A method for predicting bistable performance of a tubular deployable composite stretchable arm based on two parameters

Abstract

A method for predicting the bistable performance of tubular deployable composite stretchable arms based on two parameters, the method has three steps: step one, define the geometry and size of the tubular deployable composite stretchable arms, determine the mathematical expression of the relationship between various geometric parameters; Step two, establish a two-parameter method and determine the strain energy of the tubular deployable composite stretchable arms in any configuration, and determine the stability of the geometric configuration through the stiffness matrix of the tubular deployable composite stretchable arms; Step three, determine the stress level of the tubular deployable composite stretchable arms in the process of large deformation according to Tsai-Hill criterion, and deduce the expression of Tsai-Hill failure coefficient.

Description

Technical Field

[0001] This invention provides a method for predicting the bistable performance of a tubular deployable composite material extension arm based on two parameters, belonging to the field of manned spaceflight. Background Technology

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

[0003] This invention establishes a method for predicting the bistable properties of a tubular deployable composite extension arm based on two parameters. 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 tubular deployable composite material extension arm, and determine the mathematical expression for the relationship between the various geometric parameters.

[0005] A tubular deployable composite extender arm can achieve the transition between a deployed steady state and a folded steady state by storing and releasing elastic strain energy. The curvature directions corresponding to the two steady states of the tubular deployable composite extender arm are on the same side. The geometry of the tubular deployable composite extender arm in the deployed steady state is determined by the length L, thickness t, cross-sectional radius R, and central angle φ, as shown below. Figure 1 As shown. To characterize the geometric configuration of the tubular deployable composite extension arm in two steady states, this paper makes the following basic assumptions:

[0006] (1) The projection of the centerline of the tubular deployable composite extension arm of arbitrary configuration onto the uov plane is an Archimedean spiral, and any two loops are in close contact, such as Figure 2 As shown.

[0007] (2) The overall deformation of the tubular deployable composite extension arm can be described by the change in the shape of the neutral surface, which is bent without stretching.

[0008] (3) The Gaussian curvature of the tubular deployable composite extension arm remains unchanged under any configuration, which means that all possible configurations can be fitted to a cylindrical surface, such as Figure 3 As shown.

[0009] Based on the basic assumptions (1) and Figure 2 The projection of the centerline of an arbitrary-configuration tubular deployable composite extension arm onto the uov plane can be described by a polynomial shape function in polar coordinates (ρ, α), as follows:

[0010] ρ=aα+b,α∈0,α1 (1)

[0011] Where 'a' controls the distance between two adjacent rings, 'b' is the distance from the projection starting point to the origin of the polar coordinates, and 'α1' is the polar angle of the projection endpoint of the centerline of the tubular deployable composite material extension arm.

[0012] Based on basic assumption (1), we can obtain

[0013]

[0014] According to geometric relationships, the boundary conditions that equation (2) needs to satisfy are:

[0015]

[0016] Where r0 and r1 are the extreme radii of the projection start and end points of the centerline of the tubular deployable composite material extension arm of arbitrary configuration, respectively.

[0017] Substituting equation (2) into equation (3) and simplifying, we get

[0018]

[0019] Therefore, the geometric configuration of the centerline of the arbitrarily configured tubular deployable composite extension arm can be described using cylindrical coordinates (ρ, α, z), where...

[0020]

[0021] Based on basic assumption (2) and Figure 4 , can be obtained

[0022]

[0023] Where L is the length of the tubular deployable composite material extension arm.

[0024] Substituting equations (4) and (5) into equation (6) and integrating, we get...

[0025]

[0026] Equation (7) is an implicit function, and the correspondence between r0 and r1 can be determined by numerical solution using Newton's iteration method.

[0027] Step 2: Establish a two-parameter method and determine the strain energy of the tubular deployable composite extension arm in any configuration, and determine the stability of the verification geometry by using the stiffness matrix of the tubular deployable composite extension arm.

[0028] The spatial helix equation is used to describe all possible configurations of the tubular deployable composite extension arm. The two selected parameters are r0 and θ, where θ is the angle between the tubular deployable composite extension arm and the cylindrical axis.

[0029] The curvature at any position of a tubular deployable composite articulated arm with arbitrary configuration can be expressed as:

[0030]

[0031] according to Figure 1 It can be seen that the curvatures of the tubular deployable composite extension arm in the steady-state of deployment are respectively

[0032]

[0033] Therefore, the change in curvature of the tubular deployable composite extension arm in any configuration can be expressed as:

[0034]

[0035] The strain energy per unit area of ​​an arbitrary-configuration tubular deployable composite extension arm is

[0036]

[0037] Where D is the bending stiffness matrix of the laminate.

[0038] Substituting equation (10) into equation (12), we get

[0039]

[0040] The total strain energy of an arbitrary-configuration tubular deployable composite extension arm can be expressed as:

[0041]

[0042] Equation (13) includes two variables, r0 and θ, which determine the total strain energy of the tubular deployable composite extension arm with arbitrary configuration. Therefore, the tubular deployable composite extension arm with arbitrary configuration needs to satisfy the following condition when it is at the minimum strain energy value:

[0043]

[0044] In order to satisfy equation (14), the following conditions need to be applied, and the solution to equation (15) defines the geometric configuration under equilibrium conditions.

[0045]

[0046] The stability of the solution is determined by considering the stiffness matrix K of the tubular deployable composite extension arm in equation (16). If K is a positive definite matrix, the geometry is stable.

[0047]

[0048] When the stiffness matrix K satisfies the condition in equation (17), the stiffness matrix is ​​a positive definite matrix.

[0049]

[0050] Step 3: Determine the stress level of the tubular deployable composite extension arm during large deformation according to the Tsai-Hill criterion, and derive the Tsai-Hill failure factor expression.

[0051] The stress-strain relationship of the k-th layer of an arbitrary-configuration tubular deployable composite extension arm can be expressed as:

[0052]

[0053] According to the stress component coordinate transformation equation of the k-th layer in the laminate, as follows: Figure 5 As shown, the principal stress in the k-th layer of the laminate can be expressed as:

[0054]

[0055] Substituting equation (18) into equation (19) yields the principal stresses of the k-th layer in an arbitrary configuration of a tubular deployable composite extension arm, which are respectively... and The maximum stress in the principal direction of the obtained k-th layer is substituted into the Tsai-Hill criterion to evaluate the stress level of an arbitrary configuration tubular deployable composite extension arm. When the Tsai-Hill failure factor I... f The structure fails when the value reaches or exceeds 1; otherwise, it does not fail.

[0056] Tsai-Hill Failure Factor I f The following formula can be used to calculate

[0057]

[0058] in,

[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] Figure 6 The flowchart shows the two-parameter analytical model established for predicting the folding steady state of a tubular deployable composite extension arm for this invention.

[0063] This invention provides a method for predicting the bistable performance of a tubular deployable composite extension arm based on two parameters. The method is characterized by its ability to conveniently and quickly predict the bistable performance of the tubular deployable composite extension arm based on the material property parameters and geometric parameters of the tubular deployable composite extension arm components. Attached Figure Description

[0064] Figure 1 This is a schematic diagram of the geometric configuration of a tubular deployable composite material extension arm in its stable deployed state.

[0065] Figure 2 This is a schematic diagram of the geometric configuration of a tubular deployable composite material extension arm in a folded steady state.

[0066] Figure 3 This is a schematic diagram of a tubular deployable composite material extension arm with arbitrary configuration.

[0067] Figure 4 A schematic diagram of the centerline of an arbitrary configuration tubular deployable composite material extension arm.

[0068] Figure 5 This is a schematic diagram of stress analysis for the k-th layer of the laminate.

[0069] Figure 6 Flowchart for predicting the bistable properties of a tubular deployable composite extension arm.

[0070] The symbols in the diagram are explained as follows:

[0071] Figure 1 In the diagram, L represents the length of the tubular deployable composite material extension arm, t represents the thickness of the tubular deployable composite material extension arm, R represents the cross-sectional radius of the tubular deployable composite material extension arm, φ represents the central angle of the cross-section of the tubular deployable composite material extension arm, and x, y, and z represent the coordinate axes of the rectangular coordinate system.

[0072] Figure 2 In the figure, r0 and r1 are the polar diameters of the starting and ending points of the tubular deployable composite material extension arm under folding steady state, respectively; ρ is the polar diameter of the tubular deployable composite material extension arm under folding steady state; α is the polar angle of the tubular deployable composite material extension arm under folding steady state; and u and v are the coordinate axes of the rectangular coordinate system.

[0073] Figure 3 In this context, θ represents the angle between the tubular deployable composite extension arm and the cylindrical axis.

[0074] Figure 5 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:

[0075] Step 1: Define the geometry and dimensions of the tubular deployable composite material extension arm, and determine the mathematical expression for the relationship between the various geometric parameters.

[0076] A tubular deployable composite extender arm can achieve the transition between a deployed steady state and a folded steady state by storing and releasing elastic strain energy. The curvature directions corresponding to the two steady states of the tubular deployable composite extender arm are on the same side. The geometry of the tubular deployable composite extender arm in the deployed steady state is determined by the length L, thickness t, cross-sectional radius R, and central angle φ, as shown below. Figure 1 As shown. To characterize the geometric configuration of the tubular deployable composite extension arm in two steady states, this paper makes the following basic assumptions:

[0077] (1) The projection of the centerline of the tubular deployable composite extension arm of arbitrary configuration onto the uov plane is an Archimedean spiral, and any two loops are in close contact, such as Figure 2 As shown.

[0078] (2) The overall deformation of the tubular deployable composite extension arm can be described by the change in the shape of the neutral surface, which is bent without stretching.

[0079] (3) The Gaussian curvature of the tubular deployable composite extension arm remains unchanged under any configuration, which means that all possible configurations can be fitted to a cylindrical surface, such as Figure 3 As shown.

[0080] Based on the basic assumptions (1) and Figure 2 The projection of the centerline of an arbitrary-configuration tubular deployable composite extension arm onto the uov plane can be described by a polynomial shape function in polar coordinates (ρ, α), as follows:

[0081] ρ=aα+b,α∈0,α1 (1)

[0082] Where 'a' controls the distance between two adjacent rings, 'b' is the distance from the projection starting point to the origin of the polar coordinates, and 'α1' is the polar angle of the projection endpoint of the centerline of the tubular deployable composite material extension arm.

[0083] Based on basic assumption (1), we can obtain

[0084]

[0085] According to geometric relationships, the boundary conditions that equation (2) needs to satisfy are:

[0086]

[0087] Where r0 and r1 are the extreme radii of the projection start and end points of the centerline of the tubular deployable composite material extension arm of arbitrary configuration, respectively.

[0088] Substituting equation (2) into equation (3) and simplifying, we get

[0089]

[0090] Therefore, the geometric configuration of the centerline of the arbitrarily configured tubular deployable composite extension arm can be described using cylindrical coordinates (ρ, α, z), where...

[0091]

[0092] Based on basic assumption (2) and Figure 4 , can be obtained

[0093]

[0094] Where L is the length of the tubular deployable composite material extension arm.

[0095] Substituting equations (4) and (5) into equation (6) and integrating, we get...

[0096]

[0097] Equation (7) is an implicit function, and the correspondence between r0 and r1 can be determined by numerical solution using Newton's iteration method.

[0098] Step 2: Establish a two-parameter method and determine the strain energy of the tubular deployable composite extension arm in any configuration, and determine the stability of the verification geometry by using the stiffness matrix of the tubular deployable composite extension arm.

[0099] The spatial helix equation is used to describe all possible configurations of the tubular deployable composite extension arm. The two selected parameters are r0 and θ, where θ is the angle between the tubular deployable composite extension arm and the cylindrical axis.

[0100] The curvature at any position of a tubular deployable composite articulated arm with arbitrary configuration can be expressed as:

[0101]

[0102] according to Figure 1 It can be seen that the curvatures of the tubular deployable composite extension arm in the steady-state of deployment are respectively

[0103]

[0104] Therefore, the change in curvature of the tubular deployable composite extension arm in any configuration can be expressed as:

[0105]

[0106] The strain energy per unit area of ​​an arbitrary-configuration tubular deployable composite extension arm is

[0107]

[0108] Where D is the bending stiffness matrix of the laminate.

[0109] Substituting equation (10) into equation (12), we get

[0110]

[0111] The total strain energy of an arbitrary-configuration tubular deployable composite extension arm can be expressed as:

[0112]

[0113] Equation (13) includes two variables, r0 and θ, which determine the total strain energy of the tubular deployable composite extension arm with arbitrary configuration. Therefore, the tubular deployable composite extension arm with arbitrary configuration needs to satisfy the following condition when it is at the minimum strain energy value:

[0114]

[0115] In order to satisfy equation (14), the following conditions need to be applied, and the solution to equation (15) defines the geometric configuration under equilibrium conditions.

[0116]

[0117] The stability of the solution is determined by considering the stiffness matrix K of the tubular deployable composite extension arm in equation (16). If K is a positive definite matrix, the geometry is stable.

[0118]

[0119] When the stiffness matrix K satisfies the condition in equation (17), the stiffness matrix is ​​a positive definite matrix.

[0120]

[0121] Step 4: Determine the stress level of the tubular deployable composite extension arm during large deformation according to the Tsai-Hill criterion, and derive the Tsai-Hill failure factor expression.

[0122] The stress-strain relationship of the k-th layer of an arbitrary-configuration tubular deployable composite extension arm can be expressed as:

[0123]

[0124] According to the stress component coordinate transformation equation of the k-th layer in the laminate, as follows: Figure 5 As shown, the principal stress in the k-th layer of the laminate can be expressed as:

[0125]

[0126] Substituting equation (18) into equation (19) yields the principal stresses of the k-th layer in an arbitrary configuration of a tubular deployable composite extension arm, which are respectively... and The maximum stress in the principal direction of the obtained k-th layer is substituted into the Tsai-Hill criterion to evaluate the stress level of an arbitrary configuration tubular deployable composite extension arm. When the Tsai-Hill failure factor I... f The structure fails when the value reaches or exceeds 1; otherwise, it does not fail.

[0127] Tsai-Hill Failure Factor I f The following formula can be used to calculate

[0128]

[0129] in,

[0130]

[0131] 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.

[0132] Figure 6 The flowchart shows the two-parameter analytical model established for predicting the folding steady state of a tubular deployable composite extension arm for this invention.

[0133] This invention provides a method for predicting the bistable performance of a tubular deployable composite extension arm based on two parameters. The method is characterized by its ability to conveniently and quickly predict the bistable performance of the tubular deployable composite extension arm based on the material property parameters and geometric parameters of the tubular deployable composite extension arm components.

Claims

1. A method for predicting the bistable properties of a tubular deployable composite extension arm based on two parameters, characterized in that: The specific steps of this method are as follows: Step 1: Define the geometry and dimensions of the tubular deployable composite material extension arm, and determine the mathematical expression for the relationship between the various geometric parameters; Tubular deployable composite extension arms can achieve the interconversion between deployed steady state and folded steady state by storing and releasing elastic strain energy; The curvature directions of the two steady states of the tubular deployable composite extension arm are on the same side; the geometric configuration of the tubular deployable composite extension arm in the deployed steady state is determined by the length L, thickness t, cross-sectional radius R, and central angle φ; to characterize the geometric configuration of the tubular deployable composite extension arm in the two steady states, this paper makes the following basic assumptions: (1) The projection of the center line of the tubular deployable composite extension arm of arbitrary configuration onto the plane uov is an Archimedean spiral, and any two loops are in close contact. (2) The overall deformation of the tubular deployable composite extension arm can be described by the change in the shape of the neutral surface, which is bent without stretching. (3) The Gaussian curvature of the tubular deployable composite extension arm remains unchanged under any configuration, which means that all possible configurations can be fitted to a cylindrical surface. Based on the fundamental assumption (1), the projection of the centerline of the tubular deployable composite extension arm of arbitrary configuration onto the uov plane can be described by a polynomial shape function in polar coordinates (ρ, α), as follows: ρ=aα+b,α∈0,α1 (1) Where a controls the distance between two adjacent rings, b is the distance from the projection start point to the origin of the polar coordinates, and α1 is the polar angle of the projection end point of the center line of the tubular deployable composite material extension arm. Based on basic assumption (1), we can obtain According to geometric relationships, the boundary conditions that equation (2) needs to satisfy are: Where r0 and r1 are the polar radii of the projection start and projection end points of the centerline of the tubular deployable composite material extension arm of arbitrary configuration, respectively. Substituting equation (2) into equation (3) and simplifying, we get Therefore, the geometric configuration of the centerline of the arbitrarily configured tubular deployable composite extension arm can be described using cylindrical coordinates (ρ, α, z), where... Based on the basic assumption (2) and Figure 4, we can obtain Where L is the length of the tubular deployable composite material extension arm; Substituting equations (4) and (5) into equation (6) and integrating, we get... Equation (7) is an implicit function, and the correspondence between r0 and r1 can be determined numerically by Newton's iteration method; Step 2: Establish a two-parameter method and determine the strain energy of the tubular deployable composite extension arm in any configuration, and determine the stability of the verification geometry by using the stiffness matrix of the tubular deployable composite extension arm. The spatial helix equation is used to describe all possible configurations of the tubular deployable composite extension arm. The two selected parameters are r0 and θ, where θ is the angle between the tubular deployable composite extension arm and the cylindrical axis. The curvature at any position of a tubular deployable composite articulated arm with arbitrary configuration can be expressed as: The curvatures of the tubular deployable composite extension arm in its steady-state deployment are respectively Therefore, the change in curvature of the tubular deployable composite extension arm in any configuration can be expressed as: The strain energy per unit area of ​​an arbitrary-configuration tubular deployable composite extension arm is Where D is the bending stiffness matrix of the laminate; Substituting equation (10) into equation (12), we get The total strain energy of an arbitrary-configuration tubular deployable composite extension arm can be expressed as: Equation (13) includes two variables, r0 and θ, which determine the total strain energy of the tubular deployable composite extension arm with arbitrary configuration. Therefore, the tubular deployable composite extension arm with arbitrary configuration needs to satisfy the following condition when it is at the minimum strain energy value: In order to satisfy equation (14), the following conditions need to be applied, and the solution to equation (15) defines the geometric configuration under equilibrium conditions. The stability of the solution is determined by considering the stiffness matrix K of the tubular deployable composite extension arm in equation (16); if K is a positive definite matrix, the geometry is stable. When the stiffness matrix K satisfies the condition in equation (17), the stiffness matrix is ​​a positive definite matrix; Step 3: Determine the stress level of the tubular deployable composite extension arm during large deformation according to the Tsai-Hill criterion, and derive the Tsai-Hill failure factor expression. The stress-strain relationship of the k-th layer of an arbitrary-configuration tubular deployable composite extension arm can be expressed as: According to the stress component coordinate transformation equation of the k-th layer in a laminate, the principal stress in the k-th layer of the laminate can be expressed as: Substituting equation (18) into equation (19) yields the principal stresses of the k-th layer in an arbitrary configuration of a tubular deployable composite extension arm, which are respectively... and The maximum stress in the principal direction of the obtained k-th layer is substituted into the Tsai-Hill criterion to evaluate the stress level of the tubular deployable composite extension arm with 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, 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.

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