Interference analysis and simulation of abrasive belt grinding and polishing of blisk

Analysis of interference-free angle

The selection of the points needs to be adjusted according to the actual situation, including the size of the blades and the spacing between points. X polished points are respectively extracted from the seven sections of the concavity and convexity of the blade theoretical model, which are recorded as M, and the corresponding normal vector is recorded as N_m. The center point of the contact wheel should coincide with the cutter point when polishing the blade, and the vector of virtual cutter axis should be parallel to the spatial normal vector of each polished point. The point M′ is rotated from the polished point M through C axis during the actual polishing process. The normal vector after rotation is recorded as N_m′. The point M and the point M′ are the same point on the blade with different coordinate. Because the plane of concave virtual cutter vector N_a and convex virtual cutter vector N_t is parallel to the XOZ plane of the machine coordinate system, the normal vector N_m′ corresponding to the polished point M′ is also parallel to the XOZ plane after rotation. The interference-free angle of α, β, γ, and δ can be converted into the interference-free angles between the abrasive belt S₁, S₂, S₃, S₄, S₅, S₆, and the vector N_m′. When the belt tool are polishing at the point M′ on the blade surface, the section drawing is shown in Figure 6.

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Figure 6.

Plan of polishing interference judgment.

The center point of the contact wheel is point O, the radius of the contact wheel is r (r = 7 mm), and the thickness of the abrasive belt is h = 0.5 mm. Line L1, L2, L3, and L4 are four interference boundaries of the concavity. L1 and L4 are the critical interference boundaries between the abrasive belt and the polished blade when polishing at the point M′. L2 and L3 are the critical interference boundaries between the abrasive belt and the adjacent blade 1 when polishing at the point M′. L5 and L6 are the critical interference boundaries between the abrasive belt and the adjacent blade 2 when polishing at the point M′ on then convex surface. When polishing at the point M, point A_m′, B_m′, C_m′, D_m′, E_m′, and F_m′ are the extreme points of Z coordinate in the range of the abrasive belt where the inlet and exhaust edge points are concentrated. The calculation method will be introduced detailedly in the following section. The angles between the critical interference lines L1, L2, L3, L4, L5, L6 and the virtual cutter vector are α₁, α₂, β₁, β₂, γ₁, and δ₁, respectively. In order to ensure that the abrasive belt does not interfere with the blade, the abrasive belt in S₁ section should be between the critical interference line L1 and L2, and the abrasive belt in S₂ section should be between the critical interference line L3 and L4 when polishing the concavity of the blade. Therefore, when polishing at the point M of the concavity, the interference-free condition of the belt polishing system is

{ α 2 < α < α 1 β 1 < β < β 2

(1)

When polishing the convexity, the abrasive belt in S3 section shall not exceed the critical interference line L5, and the belt in S2 section shall not exceed the critical interference line L6. Therefore, when polishing at the point M of the convexity, the interference-free condition of the belt polishing system is

{ γ 1 < γ < 90 ° δ 1 < δ < 90 °

(2)

Extract data points of the inlet and exhaust edge

In the blisk model, data points are extracted from the inlet and exhaust edge of the polished blade, adjacent blade 1 and adjacent blade 2, as shown in Figure 7. The workpiece coordinate system is recorded as OXYZ. Y points are extracted by the equal arc length method on the exhaust edge of the polished blade, which are recorded as the point set A_i. By using the same method, the points on the exhaust edge of adjacent blade 1 are extracted as B_i, C_i, D_i, E_i, F_i. M ( x m , y m , z m ) is an arbitrary point on the blade surface, and the normal vectors at the point M is N m ( l m , m m , n m ) .

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Figure 7.

Schematic diagram of data point extraction for the theoretical model of blade.

Coordinate transformation of polished points and data points of the inlet and exhaust edge

In order to ensure that the abrasive belt polishing tool system is parallel to the space normal vector of the point M, C axis of the machine tool needs to turn an angle when polishing at the point M. Since the polishing side of the machine tool is a five-axis CNC system with swing turntable structure, the Y-axis component of the normal vector N_m′(l_m′, m_m′, n_m′) at the point M′ after rotation is zero. Let the rotation angle of C axis be θ_m.

For machine tool, the rotation transformation formula with the rotation angle θ around the Z axis is given by

Rot ( z , θ ) = [ cos θ − sin θ 0 0 sin θ cos θ 0 0 0 0 1 0 0 0 0 1 ]

(3)

Then the rotation formula is N_m′ = Rot(z, θ_m)·N_m, which gives

[ l m ′ m m ′ n m ′ 0 ] = [ cos θ m − sin θ m 0 0 sin θ m cos θ m 0 0 0 0 1 0 0 0 0 1 ] • [ l m m m n m 0 ]

(4)

And the solution of the equation is given by

{ l m ′ = l m · cos θ m − m m · sin θ m m m ′ = l m · sin θ m + m m · cos θ m n m ′ = n m

(5)

Let m_m′ = 0. When polishing at the point M, the angle that the C axis needs to rotate can be expressed as

θ m = arctan ( − m m l m )

(6)

Then, after the point M rotates around the C axis, the calculation formula of the point M′ is M′ = Rot(z, θ_m)·M, which gives

[ x m ′ y m ′ z m ′ 1 ] = [ cos θ m − sin θ m 0 0 sin θ m cos θ m 0 0 0 0 1 0 0 0 0 1 ] • [ x m y m z m 1 ]

(7)

Finally, the formula for calculating the coordinate of the point M′ is

{ x m ′ = x m · cos θ m − y m · sin θ m y m ′ = x m · sin θ m + y m · cos θ m z m ′ = z m

(8)

When calculating the interference-free angle of the point M, the point set of the polished blade, adjacent blade 1, adjacent blade 2, inlet edge, and exhaust edge must also turn the same angle. By using the same calculation method, we can calculate the coordinate transformation formulas of point set A_i′, B_i′, C_i′, D_i′, E_i′, and F_i′. The formulas are shown in Table 1.

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Table 1.

Coordinate transformation formulas of point set of the inlet and exhaust edge.

Ai′ = Rot(z, θm)·Ai

Bi′ = Rot(z, θm)·Bi

Ci′ = Rot(z, θm)·Ci

Di′ = Rot(z, θm)·Di

Ei′ = Rot(z, θm)·Ei

Fi′ = Rot(z, θm)·Fi

Selection of extreme points on the inlet and exhaust edge

When calculating the interference-free angle of the point M′ after rotation, we need to select the extreme points of the inlet and exhaust edge of the polished blade and the adjacent blade corresponding to the point M′, which are recorded as A_m′, B_m′, C_m′, D_m′, E_m′, and F_m′, from the point sets of the inlet and exhaust edge of the blade after coordinate transformation, which are A_i′, B_i′, C_i′, D_i′, E_i′, and F_i′. The width of the belt should be considered in the calculation of interference-free wrap angle. Taking the selection of the extreme point B_m′ from the exhaust edge of adjacent blade 1 as an example, we will introduce a method of selecting extreme points of the polished blades and adjacent blades. When polishing at the point M′, the point with the largest Z coordinate value in the point set B_i′ first interferes with the belt. Therefore, the point with the maximum Z coordinate value in the point set B_i′ is selected as the extreme point B_m′, as shown in Figure 8.

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Figure 8.

Schematic diagram about extreme point selection of the exhaust edge of adjacent blade 1.

According to the same method, we can calculate the extreme points A_m′, C_m′, D_m′, E_m′, and F_m′ of the inlet and exhaust edge of the polished blade, adjacent blade 1, and adjacent blade 2.

Solution of interference-free angle

As shown in Figure 9, the critical interference-free angles of the belt can be calculated by the mathematical algorithms. The concentric circle in the figure is a simplified diagram of the contact wheel and the belt. In XOZ coordinate system, taking the calculation process of α₂ as an example, this paper introduces a calculation method of critical interference-free angle of abrasive belt. In XOZ coordinate system, the coordinate of point M′ is (x_m′, z_m′) and the coordinate of the normal vector N_m′ is (l_m′, n_m′). The contact wheel radius R_j = 7 mm and the belt thickness h = 0.5 mm. The formula for the coordinate of center point O on the contact wheel can be expressed as

{ x o = x m ′ + l m ′ · ( R j + h ) z o = z m ′ + n m ′ · ( R j + h )

(9)

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Figure 9.

Calculation schematic diagram of critical interference-free angles. (a) Calculation schematic diagram of critical interference-free angles α₁, α₂, and γ₁. (b) Calculation schematic diagram of critical interference-free angles β₁, β₂, and δ₁.

Let the vector from point O to extreme point B_m′ on the exhaust edge of adjacent blade be OB_m ′. The modulus and coordinate of the vector OB_m ′ can be calculated based on the coordinate of the point O and the point B_m′, which is given by

{ | O B m ′ | = ( x bm ′ − x o ) 2 + ( z bm ′ − z o ) 2 O B m ′ = ( l ob ′ , n ob ′ ) = ( x bm ′ − x o | O B m ′ | , z bm ′ − z o | O B m ′ | )

(10)

Let the angle between the vector Nm ′ and the vector OB_m ′ be ∠Nm′ OB_m′, which is given by

∠ Nm ′ O B m ′ = arccos ( l m ′ · l ob ′ + n m ′ · n ob ′ l m ′ 2 + n m ′ 2 + l ob ′ 2 + n ob ′ 2 )

(11)

Let the angle between the vector OB_m ′ and the vector O_BB_m ′ be ∠OB_m′O_B, which is given by

∠ O B m ′ O B = arcsin ( R j | O B m ′ | )

(12)

The critical angle α₂ can be expressed as

α 2 = ∠ Nm ′ O B m ′ − ∠ O B m ′ O B

(13)

Using the same method described above, we can calculate the remaining critical interference-free angles α1, α2, β1, β2, γ1, and δ1, and summarize the calculation formulas in Table 2.

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Table 2.

Calculation formulas of the critical interference-free angles.

Angles	Formulas
α1	α 1 = ∠ Nm ′ O A m ′ − ∠ O A m ′ O A
α2	α 2 = ∠ Nm ′ O B m ′ − ∠ O B m ′ O B
β1	β 1 = ∠ Nm ′ O C m ′ − ∠ O C m ′ O C
β2	β 2 = ∠ Nm ′ O D m ′ − ∠ O D m ′ O D
γ1	γ 1 = ∠ Nm ′ O E m ′ − ∠ O E m ′ O E
δ1	δ 1 = ∠ Nm ′ O F m ′ − ∠ O F m ′ O F