A method for analyzing thick plate origami structure based on finite particle method

Abstract

The application discloses a thick plate origami structure analysis method based on a finite mass point method, which adopts a mass point-solid model fine modeling method and a mass point-rod-spring model simplified modeling, that is, the structure is discretized into mass points, thick panels are finely simulated through solid elements or are simplified simulated through rod elements and rod-rod spring elements, mass point degrees of freedom are coupled at creases, panel driving is simulated by torsional spring elements, and panel contact is simulated by contact elements. Dynamic analysis is carried out by using the finite mass point method, the deformation and internal force of the solid elements, the rod elements and the rod-rod spring elements are calculated by using virtual reverse motion, the mass point force at the creases is calculated by using motion degree of freedom coupling, the torque of the torsional spring elements and the mass point force are calculated by using dihedral angle, the contact element force is obtained by using a penalty function method, and the mass point displacement is obtained by iteratively solving the mass point motion equation. The method is simple and efficient, and can improve the accuracy of the motion analysis result of the thick plate origami structure.

Description

Technical Field

[0001] This invention belongs to the technical field of structural mechanics analysis, and particularly relates to a method for analyzing thick plate origami structures based on the finite mass method. Background Technology

[0002] Origami structures are three-dimensional structures formed by folding two-dimensional sheets. Origami structures offer advantages such as diverse shapes, excellent foldability, high fold-to-spread ratio, and high rigidity. In recent years, the mathematical principles, geometric models, and mechanical properties of origami structures have been extensively studied, leading to their widespread application in aerospace, architectural structures, robotics, medical equipment, metamaterials, and energy-absorbing structures. In practical engineering applications such as deployable roofs, foldable solar panels, and space telescopes, the origami structure, as the load-bearing main body, requires a certain degree of rigidity in its panels; therefore, its thickness cannot be ignored. Thick-plate origami structures have broad application prospects in these fields.

[0003] The thick-plate origami structure addressed in this invention has broad application prospects in spatial structures. However, due to the coupling of structural deformation and rigid body displacement involved in its folding and unfolding motion, research on the dynamic behavior of thick-plate origami structures remains lacking. Therefore, a modeling and dynamic analysis method for thick-plate origami structures based on the finite mass method is urgently needed in current scientific research and engineering applications. A reasonable and effective rapid modeling and accurate analysis method is fundamental to promoting the application of thick-plate origami structures. The purpose of this invention is to propose an effective modeling and accurate analysis method for thick-plate origami structures, thereby supporting their widespread application in scientific research and engineering.

[0004] Current methods for modeling and dynamic analysis of thick-plate origami structures often have the following shortcomings:

[0005] 1) Most existing modeling and analysis methods for origami structures focus on thin-plate origami structures. However, in practical engineering, such as deployable roofs, deployable solar panels, and space telescopes, the origami panels need to have a certain thickness to provide sufficient rigidity to withstand loads. Therefore, the panel thickness cannot be ignored. Currently, modeling and dynamic analysis methods for thick-plate origami structures are quite scarce.

[0006] 2) Existing analytical methods for thick-plate origami structures mainly rely on kinematic analysis, rarely addressing the dynamic aspects of these structures. In other words, existing kinematic analysis methods cannot effectively obtain the dynamic responses of thick-plate origami structures in practical engineering applications, such as the internal forces and strain energy of the panels.

[0007] 3) Existing modeling and analysis methods for thick-plate origami structures have certain shortcomings in handling dynamic problems. Analysis methods based on the finite element method face difficulties in analyzing variable structures due to the ill-conditioned overall stiffness matrix; analysis methods based on multibody dynamics do not consider the flexible deformation during the deformation process, making it difficult to meet the design and analysis requirements of origami structure engineering applications.

[0008] 4) Existing dynamic analysis methods for thick-plate origami structures generally treat the origami structure as a rigid body. The weight of an origami structure is usually limited, while the thickness of the panel is typically small compared to its planar dimensions, leading to non-negligible deformation of the panel during the mechanism's motion. Existing analysis methods, such as multibody dynamics, cannot effectively consider the coupling effect of panel displacement and pure deformation during the origami structure's motion.

[0009] 5) Existing modeling methods for thick-plate origami structures all use solid elements to simulate the panel. However, this type of modeling method results in a large number of solid elements in the model, making the calculation complex and computationally intensive, and difficult to effectively handle large-scale, large-scale engineering problems of thick-plate origami structures.

[0010] 6) Existing modeling and analysis methods for thick-plate origami structures do not consider the contact issue between panels. This makes it impossible to simulate the contact when the structure is fully unfolded or retracted, resulting in panel penetration and thus making it impossible to accurately simulate and analyze the deformation process of the origami structure.

[0011] In summary, existing modeling and analysis methods for thick plate origami structures suffer from a lack of modeling and analysis methods, minimal involvement of dynamic response, deficiencies in handling mechanism dynamics, failure to consider mechanism displacement and pure deformation coupling, large and complex computational load, and failure to consider panel contact. Summary of the Invention

[0012] To overcome the shortcomings of existing models and dynamic analysis methods for thick-plate origami structures, such as poor simulation effects and inaccurate analysis results, this invention provides a method for analyzing thick-plate origami structures, which can effectively improve the accuracy of motion analysis results for thick-plate origami structures.

[0013] To achieve the objectives of this invention, a method for analyzing thick-plate origami structures based on the finite mass method is provided, the method comprising:

[0014] Step 1: Discretize the thick plate origami structure into point masses to construct an origami structure model; based on the origami structure model, simulate the thick plate, creases, contact constraints between plates, and driving plate motion of the thick plate origami structure to construct corresponding simplified and refined thick plate models.

[0015] Step 2: Analyze the simplified and refined thick panel models using the virtual inverse motion of the finite mass method to obtain the panel deformation and internal forces of the mass points in the corresponding models.

[0016] Solving for the force transmission and displacement of the mass point at the crease based on the coupling of the mass degree of freedom;

[0017] The driving force of the thick plate origami structure is calculated based on the included angle between the panels, and the contact state between the panels is determined to calculate the contact reaction force of the panels.

[0018] Based on the obtained internal forces of the particle, the forces acting on the particle at the crease, the driving force, and the contact reaction force, the motion equation of the particle is constructed; according to the displacement of the particle at the previous moment and the current moment, the central difference method is used to solve the motion equation of the particle to obtain the displacement of the corresponding particle at the next moment.

[0019] Step 3: Repeat step 2 using the folding interval of the thick plate origami structure to obtain the displacement of each mass point during the folding action, thereby completing the dynamic analysis of the thick plate origami structure.

[0020] This invention employs both refined and simplified modeling of thick-plate origami structures, and utilizes the finite mass method for analysis, taking into account the contact interactions between panels, thereby accurately obtaining the analysis results.

[0021] Specifically, the simplified thick panel model adopts a mass-rod-spring model. The thick panel is simulated by rod elements along the horizontal and thickness directions obtained by meshing. The shape of the thick panel is maintained by shear stiffness provided by rod-rod spring elements. The rod-rod spring elements obtain the panel deformation and mass internal forces of the simplified thick panel model through virtual inverse motion.

[0022] The panel crease is simulated by coupling the degrees of freedom of the mass points at the crease.

[0023] The torsion spring is simulated by a torsion spring unit;

[0024] The panel contact constraint is simulated by contact elements.

[0025] Specifically, the simulation process for the rod element is as follows:

[0026] By using virtual inverse motion, the translational and rotational motions of the rigid body in reverse motion are calculated;

[0027] The pure deformation of the rod element is determined by subtracting the translational and rotational motions of the rigid body.

[0028] The axial force of the element is determined by the deformation of the element, and the internal forces of the mass points at both ends of the rod element are obtained from the force balance relationship.

[0029] Specifically, the simulation process of the rod-spring unit is as follows:

[0030] By setting mass point i as the rotation center, rods ij and ik are connected to mass point i, and rod-rod spring units are set at the angle between rod units ij and ik to provide shear stiffness;

[0031] By calculating the angle between rod ij and rod ik Where r mn (=x n -x m () represents the vector from particle m to particle n;

[0032] Determine the bending moment generated by the rod-spring unit Where k R Let be the rotational stiffness of the spring. The zero-force angle between rod ij and rod ik;

[0033] By applying the bending moment to the corresponding mass point, the force acting on mass point j is obtained. The force of particle k Where m(=r) ij ×r ik ) is the normal vector of plane ijk, n1(=r ij (×m) is The direction vector, n2(=m×r ik )yes The direction vector;

[0034] Determine the force acting on particle i by using the force balance relationship.

[0035] Specifically, the fine thick panel model adopts a mass-solid model. The thick panel is simulated by hexahedral solid elements obtained by mesh division. Each hexahedral solid element consists of 8 mass points. The panel deformation and internal forces of the mass points can be obtained through virtual inverse motion.

[0036] The creases between the panels are simulated by coupling the degrees of freedom of the mass points at the creases;

[0037] The torsion spring is simulated by a torsion spring unit;

[0038] The panel contact constraint is simulated by contact elements.

[0039] Specifically, the simulation process of the hexahedral solid unit is as follows:

[0040] Within one time step, the rigid body translational and rotational displacements of the element are subtracted by virtual inverse motion to determine the pure deformation, initial configuration, and final configuration of the element within the time step.

[0041] A local coordinate system (LCS) is established using the element pure deformation vector, and the transformation matrix Ω between the global coordinate system and the local coordinate system is determined.

[0042] The stress increment in the local coordinate system is determined using isoparametric methods, stress-strain relationships, and force balance relationships.

[0043] By increasing the stress increment Total particle stress under initial configuration at initial time Superposition yields the total particle stress under the initial configuration at the final state. By using the transformation matrix and virtual positive motion, the internal forces of the particles under the final state configuration at the final moment are determined.

[0044] Specifically, the coupling simulation of the degree of freedom of the mass point at the crease is as follows:

[0045] By defining the two overlapping particles at the crease as a sleeve and a ball head respectively, the rotation of the crease is simulated; by using Newton's laws of motion, the kinematic equilibrium equations of the contact force, internal force and external force of the two particles are established respectively.

[0046] By coupling the translational degrees of freedom of the sleeve and the ball, the contact force between the two overlapping particles at the crease is obtained, and then applied to the motion equation of the particles to realize the simulation of the crease rotation.

[0047] Specifically, the simulation process of the torsion spring unit is as follows:

[0048] Based on mass points i, j, k, l, the line jk along the crease is defined as the axis of rotation, and the adjacent panels are panel ijk and panel jkl. The torsion spring is arranged along the line jk.

[0049] By calculating the normal vector m(=r) of panel ijk ji ×r jk The normal vector n (=r) of panel jkl jk ×r lk ), where r ij (=x j -x i Let ) represent the vector pointing from particle i to particle j; calculate the dihedral angle. The actual dihedral angle was determined after correction.

[0050] The actual dihedral angle θ is obtained through calculation. F To determine the bending moment generated by the torsion spring Where K F l is the rotational stiffness of a torsion spring per unit length. F The length of the torsion spring (i.e., the length of the rod jk) It is a zero-force dihedral angle;

[0051] By drawing a perpendicular line from point i to line jk, and obtaining the foot of the perpendicular p, the internal force on point i generated by the torsional spring moment can be calculated. Where d i It is the distance from point i to the foot of the perpendicular p, and ||·|| represents the modulus operation of the vector;

[0052] By applying an equal reaction force to the foot of the perpendicular p Will Distribute the proportionate amounts to particles k and j to obtain and Among them l jp l kp These are the distances from point j and point k to the foot of the perpendicular p, respectively.

[0053] Similarly, the internal force of mass point l generated by the bending moment of the torsion spring can be calculated. And the corresponding forces acting on particles k and j and By summing the above internal forces, the resultant internal forces on particles k and j are obtained. and

[0054] Specifically, the simulation process of the touch unit is as follows:

[0055] The contact element is constructed using four mass points, and the dihedral angle θ between the two panels is calculated using the same method. C By introducing a flag with an initial value of 0, the prediction thresholds for valley contact and peak contact are set as follows: and

[0056] If flag equals 0, then determine θ C Determine whether the contact prediction zone has been entered. If it has been entered, the contact prediction zone in the valley is considered. Then set flag=1; if the target enters the mountain peak contact prediction area... Then set flag = -1;

[0057] If flag is not equal to 0, then it is determined whether the dihedral angle is greater than π (flag = 1) or less than π (flag = -1). If the conditions are met, it indicates that the panel is in contact.

[0058] If the panels come into contact, the reaction torque M generated by the contact is calculated using the penalty function method. C =K C l C Δθ, where Δθ=θ C -2π(flag = 1, valley contact) or Δθ = θ C (flag = -1, mountain peak contact);

[0059] The torque is applied to the corresponding mass point using a distribution method similar to that of a torsion spring unit;

[0060] If the dihedral angle θ C Greater than and less than This indicates that the contact prediction area has been removed, so set flag=0.

[0061] Specifically, the expression for the equation of motion of the particle is as follows:

[0062]

[0063] In the formula, m α Let α be the mass of the point mass. and These represent the external force, internal force, and damping force of particle α, respectively.

[0064] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0065] 1. During the modeling process, the panel is discretized into mass points and elements by meshing. Hexahedral solid elements (fine modeling) or rod elements and rod-rod spring elements (simplified modeling) are established in the panel plane and thickness direction. The panel thickness during the unfolding process of the thick plate origami structure is taken into account, which makes up for the shortcomings of the modeling and analysis method of thick plate origami structure.

[0066] 2. Based on the finite mass method, using hexahedral solid elements (refined modeling) or rod elements and rod-rod spring elements (simplified modeling), it can effectively analyze the dynamic response of the panel internal forces and panel strain energy during the structural motion unfolding process, providing an effective analysis method for the dynamic response analysis of thick plate origami structures.

[0067] 3. Based on the finite mass method, this method avoids the ill-conditioned problems associated with integrated global stiffness matrices and effectively analyzes the dynamic response of origami structures during motion. Furthermore, it employs virtual inverse motion to separate the rigid body displacements of the panels, obtaining their pure deformation. This avoids the shortcomings of commonly used dynamic analysis methods in analyzing thick-plate origami structures, supplementing existing methods for modeling and dynamic analysis of thick-plate origami structures.

[0068] 4. The hexahedral solid elements (refined modeling), rod elements, and rod-rod spring elements (simplified modeling) are used to simulate the panel of the thick plate origami structure. By virtual reverse motion, the translation and rotation of the rigid body of the panel during the origami structure movement are deducted to obtain the pure deformation and internal force of the panel, thus achieving the decoupling of the displacement and pure deformation of the panel mechanism during the movement of the thick plate origami structure.

[0069] 5. Simplified modeling method: The panel skeleton in the thick plate origami structure is constructed by rod elements and rod-rod spring elements. The internal force of the mass point is obtained by calculating the axial deformation of the rod elements and the rotational deformation of the rod-rod spring elements, avoiding the deformation and internal force calculation of the hexahedral solid elements, which greatly reduces the computational complexity.

[0070] 6. The contact unit calculates the dihedral angle between the panels and obtains the reaction force after the panels come into contact using the penalty function method. This can accurately simulate the panel contact during the deformation process of a thick-plate folding structure, avoiding the panel penetration phenomenon when the structure is fully unfolded or retracted, and improving the accuracy of the analysis results. Attached Figure Description

[0071] Figure 1 A flowchart of a thick plate origami structure analysis method based on the finite mass method is provided in this embodiment;

[0072] Figure 2 This is a schematic diagram of a finely detailed thick panel model of a thick origami structure provided in this embodiment;

[0073] Figure 3 This is a simplified schematic diagram of a thick panel model of a thick origami structure provided in this embodiment;

[0074] Figure 4 This is a schematic diagram of the hexahedral solid element calculation model provided in this embodiment;

[0075] Figure 5 This is a schematic diagram of the bar element calculation model provided in this embodiment.

[0076] Figure 6 This is a schematic diagram of the calculation model of the rod-rod spring unit provided in this embodiment;

[0077] Figure 7 This is a schematic diagram of the calculation model of the torsion spring unit provided in this embodiment;

[0078] Figure 8 This is a schematic diagram of the contact unit calculation model provided in this embodiment;

[0079] In the diagram: 1. Point mass; 2. Hexahedral solid element; 3. Rod element; 4. Torsion spring element; 5. Contact element; 6. Rod-rod spring element; 7. First panel; 8. Second panel. Detailed Implementation

[0080] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0081] like Figure 1 The present embodiment provides a method for analyzing thick plate origami structures based on the finite mass method, including:

[0082] Step 1: Discretize the thick plate origami structure into point masses to construct an origami structure model; based on the origami structure model, simulate the thick plate, creases, contact constraints between plates, and driving plate motion of the thick plate origami structure to construct corresponding simplified and refined thick plate models.

[0083] Step 2: Analyze the simplified and refined thick panel models using the virtual inverse motion of the finite mass method to obtain the panel deformation and internal forces of the mass points in the corresponding models.

[0084] Solving for the force transmission and displacement of the mass point at the crease based on the coupling of the mass degree of freedom;

[0085] The driving force of the thick plate origami structure is calculated based on the included angle between the panels, and the contact state between the panels is determined to calculate the contact reaction force of the panels.

[0086] Based on the obtained internal forces of the particle, the forces acting on the particle at the crease, the driving force, and the contact reaction force, the motion equation of the particle is constructed; according to the displacement of the particle at the previous moment and the current moment, the central difference method is used to solve the motion equation of the particle to obtain the displacement of the corresponding particle at the next moment.

[0087] Step 3: Repeat step 2 using the folding interval of the thick plate origami structure to obtain the displacement of each mass point during the folding action, thereby completing the dynamic analysis of the thick plate origami structure.

[0088] Furthermore, such as Figure 2 As shown, the refined modeling method divides several grid lines along the plane and thickness directions of the thick plate origami structure panel, discretizing the thick plate into mass points. The thick panel is simulated by hexahedral solid elements obtained from the grid division. Each hexahedral solid element consists of 8 mass points, and the panel deformation and internal forces of the mass points can be obtained through virtual inverse motion. The creases between the panels can be simulated by coupling the degrees of freedom of the mass points at the creases. The panel contact constraints can be simulated by contact elements to ensure that the panels do not penetrate during unfolding.

[0089] like Figure 3 As shown, the simplified modeling method divides the thick plate origami structure panel into several grid lines along the plane and thickness directions, discretizing the thick plate into mass points. The thick panel is simulated by rod elements along the horizontal and thickness directions obtained from the grid division. The shape of the thick plate is maintained by shear stiffness provided by rod-rod spring elements. The rod elements can obtain the panel deformation and mass internal forces of the simplified thick panel model through virtual inverse motion. The panel creases are simulated by coupling the degrees of freedom of the mass points at the creases. The panel contact constraints can be simulated by contact elements to ensure that the panel will not penetrate during unfolding.

[0090] like Figure 4 As shown, the hexahedral unit cell consists of 8 particles. and The representative particle at t a and t b At any given time and position, the displacement vector u of any particle i for:

[0091]

[0092] The initial configuration and the final configuration of the unit are defined as Ω. a and Ω b From the final state configuration Ω of the unit cell b Subtracting the virtual inverse translation -u1 yields the intermediate configuration Ω. ′ Then subtract the virtual out-of-plane rotation -θ op and virtual in-plane rotation -θ ip The specific calculation formulas are given to obtain the intermediate configurations Ω″ and Ω″′ in sequence:

[0093]

[0094] In the formula, n a and n b Let e ​​represent the normal vectors of planes A″B″C″ and A′B′C′, respectively. ′ c ′ i and e ′ c ′ i ′ These represent the direction vectors pointing from the center point of the intermediate plane in the hexahedral element Ω″ and Ω″′ configurations to each particle, respectively. Then, the displacements of particles 2–8 relative to particle 1 caused by the aforementioned hypothetical out-of-plane and in-plane rotations can be calculated. After subtracting the rigid body translations and rotations of the elements, the pure deformation vector of the particle can be obtained as follows:

[0095]

[0096] A local coordinate system (LCS) is established using the element pure deformation vector, and the transformation matrix Ω between the global and local coordinate systems is determined. The stress increment in the local coordinate system is then determined using the isoparametric method, stress-strain relationship, and force balance relationship of the traditional finite element model. By increasing the stress increment Apply stress to the corresponding mass point to determine the initial configuration mass point stress. The internal forces of the particles under the final configuration are determined by the transformation matrix and the virtual positive motion.

[0097] like Figure 5As shown, the rod element consists of two mass points. Through virtual inverse motion, the rigid body displacement of the inverse motion is calculated using the same method as the hexahedral element and subtracted from the total displacement of the mass points to determine the pure deformation of the rod element. Subsequently, the axial force of the element is determined through the element deformation, and the internal forces of the mass points at both ends of the rod element are obtained through the force balance relationship.

[0098] By calculating the axial deformation of the rod element and the rotational deformation of the rod-spring element, the internal forces of the mass point are obtained, avoiding the deformation and internal force calculation of the hexahedral solid element, thus greatly reducing the computational complexity. This invention replaces one hexahedral solid element with 12 rod elements and 24 spring elements. Based on the independent motion of mass points and the independent solution of element internal forces in the finite mass method, it leverages the high parallelism of the finite mass method. Utilizing advanced parallel computing technologies such as graphics processing units (GPUs) and parallel message passing interfaces (MPI), it can significantly improve the computational speed for thick-plate origami structures, solving large-scale, large-scale engineering problems related to thick-plate origami structures.

[0099] like Figure 6 As shown, two rod elements (ij, ik) connect three mass points (i, j, k), and a rod-spring element is placed at the angle between rod element ij and rod element ik. The angle θ between rod ij and rod ik is... R Calculate using the following formula:

[0100]

[0101] In the formula, r ij (=x j -x i ) represents the vector from particle i to particle j, e ik (=x k -x i () represents the vector from mass i to mass k. At time t, the bending moment M generated by the rod-spring element... R for:

[0102]

[0103] In the formula, k R Let be the rotational stiffness of the spring. Let M be the zero-force angle between rods ij and ik (i.e., the angle corresponding to zero bending moment). Then, obtain the bending moment M. R Afterwards, the forces acting between particle j and particle k and It can be calculated using the following formula:

[0104]

[0105] In the formula, m(=r ij ×r ik) is the normal vector of plane ijk, n1(=r ij (×m) is The direction vector, n2(=m×r ik )yes The direction vector. According to the force equilibrium relationship, the force acting at particle i is... It can be calculated using the following formula:

[0106]

[0107] The crease is simulated using a coupled simulation of point mass degrees of freedom. The crease is discretized into multiple point masses through mesh generation, with two point masses at the same location situated on adjacent panels. Point masses i and j, which coincide at the crease, are defined as the sleeve and the ball head, respectively. According to Newton's laws of motion, the equilibrium equations for the motion of these two point masses are:

[0108]

[0109] In the formula, m n , and These are the mass, acceleration, external force, internal force, and contact force at the crease of particle n, respectively. Since the translational degrees of freedom of the overlapping particles at the crease are coupled, that is:

[0110]

[0111] Solving the above two equations simultaneously, we obtain the contact force between the two overlapping particles at the crease:

[0112]

[0113] like Figure 7 As shown, the torsion spring unit is arranged along the rod containing the crease, consisting of 4 mass points (i, j, k, l), 5 rod units (ij, ik, ij, ik, and jk), and one torsion spring unit distributed along the crease rod jk. Based on the characteristics of the creases in the origami structure, they can be divided into two types: valley lines and peak lines. The dihedral angle at the valley line ranges from 0° to 180°, while the dihedral angle at the peak line ranges from 180° to 360°. The dihedral angle θ between adjacent surfaces of the crease... F Calculate as follows:

[0114]

[0115] In the formula, It is the dihedral angle before correction, m(=r) ji ×r jk ) is the normal vector of the triangular panel ijk, n(=r jk ×r lk ) is the normal vector of the triangular panel jkl, r ij (=x j -xi Let θ represent the vector pointing from particle i to particle j, and [x] represent the largest integer not greater than x. The corrected θ F The value range is 0° to 360°, which can uniformly handle both peak and valley lines. After obtaining the dihedral angle, the bending moment generated by the change in the dihedral angle can be calculated. The torsional spring stiffness k at the crease... F Proportional to the length of the torsion spring:

[0116] k F =K F l F

[0117] In the formula, K F l is the rotational stiffness of a torsion spring per unit length. F Let J be the length of the torsion spring (i.e., the length of rod jk). Then the bending moment generated by the torsion spring is:

[0118]

[0119] In the formula, θ F For the dihedral angle between panels, The angle is a zero-force dihedral angle. After obtaining the torsion spring moment, the force distributed to each mass point can be calculated. Draw a perpendicular line from point i to rod jk, obtaining the foot of the perpendicular p. Then the internal force at mass point i generated by the torsion spring moment is... for:

[0120]

[0121] In the formula, d i It is the distance from point i to the foot of the perpendicular p, and ||·|| denotes the modulus operation of the vector. An internal force vector of equal magnitude and opposite direction is applied to point p. And distribute them to particles k and j according to their weights:

[0122]

[0123] In the formula, l jp l kp These are the distances from particles j and k to particle p, respectively. Similarly, the internal force of particle l generated by the torsion spring moment can be obtained. And the corresponding forces acting on particles k and j and The resultant internal forces on particles k and j can be obtained by summing them, that is:

[0124]

[0125] like Figure 8As shown, the calculation of dihedral angle and internal force of mass points in the contact element is similar to that of the torsion spring element, but there are differences in the calculation of bending moment caused by contact. After calculating the dihedral angle, it is necessary to determine the contact state between the two panels; if contact occurs, the contact force is calculated and applied. The key variable for determining whether the panels are in contact is the dihedral angle between them. Since the calculated dihedral angle ranges from [0, 2π), when the panels are in contact, the dihedral angle will change abruptly, from a value slightly greater than 0 to a value slightly less than 2π (valley contact), or from a value slightly less than 2π to a value slightly greater than 0 (peak contact). To improve the versatility of the contact element, a flag with an initial value of 0 is introduced, and pre-judgment thresholds for valley contact and peak contact are set. and (The values ​​used in this invention are π / 18 and 35 / 18π, respectively).

[0126] When the valleys meet, the dihedral angle enters from 0 to... When the range is defined, set flag=1 to indicate the contact prediction area entering the valley state; with flag=1, if the dihedral angle of the next cycle exceeds π, it indicates that contact has occurred between the panels, and the dihedral angle is θ. C′ A contact force needs to be applied to limit further penetration of the panel. When contact occurs, the bending moment generated by the contact element is:

[0127]

[0128] In the formula, K C l is the stiffness of a contact element per unit length. C θ is the length of the contact unit, Δθ is the amount of penetration between the panels, and θ C This refers to the dihedral angle between the panels. The method of applying the contact force to the contact unit is similar to that of the torsion spring unit, and will not be described in detail here.

[0129] When the mountain peaks come into contact, the dihedral angle enters... When the value reaches 2π, set flag = -1 to indicate that the contact prediction area has entered the mountain peak state. With flag = -1, if the dihedral angle in the next cycle exceeds 2π, it indicates that contact has occurred between the panels, at which point the dihedral angle is θ. C′ A contact force needs to be applied to limit further penetration of the panel. When contact occurs, the bending moment generated by the contact unit can be calculated using the above formula, and the contact force is applied in the same way.

[0130] If the dihedral angle θ C Greater than and less than This indicates that the contact prediction area has been removed, so set flag=0.

[0131] The trajectory of any particle α in a paper-folding structure can be expressed as its motion at discrete times [t0, t1, t2, ..., t]. n The position representation of the particle is given by [x0, x1, x2, ..., x]. n If the time interval is small enough, the variables of the origami structure (such as stress and strain) can be considered constant within the same time interval, changing only between different time intervals. Within any given time interval, particle α is in dynamic equilibrium, and its motion follows Newton's second law:

[0132]

[0133] In the formula, m α Let α be the mass of the point mass. and Let represent the external force, internal force, and damping force of mass α, respectively, and μ be the mass damping coefficient. Each term on the right-hand side of the above equation can be expressed explicitly; therefore, using an explicit time integration scheme can avoid iterative solutions within a single step. Using the second-order central difference method, the displacement of mass α at time n+1 can be expressed as:

[0134]

[0135] In the formula, d n+1 d n and d n-1 These are the particle α at time t n+1 t n and t n-1 The displacement at time step Δt is the time integration step size, and c1 = (1 + μΔt / 2). -1 c2 = c1(1-μΔt / 2).

[0136] This invention presents a modeling and dynamic analysis method for thick-plate origami structures based on the finite mass method. A detailed model of the thick-plate origami panel is established, effectively obtaining the pure deformation of the panel. A simplified model of the thick-plate origami panel is also established, facilitating efficient calculations. Considering the thickness of the thick-plate panel and contact constraints, a complete modeling and dynamic analysis method for thick-plate origami structures is developed. The model and analysis results obtained using this method can be directly applied to the design and analysis of thick-plate origami structures in practical engineering applications, realistically simulating actual conditions, filling a gap in current engineering applications, and thus providing an important foundation for promoting the use of thick-plate origami structures.

Claims

1. A method for analyzing thick-plate origami structures based on the finite mass method, characterized in that, include: Step 1: Discretize the thick plate origami structure into point masses to construct an origami structure model; Based on the origami structure model, the thick panel, creases, contact constraints between panels, and driving panel movement of the thick panel origami structure are simulated to construct the corresponding simplified thick panel model and refined thick panel model. Step 2: Analyze the simplified and refined thick panel models using the virtual inverse motion of the finite mass method to obtain the panel deformation and internal forces of the mass points in the corresponding models. Solving for the force transmission and displacement of the mass point at the crease based on the coupling of the mass degree of freedom; The driving force of the thick plate origami structure is calculated based on the included angle between the panels, and the contact state between the panels is determined to calculate the contact reaction force of the panels. Based on the obtained internal forces of the particle, the forces acting on the particle at the crease, the driving force, and the contact reaction force, the equation of motion of the particle is constructed. Based on the displacement of the particle at the previous moment and the current moment, the central difference method is used to solve the particle's motion equation to obtain the displacement of the corresponding particle at the next moment. Step 3: Repeat step 2 using the folding interval of the thick plate origami structure to obtain the displacement of each mass point during the folding action, thereby completing the dynamic analysis of the thick plate origami structure.

2. The method for analyzing thick plate origami structures based on the finite mass method according to claim 1, characterized in that, The simplified thick panel model adopts a mass-rod-spring model. The thick panel is simulated by rod elements along the horizontal and thickness directions obtained by mesh generation. The shape of the thick panel is maintained by shear stiffness provided by rod-rod spring elements. The rod-rod spring elements obtain the panel deformation and mass internal forces of the simplified thick panel model through virtual inverse motion. The panel crease is simulated by coupling the degree of freedom of the mass at the crease. The torsion spring is simulated by a torsion spring unit, which is arranged between adjacent panels; The panel contact constraint is simulated by contact elements.

3. The method for analyzing thick plate origami structures based on the finite mass method according to claim 2, characterized in that, The simulation process for the rod element is as follows: By using virtual inverse motion, the translational and rotational motions of the rigid body in reverse motion are calculated; The pure deformation of the rod element is determined by subtracting the translational and rotational motions of the rigid body. The axial force of the element is determined by the deformation of the element, and the internal forces of the mass points at both ends of the rod element are obtained from the force balance relationship.

4. The method for analyzing thick plate origami structures based on the finite mass method according to claim 2, characterized in that, The simulation process of the rod-rod spring unit is as follows: By setting the mass point As the center of rotation, the rod And rod Connected to a point mass The rod-rod spring unit is set in the rod unit. arrive Provide shear stiffness at the corners; by calculating the rod And rod The included angle ,in Represents a point mass To the point mass The vector; determine the bending moment generated by the rod-spring unit. ,in Let be the rotational stiffness of the spring. For rod And rod The zero force angle between them; by applying the bending moment to the corresponding mass point, the mass point is obtained. force point mass force ,in It is a plane The normal vector, yes directional vector, yes The direction vector; determining the particle through the equilibrium relationship of forces. Force at the point .

5. The method for analyzing thick plate origami structures based on the finite mass method according to claim 1, characterized in that, The fine thick panel model adopts a mass-solid model. The thick panel is simulated by hexahedral solid elements obtained by mesh division. Each hexahedral solid element consists of 8 mass points. The panel deformation and internal forces of the mass points can be obtained through virtual inverse motion. The creases between the panels are simulated by coupling the degrees of freedom of the mass points at the creases; The torsion spring is simulated by a torsion spring unit, which is arranged between adjacent panels; The panel contact constraint is simulated by contact elements.

6. The method for analyzing thick plate origami structures based on the finite mass method according to claim 5, characterized in that, The simulation process of the hexahedral solid element is as follows: Within one time step, the rigid body translational and rotational displacements of the element are subtracted by virtual inverse motion to determine the pure deformation, initial configuration, and final configuration of the element within the time step. A local coordinate system is established using the element pure deformation vectors, and the transformation matrix between the global and local coordinate systems is determined. ; The stress increment in the local coordinate system is determined using isoparametric methods, stress-strain relationships, and force balance relationships. By increasing the stress increment Total particle stress under initial configuration at initial time Superposition yields the total particle stress under the initial configuration at the final state. By using the transformation matrix and virtual positive motion, the internal forces of the particles under the final state configuration at the final moment are determined.

7. The method for analyzing thick plate origami structures based on the finite mass method according to claim 2 or 5, characterized in that, The coupling simulation of the degree of freedom of the mass point at the crease is as follows: By defining the two overlapping particles at the crease as a sleeve and a ball head respectively, the rotation of the crease is simulated; by using Newton's laws of motion, the kinematic equilibrium equations of the contact force, internal force and external force of the two particles are established respectively. By coupling the translational degrees of freedom of the sleeve and the ball, the contact force between the two overlapping particles at the crease is obtained, and then applied to the motion equation of the particles to realize the simulation of the crease rotation.

8. The method for analyzing thick plate origami structures based on the finite mass method according to claim 2 or 5, characterized in that, The simulation process of the torsion spring unit is as follows: with a mass point , , , Based on this, by defining the crease along the line The pivot is the rotating part, and the adjacent panels are panels. and panel torsion spring along the line Arrangement; via calculation panel normal vector ,panel normal vector ,in Indicates a particle Pointing to the mass The vector; calculate the dihedral angle. The actual dihedral angle was determined after correction. The actual dihedral angle is obtained through calculation. To determine the bending moment generated by the torsion spring ,in The rotational stiffness of the torsion spring per unit length. The length of the torsion spring. It is a zero-force dihedral angle; passing through Particles as lines The perpendicular line is obtained to find the foot of the perpendicular. The mass generated by the bending moment of the torsion spring was calculated. internal force ,in It is a point mass to the foot distance, Represents the modulo operation of vectors; by using the foot of the perpendicular... Apply an equal reaction force ,Will Distribute to particles proportionally and point mass Up, get and ,in , They are point masses point mass to the foot The distance; similarly, the mass point generated by the torsion spring bending moment is calculated. Internal force And the corresponding action on the particle and The force on and By summing the above internal forces, we obtain the particle. and point mass The combined internal force .

9. The method for analyzing thick plate origami structures based on the finite mass method according to claim 2 or 5, characterized in that, The simulation process of the touch unit is as follows: The contact element is constructed using four mass points, and the dihedral angle between the two panels is calculated using the same method. By introducing a flag with an initial value of 0, the prediction thresholds for valley contact and peak contact are set as follows: and If flag equals 0, then the decision is made by... Determine whether the contact prediction zone has been entered. If it has been entered, the contact prediction zone in the valley is considered. Then set If you enter the mountain peak contact prediction area Then set If flag is not equal to 0, then check if the dihedral angle is greater than 0. or less If the condition is met, it indicates that the panels are in contact; if the panels are in contact, the reaction torque generated by the contact is calculated using the penalty function method. ,in or The torque is applied to the corresponding mass point using a distribution method similar to that of a torsion spring unit; if the dihedral angle Greater than and less than This indicates that the contact prediction area has been moved out of the target area. .

10. The method for analyzing thick plate origami structures based on the finite mass method according to claim 1, characterized in that, The equation of motion of the particle is expressed as follows: In the formula, For point mass quality, , and They represent point masses respectively External force, internal force, and damping force.

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