A shape perception method of phased array antenna under complex experimental modes

Abstract

The present application aims at the deficiencies of the prior art, and provides a shape perception method of phased array antenna under complex experimental modalities, which is used to solve the problem that the phased array antenna cannot be reconstructed under complex modalities. The method comprises the following steps: establishing a finite element model, extracting model features, establishing a vibration differential equation of the phased array antenna, reducing the number of degrees of freedom, calculating the real displacement modal matrix after reduction, calculating the real displacement modal matrix before reduction, obtaining modal coordinates, and reconstructing the displacement field.

Description

Technical Field

[0001] This invention belongs to the field of radar antenna technology, specifically relating to a shape sensing method for phased array antennas under complex experimental modes. Background Technology

[0002] Active phased array antennas, with their characteristics of rapid beam scanning, high-speed and flexible beam scheduling, signal energy distribution and conversion, and adaptive adjustment, are widely used in various new missile-borne, spaceborne, airborne, shipborne, and vehicle-mounted weapon platforms in my country. However, the antenna array may operate in service environments such as sunlight exposure, vibration, shock, salt spray, and humidity. This can cause changes in the relative position of the antenna array, resulting in amplitude and phase errors, and a sharp deterioration in the electrical performance of the antenna array. To ensure the reliable operation of the phased array antenna during service, it is necessary to reconstruct the displacement field of the array structure surface and perform corresponding structural deformation compensation and electrical performance compensation based on deformation information to compensate for the effects of array structure deformation.

[0003] Currently, there are two main methods for displacement field reconstruction: (1) using the modal method for displacement field reconstruction. For example, the patent application with application number CN107103111A, entitled "Displacement Field Reconstruction Method Based on Functional Surface Feature Points of Electronic Equipment with Strain Sensors", discloses a displacement field reconstruction method based on functional surface feature points of electronic equipment with strain sensors. It utilizes the modal superposition principle of the structure to represent strain and displacement using mode shapes and strain mode shapes extracted from the finite element model. It also uses the property that the generalized modal coordinates are equal to derive the relationship between strain and displacement. However, this method is based on real modes and is not suitable for reconstructing complex structures with complex modes, such as phased array antennas. (2) Displacement field reconstruction using curvature method. For example, Qiao Xiaoping, Zhu Xiaojin, Zhang Hesheng, et al. Surface reconstruction algorithm based on surface patch splicing [J]. Vibration, Testing and Diagnosis, 2013, 33(1):160-163. The surface is divided into an array of surface patches. Based on the properties of quadratic surfaces, a nonlinear equation system is established using orthogonal curvature and solved to obtain the surface patch equation. This method is extended to the entire surface to reconstruct the shape of the surface. This method requires attaching a lot of sensors, which is not convenient in experiments. Summary of the Invention

[0004] The purpose of this invention is to address the shortcomings of the existing technology by proposing a shape sensing method for phased array antennas under complex experimental modes, which solves the problem of phased array antennas being unable to reconstruct complex modes.

[0005] This invention provides a shape sensing method for phased array antennas under complex experimental modes, the method comprising the following steps:

[0006] Step 1: Establish a finite element model and extract model features.

[0007] A finite element model of the phased array antenna is established in ANSYS, and the first n-order eigenvectors of the m degrees of freedom of the surface to be reconstructed are extracted to form the eigenvector matrix {ψ}. r and the natural frequency ω corresponding to each order of eigenvector ni Damping ratio ξ i , conjugate eigenvalue λ ri , At this point, the eigenvector matrix {ψ} r ∈R m×2n It is a complex matrix.

[0008] Step 2: Establish the vibration differential equation of the phased array antenna.

[0009] For a phased array antenna system with m degrees of freedom, the most general form of its vibration differential equation is as follows:

[0010]

[0011] In the formula, [M], [C], and [K] represent the mass matrix, damping matrix, and stiffness matrix, respectively; u(t), Let be the displacement, velocity, and acceleration matrices at time t, respectively; and f(t) be the disturbance force matrix acting on the system at time t. Assume the external force f(t) is 0. The characteristic solution of the system's differential equation is... Substituting into the vibration differential equation, we get:

[0012]

[0013] Where {ψ} r Let M be the complex eigenvector matrix. Multiply both sides of the above equation by [M]. -1 Then we have:

[0014]

[0015] The extracted 2n feature values ​​λ ri , Substituting into the above equation, we get:

[0016]

[0017] Step 3: Reduce the number of degrees of freedom

[0018] When the number of degrees of freedom m is greater than the truncation order n, the above equation cannot be solved. To solve this problem, we need to use singular value decomposition to solve the m×2n {ψ}. r The matrix is ​​transformed into an n×2n matrix. When the proportional damping is small, the real parts of the eigenvectors of each order can represent all the information of the complex eigenvectors. Therefore, the complex eigenvector matrix {ψ} is used. r Construct the characteristic matrix X using the real part of the eigenvalues:

[0019] [X]=[Re{ψ}1 Re{ψ}2 … Re{ψ} n ]

[0020] Then, singular value decomposition is performed on the characteristic matrix X.

[0021] [X] m×n =[T] m×n [Σ] n×n [V] T n×n

[0022] In the above formula, [X]∈R m×n Describes the complex eigenvector matrix {ψ} r The characteristic matrix formed by the real parts of [T]∈R m×n The transformation matrix is ​​denoted by T, which is obtained by performing singular value decomposition on the eigenma matrix [X]. The transformation matrix [T] is applied to the original complex eigenvector matrix {ψ}. r After reduction, we obtain the eigenvector matrix with n degrees of freedom:

[0023]

[0024] Where [T] m×n It utilizes the complex eigenvector matrix {ψ} r The real part [X] is the transformation matrix obtained through singular value decomposition.

[0025] Step 4: Calculate the reduced real displacement mode matrix

[0026] Using the reduced Substitute into the following formula:

[0027]

[0028] Then [M] can be determined. -1 [K]. Then substitute it into... The real displacement mode matrix after reducing the degrees of freedom can then be obtained. ω r This indicates the natural frequency. At this time... It is a real matrix.

[0029] Step 5: Calculate the real displacement mode matrix before reduction.

[0030] The real displacement mode matrix after reducing the degrees of freedom Substitution The real displacement mode matrix {Φ} before the unreduced degrees of freedom can then be obtained. r , {Φ} r ∈Rm×n .

[0031] Step 6: Obtain modal coordinates

[0032] ε was extracted using the finite element model. k Ψ k , ε k Let Ψ be the strain value corresponding to k observation points in m degrees of freedom. k Let be the strain mode matrix corresponding to k observation points. Using... Find the modal coordinates q k In the formula, Represents Ψ k The false inverse, that is

[0033] Step 7: Reconstruct the displacement field

[0034] After obtaining the real displacement mode matrix {Φ} r and modal coordinates q k Later, according to q={Φ} r q k Find the full-field displacement vector q with m degrees of freedom, q∈R. m×1 Thus, the entire displacement field can be reconstructed.

[0035] The beneficial effects of this invention are:

[0036] Compared with the prior art, the significant advantages of this invention are:

[0037] This solves the problem that when using modal methods to reconstruct complex structures, the displacement mode matrix becomes complex, making it impossible to reconstruct using modal methods. Attached Figure Description

[0038] Figure 1 Flowchart of the implementation of this invention;

[0039] Figure 2 A physical image of a phased array antenna;

[0040] Figure 3 The simplification process of the finite element model of a phased array antenna;

[0041] Figure 4 Actual image of the actuator;

[0042] Figure 5 Finite element model of a phased array antenna;

[0043] Figure 6 Nine locations on the phased array finite element model where actuator constraints are applied. Detailed Implementation

[0044] The technical solutions provided by the present invention will be described in detail below with reference to specific embodiments. It should be understood that the following specific embodiments are only used to illustrate the present invention and are not intended to limit the scope of the present invention.

[0045] Reference Figure 1 This invention provides a shape sensing method for phased array antennas under complex experimental modes, the method comprising the following steps:

[0046] Step 1: Establish a finite element model and extract model features.

[0047] Establish in ANSYS Figure 2 The finite element model of the phased array antenna measures 2.7m × 1.2m. The experimental panel was modeled using shell63 elements, and the aluminum frame was modeled using SOLID92 elements. These two types of elements were connected using the MPC algorithm. The elastic modulus of both elements was set to 271 GPa, the Poisson's ratio to 0.32, and the density to 2650 kg / m³. 3 Because the heat dissipation holes on the phased array experimental panel are too dense, the computational load during mesh generation and analysis becomes excessive, causing lag and excessive time consumption. Therefore, we simplified the circular heat dissipation holes in each square area into square holes. The simplification process is as follows: Figure 3 As shown. Next, the same constraints as those applied to the physical phased array are imposed on the finite element model; that is, constraints such as those added to the finite element model are... Figure 4 The actuator constraints shown are illustrated in the final finite element model. Figure 5 As shown. Next, the first n order displacement eigenvectors of the z-axis translational degrees of freedom of the m nodes on the surface to be reconstructed are extracted to form the eigenvector matrix {ψ}. r And the intrinsic frequency ω corresponding to each order of eigenvector ni Damping ratio ξ i , conjugate eigenvalue λ ri , Because of the external friction between the antenna elements and the array surface of the phased array, the eigenvector matrix {ψ} is affected. r ∈R m×2n It is a complex matrix.

[0048] Step 2: Establish the vibration differential equations for the surface nodes of the phased array antenna.

[0049] In the previous step, after the phased array surface was meshed, there were m nodes. For each node, we only considered its z-axis translational degree of freedom. According to vibration theory, the most general form of the vibration differential equation for a phased array antenna system with m degrees of freedom is as follows:

[0050]

[0051] In the formula, [M], [C], and [K] represent the mass matrix, damping matrix, and stiffness matrix, respectively; u(t), Let f(t) be the displacement, velocity, and acceleration matrices for each degree of freedom at time t; and let f(t) be the disturbance force matrix acting on the system at time t. In a non-proportional damped system, the damping matrix cannot be decoupled in the m-dimensional principal space, therefore the external force f(t) can be set to 0. Then, the characteristic solution of the system's differential equation is: Substituting it into the vibration differential equation, we get:

[0052]

[0053] Where {ψ} r Let M be the complex eigenvector matrix. Multiply both sides of the above equation by [M]. -1 Then we have:

[0054]

[0055] The previously extracted 2n feature values ​​λ ri , Substituting into the above equation, we get:

[0056]

[0057] If we extract the m-th order feature vectors of the surface to be reconstructed, the amount of data required would be too large. Furthermore, for reconstruction, generally only the first 20 or so order feature vectors need to be superimposed; subsequent feature vectors have minimal impact on the reconstruction. Therefore, we generally only extract the first n order feature vectors. Since {ψ} at this point... r ∈R m×2n Solving the above equation at this point is problematic, so we need to change its dimension, that is, reduce the number of nodes m.

[0058] Step 3: Reduce the number of nodes

[0059] When m is generally greater than the truncation order n, the above equation cannot be solved. To solve this problem, singular value decomposition (SVD) can be used to transform the matrix {ψ} r The dimension is transformed from m×2n to n×2n. During this transformation, due to the smaller proportional damping, the real parts of the eigenvectors of each order can essentially represent all the information of the complex eigenvectors. Therefore, the complex eigenvector matrix {ψ} can be used. r Construct the characteristic matrix X using the real part of the eigenvalues:

[0060] [X]=[Re{ψ}1 Re{ψ}2 … Re{ψ} n ]

[0061] Then, singular value decomposition (SVD) is performed on the characteristic matrix X.

[0062] [X] m×n =[T] m×n [Σ] n×n [V] T n×n

[0063] In the above formula, [X]∈R m×n Describes the complex eigenvector matrix {ψ} r The matrix formed by the real parts of [T]∈R m×n The transformation matrix is ​​denoted by T, which is obtained by performing singular value decomposition on the eigenma matrix [X]. The transformation matrix [T] is applied to the original complex eigenvector matrix {ψ}. r Reduce:

[0064]

[0065] This allows us to obtain the feature vector matrix of n nodes. Where [T] m×n It utilizes the complex eigenvector matrix {ψ} r The real part [X] is the transformation matrix obtained through singular value decomposition (SVD), at which point... The purpose of singular value decomposition is to discover and extract linear correlations in eigenvectors. Then, spatial motion is incorporated into the vector of the transformation matrix [T], resulting in the transformed eigenvector matrix. Motion containing linearly independent degrees of freedom. Therefore, the eigenvector matrix... It can be used to reduce the number of nodes.

[0066] Step 4: Calculate the reduced real displacement mode matrix

[0067] At this point, the reduced complex eigenvector matrix is ​​used. Substitute into the following formula:

[0068]

[0069] Then [M] can be determined. -1 [K]. Then, the [M] that was just calculated... -1 [K] Substitute From the formula, the real displacement mode matrix after reducing the degrees of freedom can be obtained. ω r This indicates the natural frequency. At this time... It is a real matrix that can be used to reconstruct the surface of a phased array antenna.

[0070] Step 5: Calculate the real displacement mode matrix before reduction.

[0071] The displacement has m degrees of freedom, so it is also necessary to... The dimension can be transformed from n×n back to m×n. The real displacement mode matrix after reducing the degrees of freedom can be... Substitution By using the transformation matrix [T], the displacement mode matrix after reducing the degrees of freedom can be transformed back to the displacement mode matrix before reducing the degrees of freedom. From this, the real displacement mode matrix {Φ} before reducing the degrees of freedom can be obtained. r Here {Φ} r ∈R m×n It can be used to reconstruct the displacement values ​​of m nodes on the surface of a phased array antenna.

[0072] Step 6: Obtain modal coordinates

[0073] ε was extracted using the finite element model. k and Ψ k , where ε k Ψ represents the strain values ​​corresponding to k observation points selected from m nodes in the finite element model. These strain values ​​are real-time strain values ​​extracted after each load application. k This is the strain mode matrix formed by the first n order strain eigenvectors of these k observation points. The modal superposition principle also applies to strain field reconstruction; therefore, the strain values ​​ε at these k points are... k ε is available k =Ψ k q k Find the value of q here. k Here, ε represents the strain values ​​at k points in the modal coordinates. k It is known, therefore we can utilize Inversely calculate the modal coordinates q k In the formula, Represents Ψ k The false inverse, that is

[0074] Step 7: Reconstruct the displacement field

[0075] {Φ} r and modal coordinates q k Subsequently, based on the principle of modal superposition, q = {Φ} r q k That is, find the total displacement vector q of m nodes, q∈R. m×1 This allows us to reconstruct the entire displacement field on the surface of the phased array antenna.

[0076] The technical effects of the present invention will be further explained below with reference to specific experiments:

[0077] 1. Experimental conditions and contents:

[0078] Finite element analysis was performed in ANSYS 18.0, and the algorithm program was run in MATLAB R2017a. Figure 5The finite element model shown is used for verification. This experiment reconstructs the displacement values ​​of 256 points uniformly distributed across 8 aluminum frames. Simultaneously, 24 points, different from the previous 256 points, are uniformly selected across the 8 aluminum frames for calculating modal coordinates. After applying constraints, the first 20 z-axis eigenvectors of the 256 points are extracted, along with the natural frequency ω corresponding to each eigenvector. ni Damping ratio ξ i , conjugate eigenvalue λ ri , This experiment set up two effective simulation experimental conditions.

[0079] Working condition 1: In the finite element model, 6 actuators at the left and right ends are fixed, while 3 actuators in the middle move forward and backward simultaneously, making equal displacements (bending deformation in the middle). Under this working condition, deformation with two displacements (displacement 1 is 1.2 mm, displacement 2 is 2.5 mm) was carried out, and a total of 26 sets of tests were conducted.

[0080] Working condition 2: In the finite element model, 6 actuators at the left and right ends and 2 actuators at the top and bottom of the middle are fixed, while 1 actuator in the middle moves back and forth (the middle part undergoes concave-convex deformation). Under this working condition, deformation with two displacements (displacement 1 is 1.2mm and displacement 2 is 2.5mm) was carried out, and a total of 26 sets of tests were conducted.

[0081] After each applied displacement load, strain values ​​at 24 points were extracted to calculate the real-time modal coordinates. The displacement values ​​of m nodes in the finite element model extracted from ANSYS were accurate values. By comparing these values ​​with the full-field displacement reconstructed after conversion to complex modes, the percentage error (RPE), root mean square error (RMSE), and maximum error (MAE) were obtained, as shown in Table 1.

[0082] 2. Analysis of experimental results:

[0083] Table 1. Error Statistics of Reconstruction Results under Working Condition 1

[0084]

[0085] Table 2 Error Statistics of Reconstruction Results under Working Condition 2

[0086]

[0087] As shown in Table 1, this invention can transform the eigenvector matrix extracted from the surface of a phased array antenna into a real displacement mode matrix, thereby solving the problem of the inability to reconstruct phased array antenna structures with complex modes. Furthermore, it can be seen that, by comparing the reconstruction results of Condition 2 and Condition 1, the reconstruction accuracy of Condition 1 is slightly higher than that of Condition 2. This is mainly because the constraint conditions of Condition 1 are easier to simulate and analyze, resulting in more accurate constraint modes in Condition 1. Overall, the method proposed in this invention is applicable to the deformation reconstruction of non-proportional damped structures with small damping and has strong versatility.

[0088] The above description is only the best specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the protection scope of the present invention.

[0089] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A shape perception method of a phased array antenna under a plurality of experimental modes, characterized in that, The method steps are as follows: Step 1, establish a finite element model, extract model features; Step 1 is specifically: A finite element model of the phased array antenna is established in ANSYS, and the first n-order eigenvectors of the m degrees of freedom of the surface to be reconstructed are extracted to form a complex eigenvector matrix {ψ}. r and the natural frequency ω corresponding to each order of eigenvector ni Damping ratio ξ i , conjugate eigenvalue λ ri , At this point, the complex eigenvector matrix {ψ} r ∈R m×2n It is a complex matrix; Step 2, establish the vibration differential equation of the phased array antenna; Step 2 is specifically: For a phased array antenna system with m degrees of freedom, the most general form of the vibration differential equation is as follows: In the formula, [M], [C], and [K] represent the mass matrix, damping matrix, and stiffness matrix, respectively; u(t), Let f(t) be the displacement, velocity, and acceleration matrices at time t; f(t) be the disturbance force matrix acting on the system at time t; let the external force f(t) be 0; the characteristic solution of the system's differential equation is... Substituting into the vibration differential equation, we get: where {ψ} r is a complex eigenvector matrix, multiplying both sides of the above equation by [M] -1 then we have: The extracted 2n feature values ​​λ ri , Substituting into the above equation, we get: Step 3, reduce the number of degrees of freedom; Step 3 is specifically: When the proportional damping is small, the real part of each order eigenvector can represent all information of the complex eigenvector; therefore, the feature matrix X is constructed by using the real part of the complex eigenvector matrix {ψ} r : [X] = [Re{ψ}1 Re{ψ}2... Re{ψ} n ] Then singular value decomposition is performed on the characteristic matrix X; [X] m×n = [T] m×n [Σ] n×n [V] T n×n In the above formula, [X] ∈ R m×n represents a characteristic matrix composed of real parts of a complex eigenvector matrix {ψ} r , [T] ∈ R m×n represents a transformation matrix, which is a transformation matrix obtained by singular value decomposition of the characteristic matrix [X]; by reducing the original complex eigenvector matrix {ψ} r by the transformation matrix [T], an eigenvector matrix of n degrees of freedom is obtained: where [T] is a transformation matrix obtained by singular value decomposition of the real part [X] of the complex eigenvector matrix {ψ}, i.e. [T] = {ψ} -1 m×n r ​​ Step 4, calculate the real displacement modal matrix after reduction; Step 5, calculate the real displacement modal matrix before reduction; Step 6, solve the modal coordinates; Step 7, reconstruct the displacement field.

2. The method of claim 1, wherein, Step 4 is specifically: with the reduced substituted into the following equation: Then [M] can be solved -1 [K] ; and then substituting it into The real displacement modal matrix after reduction of degrees of freedom can be solved ω r ω represents the natural frequency; at this time is a real matrix.

3. The method of claim 2, wherein, Step 5 is specifically: The real displacement modal matrix after reduction of the degrees of freedom Substitute The real displacement modal matrix before reduction of the degrees of freedom {Φ} can be obtained r , {Φ} r ∈R m×n .

4. The method of claim 3, wherein, Step 6 is specifically: The ε k is extracted by a finite element model k , the ε k is a strain value corresponding to k observation points in m degrees of freedom, the Ψ k is a strain modal matrix corresponding to k observation points; the modal coordinate q is solved by using ; in the formula, the pseudo-inverse of Ψ k is represented, that is 5. The method of claim 4, wherein, Step 7 is specifically: After the real displacement modal matrix {Φ} r and the modal coordinates q k are obtained, the full-field displacement vector q of m degrees of freedom is obtained according to q = {Φ} r q k , q ∈ R m×1 , and thus the entire displacement field can be reconstructed.

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