CN113094964A - Method and device for generating blade machining coordinates - Google Patents

Method and device for generating blade machining coordinates Download PDF

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CN113094964A
CN113094964A CN202110640447.5A CN202110640447A CN113094964A CN 113094964 A CN113094964 A CN 113094964A CN 202110640447 A CN202110640447 A CN 202110640447A CN 113094964 A CN113094964 A CN 113094964A
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blade
cold
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CN113094964B (en
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刘天一
曹传军
王进春
李游
吴志青
陆晓锋
王家广
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AECC Commercial Aircraft Engine Co Ltd
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Abstract

The invention provides a method for generating blade machining coordinates, which comprises the following steps: obtaining a cold-state blade model of a blade; generating a first cold-state leaf profile by adopting a grid interpolation method; generating a second cold-state blade profile by adopting a modeling approximation method; intercepting coordinate points of the front edge and the tail edge of the blade from the second cold-state blade profile, fitting the second cold-state blade profile with the first cold-state blade profile, and replacing corresponding coordinate points in the first cold-state blade profile with the updated coordinate points to obtain a third cold-state blade profile; and smoothing the junction of the front edge and the tail edge of the third cold-state blade profile and the blade body respectively to obtain a final cold-state blade profile, wherein the coordinate point of the final cold-state blade profile is the final machining coordinate of the blade. The invention also comprises a device for generating blade machining coordinates, a storage medium and computer equipment. The method for generating the blade processing coordinate generates the blade processing coordinate on the premise of not increasing the finite element analysis grid quantity, and ensures that the processing coordinate is consistent with the finite element analysis result and the local areas of the front edge and the tail edge have correct shapes and smoothness.

Description

Method and device for generating blade machining coordinates
Technical Field
The invention relates to the field of blade machining, in particular to a method, a storage medium, computer equipment and a device for generating blade machining coordinates.
Background
When the pneumatic design of the turbine blade is carried out, the pneumatic characteristics of the blade under the design working condition are generally examined, namely the pneumatic performance of the blade when the blade bears thermal load and pneumatic load, and the designed blade shape is called as a thermal state blade profile; the shape of the blades of the turbomachinery when not in operation, i.e. when there is no thermal load and no aerodynamic load, is called the cold profile. Due to the action of thermal load and pneumatic load, the blade can generate certain deformation, and the difference exists between the hot blade profile and the cold blade profile, and the difference is more obvious in the impeller blade with high pressure and high load (such as a multistage axial flow compressor). The production blade profile used in production and processing is a cold blade profile under the condition of no thermal load and no aerodynamic load, and the cold blade profile is generally obtained by applying finite element calculation software based on the hot blade profile.
There are two currently popular methods of generating production coordinates:
1) modeling approach method: namely, blade modeling software is used for reproducing a cold finite element model obtained by cold-heat conversion. In the process, the input parameters of the iterative blade modeling are required to be continuously adjusted, so that the modeling result gradually approaches to the cold finite element model. This method is suitable for blades with simple deformation rules or small deformation amounts, such as stator blades and booster stage blades. However, for blades with complex deformation rules and large deformation amount, such as high-pressure compressor rotor blades, the method is often difficult to reproduce cold finite element models accurately enough.
2) Grid interpolation method: and (3) interpolating to generate a blade production coordinate according to the cold grid node coordinate obtained by finite element analysis. Chinese patent CN201510915701.2 provides a typical implementation of this method. The method can directly carry out processing such as sequencing, interpolation and the like on the cold-state leaf finite element grid nodes generated after the intensity hot-to-cold calculation, thereby obtaining the leaf definition file meeting the processing requirements. The mechanism of the method determines that no matter how complex the deformation rule of the blade is, the production coordinate can be smoothly output, and the production coordinate is highly consistent with the finite element analysis result. However, in order to define the shape of the leading edge and the trailing edge of the blade profile accurately enough by the cold finite element nodes, the grid density (as shown in fig. 1 a) used in the conventional finite element analysis is not enough, and the grid density near the leading edge and the trailing edge must be greatly encrypted (as shown in fig. 1 b), and when the grid density is encrypted, the grid along the height direction of the blade is also synchronously encrypted in order to meet the requirement of the length-width ratio of the grid, and finally, a high-density grid as shown in fig. 1c is formed, so that the intensity analysis time is greatly increased.
Therefore, there is a need for a method for generating blade machining coordinates, which ensures that the machining coordinates are consistent with the finite element analysis results and the local areas of the leading edge and the trailing edge have correct shapes and smoothness without increasing the amount of the cold-hot deformation finite element analysis grids of the blade.
Disclosure of Invention
The invention aims to provide a method and a device for generating blade machining coordinates, which can ensure that the machining coordinates are consistent with the finite element analysis result and the local areas of the front edge and the tail edge have correct shapes and smoothness under the condition of not increasing the cold and hot deformation finite element analysis grid quantity of a blade.
In order to solve the technical problem, the invention provides a method for generating blade machining coordinates, which comprises the following steps:
1) obtaining a cold state blade model of the blade, wherein the cold state blade model is obtained from a hot state blade profile through a hot-to-cold finite element analysis;
2) acquiring cold grid coordinate points of the cold blade model, and generating a series of elementary blade profiles on the equal height surface by adopting a grid interpolation method to form a first cold blade profile;
3) adopting a modeling approach method, wherein the modeling approach method comprises the following steps: determining a plurality of modeling conical surfaces based on the finite element analysis result, generating element blade profiles on the modeling conical surfaces, and generating a series of element blade profiles on equal-height surfaces according to element blade profile interpolation on the modeling conical surfaces to form a second cold-state blade profile, wherein the height of each equal-height surface is the same as that of each corresponding equal-height surface in the grid interpolation method in the step 2);
4) intercepting coordinate points of a leading edge and a trailing edge from the second cold-state blade profile, fitting the coordinate points with the first cold-state blade profile to obtain an updated leading edge coordinate point and an updated trailing edge coordinate point, and replacing the corresponding coordinate points of the leading edge and the trailing edge in the first cold-state blade profile with the updated leading edge coordinate point and the updated trailing edge coordinate point to obtain a third cold-state blade profile;
5) and smoothing the junction of the front edge and the tail edge of the third cold-state blade profile and the blade body respectively to obtain a final cold-state blade profile, wherein the coordinate point of the final cold-state blade profile is the final machining coordinate of the blade.
Further, the method of determining a plurality of contouring cones and generating primitive leaves on the plurality of contouring cones in step 3) comprises the steps of:
3.1) based on the finite element analysis result, selecting two points A1 and A2 of the blade tip and two points C1 and C2 of the blade root on the hot blade profile, selecting a plurality of groups of B1 and B2 which comprise two points and are positioned between the blade tip and the blade root, determining the points A '1, A' 2, C '1 and C' 2 corresponding to A1, A2, C1 and C2 on the cold blade model, and measuring the radial displacement of the points A1, A2, C1 and C2 and the corresponding points A '1, A' 2, C '1 and C' 2 in the cold blade model
Figure DEST_PATH_IMAGE002
And amount of axial displacement
Figure DEST_PATH_IMAGE004
；
For each group B1 and B2, respectively calculating the radial displacement of B '1 and B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure DEST_PATH_IMAGE006
：
Figure DEST_PATH_IMAGE008
（1）
Figure DEST_PATH_IMAGE010
（2）
Figure DEST_PATH_IMAGE012
The leaf height at B1 is the height of the leaf,
Figure DEST_PATH_IMAGE014
(ii) a leaf height of said A1 to said C1;
Figure DEST_PATH_IMAGE016
the leaf height at B2 is the height of the leaf,
Figure DEST_PATH_IMAGE018
(ii) a leaf height of said A2 to said C2;
for each group B1 and B2, respectively calculating the axial displacement quantity of the B '1 and the B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure DEST_PATH_IMAGE020
：
Figure DEST_PATH_IMAGE022
（3）
Figure DEST_PATH_IMAGE024
（4）；
3.2) according to said
Figure DEST_PATH_IMAGE026
Obtaining the positions of B '1-B' 2 modeling conical surfaces corresponding to the B1-B2 modeling conical surfaces in the cold-state blade model; elementary leaf shapes are generated on the B '1-B' 2 contouring cone using the same elementary leaf shape parameters as in the hot-state leaf shape.
Further, the method for smoothing in step 5) comprises:
5.1) selecting two control points at the transition of the profile of the leading edge portion and the profile of the blade body portion, respectively denoted D0And D;
5.2) from said D0Selecting a plurality of control points along a first direction away from the blade body, wherein the control points are away from the blade body D0The farthest control point is denoted as E0And said E0Adjacent control points are denoted as F0Selecting a plurality of control points from the D along a second direction opposite to the first direction, wherein the control point farthest from the D is marked as E, and the control point F adjacent to the E;
5.3) taking the intermediate control point G0And G, line segment E0G0And the line segment EG is respectively on the line segment E0F0And line segment EF;
5.4) according to said E0、E、G0And G, making a cubic Bezier curve;
5.5) taking a plurality of points on the cubic Bessel curve, the density of the points being similar to that of the adjacent control points, replacing the E with a plurality of the points0And E is the original coordinate point;
step 5) also comprises the following steps: two control points at the transition of the profile of the trailing edge portion and the profile of the blade body portion are selected and respectively marked as H0And H with said H0And H replaces said D in step 5.1), respectively0And D, and carrying out steps 5.1) to5.4) to smooth the junction of the trailing edge and the blade body.
Further, the control point E0The straight-line distance to the control point E is denoted as E0E, the control point D0The straight-line distance to the control point D is denoted as D0D, said E0E is at least the number D030 times of D.
Further, the control point D0To the control point E0The straight-line distance of (D) is recorded as0E0The distance from the control point D to the control point E is denoted DE, and the line segment E0G0And the line segment EG is respectively D0E00.82 times the DE.
Further, in the step 4), the ranges of the coordinate points of the intercepted leading edge and the coordinate points of the intercepted trailing edge are respectively a range of the chord length of the blade from the leading edge point by 4% -15% and a range of the chord length of the blade from the trailing edge point by 4% -15%.
Further, the blade is a high-pressure compressor rotor blade, and the ranges of the intercepted leading edge coordinate point and the intercepted trailing edge coordinate point are respectively a range of the blade chord length 5% away from the leading edge point and a range of the blade chord length 5% away from the trailing edge point.
Further, in step 4), a least square method is adopted for fitting, and when the sum of squares of distances between each coordinate point in the intercepted second cold-state blade profile and the corresponding point of the first cold-state blade profile is obtained
Figure DEST_PATH_IMAGE028
And (5) finishing fitting in a minimum mode.
Further, selecting to give different weights w to the coordinate points at different positions, and ordering
Figure DEST_PATH_IMAGE030
And when the minimum value is reached, the fitting is completed.
Further, the weight w is half of the sum of the distances between each coordinate point in the truncated second cold-state leaf form and two adjacent coordinate points.
The present invention also provides a storage medium storing a computer program which, when running, executes the above-described method of generating blade machining coordinates.
The invention also provides a computer apparatus comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor when executing the program implementing a method of generating blade machining coordinates as described above.
The invention also provides a device for generating blade machining coordinates, which comprises:
an obtaining module for obtaining a cold blade model of the blade, the cold blade model being obtained from a hot blade profile by a hot-to-cold finite element analysis;
the grid interpolation module is used for acquiring cold grid coordinate points of the cold blade model and generating a series of elementary blade profiles on an equal height surface by adopting a grid interpolation method to form a first cold blade profile;
a modeling approximation module for employing a modeling approximation, the modeling approximation comprising: determining a plurality of modeling conical surfaces based on the finite element analysis result, generating element blade profiles on the modeling conical surfaces, and generating a series of element blade profiles on the equal-height surfaces according to element blade profile interpolation on the modeling conical surfaces to form a second cold-state blade profile, wherein the heights of the equal-height surfaces are the same as the heights of the corresponding equal-height surfaces in the grid interpolation method;
the fitting module is used for intercepting coordinate points of the leading edge and the trailing edge from the second cold-state blade profile, fitting the coordinate points with the first cold-state blade profile to obtain an updated leading edge coordinate point and an updated trailing edge coordinate point, and replacing the corresponding coordinate points of the leading edge and the trailing edge in the first cold-state blade profile with the updated leading edge coordinate point and the updated trailing edge coordinate point to obtain a third cold-state blade profile;
and the blade profile fairing module is used for fairing the junction of the front edge and the tail edge of the third cold-state blade profile and the blade body respectively so as to obtain a final cold-state blade profile, and the coordinate point of the final cold-state blade profile is the final machining coordinate of the blade.
Further, the contouring approximation module employs the following steps to determine the plurality of contouring cones and generate elementary lobes on the contouring cones:
14.1) based on the cold-state blade model, selecting two points A1 and A2 of a blade tip and two points C1 and C2 of a blade root on the hot-state blade profile, selecting multiple groups of B1 and B2 between the blade tip and the blade root, determining points A '1, A' 2, C '1 and C' 2 corresponding to the points A1, A2, C1 and C2 on the cold-state blade model, and measuring the radial displacement of the points A '1, A' 2, C '1 and C' 2 corresponding to the points A '1, A' 2, C '1 and C' 2 in the cold-state blade model to obtain the radial displacement of the points A1, A2, C1 and C2
Figure DEST_PATH_IMAGE031
And amount of axial displacement
Figure 695091DEST_PATH_IMAGE004
；
For each group B1 and B2, respectively calculating the radial displacement of B '1 and B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure 403021DEST_PATH_IMAGE006
：
Figure DEST_PATH_IMAGE032
（1）
Figure DEST_PATH_IMAGE033
（2）
Figure 807458DEST_PATH_IMAGE012
The leaf height at B1 is the height of the leaf,
Figure 735094DEST_PATH_IMAGE014
(ii) a leaf height of said A1 to said C1;
Figure 453651DEST_PATH_IMAGE016
the leaf height at B2 is the height of the leaf,
Figure 32400DEST_PATH_IMAGE018
(ii) a leaf height of said A2 to said C2;
for each group B1 and B2, respectively calculating and obtaining the axial displacement quantity of B '1 and B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure 919322DEST_PATH_IMAGE020
：
Figure DEST_PATH_IMAGE034
（3）
Figure DEST_PATH_IMAGE035
（4）；
14.2) according to said
Figure DEST_PATH_IMAGE037
Obtaining the positions of B '1-B' 2 modeling conical surfaces corresponding to the B1-B2 modeling conical surfaces in the cold-state blade model; elementary leaf shapes are generated on the B '1-B' 2 contouring cone using the same elementary leaf shape parameters as in the hot-state leaf shape.
Further, the blade profile fairing module adopts the following steps to fairing the junction of the leading edge and the blade body of the third cold-state blade profile:
15.1) selecting two control points at the profile transition of the leading edge portion and the profile of the blade body portion, respectively denoted D0And D;
15.2) from said D0Selecting a plurality of control points along a first direction away from the blade body, wherein the control points are away from the blade body D0The farthest control point is denoted as E0And said E0Adjacent control points are denoted as F0Selecting a plurality of control points from the D in a second direction opposite the first direction, wherein the control furthest from the DPoint is denoted as E, and control point F adjacent to E;
15.3) taking the intermediate control point G0And G, line segment E0G0And the line segment EG is respectively on the line segment E0F0And line segment EF;
15.4) according to said E0、E、G0And G, making a cubic Bezier curve;
15.5) taking a plurality of points on said cubic Bessel curve, said points having a density similar to that of said adjacent control points, replacing said E with a plurality of said points0And E is the original coordinate point;
the steps further include: two control points at the transition of the profile of the trailing edge portion and the profile of the blade body portion are selected and respectively marked as H0And H with said H0And H replaces said D in step 15.1), respectively0And D) carrying out the steps 15.1) to 15.4) to smooth the junction of the trailing edge and the blade body.
Further, the control point E0The straight-line distance to the control point E is denoted as E0E, the control point D0The straight-line distance to the control point D is denoted as D0D, said E0E is at least the number D030 times of D;
the control point D0To the control point E0The straight-line distance of (D) is recorded as0E0The distance from the control point D to the control point E is denoted DE, and the line segment E0G0And the line segment EG is respectively D0E00.82 times the DE.
Further, the fitting module intercepts the coordinate points of the leading edge and the trailing edge within a range of 4% -15% of the chord length of the blade from the leading edge point and a range of 4% -15% of the chord length of the blade from the trailing edge point respectively.
Further, the blade is a high-pressure compressor rotor blade, and the ranges of the coordinate points of the intercepted leading edge and the coordinate points of the intercepted trailing edge are respectively a range of the chord length of the blade 5% away from the leading edge point and a range of the chord length of the blade 5% away from the trailing edge point.
Further, the fitting module adopts least square fitting, and when the sum of squares of distances between each coordinate point in the intercepted second cold-state blade profile and the corresponding point of the first cold-state blade profile is greater than the sum of squares of distances between the coordinate points and the corresponding points of the first cold-state blade profile
Figure 865413DEST_PATH_IMAGE028
And (5) finishing fitting in a minimum mode.
Further, the fitting module selects to give different weights w to the coordinate points at different positions, when the order is ordered
Figure 387661DEST_PATH_IMAGE030
When the minimum value is reached, the fitting is completed;
the weight w is half of the sum of the distances between each coordinate point in the intercepted second cold-state blade profile and two adjacent coordinate points.
The invention has the beneficial effects that:
1. according to the method, the device, the storage medium and the computer equipment for generating the blade machining coordinate, the blade machining coordinate is generated on the premise that the cold and hot deformation finite element analysis grid quantity of the blade is not increased, the machining coordinate is consistent with a finite element analysis result, and the local areas of the front edge and the tail edge have correct shapes and smoothness.
2. According to the method, the device, the storage medium and the computer equipment for generating the blade machining coordinates, when a set of cold-state blade profiles is obtained by adopting a modeling approximation method, only the geometric difference of the cold-state blade in the radial direction and the axial direction needs to be considered, the translation and the rotation of the element blade profiles in the circumferential direction do not need to be considered, and the implementation difficulty is reduced.
Drawings
The foregoing summary, as well as the following detailed description of the invention, will be better understood when read in conjunction with the appended drawings. It is to be noted that the figures are only intended as examples of the claimed solution. In the drawings, like reference characters designate the same or similar elements.
FIG. 1a is a schematic diagram of a hot to cold finite element analysis mesh meeting conventional finite element analysis criteria;
FIG. 1b is a schematic diagram of a hot to cold finite element analysis mesh meeting aerodynamic profile definition requirements;
FIG. 1c is a schematic diagram of a hot-to-cold finite element analysis mesh meeting conventional finite element analysis criteria and aerodynamic profile definition requirements;
FIG. 2 is a flow chart of a method of generating blade machining coordinates according to one embodiment of the present invention;
FIG. 3 is a flow chart of a method of determining a contouring cone and generating a second cold profile element profile in accordance with one embodiment of the present invention;
FIG. 4 is a flow chart of a fairing method of an embodiment of the invention;
FIG. 5a is a diagram of a cold grid coordinate point matrix according to an embodiment of the present invention;
FIG. 5b is an enlarged schematic view of the cold grid coordinate points at the leading/trailing edge of an embodiment of the present invention;
FIG. 5c is a schematic view of a non-isosurface primitive leaf shape and an isosurface primitive leaf shape according to an embodiment of the present invention;
FIG. 6 is a schematic diagram of solving for the location of the molding cone corresponding to the hot airfoil on the cold airfoil model in accordance with one embodiment of the present invention;
FIG. 7 is a comparative schematic diagram of the effect of intercepting different coordinate point ranges for fitting on the leading edge fit results;
FIG. 8 is a schematic illustration of the leading/trailing edge of a first cold blade profile fitting to the leading/trailing edge of a second cold blade profile in accordance with an embodiment of the present invention;
FIG. 9 is a schematic illustration of a third cold blade profile fairing, in accordance with an embodiment of the present invention;
FIG. 10 is a schematic illustration of a curvature comparison of Bezier curves for three different parameters;
FIG. 11 is a graphical illustration of aerodynamic comparison of machining coordinates generated by one embodiment of the present invention and machining coordinates generated using grid interpolation.
Detailed Description
The detailed features and advantages of the present invention are described in detail in the detailed description which follows, and will be sufficient for anyone skilled in the art to understand the technical content of the present invention and to implement the present invention, and the related objects and advantages of the present invention will be easily understood by those skilled in the art from the description, claims and drawings disclosed in the present specification.
In one embodiment, as shown in FIG. 2, a method of generating blade machining coordinates includes the steps of:
s110: and acquiring a cold-state blade model of the blade, wherein the cold-state blade model is obtained from the hot-state blade profile through hot-to-cold finite element analysis.
It should be noted that, when the hot-to-cold finite element analysis is performed, the amount of the adopted meshes meets the conventional analysis criteria, and the leading edge and the trailing edge portions do not need to be encrypted.
As shown in fig. 5a-5b, S120: and acquiring cold grid coordinate points of the cold blade model, and generating a series of elementary blade profiles on the equal-height surface by adopting a grid interpolation method to form a first cold blade profile.
It can be understood that, because the leading edge and the trailing edge are not required to be encrypted, the grid quantity of the whole blade is reduced by about half compared with the conventional grid interpolation method, and the blade strength analysis time is reduced.
The grid interpolation method comprises the following steps: the cold grid coordinate points on the surface of the cold blade model are derived in a mode of being sequentially arranged along the profile of the blade profile from the blade root to the blade tip to form a series of elementary blade profiles on unequal height surfaces, as shown in fig. 5c (the solid line is the elementary blade profile on the unequal height surface, the dotted line is the elementary blade profile on the equal height surface), a series of elementary blade profiles on the equal height surface are generated through interpolation according to the elementary blade profiles on the series of unequal height surfaces, and the combination of the elementary blade profiles on the equal height surface forms a first cold blade profile.
S130: adopting a modeling approach method, wherein the modeling approach method comprises the following steps: and determining a plurality of modeling conical surfaces based on the finite element analysis result, generating element blade profiles on the modeling conical surfaces, and generating a series of element blade profiles on the equal-height surface according to the interpolation of the element blade profiles on the series of modeling conical surfaces to form a second cold-state blade profile. The height of the equal-height surface in S130 is the same as that of the corresponding equal-height surface in S120.
It should be noted that the order of step S120 and step S130 may be exchanged.
S140: and intercepting the coordinate points of the leading edge and the trailing edge of the blade from the second cold-state blade profile, fitting the coordinate points with the first cold-state blade profile to obtain an updated leading edge coordinate point and an updated trailing edge coordinate point, and replacing the corresponding coordinate points of the leading edge and the trailing edge in the first cold-state blade profile by using the updated leading edge coordinate point and the updated trailing edge coordinate point to obtain a third cold-state blade profile.
S150: and smoothing the junction of the front edge and the tail edge of the third cold-state blade profile and the blade body respectively to obtain a final cold-state blade profile, wherein the coordinate point of the final cold-state blade profile is the final machining coordinate of the blade.
As shown in fig. 3 and 6, the method for determining a plurality of modeling cones and generating primitive leaves on the modeling cones in step S130 includes the following steps:
s131: based on finite element analysis results, two points A1 and A2 of the blade tip and two points C1 and C2 of the blade root are selected on the hot-state blade profile, multiple groups of B1 and B2 which comprise two points and are positioned between the blade tip and the blade root are selected, points A '1, A' 2, C '1 and C' 2 corresponding to A1, A2, C1 and C2 on the cold-state blade model are determined, and the radial displacement amount of the points A '1, A' 2, C '1 and C' 2 corresponding to A1, A2, C1 and C2 and the cold-state blade model is measured
Figure DEST_PATH_IMAGE038
And amount of axial displacement
Figure 289758DEST_PATH_IMAGE004
；
For each group B1 and B2, the radial displacement of B '1 and B' 2 corresponding to each group B1 and B2 in the cold-state blade model is calculated by adopting the following formula
Figure DEST_PATH_IMAGE039
：
Figure DEST_PATH_IMAGE040
（1）
Figure DEST_PATH_IMAGE041
（2）
Figure 494386DEST_PATH_IMAGE012
The leaf height at B1 (leaf heights B1 to C1),
Figure 255669DEST_PATH_IMAGE014
leaf height from a1 to C1;
Figure 706242DEST_PATH_IMAGE016
the leaf height at B2 (leaf heights B2 to C2),
Figure 869370DEST_PATH_IMAGE018
leaf height from a2 to C2;
for each group B1 and B2, the axial displacement quantity of B '1 and B' 2 corresponding to B1 and B2 in the cold-state blade model is respectively calculated by adopting the following formula
Figure 363674DEST_PATH_IMAGE020
：
Figure DEST_PATH_IMAGE042
（3）
Figure DEST_PATH_IMAGE043
（4）；
S132: according to
Figure 674569DEST_PATH_IMAGE026
Obtaining the positions of B '1-B' 2 modeling conical surfaces corresponding to the B1-B2 modeling conical surfaces in the cold-state blade model; the elementary leaf profiles are generated on the B '1-B' 2 sculpted cone using the same elementary leaf profile parameters as in the hot leaf profile to generate the second cold leaf profile.
It can be understood that, by adopting a linear interpolation method, based on the meridian projection plane of the cold-state blade model, only the radial and axial differences between the hot-state blade profile and the cold-state blade model need to be calculated, and on this basis, the elementary blade profile of the second cold-state blade profile is generated, which has satisfied the requirement of the embodiment for intercepting the coordinate points of the leading edge and the trailing edge of the second cold-state blade profile to realize the fitting with the first cold-state blade profile.
In step S140, the ranges of the coordinate points of the leading edge and the trailing edge of the blade captured are respectively a range of a chord length of the blade from the leading edge point by 4% to 15% and a range of a chord length of the blade from the trailing edge point by 4% to 15%.
Note that the leading edge point refers to an intersection point of the profile near the leading edge and the mean camber line, and the trailing edge point refers to an intersection point of the profile near the trailing edge and the mean camber line.
When the blade is a high-pressure compressor rotor blade, the ranges of the coordinate points of the intercepted blade front edge and the coordinate points of the intercepted blade tail edge can be respectively the range of the blade chord length from the front edge point by 5 percent and the range of the blade chord length from the tail edge point by 5 percent.
It can be understood that, as shown in fig. 7, the closer the first cold-state leaf profile obtained by the grid interpolation method in step S120 is to the leading edge and the trailing edge points, the more inaccurate the profile shape is, and therefore, if the range of the leading edge and the trailing edge portions used for fitting is too small, the error increases; the overall shape of the second cold blade profile obtained by the modeling approximation method in step S130 is inaccurate, and therefore the accuracy of the overall shape decreases as the range of the leading edge and the trailing edge portions used for fitting increases. Therefore, it is desirable to intercept the appropriate range of leading and trailing edge coordinate points.
It should be noted that the range of the cut coordinate points of the leading edge and the trailing edge that participate in the fitting may adopt any other suitable value according to the actual situation.
As shown in fig. 8, in step S140, a least square method is used for fitting, and when each coordinate point in the second cut-out cold-state leaf pattern (contour line marked with x) corresponds to the first cold-state leaf pattern (contour line marked with o)Sum of squares of distances of dots
Figure 413986DEST_PATH_IMAGE028
And (5) finishing fitting in a minimum mode.
This step may include using a weighted least squares method, i.e. choosing to give different weights w to coordinate points at different positions, while ordering
Figure 431621DEST_PATH_IMAGE030
And when the minimum value is reached, the fitting is completed.
It can be understood that the coordinate points at different positions are weighted, so that the fitting result is not influenced by the distribution rule of the coordinate points, and the fitting precision is improved.
The weight w may be half of the sum of the distances of each coordinate point in the truncated second cold-state leaf type from its two neighboring coordinate points.
Slight unevenness exists between the front edge part and the tail edge part of the third cold-state blade profile obtained after fitting and the blade body part, so that the third cold-state blade profile needs to be smoothed through smoothing treatment.
As shown in fig. 4 and 9, step S150 includes:
s151: two control points at the profile transition of the leading edge portion and the profile of the blade body portion are selected, and are respectively marked as D0And D;
s152: from D0Selecting a plurality of control points along a first direction away from the blade body, wherein the distance D is0The farthest control point is denoted as E0And E with0Adjacent control points are denoted as F0Selecting a plurality of control points from D along a second direction opposite to the first direction, wherein the control point farthest from D is marked as E, and the control point F adjacent to E;
s153: get intermediate control point G0And G, line segment E0G0And the line segment EG is respectively on the line segment E0F0And line segment EF;
s154: according to E0、E、G0And G, making a cubic Bezier curve;
s155: taking a plurality of points on a cubic Bessel curveDensity similar to that of adjacent control points, replacing E with a plurality of points0And E is the original coordinate point;
step S150 further includes: two control points (not shown) at the profile transition of the profile of the trailing edge portion and the profile of the blade airfoil portion are selected, and are respectively denoted as H0And H, with H0And H replace D in step S151, respectively0And D, performing steps S151 to S155 to smooth the boundary of the trailing edge and the blade body.
It will be understood that when line segment E is taken0G0And the line segment EG is respectively on the line segment E0F0And the line segment EF is on the straight line, so as to ensure the starting point and the ending point of the Bezier curve (namely E)0、E) The tangential direction of the third cold blade profile is the same as the profile of the third cold blade profile. Under normal circumstances, E can be considered0G0And EG are approximately parallel.
Control point E0The straight-line distance to the control point E is denoted as E0E, control Point D0The straight-line distance to the control point D is denoted as D0D，E0E is at least D030 times of D.
It is understood that when E0E is D0And D is 30 times or more of the D, the requirement of the smoothness of the leaf profile is met.
Control point D0To control point E0The straight-line distance of (D) is recorded as0E0The distance from control point D to control point E is denoted as DE, and line segment E0G0And the line segments EG are respectively D0E00.82 times the DE.
It will be appreciated that as shown in FIG. 10 (distance E on the abscissa)0A dimensionless distance of "0" represents a distance from E0Coincidences, "1" represents coincidences with E; ordinate is curvature), when the line segment E0G0And the line segments EG are respectively D0E00.82 times DE, the curvature peak of the bezier curve is minimal.
The accuracy of the production coordinate data obtained according to the scheme of the invention can be verified. The machining coordinates were generated by using the example and the grid interpolation method for the 2 nd rotor of a certain engine. In the method for generating blade machining coordinates according to this embodiment, the amount of the finite element mesh in the blade region used is 1/16 of mesh interpolation. As shown in fig. 11, the surface mach number distribution (the triangle is marked by the result curve of the method of this embodiment, and the circle is marked by the result curve of the grid interpolation method) of a certain element section in the middle of the blade is shown, and compared with the aerodynamic performance of the machining coordinates generated by the method of this embodiment and the grid interpolation method respectively, it can be found that the two almost completely coincide with each other, and the method of this embodiment can reduce the grid demand of the blade body part (in the flow channel) by one order of magnitude, and the finite element grid quantity of the whole blade (including the whole of the blade, the flange plate and the tenon) is reduced by about 75%. Therefore, the method for generating the blade machining coordinate of the embodiment ensures that the machining coordinate is consistent with the finite element analysis result and has correct shape and smoothness in the local areas of the leading edge and the trailing edge under the condition that the grid quantity required by the finite element analysis only needs to meet the conventional analysis criterion.
In another aspect, a storage medium is provided, which stores a computer program that executes the method of generating blade machining coordinates of the embodiment when the computer program runs.
In another aspect, a computer device is provided, comprising a memory, a processor and a computer program stored on the memory and executable on the processor, the processor implementing the method of generating blade machining coordinates of the embodiment when executing the program.
In another aspect, an apparatus for generating machining coordinates of a blade is provided, including:
the acquisition module is used for acquiring a cold-state blade model of the blade, wherein the cold-state blade model is obtained from a hot-state blade profile through a hot-to-cold finite element analysis;
the grid interpolation module is used for acquiring cold grid coordinate points of the cold blade model and generating a series of elementary blade profiles on the equal-height surface by adopting a grid interpolation method to form a first cold blade profile;
a modeling approximation module for employing a modeling approximation, the modeling approximation comprising: determining a plurality of modeling conical surfaces based on the result of finite element analysis, generating element blade profiles on the modeling conical surfaces, and generating a series of element blade profiles on the equal-height surface according to element blade profile interpolation on the series of modeling conical surfaces to form a second cold-state blade profile;
the fitting module is used for intercepting coordinate points of the leading edge and the trailing edge of the blade from the second cold-state blade profile, fitting the coordinate points with the first cold-state blade profile to obtain an updated leading edge coordinate point and an updated trailing edge coordinate point, and replacing the coordinate points of the corresponding leading edge and trailing edge parts in the first cold-state blade profile with the updated leading edge coordinate point and the updated trailing edge coordinate point to obtain a third cold-state blade profile; each equal height surface has the same height as each corresponding equal height surface in the grid interpolation method.
And the blade profile fairing module is used for fairing the junction of the front edge and the tail edge of the third cold-state blade profile and the blade body to obtain a final cold-state blade profile, and the coordinate point of the final cold-state blade profile is the final machining coordinate of the blade.
The contouring approximation module employs the same steps as the method illustrated in fig. 3 to determine a plurality of contouring cones and generate the elementary lobes thereon.
The blade profile fairing module adopts the same steps as the method shown in fig. 4 to fairing the junction of the leading edge and the blade body of the third cold-state blade profile; the blade fairing module further comprises two control points H at the transition of the profile of the selected trailing edge part and the profile of the blade body part0And H, respectively, instead of D in step S1510And D, performing steps S151 to S55 to smooth the boundary of the trailing edge and the blade body.
Control point E0The straight-line distance to the control point E is denoted as E0E, control Point D0The straight-line distance to the control point D is denoted as D0D，E0E is at least D030 times of D; control point D0To control point E0The straight-line distance of (D) is recorded as0E0The distance from control point D to control point E is denoted as DE, and line segment E0G0And the line segments EG are respectively D0E00.82 times the DE.
The range of the coordinate points of the front edge and the tail edge of the blade intercepted by the fitting module is respectively the range of the chord length of the blade from the front edge point by 4% -15% and the range of the chord length of the blade from the tail edge point by 4% -15%.
When the blade is a high-pressure compressor rotor blade, the ranges of the coordinate points of the intercepted front edge and the coordinate points of the intercepted tail edge are respectively the range of the chord length of the blade which is 5% away from the front edge point and the range of the chord length of the blade which is 5% away from the tail edge point.
The fitting module adopts least square fitting, and when the square sum of the distance between each coordinate point in the intercepted second cold-state blade profile and the corresponding point of the first cold-state blade profile is obtained
Figure 113138DEST_PATH_IMAGE028
And (5) finishing fitting in a minimum mode.
The fitting module can adopt a weighted least square method to select coordinate points at different positions to be given different weights w, and the order is given
Figure 849013DEST_PATH_IMAGE030
And when the minimum value is reached, the fitting is completed.
The weight w is half of the sum of the distances between each coordinate point in the intercepted second cold-state blade profile and the two adjacent coordinate points.
The terms and expressions which have been employed herein are used as terms of description and not of limitation. The use of such terms and expressions is not intended to exclude any equivalents of the features shown and described (or portions thereof), and it is recognized that various modifications may be made within the scope of the claims. Other modifications, variations, and alternatives are also possible. Accordingly, the claims should be looked to in order to cover all such equivalents.
Also, it should be noted that although the present invention has been described with reference to the current specific embodiments, it should be understood by those skilled in the art that the above embodiments are merely illustrative of the present invention, and various equivalent changes or substitutions may be made therein without departing from the spirit of the present invention, and therefore, it is intended that all changes and modifications to the above embodiments within the spirit of the present invention shall fall within the scope of the appended claims.

Claims (20)

1. A method of generating blade machining coordinates comprising the steps of:
1) obtaining a cold state blade model of the blade, wherein the cold state blade model is obtained from a hot state blade profile through a hot-to-cold finite element analysis;
2) acquiring cold grid coordinate points of the cold blade model, and generating a series of elementary blade profiles on the equal height surface by adopting a grid interpolation method to form a first cold blade profile;
3) adopting a modeling approach method, wherein the modeling approach method comprises the following steps: determining a plurality of modeling conical surfaces based on the finite element analysis result, generating element blade profiles on the modeling conical surfaces, and generating a series of element blade profiles on equal-height surfaces according to element blade profile interpolation on the modeling conical surfaces to form a second cold-state blade profile, wherein the height of each equal-height surface is the same as that of each corresponding equal-height surface in the grid interpolation method in the step 2);
4) intercepting coordinate points of a leading edge and a trailing edge from the second cold-state blade profile, fitting the coordinate points with the first cold-state blade profile to obtain an updated leading edge coordinate point and an updated trailing edge coordinate point, and replacing the corresponding coordinate points of the leading edge and the trailing edge in the first cold-state blade profile with the updated leading edge coordinate point and the updated trailing edge coordinate point to obtain a third cold-state blade profile;
5) and smoothing the junction of the front edge and the tail edge of the third cold-state blade profile and the blade body respectively to obtain a final cold-state blade profile, wherein the coordinate point of the final cold-state blade profile is the final machining coordinate of the blade.
2. The method of generating blade machining coordinates of claim 1, wherein said determining a plurality of contouring cones and said generating elementary airfoil shapes on said plurality of contouring cones in step 3) comprises the steps of:
3.1) based on the results of the finite element analysis, selecting two points A1 and A2 of the blade tip and two points C1 and C2 of the blade root on the thermal blade profile, and selecting a plurality of groups containing two points and positioned on the blade tip and the blade rootB1 and B2 between the blade roots, points A '1, A' 2, C '1 and C' 2 corresponding to the points A1, A2, C1 and C2 on the cold-state blade model are determined, and the radial displacement of the points A1, A2, C1 and C2 and the corresponding points A '1, A' 2, C '1 and C' 2 in the cold-state blade model are measured
Figure DEST_PATH_IMAGE001
And amount of axial displacement
Figure 506605DEST_PATH_IMAGE002
；
For each group B1 and B2, respectively calculating the radial displacement of B '1 and B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure DEST_PATH_IMAGE003
：
Figure 370394DEST_PATH_IMAGE004
（1）
Figure DEST_PATH_IMAGE005
（2）
Figure 759787DEST_PATH_IMAGE006
The leaf height at B1 is the height of the leaf,
Figure DEST_PATH_IMAGE007
(ii) a leaf height of said A1 to said C1;
Figure 857187DEST_PATH_IMAGE008
the leaf height at B2 is the height of the leaf,
Figure DEST_PATH_IMAGE009
(ii) a leaf height of said A2 to said C2;
for each group B1 and B2, respectively calculating the axial displacement quantity of the B '1 and the B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure 332031DEST_PATH_IMAGE010
：
Figure DEST_PATH_IMAGE011
（3）
Figure 671614DEST_PATH_IMAGE012
（4）；
3.2) according to said
Figure DEST_PATH_IMAGE013
Obtaining the positions of B '1-B' 2 modeling conical surfaces corresponding to the B1-B2 modeling conical surfaces in the cold-state blade model; elementary leaf shapes are generated on the B '1-B' 2 contouring cone using the same elementary leaf shape parameters as in the hot-state leaf shape.
3. The method of generating blade machining coordinates of claim 1, wherein the smoothing method in step 5) comprises:
5.1) selecting two control points at the transition of the profile of the leading edge portion and the profile of the blade body portion, respectively denoted D0And D;
5.2) from said D0Selecting a plurality of control points along a first direction away from the blade body, wherein the control points are away from the blade body D0The farthest control point is denoted as E0And said E0Adjacent control points are denoted as F0Selecting a plurality of control points from the D along a second direction opposite to the first direction, wherein the control point farthest from the D is marked as E, and the control point F adjacent to the E;
5.3) taking the intermediate control point G0And G, line segment E0G0And the line segment EG is respectively on the line segment E0F0And line segment EF;
5.4) according to said E0、E、G0And G, making a cubic Bezier curve;
5.5) taking a plurality of points on the cubic Bessel curve, the density of the points being similar to that of the adjacent control points, replacing the E with a plurality of the points0And E is the original coordinate point;
step 5) also comprises the following steps: two control points at the transition of the profile of the trailing edge portion and the profile of the blade body portion are selected and respectively marked as H0And H with said H0And H replaces said D in step 5.1), respectively0And D) carrying out the steps 5.1) to 5.4) to smooth the junction of the trailing edge and the blade body.
4. Method for generating blade machining coordinates according to claim 3, characterized in that said control point E0The straight-line distance to the control point E is denoted as E0E, the control point D0The straight-line distance to the control point D is denoted as D0D, said E0E is at least the number D030 times of D.
5. Method for generating blade machining coordinates according to claim 3, characterized in that the control point D0To the control point E0The straight-line distance of (D) is recorded as0E0The distance from the control point D to the control point E is denoted DE, and the line segment E0G0And the line segment EG is respectively D0E00.82 times the DE.
6. The method for generating blade machining coordinates of claim 1, wherein in the step 4), the range of the coordinate points of the truncated leading edge and the range of the coordinate points of the truncated trailing edge are 4% -15% of the chord length of the blade from the leading edge point and 4% -15% of the chord length of the blade from the trailing edge point respectively.
7. The method of generating blade machining coordinates of claim 6 wherein the blade is a high pressure compressor rotor blade and the ranges of the truncated leading edge coordinate points and the trailing edge coordinate points are 5% of the chord length of the blade from the leading edge point and 5% of the chord length of the blade from the trailing edge point, respectively.
8. The method for generating blade machining coordinates according to claim 1, wherein in the step 4), a least square method is adopted for fitting, and when the sum of squares of distances between each coordinate point in the truncated second cold-state blade profile and the corresponding point of the first cold-state blade profile is the sum of squares of distances between each coordinate point in the truncated second cold-state blade profile and the corresponding point of the first cold-state blade profile
Figure 915513DEST_PATH_IMAGE014
And (5) finishing fitting in a minimum mode.
9. The method of claim 8, wherein the coordinate points at different positions are chosen to be given different weights w, when ordered
Figure DEST_PATH_IMAGE015
And when the minimum value is reached, the fitting is completed.
10. The method of claim 9, wherein the weight w is half of the sum of the distances between each coordinate point in the truncated second cold state airfoil and two coordinate points adjacent to the coordinate point.
11. A storage medium storing a computer program, wherein the computer program is configured to execute the method of generating blade machining coordinates according to any one of claims 1 to 10 when running.
12. A computer arrangement comprising a memory, a processor and a computer program stored on the memory and executable on the processor, characterized in that the processor, when executing the program, carries out the method of generating blade machining coordinates according to any one of claims 1 to 10.
13. An apparatus for generating blade machining coordinates, comprising:
an obtaining module for obtaining a cold blade model of the blade, the cold blade model being obtained from a hot blade profile by a hot-to-cold finite element analysis;
the grid interpolation module is used for acquiring cold grid coordinate points of the cold blade model and generating a series of elementary blade profiles on an equal height surface by adopting a grid interpolation method to form a first cold blade profile;
a modeling approximation module for employing a modeling approximation, the modeling approximation comprising: determining a plurality of modeling conical surfaces based on the finite element analysis result, generating element blade profiles on the modeling conical surfaces, and generating a series of element blade profiles on the equal-height surfaces according to element blade profile interpolation on the modeling conical surfaces to form a second cold-state blade profile, wherein the heights of the equal-height surfaces are the same as the heights of the corresponding equal-height surfaces in the grid interpolation method;
the fitting module is used for intercepting coordinate points of the leading edge and the trailing edge from the second cold-state blade profile, fitting the coordinate points with the first cold-state blade profile to obtain an updated leading edge coordinate point and an updated trailing edge coordinate point, and replacing the corresponding coordinate points of the leading edge and the trailing edge in the first cold-state blade profile with the updated leading edge coordinate point and the updated trailing edge coordinate point to obtain a third cold-state blade profile;
and the blade profile fairing module is used for fairing the junction of the front edge and the tail edge of the third cold-state blade profile and the blade body respectively so as to obtain a final cold-state blade profile, and the coordinate point of the final cold-state blade profile is the final machining coordinate of the blade.
14. The apparatus for generating blade machining coordinates of claim 13 wherein the contouring approximation module employs the following steps to determine the plurality of contouring cones and generate elementary airfoil shapes on the contouring cones:
14.1) based on the cold-state blade model, selecting two points A1 and A2 of a blade tip and two points C1 and C2 of a blade root on the hot-state blade profile, selecting multiple groups of B1 and B2 between the blade tip and the blade root, determining points A '1, A' 2, C '1 and C' 2 corresponding to the points A1, A2, C1 and C2 on the cold-state blade model, and measuring the radial displacement of the points A '1, A' 2, C '1 and C' 2 corresponding to the points A '1, A' 2, C '1 and C' 2 in the cold-state blade model to obtain the radial displacement of the points A1, A2, C1 and C2
Figure 918236DEST_PATH_IMAGE001
And amount of axial displacement
Figure 614796DEST_PATH_IMAGE002
；
For each group B1 and B2, respectively calculating the radial displacement of B '1 and B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure 649748DEST_PATH_IMAGE003
：
Figure 794160DEST_PATH_IMAGE004
（1）
Figure 623575DEST_PATH_IMAGE005
（2）
Figure 604170DEST_PATH_IMAGE006
The leaf height at B1 is the height of the leaf,
Figure 442813DEST_PATH_IMAGE007
(ii) a leaf height of said A1 to said C1;
Figure 677616DEST_PATH_IMAGE008
the leaf height at B2 is the height of the leaf,
Figure 943512DEST_PATH_IMAGE009
(ii) a leaf height of said A2 to said C2;
for each group B1 and B2, respectively calculating and obtaining the axial displacement quantity of B '1 and B' 2 corresponding to each group B1 and B2 in the cold-state blade model by adopting the following formula
Figure 145824DEST_PATH_IMAGE010
：
Figure 53737DEST_PATH_IMAGE016
（3）
Figure DEST_PATH_IMAGE017
（4）；
14.2) according to said
Figure 381862DEST_PATH_IMAGE018
Obtaining the positions of B '1-B' 2 modeling conical surfaces corresponding to the B1-B2 modeling conical surfaces in the cold-state blade model; elementary leaf shapes are generated on the B '1-B' 2 contouring cone using the same elementary leaf shape parameters as in the hot-state leaf shape.
15. The apparatus of claim 13, wherein the blade fairing module employs the following steps to fairing the intersection of the leading edge and the blade body of the third cold blade profile:
15.1) selecting two control points at the profile transition of the leading edge portion and the profile of the blade body portion, respectively denoted D0And D;
15.2) from said D0Selecting a plurality of controls in a first direction away from the bodyA point, wherein, from said D0The farthest control point is denoted as E0And said E0Adjacent control points are denoted as F0Selecting a plurality of control points from the D along a second direction opposite to the first direction, wherein the control point farthest from the D is marked as E, and the control point F adjacent to the E;
15.3) taking the intermediate control point G0And G, line segment E0G0And the line segment EG is respectively on the line segment E0F0And line segment EF;
15.4) according to said E0、E、G0And G, making a cubic Bezier curve;
15.5) taking a plurality of points on said cubic Bessel curve, said points having a density similar to that of said adjacent control points, replacing said E with a plurality of said points0And E is the original coordinate point;
the steps further include: two control points at the transition of the profile of the trailing edge portion and the profile of the blade body portion are selected and respectively marked as H0And H with said H0And H replaces said D in step 15.1), respectively0And D) carrying out the steps 15.1) to 15.4) to smooth the junction of the trailing edge and the blade body.
16. Device for generating blade machining coordinates according to claim 15, characterized in that the control point E0The straight-line distance to the control point E is denoted as E0E, the control point D0The straight-line distance to the control point D is denoted as D0D, said E0E is at least the number D030 times of D;
the control point D0To the control point E0The straight-line distance of (D) is recorded as0E0The distance from the control point D to the control point E is denoted DE, and the line segment E0G0And the line segment EG is respectively D0E00.82 times the DE.
17. The apparatus of claim 13, wherein the fitting module intercepts the leading edge coordinate points and the trailing edge coordinate points in a range of 4% to 15% of the chord length of the blade from the leading edge point and 4% to 15% of the chord length of the blade from the trailing edge point, respectively.
18. The apparatus of claim 17, wherein the blade is a high pressure compressor rotor blade, and the range of the coordinate point at which the leading edge is truncated and the range of the coordinate point at the trailing edge are 5% of the chord length of the blade from the leading edge point and 5% of the chord length of the blade from the trailing edge point, respectively.
19. The apparatus of claim 13, wherein the fitting module uses a least squares fit when each coordinate point of the truncated second cold state airfoil is summed with the square of the distance from the corresponding point of the first cold state airfoil
Figure 943293DEST_PATH_IMAGE014
And (5) finishing fitting in a minimum mode.
20. The apparatus of claim 19, wherein the fitting module selects the coordinate points at different positions to be given different weights w, when the order is given
Figure 508267DEST_PATH_IMAGE015
When the minimum value is reached, the fitting is completed;
the weight w is half of the sum of the distances between each coordinate point in the intercepted second cold-state blade profile and two adjacent coordinate points.
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CN114818436B (en) * 2021-12-14 2024-03-29 泸州懋威科技有限公司 Centrifugal impeller and turbine blade profile cold-hot state rapid conversion method
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