Method for cooperatively determining size and machining boundary of blisk ball disc milling slotting tool

Abstract

The invention relates to a method for cooperatively determining the size and the machining boundary of a blisk ball disc milling slotting cutter, which comprises the following steps of: firstly, calculating a set of all feasible spherical centers capable of penetrating through a blisk channel without interference, and then establishing a spherical shell double-position approaching channel curved surface model which is formed by enclosing a blade basin curved surface and a blade back curved surface; according to the spherical shell double-position approaching channel curved surface model, determining a channel residual volume model after the virtual spherical shell double-position approaching leaf disc channel curved surface; and finally, under the condition that the first sphere center and the second sphere center are both located in the feasible spherical surface sphere center set, a tool size and machining boundary collaborative optimization model oriented to the maximum material removal volume is established based on the channel residual volume model, and solving is carried out. By means of the method, the cutter size and the machining boundary of the high-rigidity spherical disc milling cutter can be determined within the maximum cutter channel machinable range, and therefore the blade disc rough machining cutting efficiency is greatly improved.

Description

Technical Field

[0001] This invention relates to the field of precision machining technology for bladed disks, and in particular to a method for collaboratively determining the dimensions of milling cutters and machining boundaries for integral bladed disks. Background Technology

[0002] Open integral bladed disks (IBMs) are components with one open side, where the disk and blades are integrated. They are widely used in high-end equipment such as aero-engines. Traditionally, milling cutters used for machining open IBMs are mostly slender and have small diameters, with their upper limit being the narrowest width of the disk channel. Typically, the cutting speed of these cutters can only reach half or even one-third of the insert's maximum tolerance, resulting in low cutting efficiency. Disc milling cutters, as a new type of milling cutter, break through the limitation of the narrowest width of the disk channel and can reach more than twenty times the diameter of ordinary milling cutters. Their cutting speed can reach 100% of the insert's maximum tolerance. Disc milling cutters have greater overall rigidity than slender cutters, and therefore, when used for grooving the channels of open IBMs, their machining efficiency can be several times that of traditional slender milling cutters.

[0003] like Figure 1 As shown, the most common disc end mills are flat disc end mills and ball end mills. When placed in a twisted impeller channel, flat disc end mills are prone to interference with the blades, limiting their effective cutting area and the volume of material that can be removed, thus restricting the overall efficiency of roughing the entire impeller channel. In contrast, ball end mills overcome these difficulties. The disc surface of a ball end mill is spherical, allowing for direct machining of curved grooves. It better adapts to the twisted channel shape of the impeller, thus providing a larger machining range than flat disc end mills.

[0004] For a given geometry of an integral bladed disk part to be machined, to achieve efficient grooving with a ball end mill, it is necessary to determine the tool size and machining boundary of the ball end mill that are adapted to the channel shape of the integral bladed disk to maximize the machined range of the channel. This is to maximize the material removal capacity within the channel and improve the tool's adaptability to the bladed disk, thereby improving roughing cutting efficiency. However, there is currently a lack of a systematic and universal method for determining tool parameters and machining boundaries, which restricts the further promotion and application of ball end mills in the efficient manufacturing of integral bladed disks. Summary of the Invention

[0005] Based on this, it is necessary to provide a method for collaboratively determining the size and machining boundary of a ball milling cutter for integral impeller disks and grooving, which can determine the size and machining boundary of a ball milling cutter that maximizes the machinable range of the tool channel for a given impeller disk, thereby significantly improving the roughing cutting efficiency of the impeller disk.

[0006] This invention provides a method for collaboratively determining the tool dimensions and machining boundaries for milling and grooving integral bladed disks, comprising the following steps:

[0007] S1. Calculate the set of all feasible spherical centers that can pass through the bladed disk channel without interference. The bladed disk channel is bounded by the blade basin surface and the blade back surface.

[0008] S2. Establish a spherical shell two-position approximation channel surface model;

[0009] Specifically, a virtual spherical shell with the same thickness and mid-spherical radius as the spherical disc milling cutter and with uniform thickness is defined. The virtual spherical shell is placed at the first sphere center position so that its inner surface approximates the blade back surface, and at the same time, the virtual spherical shell is placed at the second sphere center position so that its outer surface approximates the blade basin surface. The goal is to minimize the sum of the errors of the two approximations, and a dual-position approximation channel surface model of the spherical shell is established.

[0010] S3. Determine the residual volume model of the channel after the virtual spherical shell approximates the bladed disk channel surface based on the spherical shell double-position approximation channel surface model.

[0011] S4. Under the condition that both the first and second sphere centers are located in the set of feasible spherical centers, establish and solve a collaborative optimization model of tool size and machining boundary oriented towards the maximum material removal volume based on the channel residual volume model.

[0012] The collaborative optimization model for tool size and machining range for maximizing material removal volume is as follows:

[0013]

[0014] In the formula, V res (q P ,q S ,ρ,r) represents the residual volume of the channel after the virtual spherical shell approximates the channel surface in two positions, q P q represents the first center of the virtual spherical shell. S Let represent the second center of the virtual spherical shell, ρ represent the mid-spherical radius of the ball end mill, r represent half the thickness of the ball end mill, Q represent the set of feasible spherical centers, and α represent the non-interference coefficient between the ball end mill and the channel, satisfying RR. s =αL, 1.2≤α≤2, R represents the radius of the ball end mill, R s Let L represent the radius of the ball end mill shank, L represent the radial depth of the impeller channel, and λ represent the rigidity coefficient of the ball end mill shank, satisfying RR. s =2λr, 11≤λ≤13, E max E represents the maximum of the minimum distances between the ball end mill body and the first or second ball center. min This represents the minimum of the maximum distance from the ball end mill body to the first sphere center or the maximum distance to the second sphere center.

[0015] In one embodiment, step S1 includes the following steps:

[0016] S11. Search for an initial point in the three-dimensional space where the bladed disk is located that satisfies the following formula;

[0017] max{|S P -q|}=min{|S S -q|}

[0018] In the formula, S P S represents the curved surface of the leaf back. S Let q represent the blade-plate surface, and let q represent the center of a feasible sphere that passes through the blade-plate channel without interference. The radius of q is e.

[0019] S12. Starting from the radius of the sphere at the initial point, the radius of the sphere is expanded step by step, and the lines of equal radius corresponding to different sphere radii are solved layer by layer.

[0020] The upper limit of the spherical radius step expansion is the product of the radial depth of the impeller channel and the preset maximum magnification;

[0021] S13. The three-dimensional space enclosed by the lines of equal radius obtained by solving for all different spherical radii is taken as the set of feasible spherical centers that can pass through the bladed disk channel without interference.

[0022] In one embodiment, the spherical shell dual-approximation channel surface model is as follows:

[0023] E = E P ∩E S

[0024]

[0025] in,

[0026]

[0027] In the formula, E = [E min E max ], E P The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P The distance, E S The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The distance, E P,min The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P The minimum distance, E P,max The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P Maximum distance, E S,min The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The minimum distance, E S,max The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The maximum distance.

[0028] In one embodiment, the channel residual volume model is as follows:

[0029]

[0030] In the formula, Ο P (q P ,ρ-r) represents the virtual spherical shell in q P The inner sphere at point O S (q S ,ρ+r) represents the virtual spherical shell in q S The outer spherical surface at that location.

[0031] In one embodiment, in step S12, for the spherical radius of the initial point and each spherical radius obtained by step expansion, a loop of equal radius lines is continuously traced by the tracking method.

[0032] The beneficial effects of the present invention are: the method of the present invention uses a virtual spherical shell to represent the spherical disc cutter, the thickness and mid-spherical radius of the virtual spherical shell are equal to those of the spherical disc cutter, and the thickness and mid-spherical radius of the virtual spherical shell are calculated to calculate the thickness and mid-spherical radius of the spherical disc cutter.

[0033] This invention establishes a residual volume model for the impeller channel by defining a spherical shell dual-position approximation channel surface model. After the virtual spherical shell dual-position approximation of the impeller channel surface, the residual volume of the impeller channel during rough machining is minimized, and the machinable range of the channel is maximized. Based on this, the tool size parameters and machining boundaries are integrated into a unified collaborative optimization framework to establish a collaborative optimization model for tool size and machining range oriented towards maximizing material removal volume. This model aims to minimize the residual volume. It defines the position of the virtual spherical shell, i.e., the position of the ball end mill, through a feasible spherical center set and the spherical shell dual-position approximation channel surface model. The non-interference coefficient between the ball end mill and the channel and the rigidity coefficient of the ball end mill are used as limiting conditions for the thickness of the ball end mill. Through optimization calculations, the tool size and machining boundary of the ball end mill with high rigidity under the maximum machinable range of the tool channel can be determined, thereby significantly improving the rough machining efficiency of the impeller. Attached Figure Description

[0034] Figure 1 Schematic diagram of a disc milling cutter machining an impeller;

[0035] Figure 2 This is a schematic diagram of a spherical disc milling cutter structure placed inside the impeller channel.

[0036] Figure 3 This is a flowchart illustrating the method for collaboratively determining the tool size and machining boundary of an integral bladed disk / spherical disk milling and grooving process according to an embodiment of the present invention.

[0037] Figure 4 A schematic diagram of the curved surface of the two-position approximation channel for a spherical shell;

[0038] Figure 5 A schematic diagram showing the range of tool shape parameters when the spherical shell is used to approximate the channel surface.

[0039] Figure 6 This is a schematic diagram of SFC;

[0040] Figure 7 A schematic diagram of the rough milling groove range for a certain type of open integral bladed disk channel ball disc;

[0041] Figure 8 This invention provides an optimal ball end mill shape for embodiments of the invention. Detailed Implementation

[0042] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0043] like Figure 2 As shown, the bladed disk channel is formed by the convex curved surface S on the back of the blade. P Concave leaf basin curved surface S S and the rotating hub surface S H The structure of a spherical disc milling cutter includes a cutter head base (i.e., an outer spherical surface, an inner spherical surface, and a virtual middle spherical surface, all three spheres having the same center point), a circular arc-shaped cutting edge on the disc edge, and a cylindrical tool holder. Its main structural parameters include the radius ρ of the middle spherical surface of the cutter head, the thickness 2r of the cutter head, the radius R of the cutter head, and the radius R of the tool holder. s .

[0044] The method of this invention uses a virtual spherical shell to represent a spherical disc milling cutter. By optimizing the calculation of the thickness and mid-spherical radius of the virtual spherical shell, the thickness and mid-spherical radius of the calculated spherical disc milling cutter are determined, and the machining boundary is further defined. A systematic and universal method for determining tool parameters and machining boundaries is established, which can effectively improve the roughing efficiency of integral impellers.

[0045] In one embodiment, such as Figure 3 As shown, the method for collaboratively determining the tool dimensions and machining boundaries for milling and grooving of an integral bladed disk includes the following steps:

[0046] S1. Calculate the set of all feasible spherical centers that can pass through the bladed disk channel without interference. The bladed disk channel is bounded by the blade basin surface and the blade back surface. The calculation of the range of spherical centers that can pass through the channel without interference is to reduce the amount of computation required for the virtual spherical shell position in the subsequent method.

[0047] S2. Establish a spherical shell two-position approximation channel surface model;

[0048] Specifically, a virtual spherical shell with the same thickness and mid-spherical radius as the spherical end mill is defined. The virtual spherical shell is placed at the first center position so that its inner surface approximates the blade back surface, and at the same time, it is placed at the second center position so that its outer surface approximates the blade basin surface. The goal is to minimize the sum of the errors of the two approximations, and a dual-position approximation channel surface model of the spherical shell is established.

[0049] like Figure 4 As shown, to achieve the goal of maximizing the machinable range of the channel of the spherical disc milling cutter, this embodiment proposes a spherical shell dual-position approximation channel surface strategy, that is, using a spherical shell of uniform thickness (shape parameters include the mid-spherical radius ρ and the cutter head thickness 2r) at two different spherical center positions q. P and q S Using the inner spherical surface of the spherical shell respectively P Approaching S P and the outer spherical surface of the spherical shell O S Approaching S S This optimization makes the former's O P -S P Approximation error, the latter's O S -S S Minimizing the sum of approximation errors ensures the maximum machinable range of the channel, at which point the tool size and machining boundary are optimal.

[0050] like Figure 5 As shown, for a spherical shell of a given shape (shape parameters include the spherical radius ρ and the cutter head thickness 2r) and its two different center positions q P and q S At that time, the surface model of the spherical shell two-position approximation channel is as follows:

[0051] E = E P ∩E S

[0052]

[0053] in,

[0054]

[0055] In the formula, E = [E min E max ], E P The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P The distance, E S The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The distance, E P,min The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P The minimum distance, E P,max The distance q from the cutter body to the center of the ball in a ball end mill is indicated.P Maximum distance, E S,min The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The minimum distance, E S,max The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The maximum distance.

[0056] S3. Determine the residual volume model of the channel after the virtual spherical shell approximates the bladed disk channel surface based on the spherical shell double-position approximation channel surface model.

[0057] In the spherical shell two-position approximation channel surface model and Figure 5 Based on this, we clearly have the following formula:

[0058] [ρ-r,ρ+r]∈E

[0059] Then ρ and r satisfy:

[0060]

[0061] Specifically, when r takes its maximum value, we have:

[0062]

[0063] Therefore, the residual volume model of the channel can be established as follows:

[0064]

[0065] In the formula, Ο P (q P ,ρ-r) represents the virtual spherical shell in q P The inner sphere at point O S (q S ,ρ+r) represents the virtual spherical shell in q S The outer spherical surface at that location.

[0066] When V res (q P ,q S When ,ρ,r) is at its minimum, the maximum material removal volume is achieved.

[0067] S4. Under the condition that both the first and second sphere centers are located in the set of feasible spherical centers, establish and solve a collaborative optimization model of tool size and machining boundary oriented towards the maximum material removal volume based on the channel residual volume model.

[0068] The collaborative optimization model for tool size and machining range for maximizing material removal volume is as follows:

[0069]

[0070] In the formula, V res (qP ,q S ,ρ,r) represents the residual volume of the channel after the virtual spherical shell approximates the channel surface in two positions, q P q represents the first center of the virtual spherical shell. S Let represent the second center of the virtual spherical shell, ρ represent the mid-spherical radius of the ball end mill, r represent half the thickness of the ball end mill, Q represent the set of feasible spherical centers, and α represent the non-interference coefficient between the ball end mill and the channel, satisfying RR. s =αL, 1.2≤α≤2, R represents the radius of the ball end mill, R s Let L represent the radius of the ball end mill shank, L represent the radial depth of the impeller channel, and λ represent the rigidity coefficient of the ball end mill shank, satisfying RR. s =2λr, 11≤λ≤13, E max E represents the maximum of the minimum distances between the ball end mill body and the first or second ball center. min This represents the minimum of the maximum distance from the ball end mill body to the first sphere center or the maximum distance to the second sphere center.

[0071] In this collaborative optimization model for tool size and machining range aimed at maximizing material removal volume, to ensure sufficient strength at the root of the disc milling cutter,

[0072] R s =μR

[0073] Generally, 0.25 ≤ μ < 1. The larger μ is, the better the rigidity of the cutter head. μ should be reasonably determined based on experience.

[0074] To ensure that the ball end mill does not interfere with the channel, let

[0075] RR s =αL

[0076] In the formula, 1.2≤α≤2. The larger α is, the less likely the ball end mill will interfere with the channel. α is reasonably determined based on experience.

[0077] Combining the two equations above, we can obtain:

[0078]

[0079] To ensure the rigidity of the thin-blade cutter head at high cutting speeds,

[0080]

[0081] λ is usually taken as 12. The coefficient λ is involved in the evaluation of the rigidity of complex disc milling tools. Here, only a simplified empirical model that is easy to operate in industrial production is given. In addition, the finite element method can be combined to give more accurate tool rigidity constraints.

[0082] The thickness of the cutter head must be less than or equal to the narrowest channel width E. max -E min ,Right now

[0083]

[0084] In conclusion, we get:

[0085]

[0086] In the above optimization model, the optimization variable is (q) P ,q S The optimization results include the optimal ball end mill dimensions (i.e., ρ and r) and the optimal approximation of the two-channel surface S by the tool. P and S S The position of the center of the ball at time (i.e., q) P and q S ), Maximum machining range of ball end mill groove cutting (boundary by q) P The inner spherical surface at the point O P (q P ,ρ-r) and q S The outer spherical surface at the location O S (q S (ρ+r) is determined). This optimization problem contains S P The three spatial coordinates, S S This implementation addresses the optimization problem involving three spatial coordinates, ρ, and r, totaling eight parameters. Initial uniform sampling is performed within Q, and a parallel algorithm is used to obtain an initial optimization solution. Using this initial value, the intelligent algorithm PSO (Particle Swarm Optimization) is employed for a refined solution.

[0087] It should be noted that the radius of most circular cutting blades used in industry is a standard size, meaning that r can only take certain standard values, such as r * At this point, let r = r * The aforementioned collaborative optimization model for tool size and machining range for the maximum material removal volume becomes an optimization problem with 7 parameters. The optimization results corresponding to the discrete r values ​​of all standards are obtained respectively. Among these finite sets of schemes, the scheme with the smallest residual volume of the groove is selected. The method of this invention is to adapt the optimal ball end mill tool size and corresponding machining boundary to the given shape of the impeller channel to be machined.

[0088] This invention solves the technical challenge of determining the tool size and machining boundary of a spherical disc end mill adapted to a given integral impeller channel shape. Using a spherical shell dual-position approximation channel surface model as the core, it integrates tool size parameters and machining boundaries into a unified collaborative optimization framework. Within the feasible solution space, it simultaneously optimizes key parameters such as the spherical disc end mill's center position and tool radius, ensuring maximum material removal volume while meeting interference-free constraints. This achieves efficient grooving of the integral impeller twisted channel using a spherical disc end mill, significantly improving the overall roughing efficiency of the channel compared to traditional slender end mills and straight disc end mills, demonstrating significant engineering practical value.

[0089] To reduce the computational cost of solving the above optimization problem, we first solve for all channels that can pass through the blade without interference (i.e., the blade back surface S). P Leaf basin curved surface S S The set of centers of feasible spheres is called SFC.

[0090] In one embodiment, step S1 includes the following steps:

[0091] S11. Search for an initial point in the three-dimensional space where the bladed disk is located that satisfies the following formula;

[0092] max{|S P -q|}=min{|S S -q|}

[0093] In the formula, S P S represents the curved surface of the leaf back. S Let q represent the blade-plate surface, and let q represent the center of a feasible sphere that passes through the blade-plate channel without interference. The radius of q is e.

[0094] like Figure 6 As shown, SFC is usually conical. Let q be the center of a feasible sphere that passes through the channel without interference, with radius e. The corresponding spherical representation is O(q,e). Then the set Q of all q (whose boundary is...) This is SFC. It consists of a series of iso-e lines, and The above equation applies to any point q.

[0095] Let the initial point be q e0 Its spherical radius is e0.

[0096] S12. Starting from the radius of the sphere at the initial point, the radius of the sphere is expanded step by step, and the lines of equal radius corresponding to different sphere radii are solved layer by layer.

[0097] In step S12, for each spherical radius obtained from the initial point and the step expansion, a loop of equal radii is continuously traced using a tracking method to form a line. For example, q... e0Starting from the positive direction, q is iteratively calculated along both positive and negative directions. e0 The isoline it is on is e, i.e., e = e0.

[0098] Specifically, the upper limit of the spherical radius step expansion is the product of the radial depth of the impeller channel and the preset maximum magnification. That is, the maximum value of e can be set to λ. MAX ·L, where L is the depth of the impeller channel, λ MAX λ represents the maximum magnification. MAX The value of can be, but is not limited to, 28 to 32.

[0099] When the radius of the sphere is expanded step by step, the search for adjacent iso-e lines is performed by gradually expanding from the iso-e line e = e0 to both sides, i.e., e = e0 ± Δe, Δ e Δ represents the extent of the sphere's radius expansion. e The value of is not limited to 4mm to 6mm.

[0100] S13. The three-dimensional space enclosed by the lines of equal radius obtained from solving for all different spherical radii is taken as the set of feasible spherical centers that can pass through the bladed disk channel without interference. All lines of equal radius jointly define the boundary. The three-dimensional space it encloses is Q, which is the SFC obtained.

[0101] SFC contains all feasible ball centers and provides a basic solution space for optimizing the ball center position of a ball end mill.

[0102] In a specific embodiment, taking the rough milling and grooving of a certain type of open integral bladed disk channel ball disk as an example, the specific implementation of the present invention is described. Figure 7 As shown, the radial depth L of the impeller channel is approximately 42.603 mm, the channel width ranges from approximately 13.263 mm to 18.272 mm (average width approximately 15.138 mm), and the blade width is approximately 31.901 mm. When calculating SFC, Δ... e It is 5mm, λ MAX Set the value to 30.

[0103] For the current part model, a collaborative optimization model for tool size and machining range is established to maximize material removal volume. Since the narrowest channel width is only about 13.263mm, the only selectable standard industrial circular insert diameters (2r) are 8mm, 10mm, and 12mm. The r parameter for each of the three sizes is substituted into the model for optimization. Comparison shows that the scheme with r = 4mm removes the most material. Finally, the optimal ball end mill parameters obtained using this invention are: r = 4mm, ρ = 506.838mm, R = 85.206mm, R... s =29.822mm, q P The coordinates are (311.3647 -395.7488 -350.3144), q SThe coordinates are (361.2613 -356.3650 -381.5401), obtained via q. P q S ρr and ρr can determine the machining boundary. The tool profile shape is as follows: Figure 8 As shown, the material removal percentage of the channel reaches 71.39% (its maximum processing range is referenced). Figure 7 As shown in the figure, the average height of the residual material on both sides of the channel is only 2.381 mm.

[0104] In comparison, the material removal percentage (MRP) of a conventional disc milling cutter is only 55.42%. The machinable range of the channel in this invention is increased by 28.82% compared to the conventional disc milling cutter. Under the same cutting parameters, the cutting time of this invention can be reduced by 28.82%, proving that the tool size and machining boundary determined by the method of this invention can effectively improve the cutting efficiency when cutting the entire impeller.

[0105] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the appended claims.

Claims

1. A method for collaboratively determining the dimensions of a milling tool and the machining boundary for grooving an integral bladed disk, characterized in that, Includes the following steps: S1. Calculate the set of all feasible spherical centers that can pass through the bladed disk channel without interference, wherein the bladed disk channel is enclosed by the blade basin surface and the blade back surface; S2. Establish a spherical shell two-position approximation channel surface model; Specifically, a virtual spherical shell with the same thickness and mid-spherical radius as the spherical end mill is defined, and has a uniform thickness. The virtual spherical shell is placed at the first center position so that its inner surface approximates the blade back surface, and at the same time, the virtual spherical shell is placed at the second center position so that its outer surface approximates the blade basin surface. The goal is to minimize the sum of the errors of the two approximations, and a dual-position approximation channel surface model of the spherical shell is established. S3. Determine the channel residual volume model after the virtual spherical shell double-approximation of the bladed disk channel surface based on the spherical shell double-approximation channel surface model. S4. Under the condition that both the first and second sphere centers are located in the set of feasible spherical centers, establish and solve a collaborative optimization model of tool size and machining boundary oriented towards the maximum material removal volume based on the channel residual volume model. The collaborative optimization model for tool size and machining range oriented towards maximizing material removal volume is as follows: In the formula, V res (q P ,q S ,ρ,r) represents the residual volume of the channel after the virtual spherical shell approximates the channel surface in two positions, q P q represents the first center of the virtual spherical shell. S Let represent the second center of the virtual spherical shell, ρ represent the mid-spherical radius of the ball end mill, r represent half the thickness of the ball end mill, Q represent the set of feasible spherical centers, and α represent the non-interference coefficient between the ball end mill and the channel, satisfying RR. s =αL, 1.2≤α≤2, R represents the radius of the ball end mill, R s Let L represent the radius of the ball end mill shank, L represent the radial depth of the impeller channel, and λ represent the rigidity coefficient of the ball end mill shank, satisfying RR. s =2λr, 11≤λ≤13, E max E represents the maximum of the minimum distances between the ball end mill body and the first or second ball center. min This represents the minimum of the maximum distance from the ball end mill body to the first sphere center or the maximum distance to the second sphere center.

2. The method for collaboratively determining the dimensions of the milling tool and the machining boundary for integral bladed disk spherical disc milling according to claim 1, characterized in that, Step S1 includes the following steps: S11. Search for an initial point in the three-dimensional space where the bladed disk is located that satisfies the following formula; max{|S P -q|}=min{|S S -q|} In the formula, S P S represents the curved surface of the leaf back. S Let q represent the blade-plate surface, and let q represent the center of a feasible sphere that passes through the blade-plate channel without interference. The radius of q is e. S12. Starting from the spherical radius of the initial point, the spherical radius is expanded step by step, and the lines of equal radius corresponding to different spherical radii are solved layer by layer. The upper limit of the step expansion of the spherical radius is the product of the radial depth of the bladed disk channel and the preset maximum magnification. S13. The three-dimensional space enclosed by the lines of equal radius obtained by solving for all different spherical radii is taken as the set of feasible spherical centers that can pass through the bladed disk channel without interference.

3. The method for collaboratively determining the dimensions of the milling tool and the machining boundary for integral bladed disk spherical disk milling according to claim 2, characterized in that, The spherical shell dual-position approximation channel surface model is as follows: E=E P ∩E S in, In the formula, E = [E min E max ], E P The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P The distance, E S The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The distance, E P,min The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P The minimum distance, E P,max The distance q from the cutter body to the center of the ball in a ball end mill is indicated. P Maximum distance, E S,min The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The minimum distance, E S,max The distance q from the cutter body to the center of the ball in a ball end mill is indicated. S The maximum distance.

4. The method for collaboratively determining the dimensions of the milling tool and the machining boundary for integral bladed disk spherical disc milling according to claim 3, characterized in that, The residual volume model of the channel is as follows: In the formula, Ο P (q P ,ρ-r) represents the virtual spherical shell in q P The inner sphere at point O S (q S ,ρ+r) represents the virtual spherical shell in q S The outer spherical surface at that location.

5. The method for collaboratively determining the dimensions of the milling tool and the machining boundary for integral bladed disk spherical disk milling according to claim 2, characterized in that, In step S12, for the spherical radius of the initial point and each spherical radius obtained by step expansion, a loop of equal radius lines is continuously traced by the tracking method.

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