CN104325080A - Shrinkage factor design method of precision casting die for multiple-guide vane groups - Google Patents

Abstract

The invention discloses a shrinkage factor design method of a precision casting die for multiple-guide vane groups. According to the shrinkage factor design method, a shrinkage factor is finally set by assuming parameters, assuming shrinkage directions and selecting the magnitude of the shrinkage factor; when a wax pattern die is designed and manufactured, corresponding shrinkage factors are set in three directions L, D and R (refer to the assumed shrinkage directions) greatly different in size, and therefore, the out-of-tolerance problem of the channel area of the casting of a multiple-guide vane group due to the adoption of one single shrinkage factor is solved. The shrinkage factor design method of the precision casting die is suitable for all the multiple-guide vane groups. Offset angles delta alpha 1 and delta alpha 2 are obtained by use of a computing method, and therefore, the out-of-tolerance problem of the channel area is solved, and the design concept and the design method are reasonable. The die is designed and manufactured in one step so that the product development cost is greatly reduced and the product trial production period is shortened. The shrinkage factor design method of the precision casting die for the multiple-guide vane groups can be popularized in engineering application.

Description

A kind of shrinkage factor method for designing of leafy group of stator precision casting mould

Technical field

The invention belongs to melted module precise casting technology field, particularly a kind of shrinkage factor method for designing of leafy group of stator precision casting mould.

Background technology

The establishing method of leafy group of stator precision casting mould shrinkage factor was all adopt unified comprehensive shrinkage factor in a different direction in the past.In actual production process, the shrinkage factor that the wax material of different length is corresponding is different, and conventional method is only applicable to the foundry goods that dimensional structure is in all directions more or less the same.

For leafy group of guide vane of ultra-fine length, dimensional structure differs greatly in different directions, adopts the method for single integrated shrinkage factor to be irrational, causes casting dimension overproof.Under the prerequisite ensureing pipes' analysis size and blade molding surface size, foundry goods aisle spare can be caused bigger than normal.The mould done cannot be reprocessed, and can only be redesigned die module by the method for slowly debugging or be redesigned mould to solve this kind of overproof problem, extends mould shape and product development cycle and adds development cost.

Summary of the invention

The object of the invention is to solve the problem, a kind of shrinkage factor method for designing of leafy group of stator precision casting mould is provided, when adopting the method to manufacture and design Wax mold, L, D, R (shrinkage direction see supposition) three directions that size differs greatly set corresponding shrinkage factor, solves the leafy group of overproof problem of guide vane foundry goods aisle spare caused because only adopting single shrinkage factor in the past.

To achieve these goals, the technical solution adopted in the present invention comprises the following steps:

1) parameter is supposed

Suppose, a kind of leafy group of guide vane has K blade, and the radius of supramarginal plate and inframarginal place circumference is respectively R ₂and R ₁, the width of supramarginal plate and inframarginal is D, and two interlobate angles are α, and (K-1) α < 45 °;

2) shrinkage direction is supposed

According to the dimensional structure feature of leafy group of guide vane, assuming that the shrinkage direction of leafy group of guide vane is divided into L direction, D direction and R direction, wherein, L direction is the radian direction of upper and lower listrium, D direction is the width of upper and lower listrium, and R direction is the length direction of blade;

3) shrinkage factor size is selected

According to the characteristic of wax material on-the-spot in actual production process, different length selects corresponding shrinkage factor;

4) shrinkage factor setting

When designing and producing leafy group of stator precision casting mould, the contraction in all directions sets respectively in the following manner:

On D direction: upper and lower listrium width amplifies ψ 4;

On R direction: blade length direction amplifies ψ ₁, i.e. R ₁become (1+ ψ ₁) R ₁, R ₂become (1+ ψ ₁) R ₂, interlobate angle α is constant;

On L direction: make the central axis of leafy group of guide vane overlap with reference axis Z axis before amplification, ψ is amplified in direction to supramarginal plate radian ₂, ψ is amplified in direction to inframarginal radian ₃;

If AB represents the central axis of leafy group of guide vane, overlap with reference axis Z axis; A ₁b ₁represent a blade in leafy group of guide vane after R direction is amplified; β represents A ₁b ₁the angle of the blade on direction and leafy group of guide vane central axis; A ₁' B ₁' represent blade A ₁b ₁size shape after amplifying in the l-direction; Calculate inframarginal A ₁after point amplifies in the l-direction, from A ₁to A ₁' angle of eccentricity Δ α ₁and supramarginal plate B ₁after point amplifies in the l-direction, from B ₁to B ₁' angle of eccentricity Δ α ₂.

Described step 3) in, the corresponding shrinkage factor of different length is:

Length of blade L=R ₂-R ₁, corresponding shrinkage factor is ψ ₁;

Supramarginal plate arc length L ₂=K α R ₂, corresponding shrinkage factor is ψ ₂;

Inframarginal arc length L ₁=K α R ₁, corresponding shrinkage factor is ψ ₃;

Upper and lower listrium width is D, and corresponding shrinkage factor is ψ ₄.

Described step 4) in, calculate and concrete grammar be:

Calculate Δ α ₁with Δ α ₂, by

$\frac{(\mathrm{\&beta;}+\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1)\&times;(1+{\mathrm{\&psi;}}_1)\&times;R_1-\mathrm{\&beta;}\&times;R_1}{\mathrm{\&beta;}{R}_1}={\mathrm{\&psi;}}_3---\left(1\right)$

$\frac{(\mathrm{\&beta;}+\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2)\&times;(1+{\mathrm{\&psi;}}_1)\&times;R_2-\mathrm{\&beta;}\&times;R_2}{\mathrm{\&beta;}{R}_2}={\mathrm{\&psi;}}_2---\left(2\right)$

?

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&beta;}---\left(3\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&beta;}---\left(4\right)$

When K is odd number, A ₁' B ₁' represent first blade, the then β=α of adjacent AB

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&alpha;}---\left(5\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&alpha;}---\left(6\right)$

A ₁' B ₁' when representing N number of blade of adjacent AB, β=N α

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{N\&alpha;}---\left(7\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{N\&alpha;}---\left(8\right)$

When K is even number, A ₁' B ₁' represent N number of blade of adjacent AB, then β=N α-α/2

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}(\mathrm{N\&alpha;}-\frac{\mathrm{\&alpha;}}{2})---\left(9\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}(\mathrm{N\&alpha;}-\frac{\mathrm{\&alpha;}}{2})---\left(10\right)$

Can be obtained by formula (7), (8), (9) and (10):

Work as ψ ₁=ψ ₂=ψ ₃time, namely by overall unified comprehensive shrinkage factor setting mould, △ α ₁=△ α ₂=0, at this moment blade angle there will not be biased;

Work as ψ ₂=ψ ₃time, △ α ₁=△ α ₂, to do the length of blade of mould isometric.

Compared with prior art, the present invention has following beneficial effect:

When adopting the inventive method to manufacture and design Wax mold, L, D, R (shrinkage direction see supposition) three directions that size differs greatly set corresponding shrinkage factor, solves the leafy group of overproof problem of guide vane foundry goods aisle spare caused because only adopting single shrinkage factor in the past.The present invention is applicable to leafy group of all stators.Offset angle Δ α is obtained by the method calculated ₁, Δ α ₂, solve the overproof problem of aisle spare, design concept and method more reasonable.Design and produce mould " to settle at one go ", greatly reduce product development expense, shorten the trial production cycle.And can promote the use of in engineer applied.

Accompanying drawing explanation

Fig. 1 is the shrinkage direction schematic diagram that the present invention supposes;

Fig. 2 is the bias conditions schematic diagram of supramarginal plate and inframarginal after L direction of the present invention is amplified by different shrinkage factor.

Detailed description of the invention

Below in conjunction with drawings and Examples, the present invention is further detailed explanation:

The concrete setting principle of shrinkage factor of the present invention and method as follows

(1) leafy group of stator basic condition

Suppose, a kind of leafy group of guide vane has K blade, and the radius of upper and lower listrium place circumference is respectively R ₂, R ₁, upper and lower listrium width is D, and two interlobate angles are α, and (K-1) α < 45 °.

(2) shrinkage direction is supposed

As shown in Figure 1, according to the dimensional structure feature of leafy group of guide vane, assuming that the shrinkage direction of leafy group of guide vane is divided into three direction: L (upper and lower listrium radian direction), D (upper and lower listrium width), R (length of blade direction).

The shrinkage direction schematic diagram of Fig. 1 supposition

(3) shrinkage factor size is selected

According to the characteristic of wax material on-the-spot in actual production process, different length selects different shrinkage factor:

Length of blade L=R ₂-R ₁, corresponding shrinkage factor is ψ ₁;

Supramarginal plate arc length L ₂=K α R ₂, corresponding shrinkage factor is ψ ₂;

Inframarginal arc length L ₁=K α R ₁, corresponding shrinkage factor is ψ ₃;

Upper and lower listrium width is D, and corresponding shrinkage factor is ψ 4.

(4) shrinkage establishing method:

Shrinkage factor establishing method in the past:

For leafy group of guide vane, i.e. ψ of ultra-fine length ₁> ψ ₂, ψ ₁> ψ ₃, ψ ₁> ψ ₄.

1) with ψ ₂, ψ ₃or ψ ₄as a whole during comprehensive shrinkage factor, foundry goods R direction size is less than normal, cannot ensure blade passage face V spot size and aisle spare.

2) with maximum ψ ₁set as comprehensive shrinkage factor, can meet R direction dimensional requirement, L direction size is bigger than normal, and foundry goods aisle spare is bigger than normal.

Solution:

Again make mould, according to the aisle spare pouring into a mould out foundry goods situation bigger than normal, biased blade, by less than normal for α value.

Existing shrinkage factor establishing method:

When designing and producing leafy group of stator precision casting mould, the contraction in all directions is implemented respectively in the following manner:

On D direction: upper and lower listrium width amplifies ψ ₄.

On R direction: blade length direction amplifies ψ ₁, i.e. R ₁become (1+ ψ ₁) R ₁, R ₂become (1+ ψ ₁) R ₂, interlobate angle α is constant.

On L direction:

As shown in Figure 2, must ensure before amplification that the central axis of leafy group of guide vane overlaps with reference axis Z axis, ψ is amplified in direction to supramarginal plate radian ₂, ψ is amplified in direction to inframarginal radian ₃, the result of amplification is showed by the biased of leaf angle.

If AB represents the central axis of leafy group of guide vane, overlap with reference axis Z axis.A ₁b ₁represent a blade in leafy group of guide vane after R direction is amplified.β represents A ₁b ₁the angle of the blade on direction and leafy group of guide vane central axis.A ₁' B ₁' represent blade A ₁b ₁size shape after amplifying in the l-direction.Δ α ₁represent inframarginal A ₁after point amplifies in the l-direction, from A ₁to A ₁' angle of eccentricity.Δ α ₂represent supramarginal plate B ₁after point amplifies in the l-direction, from B ₁to B ₁' angle of eccentricity.

As seen from Figure 2, Δ α ₁, Δ α ₂be calculative value, and work as ψ ₂with ψ ₃when being worth not identical, do the length of blade Length discrepancy of mould, blade pitch AB is far away, and biased larger, blade is longer.

Calculate Δ α ₁, Δ α ₂, by

$\frac{(\mathrm{\&beta;}+\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1)\&times;(1+{\mathrm{\&psi;}}_1)\&times;R_1-\mathrm{\&beta;}\&times;R_1}{\mathrm{\&beta;}{R}_1}={\mathrm{\&psi;}}_3---\left(1\right)$

$\frac{(\mathrm{\&beta;}+\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2)\&times;(1+{\mathrm{\&psi;}}_1)\&times;R_2-\mathrm{\&beta;}\&times;R_2}{\mathrm{\&beta;}{R}_2}={\mathrm{\&psi;}}_2---\left(2\right)$

?

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&beta;}---\left(3\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&beta;}---\left(4\right)$

When K is odd number, A ₁' B ₁' represent first blade, the then β=α of adjacent AB

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&alpha;}---\left(5\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&alpha;}---\left(6\right)$

A ₁' B ₁' when representing N number of blade of adjacent AB, β=N α

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{N\&alpha;}---\left(7\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{N\&alpha;}---\left(8\right)$

When K is even number, A ₁' B ₁' represent N number of blade of adjacent AB, then β=N α-α/2

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}(\mathrm{N\&alpha;}-\frac{\mathrm{\&alpha;}}{2})---\left(9\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}(\mathrm{N\&alpha;}-\frac{\mathrm{\&alpha;}}{2})---\left(10\right)$

Can be obtained by formula (7), (8), (9) and (10):

Work as ψ ₁=ψ ₂=ψ ₃time, namely by overall unified comprehensive shrinkage factor setting mould, △ α ₁=△ α ₂=0, at this moment blade angle there will not be biased;

Work as ψ ₂=ψ ₃time, △ α ₁=△ α ₂, to do the length of blade of mould isometric.

(5) comparison of shrinkage factor establishing method and existing shrinkage factor establishing method in the past, see table 1:

The comparison table of table 1 shrinkage factor establishing method and existing shrinkage factor establishing method in the past

Embodiment 1: five leaf group Pyatyi guide vane

(1) certain five leaf group guide vane overall size:

Supramarginal plate arc length L ₂be approximately 140mm, inframarginal arc length L1 is approximately 90mm, and upper and lower listrium width D is about 30mm ~ 40mm, blade length L=R ₂-R ₁be approximately 220mm, vane group total radian 4 α=16.3636 °.

(2) selected contraction size

According to the characteristic of wax material on-the-spot in actual production process, dimensions length is that 90mm ~ 140mm casting die shrinkage factor adopts 2.1%, and length is that 30mm ~ 40mm adopts 2%, and length is that 220mm adopts 2.4%.

That is, ψ ₁=2.4%, ψ ₂=ψ ₃=2.1%, ψ ₄=2%.

(3) mold shrinkage design

On D direction: upper and lower listrium width amplifies 2%.

On R direction: blade length direction amplifies 2.4%, i.e. R ₁become 1.024R ₁, R ₂become 1.024R ₂, interlobate angle 4.0909 ° is constant.

On L direction: ensure before amplification that the central axis of leafy group of guide vane overlaps with reference axis Z axis.By calculating, offset angle when N is 1, △ α=-43 ", when N is 2, △ α=-86 ".That is, the blade of central shaft both sides is positioned at, respectively to central axis direction successively biased 43 ", 86 ".

For the ease of understanding, above-mentioned shrinkage factor amplification process can describe like this: under ensureing the prerequisite that the central axis of leafy group of guide vane overlaps with reference axis Z axis, overall amplification 2.4%, and then axis both sides blade is biased 43 ", 86 " to central axis direction successively.Finally, upper and lower listrium width reduces 2.4%-2%=0.4%.

Embodiment 2: five leaf group three grades of guide vanes

(1) certain five leaf group three grades of guide vane overall sizes

Supramarginal plate arc length L ₂be approximately 145mm, inframarginal arc length L1 is approximately 90mm, and upper and lower listrium width D is about 20mm ~ 40mm, blade length L=R ₂-R ₁be approximately 180mm, vane group total radian 4 α=17.1429 °.

(2) selected contraction size

According to the characteristic of wax material on-the-spot in actual production process, dimensions length is that 90mm ~ 145mm casting die shrinkage factor adopts 2.1%, and length is that 20mm ~ 40mm adopts 2%, and length is that 180mm adopts 2.3%.

That is, ψ ₁=2.3%, ψ ₂=ψ ₃=2.1%, ψ ₄=2%.

(3) mold shrinkage design

By calculating, offset angle when N is 1, △ α=-30 ", when N is 2, △ α=-60 ".

Shrinkage factor amplification process: under ensureing the prerequisite that the central axis of leafy group of guide vane overlaps with reference axis Z axis, overall amplifies 2.3%, and then axis both sides blade is successively to central axis direction biased 30 ", 60 ".Finally, upper and lower listrium width reduces 0.3%.

Embodiment 3: five leaf group level Four guide vane

(1) certain five leaf group level Four guide vane overall size

Supramarginal plate arc length L ₂be approximately 145mm, inframarginal arc length L ₁be approximately 85mm, upper and lower listrium width D is about 20mm ~ 40mm, blade length L=R ₂-R ₁be approximately 200mm, vane group total radian 4 α=16.3636 °.

(2) selected contraction size

According to the characteristic of wax material on-the-spot in actual production process, dimensions length is that 85mm ~ 145mm casting die shrinkage factor adopts 2.1%, and length is that 20mm ~ 40mm adopts 2%, and length is that 200mm adopts 2.4%.

That is, ψ ₁=2.4%, ψ ₂=ψ ₃=2.1%, ψ ₄=2%.

(3) mold shrinkage design

By calculating, offset angle when N is 1, △ α=-43 ", when N is 2, △ α=-86 ".

Shrinkage factor amplification process: under ensureing the prerequisite that the central axis of leafy group of guide vane overlaps with reference axis Z axis, overall amplifies 2.4%, and then axis both sides blade is successively to central axis direction biased 43 ", 86 ".Finally, upper and lower listrium width reduces 0.4%.

Adopt above-mentioned three embodiments to five leaf groups three, four, the mold shrinkage of Pyatyi guide vane sets, aisle spare dimension overproof problem is obtained for solution, Design of Dies makes " settling at one go ", greatly reduces product development expense, shortens the trial production cycle.

Above content is only and technological thought of the present invention is described; protection scope of the present invention can not be limited with this; every technological thought proposed according to the present invention, any change that technical scheme basis is done, within the protection domain all falling into claims of the present invention.

Claims (3)

1. a shrinkage factor method for designing for leafy group of stator precision casting mould, is characterized in that, comprise the following steps:

1) parameter is supposed

Suppose, a kind of leafy group of guide vane has K blade, and the radius of supramarginal plate and inframarginal place circumference is respectively R ₂and R ₁, the width of supramarginal plate and inframarginal is D, and two interlobate angles are α, and (K-1) α < 45 °;

2) shrinkage direction is supposed

According to the dimensional structure feature of leafy group of guide vane, assuming that the shrinkage direction of leafy group of guide vane is divided into L direction, D direction and R direction, wherein, L direction is the radian direction of upper and lower listrium, D direction is the width of upper and lower listrium, and R direction is the length direction of blade;

3) shrinkage factor size is selected

According to the characteristic of wax material on-the-spot in actual production process, different length selects corresponding shrinkage factor;

4) shrinkage factor setting

When designing and producing leafy group of stator precision casting mould, the contraction in all directions sets respectively in the following manner:

On D direction: upper and lower listrium width amplifies ψ ₄;

On R direction: blade length direction amplifies ψ ₁, i.e. R ₁become (1+ ψ ₁) R ₁, R ₂become (1+ ψ ₁) R ₂, interlobate angle α is constant;

On L direction: make the central axis of leafy group of guide vane overlap with reference axis Z axis before amplification, ψ is amplified in direction to supramarginal plate radian ₂, ψ is amplified in direction to inframarginal radian ₃;

If AB represents the central axis of leafy group of guide vane, overlap with reference axis Z axis; A ₁b ₁represent a blade in leafy group of guide vane after R direction is amplified; β represents A ₁b ₁the angle of the blade on direction and leafy group of guide vane central axis; A ₁' B ₁' represent blade A ₁b ₁size shape after amplifying in the l-direction; Calculate inframarginal A ₁after point amplifies in the l-direction, from A ₁to A ₁' angle of eccentricity Δ α ₁and supramarginal plate B ₁after point amplifies in the l-direction, from B ₁to B ₁' angle of eccentricity Δ α ₂.

2. the shrinkage factor method for designing of leafy group of stator precision casting mould according to claim 1, is characterized in that: described step 3) in, the corresponding shrinkage factor of different length is:

Length of blade L=R ₂-R ₁, corresponding shrinkage factor is ψ ₁;

Supramarginal plate arc length L ₂=K α R ₂, corresponding shrinkage factor is ψ ₂;

Inframarginal arc length L ₁=K α R ₁, corresponding shrinkage factor is ψ ₃;

Upper and lower listrium width is D, and corresponding shrinkage factor is ψ ₄.

3. the shrinkage factor method for designing of leafy group of stator precision casting mould according to claim 1, is characterized in that: described step 4) in, calculate and concrete grammar be:

Calculate Δ α ₁with Δ α ₂, by

$\frac{(\mathrm{\&beta;}+\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1)\&times;(1+{\mathrm{\&psi;}}_1)\&times;R_1-\mathrm{\&beta;}\&times;R_1}{\mathrm{\&beta;}{R}_1}={\mathrm{\&psi;}}_3---\left(1\right)$

$\frac{(\mathrm{\&beta;}+\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2)\&times;(1+{\mathrm{\&psi;}}_1)\&times;R_2-\mathrm{\&beta;}\&times;R_2}{\mathrm{\&beta;}{R}_2}={\mathrm{\&psi;}}_2---\left(2\right)$

?

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&beta;}---\left(3\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&beta;}---\left(4\right)$

When K is odd number, A ₁' B ₁' represent first blade, the then β=α of adjacent AB

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&alpha;}---\left(5\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{\&alpha;}---\left(6\right)$

A ₁' B ₁' when representing N number of blade of adjacent AB, β=N α

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{N\&alpha;}---\left(7\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}\mathrm{N\&alpha;}---\left(8\right)$

When K is even number, A ₁' B ₁' represent N number of blade of adjacent AB, then β=N α-α/2

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_1=\frac{{\mathrm{\&psi;}}_3-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}(\mathrm{N\&alpha;}-\frac{\mathrm{\&alpha;}}{2})---\left(9\right)$

$\mathrm{\&Delta;}{\mathrm{\&alpha;}}_2=\frac{{\mathrm{\&psi;}}_2-{\mathrm{\&psi;}}_1}{1+{\mathrm{\&psi;}}_1}(\mathrm{N\&alpha;}-\frac{\mathrm{\&alpha;}}{2})---\left(10\right)$

Can be obtained by formula (7), (8), (9) and (10):

Work as ψ ₁=ψ ₂=ψ ₃time, namely by overall unified comprehensive shrinkage factor setting mould, Δ α ₁=Δ α ₂=0, at this moment blade angle there will not be biased;

Work as ψ ₂=ψ ₃time, Δ α ₁=Δ α ₂, to do the length of blade of mould isometric.