Envelope forming equipment full-spinning dynamics modeling and deformation prediction method

Abstract

The present application relates to a kind of envelope shaping equipment full rotational quantity rigid-flexible coupling dynamics modeling and deformation prediction method, link and slider are simplified as double mass spring system by lumped mass method, moving platform is simplified as single center lumped mass-six circumferential distribution lumped mass twelve spring coupled system, rigid body displacement of each connecting pair is characterized by motion rotational quantity, the deformation displacement of each spring is characterized by deformation motion rotational quantity, finally coupling mass matrix and moment of inertia is established generalized mass matrix, and then the establishment equipment full rotational quantity characterization rigid-flexible coupling dynamics equation and realize equipment deformation prediction.The present application can quickly realize the solution of equipment real-time deformation rotational quantity and space motion rotational quantity, and then predict the deformation behavior of equipment, realize equipment precision compensation control, strength check and state monitoring.

Description

Technical Field

[0001] This invention relates to the field of envelope forming, and more specifically, to a method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment. Background Technology

[0002] Enveloping forming equipment is a multi-degree-of-freedom plastic forming equipment, particularly suitable for high-performance, high-efficiency near-net-shape forming of thin-walled, high-ribbed components. This equipment provides a stable and controllable power source through six sets of servo motors, planetary reducers, and ball screws. It adopts a parallel support structure of multi-branch ball joints, ball-head connecting rods, and ball joints, and uses multi-servo motor collaborative control to achieve arbitrary spatial enveloping forming motion of the mold. Therefore, it has a large degree of freedom and good motion flexibility.

[0003] Due to the large forming load during operation, the connecting rods and moving platform undergo significant deformation, severely affecting the motion accuracy of the envelope forming equipment and consequently the geometric accuracy of the formed components. Because the envelope forming equipment is a multi-branch, multi-drive parallel system, the deformation of each branch and component is mutually restrictive and coordinated, resulting in highly complex dynamic deformation behavior. Therefore, it is essential to establish a dynamic model of the envelope forming equipment to indicate the deformation behavior of key components, which can then be used for accuracy compensation control, strength verification, and condition monitoring. Currently, there are no reports on rigid-flexible coupling dynamic modeling and deformation prediction methods for envelope forming equipment. Summary of the Invention

[0004] The technical problem to be solved by this invention is to provide a method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment. Through the established rigid-flexible coupling dynamic model, the real-time deformation spin and spatial motion spin of the equipment can be solved quickly, thereby predicting the deformation behavior of the equipment and realizing equipment accuracy compensation control, strength verification and condition monitoring.

[0005] The technical solution adopted by this invention to solve its technical problem is as follows: A method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of an envelope forming equipment is constructed. The envelope forming equipment has a six-branch parallel motion configuration. Each branch includes a motor, a slider, an upper ball joint, a connecting rod, and a lower ball joint. The six branches are connected to a moving platform. An envelope mold is installed on the moving platform. The motors on the six branches drive the corresponding sliders and connecting rods to coordinate their movements, causing the envelope mold to perform spatial envelope motion. The blank is placed in the die, and the die drives the blank to perform linear feed motion upwards. The dynamic modeling and deformation prediction method simplifies the connecting rod and slider into a dual-mass spring system using the lumped mass method, and simplifies the moving platform into a single-center lumped mass - six circumferentially distributed lumped mass twelve-spring parallel system. The motion spinor characterizes the large displacement of the rigid body of each connecting pair, and the deformation motion spinor characterizes the deformation displacement of each spring. Finally, the mass matrix and rotational inertia are coupled to establish a generalized mass matrix, thereby establishing the full-spin characterization of the rigid-flexible coupling dynamic equation of the equipment and realizing the equipment deformation prediction. This method includes the following steps:

[0006] S1. Establish the motion and force conditions of the envelope forming equipment;

[0007] S2. Establish the force-position equilibrium equations for the slider;

[0008] S3. Establish the force-position equilibrium equations for the connecting rod;

[0009] S4. Establish the force-potential equilibrium equations for the moving platform;

[0010] S5. Establish the overall rigid-flexible coupling dynamic equation of the envelope forming equipment.

[0011] According to the above scheme, in step S1, the driving force of each slider of the envelope forming equipment is the force rotation. This indicates that, because the direction of the driving force is fixed, It can be expressed as follows according to formula (1):

[0012]

[0013] In the formula, The magnitude of the driving force;

[0014] The external load force rotation W on the moving platform of the envelope forming equipment l This indicates that for a fixed-point envelope forming motion, the external load is calculated according to equation (2):

[0015]

[0016] In the formula, F l T is the total forming force vector acting on the moving platform. l Let r be the forming torque vector acting on the moving platform. l (θ) is the position vector of the equivalent point of application of the forming force, rl Let F be the radius of the equivalent point of application of the forming force, θ be the angle of the equivalent point of application of the forming force, and F be the radius of ... l F represents the magnitude of the total forming force acting on the moving platform. l The magnitude of the total forming torque acting on the moving platform;

[0017] The motions of the slider, the lower ball joint of the connecting rod, and the upper ball joint of the connecting rod are respectively expressed by the kinematic spin. This indicates that the corresponding constraint force spinors are respectively represented by force spinors. This indicates that each slider is equivalent to two lumped mass units, with masses M and M respectively. a and M ak The two mass units are connected by a spring, and the elastic force and damping force of the spring are respectively... The stiffness and deformation of a spring are expressed by its screw force. This indicates that the coordinate system centers of the two mass blocks are set at the initial point of the slider and the center of the ball joint on the slider, respectively; each link is equivalent to two lumped mass elements with masses M and M, respectively. b and M bk The two mass units are connected by a spring, and the elastic force and damping force of the connecting rod spring are respectively... The stiffness and deformation of a spring are expressed by its spin. This indicates that the moving platform of the envelope forming equipment is equivalent to a central mass element and six identical circularly distributed mass elements, with the mass set as M. ck and M c The central mass element and the circumferential mass element are connected by radial springs, and the elastic force and damping force of the radial springs are respectively... Each adjacent circumferential mass unit is connected by a circumferential spring, and the elastic force and damping force of each circumferential spring are respectively...

[0018] According to the above scheme, in step S2, the slider is equivalent to two lumped mass units, and the mass settings of the two mass units are exactly the same, both defined as m. a Both mass elements are equivalent to cylinders, with a radius equal to the cross-sectional radius r of the slider. a Its height is half the length of the slider, i.e., l a / 2, its moment of inertia relative to the center is The generalized mass matrix of the two mass elements of the slider is calculated according to equation (3):

[0019]

[0020] In the formula, I is a 3*3 identity matrix;

[0021] Coordinate system S0 is the fixed bed coordinate system. To fix it at the initial position of the slider, and An orthogonal coordinate system with its axis pointing in the direction of the slider's motion. Its relationship with the coordinate system S0 is expressed by Equation (4):

[0022]

[0023] In the formula, is the position vector of the starting point of the slider;

[0024] The coordinate systems where the two mass units of the slider are located are respectively set as and the sum of The rigid body motion of the upper mass unit of the slider is represented by the coordinate system and the screws in S0 are respectively set as and There is a conversion relationship of Equation (5) between the two screws:

[0025]

[0026] In the formula, is the magnitude of the displacement of the slider's motion, which is a scalar; is the coordinate transformation between the coordinate system and S0, and is represented by Equation (6) according to the exponential product formula of screws:

[0027]

[0028] The lower mass unit of the slider is connected to the upper mass unit of the slider through a spring and a damper. The distance between the two mass units is the length l of the slider a ; The coordinate system is set at the initial position of the lower mass unit of the slider. There is a conversion relationship of Equation (7) between the initial position of the lower mass unit of the slider and the bed coordinate system S0:

[0029]

[0030] Due to the small deformation of the slider, the finite displacement screw can be used for expression, that is, it is defined in the coordinate S a as or defined in the coordinate system S0 as There is a conversion relationship of Equation (8) between the expressions in the two coordinate systems:

[0031]

[0032] In the formula, is the deformation screw element of, which is obtained according to the elastic deformation and damping relationship in different directions. The elastic force and damping force of the upper and lower mass units of the slider are calculated according to Equation (9):

[0033]

[0034] In the formula, The generalized elastic force between the two mass units of the slider. The generalized damping force between the two mass units of the slider; The derivative of the elastic twist twistor, and The generalized stiffness matrix and the generalized damping matrix are calculated according to equation (10):

[0035]

[0036] In the formula, χ a and λ a The linear damping coefficient is given by E and u, which are the material's elastic modulus and Poisson's ratio, respectively.

[0037] Based on the elastic deformation relationship, the transformation relationship between the lower mass element of the slider and the fixed bed coordinate system S0 is given by equation (11):

[0038]

[0039] The force balance equations for the upper and lower mass elements of the slider are established using the spinor method based on the force-potential relationship, as shown in equation (12):

[0040]

[0041] In the formula, and Let g be the gravitational force acting on the upper and lower masses of the slider, and let the gravitational acceleration vector be g = [0,0,1,0,0,0]. T Therefore, and For the slider in coordinate system S a The constraint spinor in For the link in coordinate system S ak The screw of the constraint force applied to the slider is expressed by equation (13):

[0042]

[0043] According to the above scheme, in step S3, the connecting rod is equivalent to two concentrated mass elements, and the mass settings of the two mass elements are exactly the same, both defined as m. b Both mass elements are equivalent to cylinders, with a radius equal to the cross-sectional radius r of the connecting rod. a Its height is half the length of the connecting rod, i.e., l b / 2, its moment of inertia relative to the center is The generalized mass matrix of the two mass elements of the connecting rod is calculated according to equation (14):

[0044]

[0045] The coordinate systems where the two mass units of the connecting rod are located are respectively set as and The rigid body motion of the upper mass unit of the connecting rod is represented by a twist in the coordinate systems and S0 as and There is a conversion relationship of Equation (15) between the two twists:

[0046]

[0047] In the formula, is the motion twist of the spherical hinge on the connecting rod element, which defines the rotational speeds of the spherical hinge on the connecting rod around three directions. The conversion relationship between the coordinate systems and is expressed according to Equation (16):

[0048]

[0049] The lower mass unit of the connecting rod is connected to the upper mass unit of the connecting rod through a spring and a damper. The distance between the two mass units is the length l of the slider b +l bk ; The coordinate system is set at the initial position of the lower mass unit of the slider. There is a conversion relationship of Equation (17) between the initial position of the lower mass unit of the slider and the bed coordinate system S0:

[0050]

[0051] In the formula, The deformation of the connecting rod is expressed by a finite displacement twist, that is, it is defined in the coordinate S b as or defined in the coordinate system S0 as There is a conversion relationship of Equation (18) between the expressions in the two coordinate systems:

[0052]

[0053] In the formula, is the deformation twist element, which is obtained according to the elastic deformation and damping relationships in different directions. The elastic force and damping force between the upper and lower mass units of the connecting rod are expressed according to Equation (19):

[0054]

[0055] In the formula, is the generalized elastic force between the two mass units of the slider, is the generalized damping force between the two mass units of the slider; is the differential of the elastic deformation twist, Kb and D b The generalized stiffness matrix and the generalized damping matrix are calculated according to equation (20):

[0056]

[0057] In the formula, χ b and λ b The damping coefficient is linear.

[0058] Based on the elastic deformation relationship, the transformation relationship between the lower mass element of the connecting rod and the fixed bed coordinate system S0 exists as equation (21).

[0059]

[0060] The force balance equations for the upper and lower mass elements of the connecting rod are established using the spinor method based on the force-potential relationship, as shown in equation (22):

[0061]

[0062] In the formula, and Let g be the weight of the mass blocks above and below the connecting rod, then... and For the link in coordinate system S c The constraint force spinor in the equation is expressed by equation (23):

[0063]

[0064] According to the above scheme, in step S4, the moving platform is equivalent to the geometry connecting a central mass unit and six circumferential mass units, which is assumed to be a cube; the mass of the central mass unit is set as m. ck The main components are the mass of the mold and the center mass of the moving platform. The mass of the circumferential mass unit is set as m. c The main components are the mass of the ball seat and the circumferential mass of the moving platform; each mass element is equivalent to a cylindrical geometry with radii r. ck ,r ck Their heights are l ck ,l c The inertia tensors of the central mass element and the circumferential mass element are calculated based on the geometry, and the generalized mass matrix of the moving platform mass element is calculated according to equation (24):

[0065]

[0066]

[0067] Let the coordinate systems of the moving platform center and the circumferential mass element be S and S respectively. c and The rigid body motion of the circumferential mass unit of the moving platform can be represented by a screw in the coordinate systems and S0, respectively, as and There is a transformation relationship in Equation (25) between the two screws:

[0068]

[0069] Wherein, is the motion screw of the lower spherical hinge whose elements define the rotational speeds of the lower spherical hinge around three directions, and thus the transformation relationship between the coordinate systems and S0 is expressed by Equation (26):

[0070]

[0071] The central mass unit of the moving platform is connected to the circumferential mass unit of the moving platform through springs and dampers, and the distance between the two mass units is where r B is the radius of the moving platform, is the central angle corresponding to the upper spherical hinge point on the moving platform; the coordinate system is set as the initial position of the circumferential mass unit of the moving platform, and there is a transformation relationship in Equation (27) between the initial position of the circumferential mass unit of the moving platform and the bed coordinate system S0:

[0072]

[0073] Wherein, The deformation of the moving platform is expressed by a finite displacement screw, that is, it is defined in the coordinate S c S b as or defined in the coordinate system S0 as There is a transformation relationship in Equation (28) between the expressions in the two coordinate systems:

[0074]

[0075] Wherein, is the deformation screw whose elements are obtained according to the elastic deformation and damping relationships in different directions;

[0076] Each circumferential mass unit is connected to the adjacent circumferential mass through a spring, and its deformation screw is calculated according to Equation (29):

[0077]

[0078] The deformation screws of the springs on the moving platform are calculated according to Equation (30):

[0079]

[0080] In the formula, The generalized elastic force between the central mass element and the circumferential mass element of the moving platform. The generalized elastic force between adjacent circumferential mass elements of the moving platform. K is the differential of the elastic twist twistor. c ,K q D is the generalized stiffness matrix. c D q It is the generalized damping matrix;

[0081] Based on the spatial distribution relationship between the central mass element and the circumferential mass element, and their assumed cubic geometric connection form, the length of the connecting cube between the central mass element and the circumferential mass element can be obtained as r. B The width of the connected cube is equal to the thickness t of the moving platform. B Thus connecting the height w of the cube B Calculate according to formula (31):

[0082]

[0083] The connection length between adjacent circumferential mass elements is the chord length of the corresponding circumferential point, which can be solved according to equation (32):

[0084]

[0085] Its width is equal to the thickness t of the moving platform. B Its height is half the thickness of the moving platform, that is

[0086] Therefore, the stiffness matrix can be solved according to equation (33):

[0087]

[0088] The generalized damping force matrix is ​​solved according to equation (34):

[0089]

[0090] In the formula, χ c and λ c The damping coefficient is linear.

[0091] According to the elastic deformation compatibility relationship, for the same circular mass element, its deformation must be the same as that of its two adjacent circular mass elements, that is:

[0092]

[0093] The transformation relationship between the moving platform center mass element and the fixed bed coordinate system S0 is given by equation (36):

[0094]

[0095] Based on the condition that the center point is in the same position in each branch, the relationship shown in equation (37) exists:

[0096]

[0097] This formula can also be expressed as

[0098] The force balance equations for each mass element at the center and circumference of the moving platform are established using the spinor method based on the force-potential relationship, as shown in equation (38):

[0099]

[0100] In the formula, and For the gravity acting on the center and circumferential mass elements of the moving platform, there is and m ckg .

[0101] According to the above scheme, in step S5, the overall dynamic equation of the envelope forming equipment is obtained by simultaneously solving the force balance equations of each component, as shown in equation (39):

[0102]

[0103] The method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment according to the present invention has the following beneficial effects:

[0104] 1. The envelope forming equipment of the present invention adopts a configuration of six servo motors in parallel drive and six-link parallel support, which can realize envelope forming motion in any space. It has a large degree of freedom of motion and good motion flexibility, and is particularly suitable for high-performance, high-efficiency near-net-shape forming manufacturing of thin-walled, high-rib components.

[0105] 2. This invention uses the full spinor method to establish the dynamic model of the envelope forming equipment. By coupling the mass matrix and the inertial tensor, the generalized mass matrix of the envelope forming equipment is established, which has the advantages of symmetrical equation structure, unified form, convenient solution and accurate calculation.

[0106] 3. The dynamic model of the envelope forming equipment established by the present invention can accurately and quickly predict the deformation of the envelope forming equipment, thereby realizing the precision compensation control, strength verification and condition monitoring of the envelope forming equipment. Attached Figure Description

[0107] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0108] Figure 1 This is a schematic diagram of the envelope forming equipment;

[0109] Figure 2(a) is a schematic diagram of the rigid-flexible coupling dynamic system model of the envelope forming equipment;

[0110] Figure 2(b) is a diagram showing the connection relationship of the rigid-flexible coupling dynamic system of the envelope forming equipment;

[0111] Figure 3(a) is a schematic diagram of the forces acting on the slider;

[0112] Figure 3(b) is a schematic diagram of the deformation of the slider;

[0113] Figure 4(a) is a schematic diagram of the forces acting on the connecting rod;

[0114] Figure 4(b) is a schematic diagram of the deformation of the connecting rod;

[0115] Figure 5(a) is a schematic diagram of the forces acting on the moving platform;

[0116] Figure 5(b) is a schematic diagram of the deformation of the moving platform;

[0117] Figure 6 It is the displacement curve of the slider when the envelope forming equipment moves along a circular trajectory;

[0118] Figure 7 It is the deformation curve of the slider when the envelope forming equipment moves along a circular trajectory;

[0119] Figure 8 It is the deformation curve of the connecting rod when the envelope forming equipment moves in a circular trajectory;

[0120] Figure 9 It is the deformation curve of the moving platform when the envelope forming equipment moves in a circular trajectory;

[0121] Figure 10 It is the driving force curve of each slider when the envelope forming equipment moves in a circular trajectory. Detailed Implementation

[0122] 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.

[0123] like Figure 1 As shown, the method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment of the present invention includes the following steps:

[0124] S1. Envelope forming equipment configuration design; such as... Figure 1As shown, the envelope forming equipment has a 6-chain parallel motion configuration. Each chain includes a motor, a slider, an upper ball joint, a connecting rod, and a lower ball joint. The 6 chains are connected to the moving platform at specific positions. An envelope mold is mounted on the moving platform. The motors on the 6 chains drive the corresponding sliders and connecting rods to coordinate their movements, causing the envelope mold to perform complex spatial envelope motion. The blank is placed in the die cavity, which then moves the blank upwards in a linear feed motion. Under the combined action of the envelope mold and the die cavity, the blank undergoes continuous localized plastic deformation until the component is nearly net-shape formed.

[0125] S2. Establish the motion and force conditions of the envelope forming equipment; as shown in Figures 2(a) and 2(b), the driving force of each slider of the envelope forming equipment is determined by the force rotation. This indicates that, because the direction of the driving force is fixed, therefore It can be expressed as follows according to formula (1):

[0126]

[0127] In the formula, This refers to the magnitude of the driving force.

[0128] The external load force rotation W on the moving platform of the envelope forming equipment l This indicates that for a fixed-point envelope forming motion, the external load can be calculated according to equation (2):

[0129]

[0130] In the formula, F l T is the total forming force vector acting on the moving platform. l Let r be the forming torque vector acting on the moving platform. l (θ) is the position vector of the equivalent point of application of the forming force, r l Let F be the radius of the equivalent point of application of the forming force, θ be the angle of the equivalent point of application of the forming force, and F be the radius of ... l The magnitude of the total forming force on the moving platform, T l The magnitude of the total forming torque on the moving platform.

[0131] The motions of the slider, the lower ball joint of the connecting rod, and the upper ball joint of the connecting rod are respectively expressed by the kinematic spin. This indicates that the corresponding constraint force spinors are respectively represented by force spinors. This indicates that each slider is equivalent to two lumped mass units, with masses M and M respectively. a and M ak The two mass units are connected by a spring, and the elastic force and damping force of the spring are respectively... The stiffness and deformation of a spring are expressed by its screw force. This indicates that the coordinate system centers of the two mass blocks are set at the initial point of the slider and the center of the ball joint on the slider, respectively; each link is equivalent to two lumped mass elements with masses M and M, respectively.b and M bk The two mass units are connected by a spring, and the elastic force and damping force of the connecting rod spring are respectively... The stiffness and deformation of a spring are expressed by its screw force. This indicates that the moving platform of the envelope forming equipment is equivalent to one central mass element and six identical circularly distributed mass elements, with the mass set as M. ck and M c The central mass element and the circumferential mass element are connected by radial springs, and the elastic force and damping force of the radial springs are respectively... Each adjacent circumferential mass unit is connected by a circumferential spring, and the elastic force and damping force of each circumferential spring are respectively...

[0132] S3 establishes the force-position equilibrium equations for the slider; the forces and deformations of the slider are shown in Figures 3(a) and 3(b). The slider is equivalent to two lumped mass elements, and the mass settings of the two mass elements are exactly the same, both defined as m. a Both mass elements are equivalent to cylinders, with a radius equal to the cross-sectional radius r of the slider. a Its height is half the length of the slider, i.e., l a / 2, its moment of inertia relative to the center is The generalized mass matrix of the two mass elements of the slider can be calculated according to equation (3):

[0133]

[0134] In the formula, I is a 3*3 identity matrix.

[0135] Coordinate system S0 is the fixed bed coordinate system. To fix it at the initial position of the slider, and An orthogonal coordinate system with its axis pointing in the direction of slider movement, and its relationship with coordinate system S0 can be expressed by equation (4):

[0136]

[0137] In the formula, is the position vector of the slider's starting point.

[0138] The coordinate systems of the two mass elements of the slider are respectively set as and The rigid body motion of the mass element at the upper end of the slider can be represented by a coordinate system. Let the spinors in S0 be respectively set as and The transformation relationship of the two spinors existence equation (5):

[0139]

[0140] In the formula, is the magnitude of the displacement of the slider movement, which is a scalar; is the coordinate transformation between the coordinate system and S0, and can be expressed by Equation (6) according to the exponential product formula of the screw:

[0141]

[0142] The lower mass unit of the slider is connected to the upper mass unit of the slider through a spring and a damper, and the distance between the two mass units is the length l of the slider a . The coordinate system is set at the initial position of the lower mass unit of the slider. There is a conversion relationship of Equation (7) between the initial position of the lower mass unit of the slider and the bed coordinate system S0:

[0143]

[0144] Since the deformation of the slider is small, the finite displacement screw can be used to express it, that is, it is defined in the coordinate S a as or defined in the coordinate system S0 as There is a conversion relationship of Equation (8) between the expressions in the two coordinate systems:

[0145]

[0146] In the formula, is the deformation screw of the element, which is obtained according to the elastic deformation and damping relationship in different directions. The elastic force and damping force between the upper and lower mass units of the slider can be calculated according to Equation (9):

[0147]

[0148] In the formula, is the generalized elastic force between the two mass units of the slider, is the generalized damping force between the two mass units of the slider; is the differential of the elastic deformation screw, and are the generalized stiffness matrix and the generalized damping matrix, which can be calculated according to Equation (10):

[0149]

[0150] In the formula, χ a and λ a are the linear damping coefficients, and E and u are the elastic modulus of the material and the Poisson's ratio respectively.

[0151] According to the elastic deformation relationship, there is a conversion relationship of Equation (11) between the lower mass unit of the slider and the fixed bed coordinate system S0:

[0152]

[0153] The force balance equations for the upper and lower mass elements of the slider are established using the spinor method based on the force-potential relationship, as shown in equation (12):

[0154]

[0155] In the formula, and Let g be the gravitational force acting on the upper and lower masses of the slider, and let the gravitational acceleration vector be g = [0,0,1,0,0,0]. T Therefore, and For the slider in coordinate system S a The constraint spinor in For the link in coordinate system S ak The screw of the constraint force applied to the slider can be expressed by equation (13):

[0156]

[0157] S4. Establish the force-potential equilibrium equations for the connecting rod; the forces and deformations of the connecting rod are shown in Figures 4(a) and 4(b). The connecting rod is equivalent to two lumped mass elements, and the mass settings of the two mass elements are exactly the same, both defined as m. b Both mass elements are equivalent to cylinders, with a radius equal to the cross-sectional radius r of the connecting rod. a Its height is half the length of the connecting rod, i.e., l b / 2, its moment of inertia relative to the center is The generalized mass matrix of the two mass elements of the connecting rod can be calculated according to equation (14):

[0158]

[0159] The coordinate systems of the two mass elements of the connecting rod are respectively set as and The rigid body motion of the mass element at the upper end of the connecting rod can be expressed in the coordinate system by the screw. S0 and S0 are respectively set as and The transformation relation of the two spinors existence equation (15):

[0160]

[0161] In the formula, The kinematic spin of the ball joint on the connecting rod The elements define the rotational velocities of the ball joint on the link in three directions, in the coordinate system. and The transformation relationship can be expressed by equation (16):

[0162]

[0163] The lower mass unit of the connecting rod is connected to the upper mass unit of the connecting rod through a spring and a damper, and the distance between the two mass units is the length l of the slider. b +l bk . Coordinate system Set at the initial position of the lower mass unit of the slider, there is a conversion relationship of Equation (17) between the initial position of the lower mass unit of the slider and the bed coordinate system S0:

[0164]

[0165] In the formula, The deformation of the connecting rod is expressed by a finite displacement screw, that is, it is defined in the coordinate S b as or defined in the coordinate system S0 as There is a conversion relationship of Equation (18) between the expressions in the two coordinate systems:

[0166]

[0167] In the formula, is the deformation screw element of, which is obtained according to the elastic deformation and damping relationship in different directions. The elastic force and damping force of the upper and lower mass units of the connecting rod can be expressed by Equation (19):

[0168]

[0169] In the formula, is the generalized elastic force between the two mass units of the slider, is the generalized damping force between the two mass units of the slider; is the differential of the elastic deformation screw, K b and D b are the generalized stiffness matrix and the generalized damping matrix, which can be calculated by Equation (20):

[0170]

[0171] In the formula, χ b and λ b are the linear damping coefficients.

[0172] According to the elastic deformation relationship, there is a conversion relationship of Equation (21) between the lower mass unit of the connecting rod and the fixed bed coordinate system S0.

[0173]

[0174] The force balance equations for the upper and lower mass elements of the connecting rod are established using the spinor method based on the force-potential relationship, as shown in equation (22):

[0175]

[0176] In the formula, and Let g be the weight of the mass blocks above and below the connecting rod, then... and For the link in coordinate system S c The constraint force spinor in the equation can be expressed as follows:

[0177]

[0178] S5. Establish the force-potential equilibrium equations for the moving platform; as shown in Figures 5(a) and 5(b), the moving platform is equivalent to the geometry of a central mass element and six circular mass elements connected together, assumed to be a cube. Let the mass of the central mass element be m. ck The main components are the mass of the mold and the center mass of the moving platform. The mass of the circumferential mass unit is set as m. c The main components are the mass of the ball seat and the circumferential mass of the moving platform. Each mass element is equivalent to a cylindrical geometry with radii r. ck ,r ck Their heights are l ck ,l c The inertia tensors of the central mass element and the circumferential mass element can be calculated based on the geometry, and the generalized mass matrix of the moving platform mass element can be calculated according to equation (24):

[0179]

[0180]

[0181] Let the coordinate systems of the moving platform center and the circumferential mass element be S and S respectively. c and The rigid body motion of the circular mass element of the moving platform can be expressed in the coordinate system using spinors. S0 and S0 are respectively set as and The transformation relation of the two spinors existence equation (25):

[0182]

[0183] In the formula, The spinor of the lower ball joint The elements define the rotational velocities of the lower ball joint in three directions, thus defining the coordinate system. The transformation relationship between S0 and S0 can be expressed by equation (26):

[0184]

[0185] The central mass unit of the moving platform is connected to the circumferential mass unit of the moving platform through springs and dampers, and the distance between the two mass units is where r B is the radius of the moving platform, and is the central angle corresponding to the spherical hinge point on the moving platform. The coordinate system

[0186]

[0187] In the formula, the deformation of the moving platform is expressed by a finite displacement screw, that is, it is defined in the coordinate S c S b as or defined in the coordinate system S0 as The expressions in the two coordinate systems have the transformation relationship of formula (28):

[0188]

[0189] In the formula, is the deformation screw element of, which is obtained according to the elastic deformation and damping relationship in different directions.

[0190] Each circumferential mass unit is connected to the adjacent circumferential mass through a spring, and its deformation screw can be calculated according to formula (29):

[0191]

[0192] The deformation screw of each spring on the moving platform can be calculated according to formula (30):

[0193]

[0194] In the formula, is the generalized elastic force between the central mass unit and the circumferential mass unit of the moving platform, is the generalized elastic force between adjacent circumferential mass units of the moving platform, is the differential of the elastic deformation screw, K c , K q is the generalized stiffness matrix, D c , D q is the generalized damping matrix.

[0195] Based on the spatial distribution relationship between the central mass element and the circumferential mass element, and their assumed cubic geometric connection form, the length of the connecting cube between the central mass element and the circumferential mass element can be obtained as r. B The width of the connected cube is equal to the thickness t of the moving platform. B Thus connecting the height w of the cube B It can be calculated according to formula (31):

[0196]

[0197] Similarly, the connection length of each adjacent circular mass element is the chord length of the corresponding circular point, which can be solved according to equation (32):

[0198]

[0199] Its width is equal to the thickness t of the moving platform. B Its height is half the thickness of the moving platform, that is

[0200] Therefore, the stiffness matrix can be solved according to equation (33):

[0201]

[0202] The generalized damping force matrix is ​​solved according to equation (34):

[0203]

[0204] In the formula, χ c and λ c is the linear damping coefficient.

[0205] According to the elastic deformation compatibility relationship, for the same circular mass element, its deformation must be the same as that of its two adjacent circular mass elements, that is:

[0206]

[0207] The transformation relationship between the moving platform center mass element and the fixed bed coordinate system S0 is given by equation (36):

[0208]

[0209] Based on the condition that the center point is in the same position in each branch, the relationship shown in equation (37) exists:

[0210]

[0211] This formula can also be expressed as

[0212] The force balance equations for each mass element at the center and circumference of the moving platform are established using the spinor method based on the force-potential relationship, as shown in equation (38):

[0213]

[0214] In the formula, and For the gravity acting on the center and circumferential mass elements of the moving platform, there is and

[0215] S6. Establish the overall rigid-flexible coupling dynamic equation of the envelope forming equipment; by combining the force balance equations of each component, the overall mechanism dynamic equation of the envelope forming equipment can be obtained, as shown in equation (39):

[0216]

[0217] This formula has a total of (6+6+6+6+6+1+6)*6 = 222 equations. If the external force W is given... l The unknown kinetic spinor and deformation spinor include There are a total of (6+3+6+3+6)*6 = 144 variables, including unknown force variables. There are a total of (1+5+3+3)*6=72 variables. Since the rotational degree of freedom of the connecting rod about the axis does not affect the mechanism configuration, it can be set to 0. Thus, the number of variables becomes 144+72+6=222, which is the same as the number of equations. That is, the equations are solvable, and the deformation of the slider can be predicted. Deformation of the connecting rod and the deformation of the moving platform

[0218] The present invention also provides a specific example as follows:

[0219] Based on the mechanism configuration parameters in Table 1 and the dynamic parameters in Table 2, given the external load F acting on the envelope forming equipment... l =200kN,T l =500Nm, the motors of the envelope forming equipment are as follows Figure 6 The given displacement motion is, i.e. Given that, based on the above dynamic modeling method, the six-axis deformation curves of the slider, connecting rod, and moving platform (c) at different times within a single cycle of the equipment can be obtained as follows: Figures 7-9 As shown, the driving force of the slider is as follows Figure 10 As shown.

[0220] Table 1. Motion configuration parameters of the envelope forming equipment

[0221]

[0222] Table 2. Dynamic parameters of the envelope forming equipment

[0223]

[0224]

[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 full-spin rigid-flexible coupling dynamic modeling and deformation prediction of an envelope forming equipment, wherein the envelope forming equipment is a six-branch parallel motion configuration, each branch including a motor, a slider, an upper ball joint, a connecting rod, and a lower ball joint, the six branches are connected to a moving platform, an envelope mold is installed on the moving platform, the motor on the six branches drives the corresponding slider and connecting rod to move in a coordinated manner, so that the envelope mold performs spatial envelope motion, the blank is placed in the die cavity, and the die cavity drives the blank to perform linear feed motion upward, characterized in that... The dynamic modeling and deformation prediction method simplifies the connecting rod and slider into a two-mass spring system using the lumped mass method, and simplifies the moving platform into a parallel system of a single-center lumped mass and six circularly distributed lumped masses with twelve springs. The large displacement of the rigid body in each connecting pair is characterized by the rotational screw, and the deformation displacement of each spring is characterized by the deformation rotational screw. Finally, the mass matrix and moment of inertia are coupled to establish a generalized mass matrix, thereby establishing the equipment's full rotational screw characterization of the rigid-flexible coupling dynamic equations and realizing equipment deformation prediction. This method includes the following steps: S1. Establish the motion and force conditions of the envelope forming equipment; S2. Establish the force-position equilibrium equations for the slider; S3. Establish the force-position equilibrium equations for the connecting rod; S4. Establish the force-potential equilibrium equations for the moving platform; S5. Establish the overall rigid-flexible coupling dynamic equation of the envelope forming equipment. In step S1, the driving force of each slider in the envelope forming equipment is the force rotation. This indicates that, because the direction of the driving force is fixed, It can be expressed as follows according to formula (1): (1) In the formula, The magnitude of the driving force; External load force rotation on the moving platform of the envelope forming equipment This indicates that for a fixed-point envelope forming motion, the external load is calculated according to equation (2): (2) In the formula, The total forming force vector acting on the moving platform. The forming torque vector acting on the moving platform. The position vector of the equivalent point of application of the forming force. The radius of the equivalent point of application of the forming force. The angle of the equivalent point of application of the forming force. The magnitude of the total forming force on the moving platform. The magnitude of the total forming torque acting on the moving platform; The motions of the slider, the lower ball joint of the connecting rod, and the upper ball joint of the connecting rod are respectively expressed by the kinematic spin. This indicates that the corresponding constraint force spinors are respectively represented by force spinors. This indicates that each slider is equivalent to two lumped mass units, with masses of respectively... and The two mass units are connected by a spring, and the elastic force and damping force of the spring are respectively... The stiffness deformation of a spring is expressed by its spinor. This indicates that the coordinate system centers of the two mass blocks are set at the initial point of the slider and the center of the ball joint on the slider, respectively; each link is equivalent to two lumped mass elements with masses of respectively and The two mass units are connected by a spring, and the elastic force and damping force of the connecting rod spring are respectively... The stiffness deformation of a spring is expressed by its spinor. This indicates that the envelope forming equipment's moving platform is equivalent to one central mass element and six identical circularly distributed mass elements, with the mass set as... and The central mass element and the circumferential mass element are connected by radial springs, and the elastic force and damping force of the radial springs are respectively... Each adjacent circumferential mass unit is connected by a circumferential spring, and the elastic force and damping force of each circumferential spring are respectively... .

2. The method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment according to claim 1, characterized in that, In step S2, the slider is equivalent to two lumped mass units, and the mass settings of the two mass units are exactly the same, both defined as... Both mass elements are equivalent to cylinders, with radii equal to the cross-sectional radius of the slider. Its height is half the length of the slider, that is Its moment of inertia relative to the center is The generalized mass matrix of the two mass elements of the slider is calculated according to equation (3): (3) In the formula, It is a 3x3 identity matrix; coordinate system For a fixed bed coordinate system, To fix it at the initial position of the slider, and An orthogonal coordinate system with its axis pointing in the direction of slider movement, and its coordinate system... The relationship is expressed as in equation (4): (4) In the formula, The position vector of the slider's starting point; The coordinate systems of the two mass elements of the slider are respectively set as and The coordinate system for the rigid body motion of the mass element at the upper end of the slider. and The spinors in the equation are respectively set as and The transformation relationship of the two spinors in equation (5) is as follows: (5) In the formula, Let be the magnitude of the slider's displacement, which is a scalar. coordinate system and The coordinate transformation between them is expressed by equation (6) according to the exponential product formula of the spinors: (6) The lower mass element of the slider is connected to the upper mass element of the slider via a spring and a damper. The distance between the two mass elements is the length of the slider. Coordinate system Set the initial position of the mass element under the slider, and the initial position of the mass element under the slider is relative to the bed coordinate system. The transformation relation of expression (7) exists: (7) Due to the small deformation of the slider, a finite displacement screw can be used for expression, that is, it is defined in the coordinate as , or it is defined in the coordinate system as . There is a conversion relationship as shown in Equation (8) between the expressions in the two coordinate systems: (8) In the formula, For deformable spinor The elements are obtained based on the elastic deformation and damping relationship in different directions. The elastic force and damping force of the upper and lower mass elements of the slider are calculated according to equation (9): (9) In the formula, The generalized elastic force between the two mass units of the slider. The generalized damping force between the two mass units of the slider; The derivative of the elastic twist rotation, and The generalized stiffness matrix and the generalized damping matrix are calculated according to equation (10): (10) In the formula, and The linear damping coefficient is... and These are the material's elastic modulus and Poisson's ratio, respectively. Based on the elastic deformation relationship, the lower mass element of the slider and the coordinate system of the fixed bed are... The transformation relationship of equation (11) is as follows: (11) The force balance equations for the upper and lower mass elements of the slider are established using the spinor method based on the force-potential relationship, as shown in equation (12): (12) In the formula, and Let the gravitational force acting on the upper and lower masses of the slider be denoted as , and let the gravitational acceleration vector be . Therefore, and ; For the slider in the coordinate system The constraint spinor in For the link in the coordinate system The screw of the constraint force applied to the slider is expressed by equation (13): (13)。 3. The method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment according to claim 2, characterized in that, In step S3, the connecting rod is equivalent to two lumped mass elements, and the mass settings of the two mass elements are exactly the same, both defined as... Both mass elements are equivalent to cylinders, with radii equal to the cross-sectional radius of the connecting rod. Its height is half the length of the connecting rod, that is Its moment of inertia relative to the center is The generalized mass matrix of the two mass elements of the connecting rod is calculated according to equation (14): (14) The coordinate systems of the two mass elements of the connecting rod are respectively set as and The spinor of the rigid body motion of the mass element at the upper end of the connecting rod in the coordinate system and The middle is set as follows: and The transformation relationship of the two spinors existence equation (15): (15) In the formula, The kinematic spin of the ball joint on the connecting rod The elements define the rotational velocities of the ball joint on the link in three directions, in the coordinate system. and The transformation relationship is expressed by equation (16): (16) The lower mass unit of the connecting rod is connected to the upper mass unit of the connecting rod via a spring and a damper. The distance between the two mass units is the length of the slider. Coordinate system Set the initial position of the mass element under the slider, and the initial position of the mass element under the slider is relative to the bed coordinate system. The transformation relation of expression (17) exists: (17) In the formula, ; The deformation of the connecting rod is expressed by a finite displacement screw, that is, it is defined in the coordinate as , or it is defined in the coordinate system as , and there is a conversion relationship in formula (18) between the expressions in the two coordinate systems: (18) In the formula, For deformable spinor The elements are obtained based on the elastic deformation and damping relationship in different directions. The elastic force and damping force of the upper and lower mass elements of the connecting rod are expressed according to equation (19): (19) In the formula, The generalized elastic force between the two mass units of the slider. The generalized damping force between the two mass units of the slider; The derivative of the elastic twist rotation, and The generalized stiffness matrix and the generalized damping matrix are calculated according to equation (20): (20) In the formula, and The damping coefficient is linear. Based on the elastic deformation relationship, the lower mass element of the connecting rod and the coordinate system of the fixed bed are... The transformation relation of expression (21) exists. (21) The force balance equations for the upper and lower mass elements of the connecting rod are established using the spinor method based on the force-potential relationship, as shown in equation (22): (22) In the formula, and Let g be the weight of the mass blocks above and below the connecting rod, then... and ; For the link in the coordinate system The constraint force spinor in the equation is expressed by equation (23): (23)。 4. The method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment according to claim 3, characterized in that, In step S4, the moving platform is equivalent to the geometry of a connection between a central mass unit and six circumferential mass units, which is assumed to be a cube; the mass of the central mass unit is set to... The main components are the mass of the mold and the center mass of the moving platform, with the circumferential mass unit set as... The main components are the mass of the ball seat and the circumferential mass of the moving platform; each mass element is equivalent to a cylindrical geometry with radii of... Their heights are respectively The inertia tensors of the central mass element and the circumferential mass element are calculated based on the geometry, and the generalized mass matrix of the moving platform mass element is calculated according to equation (24): (24) The coordinate systems of the moving platform center and the circumferential mass element are respectively set as and The rigid body motion of the circular mass element of the moving platform can be expressed using spinors in the coordinate system. and The middle is set as follows: and The transformation relationship of the two spinors existence equation (25): (25) In the formula, The spinor of the lower ball joint The elements define the rotational velocities of the lower ball joint in three directions, thus defining the coordinate system. and The transformation relationship is expressed by equation (26): (26) The central mass element of the moving platform is connected to the circumferential mass element of the moving platform via springs and dampers, and the distance between the two mass elements is... ,in The radius of the moving platform, The central angle corresponding to the ball joint point on the moving platform; coordinate system Set the initial position of the moving platform's circumferential mass element, and the initial position of the moving platform's circumferential mass element is relative to the bed coordinate system. The transformation relation of expression (27) exists: (27) In the formula, ; The deformation of the moving platform is expressed by a finite displacement screw, that is, it is defined in the coordinate as , or is defined in the coordinate system as , and there is a conversion relationship as shown in Equation (28) between the expressions in the two coordinate systems: (28) In the formula, For deformable spinor The elements are derived from the relationship between elastic deformation and damping in different directions; Each circumferential mass unit is connected to its adjacent circumferential mass via a spring, and its deformation spin... Calculate according to formula (29): (29) The deformation rotation of each spring on the moving platform is calculated according to formula (30): (30) In the formula, The generalized elastic force between the central mass element and the circumferential mass element of the moving platform. The generalized elastic force between adjacent circumferential mass elements of the moving platform. , The derivative of the elastic twist rotation, For the generalized stiffness matrix, It is the generalized damping matrix; Based on the spatial distribution relationship between the central mass element and the circumferential mass element, and their assumed cubic geometric connection form, the length of the connecting cube between the central mass element and the circumferential mass element can be obtained as follows: The width of the connecting cube is equal to the thickness of the moving platform. Thus connecting the height of the cube Calculate according to formula (31): (31) The connection length between adjacent circumferential mass elements is the chord length of the corresponding circumferential point, which can be solved according to equation (32): (32) Its width is equal to the thickness of the moving platform. Its height is half the thickness of the moving platform, that is ; Therefore, the stiffness matrix can be solved according to equation (33): (33) The generalized damping force matrix is ​​solved according to equation (34): (34) In the formula, and The damping coefficient is linear. According to the elastic deformation compatibility relationship, for the same circular mass element, its deformation must be the same as that of its two adjacent circular mass elements, that is: (35) The coordinate system of the moving platform center mass unit and the fixed bed The transformation relation of expression (36) exists: (36) Based on the condition that the center point is in the same position in each branch, the relationship shown in equation (37) exists: (37) This formula can also be expressed as ; The force balance equations for each mass element at the center and circumference of the moving platform are established using the spinor method based on the force-potential relationship, as shown in equation (38): (38) In the formula, and For the gravity acting on the center and circumferential mass elements of the moving platform, there is and .

5. The method for full-spin rigid-flexible coupling dynamic modeling and deformation prediction of envelope forming equipment according to claim 4, characterized in that, In step S5, the overall dynamic equation of the envelope forming equipment is obtained by simultaneously solving the force balance equations of each component, as shown in equation (39): (39)。

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