CN113420393A - Method for reconstructing profile deformation field of thin-wall flat antenna - Google Patents

Method for reconstructing profile deformation field of thin-wall flat antenna Download PDF

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CN113420393A
CN113420393A CN202110751866.6A CN202110751866A CN113420393A CN 113420393 A CN113420393 A CN 113420393A CN 202110751866 A CN202110751866 A CN 202110751866A CN 113420393 A CN113420393 A CN 113420393A
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李珂翔
高国明
陈杨
宋荣贵
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Abstract

The invention discloses a method for reconstructing a profile deformation field of a thin-wall panel antenna. The algorithm is based on a plate-shell structure deformation theory, a complex structure is discretized into a limited number of units, an optimization algorithm is adopted to arrange sensing positions, a least square mathematical model of measurement strain and theoretical strain is established through a least square variational principle, and a reconstruction equation of the units is obtained. And finally, adopting the idea of Lagrange's shape function interpolation to further obtain the deformation field of the whole structure. In order to realize the purpose, firstly, a plate structure deformation reconstruction equation is established; secondly, establishing a dual-target optimization model of deformation reconstruction precision and robustness based on a finite element model of the high-fidelity antenna panel to obtain an optimal layout scheme of the sensor; and finally, verifying the technology based on a simulation analysis method. The method for reconstructing the profile deformation of the thin-wall flat antenna can measure by using a sensor arranged on one side without arranging the sensor on two sides.

Description

Method for reconstructing profile deformation field of thin-wall flat antenna
Technical Field
The technology belongs to the field of antenna structures, and particularly relates to a method for monitoring the deformation condition of an antenna array surface in real time by adopting a structural deformation reconstruction technology.
Background
Heavy-calibre, the lightweight, high integration is phased array antenna's trend of development, heavy-calibre and lightweight lead to phased array antenna structure rigidity to reduce, the flexibility increases, high integration makes the antenna array face heat sharply increase again and produces the heat load to the array face, then under environmental load and hot effect, the shape of antenna array face produces the structural deformation that surpasss the electromagnetism designer requirement, this further leads to the gain of antenna to reduce, the side lobe level increases, make the performance decline of antenna, and then influence whole system index. Therefore, the deformation of the array surface is reproduced in real time through a deformation reconstruction algorithm, real-time compensation is performed on the deformation of the antenna in advance, and the shape of the antenna array surface is guaranteed to meet requirements. Therefore, a solution to measure the deformation of the structure in real time is a feasible solution to the wavefront conformality problem.
Currently, deformation measurement can be divided into non-contact measurement and contact measurement according to different measurement modes. The non-contact measurement is based on the principle of optical imaging to measure the deformation of the structure, and has the advantages of simplicity, intuition and high measurement precision, but the method needs a certain physical distance from the measured structure, and more identification points are arranged on the surface, so that the shielding of the antenna is inevitable, and the influence of the interference of the external environment is great. For the equipment with larger size, enough measurement equipment is needed, the operation difficulty is higher, the calculated information amount is high, the real-time feedback performance is poor, and the measurement method is not suitable for the real-time deformation measurement of the antenna structure. In contact measurement, a sensor is mounted on a measured object to obtain the deformation of a structure in real time, and currently, a strain sensor is mostly adopted. Because the antenna works in a strong electromagnetic field environment, the conventional strain gauge measurement mode is greatly limited, and because the electromagnetic compatibility between the optical fiber and the microwave does not exist, and along with the rise of the optical fiber grating sensing technology, the optical fiber grating strain sensor is adopted to realize the contact method, which is accepted by the engineering community. Up to now, the structure deformation reconstruction technology is mainly divided into a KO method, a modal method and an inverse finite element method, and compared with the inverse finite element method, the KO method is mainly suitable for unidirectional deformation measurement, and the modal method needs to establish high-fidelity physics, which brings great difficulty to the application of complex structures in engineering. The inverse finite element method does not consider the property and the loading condition of the structural material, any complex structure can be discretely solved by using a finite number of elements, the topological structure is flexible, the method has better practicability, the deformation measurement of the complex structure in multiple directions can be realized, and the method has great advantages in engineering application.
The method adopts an inverse finite element method to realize the deformation measurement of the complex structure in the three-dimensional direction, and scholars at home and abroad perform a large amount of simulation analysis and experimental verification to prove the accuracy and effectiveness of the method. However, for the deformation measurement method of the phased array panel, based on the traditional inverse finite element method, a sensor needs to be installed on the lower surface of the structure, and the installation mode may affect the electrical performance of the antenna reflection unit, so the inverse finite element method based on single-side measurement strain is firstly proposed to realize the deformation reconstruction of the antenna unit on the thin-wall panel antenna panel, and meanwhile, in order to increase the applicability, an embedded fiber bragg grating measurement method is adopted, namely, the optical front grating sensor is embedded into the antenna structure.
Disclosure of Invention
Aiming at the problem that the development requirement of a thin-wall panel antenna cannot be met by depending on the traditional structure shape retention, the invention provides an on-line high-precision deformation reconstruction method based on structural strain measurement for realizing on-line real-time monitoring of the deformation of a thin-wall panel antenna array panel.
The method is based on the plate-shell structure deformation theory, the complex structure is discretized into a limited number of units, the sensing positions are distributed by adopting an optimization algorithm, and a least square mathematical model of the measurement strain and the theoretical strain is established through the least square variational principle to obtain a reconstruction equation of the units. And finally, adopting the idea of Lagrange's shape function interpolation to further obtain the deformation field of the whole structure.
In order to achieve the purpose, the invention adopts the following technical scheme: firstly, establishing a plate structure deformation reconstruction equation; secondly, establishing a dual-target optimization model of deformation reconstruction precision and robustness based on a finite element model of the high-fidelity antenna panel to obtain an optimal layout scheme of the sensor; and finally, verifying the technology based on a simulation analysis method. The method comprises the following specific steps:
step 1, taking the middle surface of the plate-shell structure as a physical coordinate system, and expressing the deformation of any point on the structure and the deformation of the middle surface by the following formula:
Figure BDA0003144982000000021
wherein (u)x,uy,uz) U (x, y), v (x, y), w (x, y) represent the deformation of a point in the neutral plane, and θ represents the deformation of a point at an arbitrary position in the platey(x,y),θx(x, y) is the angular deformation of the neutral plane about the x and y axes, and z is the distance from the point in the plate to the neutral plane.
Step 2, obtaining a mathematical expression of a surface strain field and a middle plane according to the differential relation of deformation and strain:
Figure BDA0003144982000000022
Figure BDA0003144982000000023
wherein:
e1＝u,x(x),e2＝v,y(x,y),e3＝u,y(x,y)+v,x(x, y) is translational strain;
e4＝θ,x(y),e5＝θ,y(x),e6＝θ,y(x,y)+θ,x(x, y) is bending strain;
e7＝w,x(x,z)+θy,e8＝w,y(y,z)+θxis the transverse shear strain;
step 3, arranging a sensor on the surface of the structure for measurement, and establishing the relationship between the strain at the point and the three strains;
ε(xi,yi,θi)＝m([e1(x,y),e2(x,y),e3(x,y)]T (3a)+z[e4(x,y),e5(x,y),e6(x,y)]T)
Figure BDA0003144982000000031
ε(yi,zi,γi)＝l([e1(x,y,z),e2(x,y,z),e3(x,y,z)]T (3c)+z[e4(x,y,z),e5(x,y,z),e6(x,y,z)]T)+q[e7(x,y,z),e8(x,y,z)]
wherein epsilonxx,εyy,εzz,γxy,γxz,γyzRepresenting the strain component, theta, at a certain pointi,βi,γiIndicating the strain direction angle corresponding to a certain point:
m＝[cos2θ,sin2θ,cosθsinθ]；d＝[cos2βi,0,0]；
p＝[cosβisinβi,0]；l＝[0,cos2γi,0]； (i＝1,2,..n)
q＝[0,cosγisinγi]
and 4, establishing a least square relation between theoretical strain and measured strain based on a variational principle:
Figure BDA0003144982000000032
wherein ε represents the theoretical strain εeRepresenting the measured strain, wiRepresenting the weight coefficient and n representing the number of attached sensors in a cell.
And 5, carrying out derivative on the degree of freedom u on the formula 4, and solving a minimum value of the derivative:
keue＝fe (5)
wherein, taking a four-node quadrilateral unit as an example,
ui e＝[ui vi wi θxi θyi θzi]T,i＝(1,2,3,4)
and 6, constructing a lifting spectrum unit shape function (taking a four-node quadrilateral unit as an example) based on the Lagrange shape function according to the unit node deformation obtained above.
Figure BDA0003144982000000041
Wherein the content of the first and second substances,
Figure BDA0003144982000000042
representing the shape function of the ascending spectral unit, N5And expressing the ascending function in the cell, and respectively expressing the normalized coordinates along the x direction and the y direction, and solving the node deformation of the internal cell.
And 7, adopting a method for constructing an interpolation function in the unit, and realizing the reconstruction of the deformation field in the whole unit by utilizing the deformation of the internal nodes in the step 6:
Figure BDA0003144982000000043
u(x,y)＝N1u1+N2u2+N3u3+N4u4+N5u5 (8)
wherein N is1,N2,N3,N4,N5As a new shape function, u1,u2,u3,u4,u5And (4) selecting a node deformation value in the unit.
The invention has the beneficial effects that:
1. based on the plate-shell structure deformation field theory and the minimum variation principle, the deduced mathematical model for calculating the deformation field through strain only needs to be provided with a small number of sensors on one side of the structure, and the reconstruction of the deformation field of the whole structure is realized.
2. In physical sense, the method has no relation with the structure loading condition and the material property, only needs to know the measured value of a small amount of discrete point strain, has flexible topological structure, and can be suitable for the deformation reconstruction of any complex geometric structure spliced by quadrilateral planes.
3. Based on the quadratic interpolation calculation method adopted by the invention, compared with a primary interpolation structure, the method can improve the full structure deformation reconstruction precision, and compared with the primary interpolation under the same precision requirement, the number of required plane units is reduced, thereby reducing the use number of sensors to a certain extent.
Drawings
FIG. 1 is a schematic diagram of a thin-wall planar antenna panel structure;
FIG. 2 is a simplified diagram of a thin-walled planar antenna panel force application;
FIG. 3 is a diagram of simulation measurement points of a reflecting surface unit of a thin-wall panel antenna;
FIG. 4 thin-wall panel antenna panel reflector unit experimental measurement points.
Number designation in fig. 1: 1. thin-wall panel antenna panel, 2, sensor layout route.
Number designation in fig. 2: 3. point No. 1, point No. 4 and point No. 2.
Detailed Description
The following description and the accompanying drawings are used to illustrate the implementation of the present invention, but not to limit the present invention.
The method is based on the plate-shell structure deformation theory, and discretizes the complex structure into a limited number of units; the sensing positions are distributed by adopting an optimization algorithm, so that the measurement of a single-side strain sensor can be realized; establishing a least square mathematical model of the measured strain and the theoretical strain through a least square variational principle to obtain a reconstruction equation of the unit; and adopting the Lagrange shape function interpolation idea to further obtain the deformation field of the whole structure.
Take the deformation loading of the thin-walled planar antenna panel shown in fig. 1 as an example. The invention provides a method for reconstructing a profile deformation field of a thin-wall panel antenna, which comprises the following steps:
step 1, establishing a relation between any point and a middle-surface physical coordinate system by taking a thin-wall panel antenna panel as an object;
step 2, establishing a mathematical expression of a surface strain field and a middle plane according to the differential relation of deformation and strain;
step 3, installing an optical fiber sensor on one side of the thin plate, and establishing the relationship between the strain at the position of the sensor and the translational strain, the bending strain and the transverse shear strain;
step 4, establishing a least square relation between the actually measured strain data of the sensor and the theoretically calculated strain, wherein the difference between the actually measured strain data of the sensor and the theoretically calculated strain is an optimization target;
step 5, calculating a minimum value of the strain difference, and repeating the step 4 and the step 5 to obtain an optimized sensor measuring point, namely realizing reasonable layout of the sensor;
step 6, based on the sensor position layout determined in the steps, solving the deformation of the unit node by using a Lagrange's shape function;
and 7, constructing an interpolation function in the unit, and deducing a deformation field in the whole unit by using the node deformation obtained in the step 6.
In the verification process, different forms of combined forces are applied to the force application positions respectively, so that the antenna panel deforms in different forms, and the deformation of the antenna unit is measured by adopting a dynamic measuring instrument. By adopting the interpolation calculation method, the deformation of the antenna unit is obtained, and as shown in fig. 3 and 4, the deformation is compared with the actual measurement value and the measurement value, so that the accuracy and the effectiveness of the method are verified.
Analysis results
The method is applied to the measurement of the deformation of the array antenna shown in the figure 1, different forms of force combinations are applied to the force application points 1 and 2 respectively to enable the deformation to be generated, and the simulation analysis and experiment methods are adopted to verify the deformation.
As for the simulation results, considering the factors of the antenna geometry, the displacement deformation along the x and y directions and the angular deformation along the x axis of the antenna unit are small, the following table lists the deformation along the z direction and the angular deformation along the y axis, and 24 sampling points calculated by the finite element method are extracted, as shown in fig. 3, and the calculation results are shown in tables 1 and 2. In fig. 3, the sampling point numbers are, from left to right, first rows 1, 4, 24, second rows 5, 10, 13, 16, 19, 22, third rows 6, 9, 12, 15, 18, 21, fourth rows 7, 8, 11, 14, 17, 20, and fifth rows 2, 3, 23.
For the experimental verification results, the antenna units at 36 positions of the antenna array panel are collected as check points by taking the measured data as reference values in the three-coordinate measurement, as shown in fig. 4. In fig. 4, the check point numbers are, from left to right, a first row 1, 2, 3, 4, 17, 18, 19, 20, 21, a second row 5, 6, 7, 8, 22, 23, 24, 25, 26, a third row 9, 10, 11, 12, 27, 28, 29, 30, 31, and a fourth row 13, 14, 15, 16, 32, 33, 34, 35, 36.
The calculation results are shown in tables 3 and 4. By using the RMS value index evaluation factor, and by analyzing the data in tables 1 to 2, the maximum RMS value of a reconstruction result is 0.11mm, the maximum angle deformation is 1.2 degrees, and the maximum reconstructed RMS value is 0.17 degrees when the maximum displacement deformation of the antenna unit is 5.5mm by using the finite element method calculation value as a reference; as can be seen from the data in tables 3 and 4, the maximum deformation amount is 5.0mm and the maximum RMS value of the reconstructed result is 0.14mm, using the measured value as a reference value, which proves the correctness and effectiveness of the inventive patent.
TABLE 1 comparison of measured point deformation (force applied at point 1)
Figure BDA0003144982000000061
TABLE 2 comparison of the deformation at the measurement points (simultaneous force at the application points 1 and 2)
Figure BDA0003144982000000071
TABLE 3 calculation of the deformation at the measurement points (simultaneous application of force at points 1 and 2)
Figure BDA0003144982000000072
Figure BDA0003144982000000081
TABLE 4 calculation of the deformation at the measurement points (simultaneous application of force at points 1 and 2)
Figure BDA0003144982000000082
Figure BDA0003144982000000091

Claims (1)

1. A method for reconstructing a profile deformation field of a thin-wall flat antenna is characterized by comprising the following steps:
step 1, taking the middle surface of the plate-shell structure as a physical coordinate system, and expressing the deformation of any point on the structure and the deformation of the middle surface by the following formula:
Figure FDA0003144981990000011
wherein (u)x,uy,uz) U (x, y), v (x, y), w (x, y) represent the deformation of a point in the neutral plane, and θ represents the deformation of a point at an arbitrary position in the platey(x,y),θx(x, y) is the angular deformation of the neutral plane about the x and y axes, and z is the distance from the point in the plate to the neutral plane;
step 2, obtaining a mathematical expression of a surface strain field and a middle plane according to the differential relation of deformation and strain:
Figure FDA0003144981990000012
Figure FDA0003144981990000013
wherein:
e1＝u,x(x),e2＝v,y(x,y),e3＝u,y(x,y)+v,x(x, y) is translational strain;
e4＝θ,x(y),e5＝θ,y(x),e6＝θ,y(x,y)+θ,x(x, y) is bending strain;
e7＝w,x(x,z)+θy,e8＝w,y(y,z)+θxis the transverse shear strain;
and 3, arranging sensors on the surface of the structure to measure the positions, and establishing the relationship between the strain at the point and three strains:
Figure FDA0003144981990000014
Figure FDA0003144981990000015
Figure FDA0003144981990000016
wherein epsilonxx,εyy,εzz,γxy,γxz,γyzRepresenting the strain component, theta, at a certain pointi,βi,γiRepresenting the strain direction angle corresponding to a certain point;
Figure FDA0003144981990000017
and 4, establishing a least square relation between theoretical strain and measured strain based on a variational principle:
Figure FDA0003144981990000021
wherein ε represents the theoretical strain εeRepresenting the measured strain; w is aiRepresenting a weight coefficient; n represents the number of attached sensors in a cell;
and 5, carrying out derivative on the degree of freedom u, and solving a minimum value:
keue＝fe；
step 6, constructing a raised order spectrum unit shape function based on a Lagrange shape function according to the unit node deformation obtained above;
Figure FDA0003144981990000022
wherein the content of the first and second substances,
Figure FDA0003144981990000023
representing the shape function of the ascending spectral unit, N5Expressing the ascending order function in the unit, and respectively expressing the normalized coordinates along the x direction and the y direction by s and t, and solving the node deformation of the internal unit;
step 7, adopting a method of constructing an interpolation function in the unit, and realizing the reconstruction of a deformation field in the whole unit by utilizing the deformation of the internal nodes in the step 6;
Figure FDA0003144981990000024
u(x,y)＝N1u1+N2u2+N3u3+N4u4+N5u5
wherein N is1,N2,N3,N4,N5As a new shape function, u1,u2,u3,u4,u5And (4) selecting a node deformation value in the unit.
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CN111931395A (en) * 2020-06-22 2020-11-13 江苏理工学院 Sensor measuring point optimization method for reducing strain field reconstruction errors

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Publication number Priority date Publication date Assignee Title
CN107389284A (en) * 2017-06-05 2017-11-24 西安电子科技大学 A kind of measuring method of the frame structure elastic deformation based on strain
CN108895974A (en) * 2018-05-08 2018-11-27 航天东方红卫星有限公司 A kind of malformation fiber-optic monitoring and reconstructing method and system
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