A high-order polynomial topology modification method for generating a tooth surface of a gear

Abstract

The application relates to a high-order polynomial topology modification method for generating a tooth surface of a gear, and belongs to the technical field of gear machining and manufacturing. The operation steps are as follows: a continuous generating grinding mathematical model of a worm grinding wheel is established, a position function additional theoretical high-order polynomial of a radial feed axis X1 of the grinding wheel, a tangential feed axis Y1 of the grinding wheel and an axial feed axis Z1 of the grinding wheel is established, a normal deviation between an actual machined tooth surface and a standard tooth surface is taken as a judgment standard, polynomial coefficients of machine tool axis motion functions are optimized through a sensitivity matrix (SM) algorithm, the axis motion functions are converted into machine tool interpolation polynomials which can be run on a gear grinding machine through a polynomial conversion mode proposed in the application, and axis motion setting experiments are carried out on the YW7232 CNC gear grinding machine according to a set target function by using a POLY code of a polynomial interpolation function of a Siemens 840Dsl numerical control system. Through analysis and experiments on the polynomial interpolation program of the Siemens 840Dsl gear grinding machine numerical control system, the feasibility of the high-order polynomial topology modification method is proved.

Description

Technical Field

[0001] This invention belongs to the field of gear processing and manufacturing technology, and specifically relates to a high-order polynomial implementation method for topological modification of the tooth surface of generating ground teeth. Background Technology

[0002] With globalization and the continuous transformation and development of the manufacturing industry, various sectors have placed higher demands on gear surface quality and meshing performance. To meet these demands, topological modification of the gear tooth surface is employed to improve its surface quality and meshing performance. Traditional machining modification typically involves adjusting the grinding wheel profile and grinding wheel mounting parameters. However, this method requires special customization of the grinding wheel and is not suitable for small-batch processing where tooth surface modification parameters change, resulting in high processing costs, low efficiency, and poor flexibility. Advanced modification technology, on the other hand, does not require modification of the cutting tool or dressing wheel when tooth surface parameters change, offering the advantage of high efficiency. Furthermore, current research on advanced modification technology is largely confined to theory and has not been integrated with practical applications. Currently, most worm gear grinding machines in China use the Siemens 840Dsl CNC system. In the implementation of advanced modification technology, there is a problem of incompatibility between the fitted theoretical axis function polynomial and the polynomial interpolation function of the Siemens 840Dsl CNC system, causing most advanced modification technologies to remain at the theoretical stage. Summary of the Invention

[0003] In order to achieve compatibility between the fitted theoretical axis function polynomial and the polynomial interpolation function of the Siemens 840Dsl CNC system, this invention provides a high-order polynomial topology modification method for generating and grinding tooth surfaces.

[0004] A high-order polynomial topology shaping method for generating and grinding tooth surfaces is applicable to worm gear grinding machines based on the Siemens 840Dsl CNC system. The method includes nine CNC axes and an electronic gearbox. The nine CNC axes are: grinding wheel head rotation axis A1, grinding wheel spindle B1, outer support moving axis Z2, worktable rotation axis C1, dressing wheel rotation axis B2, dressing mechanism rotation axis C2, grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1, and grinding wheel axial feed axis Z1.

[0005] Under the condition of satisfying the generating relationship of the electronic gearbox, the position functions of the radial feed axis X1, tangential feed axis Y1, and axial feed axis Z1 of the worm gear grinding machine are defined as higher-order polynomials, and the motion is performed according to the set higher-order polynomials. The operation steps are as follows:

[0006] (1) Establish a mathematical model for continuous generating grinding of worm gear grinding wheels

[0007] The mathematical model for continuous generating grinding of worm gear grinding wheel is as follows (3):

[0008]

[0009] In equation (3), ξ is the involute parameter, which is dimensionless; τ is the helical parameter, which is dimensionless. This refers to the rotation angle of the worm gear grinding wheel, measured in rad. This indicates the position of the machine tool's Z1 axis, in mm. This refers to the position of the machine tool's Y1 axis, in mm; n p In the gear coordinate system S p Normal vector in coordinate system; R p It is in S p Biparametric surface of a worm gear grinding wheel in a coordinate system; t is time, in seconds;

[0010] (2) Establish the position functions of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1, and grinding wheel axial feed axis Z1, and supplement the theoretical higher-order polynomials.

[0011] In the mathematical model of continuous generating grinding of worm wheel in step (1), the feed amount of the radial feed axis X1 of the grinding wheel, the feed amount of the tangential feed axis Y1 of the grinding wheel and the feed amount of the axial feed axis Z1 of the grinding wheel are respectively expressed as functions that change with time, as shown in formula (4).

[0012]

[0013] In equation (4), t is the time variable, in seconds; A is the center distance between the cylindrical gear and the worm wheel, in mm. This represents the axial feed rate of the worm gear grinding wheel, expressed in mm / s. This represents the tangential feed rate of the worm gear grinding wheel, expressed in mm / s. This refers to the rotation angle of the worm gear grinding wheel, measured in rad. The rotation angle of the workpiece gear is expressed in rad; n B This indicates the rotational speed of the worm gear grinding wheel, measured in rad / min (N). g N represents the number of teeth on a gear, dimensionless; w p represents the number of grinding wheel heads, dimensionless; w This refers to the lead parameter of the grinding wheel, in mm; p g The helix parameter of the gear is dimensionless. This indicates the position of the machine tool's Z1 axis, in mm. This refers to the position of the Y1 axis of the machine tool, in mm. This refers to the position of the X1 axis of the machine tool, in mm.

[0014] In order to achieve continuous generating topology modification of the gear grinding and realize the high-order polynomial motion of the three axes of the worm gear grinding machine, namely the grinding wheel axial feed axis Z1, the grinding wheel tangential feed axis Y1, and the grinding wheel radial feed axis X1, a fourth-order polynomial motion is added to the standard motion of the grinding wheel axial feed axis Z1, the grinding wheel tangential feed axis Y1, and the grinding wheel radial feed axis X1, as shown in formula (5):

[0015]

[0016] This yields three theoretical axis functions of higher order polynomials;

[0017] In equation (5), The ratio of the Z1 axis position to the tooth width without modification is dimensionless; as can be seen from equation (4), Therefore, the variable in equation (5) is time t; from equation (4), we know that F N (N = X1, Y1, Z1) represents the position of each axis under standard motion; F' N (N = X1, Y1, Z1) represents the position of each axis after adding a fourth-order polynomial motion to the standard motion; λ1~λ4 are the polynomial coefficient values ​​of the higher-order polynomial of the radial feed axis X1 of the grinding wheel, dimensionless; λ5~λ8 are the polynomial coefficient values ​​of the higher-order polynomial of the tangential feed axis Y1 of the grinding wheel, dimensionless; λ9~λ 12 The polynomial coefficients of the higher-order polynomial of the grinding wheel axial feed axis Z1 are dimensionless.

[0018] (3) Sensitivity matrix algorithm fits the numerical values ​​of higher-order polynomial coefficients of theoretical axis functions

[0019] By modifying the polynomial coefficients of the higher-order polynomial motion of the fourth-order theoretical axis function shown in equation (5), the topological modification of the gear generating grinding is theoretically realized; taking the tooth surface normal deviation between the topological modified tooth surface and the standard tooth surface as the evaluation criterion, the sensitivity matrix algorithm iterates out the 12 polynomial coefficient values ​​of the three theoretical axis function higher-order polynomials of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1 that satisfy the target normal deviation;

[0020] (4) Selecting polynomial transformation sample points

[0021] The 12 polynomial coefficient values ​​of the three theoretical axis function higher-order polynomials obtained are written into equation (5) of the three theoretical axis function higher-order polynomials of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1, respectively, to obtain three theoretical axis function higher-order polynomials with coefficient values; in order to realize the operation of the three theoretical axis function higher-order polynomials on the Siemens 840Dsl CNC system, the three theoretical axis function higher-order polynomials are converted into machine tool interpolation polynomials that can be directly input into the code on the worm gear grinding machine. The machine tool interpolation polynomial of the worm gear grinding machine is a fifth-order polynomial; in the process of converting the three theoretical axis function higher-order polynomials, in order to ensure the global accuracy of the tooth surface, the method of selecting the axis coordinate points corresponding to the time of the tooth surface grid points is adopted as the sample points to ensure that the sample points of the entire tooth surface are evenly distributed, and to avoid the problem of local deviation of the tooth surface caused by the sample points being too concentrated in a certain area;

[0022] The machine tool interpolation polynomial of the worm gear grinding machine is expressed as shown in equation (8):

[0023] f(p) = a0 + a1p + a2p 2 +a3p 3 +a4p 4 +a5p 5 (8)

[0024] In equation (8), f(p) is the endpoint position of the interpolation code, representing the endpoint position parameter of each axis in mm; p is the machine tool interpolation polynomial parameter variable, which ranges from 0 to PL = pl and is dimensionless; a0 represents the axis position at the end of the program segment being executed, which the machine tool can automatically obtain during operation, in mm, and is a known parameter.

[0025] a2, a3, a4, a5 represent the coefficients of a given polynomial, which are unknown variables to be determined and are dimensionless.

[0026] a1 is calculated as the difference between the axis position at the end of the defined parameter range (PL) and the starting position. It is dimensionless, and the calculation formula is shown in equation (9):

[0027] a1=(x e -a0-g(pl)) / pl (9)

[0028] In equation (9), g(pl) = a2pl 2 +…+a m pl m (m=3,5), a2~a m represents the coefficients of a given polynomial, which are dimensionless; m is the polynomial order of the machine tool interpolation polynomial defined in formula (8), which is dimensionless;

[0029] (5) Calculate the number of segments in the polynomial transformation process.

[0030] To achieve high-precision conversion of three theoretical axis function higher-order polynomials to machine tool interpolation polynomials, the three theoretical axis function higher-order polynomials are divided into segments, and each segment of the theoretical axis function higher-order polynomial is converted separately. The number of sample points selected for each segment of the theoretical axis function higher-order polynomial is n, where n is 5, 7, or 9.

[0031] The operation of the radial feed axis X1 of the grinding wheel is as follows: According to step (1), by using the meshing condition of equation (3) and giving the same time interval, the tooth surface coordinate point cloud is obtained, and the tooth surface mesh is divided by the tooth surface coordinate point cloud to obtain the tooth surface mesh points and the number of tooth surface mesh points; according to the number of tooth surface mesh points and the number of more than 5 sample points selected by the higher-order polynomial of the theoretical axis function of each segment of the radial feed axis X1 of the grinding wheel, the number of tooth surface mesh points and the number of sample points selected by the higher-order polynomial of the theoretical axis function of each segment of the radial feed axis X1 of the grinding wheel are simultaneously reduced by one, and then the quotient is calculated to determine the number of segments to be divided by the transformation of the higher-order polynomial of the theoretical axis function of the radial feed axis X1 of the grinding wheel;

[0032] The grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1 are respectively divided into the number of segments by the theoretical axis function polynomial transformation of the grinding wheel tangential feed axis Y1 and the number of segments by the theoretical axis function polynomial transformation of the grinding wheel axial feed axis Z1 according to the above operation.

[0033] (6) Sample point parameter positions of the piecewise computer bed interpolation polynomial

[0034] The operation of the radial feed axis X1 of the grinding wheel is as follows: According to the number of polynomial segments in the process of converting the theoretical axis function high-order polynomial of the radial feed axis X1 of the grinding wheel to the machine tool interpolation polynomial in step (5), the theoretical axis function high-order polynomial of the radial feed axis X1 of the grinding wheel is segmented, and in each polynomial conversion, the corresponding time of each sample point is divided with the total time of the corresponding theoretical axis function high-order polynomial, and multiplied by the parameter interval length of the machine tool interpolation polynomial. The parameter interval length is the variable in the polynomial interpolation code of the Siemens 840Dsl CNC system, so as to obtain the parameter position of each sample point of the machine tool interpolation polynomial of the radial feed axis X1 of the grinding wheel of the worm gear grinding machine.

[0035] The specific operation is as follows: In the process of converting the theoretical axis function high-order polynomial of each segment of the grinding wheel radial feed axis X1 to the machine tool interpolation polynomial, the theoretical axis function high-order polynomial of the grinding wheel radial feed axis X1 with parameter t is linked with the machine tool interpolation polynomial of the grinding wheel radial feed axis X1 with parameter p through the proportional relationship, and the position of parameter p of each sample point of the machine tool interpolation polynomial of the grinding wheel radial feed axis X1 of the worm gear grinding machine is calculated. The formula (12) for calculating the position of parameter p of each sample point is as follows:

[0036]

[0037] In equation (12), T = t (n-1)k+1 -t (k-1)(n-1)+1 Δt represents the total time in seconds. i =t (n-1)(k-1)+i -t (k-1)(n-1)+1 , where is the time interval between the sample point and the start time of the polynomial segment, in seconds; k is the polynomial segment number, dimensionless; i is the sample point number of a single polynomial segment, dimensionless; n is the number of sample points selected for each polynomial segment, dimensionless; p1~p n The parameter position of the machine tool interpolation polynomial for the radial feed axis X1 of the grinding wheel; pl represents the dimensionless numerical value of the parameter interval length defining the polynomial;

[0038] The parameter positions of each sample point of the machine interpolation polynomial for the tangential feed axis Y1 of the grinding wheel and the parameter positions of each sample point of the machine interpolation polynomial for the axial feed axis Z1 of the grinding wheel are calculated by performing the same operation as described above for calculating the parameter positions of each sample point of the machine interpolation polynomial for the radial feed axis X1 of the grinding wheel.

[0039] (7) Computer bed polynomial coefficients

[0040] The operation of the radial feed axis X1 of the grinding wheel is as follows: In the conversion from the theoretical axis function high-order polynomial of each segment of the radial feed axis X1 to the machine interpolation polynomial, the parameter positions of more than 5 sample points in the theoretical axis function high-order polynomial of the radial feed axis X1 and the parameter positions of the corresponding sample points in the machine interpolation polynomial of the radial feed axis X1 are known. Based on the equation relationship between the theoretical axis function high-order polynomial and the machine interpolation polynomial at each sample point, more than 5 equations are listed to form a system of equations. The unknown coefficients of each segment of the machine interpolation polynomial of the worm gear grinding machine are solved. The least squares method is used to solve the system of equations to obtain the unknown coefficients of each segment of the machine interpolation polynomial of the radial feed axis X1 of the grinding wheel.

[0041] The grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1 obtain the unknown coefficients of each segment of the machine tool interpolation polynomial of the grinding wheel tangential feed axis Y1 and the unknown coefficients of each segment of the machine tool interpolation polynomial of the grinding wheel axial feed axis Z1 respectively according to the above operation, and perform polynomial interpolation to realize the topology modification machining of the gear tooth surface of generating grinding based on the Siemens 840Dsl CNC system.

[0042] The further defined technical solution is as follows:

[0043] In step (1), the worm grinding wheel is a ZI-type worm, whose overall geometry is the same as that of the involute worm for gear hobbing; the two-parameter curved surface of the left tooth surface of the worm grinding wheel is shown in formula (1):

[0044]

[0045] In equation (1), R w2 For the ZI type worm gear grinding wheel expression; r w The reference circle radius for the worm gear grinding wheel is in mm; Δz w These are key parameters, specifically the tooth cogging angle parameter, in mm. This parameter ensures that the axial section profile of the worm grinding wheel contacts the left and right tooth surfaces of the gear simultaneously. ξ is the involute parameter, dimensionless; τ is the helical parameter, dimensionless; p w These are the lead parameters of the grinding wheel, in mm.

[0046] Given the biparametric surface of the worm grinding wheel from equation (1), the coordinate transformation matrix M... pw2 The worm gear grinding wheel is determined from the grinding wheel coordinate system S. w2 To gear coordinate system S p The transformation yields the worm grinding wheel in the gear coordinate system S. p The trajectory is shown in formula (2):

[0047]

[0048] In equation (2), R w2 It is in S w2 The biparametric surface of the worm grinding wheel in the coordinate system, n w2 It is in S w2 Normal vector in coordinate system; R p It is in S p The biparametric surface of the worm grinding wheel in the coordinate system, n p In the gear coordinate system S p Normal vector in coordinate system; M pw2 Let S be the coordinate system of the grinding wheel. w2 To gear coordinate system S p The transformation matrix L is dimensionless; pw2 For M pw2 The top-left matrix is ​​dimensionless;

[0049] The mathematical model of continuous generating grinding of worm wheel is obtained from equation (2) and the meshing principle, namely formula (3).

[0050] In step (3), based on the fundamental principle of gear tooth surface modification, given the target normal deviation value of the tooth surface grid points, the coefficient values ​​of the higher-order polynomial of the theoretical axis function are obtained using the following formula (6):

[0051] {δε j} = M S {δλ i} (6)

[0052] In equation (6), δε j δλ represents the target normal deviation value, in mm. i M represents the coefficients of the higher-order polynomial of the theoretical axis function, which are dimensionless; S The sensitivity matrix is ​​dimensionless.

[0053] Since the sensitivity matrix is ​​ill-conditioned, it is almost singular in most cases. To prevent the risk of numerical solutions diverging, the Levenberg-Marquardt algorithm combined with the sensitivity matrix is ​​used to solve equation (6) for corrected coefficients.

[0054] {δλ i}=(M S T M S +ρI) -1 M S T {δε i} (7)

[0055] In equation (7), ρ is the damping coefficient adjusted during iteration, and the value of the first iteration can be selected from matrix M. S T M S The absolute value of the largest element in the matrix, dimensionless; I is the identity matrix, dimensionless; M S The sensitivity matrix is ​​dimensionless; δλ i The coefficients of the higher-order polynomials of the theoretical axis functions are given, and are dimensionless; δε j This represents the target normal deviation value, in mm.

[0056] Theoretically, by iterating 2 to 3 times, the values ​​of 12 polynomial coefficients of the three theoretical axis functions of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1 that satisfy the target normal deviation can be obtained.

[0057] In step (4), given the machine tool interpolation polynomial, the polynomial interpolation is implemented using the POLY code of the Siemens 840Dsl CNC system. Taking the radial feed axis X1 of the grinding wheel as an example, the input rules for the POLY code of the Siemens 840Dsl CNC system are as follows:

[0058] X1 = PO(x e ,a2,a3,a4,a5)PL=pl (10)

[0059] In equation (10), X1 represents the name of the radial feed axis X1 of the grinding wheel. When performing polynomial interpolation of the tangential feed axis Y1 and the axial feed axis Z1 of the grinding wheel, the X1 axis can be replaced with Y1 and Z1; e The endpoint position parameters of each axis are in mm; a2, a3, a4, a5 represent the coefficients of the polynomial program defined by this program, which are dimensionless; PL represents the parameter interval length of the defined polynomial, which is dimensionless; if PL is not programmed, then PL = 1.

[0060] In step (5), during the calculation of the number of segments of the theoretical axis function higher-order polynomial of the grinding wheel radial feed axis X1, the number of grid points s on the tooth surface and the number of sample points n selected for each segment of the theoretical axis function higher-order polynomial of the grinding wheel radial feed axis X1 are simultaneously reduced by one, and then the quotient is calculated to determine the number of segments to be divided by the transformation of the theoretical axis function higher-order polynomial of the grinding wheel radial feed axis X1, as shown in the following formula (11):

[0061]

[0062] In equation (11), k a The total number of segments to divide the curve is dimensionless; s is the number of grid points on the tooth surface, dimensionless; n is the number of sample points selected for each polynomial target segment, dimensionless.

[0063] The calculation of the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel tangential feed axis Y1 and the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel axial feed axis Z1 are performed in accordance with the above-described operation for calculating the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel radial feed axis X1.

[0064] In step (7), during the conversion of the theoretical axis function high-order polynomial of each segment of the grinding wheel radial feed axis X1 to the machine tool interpolation polynomial, when the number of sample points n = 5, the time variable values ​​t1~t5 corresponding to each sample point of the theoretical axis function high-order polynomial of the grinding wheel radial feed axis X1 and the variable values ​​p1~p5 corresponding to each sample point of the machine tool interpolation polynomial of the grinding wheel radial feed axis X1 are known; at each sample point, the equation relationship between the theoretical axis function high-order polynomial and the machine tool interpolation polynomial is: the formula (5) in step (2) is placed on the left side of the equation, and the formula (8) in step (4) is placed on the right side of the equation, resulting in formula (13); the parameters calculated by formula (9) in step (4) and the parameters calculated by formula (12) in step (6) are substituted into formula (13) to solve for the unknown coefficients a2~a of the machine tool interpolation polynomial of the grinding wheel radial feed axis X1. m Equation (13) is as follows:

[0065]

[0066] In equation (13), d = λ·v Z1 / b g , where is the coefficient of each order of the higher-order polynomial in equation (5) with time t as the variable, and is dimensionless; from equation (4), it can be seen that F N (N = X1, Y1, Z1) represents the position of each axis under standard motion; p1~p n The parameter variables of the machine tool interpolation polynomial for the radial feed axis X1 of the grinding wheel are dimensionless; a0~a m t is the coefficient of the machine tool interpolation polynomial, dimensionless; t is time, in seconds; m is the polynomial order of the machine tool interpolation polynomial defined in formula (8), dimensionless; n is the number of sample points selected for each polynomial segment, dimensionless; k is the polynomial segment number, dimensionless; i is the sample point number of a single polynomial segment, dimensionless.

[0067] Solving the above equations using the least squares method ultimately yields the unknown coefficients a2 to a1 of the machine tool interpolation polynomial for each segment of the grinding wheel's radial feed axis X1. m ;

[0068] The unknown coefficients of the machine interpolation polynomial for the tangential feed axis Y1 of each grinding wheel segment and the unknown coefficients of the machine interpolation polynomial for the axial feed axis Z1 of each grinding wheel segment are calculated by performing the same operation as described above for calculating the unknown coefficients of the machine interpolation polynomial for the radial feed axis X1 of each grinding wheel segment.

[0069] The beneficial effects of this invention are reflected in the following aspects:

[0070] 1. This invention presents a high-order polynomial topology modification method for generating ground gear teeth. By combining a sensitivity algorithm and the LM algorithm, the high-order polynomial coefficients of the high-order modification technique are calculated, effectively enabling topology modification on standard cylindrical helical gears without the need for grinding wheel dressing. This solves the problems of high cost and poor flexibility in traditional modification processes. Furthermore, the combined sensitivity and LM algorithm offers the advantage of fast iteration speed. This invention uses a proposed polynomial transformation method to compare the theoretical axis function polynomial with the transformed machine tool interpolation polynomial using a polynomial similarity evaluation method. The similarity of each segment of the transformed polynomial is found to be in the range of 9.04E-10 to 1.91E-04, far less than 1. The mathematical model for similarity evaluation shows that a similarity value far less than 1 indicates a high similarity between the two polynomials. Therefore, the above numerical examples demonstrate the effectiveness of this polynomial transformation method within a certain accuracy range. To meet the machine tool travel requirements, the above transformation polynomial was fine-tuned and then written into NC code, which was input into the YW7232CNC gear grinding machine. Using the secondary development interface for the Siemens 840Dsl, data from each axis of the machine tool was collected, and curves were plotted as follows: Figure 5 As shown. By Figure 5 It can be observed that the numerical curves of the machine tool's X, Y, and Z axes largely coincide with the X, Y, and Z coordinates of the theoretical sample points in both trend and value. This verifies the feasibility of the polynomial transformation method proposed in this invention on the Siemens 840Dsl gear grinding machine. Therefore, it provides a theoretical basis for the implementation of advanced profile modification technology on the Siemens 840Dsl gear grinding machine.

[0071] 2. The present invention provides a method for high-order polynomial topology modification of the tooth surface of generated grinding gears. By comparing the similarity between the machine tool interpolation polynomial with parameter p and the theoretical axis function polynomial with parameter t, the effectiveness of the polynomial transformation method proposed in this invention is verified within a certain accuracy range. This allows the high-order modification technology to be combined with actual conditions, no longer limited to the theoretical stage, and is feasible on the Siemens 840Dsl CNC system. Attached Figure Description

[0072] Figure 1 This is a picture of a gear grinding machine.

[0073] Figure 2 This is a mesh diagram of the tooth surface.

[0074] Figure 3 Flowchart for closed-loop topology modification of tooth surface;

[0075] Figure 4 This is a diagram illustrating polynomial transformation;

[0076] Figure 5 This is a comparison chart of the machine tool experimental curve and the theoretical sample points. Detailed Implementation

[0077] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0078] Example

[0079] A high-order polynomial topology modification method for generating and grinding tooth surfaces is applicable to the worm gear grinding machine of the Siemens 840Dsl CNC system. See [link to relevant documentation]. Figure 1 Taking the YW7232CNC gear grinding machine as an example. The number of teeth N of the gear being machined. g The normal modulus is 48, and the normal modulus is m. n The normal pressure angle is 4. n It is 20 degrees, and the tooth tip height coefficient is... The porosity coefficient c is 1. * The value is 0.25, the tooth width B is 40mm, the helix angle is 30 degrees, and the direction is right-handed.

[0080] The specific operation steps of the high-order polynomial implementation method of this invention are as follows:

[0081] (1) Establish a mathematical model for continuous generating grinding of worm gear grinding wheels

[0082] The worm grinding wheel uses a ZI type worm, whose overall geometry is the same as that of the involute worm for gear hobbing; the two-parameter curved surface of the left tooth surface of the worm grinding wheel is represented by the following formula (1):

[0083]

[0084] In equation (1), R w2 For the ZI type worm gear grinding wheel expression; r w The reference circle radius for the worm gear grinding wheel is in mm; Δz w These are key parameters, specifically the tooth cogging angle parameter, in mm. This parameter ensures that the axial section profile of the worm grinding wheel contacts the left and right tooth surfaces of the gear simultaneously. ξ is the involute parameter, dimensionless; τ is the helical parameter, dimensionless; p w These are the lead parameters for the worm gear grinding wheel, in mm.

[0085] The reference circle radius r of the worm grinding wheel w The lead parameter p of the worm gear grinding wheel is 16.3517 mm. w The tooth cogging angle parameter Δz is 6.0065. w Substituting the data of -22.2779 into equation (1) is as follows:

[0086]

[0087] A biparametric surface with specific numerical values ​​is obtained on the left tooth surface of the worm gear grinding wheel.

[0088] Given the biparametric surface of the worm grinding wheel from equation (1), the coordinate transformation matrix M... pw2 The worm gear grinding wheel is determined from the grinding wheel coordinate system S. w2 To gear coordinate system S p The transformation yields the worm grinding wheel in the gear coordinate system S. p The trajectory is as follows (2):

[0089]

[0090] In equation (2), R w2 It is in S w2 The biparametric surface of the worm grinding wheel in the coordinate system, n w2 It is in S w2 Normal vector in coordinate system; R p It is in S p The biparametric surface of the worm grinding wheel in the coordinate system, n p In the gear coordinate system S p Normal vector in coordinate system; M pw2 Let S be the coordinate system of the grinding wheel. w2 To gear coordinate system S p The transformation matrix L is dimensionless; pw2 For M pw2 The top-left matrix is ​​dimensionless;

[0091] Based on equation (2) and the meshing principle, the mathematical model for continuous generating grinding of worm gear grinding wheels is obtained as shown in equation (3):

[0092]

[0093] In equation (3), ξ is the involute parameter, which is dimensionless and ξ∈[8.2773,9.5295]; τ is the helical parameter, which is dimensionless and τ∈[-π,3π]; This refers to the rotation angle of the worm gear grinding wheel, measured in rad. This indicates the position of the machine tool's Z1 axis, in mm. This refers to the position of the machine tool's Y1 axis, in mm; n p In the gear coordinate system S p Normal vector in coordinate system; R p It is in S p A biparametric surface of a worm gear grinding wheel in a coordinate system; t represents time in seconds.

[0094] (2) Establish the position functions of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1, and grinding wheel axial feed axis Z1, and supplement the theoretical higher-order polynomials.

[0095] In the mathematical model of continuous generating grinding of the worm wheel in step (1), the feed amount of the radial feed axis X1, the feed amount of the tangential feed axis Y1, and the feed amount of the axial feed axis Z1 are expressed as functions that change with time, as shown in formula (4):

[0096]

[0097] In equation (4), t is the time variable, in seconds; A is the center distance between the cylindrical gear and the worm wheel, in mm. This represents the axial feed rate of the worm gear grinding wheel, expressed in mm / s. This represents the tangential feed rate of the worm gear grinding wheel, expressed in mm / s. This refers to the rotation angle of the worm gear grinding wheel, measured in rad. The rotation angle of the workpiece gear is expressed in rad; n B This indicates the rotational speed of the worm gear grinding wheel, measured in rad / min (N). g N represents the number of teeth on a gear, dimensionless; w p represents the number of grinding wheel heads, dimensionless; w This refers to the lead parameter of the grinding wheel, in mm; p g The helix parameter of the gear is dimensionless. This indicates the position of the machine tool's Z1 axis, in mm. This refers to the position of the Y1 axis of the machine tool, in mm. This refers to the position of the X1 axis of the machine tool, in mm.

[0098] With the center distance A between the cylindrical gear and the worm grinding wheel set to 250.8513 mm, and the tangential feed rate of the worm grinding wheel set to... The axial feed rate of the worm gear grinding wheel is 0 mm / s. for The rotational speed n of the worm grinding wheel B 3000 rad / min, number of teeth N of the gear g The number of grinding wheel heads is 48, N. w 3. The lead parameter p of the grinding wheel w The helix parameter p of the gear is 6.0065mm. g Substituting the data of 192 into equation (4):

[0099]

[0100] The position function of the grinding wheel radial feed axis X1 with relevant data is obtained by adding a theoretical higher-order polynomial.

[0101] In order to achieve continuous generating topology modification of the gear grinding and realize the high-order polynomial motion of the three axes of the worm gear grinding machine, namely the grinding wheel axial feed axis Z1, the grinding wheel tangential feed axis Y1, and the grinding wheel radial feed axis X1, a fourth-order polynomial motion is added to the standard motion of the grinding wheel axial feed axis Z1, the grinding wheel tangential feed axis Y1, and the grinding wheel radial feed axis X1, as shown in formula (5):

[0102]

[0103] This yields three theoretical axis functions of higher order polynomials;

[0104] In equation (5), The ratio of the Z1 axis position to the tooth width without modification is dimensionless; as can be seen from equation (4), Therefore, the variable in equation (5) is time t; from equation (4), we know that F N (N = X1, Y1, Z1) represents the position of each axis under standard motion, in mm; F' N (N = X1, Y1, Z1) represents the position of each axis after adding a fourth-order polynomial motion to the standard motion; λ1~λ4 are the polynomial coefficient values ​​of the higher-order polynomial of the radial feed axis X1 of the grinding wheel, dimensionless; λ5~λ8 are the polynomial coefficient values ​​of the higher-order polynomial of the tangential feed axis Y1 of the grinding wheel, dimensionless; λ9~λ 12 The polynomial coefficients of the higher-order polynomial of the grinding wheel axial feed axis Z1 are dimensionless.

[0105] Position of the grinding wheel radial feed axis X1 under standard motion The position is 250.8513mm, under the standard motion of the grinding wheel tangential feed axis Y1. The position of the grinding wheel axial feed axis Z1 under standard motion is 0.000000mm. The axial feed rate of the worm gear grinding wheel is -2.1333tmm. With tooth width b g ratio Substituting the data of 0.0053 into equation (5) is as follows:

[0106]

[0107] This yields three higher-order polynomials of theoretical axis functions with data.

[0108] (3) Sensitivity matrix algorithm fits the numerical values ​​of higher-order polynomial coefficients of theoretical axis functions

[0109] By modifying the polynomial coefficients of the higher-order polynomial motion of the fourth-order theoretical axis function shown in equation (5), the topological modification of the gear generating grinding is theoretically realized; taking the tooth surface normal deviation between the topological modified tooth surface and the standard tooth surface as the evaluation criterion, the sensitivity matrix algorithm iterates out the 12 polynomial coefficient values ​​of the three theoretical axis function higher-order polynomials of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1 that satisfy the target normal deviation;

[0110] The specific steps are as follows:

[0111] Based on the fundamental principle of gear tooth surface modification, given the target normal deviation value of the tooth surface grid points, the coefficient values ​​of the higher-order polynomial of the theoretical axis function are obtained using the following formula (6):

[0112] {δε j} = M S {δλ i} (6)

[0113] In equation (6), δε j δλ represents the target normal deviation value, in mm. i M represents the coefficients of the higher-order polynomial of the theoretical axis function, which are dimensionless; S This is the sensitivity matrix, which is dimensionless.

[0114] Since the sensitivity matrix is ​​ill-conditioned and almost singular in most cases, to prevent the risk of numerical solutions diverging, the following approach combines the Levenberg-Marquardt algorithm with the sensitivity matrix, as follows: Figure 3 As shown, the correction coefficients of equation (6) are solved, and equation (7) is the formula for the Levenberg–Marquardt algorithm:

[0115] {δλ i}=(M S T M S +ρI) -1 M S T {δε i} (7)

[0116] In equation (7), ρ is the damping coefficient adjusted during iteration, and the value of the first iteration can be selected from matrix M. S T M S The absolute value of the largest element in the matrix, dimensionless; I is the identity matrix, dimensionless; M S The sensitivity matrix is ​​dimensionless; δλ i The coefficients of the theoretical axis function polynomial are dimensionless; δε j This represents the target normal deviation value, in mm.

[0117] Theoretically, by iterating 2 to 3 times, the values ​​of 12 polynomial coefficients of the three theoretical axis functions of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1 that satisfy the target normal deviation can be obtained.

[0118] The coefficients of the 12 polynomials of the theoretical axis function polynomial are obtained through the above process and are shown in Table 1 below:

[0119] Table 1

[0120] numerical values -2.193 35.1881 -200.4 279.949 -4.241 …… -0.022 -0.017

[0121] (4) Selecting polynomial transformation sample points

[0122] The 12 polynomial coefficient values ​​of the three theoretical axis function higher-order polynomials obtained in step (3) are written into equation (5) of the three theoretical axis function higher-order polynomials of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1, and grinding wheel axial feed axis Z1, respectively, to obtain three theoretical axis function higher-order polynomials with coefficient values. In order to realize the operation of the three theoretical axis function higher-order polynomials on the Siemens 840Dsl CNC system, the three theoretical axis function higher-order polynomials are converted into machine tool interpolation polynomials that can be directly input into the code on the worm gear grinding machine. The machine tool interpolation polynomial of the worm gear grinding machine is a fifth-order polynomial. In the process of converting the three theoretical axis function higher-order polynomials, in order to ensure the global accuracy of the tooth surface, the method of selecting the axis coordinate points corresponding to the time of the tooth surface grid points is adopted as the sample points to ensure that the sample points of the entire tooth surface are evenly distributed, avoiding the problem of local deviation of the tooth surface caused by the sample points being too concentrated in a certain area.

[0123] The higher-order polynomials of the three theoretical axis functions with coefficient values ​​are as follows:

[0124]

[0125] The expression of the machine tool interpolation polynomial of the worm gear grinding machine is shown in equation (8):

[0126] f(p) = a0 + a1p + a2p 2 +a3p 3 +a4p 4 +a5p 5 (8)

[0127] In equation (8), f(p) represents the endpoint position of the interpolation code, indicating the endpoint position parameter of each axis in mm; p is the machine tool interpolation polynomial parameter variable, ranging from 0 to PL = pl, dimensionless; a0 represents the axis position at the end of the currently executing program segment, which the machine tool can automatically obtain during operation, in mm, and is a known parameter; a2, a3, a4, a5 represent the given polynomial coefficients, which are unknown variables to be determined, dimensionless; a1 is calculated from the difference between the axis position at the endpoint of the defined parameter range (PL) and the starting position, dimensionless, and the calculation formula is shown below:

[0128] a1=(x e -a0-g(pl)) / pl (9)

[0129] In equation (9), g(pl) = a2pl 2 +...+a m pl m (m=3,5), a2~a m represents the coefficients of a given polynomial, which are dimensionless; m is the polynomial order of the machine tool interpolation polynomial defined in formula (8), which is dimensionless.

[0130] Given the machine tool interpolation polynomial, polynomial interpolation is implemented using the POLY code on the Siemens 840Dsl machine tool. Taking the radial feed axis X1 of the grinding wheel as an example, the code input rule is as follows:

[0131] X1 = PO(x e ,a2,a3,a4,a5)PL=pl (10)

[0132] In equation (10), X1 represents the name of the radial feed axis X1 of the grinding wheel. When performing polynomial interpolation of the tangential feed axis Y1 and the axial feed axis Z1 of the grinding wheel, the X1 axis can be replaced with Y1 and Z1; e The endpoint position parameters of each axis are in mm; a2, a3, a4, a5 represent the coefficients of the polynomial program defined by this program, which are dimensionless; PL represents the parameter interval length of the defined polynomial, which is dimensionless; if PL is not programmed, then PL = 1.

[0133] In the process of transforming the three theoretical axis functions into higher-order polynomials, in order to ensure the global accuracy of the tooth surface, the method of selecting the axis coordinate points corresponding to the time of the tooth surface grid points as sample points is adopted. This ensures that the sample points are evenly distributed throughout the tooth surface and avoids the problem of local deviations caused by the sample points being too concentrated in a certain area.

[0134] The coordinate points of each axis corresponding to the selected sample point time are shown in Table 2 below:

[0135] Table 2

[0136]

[0137]

[0138] Table 2 above shows the coordinate data of the three axes corresponding to the time of the selected 45 sample points.

[0139] (5) Calculate the number of segments in the polynomial transformation process.

[0140] To achieve high-precision conversion of three theoretical axis function higher-order polynomials to machine tool interpolation polynomials, the three theoretical axis function higher-order polynomials are divided into segments, and each segment of the theoretical axis function higher-order polynomial is converted separately. The number of sample points selected for each segment of the theoretical axis function higher-order polynomial is n, where n is 5, 7, or 9.

[0141] The operation of the radial feed axis X1 of the grinding wheel is as follows: According to step (1), by using the meshing condition of equation (3) and giving the same time interval, the tooth surface coordinate point cloud is obtained, and the tooth surface mesh is divided using this tooth surface coordinate point cloud, such as Figure 2 As shown, the number of tooth surface mesh points and the number of tooth surface mesh points are obtained. Based on the number of tooth surface mesh points and the number of more than 5 sample points selected by the higher-order polynomial of the theoretical axis function for each segment of the grinding wheel radial feed axis X1, the number of tooth surface mesh points and the number of sample points selected by the higher-order polynomial of the theoretical axis function for each segment of the grinding wheel radial feed axis X1 are simultaneously reduced by one, and then the quotient is calculated to determine the number of segments to be divided by the transformation of the higher-order polynomial of the theoretical axis function of the grinding wheel radial feed axis X1.

[0142] In calculating the number of segments of the theoretical axis function higher-order polynomial of the grinding wheel radial feed axis X1, the number of grid points s on the tooth surface and the number of sample points n selected for each segment of the theoretical axis function higher-order polynomial of the grinding wheel radial feed axis X1 are both reduced by one, and then the quotient is calculated to determine the number of segments to be divided by the transformation of the theoretical axis function higher-order polynomial of the grinding wheel radial feed axis X1, as shown in the following formula (11):

[0143]

[0144] In equation (11), k a The total number of segments to divide the curve is dimensionless; s is the number of grid points on the tooth surface, dimensionless; n is the number of sample points selected for each polynomial target segment, dimensionless.

[0145] Substituting the number of mesh points on the tooth surface (45) and the number of sample points selected for each polynomial target segment (5) into equation (11) as follows:

[0146]

[0147] The total number of curve segments k to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel radial feed axis X1 is obtained. a It is 11.

[0148] Following the same procedure described above for calculating the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel radial feed axis X1, the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel tangential feed axis Y1 and the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel axial feed axis Z1 are calculated to be 11.

[0149] (6) Sample point parameter positions of the piecewise computer bed interpolation polynomial

[0150] The operation of the radial feed axis X1 of the grinding wheel is as follows: According to the number of polynomial segments in the process of converting the theoretical axis function high-order polynomial of the radial feed axis X1 of the grinding wheel to the machine tool interpolation polynomial in step (5), the theoretical axis function high-order polynomial of the radial feed axis X1 of the grinding wheel is segmented, and in each polynomial conversion, the corresponding time of each sample point is divided with the total time of the corresponding theoretical axis function high-order polynomial, and multiplied by the parameter interval length of the machine tool interpolation polynomial. The parameter interval length is the variable in the polynomial interpolation code of the Siemens 840Dsl CNC system, so as to obtain the parameter position of each sample point of the machine tool interpolation polynomial of the radial feed axis X1 of the grinding wheel of the worm gear grinding machine.

[0151] The specific operation is as follows: In the process of converting the theoretical axis function high-order polynomial of each segment of the grinding wheel radial feed axis X1 to the machine tool interpolation polynomial, the theoretical axis function high-order polynomial of the grinding wheel radial feed axis X1 with parameter t is linked with the machine tool interpolation polynomial of the grinding wheel radial feed axis X1 with parameter p through the proportional relationship, and the position of parameter p of each sample point of the machine tool interpolation polynomial of the grinding wheel radial feed axis X1 of the worm gear grinding machine is calculated. The formula (12) for calculating the position of parameter p of each sample point is as follows:

[0152]

[0153] In equation (12), T = t (n-1)k+1 -t (k-1)(n-1)+1 Δt represents the total time in seconds. i =t (n-1)(k-1)+i -t (k-1)(n-1)+1 , where is the time interval between the sample point and the start time of the polynomial segment, in seconds; k is the polynomial segment number, dimensionless; i is the sample point number of a single polynomial segment, dimensionless; n is the number of sample points selected for each polynomial segment, dimensionless; p1~p n The parameter position of the machine tool interpolation polynomial for the radial feed axis X1 of the grinding wheel; pl represents the numerical value of the parameter interval length of the defined polynomial, which is dimensionless.

[0154] In the process of converting the theoretical axis function high-order polynomial of the grinding wheel radial feed axis X1 to the machine tool interpolation polynomial, taking an example with n=5 sample points per segment and p1=1, the theoretical axis function high-order polynomial of the grinding wheel radial feed axis X1 with parameter t and the machine tool interpolation polynomial of the grinding wheel radial feed axis X1 with parameter p are linked by a proportional relationship, such as... Figure 4 As shown, Figure 4 In the diagram, A represents the process of converting an 11-segment theoretical axis function high-order polynomial to a machine tool interpolation polynomial. The sixth segment of the conversion process, circled in the diagram, is shown in the attached diagram. Figure 4 In B, the parameters t of the higher-order polynomial of the theoretical axis function for each sample point, and the parameters p of the machine tool interpolation polynomial, are shown in the reference. Figure 4 Table 3 shows the polynomial sample points for the first segment of the higher-order polynomial of the theoretical axis function of the grinding wheel radial feed axis X1, with times t1 to t5 as follows:

[0155] Table 3

[0156] numerical values 6.593e-04 2.5609 2.5673 4.4923 5.1211

[0157] The time interval between the sample point and the start time of the polynomial in this segment:

[0158] Δt1=(6.593e-4)-(6.593e-4)=0

[0159] Δt2=(2.5609)-(6.593e-4)=2.5602

[0160] Δt3=(2.5673)-(6.593e-4)=2.5666

[0161] Δt3=(4.4923)-(6.593e-4)=4.4916

[0162] Δt5=(5.1211)-(6.593e-4)=5.1204

[0163] The total time is:

[0164] T = 5.1211 - 6.593e-4 ≈ 5.1204

[0165] Substituting the data of the interval time Δt between the above sample points, the total time T of the higher-order polynomial of each theoretical axis function being 5.1204s, and the parameter interval length pl of the defined polynomial being 1 into formula (12):

[0166]

[0167] The sample point parameter positions p1 to p5 of the five machine tool interpolation polynomials are shown in Table 4 below:

[0168] Table 4

[0169] numerical values 0 0.5000 0.5012 0.8772 1

[0170] The conversion of the theoretical axis function high-order polynomial of the remaining 10 segments of the grinding wheel radial feed axis X1 to the machine tool interpolation polynomial is performed according to the above operation.

[0171] The calculation of the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel tangential feed axis Y1 and the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel axial feed axis Z1 are performed in accordance with the above-described operation for calculating the number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function of the grinding wheel radial feed axis X1.

[0172] (7) Computer bed polynomial coefficients

[0173] The operation of the radial feed axis X1 of the grinding wheel is as follows: In the conversion from the theoretical axis function high-order polynomial of each segment of the radial feed axis X1 to the machine interpolation polynomial, the parameter positions of more than 5 sample points in the theoretical axis function high-order polynomial of the radial feed axis X1 and the parameter positions of the corresponding sample points in the machine interpolation polynomial of the radial feed axis X1 are known. Based on the equation relationship between the theoretical axis function high-order polynomial and the machine interpolation polynomial at each sample point, more than 5 equations are listed to form a system of equations. The unknown coefficients of each segment of the machine interpolation polynomial of the worm gear grinding machine are solved. The least squares method is used to solve the system of equations to obtain the unknown coefficients of each segment of the machine interpolation polynomial of the radial feed axis X1 of the grinding wheel.

[0174] In the process of converting the theoretical axis function high-order polynomial of each segment of the grinding wheel radial feed axis X1 to the machine tool interpolation polynomial, when the number of sample points n = 5, the time variable values ​​t1~t5 corresponding to each sample point of the theoretical axis function high-order polynomial of the grinding wheel radial feed axis X1 and the variable values ​​p1~p5 corresponding to each sample point of the machine tool interpolation polynomial of the grinding wheel radial feed axis X1 are known; at each sample point, the equation relationship between the theoretical axis function high-order polynomial and the machine tool interpolation polynomial is: the formula (5) in step (2) is placed on the left side of the equation, and the formula (8) in step (4) is placed on the right side of the equation, resulting in formula (13); the parameters calculated by formula (9) in step (4) and the parameters calculated by formula (12) in step (6) are substituted into formula (13) to solve for the unknown coefficients a2~a5 of the machine tool interpolation polynomial of the grinding wheel radial feed axis X1; formula (13) is as follows:

[0175]

[0176] In equation (13), d = λ·v Z1 / b g, where is the coefficient of each order of the higher-order polynomial in equation (5) with time t as the variable, and is dimensionless; from equation (4), it can be seen that F N (N = X1, Y1, Z1) represents the position of each axis under standard motion; p1 to p5 represent the sample point parameter positions of the machine tool interpolation polynomial for the radial feed axis X1 of the grinding wheel, dimensionless; a0 to a5 represent the coefficients of the machine tool interpolation polynomial, dimensionless; t represents time, in seconds; n represents the number of sample points selected for each polynomial segment, dimensionless; k represents the polynomial segment number, dimensionless; i represents the sample point number of a single polynomial segment, dimensionless.

[0177] Position of the grinding wheel radial feed axis X1 under standard motion The axial feed rate of the worm wheel is 250.8513 mm, and the time t for each sample point is shown in Table 3. With tooth width b g ratio The product of the polynomial coefficients λ of the higher-order polynomial of the theoretical axis function is d = λ·v Z1 / b g The polynomial coefficients λ of the higher-order polynomials of the theoretical axis function are shown in Table 1. The axial feed rate of the worm grinding wheel... With tooth width b g ratio The value is 0.0053; the sample point parameter positions of the machine tool interpolation polynomial for the radial feed axis X1 of the grinding wheel are p1 to p5, and the data are shown in Table 4; the data are substituted into formula (13):

[0178]

[0179] Solving the above equations using the least squares method yields the unknown coefficients a2 to a5 of the machine interpolation polynomial for each segment of the grinding wheel radial feed axis X1. The polynomial coefficients of the machine interpolation polynomials for the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1, and grinding wheel axial feed axis Z1 are shown in Table 5 below:

[0180] Table 5

[0181]

[0182] The unknown coefficients of the machine interpolation polynomial for the tangential feed axis Y1 of each grinding wheel segment and the unknown coefficients of the machine interpolation polynomial for the axial feed axis Z1 of each grinding wheel segment are calculated by performing the same operation as described above for calculating the unknown coefficients of the machine interpolation polynomial for the radial feed axis X1 of each grinding wheel segment.

[0183] After obtaining the coefficient values ​​of all segments of the machine tool interpolation polynomial, in order to verify the polynomial conversion effect, the similarity between the higher-order polynomials of the theoretical axis functions of all segments of the grinding wheel radial feed axis X1, grinding wheel tangential feed axis Y1, and grinding wheel axial feed axis Z1 and the machine tool interpolation polynomial can be calculated according to the polynomial similarity model, as shown in Table 6 below:

[0184] Table 6

[0185]

[0186]

[0187] After transformation using the proposed polynomial transformation method, the similarity between the higher-order polynomial of each theoretical axis function and the transformed machine tool interpolation polynomial is far less than 1. The mathematical model for similarity evaluation shows that a similarity value far less than 1 indicates a high degree of similarity between the two polynomials. Therefore, the numerical examples above demonstrate the effectiveness of this polynomial transformation method within a certain accuracy range. This method allows for polynomial interpolation on CNC machine tools, theoretically enabling topological shaping of the gear tooth surface during generating and grinding.

[0188] To meet the machine tool travel requirements, the above transformation polynomial was fine-tuned and then written into NC code, which was input into the YW7232CNC gear grinding machine for polynomial interpolation. Using the secondary development interface of the Siemens 840Dsl CNC system, communication with the machine tool PLC was established to collect data from each axis of the machine tool. The axis coordinates of the radial feed axis X1, tangential feed axis Y1, and axial feed axis Z1 of the grinding wheel were obtained and compared with the theoretical axis coordinates corresponding to the sample points of the three theoretical axis functions at the corresponding times, as shown in Table 7 below.

[0189] Table 7

[0190]

[0191] From the above table 7 and Figure 5 It can be observed that the axis coordinate curves of the machine tool's grinding wheel tangential feed axis Y1 and grinding wheel axial feed axis Z1 basically coincide with the theoretical coordinates corresponding to the sample points of the three theoretical axis functions' higher-order polynomials in terms of both trend and value. This verifies the feasibility of the polynomial transformation method proposed in this invention on the Siemens 840Dsl gear grinding machine. Therefore, it provides a theoretical basis for the implementation of topology modification technology on the Siemens 840Dsl gear grinding machine.

Claims

1. A high-order polynomial topology modification method for generating and grinding tooth surfaces, the high-order polynomial topology modification method being applicable to worm gear grinding machines based on the Siemens 840Dsl CNC system, comprising nine CNC axes and an electronic gearbox, wherein the nine CNC axes are respectively the grinding wheel head rotation axes. Grinding wheel spindle External support moving axis Worktable rotary axis Repairing the rotating shaft , Dressing mechanism rotating shaft Grinding wheel radial feed axis 1. Grinding wheel tangential feed axis and grinding wheel axial feed axis Its features are: Under the condition of satisfying the generating relationship of the electronic gearbox, the radial feed shaft of the grinding wheel in the worm gear grinding machine is...

1. Grinding wheel tangential feed axis and grinding wheel axial feed axis The position function is defined as a higher-order polynomial, and the movement is performed according to the set higher-order polynomial. The operation steps are as follows: (1) Establish a mathematical model for continuous generating grinding of worm gear grinding wheel The mathematical model for continuous generating grinding of worm wheel is as follows (3): （3） In equation (3), These are involute parameters, dimensionless; The parameters for the helix are dimensionless. This refers to the rotation angle of the worm gear grinding wheel, measured in rad. machine tool Shaft position, in mm; For machine tools Shaft position, in mm; In the gear coordinate system Normal vector in the coordinate system; Is Biparametric surface of worm grinding wheel in coordinate system; Time, in seconds; (2) Establish the radial feed axis of the grinding wheel 1. Grinding wheel tangential feed axis and grinding wheel axial feed axis Position function additional theory higher-order polynomials In step (1) of the mathematical model for continuous generating grinding of worm wheel, the radial feed axis of the grinding wheel is... Feed rate, grinding wheel tangential feed axis feed rate and grinding wheel axial feed axis The feed rates are expressed as functions of time, as shown in formula (4); （4） In equation (4), The variable is time, and the unit is seconds (s). This is the center distance between the cylindrical gear and the worm wheel, in mm; This represents the axial feed rate of the worm gear grinding wheel, expressed in mm / s. This represents the tangential feed rate of the worm gear grinding wheel, expressed in mm / s. This refers to the rotation angle of the worm gear grinding wheel, measured in rad. The rotation angle of the workpiece gear is expressed in rad. This indicates the rotational speed of the worm gear grinding wheel, expressed in rad / min. Indicates the number of teeth on a gear; dimensionless. Indicates the number of grinding wheel heads; dimensionless. This refers to the lead parameter of the grinding wheel, in mm. The helix parameter of the gear is dimensionless. machine tool Shaft position, in mm; For machine tools Shaft position, in mm; For machine tools Shaft position, in mm; To achieve continuous generating topology modification in gear grinding, and to realize the axial feed axis of the grinding wheel in a worm gear grinding machine...

1. Grinding wheel tangential feed axis and grinding wheel radial feed axis The high-order polynomial motion of the three axes, along the feed axis of the grinding wheel in the worm gear grinding machine.

1. Grinding wheel tangential feed axis and grinding wheel radial feed axis The standard motion of formula (4) is supplemented with a fourth-order polynomial motion, as shown in formula (5): （5） This yields three theoretical axis functions of higher order polynomials; In equation (5), For the unmodified form The ratio of shaft position to tooth width is dimensionless. From equation (4), we can see that Therefore, the variable in equation (5) is time. From equation (4), we can see that The positions of each axis under standard motion; The position of each axis after adding fourth-order polynomial motion to the standard motion; Radial feed axis of grinding wheel The polynomial coefficients of a higher-order polynomial are dimensionless. Tangential feed axis of the grinding wheel The polynomial coefficients of a higher-order polynomial are dimensionless. For the axial feed axis of the grinding wheel The polynomial coefficients of a higher-order polynomial are dimensionless. (3) Sensitivity matrix algorithm fits the numerical values ​​of higher-order polynomial coefficients of theoretical axis functions By modifying the polynomial coefficients of the higher-order polynomial motion of the fourth-order theoretical axis function shown in equation (5), the topological modification of the gear grinding is theoretically realized; using the tooth surface normal deviation between the topologically modified tooth surface and the standard tooth surface as the evaluation criterion, the grinding wheel radial feed axis that satisfies the target normal deviation is iterated through the sensitivity matrix algorithm.

1. Grinding wheel tangential feed axis and grinding wheel axial feed axis The numerical values ​​of the 12 polynomial coefficients of the higher-order polynomials of the three theoretical axis functions; (4) Select polynomial transformation sample points The 12 polynomial coefficients of the three theoretical axis functions obtained are written into the radial feed axis of the grinding wheel.

1. Grinding wheel tangential feed axis and grinding wheel axial feed axis In equation (5) of the three theoretical axis function higher-order polynomials, we obtain three theoretical axis function higher-order polynomials with coefficient values; in order to realize the operation of the three theoretical axis function higher-order polynomials on the Siemens 840Dsl CNC system, the three theoretical axis function higher-order polynomials are converted into machine tool interpolation polynomials that can be directly input into the code on the worm gear grinding machine. The machine tool interpolation polynomial of the worm gear grinding machine is a fifth-order polynomial; in the process of converting the three theoretical axis function higher-order polynomials, in order to ensure the global accuracy of the tooth surface, the method of selecting the axis coordinate points corresponding to the time of the tooth surface grid points is adopted as the sample points to ensure that the sample points of the entire tooth surface are evenly distributed, and to avoid the problem of local deviation of the tooth surface caused by the sample points being too concentrated in a certain area. The machine tool interpolation polynomial of the worm gear grinding machine is expressed as shown in equation (8): （8） In equation (8), To indicate the endpoint position of the interpolation code, the unit of the endpoint position parameter for each axis is mm; The interpolation polynomial parameter variables for the machine tool range from 0 to... Within the range, dimensionless; This indicates the axis position at the end of the currently executing program segment. The machine tool can automatically obtain this position during operation. The unit is mm, and it is a known parameter. The coefficients of a given polynomial are dimensionless and represent the unknown variables to be determined. By defining the parameter range The difference between the axis position at the endpoint and the starting position is calculated and is dimensionless. The calculation formula is shown in equation (9): （9） In equation (9), , Represents the coefficients of a given polynomial, which are dimensionless; Let be the polynomial order of the machine tool interpolation polynomial defined in formula (8), which is dimensionless; (5) Calculate the number of segments in the polynomial transformation process. To achieve high-precision conversion between three theoretical axis function high-order polynomials and machine tool interpolation polynomials, the three theoretical axis function high-order polynomials are segmented, and each segment is converted separately. The number of sample points selected for each segment is... n is 5, 7, or 9; radial feed axis of grinding wheel The operation is as follows: According to step (1), by using the meshing condition of equation (3) and giving the same time interval, the tooth surface coordinate point cloud is obtained, and the tooth surface mesh is divided by the tooth surface coordinate point cloud to obtain the tooth surface mesh points and the number of tooth surface mesh points; according to the number of tooth surface mesh points and the radial feed axis of the grinding wheel The number of sample points selected for each theoretical axis function high-order polynomial, the number of tooth surface grid points, and the number of grinding wheel radial feed axes. The number of sample points selected for each segment of the theoretical axis function's higher-order polynomial is simultaneously reduced by one, and then the quotient is calculated to determine the radial feed axis of the grinding wheel. The number of segments to be divided by the transformation of the theoretical axis function into a higher-order polynomial; The grinding wheel tangential feed axis and grinding wheel axial feed axis The above operations are used to obtain the tangential feed axis of the grinding wheel. The number of segments to be divided by the theoretical axis function polynomial transformation and the axial feed axis of the grinding wheel The number of segments to be divided by the theoretical axis function polynomial transformation; (6) Sample point parameter positions of piecewise computer bed interpolation polynomial radial feed axis of grinding wheel The operation is as follows: According to the radial feed axis of the grinding wheel in step (5) The number of polynomial pieces in the process of transforming the theoretical axis function high-order polynomial to the machine tool interpolation polynomial, and the radial feed axis of the grinding wheel. The theoretical axis function high-order polynomial is segmented, and in the polynomial transformation of each segment, the corresponding time of each sample point is divided by the total time of the corresponding theoretical axis function high-order polynomial, and then multiplied by the parameter interval length of the machine tool interpolation polynomial. The parameter interval length is a variable in the polynomial interpolation code of the Siemens 840Dsl CNC system, thus obtaining the radial feed axis of the worm gear grinding machine. The position of each sample point parameter in the machine tool interpolation polynomial; The specific operation is as follows: On the radial feed axis of each grinding wheel segment... In the process of converting the theoretical axis function high-order polynomial to the machine tool interpolation polynomial, the parameters are converted through proportional relationships. The radial feed axis of the grinding wheel The theoretical axis function higher-order polynomial and parameters are The radial feed axis of the grinding wheel By relating the machine tool interpolation polynomial, the radial feed axis of the worm gear grinding machine can be calculated. The parameters of each sample point of the machine tool interpolation polynomial The location is used to calculate the parameters for each sample point. The formula (12) for the position is as follows: （12） In equation (12), , where is the total time, in seconds; , is the time interval between the sample point time and the start time of the polynomial in this segment, in seconds; The segment number of the polynomial is dimensionless. is the sample point number of a single-segment polynomial, dimensionless; The number of sample points selected for each polynomial objective, dimensionless; Radial feed axis of grinding wheel The parameter positions of the machine tool interpolation polynomial; This represents the numerical value of the parameter interval length in the polynomial definition; it is dimensionless. Calculate the tangential feed axis of the grinding wheel The parameter position of each sample point of the machine tool interpolation polynomial and the axial feed axis of the grinding wheel. The parameter positions of each sample point of the machine tool interpolation polynomial are calculated according to the above method for the radial feed axis of the grinding wheel. The operation of the parameter position of each sample point of the machine tool interpolation polynomial is performed; (7) Computer bed polynomial coefficients radial feed axis of grinding wheel The operation is as follows: on the radial feed axis of the grinding wheel In the transformation from the higher-order polynomial of each theoretical axis function to the machine tool interpolation polynomial, the radial feed axis of the grinding wheel is known. The positions of parameters at more than 5 sample points in the higher-order polynomial of the theoretical axis function are related to the radial feed axis of the grinding wheel. The parameter positions of the sample points corresponding to the machine tool interpolation polynomial are determined, and based on the equation relationship between the theoretical axis function higher-order polynomial and the machine tool interpolation polynomial at each sample point, more than five equations are listed to form a system of equations. The unknown coefficients of each segment of the machine tool interpolation polynomial of the worm gear grinding machine are solved. The system of equations is solved using the least squares method to obtain the radial feed axis of the grinding wheel. The unknown coefficients of each segment of the machine tool interpolation polynomial; The grinding wheel tangential feed axis and grinding wheel axial feed axis The above operations are used to obtain the tangential feed axis of the grinding wheel. The unknown coefficients of each segment of the machine tool interpolation polynomial and the axial feed axis of the grinding wheel The unknown coefficients of each segment of the machine tool interpolation polynomial are used to perform polynomial interpolation, thereby realizing the topology modification machining of the generating grinding tooth surface based on the Siemens 840Dsl CNC system.

2. The method for high-order polynomial topology modification of generated grinding tooth surfaces according to claim 1, characterized in that: In step (1), the worm grinding wheel is a ZI-type worm, whose overall geometry is the same as that of the involute worm for gear hobbing; the two-parameter curved surface of the left tooth surface of the worm grinding wheel is shown in formula (1): （1） In equation (1), For the ZI type worm gear grinding wheel expression; The reference circle radius for the worm gear grinding wheel is in mm; This is a key parameter, the tooth cogging angle, in mm. This parameter ensures that the axial profile of the worm grinding wheel contacts the left and right tooth surfaces of the gear simultaneously. These are involute parameters, dimensionless; The parameters for the helix are dimensionless. These are the lead parameters for the worm gear grinding wheel, in mm. Given the biparametric surface of the worm grinding wheel from equation (1), the coordinate transformation matrix can be used to determine the relationship between the two parameters. Determine the worm gear grinding wheel using the grinding wheel coordinate system. To the gear coordinate system The transformation yields the worm grinding wheel in the gear coordinate system. The trajectory is shown in formula (2): （2） In equation (2), Is Two-parameter surface of worm grinding wheel in coordinate system Is Normal vector in the coordinate system; Is Two-parameter surface of worm grinding wheel in coordinate system In the gear coordinate system Normal vector in the coordinate system; For the grinding wheel coordinate system To the gear coordinate system The transformation matrix is ​​dimensionless; for The top-left matrix is ​​dimensionless; The mathematical model of continuous generating grinding of worm wheel is obtained from equation (2) and the meshing principle, namely formula (3).

3. The high-order polynomial topology modification method for generating and grinding tooth surfaces according to claim 1, characterized in that: In step (3), based on the fundamental principle of gear tooth surface modification, given the target normal deviation value of the tooth surface grid points, the coefficient values ​​of the higher-order polynomial of the theoretical axis function are obtained using the following formula (6): （6） In equation (6), This represents the target normal deviation value, in mm. The coefficients of the higher-order polynomial of the theoretical axis function are dimensionless. The sensitivity matrix is ​​dimensionless. Since the sensitivity matrix is ​​ill-conditioned, it is almost singular in most cases. To prevent the risk of numerical solutions diverging, the following uses the Levenberg-Marquardt algorithm combined with the sensitivity matrix to solve for the correction coefficients in equation (6): （7） In equation (7), The damping coefficient is adjusted during iteration; the values ​​for the first iteration can be obtained using a matrix. The absolute value of the largest element in the set, dimensionless; It is an identity matrix, dimensionless; The sensitivity matrix is ​​dimensionless. The coefficients of the higher-order polynomial of the theoretical axis function are dimensionless. This represents the target normal deviation value, in mm. Theoretically, the radial feed axis of the grinding wheel that satisfies the target normal deviation can be obtained through 2 to 3 iterations.

1. Grinding wheel tangential feed axis and grinding wheel axial feed axis The numerical values ​​of the 12 polynomial coefficients of the higher-order polynomials of the three theoretical axis functions.

4. The high-order polynomial topology modification method for generating and grinding tooth surfaces according to claim 1, characterized in that: In step (4), given the machine tool interpolation polynomial, polynomial interpolation is implemented using the POLY code of the Siemens 840Dsl CNC system, with the grinding wheel radial feed axis as the reference. For example, the POLY code input rule for the Siemens 840Dsl CNC system is as follows: (Formula 10) （10） In equation (10), Indicates the radial feed axis of the grinding wheel Shaft name, the shaft used for tangential feed of the grinding wheel. and grinding wheel axial feed axis When performing polynomial interpolation, one can... Change at the shaft and ; The unit for the endpoint position parameter of each axis is mm; This represents the coefficients of the polynomial program defined by the program; it is dimensionless. This represents the length of the parameter interval in the defining polynomial; it is dimensionless. Without programming but .

5. The method for high-order polynomial topology modification of generated grinding tooth surfaces according to claim 1, characterized in that: In step (5), the radial feed axis of the grinding wheel is calculated. In the process of piecewise division of the higher-order polynomial of the theoretical axis function, the number of mesh points on the tooth surface With each section of the grinding wheel radial feed axis The number of sample points selected for the higher-order polynomial of the theoretical axis function Simultaneously subtract one, then perform the quotient calculation to determine the radial feed axis of the grinding wheel. The number of segments to be divided by the transformation of the theoretical axis function into a higher-order polynomial is shown in the following formula: (11) In equation (11), The total number of segments to divide the curve, dimensionless; The number of grid points on the tooth surface is dimensionless. The number of sample points selected for each polynomial objective, dimensionless; Calculate the tangential feed axis of the grinding wheel The number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function and the axial feed axis of the grinding wheel. The number of segments to be divided by the higher-order polynomial transformation of the theoretical axis function is calculated as described above for the radial feed axis of the grinding wheel. The operation is performed to divide the theoretical axis function into segments by transforming the higher-order polynomial.

6. The method for high-order polynomial topology modification of generated grinding tooth surfaces according to claim 1, characterized in that: In step (7), on each section of the grinding wheel radial feed axis During the conversion from theoretical axis function high-order polynomials to machine tool interpolation polynomials, when the number of sample points in each segment... At that time, the radial feed axis of the grinding wheel is known. The theoretical axis function of the higher-order polynomial, the time variable value corresponding to each sample point. , with the radial feed axis of the grinding wheel The variable value corresponding to each sample point in the machine tool interpolation polynomial. At each sample point, the equation relationship between the theoretical axis function higher-order polynomial and the machine tool interpolation polynomial is as follows: Equation (5) in step (2) is placed on the left side of the equation, and Equation (8) in step (4) is placed on the right side of the equation, resulting in equation (13); Substitute the parameters calculated by equation (9) in step (4) and the parameters calculated by equation (12) in step (6) into equation (13) to solve for the radial feed axis of the grinding wheel. Unknown coefficients of machine tool interpolation polynomial Equation (13) is as follows: （13） In equation (13), Equation (5) is based on time. When the variable is used, the coefficients of higher-order polynomials are dimensionless. From equation (4), we can see that The positions of each axis under standard motion; Radial feed axis of grinding wheel The parameter variables of the machine tool interpolation polynomial are dimensionless; The interpolation polynomial coefficients for the machine tool are dimensionless. Time, in seconds; Let be the polynomial order of the machine tool interpolation polynomial defined in formula (8), which is dimensionless; The number of sample points selected for each polynomial objective, dimensionless; The segment number of the polynomial is dimensionless. is the sample point number of a single-segment polynomial, dimensionless; Solving the above equations using the least squares method ultimately yields the radial feed axis of each grinding wheel segment. Unknown coefficients of machine tool interpolation polynomial ; Calculate the tangential feed axis of each grinding wheel segment The unknown coefficients of the machine tool interpolation polynomial and the axial feed axis of each grinding wheel segment The unknown coefficients of the machine tool interpolation polynomial should be calculated according to the above for each segment of the grinding wheel radial feed axis. The operation of the unknown coefficients of the machine tool interpolation polynomial is performed.

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