left(\sin {\mathrm{\&rho;}}_{\min }\cos {\mathrm{\&alpha;}}_n\right)\left]\cos {\mathrm{\&alpha;}}_n\right\};$

Wherein ρ is the pitch cone angle of the osculating circle conical surface of the conical surface, satisfies:

ρ _min＝min{|ρ|，(θ∈(0，2π))}。

4, the secondary geometric design method of variable ratio concurrent aces straight bevel gear according to claim 1, it is characterized in that: the gear teeth curve calculation method of design is in the described step (3):

(1) addendum cone γ=γ (θ _γ) and tooth base cone β=β (θ _β) the parametric equation computational methods be:

Tangent line azimythal angle δ is:

cosγ＝coscosα _a-sinsinα _asinδ

cosβ＝coscosα _f+sinsinα _f?sinδ

When $\mathrm{\&delta;}<\frac{\mathrm{\&pi;}}{2}$ The time, get: θ _γ=θ-Δ θ _γ, θ _β=θ+Δ θ _β

When $\mathrm{\&delta;}>\frac{\mathrm{\&pi;}}{2}$ The time, get: θ _γ=θ+Δ θ _γ, θ _β=θ-Δ θ _β

(2) parametric equation of bevel-gear spherical flank profil is:

On the Pitch surfaces from round angle θ ₀To the θ motion, the central angle of the arc correspondence that is unfolded is:

Angle parameter is:

τ＝arctan(tanα _nsecγ)

α＝arcsin(sinγcosα _n)

Concern by the position of profile of tooth, must the profile of tooth angle of distribution be at pitch curve:

$\left\{\begin{array}{cc}{\mathrm{\&lambda;}}_{\mathrm{lu}}=\mathrm{\&pi;}-\mathrm{\&delta;}+\mathrm{\&tau;},{\mathrm{\&lambda;}}_{\mathrm{ld}}=\mathrm{\&delta;}-\mathrm{\&tau;}& \\ {\mathrm{\&lambda;}}_{\mathrm{rd}}=\mathrm{\&pi;}-\mathrm{\&delta;}-\mathrm{\&tau;},{\mathrm{\&lambda;}}_{\mathrm{ru}}=\mathrm{\&delta;}+\mathrm{\&tau;}& \end{array}\right.$

Wherein: l represents the left side profile of tooth, and r represents the right side profile of tooth, and u represents the profile of tooth that pitch cone is above, and d represents the profile of tooth that pitch cone is following, and then the parametric equation of the flank of tooth is:

Wherein, lu, rd get "+", and ld, ru get "-".

5, the secondary geometric design method of variable ratio concurrent aces straight bevel gear according to claim 1, it is characterized in that: described step (4) is: design is generating cutting tools with the cone gear, according to the location parameter of sphere meshing condition decision cutter, circulation is done Boolean calculation and is generated entity.

6, the secondary geometric design method of variable ratio concurrent aces straight bevel gear according to claim 5, it is characterized in that: described design is that generating cutting tools are to determine that cutter is corresponding to any position on the pitch curve with the cone gear, this bevel pinion shaft of initial position overlaps with the x axle, geared surface is in the face of the negative semiaxis of x, the summit is positioned at coordinate round dot O, be reference always, for last 1 the P| θ=θ of pitch cone = (θ) with θ=0 ₀, the position of cutter correspondence is:

(1) cutter rotation (around the rotation of x axle), angle is

(2) cutter teeth face tilt (around the rotation of y axle), angle is

(3) cutter rotates (around the rotation of z axle) around Gear axis, and angle is θ ₀

Ask

And get δ in δ ∈ (0, the value in π) interval, cutter is around the rotation of OP axle, angle is:

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