Modeling method of multi-degree-of-freedom forming equipment kinetic model considering connecting rod deformation and regenerative forming force

Abstract

The invention relates to a modeling method of a multi-degree-of-freedom forming equipment dynamic model considering connecting rod deformation and regenerative forming force. The modeling method comprises the following steps: S1, establishing a parallel mechanism dynamic model considering connecting rod deformation; s2, establishing a model of regeneration forming force in multi-degree-of-freedom forming; and S3, a dynamic model of multi-degree-of-freedom forming equipment is established according to the dynamic model of the parallel mechanism and the model of the regeneration forming force. According to the dynamic modeling method of the multi-degree-of-freedom forming equipment, the calculation error can be remarkably reduced, and the dynamic behavior of the multi-degree-of-freedom forming equipment in the forming process can be accurately predicted, so that the equipment provides an accurate mathematical model for subsequent research and development.

Description

TECHNICAL FIELD

[0001] The present application relates to a multi-degree-of-freedom forming process, more particularly, to a modeling method of a multi-degree-of-freedom forming equipment dynamics model considering link deformation and regenerated forming force. BACKGROUND

[0002] The multi-degree-of-freedom forming process is an advanced manufacturing technology that can flexibly adjust the shape and geometric features of materials during the manufacturing process. It enables precise forming of workpieces under multi-degree-of-freedom (usually referring to movement in multiple directions) through multi-axis control and multi-station coordination. This process is commonly used in the manufacturing of complex parts, particularly in the fields of aerospace, automotive, precision instruments, etc. Compared with traditional cutting processes, it has the advantages of high material utilization, strong process flexibility, good microstructure performance and mechanical properties, etc.

[0003] For this process, the applicant has developed a new 6-PSS heavy-load parallel machine tool for multi-degree-of-freedom forming, which realizes the multi-degree-of-freedom movement of the upper die of the equipment through the coupling movement of multiple motion branches. Currently, multi-degree-of-freedom parallel machine tools are mainly used for light-load cutting machining, and deformation mainly occurs in the tool system, which is usually established as a three-degree-of-freedom linear system. The cutting force is derived from the removal of materials. However, in the multi-degree-of-freedom forming process, the rigidity of the upper die is large, and the deformation mainly occurs in the links, and there are complex geometric constraints between the links, resulting in coupling of the link deformation. At the same time, in the multi-degree-of-freedom forming process, the forming force is derived from the complex metal flow and the contact between the die and the blank. These two factors result in a dynamics model of the multi-degree-of-freedom forming equipment that is inconsistent with the current cutting equipment. The mechanical model of the cutting equipment in the prior art cannot be directly applied to the dynamics model of the multi-degree-of-freedom forming equipment, and cannot accurately predict the dynamic behavior in the forming process. SUMMARY

[0004] The technical problem to be solved by the present application is to provide a modeling method of a multi-degree-of-freedom forming equipment dynamics model considering link deformation and regenerated forming force, which can accurately predict the dynamic behavior of the equipment in the forming process.

[0005] The technical solution adopted by the present application to solve the technical problem is: a modeling method of a multi-degree-of-freedom forming equipment dynamics model considering link deformation and regenerated forming force is constructed, comprising the following steps: S1, establishing a parallel mechanism dynamics model considering link deformation; S2, establishing a model of regenerated forming force in multi-degree-of-freedom forming; S3, establishing a dynamics model of the multi-degree-of-freedom forming equipment according to the parallel mechanism dynamics model and the model of regenerated forming force.

[0006] According to the above scheme, in the step S1, the deformation of the connecting rod is decoupled, and a dynamic model of the parallel mechanism is established based on the Lagrange method.

[0007] According to the above scheme, the contact area and pressure distribution of the blank and the upper die are calculated, the influence of vibration on the forming force is analyzed, and a regenerated forming force model is established.

[0008] According to the above scheme, in the step S1, a coordinate system is established on the upper die , wherein is fixed at the center of the upper die, the axis is parallel to the surface of the blank, the axis is parallel to the axis of the upper die; a coordinate system is established on the blank , wherein is fixed at the center of the upper surface of the blank, the axis is on the upper surface of the blank, the axis is parallel to ; is a six-position vector of the moving platform, and is set as , wherein is the position of the upper die, is the Euler angle thereof; The dynamic model of the parallel mechanism is:

[0009] In the formula, is a transformation matrix, is a moving platform position vector when there is a load, wherein is the position of the moving platform, is the Euler angle, is the first-order differential of is the force applied on the upper die along the axis direction respectively, is the corresponding torque generated respectively, is the inertia matrix of the platform, is the position vector of the center point of the upper spherical hinge and the lower spherical hinge, is a coordinate transformation matrix, is the position change vector of the lower S joint, is the first-order differential of the connecting rod deformation vector, is the corresponding elastic stiffness coefficient of the six connecting rods.

[0010] According to the above scheme, in the step S1, is set as the center point of the lower spherical hinge, is the center of the upper spherical hinge; in the coordinate system , the position vector of is set as , and is calculated as follows:

[0011] wherein, , are the polar angle and polar radius of the lower spherical hinge respectively, is the distance between the lower mold center point and the moving platform; In the coordinate system , the position vector of the point is set as , which is calculated as follows: .

[0012] According to the above scheme, in the step S1, in the coordinate system , the position vector of the point is set as , which is calculated as:

[0013] wherein is the coordinate transformation matrix from to :

[0014] wherein and represent sin and cos functions respectively.

[0015] According to the above scheme, in the step S1, according to the principle of constant rod length , the vertical displacement of the slider can be obtained by calculation: .

[0016] According to the above scheme, in the step S1, the transformation relationship between the motion speed of the moving platform and the first-order differential of the pose parameters is represented as:

[0017] wherein is the angular velocity corresponding to the platform, is the transformation matrix, is the first-order differential of the corresponding element, is the first-order differential of .

[0018] According to the above scheme, the regenerated shaping force model of the multi-degree-of-freedom shaping in the step S2 is:

[0019] wherein, ​This refers to the actual position of the upper mold in the previous cycle. These are the polar coordinates of the point of application of the concentrated force.

[0020] According to the above scheme, the dynamic model of the multi-degree-of-freedom forming equipment established in step S3 is as follows: .

[0021] The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment that considers link deformation and regenerative forming force, as described in this invention, has the following beneficial effects: In multi-degree-of-freedom forming processes, deformation mainly occurs in the connecting rods, and due to complex geometric constraints, coupling exists, requiring decoupling. Simultaneously, the forming force originates from complex metal flow and the contact between the die and the blank, necessitating the calculation of the contact area and pressure distribution between the upper die and the blank, and using this information to analyze the regenerative effect in the multi-degree-of-freedom forming process. The dynamic modeling method for multi-degree-of-freedom forming equipment proposed in this invention can significantly reduce calculation errors and accurately predict the dynamic behavior of multi-degree-of-freedom forming equipment during the forming process, thus providing an accurate mathematical model for subsequent research and development. Attached Figure Description

[0022] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings: Figure 1 This is a structural diagram of a multi-degree-of-freedom forming device based on a parallel mechanism; Figure 2 This is a rigid-flexible coupling model for a parallel mechanism. Figure 3 A schematic diagram of the deformation of a connecting rod in a multi-degree-of-freedom forming equipment. Figure 4 This is a schematic diagram showing the contact between the upper mold and the blank; Figure 5 The principle of feed rate variation in the radial development diagram of the blank; Figure 6 Setup for vibration acquisition experiments of multi-degree-of-freedom forming equipment; Figure 7 The upper mold vibrates along the x-axis and the corresponding spectrum; Figure 8 The upper mold vibrates along the y-axis and the corresponding spectrum; Figure 9 The upper mold vibrates along the z-axis and the corresponding spectrum; Figure 10 This is a measured vibration spectrum diagram; Figure 11 To calculate the vibration spectrum considering the regenerative forming force dynamics model; Figure 12 The vibration spectrum was calculated for a dynamic model that does not consider regenerative forming forces. Detailed Implementation

[0023] 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.

[0024] The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force of the present invention includes the following steps: S1. Establish a dynamic model of the parallel mechanism that considers the deformation of the connecting rod.

[0025] like Figure 1 As shown, a coordinate system is established on the upper model. ,in Fixed at the center of the upper mold, The axis is parallel to the surface of the blank. The axis is parallel to the upper mold axis. Similarly, establish a coordinate system on the blank. ,in Fixed at the center of the upper surface of the blank. The shaft is located on the upper surface of the blank. The axis is parallel to . Let the six-bit pose vector of the moving platform be set as follows: ,in This is the position of the upper mold. Let its Euler angles be given. In the coordinate system... If the six pose vectors of the moving platform are known, then the displacements of the six driving sliders can be calculated.

[0026] Let the center point of the lower ball joint be set. The center of the upper ball joint. In the coordinate system... middle, The position vector should be set accordingly. The calculation is as follows: (1) In the formula, , Let be the polar angle and polar radius of the lower ball joint, respectively. This is the distance between the center point of the lower mold and the moving platform.

[0027] In coordinate system middle, The position vector of the point should be set accordingly. The calculation formula is as follows: (2) In the formula, These are the polar angle and polar radius of the upper ball joint, respectively. This represents the slider displacement parallel to the z-axis.

[0028] In coordinate system middle, The position vector of the point is set to It can be calculated as: (3) In the formula for arrive Coordinate transformation matrix: (4) In the formula and Let these represent the sin and cos functions, respectively. According to the principle that the rod length remains constant ( The vertical displacement of the slider. It can be obtained through calculation: (5) like Figure 2 As shown, the six links are configured as a spring-damped system, and a rigid-flexible coupling model of the parallel mechanism is established. respectively applied to the upper edge of the upper mold Force in the axial direction, These represent the corresponding torques generated. To simplify the dynamic model, the following three assumptions are proposed: 1) The mass of the connecting rod is mainly concentrated at both ends, and the mass in the middle of the connecting rod can be ignored in the calculation. Therefore, the mass of the connecting rod can be equivalent to that of the slider and the moving platform. 2) The moving platform and machine tool have high rigidity and can be regarded as rigid bodies, while the ball joint has relatively low rigidity. Therefore, the ball joint can be regarded as a flexible body, and the friction at its connection is not considered. Only the axial deformation of the joint is considered.

[0029] 3) The displacement of the 6 sliders is set to be determined solely by the servo motor input and is not affected by platform vibration.

[0030] Based on the assumption that the connecting rod, as a flexible body in the system, will inevitably deform during the forming process, and due to the geometric constraints between the connecting rods, the deformation of the connecting rods is coupled, so it is necessary to decouple the deformation of the connecting rods first. For example... Figure 3 As shown, the black outline represents the pose of the link and platform when there is no load, and the red outline represents the pose of the link and platform when there is a load. This is the position of the lower ball joint when it is under load. Let the theoretical lower ball joint position (i.e., under no load) be given, and the pose vector of the moving platform under load be set as follows: ,in For the position of the moving platform, For Euler angles, the theoretical position vector of the moving platform under no load is: . Position vector is expressed as .

[0031] Since the change of the link posture in deformation is small, i.e. the included angle between the link under no load and the link under load is negligible, i.e. , , are collinear. Based on this, the vector of deformation of each link can be expressed as: (6) The dynamics model of the multi-DOF forming equipment is established by using the Lagrange method: (7) First, the kinetic energy of the system is analyzed. The components of the entire equipment having kinetic energy are the slider, the link, and the moving platform. The velocity of the slider is determined by the input of the motor according to assumption (3) and is independent of the vibration of the moving platform, so the kinetic energy of the slider part does not need to be analyzed in the kinetic energy analysis. According to assumption (1), the mass of the link is equivalent to the slider and the moving platform, so the kinetic energy of the link does not need to be analyzed separately. The kinetic energy of the moving platform needs to be considered in the entire multi-DOF forming equipment.

[0032] The rotational velocity of the moving platform cannot be directly obtained by directly differentiating the rotation angle, and needs to be obtained according to the Coriolis force. The transformation relationship between the motion velocity of the moving platform and the first-order differential of the posture parameters is expressed as: (8) In the formula, is the angular velocity of the platform corresponding to the angular velocity, is the transformation matrix, is the first-order differential of the corresponding element, is the first-order differential of .

[0033] The dynamic energy of the system is only the dynamic energy of the platform, which can be expressed as: (9) In the formula, the inertia matrix of the platform is: (10) In the formula, is the mass of the platform, is the corresponding moment of inertia of the platform around the X, Y, and Z axes.

[0034] ​The potential energy of the system has two parts: the potential energy of the moving platform (gravity potential energy) and the elastic potential energy of the links (elastic potential energy). The gravity potential energy of the system is the same as the kinetic energy of the system, and only the potential energy of the moving platform needs to be considered. First, set the plane as the potential energy surface. The gravity potential energy of the moving platform can be expressed as: (11) According to assumption (2), only the links are flexible bodies in the entire system, and therefore the elastic potential energy of the system is only the energy of the deformation of the links. According to equation (6), the elastic potential energy of the deformation of the links is: (12) In the equation, is the elastic stiffness coefficient corresponding to the six links.

[0035] Finally, the potential energy of the entire system is obtained by adding equation (11) and equation (12): (13) The dissipation energy of the system is the energy consumed by the damping of the six links in the deformation process. It can be expressed as: (14) In the equation, is the position change vector of the lower S joint, which is expressed as , is the damping coefficient of the link, is the first-order derivative of the deformation vector of the link, is the first-order derivative of .

[0036] The dynamics model of the parallel mechanism is obtained by bringing equation (9), equation (13), and equation (14) into equation (7): (15) S2, a model of reshaping force in multi-degree-of-freedom forming is established.

[0037] As shown in Figure 4 , the included angle between the center line of the upper die and the center line of the blank is the swing angle , is the feed rate of each circle, and the red and blue shaded areas are the contact areas , and the forming force applied on the upper die can be expressed as a concentrated force at point . Since the curvature of the upper die is large, and the forming process of the upper die on the blank is downward, the contact area between the upper die and the blank is almost horizontal, and the forming force can also be regarded as a vertical upward force , which can be ignored, that is, Therefore, the forming force acting on the upper die can be expressed as: (16) When the device is working, the six sliders push the moving platform and drive the upper die to rotate around the axis, generating a swing motion. At the same time, the upper die is fed under the action of the servo motor, continuously rolling on the workpiece, locally and orderly pressurizing the workpiece, causing the blank to deform.

[0038] Taking a circular trajectory as an example, the theoretical position of the upper die is determined by the swing angle , the theoretical feed rate , and the motion frequency , i.e. . In order to calculate the contact area between the upper die and the blank, the specific situation of the contact surface is described in the coordinate system Figure 4 on the surface of the blank as shown. The shaded area in the figure is the contact area between the blank and the upper die . Among them, is a point on the boundary of the contact area, and the position vectors are respectively set as The contact area is divided into red and blue shaded areas by the straight line between and . The forming force can be regarded as a concentrated force, and the action point is , and the polar coordinate position corresponding to the action point is .

[0039] In the coordinate system , the equation of the surface of the upper die can be expressed as: (17) where is the position vector in the coordinate system .

[0040] The equation of the surface of the blank is , and the equation of the curve is: (18) The boundary of the blank is a circle, and its equation in the coordinate system is: (19) where is the radius of the blank.

[0041] Combining equations (3-3) and (3-4), the axis coordinates of can be calculated as: ​ (20) Substituting equation (20) into equation (19), the contact area ratio is obtained as Axial coordinate: (21) Adding the red and blue shadow parts, the contact area is obtained as (22) The result of equation (22) is calculated as (23) where

[0042] Equation (23) can be simplified as (24) where is the relative feed rate, is the contact area ratio:

[0043] (25) In multi-degree-of-freedom forming, the change of feed rate is caused by vibration. As shown in Fig. 5, the black thin solid line is the real upper die trajectory of the last cycle, and the black thick solid line is the real upper die trajectory of the current cycle. In addition, the red dashed line is the theoretical upper die trajectory. The area between the red dashed lines is the theoretical forming area, and the black shadow area is the actual forming area. The red solid line is the forming moment Figure 5 , is the theoretical feed rate, is the Z-axis position of the upper die at time , is the Z-axis position of the upper die in the last cycle, is the actual feed rate. is the moving period of the parallel mechanism, which is calculated as .

[0044] In multi-degree-of-freedom forming, the feed motion makes the blank height decrease and the blank radius increase. Assuming that the overall volume is constant, the real-time height and radius of the blank are obtained as (26) where are the initial height and radius of the blank, respectively.

[0045] Concentrated force can be calculated by solving the following equations: (27) where ​​​​Pave is the average pressure of the contact surface, Plobe is the lobe parameter, Pyield is the yield strength of the blank material, Pave is the average pressure coefficient Pc is the polar coordinate of the point of action of the concentrated force Pave can be expressed as (28) The relationship between them can be expressed as: (29) In the formula Pprev is the actual position of the upper die in the previous cycle Based on the above, the regenerative forming force model of multi-degree-of-freedom forming can be established: (30) S3, a dynamic model of a multi-degree-of-freedom forming equipment is established.

[0046] By combining equation (15) with equation (30), a dynamic model of a multi-degree-of-freedom forming equipment can be established: (31) In order to solve this dynamic equation, the boundary conditions of the equation need to be determined. The boundary conditions are the first and second order parameters of the position and attitude vector of the moving platform , which are set to , i.e. (32) Therefore, by combining the dynamic equation (equation (31)) and the boundary conditions (equation (32)), the dynamic differential equation set has a total of 36 variables, among which there are theoretical motion variables describing the position and attitude of the moving platform under no load, including the theoretical position , first order differential and second order differential of the upper die, and actual motion variables describing the position and attitude of the moving platform under load, including the actual position , first order differential and second order differential of the upper die.

[0047] In this case, the number of equations is 6+12=18, and the number of output variables is 36. The number of variables is 18 more than the number of equations. This means that in order to solve the dynamic problem, at least 18 variables need to be known. In general, in the inverse kinematics model, the theoretical motion of the upper die , , , a total of 18 variables) is determined by setting the circular trajectory motion parameters The given, thus the entire differential equation equation number is equal to the number of variables that need to be solved, the actual attitude displacement of the moving platform can be solved , , , the actual motion of the moving platform is predicted.

[0048] Embodiment Based on the equipment structure parameters given in Table 1, the dynamic model of the multi-degree-of-freedom forming equipment can be established according to the above method, and the vibration of the upper die in the forming process is calculated. Compared with the results collected by the experiment, the accuracy of the dynamic model can be verified.

[0049] Table 1 Configuration parameters of six-link multi-degree-of-freedom equipment

[0050] As shown in Figure 6 , an experiment is arranged to process a circular blank with a radius of 30 mm and a height of 25 mm in a circular trajectory, and the vibration is measured by an acceleration sensor installed at the edge of the upper die and collected by an LMS signal collection system. The circular trajectory parameters are set to , , , and the theoretical position of the trajectory moving platform is represented as: .

[0051] In the multi-degree-of-freedom forming process, the displacement of the upper die is monitored, and the vibration of the upper die in the forming process is calculated by . In analyzing the dynamic performance of the multi-degree-of-freedom forming equipment, the angular vibration can be ignored, and only the vibration along the x, y, and z axes (i.e. , , ) is considered.

[0052] As shown in Figure 7 , 8 , 9, all subgraphs marked (a) represent the vibration of the tool along the x, y, and z axes; the subgraph marked (b) is a partial enlarged view of subgraph (a). In subgraphs (a) and (b), the black curve represents the measured vibration of the upper die in the experiment, the blue curve represents the vibration calculated by the dynamic model considering the regenerative forming force (i.e. calculated by formula (30)), and the red curve represents the vibration calculated by the dynamic model without considering the regenerative forming force (i.e. the real-time feed rate is set to a constant, ).

[0053] As shown in Figure 7As shown in FIGS. 8a, 9a, the vibration along the x-axis and the y-axis is greater than the vibration along the z-axis. This is due to the lower stiffness of the equipment along the x-axis and the y-axis, and the higher stiffness along the z-axis. The vibration of the red curve is smaller, and the vibration of the blue curve is greater, which is closer to the actual measured vibration. Compared with the measured vibration value, the vibration error in the x-axis direction is 21%, the vibration error in the y-axis direction is 16%, and the vibration error in the z-axis direction is 18% in the dynamic model calculation result considering the re-shaping force. In the dynamic model calculation result without considering the re-shaping force, the vibration error in the x-axis direction is 92%, the vibration error in the y-axis direction is 91%, and the vibration error in the z-axis direction is 95%.

[0054] Figure 10 、 Figure 11 、 Figure 12 The frequency spectrum of the actual measurement result, the dynamic model calculation result considering the re-shaping force, and the dynamic model calculation result without considering the re-shaping force is shown respectively. From the vibration component analysis, the vibration amplitude of the dynamic model calculation result without considering the re-shaping force is lower than the measured value, especially the vibration of the second-order mode in the z-axis direction is significantly lower than the measured value. The dynamic model calculation result considering the re-shaping force is more consistent with the measured value in each vibration component. This is because in the actual forming process, the re-shaping force in the z-axis direction causes the upper die to vibrate self-excitedly in the second-order mode, so that the vibration of the upper die is more intense.

[0055] The embodiments of the present application are described above in combination with the drawings, but the present application is not limited to the above specific embodiments, and the above specific embodiments are only illustrative, but not restrictive. Those skilled in the art can make many forms under the inspiration of the present application without departing from the purpose of the present application and the scope protected by the claims, which are all within the protection of the present application.

Claims

1. A modeling method for the dynamics of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force, characterized in that, Includes the following steps: S1. Establish a dynamic model of the parallel mechanism that considers the deformation of the connecting rod; S2. Establish a model of regenerative forming force in multi-degree-of-freedom forming; S3. Establish a dynamic model of the multi-degree-of-freedom forming equipment based on the dynamic model of the parallel mechanism and the model of the regenerative forming force.

2. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force according to claim 1, characterized in that, In step S1, the deformation of the connecting rod is decoupled, and a dynamic model of the parallel mechanism is established based on the Lagrange method.

3. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force as described in claim 1, characterized in that, In step S2, the contact area and pressure distribution between the blank and the upper mold are calculated, the influence of vibration on the forming force is analyzed, and a regenerative forming force model is established.

4. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force according to claim 1, characterized in that, In step S1, a coordinate system S is established on the upper model. A (O A -x A y A z A ), where O A Fixed at the center of the upper mold, x A ,y A The axis is parallel to the surface of the blank, z A The axis is parallel to the upper mold axis; establish a coordinate system S on the blank. B (O B -x B y B z B ), where O B Fixed at the center of the upper surface of the billet, x B ,y B The shaft is located on the upper surface of the blank, z B The axis is parallel to z. A ; u is the six-position pose vector of the moving platform, denoted as [x0, y0, z0, α0, β0, γ0] T Where x0, y0, z0 are the positions of the upper mold, and α0, β0, γ0 are its Euler angles; The dynamic model of the parallel mechanism is as follows: In the formula, R(q) is the transformation matrix, q=[x,y,z,α,β,γ] T Let be the pose vector of the moving platform under load, where x, y, z are the positions of the moving platform, and α, β, γ are the Euler angles. F is the first differential of q. x ,F y ,F z The x-axis is applied to the upper mold respectively. B ,y B ,z B Force in the axial direction, M x M y M z The corresponding torques M are respectively generated. p For the platform's inertia matrix, Let be the position vector of the center point of the upper and lower ball joints. Let ΔL be the coordinate transformation matrix. i Let S be the position change vector of the lower S joint. Let be the first derivative of the link deformation vector, and k be the elastic stiffness coefficients corresponding to the six links.

5. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force according to claim 4, characterized in that, In step S1, A 1 A 2 A 3 A 4 A 5 A 6 Let B be the center point of the lower ball joint. 1 B 2 B 3 B 4 B 5 B 6 The center of the upper ball joint; in coordinate system S A In the middle, A 1 A 2 A 3 A 4 A 5 A 6 The position vector is set accordingly. The calculation is as follows: In the formula, r A These are the polar angle and polar radius of the lower ball joint, respectively, h. A This is the distance between the center point of the lower mold and the moving platform. In coordinate system S B In the middle, A 1 A 2 A 3 A 4 A 5 A 6 The position vector of the point is set to The calculation is as follows:

6. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force according to claim 5, characterized in that, In step S1, in coordinate system S B In the middle, A 1 A 2 A 3 A 4 A 5 A 6 The position vector of the point is set to The calculation is as follows: In the formula For S A To S B Coordinate transformation matrix: In the formula, s and c represent the sin and cos functions, respectively.

7. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force as described in claim 6, characterized in that, In step S1, according to the principle of constant rod length Vertical displacement of the slider It can be obtained through calculation:

8. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force according to claim 7, characterized in that, In step S1, the transformation relationship between the motion velocity of the motion platform and the first derivative of the pose parameters is expressed as: In the formula ω x ,ω y ,ω z R(q) represents the angular velocity corresponding to the platform, and R(q) is the transformation matrix. Let q be the first derivative of the element corresponding to q. It is the first differential of q.

9. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force as described in claim 8, characterized in that, The regenerative forming force model for multi-degree-of-freedom forming established in step S2 is as follows: Q(q,q τ )=[0,0,F z (q,q τ ),F z (q,q τ )R e (t)sin(θ e (t)),F z (q,q τ )R e (t)cos(θ e (t)),0] T In the formula, q τ R represents the actual position of the upper mold in the previous cycle. e ,θ e These are the polar coordinates of the point of application of the concentrated force.

10. The modeling method for the dynamic model of a multi-degree-of-freedom forming equipment considering link deformation and regenerative forming force according to claim 9, characterized in that, The dynamic model of the multi-degree-of-freedom forming equipment established in step S3 is as follows:

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