CN110414056A - A Cycloidal Tooth Profile Modification Method Compensating for Elastic Deformation - Google Patents

Abstract

The invention discloses a kind of Cycloid tooth profile correction methods for compensating flexible deformation, by design based on equidistant plus modification of moved distance flank profil, and cycloidal gear teeth profile is sought using optimization algorithm, this method considers influence of the flexible deformation to Cycloid tooth profile, can improve Cycloidal Wheel performance；Analog simulation is carried out convenient for exploitation mathematics software, facilitates Cycloid tooth profile correction of the flank shape, there is very high use value；And method can be embedded into RV Gear Reducer Optimal Design software, have good social value and economic value.

Description

A Cycloidal Tooth Profile Modification Method Compensating for Elastic Deformation

technical field

The invention belongs to the technical field of mechanical design and manufacture, and relates to a cycloid gear tooth profile modification optimization design method, which is suitable for RV reducer cycloid gear tooth profile modification.

Background technique

As the core component of the RV reducer, the cycloidal wheel directly affects the transmission accuracy, efficiency, carrying capacity, life, reliability and other important performances of the RV reducer. The tooth profile modification of cycloidal gear is an important part in the design and manufacture of cycloidal gear. It is of great significance and practical value to study and determine the optimal modification amount of cycloidal gear to improve the performance of cycloidal gear. Scholars at home and abroad have done a lot of research on the profile modification of cycloidal gears. He Weidong et al. used the combined optimization modification method of negative equidistance plus negative displacement to obtain the tooth shape of multi-tooth conjugate meshing and reduce the backlash; Nie Shaowen et al. proposed a method based on equidistance, displacement and tooth height. Compared with other modification methods, it has a higher fitting degree to the tooth profile of the corner modification; Zhao Bo et al. proposed a parabolic tooth profile modification method, in the working tooth profile of the cycloid wheel, the modification The tooth profile after shaping is closer to the conjugate tooth profile; Chmurawa et al. achieved better transmission performance through the optimization of the cycloid tooth profile, and studied the influence of the modification parameters on the load distribution and stress.

Modification theory of cycloid gear tooth profile

Figure 1 is the standard theoretical tooth profile of the cycloidal wheel, and the standard theoretical tooth profile equation of the cycloidal wheel is:

In the formula: r _p is the radius of the center circle of the pin tooth, r _rp is the radius of the pin tooth, z _p is the number of pin teeth, i is the relative transmission ratio of the cycloid wheel and the pin wheel, is the meshing phase angle of the cycloidal wheel, a is the eccentricity of the cycloidal wheel, and K is the short-width coefficient of the cycloidal wheel.

The standard cycloidal wheel meshes with the pin teeth without clearance, which cannot compensate for manufacturing errors and meet the lubrication conditions. Therefore, in actual use, the standard cycloidal wheel cannot be used directly, and the tooth profile of the cycloidal wheel must be modified. Common modification methods include isometric modification, shift modification, corner modification and combination modification. Corner modification is the most ideal modification method. The tooth shape after modification is a conjugate meshing tooth shape with uniform backlash, but there is no meshing gap between the tooth top and the tooth root. Due to the complex processing technology of corner plus equidistant or corner plus displacement modification, the modification time is long, and the modification method of equidistant plus displacement is usually used to fit the corner modified tooth profile to obtain the tooth tip and root Conjugate meshing tooth profile with reasonable meshing clearance.

The modification amount is δ rotation angle and the modified tooth profile equation is:

The equal distance plus shift distance modified tooth profile equation is:

In the formula, Δr _p , Δr _rp are the displacement modification amount and the isometric modification amount respectively, K'=az _p /(r _p +Δr _p ).

At no-load, the cycloidal wheel meshes with the single tooth of the needle tooth. After the torque is applied, the meshing cycloidal wheel and the needle tooth produce elastic deformation. Considering the influence of elastic deformation on the tooth profile, the tooth profile of the cycloidal wheel will be concave, no longer It is the theoretical tooth profile, and the conjugate meshing relationship between the cycloid wheel and the pin teeth is destroyed, which affects the transmission characteristics of the RV reducer and reduces the transmission accuracy. The heavy-duty RV reducer transmits a large torque, and the elastic deformation is larger than that of the general RV reducer, which seriously affects the performance of the RV reducer. Therefore, it is very necessary to consider the elastic deformation during the modification design process of the RV reducer.

Contents of the invention

The present invention aims at the deficiencies of the prior art, and provides a cycloid gear tooth profile modification optimization design method, which improves the performance of the cycloid gear.

In order to achieve the above object, a method for modifying the tooth profile of a cycloid gear designed to compensate for elastic deformation in the present invention is special in that:

S1 modify the tooth profile based on equidistant plus shift distance, and design cycloid gear tooth profile:

Rotate the meshing point of the cycloidal wheel to the same profile, and use the principle of coordinate transformation to convert the meshing point of the cycloidal wheel to the first tooth, and fit the meshing point to obtain the tooth profile of the cycloidal wheel after contact deformation. The coordinates are :

Among them, the coordinates of the meshing point (x _j , y _j ), Δx, Δy are the components of the contact deformation along the x and y directions, z _c is the number of teeth of the cycloidal wheel, the meshing point is the jth pin tooth, and the center C of the pin tooth is opposite to The angle of rotation of the arm is

S2 calculates the meshing characteristics of the cycloid wheel by using the deviation degree between the actual modified tooth profile and the angle modified tooth profile with equal distance plus shift distance, and the calculation formula is as follows:

In the formula, Δr _p , Δr _rp are the displacement modification amount and equidistant modification amount respectively, m is the number of meshing teeth at the same time, i=1,2,3...m;

S3 designs an optimization model and uses the model to find the optimal solution:

Wherein, the objective function of the optimization model: the degree of deviation between the actual modified tooth profile and the rotational angle modified tooth profile is used as the objective function, namely

z=min f(Δr _rp ,Δr _p )

The design variables of the optimization model:

The objective function is a function about Δr _p Δr _rp . In order to ensure a reasonable radial clearance Δr between the dedendum and tooth tip of the cycloidal gear, let Δr=Δr _rp -Δr _p , then Δr _p = Δr _rp -Δr, as Equidistant modification Δr _rp and radial clearance Δr are design variables;

The constraints of the optimization model:

Further, in the step s3, a genetic algorithm is used to calculate the optimal solution, and the setting of the algorithm is as follows:

The population size is N=50, the number of iterations is n=100, the crossover rate is 0.9, and the mutation rate is 0.1.

The advantages of the present invention are:

Considering the contact deformation between the cycloidal wheel and the needle teeth, the tooth profile of the cycloidal wheel is closer to the corner modified tooth profile in actual work, and the meshing between the cycloidal wheel and the needle teeth is closer to the conjugate meshing, and the transmission is more stable. The distance modification method is convenient for processing, and the genetic algorithm is used to find the optimal solution, which improves the optimization efficiency.

Description of drawings

Figure 1 is a schematic diagram of the standard theoretical tooth profile of a cycloidal wheel.

Figure 2 is a schematic diagram of cycloid wheel meshing.

Figure 3 is the initial meshing clearance of the cycloid wheel.

Fig. 4 is a flow chart of the genetic algorithm in the present invention.

Fig. 5 is an iteration diagram of the genetic algorithm in the present invention.

Fig. 6 is a curve diagram of the theoretical tooth profile of the cycloidal wheel.

Figure 7 is a curve diagram of the initial meshing gap and elastic deformation of the modified cycloidal wheel.

Figure 8 is a curve diagram of the contact force distribution on the tooth surface of the cycloidal wheel.

Figure 9 is a comparison diagram of the theoretical tooth profile considering elastic deformation and angle modification.

Figure 10 is a comparison diagram of the actual tooth profile considering elastic deformation and angle modification.

Figure 11 is a partially enlarged view of the actual working tooth profile of the cycloidal wheel.

Detailed ways

Below in conjunction with accompanying drawing and specific embodiment the present invention will be described in further detail:

In order to reduce the impact of the elastic deformation caused by the meshing of the cycloidal wheel and the pin teeth on the performance of the RV reducer, the elastic deformation is compensated when the cycloidal tooth profile is modified, and the equal distance and displacement modification are optimized so that the pendulum The modified tooth profile of the wire wheel approaches the theoretical conjugate meshing tooth profile after producing elastic deformation.

S1 modify the tooth profile based on equidistant plus shift distance, and design cycloid gear tooth profile:

Since the cycloidal wheel and the needle teeth are meshed with multiple teeth, contact deformation exists on multiple cycloidal wheel teeth. In order to directly reflect the influence of contact deformation on the profile of the cycloidal wheel, the meshing point of the cycloidal wheel is rotated to the same profile to obtain The tooth profile of the cycloid gear after the contact deformation is obtained by fitting the meshing points. First determine the position of the contact point between the cycloidal wheel and the pin tooth, the meshing point or the point to be meshed of the cycloidal wheel intersects with the center line of the pin tooth at node P, as shown in Figure 2, the intersection point D is one of the cycloidal wheel and the pin tooth The meshing point is assumed to be the jth pin tooth, and the rotation angle of the pin tooth center C relative to the rotating arm is The angle γ between DO _c and O _c P can be obtained from the law of cosines, then the meshing phase angle α=z _c γ of point D on the cycloid wheel, and bringing α into formula (6)(7) is the meshing point Coordinates (x _j , y _j ); considering the influence of contact deformation, the contact deformation along the common normal direction (the line connecting the node P and the center of the pin tooth) will occur after the cycloid wheel meshes with the pin tooth, and the cycloid after contact deformation The coordinates of the point on the wheel are (x _j -Δx, y _j -Δy), Δx and Δy are the components of the contact deformation along the x and y directions, which can be known from the geometric conditions

Δx=εsin(π-γ) (9)

Δy=εcos(π-γ) (10)

Where ε is the contact deformation along the common normal direction of the cycloidal wheel meshing with the pin teeth; finally, using the coordinate transformation principle, the contact point of the cycloidal wheel is converted to the first gear tooth, and its coordinates are

Among them, the relevant parameters of contact deformation in equations (9) (10) are calculated as follows:

After the tooth profile of the cycloidal wheel is modified by equidistant plus shift distance, there is a certain gap between the cycloidal wheel and the pin teeth, and it can only mesh with the pin teeth after turning a rotation angle β _t relative to the pin teeth. Have The needle teeth at the center are in contact with the cycloid wheel, and there are gaps of different sizes between the rest of the needle teeth and the cycloid wheel, as shown in Figure 3, using the initial meshing gap Indicates the gap between the cycloid wheel and the needle teeth in the direction of the theoretical common normal.

The first point of contact between the cycloidal wheel and the pin teeth at no load is Then the relative rotation angle there is

In the formula, z _c is the number of cycloid gear teeth,

At this time, the initial meshing clearance of the cycloid wheel

After loading, considering the compensation effect of the elastic deformation of the cycloid wheel and the pin tooth, when the contact deformation between the cycloid wheel and the pin tooth is greater than the initial meshing gap, the pin tooth meshes with the cycloid wheel, otherwise it does not mesh. Let the pin tooth contact force at the first contact point be F _T , the contact deformation be ε _T , at any position At , the pin tooth contact force is F _i , the contact deformation is ε _i , and the pin tooth contact force has a linear relationship with the difference between the deformation and the initial meshing gap, that is

in,

by moment balance

In the formula, T is the output torque, m and n are the tooth number of the start mesh and the tooth number of the end mesh respectively, and L _i is the distance from the common normal of the i-th pin tooth mesh point to the center O _c of the cycloid wheel.

The deformation includes two parts, one is the contact deformation between the cycloid wheel and the pin teeth, and the other is the contact deformation between the pin teeth and the pin gear housing. The amount of deformation can be calculated by the Hertz formula

Among them, u and E are the Poisson's ratio and elastic modulus of the cycloidal wheel material (E=2.06*10^5MPa, u=0.3), b is the width of the cycloidal wheel, and R1 and R2 are the curvatures of the two contact cylinders Radius, when the cycloid wheel is in contact with the pin teeth, the equivalent radius of curvature ρ _D =2|ρ|r _rp /(|ρ|+r _rp ), when the pin teeth are in contact with the pin gear housing, the equivalent curvature radius ρ _D =2| ρ|r _rp /(|ρ|-r _rp ), ρ is the cycloid wheel The radius of curvature at .

When calculating ε _T , you need to know _FT , and when calculating _{FT, you need to know ε T ,} so the maximum contact force cannot be obtained directly. Usually, an iterative method is used for calculation. First, an initial value _FT0 is given, and ε _T0 is calculated, and ε _T0 is calculated. Calculate F _T1 , and compare F _T0 and F _T1 , if the absolute value of the difference between the two is greater than 0.1% F _T1 , continue the iterative calculation until |F _Tk -F _Tk-1 |<0.1% F _Tk and take F _T = (F _Tk +F _{Tk -1} )/2, after calculating FT, the distribution of contact force and contact deformation on the tooth surface of the cycloid can be obtained.

S2 calculates the meshing characteristics of the cycloid wheel by using the deviation degree between the actual modified tooth profile and the angle modified tooth profile with equal distance plus shift distance, and the calculation formula is as follows:

The actual working tooth profile of the cycloidal wheel is obtained by fitting the coordinates of the contact point of the cycloidal wheel. In order to measure the fitting degree of the actual working tooth profile and the modified tooth profile of the rotation angle with different modification amounts, the meshing interval of the cycloidal wheel is obtained from the meshing start and end points ( A, B), divide (A, B) into points equally, m is the number of meshing teeth at the same time, that is Substituting formula (4) (5) to get the corner modified tooth profile coordinates ( _xi , y _i ), i=1, 2, 3...m, let y _i = y _i ', equidistant plus displacement distance actual modified tooth The degree of deviation between the tooth profile and the angle modified tooth profile is calculated by formula (12)

The smaller the deviation from the angle modification curve, that is, the higher the fitting degree between the actual tooth profile of the equidistant plus displacement modification and the tooth profile of the rotation angle modification, the closer the actual tooth profile curve of the equidistant plus displacement modification is to the rotation angle modification The closer the tooth profile curve is to the conjugate meshing tooth shape, the smoother the transmission and the better the meshing characteristics of the cycloidal wheel.

S3 designs an optimization model and uses the model to find the optimal solution:

3.1 The objective function of the optimization model:

The modification method of equal distance plus displacement is adopted, and the degree of deviation between the tooth profile curve of equidistant plus displacement modification after elastic deformation and the tooth profile curve of rotation angle modification is taken as the objective function, that is,

z=min f(Δr _rp ,Δr _p ) (21)

3.2 Design variables of the optimization model:

The objective function is a function about Δr _p Δr _rp . In order to ensure a reasonable radial clearance Δr between the dedendum and tooth tip of the cycloidal gear, let Δr=Δr _rp -Δr _p , then Δr _p = Δr _rp -Δr, as Equidistant modification Δr _rp and radial clearance Δr are design variables.

3.3 Constraints of the optimization model:

On the one hand, in order to give full play to the characteristics of the wide range of optimization of the genetic algorithm, the value range of the equidistant modification amount should be expanded as much as possible; Therefore, the amount of modification should not be too large. Constraining isometric modifiers

|Δr _rp |＜0.2mm (22)

In addition to constraining the equidistant modification amount, the radial clearance also needs to be constrained:

0.01≤Δr≤0.1mm (23)

Therefore, the constraints in this paper are

3.4 The optimization algorithm adopted by the optimization model:

In this paper, the genetic algorithm is used to find the optimal solution. The algorithm parameters are set as follows: population size N=50, number of iterations n=100, crossover rate 0.9, and mutation rate 0.1. The algorithm flow is shown in Figure 4.

Specific examples:

The RV-550E reducer is a typical precision heavy-duty RV reducer, which has the characteristics of large load capacity, high transmission precision and long life. It is widely used in robots, machine tools and other fields. The RV-550E reducer cycloid The basic parameters are shown in Table 1. In this paper, the genetic algorithm is used to optimize the modification design of the RV-550E reducer cycloidal wheel, and the optimization algorithm is written by MATLAB to solve it, and compared with the optimal modification method that does not consider elastic deformation.

Table 1 Cycloid wheel parameter table

Theoretical tooth profile: the tooth profile without elastic deformation of the working tooth segment;

Actual tooth profile: the tooth profile with elastic deformation in the working tooth segment;

Theoretical modification method: Regardless of elastic deformation, the degree of deviation between the theoretical tooth profile of equidistant plus displacement modification and the tooth profile of rotation angle modification is the smallest;

Compensation for elastic deformation modification method: considering elastic deformation, the deviation between the actual tooth profile and the corner modified tooth profile is the smallest.

Figure 5 is the algorithm iteration diagram. As the number of iterations increases, the degree of deviation between the actual tooth profile of the cycloidal wheel and the angle modification curve becomes smaller and smaller. After 25 iterations, it tends to converge, and the optimal solution is equidistant modification. Measure 0.0886mm, radial clearance 0.0105mm. Figure 6 shows the theoretical tooth profile of the cycloid wheel modification. It can be seen that there is a certain radial gap between the tooth top and the tooth root, which is used to compensate for manufacturing errors and meet the lubrication conditions. The outer side is concave after being deformed by force, and the actual tooth profile is closer to the corner modification curve, which is close to the conjugate meshing relationship; Figure 7 is the distribution diagram of the initial meshing gap and elastic deformation of the cycloidal wheel. The thread wheel meshes with the pin teeth, otherwise, it is not meshed. The intersection of the two curves is the start meshing point and the end meshing point of the cycloid wheel. 30, more than the modified cycloidal wheel, so after the cycloidal wheel is modified, the maximum tooth surface contact force will increase, as shown in Figure 8, but the tooth profile of the modified cycloidal wheel is "reverse Bow" tooth profile, the distribution of contact force on the tooth surface is more uniform, compared with the general modification method, it can effectively reduce the contact force on the tooth surface.

Figure 9 shows the theoretical tooth profile curves of the two modification methods, that is, there is no contact deformation. It can be seen from the figure that when the elastic deformation of the cycloid wheel is not considered, the tooth profile curve of the theoretical modification method is closer to that of the optimal method of compensating elastic deformation. Angle modification tooth profile curve; Fig. 10 is the actual tooth profile curves of the two modification methods, and Fig. 11 is a partial enlarged view of the actual working tooth profile of the cycloid wheel. It can be seen that the compensation elastic deformation optimization method is closer to the corner modification curve . This is mainly because the theoretical tooth profile curve of the compensation elastic deformation optimization method is outside the corner modification curve. When the cycloid wheel meshes with the pin teeth, contact deformation along the normal direction occurs, and the tooth profile is concave, exceeding the corner modification tooth profile curve. The elastic deformation is partially compensated, and the actual working tooth profile is closer to the corner modified tooth profile; while the tooth profile curve of the theoretical modification method is located inside the corner modified curve, when the cycloid wheel meshes with the pin teeth, not only cannot compensate the elastic deformation, but due to the elastic The deformation is concave, away from the corner modification curve.

Table 2 is the parameter table of the tooth profile of the cycloidal wheel. From the numerical point of view, compared with the theoretical modification method, the degree of deviation between the actual tooth profile and the rotation angle modified tooth profile of the compensation elastic deformation modification method is reduced by 34.38%. At the same time, the maximum contact force of the tooth surface of the cycloid wheel with the compensation elastic modification method is reduced by 8.27% compared with the theoretical modification method, mainly because the theoretical modification method has a large amount of modification and radial clearance, so that the number of meshing teeth is less at the same time. The reduction in the number of load-bearing teeth leads to an increase in the maximum contact force of the tooth surface of the cycloidal gear; while the modification amount and the radial clearance of the method of compensating for elastic deformation are relatively small, and at the same time, the number of meshing teeth is large, and the number of load-bearing teeth increases, which reduces the cycloidal gear. The maximum contact force on the tooth surface.

Table 2 Cycloid gear tooth profile parameter table

The beneficial effects of the patent of the present invention:

1) The present invention proposes a cycloid gear tooth profile modification optimization design method for compensating elastic deformation, which takes into account the influence of elastic deformation on the cycloid gear tooth profile, and can improve the performance of the cycloid gear;

2) The method provided by the present invention is convenient for developing computer mathematics software to carry out simulation simulation, contributes to cycloid gear profile modification, and has very high use value;

3) The method provided by the invention can be embedded in the optimal design software of the RV reducer, and has good social value and economic value.

The above embodiments are only used to illustrate the design concept and characteristics of the present invention, and its purpose is to enable those skilled in the art to understand the content of the present invention and implement it accordingly. The protection scope of the present invention is not limited to the above embodiments. Therefore, all equivalent changes or modifications based on the principles and design ideas disclosed in the present invention are within the protection scope of the present invention.

Claims (2)

1. A cycloidal tooth profile modification method for compensating elastic deformation, characterized in that: S1 modify the tooth profile based on equidistant plus shift distance, and design cycloid gear tooth profile: Rotate the meshing point of the cycloidal wheel to the same profile, use the principle of coordinate transformation, convert the contact point of the cycloidal wheel to the first gear tooth, and fit the meshing point to obtain the tooth profile of the cycloidal wheel after contact deformation, and its coordinates are : Among them, the coordinates of the meshing point (x _j , y _j ), Δx, Δy are the components of the contact deformation along the x and y directions, and the meshing point is set as the jth pin tooth, and the rotation angle of the pin tooth center C relative to the rotating arm is z _c is the number of cycloid gear teeth; S2 calculates the meshing characteristics of the cycloid wheel by using the deviation degree between the actual modified tooth profile and the angle modified tooth profile with equal distance plus shift distance, and the calculation formula is as follows: In the formula, Δr _p , Δr _rp are the displacement modification amount and equidistant modification amount respectively, m is the number of meshing teeth at the same time, i=1,2,3...m; S3 designs an optimization model and uses the model to find the optimal solution: Wherein, the objective function of the optimization model: the degree of deviation between the actual modified tooth profile and the rotational angle modified tooth profile is used as the objective function, namely z=min f(Δr _rp ,Δr _p ) The design variables of the optimization model: The objective function is a function about Δr _p Δr _rp . In order to ensure a reasonable radial clearance Δr between the dedendum and tooth tip of the cycloidal gear, let Δr=Δr _rp -Δr _p , then Δr _p = Δr _rp -Δr, as Equidistant modification Δr _rp and radial clearance Δr are design variables; The constraints of the optimization model: 2. the cycloid tooth profile modification method of compensating elastic deformation according to claim 1, is characterized in that: adopts genetic algorithm to carry out calculating of optimal solution in the described step s3, and the setting of this algorithm is as follows: The population size is N=50, the number of iterations is n=100, the crossover rate is 0.9, and the mutation rate is 0.1.