CN110377932B - A method of expressing the profile of the cycloid disk of the steel ball reducer - Google Patents

Abstract

The invention discloses a method for representing the profile of a cycloid disk of a steel ball reducer, which includes the following steps: obtaining a data model of the cycloid disk; determining the node vector of the profile data points of the cycloid disk according to the data model; using the data points The node vector is reversely calculated to obtain the control vertex of the NURBS interpolation curve; the output result of the interpolation spline curve is obtained; the curve fitting accuracy error is verified. The invention has the beneficial effects: it can locally modify the profile of the cycloid disk, and improve the local characteristics of the profile without affecting the performance in other intervals, thereby optimizing the profile of the cycloid disk.

Description

A method of expressing the profile of the cycloid disk of the steel ball reducer

Technical field

The present invention relates to the technical field of cycloidal steel ball reducers, and in particular to a method of expressing the profile of a cycloidal disc of a steel ball reducer.

Background technique

In recent years, the cycloid disk has been one of the important components of the cycloidal steel ball reducer, and its profile directly affects the performance of the reducer. The profile design parameters of the cycloid disk are closely related to whether undercutting occurs in the cycloid groove, which has a significant impact on the efficiency and load-bearing capacity of the reducer.

The existing research is to analyze the impact of the value of the short-amplitude coefficient on the transmission performance of the steel ball reducer, and to study the value range of the short-amplitude coefficient. However, in this study, the value of K is usually set to reduce the efficiency to ensure that the gear The load-bearing profile cannot realize the joint optimization of efficiency and load-bearing. At the same time, some researchers have conducted an optimization design based on genetic algorithms on the efficiency, volume and load-bearing capacity of the steel ball reducer. This research has optimized the design parameters of the steel ball reducer to a certain extent, but this research is within a certain range. Achieving relative optimization of the objective function by adjusting the values of each parameter cannot achieve the optimal value for each objective.

The theoretical tooth profile curve when the deceleration mechanism is working is a pair of short-width inner and outer cycloids that mesh with each other. The formation methods of the short-width inner and outer cycloids include the wrapping method and the non-wrapping method to derive the inner and outer cycloid equations, but for Modification of a certain design parameter of the profile will lead to changes in the overall profile, which avoids undercutting while reducing efficiency and load-bearing capacity, and cannot optimize the overall performance of the reducer.

Contents of the invention

The purpose of this section is to outline some aspects of embodiments of the invention and to briefly introduce some preferred embodiments. Some simplifications or omissions may be made in this section, the abstract and the title of the invention to avoid obscuring the purpose of this section, the abstract and the title of the invention, and such simplifications or omissions cannot be used to limit the scope of the invention.

In view of the above-mentioned existing problems, the present invention is proposed.

Therefore, the technical problem solved by the present invention is: since the inner and outer cycloid equations cannot complete the local modification of the profile, the NURBS curve design method is introduced to realize the local adjustment of the profile of the cycloidal disk.

In order to solve the above technical problems, the present invention provides the following technical solution: a method for representing the profile of a cycloid disk of a steel ball reducer, including the following steps: obtaining a data model of the cycloid disk; determining the profile line of the cycloid disk according to the data model The node vector of the data point; use the node vector of the data point to obtain the control vertex of the NURBS interpolation curve; obtain the output result of the interpolation spline curve; verify the curve fitting accuracy error.

As a preferred solution of the method of expressing the profile of the cycloidal plate of the steel ball reducer of the present invention, the acquisition of the cycloidal plate data model includes the following steps. It is known that the profile design of the cycloidal plate in the reducer is Parameters; input the design parameters into the parametric equation to obtain the cycloidal plate profile of the reducer; discretize and sample the cycloidal plate profile to obtain a data model.

As a preferred solution of the method of expressing the profile of the cycloid plate of the steel ball reducer of the present invention, the parameter equation is the expression equation of the cycloid plate, as follows:

The equation of the hypocycloid is:

The epicycloid equation is:

Among them, d _s is the diameter of the distribution circle of the steel ball, and its calculation formula is: d _s =r _d /sin (180/n _h ), and r _d is the radius of the steel ball, and θ is the pure rolling of the non-centered rolling circle on the base circle. The angle at which the time base circle is rolled, c is half of the eccentricity, n _h is the number of steel balls, and n _b is the number of steel balls.

As a preferred solution for the profile representation method of the steel ball reducer cycloidal disk of the present invention, the cumulative chord length parameter method is used to uniformly parameterize the data points of the profile of the data model, and also include,

Let L be the total length of the curve, Let u ₀ =0u _n =1; then/> i=1,...,n-1.

As a preferred solution for the profile representation method of the steel ball reducer cycloid disk of the present invention, the control vertex of the NURBS curve is back-calculated based on the node vector determined by the profile value point on the curve, including the following Step: Use the type value points as the data points to back-calculate the control vertices; first set the weight factors of all the control vertices to 1, and after obtaining the corresponding control vertices, perform the corresponding weight factors on the curve as needed. Adjustment; where the cubic non-uniform B-spline equation used to interpolate n+1 data points is:

Substitute Q _i and the corresponding node values into the above equation to obtain a linear equation system composed of n+1 vector equations, as follows:

In addition, two additional equations given by the boundary conditions are added:

C ₀ ', C _n ' are the tangent vectors at the end points, and the following system of equations is obtained:

in

As a preferred method of representing the profile of the cycloidal plate of the steel ball reducer of the present invention, the interpolation spline output results include the following sampling methods. The first type is based on the profile of the cycloidal plate. The results obtained by uniform sampling are obtained by taking any section of the waveform as an example; the second type is obtained by uniformly sampling half of the waveforms at any two ends of the cycloidal disk-shaped line as an example; the third type is obtained by uniform sampling of any waveform on the cycloidal disk-shaped line. Take half the waveform at each end as an example to uniformly sample and increase the number of control points.

As a preferred solution for the profile representation method of the steel ball reducer cycloid disk of the present invention, the fitting accuracy error verification includes the following steps, calculating the distance from the data point to the curve, including calculating the average distance and Maximum distance; perform accuracy error analysis on the curve fitting results by citing the two fitting curve error analysis methods of MATLAB tools, including calculation and variance and coefficient of determination.

As a preferred solution for the profile representation method of the steel ball reducer cycloid disk according to the present invention, wherein: wherein the sum variance is SSE: It calculates the sum of squares of the errors between the fitted data and the corresponding points of the original data. When the SSE is closer to 0, it means better model selection and fitting, and more successful data prediction.

As a preferred solution for the profile representation method of the steel ball reducer cycloid plate according to the present invention, the determination coefficient is R-square: It represents the quality of a fit through changes in data. The normal range of the determination coefficient is [0, 1]. When it is closer to 1, it indicates that the variables of the equation have a stronger ability to explain y, and the model fits the data better. Also better.

As a preferred solution for the profile representation method of the steel ball reducer cycloid disk of the present invention, it also includes: the sampling method of the interpolation spline output result is verified corresponding to three types of fitting accuracy errors; The first type is the fitting accuracy error of the fitting curve obtained by uniform sampling using the first sampling method by taking any waveform on the cycloid line as an example; the second type is taking any two ends of the cycloid line The fitting accuracy error of the fitting curve obtained by taking half of the waveform as an example for uniform sampling; the third category is taking half of the waveform at each end of the cycloid line as an example for uniform sampling and increasing the number of control points The accuracy error of the obtained fitting curve; the third type error reaches 0.1 micron, and the coefficient of determination reaches 0.999.

The beneficial effects of the present invention are: First, by proposing the NURBS representation method of the profile, the representation of the profile is increased; in view of the problem that the construction freedom of the traditional regular curve cannot meet the flexibility of the profile design, it is proposed to use the NURBS curve method to obtain Cycloidal disc profile with higher degree of freedom and curve continuity; secondly, it can locally modify the cycloidal disc profile to improve the local characteristics of the profile without affecting the performance in other intervals, thereby optimizing the cycloid. The shape of the plate.

Description of the drawings

In order to explain the technical solutions of the embodiments of the present invention more clearly, the drawings needed to be used in the description of the embodiments will be briefly introduced below. Obviously, the drawings in the following description are only some embodiments of the present invention. Those of ordinary skill in the art can also obtain other drawings based on these drawings without exerting any creative effort. in:

Figure 1 is a schematic diagram of the epicycloid generation principle according to the first embodiment of the present invention;

Figure 2 is a schematic diagram of the principle of hypocycloid generation according to the first embodiment of the present invention;

Figure 3 is a schematic diagram of the NURBS curve design process according to the first embodiment of the present invention;

Figure 4 is a schematic diagram of the BR85us-10G-6 hypocycloid disk profile according to the first embodiment of the present invention;

Figure 5 is a line diagram of the first type of NURBS curve fitting according to the first embodiment of the present invention;

Figure 6 is a line diagram of the second type of NURBS curve fitting according to the first embodiment of the present invention;

Figure 7 is an overall shape diagram of the hypocycloid of the third type of NURBS curve fitting according to the first embodiment of the present invention;

Figure 8 is a partially enlarged schematic diagram of the overall shape of the hypocycloid curve fitted by the third type of NURBS curve fitting according to the first embodiment of the present invention;

Figure 9 is an error diagram of the NURBS curve fitting profile according to the first embodiment of the present invention.

Detailed ways

In order to make the above objects, features and advantages of the present invention more obvious and easy to understand, the specific embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings. It is obvious that the described embodiments are part of the embodiments of the present invention, not all of them. Example. Based on the embodiments of the present invention, all other embodiments obtained by ordinary people in the art without creative efforts should fall within the protection scope of the present invention.

Many specific details are set forth in the following description to fully understand the present invention. However, the present invention can also be implemented in other ways different from those described here. Those skilled in the art can do so without departing from the connotation of the present invention. Similar generalizations are made, and therefore the present invention is not limited to the specific embodiments disclosed below.

Second, reference herein to "one embodiment" or "an embodiment" refers to a specific feature, structure, or characteristic that may be included in at least one implementation of the present invention. “In one embodiment” appearing in different places in this specification does not all refer to the same embodiment, nor is it a separate or selective embodiment that is mutually exclusive with other embodiments.

The present invention will be described in detail with reference to schematic diagrams. When describing the embodiments of the present invention in detail, for the convenience of explanation, the cross-sectional diagrams showing the device structure will be partially enlarged according to the general scale. Moreover, the schematic diagrams are only examples and shall not limit the present invention. scope of protection. In addition, the three-dimensional dimensions of length, width and depth should be included in actual production.

At the same time, in the description of the present invention, it should be noted that the orientation or positional relationship indicated by the terms "upper, lower, inner and outer" are based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present invention. The invention and simplified description are not intended to indicate or imply that the devices or elements referred to must have a specific orientation, be constructed and operate in a specific orientation, and therefore are not to be construed as limitations of the invention. Furthermore, the terms "first, second or third" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance.

Unless otherwise clearly stated and limited in the present invention, the terms "installation, connection, and connection" should be understood in a broad sense. For example, it can be a fixed connection, a detachable connection, or an integrated connection; it can also be a mechanical connection, an electrical connection, or a direct connection. A connection can also be indirectly connected through an intermediary, or it can be an internal connection between two components. For those of ordinary skill in the art, the specific meanings of the above terms in the present invention can be understood on a case-by-case basis.

Example 1

Regarding the research on steel ball reducers, the existing research is to analyze the impact of the value of the short-amplitude coefficient on the transmission performance of the steel ball reducer, and to study the value range of the short-amplitude coefficient. However, in this study, K is usually The value is selected to reduce the efficiency to ensure the tooth profile load-bearing, and cannot achieve the joint optimization of efficiency and load-bearing. According to existing research and analysis, it can be seen that the efficiency increases as the short-width coefficient K increases, and the load-bearing capacity first decreases and then increases as k increases. Therefore, it is impossible to achieve optimal values for efficiency and load-bearing at the same time by adjusting the value of K. .

In this embodiment, the nurbs curve is used to redesign the cycloidal disc profile. Without changing the short width coefficient of the profile, that is, it ensures that the cycloidal disc profile does not change as a whole. Local modifications reduce stress concentration, thereby improving load-bearing capacity.

Specifically, refer to Figure 3, which illustrates a schematic diagram of the NURBS curve design process in the profile representation method of the steel ball reducer cycloid disk in this embodiment. The figure illustrates that the method includes using known curves to calculate inversely Control points, obtain the interpolation spline curve, and judge whether the obtained spline curve meets the accuracy requirements, constantly modify the parameters, and finally obtain the interpolation spline curve that meets the accuracy requirements. More specifically, the method includes obtaining a cycloid disk data model, determining the node vectors of the cycloidal disk profile line data points according to the data model, using the node vectors of the data points to obtain the control vertices of the NURBS interpolation curve, and obtaining Interpolation spline curve output results and curve fitting accuracy error verification. in,

Obtaining the cycloid disk data model includes:

The design method of the inner and outer cycloid disc profiles of the reducer is the same. Therefore, in this embodiment, the hypocycloid disc profile is used as an example to design, so that the outer cycloid disc profile design can be completed by referring to the hypocycloid disc profile design process. In this embodiment, referring to the diagram in Figure 4, the design parameters of the cycloid plate profile of the BR85us-10G-6 reducer are known. According to the parameter equation, the cycloidal plate profile of the reducer can be obtained, and the cycloidal plate profile is discretized and sampled. Get the data model.

It should also be noted that the currently widely used method for designing the profile of the cycloidal disc profile is to draw the inner and outer cycloidal disc profiles based on the cycloid equation and given condition parameters. The theoretical tooth profile curve when the deceleration mechanism is working is a pair of short-width inner and outer cycloids that mesh with each other. The formation methods of the short-width inner and outer cycloids include the wrapping method and the non-wrapping method. In this embodiment, the non-wrapping method is used. Take an example to derive the inner and outer cycloidal equations. as follows:

Figure 1 shows the principle diagram of epicycloid generation. In the figure, O ₀ performs pure rolling around O _1. M is a point inside the generating circle. The eccentricity e=O ₀ M. The number of teeth of the generated epicycloid is Z ₁ = R ₁ /R ₀ , the motion trajectory of point M is the epicycloid;

Figure 2 shows the schematic diagram of hypocycloid generation. In the figure, O ₀ purely rolls around O ₂ , M' is a point inside the generating circle, the eccentricity A=O ₀ M', and the number of teeth of the generated hypocycloid is Z ₁ =R ₂ /R ₀ , the motion trajectory of point M' is the hypocycloid. The expression equation of the cycloidal disk in the traditional design method is as follows.

The equation of the hypocycloid is:

The epicycloid equation is:

d _s is the diameter of the steel ball distribution circle, and its calculation formula is as follows: d _s = r _d /sin (180/n _h ), where r _d is the radius of the steel ball; θ is the pure circle without center on the base circle When rolling, the angle at which the base circle is rolled; c is half of the eccentricity; n _h is the number of steel balls; n _b is the number of steel balls.

This embodiment draws the inner and outer cycloid disk profile lines based on the cycloid equation and given condition parameters. However, modifying a certain design parameter of the profile line will cause the overall profile line to change, which avoids undercutting and reduces efficiency and load-bearing capacity. The overall performance of the reducer cannot be optimized.

The node vectors that determine the data points of the cycloid disk shape include:

The parameterization of data points is an important step in curve construction and has an important impact on the smoothness of the curve. In order to obtain better smoothness, this embodiment uses the cumulative chord length parameter method to uniformly parameterize the profile data points, including the following steps:

Let L be the total length of the curve, Let u ₀ =0u _n =1; then/> i=1,...,n-1. Using the above parameterization faithfully reflects the distribution of data points according to chord length, and the resulting interpolation curve has good smoothness.

The control vertices of the reverse NURBS interpolation curve include:

In this implementation, if a cubic NURBS curve is to be used to express a series of data points on the measured cycloidal disk shape line, the control points of the NURBS curve must first be back-calculated based on the node vectors determined by the shape value points on the curve. It should be noted that the shape value points are used as data points to determine the node vectors, and the data points are obtained by obtaining the data model through discretization sampling of the cycloidal disk shape line.

The process of obtaining control vertices is as follows:

Use the same processing method as the back calculation of the non-uniform B-spline curve, that is, use the type value points as data points to back calculate the control points. During calculation, first take the weight factors of all control vertices to 1, and then obtain the corresponding preliminary control After the vertex is reached, the curve is adjusted according to the corresponding weight factor as needed to obtain the final control vertex corresponding to the demand.

This embodiment defines the cubic non-uniform B-spline curve equation used to interpolate n+1 data points as:

Substitute Q _i and the corresponding node values into the above equation to obtain a linear equation system composed of n+1 vector equations, as follows:

In addition, two additional equations given by the boundary conditions are added:

C ₀ ', C _n ' are the tangent vectors at the end points, and the following system of equations is obtained:

in

Interpolation spline output results:

Figures 5 and 6 respectively adopt two different sampling methods. Figure 5 illustrates the first method using an arbitrary section of the waveform on the cycloid disk as an example to obtain uniform sampling results; Figure 6 illustrates the second method using a cycloidal disk. Take half of the waveforms at each end of any two ends of the cycloid line as an example to perform uniform sampling and obtain the results; Figure 7 The third method takes half of the waveforms at any two ends of the cycloid line as an example to perform uniform sampling and increase the number of control points. number. Among them, the fitting curve obtained by the first sampling method has a relatively fast curvature change speed at the left and right endpoints, which causes the fitted curve to deviate greatly from the original curve.

In order to avoid the impact of rapid changes in curvature of the curve at the fitting endpoints, the sampling method shown in Figure 6 is adopted to make the curves at both endpoints gentle, and increase the sampling frequency in the middle section, thereby reducing the impact of rapid changes in curvature on the fitted curve. influence of trends.

Figure 7 is a schematic diagram of the overall shape of the hypocycloid obtained by NURBS curve fitting after rotation transformation, which is a schematic diagram of the third type of sampling method in this embodiment. For a clearer illustration, Figure 8 is a diagram of the third type of sampling method of the present invention. A partially enlarged schematic diagram of the overall shape of the hypocycloid fitted by the NURBS curve, that is, the partially enlarged schematic diagram of Figure 7.

Curve fitting accuracy error verification:

The basic method for calculating the fitting error between the fitting curve and the data points is to calculate the distance from the data point to the curve. The average distance and the maximum distance are the main evaluation indicators. In this embodiment, in order to better compare various fitting methods To determine the size of the existing error, the accuracy error analysis of the curve fitting results of the previous stage is carried out by citing the two fitting curve error analysis methods of the MATLAB tool. The two methods are sum variance and coefficient of determination.

where the sum and variance are expressed as SSE: This statistical parameter calculates the sum of squares of the errors between the fitted data and the corresponding points of the original data. When the SSE is closer to 0, it means that the model selection and fitting are better, and the data prediction is more successful.

The coefficient of determination is expressed as R-square: In this embodiment, the coefficient of determination represents the quality of a fit through changes in data, and the normal range of the coefficient of determination is [0, 1]. When it is closer to 1, it indicates that the variables of the equation have a stronger ability to explain y. This The model also fits the data well.

Example 2

Referring to FIG. 9 , the sampling method of the interpolation spline output result in this embodiment corresponds to three types (three types) of fitting accuracy errors for verification. Among them, error 1 in Figure 9 is the fitting accuracy error of the fitting curve obtained by the first sampling method, error 2 is the fitting accuracy error of the fitting curve obtained by the second sampling method, and error 3 is the basis of the second sampling method. The accuracy error of the fitting curve obtained by increasing the number of control points. In the three cases, the accuracy error is basically at the same order of magnitude. This also reflects the correctness of the design method. However, in the third case, the accuracy error is more evenly distributed. , the error reaches 0.1 micron, and the coefficient of determination R-square reaches 0.999, which can more realistically reproduce the cycloid disk shape.

It should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solution of the present invention can be carried out. Modifications or equivalent substitutions without departing from the spirit and scope of the technical solution of the present invention shall be included in the scope of the claims of the present invention.

Claims (1)

1. A method of expressing the profile of the cycloid disk of a steel ball reducer, which is characterized by: including the following steps: Get the cycloid disk data model; Determine the node vectors of the cycloidal disk profile line data points according to the data model; Use the node vector of the data points to obtain the control vertex of the NURBS interpolation curve; Get the interpolation spline output result; Curve fitting accuracy error verification; The acquisition of the cycloid disk data model includes the following steps. The design parameters of the cycloidal disk profile inside the reducer are known; Input the design parameters into the parametric equation to obtain the inner cycloid plate profile of the reducer; The data model is obtained by discretizing and sampling the cycloid disk profile; The parametric equation is the expression equation of the cycloidal disk, as follows: The equation of the hypocycloid is: The epicycloid equation is: Among them, ds is the diameter of the steel ball distribution circle, and its calculation formula is: d _s = r _d /sin (180/nh), and r _d is the radius of the steel ball, and θ is the time base for pure rolling on the base circle of a non-centered rolling circle. The angle at which the circle is rolled, c is half of the eccentricity, n _b is the number of steel balls; Using the cumulative chord length parameter method to uniformly parameterize the data points of the profile line of the data model, it also includes: Let L be the total length of the curve, Let u ₀ =0, u _n =1; then/> Back-calculating the control vertices of the NURBS curve based on the node vectors determined by the type value points on the curve includes the following steps: Use the type value points as the data points to back-calculate the control vertices; First, set the weight factors of all the control vertices to 1. After obtaining the corresponding control vertices, adjust the curve according to the corresponding weight factors as needed; The cubic non-uniform B-spline curve equation used to interpolate n+1 data points is: Substitute Q _i and the corresponding node values into the above equation to obtain a linear equation system composed of n+1 vector equations, as follows: In addition, two additional equations given by the boundary conditions are added: C ₀ ', C _n ' are the tangent vectors at the end points, and the following system of equations is obtained: in, The interpolation spline output results include the following sampling methods. The first type uses any section of the waveform on the cycloid line as an example to obtain uniform sampling results; the second type uses any two ends of the cycloid line as an example. The results are obtained by taking half of the waveform as an example for uniform sampling; the third type is based on taking half of the waveform at each end of the cycloid line as an example for uniform sampling and increasing the number of control points; The fitting accuracy error verification includes the following steps: Calculate the distance from the data point to the curve, including calculating the average distance and maximum distance; Perform precision error analysis on the curve fitting results by citing the two fitting curve error analysis methods of MATLAB tools, including calculation and variance and coefficient of determination; where the sum and variance is SSE: It calculates the sum of squares of the errors between the fitted data and the corresponding points of the original data, and when the SSE is closer to 0, it means better model selection and fitting, and more successful data prediction; The coefficient of determination is R-square: It represents the quality of a fit through changes in data. The normal range of the determination coefficient is [0, 1]. When it is closer to 1, it indicates that the variables of the equation have a stronger ability to explain y, and the model fits the data better. also better; It also includes verification that the sampling method of the interpolation spline output result corresponds to three types of fitting accuracy errors; The first type is the fitting accuracy error of the fitting curve obtained by uniform sampling using the first sampling method by taking any waveform on the cycloid line as an example; the second type is taking any two ends of the cycloid line The fitting accuracy error of the fitting curve obtained by taking half of the waveform as an example for uniform sampling; the third category is taking half of the waveform at each end of the cycloid line as an example for uniform sampling and increasing the number of control points The accuracy error of the obtained fitting curve; the third type error reaches 0.1 micron, and the coefficient of determination reaches 0.999.