A rigid-flexible coupling modal modeling method for a spatial solar panel servo motor system

Abstract

The present application relates to a kind of space solar panel servo motor system rigid-flexible coupling modal modeling method, belong to spacecraft dynamics modeling technical field.This method aims to solve the failure of existing analytical model under single-axis large angle rotation and the problem that finite element model lacks explicit mathematical equation and is difficult to be directly used for control system design.The technical scheme is:establishing system geometry model and coordinate system;Adopting characteristic orthogonal polynomial to construct solar panel vibration displacement function;System kinetic energy, strain energy and hinge potential energy function are constructed;Based on Rayleigh-Ritz method, the system modal characteristic equation is established and the natural frequency and vibration mode are solved numerically.This method can realize the accurate modeling of single-axis large range rotation, and directly obtain low-dimensional high-precision explicit mathematical model, which is convenient for docking and co-simulation with control software.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft dynamics modeling technology, and relates to a rigid-flexible coupling modal modeling method for a space solar panel servo motor system. Background Technology

[0002] The on-orbit attitude control and energy supply of spacecraft are highly dependent on their large space solar panels. To ensure solar orientation and maximize energy harvesting, or to meet specific mission attitude requirements, space solar panels are typically driven by servo motors to perform large-scale continuous rotation along a single axis. During this motion, a strong dynamic coupling occurs between the rigid rotation of the drive motor and the elastic deformation of the large-area flexible solar panel, known as rigid-flexible coupling. This effect can excite complex vibrations in the solar panel, affecting not only pointing accuracy and stability but also potentially threatening structural safety. Therefore, establishing a system dynamics model that accurately reflects this rigid-flexible coupling characteristic, especially obtaining its precise modal parameters (natural frequencies and mode shapes), is a crucial prerequisite for system dynamics analysis, vibration suppression controller design, and high-fidelity electromechanical co-simulation.

[0003] Currently, dynamic modeling methods for such systems are mainly divided into two categories: analytical modeling methods based on simplification assumptions and finite element methods based on numerical discretization. However, both have significant limitations when dealing with large-range rotational conditions on a single axis.

[0004] On the one hand, traditional analytical modeling methods (such as hypothetical modal methods or continuum dynamics methods) are mostly derived from the background of spacecraft performing small-angle attitude maneuvers. These methods typically linearize the system's equations of motion, assuming that structural deformation and attitude angles are small. However, when servo motors drive solar panels to rotate continuously around a single axis by several degrees, tens of degrees, or even hundreds of degrees, the system's dynamics exhibit strong geometric nonlinear characteristics, and the traditional linearized small-angle assumption no longer holds. Models built based on this assumption cannot accurately capture the nonlinear dynamic behavior generated by the coupling of large-scale rigid body motion and elastic vibration, leading to a severe decrease in model accuracy or even model failure under large-angle rotation conditions, making them unsuitable for high-precision control design and simulation.

[0005] On the other hand, while commercial finite element analysis software (such as ANSYS and Abaqus) can perform numerical calculations on complex structures at high degrees of freedom through fine mesh discretization and obtain relatively accurate modal results, the finite element method creates models with extremely high degrees of freedom, essentially making them a "numerical black box." Although it can output numerical solutions, it is difficult to extract low-dimensional, explicit overall system dynamic equations (such as second-order matrix differential equations) from them. This "black box" characteristic makes it difficult for finite element models to be directly and efficiently integrated with control system design software (such as Matlab / Simulink) for joint simulation, and it cannot provide a direct mathematical equation foundation for model-based control algorithm design (such as optimal control and robust control). Furthermore, the extremely high-dimensionality of the model is also not conducive to rapid parametric research and optimization design.

[0006] In summary, existing technologies lack a method specifically designed for space solar panel servo motor systems operating under single-axis, large-range continuous rotation conditions. This method not only overcomes the small-angle linearization assumption to accurately characterize nonlinear rigid-flexible coupling effects but also constructs a low-degree-of-freedom, high-precision modal model with explicit mathematical expressions. This has become a bottleneck restricting the development of high-performance control and high-fidelity simulation technologies for such systems. Summary of the Invention

[0007] This invention aims to overcome the problems of existing analytical models failing under large-angle rotation on a single axis, and the lack of explicit mathematical equations in finite element models, making them difficult to directly apply to control system design. It provides a rigid-flexible coupling modal modeling method suitable for servo motor systems of space solar panels with large-range continuous rotation on a single axis. Its purpose is to directly obtain the explicit analytical equations of the rigid and flexible modes of the system under large-amplitude motion, providing a low-dimensional, high-precision mathematical model foundation for rigid-flexible coupling mechanism analysis, electromechanical co-simulation, and control algorithm development under servo drive.

[0008] To achieve the above objectives, the present invention provides the following technical solution: A method for rigid-flexible coupling modal modeling of a space solar panel servo motor system includes the following steps: S1: Establish the geometric model and coordinate system of the space solar panel servo motor system: The servo motor is simplified as a central rigid body, and the hinge between the servo motor and the solar panel, as well as the hinge between multiple substrates in the solar panel, are simplified as torsion spring models. An inertial coordinate system is established with the center of mass O of the central rigid body as the origin. O - xyz ; Establish a follower coordinate system fixed to the center of mass O of the central rigid body. O 0- x 0y 0 z 0; Establish a body coordinate system on the i-th solar panel substrate O i - x i y i z i , where i=1,2,...N, and N is the number of single-sided substrates; Assume that the servo motor only carries the solar panel around y The shaft rotates on a single axis with an angle of θ. θ And since the central rigid body itself has no translational motion, then from the following coordinate system O 0- x 0 y 0 z 0 to the inertial coordinate system O - xyz coordinate transformation matrix A for:

[0009] S2: Constructing the solar panel vibration displacement function: Assuming the solar panel operates at its natural angular frequency ω Performing simple harmonic motion, any point on the solar panel ( x i , y i , z i vibration displacement function Represented as:

[0010] in, W i ( x i , y i ) is the first i Mode shape function of the block substrate; The mode shape function W i ( x i , y i It is represented by a linear combination of characteristic orthogonal polynomial basis functions:

[0011] in, φ m (i) (x i )and φ n (i) ( y i ) are respectively the solar panel substrate in x i and y i Characteristic orthogonal polynomial basis functions of the direction, m i and n i It is the number of basis functions of the eigenorthogonal polynomials that are truncated. A mn (i) It is the mode shape factor; S3: Construct the energy function of the system: The total kinetic energy of the system T for:

[0012] in, J m It is the moment of inertia of the central rigid body about the y-axis. ρ It is the bulk density of the windsurfing material. and These are located in the following coordinate system x The first half of the negative half-axis and the second half of the positive half-axis i The volume of the gasket substrate. and These are the velocity vectors of a point on the left and right sides of the sail in the inertial coordinate system, respectively. Strain energy of solar panel vibration deformation U p for:

[0013] in, E It is the Young's modulus of the windshield material. μ It is the Poisson's ratio of the windsurfing material. h It's the thickness of the windsurfing board. a It is the length of a single substrate, 2 b It is the width of a single substrate; The sum of the elastic potential energies of all hinges U j for:

[0014] in, k It refers to the torsional stiffness at the hinge. θ ALi, θ BLi , θ ARi , θ BRi These are the hinges on the left and right sides of the sail. A i and B i The angle of torsion; S4: Establish the modal characteristic equations of the system: The motion of the servo motor is decomposed into a wide range of rigid body rotation and elastic vibration coupled with the vibration of the solar panel, and its rotation angle is... Represented as:

[0015] in, Indicates the rigid body motion. Indicates the elastic vibration component; The elastic vibration part Represented as:

[0016] in, The coefficients to be determined; According to the Rayleigh-Ritz method, a Lagrangian function Π containing displacement constraints at the hinge is constructed, and the Lagrangian function Π is allowed to take stationary values ​​for all unknown coefficients, thus deriving the characteristic equation of the system as follows:

[0017] Where X is the coefficient of all unknown mode shapes. ,coefficient And the eigenvectors of the Lagrange multipliers, K is the generalized stiffness matrix, M is the generalized mass matrix, and Λ is the matrix related to the Lagrange multipliers; S5: Numerically solve the modal characteristic equations of the system to obtain the natural frequencies and corresponding mode shapes of the system.

[0018] Furthermore, in S1, the body coordinate system O i - x i y i z i The origin O i Located in the following coordinate system x On the 0 axis, and located on the solar panel at a distance from the center of mass of the central rigid body.O The center of the closer side.

[0019] Furthermore, in S1, the substrate of the solar panel is considered as a thin plate structure, and the following assumption is made: the thin plate is... z i There is no deformation or stress in the axial direction; the normal to the middle surface of the thin plate is always perpendicular to the middle surface when the plate deforms; there is no displacement of any point on the middle surface of the thin plate in the direction parallel to the middle surface; each of the substrates is isotropic; the flexible hinge is simplified to a hinge with an additional torsion spring, and the mass, size, damping and friction of the torsion spring are ignored.

[0020] Furthermore, in S2, the characteristic orthogonal polynomial basis functions are generated through the Gram-Schmidt orthogonalization process.

[0021] Furthermore, in S3, the hinge A i and B i The torsion angle at the junction is determined by the difference in the angles of the two substrates connected to it at the junction.

[0022] Furthermore, in S4, the specific elements of the generalized stiffness matrix K, the generalized mass matrix M, and the matrix Λ are determined by the mode shape function. W i ( x i , y i The characteristic orthogonal polynomial basis functions φ m (i) ( x i )and φ n (i) ( y i The system's geometric parameters, material parameters, and hinge stiffness parameters are calculated.

[0023] Furthermore, in S5, the characteristic equation is solved using numerical methods. A series of eigenvalues ​​were obtained. and its corresponding eigenvector X, the eigenvalues The square root is the natural frequency of the system. .

[0024] Furthermore, the mode shape coefficients are extracted from the obtained eigenvector X. and the mode shape coefficient Substitute the mode shape function expression in S2 The mode shapes of the system corresponding to each natural frequency are obtained.

[0025] Furthermore, following S5, S6 is also included: comparing the system's natural frequencies and mode shapes calculated by the method with the modal analysis results of the same system model established based on Finite Element Analysis (FEA) software, in order to verify the correctness of the modeling method.

[0026] Furthermore, the FEA software is ANSYS software.

[0027] The beneficial effects of this invention are as follows: (1) The method of this invention does not adopt the small-angle linearization assumption commonly found in traditional spacecraft attitude dynamics, but directly establishes the kinetic and potential energy expressions of the system based on the geometric relationship of a large range of rotation. Therefore, this method can accurately capture and describe the strong nonlinear rigid-flexible coupling dynamic characteristics caused by the continuous rotation of the solar panel around a single axis by a servo motor at tens or even hundreds of degrees, and solves the problem of the failure of existing analytical models under such actual working conditions.

[0028] (2) This invention uses characteristic orthogonal polynomials as basis functions to construct the mode shape functions of the sail, and derives the characteristic equation of the system using the Rayleigh-Ritz method. This method can achieve this with extremely low degrees of freedom (due to the limited number of polynomials truncated). m i and n i By determining the optimal frequency (m=n=6), the dominant modes of the system can be characterized with high precision. As shown in Table 1, when m=n=6, the error between the calculated first 11 natural frequencies and the high-precision finite element results does not exceed 0.8%. More importantly, the final model is an explicit matrix equation, with all matrix elements having clear analytical expressions. This low-dimensional, explicit mathematical model greatly facilitates direct integration with control system design software (such as Matlab / Simulink), providing an efficient mathematical foundation for model-based controller design, stability analysis, and real-time simulation.

[0029] (3) The method of this invention can systematically solve for a complete set of modes from order 0 (rigid body mode) to higher orders without any mode loss. For example Figures 4 to 15 As shown in Table 1, the solved vibration modes are highly consistent with the finite element results in terms of deformation region, symmetry, and number of nodes. In particular, its low-order modes (such as the overall bending mode) are in extremely good agreement with the finite element results, which provides a reliable low-order model basis for the design of control systems aimed at suppressing low-frequency vibrations and improving pointing accuracy.

[0030] (4) By adjusting the number of cutoffs in the characteristic orthogonal polynomial basis functions mi and n i The method of this invention can achieve a flexible trade-off between model accuracy and computational complexity. As shown in Table 1, even with a relatively small number of basis functions (e.g., m=n=4), the capture of the main low-frequency modes is quite accurate; as the number of basis functions increases, the calculation results quickly converge to a high-precision solution. This provides convenience for users to select appropriate model dimensions according to different accuracy requirements and computational resource conditions.

[0031] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0032] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein: Figure 1 A rigid-flexible coupling geometric model for a space solar panel servo motor system; Figure 2 It is a multi-plate hinged structure for space solar panels; Figure 3 ANSYS finite element model of a space solar panel servo motor system Figure 4 It is the zeroth mode shape of a rigid body. Figure 5 It is a first-order vibration mode; Figure 6 It is a second-order vibration mode; Figure 7 It is a third-order vibration mode; Figure 8 It is a fourth-order vibration mode; Figure 9 It is a 5th order vibration mode; Figure 10 It is a 6th order vibration mode; Figure 11 It is a 7th order vibration mode; Figure 12 It is an 8th order vibration mode; Figure 13 It is a 9th order vibration mode; Figure 14 It is a 10th order vibration mode; Figure 15 It is an 11th-order vibration mode; Detailed Implementation The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0033] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0034] In the accompanying drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components. In the description of the present invention, it should be understood that if terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.

[0035] This invention proposes a rigid-flexible coupling modal modeling method for a space solar panel servo motor system, which establishes the system modal model through the following steps: 1. Establish the system's geometric model and coordinate system. The rigid-flexible coupling geometric model of the space solar panel servo motor system is as follows: Figure 1 As shown, the motor is simplified as a central rigid body, and the hinge between the motor and the sailboard, as well as the hinge between multiple sailboards, are simplified as torsion spring models.

[0036] (1) Establish an inertial coordinate system with the center mass of the rigid body as the origin. O - xyz ; (2) Establish a moving coordinate system fixed at the center of the rigid body. O 0- x 0 y 0 z 0; (3) with x Distance on the 0 axis and on the windsurf board OThe origin is the center of the side closer to the point. O i , i The first part, represented as a multi-plate structure i Block substrate, i =1, 2, 3… N , N To determine the number of substrates on one side, establish a body coordinate system fixed to each solar panel substrate. O i - x i y i z i .

[0037] Windsurfing y i The coordinates at both ends of the axis are - b and b ,exist z i The coordinates at both ends of the axis are - h and h The length of each substrate is a Considering only the motor carrying the sailboard rotating around the y-axis, that is... θ x =θ z = 0, θ y =θ Furthermore, the rigid body itself has no translational motion, therefore O 0 and O The points coincide. Therefore, from the moving coordinate system... O 0- x 0 y 0 z From 0 to the inertial coordinate system O - xyz The coordinate transformation matrix A is (1) Angular velocity vector of a rigid body relative to an inertial frame ω For [0, , 0] T Its corresponding antisymmetric matrix As shown in equation (2), and we have .

[0038] (2) Space solar panel multi-plate hinged structure, such as Figure 2 As shown, b 0 represents the hinge spacing, hinge A i ,B i The ordinates are respectively y a = - b 0 / 2 and y b =b 0 / 2.

[0039] The solar panel substrate is considered to be a thin plate structure and the following assumptions are made: (1) The thin plate is in z i (1) No deformation or stress in the axial direction; (2) The thin plate deforms under the action of external force, and the normal of the middle surface is always perpendicular to the middle surface; (3) There is no displacement of each point on the middle surface in the direction parallel to the middle surface; (4) Each substrate is isotropic; (5) The flexible hinge is simplified to a hinge with an additional torsion spring. The mass, size, damping and friction of the torsion spring are ignored. The solar wing is in a fully deployed state and the hinge is locked.

[0040] 2. Construct the vibration displacement function of the solar panel. Assuming the solar panels vibrate together at a certain frequency, due to the coupling of vibrations between the panels, any point on the solar panel ( x i , y i , z i The displacement of ) can be expressed by equation (3).

[0041] (3) in, ω Let be the natural angular frequency of the solar panel. W i ( x i , y i ) is the mode shape function of each substrate of the solar panel, which can be represented by a linear combination of modal basis functions that satisfy specific boundary conditions.

[0042] This is represented by a linear combination of characteristic orthogonal polynomial basis functions. W i ( x i , y i As shown in equation (4).

[0043] (4) in, φ m (i) ( x i )and φn (i) ( y i These are the solar panels in... x and y The characteristic orthogonal polynomials of the directions can be generated through the Gram-Schmidt orthogonalization process; m t and n t The number of these two types of polynomials truncated during actual calculations is determined by the required model accuracy. A mn (i) It is the mode shape factor.

[0044] Construct mode shape functions for each of the left and right base plates of the sail. and .

[0045] 3. Construct the system's energy function A little on the right side of the solar panel substrate In coordinate system O 0- x 0 y 0 z The position vector in 0 can be represented as: (5) Then point In an inertial frame O - xyz The position vector in the vector can be expressed as: (6) point In an inertial frame O - xyz The speed in is: (7) For the point on the left Simply put r 0+ a ( i -1) becomes - r 0- a ( i -1), subscript R i Become L i .

[0046] The kinetic energy of the rigid-flexible coupled system T As shown in equation (8): (8) in J mFor the rigid body part of the winding y Moment of inertia of the shaft, ρ The bulk density of the windshield is [the mass of the windshield]. , They represent x The volume of the sail on the negative half-axis and the positive half-axis of the 0 axis.

[0047] Strain energy of solar panel vibration deformation U p As shown in equation (9): (9) , , For along x , y Linear strain in the direction and along z Angular strain in direction, , , The corresponding stress is given.

[0048] Substituting the displacement function, we get: (10) In the formula, E and μ These are Young's modulus and Poisson's ratio, respectively.

[0049] The sum of the elastic potential energy of the hinge U j for: (11) In the formula, θ ALi , θ BLi , θ ARi , θ BRi Indicates hinge A Li , B Li , A Ri , B Ri The angle of twist, k Let be the torsional stiffness at the hinge.

[0050] When the spacecraft is operating normally in orbit, the solar panels are in the deployed and locked state; therefore, it is assumed that the hinge is firmly connected to the base plate. Based on geometric relationships, the hinge's rotation angle... θAi and θ Bi It should be equal to the angle difference at the connection between the two substrates. Since the hinge's vibration is coupled with the components of the system, the hinge's displacement can also be expressed as... θ Ai = θ Ai sin ωt In the form of, the matching conditions at the hinge are as follows: (12) In the formula, y a =b 0 / 2, y b = - b 0 / 2, i = 2, 3, ... N .

[0051] 4. Establish the modal characteristic equations of the system. The motion of the space solar panel servo motor system can be represented by two parts: large-scale rigid body rotation and elastic vibration coupled with the solar panel's vibration. Therefore, the motor's motion can be represented as: (13) In the formula, the subscript r and v These represent rigid body motion and elastic vibration, respectively.

[0052] Based on the concept of global modes, the vibration of the spacecraft's motors and the vibration of the flexible solar array are synchronous and coupled. Therefore, the elastic vibration of the motor's rotation angle can also be expressed as the product of a constant and a time-dependent term, as shown below: (14) In the formula, θ 0 represents an unknown coefficient.

[0053] Ignoring nonlinear coupling terms of degree three or higher, the maximum kinetic energy of the system is... T max With maximum potential energy U max As shown in equations (15) and (16) respectively.

[0054] (15) (16) Because the hinges are locked during spacecraft operation and the hinge dimensions are not considered in the modeling, there is no relative displacement between the two base plates. Therefore, Figure 2 The displacement relationship at the middle hinge is as follows: (17) In the formula, i = 2, 3, ... N , , , , Points on the left and right substrates A i and B i The relative displacement.

[0055] Introducing Lagrange multipliers , , , Considering the matching conditions at the hinge, the Lagrangian function Π with constraints is constructed as shown in equation (18).

[0056] (18) According to the Rayleigh-Ritz method, Π takes stationary values ​​with respect to the unknown coefficients, satisfying the following condition: (19) Based on this, the characteristic equation of the system can be derived as follows: (20) Where X is the eigenvector with constraints, K and M are the generalized stiffness matrix and generalized mass matrix with constraints, respectively, and Λ is the matrix related by Lagrange multipliers. K, M, and Λ are all (2 Nm t n t +4 N +1)×(2 Nm t n t +4 N A +1) dimensional matrix. The specific form is as follows: (twenty one) (twenty two) In the formula,

[0057] M 21 2 Nm t n tA 1×1 dimensional matrix, and M 12 =M 21 T .

[0058] And M 21 (Li) With M 21 (Ri) All m t n t The specific expression for a ×1 matrix is ​​as follows.

[0059]

[0060]

[0061]

[0062] Among them, M 22 2 Nm t n t ×2 Nm t n t A matrix of dimension M 22 (Li) With M 22 (Ri) All m t n t × m t n t 3D matrix, M 22 (Li) The specific expressions for each matrix element are as follows:

[0063]

[0064] in, j , m = 1, 2, …, m t ; k , n = 1, 2, …, n t .

[0065] M 22 (Ri) Matrix form and M 22 (Li)They are identical, and the specific expressions for each element are as follows:

[0066] (twenty three) Matrix K Lii With K Rii ( i =1,2,..., N -1) construction form and M 22 (Li) Similarly, the specific expressions for its matrix elements are as follows:

[0067]

[0068] Matrix K LNN With K RNN The construction form and M 22 (Li) Similarly, the specific expressions for its matrix elements are as follows:

[0069]

[0070] Matrix K Li(i+1) With K Ri(i+1) ( i =1,2,..., N -1) construction form and M 22 (Li) Similarly, the specific expressions for its matrix elements are as follows:

[0071]

[0072] K L(i+1)i = K Li(i+1) T K R(i+1)i = K Ri(i+1) T ( i =1,2,..., N -1).

[0073] (twenty four) In the formula, Λ 12 Λ 21 The matrix is ​​a block matrix of Λ. Λ 21 = Λ 12 T Λ 12 (Li) Λ 12 (Li) All m t n t ×4 N 3D matrix, when i =1, 2, ..., N -1, Λ 12 (Li) The 2nd i -1~2 i +2 columns are respectively Λ 12 (Li) (0, y a ), Λ 12 (Li) (0, y b ), Λ 12 (Li) (- a , y a ), Λ 12 (Li) (- a , y b ), Λ 12 (Ri) The 2nd N +2 i -1~2 N +2 i +2 columns are respectively Λ 12 (Ri) (0, y a ), Λ 12 (Ri) (0, y b ), Λ 12 (Ri) ( a , y a ), Λ 12 (Ri) ( a , y b ), and all other columns are 0. When i = N Λ 12 (LN) Except for the 2nd N -1, 2N Listed as Λ 12 (LN) (0, y a ), Λ 12 (LN) (0, y b In addition, Λ 12 (RN) Except for the 4th N -1, 4 N Listed as Λ 12 (LN) (0, y a ), Λ 12 (LN) (0, y b Except for Λ, all other elements are 0. 12 (Li) ( α , β ), Λ 12 (Ri) ( α , β The construction method of ) is as follows ( i =1, 2, ..., N ): ,

[0074] 5. Modal solution of the system For the characteristic equation of equation (20), when the matrix dimension is less than 5, there is theoretically an analytical expression with roots. However, for multi-degree-of-freedom systems, there are generally hundreds of degrees, so there is usually no radical solution unless the coefficients have some special characteristics. Therefore, numerical solutions are usually used. Equation (20) is a generalized eigenvalue problem. For numerical solutions to such problems, the generalized eigenvalue problem can be transformed into a standard eigenvalue problem through Cholesky decomposition, and then solved using the Jacobi method. The development of modern computers and programming has made numerical solutions very convenient. The natural frequencies and corresponding eigenvectors of the characteristic equation can be easily solved using the eig function in MATLAB. The natural frequencies of the system are obtained by solving equation (20). ω The value is obtained, and then substituted into equation (20) to obtain the eigenvector X corresponding to the natural frequency. The obtained mode shape coefficients are then used to calculate the mode shape coefficients. , Substituting into equation (4) yields the system mode shape.

[0075] To verify the effect, an ANSYS finite element model of a spatial solar panel servo motor system with a substrate length of 1.5m, a width of 3m, and three substrates hinged on one side was established, as shown below. Figure 3As shown.

[0076] The method used in the ANSYS modal analysis module to solve for the first 11 modal natural frequencies of the system differs from the method proposed in this invention. m t , n t The numerical solution results are compared below, as shown in Table 1.

[0077] Table 1. Comparison of the first 11 natural frequencies obtained by the method of this invention and the finite element method.

[0078] As can be seen from the results in Table 1: (1) The 0th order frequency of this method is 0, which can accurately capture rigid body modes; (2) m t , n t The error is larger when the size is small, but it increases with... m t , n t Increases and converges rapidly, when m t = n t When the value is 6, the calculation error of each natural frequency does not exceed 0.8%, which combines calculation accuracy and calculation efficiency. (3) All modes were calculated completely, and no mode loss occurred; (4) The low-frequency modes are highly compatible and suitable for control requirements.

[0079] Comparison of the ANSYS finite element method calculation results with the method of this invention for the mode shapes corresponding to the first 11 natural frequencies is shown below. Figures 4-15 As shown, the method of the present invention has the following advantages: (1) Overall consistency: The two are highly consistent in terms of main deformation region, symmetry and number of nodes, indicating that the theoretical model can reliably characterize the dominant dynamic characteristics of the structure; (2) Low-order modes are consistent: the 0th order mode shows overall motion and weak elastic deformation, with obvious rigid body mode characteristics; the 1st and 2nd orders show typical overall bending dominance, and the symmetry family is clear (symmetric / antisymmetric characteristics are consistent), indicating that the global response in the low-frequency band is highly consistent. (3) High-order mode matching: The 6th order transitions from pure bending to bending-turn coupling and shows obvious torsional features. The 9th order is consistent in terms of the increase of high-order nodes and the enhancement of localized deformation. The location and morphological features of the main deformation zone correspond well. The 11th order is a high-order coupling mode with multiple antinodes / nodes. It can still maintain the consistency of key topological features under complex morphology, which reflects the effectiveness of high-order prediction.

[0080] By comparing the modal solution results with those of ANSYS finite element method, the correctness of the modal modeling method for the space solar panel servo motor system of this invention can be verified.

[0081] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A rigid-flexible coupling modal modeling method for a space solar panel servo motor system, characterized in that: Includes the following steps: S1: Establish the geometric model and coordinate system of the space solar panel servo motor system: The servo motor is simplified as a central rigid body, and the hinge between the servo motor and the solar panel, as well as the hinge between multiple substrates in the solar panel, are simplified as torsion spring models. An inertial coordinate system is established with the center of mass O of the central rigid body as the origin. O - xyz ; Establish a follower coordinate system fixed to the center of mass O of the central rigid body. O 0- x 0 y 0 z 0; Establish a body coordinate system on the i-th solar panel substrate O i - x i y i z i , where i=1,2,...N, and N is the number of single-sided substrates; Assume that the servo motor only carries the solar panel around y The shaft rotates on a single axis with an angle of θ. θ And since the central rigid body itself has no translational motion, then from the following coordinate system O 0- x 0 y 0 z 0 to the inertial coordinate system O - xyz coordinate transformation matrix A for: S2: Constructing the solar panel vibration displacement function: Assuming the solar panel operates at its natural angular frequency ω Performing simple harmonic motion, any point on the solar panel ( x i , y i , z i vibration displacement function Represented as: in, W i ( x i , y i ) is the first i Mode shape function of the block substrate; The mode shape function W i ( x i , y i It is represented by a linear combination of characteristic orthogonal polynomial basis functions: in, φ m (i) ( x i )and φ n (i) ( y i ) are respectively the solar panel substrate in x i and y i Characteristic orthogonal polynomial basis functions of the direction, m i and n i It is the number of basis functions of the eigenorthogonal polynomials that are truncated. A mn (i) It is the mode shape factor; S3: Construct the energy function of the system: The total kinetic energy of the system T for: in, J m It is the moment of inertia of the central rigid body about the y-axis. ρ It is the bulk density of the windsurfing material. and These are located in the following coordinate system x The first half of the negative half-axis and the second half of the positive half-axis i The volume of the gasket substrate. and These are the velocity vectors of a point on the left and right sides of the sail in the inertial coordinate system, respectively. Strain energy of solar panel vibration deformation U p for: in, E It is the Young's modulus of the windshield material. μ It is the Poisson's ratio of the windsurfing material. h It's the thickness of the windsurfing board. a It is the length of a single substrate, 2 b It is the width of a single substrate; The sum of the elastic potential energies of all hinges U j for: in, k It refers to the torsional stiffness at the hinge. θ ALi , θ BLi , θ ARi , θ BRi These are the hinges on the left and right sides of the sail. A i and B i The angle of torsion; S4: Establish the modal characteristic equations of the system: The motion of the servo motor is decomposed into a wide range of rigid body rotation and elastic vibration coupled with the vibration of the solar panel, and its rotation angle is... Represented as: in, Indicates the rigid body motion. Indicates the elastic vibration component; The elastic vibration part Represented as: in, The coefficients to be determined; According to the Rayleigh-Ritz method, a Lagrangian function Π containing displacement constraints at the hinge is constructed, and the Lagrangian function Π is allowed to take stationary values ​​for all unknown coefficients, thus deriving the characteristic equation of the system as follows: Where X is the coefficient of all unknown mode shapes. ,coefficient And the eigenvectors of the Lagrange multipliers, K is the generalized stiffness matrix, M is the generalized mass matrix, and Λ is the matrix related to the Lagrange multipliers; S5: Numerically solve the modal characteristic equations of the system to obtain the natural frequencies and corresponding mode shapes of the system.

2. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 1, characterized in that: In S1, the body coordinate system O i - x i y i z i The origin O i Located in the following coordinate system x On the 0 axis, and located on the solar panel at a distance from the center of mass of the central rigid body. O The center of the closer side.

3. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 1, characterized in that: In S1, the substrate of the solar panel is considered as a thin plate structure, and the following assumption is made: the thin plate is... z i There is no deformation or stress in the axial direction; the normal to the middle surface of the thin plate is always perpendicular to the middle surface when the plate deforms; there is no displacement of any point on the middle surface of the thin plate in the direction parallel to the middle surface; each of the substrates is isotropic; the flexible hinge is simplified to a hinge with an additional torsion spring, and the mass, size, damping and friction of the torsion spring are ignored.

4. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 1, characterized in that: In S2, the characteristic orthogonal polynomial basis functions are generated through the Gram-Schmidt orthogonalization process.

5. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 1, characterized in that: In S3, hinge A i and B i The torsion angle at the junction is determined by the difference in the angles of the two substrates connected to it at the junction.

6. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 1, characterized in that: In S4, the specific elements of the generalized stiffness matrix K, the generalized mass matrix M, and the matrix Λ are determined by the mode shape function. W i ( x i , y i The characteristic orthogonal polynomial basis functions φ m (i) ( x i )and φ n (i) ( y i The system's geometric parameters, material parameters, and hinge stiffness parameters are calculated.

7. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 1, characterized in that: In S5, the characteristic equation is solved using numerical methods. A series of eigenvalues ​​were obtained. and its corresponding eigenvector X, the eigenvalues The square root is the natural frequency of the system. .

8. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 7, characterized in that: The mode shape coefficients are extracted from the eigenvector X obtained from the solution. and the mode shape coefficient Substitute the mode shape function expression in S2 The mode shapes of the system corresponding to each natural frequency are obtained.

9. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to any one of claims 1 to 8, characterized in that: Following S5, S6 is also included: comparing the system's natural frequencies and mode shapes calculated by the method with the modal analysis results of the same system model established based on finite element analysis (FEA) software, in order to verify the correctness of the modeling method.

10. The rigid-flexible coupling modal modeling method for a space solar panel servo motor system according to claim 9, characterized in that: The FEA software is ANSYS software.

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