Method for acquiring contour line of grinding sand for forming screw rod

Abstract

The invention discloses a method for obtaining a screw forming grinding sand contour line, which comprises the following steps of: (1) limiting a solution interval of a transcendental equation about a spiral rotation angle in a specific interval, and obtaining all solutions of the transcendental equation in the interval; (2) the change rate of two adjacent solutions is calculated, and the solutions larger than or equal to a set threshold value are removed; and (3) substituting the screened solution into a grinding wheel contour equation to obtain grinding wheel contour discrete points, and adopting a fitting algorithm for an obtained discrete point set to obtain a smooth grinding wheel contour line. According to the method, the solution interval of the transcendental equation is limited in the peak period, close to the original point, of the screw spiral line, the complexity of multiple solutions is reduced, it is guaranteed that the radius of the grinding wheel is small, and machining is more convenient; a reasonable solution is screened out through the change rate, the continuity of the sand contour line is ensured, and the discontinuity is avoided by smoothing points on the sand contour line, so that high-precision machining is realized.

Description

Technical Field

[0001] The invention relates to the technical field of grinding processing, in particular to a method for obtaining a screw forming grinding sand contour line. Background Art

[0002] As the core component of screw machinery, the manufacturing accuracy 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 prior art, the screw rotor is generally processed by grinding. Traditional grinding wheels have problems such as easy shedding of abrasive grains, high grinding specific energy, large ratio of normal force to tangential force, need for regular sharpening, and easy clogging when processing plastic metal materials. Domestic and foreign scholars have made innovations and improvements in the structure of grinding wheels. By installing petal grinding wheels or radially grooving the grinding wheels to prepare slotted grinding wheels, intermittent grinding is achieved to improve heat dissipation and cooling effects, reduce grinding thermal damage and improve grinding efficiency (Nguyen T, Zhang L C. The coolant Penetration in Grinding with a Segmented Wheel-Part 1: Mechanism and Comparison with Conventional Wheels [J]. International Journal of Machine Tools and Manufacture, 2005, 45 (11): 1412-1420). However, previous studies have only focused on optimizing the material and structure of the grinding wheel, and have not fundamentally solved the problem of the accuracy of the grinding contour line. In the prior art, the CAM system generally determines the installation parameters and shape of the grinding wheel based on the shape of the screw and the relevant parameters of the machine tool, and gives the grinding process path. Regarding the determination of the grinding wheel shape, the prior art generally uses an analytical-based algorithm.

[0003] The first step of the analytical method is to establish the conversion relationship between the screw and the grinding wheel coordinate system according to the installation parameters. Since the two can be regarded as relatively static during grinding, the contact between the two is a fixed contact line in space. A transcendental equation is solved for each discrete point on the screw profile; each solution obtained corresponds to a point on the screw surface that can be ground on the profile point. All these points, that is, the discrete point series of the contact line, are rotated around the grinding wheel axis to obtain the shape of the grinding wheel. However, due to the complexity of the transcendental equation, the solution becomes difficult to predict. There may be multiple solutions, or even countless solutions, which requires a lot of time and computing resources to analyze and screen these solutions, resulting in a greatly increased workload and difficulty of calculation. Since the optimal solution cannot be determined, it is easy to cause the grinding wheel shape to not match the screw spiral shape, which affects the processing accuracy and increases the scrap rate. Therefore, it is urgent to develop a method for (calculating and) screening the optimal solution of the screw forming grinding sand contour line. Summary of the invention

[0004] The object of the present invention is to provide a method for obtaining a screw profile grinding sand contour line to solve the problems raised in the above background technology.

[0005] To achieve the above-mentioned purpose, the present invention provides the following technical scheme: a method for (calculating and) screening the optimal solution of the screw forming grinding sand contour line, comprising the following steps: step one: limiting the solution interval; step two: calculating the rate of change; step three: screening the solution; step four: smoothing treatment.

[0006] Furthermore, a method for obtaining a screw forming grinding sand contour line comprises:

[0007] (1) limiting the solution interval of the transcendental equation about the helical angle to a specific interval, and obtaining all solutions of the transcendental equation within the interval;

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

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

[0010] Before performing step (1), according to the prior art, 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 static during grinding, the contact between the two is a fixed spatial contact line in space, and a transcendental equation is solved for each discrete point on the screw profile.

[0011] More specifically, the transcendental equation for the helical 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 ) is a series of discrete points on the screw end tooth profile, θ is the helical angle, and K represents the discrete points (x u ,y u ), p is the spiral parameter, T is the installation center distance between the grinding wheel and the screw rotor, Y = x u sinθ+y u cosθ.

[0016] The above K represents the discrete point (x u ,y u ) can be obtained by MATLAB software. Solving the above equations can obtain a series of θ, obtain the spatial contact points, and the cross-section equation of the formed grinding wheel is:

[0017]

[0018] That is, by obtaining a series of θ, we can get the corresponding (Z c , R c ), that is, discrete points on the grinding wheel surface, R c The grinding wheel width is Z c The radius at Z c is the coordinate along the axial direction of the grinding wheel in the grinding wheel coordinate system. c , R c ) can be further fitted to obtain the grinding wheel profile function, that is, the function Z(R) about R, and the fitting algorithm is used to smooth Z(R).

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

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

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

[0022] Preferably, in the above step (2), a threshold of 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; and 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 eliminated, 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 change rate calculation formula is as follows:

[0024]

[0025] Where H is the rate of change, θ is the solution of the transcendental equation, where θ i is the i-th solution, θ i+1 is the i+1th solution.

[0026] The screened θ solution is substituted into the grinding wheel profile equation to obtain the grinding wheel profile discrete point data, and the obtained grinding wheel profile discrete point data is fitted using a fitting algorithm to obtain the grinding wheel profile function.

[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., sand contour line) is specifically:

[0029] Z=a 0 +a 1 R+a 2 R 2 +…+a m R m

[0030] Where (Z, R) represents a discrete point on the grinding wheel profile, whose value is determined by the helical angle θ; m is the degree of the polynomial; a 0 ,a 1 ,…a m is the coefficient to be determined.

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

[0032] Preferably, in the process of solving the linear equations by the least squares method, the error function E is:

[0033]

[0034]

[0035] In order to solve for the coefficient a 0 ,a 1 ,…a m , the error function E is respectively about a 0 ,a 1 ,…a m Find the partial derivative and set it to zero:

[0036]

[0037] Thus, we get a linear system of equations consisting of m+1 equations, and the coefficient a can be solved by matrix operation. 0 ,a 1 ,…a m .

[0038] Compared with the prior art, the beneficial effects of the present invention are as follows: the present invention reduces the complexity of multiple solutions by limiting the solution interval of the transcendental equation to the peak period of the screw helix close to the origin, while ensuring that the grinding wheel radius is small. Because for a spiral line, the closer the part closer to the origin is to the grinding wheel axis, the smaller the required grinding radius is, making processing more convenient. By defining a threshold value of the rate of change between adjacent solutions, calculating the rate of change between adjacent solutions, and comparing it with the defined threshold value, reasonable solutions can be effectively screened out, thereby ensuring the continuity of the sand contour line, and bringing the screened solutions into the grinding wheel profile equation for smoothing, effectively avoiding the discontinuity introduced by solutions of different periods, and ensuring that the grinding wheel can perfectly fit the screw surface during processing, thereby achieving high-precision processing. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 is a flow chart of the method of the present invention;

[0040] Figure 2 is a flow chart of the steps of the present invention;

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

[0042] Figure 4 It is the period between two adjacent lowest points of the sine curve projected by the spiral line in the vertical plane;

[0043] Figure 5 The model value point data (x u ,y u );

[0044] Figure 6 is the coordinate of the sand contour obtained by solving the θ coordinate;

[0045] Figure 7 The obtained sand contour line. DETAILED DESCRIPTION

[0046] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0047] Please refer to the attached Figure 1 -Attached Figure 2 , an embodiment provided by the present invention: a method for accurately calculating the contour line of screw forming grinding sand, comprising the following steps: step one, limiting the solution interval; step two, calculating the change rate; step three, screening the solution; step four, smoothing treatment;

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

[0049] In order to study the screw rotor forming processing principle and the design method of the forming tool, it is necessary to first study the spatial position relationship between the screw rotor and the forming tool (see Appendix). Figure 3 ).

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

[0051]

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

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

[0054]

[0055] In the formula, X c ,Y c ,Z c is the grinding wheel profile equation, R c The grinding wheel width is Z cThe radius of the line R. c With face Y c O c Z c The angle between.

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

[0057] (k c ×R)·n=0

[0058] In the formula, k c It's 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 on the three coordinate axes for any point on the screw surface can be calculated by the following formula:

[0060] have to:

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

[0062]

[0063] Substituting ②③ into ① and simplifying it, we can get:

[0064]

[0065] Further simplify the formula, let:

[0066]

[0067] We get the transcendental equation about 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] Where K represents the discrete point (x u ,yu ) can be obtained by MATLAB software. Solving the above equations can obtain a series of θ, obtain the spatial contact points, and the cross-section equation of the formed grinding wheel is:

[0072]

[0073] ④ In the above step 1, the solution interval of the transcendental equation for solving the helical angle is limited to a peak period of the screw helix, and all solutions of the transcendental equation within the peak period are obtained; wherein the peak period is the period between two adjacent lowest points of the sine curve projected on the vertical plane of the helix; the peak period is the peak period closest to the origin;

[0074] In the above step 2, a threshold value of the change rate between adjacent solutions is defined, the solution obtained in step 1 is taken, and the change rate between adjacent solutions is calculated; the change rate calculation formula is as follows:

[0075]

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

[0077] In the above step 3, the change rate calculated in step 2 is compared with the defined threshold. If the change rate is greater than or equal to the threshold, the solution is eliminated. If the change rate is less than the threshold, the solution is retained to obtain a set of solutions.

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

[0079] Z=a 0 +a 1 R+a 2 R 2 +…+a m R m

[0080] Where (Z, R) represents a discrete point on the grinding wheel profile, whose value is determined by the helical angle θ; m is the degree of the polynomial; a 0 ,a 1 ,…a m is the coefficient to be determined;

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

[0082]

[0083]

[0084] In order to solve for the coefficient a 0 ,a 1 ,…a m , the error function E is respectively about a 0 ,a 1 ,…a m Find the partial derivative and set it to zero:

[0085]

[0086] Thus, we get a linear system of equations consisting of m+1 equations, and the coefficient a can be solved by matrix operation. 0 ,a 1 ,…a m , and finally obtain the functional formula about (Z, R).

[0087] In order to further verify the effect of the present invention, the following experimental simulation was performed.

[0088] The following is a detailed calculation simulation process designed for the C1313 female rotor of the present invention, which is used to calculate the value point (x u ,y u ) Calculate the screw profile, screen the rotation angle θ, calculate the sand contour line (Z c , R c ) and perform polynomial fitting.

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

[0090] 1.1 Import data

[0091] Import the attached file 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 fitting curve is drawn. 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. Screening rotation angle θ

[0097] First, limit θ to the interval [0, π], set the change threshold H < 0.1, and solve θ according to 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 contour line

[0103] According to the above 3, the θ coordinate is obtained, and the sand contour coordinate is solved according to the following formula (see attached) Figure 6 :

[0104]

[0105] The three columns of coordinates are (X c , Y c , Z c ) coordinates, and satisfy the formula Xc 2 +Yc 2 =Rc 2 , and obtain a series of coordinates (R c , Z c ), and the least squares polynomial fitting method is used to fit Z c About R c Function Z c =f(R c ) and can be obtained with Figure 7 The sand contour line (white part).

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

[0107] Based on the above, the advantage of the present invention is that when the present invention is used, firstly, by limiting the solution interval of the transcendental equation for solving the spiral angle to the peak period of the screw helix close to the origin, the complexity of multiple solutions can be effectively reduced, and the radius of the grinding wheel can be ensured to be small, making processing more convenient; then, by defining a change rate threshold, the solution is gradient screened, thereby ensuring the continuity of the sand contour line; finally, by adopting a polynomial fitting method, the contour of the grinding wheel can be represented as a smooth curve through polynomial fitting, avoiding mutations caused by different periodic solutions, thereby obtaining a simply connected smooth curve as the contour of the grinding wheel.

[0108] It will be apparent to those skilled in the art that the 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 the spirit or essential features of the invention. Therefore, the embodiments should be considered exemplary and non-limiting in all respects, and the scope of the invention is defined by the appended claims rather than the foregoing description, and it is intended that all variations falling within the meaning and scope of the equivalent elements of the claims be included in the invention. Any reference numeral in a claim should not be considered as limiting the claim to which it relates.

Claims

1. A method for obtaining a screw forming grinding sand contour line, the characteristics of which include: (1) limiting the solution interval of the transcendental equation about the helical angle to a specific interval, and obtaining all solutions of the transcendental equation within the interval; (2) Calculate the change rate of two adjacent solutions and eliminate the solutions that are greater than or equal to the set threshold; (3) The screened solution is substituted into the grinding wheel profile equation to obtain a series of discrete points on the sand contour line, and the obtained discrete points are fitted to obtain the sand contour line.

2. The method for obtaining the screw forming grinding sand contour line according to claim 1, characterized in that: In step (1), the solution interval of the transcendental equation of the helical angle is limited to a peak period of the screw helix.

3. The method for obtaining the screw forming grinding sand contour line according to claim 2, characterized in that: The peak period is the peak period corresponding to two adjacent lowest points of a sine curve obtained by projecting the spiral line in the vertical direction.

4. The method for obtaining the screw forming grinding sand contour line according to claim 2, characterized in that: The peak period is 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.

5. The method for obtaining the screw forming grinding sand contour line according to claim 1, characterized in that: In step (2), the change rate calculation formula is as follows: Where H is the rate of change and θ is the solution to the transcendental equation.

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

7. The method for obtaining the screw forming grinding sand contour line according to claim 6, characterized in that: The sand contour obtained by fitting is expressed as follows: Z=a0+a1R+a2R 2 +…+a m R m Where (Z, R) represents a discrete point on the grinding wheel surface, and its value is determined by the helical angle θ; m is the degree of the polynomial; a0, a1, … a m is the coefficient to be determined.

8. The method for obtaining the screw forming grinding sand contour line according to claim 6, characterized in that: In the polynomial fitting method, the least square method is used to solve the linear equations to determine the polynomial coefficients.

9. The method for obtaining the screw forming grinding sand contour line according to any one of claims 1 to 8, characterized in that: The transcendental equation expression for the helical angle is as follows: [(x u cosθ-y u sinθ)(cosθ-Ksinθ)+(x u sinθ+y u cosθ)(sinθ+Kcosθ)] ·[YT-pcotω]+[p(cosθ-Ksinθ)]·pθ +[p(sinθ+Kcosθ)]·Tcotω=0 Among them, (x u ,y u ) is a series of discrete points on the screw end tooth profile, θ is the helical angle, and K represents the discrete points (x u ,y u ), p is the spiral parameter, T is the installation center distance between the grinding wheel and the screw rotor, Y = x u sinθ+y u cosθ.

10. The method for obtaining the screw forming grinding sand contour line according to claim 9, characterized in that: The grinding wheel profile equation is as follows: (Z c , R c ) represents the discrete points on the grinding wheel surface, R c The grinding wheel width is Z c The radius at this time, ω is the installation angle between the grinding wheel and the screw rotor.

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