CN112541235A - Universal design method for hypoid gear pair - Google Patents

Abstract

The invention relates to the technical field of gear pair design, in particular to a general design method of a hypoid gear pair, which provides a hypoid gear space pitch cone parameter design and calculation flow under the working conditions of positive axis intersection angle and small axis intersection angle based on a space meshing theory; solving basic geometric parameters of the hypoid gear based on the geometric characteristics of the hypoid gear blank by the calculated parameters; the method optimizes a local synthesis method, provides a hypoid gear pair meshing characteristic presetting method and a processing parameter calculation flow under the working conditions of positive axis intersection angle and small axis intersection angle, and overcomes the defects that the traditional design method is poor in convergence, complex in design, required to be reversely adjusted for multiple times, only aiming at a single orthogonal working condition and the like; the method has high integration and high calculation efficiency, can cope with any target working condition, and realizes the integrated design and calculation of the geometric parameters of the gear, the basic parameters of the gear blank and the gear tooth processing parameters.

Description

A general design method of hypoid gear pair

technical field

The invention relates to the technical field of gear pair design, and relates to a general design method of a hypoid gear pair, in particular to a general design method of a hypoid gear pair considering the working conditions of positive axis intersecting angle and small axis intersecting angle.

Background technique

As one of the hot issues in the field of space gear transmission, hypoid gear transmission has the advantages of large coincidence, strong tolerance, high meshing stability, and excellent meshing performance. It is widely used in high-speed ship power transmission, four-wheel drive vehicle power transmission. Drives, aero-engine accessory drives, helicopter main rotor drives, automation equipment steering drives and other fields. Due to the complexity of its own geometric structure, the hypoid gear pair has high requirements for designers and design methods. At present, the existing hypoid gear pair design methods have the following characteristics:

1) Single scope of application: The existing hypoid gear pair design method is based on the orthogonal pitch cone tangent contact theory, and uses nonlinear equations to solve the unique gear geometric design parameter in the space that satisfies all the conditions of the tangent contact point. Due to the limitation of the design method itself, this condition is only for the case of positive axis intersection (axis intersection angle of 90°), and cannot be applied to other axis intersection angles;

2) The inversion of meshing characteristics is complex: the initial processing parameters of the hypoid gear pair are generally set first. Based on the mismatch design method, the processing parameters are inversely adjusted to achieve the target contact characteristics. This method has high accuracy and convergence, but requires more inversions. Second, the designer has higher requirements for anti-adjustment experience;

3) The calculation efficiency is not high: the existing hypoid gear pair design method is based on the Newton iteration method to solve the nonlinear equations of space tangent contact. Due to the complexity of the hypoid gear geometry, it contains many parameters, so the equations are solved. The time is long and the overall computing efficiency is low;

Therefore, based on the defects of the existing hypoid gear pair design methods and the characteristics of the hypoid gear pair, researchers in the field should invent new design methods to overcome the above-mentioned defects.

SUMMARY OF THE INVENTION

In view of this, in order to solve the problems that the original design method of the hypoid gear pair has a single application range, complex meshing characteristics reverse adjustment, and low calculation efficiency, the present invention provides a method that considers the working conditions of the normal axis intersection angle and the small axis intersection angle. A general design method for hypoid gear pairs.

In order to achieve the above object, the present invention provides a general design method for hypoid gear pairs considering the working conditions of the normal axis intersection angle and the minor axis intersection angle, including the following steps:

Step S1: For the working condition of space staggered axis orthogonal or small axis intersection, in the case of known shaft intersection angle Σ, offset distance E, transmission ratio i ₁₂ and large wheel outer end diameter d _ae2 , solve the problem that satisfies the two-section cone in Geometric parameters tangent to the known tangent contact point M in space: obtain the conical parameters of the hypoid gear pitch under the target condition, namely the pitch angle γ _m2 and γ _m1 of the large and small gears, and the installation distance of the large and small gears A _m2 , A _m1 and the helix angles β _m2 and β _m1 of the large and small wheels;

Step S2: According to the above-mentioned basic geometric parameters of the hypoid gear pitch cone that meet the conditions, select the shrinkage tooth system for the working condition, and calculate the basic parameters of the hypoid gear blank that meet the requirements of the working condition;

与第一、第二主方向

依据局部综合法，小轮与大轮在该点完全共轭，因此在该点设置准双曲面齿轮啮合特性参数，包括：接触迹线与齿面第一主方向夹角θ_cr、接触椭圆半长轴长度L_ce、传递误差一阶导数m₁₂。由此推算出满足啮合特性条件的小轮在该点的第一、第二主曲率

与第一、第二主方向

基于面铣展成法小轮加工原理，求解出满足该啮合特性条件下的唯一准双曲面齿轮小轮加工参数，包括径向刀位S_r1、角向刀位Q_r1、机床根锥角M_r1、水平轮位ΔX_B1、床位ΔX_A1、刀倾角I₁、刀转角J₁、滚比C₁₂和垂直轮位E_m1；Step S3: After determining all the geometric parameters of the gear that meet the working condition, based on the machining principle of the face milling method, the machining parameters of the hypoid gear can be solved, including: horizontal tool position H ₂ , radial tool position V ₂ , the machine tool root taper angle M _r2 , the horizontal wheel position ΔX _B2 and the bed position ΔX _A2 , select the contact reference point F, and calculate the first and second principal curvatures of the contact reference point F based on the Rogrids formula

with the first and second principal directions

According to the local synthesis method, the small wheel and the large wheel are completely conjugated at this point, so the meshing characteristic parameters of the hypoid gear are set at this point, including: the angle θ _cr between the contact trace and the first main direction of the tooth surface, the contact ellipse half The major axis length L _ce and the first derivative of the transmission error m ₁₂ . From this, the first and second principal curvatures of the small wheel at this point that satisfy the meshing characteristic conditions are calculated.

with the first and second principal directions

Based on the pinion machining principle of face milling generation method, the unique hypoid gear pinion machining parameters under the condition of meshing characteristics are solved, including radial tool position S _r1 , angular tool position Q _r1 , machine tool root taper angle M _r1 , horizontal wheel position ΔX _B1 , bed position ΔX _A1 , knife inclination angle I ₁ , knife rotation angle J ₁ , rolling ratio C ₁₂ and vertical wheel position E _m1 ;

Step S4: After completing the derivation of the geometric parameters of the hypoid gear, the basic parameters of the gear blank and the processing parameters for the above working conditions, a three-dimensional model of the hypoid gear pair can be established based on the reverse engineering of the tooth surface, considering the orthogonal and small axis intersection angles Due to the complex geometric characteristics of the hypoid gear tooth surface, it is necessary to check the geometric characteristic index of the gear pair tooth surface to ensure the accuracy of the gear tooth surface under this parameter. The three major tooth surface index detections include root undercut analysis, tooth tip change tip inspection, and tooth root smooth transition analysis to ensure the accuracy and quality of the tooth surface.

Further, in step S1, according to the known axis intersection angle Σ, offset distance E, diameter d _ae2 of the outer end of the large wheel, and three-dimensional coordinates (X _m , Y _m , Z _m ) of the spatial tangent contact point M, the following three geometrical Relationship:

In the formula, i ₁₂ represents the transmission ratio of the gear pair, and β _m12 is the difference between the helix angles of the large and small wheels.

Based on the forming principle and generation principle of the hypoid gear tooth surface, two constraints on the geometric characteristics of the hypoid gear pair are obtained:

和α_nlim分别表示极限压力角和极限法曲率。In the formula,

and α _nlim represent the ultimate pressure angle and ultimate normal curvature, respectively.

Based on the working condition of space staggered axis orthogonal or small axis intersection angle, according to the three geometric relations of gears and two limiting conditions, it is proposed to consider the helix angle of the large wheel and the auxiliary angle of the small wheel as the iterative variables, and the offset distance and the limit method curvature as the judgment How to iterate the condition. Among them, the inner loop is the offset distance loop, and the judgment conditions are:

|EE ^* |<0.001

where E ^* represents the offset distance of the hypoid gear after each iteration of the inner loop.

When the inner loop judgment condition is met, the inner loop ends, and the outer loop is performed. The judgment condition is:

表示选定的刀盘半径，

表示外循环每次迭代后准双曲面齿轮齿面极限法曲率。In the formula,

Indicates the radius of the selected cutter head,

Represents the ultimate curvature of the hypoid gear tooth surface after each iteration of the outer loop.

When the outer cycle judgment condition is satisfied, the cycle ends, and the unique hypoid gear pitch cone parameter that satisfies the target working condition is output.

Further, the general formula for the shrinkage mode of the face milling hypoid gear in step S2 is as follows:

In the formula, m _et is the modulus of the gear end face, R _e2 is the taper distance at the big end of the big wheel, and R _m2 is the taper distance at the reference point of the big wheel.

Further, the calculation formula of the machining parameters of the hypoid gear wheel in step S3 is as follows:

Based on the relationship between the basic parameters of the local synthesis method, the following equation can be obtained:

in,

In the formula, the angle between the first main direction of the meshing tooth surface of the small wheel and the large wheel is σ ₁₂ , and a ₁₁ , a ₁₂ and a ₂₂ are the basic equation coefficients of the local comprehensive method.

The equation for examining the three unknown coefficients is as follows:

in,

According to the local synthesis method, the following expressions can be obtained:

in,

From this, it can be deduced that the expression of the angle included in the first main direction of the meshing tooth surface of the small wheel and the large wheel is:

Among them, the expressions of the first principal curvature K _Ip and the second principal curvature K _IIp of the pinion at point F of the tooth surface are as follows:

According to the above equation, the first and second main directions of the pinion tooth surface at point F of the tooth surface can be obtained as follows:

Based on the main curvature and main direction of the pinion tooth surface at point F on the tooth surface, three meshing characteristic parameters are jointly preset: the angle θ _cr between the contact trace and the first main direction of the tooth surface, the length of the semi-major axis of the contact ellipse L _ce , the transmission The first derivative of the error m ₁₂ is used to solve the machining parameters of the hypoid gear pair by the face milling method: radial tool position S _r1 , angular tool position Q _r1 , machine tool root taper angle M _r1 , horizontal wheel position ΔX _B1 , The bed position ΔX _A1 , the knife inclination angle I ₁ , the knife rotation angle J ₁ , the rolling ratio C ₁₂ and the vertical wheel position E _m1 .

The beneficial effects of this program are:

1. The general design method of the hypoid gear pair disclosed in the present invention considering the working conditions of the normal axis intersection angle and the minor axis intersection angle is based on the orthogonal or minor axis intersecting angle working conditions of the space staggered axis, according to the three major geometric relationships of the gear and the two major constraints conditions, an iterative method is proposed in which the helix angle of the large wheel and the auxiliary angle of the small wheel are the iterative variables, and the offset distance and the limit method curvature are the judgment conditions. Angled hypoid gear.

2. The general design method of the hypoid gear pair disclosed in the present invention considering the working conditions of the normal axis intersecting angle and the minor axis intersecting angle, and based on the known spatial staggered axis working conditions, the quasi-hypoid gear pair considering the working conditions of the positive axis intersecting angle and the minor axis intersecting angle is determined. The general design process of the hyperboloid gear pair can efficiently solve the unique gear geometry and processing parameters that meet the working conditions, the preset meshing characteristic parameters, and the tooth surface shrinkage. Moreover, this method avoids the defects of traditional design methods, such as poor convergence, complex design, requiring multiple inversions, and only for a single orthogonal working condition. , the integrated design and calculation of the basic parameters of the gear blank and the machining parameters of the gear teeth.

3. The general design method of the hypoid gear pair disclosed in the present invention considering the working conditions of the intersection angle of the positive axis and the intersection angle of the minor axis is based on the theory of space section conical tangent contact, the local synthesis method and the machining principle of face milling hypoid gears, and is satisfied. Geometry and machining parameters of hypoid gear pairs for orthogonal and extreme conditions. The method overcomes the defects of the original technical object being single and complex inverse adjustment, and can not only be used for gear pairs in conventional working conditions, but also suitable for hypoid gear transmission under small axis intersection angles and extreme working conditions; at the same time, it improves the original In the technology, in order to achieve the target contact characteristics, it is necessary to reverse the characteristics of multiple times. The machine tool processing parameters are inversely calculated through the preset contact characteristics, which has high calculation efficiency and calculation accuracy, and can meet the target contact characteristics at one time without multiple reverse adjustments. Experience is not too demanding. This method greatly improves the defects of the original technology.

Other advantages, objects and features of the present invention will be set forth in the description which follows, to the extent that will be apparent to those skilled in the art based on a study of the following, or may be learned from is taught in the practice of the present invention. The objectives and other advantages of the present invention may be realized and attained by the following description.

Description of drawings

Fig. 1 is the basic geometric design parameter calculation flow chart of the hypoid gear pair of the orthogonal axis intersection angle and the minor axis intersection angle of the present invention;

Fig. 2 is the definition diagram of the geometric parameters of the hypoid gear pair blank of the present invention;

Fig. 3 is the calculation flow chart of the machining parameters of the hypoid gear pair of the orthogonal axis intersecting angle and the minor axis intersecting angle of the present invention;

Figure 4(a) is a schematic diagram 1 of the tooth surface defect of the hypoid gear pair of the present invention;

Figure 4(b) is a schematic diagram 2 of the tooth surface defect of the hypoid gear pair of the present invention;

Figure 5(a) is a schematic diagram 1 of the detection of the smooth transition of the tooth root of the hypoid gear pair according to the present invention;

Figure 5(b) is a schematic diagram 2 of the detection of the smooth transition of the tooth root of the hypoid gear pair according to the present invention;

Figure 5(c) is a schematic diagram 3 of the detection of the smooth transition of the tooth root of the hypoid gear pair according to the present invention;

FIG. 6 is a schematic diagram of detecting the tooth root tip width of a hypoid gear pair according to the present invention;

FIG. 7 is the meshing model of the hypoid gear pair in the orthogonal state of the present invention;

FIG. 8 is the meshing model of the hypoid gear pair under the state of the small axis intersection angle of the present invention;

Fig. 9 is the geometrical feature details and meshing characteristics of the tooth surface of the hypoid gear pair in the orthogonal state of the present invention;

Fig. 10 shows the details of the tooth surface geometry and meshing characteristics of the hypoid gear pair in the state of the small axis intersecting angle of the present invention.

Detailed ways

The following is further described in detail by specific embodiments:

A general design method of hypoid gear pair considering the working conditions of the intersection angle of the positive axis and the intersection angle of the minor axis, including the following steps:

Step S1: For the working condition of space staggered axis orthogonal or small axis intersection, in the case of known shaft intersection angle Σ, offset distance E, transmission ratio i ₁₂ and large wheel outer end diameter d _ae2 , solve the problem that satisfies the two-section cone in Geometric parameters of the known tangent point M in space: obtain the conical parameters of the hypoid gear pitch under the target working condition, namely the pitch angles of the large and small wheels γ _m2 , γ _m1 , the installation distances of the large and small wheels A _m2 , A _m1 and the helix angles β _m2 and β _m1 of the large and small wheels;

Among them, according to the characteristic that the space staggered shaft pitch cones are tangent to the known point M in space, the three geometric relations of the conical hypoid gear with arbitrary intersection angle pitches can be deduced: 1) the geometric relation of the center distance, 2) the geometric relation of the offset angle 3) The geometric relationship of the transmission ratio, the above three relationship have a direct geometric relationship with the basic parameters of the pitch cone.

According to the known shaft intersection angle Σ, offset distance E, diameter d _ae2 of the outer end of the big wheel, and three-dimensional coordinates (X _m , Y _m , Z _m ) of the spatial tangent contact point M, the following three relational expressions can be obtained:

In the formula, i ₁₂ represents the transmission ratio of the gear pair, and β _m12 is the difference between the helix angles of the large and small wheels.

Based on the forming principle and generation principle of the hypoid gear tooth surface, two constraints on the geometric characteristics of the hypoid gear pair are obtained:

和α_nlim分别表示极限压力角和极限法曲率。In the formula,

and α _nlim represent the ultimate pressure angle and ultimate normal curvature, respectively.

限定条件，齿轮齿向弧度又可通过极限法曲率

衡量。因此，求解准双曲面齿轮除了满足三大几何关系式，还需满足两大限定条件：极限压力角限定条件和极限法曲率限定条件；The ultimate pressure angle α _nlim is the only index to measure the formation of the concave-convex tooth surface for hypoid gears. There is a direct geometric relationship between the concave-convex tooth surface pressure angles α _cc and α _cv and the ultimate pressure angle α _nlim —limiting condition of the ultimate pressure angle α _nlim , if the pressure angle of the concave-convex tooth surface does not meet the above-mentioned limited conditions, the tooth surface will generate a singular point, the meshing equation cannot be solved, and the tooth surface will be distorted, resulting in undercutting. At the same time, based on the machining characteristics of the face milling generation method, there is a unique geometric relationship between the cutter head radius and the gear tooth radian—the limit method curvature

Constraints, the gear tooth radian can pass the limit method curvature

measure. Therefore, in order to solve the hypoid gear, in addition to satisfying the three geometric relations, it also needs to meet two constraints: the limit pressure angle limit and the limit method curvature limit;

Based on the working condition of space staggered axis orthogonal or small axis intersection angle, according to the three geometric relations of gears and two limiting conditions, it is proposed to consider the helix angle of the large wheel and the auxiliary angle of the small wheel as the iterative variables, and the offset distance and the limit method curvature as the judgment How to iterate the condition. Among them, the inner loop is the offset distance loop, and the judgment conditions are:

|EE ^* |<0.001

where E ^* represents the offset distance of the hypoid gear after each iteration of the inner loop.

When the inner loop judgment condition is met, the inner loop ends, and the outer loop is performed. The judgment condition is:

表示选定的刀盘半径，

表示外循环每次迭代后准双曲面齿轮齿面极限法曲率。In the formula,

Indicates the radius of the selected cutter head,

Represents the ultimate curvature of the hypoid gear tooth surface after each iteration of the outer loop.

When the outer cycle judgment condition is satisfied, the cycle ends, and the hypoid gear pitch cone parameters for the target working condition are output.

Step S2: According to the above-mentioned basic geometric parameters of hypoid gear sub-pitch cones that meet this condition, select the shrinkage tooth system for this working condition, and calculate the geometric parameters of the hypoid gear that meet the requirements of this working condition; the gear tooth surface shrinkage method is: The very important geometric index of the arc hypoid gear is different from the existing standard, the present invention considers the influence of different axis intersection angles on the pitch cone distance, and deduces the hypoid gear shrinkage mode suitable for all intersection angles.

Step B2: The four parameters of the hypoid gear pair assembly relationship are: offset distance E, axis intersection angle Σ, large wheel installation distance A _m2 , small wheel installation distance A _m1 , the above four parameters fully constrain the hypoid gear assembly feature. The geometry parameters of fully constrained tooth surface mainly include: pitch cone angle γ _mi , face cone angle γ _ai , root cone angle γ _fi , distance Z _mi from pitch cone vertex to shaft intersection, Z _ai from face cone vertex to shaft intersection, root The distance Z _fi from the cone vertex to the axis intersection. Among them, when i=1, it means a small wheel, and when i=2, it means a big wheel. The above parameters are derived from the geometric relationship of the staggered-axis hypoid gear pair.

The general formula for the shrinkage mode of face milling hypoid gear is as follows:

In the formula, m _et is the modulus of the gear end face, R _e2 is the taper distance at the big end of the big wheel, and R _m2 is the taper distance at the reference point of the big wheel.

与第一、第二主方向

依据局部综合法，小轮与大轮在该点完全共轭，因此在该点设置准双曲面齿轮啮合特性参数，包括：接触迹线与齿面第一主方向夹角θ_cr、接触椭圆半长轴长度L_ce、传递误差一阶导数m₁₂。由此推算出满足啮合特性条件的小轮在该点的第一、第二主曲率

与第一、第二主方向

基于面铣展成法小轮加工原理，可求解出唯一准双曲面齿轮小轮加工参数，包括径向刀位S_r1、角向刀位Q_r1、机床根锥角M_r1、水平轮位ΔX_B1、床位ΔX_A1、刀倾角I₁、刀转角J₁、滚比C₁₂和垂直轮位E_m1；Step S3: After determining all the gear parameters that meet the working condition, based on the machining principle of the face milling method, the machining parameters of the large wheel of the hypoid gear can be solved, including: horizontal tool position H ₂ , radial tool position V ₂ , The machine tool root taper angle M _r2 , the horizontal wheel position ΔX _B2 and the bed position ΔX _A2 , the contact reference point F is selected, and the first and second principal curvatures of the contact reference point F are calculated based on the Rogrids formula

with the first and second principal directions

According to the local synthesis method, the small wheel and the large wheel are completely conjugated at this point, so the meshing characteristic parameters of the hypoid gear are set at this point, including: the angle θ _cr between the contact trace and the first main direction of the tooth surface, the contact ellipse half The major axis length L _ce and the first derivative of the transmission error m ₁₂ . From this, the first and second principal curvatures of the small wheel at this point that satisfy the meshing characteristic conditions are calculated.

with the first and second principal directions

Based on the pinion machining principle of face milling generation method, the unique hypoid gear pinion machining parameters can be solved, including radial tool position S _r1 , angular tool position Q _r1 , machine tool root taper angle M _r1 , horizontal wheel position ΔX _B1 , bed position ΔX _A1 , knife inclination angle I ₁ , knife rotation angle J ₁ , rolling ratio C ₁₂ and vertical wheel position E _m1 ;

Among them, (1) the machining parameters of the face milling generation method hypoid gear large wheel machine tool can be derived based on the face milling generation method hypoid gear machining process; (2) the preset contact reference point F and the tangent contact point M The inter-axial and radial distances can be represented by ΔX and ΔY, respectively. The distance can be defined as an offset, which represents the deviation between the center of the preset contact position and the tangent contact point; (3) According to the local comprehensive theory, there is a unique complete conjugate contact reference point F for the hypoid gear pair, and the tooth surface of the large gear The main direction and main curvature of the pinion tooth surface are equal only at this point. Based on this characteristic, three parameters of the meshing characteristic are preset: the angle θ _cr between the contact trace and the first main direction of the tooth surface, the length of the semi-major axis of the contact ellipse L _ce , and the first derivative of the transmission error m ₁₂ , so as to achieve complete control of meshing characteristics. the goal of. The above three parameters respectively control the width of the contact imprint, the direction of the contact imprint and the magnitude of the transmission error; (4) Based on the above preset parameters and the main direction and main curvature of the large wheel in the global coordinate system, the hypoid gear pair can be solved for small The main direction and main curvature of the wheel, based on the main direction and main curvature of the pinion, considering the machining principle of the hypoid gear by the face milling method, the machining parameters of the hypoid gear pinion are solved.

The formula for calculating the machining parameters of the hypoid gear wheel is as follows:

Based on the relationship between the basic parameters of the local synthesis method, the following equation can be obtained:

in,

In the formula, the angle between the first main direction of the meshing tooth surface of the small wheel and the large wheel is σ ₁₂ , and a ₁₁ , a ₁₂ and a ₂₂ are the basic equation coefficients of the local comprehensive method.

The equation for examining the three unknown coefficients is as follows:

in,

According to the local synthesis method, the following expressions can be obtained:

in,

From this, it can be deduced that the expression of the angle included in the first main direction of the meshing tooth surface of the small wheel and the large wheel is:

Among them, the expressions of the first principal curvature K _Ip and the second principal curvature K _IIp of the pinion at point F of the tooth surface are as follows:

According to the above equation, the first and second main directions of the pinion tooth surface at point F of the tooth surface can be obtained as follows:

Based on the main curvature and main direction of the pinion tooth surface at point F on the tooth surface, three meshing characteristic parameters are jointly preset: the angle θ _cr between the contact trace and the first main direction of the tooth surface, the length of the semi-major axis of the contact ellipse L _ce , the transmission The first derivative of the error m ₁₂ can be used to solve the machining parameters of the small wheel of the hypoid gear pair by the face milling method: radial tool position S _r1 , angular tool position Q _r1 , machine tool root taper angle M _r1 , horizontal wheel position ΔX _B1 , the bed position ΔX _A1 , the knife inclination angle I ₁ , the knife rotation angle J ₁ , the rolling ratio C ₁₂ and the vertical wheel position E _m1 .

Step S4: After completing the derivation of the geometric parameters of the hypoid gear, the basic parameters of the gear blank and the processing parameters for the above working conditions, a three-dimensional model of the hypoid gear pair can be established based on the reverse engineering of the tooth surface, considering the orthogonal and small axis intersection angles Due to the complex geometric characteristics of the hypoid gear tooth surface, it is necessary to check the geometric characteristics of the tooth surface to ensure the accuracy of the tooth surface under this parameter. The three major tooth surface index detections include root undercut analysis, tooth tip change tip inspection, and tooth root smooth transition analysis to ensure the accuracy and quality of the tooth surface.

Among them (1) the hypoid gear tooth surface is extremely sensitive to the geometric design parameters, slight parameter changes will cause changes in the geometric characteristics of the tooth surface, resulting in the change of the critical value of the tooth profile at the singular point of the tooth surface. Based on this feature, an undercut inspection is proposed. To ensure the quality of the tooth surface; (2) The tooth surface of the hypoid gear usually adopts high and low displacement, which also causes the tip width to become sharper, which affects the geometric characteristics, and the tip tip inspection can avoid this problem; (3) hypoid The tooth root part of the gear tooth surface has a unique direct relationship with the machine tool root taper angle. Based on the spatial projection relationship, iteratively satisfies the only parameter of the machine tool root taper angle that the hypoid gear tooth root segment is over-smooth.

As shown in Figure 1, the calculation flow chart of the basic geometric design parameters of the hypoid gear pair for the intersection angle of the positive axis and the intersection angle of the minor axis is, in order to facilitate the understanding of the present invention, the essence of the technical route is: based on the given input parameters, through iteration Key geometric variables, solve the basic geometric parameters of gear pairs suitable for a given working condition.

It should be noted that the input parameters of hypoid gear with small intersection angle are divided into two categories: given input parameters and calculated input parameters. Among them, the given input parameters include: the shaft intersection angle Σ, the offset distance E, the nominal radius of the cutter head r _co , the transmission ratio i ₁₂ and the pitch circle diameter d _ae2 at the outer end of the big wheel, these five parameters are determined by the design conditions, Once the design case is decided, it cannot be changed.

In addition, the calculation output parameters include: the number of teeth of the pinion N ₁ , the number of teeth of the large wheel N ₂ , the tooth width b ₂ of the large wheel and the helix angle β _m2 of the large wheel (this value is a given parameter, and the detailed solution process will be described later). Two iterative variables are included in the design flow: the large wheel helix angle γ _m2 and the large wheel auxiliary bias angle ε _m2 . The calculation formula of the initial value of the big wheel helix angle γ _m2 is as follows:

The calculation formula of the initial value of the auxiliary offset angle ε _m2 of the big wheel is as follows

Loop A represents the initial value calculation process of the two-iteration variables. The purpose of this loop is to provide the initial value for the entire design calculation process, so it is only used once in the entire loop. Based on the initial variables of the two iterations and the geometric relationship in Fig. 1, the basic geometric parameters of the pitch cone can be solved: the pitch angle of the pinion γ _m1 , the helix angle of the pinion β _m1 and the pitch radius r _m1 of the pitch reference point of the pinion. The paranoia distance E ^* under the iterative parameter can be obtained through the above parameters and the third geometric relationship of the hypoid gear with the small axis intersection angle. Therefore, the first judgment condition is as follows:

|EE ^* |≤A _E (3) In the formula, A _E represents the accuracy of the first judgment condition. When the difference between the calculated paranoia distance and the given offset distance is less than the judgment accuracy, proceed to the next step of calculation, otherwise, iterate the big round Auxiliary offset angle ε′ _m2 , the specific iterative equation is as follows:

Further, when the first judgment condition is established, according to the ultimate method curvature equation and the ultimate pressure angle formula, the ultimate method curvature and pressure angle of the large gear tooth surface can be obtained, and the second judgment condition is as follows:

In the formula, A _R represents the accuracy of the second judgment condition. When the calculation of the limit method curvature and the nominal radius of the cutter head meet the judgment accuracy, the final value of all the iterative variables that satisfy the two judgment conditions can be obtained. Otherwise, if the judgment conditions are met Under the condition that the auxiliary offset angle of the first large wheel remains unchanged, the pitch angle of the large wheel is iterated, and the specific iterative equation is as follows:

γ′ _m2 =′ _m2 +Δγ _m2 (6)

In the formula, Δγ _m2 is the iteration variable. All basic geometric parameters of pitch cones can be solved through the above iterations, including: pinion pitch angle γ _m1 , pinion helix angle β _m1 , pitch pitch r _m1 , pitch angle γ _m2 , pitch helix angle β _m2 and pinion pitch r _m2 . Among them, the iteration accuracy and the judgment accuracy can be freely controlled. In theory, the higher the iteration accuracy and the judgment accuracy, the more accurate the final iteration value, and the longer the calculation time of the whole loop. Here, the recommended iteration accuracy and judgment accuracy are 10 ^-3 mm.

FIG. 2 and FIG. 3 are respectively the definition diagram of the geometry parameters of the hypoid gear pair gear blank according to the present invention and the calculation flow chart of the hypoid gear pair machining parameters of the present invention for the normal axis intersection angle and the small axis intersection angle.

In order to facilitate the understanding of the present invention, the essence of the technical route is: based on the pre-set theoretical contact point and meshing characteristic parameters based on the conical reference point of the large wheel segment, and a method for solving the unique small wheel machining parameters that satisfy the meshing conditions. Among them, the entire tooth surface is only completely conjugated at this theoretical contact point, and the purpose of completely controlling the contact characteristics is achieved by presetting the contact characteristic parameters.

Based on the hypoid gear machining process, the local synthesis method and the tooth root traversal cycle model, the machining parameter calculation and meshing characteristics design process of the hypoid gear with small axis intersection angle are proposed, as shown in Figure 3.

Specifically, by solving the pitch cone model of the hypoid gear with the small axis angle and the basic parameters of the tooth surface, the basic geometric parameters of the tooth surface can be obtained. parameter. Through the above-mentioned cutter head parameters and processing parameters, a unique large wheel mathematical model can be established. By establishing a preset tooth flank meshing point, considering the distance from the flank reference point, a first loop variable is proposed to control the position of the contact imprint. According to the Rodriguez space curvature theorem, the principal curvature and principal direction at the reference point of the big wheel are solved. Based on the above process, a second cyclic variable is proposed, which is the machine tool root taper angle between the concave and convex surfaces of the gear, so as to control the excessive smoothness at the root of the concave and convex surfaces. The third, fourth and fifth cycle variables are the basic input variables of the local synthesis method, including the length of the long axis of the contact imprint, the angle between the contact trajectory and the main direction of the tooth surface, and the preset peak-to-peak value of the transmission error. The size and direction of the contact imprint are controlled by the above three variables. and transmission error size.

Further, based on the above-mentioned parameters of the local synthesis method and the assembly mathematical model of the hypoid gear with small intersection angle, the main curvature and main direction of the pinion tooth surface at the meshing point can be obtained. According to the machining parameters of the machine tool and the parameters of the cutter head, the tooth surface of the pinion is established according to the mathematical model of the pinion by the generation method.

Based on the above-established tooth surfaces of the large and small gears, the present invention simulates the gear meshing process through tooth surface contact analysis, and obtains the meshing characteristics of the hypoid gear pair with the small axis intersection angle, including contact marks and transmission errors. Considering the fine-tuning of meshing characteristics, a closed-loop tooth surface contact characteristic design process is established. According to different meshing characteristic indicators, a method of fine-tuning the corresponding cyclic variables and inversely calculating the machining parameters of large and small wheel machine tools is proposed.

On the other hand, for the hypoid gear pair with the small axis intersecting angle, the undercut is particularly prone to occur at the root of the pinion tooth surface, that is, the working tooth surface segment and the excessive arc segment cannot smoothly transition, as shown in Fig. 4(a) and Fig. 4(a). 4(b). It can be seen from the figure that when undercut occurs, the tooth surface cannot reach the tooth root segment smoothly, and the convergence of the meshing equation until the tooth root segment is solved becomes poor, and a high-quality solution cannot be found. According to the principle of gear meshing, the point where the meshing equation cannot be solved is defined as the singular point. In the process of tooth surface generation, the singular point can be expressed as the following equation:

In the formula, r ₁ represents the position vector of any point on the pinion tooth surface.

Based on the principle of gear meshing, the above equations can be converted into four matrices:

Based on the principle of gear meshing, the critical condition for the existence of singular points on the pinion tooth surface is:

The set of singular points on the pinion tooth surface can be expressed by the following equation:

表示齿面啮合方程，u和β分别表示小轮齿面齿廓参数和齿向参数，

表示齿轮展成过程中的转角。通过上述方程组可以求解出奇异点的边界u_lim。In the formula,

represents the tooth surface meshing equation, u and β represent the pinion tooth profile parameters and tooth direction parameters, respectively,

Indicates the rotation angle during gear generation. The boundary u _lim of the singular point can be solved by the above equation system.

Based on the contact characteristics of the tooth surface, it can be known that the minimum value u _min of the tooth profile of the working tooth surface is expressed as follows:

表示刀盘压力角，其中，i＝cc表示凹面，i＝cv表示凸面。小轴交角准双曲面齿轮齿面不产生根切的条件如下所示：In the formula, R _Fi is the radius of the transition arc segment,

represents the pressure angle of the cutter head, where i=cc represents the concave surface and i=cv represents the convex surface. The condition that the tooth surface of the hypoid gear with the small axis intersecting angle does not produce undercut is as follows:

u _lim < u _min (14)

It should be noted here that when the minimum value of the tooth profile parameter is greater than the minimum u value that produces an undercut, the undercut will not be generated. At the same time, there are many parameters that affect the undercut of the tooth surface. Almost all the basic design parameters of the tooth surface will have an effect on the undercut of the tooth surface. Therefore, it is necessary to discuss the basic design parameters of the tooth surface of the small-axis cross-angle gear.

与

的交点定义为B，B在平面Q中的坐标表达式为：Based on the goal of over-smooth hypoid gear root, an iterative method is proposed, as shown in Fig. 5(a), Fig. 5(b) and Fig. 5(c). The hypoid gear with small axis intersection angle is discretized into m×n points, where m represents the discrete points in the tooth profile direction of the tooth surface, and n represents the discrete points in the tooth surface and tooth direction. R _Avk and R _Ack represent the kth point of the root segment, where k ranges from 1 to n. A plane Q is established through the axis of rotation of the gear. The projections of the concave and convex surfaces on the plane Q are shown in the figure. The projection points of the convex and concave surfaces of the tooth surface on the R _Avk and R _Ack projection planes are C _k and V _k respectively, line segment

and

The intersection point of is defined as B, and the coordinate expression of B in the plane Q is:

与

的辅助参数，k₁和k₂为线段

与

的斜率，上述四个参数表达式如下所示：In the formula, x _b and y _b represent the abscissa and ordinate values of B in the plane Q, and b ₁ and b ₂ are the tooth root line segments

and

The auxiliary parameters of , k ₁ and k ₂ are line segments

and

The slope of the above four parameters is as follows:

和

的夹角σ₁表达式如下：In the formula, (x _c1 , y _c1 , z _c1 ), (x _cn , y _cn , z _cn ), (x _v1 , y _c1 , z _c1 ) and (x _vn , y _vn , z _vn ) represent the point R respectively Coordinate values of _Ac1 , R _Av1 , R _Acn and R _Avn in the big wheel coordinate system, after projection, the line segment

and

The included angle σ ₁ is expressed as follows:

和

的夹角σ_n表达式如下：line segment

and

The included angle σ _n is expressed as follows:

The sum of the two angles is:

Σσ=σ ₁ +σ _n (23)

When the sum of the two angles is closer to 0, the transition of the tooth surface is also smoother. As shown in Figure 5(b), when the intersection point B is not within the range of the tooth surface, although the angle between the concave surface and the convex surface is close to or even equal to 0 degrees, the root of the tooth still cannot transition smoothly, so it is necessary to limit the coordinate axis of point B ,details as follows:

x _n < x _b < x ₁ (24)

y _n <y _b <y ₁ (25)

The above conditions limit the intersection point B within the range of the tooth surface, avoiding the above situation.

Further, the geometry of the hypoid gear pair with the small axis angle is extremely sensitive to almost all gear parameters. Based on this, a detection method for the tip change of the tooth surface of the hypoid gear with the small axis angle is proposed, as shown in Figure 6. . In the figure, the tooth surface is discretized into a plurality of points, where m represents the discrete points in the tooth profile direction of the tooth surface, and n represents the dispersion in the tooth direction of the tooth surface. As shown in the figure, the first point near the tip of the small end is defined as (1, 1), the first point near the root of the small end is defined as (m, 1), and the first point near the tip of the big end is defined as (m, 1). It is defined as (1,n), and the first point near the root of the big end is defined as (m,n). The gear top points are defined as TA _Cj and TA _Vj , TA _Cj is the concave tooth vertex, TA _Vj is the convex tooth vertex, and the tooth tip width is defined as DT _CVj , where j represents the number of the discrete points of the hypoid gear tooth tip, The range is 1 to n. The addendum width expression is as follows:

In the formula, (x _cj , y _cj , z _cj ) and (x _vj , y _vj , z _vj ) represent the Cartesian coordinate system of the tooth vertex sum, respectively.

Here, in order to ensure sufficient contact strength of the tooth surface, based on the gear manual, the tooth tip width must meet the following conditions:

DT _CVj >0.3m _n (27)

In the formula, m _n represents the normal modulus of the gear.

Example 1 is a meshing model of a hypoid gear pair with a positive axis intersection angle, and Example 2 is a hypoid gear pair meshing model with a small axis intersection angle.

Table 1 Hypoid gear counter pitch cone design parameters

Table 2 Basic parameters of tooth surface geometry of hypoid gear pair

Table 3 Processing parameters of hypoid gear pair large wheel

Table 4 Preset contact characteristic parameters of hypoid gear pair

Table 5. Processing parameters of hypoid gear pinion

Based on the data in Tables 1 to 5 above, the meshing model of the hypoid gear pair with the orthogonal angle and the minor axis angle can be established, as shown in Figures 7 and 8.

Figure 9 and Figure 10 respectively show the geometrical details and meshing characteristics of the hypoid gear pair model with orthogonal angle and small cross angle established after geometric inspection. The contact imprint and transmission error are close to the preset parameters. Contact analysis verified the reliability of the method.

To sum up, compared with the existing methods, the implementation method of the present invention has target diversity, high computational efficiency, and accuracy. At the same time, the meshing characteristic parameters can be designed in advance through the preset method, without the need for reverse adjustment. The above examples have all verified the effectiveness and practicability of the method, so the present invention has high practical significance and promotion value in the field of the industry.

The above descriptions are only examples of the present invention, and common knowledge such as well-known specific structures and characteristics in the solution are not described too much here. It should be pointed out that for those skilled in the art, on the premise of not departing from the structure of the present invention, several modifications and improvements can also be made, which should also be regarded as the protection scope of the present invention, and these will not affect the implementation of the present invention. Effects and applicability of the present invention.

Claims (4)

1. a general design method of hypoid gear pair, is characterized in that: comprise the following steps: Step S1: For the working condition of space staggered axis orthogonal or small axis intersection, in the case of known shaft intersection angle ∑, offset distance E, transmission ratio i ₁₂ and diameter of the outer end of the large wheel d _ae2 , solve the two-section cone in the Geometric parameters tangent to the known tangent contact point M in space: obtain the conical parameters of the hypoid gear pitch under the target condition, namely the pitch angle γ _m2 and γ _m1 of the large and small gears, and the installation distance of the large and small gears A _m2 , A _m1 and the helix angles β _m2 and β _m1 of the large and small wheels; Step S2: According to the above-mentioned basic geometric parameters of the hypoid gear pitch cone that meet the requirements, select the shrinkage tooth system for the working condition, and calculate the basic parameters of the hypoid gear blank that meets the requirements of the working condition; Step S3: After determining all the geometrical parameters of the gear that meet the working condition, based on the machining principle of the face milling method, the machining parameters of the large wheel of the hypoid gear are solved, including: horizontal tool position H ₂ , radial tool position V ₂ , The machine tool root taper angle M _r2 , the horizontal wheel position ΔX _B2 and the bed position ΔX _A2 , the contact reference point F is selected, and the first and second principal curvatures of the contact reference point F are calculated based on the Rogrids formula

with the first and second principal directions

According to the local synthesis method, the small wheel and the large wheel are completely conjugated at this point, so the meshing characteristic parameters of the hypoid gear are set at this point, including: the angle θ _cr between the contact trace and the first main direction of the tooth surface, the contact ellipse half The length of the major axis L _ce , the first derivative of the transmission error m ₁₂ ; from this, the first and second principal curvatures of the small wheel at this point that meet the meshing characteristic conditions are calculated

with the first and second principal directions

Based on the pinion machining principle of face milling generation method, the unique hypoid gear pinion machining parameters under the condition of meshing characteristics are solved, including radial tool position S _r1 , angular tool position Q _r1 , machine tool root taper angle M _r1 , horizontal wheel position ΔX _B1 , bed position ΔX _A1 , knife inclination angle I ₁ , knife rotation angle J ₁ , rolling ratio C ₁₂ and vertical wheel position E _m1 ; Step S4: After completing the derivation of the geometric parameters of the hypoid gear, the basic parameters of the gear blank and the processing parameters for the above working conditions, a three-dimensional model of the hypoid gear pair is established based on the reverse engineering of the tooth surface, considering the orthogonal and small axis intersection angles. Due to the complex geometric characteristics of the tooth surface of the hyperboloid gear, it is necessary to check the geometric characteristic index of the tooth surface of the gear pair to ensure the accuracy of the gear tooth surface under this parameter; the three major tooth surface index inspections include root undercut analysis, tooth tip The point change inspection and the smooth transition analysis of the tooth root ensure the accuracy and quality of the tooth surface. 2. The general design method of hypoid gear pair according to claim 1, characterized in that: in step S1, according to the known shaft intersection angle ∑, offset distance E, diameter d _ae2 of the outer end of the large wheel, space tangent contact point M three-dimensional coordinates (X _m , Y _m , Z _m ), the following three geometric relations can be obtained:

In the formula, i ₁₂ represents the transmission ratio of the gear pair, and β _m12 is the difference between the helix angles of the large and small wheels; Based on the forming principle and generating principle of the hypoid gear tooth surface, two constraints on the geometric characteristics of the hypoid gear pair are obtained:

In the formula,

and α _nlim represent the ultimate pressure angle and ultimate normal curvature, respectively; Based on the working condition of space staggered axis orthogonal or small axis crossing angle, according to the three geometric relations of gears and two limiting conditions, it is proposed to consider the helix angle of the large wheel and the auxiliary angle of the small wheel as iterative variables, and the offset distance and the limit method curvature as the judgment The iterative way of the condition; Among them, the inner loop is the offset distance loop, and the judgment conditions are: |EE ^* |＜0.001 In the formula, E ^* represents the offset distance of the hypoid gear after each iteration of the inner cycle; When the inner loop judgment condition is met, the inner loop ends, and the outer loop is performed. The judgment condition is:

In the formula,

Indicates the radius of the selected cutter head,

represents the limit method curvature of the hypoid gear tooth surface after each iteration of the outer cycle; When the outer cycle judgment condition is satisfied, the cycle ends, and the unique hypoid gear pitch cone parameter that satisfies the target working condition is output. 3. the general design method of hypoid gear pair according to claim 2, is characterized in that: in step S2, the general formula of shrinkage mode of face milling hypoid gear is as follows:

In the formula, m _et is the modulus of the gear end face, R _e2 is the taper distance at the big end of the big wheel, and R _m2 is the taper distance at the reference point of the big wheel. 4. the general design method of hypoid gear pair according to claim 3, is characterized in that: in step S3, the calculation formula of the machining parameter of hypoid gear large wheel is as follows:

Based on the relationship between the basic parameters of the local synthesis method, the following equation is obtained:

in,

In the formula, the angle between the first main direction of the meshing tooth surface of the small wheel and the large wheel is σ ₁₂ , a ₁₁ , a ₁₂ and a ₂₂ are the basic equation coefficients of the local comprehensive method;

The equation for examining the three unknown coefficients is as follows:

in,

According to the local synthesis method, the following expressions are obtained:

in,

From this, the expression of the included angle in the first main direction of the meshing tooth surfaces of the small wheel and the large wheel is derived as:

Among them, the expressions of the first principal curvature K _Ip and the second principal curvature K _IIp of the pinion at point F of the tooth surface are as follows:

According to the above equation, the expressions of the first and second principal directions of the pinion tooth surface at point F of the tooth surface are obtained as follows:

Based on the main curvature and main direction of the pinion tooth surface at point F of the tooth surface, three meshing characteristic parameters are jointly preset: the angle θ _cr between the contact trace and the first main direction of the tooth surface, the length of the semi-major axis of the contact ellipse L _ce , the transmission The first derivative of the error m ₁₂ is used to solve the machining parameters of the small wheel of the hypoid gear pair by the face milling generation method: radial tool position S _r1 , angular tool position Q _r1 , machine tool root taper angle M _r1 , horizontal wheel position ΔX _B1 , The bed position ΔX _A1 , the knife inclination angle I ₁ , the knife rotation angle J ₁ , the rolling ratio C ₁₂ and the vertical wheel position E _m1 .

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