Equivalent analysis method for maximum paraboloid assembled antenna module

Abstract

The invention discloses an equivalent analysis method for a maximum paraboloid assembled antenna, which greatly reduces the analysis cost of a module assembled antenna structure, and can quickly and accurately realize the evaluation of natural vibration characteristics and static deformation characteristics of module assembled antenna structures with different calibers. Two analysis methods of theoretical derivation and numerical simulation are combined when the scaling rule is constructed, full verification is carried out, and the scaling rule is accurate and reliable. According to the scaling rule, the mechanical property of the full-size on-orbit operation structure can be reflected through a mechanical test of a ground small-caliber prototype, and the method is suitable for industrial large-scale application and popularization.

Description

An equivalent analysis method for a maximally parabolic assembled antenna module Technical Field

[0001] This invention belongs to the field of large space antenna analysis, specifically relating to an equivalent analysis method for a maximally parabolic assembled antenna. Background Technology

[0002] Large space antennas are core aerospace equipment for missions such as deep space exploration, Earth observation, and security and defense, with sizes reaching hundreds of meters or even kilometers. Due to the size limitations of launch vehicle fairings, these antennas typically need to be loaded into orbit from the ground in a highly retractable folded or segmented manner, and deployed in orbit through a controllable deployment mechanism. For even larger aperture requirements, advanced technologies such as on-orbit assembly, modular splicing, and robot-assisted construction are needed to achieve high-precision structural forming and stable operation. As a cutting-edge direction in global aerospace engineering, extremely large-aperture space antennas not only embody integrated innovation across multiple disciplines such as structure, materials, and control, but also represent the highest level of future space infrastructure construction.

[0003] The on-orbit implementation of extremely large-aperture antennas in space can be mainly divided into three categories: first, independently deployable antennas in space; second, large antennas assembled by astronauts using a space station platform; and third, large antennas constructed autonomously in orbit without human intervention. The first two methods are limited by antenna aperture and space station platform dimensions, making it difficult to meet the on-orbit construction requirements of extremely large-aperture antennas. The third method, autonomous construction without human intervention, involves launching deployable antenna modules via launch vehicles and autonomously combining and assembling multiple modules using a spacecraft platform, and is considered the optimal path to achieving extremely large-aperture antennas. The assembly structure of large-aperture space antennas requires close attention to its inherent vibration characteristics to assess its stiffness and stability during on-orbit service. Furthermore, the static deformation behavior of the antenna during assembly needs to be monitored to assess the impact of assembly loads on the overall structural deformation behavior and surface accuracy. Due to the large size of the space module-assembled antenna structure, it is difficult to conduct full-scale mechanical testing under existing ground unloading schemes and experimental conditions. Summary of the Invention

[0004] The purpose of this invention is to provide an equivalent analysis method for a maximally parabolic assembled antenna, so as to solve the problem that it is difficult to achieve full-size mechanical testing of space module assembled antennas in the existing technology.

[0005] To solve the above-mentioned technical problems, the present invention adopts the following technical solution: an equivalent analysis method for a maximally parabolic assembled antenna module, comprising the following steps: Step 0, determining the number of modular antennas and the spatial geometric position of each modular antenna in the maximally parabolic assembled antenna module to be equivalently analyzed; Step 1, arbitrarily selecting a modular antenna, equating the modular antenna to multiple rods, establishing a finite element model of each rod, forming a geometric model of the modular antenna; Step 2, using the method in Step 1 to traverse each modular antenna, distributing multiple geometric models according to the spatial geometric positions determined in Step 0, forming an assembled model composed of multiple modular antennas; Step 3, constructing the inherent vibration equation of the assembled model based on the mass matrix, displacement matrix, and stiffness matrix of the assembled model, thereby realizing the equivalent analysis of the inherent vibration characteristics of the assembled model; Step 4: Equivalently represent the assembly model as a cantilever beam structure, establish the relationship between the natural frequency and structural length of the assembly model, and determine the ratio between the aperture and natural frequency of the assembly model; based on the ratio between the aperture and natural frequency of the assembly model, establish the scaling law between the antenna aperture structure and the natural frequency in the assembly model; Step 5: Establish the equilibrium equation of the assembly model under external load; apply a concentrated load to the free end of the module antenna, solve the equilibrium equation to obtain the deformation distribution of the assembly model, and then obtain the maximum deformation to achieve the static structural characteristics analysis of the assembly model; Step 6: Equivalently represent the assembly model as a cantilever beam structure, establish the relationship between static deformation and structural length, and achieve the equivalent analysis of static deformation of the module antenna and structural length; Step 7: Based on the relationship between the static deformation and structural length of the cantilever beam structure, establish the scaling law between the antenna aperture and the static deformation of the structure.

[0006] The present invention also has the following features: Further, step 2 includes the following sub-steps: step 21, using the method of step 1 to re-establish the geometric model of each module antenna; step 22, distributing the geometric models of multiple module antennas according to their spatial geometric positions; in the assembly model, two adjacent module antenna components are assembled through a mating surface, and each mating surface is configured with three mating nodes; step 23, applying degree-of-freedom binding constraints to each mating node to form an assembly model composed of multiple module antennas.

[0007] Further, step 3 includes the following sub-steps: Step 31, set the translational and rotational degrees of freedom at the six corner points of the assembled model to 0; Step 32, construct the natural vibration equations and principal vibration equations of the assembled model based on the mass matrix, displacement matrix, and stiffness matrix of the assembled model, as shown below:

[0008]

[0009] Where M represents the mass matrix of the assembled model; K represents the acceleration matrix of the assembled model; K represents the stiffness matrix of the assembled model; u represents the displacement matrix of the assembled model. Represents the eigenvector; Indicates the principal vibration frequency of the assembled model; The phase angle of the principal vibration of the assembled model is represented by t; time is represented by t; step 33, substitute the principal vibration equation into the natural vibration equation to obtain the algebraic secondary equation, as shown below:

[0010] in, The eigenvalues ​​are ω1 to ω6. In step 34, the arithmetic square roots of the first 6 eigenvalues ​​of the second algebraic equation are arranged in ascending order to obtain the first to sixth natural frequencies of the assembled antenna and the corresponding mode shapes, thus completing the equivalent analysis of the inherent vibration characteristics of the assembled model.

[0011] Furthermore, the relationship between the natural frequency of the assembled model in step 4 and the structural length dimension is as follows:

[0012]

[0013]

[0014]

[0015] Where EI represents the bending stiffness of the equivalent cantilever beam structure; L represents the length of the equivalent cantilever beam structure; ρ represents the density of the equivalent cantilever beam structure; and A represents the cross-sectional area of ​​the equivalent cantilever beam structure. This represents the order of the natural vibration; further, the scaling law of the natural frequency of the antenna aperture structure obtained in step 4 is as follows:

[0016] Where a and b represent any two types of antennas; These represent the r-th natural frequencies of the assembly models corresponding to antenna apertures a and b, respectively. These represent the equivalent cantilever beam lengths of the assembly models corresponding to antenna apertures a and b, respectively. These represent the apertures of antennas a and b, respectively.

[0017] Furthermore, step 6 specifically includes the following sub-steps: Step 61, the assembled model is equivalent to a cantilever beam structure, and the bending moment equation of the assembled model is expressed using the following formula:

[0018] This represents the bending moment at any section of the cantilever beam. Concentrated load; Indicates bending moment; Represents the length of the equivalent cantilever beam; Step 62, use the following formula to express the elastic curve equation of the cantilever beam:

[0019] in, Indicates the deflection of a cantilever beam For axial position The second derivative of .

[0020] Step 63: Substitute the elastic curve equation of the cantilever beam into the bending moment equation, and perform two differentials to obtain the formula for calculating the deflection of the cantilever beam, as follows:

[0021] Where C1 and C2 are integration constants; in step 64, substitute the boundary conditions to calculate the values ​​of constants C1 and C2, and obtain the maximum deformation at the free end of the cantilever beam; specifically, the boundary conditions are as follows:

[0022]

[0023] Maximum deformation at the free end of the cantilever beam As shown in the following formula: .

[0024] Furthermore, the scaled-down formula for the static deformation ratio of the module antenna structure in step 7 is as follows:

[0025] in, These represent the maximum static deformation of the assembly model corresponding to antenna apertures a and b, respectively. These represent the equivalent cantilever beam lengths of the assembly models corresponding to antenna apertures a and b, respectively. These represent antenna apertures a and b, respectively.

[0026] Compared with existing technologies, this invention has the following technical advantages: The equivalent analysis method for maximally parabolic assembled antennas significantly reduces the cost of analyzing modular assembled antenna structures, enabling rapid and accurate evaluation of the inherent vibration characteristics and static deformation characteristics of modular assembled antenna structures of different apertures. By combining theoretical derivation and numerical simulation in constructing the scaling law, and after thorough verification, the scaling law is accurate and reliable. Based on this scaling law, the mechanical characteristics of full-size on-orbit operating structures can be reflected through mechanical testing of small-aperture ground prototypes, making it suitable for large-scale industrial use and promotion. Attached Figure Description

[0027] Figure 1 is a flowchart of the equivalent analysis method of the maximally parabolic assembled antenna module of the present invention; Figure 2 is a schematic diagram of the module and assembly structure of the present invention; Figure 3 is the inherent vibration characteristics of a typical module assembly structure of the present invention; Figure 4 is the static equilibrium state of a typical module assembly structure of the present invention. Detailed Implementation

[0028] It should be noted that, unless otherwise specified, all components in this invention are components known in the prior art.

[0029] The following are specific embodiments of the present invention. It should be noted that the present invention is not limited to the following specific embodiments. All equivalent modifications made based on the technical solutions of this application fall within the protection scope of the present invention.

[0030] An equivalent analysis method for a maximally parabolic assembled antenna module includes the following steps: Step 1, the modular antenna is equivalent to multiple rods, and a finite element model of each rod is established to form the geometric model of the modular antenna; Step 2, using the method in Step 1, multiple geometric models of the modular antennas are established; the geometric models of the multiple modular antennas are distributed according to their spatial geometric positions to form an assembled model composed of multiple modular antennas; Step 3, the natural vibration equations of the assembled model are constructed based on the mass matrix, displacement matrix, and stiffness matrix of the assembled model, thereby realizing the equivalent analysis of the natural vibration characteristics of the assembled model; Step 4, the assembled model is equivalent to a cantilever beam structure, and the relationship between the natural frequency of the assembled model and the length dimension of the structure is established. The steps are as follows: Step 5: Establish the equilibrium equation of the assembled model under external load; Apply a concentrated load to the free end of the modular antenna, solve the equilibrium equation to obtain the deformation distribution of the assembled model, and then obtain the maximum deformation to achieve the static structural characteristics analysis of the assembled model; Step 6: Equivalent the assembled model to a cantilever beam structure, establish the relationship between static deformation and structural length, and achieve the equivalent analysis of static deformation and structural length of the modular antenna; Step 7: Based on the relationship between static deformation and structural dimensions of the cantilever beam structure, establish the scaling law of antenna aperture and static deformation of structure.

[0031] In a more specific implementation, step 2 establishes models of multiple modular antennas based on their spatial geometric positions. Assembly between adjacent modular antennas is achieved through mating surfaces, each with three mating nodes. To ensure continuous stiffness transmission at the connection points after assembly, freedom constraints need to be applied to the mating nodes. Figure 1 shows the application of freedom binding constraints between paired mating nodes on the mating surfaces of adjacent modular antennas. Specifically, this includes the following steps: Step 21, using the method in step 1, establish geometric models of multiple modular antennas; Step 22, distribute the geometric models of the multiple modular antennas according to their spatial geometric positions; in the assembly model, adjacent modular antenna components are assembled through mating surfaces, each with three mating nodes; Step 23, apply freedom binding constraints to each mating node to form an assembly model composed of multiple modular antennas. The freedom binding constraints are as follows:

[0032]

[0033] By binding the degrees of freedom, the translational and rotational degrees of freedom of the slave nodes are always consistent with those of the master node, thus achieving an ideal rigid connection between the various module antennas. Finally, a structural model of the three-module assembled antenna is established for mechanical analysis.

[0034] Further, step 3 includes the following sub-steps: Step 31, set the translational and rotational degrees of freedom of the six corner points of the central module antenna of the assembly model to 0; here, six corner points are selected, which correspond to the fixed positions of the entire assembly model.

[0035] Step 32: Construct the natural vibration equations and principal vibration equations of the assembled model, as follows:

[0036] Where M represents the mass matrix of the assembled model; K represents the stiffness matrix of the assembled model; and u represents the displacement matrix of the assembled model. Represents the eigenvector; Represents the acceleration matrix of the assembled model; Indicates the principal vibration frequency of the assembled model; The phase angle of the principal vibration of the assembled model is represented by t; time is represented by t; step 33, the algebraic secondary equations are obtained based on the natural vibration equation and the principal vibration equation, as follows:

[0037] The aforementioned system of algebraic equations has nonzero solutions. The necessary and sufficient condition is:

[0038] in, ω is the eigenvalue; the above equation is the characteristic equation of the assembled model, where ω 2 These are called eigenvalues ​​or eigenvalues, and their values ​​depend only on the system's physical parameters such as stiffness and mass. The arithmetic square root ω of all solutions to the above equation is... i Arranged in ascending order, corresponding to the i-th natural frequency of the system. Let ω... i Substituting into the second algebraic equation system and solving it yields the corresponding... This is called the eigenvector, which represents the i-th mode shape of the system.

[0039] Using the above method, the inherent vibration characteristics of the modular antenna structure are calculated. Since the lower-order modes contribute the most to the overall dynamic response of the structure and can reflect the main vibration characteristics, while the higher-order modes have a smaller impact on the global response, the first six natural frequencies and their mode shapes of the modular antenna structure are extracted to complete the analysis of the inherent vibration characteristics of the system, as shown in Figure 3.

[0040] Step 34: Arrange the arithmetic square roots ω1 to ω6 of the first 6 eigenvalues ​​of the characteristic equation of the assembled model in ascending order to obtain the first to sixth natural frequencies of the assembled antenna and the corresponding mode shapes, thereby realizing the natural vibration characteristic analysis.

[0041] Furthermore, the relationship between the natural frequency of the assembled model in step 4 and the structural length dimension is as follows:

[0042]

[0043]

[0044]

[0045] Where EI represents the bending stiffness of the equivalent cantilever beam structure; L represents the length of the equivalent cantilever beam structure; ρ represents the density of the equivalent cantilever beam structure; and A represents the cross-sectional area of ​​the equivalent cantilever beam structure. It indicates the order of the natural vibration.

[0046] Furthermore, the ratio of the antenna aperture to the natural frequency of the assembled model in step 4 is as follows:

[0047] in, These represent the r-th natural frequencies of the assembly models corresponding to antenna apertures a and b, respectively. These represent the equivalent cantilever beam lengths of the assembly models corresponding to antenna apertures a and b, respectively. These represent antenna apertures a and b, respectively.

[0048] The above work shows that, compared to increasing the aperture D of the modular antenna by a factor of n, the equivalent cantilever beam length of the assembled structure increases by a factor of n, and each natural frequency of the system becomes 1 / n² of the initial modular antenna aperture design. Calculation models of modular antenna structures with different apertures are established, and this step is repeated to obtain and compare the natural frequencies of the assembled modules, verifying the established scaling law of the natural frequencies of the antenna aperture structure.

[0049] Based on the finite element model of the modular antenna structure and the stiffness matrix of the assembled structure, the equilibrium equations of the system under external loads are constructed according to the generalized Hooke's law. A concentrated load is applied to the free end of the modular antenna structure, and the deformation distribution of the system is solved by solving the equilibrium equations, as shown in Figure 3. The maximum deformation is further calculated, thus completing the static characteristic analysis of the modular antenna structure.

[0050] Based on the deformation morphology of the modular antenna structure calculated in the previous step, it is analyzed that the modular antenna structure can be approximately equivalent to a cantilever beam structure under its boundary conditions. The calculation equation for the maximum deflection, i.e., static deformation, of the free end of the cantilever beam structure under concentrated load is derived. Step 6 specifically includes the following sub-steps: Step 61, the assembly model is equivalent to a cantilever beam structure, and the bending moment equation of the assembly model is expressed by the following formula:

[0051] This represents the bending moment at any section of the cantilever beam. Concentrated load; Indicates bending moment; This indicates the equivalent cantilever beam length.

[0052] Step 62, use the following formula to express the elastic curve equation of the cantilever beam:

[0053] in This represents the second derivative of the cantilever beam deflection y with respect to the axial position x.

[0054] Step 63: Substitute the elastic curve equation of the cantilever beam into the bending moment equation, and perform two differentials to obtain the formula for calculating the deflection of the cantilever beam, as follows:

[0055] Where C1 and C2 are integration constants; in step 64, substitute the boundary conditions to calculate the values ​​of integration constants C1 and C2, and obtain the maximum deformation at the free end of the cantilever beam; specifically, the boundary conditions are as follows:

[0056]

[0057] The maximum deformation at the free end of the cantilever beam is as follows: .

[0058] Specifically, without changing the material properties, cross-sectional shape, and geometric configuration of the modular antenna members, the equivalent cantilever beam length L of the assembled structure changes accordingly after the aperture D of the modular antenna is changed. Based on the relationship between the static deformation and structural dimensions of the cantilever beam structure established in step 8, the proportional relationship of static deformation of the modular antenna assembly structure with different apertures is further derived and extended, forming a scaled-down formula for the proportional relationship of static deformation of the modular antenna structure, as follows:

[0059] in, These represent the maximum static deformation of the assembly model corresponding to antenna apertures a and b, respectively. These represent the equivalent cantilever beam lengths of the assembly models corresponding to antenna apertures a and b, respectively. These represent antenna apertures a and b, respectively.

[0060] The above work shows that, compared to increasing the aperture of the modular antenna by a factor of n, the equivalent cantilever beam length of the assembled structure increases by a factor of n, and under the same static load, the maximum static deformation of the system increases by n. 3 The calculation model of the antenna structure assembled with modules of different apertures is established. This step is repeated to verify the scaling law of the antenna aperture and the static deformation of the structure.

Claims

1. An equivalent analysis method for a maximally parabolic assembled antenna module, characterized in that, Includes the following steps: Step 0: Determine the number of modular antennas and the spatial geometric position of each modular antenna in the maximally parabolic assembled antenna module to be equivalently analyzed; Step 1: Randomly select a modular antenna, and treat it as multiple rods, establishing a finite element model for each rod to form the geometric model of the modular antenna; Step 2: Use the method in Step 1 to traverse each modular antenna, distributing multiple geometric models according to the spatial geometric positions determined in Step 0, forming an assembled model composed of multiple modular antennas; Step 3: Construct the natural vibration equations of the assembled model based on the mass matrix, displacement matrix, and stiffness matrix, thereby achieving equivalent analysis of the natural vibration characteristics of the assembled model; Step 4: Treat the assembled model as an equivalent cantilever beam structure, establish the relationship between the natural frequency and the structural length of the assembled model, and determine the ratio between the aperture and the natural frequency of the assembled model; based on the ratio between the aperture and the natural frequency of the assembled model, establish the scaling law between the antenna aperture structure and the natural frequency in the assembled model; Step 5: Establish the equilibrium equations of the assembled model under external loads; A concentrated load is applied to the free end of the modular antenna, and the equilibrium equation is solved to obtain the deformation distribution of the assembly model, thereby obtaining the maximum deformation and realizing the structural static characteristics analysis of the assembly model; Step 6, the assembly model is equivalent to a cantilever beam structure, and the relationship between static deformation and structural length is established to realize the equivalent analysis of static deformation and structural length of the modular antenna; Step 7, based on the relationship between static deformation and structural length of the cantilever beam structure, the scaling law between antenna aperture and structural static deformation is established.

2. The equivalent analysis method for a maximally parabolic assembled antenna module as described in claim 1, characterized in that, Step 2 includes the following sub-steps: Step 21: Use the method in Step 1 to re-establish the geometric model of each module antenna; Step 22: Distribute the geometric models of multiple module antennas according to their spatial geometric positions; In the assembly model, two adjacent module antennas are assembled through a mating surface, and each mating surface is configured with three mating nodes; Step 23: Apply degree-of-freedom binding constraints to each mating node to form an assembly model composed of multiple module antennas.

3. The equivalent analysis method for a maximally parabolic assembled antenna module as described in claim 2, characterized in that, Step 3 The process includes the following steps: Step 31, set the translational and rotational degrees of freedom at the six corner points of the assembled model to 0; Step 32, construct the natural vibration equations and principal vibration equations of the assembled model based on its mass matrix, displacement matrix, and stiffness matrix, as shown below: Where M represents the mass matrix of the assembled model; K represents the acceleration matrix of the assembled model; K represents the stiffness matrix of the assembled model; u represents the displacement matrix of the assembled model. Represents the eigenvector; Indicates the principal vibration frequency of the assembled model; The phase angle of the principal vibration of the assembled model is represented by t; time is represented by t; step 33, substitute the principal vibration equation into the natural vibration equation to obtain the algebraic secondary equation, as shown below: in, The eigenvalues ​​are ω1 to ω6. In step 34, the arithmetic square roots of the first 6 eigenvalues ​​of the second algebraic equation are arranged in ascending order to obtain the first to sixth natural frequencies of the assembled antenna and the corresponding mode shapes, thus completing the equivalent analysis of the inherent vibration characteristics of the assembled model.

4. The equivalent analysis method for a maximally parabolic assembled antenna module as described in claim 3, characterized in that, The natural frequency of the assembled model in step 4 is related to the structural length dimension as follows: Where EI represents the bending stiffness of the equivalent cantilever beam structure; L represents the length of the equivalent cantilever beam structure; ρ represents the density of the equivalent cantilever beam structure; and A represents the cross-sectional area of ​​the equivalent cantilever beam structure. It indicates the order of the natural vibration.

5. The equivalent analysis method for a maximally parabolic assembled antenna module as described in claim 3, characterized in that, The scaling law of the natural frequency of the antenna aperture structure obtained in step 4 is as follows: Where a and b represent any two types of antennas; These represent the r-th natural frequencies of the assembly models corresponding to antenna apertures a and b, respectively. These represent the equivalent cantilever beam lengths of the assembly models corresponding to antenna apertures a and b, respectively. These represent the apertures of antennas a and b, respectively.

6. The equivalent analysis method for a maximally parabolic assembled antenna module as described in claim 5, characterized in that, Step 6 specifically includes the following sub-steps: Step 61, the assembled model is equivalent to a cantilever beam structure, and the bending moment equation of the assembled model is expressed by the following formula: This represents the bending moment at any section of the cantilever beam; Concentrated load; Indicates bending moment; Represents the length of the equivalent cantilever beam; Step 62, use the following formula to express the elastic curve equation of the cantilever beam: in, Indicates the deflection of a cantilever beam For axial position The second derivative of .

7. Step 63: Substitute the elastic curve equation of the cantilever beam into the bending moment equation, perform two differentials, and obtain the formula for calculating the deflection of the cantilever beam, as follows: in, C1 and C2 are integration constants; in step 64, substitute the boundary conditions to calculate the values ​​of constants C1 and C2, and obtain the maximum deformation at the free end of the cantilever beam; specifically, the boundary conditions are as follows: Maximum deformation at the free end of the cantilever beam As shown in the following formula: 。 8. The equivalent analysis method for a maximally parabolic assembled antenna module as described in claim 6, characterized in that, The scaled-down formula for the static deformation ratio of the module antenna structure in step 7 is as follows: in, These represent the maximum static deformation of the assembly model corresponding to antenna apertures a and b, respectively. These represent the equivalent cantilever beam lengths of the assembly models corresponding to antenna apertures a and b, respectively. These represent antenna apertures a and b, respectively.

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