CN105134907A - Large-overlap-ratio inner engaged gear tooth form design method based on arc line of action - Google Patents

Abstract

The invention discloses a large-overlap-ratio inner engaged gear tooth form design method based on an arc line of action. The line of action designed through the method is an arc connecting an intersection of addendum circles of an inner gear and an outer gear and an intersection of nodal circles, and on the basis of a plane engagement theory, a conjugation equation expressing the corresponding relation between the line of action and an inner engaged gear conjugate profile curve is established; conjugate profile curves according with the arc line of action are established, wherein the conjugate profile curves are the inner gear addendum profile curve and the outer gear addendum profile curve; the conjugation principle is used for establishing dedendum profiles conjugated with the addendum profile curves and the inner gear dedendum profile is shaped; and transition curves between the gear dedendum profiles and dedendum circles are designed. The method has the beneficial effects that the overlap ratio is high, compared with an involute tooth form, the overlap ratio is increased more than twice, multiple gear teeth are involved in engagement, bearing capacity is high, dynamic impact is small, the relative sliding ratio is low, and the lubricating condition is good; and the profile interference problem is avoided and gear parameters are designed more freely.

Description

A kind of large contact ratio inside engaged gear tooth Shape Design method based on circular arc line of contact

Technical field

The invention belongs to driving gear design field, relate to a kind of large contact ratio inside engaged gear tooth Shape Design method based on circular arc line of contact.

Background technique

Inside engaged gear transmission is gear-driven important type of belt drive, transmit the dynamic force and motion of equidirectional, compared with external gear pump, the power loss because conjugate curve relative sliding causes can be reduced, there is the advantages such as transmission efficiency is high, contact ratio large, bearing capacity intensity, be widely used in the transmission systems such as vehicle.The profile of tooth of inside engaged gear is mainly involute tooth and cycloid tooth, wherein involute tooth easily manufactures and interchangeability with it, obtain and apply the most widely, define industrial standard, but involute is subject to the restriction of profile of tooth in design, easily interferes, and defines minimum number of teeth in the design, sometimes have to carry out displacement processing avoid interfering, contact ratio is also by the restriction at the number of teeth, design pressure angle.

Along with the appearance of novel drive structure and the raising to the requirement of inside engaged gear transmission performance, the design of inside engaged gear is had higher requirement: higher bearing capacity and design freedom.The bearing capacity improving gear is realized by two aspects: improve single-tooth strength or increase the number of teeth simultaneously participating in engaging, when adopting involute tooth or cycloid tooth, the limit that single-tooth strength or contact ratio have reached inside engaged gear performance boost is improved by geartooth correction, simultaneously, be subject to the restriction of existing profile of tooth, in inside engaged gear design, involute can only be avoided interfering by change pressure angle and modification coefficient, reduces the degrees of freedom of inside engaged gear design.Current scholar has carried out large quantifier elimination to external gear pump design, proposes many tooths, but at inside engaged gear design aspect, lacks corresponding research.

Summary of the invention

The present invention is directed to the number of teeth in the design of involute inside engaged gear, modification coefficient limits, the problems such as contact ratio is little, provide a kind of large contact ratio inside engaged gear tooth Shape Design method based on circular arc line of contact, the method is for realizing the large contact ratio of inside engaged gear, using connecting internal gear and the external gear teeth tip circle intersection point circular arc to node as designing line of contact, according to the face gear theory of engagement, establish the Conjugate Equation about line of contact and inside engaged gear conjugate curve corresponding relation, thus be met the conjugate curve equation representation of circular arc line of contact, this representation describes the tooth top tooth curve of internal gear and external gear, then, according to conjugate principle, the tooth root tooth curve of structure and tooth top tooth curve conjugation, for avoiding contact at 2 simultaneously, internal Gear Root tooth curve has carried out correction of the flank shape, finally, the transition arc of internal gear and external gear teeth rooted tooth exterior feature and root circle is devised.The inside engaged gear line of contact that the method obtains is made up of two-part, the circular arc line of contact of part design, internal gear teeth top tooth curve and external gear teeth top tooth curve participate in engaging in this line of contact, another part is the line of contact that tooth top flank profil and the external gear teeth root tooth curve of internal gear participates in engaging, compared with involute internal messing flank profil, its contact ratio is larger, simultaneously due to internal-external gear tooth top flank profil conjugation, the wide conjugation wide with internal gear teeth apical tooth of external gear teeth rooted tooth, internal gear teeth rooted tooth exterior feature have passed through correction of the flank shape, gear tooth profile can not be produced interfere, degrees of freedom is larger in design.

The technical solution used in the present invention is: a kind of large contact ratio inside engaged gear tooth Shape Design method based on circular arc line of contact, it is characterized in that: the method connects the design line of contact of circular arc as inside engaged gear of internal gear, external gear teeth tip circle intersection point and pitch circle intersection point using one section, by theory of plane engagement, set up the Conjugate Equation about corresponding relation between line of contact and inside engaged gear conjugate curve; According to Conjugate Equation, structure meets the conjugate curve of circular arc line of contact, and this conjugate curve is internal gear teeth top tooth curve and external gear teeth top tooth curve; Utilize conjugate principle to construct the tooth root flank profil with tooth top tooth curve conjugation, and internally Gear Root flank profil carry out correction of the flank shape; Transition curve between design gear teeth tooth root flank profil and root circle.The method concrete steps are as follows:

Step one, selection connect the design line of contact of circular arc as inside engaged gear of internal gear, external gear teeth tip circle intersection point and pitch circle intersection point, according to theory of plane engagement, set up the Conjugate Equation of inside engaged gear transmission, this equation has reacted the corresponding relation on point in line of contact and inside engaged gear conjugate curve between conjugate point;

Step 2, structure meet the internal-external gear conjugation tooth top tooth curve of design circular arc line of contact.Equation of engaged line is brought into Conjugate Equation, obtain the corresponding relation formula of circular arc line of contact and conjugate curve, by transformation of coordinates and calculating, be met the conjugate curve of circular arc line of contact, because circular arc line of contact is between top circle and pitch circle, only have tooth top flank profil to participate in engagement, the conjugate curve therefore obtained is internal gear teeth top tooth curve and external gear teeth top tooth curve;

The tooth root tooth curve of step 3, structure internal gear and external gear.The tooth top tooth curve of known internal gear and external gear and relative motion relation between them, according to conjugate principle, structure and the tooth root tooth curve of the external gear of internal gear teeth top tooth curve conjugation and the internal gear teeth rooted tooth exterior feature with the wide conjugation of external gear teeth apical tooth respectively; Engage with the tooth top flank profil of external gear when tooth top flank profil and the tooth root flank profil of internal gear while, generation 2 contacted, internal Gear Root flank profil carries out correction of the flank shape, avoids two-point contact in the engagement process of flank profil simultaneously;

Step 4, design internal-external gear gear teeth tooth root and root circle between transition curve.Determine tangential at end points place of gear teeth tooth root tooth curve, structure and root circle and tooth root tooth curve are at the tangent arc transition curve of end points.

The present invention's advantage is compared with prior art:

Proposed by the invention is a kind of tooth, outstanding feature is one section of line of contact is wherein the circular arc connecting top circle intersection point and pitch circle intersection point, there is the advantage that contact ratio is large, large more than 2 times of contact ratio compared with involute profile, multiple gear teeth participate in engagement, and bearing capacity is strong, and dynamic impulsion is little, relative slippage is low simultaneously, and lubricating condition is good; Due to the wide conjugation wide with external gear teeth apical tooth of internal gear teeth apical tooth, the wide conjugation wide with external gear teeth rooted tooth of internal gear teeth apical tooth, interior flank profil tooth root flank profil, through correction of the flank shape, can not produce the problem of interference, and gear parameter design is freer.Along with the raising of multi-axis linkage numerical control milling, grinding accuracy, for the manufacturing of tooth provides solution.

Accompanying drawing explanation

Fig. 1 is a kind of novel large contact ratio inside engaged gear tooth Shape Design method flow diagram of the present invention;

Fig. 2 is the design diagram of circular arc line of contact;

PP ₀for connecting the circular arc of top circle intersection point and pitch circle intersection point;

The line of contact schematic diagram of internal-external gear before Fig. 3 practices Buddhism or Taoism;

PP ₀for the wide line of contact wide with internal gear teeth apical tooth of external gear teeth apical tooth, P ₀p' is the wide line of contact wide with internal gear teeth apical tooth of external gear teeth rooted tooth, P ₀p " be the wide line of contact wide with external gear teeth apical tooth of internal gear teeth rooted tooth before correction of the flank shape;

Fig. 4 internal gear teeth root profile modification schematic diagram;

Γ _a2for internal gear teeth apical tooth is wide, Γ _f2for internal gear teeth rooted tooth before correction of the flank shape is wide, Γ ' _f2for internal gear teeth rooted tooth after correction of the flank shape is wide;

Fig. 5 tooth root flank profil is to the transition arc organigram of root circle;

Circle O ₀for with tooth root flank profil Γ _fthe circle tangent with flank profil circle, T ₀, T _fbe respectively point of contact, t is at T ₀that locates is tangential;

Fig. 6 external gear and internal gear monodentate shape;

Reference character implication in figure:

1,6-top circle; 4,9-transition arc curve; 5,10-root circle; 2-external gear teeth apical tooth is wide; 3-external gear teeth rooted tooth is wide; 7-internal gear teeth apical tooth is wide; 8-internal gear teeth rooted tooth is wide;

Fig. 7 internal-external gear global shape and line of contact.

Embodiment

Elaborate below in conjunction with the mode of execution of accompanying drawing to the inventive method.

The flow chart of a kind of large contact ratio inside engaged gear tooth Shape Design method based on circular arc line of contact of the present invention as shown in Figure 1, comprise the design line of contact of circular arc as inside engaged gear that (1) selects to connect internal gear, external gear teeth tip circle intersection point and pitch circle intersection point, utilize theory of plane engagement, set up the Conjugate Equation about line of contact and inside engaged gear conjugate curve corresponding relation; (2) structure meets internal gear teeth top tooth curve and the external gear teeth top tooth curve of circular arc line of contact; (3) structure with the tooth root flank profil of tooth top tooth curve conjugation and internally Gear Root flank profil carry out correction of the flank shape; (4) transition curve of gear teeth tooth root flank profil and root circle is designed.The method concrete steps are as follows:

Step one, design connect the circular arc line of contact of internal gear, external gear teeth tip circle intersection point and pitch circle intersection point, set up Conjugate Equation and about line of contact and inside engaged gear conjugate curve corresponding relation.

In gear transmission process, along with point of contact slide in line of contact time, the track that gear system of coordinates marks participates in meshed gears tooth curve exactly.For improving the contact ratio of inside engaged gear, using the line of contact of circular arc as inside engaged gear connecting internal gear, external gear teeth tip circle intersection point and pitch circle intersection point.As shown in Figure 2, P is internal gear and external gear teeth tip circle intersection point, P ₀internal gear and external gear pitch circle intersection point, PP ₀that the center of circle is at Y _ftie point P and P on axle ₀one section of circular arc, according to geometrical principle, circular arc PP can be solved easily ₀radius R.The circular arc line of contact obtained is at system of coordinates S _fon equation representation be:

$\{\begin{array}{c}{x}_f=R\·\sin \mathrm{\θ}\\ y_f=-R\·\cos \mathrm{\θ}+R\end{array}---\left(1\right)$

Wherein, x _f, y _ffor contact points P is at rectangular coordinate system S _fon coordinate value.Because circular arc line of contact is between the top circle and pitch circle of internal gear and external gear, tooth top flank profil is only had to participate in engagement, so be the tooth top flank profil of internal gear and external gear by the conjugate profiles of this circular arc line of contact structure.This line of contact is the line of contact of flank profil on the left of flank profil and internal gear on the right side of external gear, and generally speaking the gear teeth of gear are symmetrical, and its line of contact is about coordinate axes Y _fsymmetrical.

Set up rectangular coordinate system S ₁: { O ₁, X ₁, Y ₁, S2:{O ₂, X ₂, Y ₂be connected with external gear and internal gear respectively, system of coordinates S _f: { O _f, X _f, Y _fbe fixed coordinate system, and O _ffor node, then O _fon gears meshing line of action.If be respectively the rotation angle of external gear and internal gear, velocity ratio is z ₂, z ₁be respectively the number of teeth of external gear and internal gear, R ₁, R ₂be respectively the Pitch radius of external gear and internal gear.

If line of contact is at rectangular coordinate system S _fon vector expression be:

$r_f=\left[\begin{array}{c}{x}_f\left(\mathrm{\θ}\right)\\ y_f\left(\mathrm{\θ}\right)\\ 1\end{array}\right]---\left(2\right)$

In Meshing Process of Spur Gear, point of contact is in line of contact and along line of contact slip, point of contact is at rectangular coordinate system S ₁and S ₂the track of upper slip is exactly the tooth curve of external gear and internal gear.Conjugate point in line of contact is at rectangular coordinate system S ₁on the vector representation of track be:

Wherein, M _1ffor rectangular coordinate system S _fto S ₁transformation matrix of coordinates.

Conjugate point in line of contact is at rectangular coordinate system S ₂on the vector representation of track be:

Wherein, M _2ffor rectangular coordinate system S _fto S ₁transformation matrix of coordinates.

According to face gear mesh theory, the common normal line of conjugate tooth profile curve at point of contact place and gyration center line O ₁o ₂intersect and this line is divided into two sections of O ₂p and O ₁p, the ratio of two line segments is gear ratio.Then obtain line of contact is put the normal vector of P at rectangular coordinate system S _fon vector representation be:

$n_f=\frac{1}{\sqrt{x_f^2+y_f^2}}\·\left[\begin{array}{c}{x}_f\\ y_f\\ 0\end{array}\right]---\left(7\right)$

At rectangular coordinate system S ₁on the normal vector of some P be expressed as:

The mesh equation of the external gear obtained according to formula (7), (8) is:

Draw from above formula:

Formula (10) integration is obtained:

The representation obtaining the angular displacement of internal gear according to formula (2) is:

Circular arc Equation of engaged line formula (1) is brought into formula (11) and formula (12) obtains:

Formula (13), (14) represent the corresponding relation on point in circular arc line of contact and conjugate curve between point of contact.

Step 2, structure meet the internal gear of circular arc line of contact and the conjugation tooth top tooth curve of external gear

Wushu (13), (14) bring formula (3) into, (5) obtain the representation of internal gear and external gear teeth top tooth curve:

${\mathrm{\Γ}}_{a1}:r_{a1}\left(\mathrm{\θ}\right)=\left[\begin{array}{c}R\·sin(\mathrm{\θ}\·(1-R/R_1))+(R-R_1)\·sin(\mathrm{\θ}\·R/R_1)\\ -R\·cos(\mathrm{\θ}\·(1-R/R_1))+(R-R_1)\·cos(\mathrm{\θ}\·R/R_1)\\ 1\end{array}\right]---\left(15\right)$

${\mathrm{\Γ}}_{a2}:r_{a2}\left(\mathrm{\θ}\right)=\left[\begin{array}{c}R\·sin(\mathrm{\θ}\·(1-R/R_2))+(R-R_2)\·sin(\mathrm{\θ}\·R/R_2)\\ -R\·cos(\mathrm{\θ}\·(1-R/R_2))+(R-R_2)\·cos(\mathrm{\θ}\·R/R_2)\\ 1\end{array}\right]---\left(16\right)$

Step 3, structure are with the internal-external gear tooth root flank profil of tooth top flank profil conjugation and internal Gear Root flank profil carries out correction of the flank shape

The tooth top tooth curve of known external gear and internal gear is respectively Γ _a1, Γ _a2, suppose and internal gear teeth top tooth curve Γ _a2the external gear teeth root tooth curve of conjugation is Γ _f1, with external gear teeth top tooth curve Γ _a1the external gear teeth root tooth curve of conjugation is Γ _f2, then external gear teeth root tooth curve Γ _f1equation can be expressed as:

Wherein, M ₁₂for system of coordinates S ₁to system of coordinates S ₂transformation matrix of coordinates, formula (18) is mesh equation, and according to Principles of Gear Connection, mesh equation can be expressed as:

Wherein, N _ffor at fixed coordinate system S _fin unit normal vector, for conjugate profiles is at the relative velocity at point of contact place, their equation representation is:

Wushu (19), (20) are brought equation of meshing (18) into and are carried out abbreviation and obtain:

Formula (21) has two solutions:

First solution identical with formula (14), demonstrating internal gear teeth apical tooth wide is conjugation with the tooth top tooth curve of external gear;

Second solution provide the tooth root tooth curve of the external gear of the wide conjugation with internal gear teeth apical tooth. formula (24) being brought into the equation representation that formula (15) obtains the external gear teeth root tooth curve of the wide conjugation with internal gear teeth apical tooth is:

${\mathrm{\Γ}}_{f1}:r_{f1}\left(\mathrm{\θ}\right)=\left[\begin{array}{c}(R_1-R_2+R)\·sin\left(\right(R_2-R)/R_1\·\mathrm{\θ})-(R_2-R)\·sin\left(\right(R_1-R_2+R)/R_1\·\mathrm{\θ})\\ -(R_1-R_2+R)\·cos\left(\right(R_2-R)/R_1\·\mathrm{\θ})-(R_2-R)\·\cos \left(\right(R_1-R_2+R)/R_1\·\mathrm{\θ})\\ 1\end{array}\right]---\left(25\right)$

Internal gear teeth apical tooth is wide with the line of contact of external gear teeth root tooth curve is:

Then this line of contact is one section of radius is R-R ₂circular arc, parameter θ (0< θ < θ _max), then this line of contact is tangent in the circular arc line of contact of Nodes and design.

Equally, with the internal gear teeth root tooth profile curve equation representation of external gear teeth top tooth curve conjugation be:

${\mathrm{\&Gamma;}}_{f2}:r_{f2}\left(\mathrm{\&theta;}\right)=\left[\begin{array}{c}(R_2-R_1+R)\&CenterDot;sin\left(\right(R_1-R)/R_2\&CenterDot;\mathrm{\&theta;})-(R_1-R)\&CenterDot;sin\left(\right(R_2-R_1+R)/R_2\&CenterDot;\mathrm{\&theta;})\\ -(R_2-R_1+R)\&CenterDot;cos\left(\right(R_1-R)/R_2\&CenterDot;\mathrm{\&theta;})-(R_1-R)\&CenterDot;\cos \left(\right(R_2-R_1+R)/R_2\&CenterDot;\mathrm{\&theta;})\\ 1\end{array}\right]---\left(27\right)$

External gear teeth apical tooth is wide with the line of contact of internal gear teeth root tooth curve is:

Shown in Fig. 3, PP ₀the connection top circle intersection point of design and the circular arc line of contact of pitch circle intersection point, the wide Γ of internal gear teeth apical tooth _a2Γ wide with external gear teeth apical tooth _a1at line of contact PP ₀upper engagement, P ₀p' is the wide Γ of internal gear teeth apical tooth _a2Γ wide with external gear teeth rooted tooth _f1line of contact, P ₀p " be the wide Γ of internal gear teeth rooted tooth _f2Γ wide with external gear teeth apical tooth _a1line of contact, as the tooth root flank profil Γ of internal gear _f2Γ is engaged with external gear teeth apical tooth exterior feature _a1at P ₀p " on when engaging, the tooth top flank profil Γ of internal gear _a2with the tooth top flank profil Γ of external gear _a1also at line of contact PP ₀upper engagement, for avoiding engage at 2 simultaneously, needs internal Gear Root flank profil Γ _f2carry out correction of the flank shape.

Internal gear teeth root profile modification is at Γ _f2correction of the flank shape is carried out on basis, as shown in Figure 4, if T is (x _f2, y _f2) be internal gear teeth root tooth curve Γ _f2on a bit, it is to internal gear center O ₂line and coordinate axes Y _fangle be δ, then have relation:

$\mathrm{\&delta;}=arccos\left(\frac{R_2-y_{f2}}{x_{f2}^2+{(R_2-y_{f2})}^2}\right)---\left(29\right)$

Then for internal gear teeth root tooth curve Γ _f2on node P ₀have δ=0, the T that sets up an office is around internal gear center O ₂be rotated counterclockwise an angle to some T', T' to internal gear center O ₂line and coordinate axes Y _fangle δ ' (δ) be monotonic increase continuous function about δ, and δ ' (0)=0, then the tooth root tooth curve Γ ' formed by T' _f2be continuous print and at node P ₀with tooth top tooth curve Γ _a2tangent, for the sake of simplicity, δ ' (δ)=δ/2 can be got.

Step 4, design internal-external gear tooth root flank profil and root circle between transition curve

Determine that gear teeth tooth root tooth curve is at end points T ₀the tangential t at place, structure and root circle and tooth root tooth curve Γ _fat the arc transition curve T that end points is tangent ₀t _f., as shown in Figure 5.

Concrete example illustrates below:

With the inside engaged gear of number of teeth z1=35, z2=53, m=4 for example, the mode of execution of brief description this method.Other parameters are ha1=4mm, hf1=5mm, ha2=4mm, hf2=5mm.

The circular arc line of contact of step one, design is the one section of circular arc connecting external gear and internal gear teeth tip circle intersection point and pitch circle intersection point, and the radius obtaining circular arc according to geometrical principle is R=84.7273mm, radian θ _max=0.6932rad, its equation representation is:

$\{\begin{array}{c}{x}_f=84.7273\&CenterDot;\sin \mathrm{\&theta;}\\ y_f=-84.7273\&CenterDot;\cos \mathrm{\&theta;}+84.7273\end{array}---\left(30\right)$

Wherein, the span of parameter θ is 0≤θ≤0.6932.

The Pitch radius of external gear and internal gear is R ₁=z ₁× m/2, R ₂=z ₂× m/2, the corresponding relation on the point and conjugate curve that obtain in circular arc line of contact according to deriving between point of contact:

Wherein, formula (31), (32) illustrate when external gear and internal gear conjugate profiles rotate respectively from initial position radian, by the point (x in the circular arc line of contact of design _f, y _f) engagement.

The structure of step 2, internal gear and external gear teeth top tooth curve

According to formula (15) and (16), the tooth top tooth profile curve equation of internal gear and external gear is,

${\mathrm{\&Gamma;}}_{a1}:r_{a1}\left(\mathrm{\&theta;}\right)=\left[\begin{array}{c}84.7273\&CenterDot;sin(-0.2104\&CenterDot;\mathrm{\&theta;})+14.7273\&CenterDot;sin(1.2104\&CenterDot;\mathrm{\&theta;})\\ -84.7273\&CenterDot;cos(-0.2104\&CenterDot;\mathrm{\&theta;})+14.7273\&CenterDot;cos(1.2104\&CenterDot;\mathrm{\&theta;})\end{array}\right]---\left(33\right)$

${\mathrm{\&Gamma;}}_{a2}:r_{a2}\left(\mathrm{\&theta;}\right)=\left[\begin{array}{c}84.7273\&CenterDot;sin(0.2007\&CenterDot;\mathrm{\&theta;})-21.2727\&CenterDot;sin(0.7993\&CenterDot;\mathrm{\&theta;})\\ -84.7273\&CenterDot;cos(0.2007\&CenterDot;\mathrm{\&theta;})-21.2727\&CenterDot;cos(0.7993\&CenterDot;\mathrm{\&theta;})\end{array}\right]---\left(34\right)$

Wherein, the span of θ is 0≤θ≤0.6932.Profile of tooth point is above as shown in table 1.

The profile of tooth point of table 1 external gear teeth apical tooth exterior feature

The profile of tooth point of table 2 internal gear teeth apical tooth exterior feature

The tooth root tooth curve of step 3, structure internal gear and external gear

According to formula (25), be expressed as with the external gear teeth root tooth profile curve equation of internal gear teeth top tooth curve conjugation:

${\mathrm{\&Gamma;}}_{f1}:r_{f1}\left(\mathrm{\&theta;}\right)=\left[\begin{array}{c}48.7273\&CenterDot;sin(0.3039\&CenterDot;\mathrm{\&theta;})-21.2727\&CenterDot;sin(0.6961\&CenterDot;\mathrm{\&theta;})\\ -48.7273\&CenterDot;cos(0.3039\&CenterDot;\mathrm{\&theta;})-21.2727\&CenterDot;cos(0.6961\&CenterDot;\mathrm{\&theta;})\end{array}\right]---\left(35\right)$

Internal gear teeth apical tooth is wide with the line of contact of external gear teeth root tooth curve is:

$r_{a2-f1}\left(\mathrm{\&theta;}\right)=\left[\begin{array}{c}-21.2727\&CenterDot;sin\mathrm{\&theta;}\\ -21.2727\&CenterDot;\cos \mathrm{\&theta;}+106\end{array}\right]---\left(36\right)$

With the internal gear teeth root tooth profile curve equation representation of external gear teeth top tooth curve conjugation be:

${\mathrm{\&Gamma;}}_{f2}:r_{f2}\left(\mathrm{\&theta;}\right)=\left[\begin{array}{c}{x}_{f2}\\ y_{f2}\end{array}\right]=\left[\begin{array}{c}120.7273\&CenterDot;\sin (-0.1389\&CenterDot;\mathrm{\&theta;})+120.7273\&CenterDot;\sin (1.1389\&CenterDot;\mathrm{\&theta;})\\ -120.7273\&CenterDot;\cos (-0.1389\&CenterDot;\mathrm{\&theta;})+120.7273\&CenterDot;\cos (1.1389\&CenterDot;\mathrm{\&theta;})\end{array}\right]---\left(37\right)$

Wherein, the span of θ is 0≤θ≤0.6932.Point T (x on internal gear teeth rooted tooth exterior feature _f2, y _f2) to internal gear center O ₂line and coordinate axes Y _fangle be δ:

$\mathrm{\&delta;}=arccos\left(\frac{106-y_{f2}}{\sqrt{x_{f2}^2+{(106-y_{f2})}^2}}\right)---\left(38\right)$

Point T is to internal gear center O ₂distance r ₀for:

$r_0=\sqrt{x_{f2}^2+{(106-y_{f2})}^2}---\left(39\right)$

When a T is around internal gear center O ₂rotate an angle to some T', this time point T' and internal gear center O ₂line and coordinate axes X _fthe angle formed be δ ', δ ' be the monotonic increase continuous function of δ:

δ′(δ)＝5δ ²(40)

Then after correction of the flank shape, the wide curve of internal gear teeth rooted tooth is expressed as:

${\mathrm{\&Gamma;}}_{f2}^{\&prime;}:r_{f2}\left(\mathrm{\&theta;}\right)=\left[\begin{array}{c}{x}_{f2}^{\&prime;}\\ y_{f2}^{\&prime;}\end{array}\right]=\left[\begin{array}{c}-r_0\&CenterDot;sin{\mathrm{\&delta;}}^{\&prime;}\\ -r_0\&CenterDot;\cos {\mathrm{\&delta;}}^{\&prime;}\end{array}\right]---\left(41\right)$

Internal gear teeth rooted tooth exterior feature after practicing Buddhism or Taoism will not engage with external gear teeth apical tooth exterior feature.The wide profile of tooth point of the external gear teeth rooted tooth obtained is as shown in table 2, and the profile of tooth point before and after internal gear teeth root profile modification is as shown in table 3.

The wide profile of tooth point of table 3 external gear teeth rooted tooth

Profile of tooth point before and after table 4 internal gear teeth root profile modification

Step 4, design gear teeth tooth root and root circle between transition curve

The end points of external gear teeth root tooth curve is T ₁₀(0.3180 ,-66.4937), at end points T ₁₀tangent direction t, the root circle center O of external gear ₁(0,0), radius r _f1=65mm, by geometry solving, obtains at a T ₁₀tangent direction be t=(0.1349,0.9909) and with external gear root circle outside be cut in a T _f1one section of circular arc T ₁₀t _f1center O ₁₀(2.0653 ,-66.7315), radius r ₁₀=1.7634mm, T ₁₀with external gear center O ₁line and the included angle of coordinate axes X1 _min=1.5335rad, T _f1with external gear center O ₁line and the included angle of coordinate axes X1 _maax=3.0063rad, then the wide transition arc equation representation to root circle of external gear teeth rooted tooth is:

$\begin{array}{cc}\left\{\begin{array}{c}{x}_{fillet1}=1.7634\&CenterDot;\cos \mathrm{\&phi;}+2.0653\\ y_{fillet1}=1.7634\&CenterDot;\sin \mathrm{\&theta;}-66.7315\end{array}\right.& (1.5335\&le;\mathrm{\&phi;}\&le;3.0063)\end{array}---\left(41\right)$

The end points of internal gear teeth root tooth curve is T ₂₀(-1.092 ,-109.7977), at end points T ₂₀tangent direction t, the root circle center O of external gear ₂(0,0), radius r _f2=111mm, by geometry solving, obtains at a T ₂₀tangent direction is t=(-0.3896 ,-0.9210) and is inscribed within a T with internal gear root circle _f2one section of circular arc T ₂₀t _f2center O ₂₀(-2.8503 ,-109.0541), radius r ₂₀=1.9087mm, T ₂₀with internal gear center O ₂line and the included angle of coordinate axes X2 _min=-1.5969rad, T _f2with internal gear center O ₂line and the included angle of coordinate axes X2 _maax=-0.4002rad, then the wide transition arc equation representation to root circle of internal gear teeth rooted tooth is:

$\begin{array}{cc}\left\{\begin{array}{c}{x}_{fillet2}=1.9087\&CenterDot;\cos \mathrm{\&phi;}-2.8503\\ y_{fillet2}=1.9087\&CenterDot;\sin \mathrm{\&theta;}-109.0541\end{array}\right.& (-1.5969\&le;\mathrm{\&phi;}\&le;-0.4002)\end{array}---\left(41\right)$

External gear and internal gear teeth rooted tooth are wide as shown in table 5 to the point on the transition curve of root circle.

Table 5 external gear and internal gear transition curve profile of tooth point

The external gear obtained and internal gear teeth are as shown in Figure 6 and Figure 7, Fig. 6 is external gear and internal gear monodentate shape, be made up of top circle, tooth top flank profil, tooth root flank profil, transition arc curve and root circle respectively, wherein external gear teeth apical tooth is wide eats apical tooth exterior feature with internal gear and engages, and its line of contact is the circular arc PP designed ₀, external gear teeth rooted tooth exterior feature and internal gear teeth top gears meshing, line of contact is circular arc P ₀p ", the overall profile of tooth of external gear and internal gear and line of contact as shown in Figure 7.

Non-elaborated part of the present invention belongs to the known technology of related domain.

The above; be only part embodiment of the present invention, but protection scope of the present invention is not limited thereto, any those skilled in the art are in the technical scope that the present invention discloses; the change that can expect easily or replacement, all should be encompassed within protection scope of the present invention.

Claims (1)

1. the large contact ratio inside engaged gear tooth Shape Design method based on circular arc line of contact, it is characterized in that: the method connects the design line of contact of circular arc as inside engaged gear of internal gear, external gear teeth tip circle intersection point and pitch circle intersection point using one section, by theory of plane engagement, set up the Conjugate Equation about corresponding relation between line of contact and inside engaged gear conjugate curve; According to Conjugate Equation, structure meets the conjugate curve of circular arc line of contact, and this conjugate curve is internal gear teeth top tooth curve and external gear teeth top tooth curve; Utilize conjugate principle to construct the tooth root flank profil with tooth top tooth curve conjugation, and internally Gear Root flank profil carry out correction of the flank shape; Transition curve between design gear teeth tooth root flank profil and root circle, the method concrete steps are as follows:

Step one, selection connect the design line of contact of circular arc as inside engaged gear of internal gear, external gear teeth tip circle intersection point and pitch circle intersection point, according to theory of plane engagement, set up the Conjugate Equation of inside engaged gear transmission, this equation has reacted the corresponding relation on point in line of contact and inside engaged gear conjugate curve between conjugate point;

Step 2, structure meet the internal-external gear conjugation tooth top tooth curve of design circular arc line of contact, Equation of engaged line is brought into Conjugate Equation, obtain the corresponding relation formula of circular arc line of contact and conjugate curve, by transformation of coordinates and calculating, be met the conjugate curve of circular arc line of contact, because circular arc line of contact is between top circle and pitch circle, only have tooth top flank profil to participate in engagement, the conjugate curve therefore obtained is internal gear teeth top tooth curve and external gear teeth top tooth curve;

The tooth root tooth curve of step 3, structure internal gear and external gear, the tooth top tooth curve of known internal gear and external gear and relative motion relation between them, according to conjugate principle, structure and the tooth root tooth curve of the external gear of internal gear teeth top tooth curve conjugation and the internal gear teeth rooted tooth exterior feature with the wide conjugation of external gear teeth apical tooth respectively; Engage with the tooth top flank profil of external gear when tooth top flank profil and the tooth root flank profil of internal gear while, generation 2 contacted, internal Gear Root flank profil carries out correction of the flank shape, avoids two-point contact in the engagement process of flank profil simultaneously;

Step 4, transition curve between design internal-external gear gear teeth tooth root and root circle, determine tangential at end points place of gear teeth tooth root tooth curve, and structure and root circle and tooth root tooth curve are at the tangent arc transition curve of end points.