Asymmetric flexible spacecraft equivalent dynamic modeling method considering thermal deformation

Abstract

The invention discloses an asymmetric flexible spacecraft equivalent dynamic modeling method considering thermal deformation. The method comprises the following steps: establishing an equivalent beam model for a space truss structure, and establishing a coordinate system; describing the displacement of any point on the periodic units of the truss structure; analyzing stress, strain and structural volume of the periodic unit along the neutral axis direction to obtain thermal deformation strain energy and axial strain energy of the periodic unit after thermal deformation; based on the total strain energy decomposition, obtaining an elastic matrix and a stress vector; superposing the kinetic energy of each rod piece in the periodic unit to obtain the total kinetic energy of the periodic unit, and sorting and decomposing an inertia matrix of the periodic unit; based on a finite element method, obtaining a mass element matrix, a linear stiffness element matrix and a geometric stiffness element matrix of each periodic element; and assembling to obtain a corresponding total mass matrix, a total linear stiffness matrix and an initial stress stiffness matrix so as to obtain an equivalent beam damping-free vibration equation. According to the method, the existing equivalent continuum modeling theory is perfected, and the precision of the equivalent model is improved.

Description

Technical Field

[0001] The present invention relates to a method for equivalent dynamic modeling of an asymmetric flexible spacecraft considering thermal deformation, belonging to the field of aerospace vibration control technology. Background Technique

[0002] For the large flexible space antenna truss structure with a radar array surface, establishing a dynamic model is of great significance for its on-orbit vibration control. At present, establishing an accurate and efficient equivalent model for dynamic analysis is one of the key technologies for the vibration controller of a large flexible space deployable antenna structure.

[0003] In the existing equivalent beam modeling method, the connection line of the centroids of the truss cross-sections is used as the neutral axis, and the strain energy of the whole truss is expressed by the strain components on the neutral axis, and it is considered that the equivalent beam coincides with the neutral axis of the truss structure. Since the neutral axis of a symmetric truss structure is the connection line of the centroids of the cross-sections, the above treatment method is applicable to symmetric structures. However, the determination of the position of the neutral axis of an asymmetric structure is more complicated and is no longer at the centroid. Therefore, the existing treatment method cannot establish an accurate equivalent dynamic model for an asymmetric truss structure. In addition, under the action of internal heat-generating components and solar radiation heat flux on the satellite antenna, the temperature distribution of the structure changes after heating, resulting in deformation. The existing equivalent modeling method does not consider the influence of the thermal deformation of the antenna on the dynamic characteristics. Summary of the Invention

[0004] The technical problem to be solved by the present invention is: to overcome the deficiencies of the prior art, and provide a method for equivalent dynamic modeling of an asymmetric flexible spacecraft considering thermal deformation, incorporating the deformation potential energy of the structure into the total energy of the structure, and based on the energy equivalence principle, establishing a more complete equivalent dynamic modeling method of a flexible spacecraft under the influence of thermal loads to achieve rapid and accurate solution of the dynamic response of the spacecraft.

[0005] The technical solution of the present invention is: a method for equivalent dynamic modeling of an asymmetric flexible spacecraft considering thermal deformation, the asymmetric flexible spacecraft being a space truss structure including longitudinal beams, cross beams, stay cables, and a bottom plate, the method comprising:

[0006] Establish an equivalent beam model for the space truss structure, determine the position of the neutral axis of a periodic unit in the equivalent beam model, and establish a coordinate system;

[0007] Describe the displacement of any point on the periodic unit of the truss structure in the established coordinate system;

[0008] Analyze the stress, strain, and structure volume of the periodic unit of the space truss structure along the neutral axis direction to obtain the thermal deformation strain energy and axial strain energy after thermal deformation of the periodic unit;

[0009] The total strain energy after thermal deformation of the periodic unit is obtained from the thermal deformation strain energy and the axial strain energy. After sorting it out, the elastic matrix and the stress matrix are decomposed;

[0010] The velocities of both ends of each rod in the periodic unit are obtained by differentiating the displacements, and based on this, the kinetic energy of each rod in the periodic unit is obtained. After superposition, the total kinetic energy of the periodic unit is obtained. After sorting it out, the inertia matrix of the periodic unit is decomposed;

[0011] Based on the finite element method, the equivalent beam model is divided by the length of the periodic unit to obtain the mass element matrix, the linear stiffness element matrix, and the geometric stiffness element matrix of each periodic unit;

[0012] The mass element matrix, the linear stiffness element matrix, and the geometric stiffness element matrix are assembled respectively to obtain the corresponding overall mass matrix, overall linear stiffness matrix, and overall geometric stiffness matrix; furthermore, the undamped vibration equation of the equivalent beam under the initial thermal stress is obtained, and based on this, the dynamic response problem of the space truss structure after deformation under the influence of thermal load can be calculated.

[0013] Preferably, the position of the neutral axis of a periodic unit is represented by the distance from the bottom surface of the space truss:

[0014]

[0015] In the formula, M T is the total number of rods in the periodic unit, k is the serial number index of each rod in the periodic unit; E (k) , S (k) and A (k) are the elastic modulus, static moment, and cross-sectional area of the kth component respectively; z c is the distance from the neutral axis of the overall structure to the bottom surface of the space truss structure, and z ck is the distance from the centroid of the kth component to the bottom surface of the space truss structure.

[0016] Preferably, when establishing the coordinate system, the direction perpendicular to the bottom surface of the space truss structure is defined as the Z-axis direction, the X-axis direction is parallel to the longitudinal beam of the space truss, and the Y-axis is perpendicular to the plane formed by the Z-axis and the X-axis; the origin position is determined according to the actual situation.

[0017] Preferably, the displacement of any point on the periodic unit of the truss structure is described in the established coordinate system, specifically:

[0018]

[0019] In the formula, the superscript c represents the variable corresponding to the origin of the coordinate system. The components in (x, y, z) respectively represent the coordinates of a point on the periodic unit in the X, Y, and Z directions. u(x, y, z) represents the displacement in the X direction, v(x, y, z) represents the displacement in the Y direction, w(x, y, z) represents the displacement in the Z direction, and φ x (x, y, z) represents the rotation angle about the X-axis, and φ y (x, y, z) represents the rotation angle about the Y-axis, and φ z (x, y, z) represents the rotation angle about the Z-axis, and u c represents the displacement in the X direction at the origin of the periodic unit, and v c represents the displacement in the Y direction at the origin, and w c represents the displacement in the Z direction at the origin, represents the rotation angle about the X-axis at the origin of the periodic unit, represents the rotation angle about the Y-axis at the origin of the periodic unit, represents the rotation angle about the Z-axis at the origin of the periodic unit, represents the strain along the X-axis at the origin, represents the curvature in the Y direction at the origin, represents the curvature in the Z direction at the origin, represents the shear strain in the XY plane at the origin, represents the strain along the Y-axis at the origin, represents the curvature in the X direction at the origin, represents the shear strain in the YZ plane at the origin, represents the shear strain in the XZ plane at the origin, represents the strain along the Z-axis at the origin.

[0020] Preferably, after the thermal deformation of the periodic unit of the space truss structure, the thermal deformation strain energy U of the longitudinal beam T is:

[0021]

[0022] In the formula, the superscript (k) represents the member number, and M z is the number of longitudinal beams, is the axial strain of the member, and A (k) represents the cross-sectional area of the member, is along the axis direction of member k; σ T is the thermal stress of the longitudinal beam of the space truss structure along the neutral axis direction;

[0023] The elastic strain energy U of the periodic unit of the space truss structure after thermal deformation p is:

[0024]

[0025] In the formula, E(k) is the elastic modulus of the rod k.

[0026] Preferably, the total strain energy U after thermal deformation of the periodic unit is U = U T +U p , which is arranged as:

[0027]

[0028] In the formula, l c represents the length of the equivalent beam model of the periodic unit, and Γ is the strain component on the neutral axis:

[0029]

[0030] D is the elastic matrix, and its non-zero elements are:

[0031]

[0032] Among them, A1 represents the cross-sectional area of the cross beam, A2 represents the cross-sectional area of the longitudinal beam, A3 represents the cross-sectional area of the stay cable, E1 represents the elastic modulus of the cross beam, E2 represents the elastic modulus of the longitudinal beam, E3 represents the elastic modulus of the stay cable, l1 represents the length of the cross beam, l2 represents the length of the longitudinal beam, and l3 represents the length of the stay cable;

[0033] S is the stress matrix, and its non-zero elements are:

[0034]

[0035] In the formula, represents the axial thermal stress of the rod k, and the specific expression is:

[0036]

[0037] In the formula, E (k) is the elastic modulus of the rod k, is the thermal expansion coefficient of the rod k, T (k ) is the current temperature of the rod k, is the initial temperature of the rod k.

[0038] Preferably, the total kinetic energy T of the periodic unit p is:

[0039]

[0040] In the formula, M T is the total number of rods in the periodic unit, M B represents the number of bottom plates in the periodic unit, T (k) represents the kinetic energy of the rod k:

[0041]

[0042] Denote the bottom plate as k b The kinetic energy of which is specifically:

[0043]

[0044] where the subscripts 1 and 2 are the two endpoints of the rod, and ρ (k) denotes the density of the rod k, l (k) denotes the length of the rod k, A (k) denotes the area of the rod k, V x,1 and V x,2 respectively denote the X-direction velocities of the two endpoints of the rod k, V y,1 and V y,2 respectively denote the Y-direction velocities of the two endpoints of the rod k, V z,1 and V z,2 respectively denote the Z-direction velocities of the two endpoints of the rod k; m b is the mass of the bottom plate, is the X-direction velocity of the i-th vertex among the four vertices of the bottom plate, is the Y-direction velocity of the i-th vertex among the four vertices of the bottom plate, is the Z-direction velocity of the i-th vertex among the four vertices of the bottom plate.

[0045] Preferably, after organizing the total kinetic energy T p of the periodic unit, the inertia matrix of the periodic unit is decomposed out Specifically:

[0046]

[0047] where υ T is the transpose of υ:

[0048]

[0049] denotes the rotation angle of the periodic unit about the X-axis at the origin, denotes the rotation angle of the periodic unit about the Y-axis at the origin, denotes the rotation angle of the periodic unit about the Z-axis at the origin, u c denotes the X-direction displacement of the periodic unit at the origin, v c denotes the Y-direction displacement at the origin, w c denotes the Z-direction displacement at the origin;

[0050] is the inertia matrix, and its non-zero elements are:

[0051]

[0052] Where: A1 represents the cross-sectional area of the cross beam, A2 represents the cross-sectional area of the longitudinal beam, A3 represents the cross-sectional area of the stay cable, m b is the mass of the bottom plate, l1 represents the length of the cross beam, l2 represents the length of the longitudinal beam, l3 represents the length of the stay cable; ρ1 represents the density of the cross beam, ρ2 represents the density of the longitudinal beam, ρ3 represents the density of the stay cable; l c represents the length of the equivalent beam model of the periodic unit.

[0053] Preferably, based on the finite element method, when dividing the equivalent beam model by the length of the periodic unit, a 2-node 8-degree-of-freedom element is used to simulate the structural deformation of the equivalent beam model, and the mass element matrix M e of each periodic unit, the linear stiffness element matrix and the geometric stiffness element matrix are respectively obtained by using the elastic matrix, stress vector and inertia matrix of the periodic unit:

[0054]

[0055] Where: l e represents the length of the periodic unit, l e =l c , D is the elastic matrix, S is the stress matrix, L and H are differential operators, and N is the parameter matrix when the displacement u at any point x in the periodic unit is expressed by the nodal displacement u e in the periodic unit; specifically:

[0056]

[0057] N1 = 1 - ξ, N2 = ξ, N3 = 1 - 3ξ 2 + 2ξ 3 , N4 = (ξ - 2ξ 2 + ξ 3 )l e , N5 = 3ξ 2 - 2ξ 3 , N6 = (ξ 3 - ξ 2 )l e

[0058] l e is the beam element length, ξ = x / l e , x ∈ (0, l e ).

[0059] Preferably, when assembling the mass element matrix, the linear stiffness element matrix and the geometric stiffness element matrix, the left-end fixed boundary condition is considered to obtain the corresponding overall mass matrix M, overall linear stiffness matrix K L and overall geometric stiffness matrix K S ;

[0060] The undamped vibration equation of the equivalent beam under initial thermal stress is as follows:

[0061]

[0062] w is the vector composed of the displacements of all nodes in the equivalent beam model, and f is the external load.

[0063] The present invention has the following advantages compared with the prior art:

[0064] (1) The present invention preprocesses the asymmetric antenna structure to determine the position of the neutral axis, enabling the equivalent dynamic modeling of the asymmetric structure. In addition, the influence of structural thermal deformation on the dynamic response is considered as an initial stress and introduced into the equivalent dynamic equation, realizing the construction of an equivalent dynamic model after the structure is thermally deformed, and being able to more realistically simulate the influence of the thermal load during the on-orbit operation of the spacecraft.

[0065] (2) The present invention simplifies the asymmetric satellite antenna with thermal deformation into a beam model and establishes a low-dimensional dynamic equation considering the influence of thermal stress, which not only simulates the influence of thermal deformation on the dynamic response but also greatly reduces the solution time of the dynamic equation, realizing the efficient and accurate dynamic solution of the spacecraft under the influence of thermal deformation. Description of the Drawings

[0066] Figure 1 It is a schematic diagram of the space truss structure described in the present invention;

[0067] Figure 2 It is a schematic cross-sectional view of the space truss structure described in the present invention;

[0068] Figure 3 It is a schematic diagram of a periodic unit of the equivalent beam model described in the present invention;

[0069] Figure 4 It is a schematic diagram of the finite element equivalence described in the present invention;

[0070] Figure 5 It is a flowchart of the method of the present invention. Detailed Embodiments

[0071] The satellite antenna truss with a bottom plate is an asymmetric structure, and the existing methods cannot establish a relatively accurate equivalent model. To make up for the deficiencies of the existing equivalent methods, the present invention provides a method for equivalent dynamic modeling of an asymmetric flexible spacecraft, further improving the accuracy of the equivalent model and having high computational efficiency, which can be used for the design of the on-orbit vibration control system of the spacecraft. The asymmetric flexible spacecraft is a space truss structure, including longitudinal beams, cross beams, stay cables, and a bottom plate.

[0072] To achieve the above objectives, the technical solution adopted by the present invention is as follows: an equivalent dynamic modeling method for an asymmetric flexible spacecraft considering thermal deformation, as Figure 5 shown, the steps are as follows:

[0073] Step 1: Determine the position of the neutral axis of the space truss structure and establish a coordinate system

[0074] As Figure 1 shown, the satellite antenna truss with a bottom plate can be considered as a composite beam structure, including longitudinal beams, cross beams, stay cables, and a bottom plate, and different components are made of different materials. Figure 2 Given the schematic diagram of the cross-section of the truss part, it can be seen that the truss cross-section changes along the axial direction. The equivalent cross-section method can be used to determine the position of the neutral axis. This method regards the composite beam as an overall structure. The total static moment of the structure is the sum of the static moments of each component multiplied by the corresponding weights, and the total area is the sum of the areas of each component multiplied by the corresponding weights, where the weight is the elastic modulus of each component. Therefore, the calculation formula for the position of the neutral axis of the overall structure is as follows:

[0075]

[0076] In the formula, M T is the total number of members in a periodic unit of the space truss structure, k is the serial number index of each member in a periodic unit of the space truss structure; E (k) , S (k) and A (k) are the elastic modulus, static moment, and cross-sectional area of the k-th component respectively; z c is the distance from the neutral axis of the overall structure to the bottom surface of the space truss structure, and z ck is the distance from the centroid of the k-th component to the bottom surface of the space truss structure.

[0077] When establishing the coordinate system, define the direction perpendicular to the bottom surface of the space truss structure as the Z-axis direction. At the same time, since the space truss structure is only asymmetric in the Z-axis direction, the X-axis direction is parallel to the longitudinal beam of the space truss, and the Y-axis is perpendicular to the plane formed by the Z-axis and the X-axis; the origin position is not specified and is determined according to the actual situation.

[0078] Establish a rectangular coordinate system based on the neutral axis, where the x-axis coincides with the neutral axis and the coordinate origin is located at the end of the truss.

[0079] Step 2: Establish the displacement components of any point on the space truss structure

[0080] Based on the neutral axis obtained in the previous step and based on the space beam theory, describe the displacement of any point (x, y, z) on the truss periodic unit (as Figure 3 shown) on the established coordinate system (using the displacement and strain components on the neutral axis):

[0081]

[0082] In the formula, the superscript c represents the variable corresponding to the origin of the coordinate system. The components in (x, y, z) respectively represent the coordinates of a point on the structure in the X, Y, and Z directions. u(x, y, z) represents the displacement in the X direction, v(x, y, z) represents the displacement in the Y direction, w(x, y, z) represents the displacement in the Z direction, and φ x (x, y, z) represents the rotation angle about the X-axis, and φ y (x, y, z) represents the rotation angle about the Y-axis, and φ z (x, y, z) represents the rotation angle about the Z-axis, and u c represents the displacement of the structure (periodic unit) in the X direction at the origin, and v c represents the displacement in the Y direction at the origin, and w c represents the displacement in the Z direction at the origin. represents the rotation angle about the X-axis of the structure (periodic unit) at the origin. represents the rotation angle about the Y-axis of the structure (periodic unit) at the origin. represents the rotation angle about the Z-axis of the structure (periodic unit) at the origin. represents the strain along the X-axis at the origin. represents the curvature in the Y direction at the origin. represents the curvature in the Z direction at the origin. represents the shear strain in the XY plane at the origin. represents the strain along the Y-axis at the origin. represents the curvature in the X direction at the origin. represents the shear strain in the YZ plane at the origin. represents the shear strain in the XZ plane at the origin. represents the strain along the Z-axis at the origin.

[0083] Step 3: Determine the strain energy induced by the thermal load

[0084] Under the action of the thermal load, thermal stresses will be generated in the longitudinal and cross beams of the satellite antenna. Since the cross-sectional area of the cross beam is small, the structural deformation is not obvious, and the deformation strain energy caused by heat in the cross beam can be ignored. However, the dimension of the antenna along the longitudinal beam direction is large, and the thermal expansion deformation is relatively large and cannot be ignored. For the thermal stress σ T along the neutral axis of the longitudinal beam is:

[0085] σ T = Eα T (T - T0)(2)

[0086] In the formula, E is the elastic modulus, α T is the coefficient of thermal expansion, T0 is the initial temperature, and T is the temperature of the structure after heating.

[0087] The thermal deformation strain energy U of the longitudinal beam in the antenna T can be obtained by calculating the stress-strain and the structural volume

[0088]

[0089] In the formula, the superscript (k) represents the member number, and M z is the number of longitudinal beams, is the axial strain of the member, and A (k) represents the cross-sectional area of the member, is along the axis direction of member k.

[0090] Step 4: Determine the elastic matrix of the equivalent beam model

[0091] Since the elastic modulus of the bottom plate is large, only the kinetic energy of the bottom plate can be considered, and the strain energy generated by the deformation of the bottom plate can be ignored. Therefore, the elastic strain energy U of the periodic unit p is composed of the axial strain energies of the longitudinal beams, cross beams, and stay cables, specifically as follows:

[0092]

[0093] Among them, the superscript (k) represents the member number, and E (k) is the elastic modulus of member k, and A (k) is the cross-sectional area of member k, is the axial strain of member k.

[0094] Consider the total strain energy after thermal deformation of the periodic unit:

[0095] U = U T + U p (5)

[0096] The matrix expression obtained after reorganizing formula (5) is:

[0097]

[0098] In the formula, l c represents the length of the equivalent beam model, and the strain component Γ on the neutral axis is:

[0099]

[0100] And, D is the elastic matrix, S is the stress matrix, and the non-zero elements are:

[0101]

[0102] Wherein, A1 represents the cross-sectional area of the cross beam, A2 represents the cross-sectional area of the longitudinal beam, A3 represents the cross-sectional area of the stay cable, E1 represents the elastic modulus of the cross beam, E2 represents the elastic modulus of the longitudinal beam, E3 represents the elastic modulus of the stay cable, l1 represents the length of the cross beam, l2 represents the length of the longitudinal beam, and l3 represents the length of the stay cable; represents the axial thermal stress of member k.

[0103] represents the axial thermal stress of member k, and the specific expression is:

[0104]

[0105] In the formula, E (k) is the elastic modulus of member k, is the coefficient of thermal expansion of member k, T (k) is the current temperature of member k, is the initial temperature of member k.

[0106] Step 5: Determine the inertia matrix of the equivalent beam model

[0107] The kinetic energy of each member includes translational and rotational kinetic energies, and the kinetic energy is expressed by the velocities at both ends of the member as:

[0108]

[0109] The kinetic energy of each base plate can be expressed as:

[0110]

[0111] In the formula, the subscripts 1 and 2 are the two ends of the member. ρ (k) represents the density of member k, l (k) represents the length of member k, A (k) represents the area of member k, V x,1 、V x,2 respectively represent the velocities in the X direction at both ends of member k, V y,1 、V y,2 respectively represent the velocities in the Y direction at both ends of member k, V z,1 、V z,2 respectively represent the velocities in the Z direction at both ends of member k; m b is the mass of the base plate, is the velocity in the X direction of the i-th vertex among the four vertices of the base plate, is the velocity in the Y direction of the i-th vertex among the four vertices of the base plate, is the velocity in the Z direction of the i-th vertex among the four vertices of the base plate.

[0112] The expression (7) of the structural velocity is obtained by taking the derivative of formula (1) with respect to time, and the total kinetic energy T of the periodic unit is obtained by substituting it into the following formula p :

[0113]

[0114] In the formula, is the result of sorting , where:

[0115]

[0116] And, is the inertia matrix (a 6×6 matrix), and the non-zero elements are

[0117]

[0118] In the formula: A1 represents the cross-sectional area of the cross beam, A2 represents the cross-sectional area of the longitudinal beam, A3 represents the cross-sectional area of the stay cable, m b is the mass of the bottom plate, l1 represents the length of the cross beam, l2 represents the length of the longitudinal beam, l3 represents the length of the stay cable; ρ1 represents the density of the cross beam, ρ2 represents the density of the longitudinal beam, ρ3 represents the density of the stay cable; l c represents the length of the equivalent beam model of the periodic unit.

[0119] Step 6: Finite element modeling

[0120] Based on the finite element method, the equivalent beam model is meshed with the length of the truss periodic structure as the mesh size, and a 2-node 8-degree-of-freedom element is used to simulate the structural deformation. The element mass matrix, element linear stiffness matrix, and element initial stress stiffness matrix are derived according to the inertia matrix, elastic matrix, and stress vector, and the element matrices are assembled to generate the global mass matrix M, linear stiffness matrix K L and initial stress stiffness matrix K σ . Thus, the dynamic equation of the equivalent beam structure with initial thermal stress can be written as where w is the displacement vector of all nodes, and f is the external force applied to the structure.

[0121] Specifically, according to Step 2, the external dimension of the equivalent beam model can be obtained, and the equivalent beam is meshed with a 2-node 8-degree-of-freedom element. The finite element model is as Figure 4 shown.

[0122] The displacement vector of the element nodes is:

[0123]

[0124] In the formula, the subscripts 1 and 2 represent the two end nodes of the element, the subscripts x, y, and z represent the displacements corresponding to the three directions respectively, and the superscripts b and s represent the displacements caused by bending and shear. Each node will generate displacements in the translational and rotational directions, and there are many forms of expression for the node displacement vector. Formula (11) is one of them.

[0125] The displacement u at any point x within the periodic unit can be expressed through the element nodal displacements u e as follows:

[0126] u = Nu e (12)

[0127] That is,

[0128]

[0129] where N is the parameter matrix when the displacement u at any point x within the periodic unit is expressed through the element nodal displacements u e and the non - zero elements therein are:

[0130]

[0131] In the formula, l e is the length of the beam element, ξ = x / l e , and x ∈ (0, l e ).

[0132] Define the differential operators L and H

[0133]

[0134]

[0135] The mass element matrix M e , linear stiffness element matrix and geometric stiffness element matrix of the periodic unit of a space truss structure considering shear deformation can be expressed as:

[0136]

[0137] where: l e represents the length of the periodic unit, l e = l c ;

[0138] Assemble the element matrices and add the left - end fixed boundary conditions to obtain the global mass matrix M, global linear stiffness matrix K L and initial stress stiffness matrix K S . The undamped vibration equation of the equivalent beam under initial thermal stress can be expressed as:

[0139]

[0140] where w is the vector composed of all nodal displacements in the equivalent beam model, and f is the external load.

[0141] Based on the structural temperature distribution, the dynamic response problem after considering the deformation of the structure under the influence of thermal loads can be calculated using the above formula.

[0142] An equivalent dynamic modeling method for an asymmetric flexible spacecraft considering thermal deformation proposed by the present invention determines the position of the neutral axis of the asymmetric flexible spacecraft by the weighted average method, considers the structural thermal deformation as the initial thermal stress, and adopts the energy equivalence method to equivalent the spacecraft to a beam model located at the neutral axis, and the two maintain the same mechanical properties, and establish an equivalent beam dynamic equation containing the initial thermal stress. It can solve the influence of the thermal deformation caused by thermal radiation on the dynamic response of the asymmetric flexible structure of the spacecraft during the on-orbit operation of large spacecraft, as well as the problem that the structural vibration equation cannot be accurately and efficiently solved.

[0143] In summary, an equivalent dynamic modeling method for an asymmetric flexible spacecraft considering thermal deformation proposed by the present invention is based on the equivalent continuum theory, introduces the influence of the thermal deformation of the satellite antenna on the structural dynamic characteristics, equivalent the antenna structure after thermal deformation to a beam model, and establish an equivalent dynamic model of the satellite antenna considering thermal deformation to improve the solving speed of the dynamic response. First, for the asymmetric truss structure of the satellite antenna with a radar array surface, the equivalent cross-section method is used to determine the position of the neutral axis; then, a rectangular coordinate system with the x-axis coinciding with the neutral axis is established, and based on the classical continuum theory, the spatial displacement expression related to the parameters on the x-axis is derived; secondly, to consider the influence of the structural thermal deformation on the structural dynamic characteristics, the strain energy generated by the thermal stress is introduced into the structural energy expression, and since the structural thermal deformation is a slow-varying process, the thermal stress can be updated according to the position of the spacecraft during on-orbit operation after a period of time; thirdly, the strain energy and kinetic energy expressions of the truss periodic structure are established, and based on the energy equivalence principle, the equivalent inertia and elastic parameters of the satellite antenna under the influence of thermal loads are constructed; finally, the finite element method is used to establish an equivalent dynamic equation containing thermal stress for solving the dynamic response of the asymmetric satellite truss considering thermal deformation. The equivalent beam model established by the present invention fully considers the asymmetry of the structure and the influence of thermal loads on the structural dynamic characteristics, improves the existing equivalent continuum modeling theory, improves the accuracy of the equivalent model, and has important significance for the design of the vibration control system.

[0144] The content not described in detail in the specification of the present invention belongs to the prior art well-known to those skilled in the art.

Claims

1. An equivalent dynamic modeling method for an asymmetric flexible spacecraft considering thermal deformation. The asymmetric flexible spacecraft is a space truss structure, including longitudinal beams, cross beams, stay cables, and a bottom plate, characterized in that Including: Establish an equivalent beam model for the space truss structure, determine the position of the neutral axis of a periodic unit in the equivalent beam model, and establish a coordinate system; Describe the displacement of any point on the periodic unit of the truss structure in the established coordinate system; Analyze the stress, strain and structural volume of the periodic unit of the space truss structure along the direction of the neutral axis to obtain the thermal deformation strain energy and axial strain energy after thermal deformation of the periodic unit; Obtain the total strain energy after thermal deformation of the periodic unit from the thermal deformation strain energy and axial strain energy, and decompose the elastic matrix and stress matrix after sorting it out; Derive the velocity of the displacements at both ends of each member of the periodic unit to obtain the kinetic energy of each member in the periodic unit, and after superimposing, obtain the total kinetic energy of the periodic unit, and decompose the inertia matrix of the periodic unit after sorting it out; Based on the finite element method, divide the equivalent beam model by the length of the periodic unit to obtain the mass element matrix, linear stiffness element matrix and geometric stiffness element matrix of each periodic unit; Assemble the mass element matrix, linear stiffness element matrix and geometric stiffness element matrix respectively to obtain the corresponding overall mass matrix, overall linear stiffness matrix and overall geometric stiffness matrix; further obtain the undamped vibration equation of the equivalent beam under the initial thermal stress, and based on this, the dynamic response problem of the space truss structure after deformation under the influence of thermal load can be calculated.

2. The equivalent dynamic modeling method of an asymmetric flexible spacecraft considering thermal deformation according to claim 1, characterized in that: The position of the neutral axis of a periodic unit is represented by the distance from the bottom surface of the space truss: Where M T is the total number of members in the periodic unit, and k is the serial number index of each member in the periodic unit; E (k) , S (k) and A (k) are respectively the elastic modulus, static moment and cross-sectional area of the k-th component; z c is the distance from the neutral axis of the overall structure to the bottom surface of the space truss structure, and z ck is the distance from the centroid of the k-th component to the bottom surface of the space truss structure.

3. The equivalent dynamic modeling method of an asymmetric flexible spacecraft considering thermal deformation according to claim 2, characterized in that: When establishing the coordinate system, define the direction perpendicular to the bottom surface of the space truss structure as the Z-axis direction, the X-axis direction is parallel to the longitudinal beam of the space truss, and the Y-axis is perpendicular to the plane formed by the Z-axis and the X-axis; the origin position is determined according to the actual situation.

4. A method for equivalent dynamic modeling of an asymmetric flexible spacecraft considering thermal deformation according to claim 1, characterized in that: Describe the displacement of any point on the periodic unit of the truss structure in the established coordinate system, specifically: In the formula, the superscript c represents the variable corresponding to the origin of the coordinate system. The components in (x, y, z) respectively represent the coordinates of a point on the periodic cell in the X, Y, and Z directions. u(x, y, z) represents the displacement in the X direction, v(x, y, z) represents the displacement in the Y direction, w(x, y, z) represents the displacement in the Z direction, and φ x (x, y, z) represents the rotation angle about the X axis, and φ y (x, y, z) represents the rotation angle about the Y axis, and φ z (x, y, z) represents the rotation angle about the Z axis, and u c represents the displacement in the X direction at the origin of the periodic cell, and v c represents the displacement in the Y direction at the origin, and w c represents the displacement in the Z direction at the origin, represents the rotation angle about the X axis at the origin of the periodic cell, represents the rotation angle about the Y axis at the origin of the periodic cell, represents the rotation angle about the Z axis at the origin of the periodic cell, represents the strain along the X axis at the origin, represents the curvature in the Y direction at the origin, represents the curvature in the Z direction at the origin, represents the shear strain in the XY plane at the origin, represents the strain along the Y axis at the origin, represents the curvature in the X direction at the origin, represents the shear strain in the YZ plane at the origin, represents the shear strain in the XZ plane at the origin, represents the strain along the Z axis at the origin.

5. An equivalent dynamics modeling method for an asymmetric flexible spacecraft considering thermal deformation according to claim 1, characterized in that: After the thermal deformation of the periodic unit of the space truss structure, the thermal deformation strain energy U of the longitudinal beam T is as follows: In the formula, the superscript (k) represents the member number, M z is the number of stringers, is the axial strain of the member, A (k) represents the cross-sectional area of the member, is along the axis direction of member k; σ T is the thermal stress of the stringer of the space truss structure along the neutral axis direction. The elastic strain energy U of the periodic unit of the space truss structure after thermal deformation p is as follows: where E (k) is the elastic modulus of member k.

6. The equivalent dynamic modeling method of an asymmetric flexible spacecraft considering thermal deformation according to claim 5, characterized in that: The total strain energy U after the thermal deformation of the periodic unit is U = U T + U p , and it is sorted out as follows: where \(l\) c represents the length of the equivalent beam model of the periodic cell, and \(\Gamma\) is the strain component on the neutral axis: D is the elastic matrix, and its non-zero elements are: Where, A1 represents the cross-sectional area of the cross beam, A2 represents the cross-sectional area of the longitudinal beam, A3 represents the cross-sectional area of the cable-stayed cable, E1 represents the elastic modulus of the cross beam, E2 represents the elastic modulus of the longitudinal beam, E3 represents the elastic modulus of the cable-stayed cable, l1 represents the length of the cross beam, l2 represents the length of the longitudinal beam, and l3 represents the length of the cable-stayed cable; S is the stress matrix, and its non-zero elements are: In the formula, represents the axial thermal stress of rod k, and the specific expression is: Where, E (k) is the elastic modulus of member k, is the coefficient of thermal expansion of member k, T (k) is the current temperature of member k, is the initial temperature of member k.

7. An equivalent dynamics modeling method for an asymmetric flexible spacecraft considering thermal deformation according to claim 1, characterized in that: The total kinetic energy T of the periodic unit p is as follows: where M T is the total number of bars in the periodic unit, and M B represents the number of bottom plates in the periodic unit, and T (k) represents the kinetic energy of bar k: Indicates the kinetic energy of the bottom plate k b Specifically: Among them, the subscripts 1 and 2 are the two endpoints of the rod, and ρ (k) represents the density of the rod k, l (k) represents the length of the rod k, A (k) represents the cross-sectional area of the rod k, V x,1 and V x,2 respectively represent the X-direction velocities of the two endpoints of the rod k, V y,1 and V y,2 respectively represent the Y-direction velocities of the two endpoints of the rod k, V z,1 and V z,2 respectively represent the Z-direction velocities of the two endpoints of the rod k; m b is the mass of the bottom plate, is the X-direction velocity of the i-th vertex among the four vertices of the bottom plate, is the Y-direction velocity of the i-th vertex among the four vertices of the bottom plate, is the Z-direction velocity of the i-th vertex among the four vertices of the bottom plate.

8. An equivalent dynamic modeling method for an asymmetric flexible spacecraft considering thermal deformation according to claim 7, characterized in that: The total kinetic energy T of the periodic unit p After rearrangement, the inertia matrix of the periodic unit is decomposed Specifically: where, υ T is the transpose of υ: represents the rotation angle of the periodic unit around the X-axis at the origin, represents the rotation angle of the periodic unit around the Y-axis at the origin, represents the rotation angle of the periodic unit around the Z-axis at the origin, u c represents the displacement of the periodic unit in the X direction at the origin, v c represents the displacement in the Y direction at the origin, w c represents the displacement in the Z direction at the origin; is the inertia matrix, and its non-zero elements are: Where: A1 represents the cross-sectional area of the cross beam, A2 represents the cross-sectional area of the longitudinal beam, A3 represents the cross-sectional area of the stay cable, m b is the mass of the bottom plate, l1 represents the length of the cross beam, l2 represents the length of the longitudinal beam, l3 represents the length of the stay cable; ρ1 represents the density of the cross beam, ρ2 represents the density of the longitudinal beam, ρ3 represents the density of the stay cable; l c represents the length of the equivalent beam model of the periodic unit.

9. An equivalent dynamic modeling method for an asymmetric flexible spacecraft considering thermal deformation according to claim 1, characterized in that: When dividing the equivalent beam model based on the finite element method with the length of the periodic unit, a 2-node 8-degree-of-freedom element is used to simulate the structural deformation of the equivalent beam model. Using the elastic matrix, stress vector, and inertia matrix of the periodic unit, the mass element matrix M of each periodic unit is obtained. e , the linear stiffness element matrix and the geometric stiffness element matrix are respectively: Where: l e represents the length of the periodic unit, l e = l c , D is the elastic matrix, S is the stress matrix, L and H are differential operators, and N is the parameter matrix when the displacement u at any point x in the periodic unit is represented by the nodal displacement u e in the periodic unit; specifically: l e is the length of the beam element, ξ = x / l e , x ∈ (0, l e ).

10. A method for equivalent dynamic modeling of an asymmetric flexible spacecraft considering thermal deformation according to claim 1, characterized in that: When assembling the mass unit matrix, the linear stiffness unit matrix, and the geometric stiffness unit matrix, consider adding the fixed boundary condition at the left end to obtain the corresponding global mass matrix M, global linear stiffness matrix K L and global geometric stiffness matrix K S ; The undamped vibration equation of the equivalent beam under the initial thermal stress is: w is the vector composed of the displacements of all nodes in the equivalent beam model, and f is the external load.

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