A method for obtaining a profile of a screw forming grinding wheel

Abstract

The application discloses a kind of screw forming grinding wheel profile line obtaining method, comprising: (1) the solving interval of transendental equation about helical rotation angle is limited in certain interval, and all solutions of transendental equation in the interval are obtained;(2) the change rate of two adjacent solutions is calculated, and the solution greater than or equal to the set threshold is eliminated;(3) the screened solution is brought into grinding wheel profile equation to obtain grinding wheel profile discrete points, and the obtained discrete point set is fitted to obtain smooth grinding wheel profile line.The application limits the solving interval of transendental equation in the peak cycle close to the origin of screw helix, reduces the complexity of multiple solutions, ensures that the grinding wheel radius is small, so that processing is more convenient;Reasonable solution is screened by change rate, the continuity of grinding wheel profile line is guaranteed, the points on grinding wheel profile line are smoothed, discontinuity is avoided, so that high-precision processing is realized.

Description

Technical Field

[0001] This invention relates to the field of grinding technology, specifically to a method for obtaining the profile of a screw forming grinding sand. Background Technology

[0002] As the core component of screw machinery, the manufacturing precision and surface quality of the screw rotor have a crucial impact on the overall performance, service life, vibration and noise of the product. In the existing technology, the screw rotor is generally processed by grinding. Traditional grinding wheels have problems such as easy abrasive grain shedding, high grinding specific energy, large ratio of normal force to tangential force, need for regular sharpening, and easy clogging when processing ductile metal materials. Domestic and foreign scholars have made innovations and improvements in the structure of grinding wheels. By installing petal grinding wheels or making radial grooves on the grinding wheels, grooved grinding wheels are prepared to achieve intermittent grinding, so as to improve heat dissipation and cooling effect, reduce thermal damage in grinding and improve grinding efficiency (Nguyen T, Zhang L C. The coolant Penetration in Grinding with a Segmented Wheel-Part1:Mechanism and Comparison with Conventional Wheels[J].InternationalJournal of Machine Tools andManufacture,2005,45(11):1412-1420). However, previous research has only focused on optimizing the grinding wheel material and structure, without fundamentally solving the problem of grinding wheel profile accuracy. In existing technologies, CAM systems typically determine the grinding wheel's installation parameters and shape based on the screw shape and machine tool parameters, and then provide the grinding path. Regarding the determination of the grinding wheel shape, existing technologies generally employ analytical algorithms.

[0003] The first step in the analytical method is to establish the transformation relationship between the screw and grinding wheel coordinate systems based on the installation parameters. Since both can be considered relatively stationary during grinding, their contact is a fixed spatial contact line. A transcendental equation is solved for each discrete point on the screw profile. Each solution corresponds to a point on the screw surface where that profile point can be ground. All these points, i.e., the discrete point sequence of the contact line, are rotated around the grinding wheel axis to obtain the shape of the grinding wheel. However, the complexity of the transcendental equation makes the solutions unpredictable, potentially resulting in multiple or even countless solutions. This necessitates significant time and computational resources to analyze and select these solutions, greatly increasing the workload and difficulty of the calculations. The inability to determine the optimal solution can easily lead to poor matching between the grinding wheel shape and the screw helical shape, affecting machining accuracy and increasing the scrap rate. Therefore, there is an urgent need to develop a method for calculating and selecting the optimal solution for the grinding wheel profile of the screw forming process. Summary of the Invention

[0004] The purpose of this invention is to provide a method for obtaining the profile of the grinding sand used for screw forming, so as to solve the problems mentioned in the background art.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a method for (calculating) and screening the optimal solution of the profile of the screw forming grinding sand, comprising the following steps: Step 1: limiting the solution interval; Step 2: calculating the rate of change; Step 3: screening the solution; Step 4: smoothing.

[0006] Furthermore, a method for obtaining the profile of a screw forming grinding wheel includes:

[0007] (1) Limit the solution interval of the transcendental equation concerning the spiral rotation angle to a specific interval, and obtain all solutions of the transcendental equation within that interval;

[0008] (2) Calculate the rate of change between two adjacent solutions and remove solutions that are greater than or equal to a set threshold;

[0009] (3) Substitute the filtered solution into the grinding wheel profile equation to obtain the coordinates of discrete points on the sand profile line, and fit the obtained discrete points to obtain the sand profile line.

[0010] Before proceeding to step (1), according to the existing technology, the transformation relationship between the screw and grinding wheel coordinate systems is first established based on the installation parameters. Since the two can be regarded as relatively stationary during grinding, the contact between the two is a fixed spatial contact line in space. A transcendental equation is solved for each discrete point on the screw profile.

[0011] More specifically, the transcendental equation expression for the helical rotation angle is as follows:

[0012] [(x ucosθ-y u sinθ)(cosθ-K sinθ)+(x u sinθ+y u cosθ)(sinθ+K cosθ)]

[0013] ·[-Tp cotω]+[p(cosθ-K sinθ)]·pθ

[0014] +[p(sinθ+K cosθ)]·T cotω=0

[0015] Among them, (x u y u ) represents a series of discrete points on the tooth profile of the screw end face, θ is the helical rotation angle, and K represents the discrete point (x u y u The slope of the curve corresponding to ) is given by p, where p is the helical parameter, T is the center distance between the grinding wheel and the screw rotor, and Y = x. u sinθ+y u cosθ.

[0016] The above K represents the discrete point (x) u y u The slope of the curve corresponding to the given point can be obtained using MATLAB software. Solving the above equation yields a series of θ values, thus identifying the spatial contact point. The equation for the forming grinding wheel cross-section is:

[0017]

[0018] That is, by finding a series of θ, the corresponding (Z) can be obtained. c R c ), that is, discrete points on the grinding wheel profile, R c The width of the grinding wheel is Z. c The radius of time, Z c This represents the coordinate along the grinding wheel axis in the grinding wheel coordinate system. (Z) c R c The grinding wheel profile function can be further fitted to obtain the function Z(R) with respect to R. The fitting algorithm is then used to smooth Z(R).

[0019] Preferably, in step (1), the solution range of the transcendental equation for the helical rotation angle is limited to one peak period of the screw helix, and all solutions of the transcendental equation within that peak period are obtained.

[0020] Furthermore, the peak period is the period between two adjacent lowest points of the sine curve obtained by projecting the helix in the vertical direction (the corresponding peak period) (see appendix). Figure 4 ).

[0021] Furthermore, the peak period is selected from the peak period closest to the origin of the screw coordinate system (or the peak period including the origin), where the origin is the point on the screw axis closest to the grinding wheel axis.

[0022] Preferably, in step (2) above, a threshold for the rate of change between adjacent solutions is defined, the solution obtained in step (1) is taken, and the rate of change between adjacent solutions is calculated; the calculated rate of change is compared with the defined threshold. If the rate of change is greater than or equal to the threshold, the solution is removed; if the rate of change is less than the threshold, the solution is retained, and a set of solutions is obtained.

[0023] Furthermore, in step (2), the formula for calculating the rate of change is as follows:

[0024]

[0025] Where H is the rate of change, and θ is the solution to the transcendental equation, where θ i For the i-th solution, θ i+1 This is the (i+1)th solution.

[0026] The filtered θ solution is substituted into the grinding wheel profile equation to obtain the discrete point data of the grinding wheel profile. The obtained discrete point data of the grinding wheel profile is fitted by a fitting algorithm to obtain the function expression of the grinding wheel profile.

[0027] Preferably, in step (3), a polynomial fitting method is used for fitting.

[0028] Preferably, in the polynomial fitting method, the fitting function (i.e., the sand contour line) is specifically:

[0029] Z = a0 + a1R + a2R 2 +…+a m R m

[0030] Where (Z, R) represent discrete points on the grinding wheel profile, and their values ​​are determined by the helical rotation angle θ; m is the degree of the polynomial; a0, a1, ... a m The coefficients are to be determined.

[0031] Preferably, in the polynomial fitting method, the least squares method is used to solve the linear equation system and determine the polynomial coefficients.

[0032] Preferably, in the process of solving the linear equation system using the least squares method, the error function E is:

[0033]

[0034]

[0035] To solve for the coefficients a0, a1, ... am For the error function E with respect to a0, a1, ... a m Find the partial derivatives and set them to zero:

[0036]

[0037] This results in a system of linear equations consisting of m+1 equations, from which the coefficients a0, a1, ... a1 can be solved using matrix operations. m .

[0038] Compared with the prior art, the beneficial effects of the present invention are as follows: By limiting the solution interval of the transcendental equation to the peak period of the screw helix near the origin, the present invention reduces the complexity of multiple solutions and ensures a small grinding wheel radius. This is because for a helix, the closer it is to the origin, the closer it is to the grinding wheel axis, and the smaller the required grinding radius, making the machining process more convenient. By defining a threshold for the rate of change between adjacent solutions, calculating the rate of change between adjacent solutions, and comparing it with the defined threshold, reasonable solutions can be effectively selected, thus ensuring the continuity of the grinding wheel profile. The selected solutions are then substituted into the grinding wheel profile equation for smoothing, effectively avoiding discontinuities introduced by solutions of different periods, ensuring that the grinding wheel can perfectly fit the screw surface during machining, thereby achieving high-precision machining. Attached Figure Description

[0039] Figure 1 This is a flowchart of the method of the present invention;

[0040] Figure 2 This is a flowchart of the steps of the present invention;

[0041] Figure 3 The spatial relationship between the screw rotor and the grinding wheel;

[0042] Figure 4 The period is the interval between two adjacent lowest points of the sine curve projected onto the vertical plane by the spiral.

[0043] Figure 5 The simulation in this embodiment uses the model value point data (x) u y u );

[0044] Figure 6 The coordinates of the sand profile obtained from the θ coordinates;

[0045] Figure 7 The obtained sand outline. Detailed Implementation

[0046] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0047] Please see the appendix Figure 1 - Appendix Figure 2 The present invention provides an embodiment of a method for accurately calculating the profile of a screw forming grinding sand, comprising the following steps: Step 1, defining the solution interval; Step 2, calculating the rate of change; Step 3, filtering solutions; Step 4, smoothing the surface.

[0048] Before proceeding to step one, the transformation relationship between the screw and grinding wheel coordinate systems is first established based on the installation parameters. Since the two can be considered to be relatively stationary during grinding, their contact is a fixed spatial contact line in space. A transcendental equation is solved for each discrete point on the screw profile.

[0049] To study the forming process principle of screw rotors and the design method of forming tools, it is first necessary to investigate the spatial relationship between the screw rotor and the forming tool (see appendix). Figure 3 ).

[0050] In the screw profile grinding process, the screw rotor and the profile grinding wheel mesh in space to grind the screw profile. It is known that the screw end face tooth profile is formed by a series of discrete points (x... u， y u If it is composed of ), then its helical surface equation is:

[0051]

[0052] Where X, Y, and Z are the equations of the screw's helical surface; p is the helical parameter; and θ is the helical rotation angle.

[0053] Assume the grinding wheel profile consists of a series of discrete points (Z). c ,R c Composed of ) The surface equation can be expressed as:

[0054]

[0055] In the formula X c ,Y c Z c It is the equation of the grinding wheel profile, R c The width of the grinding wheel is Z. c The radius at time. φ is a parameter, representing the radius of the line R. c With face Y c O c Z cThe angle between them.

[0056] When solving for the sand profile given the screw profile, point M is one of the contact points, and the equation of the contact line can be expressed as:

[0057] (k c ×R)·n=0

[0058] In the formula, k c It is OX c Y c Z c Z in the coordinate system c The unit vector in the coordinate direction, R is the polar coordinate form of the grinding wheel surface equation, and n is the normal of point M in the coordinate system O-XYZ.

[0059] The components of the normal vector at any point on the screw surface along the three coordinate axes can be obtained by the following formula.

[0060] have to:

[0061] Taking the partial derivatives of u and θ of the first equation, we get:

[0062]

[0063] Substituting ② and ③ into ① and simplifying, we get:

[0064]

[0065] To further simplify the formula, let:

[0066]

[0067] We obtain the transcendental equation for θ:

[0068] [(x u cosθ-y u sinθ)(cosθ-K sinθ)+(x u sinθ+y u cosθ)(sinθ+K cosθ)]

[0069] ·[YTp cotω]+[p(cosθ-K sinθ)]·pθ

[0070] +[p(sinθ+K cosθ)]·T cotω=0

[0071] In the formula, K represents the discrete point (x) u y u The slope of the curve corresponding to the given point can be obtained using MATLAB software. Solving the above equation yields a series of θ values, thus identifying the spatial contact point. The equation for the forming grinding wheel cross-section is:

[0072]

[0073] ④ In step one above, the solution interval of the transcendental equation for solving the helical rotation angle is limited to one peak period of the screw helix, and all solutions of the transcendental equation within that peak period are obtained; wherein, the peak period is the period between two adjacent lowest points of the sine curve projected by the helix in the vertical plane; the peak period is selected as the one closest to the origin.

[0074] In step two above, a threshold for the rate of change between adjacent solutions is defined. The solutions obtained in step one are taken, and the rate of change between adjacent solutions is calculated. The formula for calculating the rate of change is as follows:

[0075]

[0076] Where H is the rate of change and θ is the solution to the transcendental equation;

[0077] In step three above, the rate of change calculated in step two is compared with a defined threshold. If the rate of change is greater than or equal to the threshold, the solution is removed; if the rate of change is less than the threshold, the solution is retained, thus obtaining a set of solutions.

[0078] Then, the obtained θ is input into the grinding wheel cross-section equation to obtain multiple sets of (Z, R). These (Z, R) data sets are then used for smoothing. The fitting algorithm uses a polynomial fitting method, and the fitting function is as follows:

[0079] Z = a0 + a1R + a2R 2 +…+a m R m

[0080] Where (Z, R) represent discrete points on the grinding wheel profile, and their values ​​are determined by the helical rotation angle θ; m is the degree of the polynomial; a0, a1, ... a m The coefficients are to be determined.

[0081] In polynomial fitting, the least squares method is used to solve the system of linear equations and determine the polynomial coefficients. During the process of solving the system of linear equations using the least squares method, the error function E is:

[0082]

[0083]

[0084] To solve for the coefficients a0, a1, ... a m For the error function E with respect to a0, a1, ... a m Find the partial derivatives and set them to zero:

[0085]

[0086] This results in a system of linear equations consisting of m+1 equations, from which the coefficients a0, a1, ... a1 can be solved using matrix operations. m Finally, we obtain a function relating (Z, R).

[0087] To further verify the effectiveness of the present invention, the following experimental simulations were conducted.

[0088] The following is a detailed calculation and simulation process for the design of the C1313 female rotor in this invention, used to analyze the model value point (x). u y u Calculate the screw profile, screening rotation angle θ, and sand profile (Z). c R c And perform polynomial fitting.

[0089] 1. Data import and B-spline fitting

[0090] 1.1 Importing Data

[0091] Importing the appendix into Matlab Figure 5 The type value point data (x) u y u ), where p = 36.63 mm, ω = 42°, and T = 200 mm.

[0092] 1.2B Spline Fitting

[0093] The screw profile is generated using B-spline fitting, and the fitted curve is plotted. The specific fitting process is implemented in Matlab.

[0094] 2. Calculate the slope K

[0095] Calculate the slope based on the B-spline fitting results. The specific fitting process is implemented in Matlab.

[0096] 3. Filter by rotation angle θ

[0097] First, limit θ to the interval [0, π], with a change threshold H < 0.1, and solve for θ using the following two formulas.

[0098] [(x u ucosθ-y u sinθ)(cosθ-K sinθ)+(x u sinθ+y u cosθ)(sinθ+K cosθ)]

[0099] ·[YTp cotω]+[p(cosθ-K sinθ)]·pθ

[0100] +[p(sinθ+K cosθ)]·T cotω=0

[0101]

[0102] 4. Obtain the coordinates of the sand outline.

[0103] Based on the above 3, the θ coordinate is obtained, and the coordinates of the sand contour line are solved according to the following formula (see appendix). Figure 6 :

[0104]

[0105] The three columns of coordinates are (X) c Y c Z c The coordinates satisfy the formula Xc 2 +Yc 2 =Rc 2 A series of coordinates (R) are obtained. c Z c Z was fitted using the least squares polynomial fitting method. c Regarding R c function Z c =f(R) c ), and can obtain attachments Figure 7 The outline of the sand (white part).

[0106] Depend on Figure 7 The results show that the sand contour line after fitting using the method of the present invention is smooth and gentle, which better meets the requirements of industrial processing and reduces the processing difficulty.

[0107] Based on the above, the advantages of this invention are as follows: First, by limiting the solution interval of the transcendental equation for solving the helical rotation angle to the peak period near the origin of the screw helix, the complexity of multiple solutions can be effectively reduced, ensuring a smaller grinding wheel radius and making processing more convenient. Second, by defining a rate of change threshold, gradient screening of solutions is performed, thereby ensuring the continuity of the grinding wheel profile. Finally, by employing a polynomial fitting method, the profile of the grinding wheel can be represented as a smooth curve, avoiding abrupt changes caused by solutions of different periods, thus obtaining a simply connected smooth curve as the profile of the grinding wheel.

[0108] It will be apparent to those skilled in the art that the present invention is not limited to the details of the exemplary embodiments described above, and that the invention can be implemented in other specific forms without departing from its spirit or essential characteristics. Therefore, the embodiments should be considered in all respects as exemplary and non-limiting, and the scope of the invention is defined by the appended claims rather than the foregoing description. Thus, all variations falling within the meaning and scope of equivalents of the claims are intended to be included within the present invention. No reference numerals in the claims should be construed as limiting the scope of the claims.

Claims

1. A method for obtaining the profile of a screw forming grinding wheel, characterized by: (1) Limit the solution interval of the transcendental equation concerning the spiral rotation angle to a specific interval, and obtain all solutions of the transcendental equation within that interval; (2) Calculate the rate of change between two adjacent solutions and remove solutions that are greater than or equal to a set threshold; (3) Substitute the filtered solution into the grinding wheel profile equation to obtain a series of discrete points on the sand profile line, and fit the obtained discrete points to obtain the sand profile line. In step (1), the solution range of the transcendental equation for the helical rotation angle is limited to one peak period of the screw helix; The peak period is selected from the peak period closest to the origin of the screw coordinate system, where the origin is the point on the screw axis that is closest to the grinding wheel axis. In step (2), the formula for calculating the rate of change is as follows: Where H is the rate of change and θ is the solution to the transcendental equation.

2. The method for obtaining the profile of the screw forming grinding sand according to claim 1, characterized in that, The peak period is the peak period between two adjacent lowest points of a sine curve obtained by projecting the helix in the vertical direction.

3. The method for obtaining the profile of the screw forming grinding sand according to claim 1, characterized in that, In step (3), a polynomial fitting method is used for fitting.

4. The method for obtaining the profile of screw forming grinding sand according to claim 3, characterized in that, The sand profile obtained by fitting is represented as follows: Where (Z, R) represents discrete points on the grinding wheel profile, and its value is determined by the helix angle θ; m is the degree of the polynomial; The coefficients are to be determined.

5. The method for obtaining the profile of the screw forming grinding sand according to claim 3, characterized in that, In the polynomial fitting method, the least squares method is used to solve the linear equation system and determine the polynomial coefficients.

6. The method for obtaining the profile of screw forming grinding sand according to any one of claims 1 to 5, characterized in that, The transcendental equation expression for the helical rotation angle is as follows: Among them, (x) u y u ) represents a series of discrete points on the tooth profile of the screw end face, θ is the helical angle, and K represents the discrete point (x). u y u The slope of the curve corresponding to ) is p, where p is the helical parameter and T is the center distance between the grinding wheel and the screw rotor. .

7. The method for obtaining the profile of screw forming grinding sand according to claim 6, characterized in that, The grinding wheel profile equation is as follows: (Z) c R c R represents discrete points on the grinding wheel profile. c The width of the grinding wheel is Z. c radius at time, The installation angle between the grinding wheel and the screw rotor.

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