A method of predicting the folded stable state of a bistable deployable composite stretchable arm with a flat section

Abstract

A method for predicting the folded stable state of bistable deployable composite stretchable arms with flat segments, the method has four steps: step one, define the geometry and size of the bistable deployable composite stretchable arms with flat segments, and determine the mathematical expression of the relationship between various geometric parameters; step two, determine the strain energy of the bistable deployable composite stretchable arms with flat segments in any configuration, and determine the stability of the geometric configuration through the stiffness matrix of the bistable deployable composite stretchable arms with flat segments; step three, derive the bistable criterion by checking whether the stiffness matrix of the above folded configuration is a positive definite matrix, so as to determine the folded stable state of the bistable deployable composite stretchable arms with flat segments; step four, determine the stress level of the bistable deployable composite stretchable arms with flat segments in the process of large deformation according to Tsai-Hill criterion, and derive the expression of Tsai-Hill failure coefficient.

Description

Technical Field

[0001] This invention provides a method for predicting the folding steady state of a bistable deployable composite material extension arm with a straight section, belonging to the field of manned spaceflight. Background Technology

[0002] Due to their lightweight, high stiffness, high folding efficiency, and reliable deployment process, bistable deployable composite material extendable arms have received widespread attention and research in the aerospace field, showing promising application prospects. Bistable deployable composite material extendable arms with straight sections are typically made of carbon fiber resin-based composite materials and are thin-walled tubular rod structures capable of folding and unfolding. Folding steady-state is a key mechanical performance indicator for bistable deployable composite material extendable arms with straight sections, making its analysis necessary. Directly measuring the folding steady-state of bistable deployable composite material extendable arms with straight sections using experimental methods is costly and susceptible to many accidental factors during testing. Finite element numerical simulation methods require establishing complex finite element models, resulting in complex calculations, low computational efficiency, and difficulty in guaranteeing computational accuracy. Therefore, this paper establishes an effective method for predicting the folding steady-state of bistable deployable composite material extendable arms with straight sections. Only a small number of component material performance parameters and geometric parameters are needed to quickly and accurately predict the folding steady-state of bistable deployable composite material extendable arms with straight sections, demonstrating the significant academic value and broad engineering application prospects of this invention. Summary of the Invention

[0003] This invention establishes a method for predicting the folding steady state of a bistable deployable composite material extension arm with a straight section. This method has the advantages of simple calculation and high accuracy. The technical solution is as follows:

[0004] Step 1: Define the geometry and dimensions of the bistable deployable composite extension arm with a straight section, and determine the mathematical expression for the relationship between the various geometric parameters.

[0005] The geometric parameters of a bistable deployable composite extendable arm with a straight section in its deployed steady state include length L, thickness t, width w of the straight section, radius R of the arc section, and central angle α of the arc section, such as... Figure 1 and 2 As shown. To characterize the geometric properties of the bistable deployable composite extension arm with a straight section in both steady states, this paper makes the following basic assumptions:

[0006] (1) The overall deformation of a bistable deployable composite extension arm with a straight section can be described by the change in the shape of the neutral surface, which is curved without stretching. This means that all deformation is uniform and non-stretched.

[0007] (2) The bistable deployable composite extension arm with a straight section can fit all possible folding steady-state configurations to the cylindrical surface, such as Figure 3 As shown.

[0008] Based on the two assumptions above, all possible configurations can be defined by two parameters: the curvature C of the cylinder and the direction θ of the bistable deployable composite extension arm relative to the cylinder axis. The relationship between the curvature of any configuration and the two parameters can be expressed as follows:

[0009]

[0010] Among them, κ x Let κ be the curvature of the laminate in the x-direction. y Let κ be the curvature of the laminate in the y-direction. xy The curvature of the laminate in the x and y directions.

[0011] Step 2: Determine the strain energy of the bistable deployable composite extension arm with a straight section in any configuration, and determine the stability of the verification geometry by using the stiffness matrix of the bistable deployable composite extension arm with a straight section.

[0012] according to Figure 1 It can be seen that the curvatures of the straight and circular segments of the bistable deployable composite material extension arm in the deployed steady state are respectively...

[0013]

[0014]

[0015] The subscript "flat" indicates a straight segment, and the subscript "arc" indicates a circular arc segment.

[0016] Therefore, the change in curvature of a bistable deployable composite extension arm with a straight section in any configuration can be expressed as:

[0017]

[0018]

[0019] The strain energy per unit area of ​​the bistable deployable composite extension arm with a straight section after deformation is:

[0020]

[0021] Where D is the bending stiffness matrix of the laminate.

[0022] Substituting equations (4) and (5) into equation (6), we obtain the strain energy per unit area expressed in terms of parameters C and θ, as follows:

[0023]

[0024] Therefore, the strain energy of a bistable deployable composite extension arm with a straight section in any configuration can be expressed as:

[0025]

[0026] A bistable deployable composite extension arm with a straight section needs to satisfy the following condition when it is at the minimum strain energy:

[0027]

[0028] The specific expansion result of the first-order partial derivative of U is as follows:

[0029]

[0030]

[0031] In order to satisfy equation (9), the following conditions need to be applied, and the solution to equation (12) defines the geometric configuration under equilibrium conditions.

[0032]

[0033] The stability of the solution is determined by considering the stiffness matrix K of the bistable deployable composite extension arm with a straight section in equation (13). If K is a positive definite matrix, the geometry is stable.

[0034]

[0035] The specific expansion result of the second-order partial derivative of U is as follows:

[0036]

[0037]

[0038]

[0039] Step 3: By checking whether the stiffness matrix of the above folding configuration is a positive definite matrix, the bistable criterion is derived, thereby determining the folding steady state of the bistable deployable composite extension arm with a straight section.

[0040] To directly determine the folding steady state of a bistable deployable composite extension arm with a straight section without analyzing the strain energy minimum for each design case in a parametric study, this invention derives a bistable criterion. The parameters C and θ corresponding to the folding configuration can be obtained by solving equation (12), as follows:

[0041]

[0042] The bistable criterion is obtained by checking whether the stiffness matrix of the above folded configuration is a positive definite matrix, as follows:

[0043]

[0044] Substituting equation (17) into equations (14) to (16) and simplifying, we get

[0045]

[0046]

[0047]

[0048] zero and It is a positive number. Therefore, when When the value is greater than zero, the stiffness matrix corresponding to the above folded configuration is a positive definite matrix. The above bistable deployable composite extension arm with a straight section exhibits a folded steady state.

[0049] Step 4: Determine the stress level of the bistable deployable composite extension arm with a straight section during large deformation according to the Tsai-Hill criterion, and derive the Tsai-Hill failure factor expression.

[0050] First, we analyze the stress level in the straight section. The stress-strain relationship of the k-th layer in the laminated plate of the straight section is:

[0051]

[0052] According to the stress component coordinate transformation equation of the k-th layer in the laminate, as follows: Figure 4 As shown, the maximum stress in the principal direction of the k-th layer of the laminate can be expressed as:

[0053]

[0054] Substituting equations (3) and (22) into equation (23) yields the maximum stress in the principal direction of the k-th layer in the straight section, which are respectively and The maximum stress in the principal direction of the k-th layer of the circular arc segment was obtained using the same method, namely: and The maximum stress in the principal direction of the k-th layer obtained from the straight and circular arc segments were respectively substituted into the Tsai-Hill criterion to evaluate the stress level of a bistable deployable composite extension arm with a straight segment of arbitrary configuration. When the Tsai-Hill failure factor I... f The structure fails when the value reaches or exceeds 1; otherwise, it does not fail.

[0055] Tsai-Hill Failure Factor I f The following formula can be used to calculate

[0056]

[0057] in,

[0058]

[0059]

[0060]

[0061] in, and These are the principal stresses in the longitudinal and transverse directions of the k-th laminate, respectively. Let X be the shear stress in the longitudinal and transverse directions of the k-th layer of the laminate. t and X c These are the longitudinal tensile strength and compressive strength of the composite material, respectively, with X1 and X2 as intermediate variables, and Y... t and Y c These are the transverse tensile strength and compressive strength of the composite material, respectively, with Y being an intermediate variable.

[0062] This invention provides a method for predicting the folding steady state of a bistable deployable composite material extension arm with a straight section. The method is characterized by its ability to conveniently and quickly predict the folding steady state of the bistable deployable composite material extension arm with a straight section based on the component material performance parameters and geometric parameters. Attached Figure Description

[0063] Figure 1 This is a schematic diagram of the geometric configuration of a bistable deployable composite extension arm with a straight section in its stable deployment state.

[0064] Figure 2 This is a schematic cross-sectional view of a bistable deployable composite extension arm with a straight section in its deployed steady state.

[0065] Figure 3 This is a schematic diagram of a tubular deployable composite material extension arm with a straight section of arbitrary configuration.

[0066] Figure 4 This is a schematic diagram of stress analysis for the k-th layer of the laminate.

[0067] The symbols in the diagram are explained as follows:

[0068] Figure 1 In the figure, L represents the length of the bistable deployable composite extension arm with a straight section, and x, y, and z are the coordinate axes of the Cartesian coordinate system.

[0069] Figure 2 In this context, w represents the width of the straight segment, R represents the radius of the arc segment, and α represents the central angle of the arc segment.

[0070] Figure 3 In this context, C represents the curvature of the cylinder, and θ represents the angle between the bistable deployable composite extension arm with a straight section and the cylinder axis.

[0071] Figure 4 middle and Let be the normal stresses in the x and y directions of the k-th composite material layer in the laminate, respectively. Let denot be the shear stress of the k-th layer of composite material in the laminate, and β be the layup angle, 1. Principal direction 1, 2. Principal direction 2. Detailed implementation method:

[0072] Step 1: Define the geometry and dimensions of the bistable deployable composite extension arm with a straight section, and determine the mathematical expression for the relationship between the various geometric parameters.

[0073] The geometric parameters of a bistable deployable composite extendable arm with a straight section in its deployed steady state include length L, thickness t, width w of the straight section, radius R of the arc section, and central angle α of the arc section, such as... Figure 1 and 2 As shown. To characterize the geometric properties of the bistable deployable composite extension arm with a straight section in both steady states, this paper makes the following basic assumptions:

[0074] (1) The overall deformation of a bistable deployable composite extension arm with a straight section can be described by the change in the shape of the neutral surface, which is curved without stretching. This means that all deformation is uniform and non-stretched.

[0075] (2) The bistable deployable composite extension arm with a straight section can fit all possible folding steady-state configurations to the cylindrical surface, such as Figure 3 As shown.

[0076] Based on the two assumptions above, all possible configurations can be defined by two parameters: the curvature C of the cylinder and the direction θ of the bistable deployable composite extension arm relative to the cylinder axis. The relationship between the curvature of any configuration and the two parameters can be expressed as follows:

[0077]

[0078] Among them, κ x Let κ be the curvature of the laminate in the x-direction. y Let κ be the curvature of the laminate in the y-direction. xy The curvature of the laminate in the x and y directions.

[0079] Step 2: Determine the strain energy of the bistable deployable composite extension arm with a straight section in any configuration, and determine the stability of the verification geometry by using the stiffness matrix of the bistable deployable composite extension arm with a straight section.

[0080] according to Figure 1 It can be seen that the curvatures of the straight and circular segments of the bistable deployable composite material extension arm in the deployed steady state are respectively...

[0081]

[0082]

[0083] The subscript "flat" indicates a straight segment, and the subscript "arc" indicates a circular arc segment.

[0084] Therefore, the change in curvature of a bistable deployable composite extension arm with a straight section in any configuration can be expressed as:

[0085]

[0086]

[0087] The strain energy per unit area of ​​the bistable deployable composite extension arm with a straight section after deformation is:

[0088]

[0089] Where D is the bending stiffness matrix of the laminate.

[0090] Substituting equations (4) and (5) into equation (6), we obtain the strain energy per unit area expressed in terms of parameters C and θ, as follows:

[0091]

[0092] Therefore, the strain energy of a bistable deployable composite extension arm with a straight section in any configuration can be expressed as:

[0093]

[0094] A bistable deployable composite extension arm with a straight section needs to satisfy the following condition when it is at the minimum strain energy:

[0095]

[0096] The specific expansion result of the first-order partial derivative of U is as follows:

[0097]

[0098]

[0099] In order to satisfy equation (9), the following conditions need to be applied, and the solution to equation (12) defines the geometric configuration under equilibrium conditions.

[0100]

[0101] The stability of the solution is determined by considering the stiffness matrix K of the bistable deployable composite extension arm with a straight section in equation (13). If K is a positive definite matrix, the geometry is stable.

[0102]

[0103] The specific expansion result of the second-order partial derivative of U is as follows:

[0104]

[0105]

[0106]

[0107] Step 3: By checking whether the stiffness matrix of the above folding configuration is a positive definite matrix, the bistable criterion is derived, thereby determining the folding steady state of the bistable deployable composite extension arm with a straight section.

[0108] To directly determine the folding steady state of a bistable deployable composite extension arm with a straight section without analyzing the strain energy minimum for each design case in a parametric study, this invention derives a bistable criterion. The parameters C and θ corresponding to the folding configuration can be obtained by solving equation (12), as follows:

[0109]

[0110] The bistable criterion is obtained by checking whether the stiffness matrix of the above folded configuration is a positive definite matrix, as follows:

[0111]

[0112] Substituting equation (17) into equations (14) to (16) and simplifying, we get

[0113]

[0114]

[0115]

[0116] zero and It is a positive number. Therefore, when When the value is greater than zero, the stiffness matrix corresponding to the above folded configuration is a positive definite matrix. The above bistable deployable composite extension arm with a straight section exhibits a folded steady state.

[0117] Step 4: Determine the stress level of the bistable deployable composite extension arm with a straight section during large deformation according to the Tsai-Hill criterion, and derive the Tsai-Hill failure factor expression.

[0118] First, we analyze the stress level in the straight section. The stress-strain relationship of the k-th layer in the laminated plate of the straight section is:

[0119]

[0120] According to the stress component coordinate transformation equation of the k-th layer in the laminate, as follows: Figure 4 As shown, the maximum stress in the principal direction of the k-th layer of the laminate can be expressed as:

[0121]

[0122] Substituting equations (3) and (22) into equation (23) yields the maximum stress in the principal direction of the k-th layer in the straight section, which are respectively and The maximum stress in the principal direction of the k-th layer of the circular arc segment was obtained using the same method, namely: and The maximum stress in the principal direction of the k-th layer obtained from the straight and circular arc segments were respectively substituted into the Tsai-Hill criterion to evaluate the stress level of a bistable deployable composite extension arm with a straight segment of arbitrary configuration. When the Tsai-Hill failure factor I... f The structure fails when the value reaches or exceeds 1; otherwise, it does not fail.

[0123] Tsai-Hill Failure Factor I f The following formula can be used to calculate

[0124]

[0125] in,

[0126]

[0127]

[0128]

[0129] in, and These are the principal stresses in the longitudinal and transverse directions of the k-th laminate, respectively. Let X be the shear stress in the longitudinal and transverse directions of the k-th layer of the laminate. t and X c These are the longitudinal tensile strength and compressive strength of the composite material, respectively, with X1 and X2 as intermediate variables, and Y... t and Y c These are the transverse tensile strength and compressive strength of the composite material, respectively, with Y being an intermediate variable.

[0130] This invention provides a method for predicting the folding steady state of a bistable deployable composite material extension arm with a straight section. The method is characterized by its ability to conveniently and quickly predict the folding steady state of the bistable deployable composite material extension arm with a straight section based on the component material performance parameters and geometric parameters.

Claims

1. A method for predicting the folding steady state of a bistable deployable composite extension arm with a straight section, characterized in that: The specific steps of this method are as follows: Step 1: Define the geometry and dimensions of the bistable deployable composite extension arm with a straight section, and determine the mathematical expression for the relationship between the various geometric parameters. The geometric parameters of the bistable deployable composite arm with a straight section in the deployed steady state include length L, thickness t, width w of the straight section, radius R of the arc section, and central angle α of the arc section. To characterize the geometric properties of the bistable deployable composite arm with a straight section in the two steady states, this paper makes the following basic assumptions: (1) The overall deformation of a bistable deployable composite extension arm with a straight section can be described by the change in the shape of the neutral surface, which is curved without stretching; this means that all deformation is uniform and unstretched. (2) All possible folding steady-state configurations of the bistable deployable composite extension arm with a straight section can be fitted to the cylindrical surface. Based on the above two assumptions, all possible configurations can be defined by two parameters: the curvature C of the cylinder and the direction θ of the bistable deployable composite extension arm relative to the cylinder axis. The relationship between the curvature of any configuration and the two parameters can be expressed as follows: Among them, κ x Let κ be the curvature of the laminate in the x-direction. y Let κ be the curvature of the laminate in the y-direction. xy The curvature of the laminate in the x and y directions; Step 2: Determine the strain energy of the bistable deployable composite extension arm with a straight section in any configuration, and determine the stability of the verification geometry by using the stiffness matrix of the bistable deployable composite extension arm with a straight section. When the bistable deployable composite extendable arm is in a stable deployed state, the curvatures of its straight and circular segments are respectively... In this context, the subscript "flat" indicates a straight segment, and the subscript "arc" indicates a circular arc segment; Therefore, the change in curvature of a bistable deployable composite extension arm with a straight section in any configuration can be expressed as: The strain energy per unit area of ​​the bistable deployable composite extension arm with a straight section after deformation is: Where D is the bending stiffness matrix of the laminate; Substituting equations (4) and (5) into equation (6), we obtain the strain energy per unit area expressed in terms of parameters C and θ, as follows: Therefore, the strain energy of a bistable deployable composite extension arm with a straight section in any configuration can be expressed as: A bistable deployable composite extension arm with a straight section needs to satisfy the following condition when it is at the minimum strain energy: The specific expansion result of the first-order partial derivative of U is as follows: In order to satisfy equation (9), the following conditions need to be applied, and the solution to equation (12) defines the geometric configuration under equilibrium conditions. The stability of the solution is determined by considering the stiffness matrix K of the bistable deployable composite extension arm with a straight section in equation (13); if K is a positive definite matrix, the geometry is stable. The specific expansion result of the second-order partial derivative of U is as follows: Step 3: By checking whether the stiffness matrix of the above folding configuration is a positive definite matrix, the bistable criterion is derived, thereby determining the folding steady state of the bistable deployable composite extension arm with a straight section. To directly determine the folding steady state of a bistable deployable composite extension arm with a straight section without analyzing the minimum strain energy for each design case in the parametric study, this invention derives a bistable criterion; by solving equation (12), the parameters C and θ corresponding to the folding configuration can be obtained as follows: The bistable criterion is obtained by checking whether the stiffness matrix of the above folded configuration is a positive definite matrix, as follows: Substituting equation (17) into equations (14) to (16) and simplifying, we get zero and It is a positive number; therefore, when When the value is greater than zero, the stiffness matrix corresponding to the above folded configuration is a positive definite matrix; the above bistable deployable composite extension arm with a straight section has a folded steady state; Step 4: Determine the stress level of the bistable deployable composite extension arm with a straight section during large deformation according to the Tsai-Hill criterion, and derive the Tsai-Hill failure factor expression. First, we analyze the stress level in the straight section. The stress-strain relationship of the k-th layer in the laminated plate of the straight section is: According to the stress component coordinate transformation equation of the k-th layer in a laminate, the maximum stress in the principal direction of the k-th layer can be expressed as: Substituting equations (3) and (22) into equation (23) yields the maximum stress in the principal direction of the k-th layer in the straight section, which are respectively and The maximum stress in the principal direction of the k-th layer of the circular arc segment was obtained using the same method, namely: and The maximum stress in the principal direction of the k-th layer obtained from the straight and circular arc segments is substituted into the Tsai-Hill criterion to evaluate the stress level of a bistable deployable composite extension arm with a straight segment of arbitrary configuration; when the Tsai-Hill failure factor I f The structure fails when the value reaches or exceeds 1; otherwise, it does not fail. Tsai-Hill Failure Factor I f The following formula can be used to calculate in, Where, σ1 k and σ2 k These are the principal stresses in the longitudinal and transverse directions of the k-th laminate, τ1 and τ2, respectively. k 2 represents the shear stress in the longitudinal and transverse directions of the k-th laminate, X t and X c These are the longitudinal tensile strength and compressive strength of the composite material, respectively, with X1 and X2 as intermediate variables, and Y... t and Y c These are the transverse tensile strength and compressive strength of the composite material, respectively, with Y being an intermediate variable.

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