Rotary multistable structure configuration design method and rotary multistable structure configuration

Abstract

The invention discloses a rotary multistable structure configuration design method and a rotary multistable structure configuration, and the design method comprises the steps: calculating the theoretically required minimum crease number according to the number of target steady-state angles; according to the determined minimum crease number, calculating to obtain a corresponding steady-state total number; constructing a steady-state angle coefficient matrix according to the minimum crease number and the corresponding steady-state total number; n rows are selected from the steady-state angle coefficient matrix to serve as sub-matrixes A, and a linear equation set containing N equations is formed and solved; and verifying the solution of the system of linear equations to find the crease included angles which meet all target steady-state angle requirements and conform to all constraint conditions. According to the method, the design efficiency of the multistable paper-cut structure is remarkably improved.

Description

Technical Field

[0001] This invention relates to the field of paper-cutting structure design, and specifically to a reverse design method for a rotationally multistable paper-cutting structure configuration. Background Technology

[0002] In recent years, structural design methods derived from traditional paper-cutting art have shown broad application prospects in fields such as programmable metamaterials, smart structures, and flexible devices due to their unique advantages in geometric configuration transformation and mechanical property control. In particular, rotational multistable paper-cutting structures, which integrate rotational lifting motion characteristics with multistable mechanical behavior, can achieve controllable transitions between different stable configurations, and have significant potential in energy absorption, reconfigurable mechanisms, mechanical storage, and optical control.

[0003] Currently, research on rotationally multistable paper-cutting structures mainly focuses on experimental exploration and performance analysis of specific configurations. For example, rotational upright systems and their variants achieve vertically driven rotational motion and multistable locking of the central platform through preset cutting and folding patterns. However, existing design methods still have the following problems: 1. Lack of systematic design methodology and reverse engineering capability: Existing configuration designs mostly rely on empirical trial and error or forward parameter scanning, and have not yet established a reverse design theory that starts from the target steady-state performance and systematically reverse-engineers geometric parameters. Designers find it difficult to quickly and accurately determine the number of creases, the included angles, and their spatial layout based on the required steady-state quantity, steady-state angles, and other target performance parameters.

[0004] 2. Difficulty in controlling the number and angle of steady states in structural design: Although studies have shown that increasing the number of creases can increase the number of steady states, the quantitative relationship between the number of steady states and the number of creases is still unclear. Furthermore, the mapping relationship between the platform rotation angle (steady-state angle) corresponding to different steady states and the crease angle is complex and lacks a general mathematical model to support it. 3. Lack of theoretical support for composite structure design: Existing research focuses on single-unit structures, with insufficient research on the combination mechanism of multiple units in the vertical or planar direction. It has failed to effectively utilize the combination effect to expand the steady-state quantity of the system or to achieve the coordinated motion and functional integration of the overall structure.

[0005] Therefore, there is an urgent need to develop a systematic and reversible method for designing rotational multistable paper-cutting structures that can automatically solve crease parameters based on target steady-state performance (such as the number of steady states and the steady-state angle matrix) and support the design and performance prediction of complex combined structures to meet the growing demand for high-functionality intelligent structures and devices. Summary of the Invention

[0006] Objective of the Invention: The objective of this invention is to provide a reverse design method for a rotationally multistable paper-cutting structure. By increasing the number of creases, the design is extended to a multistable structure, and a relationship is established between the number of stable states Sn and the number of creases Cn. S n =2 Cn-1 The relationship between the steady-state angle θs and the crease angle θ was analyzed, and the maximum steady-state angle of different polygonal reference figures was determined. Three geometric constraints were proposed; finally, by introducing the steady-state angle coefficient matrix, a design algorithm was established to inversely calculate the crease angle from the target steady-state angle.

[0007] Technical solution: A design method for a rotationally multistable paper-cutting structure configuration includes: Step 1: Calculate the theoretically minimum number of creases required based on the target steady-state angles; Step 2: Calculate the corresponding steady-state total number based on the determined minimum number of creases; construct the steady-state angle coefficient matrix based on the minimum number of creases and the corresponding steady-state total number. Step 3: Select N rows from the steady-state angle coefficient matrix as submatrix A. There are a total of N ways to select these N rows. kind, S n,min Let A be the total number of steady states; using one of the combinations of submatrices A, a matrix containing... N The system of linear equations consisting of ...

[0008] in, for The vector of the included angle of the crease to be solved. , ; Let be the target steady-state angle column vector. , The target steady-state angle with the smallest angle. The target steady-state angle is the angle with the largest angle, and the steady-state angles in the vector are arranged in ascending order; Step 4: Verify the solution to the linear equation system from Step 3. If the linear equation system has a unique solution... Further verification is needed to determine if the solution satisfies all constraints. If it does, it indicates that the submatrix of the combination is valid. A and the calculated angles of this set of creases Capable of achieving all N One target steady-state angle, group This is the simplest crease design solution that meets the requirements; Step 5: If you have tried all If none of the combinations satisfies the constraints, the number of creases is increased, and the total number of steady states corresponding to the new value is recalculated. Based on the increased number of creases and the recalculated total number of steady states, a new steady-state angle coefficient matrix is ​​constructed. Step 3 is returned, and the combination selection, solution, and verification process is repeated until a crease angle that satisfies all target steady-state angle requirements and meets all constraints is found. .

[0009] In step 4, the constraints include: Initial crease constraint: Assuming the vertical displacement of the rigid platform is always positive, then the first crease state is either no crease or valley crease; Adjacent crease state constraint: To avoid unacceptable geometric interference between adjacent folds, any two adjacent creases cannot be in the same type of fold state at the same time. Crease type quantity balance constraint: For a complete steady-state configuration, the total number of valley creases is equal to the total number of mountain creases.

[0010] In step 1, the theoretically minimum number of creases required is calculated as follows: Calculate the theoretically required minimum number of creases based on the target steady-state angle number N. C n,min ;

[0011] In the formula: C n,min The minimum number of creases required in theory; ceil(x) represents rounding up the variable x; N is the target steady-state angle.

[0012] In step 2, the corresponding total number of steady states is calculated as follows: S n,min =2 Cn,min-1 In the formula: S n,min This represents the total steady-state number.

[0013] In step 2, the constructed steady-state angle coefficient matrix is: F=

[0014] Where F is the steady-state angle coefficient matrix, the matrix has a dimension of Sn×(Cn-1), and Cn is the number of creases in the target configuration. S n F represents the total number of steady-state components of the target assembly. pt F is the steady-state angle coefficient of the p-th steady state and the t-th steady state. pt =2×(S1+S2+S3+...+S t ), S tThis represents the crease state of the t-th crease when the p-th steady state is reached.

[0015] The present invention also provides a rotational multistable paper-cutting structure configuration, which is designed according to the design method of the rotational multistable paper-cutting structure configuration provided above.

[0016] Beneficial effects: The rotational multistable paper-cutting structure designed using the above reverse design method has the following advantages: (1) High efficiency in structural design: This invention realizes the automated and systematic reverse design from the target steady-state perspective to the geometric parameters of the paper-cutting structure, which significantly improves the design efficiency of multi-steady-state paper-cutting structures.

[0017] (2) Wide applicability of the design method: The design method proposed in this invention is applicable to rotationally multistable paper-cutting structures of various sizes and complex geometric parameters. The construction of the steady-state angle coefficient matrix adopts a recursive algorithm to ensure its completeness and consistency. The design method proposed in this invention is applicable to various reference shapes such as quadrilaterals and hexagons. Attached Figure Description

[0018] Figure 1 This is a design method for rotationally multistable paper-cutting structures.

[0019] Figure 2 This is a schematic diagram of a four-fold crease configuration.

[0020] Figure 3 Steady-state definition of four-fold configuration.

[0021] Figure 4 It is a matrix form of crease state combination for a four-crease configuration.

[0022] Figure 5 represents the steady-state angle coefficient matrix for different orders.

[0023] Figure 6 This is the steady-state angle coefficient matrix for the four-fold configuration.

[0024] Figure 7 The program interface for solving the crease angle.

[0025] Figure 8 This is the configuration design diagram for example three.

[0026] Figure 9 Paper models under different steady states. Detailed Implementation

[0027] The design method will be further explained below with reference to specific embodiments and the accompanying drawings: Define the number N of steady-state angles that the rotationally multistable paper-cutting structure needs to possess and their specific values, and construct the target steady-state angle vector. ,in This must be included, representing the initial planar steady state of the structure when it is fully unfolded. Simultaneously, a maximum allowed number of creases, Cn,max, is set as an upper limit for algorithm iterations to prevent the algorithm from getting stuck in an infinite loop when there is no solution.

[0028] For the design method of this invention, see [link to invention]. Figure 1 Specifically: Step 1: Calculate the theoretically required minimum number of creases based on the target steady-state angle number N. C n,min This quantity is determined by the relationship between the steady-state quantity and the number of creases. S n =2 Cn-1 By reverse engineering, we can ensure Sn ≥ N The calculation formula is:

[0029] In the formula: ceil(x) represents rounding up the variable x, such as

[0030] Step 2: Based on the determined minimum number of creases C n,min The corresponding steady-state total number is calculated. S n,min =2 Cn,min-1 Based on the minimum number of creases C n,min and the corresponding steady-state total S n,min Construct the steady-state angle coefficient matrix F, F ∈ R Sn ,min×(Cn,min-1) .

[0031] If a given configuration has Cn creases, corresponding to Cn-1 crease angles, denoted as θ1, θ2…θCn-1 respectively, each crease has three basic states: no crease, valley crease, and mountain crease (defined as 0, 1, and -1 respectively). To ensure the configuration is geometrically valid, the designed crease angles θ1, θ2…θCn-1 must first satisfy the geometric compatibility condition (each crease angle is smaller than the interior angle of the inner polygon). γ).To achieve this reverse design, the concept of "steady-state angle coefficient" is introduced to quantitatively describe the contribution of the crease angle to the overall steady-state angle. The steady-state angle coefficient is defined as the degree of influence of a change in a crease angle on the overall steady-state angle of the configuration. This relationship is systematically represented by a steady-state angle coefficient matrix F. This matrix has a dimension of Sn×(Cn-1). For each row of the matrix, its elements correspond to the weighted coefficients of each crease angle θ1, θ2…θCn-1. By summing the weighted coefficients of this row with the corresponding crease angles (i.e., multiplying each row element by its corresponding crease angle and then summing the results), the steady-state angle of the configuration can be obtained. The steady-state angle coefficient matrix F consists of all permutations of three binary variables (each column taking values ​​of 0 or 2), with a dimension of Sn,min×(Cn,min-1). It does not change with the size of the crease angle but is only related to the number of creases, exhibiting significant iterative characteristics. That is, as the number of creases increases, the new matrix is ​​an extension of the previous matrix. Figure 5 The dashed box in the middle marks the previous order matrix, which is completely embedded in the new matrix.

[0032] Step 3: From the steady-state angle coefficient matrix F of S n,min Selecting from rows N The row is chosen as a submatrix A. N There are a total of ways to combine rows Seed; using submatrix A, a structure containing N The system of equations is formed by solving the system of equations:

[0033] in, for The vector of the included angle of the crease to be solved. , ; Let be the target steady-state angle column vector. , The target steady-state angle with the smallest angle. The target steady-state angle is the angle with the largest angle, and the steady-state angles in the vector are arranged in ascending order.

[0034] Step 4: Verification of the solution: If the system of linear equations has a unique solution. Further verification is needed to determine if the solution satisfies all constraints. If it does, then the currently selected solution is... N Row steady-state angle coefficients (i.e., submatrices) A and the calculated angles of this set of creases Capable of achieving all N One target steady-state angle. This set This means finding the simplest crease design that meets the requirements by iterating through all combinations.

[0035] The constraints include: Initial crease constraint: Assuming the vertical displacement of the rigid platform is always positive, crease 1 cannot be in a mountain crease state; that is, its state can only be 0 or 1. Let the total number of creases in the paper-cutting structure be C. n There are creases, and the state of each crease is represented by the variable S. i It indicates that S i =1 indicates a valley break, S i =-1 indicates a mountain break, S i =0 indicates that it is not collapsed.

[0036] Adjacent Crease State Constraint: To avoid unacceptable geometric interference between adjacent folded surfaces (i.e., one panel penetrating another), any two adjacent creases cannot simultaneously be in the same type of fold state, i.e., they cannot both be valley creases or mountain creases. This condition can be further stated as follows: folding directions must alternate, with any number of folded creases in between. That is, for two adjacent creases i and i+1 (i=1, 2, 3, ..., C...), ... n-1 The constraint condition is: S i +S i+1 =0 (S i S i+1 Not equal to 0).

[0037] Crease type quantity balance constraint: For a complete steady-state configuration, the total number of valley creases must be strictly equal to the total number of mountain creases. Let N... v N represents the number of valley creases. m If N represents the number of mountain folds, then this constraint can be expressed as: v =N m .

[0038] Step 6: Iterative increment: If all have been tried If none of the combinations satisfy the constraints, then increase the number of creases, making... C n = C n,min +1, recalculate the total number of steady states corresponding to the new value. S n =2 Cn-1 Then construct a new steady-state angle coefficient matrix, return to step 3, and repeat the combination selection, solution, and verification process.

[0039] Step 7: Continue this iterative process until a crease angle that satisfies all target steady-state angle requirements and meets all constraints is found. .

[0040] Taking a configuration with crease count Cn=4 as an example, according to the design method of the rotational multistable paper-cutting structure of the present invention, the steps of solving the steady-state angle coefficient matrix and the design method of inversely solving the crease angle based on the target steady-state angle are as follows: Taking the configuration with crease count Cn=4 as an example, the corresponding steady-state total number is calculated. S n =2 Cn-1 =8; based on the number of creases C n and the corresponding steady-state total S n Construct the steady-state angle coefficient matrix F, F ∈ R Sn×(Cn-1) F is an 8×3 matrix.

[0041] Steady-state angle coefficient matrix F The elements in the equation are steady-state angle coefficients, which are used to quantitatively describe the contribution of the crease angle to the overall steady-state angle. The steady-state angle coefficient is defined as the degree of influence of a change in a certain crease angle on the overall steady-state angle of the configuration.

[0042] To verify this through examples, a specific set of crease angles is defined, with crease angles of θ1=10°, θ2=20°, and θ3=30°. The design diagram is shown below. Figure 2 As shown. According to the initial crease constraint, crease one cannot be in a mountain crease state, so crease one is in an unfolded or valley crease state; according to the adjacent crease state constraint, any two adjacent creases cannot be in the same type of fold state at the same time, that is, they cannot be both valley creases or mountain creases at the same time; according to the crease type quantity balance constraint, the total number of valley creases must be strictly equal to the total number of mountain creases. Finally, all 8 valid steady-state configurations that meet the conditions can be determined ( Figure 3 ): Steady state 1: All four creases are in an unfolded state, the structure is completely flat, and the rotation angle θr = 0°; Steady state 2: Crease 1 and crease 2 are in an undisturbed state, crease 3 is in a valley crease state, and crease 4 is in a mountain crease state, with a rotation angle of θr = 60°; Steady state 3: Crease 1 and crease 3 are in an undisturbed state, crease 2 is in a valley crease state, and crease 4 is in a mountain crease state, with a rotation angle θr = 100°; Steady state 4: Crease 1 and crease 4 are in an unfolded state, crease 2 is in a valley fold state, crease 3 is in a mountain fold state, and the rotation angle θr = 40°; Steady state 5: Creases 2 and 3 are in an undisturbed state, crease 1 is in a valley fold state, crease 4 is in a mountain fold state, and the rotation angle θr = 120°; Steady state 6: Crease 2 and crease 4 are in an undisturbed state, crease 1 is in a valley crease state, crease 4 is in a mountain crease state, rotation angle θr = 60°; Steady state 7: Creases 3 and 4 are in an undisturbed state, crease 1 is in a valley fold state, crease 2 is in a mountain fold state, and the rotation angle θr = 20°; Steady state 8: Creases 1 and 3 are in a valley fold state, creases 2 and 4 are in a mountain fold state, rotation angle θr = 80° The state of each crease is represented by the variable S. i It indicates that S i =1 indicates a valley break, S i =-1 indicates a mountain break, S i =0 indicates folding (i=1, 2, 3, ..., C) n Example C n =4. The crease state combinations of the above 8 effective steady-state configurations are represented in matrix form. The dimension of this state matrix is ​​Sn×Cn. Let this matrix be the crease state combination matrix, defined as matrix A. Each row represents a feasible steady-state configuration, containing the state information of all creases under that steady state. Each column corresponds to the state of a crease with a specific number. Each crease has three basic states: no crease, valley crease, and mountain crease (defined as 0, 1, and -1, respectively). A ij Let A be an element in matrix A (where i represents the steady state number of the element and j represents the j-th crease state of the steady state of the element), then A can be obtained. 11 =0,A 12 =0,A 13 =0,A 14 =0;A 21 =0,A 22 =0,A 23 =1,A 24 =-1;A 31 =0,A 32 =1,A 33 =0,A 34 =-1;A 41 =0,A 42 =1,A 43 =-1,A 44 =0;A 51 =1,A 52 =0,A 53 =0,A 54 =-1;A 61 =1,A 62 =0,A 63 =-1,A 64 =0;A 71 =1,A 72 =-1,A 73=0,A 74 =0;A 81 =1,A 82 =-1,A 83 =1,A 84 =-1. The matrix A is shown below.

[0043]

[0044] The concept of "steady-state angle coefficient" is used to quantitatively describe the contribution of the crease angle to the overall steady-state angle. The number of rows in the steady-state angle coefficient matrix represents the number of steady states, and the number of columns represents the number of independent crease states. Therefore, the contribution of the folded state to the overall steady-state angle is 0, while the contributions of mountain creases and valley creases are both 1. Thus, two binary variables, 0 and 2, are defined for weighted summation. Since the steady-state angle coefficient is obtained through linear changes between crease states, specifically, for a four-crease configuration, the crease state combination matrix has four crease states (S1, S2, S3, S4). Because these states need to satisfy the constraint of crease type quantity balance, S1 + S2 + S3 + S4 = 0, therefore there are only three independent state variables (usually S1, S2, S3). Therefore, the number of columns in the steady-state angle coefficient matrix is ​​reduced by one column compared to matrix A (i.e., removing the last crease state), resulting in Cn-1 crease states. The Cn-1 elements F in each row of the Sn×(Cn-1) order steady-state angle coefficient matrix t (t=1, 2, 3, ..., Cn-1) is the result of the row S1, S2, S3, ..., S n-1 The following formula is used to obtain the steady-state angle coefficient matrix. Next, the p-th row (p=1, 2, 3, ..., S) is solved. n The steady-state angle coefficient at time p, and the crease states of each crease corresponding to the p-th row are S1, S2, S3, ..., S n-1 : F p1 =2×S1 F p2 =2×(S1+S2) F p3 =2×(S1+S2+S3) ... F pn-1 =2×(S1+S2+S3+...+S n-1 ) Next, we calculate the steady-state angle coefficient matrix F (composed of all permutations of three binary variables, with each column taking a value of 0 or 2) when Cn=4. The elements of this matrix F... pt (p is the p-th steady state of the element, and t is the t-th steady-state angle coefficient of the element's steady state), the matrix is ​​an 8×3 matrix: Steady state 1: F 11 : 2×0=0; F 12 : 2 × (0 + 0) = 0; F 13 2 × (0 + 0 + 0) = 0 Steady state two: F 21 : 2×0=0; F 22 : 2 × (0 + 0) = 0; F 23 2 × (0 + 0 + 1) = 2 Steady state three: F 31 : 2×0=0; F 32 : 2 × (0 + 1) = 2; F 33 2 × (0 + 1 + 0) = 2 Steady state four: F 41 : 2×0=0; F 42 : 2 × (0 + 1) = 2; F 43 2 × (0 + 1 - 1) = 0 Steady state five: F 51 : 2×1=2; F 52 : 2 × (1 + 0) = 2; F 53 2 × (1 + 0 + 0) = 2 Steady state six: F 61 : 2×1=2; F 62 : 2 × (1 + 0) = 2; F 63 2 × (1 + 0 - 1) = 0 Steady state seven: F 71 : 2×1=2; F 72 : 2×(1-1)=0; F 73 2 × (1 - 1 + 0) = 0 Steady-state eight; F 81 : 2×1=2; F 82 : 2×(1-1)=0; F 83 2 × (1 - 1 + 1) = 2 The matrix F is shown in the figure below:

[0045] The above steps solved the steady-state angle coefficient matrix of the four-fold configuration, and the calculation results are as follows: Figure 6 As shown, for each row of the matrix, its elements correspond to the weighted coefficients of the crease angles θ1, θ2, ..., θCn-1. By summing the weighted coefficients of this row with the corresponding crease angles (i.e., multiplying each row element by its corresponding crease angle and then summing the results), the steady-state angles of this steady-state configuration can be obtained.

[0046] C can be obtained using the same method. n The steady-state angle coefficient matrix when ω = 3 (the dimension of this matrix is ​​4×2) is shown below:

[0047] When C n The steady-state angle coefficient matrix when ω = 5 (the matrix has a dimension of 16×4) is shown in the figure below:

[0048] The steady-state angle coefficient matrix can be constructed using a recursive algorithm. For example... Figure 5 As shown, the steady-state angle coefficient matrix is ​​given when the number of creases Cn=3. When Cn=4, the first column of the matrix always takes the value 0 or 2, while the remaining columns are formed by combining the complete matrix when the number of creases Cn=3. That is, when the number of creases increases, the new matrix is ​​extended from the previous matrix, which is a recursive increase. The dashed box in the figure marks the previous matrix, which is completely embedded in the new matrix. The first column of the new matrix always takes the value 0 or 2.

[0049] Example 1 Based on the reverse design framework of this invention, the following detailed algorithm flow for solving the effective crease state combinations and corresponding crease angle vectors θ∈RCn,min×1 that satisfy the five target steady-state angle column vectors θs,N∈RN×1 is as follows, with specific embodiments: Step 1. The five target steady-state angles are 0°, 15°, 25°, 40°, and 45°, forming a target steady-state angle column vector. The range is [0, 15, 25, 40, 45]. T Based on the target steady-state angle count N=5, the theoretically minimum number of creases required is... C n,min =4. This quantity is derived from the relationship between the steady-state quantity and the number of creases. S n =2 Cn-1 By reverse engineering, we can ensure Sn ≥ N The calculation formula is:

[0050] Step 2. From the determined minimum number of creases C n,min Initially, the corresponding total steady-state count is calculated. S n,min =2 Cn ,min-1 The corresponding steady-state total S n,min =8. When Cn=4, its corresponding steady-state angle coefficient matrix F ∈ R Sn,min×(Cn,min-1) As shown below.

[0051]

[0052] Step 3. From the steady-state angle coefficient matrix F Five rows are selected from the eight rows to form a system of five equations, each corresponding to one of the five target steady-state angles. There are a total of [number] ways to combine these five rows. Species. The above. For each of the possible combinations, construct a submatrix. A ∈ R N×(Cn,min-1) In this embodiment, a selected combination and constructed submatrix A is shown below.

[0053] Then establish the vector of the crease angle to be solved. , And construct a system of linear equations. ,in Let be the vector of the included angle of the crease to be solved, and the equation is as follows: . =

[0054] Step 4. Verify the solution obtained from solving the linear equation system in Step 3.

[0055] Calculations show that this system of linear equations has a unique solution. Further verification shows that the solution satisfies all constraints: nonnegativity constraint; geometric compatibility constraint; and crease state validity constraint. If the solution satisfies all constraints, it indicates that the currently selected 5 rows of steady-state angle coefficients (i.e., submatrices) are effective. A and the calculated angles of this set of creases It can achieve all five target steady-state angles. This set This is a simplified crease design that meets the requirements. The reliability of this reverse engineering method is demonstrated through examples.

[0056] Based on the above reverse design process, design a tool for calculating crease angles, with a user interface (such as...). Figure 7The tool is mainly divided into four functional areas: the input area allows users to define the target steady-state angle vector θs,N (in degrees) and the maximum allowed number of creases, which serve as the termination condition for algorithm iteration, preventing the program from entering an infinite loop when there is no solution. Since θs=0 is an inherent steady state for all rotational multistable paper-cutting configurations, corresponding to the initial state of full unfolding, θs=0 must be included when setting the target steady-state angle; the solution control area contains the solution button to start the calculation process and auxiliary control functions; the output display area displays all crease angle combinations [θ] that meet the design requirements and draws corresponding visualization charts; the status information bar is used to update and display the execution status of the algorithm in real time. The pseudocode of the core solution algorithm of this tool is shown in Tables 1 and 2.

[0057] Table 1:

[0058] Table 2:

[0059] Continued from Table 2:

[0060] By verifying and solving the target steady-state angles in Table 3 in turn, it can be seen that the program can solve multiple sets of numerical combinations of crease angles for different target steady-state angles.

[0061] Table 3:

[0062] To verify the solution, Solution 1 from Example 3 was used for verification. The target steady-state angles [0, 10, 30, 50, 70, 90, 110, 130] were given. The ratio of the inner and outer quadrilateral side lengths b / B = 0.5 and the relative rotation angle α = 10° were taken. Using the above calculation procedure, five creases were obtained, with crease angles θ1 = 5°, θ2 = 10°, θ3 = 15°, and θ4 = 35°. The crease design is as follows: Figure 8 As shown.

[0063] This configuration has 16 steady states. There is a linear relationship between the crease angles: θ1 + θ2 = θ3, and θ1 + θ2 + θ4 = θ3 + θ4. Therefore, it can be considered that there are two linear relationships. The formula shows that the combination of crease angles can achieve 14 steady-state angles, namely 0°, 10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90°, 100°, 110°, 120°, and 130°. Among them, 30° and 100° correspond to two configurations respectively, and the remaining steady-state angles correspond to one configuration. The steady-state characteristic parameters of the configuration in Example 3 are shown in Table 4.

[0064] Table 4:

[0065] Continued from Table 4:

[0066] The paper model corresponding to Example 3 is as follows: Figure 9 As shown, it can be seen that it can achieve the target steady-state angle matrix, indicating that the above method is reasonable.

[0067] In summary, this paper proposes a reverse design method that can construct rotationally multistable paper-cutting structures with expected functions for various cases with different initial geometric parameters and different boundary shapes, fully demonstrating the effectiveness and wide applicability of the design method.

[0068] This design method is not limited to the embodiments described above. The above description of specific embodiments is intended to illustrate the technical solution of this design, and the above specific embodiments are merely illustrative and not restrictive. Without departing from the scope of protection of the claims, those skilled in the art can make many specific modifications based on the teachings of this design method, and these modifications all fall within the protection scope of this invention.

Claims

1. A design method for a rotationally multistable paper-cutting structure configuration, characterized in that, include: Step 1: Calculate the theoretically minimum number of creases required based on the target steady-state angles; Step 2: Calculate the corresponding steady-state total number based on the determined minimum number of creases; construct the steady-state angle coefficient matrix based on the minimum number of creases and the corresponding steady-state total number. Step 3: Select N rows from the steady-state angle coefficient matrix as submatrix A. There are a total of N ways to select these N rows. kind, S n,min Let A be the total number of steady states; using one of the combinations of submatrices A, a matrix containing... N The system of linear equations consisting of ... in, for The vector of the included angle of the crease to be solved. , ; Let be the target steady-state angle column vector. , The target steady-state angle with the smallest angle. The target steady-state angle is the angle with the largest angle, and the steady-state angles in the vector are arranged in ascending order; Step 4: Verify the solution to the linear equation system from Step 3. If the linear equation system has a unique solution... Further verification is needed to determine if the solution satisfies all constraints. If it does, it indicates that the submatrix of the combination is valid. A and the calculated angles of this set of creases Capable of achieving all N One target steady-state angle, group This is the simplest crease design solution that meets the requirements; Step 5: If you have tried all If none of the combinations satisfies the constraints, the number of creases is increased, and the total number of steady states corresponding to the new value is recalculated. Based on the increased number of creases and the recalculated total number of steady states, a new steady-state angle coefficient matrix is ​​constructed. Step 3 is returned, and the combination selection, solution, and verification process is repeated until a crease angle that satisfies all target steady-state angle requirements and meets all constraints is found. .

2. The design method for the rotational multistable paper-cutting structure configuration according to claim 1, characterized in that, In step 4, the constraints include: Initial crease constraint: Assuming the vertical displacement of the rigid platform is always positive, then the first crease state is either no crease or valley crease; Adjacent crease state constraint: To avoid unacceptable geometric interference between adjacent folds, any two adjacent creases cannot be in the same type of fold state at the same time. Crease type quantity balance constraint: For a complete steady-state configuration, the total number of valley creases is equal to the total number of mountain creases.

3. The design method for the rotational multistable paper-cutting structure configuration according to claim 1, characterized in that, In step 1, the theoretically minimum number of creases required is calculated as follows: Calculate the theoretically required minimum number of creases based on the target steady-state angle number N. C n,min ; In the formula: C n,min The minimum number of creases required in theory; ceil(x) represents rounding up the variable x; N is the target steady-state angle.

4. The design method for the rotational multistable paper-cutting structure configuration according to claim 3, characterized in that, In step 2, the corresponding total number of steady states is calculated as follows: S n,min =2 Cn,min-1 In the formula: S n,min This represents the total steady-state number.

5. The design method for the rotational multistable paper-cutting structure configuration according to claim 3, characterized in that, In step 2, the constructed steady-state angle coefficient matrix is: F= Where F is the steady-state angle coefficient matrix, the matrix has a dimension of Sn×(Cn-1), and Cn is the number of creases in the target configuration. S n F represents the total number of steady-state components of the target assembly. pt F is the steady-state angle coefficient of the p-th steady state and the t-th steady state. pt =2×(S1+S2+S3+...+S t ), S t This represents the crease state of the t-th crease when the p-th steady state is reached.

6. A rotationally multistable paper-cutting structure configuration, characterized in that, The design method for the rotational multistable paper-cutting structure configuration according to any one of claims 1-5 is used to design the structure.

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