A method of predicting the folding performance of a deployable composite stretchable arm

Abstract

A method for predicting the folding performance of deployable composite stretchable arms, the method has three steps: step one, define the geometry and size of the deployable composite stretchable arms, determine the mathematical expression of the relationship between various geometric parameters; step two, according to the classical laminated plate theory and energy method, determine the folding torque required by the deployable composite stretchable arms in the process of forward folding and reverse folding; step three, according to Tsai-Hill criterion and maximum stress criterion, the failure coefficient expression of Tsai-Hill criterion and the failure coefficient expression of maximum stress criterion are derived.

Description

TECHNICAL FIELD

[0001] The application provides a method for predicting folding performance of deployable composite stretchable arm, belonging to the field of manned spaceflight. BACKGROUND

[0002] Due to the characteristics of light weight, large stiffness, high folding efficiency, reliable deployment process, etc., the deployable composite stretchable arm has been widely concerned and researched in the field of spaceflight and has good application prospect. The deployable composite stretchable arm is usually made of carbon fiber resin-based composite material and is a thin-walled tubular rod structure that can realize folding and unfolding functions. When folding, a folding torque is applied at both ends of the reel to wind up the deployable composite stretchable arm to form a folded state; and when unfolding, the deployable composite stretchable arm can recover from the folded state to the unfolded state by relying on its own elastic strain energy. It is necessary to analyze the folding performance as a key mechanical performance index of the deployable composite stretchable arm. The experimental method directly measures the folding performance of the deployable composite stretchable arm at a high cost, and the test process is easily affected by many accidental factors. The finite element numerical simulation method needs to establish a complex finite element model, and the calculation is complex, the calculation efficiency is low, and the calculation accuracy is difficult to guarantee. Therefore, the application establishes a method for effectively predicting the folding performance of the deployable composite stretchable arm. Only a small amount of component material performance parameters and geometric parameters are needed to quickly and accurately predict the folding performance of the deployable composite stretchable arm, so the application has important academic value and broad engineering application prospect. SUMMARY

[0003] The application establishes a method for predicting the folding performance of the deployable composite stretchable arm, which has the advantages of simple calculation and high precision, and the technical scheme is as follows:

[0004] Step one, define the geometric shape and size of the deployable composite stretchable arm, and determine the mathematical expression of the relationship between each geometric parameter.

[0005] The deployable composite stretchable arm has two types of folding modes. When the winding direction of the deployable composite stretchable arm faces the opening direction, it is called forward folding; when the winding direction of the deployable composite stretchable arm faces away from the opening direction, it is called reverse folding. The geometric configuration of the deployable composite stretchable arm in the initial state is determined by the length L, the thickness t, the cross-sectional radius R and the central angle θ, as shown in the formula (1). Figure 1 In order to characterize the geometric model of the deployable composite stretchable arm in the deformation process of forward folding and reverse folding, the following basic assumptions are made in this paper:

[0006] (1) The whole folding process of the deployable composite stretchable arm is divided into two stages. The first stage includes the coiled section, the transition section and the free section. When the deployable composite stretchable arm is folded to a certain extent, the free section disappears, and the folding process enters the second stage, which only has the coiled section and the transition section, as shown in FIG. 1. Figure 2

[0007] (2) The center line shape of the longitudinal section of the deployable composite stretchable arm in the coiled state is an Archimedes spiral, as shown in FIG. 2. Figure 3

[0008] (3) The wall thickness change of the deployable composite stretchable arm in the folding deformation process is ignored, so the overall deformation can be described by the shape and curvature radius of the neutral surface, and the neutral surface is not stretched.

[0009] According to the basic assumption (2) and Figure 3 , the coiled deformation of the deployable composite stretchable arm in the folded state can be described by a polynomial shape function in polar coordinates (ρ1, α)

[0010] ρ1=aα+b, α∈α0,α1 (1)

[0011] According to the geometric relationship, the boundary conditions that formula (1) needs to satisfy are

[0012]

[0013] Where r0 and r1 are the polar radii of the starting point and the ending point of the fully coiled state of the deployable composite stretchable arm in the folded state, respectively.

[0014] Usually, α0=0, then formula (2) can be simplified as

[0015]

[0016] According to the basic assumption (3), we can get

[0017]

[0018] Substitute formulas (1)-(3) into formula (4) and integrate to get

[0019]

[0020] Obviously, formula (5) is an implicit function, and the unknown quantity a can be solved by Newton iteration method.

[0021] According to the definition of the curvature radius, the curvature in the x direction during the coiling process of the deployable composite stretchable arm can be expressed as

[0022] ​​

[0023] where p2 is the radius of curvature in x direction during the crimping process of the deployable composite stretchable boom.

[0024] Step two, determine the folding moment of the deployable composite stretchable boom during the forward and reverse folding processes according to the classical laminate theory and energy method.

[0025] In order to establish the strain energy generated by the deployable composite stretchable boom during the forward and reverse folding processes, the forward and reverse folding processes are divided into two steps, as shown in Figure 4 and 5 First, apply a bending moment M y to the edge, which changes the curvature of the deployable composite stretchable boom in the y direction from 1 / R to 0, and in this step the external work is converted into bending strain energy; then, apply a bending moment M x to the ends of the deployable composite stretchable boom, which changes the curvature of the deployable composite stretchable boom in the x direction from 0 to K x , and in this step the external work is converted into bending strain energy.

[0026] Using the ABD matrix in the classical laminate theory, the elastic deformation of the deployable composite stretchable boom during the folding process can be expressed as

[0027]

[0028] The deployable composite stretchable boom usually chooses a symmetric laminate, B = 0, and due to the restriction of the constraint mechanism, the torsional deformation K xy can be ignored, so formula (7) can be simplified as

[0029]

[0030] The strain energy generated by the unit length of the deployable composite stretchable boom from the free state to the crimped state can be expressed as

[0031]

[0032] where R and θ are the radius and center angle of the cross section of the deployable composite stretchable boom, respectively. ΔK x and ΔK y can be derived as

[0033]

[0034] where the value of ΔK x depends on the folding direction of the deployable composite stretchable boom. When forward folding, ΔK x takes a positive value, and when reverse folding, ΔK x takes a negative value, as Figure 4 and 5shown.

[0035] Substituting equation (10) into equation (9), we have

[0036]

[0037] In the actual folding process, the first step is to flatten one end, and the second step is to complete the forward and reverse folding process through the reel. Therefore, there is a transition section between the crimped section and the free section. For any microelement obtained from the transition section, there is an internal moment and an out-of-plane shear force on the edge, as shown in Figure 6 x1, y1 and z1 are the local coordinate system located on the center line of the transition section. The balance equation satisfied by the resultant force and the resultant moment on the arbitrary microelement of the transition section is

[0038]

[0039] The main deformation of the deployable composite stretchable arm is the curvature change caused by bending, and the influence of in-plane force on normal force is small, so the influence of in-plane force on normal force balance equation is ignored. Therefore, equation (12) can be simplified as

[0040]

[0041] According to the classical laminated plate theory, the internal moment and the curvature change have the following relationship:

[0042]

[0043] and

[0044]

[0045] where κ x1,0 , κ y1,0 and κ x1y1,0 are the initial curvatures of the deployable composite stretchable arm in the free state.

[0046]

[0047] The κ x1 of the transition section is 0, and the deformation compatibility equation satisfied between the bending curvature and the torsion curvature is

[0048]

[0049] Substituting equations (14)-(17) into equation (13) and simplifying, we have

[0050]

[0051] Obviously, equation (18) is a second-order partial differential equation, and the boundary conditions satisfied are

[0052]

[0053] where L1 is the length of the transition segment, as shown in Figure 7 .

[0054] The approximate solution of the second-order partial differential equation satisfying the boundary conditions is obtained using the method of separation of variables as

[0055]

[0056] where H1, H2, H3, and H4 are intermediate variables, which are

[0057]

[0058]

[0059]

[0060]

[0061] Substituting equations (18)-(24) into equation (17) can derive

[0062]

[0063] Therefore, the strain energy of the transition segment can be expressed as

[0064]

[0065] Substituting equations (20) and (25) into equation (26) and using the principle of minimum strain energy can obtain the length L1 of the transition segment.

[0066]

[0067] Obviously, equation (27) is an implicit function, which can be solved numerically by the Newton iteration method. Substituting the obtained L1 into equation (26) can obtain the strain energy of the transition segment.

[0068] When the deployable composite stretchable arm is folded to a certain extent, the free segment disappears, and the folding process becomes the second stage, which is divided into a crimping segment and a transition segment, as shown in Figure 7 . The critical polar angle of the first stage and the second stage can be derived as

[0069]

[0070] Similarly, equation (28) is an implicit function, which is solved numerically by the Newton iteration method to obtain the critical polar angle a c .

[0071] Therefore, the increment of strain energy stored by the deployable composite stretchable arm in the first stage at any folded state relative to the initial state can be expressed as

[0072]

[0073] By Newton-Leibnitz formula, the folding moment of the deployable composite stretchable arm in the first stage can be derived as

[0074]

[0075] In the second stage, the length of the transition section gradually decreases as the folding process proceeds. The reduced length of the transition section can be expressed as

[0076]

[0077] The strain energy stored by the reduced length of the transition section can be derived as

[0078]

[0079] Therefore, the increment of strain energy stored by the deployable composite stretchable arm in the second stage relative to the initial state of the second stage can be expressed as

[0080]

[0081] By Newton-Leibnitz formula, the folding moment of the deployable composite stretchable arm in the second stage can be derived as

[0082]

[0083] In summary, the folding moment of the deployable composite stretchable arm in the entire folding process can be expressed as

[0084]

[0085] Step three, according to Tsai-Hill criterion and maximum stress criterion, the Tsai-Hill criterion failure coefficient expression and the maximum stress criterion failure coefficient expression are derived.

[0086] According to the basic assumption (2), the radius of curvature in the x direction of the deployable composite stretchable arm gradually increases and the curvature gradually decreases in the forward and reverse folding processes, so the deployable composite stretchable arm has the maximum strain and stress in the x direction at the starting position. The maximum value of the curvature change in the x direction of the deployable composite stretchable arm in the forward and reverse folding processes is

[0087]

[0088] The maximum value of the curvature change in the y direction during folding of the deployable composite stretchable arm is

[0089]

[0090] The stress-strain relationship of the kth layer of the dangerous point in the laminate is

[0091]

[0092] Substituting equations (36) and (37) into equation (38) and simplifying, we obtain

[0093]

[0094]

[0095]

[0096] According to the stress component coordinate transformation equation of the kth layer in the laminate, the maximum principal direction stress of the kth layer in the laminate can be expressed as

[0097]

[0098] Substituting equations (39) to (41) into equation (42), we obtain

[0099]

[0100]

[0101]

[0102] The laminate failure criteria can be divided into interaction theory and limit theory. In the interaction theory, such as Tsai-Hill criterion, the interaction between stress components is considered, usually by considering the contribution of each component to the total strain energy in the structure. In the limit theory, such as the maximum stress failure theory, the local stress and local strain components are compared with their corresponding strengths, ignoring the interaction between the components. Based on the above two failure criteria, the failure coefficient is used to analyze the stress level of the deployable composite stretchable arm during forward folding and reverse folding. When the failure coefficient I f reaches or exceeds 1, the deployable composite stretchable arm fails, otherwise, there is no failure.

[0103] The Tsai-Hill failure coefficient I f,1 can be calculated by the following formula

[0104]

[0105] wherein,

[0106]

[0107]

[0108]

[0109] where X t and X c are the longitudinal tensile and compressive strengths of the composite material, X1 and X2 are the longitudinal strengths of the composite material, Y t and Y c are the transverse tensile and compressive strengths of the composite material, Y is the transverse strength of the composite material. When the failure coefficient reaches or exceeds 1, the deployable composite cabin section fails, otherwise, it does not fail.

[0110] Maximum stress criterion failure coefficient I f,2 can be calculated by the following formula

[0111]

[0112] The present application is a method for quickly predicting the folding performance of a deployable composite stretchable arm, geometric equations of a curling section and a transition section of the deployable composite stretchable arm in a folding process are established, an analytical model for predicting a folding torque and a torsion displacement curve of the deployable composite stretchable arm is derived, and the maximum failure coefficient of the deployable composite stretchable arm in the folding process is analyzed. With the aid of the analytical model proposed in the present application and the folding performance of the deployable composite stretchable arm with different geometric parameters, it is proved that the geometric parameters are the key factors affecting the folding performance of the deployable composite stretchable arm. BRIEF DESCRIPTION OF DRAWINGS

[0113] Figure 1 It is a geometric configuration schematic diagram of the deployable composite stretchable arm.

[0114] Figure 2 It is a geometric configuration schematic diagram of the deployable composite stretchable arm in the first stage and the second stage.

[0115] Figure 3 It is a geometric configuration schematic diagram of the deployable composite stretchable arm in a completely folded state.

[0116] Figure 4 It is a force analysis schematic diagram of the deployable composite stretchable arm in the first stage.

[0117] Figure 5 It is a force analysis schematic diagram of the deployable composite stretchable arm in the second stage.

[0118] Figure 6Fig. 1 is a schematic diagram of a stress analysis of a micro-element of a transition section of a deployable composite stretchable arm.

[0119] Figure 7 Fig. 2 is a schematic diagram of a folding deformation process of a deployable composite stretchable arm.

[0120] Fig. 3 is a symbol explanation of the figures.

[0121] Figure 1 In the figure, L0 is the length of the deployable composite stretchable arm, t is the thickness of the deployable composite stretchable arm, R is the cross-sectional radius of the deployable composite stretchable arm, θ is the central angle of the deployable composite stretchable arm, x, y and z are the coordinate axes of the rectangular coordinate system, and 1 is the deployable composite stretchable arm.

[0122] Figure 2 In the figure, 2 is a free section, 3 is a transition section, and 4 is a coiled section.

[0123] Figure 3 In the figure, r0 and r1 are the polar radii of the starting point and the ending point of the fully coiled state of the deployable composite stretchable arm in the folded state, respectively, ρ1 is the polar radius of the fully coiled state of the deployable composite stretchable arm in the folded state, and u and v are the coordinate axes of the rectangular coordinate system.

[0124] Figure 4 In the figure, M x and M y are the bending moments of the deployable composite stretchable arm in the x and y directions in the first stage, respectively, κ x is the curvature of the deployable composite stretchable arm in the x direction, κ y is the curvature of the deployable composite stretchable arm in the y direction.

[0125] Figure 6 In the figure, and are the curvatures of the four boundaries of the transition section of the deployable composite stretchable arm in the y direction, x1, y1 and z1 are the coordinate axes of the rectangular coordinate system, and are the force and bending moment received by an arbitrary micro-element of the transition section of the deployable composite stretchable arm.

[0126] Figure 7 In the figure, L1 is the length of the transition section, and L2 is the length reduced in the second stage. DETAILED DESCRIPTION

[0127] Step one, define the geometry and size of the deployable composite stretchable arm, and determine the mathematical expressions of the relationships between the geometric parameters.

[0128] There are two types of folding modes for the deployable composite stretchable arms, which are called forward folding and reverse folding. The geometry of the deployable composite stretchable arm in the initial state is determined by the length L, the thickness t, the cross-sectional radius R and the central angle θ, as shown in FIG. 1. In order to characterize the geometric model of the deployable composite stretchable arm in the process of forward folding and reverse folding deformation, the following basic assumptions are made in this paper: Figure 1

[0129] (1) The entire folding process of the deployable composite stretchable arm is divided into two stages, the first stage includes the winding section, the transition section and the free section. When the deployable composite stretchable arm is folded to a certain extent, the free section disappears, and the folding process enters the second stage, which only has the winding section and the transition section, as shown in FIG. 2. Figure 2

[0130] (2) The center line shape of the longitudinal section of the deployable composite stretchable arm in the winding state is an Archimedes spiral, as shown in FIG. 3. Figure 3

[0131] (3) The thickness change of the deployable composite stretchable arm in the folding deformation process is ignored, so the overall deformation can be described by the shape and curvature radius of the neutral surface, and the neutral surface is not stretched.

[0132] According to the basic assumptions (2) and Figure 3 , the winding deformation of the folded state of the deployable composite stretchable arm can be described by a polynomial shape function in polar coordinates (ρ1, α)

[0133] ρ1=aα+b, α∈α0,α1 (1)

[0134] According to the geometric relationship, the boundary conditions that formula (1) needs to meet are

[0135]

[0136] Where r0 and r1 are the polar radii of the starting point and the ending point of the complete winding state of the deployable composite stretchable arm in the folded state, respectively.

[0137] Usually, α0=0, then formula (2) can be simplified as

[0138]

[0139] According to the basic assumption (3), it can be obtained that

[0140]

[0141] ​​​Substitute equation (1) - (3) into equation (4) and integrate, we get

[0142]

[0143] Obviously, equation (5) is an implicit function, which can be solved by Newton iteration method.

[0144] According to the definition of the radius of curvature, the curvature in x direction during the rolling process of the deployable composite stretchable arm can be expressed as

[0145]

[0146] where, ρ2 is the radius of curvature in x direction during the rolling process of the deployable composite stretchable arm.

[0147] Step two, according to the classical laminated plate theory and energy method, determine the folding torque required for the deployable composite stretchable arm during the forward and reverse folding processes.

[0148] In order to establish the strain energy generated by the deployable composite stretchable arm during the forward and reverse folding processes, the forward and reverse folding processes are divided into two steps, as shown in Figure 4 and 5 First, apply bending moment M y to the edge, which changes the curvature of the deployable composite stretchable arm in y direction from 1 / R to 0, and in this step, the external work is converted into bending strain energy; then, apply bending moment M x to the two ends of the deployable composite stretchable arm, which changes the curvature of the deployable composite stretchable arm in x direction from 0 to κ x , and in this step, the external work is converted into bending strain energy.

[0149] Using the ABD matrix in the classical laminated plate, the elastic deformation of the deployable composite stretchable arm during the folding process can be expressed as

[0150]

[0151] The deployable composite stretchable arm usually chooses symmetric laminated plate, B = 0, and due to the restriction of the constraint mechanism, the torsional deformation κ xy can be ignored, so equation (7) can be simplified as

[0152]

[0153] The strain energy generated by the unit length of the deployable composite stretchable arm from the free state to the rolled state can be expressed as

[0154]

[0155] where R and θ are the radius and the central angle of the cross section of the deployable composite stretchable arm, respectively. x and Δκ y can be derived as

[0156]

[0157] where Δκ x is the difference between the initial curvature and the curvature after folding. The value of Δκ x depends on the folding direction of the deployable composite stretchable arm. When folding in the positive direction, Δκ x takes a positive value, and when folding in the negative direction, Δκ Figure 4 and 5 takes a negative value, as shown in

[0158] Substituting equation (10) into equation (9), we have

[0159]

[0160] In the actual folding process of the deployable composite stretchable arm, the first step is to flatten one end, and the second step is to complete the positive and negative folding process through the reel. Therefore, there is a transition section between the winding section and the free section. For any microelement obtained from the transition section, there is an in-moment and an out-of-plane shear force on the edge, as shown in Figure 6 x1, y1, and z1 are the local coordinate system located on the center line of the transition section. The balance equation satisfied by the resultant force and the resultant moment on the arbitrary microelement of the transition section is

[0161]

[0162] The main deformation of the deployable composite stretchable arm is the curvature change caused by bending, and the influence of in-plane force on normal force is small, so the influence of in-plane force on the normal force balance equation is ignored. Therefore, equation (12) can be simplified as

[0163]

[0164] According to the classical laminated plate theory, the internal moment and the curvature change have the following relationship:

[0165]

[0166] and

[0167]

[0168] where and are the initial curvatures of the deployable composite stretchable arm in the free state.

[0169]

[0170] The For 0, the deformation compatibility equation satisfied between the bending curvature and the twisting curvature is

[0171]

[0172] Substituting equations (14)-(17) into equation (13) and simplifying, we have

[0173]

[0174] Obviously, equation (18) is a second-order partial differential equation, and the boundary conditions satisfied are

[0175]

[0176] where L1 is the length of the transition section, as shown in Figure 7

[0177] Using the method of separation of variables, an approximate solution of the second-order partial differential equation satisfying the boundary conditions is obtained as

[0178]

[0179] where H1, H2, H3, and H4 are intermediate variables, respectively

[0180]

[0181]

[0182]

[0183]

[0184] Substituting equations (18)-(24) into equation (17), we can derive

[0185]

[0186] Therefore, the strain energy of the transition section can be expressed as

[0187]

[0188] Substituting equations (20) and (25) into equation (26) and using the principle of minimum strain energy, we can obtain the length L1 of the transition section.

[0189]

[0190] Obviously, equation (27) is an implicit function, which can be numerically solved by the Newton iteration method. Substituting the obtained L1 into equation (26), we can obtain the strain energy generated by the transition section.

[0191] ​When the deployable composite stretchable arm is folded to a certain extent, the free segment disappears, and the folding process becomes the second stage, which is divided into a crimping segment and a transition segment, as shown in FIG. 8. Figure 7 The critical polar angle of the first stage and the second stage can be derived as

[0192]

[0193] Similarly, formula (28) is an implicit function, and the numerical solution of the critical polar angle a c .

[0194] Therefore, the strain energy increment stored by the deployable composite stretchable arm in the first stage at any folding state relative to the initial state can be represented as

[0195]

[0196] By the Newton-Leibniz formula, the folding torque of the deployable composite stretchable arm in the first stage can be derived as

[0197]

[0198] In the second stage, the length of the transition segment gradually decreases as the folding process proceeds. The length of the transition segment that decreases can be represented as

[0199]

[0200] The strain energy stored by the part of the transition segment that decreases can be derived as

[0201]

[0202] Therefore, the strain energy increment stored by the deployable composite stretchable arm in the second stage relative to the initial state of the second stage can be represented as

[0203]

[0204] By the Newton-Leibniz formula, the folding torque of the deployable composite stretchable arm in the second stage can be derived as

[0205]

[0206] In summary, the folding torque of the deployable composite stretchable arm in the entire folding process can be represented as

[0207]

[0208] Step three, according to the Tsai-Hill criterion and the maximum stress criterion, the Tsai-Hill criterion failure coefficient expression and the maximum stress criterion failure coefficient expression are derived.

[0209] According to the basic assumption (2), the radius of curvature of the deployable composite stretchable arm in the x direction gradually increases and the curvature gradually decreases during the forward and reverse folding processes, so the deployable composite stretchable arm has the maximum strain and stress in the x direction at the starting position. The maximum value of the change in the curvature of the deployable composite stretchable arm in the x direction during the folding process is

[0210]

[0211] The maximum value of the change in the curvature of the deployable composite stretchable arm in the y direction during the folding process is

[0212]

[0213] The stress-strain relationship of the kth layer of the dangerous point in the laminate is

[0214]

[0215] Substituting equations (36) and (37) into equation (38) and simplifying can obtain

[0216]

[0217]

[0218] According to the stress component coordinate transformation equation of the kth layer in the laminate, the maximum principal direction stress of the kth layer in the laminate can be expressed as

[0219]

[0220] Substituting equations (39) to (41) into equation (42) can obtain

[0221]

[0222]

[0223]

[0224] The laminate failure criterion can be divided into interaction theory and limit theory. In the interaction theory, for example, Tsai-Hill criterion, the interaction between stress components is considered, usually by considering the contribution of each component to the total strain energy in the structure. In the limit theory, for example, the maximum stress failure theory, the local stress and local strain components are compared with their corresponding strengths, ignoring the interaction between the components. Based on the above two failure criteria, the failure coefficient is used to analyze the stress level of the deployable composite stretchable arm during the forward folding and reverse folding processes. When the failure coefficient I fWhen the failure coefficient reaches or exceeds 1, the deployable composite stretchable arm fails, otherwise, it does not fail.

[0225] Tsai-Hill failure coefficient I f,1 The Tsai-Hill failure coefficient I can be calculated by the following formula

[0226]

[0227] Wherein,

[0228]

[0229]

[0230]

[0231] Wherein, X t and X c are the longitudinal tensile strength and compressive strength of the composite material respectively, X1 and X2 are the longitudinal strength of the composite material respectively, Y t and Y c are the transverse tensile strength and compressive strength of the composite material respectively, Y is the transverse strength of the composite material. When the failure coefficient reaches or exceeds 1, the deployable composite cabin section fails, otherwise, it does not fail.

[0232] Maximum stress criterion failure coefficient I f,2 The maximum stress criterion failure coefficient I can be calculated by the following formula

[0233]

[0234] The present application is a kind of fast prediction deployable composite stretchable arm folding performance method, the geometric equation of the deployable composite stretchable arm in the folding process is established, the analytical model for predicting the folding torque and torsion displacement curve of the deployable composite stretchable arm is deduced, and the maximum failure coefficient of the deployable composite stretchable arm in the folding process is analyzed. With the analytical model proposed in this paper, and the folding performance of the deployable composite stretchable arm with different geometric parameters, it is proved that the geometric parameter is the key factor affecting the folding performance of the deployable composite stretchable arm.

Claims

1. A method of predicting the folding performance of a deployable composite stretchable arm, characterized by: The method comprises the following steps: Step one, defining the geometry and size of the deployable composite stretchable arm, and determining the mathematical expression of the relationship between the geometric parameters; There are two types of folding modes for the deployable composite stretchable arm, when the coiling direction of the deployable composite stretchable arm faces the opening direction, it is called forward folding; when the coiling direction of the deployable composite stretchable arm faces away from the opening direction, it is called reverse folding; The geometric configuration of the deployable composite stretchable arm in the initial state is determined by the length L, the thickness t, the cross-sectional radius R and the central angle θ; in order to characterize the geometric model of the deployable composite stretchable arm in the process of forward folding and reverse folding deformation, the following basic assumptions are made: (1) the whole folding process of the deployable composite stretchable arm is divided into two stages, the first stage includes the curling section, the transition section and the free section; when the deployable composite stretchable arm is folded to a certain extent, the free section disappears, and the folding process enters the second stage, and the second stage only has the curling section and the transition section; (2) the center line shape of the deployable composite stretchable arm in the curling state is an Archimedes spiral; (3) the thickness change of the deployable composite stretchable arm in the folding deformation process is ignored, the overall deformation is described by the shape and curvature radius of the neutral surface, and the neutral surface is not stretched; According to the basic assumption (2), the curling deformation of the deployable composite stretchable arm in the folded state can be described by a polynomial shape function in polar coordinates (ρ1, α), that is ρ1=aα+b, (α∈[α0, α1]) (1) According to the geometric relationship, the boundary conditions that formula (1) needs to meet are Wherein, r0 and r1 are the polar radii of the starting point and the ending point of the completely curled state of the deployable composite stretchable arm in the folded state respectively; Usually, α0=0, then formula (2) is simplified as According to the basic assumption (3), it can be obtained Substitute formulas (1)-(3) into formula (4) and integrate to obtain Formula (5) is an implicit function, and the unknown quantity a is solved by Newton iteration method; According to the definition of the curvature radius, the curvature in the x direction of the deployable composite stretchable arm in the curling process is Wherein, ρ2 is the curvature radius of the deployable composite stretchable arm in the x direction in the curling process; Step two, according to the classical laminated plate theory and energy method, the folding torque required by the deployable composite stretchable arm in the process of forward folding and reverse folding is determined; To establish the strain energy generated by the deployable composite stretchable arm during the forward and reverse folding processes, the forward and reverse folding processes are divided into two steps; first, a bending moment M is applied at the edge y The curvature of the deployable composite stretchable arm in the y direction is changed from 1 / R to 0, and in this step the external work is converted into bending strain energy; then, a bending moment M is applied at both ends of the deployable composite stretchable arm x The curvature of the deployable composite stretchable arm in the x direction is changed from 0 to κ x In this step, the external work is converted into bending strain energy; Using the ABD matrix in the classical laminated plate, the elastic deformation of the deployable composite stretchable arm in the folding process is represented The deployable composite stretchable arms usually choose symmetric laminates, B = 0, and due to the restriction of the constraint mechanism, the torsional deformation κ xy is ignored, equation (7) is simplified as The strain energy generated by the unit length deployable composite stretchable arm from the free state to the curled state is represented as where R and θ are the radius and the central angle of the cross section of the deployable composite stretchable arm, respectively; Δκ x and Δκ y are derived as wherein the value of Δκ x depends on the folding direction of the deployable composite stretchable arm; when folding in the positive direction, Δκ x takes a positive value, when folding in the negative direction, Δκ x takes a negative value; Substitute formula (10) into formula (9) to obtain In the actual folding process of the deployable composite stretchable arm, the first step is to flatten one end, and the second step is to complete the forward folding and reverse folding process through a reel; there is a transition section between the curling section and the free section; for any microelement obtained from the transition section, there is an internal moment and an out-of-plane shear force on the edge; x1, y1 and z1 are local coordinate systems located on the center line of the transition section; the balance equation satisfied by the resultant force and the resultant moment on the transition section is The main deformation of the deployable composite stretchable arm is the curvature change caused by bending, and the influence of in-plane force on normal force is small, so the influence of in-plane force on normal force balance equation is ignored; formula (12) is simplified as According to the classical laminated plate theory, the internal moment and the curvature change exist the following relationship: and wherein, and is the initial curvature of the deployable composite stretchable arm in free state. The transition section For 0, the deformation compatibility equation satisfied between the bending curvature and the twisting curvature is Substitute formula (14)-(17) into formula (13) and simplify to obtain Formula (18) is a second-order partial differential equation, and the boundary conditions that satisfy are Wherein, L1 is the length of the transition section; The approximate solution of the second-order partial differential equation that satisfies the boundary conditions is obtained by using the separation of variables method, which is Wherein, H1, H2, H3 and H4 are intermediate variables, which are respectively Substitute formula (18)-(24) into formula (17) to derive Therefore, the strain energy of the transition section can be expressed as Substitute formula (20) and (25) into formula (26), and use the principle of minimum strain energy to obtain the length L1 of the transition section; Formula (27) is an implicit function, which is numerically solved by Newton iteration method; Substitute the obtained L1 into formula (26) to obtain the strain energy generated by the transition section; When the deployable composite stretchable arm is folded to a certain extent, the free section will disappear, and the folding process becomes the second stage, which is divided into the winding section and the transition section; The critical polar angle of the first stage and the second stage is derived as Similarly, equation (28) is an implicit function, which is solved numerically for the critical polar angle a by Newton's iteration method c ; The strain energy increment stored by the deployable composite stretchable arm in the first stage at any folding state relative to the initial state is expressed as Through the Newton-Leibniz formula, the folding moment of the deployable composite stretchable arm in the first stage is derived as In the second stage, with the progress of the folding process, the length of the transition section gradually decreases; The length of the transition section that is reduced is expressed as The strain energy stored by the part of the length of the transition section that is reduced is derived as The strain energy increment stored by the deployable composite stretchable arm in the second stage relative to the initial state of the second stage is expressed as Through the Newton-Leibniz formula, the folding moment of the deployable composite stretchable arm in the second stage is derived as The folding moment of the deployable composite stretchable arm in the whole folding process is expressed as Step three, according to Tsai-Hill criterion and maximum stress criterion, the expression of Tsai-Hill criterion failure coefficient and the expression of maximum stress criterion failure coefficient are derived; According to the basic assumption (2), the curvature radius of the deployable composite stretchable arm in the x direction gradually increases and the curvature gradually decreases in the forward and reverse folding processes, so the deployable composite stretchable arm has the maximum strain and stress in the x direction at the starting position; The maximum value of the curvature change of the deployable composite stretchable arm in the x direction in the forward and reverse folding processes is The maximum value of the curvature change of the deployable composite stretchable arm in the y direction in the folding process is The stress-strain relationship formula of the kth layer of the dangerous point in the laminated plate is Substitute formula (36) and (37) into formula (38) and simplify to obtain According to the stress component coordinate transformation equation of the kth layer in the laminated plate, the maximum principal direction stress of the kth layer in the laminated plate is expressed as Substitute formula (39) to formula (41) into formula (42) to obtain Based on the interaction theory and limit theory in the laminate failure criteria, the failure coefficient is used to analyze the stress level of deployable composite stretchable arms during the forward folding and reverse folding processes; when the failure coefficient I f reaches or exceeds 1, the deployable composite stretchable arms fail, otherwise, there is no failure; Tsai-Hill failure coefficient I f,1 may be calculated with the following equation Wherein, wherein X t and X c are the longitudinal tensile and compressive strength of the composite material, respectively, and X1 and X2 are the longitudinal strength of the composite material, respectively, Y t and Y c are the composite material tensile and compressive strengths in the cross direction, respectively, and Y is the composite material cross-direction strength; when the failure factor reaches or exceeds 1, the deployable composite material section fails, otherwise, it does not fail; Maximum stress criterion failure factor I f,2 is calculated using the following equation

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