阅读说明：本技术 一种确保矩形环件轧制过程环件刚度的方法 (Method for ensuring ring rigidity in rolling process of rectangular ring ) 是由 谢丹 徐戊矫 王雨 陈锐 于 2021-01-14 设计创作，主要内容包括：本发明公开了一种确保矩形环件轧制过程环件刚度的方法,步骤包括：1)确定环件的径向轧制工艺参数；2)建立环件径向进给速度与环件几何形状和尺寸关系；3)基于流函数计算环件自由侧表面形状；4)建立矩形环件刚度模型；5)建立环件轧制过程中径轴向协调进给策略,计算出环件轧制当前转数对应的环件轴向每转进给量ΔB-n；6)输出环件轧制当前转数对应的环件轴向每转进给量ΔB-n,对环件进行轧制；本发明通过建立矩形环件刚度模型,建立环件轧制过程中径轴向协调进给策略,计算出环件轧制当前转数对应的环件轴向每转进给量,保证了环件在轧制过程中的刚度和精度。(The invention discloses a method for ensuring ring rigidity in a rolling process of a rectangular ring, which comprises the following steps: 1) determining the radial rolling technological parameters of the ring piece; 2) establishing the relation between the radial feeding speed of the ring piece and the geometric shape and the size of the ring piece; 3) calculating the shape of the free side surface of the ring part based on the flow function; 4) establishing a rectangular ring rigidity model; 5) establishing a radial and axial coordinated feeding strategy in the ring rolling process, and calculating the axial feeding quantity delta B per revolution of the ring corresponding to the current revolution of the ring rolling n (ii) a 6) Axial feed amount delta B per revolution of ring corresponding to current revolution of rolling output ring n Rolling the ring piece; according to the invention, the radial and axial coordinated feeding strategy in the ring rolling process is established by establishing the rectangular ring rigidity model, and the axial feeding amount per revolution of the ring corresponding to the current revolution of the ring rolling is calculated, so that the rigidity and the precision of the ring in the rolling process are ensured.)

1. A method for ensuring the rigidity of a ring in a rolling process of a rectangular ring is characterized by comprising the following steps:

1) determining the radial rolling technological parameters of the ring piece;

2) establishing the relation between the radial feeding speed of the ring piece and the geometric shape and the size of the ring piece;

3) and calculating the shape of the free side surface of the ring part based on the flow function.

4) Establishing a rectangular ring rigidity model;

5) establishing a radial and axial coordinated feeding strategy in the ring rolling process, and calculating the axial feeding quantity delta B per revolution of the ring corresponding to the current revolution of the ring rolling_n；

6) Axial feed amount delta B per revolution of ring corresponding to current revolution of rolling output ring_nAnd rolling the ring piece.

2. The method for ensuring the rigidity of the ring in the rolling process of the rectangular ring according to claim 1, wherein the method comprises the following steps: in the step 1), the radial rolling process parameters of the ring piece comprise the initial outer diameter D of the ring blank₀Initial inner diameter d of ring blank₀Initial wall thickness H of ring blank₀Radial feeding speed v (t) of the core roller, shear yield strength k of the material, friction coefficient mu of the ring and the roller, and yield stress sigma of the material_sFriction factor m between the ring piece and the roller, conical roller vertex angle gamma, included angle theta between the guide roller and the z axis, and conical roller radius R corresponding to the contact between the conical roller and the outer surface of the ring piece₆The linear velocity V at which the drive roller rotates.

3. Method for ensuring ring stiffness in a rectangular ring rolling process according to claim 3, wherein in step 2), the calculation of the ring radial feed speed versus ring geometry and dimensions specifically comprises the steps of:

2.1) calculating the rotation time of the ring piece per revolution according to the rotation linear speed of the driving roller:

in the formula (1), n is the rotation number of the current ring piece; t is_nThe rotating time of the ring piece at the current rotating speed is taken as the rotating time of the ring piece at the current rotating speed; d_n-1The outer diameter of the ring piece corresponding to the previous revolution;

2.2) calculating the rolling time of the ring according to the rotation time of the ring per revolution:

in the formula (2), t_nThe rolling time of the ring piece when the current revolution is finished is taken as the rolling time of the ring piece;

2.3) calculating the radial feed per revolution according to the radial feed speed of the core roller:

in the formula (3), t_n-1The rolling time of the ring piece when the previous revolution is started;

2.4) feed quantity Deltah per revolution according to radial direction_nCalculating the wall thickness of the ring deformation zone per revolution:

in the formula (4), H_n(0) The wall thickness of the ring piece corresponding to the position where x is 0 at the current revolution;

2.5) calculating the inner diameter and the outer diameter of the ring piece according to the feeding speed of the core roller:

in formulae (5) to (7), v (t)_n) The core roller feeding speed when the current revolution is finished; d_nThe outer diameter of the ring piece corresponding to the current revolution; dn is the inner diameter of the ring corresponding to the current revolution; d_a，nThe average diameter of the ring piece corresponding to the current revolution;

2.6) calculating the inner and outer radiuses of the ring piece according to the inner and outer diameters of the ring piece:

in formulae (8) to (10), R_nThe initial outer radius of the ring piece corresponding to the current revolution; r is_nThe initial inner radius of the ring piece corresponding to the current revolution; r_a，nThe average radius of the ring piece corresponding to the forward rotation number;

2.7) calculating the contact arc length of the radial deformation zone according to the radial feed per revolution and the inner and outer radii of the ring piece:

in the formula (11), R₁Is the radius of the drive roller; r₂Is the radius of the core roll; l is_j，nJ is (1,2) and respectively represents a deformation zone I and a deformation zone II in the radial deformation zone of the ring piece, wherein the contact arc length of the roll and the ring piece corresponds to the current deformation zone and the number of revolutions;

2.8) calculating the widths of different deformation zones according to the geometrical relation of ring rolling feed:

H_n(0)＝h_1，n(0)+h_2，n(0) (14)

h_1，n(L_1，n)＝h_1，n-1(0) (15)

h_2，n(L_1，n)＝h_2，n-1(0) (16)

in formulae (12) to (16), Δ h_1，n(0) Feeding amount per revolution of a deformation area I corresponding to the position where x is 0 at the current revolution; Δ h_2，n(0) Feeding amount per revolution of a deformation area II corresponding to the position where x is 0 at the current revolution; h is_1，n(0) The width of a deformation area I corresponding to the position where x is 0 at the current revolution; h is_2，n(0) The width of a deformation area II corresponding to the position where x is 0 at the current revolution; h is_1，n-1(0) The width of a deformation area I corresponding to the position where x is 0 in the previous rotation; h is_2，n-1(0) The width of a deformation area II corresponding to the position where x is 0 in the previous rotation; h is_1，n(L_1，n) For x ═ L at the current revolution_1，nThe width of a deformation area I corresponding to the position; h is_2，n(L_1，n) For x ═ L at the current revolution_1，nThe width of a deformation area II corresponding to the position;

2.9) establishing equations of the width of different deformation zones of the ring, the height of an outlet and the free side surface:

in formulae (17) to (19), h_j，n(x) A width equation corresponding to the current deformation area and the revolution number; k_j，n(z) is a height equation of the current deformation zone and the outlet corresponding to the position where x is 0 in the number of revolutions; b is_n，minThe minimum height of an outlet of a deformation area corresponding to the current revolution; c. C_j，nThe coefficients corresponding to the front deformation zone and the number of revolutions;a free side surface equation corresponding to the current deformation zone and the number of revolutions; k_j，n-1(z) is an outlet height equation corresponding to the current deformation zone and the previous rotation; a is_j，nThe coefficient corresponding to the current deformation zone and the revolution number; k (z) is an equation of cross-sectional height at the outlet corresponding to the position where x is 0; h is_j，n(L_j，n) For x ═ L at the current revolution_1，nThe width of the deformation zone corresponding to the position.

4. The method for ensuring the ring stiffness in the rectangular ring rolling process according to claim 1, wherein in the step 3), the calculating the ring free-side surface shape based on the flow function specifically comprises the following steps:

3.1) establishing a flow function of the ring section according to the section shape of the radial deformation zone of the middle ring:

in the formula (20), A_j，nThe coefficient corresponding to the current deformation zone and the revolution number;

3.2) calculating the speed field of ring rolling based on the incompressible condition of the speed field and the orthogonal of the streamline on the flow function and the normal of the curve:

in the formula (21), u_j，n，xThe material flow speed in the x direction corresponding to the current deformation zone and the revolution; u. of_j，n，yThe material flow speed in the y direction corresponding to the current deformation area and the revolution number; u. of_j，n，zThe material flow speed in the z direction corresponding to the current deformation zone and the number of revolutions;

3.3) establishing a total power equation of a radial deformation zone of the ring rolling:

W_n＝W_1，n+W_2，n (22)

W_j，n＝W_1，j，n+W_2，j，n+W_3，j，n (23)

in formulae (22) to (23), W_nThe total power of the radial deformation zone for rolling the ring; w_1，nThe total power of a radial I deformation zone for rolling the ring piece; w_2，nThe total power of a radial II deformation zone for rolling the ring piece; w_1，j，nThe plastic deformation power corresponding to the current deformation area and the revolution; w_2，j，nThe friction power of the inner surface and the outer surface of the ring piece corresponding to the current deformation zone and the revolution number; w_3，j，nShearing power for the speed discontinuity of the inlet and outlet of the ring part corresponding to the current deformation zone and the revolution;

3.4) calculating the power in step 3.3):

in the formulae (24) to (26), V' is the volume of the radial deformation region, ε is the strain of the deformation region, S is the cross-sectional area of the radial deformation region, and S_j，n，0The area of the outlet section corresponding to the current deformation zone and the revolution; s_j，n，1The cross section area of the inlet corresponding to the current deformation area and the revolution; Δ v_j，n，0The outlet section speed corresponding to the current deformation zone and the revolution; Δ v_j，n，1The inlet section speed corresponding to the current deformation zone and the revolution; s_j，n，fThe contact area of the roller and the ring piece corresponding to the current deformation zone and the revolution; Δ v_j，n，fThe contact surface speed of the roller and the ring piece is corresponding to the current deformation zone and the revolution;

3.5) calculating the speed in step 3.4):

3.6) solving a coefficient corresponding to the minimum power based on an energy minimum principle so as to obtain an equation of the free side surface of the ring part:

in the formula (30), K_1，n(0) Is composed ofThe section height of the current revolution at the outlet of the deformation I area corresponding to the position where z is 0; k_2，n(0) The cross-sectional height at the exit of the deformation II zone corresponding to the position z-0 is given for the current revolution.

5. The method for ensuring the ring stiffness in the rectangular ring rolling process according to claim 1, wherein in the step 4), establishing a rectangular ring stiffness model specifically comprises the following steps:

4.1) calculating the force of the roller on the ring in the ring rolling process:

in formulae (31) to (34), P_1，nThe force of the driving roller corresponding to the current revolution to the ring piece is obtained; p_2，nThe force of the core roller to the ring piece corresponding to the current revolution; h_n(L_j，n) For x ═ L at the current revolution_j，nWidth of radial deformation zone, P, corresponding to position_5，nThe force of the upper conical roller corresponding to the current revolution to the ring piece is obtained; p_6，nThe force of the lower conical roller corresponding to the current revolution to the ring piece is obtained; p_3，nThe force of the guide roll on the ring piece at the outlet side corresponding to the current revolution is obtained; p_4，nThe force of the guide roll on the ring piece at the inlet side corresponding to the current revolution number; s_nThe contact area of the conical roller corresponding to the current revolution and the end face of the ring piece is obtained; delta B_n-1Corresponds to the previous rotationThe axial feed per rotation of the conical roller; alpha is alpha_1，nThe contact angle of the driving roller and the ring piece corresponding to the current revolution number; alpha is alpha_2，nThe contact angle between the core roller and the ring piece is corresponding to the current revolution; alpha is alpha_5，nThe contact angle between the upper conical roller and the ring piece is corresponding to the current revolution; alpha is alpha_6，nThe contact angle between the lower conical roller and the ring piece is corresponding to the current revolution;

4.2) angle calculation:

in formulae (35) to (38), R₆The radius of the conical roller corresponding to the contact between the conical roller and the outer surface of the ring piece; r₅The radius of the conical roller corresponding to the contact between the conical roller and the inner surface of the ring piece, q_1，nthe contact width of the ring piece in the I area of the outer deformation area of the ring piece corresponding to the current revolution with the conical roller is set; q. q.s_2，nThe contact width of the ring piece in the deformation zone II in the ring piece corresponding to the current revolution and the conical roller is set;

4.3) determining the ring rolling rigidity condition:

in formula (39), M_nBending moment of a radial deformation zone of the ring corresponding to the current revolution; w_nThe bending resistance section coefficient of the section of the ring piece corresponding to the current revolution, wherein,

M_n＝P_3，nR_a.nsinθ-P_4，nR_a.nsinθ+μP_5，nD_a，n-P_5.nD_a，n-P_6，nD_a，n (41)

in formulae (39) to (41), K_1，n(z) is an outlet section height equation of a deformation I area corresponding to the current revolution; k_2，n(z) is a section height equation at the outlet of the deformation II area corresponding to the current revolution.

6. The method for ensuring the rigidity of the ring in the rectangular ring rolling process according to claim 1, wherein in the step 5), a radial and axial coordinated feeding strategy in the ring rolling process is determined, and the axial feeding amount per revolution Δ B of the ring corresponding to the current number of revolutions of the ring rolling is calculated_nThe method specifically comprises the following steps:

5.1) calculating the axial contact area of the conical roller and the ring piece based on the shape of the free side surface of the ring piece:

5.2) calculating the axial feed per revolution; the method specifically comprises the following steps:

5.2.1) limiting rigidity conditions according to ring rollingCalculating the current number of revolutionsAxial maximum contact area S_n，maxAnd has the following components:

S_n≤S_n，max (43)

5.2.2) the maximum limit heights of the ring pieces in the axial deformation zones corresponding to different deformation zones under the current revolution are equal:

Y_1，n，max＝Y_2，n，max (44)

K_n(h_j，n(0)-q_j，n，max)＝K_n(h_j，n(0)-q_j，n，max) (45)

5.2.3) calculating a formula (42) according to the axial contact area, a formula (45) of the highest limit height of the ring piece in the axial deformation zone and the axial maximum contact area S_n，maxCalculating the maximum value q of the contact width between the ring piece and the conical roller corresponding to the current revolution_1，n，maxAnd q is_2，n，max；

5.2.4) calculating the axial maximum feeding amount of the ring corresponding to the current revolution:

ΔB_n，max＝Y_1，n，max-Y_{1，n-1，max} (46)

in the formula (46), Y_1，n，maxFor the current number of revolutions and the maximum axial height, Y, of the ring corresponding to zone I of the deformation zone_{1，n-1，max}The axial maximum height of the ring piece corresponding to the previous revolution and the deformation zone I is obtained;

5.2.5) calculating the maximum feeding amount per axial rotation of the ring piece as follows:

ΔB_n＝gΔB_n，max (47)

in the formula (47), g represents a safety factor, and g is 0.5 to 0.8.

Technical Field

The invention belongs to the technical field of rolling processes, and particularly relates to a method for ensuring the rigidity of a ring piece in a rectangular ring piece rolling process.

Background

The large ring piece is widely applied to the industrial fields of aerospace, energy, automobiles, ships, chemical engineering and the like, and because the large ring piece is large in size and large in required rolling force, the ring piece rolling needs to meet the more extreme rigidity condition, and the rigidity of the ring piece in the ring piece rolling process cannot be ensured by the conventional rolling process.

Therefore, a method capable of ensuring the rigidity of the ring in the rolling process of the rectangular ring is needed in the prior art.

Disclosure of Invention

The technical solution adopted for the purpose of the present invention is that a method for ensuring the rigidity of a ring in a rolling process of a rectangular ring comprises the following steps:

1) and determining the radial rolling process parameters of the ring.

2) The ring radial feed speed is related to the ring geometry and size.

3) And calculating the shape of the free side surface of the ring part based on the flow function.

4) And establishing a rectangular ring rigidity model.

5) Establishing a radial and axial coordinated feeding strategy in the ring rolling process, and calculating the axial feeding quantity delta B per revolution of the ring corresponding to the current revolution of the ring rolling_n。

6) Axial feed amount delta B per revolution of ring corresponding to current revolution of rolling output ring_nAnd rolling the ring piece.

Further, in the step 1), the radial rolling process parameters of the ring piece comprise the initial outer diameter D of the ring blank₀Initial inner diameter d of ring blank₀Initial wall thickness H of ring blank₀Radial feeding speed v (t) of the core roller, shear yield strength k of the material, friction coefficient mu of the ring and the roller, and yield stress sigma of the material_sFriction factor m between the ring piece and the roller, conical roller vertex angle gamma, included angle theta between the guide roller and the z axis, and conical roller radius R corresponding to the contact between the conical roller and the outer surface of the ring piece₆The linear velocity V at which the drive roller rotates.

Further, in step 2), the step of calculating the relationship between the radial feed speed of the ring and the geometry and the size of the ring specifically comprises the following steps:

2.1) calculating the rotation time of the ring piece per revolution according to the rotation linear speed of the driving roller:

in the formula (1), n is the number of rotation of the current ring. T is_nThe rotating time of the ring piece at the current rotating speed is shown. D_n-1The outer diameter of the ring corresponding to the previous revolution.

2.2) calculating the rolling time of the ring according to the rotation time of the ring per revolution:

in the formula (2), t_nThe ring rolling time at the end of the current revolution.

2.3) calculating the radial feed per revolution according to the radial feed speed of the core roller:

in the formula (3), t_n-1The ring rolling time at the beginning of the previous revolution.

2.4) feed quantity Deltah per revolution according to radial direction_nCalculating the wall thickness of the ring deformation zone per revolution:

in the formula (4), H_n(0) And the ring wall thickness corresponding to the position where x is 0 at the current revolution.

2.5) calculating the inner diameter and the outer diameter of the ring piece according to the feeding speed of the core roller:

in formulae (5) to (7), v (t)_n) The core roller feed speed at the end of the current revolution. D_nThe outer diameter of the ring corresponding to the current revolution. d_nThe inner diameter of the ring corresponding to the current number of revolutions. D_a,nThe average diameter of the ring corresponding to the current number of revolutions.

2.6) calculating the inner and outer radiuses of the ring piece according to the inner and outer diameters of the ring piece:

in formulae (8) to (10), R_nThe initial outer radius of the ring corresponding to the current number of revolutions. r is_nThe initial inner radius of the ring corresponding to the current revolution. R_a,nThe average radius of the ring corresponding to the number of turns before.

2.7) calculating the contact arc length of the radial deformation zone according to the radial feed per revolution and the inner and outer radii of the ring piece:

in the formula (11), R₁Is the radius of the drive roller. R₂The radius of the core roll. L is_j,nJ is (1,2) and respectively indicates a deformation zone I and a deformation zone II in the radial deformation zone of the ring member, wherein the contact arc length of the roller and the ring member corresponds to the current deformation zone and the rotation number.

2.8) calculating the widths of different deformation zones according to the geometrical relation of ring rolling feed:

H_n(0)＝h_1,n(0)+h_2,n(0) (14)

h_1,n(L_1,n)＝h_1,n-1(0) (15)

h_2,n(L_1,n)＝h_2,n-1(0) (16)

in formulae (12) to (16), Δ h_1,n(0) And feeding per revolution of the deformation zone I corresponding to the position where x is 0 at the current revolution. Δ h_2,n(0) And feeding amount per revolution of a deformation zone II corresponding to the position where x is 0 at the current revolution. h is_1,n(0) And the width of the deformation region I corresponding to the position where x is 0 at the current revolution. h is_2,n(0) And the width of the deformation zone II corresponding to the position where x is 0 at the current revolution. h is₁,_n-1(0) The width of the deformation zone I corresponding to the position where x is 0 in the previous rotation. h is_2,n-1(0) The width of the deformation zone II corresponding to the position where x is 0 in the previous rotation is obtained. h is_1,n(L_1,n) For x ═ L at the current revolution_1,nAnd the width of the deformation zone I corresponding to the position. h is_2,n(L_1,n) For x ═ L at the current revolution_1,nAnd the width of the deformation zone II corresponding to the position.

2.9) establishing equations of the width of different deformation zones of the ring, the height of an outlet and the free side surface:

in formulae (17) to (19), h_j,n(x) The width equation corresponding to the current deformation zone and the number of revolutions. K_j,n(z) is the height equation of the current deformation zone and the outlet corresponding to the position where x is 0. B is_n,_minThe minimum height of the outlet of the deformation zone corresponding to the current revolution. c. C_j,nThe coefficient corresponding to the front deformation zone and the number of revolutions.The equation for the free-side surface for the current deformation zone and the number of revolutions. K_j,n-1(z) is the exit height equation for the current deformation zone and the previous revolution. a is_j,nThe current deformation zone is the coefficient corresponding to the number of revolutions. K (z) is an equation for the cross-sectional height at the outlet corresponding to the position where x is 0. h is_j,n(L_j,n) For x ═ L at the current revolution₁,_nThe width of the deformation zone corresponding to the position.

Further, in step 3), the calculation of the free-side surface shape of the ring based on the flow function specifically comprises the following steps:

3.1) establishing a flow function of the ring section according to the shape of the ring radial deformation zone section:

in the formula (20), A_j，nThe current deformation zone is the coefficient corresponding to the number of revolutions.

3.2) calculating the speed field of ring rolling based on the incompressible condition of the speed field and the orthogonal of the streamline on the flow function and the normal of the curve:

in the formula (21), u_j，n，xThe x-direction material flow rate corresponding to the current deformation zone and the number of revolutions. u. of_j，_n，yThe current deformation zone and the y-direction material flow velocity corresponding to the number of revolutions. u. of_j，n，zThe z-direction material flow velocity corresponding to the current deformation zone and the number of revolutions.

3.3) establishing a total power equation of a radial deformation zone of the ring rolling:

W_n＝W_1,n+W_2,n (22)

W_j,n＝W_1，j,n+W_2,j,n+W_3,j,n (23)

in formulae (22) to (23), W_nThe total power of the radial deformation zone for rolling the ring. W_1,nThe total power of the radial I deformation zone for rolling the ring. W_2,nThe total power of the radial II deformation zone of the ring rolling is obtained. W_1，j,nThe plastic deformation power corresponding to the current deformation zone and the number of revolutions. W_2,j,nAnd the friction power of the inner surface and the outer surface of the ring corresponding to the current deformation zone and the revolution number. W_3,j,nAnd the shearing power of the speed discontinuity of the inlet and outlet of the ring corresponding to the current deformation zone and the revolution.

3.4) calculating the power in step 3.3):

in the formulae (24) to (26), V' is the volume of the radial deformation region. ε is the strain in the deformation zone. S is the cross-sectional area of the radial deformation zone. s_j,n,0The outlet cross-sectional area corresponding to the current deformation zone and the number of revolutions. s_j,n,1The cross-sectional area of the inlet corresponding to the current deformation zone and the number of revolutions. Δ v_j,n,0The exit cross-sectional velocity is the current deformation zone and the corresponding number of revolutions. Δ v_j,n,1The inlet cross-sectional velocity is the current deformation zone and the corresponding revolution. s_j，n，fRolls corresponding to the current deformation zone and number of revolutionsArea of contact with the ring. Δ v_j，n，fThe contact surface speed of the roller and the ring corresponding to the current deformation zone and the number of revolutions.

3.5) calculating the speed in step 3.4):

3.6) solving a coefficient corresponding to the minimum power based on an energy minimum principle so as to obtain an equation of the free side surface of the ring part:

in the formula (30), K_1，n(0) The section height at the outlet of the deformation I zone corresponding to the position where z is 0 is the current revolution. K_2,n(0) The section height at the outlet of the deformation II zone corresponding to the position where z is 0 is taken as the current revolution.

Further, in step 4), establishing a rectangular ring stiffness model specifically includes the following steps:

4.1) calculating the force of the roller on the ring in the ring rolling process:

in formulae (31) to (34), P_1,nThe force of the driving roller corresponding to the current revolution to the ring member. P_2,nThe force of the core roller to the ring corresponding to the current revolution. H_n(L_j,n) For x ═ L at the current revolution_j,_nWidth of radial deformation zone, P, corresponding to position_5,nThe force of the upper conical roller to the ring piece corresponding to the current revolution. P_6,nThe force of the lower conical roller to the ring piece is corresponding to the current revolution. P_3,nAnd the force of the guide roll on the ring piece on the outlet side corresponding to the current revolution is obtained. P_4,nThe force of the guide roll to the ring piece on the inlet side corresponding to the current revolution. S_nThe contact area of the conical roller corresponding to the current revolution and the end face of the ring piece. Delta B_n-1The axial feed amount per revolution of the conical roller corresponding to the previous revolution. Alpha is alpha_1,nThe contact angle of the driving roller and the ring member corresponding to the current number of revolutions. Alpha is alpha_2,nThe contact angle of the core roller and the ring corresponding to the current revolution number. Alpha is alpha₅,_nThe contact angle of the upper conical roller and the ring piece corresponding to the current revolution is shown. Alpha is alpha_6，nThe contact angle of the lower conical roller and the ring piece corresponding to the current revolution is shown.

4.2) angle calculation:

in formulae (35) to (38), R₆The radius of the conical roller is the radius of the conical roller corresponding to the contact between the conical roller and the outer surface of the ring piece. R₅The radius of the conical roller corresponding to the contact between the conical roller and the inner surface of the ring piece, q_1,nthe contact width of the ring in the outer deformation zone I of the ring corresponding to the current revolution with the conical roller. q. q.s_2,nThe contact width of the ring piece in the deformation zone II corresponding to the current revolution with the conical roller.

4.3) determining the ring rolling rigidity condition:

in formula (39), M_nThe bending moment of the radial deformation zone of the ring corresponding to the current revolution is obtained. W_nThe bending resistance section coefficient of the section of the ring piece corresponding to the current revolution, wherein,

M_n＝P_3,nR_a.nsinθ-P_4,nR_a.nsinθ+μP_5,nD_a,n-P_5.nD_a,n-P_6,nD_a,n (41)

in formulae (39) to (41), K_1,n(z) is a section height equation at the outlet of the deformation I area corresponding to the current revolution. K_2,n(z) is a section height equation at the outlet of the deformation II area corresponding to the current revolution.

Further, in step 5), a radial and axial coordinated feeding strategy in the ring rolling process is determined, and the axial feeding amount per revolution delta B of the ring corresponding to the current revolution of ring rolling is calculated_nConcrete bagThe method comprises the following steps:

5.1) calculating the axial contact area of the conical roller and the ring piece based on the shape of the free side surface of the ring piece:

5.2) calculating the axial feed per revolution. The method specifically comprises the following steps:

5.2.1) limiting rigidity conditions according to ring rollingCalculating the axial maximum contact area S corresponding to the current revolution_n,maxAnd has the following components:

S_n≤S_n,max (43)

5.2.2) the maximum limit heights of the ring pieces in the axial deformation zones corresponding to different deformation zones under the current revolution are equal:

Y_1,n,max＝Y_2,n,max (44)

K_n(h_j,n(0)-q_j,n,max)＝K_n(h_j,n(0)-q_j,n,max) (45)

5.2.3) calculating a formula (42) according to the axial contact area, a formula (45) of the highest limit height of the ring piece in the axial deformation zone and the axial maximum contact area S_n,maxCalculating the maximum value q of the contact width between the ring piece and the conical roller corresponding to the current revolution_1,n,maxAnd q is_2,n,max。

5.2.4) calculating the axial maximum feeding amount of the ring corresponding to the current revolution:

ΔB_n,max＝Y_1,n,max-Y_1,n-1,max (46)

in the formula (46), Y_1,n,maxFor the current number of revolutions and the maximum axial height, Y, of the ring corresponding to zone I of the deformation zone_1,n-1,maxThe axial maximum height of the ring corresponding to the previous revolution and the deformation zone I.

5.2.5) calculating the maximum feeding amount per axial rotation of the ring piece as follows:

ΔB_n＝gΔB_n,max (47)

in the formula (47), g represents a safety factor, and g is 0.5 to 0.8.

The method has the advantages that the relation between the radial feeding speed of the ring piece and the geometric shape and the size of the ring piece is established according to the determined radial rolling process parameters of the ring piece, the shape of the free side surface of the ring piece is calculated based on a flow function, a radial and axial coordinated feeding strategy in the ring piece rolling process is established by establishing a rectangular ring piece rigidity model, the axial feeding amount per revolution of the ring piece corresponding to the current revolution of the ring piece rolling is calculated, and the rigidity and the precision of the ring piece in the rolling process are directly guaranteed.

Drawings

Fig. 1 is a schematic view of the stress analysis (a) of the ring rolling and the geometric shapes of the radial deformation zone (b) and the axial deformation zone (c).

Detailed Description

The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.

Example 1:

the embodiment discloses a method for ensuring the rigidity of a ring piece in a rolling process of a rectangular ring piece, wherein a rolling device adopts a horizontal ring rolling machine, and the method comprises the following steps:

1) and determining the radial rolling process parameters of the ring. Wherein the radial rolling technological parameters of the ring piece comprise the initial outer diameter D of the ring blank₀Initial inner diameter d of ring blank₀Initial wall thickness H of ring blank₀Radial feeding speed v (t) of the core roller, shear yield strength k of the material, friction coefficient mu of the ring and the roller, and yield stress sigma of the material_sFriction factor m between the ring piece and the roller, conical roller vertex angle gamma, included angle theta between the guide roller and the z axis, and conical roller radius R corresponding to the contact between the conical roller and the outer surface of the ring piece₆The linear velocity V at which the drive roller rotates.

2) The ring radial feed speed is calculated in relation to the ring geometry and size. The method comprises the following steps:

2.1) calculating the rotation time of the ring piece per revolution according to the rotation linear speed of the driving roller:

in the formula (1), n is the number of rotation of the current ring. T is_nThe rotating time of the ring piece at the current rotating speed is shown. D_n-1The outer diameter of the ring corresponding to the previous revolution.

2.2) calculating the rolling time of the ring according to the rotation time of the ring per revolution:

in the formula (2), t_nThe ring rolling time at the end of the current revolution.

2.3) calculating the radial feed per revolution according to the radial feed speed of the core roller:

in the formula (3), t_n-1The ring rolling time at the beginning of the previous revolution.

2.4) feed quantity Deltah per revolution according to radial direction_nCalculating the wall thickness of the ring deformation zone per revolution:

in the formula (4), H_n(0) And the ring wall thickness corresponding to the position where x is 0 at the current revolution.

2.5) calculating the inner diameter and the outer diameter of the ring piece according to the feeding speed of the core roller:

in formulae (5) to (7), v (t)_n) The core roller feed speed at the end of the current revolution. D_nThe outer diameter of the ring corresponding to the current revolution. d_nThe inner diameter of the ring corresponding to the current number of revolutions. D_a,nThe average diameter of the ring corresponding to the current number of revolutions.

2.6) calculating the inner and outer radiuses of the ring piece according to the inner and outer diameters of the ring piece:

in formulae (8) to (10), R_nThe initial outer radius of the ring corresponding to the current number of revolutions. r is_nThe initial inner radius of the ring corresponding to the current revolution. R_a,nThe average radius of the ring corresponding to the number of turns before.

2.7) calculating the contact arc length of the radial deformation zone according to the radial feed per revolution and the inner and outer radii of the ring piece:

in the formula (11), R₁Is the radius of the drive roller. R₂The radius of the core roll. L is_j,nRolls and rings corresponding to the current deformation zone and number of revolutionsJ ═ 1,2, which respectively represent the zone I and the zone ii of deformation in the radial deformation zone of the ring in fig. 1.

2.8) calculating the widths of different deformation zones according to the geometrical relation of ring rolling feed:

H_n(0)＝h_1,n(0)+h_2,n(0) (14)

h_1,n(L_1,n)＝h_1,n-1(0) (15)

h_2,n(L_1，n)＝h_2，n-1(0) (16)

in formulae (12) to (16), Δ h_1,n(0) And feeding per revolution of the deformation zone I corresponding to the position where x is 0 at the current revolution. Δ h_2，n(0) And feeding amount per revolution of a deformation zone II corresponding to the position where x is 0 at the current revolution. h is_1，n(0) And the width of the deformation region I corresponding to the position where x is 0 at the current revolution. h is_2，n(0) And the width of the deformation zone II corresponding to the position where x is 0 at the current revolution. h is_1，n-1(0) The width of the deformation zone I corresponding to the position where x is 0 in the previous rotation. h is_2，n-1(0) The width of the deformation zone II corresponding to the position where x is 0 in the previous rotation is obtained. h is_1,n(L_1,n) For x ═ L at the current revolution_1,nAnd the width of the deformation zone I corresponding to the position. h is_2,n(L_1,n) For x ═ L at the current revolution_1,nAnd the width of the deformation zone II corresponding to the position.

2.9) establishing equations of the width of different deformation zones of the ring, the height of an outlet and the free side surface:

in formulae (17) to (19), h_j,n(x) The width equation corresponding to the current deformation zone and the number of revolutions. K_j,n(z) is the height equation of the current deformation zone and the outlet corresponding to the position where x is 0. B is_n,minThe minimum height of the outlet of the deformation zone corresponding to the current revolution. c. C_j，nThe coefficient corresponding to the front deformation zone and the number of revolutions.The equation for the free-side surface for the current deformation zone and the number of revolutions. K_j,n-1(z) is the exit height equation for the current deformation zone and the previous revolution. a is_j,nThe current deformation zone is the coefficient corresponding to the number of revolutions. K (z) is the cross-sectional height equation at the exit corresponding to the x-0 position. h is_j,n(L_j,n) For x ═ L at the current revolution_1,nThe width of the deformation zone corresponding to the position.

3) And calculating the shape of the free side surface of the ring part based on the flow function. The method specifically comprises the following steps:

3.1) establishing a flow function of the ring section according to the ring radial deformation zone section shape in FIG. 1 (b):

in the formula (20), A_j,nThe current deformation zone is the coefficient corresponding to the number of revolutions.

3.2) calculating the speed field of ring rolling based on the incompressible condition of the speed field and the orthogonal of the streamline on the flow function and the normal of the curve:

in the formula (21), u_j，n，xThe x-direction material flow rate corresponding to the current deformation zone and the number of revolutions. u. of_j，n,yThe current deformation zone and the y-direction material flow velocity corresponding to the number of revolutions. u. of_j,n，zThe z-direction material flow velocity corresponding to the current deformation zone and the number of revolutions.

3.3) establishing a total power equation of a radial deformation zone of the ring rolling:

W_n＝W_1,n+W_2,n (22)

W_j，n＝W_1,j,n+W_2，j，n+W_3,j,n (23)

in formulae (22) to (23), W_nThe total power of the radial deformation zone for rolling the ring. W_1,nThe total power of the radial I deformation zone for rolling the ring. W_2,nThe total power of the radial II deformation zone of the ring rolling is obtained. W_1,j，nThe plastic deformation power corresponding to the current deformation zone and the number of revolutions. W_2，j，nAnd the friction power of the inner surface and the outer surface of the ring corresponding to the current deformation zone and the revolution number. W_3，j，nAnd the shearing power of the speed discontinuity of the inlet and outlet of the ring corresponding to the current deformation zone and the revolution.

3.4) calculating the power in step 3.3):

in the formulae (24) to (26), V' is the volume of the radial deformation region. ε is the strain in the deformation zone. S is the cross-sectional area of the radial deformation zone. s_j，n,0The outlet cross-sectional area corresponding to the current deformation zone and the number of revolutions. s_j,n,1The cross-sectional area of the inlet corresponding to the current deformation zone and the number of revolutions. Δ v_j,n，0The exit cross-sectional velocity is the current deformation zone and the corresponding number of revolutions. Δ v_j,n,1The inlet cross-sectional velocity is the current deformation zone and the corresponding revolution. s_j,n,fThe contact area of the roller and the ring corresponding to the current deformation zone and the number of revolutions. Δ v_j,n,fThe contact surface speed of the roller and the ring corresponding to the current deformation zone and the number of revolutions.

3.5) calculating the speed in step 3.4):

3.6) solving a coefficient corresponding to the minimum power based on an energy minimum principle so as to obtain an equation of the free side surface of the ring part:

in the formula (30), K_1,n(0) The section height at the outlet of the deformation I zone corresponding to the position where z is 0 is the current revolution. K_2,n(0) The section height at the outlet of the deformation II zone corresponding to the position where z is 0 is taken as the current revolution.

4) And establishing a rigidity model of the large rectangular ring piece. The method specifically comprises the following steps:

4.1) calculating the force of the roller on the ring in the ring rolling process:

in formulae (31) to (34), P_1,nThe force of the driving roller corresponding to the current revolution to the ring member. P_2,nThe force of the core roller to the ring corresponding to the current revolution. H_n(L_j,n) For x ═ L at the current revolution_j,_nWidth of radial deformation zone, P, corresponding to position_5,nThe force of the upper conical roller to the ring piece corresponding to the current revolution. P_6,nThe force of the lower conical roller to the ring piece is corresponding to the current revolution. P_3,nAnd the force of the guide roll on the ring piece on the outlet side corresponding to the current revolution is obtained. P_4,nThe force of the guide roll to the ring piece on the inlet side corresponding to the current revolution. S_nThe contact area of the conical roller corresponding to the current revolution and the end face of the ring piece. Delta B_n-1The axial feed amount per revolution of the conical roller corresponding to the previous revolution. Alpha is alpha_1,nThe contact angle of the driving roller and the ring member corresponding to the current number of revolutions. Alpha is alpha_2,nThe contact angle of the core roller and the ring corresponding to the current revolution number. Alpha is alpha₅,_nThe contact angle of the upper conical roller and the ring piece corresponding to the current revolution is shown. Alpha is alpha_6,nThe contact angle of the lower conical roller and the ring piece corresponding to the current revolution is shown.

4.2) angle calculation:

in formulae (35) to (38), R₆The radius of the conical roller is the radius of the conical roller corresponding to the contact between the conical roller and the outer surface of the ring piece. R₅The radius of the conical roller corresponding to the contact between the conical roller and the inner surface of the ring piece, q_1,nthe contact width of the ring in the outer deformation zone I of the ring corresponding to the current revolution with the conical roller. q. q.s_2,nThe contact width of the ring piece in the deformation zone II corresponding to the current revolution with the conical roller.

4.3) determining the ring rolling rigidity condition:

in order to ensure the rigidity of the ring in the ring rolling process, the ring rolling needs to meet the rigidity condition:

in formula (39), M_nThe bending moment of the radial deformation zone of the ring corresponding to the current revolution is obtained. W_nThe bending resistance section coefficient of the section of the ring piece corresponding to the current revolution, wherein,

M_n＝P_3,nR_a.nsinθ-P_4，nR_a.nsinθ+μP_5,nD_a,n-P_5.nD_a，n-P_6，nD_a,n (41)

in formulae (39) to (41), K_1，n(z) is a section height equation at the outlet of the deformation I area corresponding to the current revolution. K_2,n(z) is a section height equation at the outlet of the deformation II area corresponding to the current revolution.

5) Determining a radial and axial coordinated feeding strategy in the ring rolling process, and calculating the axial feeding quantity delta B per revolution of the ring corresponding to the current revolution of the ring rolling_n. According to the formulas (36) to (38), the force of the roller on the ring and the geometry of the ring deformation zone are important factors influencing the rigidity of the ring. According to the relations between the radial feeding speed of the core roller and the geometric shape and the size of the ring, the radial feeding speed of the ring is an important factor influencing the geometric shape and the size of the deformation zone of the ring. According to the calculation formulas (29) to (35) of the force of the roller to the ring, the feeding amount per radial rotation and the feeding amount per axial rotation are key factors for determining the force of the roller to the ring. In order to meet the ring stiffness condition, the axial feed per revolution needs to be determined according to the radial feed per revolution. The method specifically comprises the following steps:

5.1) when the radial feeding speed of the ring member is determined, the contact area of the conical roller and the ring member is a key factor for determining whether the rigidity condition of the ring member is met. Calculating the axial contact area of the conical roller and the ring piece based on the shape of the free side surface of the ring piece:

5.2) calculating the axial feed per revolution. The method specifically comprises the following steps:

5.2.1) limiting rigidity conditions according to ring rollingCalculating the axial maximum contact area S corresponding to the current revolution_n,maxAnd has the following components:

S_n≤S_n,max (43)

5.2.2) the maximum limit heights of the ring pieces in the axial deformation zones corresponding to different deformation zones under the current revolution are equal:

Y_1,n,max＝Y_2,n,max (44)

K_n(h_j,n(0)-q_j,n,max)＝K_n(h_j,n(0)-q_j,n,max) (45)

5.2.3) calculating a formula (42) according to the axial contact area, a formula (45) of the highest limit height of the ring piece in the axial deformation zone and the axial maximum contact area S_n,maxCalculating the maximum value q of the contact width between the ring piece and the conical roller corresponding to the current revolution_1,n,maxAnd q is_2,n,max。

5.2.4) calculating the axial maximum feeding amount of the ring corresponding to the current revolution:

ΔB_n,max＝Y_1,n,max-Y_1,n-1,max (46)

in the formula (46), Y_1,n,maxFor the current number of revolutions and the maximum axial height, Y, of the ring corresponding to zone I of the deformation zone_1,n-1,maxThe axial maximum height of the ring corresponding to the previous revolution and the deformation zone I.

5.2.5) the ring member jumping condition caused by instability in the ring member rolling process is considered, a safety factor g is required to be set for the axial feeding amount of the ring member, and the maximum feeding amount of the ring member per axial rotation is calculated as follows:

ΔB_n＝gΔB_n,max (47)

in the formula (47), g represents a safety factor, and g is 0.5 to 0.8.

6) Axial feed amount delta B per revolution of ring corresponding to current revolution of rolling output ring_nAnd rolling the ring piece.

According to the method for ensuring the rigidity of the ring piece in the rectangular ring piece rolling process, the relation between the radial feeding speed of the ring piece and the geometric shape and the size of the ring piece is established according to the determined radial rolling process parameters of the ring piece, the shape of the free side surface of the ring piece is calculated based on a flow function, a radial and axial coordinated feeding strategy in the ring piece rolling process is established by establishing a rectangular ring piece rigidity model, the axial feeding amount per revolution of the ring piece corresponding to the current revolution of the ring piece rolling is calculated, and the rigidity and the precision of the ring piece in the rolling process are directly ensured.

Example 2:

the embodiment discloses a method for ensuring ring rigidity in a rectangular ring rolling process, which comprises the following steps:

1) the axial feeding amount per rotation of the rolling of the 5m 2219 aluminum alloy ring piece is determined.

Initial outer diameter D of ring blank₀3600mm, initial inner diameter d of ring blank₀3140mm, initial wall thickness H of ring blank₀230mm, initial height B of ring blank₀500mm core feed speedThe shear yield strength k of the 2219 aluminum alloy material corresponding to the rolling temperature is 20.2Mpa, the friction coefficient mu of the ring piece and the roller is 0.3, and the yield stress of the 2219 aluminum alloy material corresponding to the rolling temperature is sigma_s35Mpa, the friction factor m between the ring piece and the roller is 0.52, the vertex angle gamma of the conical roller is 17.5 degrees, the included angle theta between the guide roller and the z axis is 45 degrees, and the radius R of the conical roller corresponding to the contact between the conical roller and the outer surface of the ring piece₆The linear speed V of the rotation of the driving roller is 1200mm/s, which is 450 mm.

2) Axial feed calculation for the first revolution of the ring

2.1) calculating the relation between the radial feeding speed of the ring and the geometric shape and the size of the ring

Obtaining the first rotation time of the ring piece according to the rotation linear speed of the driving roller:

obtaining the rolling time of the ring piece according to the first rotation time of the ring piece:

t₁＝T₁＝9.42s (2-2)

obtaining a radial first rotary feeding amount according to the radial feeding speed of the core roller:

obtaining the first rotary wall thickness of the ring deformation zone according to the radial first rotary feeding amount and the initial wall thickness of the ring:

H₁(0)＝H₀-Δh₁＝230-9.1＝220.9mm (2-4)

obtaining the inner diameter and the outer diameter of the first rotating ring piece of the ring piece rotation according to the feeding speed of the core roller:

obtaining the inner and outer radiuses of the first rotating ring piece of the ring piece according to the inner and outer diameters of the ring piece:

obtaining the contact arc length of the ring piece in the first rotation according to the radial feeding amount of each rotation and the inner and outer radiuses of the ring piece:

wherein: j is 1, 2. Respectively showing an outer deformation zone I of the ring member and an outer deformation zone II of the ring member in figure 1; l is_j,1And the contact arc length of the roller corresponding to the first rotation of the ring piece and the ring piece is long.

Obtaining the width of different deformation zones of the first rotation of the ring according to the geometric relation of the rolling feeding of the ring:

h_1,1(0)+h_2,1(0)＝220.9mm (2-14)

h is obtained by solving the equations (13) and (14)_1,1(0)＝48.33mm,h_2,1(0) 172.57 mm. Since the width of the ring varies in the radial deformation zone, the wall thickness at the inlet of the ring is the wall thickness at the outlet of the rotor on the ring:

h_1,1(L_1,1)＝h_1,0(0)＝115mm (2-15)

h_2,1(L_2,1)＝h_2,0(0)＝115mm (2-16)

establishing width equations of different deformation zones corresponding to the first rotation of the ring piece:

establishing a height equation of different deformation zones corresponding to the first rotation of the ring piece:

establishing equations of free side surfaces of different deformation zones corresponding to the first rotation of the ring piece:

2.2) obtaining the shape of the free side surface of the ring based on the flow function.

Establishing a flow function of the ring section of the I deformation zone corresponding to the first revolution of the ring according to the ring section shape of FIG. 1:

establishing a flow function of the cross section of the ring in a deformation zone II corresponding to the first rotation of the ring according to the cross section shape of the ring in the figure 1:

obtaining the velocity field of the I deformation zone corresponding to the first rotation of the ring member based on the incompressible condition of the velocity field and the orthogonality of the streamline on the flow function and the normal of the curve:

and obtaining the velocity field of the II deformation zone corresponding to the first rotation of the ring piece based on the incompressible condition of the velocity field and the orthogonality of the streamline on the flow function and the normal of the curve:

calculating the total power of a radial deformation zone corresponding to the first rotation of the ring piece:

W_j,1＝W_1,1+W_2,1 (2-31)

and (3) carrying the speed fields of the radial deformation zones corresponding to the first rotation of the ring member in the formulas (2-25) - (2-27) and (2-28) - (2-30) into the formulas (27) - (29) to calculate the total power of the radial deformation zones corresponding to the first rotation of the ring member carried into the formulas (2-31). Based on the energy minimization principle, the coefficient c corresponding to the minimum power corresponding to the first rotation of the member is obtained according to the formula (30)_1,1、c_2,1And a minimum height B_j,minSo as to obtain a height equation of an I deformation zone corresponding to the first rotation of the ring member And equation of height of the deformation zone II

2.3) the axial feed corresponding to the first revolution of the ring.

And (3) bringing the formula (2-11) into the formulas (35) to (36) to obtain the corresponding angle of the ring member in the first rotation:

calculating the force of the driving roller corresponding to the first rotation of the ring member on the ring member:

calculating the force of the core roller corresponding to the first rotation of the ring piece on the ring piece:

calculating the force of the conical roller corresponding to the first rotation of the ring piece on the ring piece:

P_5，1＝P_6，1＝S₁σ_s＝35S₁ (2-38)

calculating the force of the guide roll corresponding to the first rotation of the ring piece on the ring piece:

P_3,1＝10.5S₁*cos(α_5，n)-35S₁*sin(α_5，n)-35S₁*sin(α_6，n)+1.8272e6 (2-39)

P_4，1＝13.726S₁*cos(α_5,n)-45.75S₁*sin(α_5,n)-45.75S₁*sin(α_6,n)-743500 (2-40)

calculating the bending moment of the radial deformation zone of the ring piece corresponding to the first rotation of the ring piece according to the force of the roller corresponding to the first rotation of the ring piece on the ring piece:

M₁＝13262.0*S₁*sin(α_5，n)-3979.0*S₁*cos(α_5,n)-207600.0*S₁+13262.0*S₁*sin(α_6,n)+3.1707e9 (2-41)

calculating the bending section coefficient of the section of the ring piece corresponding to the first rotation of the ring piece:

according to the condition of ultimate rigidityThe relation formula (42) of the axial contact area of the ring piece and the maximum contact width of the ring piece and the conical roller is equal to the maximum limit heights of the deformation zones I and I, and the formula (44) -formula (45) are used for calculating the axial maximum contact area S corresponding to the first rotation of the ring piece_1,maxAnd the maximum contact width q of the ring member and the conical roller_1,1,maxAnd q is_2,1,max：

q_1,1,max＝47.88mm (2-43)

q_2,1,max＝168.22mm (2-44)

Obtaining the highest limit height of the ring in the axial deformation zone according to the maximum contact width between the ring corresponding to the first rotation of the ring and the conical roller:

Y_1,1,max＝Y_2,1,max＝K₁(h_1，1(0)-q_1,1，max)＝251.23mm (2-45)

the maximum feed per axial rotation of the ring is thus obtained:

ΔB_1,max＝Y_1,1,max-Y_1,0,max＝251.23-250＝1.23mm (2-46)

setting the safety factor g to be 0.8, and setting the axial feed corresponding to the first rotation of the ring:

ΔB₁＝0.8ΔB_1,max＝0.984mm (2-47)

3) axial feed per revolution in ring rolling process

According to the axial feeding amount calculation method of the first rotation of the ring piece, the axial feeding amount of the second rotation shaft of the ring piece is obtained, and similarly, the axial feeding amount delta B of the third rotation shaft is calculated₃N. the axial feed amount of the shaft Δ B_n。