A method for solving the grinding trajectory of a conical end mill helical groove
By using particle swarm optimization algorithm and grinding wheel pose solution method, the uncertainty of the helical groove grinding process of conical end mills was solved, the precision grinding of helical grooves was realized, and the machining stability and accuracy were improved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- SOUTHWEST JIAOTONG UNIV
- Filing Date
- 2024-05-16
- Publication Date
- 2026-05-26
AI Technical Summary
There is limited research on grinding processes for helical grooves in conical end mills in the current technology, which leads to a reliance on experience in the manufacturing process and affects the stability and accuracy of machining.
A particle swarm optimization algorithm combined with a grinding wheel pose solution method is adopted. By defining the process characteristics of the spiral groove and modeling the grinding motion of the grinding wheel, constraint equations for the rake angle and core diameter of the spiral groove are established. The grinding wheel pose is then iteratively solved using the particle swarm optimization algorithm to meet the grinding accuracy requirements.
Precision grinding of the helical groove of the conical end mill was achieved, improving machining stability and accuracy, and meeting the machining accuracy requirements for rake angle and core diameter.
Smart Images
Figure CN118305651B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of integral CNC end mill machining technology, specifically relating to a method for solving the grinding trajectory of a conical end mill helical groove. Background Technology
[0002] Solid CNC end mills, as cutting tools used in mechanical manufacturing, are characterized by low manufacturing cost and high material removal rate. The helical flute, as one of the structural features of the end mill, has its grinding process directly determining parameters such as the rake angle and core diameter, thus playing a crucial role in the tool's cutting performance, chip removal, and strength.
[0003] Designing a helical groove first requires establishing a motion model of its helical cutting edge and the grinding wheel, followed by setting the grinding process for the helical groove surface. Currently, there is considerable research on helical groove grinding technology, but most studies focus on the grinding process of helical grooves for cylindrical CNC end mills. Research on the grinding process of helical grooves for conical end mills is scarce. This leads to the common reliance on experience for grinding helical grooves when manufacturing conical CNC end mills in China. The helical groove is the largest structure in an end mill, and it has a crucial impact on the stability and accuracy of tool manufacturing and machining, playing an indispensable role in high-speed precision machining. Summary of the Invention
[0004] To address the problem of grinding helical grooves in integral conical CNC end mills, this invention provides a method for solving the grinding trajectory of helical grooves in conical end mills.
[0005] This invention discloses a method for solving the grinding trajectory of a helical groove in a conical end mill. First, the process characteristics of the helical groove are defined, including tool helical cutting edge modeling, tool parameter definition, and grinding wheel motion modeling. Second, a solution equation for the grinding wheel pose is established based on the tool parameters. Finally, a particle swarm optimization algorithm is used to solve for the grinding wheel pose. Specifically, the method includes the following steps:
[0006] Step 1: Modeling the grinding motion of the spiral groove and grinding wheel.
[0007] Establish workpiece coordinate system O W -X W Y W Z W The origin O is the coordinate system. W With X W Y W The plane is located on the end face of the tool, Z W The axis and the tool's axis vector coincide. Point P is defined as any point on the helical cutting edge, k is the tool taper, β is the tool's helix angle, and r is the tool's position on the X-axis. W Y W Let the radius in the plane be L, the blade length be L, and ζ be the rotation angle corresponding to point P. The expression for point P is as follows:
[0008] (1)
[0009] In the formula This represents the initial rotation angle corresponding to the starting point of the spiral blade.
[0010] With X W Y W Using the radial cross-section curve of the helical groove in the plane as a reference, the rake angle parameter γ of the helical groove is expressed. Point P r1 The vector T is the starting point on the radial cross-section curve of the helical groove. Pr1 Let P be the point r1 The tangent vector of the cross curve at point T, vector T OP O is the origin of the coordinate system. W Point P r1 The vector, vector T Pr1 and vector T OP The included angle between them is the rake angle parameter of the helical groove, and the specific expression is as follows:
[0011] (2)
[0012] With X W Y W Using the radial cross-sectional curve of the helical groove in the plane as a reference, the core diameter rc of the helical groove is expressed. Point P ri The radial section curve of the spiral groove is located at a distance O from the origin. W The closest point is the point that is closest to the origin O. W The distance between them is the core diameter, and the specific expression is as follows:
[0013] (3)
[0014] Define the grinding wheel coordinate system O G -X G Y G Z G coordinate axis Z G The axis coincides with the axis of the grinding wheel, and h is defined as the axial distance parameter of the grinding wheel, R g r is the maximum radius of the grinding wheel g For the radius of the grinding wheel fillet, k g Where φ is the grinding wheel taper, H is the grinding wheel thickness, and R is the grinding wheel taper. s (h, φ) is the equation of the grinding wheel's rotating surface, and R(h) is the equation of the grinding wheel's profile. The equations of the grinding wheel's profile and rotating surface are expressed as follows:
[0015] (4)
[0016] (5)
[0017] Define the grinding wheel pose parameters (a) x , a y , a z O is the origin of the grinding wheel coordinate system. G In the workpiece coordinate system, the installation angle α is the angle between the grinding wheel axis and the tool axis, δ is the rotational motion angle of the grinding wheel, and M is the motion matrix of the grinding wheel in the grinding motion. e Represented as:
[0018] (6)
[0019] Step 2: Calculation of grinding wheel trajectory.
[0020] Constraint equations for the rake angle and core diameter of the helical groove and the grinding wheel trajectory are established respectively. Using the axial distance parameter h and the installation angle α of the grinding wheel as input parameters, the grinding wheel trajectory is calculated through the rake angle constraint equation. It is then determined whether this trajectory meets the core diameter machining requirements of the helical groove. Finally, the input parameters are iteratively solved using a particle swarm optimization algorithm to calculate the grinding wheel trajectory that simultaneously satisfies the grinding accuracy requirements of both the rake angle and core diameter. The details are as follows:
[0021] (1) Front angle constraint equation.
[0022] Based on the parameterized equations and motion equations of the grinding wheel, the rotating surface R of the grinding wheel during the grinding process... s (h,φ) is represented as:
[0023] (7)
[0024] By taking the partial derivatives of parameters h and φ in equation (7) respectively, and then performing the cross product of the two partial derivatives, the expression for the normal vector on the rotating surface of the grinding wheel can be obtained, as shown below:
[0025] (8)
[0026] In the formula, , .
[0027] Vector T pr1 Let P be the point r1 The tangent vector τ of the helical groove profile curve at the location. Pr1 Let P be the point r1 The tangent vector on the helical edge, vector n pr1 Let P be the point r1 The normal vector of the rake face at the location, vector T pr1 and vector τ Pr1 The two are represented as follows:
[0028] (9)
[0029] Point n Pr1 The normal vector of the rake face at the location is perpendicular to vector T. pr1 and vector τ Pr1 Therefore, point P r1 The normal vector of the rake face at the location can be obtained by adjusting the vector n. Pr1 and vector τ Pr1 The cross product yields the result, as shown in the following expression:
[0030] (10)
[0031] In the formula, .
[0032] The normal vector of the rake face at any point on the helical cutting edge can be obtained through P. r1 The normal vector of the rake face at the point is obtained by rotating it by an angle ξ. Therefore, the normal vector of the rake face on the helical cutting edge is expressed as:
[0033] (11)
[0034] Since the coordinates of the grinding wheel and the spiral groove are equal at the contact point, the following equation can be obtained:
[0035] (12)
[0036] The normal vectors of the grinding wheel and the spiral groove are equal at the contact point. By making the normal vector of the grinding wheel's rotating surface in equation (8) equal to the normal vector of the rake face in equation (10), the following equation can be obtained:
[0037] (13)
[0038] In the formula, .
[0039] Therefore, by setting the axial distance parameter h and the grinding wheel installation angle α as input parameters, the position of the grinding wheel during the grinding process can be solved. First, solve equation (13). Considering that φ must satisfy the condition 90° < φ < 270°, φ is expressed as:
[0040] (14)
[0041] In the formula, .
[0042] Substituting equation (14) into equation (13), we obtain the following expression:
[0043] (15)
[0044] In the formula,
[0045] (16)
[0046] Substituting equations (15) and (16) into equations (12) and (13), the grinding posture of the grinding wheel can be obtained based on the rake angle γ. The expression for the grinding wheel posture is shown in the following equation:
[0047] (17)
[0048] (2) Core diameter constraint equation.
[0049] The core diameter of the tool will change in different radial sections. Using the designed core diameter as the dependent variable and the axial length z as the independent variable, an equation governing the change in the core diameter is established:
[0050] (18)
[0051] In the formula, r c0 For the spiral groove in X W Y W Core diameter within the cross-section.
[0052] Point P Q For any point on the contact line, according to the motion equation of the grinding wheel's rotating surface in equation (7), the expression of this point in the workpiece coordinate system is:
[0053] (19)
[0054] In the formula, h PQ and φ PQ Contact point P Q The corresponding axial distance parameters of the grinding wheel and the rotation angle of the grinding wheel.
[0055] According to equation (19), any point P on the contact surface can be obtained. Q With the tool axis Z W The distance between axes, where the distance from axis Z is... W The closest distance between the shafts is the actual core diameter r' of the spiral groove. c The specific expression is as follows:
[0056] (20)
[0057] (3) Calculation of grinding wheel trajectory.
[0058] The grinding posture of the grinding wheel is calculated based on the designed rake angle and substituted into equations (18) and (20). If the obtained core diameter value is closer to the designed core diameter parameter, it indicates that the grinding accuracy of the core diameter is higher and the grinding posture is closer to the optimal solution. Taking the error of the spiral groove core diameter parameter as the optimization target and the contact between the grinding wheel and the tool as the constraint condition, the input parameter optimization objective function is established as shown in the following equation:
[0059] (twenty one)
[0060] The input parameters of the grinding wheel are defined as a two-dimensional particle, denoted as In(h, α). The number of particles N is determined, and the initial value In of each particle is initialized within the search space of the input parameter particles. i_j (h i_j , α i_j ) and velocity V i_j (1 ≤ j ≤ N) and perform iterative calculations, with the specific iteration rules as follows:
[0061] (twenty two)
[0062] In the formula, ω is the inertia weight, c1 and c2 are the individual learning factor and the group learning factor, respectively, r1 and r2 are random numbers between [0,1], and p i_j (h i_j , α i_j ) represents the optimal individual position of the particle, p global_i_j (h i_j , α i_j ) represents the particle's current global optimal position.
[0063] Furthermore, the specific calculation process for the grinding wheel trajectory is as follows:
[0064] (1) Determine the geometric parameters (R) of the grinding wheel. g H, r g k g ), tool design parameters (r, β, k, γ, r c The parameters of the PSO algorithm are (ω, c1, c2, r1, r2), the maximum number of iterations M, and the minimum error.
[0065] (2) Randomly initialize the particle position In according to the given range of h and α values. i_j (h i_j , α i_j ) and particle velocity V i_j .
[0066] (3) Substitute the initialized particle and particle velocity into the formula to calculate the fitness value of each particle, and then update the individual optimal position p of each particle. i_j and the global optimal position p global_i_j .
[0067] (4) In each iteration, update the particle In according to equation (22). i_j and particle velocity V i_j And update the optimal position p of the individual particles.i_j and the global optimal position p global_i_j .
[0068] (5) Set the global optimal position p of the particle global_i_j Substituting into equation (21), if the result satisfies the condition Fit(h, α) ≤ error, or if the number of iterations reaches the preset maximum number of iterations, the iteration stops and the global optimal position p is output. global The value is used as the optimization result.
[0069] (6) The optimization result p obtained through iterative calculation in steps (1)-(5) global Substituting (h, α) into equations (15)-(17) yields the grinding wheel position that simultaneously satisfies the helical groove rake angle and the core diameter.
[0070] The beneficial technical effects of this invention are as follows:
[0071] This invention addresses the problem of varying helical groove parameters and complex pose calculations for conical end mills under different cross-sections. First, a constraint equation for the grinding wheel pose is established based on the rake angle of the helical groove, and the grinding wheel pose is calculated based on this equation. It then determines whether the machining accuracy requirements of the helical groove core diameter can be met under this pose. Finally, the PSO algorithm is used to iteratively solve the input parameters, so that the final calculated grinding wheel pose can simultaneously meet the machining accuracy requirements of both the rake angle and the core diameter.
[0072] This invention conducts in-depth research on the grinding method of helical grooves for conical end mills, and proposes a grinding algorithm that can meet the grinding accuracy requirements of rake angle and core diameter. Based on actual machining results, it is shown that this method can be effectively used for machining helical grooves, proving the accuracy and feasibility of this grinding method, and providing a theoretical reference for the actual production and machining of cutting tools. Attached Figure Description
[0073] Figure 1 This is a schematic diagram of the helical cutting edge of a conical end mill.
[0074] Figure 2 This is a schematic diagram of the rake angle parameters of the spiral groove.
[0075] Figure 3 This is a schematic diagram of the core diameter parameters of the spiral groove.
[0076] Figure 4 It is a schematic diagram of the grinding wheel profile and the grinding wheel rotation surface.
[0077] Figure 5 This is a schematic diagram of the motion equation of a grinding wheel.
[0078] Figure 6 This is a schematic diagram of the normal vector of the rotating surface of the grinding wheel.
[0079] Figure 7 This is a schematic diagram of the tangent vector of the spiral groove and the tangent vector of the spiral blade.
[0080] Figure 8 This is a schematic diagram illustrating the variation pattern of the core diameter.
[0081] Figure 9 This is a schematic diagram of the actual core diameter during the spiral groove grinding process.
[0082] Figure 10 This is a diagram showing the machining effect of a spiral groove. Detailed Implementation
[0083] The present invention will be further described in detail below with reference to the accompanying drawings and specific implementation methods.
[0084] This invention discloses a method for solving the grinding trajectory of a helical groove in a conical end mill. First, the process characteristics of the helical groove are defined, including tool helical cutting edge modeling, tool parameter definition, and grinding wheel motion modeling. Second, a solution equation for the grinding wheel pose is established based on the tool parameters. Finally, a particle swarm optimization algorithm is used to solve for the grinding wheel pose. Specifically, the method includes the following steps:
[0085] Step 1: Modeling the grinding motion of the spiral groove and grinding wheel.
[0086] Establish workpiece coordinate system O W -X W Y W Z W ,like Figure 1 As shown, the origin O is... W With X W Y W The plane is located on the end face of the tool, Z W The axis and the tool's axis vector coincide. Point P is defined as any point on the helical cutting edge, k is the tool taper, β is the tool's helix angle, and r is the tool's position on the X-axis. W Y W Let the radius in the plane be L, the blade length be L, and ζ be the rotation angle corresponding to point P. The expression for point P is as follows:
[0087] (1)
[0088] In the formula This represents the initial rotation angle corresponding to the starting point of the spiral blade.
[0089] like Figure 2 As shown, with X W Y W Using the radial cross-section curve of the helical groove in the plane as a reference, the rake angle parameter γ of the helical groove is expressed. Point P r1 The vector T is the starting point on the radial cross-section curve of the helical groove. Pr1 Let P be the point r1The tangent vector of the cross curve at point T, vector T OP O is the origin of the coordinate system. W Point P r1 The vector, vector T Pr1 and vector T OP The included angle between them is the rake angle parameter of the helical groove, and the specific expression is as follows:
[0090] (2)
[0091] like Figure 3 As shown, with X W Y W Using the radial cross-sectional curve of the helical groove in the plane as a reference, the core diameter rc of the helical groove is expressed. Point P ri The radial section curve of the spiral groove is located at a distance O from the origin. W The closest point is the point that is closest to the origin O. W The distance between them is the core diameter, and the specific expression is as follows:
[0092] (3)
[0093] like Figure 4 As shown, the grinding wheel coordinate system O is defined. G -X G Y G Z G coordinate axis Z G The axis coincides with the axis of the grinding wheel, and h is defined as the axial distance parameter of the grinding wheel, R g r is the maximum radius of the grinding wheel g For the radius of the grinding wheel fillet, k g Where φ is the grinding wheel taper, H is the grinding wheel thickness, and R is the grinding wheel taper. s (h, φ) is the equation of the grinding wheel's rotating surface, and R(h) is the equation of the grinding wheel's profile. The equations of the grinding wheel's profile and rotating surface are expressed as follows:
[0094] (4)
[0095] (5)
[0096] like Figure 5 As shown, the grinding wheel pose parameters (a) are defined. x , a y , a z O is the origin of the grinding wheel coordinate system. G In the workpiece coordinate system, the installation angle α is the angle between the grinding wheel axis and the tool axis, δ is the rotational motion angle of the grinding wheel, and M is the motion matrix of the grinding wheel in the grinding motion. e Represented as:
[0097] (6)
[0098] Step 2: Calculation of grinding wheel trajectory.
[0099] Constraint equations for the rake angle and core diameter of the helical groove and the grinding wheel trajectory are established respectively. Using the axial distance parameter h and the installation angle α of the grinding wheel as input parameters, the grinding wheel trajectory is calculated through the rake angle constraint equation. It is then determined whether this trajectory meets the core diameter machining requirements of the helical groove. Finally, the input parameters are iteratively solved using a particle swarm optimization algorithm to calculate the grinding wheel trajectory that simultaneously satisfies the grinding accuracy requirements of both the rake angle and core diameter. The details are as follows:
[0100] (1) Front angle constraint equation.
[0101] like Figure 6 As shown, based on the parameterized equations and motion equations of the grinding wheel, the rotating surface R of the grinding wheel during the grinding process... s (h, φ) is represented as:
[0102] (7)
[0103] By taking the partial derivatives of parameters h and φ in equation (7) respectively, and then performing the cross product of the two partial derivatives, the expression for the normal vector on the rotating surface of the grinding wheel can be obtained, as shown below:
[0104] (8)
[0105] In the formula, , .
[0106] like Figure 7 As shown, vector T pr1 Let P be the point r1 The tangent vector τ of the helical groove profile curve at the location. Pr1 Let P be the point r1 The tangent vector on the helical edge, vector n pr1 Let P be the point r1 The normal vector of the rake face at the location, vector T pr1 and vector τ Pr1 The two are represented as follows:
[0107] (9)
[0108] Point n Pr1 The normal vector of the rake face at the location is perpendicular to vector T. pr1 and vector τ Pr1 Therefore, point P r1 The normal vector of the rake face at the location can be obtained by adjusting the vector n. Pr1 and vector τ Pr1The cross product yields the result, as shown in the following expression:
[0109] (10)
[0110] In the formula, .
[0111] The normal vector of the rake face at any point on the helical cutting edge can be obtained through P. r1 The normal vector of the rake face at the point is obtained by rotating it by an angle ξ. Therefore, the normal vector of the rake face on the helical cutting edge is expressed as:
[0112] (11)
[0113] Since the coordinates of the grinding wheel and the spiral groove are equal at the contact point, the following equation can be obtained:
[0114] (12)
[0115] The normal vectors of the grinding wheel and the spiral groove are equal at the contact point. By making the normal vector of the grinding wheel's rotating surface in equation (8) equal to the normal vector of the rake face in equation (10), the following equation can be obtained:
[0116] (13)
[0117] In the formula, .
[0118] Therefore, by setting the axial distance parameter h and the grinding wheel installation angle α as input parameters, the position of the grinding wheel during the grinding process can be solved. First, solve equation (13). Considering that φ must satisfy the condition 90° < φ < 270°, φ is expressed as:
[0119] (14)
[0120] In the formula, .
[0121] Substituting equation (14) into equation (13), we obtain the following expression:
[0122] (15)
[0123] In the formula,
[0124] (16)
[0125] Substituting equations (15) and (16) into equations (12) and (13), the grinding posture of the grinding wheel can be obtained based on the rake angle γ. The expression for the grinding wheel posture is shown in the following equation:
[0126] (17)
[0127] (2) Core diameter constraint equation.
[0128] like Figure 8 As shown, the core diameter of the tool changes in different radial sections. Using the designed core diameter as the dependent variable and the axial length z as the independent variable, an equation governing the change in the core diameter is established:
[0129] (18)
[0130] In the formula, r c0 For the spiral groove in X W Y W Core diameter within the cross-section.
[0131] Point P Q For any point on the contact line, according to the motion equation of the grinding wheel's rotating surface in equation (7), the expression of this point in the workpiece coordinate system is:
[0132] (19)
[0133] In the formula, h PQ and φ PQ Contact point P Q The corresponding axial distance parameters of the grinding wheel and the rotation angle of the grinding wheel.
[0134] According to equation (19), any point P on the contact surface can be obtained. Q With the tool axis Z W The distance between axes, where the distance from axis Z is... W The closest distance between the shafts is the actual core diameter r' of the spiral groove. c ,like Figure 9 As shown, the specific expression is as follows:
[0135] (20)
[0136] (3) Calculation of grinding wheel trajectory.
[0137] The grinding posture of the grinding wheel is calculated based on the designed rake angle and substituted into equations (18) and (20). If the obtained core diameter value is closer to the designed core diameter parameter, it indicates that the grinding accuracy of the core diameter is higher and the grinding posture is closer to the optimal solution. Taking the error of the spiral groove core diameter parameter as the optimization target and the contact between the grinding wheel and the tool as the constraint condition, the input parameter optimization objective function is established as shown in the following equation:
[0138] (twenty one)
[0139] The input parameters of the grinding wheel are defined as a two-dimensional particle, denoted as In(h, α). The number of particles N is determined, and the initial value In of each particle is initialized within the search space of the input parameter particles. i_j (h i_j , α i_j ) and velocity V i_j (1 ≤ j ≤ N) and perform iterative calculations, with the specific iteration rules as follows:
[0140] (twenty two)
[0141] In the formula, ω is the inertia weight, c1 and c2 are the individual learning factor and the group learning factor, respectively, r1 and r2 are random numbers between [0,1], and p i_j (h i_j , α i_j ) represents the optimal individual position of the particle, p global_i_j (h i_j , α i_j ) represents the particle's current global optimal position.
[0142] Furthermore, the specific calculation process for the grinding wheel trajectory is as follows:
[0143] (1) Determine the geometric parameters (R) of the grinding wheel. g H, r g k g ), tool design parameters (r, β, k, γ, r c The parameters of the PSO algorithm are (ω, c1, c2, r1, r2), the maximum number of iterations M, and the minimum error.
[0144] (2) Randomly initialize the particle position In according to the given range of h and α values. i_j (h i_j , α i_j ) and particle velocity V i_j .
[0145] (3) Substitute the initialized particle and particle velocity into the formula to calculate the fitness value of each particle, and then update the individual optimal position p of each particle. i_j and the global optimal position p global_i_j .
[0146] (4) In each iteration, update the particle In according to equation (22). i_j and particle velocity V i_j And update the optimal position p of the individual particles. i_j and the global optimal position p global_i_j .
[0147] (5) Set the global optimal position p of the particle global_i_j Substituting into equation (21), if the result satisfies the condition Fit(h, α) ≤ error, or if the number of iterations reaches the preset maximum number of iterations, the iteration stops and the global optimal position p is output. global The value is used as the optimization result.
[0148] (6) The optimization result p obtained through iterative calculation in steps (1)-(5) global Substituting (h, α) into equations (15)-(17) yields the grinding wheel position that simultaneously satisfies the helical groove rake angle and the core diameter.
[0149] Based on the above grinding algorithm, an algorithm module was developed in the VC++ environment. After inputting the relevant parameters shown in Table 1, the tool path can be obtained. Some calculation results are shown in Table 2.
[0150] Table 1. Process parameters for the circular arc cutting edge rake face
[0151]
[0152] Table 2. Results of partial toolpath calculations
[0153]
[0154] Finally, a five-axis CNC grinding machine was used for actual machining verification. The tool obtained after actual machining is shown below. Figure 10 As shown.
Claims
1. A method for solving the grinding trajectory of a helical groove in a conical end mill, characterized in that, First, the process characteristics of the helical groove are defined, including tool helical cutting edge modeling, tool parameter definition, and grinding wheel motion modeling. Second, the solution equation for the grinding wheel pose is established based on the tool parameters. Finally, the particle swarm optimization algorithm is used to solve the grinding wheel pose, specifically including the following steps: Step 1: Modeling the grinding motion of the spiral groove and grinding wheel; Establish workpiece coordinate system O W -X W Y W Z W The origin O is the coordinate system. W With X W Y W The plane is located on the end face of the tool, Z W The axis and the tool's axis vector coincide. Point P is defined as any point on the helical cutting edge, k is the tool taper, β is the tool's helix angle, and r is the tool's position on the X-axis. W Y W Let the radius in the plane be L, the blade length be L, and ξ be the rotation angle corresponding to point P. The expression for point P is as follows: （1） In the formula This represents the initial rotation angle corresponding to the starting point of the spiral blade. With X W Y W Using the radial cross-section curve of the helical groove in the plane as a reference, the rake angle parameter γ of the helical groove is expressed; point P r1 The vector T is the starting point on the radial cross-section curve of the helical groove. Pr1 Let P be the point r1 The tangent vector of the cross curve at the point, vector T OP O is the origin of the coordinate system. W Point P r1 The vector, vector T Pr1 and vector T OP The included angle between them is the rake angle parameter of the helical groove, and the specific expression is as follows: （2） With X W Y W Using the radial cross-sectional curve of the helical groove in the plane as a reference, the core diameter rc of the helical groove is expressed; point P ri The radial section curve of the spiral groove is located at a distance O from the origin. W The closest point is the point that is closest to the origin O. W The distance between them is the core diameter, and the specific expression is as follows: （3） Define the grinding wheel coordinate system O G -X G Y G Z G coordinate axis Z G The axis coincides with the axis of the grinding wheel, and h is defined as the axial distance parameter of the grinding wheel, R g r is the maximum radius of the grinding wheel g For the radius of the grinding wheel fillet, k g Where φ is the grinding wheel taper, H is the grinding wheel thickness, and R is the grinding wheel taper. s (h,φ) is the equation of the grinding wheel's rotating surface, and R(h) is the equation of the grinding wheel's profile. The equations of the grinding wheel's profile and rotating surface are expressed as follows: （4） （5） Define the grinding wheel pose parameters (a x , a y , a z O is the origin of the grinding wheel coordinate system. G In the workpiece coordinate system, the installation angle α is the angle between the grinding wheel axis and the tool axis, δ is the rotational motion angle of the grinding wheel, and M is the motion matrix of the grinding wheel in the grinding motion. e Represented as: （6） Step 2: Calculation of grinding wheel trajectory; Constraint equations for the rake angle and core diameter of the helical groove and the grinding wheel trajectory are established respectively. Using the axial distance parameter h and the installation angle α of the grinding wheel as input parameters, the grinding wheel trajectory is calculated through the rake angle constraint equation. It is then determined whether this trajectory meets the core diameter machining requirements of the helical groove. Finally, the input parameters are iteratively solved using a particle swarm optimization algorithm to calculate the grinding wheel trajectory that simultaneously satisfies the grinding accuracy requirements of both the rake angle and core diameter, as detailed below: (1) Front Angle Constraint Equation Based on the parameterized equations and motion equations of the grinding wheel, the rotating surface R of the grinding wheel during the grinding process... s (h, φ) is represented as: （7） By taking the partial derivatives of parameters h and φ in equation (7) respectively, and then performing the cross product of the two partial derivatives, the expression for the normal vector on the rotating surface of the grinding wheel can be obtained, as shown below: （8） In the formula, , ; Vector T pr1 Let P be the point r1 The tangent vector τ of the helical groove profile curve at the location. Pr1 Let P be the point r1 The tangent vector on the helical edge, vector n pr1 Let P be the point r1 The normal vector of the rake face at the location, vector T pr1 and vector τ Pr1 The two are represented as follows: （9） Point n Pr1 The normal vector of the rake face at the location is perpendicular to vector T. pr1 and vector τ Pr1 Therefore, point P r1 The normal vector of the rake face at the location can be obtained by adjusting the vector n. Pr1 and vector τ Pr1 The cross product yields the result, as shown in the following expression: （10） In the formula, ; The normal vector of the rake face at any point on the helical cutting edge can be obtained through P. r1 The normal vector of the rake face at the point is obtained by rotating it by an angle ξ. Therefore, the normal vector of the rake face on the helical cutting edge is expressed as: （11） Since the coordinates of the grinding wheel and the spiral groove are equal at the contact point, the following equation can be obtained: （12） The normal vectors of the grinding wheel and the spiral groove are equal at the contact point. By making the normal vector of the grinding wheel's rotating surface in equation (8) equal to the normal vector of the rake face in equation (10), the following equation can be obtained: （13） In the formula, ; Therefore, by setting the axial distance parameter h and the grinding wheel installation angle α as input parameters, the position of the grinding wheel during the grinding process can be solved. First, solve equation (13). Considering that φ must satisfy the condition 90° < φ < 270°, φ is expressed as: （14） In the formula, ; Substituting equation (14) into equation (13), we obtain the following expression: （15） In the formula, （16） Substituting equations (15) and (16) into equations (12) and (13), the grinding posture of the grinding wheel can be obtained based on the rake angle γ. The expression for the grinding wheel posture is shown in the following equation: （17） (2) Core diameter constraint equation The core diameter of the tool will change in different radial sections. Using the designed core diameter as the dependent variable and the axial length z as the independent variable, an equation governing the change in the core diameter is established: （18） In the formula, r c0 For the spiral groove in X W Y W Core diameter within the cross-section; Point P Q For any point on the contact line, according to the motion equation of the grinding wheel's rotating surface in equation (7), the expression of this point in the workpiece coordinate system is: （19） In the formula, h PQ and φ PQ Contact point P Q The corresponding axial distance parameters of the grinding wheel and the rotation angle of the grinding wheel; According to equation (19), any point P on the contact surface can be obtained. Q With the tool axis Z W The distance between axes, where the distance from axis Z is... W The closest distance between the shafts is the actual core diameter r' of the spiral groove. c The specific expression is as follows: （20） (3) Calculation of grinding wheel trajectory The grinding posture of the grinding wheel is calculated based on the designed rake angle and substituted into equations (18) and (20). If the obtained core diameter value is closer to the designed core diameter parameter, it indicates that the grinding accuracy of the core diameter is higher and the grinding posture is closer to the optimal solution. Taking the error of the spiral groove core diameter parameter as the optimization target and the contact between the grinding wheel and the tool as the constraint condition, the input parameter optimization objective function is established as shown in the following equation: （21） The input parameters of the grinding wheel are defined as a two-dimensional particle, denoted as In(h, α). The number of particles N is determined, and the initial value In of each particle is initialized within the search space of the input parameter particles. i_j (h i_j , α i_j ) and velocity V i_j (1 ≤ j ≤ N) and perform iterative calculations, with the specific iteration rules as follows: （22） In the formula, ω is the inertia weight, c1 and c2 are the individual learning factor and the group learning factor, respectively, r1 and r2 are random numbers between [0, 1], and p... i_j (h i_j , α i_j ) represents the optimal individual position of the particle, p global_i_j (h i_j , α i_j ) represents the particle's current global optimal position.
2. The method for solving the grinding trajectory of a conical end mill helical groove according to claim 1, characterized in that, The specific calculation process for the grinding wheel trajectory is as follows: (1) Determine the geometric parameters R of the grinding wheel g H, r g k g， Tool design parameters r, β, k, γ, r c， PSO algorithm parameters ω, c1, c2, r1, r 2， The algorithm's maximum number of iterations M and minimum error; (2) Randomly initialize the particle position In according to the given range of h and α values. i_j (h i_j , α i_j ) and particle velocity V i_j ; (3) Substitute the initialized particle and particle velocity into the formula to calculate the fitness value of each particle, and then update the individual optimal position p of each particle. i_j and the global optimal position p global_i_j ; (4) In each iteration, update the particle In according to equation (22). i_j and particle velocity V i_j And update the optimal position p of the individual particles. i_j and the global optimal position p global_i_j ; (5) Set the global optimal position p of the particle global_i_j Substituting into equation (21), if the result satisfies the condition Fit(h, α) ≤ error, or if the number of iterations reaches the preset maximum number of iterations, the iteration stops and the global optimal position p is output. global The value is used as the optimization result; (6) The optimization result p obtained through iterative calculation in steps (1)-(5) global Substituting (h, α) into equations (15)-(17) yields the grinding wheel position that simultaneously satisfies the helical groove rake angle and the core diameter.