A method for predicting nonlinear stiffness softening effect of rigid-flexible coupling spacecraft based on improved multi-scale method

Abstract

The application provides a kind of rigid-flexible coupling spacecraft nonlinear stiffness softening effect prediction method based on improved multiscale method, the core is to improve the solution format of traditional multiscale method, for accurately predicting the nonlinear dynamic response of rigid-flexible coupling spacecraft, especially reveal the stiffness softening effect caused by coupling nonlinear inertial force and its influence on system stability.The method first establishes the dynamic model of the central rigid body-flexible beam-additional mass system, then obtains the accurate analytical expression through the reconstruction of multiscale method perturbation solution format, not only can clearly reveal that nonlinear coupling inertial force (Coriolis force, tangential inertial force, centrifugal inertial force) is the physical root of the overall stiffness softening of the system, and can quantitatively analyze the contribution of each inertial force, while having the advantages of strong applicability and comprehensive prediction ability.

Description

Technical Field

[0001] This invention relates to the field of spacecraft dynamics and control technology, and in particular to a method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on an improved multi-scale method. Background Technology

[0002] Nonlinear stiffness soft / hardening effect refers to the excitation frequency shift that occurs when a spacecraft system experiences principal resonance under the influence of nonlinear factors (coupling nonlinearity between components, nonlinearity of the structure itself, etc.). In modern aerospace structures, spacecraft cabins, truss bases, solar panels, and robotic arms constitute typical rigid-flexible coupled complex structural systems. Among them, truss-based solar panel structures typically exhibit significant flexible characteristics. However, under the influence of spacecraft control forces and external extreme space environment excitation forces, a strong nonlinear rigid-flexible coupling effect will occur between its flexible vibration and cabin attitude motion. At the same time, because the structure's natural frequency is low, it is easy to overlap with the external excitation frequency, inducing system resonance and seriously affecting the stable operation of the spacecraft. Therefore, accurately predicting the nonlinear stiffness soft / hardening effect of rigid-flexible coupled spacecraft systems is of great significance for the dynamic analysis and stability prediction of the structure.

[0003] In terms of analytical methods, current approaches mainly rely on linearized model analysis and purely numerical nonlinear model analysis. The former simplifies the analysis by ignoring or linearizing nonlinear terms in the system. While computationally simple, this method cannot accurately predict long-term on-orbit dynamic behavior or capture the stiffness softening effect caused by coupled inertial forces. The latter, while highly accurate, is computationally expensive and cannot reveal the intrinsic analytical relationship between system parameters (such as damping, stiffness, and excitation amplitude) and dynamic responses (such as resonant frequency and stability), making it difficult to guide controller design and parameter optimization. Among these methods, the multi-scale method, as a representative of analytical methods for nonlinear system analysis, can reveal the bifurcation mechanism and dependence of key parameters. However, for the highly complex coupled nonlinear equations of spacecraft systems, directly applying traditional multi-scale method solution formats can lead to solution biases, limiting its applicability.

[0004] In summary, there is an urgent need to develop a unified and high-precision analytical framework to systematically analyze the stiffness softening effect of rigid-flexible coupled spacecraft, in order to further reveal the dynamic behavior mechanisms such as nonlinear response and stability. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a high-precision and high-efficiency analytical calculation method for nonlinear dynamic modeling, response analysis and stability prediction of spacecraft with large flexible attachments. This method is used to accurately predict the nonlinear dynamic response of rigid-flexible coupled spacecraft, and in particular, to reveal the stiffness softening effect caused by coupled nonlinear inertial forces and its impact on system stability. The core of this invention lies in improving the solution format of the traditional multi-scale method.

[0006] The technical solution of this invention is as follows:

[0007] A method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on an improved multi-scale method, characterized by the following specific steps:

[0008] Step 1: Establish a dynamic model of the central rigid body-flexible beam-additional mass system and linearize it. Then, based on the linear system, calculate the natural frequencies and modal functions of the system. Finally, obtain the final response equation based on the orthogonality condition between the modes.

[0009] Step 2: Introduce the multi-scale method to solve the response equation obtained in Step 1, and then explain the stiffness softening effect of the system according to the non-resonance and main resonance cases respectively.

[0010] Step 3: Following the process in Step 2, solve the nonlinear correlation parameters under different load conditions and main resonance, and quantitatively analyze the degree of nonlinear stiffness softening of the system.

[0011] Furthermore, step 1 specifically includes:

[0012] Sub-step 1.1: Define a physical model of a rigid-flexible coupled spacecraft system based on the hybrid coordinate method, specify all parameters of the system, calculate the coupled dynamic equations of the spacecraft system, and simultaneously give the boundary conditions of the system;

[0013] Sub-step 1.2: Linearize the spacecraft system coupled dynamic equations and boundary conditions obtained in sub-step 1.1, and then calculate the first-order natural frequencies of the system. and modal functions;

[0014] Sub-step 1.3: Obtain the orthogonality condition of the system based on the linearized equation and boundary conditions obtained in sub-step 1.2, and then obtain the response equation of the system based on the orthogonality condition and the equation and boundary conditions obtained in sub-step 1.1.

[0015] Furthermore, in sub-step 1.1:

[0016] The physical model of the rigid-flexible coupled spacecraft system includes: a central rigid body with rotational degrees of freedom, solar panels that are symmetrical on both sides and are respectively regarded as flexible beam structures, and a modular tuning device at the free end of the flexible beam.

[0017] All parameters of the system include:

[0018] The length of the central rigid body is The moment of inertia is ;

[0019] The length, Young's modulus, linear density, moment of inertia, and cross-sectional area of ​​the flexible beam structure are respectively... , ;

[0020] The mass of the module tuning device is ;

[0021] The central rigid body platform is subject to time. Related control bending moment ,in and To control the torque amplitude and frequency;

[0022] The system consists of a moving coordinate system O-xy and a globally fixed coordinate system O-XY.

[0023] Angular displacement of the central rigid body at time t And the flexible beam (arbitrary) Lateral displacement at the location ,in and These are the first flexible beams First-order mode function and response coordinates This refers to the number of modes selected for the flexible beam;

[0024] The motion of the flexible beam in different coordinate systems is described using the mixed coordinate method, and the coupled dynamic equations of the spacecraft system in continuous form are obtained based on Hamilton's principle:

[0025] (1)

[0026] (2)

[0027] Among them, generalized moment of inertia and defining functions The expressions are as follows:

[0028] (3)

[0029] (4)

[0030] The boundary conditions of the system are:

[0031] (5).

[0032] Furthermore, in sub-step 1.2:

[0033] The equations and boundary conditions after linearization are as follows:

[0034] (6)

[0035] (7)

[0036] (8)

[0037] Substituting equations (6) and (7) into equation (8) yields the frequency equation to be solved. ,in To match the system frequency The relevant coefficient matrix, This is the modal coefficient vector; then Newton's method is used to analyze the characteristic equation. Solve to determine the first-order natural frequency of the system. Then, substituting the obtained frequency back into the frequency equation, we can directly solve for the solution. ;

[0038] The mode function is:

[0039] (9)

[0040] in, The expression is:

[0041] (10)

[0042] In the formula ,in Let be the i-th natural frequency of the system.

[0043] Furthermore, in sub-step 1.3:

[0044] The orthogonality condition of the system is:

[0045] (11)

[0046] (12)

[0047] in, For the system's first First mode; For the relevant functions of the system, its expression is: ; and It is a constant;

[0048] Considering only the first-order bending mode, the response equation of the system is:

[0049] (13)

[0050] in, This is the first-order response function of the flexible beam; ; ; ; For structural damping; and The damping coefficient is related to the structural mass and stiffness.

[0051] Further dimensionless processing of equation (13) yields:

[0052] (14)

[0053] The response equation of the system then simplifies to the following form:

[0054] (15)

[0055] in,

[0056] (16).

[0057] Furthermore, step 2 specifically includes:

[0058] Introducing time scale ,in For small parameters, these time scales are treated as independent variables, and the expression for the response function is:

[0059] (17)

[0060] The derivative operator is defined as follows:

[0061] (18)

[0062] (19)

[0063] Next, the stiffness softening effect of the system is explained in two cases: non-resonance and main resonance.

[0064] Furthermore, the steps for explaining the stiffness softening effect of the system under the non-resonance condition include:

[0065] First, the dimensionless external excitation frequency in the non-resonance case. Far from the first-order dimensionless natural frequency At this point, equation (15) can be written as:

[0066] (20)

[0067] in,

[0068] (twenty one)

[0069] Next, substitute equations (17), (18), and (19) into equation (20) and compare. The same power yields the following series of linear partial differential equations:

[0070] (twenty two)

[0071] (twenty three)

[0072] Solving equation (22) yields:

[0073] (twenty four)

[0074] in, ; and All are time scales Real functions; ; ;

[0075] Substituting equation (24) into equation (23), we get:

[0076] (25)

[0077] (26)

[0078] in, These are the correlation coefficients of the harmonic order term in the equation;

[0079] To eliminate the effect of the long-term term, the coefficients of some exponential terms in equations (25) and (26) are required to be 0, as shown below:

[0080] (27)

[0081] (28)

[0082] According to equation (27), we can solve for... Let these be constants related to the initial conditions, and then... Substituting the expression into equation (28), the magnitude of the system response can be obtained by separating the real and imaginary parts and solving for them. With phase The expression, further combined with the initial conditions, yields the final first-order multi-scale perturbation solution expression for the undamped nonlinear system as follows:

[0083] (29)

[0084] Then, ignore the part in equation (20) The direct integration yields the response expression for the undamped linear system as follows:

[0085] (30)

[0086] in, These are the angular displacement of the central rigid body, the lateral displacement of the flexible beam, the displacement amplitude, and the phase calculated using the linear model. The correlation coefficient was calculated based on the initial conditions;

[0087] Finally, by comparing and analyzing the solutions of the nonlinear system and the linear system, the occurrence of the nonlinear softening effect is explained.

[0088] Furthermore, the steps for explaining the stiffness softening effect of the system under the main resonance condition include:

[0089] First, the dimensionless external excitation frequency under the main resonance condition. First-order dimensionless natural frequency Nearby and meets ,in For the tuning parameters, equation (15) can be written as:

[0090] (31)

[0091] in, ;

[0092] Next, substitute equations (17), (18), and (19) into equation (31) and compare. The same power yields the following series of linear partial differential equations:

[0093] (32)

[0094] (33)

[0095] Solving equation (31) yields:

[0096] (34)

[0097] Substituting equation (33) into equation (32), we obtain the condition for eliminating the long-term term:

[0098] (35)

[0099] Will Substituting the expression into equation (34), separating the real and imaginary parts, we have:

[0100] (36)

[0101] Among them, generalized amplitude ;

[0102] To obtain the steady-state oscillation solution of the rigid-flexible coupled spacecraft system, let , , the amplitude of the steady solution and phase The following relationship must be satisfied:

[0103] (37)

[0104] (38)

[0105] According to equations (37) and (38), the amplitude-frequency response equation is:

[0106] (39)

[0107] Linearizing equation (36) yields information about the perturbation. and Autonomous differential equation:

[0108] (40)

[0109] eliminate The characteristic equation is obtained as follows:

[0110] (41)

[0111] At this point, based on the Lyapunov stability theory, the stability of the steady-state amplitude as a function of the external excitation frequency is analyzed. When the multi-scale solution curve shows asymmetry and resonance peak shift, it indicates the occurrence of stiffness softening effect.

[0112] Furthermore, in step 3, the quantitative analysis of the nonlinear stiffness softening degree of the system requires: according to equation (20), whether to consider... Representing different nonlinear inertial force conditions, the maximum amplitude error of the principal resonance steady-state vibration under different nonlinear inertial force conditions is defined. and the maximum amplitude frequency offset .

[0113] The beneficial effects of this invention are as follows:

[0114] 1. The method proposed in this invention can obtain an accurate analytical expression by reconstructing the multi-scale perturbation solution scheme, and at the same time, by accurately describing the motion state of the system, the established dynamic model is more accurate;

[0115] 2. The multi-scale method for analyzing nonlinear spacecraft systems provides a stronger physical insight. It can not only clearly reveal that nonlinear coupled inertial forces (Coriolis force, tangential inertial force, and centrifugal inertial force) are the physical root cause of the softening of the overall system stiffness, but also quantitatively analyze the contribution of each inertial force.

[0116] 3. The method proposed in this invention has strong applicability and comprehensive predictive ability. It can not only calculate the non-resonance response, but also accurately predict the amplitude-frequency response curve, multi-valued solution region, jump and hysteresis phenomena under the main resonance, as well as the stable and unstable regions of the system. Attached Figure Description

[0117] Figure 1 A physical model diagram of a rigid-flexible coupled spacecraft system defined by the method proposed in this invention;

[0118] Figure 2 Comparison of dynamic responses of undamped spacecraft systems under different models and methods, where (a) is the response at the free end of a flexible beam; and (b) is the response of a rigid body during rotational angular displacement.

[0119] Figure 3 The amplitude-frequency response curve of the main resonance of the flexible beam;

[0120] Figure 4 The amplitude-frequency curves of the main resonance under different combinations of nonlinear inertial forces;

[0121] Figure 5 This is a flowchart of the method proposed in this invention. Detailed Implementation

[0122] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. These embodiments are only used to illustrate the present invention and are not intended to limit the scope of protection of the present invention.

[0123] This invention proposes a method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on an improved multi-scale method. This method can accurately and quickly obtain the multi-scale solution of the structural vibration response, and, for the main resonance condition, derive the amplitude-frequency characteristic equation and related nonlinear dynamic parameters under steady-state vibration of the system, such as... Figure 5 As shown, the specific steps are as follows:

[0124] Step 1: Establish a dynamic model of the central rigid body-flexible beam-additional mass system and linearize it. Then, based on the linear system, calculate the natural frequencies and modal functions of the system. Finally, obtain the final response equation according to the orthogonality condition between modes. This includes the following sub-steps:

[0125] Sub-step 1.1: Define a physical model of a rigid-flexible coupled spacecraft system based on the hybrid coordinate method, specify all system parameters, calculate the coupled dynamic equations of the spacecraft system, and simultaneously provide the system's boundary conditions. Specifically:

[0126] See Figure 1 The physical model of the spacecraft system includes a central rigid body with rotational degrees of freedom, and two parts that can be considered as having lengths of... The flexible beam structure consists of symmetrical solar panels on both sides and modular tuning devices at the free ends of the flexible beam, wherein the length of the central rigid body is... The moment of inertia is The Young's modulus, linear density, moment of inertia, and cross-sectional area of ​​the flexible beam structure are respectively... The modular tuning device can be considered as having a mass of Concentrated mass;

[0127] In addition, the action on a rigid body platform is related to time. Related control bending moment ,in and To control the torque amplitude and frequency; O-xy and O-XY represent the servo coordinate system fixed to the system and the global fixed coordinate system, respectively; and Represent Angular displacement of rigid body rotation at time and arbitrary flexible beam Lateral displacement at the location;

[0128] The motion of the flexible beam in different coordinate systems is described using the mixed coordinate method, and the continuous form of the coupled dynamic equations of the spacecraft system can be obtained based on Hamilton's principle:

[0129] (1)

[0130] (2)

[0131] Among them, generalized moment of inertia and defining functions The expressions are as follows:

[0132] (3)

[0133] (4)

[0134] The system boundary conditions are given below:

[0135] (5)

[0136] Sub-step 1.2: Linearize the spacecraft system coupled dynamic equations and boundary conditions obtained in sub-step 1.1, and then calculate the first-order natural frequencies of the system. and mode functions , specifically:

[0137] According to the modal superposition method It can be written as ,in, and These are the first flexible beams First-order mode function and response coordinates; This refers to the number of modes selected for the flexible beam;

[0138] Ignoring the nonlinear terms and external excitations in equations (1), (2), and (5), the linearized equations and boundary conditions can be written as follows:

[0139] (6)

[0140] (7)

[0141] (8)

[0142] The expression for the modal function can be written as:

[0143] (9)

[0144] Among them, coefficient It needs to be determined through boundary conditions; The expression is:

[0145] (10)

[0146] In the formula ,in Let i be the natural frequency of the system. Substituting equations (6) and (7) into equation (8) yields the frequency equation to be solved. ,in To and The relevant coefficient matrix, Given the modal coefficient vector, the first-order frequencies can be numerically solved using Newton's method. ;Will Substitution The eigenvectors obtained from this are the coefficients of the first-order mode function. .

[0147] Sub-step 1.3: Based on the linearized equations and boundary conditions obtained in sub-step 1.2, the orthogonality condition of the system is obtained. Then, based on this orthogonality condition and the equations and boundary conditions obtained in sub-step 1.1, the response equation of the system is obtained. Specifically:

[0148] The first introduction of the system First mode and related functions According to equations (6), (7), and (8), the orthogonality condition of the system is:

[0149] (11)

[0150] (12)

[0151] in, ; and It is a constant;

[0152] Based on equations (1)(2)(5)(11)(12), and considering only the first-order bending mode, the response equation of the system is obtained:

[0153] (13)

[0154] in, This is the first-order response function of the flexible beam; ; ; ; For structural damping; and The damping coefficient is related to the structural mass and stiffness.

[0155] Dimensionless processing of equation (13) yields:

[0156] (14)

[0157] Then equation (13) can be simplified to the following form:

[0158] (15)

[0159] in,

[0160] (16)

[0161] Step 2: Solve the response equation obtained in Step 1 using the multi-scale method, and then explain the stiffness softening effect of the system according to the non-resonance and main resonance cases respectively. The specific steps are as follows:

[0162] Introducing time scale ,in For small parameters, treating these time scales as independent variables, the response function can be expressed as:

[0163] (17)

[0164] The derivative operator is defined as follows:

[0165] (18)

[0166] (19)

[0167] Scenario 1: Explain the stiffness softening effect of the system based on the non-resonance case, specifically:

[0168] First, the dimensionless external excitation frequency in the non-resonance case. Far from the first-order dimensionless natural frequency At this point, equation (15) can be written as:

[0169] (20)

[0170] in,

[0171] (twenty one)

[0172] Then, substitute equations (17), (18), and (19) into equation (20) and compare. The same power can yield the following series of linear partial differential equations:

[0173] (twenty two)

[0174] (twenty three)

[0175] Solving equation (22) yields:

[0176] (twenty four)

[0177] in, ; and All are time scales Real functions; ; ;

[0178] Substituting equation (24) into equation (23) yields the following:

[0179] (25)

[0180] (26)

[0181] in, These are the correlation coefficients of the harmonic order term in the equation;

[0182] To eliminate the effect of the long-term term, the coefficients of some exponential terms in equations (25) and (26) are required to be 0, as shown below:

[0183] (27)

[0184] (28)

[0185] According to equation (27), we can solve for... Let these be constants related to the initial conditions, and then... Substituting the expression into equation (28), the magnitude of the system response can be obtained by separating the real and imaginary parts and solving for them. With phase The expression, further combined with the initial conditions, yields the final first-order multi-scale perturbation solution expression for the undamped case as follows:

[0186] (29)

[0187] Finally, to explain the occurrence of the nonlinear softening effect, it is necessary to solve and compare the linear system. The solution of the linear system can be obtained by solving equation (20). Ignoring direct integration, the expression for the response of the linear system in the undamped case is:

[0188] (30)

[0189] in, These represent the angular displacement of the central rigid body, the lateral displacement of the flexible beam, the displacement amplitude, and the phase calculated using the linear model. The correlation coefficient was calculated based on the initial conditions;

[0190] The calculated response of the undamped system is as follows: Figure 2 As shown, (a) is the transverse vibration displacement curve of the free end of the flexible beam within 200s, and (b) is the angular velocity curve of the central rigid body within 200s. The solid line, dashed line, and dotted line are respectively the result curves obtained from the multi-scale solution of the nonlinear model (29), the curve calculated by the Runge-Kutta method of the nonlinear model, and the curve obtained from the solution of the linear model (30). The results not only verify the accuracy of the multi-scale solution, but also show that the period of the nonlinear model solution is higher than that of the linear model solution under long-term vibration, indicating the occurrence of stiffness softening behavior.

[0191] Scenario 2: Explain the stiffness softening effect of the system based on the main resonance situation, specifically:

[0192] First, the dimensionless external excitation frequency under the main resonance condition. First-order dimensionless natural frequency Nearby and meets ,in For the tuning parameters, equation (15) can be written as:

[0193] (31)

[0194] in, ;

[0195] Then, substitute equations (17), (18), and (19) into equation (31) and compare. The same power can yield the following series of linear partial differential equations:

[0196] (32)

[0197] (33)

[0198] Solving equation (31) yields:

[0199] (34)

[0200] Substituting equation (33) into equation (32), we obtain the condition for eliminating the long-term term:

[0201] (35)

[0202] Will Substituting the expression into equation (34), separating the real and imaginary parts, we have:

[0203] (36)

[0204] Among them, generalized amplitude ;

[0205] To obtain the steady-state oscillation solution of the rigid-flexible coupled spacecraft system, let , , the amplitude of the steady solution and phase The following relationship must be satisfied:

[0206] (37)

[0207] (38)

[0208] According to equations (37) and (38), the amplitude-frequency response equation can be obtained as follows:

[0209] (39)

[0210] The curve showing the relationship between steady-state amplitude and external excitation can be obtained from equation (39);

[0211] Linearizing equation (36) yields information about the perturbation. and Autonomous differential equation:

[0212] (40)

[0213] eliminate The characteristic equation can then be obtained as:

[0214] (41)

[0215] At this point, based on Lyapunov stability theory, the stability of the steady-state amplitude as a function of the external excitation frequency can be analyzed, see... Figure 3 The areas marked with solid and dashed lines;

[0216] To demonstrate the accuracy of the steady-state solution, Figure 3 The discrete dots in the figure give the steady-state response amplitudes at different external excitation frequencies calculated using the Runge-Kutta method. It can be seen that the numerical solution and the multi-scale solution are in good agreement. Meanwhile, the asymmetry of the multi-scale solution curve and the leftward shift of the resonance peak indicate the occurrence of stiffness softening effect.

[0217] Step 3: Following the process in Step 2, solve for the nonlinear correlation parameters under different load conditions and principal resonance, and quantitatively analyze the degree of nonlinear stiffness softening of the system. Specifically:

[0218] According to equation (20), should we consider...? Representing different nonlinear inertial force conditions, the maximum amplitude error of the principal resonance steady-state vibration under different nonlinear inertial force conditions is defined. and the maximum amplitude frequency offset The degree of nonlinear stiffness softening of the system can be quantitatively analyzed.

[0219] The spacecraft structure and external excitation parameters in the embodiments adopted in this invention are as follows: , , , , , , , , , , , The implementation case was verified and analyzed, and the analysis results under different nonlinear inertial force conditions are shown in Table 1 and... Figure 4As shown.

[0220] Table 1 Comparison of nonlinear characteristics of the system under different inertial force conditions

[0221]

[0222] As can be seen from Table 1, the presence of nonlinear inertial force terms can cause the stiffness of spacecraft to soften or harden, and the degree of softening or hardening depends on the magnitude of the inertial force. Figure 4 The results show that the centrifugal inertial force ( (terms) and tangential inertial force ( The term will cause the system's stiffness to soften, while the Coriolis inertial force ( The item will cause stiffness hardening, and under the speed limit in the previous section, the strength of the three factors on the system is Coriolis inertial force > centrifugal inertial force > tangential inertial force.

Claims

1. A method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on an improved multi-scale method, characterized in that, The specific steps include: Step 1: Establish a dynamic model of the central rigid body-flexible beam-additional mass system and linearize it. Then, based on the linear system, calculate the natural frequencies and modal functions of the system. Finally, obtain the final response equation based on the orthogonality condition between the modes. Step 2: Introduce a multi-scale method to solve the response equation obtained in Step 1, that is, introduce a time scale. ,in For small parameters, these time scales are treated as independent variables, and the expression for the response function is: (17) in, for The angular displacement of the central rigid body at time t; for Modal participation factor of angular displacement over time scale; for Modal participation factor of angular displacement over time scale; These are the first-order response coordinates of the flexible beam; for Modal participation coefficients of the first-order response coordinates over a given time scale; for Modal participation coefficients of the first-order response coordinates over a given time scale; Define the derivative operator as follows: (18) (19) in, It is a partial differential form on a time scale; Both represent variable symbols; The stiffness softening effect of the system is then explained based on the non-resonance and main resonance cases respectively. Step 3: Following the process in Step 2, solve the nonlinear correlation parameters under different load conditions and main resonance, and quantitatively analyze the degree of nonlinear stiffness softening of the system.

2. The method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on the improved multi-scale method as described in claim 1, characterized in that, Step 1 specifically includes: Sub-step 1.1: Define a physical model of a rigid-flexible coupled spacecraft system based on the hybrid coordinate method, specify all parameters of the system, calculate the coupled dynamic equations of the spacecraft system, and simultaneously give the boundary conditions of the system; Sub-step 1.2: Linearize the spacecraft system coupled dynamic equations and boundary conditions obtained in sub-step 1.1, and then calculate the first-order natural frequencies of the system. and modal functions; Sub-step 1.3: Obtain the orthogonality condition of the system based on the linearized equation and boundary conditions obtained in sub-step 1.2, and then obtain the response equation of the system based on the orthogonality condition and the equation and boundary conditions obtained in sub-step 1.

1.

3. The method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on the improved multi-scale method as described in claim 2, characterized in that, In sub-step 1.1: The physical model of the rigid-flexible coupled spacecraft system includes: a central rigid body with rotational degrees of freedom, solar panels that are symmetrical on both sides and are respectively regarded as flexible beam structures, and a modular tuning device at the free end of the flexible beam. All parameters of the system include: The length of the central rigid body is The moment of inertia is ; The length, Young's modulus, linear density, moment of inertia, and cross-sectional area of ​​the flexible beam structure are respectively... , ; The mass of the module tuning device is ; The central rigid body platform is subject to time. Related control bending moment ,in and To control the torque amplitude and frequency; The system consists of a moving coordinate system O-xy and a globally fixed coordinate system O-XY. Angular displacement of the central rigid body at time t And the flexible beam (arbitrary) Lateral displacement at the location ,in and These are the first flexible beams First-order mode function and response coordinates This refers to the number of modes selected for the flexible beam; The motion of the flexible beam in different coordinate systems is described using the mixed coordinate method, and the coupled dynamic equations of the spacecraft system in continuous form are obtained based on Hamilton's principle: (1) (2) in, Generalized moment of inertia and defining functions The expressions are as follows: (3) (4) The boundary conditions of the system are: (5)。 4. The method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on the improved multi-scale method as described in claim 3, characterized in that, In sub-step 1.2: The equations and boundary conditions after linearization are as follows: (6) (7) (8) Substituting equations (6) and (7) into equation (8) yields the frequency equation to be solved. ,in To match the system frequency The relevant coefficient matrix, This is the modal coefficient vector; then Newton's method is used to analyze the characteristic equation. Solve to determine the first-order natural frequency of the system. Then, substituting the obtained frequency back into the frequency equation, we can directly solve for the solution. ; The mode function is: (9) in, The expression is: (10) In the formula ,in Let be the i-th natural frequency of the system.

5. The method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on the improved multi-scale method as described in claim 4, characterized in that, In sub-step 1.3: The orthogonality condition of the system is: (11) (12) in, For the system's first First mode; For the relevant functions of the system, its expression is: ; and It is a constant; Considering only the first-order bending mode, the response equation of the system is: (13) in, This is the first-order response function of the flexible beam; ; ; ; For structural damping; and The damping coefficient is related to the structural mass and stiffness. Further dimensionless processing of equation (13) yields: (14) The response equation of the system then simplifies to the following form: (15) in, (16)。 6. The method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on the improved multi-scale method as described in claim 5, characterized in that, The steps for explaining the stiffness softening effect of the system under non-resonance conditions include: First, the dimensionless external excitation frequency in the non-resonance case. Far from the first-order dimensionless natural frequency At this point, equation (15) can be written as: (20) in, (21) Next, substitute equations (17), (18), and (19) into equation (20) and compare. The same power yields the following series of linear partial differential equations: (22) (23) Solving equation (22) yields: (24) in, ; and All are time scales Real functions; ; ;cc represents the conjugate of the preceding term; Substituting equation (24) into equation (23), we get: (25) (26) in, These are the correlation coefficients of the harmonic order term in the equation; To eliminate the effect of the long-term term, the coefficients of some exponential terms in equations (25) and (26) are required to be 0, as shown below: (27) (28) According to equation (27), we can solve for... Let these be constants related to the initial conditions, and then... Substituting the expression into equation (28), the magnitude of the system response can be obtained by separating the real and imaginary parts and solving for them. With phase The expression, further combined with the initial conditions, yields the final first-order multi-scale perturbation solution expression for the undamped nonlinear system as follows: (29) Then, ignore the part in equation (20) The direct integration yields the response expression for the undamped linear system as follows: (30) in, These are the angular displacement of the central rigid body, the lateral displacement of the flexible beam, the displacement amplitude, and the phase calculated using the linear model. The correlation coefficient was calculated based on the initial conditions; Finally, a comparative analysis of the solutions of nonlinear and linear systems demonstrates the occurrence of the nonlinear softening effect.

7. The method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on the improved multi-scale method as described in claim 6, characterized in that, The steps for explaining the stiffness softening effect of the system under the main resonance condition include: First, the dimensionless external excitation frequency under the main resonance condition. First-order dimensionless natural frequency Nearby and meets ,in For the tuning parameters, equation (15) can be written as: (31) in, , This refers to the external excitation amplitude after introducing small parameters; Next, substitute equations (17), (18), and (19) into equation (31) and compare. The same power yields the following series of linear partial differential equations: (32) (33) Solving equation (31) yields: (34) Substituting equation (33) into equation (32), we obtain the condition for eliminating the long-term term: (35) Will Substituting the expression into equation (34), separating the real and imaginary parts, we have: (36) Among them, generalized amplitude ; To obtain the steady-state oscillation solution of the rigid-flexible coupled spacecraft system, let , , the amplitude of the steady solution and phase The following relationship must be satisfied: (37) (38) According to equations (37) and (38), the amplitude-frequency response equation is: (39) Linearizing equation (36) yields information about the perturbation. and Autonomous differential equation: (40) eliminate The characteristic equation is obtained as follows: (41) At this point, based on the Lyapunov stability theory, the stability of the steady-state amplitude as a function of the external excitation frequency is analyzed. When the multi-scale solution curve shows asymmetry and resonance peak shift, it indicates the occurrence of stiffness softening effect.

8. The method for predicting the nonlinear stiffness softening effect of rigid-flexible coupled spacecraft based on the improved multi-scale method as described in claim 7, characterized in that, In step 3, the degree of nonlinear stiffness softening of the system needs to be quantitatively analyzed: according to equation (20), should the following be considered? Representing different nonlinear inertial force conditions, the maximum amplitude error of the principal resonance steady-state vibration under different nonlinear inertial force conditions is defined. and the maximum amplitude frequency offset .

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