Multi-dof forming workspace calculation method considering mechanism deformation and motor loading

Abstract

The present application relates to a kind of multi-degree-of-freedom forming workspace calculation method considering mechanism deformation and motor loading, for the workspace of multi-degree-of-freedom forming equipment, comprising the following steps: S1, the geometric workspace of equipment is calculated based on its geometric configuration;S2, the workspace of equipment considering mechanism deformation is calculated by establishing equipment deformation error model;S3, the workspace of equipment considering motor loading characteristics is calculated based on statics model;S4, the comprehensive workspace of equipment is calculated while considering mechanism deformation and motor loading characteristics.The present application can calculate the effective working area of multi-degree-of-freedom forming equipment, so that the equipment realizes complex multi-degree-of-freedom motion under heavy load working condition.

Description

Technical Field

[0001] This invention relates to the field of multi-degree-of-freedom forming, and more specifically, to a method for calculating the working space of multi-degree-of-freedom forming that takes into account mechanism deformation and motor loading. Background Technology

[0002] Multi-degree-of-freedom forming is a process in which the blank is forced to undergo local near-net-shape plastic deformation under multi-degree-of-freedom loading of the upper die, and the metal flows in multiple directions to fill the cavity of the upper die until the target shape of a complex part is achieved. Compared with ordinary single-degree-of-freedom integral forming processes, it has technical advantages such as lower forming force, more complex geometric shapes of formed parts, and better microstructure and mechanical properties. For this process, a heavy-duty parallel machine tool for multi-degree-of-freedom forming has been developed, which achieves multi-degree-of-freedom movement of the upper die through the coupling motion of multiple motion chains. The workspace of the parallel machine tool is usually calculated based on the geometric constraints of the mechanism, and this method is effective for parallel machine tools under light load conditions. However, under heavy load conditions, the parallel machine tool may experience excessive deformation in certain special positions, weakening its stiffness. At the same time, heavy load conditions affect the motor's loading performance, and insufficient motor loading may occur in certain special positions. That is, in addition to geometric constraints, the workspace of the heavy-duty parallel machine tool is also constrained by the deformation of the mechanism and the loading characteristics of the motor. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a method for calculating the working space of a multi-degree-of-freedom forming equipment that takes into account the deformation of the mechanism and the loading of the motor. This method can calculate the effective working area of ​​the multi-degree-of-freedom forming equipment, thereby enabling the equipment to achieve complex multi-degree-of-freedom motion under heavy load conditions.

[0004] The technical solution adopted by this invention to solve its technical problem is: to construct a multi-degree-of-freedom forming workspace calculation method that considers mechanism deformation and motor loading, for calculating the workspace of multi-degree-of-freedom forming equipment, including the following steps:

[0005] S1. Calculate the geometric workspace based on the equipment's geometric configuration;

[0006] S2. By establishing an equipment deformation error model, calculate the working space of the equipment considering the deformation of the mechanism;

[0007] S3. Calculate the working space of the equipment based on the static model, taking into account the motor loading characteristics;

[0008] S4. Simultaneously consider the deformation of the mechanism and the loading characteristics of the motor to calculate the comprehensive workspace of the equipment.

[0009] According to the above scheme, step S1 includes:

[0010] Let the centers of the ball joints on the static platform of the equipment be A1, A2, A3, A4, A5, A6, and the centers of the ball joints on the moving platform of the equipment be B1, B2, B3, B4, B5, B6. Simultaneously, establish a global coordinate system S at the center of the static platform of the equipment. A (O A -x A y A z A ), where z A The axis is perpendicular to the static platform, z A The axis does not change with the movement of the equipment; a dynamic coordinate system S is established on the moving platform of the equipment. B (O B -x B y B z B ), S B The origin and S A The distance between the origins is d, S B The origin and S A The origin is located on the perpendicular line from the center of the static platform, i.e., z B axis and z A The axes are in the same direction; θ Ai (i = 1-6) represents the distribution angle of the ball joint points on the static platform, θ Bi (i = 1-6) represents the distribution angle of the ball joint points on the moving platform;

[0011] Therefore A i In the global coordinate system S A Position vector a i Determined by equation (1):

[0012] a i (a ix )=[a ix ,a ix tanθ Ai ,0] T (35)

[0013] In the formula, a ix The position vector of the ball joint point of the static platform in the global coordinate system is in the x-axis. A Projection on the axis;

[0014] Furthermore, B i In the moving coordinate system S B Position vector b i Determined by equation (2):

[0015] b i '=[r B cosθ Bi ,r B sinθ Bi ,d] T(36)

[0016] In the formula, r B d is the radius of the circumcircle of the ball joint of the moving platform, and d is the height of the upper mold;

[0017] The motion of the equipment platform can be represented by six parameters: x, y, z represent the translational motion of the upper die in three directions, and α, β, γ represent the rotational angles of the upper die in three directions. In the multi-degree-of-freedom forming process, the motion equation of the upper die is:

[0018]

[0019] In the formula, ω is the swing angle of the upper die movement, k is the rotational speed of the upper die movement, h is the feed rate of the upper die movement, and h is the feed distance of the upper die movement.

[0020] Therefore S B Compared to S A rotation matrix Determined by equation (4):

[0021]

[0022] Furthermore, B i In the global coordinate system S A Position vector b i Determined by equation (5):

[0023]

[0024] In the formula, p OB It is S B The origin of the coordinate system relative to S A Position vector; p OB It can be determined by equation (6):

[0025] p OB =[x,y,z+Z P ] T (40)

[0026] In the formula, Z P It is the distance between the end of the upper mold and the center of the stationary platform;

[0027] Based on the constraint that the link length remains constant, a ix Determined by equation (7):

[0028]

[0029] In the formula, l is the length of the connecting rod.

[0030] b ix=d(sinαsinγ+cosαcosγsinβ)-r B sinθ Bi (cosαsinγ-cosγsinαsinβ)+

[0031] r B cosβcosγcosθ Bi

[0032] b iy =r B sinθ Bi (cosαcosγ+sinαsinβsinγ)-d(cosγsinα-cosαsinβsinγ)+

[0033] r B cosβsinγcosθ Bi

[0034] b iz =h+Z P -r B sinβcosθ Bi +cosαcosβd+r B cosβsinαsinθ Bi

[0035] The displacement s of each slider is equipped i Determined by equation (8):

[0036]

[0037] In the formula, r A It is the distance between the initial position of the driving slider and the origin of the global coordinate system;

[0038] The maximum and minimum values ​​of the slider displacement are represented by s. max and s min This means that the geometric constraints satisfying equation (9) are:

[0039] s min ≤s i ≤s max (43)

[0040] When the moving platform of the equipment rotates, the velocity of any point on the moving platform is equal to the velocity of the center of the moving platform plus the angular velocity of that point rotating around the center of the moving platform, and the velocity v of the lower ball joint of the connecting rod. Bi Solve using equation (10):

[0041]

[0042] In the formula, r i It is a vector Position vector in the global coordinate system;

[0043] Based on the velocity projection theorem, the velocity v of the ball joint on the connecting rod is obtained. Ai and the speed of the ball head v Bi The relationship is shown in equation (11):

[0044]

[0045] The velocity v of the ball head under the rod obtained by equation (10) Bi Substituting into equation (11), we get:

[0046]

[0047] Among them, v i It refers to the speed of each driving slider;

[0048] Equation (12) can be expressed in matrix form as shown in equation (13):

[0049] J s [v1,L,v6] T =J p [v p ,ω p ] T (47)

[0050] In the formula

[0051]

[0052]

[0053] The workspace of the multi-degree-of-freedom forming equipment does not contain the upper mold pose as a singular configuration, that is, it satisfies the geometric constraint condition of equation (14):

[0054]

[0055] In a multi-degree-of-freedom forming machine, the moving platform and connecting rods, as well as the drive slider and connecting rods, are all connected by spherical joints. The rotation range of these spherical joints is limited; let the maximum rotation angle of the spherical joints be θ. max The rotation angle θ of the spherical pair is equipped with ai and the rotation angle θ of the lower spherical pair bi The geometric constraints that should be satisfied by equation (15) are as follows:

[0056]

[0057] The upper die feed distance and swing angle are used as process parameters to express the workspace of the equipment. The range of values ​​for each process parameter is calculated, which is the workspace of the equipment. Based on the geometric constraints of formulas (9), (14) and (15) and the upper die motion equation of formula (3), the swing angle of the upper die is calculated. Upper die feed distance

[0058] According to the above scheme, step S2 includes:

[0059] Let the forming load applied to the upper die of the multi-degree-of-freedom envelope forming equipment be (f, m), where f is the forming force and m is the forming torque; under the action of the external load, the comprehensive deformation of the upper die is set as (ΔX, Δθ), where ΔX is the linear deformation along the x, y, and z axes, and Δθ is the angular deformation along the x, y, and z axes; the relationship between the comprehensive deformation and the forming load is determined by equation (16):

[0060]

[0061] In the formula, K is the stiffness matrix;

[0062] According to Hooke's law, the axial force of the connecting rod can be expressed as:

[0063]

[0064] In the formula, Δl i and k i These represent the axial deformation and axial stiffness of the connecting rod, respectively; E is the elastic modulus of the connecting rod material; and A represents the cross-sectional area of ​​the connecting rod.

[0065] Integrating the axial forces and axial deformations of the six connecting rods into the matrix expression, we get:

[0066] F l =diag([k1L k6])Δl (52)

[0067] In the formula, F l =[f1,f2,f3,f4,f5,f6] T ,Δl=[Δl1,Δl2,Δl3,Δl4,Δl5,Δl6] T

[0068] According to the principle of virtual work, we get:

[0069]

[0070] The force equilibrium condition of the moving platform is:

[0071]

[0072] In the formula,

[0073] According to equations (16), (18), (19), and (20), the stiffness matrix K is determined by equation (21):

[0074] K = J f diag([k1L k6])J f T (55)

[0075] The deformation of the upper die under forming load is determined by equation (22):

[0076]

[0077] Because the deformation of the upper mold will cause a certain error between its actual trajectory and the theoretical trajectory, the upper mold represented by spherical joint B1L B6 is the theoretical position without considering the deformation of the connecting rod, while the upper mold represented by spherical joint B1'L B6' is the actual position with deformation error; C i C represents a point on the upper mold without deformation error. i ' is a point on the upper mold considering deformation error; A1 represents the ball joint point on the static platform; S B It is the moving coordinate system of the upper mold that does not consider deformation errors, S B ' is the moving coordinate system of the upper die considering deformation error; under heavy load conditions, the deformation error of the multi-degree-of-freedom forming equipment is the difference between the theoretical trajectory and the actual trajectory of the reference point of the upper die, and Δe is the deformation error of the upper die; according to equation (22), the deformation error of the multi-degree-of-freedom envelope forming equipment is expressed as:

[0078] Δe=|ΔX+(R(α+Δθx,β+Δθy,γ+Δθz)-R(α,β,γ))ci| (57)

[0079] In the formula, c i =[r t cosφ,r t sinφ,1](r=0:r t ,φ=0:2π) is the upper reference point C of the upper model. i Position vector, r t φ is the radius of the circumcircle of the upper die, and φ is the polar angle of the upper die reference point.

[0080] When the upper mold of the equipment moves in the workspace, it satisfies the geometric constraint condition and deformation error constraint condition shown in equation (24);

[0081]

[0082] Considering deformation error, workspace utilization λ s Calculated using equation (25):

[0083]

[0084] According to the above scheme, step S3 includes:

[0085] The slider of the multi-degree-of-freedom forming equipment only translates along the guide rail direction; therefore, its force equilibrium condition is expressed as:

[0086]

[0087] In the formula, τ i It is the driving force of the slider, f i It is the axial force of the connecting rod;

[0088] The axial force of the connecting rod is determined by equation (27):

[0089]

[0090] Substituting equation (27) into equation (26), we get:

[0091]

[0092] In the formula, τ=[τ1,L,τ6] T ;

[0093] When the multi-degree-of-freedom forming equipment is working, the rotational motion of the servo motor is converted into the linear reciprocating motion of the slider through the ball screw; the relationship between the torque output by the servo motor and the driving force of the slider is determined by equation (29):

[0094] T i =τ i h s / (2πη) (63)

[0095] In the formula, η is the transmission efficiency of the ball screw, and T i It is the torque output by the motor;

[0096] The relationship between the slider speed and the servo motor speed is expressed as follows:

[0097] n i =60v si / h s (64)

[0098] In the formula, n i It refers to the rotational speed of each motor, h. s It is the lead of the ball screw;

[0099] Once the motion equation of the upper mold is determined, the speed of each slider can be calculated according to equation (13); by substituting the slider speed into equation (30), the rotational speed of the servo motor can be calculated, and the maximum torque [T(n] that the servo motor can output at the current rotational speed can be obtained according to the characteristic curve of the servo motor. i ]; Given the forming load of the equipment, the driving force of each slider can be calculated according to equation (28). Substituting the driving force of each slider into equation (29) yields the torque T(n) of each servo motor. i When the torque T(n) of the servo motor i The torque is less than the maximum torque that the motor can output [T(n)]. i When the multi-degree-of-freedom forming equipment is in operation, the motor loading characteristic constraint and geometric constraint of the multi-degree-of-freedom forming equipment should satisfy equation (31).

[0100]

[0101] To quantitatively analyze the impact of forming load on the equipment workspace, a workspace utilization rate λ considering motor loading characteristics is introduced. m As shown in equation (32):

[0102]

[0103] According to the above scheme, step S4 includes:

[0104] The feed distance, swing angle and rotation speed of the upper die are used as three process parameters to describe the comprehensive workspace of the multi-degree-of-freedom envelope forming equipment; the comprehensive workspace of the multi-degree-of-freedom envelope forming equipment satisfies the constraint condition of equation (33).

[0105]

[0106] Based on the constraints in equation (33), the feed distance of the upper die under different working conditions is calculated. A workspace utilization λ is introduced that takes into account mechanism deformation and motor loading characteristics. t As shown in equation (34):

[0107]

[0108] The multi-degree-of-freedom forming workspace calculation method of the present invention, which considers mechanism deformation and motor loading, has the following beneficial effects:

[0109] This invention provides a multi-degree-of-freedom forming workspace calculation method that considers mechanism deformation and motor loading. It can avoid insufficient motor loading and excessive deformation of multi-degree-of-freedom forming equipment under certain special poses, while finding the effective working area of ​​the equipment, thereby enabling the equipment to achieve complex multi-degree-of-freedom motion under heavy load conditions. Attached Figure Description

[0110] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0111] Figure 1 These are the inverse kinematics principle diagram and ball joint point distribution diagram of multi-degree-of-freedom forming equipment;

[0112] Figure 2 This is a force diagram of the moving platform of a multi-degree-of-freedom forming equipment;

[0113] Figure 3 This is a schematic diagram of the deformation error of a multi-degree-of-freedom forming equipment;

[0114] Figure 4 It is the characteristic curve of the motor of multi-degree-of-freedom forming equipment;

[0115] Figure 5 It is the workspace of a multi-degree-of-freedom forming equipment that takes into account the deformation of the mechanism;

[0116] Figure 6 It takes into account the utilization rate of the workspace when the mechanism is deformed;

[0117] Figure 7 It is the workspace of a multi-degree-of-freedom forming equipment that takes into account the motor loading characteristics;

[0118] Figure 8 It takes into account the workspace utilization rate considering the motor loading characteristics;

[0119] Figure 9 It is a comprehensive workspace for multi-degree-of-freedom forming equipment;

[0120] Figure 10 It is the overall workspace utilization rate of multi-degree-of-freedom forming equipment. Detailed Implementation

[0121] To provide a clearer understanding of the technical features, objectives, and effects of the present invention, specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0122] The multi-degree-of-freedom forming workspace calculation method of the present invention, considering mechanism deformation and motor loading, includes:

[0123] S1. Workspace of multi-degree-of-freedom forming equipment considering geometric constraints.

[0124] like Figure 1 As shown, let the centers of the ball joints on the static platform of the equipment be A1, A2, A3, A4, A5, A6, and the centers of the ball joints on the moving platform of the equipment be B1, B2, B3, B4, B5, B6. Simultaneously, establish a global coordinate system S at the center of the static platform of the equipment. A (O A-x A y A z A ), where z A The axis is perpendicular to the static platform and does not change with the movement of the equipment. A dynamic coordinate system S is established on the moving platform of the equipment. B (O B -x B y B z B ), S B The origin and S A The distance between the origins is d, S B The origin and S A The origin is located on the perpendicular line from the center of the static platform, i.e., z B axis and z A The axes are in the same direction. θ Ai (i = 1-6) represents the distribution angle of the ball joint points on the static platform, θ Bi (i = 1-6) represents the distribution angle of the ball joint points on the moving platform.

[0125] Therefore A i In the global coordinate system S A Position vector a i Determined by equation (1):

[0126] a i (a ix )=[a ix ,a ix tanθ Ai ,0] T (69)

[0127] In the formula, a ix The position vector of the ball joint point of the static platform in the global coordinate system is in the x-axis. A Projection on the axis.

[0128] Furthermore, B i In the moving coordinate system S B Position vector b i Determined by equation (2):

[0129] b i '=[r B cosθ Bi ,r B sinθ Bi ,d] T (70)

[0130] In the formula, r B d is the radius of the circumscribed circle of the ball joint of the moving platform, and d is the height of the upper mold.

[0131] The motion of the equipment platform can be represented by six parameters: x, y, and z represent the translational motion of the upper die in three directions, and α, β, and γ represent the rotational angles of the upper die in three directions. In the multi-degree-of-freedom forming process, the motion equation of the upper die is:

[0132]

[0133] In the formula, ω is the swing angle of the upper die movement, k is the rotational speed of the upper die movement, h is the feed rate of the upper die movement, and h is the feed distance of the upper die movement.

[0134] Therefore S B Compared to S A rotation matrix Determined by equation (4):

[0135]

[0136] Furthermore, B i In the global coordinate system S A Position vector b i Determined by equation (5):

[0137]

[0138] In the formula, p OB It is S B The origin of the coordinate system relative to S A The position vector. p OB It can be determined by equation (6):

[0139] p OB =[x,y,z+Z P ] T (74)

[0140] In the formula, Z P It is the distance between the end of the upper mold and the center of the static platform.

[0141] Based on the constraint that the link length remains constant, a ix Determined by equation (7):

[0142]

[0143] In the formula, l is the length of the connecting rod.

[0144] b ix =d(sinαsinγ+cosαcosγsinβ)-r B sinθ Bi (cosαsinγ-cosγsinαsinβ)+

[0145] r B cosβcosγcosθ Bi b iy =r B sinθ Bi (cosαcosγ+sinαsinβsinγ)-d(cosγsinα-cosαsinβsinγ)+

[0146] r B cosβsinγcosθ Bi b iz =h+Z P -r B sinβcosθ Bi +cosαcosβd+r B cosβsinαsinθ Bi

[0147] Furthermore, the displacement s of each slider is equipped. i Determined by equation (8):

[0148]

[0149] In the formula, r A It is the distance between the initial position of the driving slider and the origin of the global coordinate system.

[0150] Due to the limitation of the guide rail length, the displacement of the drive slider in its respective direction of motion cannot exceed the allowable range. The maximum and minimum values ​​of the slider displacement are denoted by s. max and s min This means that the geometric constraints satisfying equation (9) are:

[0151] s min ≤s i ≤s max (77)

[0152] When the moving platform of the equipment rotates, the velocity of any point on the moving platform is equal to the velocity of the center of the moving platform plus the angular velocity of that point about the center of the moving platform. Therefore, the velocity v of the lower ball joint of the connecting rod is... Bi The solution can be obtained using equation (10):

[0153]

[0154] In the formula, r i It is a vector Position vector in the global coordinate system.

[0155] Based on the velocity projection theorem, the velocity v of the ball joint on the connecting rod can be obtained. Ai and the speed of the ball head v Bi The relationship is shown in equation (11):

[0156]

[0157] The velocity v of the ball head under the rod obtained by equation (10) Bi Substituting into equation (11), we get:

[0158]

[0159] Among them, v i It refers to the speed of each driving slider.

[0160] Equation (12) can then be expressed in matrix form, as shown in equation (13):

[0161] J s [v1,L,v6] T =J p [v p ,ω p ] T (81)

[0162] In the formula

[0163]

[0164]

[0165] When a multi-degree-of-freedom forming equipment is in a singular configuration during motion, it may experience loss of control or changes in the degree of freedom. Therefore, in order to ensure the stability of the upper mold's motion, the working space of the multi-degree-of-freedom forming equipment should not contain upper mold poses in singular configurations, that is, satisfy the geometric constraint condition of equation (14):

[0166]

[0167] In multi-degree-of-freedom forming equipment, the moving platform and connecting rods, as well as the drive slider and connecting rods, are all connected by spherical joints. The rotation range of the spherical joints is limited. Let the maximum rotation angle of the spherical joints of the equipment be θ. max Therefore, the rotation angle θ of the spherical pair is equipped with ai and the rotation angle θ of the lower spherical pair bi The geometric constraints of equation (15) should be satisfied:

[0168]

[0169] The upper die feed distance and swing angle are used as important process parameters to express the workspace of the equipment. The range of values ​​for each process parameter is calculated, which is the workspace of the equipment. Based on the geometric constraints of formulas (9), (14), and (15) and the upper die motion equation of formula (3), the swing angle of the upper die can be calculated. Upper die feed distance

[0170] S2, Workspace of multi-degree-of-freedom forming equipment considering mechanism deformation.

[0171] Let the forming load applied to the upper die of the multi-degree-of-freedom envelope forming equipment be (f, m), where f is the forming force and m is the forming torque. Under the action of the external load, the comprehensive deformation of the upper die is set as (ΔX, Δθ), where ΔX is the linear deformation along the x, y, and z axes, and Δθ is the angular deformation along the x, y, and z axes. The relationship between the comprehensive deformation and the forming load is determined by equation (16):

[0172]

[0173] In the formula, K is the stiffness matrix.

[0174] When a multi-degree-of-freedom forming machine is subjected to a large load, the six connecting rods will undergo axial deformation. The stiffness of the moving platform and the stationary platform is much higher than that of the connecting rods, so they can be considered as rigid bodies. Therefore, the deformation of the upper die can be considered as being caused by the deformation of the connecting rods. According to Hooke's Law, the axial force of the connecting rods can be expressed as:

[0175]

[0176] In the formula, Δl i and k i These represent the axial deformation and axial stiffness of the connecting rod, respectively; E is the elastic modulus of the connecting rod material; and A represents the cross-sectional area of ​​the connecting rod.

[0177] By integrating the axial forces and axial deformations of the six connecting rods into the matrix expression, we can obtain:

[0178] F l =diag([k1L k6])Δl (86)

[0179] In the formula, F l =[f1,f2,f3,f4,f5,f6] T ,Δl=[Δl1,Δl2,Δl3,Δl4,Δl5,Δl6] T

[0180] According to the principle of virtual work, we can obtain:

[0181]

[0182] according to Figure 2 From the force diagram of the moving platform shown, the force equilibrium condition of the moving platform can be obtained as follows:

[0183]

[0184] In the formula,

[0185] According to equations (16), (18), (19), and (20), the stiffness matrix K is determined by equation (21):

[0186] K = J f diag([k1L k6])J f T (89)

[0187] The deformation of the upper die under forming load is then determined by equation (22):

[0188]

[0189] Because the deformation of the upper mold of the equipment will cause a certain error between its actual trajectory and the theoretical trajectory, such as Figure 3 As shown, the upper mold represented by spherical joints B1L and B6 is the theoretical position without considering the deformation of the connecting rod, while the upper mold represented by spherical joints B1'L and B6' is the actual position with deformation error. i C represents a point on the upper mold without deformation error. i ' is a point on the upper mold considering deformation errors. A1 represents the ball joint point on the static platform. S B It is the moving coordinate system of the upper mold that does not consider deformation errors, S B This refers to the moving coordinate system of the upper die, taking into account deformation errors. Under heavy-load conditions, the deformation error of a multi-degree-of-freedom forming machine is the difference between the theoretical trajectory and the actual trajectory of the upper die's reference point. Figure 3 The Δe shown is the deformation error of the upper die. According to equation (22), the deformation error of the multi-degree-of-freedom envelope forming equipment can be expressed as:

[0190] Δe=|ΔX+(R(α+Δθ x ,β+Δθ y ,γ+Δθ z )-R(α,β,γ))c i | (91)

[0191] In the formula, c i =[r t cosφ,r t sinφ,1](r=0:r t ,φ=0:2π) is the upper reference point C of the upper model. iPosition vector, r t φ is the circumcircle radius of the upper die, and φ is the polar angle of the upper die reference point.

[0192] When the deformation error of a multi-degree-of-freedom forming equipment is large, it means that the actual trajectory of the upper mold during the movement process deviates significantly from the theoretical trajectory. This leads to a reduction in the accuracy of the heavy-duty multi-degree-of-freedom forming equipment, thus failing to meet the requirements of the multi-degree-of-freedom forming process. Therefore, when the upper mold of the equipment moves in the workspace, it should satisfy the geometric constraint conditions and deformation error constraint conditions shown in equation (24).

[0193]

[0194] To quantitatively analyze the impact of forming load on the equipment's workspace, a workspace utilization rate λ considering deformation error is introduced. s As shown in equation (25):

[0195]

[0196] S3. Workspace of multi-degree-of-freedom forming equipment considering motor loading characteristics.

[0197] The slider of the multi-degree-of-freedom forming equipment only translates along the guide rail direction; therefore, its force equilibrium condition can be expressed as:

[0198]

[0199] In the formula, τ i It is the driving force of the slider, f i It is the axial force of the connecting rod.

[0200] The axial force of the connecting rod is determined by equation (27):

[0201]

[0202] Substituting equation (27) into equation (26), we get:

[0203]

[0204] In the formula, τ=[τ1,L,τ6] T

[0205] When the multi-degree-of-freedom forming equipment is working, the rotational motion of the servo motor is converted into the linear reciprocating motion of the slider through the ball screw. The relationship between the torque output by the servo motor and the driving force of the slider is determined by equation (29):

[0206] T i =τ i h s / (2πη) (97)

[0207] In the formula, η is the transmission efficiency of the ball screw, and T i It is the torque output by the motor.

[0208] The relationship between the slider speed and the servo motor speed is expressed as follows:

[0209] n i =60v si / h s (98)

[0210] In the formula, n i It refers to the rotational speed of each motor, h. s It is the lead of the ball screw.

[0211] Once the motion equation of the upper mold is determined, the speed of each slider can be calculated according to equation (13). Substituting the slider speed into equation (30) yields the rotational speed of the servo motor. Figure 4 The servo motor characteristic curve shown can be used to obtain the maximum torque [T(n)] that the servo motor can output at the current speed. i Given the forming load of the equipment, the driving force of each slider can be calculated according to equation (28). Substituting the driving force of each slider into equation (29) yields the torque T(n) of each servo motor. i When the torque T(n) of the servo motor i The torque is less than the maximum torque that the motor can output [T(n)]. i Only when the [condition] is met can the multi-degree-of-freedom forming equipment function normally. Therefore, the motor loading characteristic constraints and geometric constraints of the multi-degree-of-freedom forming equipment should satisfy equation (31).

[0212]

[0213] Variations in the rotational speed of the upper die in a multi-degree-of-freedom forming machine cause changes in the slider speed, which in turn affects the servo motor's rotational speed. Therefore, the shape and volume of the workspace satisfying the motor loading characteristics constraints differ depending on the upper die's rotational speed. Thus, the upper die rotational speed is also considered as a parameter representing the machine's workspace. To quantitatively analyze the impact of forming load on the machine's workspace, a workspace utilization rate λ considering the motor loading characteristics is introduced. m As shown in equation (32):

[0214]

[0215] S4, Multi-degree-of-freedom envelope forming equipment integrated workspace.

[0216] The forming equipment of this invention processes thin-walled, high-rib components through a multi-degree-of-freedom envelope forming process. Therefore, the feed distance, swing angle, and rotational speed of the upper die are used as three process parameters to describe the comprehensive workspace of the multi-degree-of-freedom envelope forming equipment. First, based on the geometric configuration of the equipment, the sliding range, the spherical joint rotation angle range, and the singular configuration of the mechanism are used as constraints for calculating the comprehensive workspace. Second, in order for the equipment to complete the multi-degree-of-freedom envelope forming process under heavy load conditions, the motor must be able to provide the driving force required for the drive joints of the six motion chains. Therefore, the motor loading characteristics are also used as constraints for calculating the comprehensive workspace. When the motion trajectory of the upper die deviates too much from the desired trajectory during the forming process, the multi-degree-of-freedom forming equipment cannot process components that meet the requirements. Therefore, in order to ensure the processing accuracy and efficiency of the equipment, the deformation error of the mechanism is used as a constraint for calculating the comprehensive workspace. In summary, in addition to considering geometric constraints, the constraints of mechanism deformation and motor loading performance should also be considered when calculating the comprehensive workspace of the equipment. Therefore, the comprehensive workspace of the multi-degree-of-freedom envelope forming equipment should satisfy the constraint conditions of equation (33).

[0217]

[0218] Based on the constraints in equation (33), the feed distance of the upper die under different working conditions can be calculated. To quantitatively analyze the impact of forming load on the overall workspace of the equipment, a workspace utilization rate λ considering mechanism deformation and motor loading characteristics is introduced. t As shown in equation (34):

[0219]

[0220] Based on equations (24) and (25) and the design constraints given in Table 1, the working space and working space utilization rate of a multi-degree-of-freedom forming equipment considering mechanism deformation under different working conditions can be calculated using the method provided by this invention. Figure 5 , Figure 6 As shown. Figure 5 The colors in the diagram represent the distribution of the upper mold deformation error within the deformation working space of the equipment considering the mechanism. Figure 6 The color of the 3D surface and the height of the bar chart represent the workspace utilization rate under the constraint of mechanism deformation. Figure 5 , Figure 6 It can be seen that as the forming load or the forming load lever arm increases, the working space of the equipment considering mechanism deformation gradually decreases. Under the condition of a forming load of 6000kN and a forming load lever arm of 200mm, the utilization rate of the working space considering mechanism deformation is 21.7%. Therefore, as the forming load increases, the working space of the multi-degree-of-freedom forming equipment considering mechanism deformation decreases to 1 / 5 of the theoretical geometric working space.

[0221] Table 1 Design Constraints of Equipment

[0222]

[0223] Based on equations (31) and (32) and the design constraints given in Table 1, the workspace and workspace utilization rate of the multi-degree-of-freedom forming equipment considering the motor loading characteristics under different working conditions can be calculated, such as... Figure 7 , Figure 8 As shown. Figure 7 The colors in the diagram represent the distribution of upper mold deformation error within the working space of the equipment, taking into account the motor loading characteristics. Figure 8 The color of the 3D surface and the height of the bar chart represent the workspace utilization rate under the constraints of motor loading characteristics. Figure 7 , Figure 8 It can be seen that as the forming load or forming load lever arm increases, the workspace of the equipment considering the motor loading characteristics gradually decreases. Under the condition of a forming load of 6000kN and a forming load lever arm of 200mm, the workspace utilization rate considering the motor loading characteristics is 28.3%. Therefore, as the forming load increases, the workspace of the multi-degree-of-freedom forming equipment considering the motor loading characteristics decreases to 3 / 10 of the theoretical geometric workspace.

[0224] Based on equations (33) and (34) and the design constraints given in Table 1, the comprehensive workspace and comprehensive workspace utilization rate of the equipment under different working conditions can be obtained by following the method provided by this invention. Figure 9 and Figure 10 As shown, the comprehensive workspace of the equipment is expressed using process parameters for multi-degree-of-freedom forming, i.e. Figure 9 The X-axis represents the feed distance of the upper die, the Y-axis represents the swing angle of the upper die, and the Z-axis represents the rotational speed of the upper die. Figure 9 The colors in the diagram represent the distribution of upper mold deformation error in the integrated workspace. Figure 10 The color of the 3D surface and the height of the bar chart represent the overall workspace utilization rate. Figure 9 , Figure 10 It can be seen that as the forming load or forming load lever arm increases, the overall working space of the equipment gradually decreases. Under the condition of a forming load of 6000kN and a forming load lever arm of 200mm, the utilization rate of the overall working space is 9.9%. Therefore, as the forming load increases, the overall working space of the multi-degree-of-freedom forming equipment decreases to 1 / 10 of the theoretical geometric working space.

[0225] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A method for calculating the workspace of a multi-degree-of-freedom forming machine that considers mechanism deformation and motor loading, used for calculating the workspace of a multi-degree-of-freedom forming equipment, characterized in that... Includes the following steps: S1. Calculate the geometric workspace based on the equipment's geometric configuration; S2. By establishing an equipment deformation error model, calculate the working space of the equipment considering the deformation of the mechanism; S3. Calculate the working space of the equipment based on the static model, taking into account the motor loading characteristics; S4. Simultaneously consider the deformation of the mechanism and the loading characteristics of the motor to calculate the comprehensive workspace of the equipment. Step S1 includes: Let the centers of the ball joints on the static platform of the equipment be A1, A2, A3, A4, A5, A6, and the centers of the ball joints on the moving platform of the equipment be B1, B2, B3, B4, B5, B6. Simultaneously, establish a global coordinate system S at the center of the static platform of the equipment. A (O A -x A y A z A ), where z A The axis is perpendicular to the static platform, z A The axis does not change with the movement of the equipment; a dynamic coordinate system S is established on the moving platform of the equipment. B (O B -x B y B z B ), S B The origin and S A The distance between the origins is d, S B The origin and S A The origin is located on the perpendicular line from the center of the static platform, i.e., z B axis and z A The axes are in the same direction; θ Ai The θ represents the angles representing the distribution of the ball joint points on the static platform, i = 1 - 6; Bi The angle representing the distribution of the ball joint points on the moving platform is i = 1-6; Therefore A i In the global coordinate system S A Position vector a i Determined by equation (1): a i (a ix )=[a ix ,a ix tanθ Ai ,0] T (1) In the formula, a ix The position vector of the ball joint point of the static platform in the global coordinate system is in the x-axis. A Projection on the axis; Furthermore, B i In the moving coordinate system S B Position vector b i Determined by equation (2): b i '=[r B cosθ Bi ,r B sinθ Bi ,d] T (2) In the formula, r B d is the radius of the circumcircle of the ball joint of the moving platform, and d is the height of the upper mold; The motion of the equipment platform is represented by six parameters: x, y, and z represent the translational motion of the upper die in three directions, and α, β, and γ represent the rotational angles of the upper die in three directions. In the multi-degree-of-freedom forming process, the motion equation of the upper die is: (3) In the formula, ω is the swing angle of the upper die movement, k is the rotational speed of the upper die movement, h is the feed rate of the upper die movement, and h is the feed distance of the upper die movement. s i To measure the displacement of each slider, the maximum and minimum slider displacements are represented by s. max and s min Indicates; based on the constraint that the link length remains constant, a ix Determined by equation (7): In the formula, It is the length of the connecting rod. b ix =d(sinαsinγ+cosαcosγsinβ)-r B sinθ Bi (cosαsinγ-cosγsinαsinβ)+r B cosβcosγcosθ Bi b iy =r B sinθ Bi (cosαcosγ+sinαsinβsinγ)-d(cosγsinα-cosαsinβsinγ)+r B cosβsinγcosθ Bi b iz =h+Z P -r B sinβcosθ Bi +cosαcosβd+r B cosβsinαsinθ Bi Z P It is the distance between the end of the upper mold and the center of the stationary platform; , r i It is a vector Position vector in the global coordinate system; Step S4 includes: The feed distance, swing angle and rotation speed of the upper die are used as three process parameters to describe the comprehensive workspace of the multi-degree-of-freedom envelope forming equipment; the comprehensive workspace of the multi-degree-of-freedom envelope forming equipment satisfies the constraint condition of equation (33). ； Among them: the maximum value of the spherical sub-rotation angle is θ. max The rotation angle of the spherical pair is θ. ai The rotation angle of the lower spherical joint is θ. bi； Δe is the deformation error of the upper mold; n i These are the rotational speeds of each motor; substituting the slider speed into the equation yields the servo motor speed, and based on the servo motor characteristic curve, the maximum torque [T(n] that the servo motor can output at the current speed can be obtained. i Given the forming load of the equipment, the driving force of each slider is calculated, and the torque T(n) of each servo motor is obtained from the driving force of each slider. i ); Based on the constraints in equation (33), the feed distance of the upper die under different working conditions is calculated. A workspace utilization λ is introduced that takes into account mechanism deformation and motor loading characteristics. t As shown in equation (34): (34)。 2. The multi-degree-of-freedom forming workspace calculation method considering mechanism deformation and motor loading according to claim 1, characterized in that, Step S1 further includes: S B Compared to S A rotation matrix Determined by equation (4): Furthermore, B i In the global coordinate system S A Position vector b i Determined by equation (5): In the formula, p OB It is S B The origin of the coordinate system relative to S A Position vector; p OB It can be determined by equation (6): p OB =[x,y,z+Z P ] T (6) In the formula, Z P It is the distance between the end of the upper mold and the center of the stationary platform; The displacement s of each slider is equipped i Determined by equation (8): In the formula, r A It is the distance between the initial position of the driving slider and the origin of the global coordinate system; The maximum and minimum values ​​of the slider displacement are represented by s. max and s min This means that the geometric constraints satisfying equation (9) are: s min ≤s i ≤s max (9) When the moving platform of the equipment rotates, the velocity of any point on the moving platform is equal to the velocity of the center of the moving platform plus the angular velocity of that point about the center of the moving platform, and the velocity v of the lower ball joint of the connecting rod. Bi Solve using equation (10): In the formula, r i It is a vector Position vector in the global coordinate system; Based on the velocity projection theorem, the velocity v of the ball joint on the connecting rod is obtained. Ai and the speed of the ball head v Bi The relationship is shown in equation (11): The velocity v of the ball head under the rod obtained by equation (10) Bi Substituting into equation (11), we get: Among them, v i It refers to the speed of each driving slider; Equation (12) can be expressed in matrix form as shown in equation (13): (13) ； The workspace of the multi-degree-of-freedom forming equipment does not contain the upper mold pose as a singular configuration, that is, it satisfies the geometric constraint condition of equation (14): In a multi-degree-of-freedom forming machine, the moving platform and connecting rods, as well as the drive slider and connecting rods, are all connected by spherical joints. The rotation range of these spherical joints is limited; let the maximum rotation angle of the spherical joints be θ. max The rotation angle θ of the spherical pair is equipped with ai and the rotation angle θ of the lower spherical pair bi The geometric constraints that should be satisfied by equation (15) are as follows: The upper die feed distance and swing angle are used as process parameters to express the workspace of the equipment. The range of values ​​for each process parameter is calculated, which is the workspace of the equipment. Based on the geometric constraints of formulas (9), (14) and (15) and the upper die motion equation of formula (3), the swing angle of the upper die is calculated. Upper die feed distance .

3. The multi-degree-of-freedom forming workspace calculation method considering mechanism deformation and motor loading according to claim 2, characterized in that, Step S2 includes: Let the forming load applied to the upper die of the multi-degree-of-freedom envelope forming equipment be (f, m), where f is the forming force and m is the forming torque; under the action of the external load, the comprehensive deformation of the upper die is set as (ΔX, Δθ), where ΔX is the linear deformation along the x, y, and z axes, and Δθ is the angular deformation along the x, y, and z axes; the relationship between the comprehensive deformation and the forming load is determined by equation (16): In the formula, K is the stiffness matrix; According to Hooke's law, the axial force of the connecting rod can be expressed as: （17） In the formula, i = 1…6; Δl i and k i These represent the axial deformation and axial stiffness of the connecting rod, respectively; E is the elastic modulus of the connecting rod material; and A represents the cross-sectional area of ​​the connecting rod. Integrating the axial forces and axial deformations of the six connecting rods into the matrix expression, we get: F l =diag([k1…k6])Δl(18) In the formula, F l = [f1, f2, f3, f4, f5, f6] T , Δl = [Δl1, Δl2, Δl3, Δl4, Δl5, Δl6] T According to the principle of virtual work, we get: The force equilibrium condition of the moving platform is: In the formula, According to equations (16), (18), (19), and (20), the stiffness matrix K is determined by equation (21): K=J f diag([k1…k6])J f T (21) The deformation of the upper die under forming load is determined by equation (22): Because the deformation of the upper mold will cause a certain error between its actual trajectory and the theoretical trajectory, the upper mold positions represented by spherical joints B1…B6 are theoretical positions without considering the deformation of the connecting rod, while the upper mold positions represented by spherical joints B1'…B6' are actual positions with deformation errors; C i C represents a point on the upper mold without deformation error. i ' is a point on the upper mold considering deformation error; A1 represents the ball joint point on the static platform; S B It is the moving coordinate system of the upper mold that does not consider deformation errors, S B ' is the moving coordinate system of the upper die considering deformation error; under heavy load conditions, the deformation error of the multi-degree-of-freedom forming equipment is the difference between the theoretical trajectory and the actual trajectory of the reference point of the upper die, and Δe is the deformation error of the upper die; according to equation (22), the deformation error of the multi-degree-of-freedom envelope forming equipment is expressed as: (23) In the formula, Reference point C on the upper mold i Position vector, r t It is the circumcircle radius of the upper mold. It is the polar angle of the upper mold reference point; When the upper mold of the equipment moves in the workspace, it satisfies the geometric constraint condition and deformation error constraint condition shown in equation (24); Considering deformation error, workspace utilization λ s Calculated using equation (25): 。 4. The multi-degree-of-freedom forming workspace calculation method considering mechanism deformation and motor loading according to claim 1, characterized in that, Step S3 includes: The slider of the multi-degree-of-freedom forming equipment only translates along the guide rail direction; therefore, its force equilibrium condition is expressed as: In the formula, τ i It is the driving force of the slider, f i It is the axial force of the connecting rod; The axial force of the connecting rod is determined by equation (27): (27) Substituting equation (27) into equation (26), we get: (28) In the formula, τ=[τ1,…,τ6] T ; When the multi-degree-of-freedom forming equipment is working, the rotational motion of the servo motor is converted into the linear reciprocating motion of the slider through the ball screw; the relationship between the torque output by the servo motor and the driving force of the slider is determined by equation (29): T i =t i h s / (2π)(29) In the formula, η is the transmission efficiency of the ball screw, and T i It is the torque output by the motor; The relationship between the slider speed and the servo motor speed is expressed as follows: n i =60v si / h s (30) In the formula, n i It refers to the rotational speed of each motor, h. s It is the lead of the ball screw; Once the motion equation of the upper mold is determined, the speed of each slider is calculated according to equation (13); by substituting the slider speed into equation (30), the rotational speed of the servo motor can be calculated, and the maximum torque [T(n] that the servo motor can output at the current rotational speed can be obtained according to the characteristic curve of the servo motor. i ]; Given the forming load of the equipment, the driving force of each slider can be calculated according to equation (28). Substituting the driving force of each slider into equation (29) yields the torque T(n) of each servo motor. i When the torque T(n) of the servo motor i The torque is less than the maximum torque that the motor can output [T(n)]. i When the multi-degree-of-freedom forming equipment is in operation, the motor loading characteristic constraint and geometric constraint of the multi-degree-of-freedom forming equipment should satisfy equation (31). To quantitatively analyze the impact of forming load on the equipment workspace, a workspace utilization rate λ considering motor loading characteristics is introduced. m As shown in equation (32): (32)。

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