An online measurement method and system for dynamic deformation of distributed wing aperture antenna main beam structure based on FBG strain sensor

Abstract

The invention discloses an online measurement method and system for the dynamic deformation of a distributed wing aperture antenna main beam structure based on an FBG strain sensor. The method comprises: determining the strain of any point on the surface of a wing main beam according to a constructed nine-degree-of-freedom displacement field to obtain a constitutive relationship; determining the order of an interpolation shape function according to the constitutive relationship, and establishing an interpolation function of an inverse unit displacement field of the wing main beam; determining the neutral axis strain function of the wing main beam based on the measured strain detected by the FBG strain sensor, and obtaining a measured neutral axis strain function of any point on the wing main beam; then constructing an online evaluation model for the deformation of the wing main beam, and obtaining unknown node degrees of freedom of the wing main beam through the online evaluation model for the deformation of the wing main beam; the invention ensures that the measurement result is more accurate by considering the influence of the in-plane distortion and out-of-plane warping deformation of the wing main beam on the spatial posture of the antenna unit of an airborne platform, and realizes the nine-degree-of-freedom online perception of the wing main beam based on the online measurement of FBG strain.

Description

Technical Field

[0001] The present invention belongs to the technical field of online measurement of wing deformation, and in particular relates to an online measurement method and system for dynamic deformation of a distributed wing aperture antenna main beam structure based on an FBG strain sensor. Background Art

[0002] The AWACS is the core hub of modern systemized operations, with a series of functions such as long-range detection, battlefield command and communication, friend-or-foe identification, and electronic countermeasures. The integrated design of radar, antenna electronic equipment and airborne platform can not only enhance the detection power of AWACS, but also improve the aerodynamic performance of the aircraft. However, in actual service, when the wing is subjected to aerodynamic loads, inertial loads, vibration and other loads during high-speed flight, the antenna structure of the airborne platform is deformed, resulting in the attenuation of antenna performance. To this end, it is necessary to measure the deformation of the array surface in real time and compensate for the real-time phase through the deformation data. At the same time, advanced transformable aircraft achieve the function of "deformation on demand" through built-in sensors, actuators and control systems. During the flight, the aircraft will make the wing adaptively and actively deform according to the mission requirements and the current deformation of the wing. This not only helps to improve the aerodynamic efficiency to improve the aircraft's endurance, but also can control flutter to improve the aircraft's safety performance and reduce the radar mark to improve the aircraft's stealth performance. Therefore, whether it is electromagnetic compensation of the airborne platform or the flight attitude control of the transformable body, the premise is the accurate online measurement of the wing deformation.

[0003] At present, the wing deformation measurement methods are mainly divided into contact and non-contact methods. The non-contact method uses auxiliary devices such as stereo cameras and scanners to obtain the surface image of the measured structure to achieve high-precision deformation measurement. However, this method has problems such as poor portability of auxiliary test equipment, limited deformation measurement range, and measurement results are easily affected by the environment. The contact method is to attach or implant the sensor to the object to be measured for contact measurement, and use real-time acquisition of sensor parameters such as strain, acceleration, pressure, etc., combined with various advanced deformation measurement algorithms to achieve high-precision real-time measurement of wing deformation. The main methods for establishing strain-displacement deformation perception models currently include the following: curvature method, neural network method, modal method and inverse finite element method; curvature method: based on classical beam theory, a large number of strain sensor networks are used to measure discrete point strains, and the functional relationship between the measured strain and deformation displacement is established based on the strain integral method or the strain-displacement linear relationship based on the finite element. This method is only suitable for unidirectional deflection deformation monitoring. Neural network method: Based on the generalization ability of neural and fuzzy networks, the strain-displacement deformation perception relationship is constructed. This method is prone to the "one-to-many problem" when applied to the deformation perception of the conformal load-bearing antenna unit of a large aspect ratio wing. That is, a set of measured strains corresponds to multiple deformation displacements. Modal method: Based on segmented or global shape functions, a small amount of discrete point strain information is used to reconstruct the entire deformation field. The modal vibration shape of this method is heavily dependent on the finite element model. Inverse finite element method: The neutral axis strain field is solved by a small number of discrete point strain values, and then the functional relationship between the measured strain and the deformation displacement is established using functionals. This method is only related to the measured strain, and has nothing to do with information such as structural material properties, inertia / damping coefficients, and external loads.

[0004] The distributed wing aperture antenna structure usually consists of skin, transverse ribs and main beam (longitudinal I-beam), such as Figure 1 As shown in the figure. During the service life of the wing, the main beam bears the main load and the cross section undergoes translational deformation, distortion deformation and warping deformation. However, the existing deformation-aware modeling methods all treat the cross section of the main beam as a rigid body without considering cross-sectional distortion and warping deformation, resulting in low modeling reconstruction accuracy and poor compensation effect of the antenna's comprehensive electrical performance. Summary of the invention

[0005] In order to overcome the shortcomings of the above-mentioned prior art, the purpose of the present invention is to provide an online measurement method and system for the dynamic deformation of the main beam structure of a distributed wing aperture antenna based on an FBG strain sensor. By considering the influence of the in-plane distortion and out-of-plane warping deformation of the wing main beam (I-beam) on the spatial posture of the airborne platform antenna unit, the measurement result can be guaranteed to be more accurate, and the nine-degree-of-freedom online perception of the wing main beam based on the online measurement of FBG strain can be realized, laying the foundation for the electrical performance compensation of the antenna.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] An online measurement method for dynamic deformation of a distributed wing aperture antenna main beam structure based on an FBG strain sensor comprises the following steps:

[0008] Step 1: Construct a nine-degree-of-freedom displacement field, deduce the strain at any point on the surface of the wing main beam based on the nine-degree-of-freedom displacement field, and obtain the constitutive relationship;

[0009] Step 2: According to the constitutive relationship in step 1, the interpolation shape function order is derived and the interpolation function of the inverse unit displacement field of the wing main beam is established;

[0010] Step 3: Determine the measured neutral axis strain function of the wing main beam based on the measured strain detected by the FBG strain sensor, and obtain the measured neutral axis strain function of any point of the wing main beam;

[0011] Step 4: According to the interpolation function of the inverse unit displacement field of the wing main beam in step 2 and the measured neutral axis strain function of any point of the wing main beam in step 3, an online evaluation model of the wing main beam deformation is constructed. Through the online evaluation model of the wing main beam deformation, the unknown node degrees of freedom of the wing main beam are obtained.

[0012] When constructing the nine-degree-of-freedom displacement field in step 1, the following basic assumptions are made for the wing main beam:

[0013] a. Ignore the mid-surface shear strain of the upper flange, lower flange and middle web;

[0014] b. Ignore the shear strain of the middle web ( and ), shear strain of upper flange and lower flange ( and );

[0015] c. Assume that the upper flange and the lower flange are The plane is rigid;

[0016] d. The middle web is The plane is deformable and the axial strain caused by cross-sectional distortion is neglected.

[0017] The nine degrees of freedom of the wing main beam in step 1 include along the coordinate axis The translational degrees of freedom , around the coordinate axis The angular freedom and the distortion degree of freedom, which consists of a warping degree of freedom and two cross-sectional distortion degrees of freedom, the cross-sectional distortion degrees of freedom including the symmetric distortion degrees of freedom and antisymmetric distortion degrees of freedom .

[0018] The expression of the nine-degree-of-freedom displacement field in step 1 is:

[0019] (2a)

[0020] (2b)

[0021] (2c)

[0022] In the formula, They represent the points on the surface of the wing main beam along Direction translation displacement, operator Represents the coordinate axis Derivation , the three lines in the curly brackets represent the cross-sectional deformation of the upper flange, the middle web and the lower flange from top to bottom. represents the neutral axis shear displacement, represents the area function of the cross section of the wing main beam, represents the warping degree of freedom, Indicates the distance between the upper flange and the lower flange mid-surface. represents the first-order derivative of the symmetric distortion degree of freedom, represents the first-order derivative of the antisymmetric distortion degree of freedom, The deformation shape function of the middle web of the wing main beam during symmetrical deformation, represents the deformation shape function of the middle web of the wing main beam during antisymmetric deformation, Representation function About variables The first derivative of Representation function About variables The first derivative of represents the symmetrical distortion degrees of freedom of the wing main beam, represents the antisymmetric distortion degree of freedom of the wing main beam, Respectively express about The angular degrees of freedom of the coordinate axes, They respectively represent the displacement of any point on the neutral axis along the coordinate axis.

[0023] The order of the interpolation shape function in step 2 is: neutral axis strain and is a constant, is the linear neutral axis strain, the neutral axis strain and As a function of the same order, that is, a quadratic function; the neutral axis strain and Neutral axis strain and Combined with the boundary conditions, the interpolation function of the inverse unit displacement field of the wing main beam obtained by the interpolation shape function order is expressed as follows:

[0024] , , (10a)

[0025] , , (10b)

[0026] , , (10c)

[0027] in, They represent the displacement of any point on the neutral axis along the coordinate axis, Respectively express about The angular degrees of freedom of the coordinate axes, They represent the antisymmetric distortion degree of freedom, the symmetric distortion degree of freedom and the warping degree of freedom respectively. denote the first-order and third-order interpolation shape functions, respectively. They represent the unknown node variables displaced along the coordinate axis, Respectively represent the The unknown nodal variables of the axis rotational degrees of freedom, the antisymmetric distortion degrees of freedom, and the symmetric distortion degrees of freedom, For the coordinate axis Derivative.

[0028] The construction process of the neutral axis strain function of the wing main beam in step 3 is as follows:

[0029] Step 3.1: Calculate the neutral axis strain of discrete points on the wing main beam surface. The expressions are:

[0030] , (12b)

[0031] , (12e) (13c)

[0032] (13d)

[0033] (14b)

[0034] in, and Represent the surface strain measurement values ​​of the upper flange and the lower flange, Indicates the distance between the upper flange and the lower flange mid-surface. Indicates the thickness of the upper flange, represents the thickness of the middle web, Indicates that The calculated axial strain of the middle web is, Represent the measured strains of the upper and lower flanges, Indicates the installation position of the FBG strain sensor Towards coordinates, Indicates the position of the upper flange surface Towards coordinates, represents Poisson's ratio, Indicates the axial position The warping degrees of freedom at Indicates FBG strain sensors installed Axis coordinates, Indicates the FBG strain sensor and the coordinate axis The angle of

[0035] Step 3.2: According to the neutral axis strain of discrete points on the surface of the wing main beam and the order of the neutral axis strain function, establish the measured neutral axis strain function of the wing main beam;

[0036] The measured neutral axis function of the wing main beam is:

[0037] (15a)

[0038] The parameter constraints are:

[0039] , , (15c)

[0040] In the formula, Indicates unknown parameters.

[0041] The measured neutral axis strain function at any point of the wing main beam is:

[0042] (16b)

[0043] In the formula, represents the unknown parameter vector, , , represents the FBG strain sensor position matrix;

[0044] in, The subscripts in the Row and The elements of the column, The subscripts in the to Row and to Elements of a column.

[0045] The unknown node degrees of freedom of the wing main beam in step 4 ,matrix is a function of the coordinates of the displacement points to be evaluated on the wing main beam, and is related to the interpolation function of the inverse unit displacement field of the wing main beam; the matrix Related to the measured neutral axis strain at any point on the wing spar.

[0046] The present invention also provides an online measurement system for dynamic deformation of a distributed wing aperture antenna main beam structure based on a FBG strain sensor, comprising:

[0047] The first calculation module is used to construct a nine-degree-of-freedom displacement field, and derive the strain of any point on the surface of the wing main beam according to the nine-degree-of-freedom displacement field to obtain the constitutive relationship;

[0048] The second calculation module is used to derive the interpolation shape function order according to the constitutive relationship and establish the interpolation function of the inverse unit displacement field of the wing main beam;

[0049] The third calculation module is used to determine the neutral axis strain function of the wing main beam based on the measured strain detected by the FBG strain sensor, and obtain the measured neutral axis strain function of any point of the wing main beam;

[0050] The fourth calculation module: According to the interpolation function of the inverse unit displacement field of the wing main beam and the measured neutral axis strain function of any point of the wing main beam, an online evaluation model of the wing main beam deformation is constructed, and the unknown node degrees of freedom of the wing main beam are obtained through the online evaluation model of the wing main beam deformation.

[0051] Compared with the prior art, the present invention has the following beneficial effects:

[0052] Based on the thin-wall deformation hypothesis of the wing main beam, the present invention integrates the cross-sectional distortion and warping factors into the Timoshenko beam theory, and establishes a more accurate displacement field function; by establishing the relationship between the external load and the neutral axis strain, the order of the displacement field interpolation function is determined; an algorithm for solving the neutral axis strain using surface discrete strain and a surface strain sensor layout scheme are proposed, and the neutral axis strain function is solved using a minimum number of strain sensors. The nine-degree-of-freedom wing main beam deformation online monitoring model established by the present invention does not depend on the structural material properties, and the measurement result is more accurate, realizing the nine-degree-of-freedom online perception of the wing main beam based on FBG strain online measurement, laying a foundation for antenna electrical performance compensation. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 Schematic diagram of the structure of the distributed wing aperture antenna.

[0054] Figure 2a It is a three-dimensional diagram of the I-beam of the present invention.

[0055] Figure 2b The I-beam of the present invention Floor plan.

[0056] Figure 3a It is a schematic diagram of the deformation of the cross section of the present invention with symmetrical bending.

[0057] Figure 3b It is a schematic diagram of the deformation of the cross-section antisymmetric bending of the present invention.

[0058] Figure 4 It is a schematic diagram of the axial force, cross-sectional force, moment and stress of the I-beam of the present invention.

[0059] Figure 5a It is a schematic diagram of installing the FBG strain sensor of the present invention on the upper flange surface.

[0060] Figure 5b It is a schematic diagram of the installation of the FBG strain sensor of the present invention on the side of the middle web. DETAILED DESCRIPTION

[0061] The technical solution of the present invention is further described below in conjunction with the accompanying drawings.

[0062] An online measurement method for dynamic deformation of a distributed wing aperture antenna main beam structure based on an FBG strain sensor comprises the following steps:

[0063] Step 1: Construct a nine-degree-of-freedom displacement field, deduce the strain at any point on the surface of the wing main beam based on the nine-degree-of-freedom displacement field, and obtain the constitutive relationship;

[0064] Step 1.1: The wing main beam is usually an I-beam composed of an upper flange, a lower flange and an intermediate web. In order to accurately describe the deformation state of the wing main beam, the following basic assumptions are made for the I-beam:

[0065] a. Ignore the mid-surface shear strain of the upper flange, lower flange and middle web;

[0066] b. Ignore the shear strain of the middle web ( and ), shear strain of upper flange and lower flange ( and );

[0067] c. Assume that the upper flange and the lower flange are The plane is rigid;

[0068] d. The middle web is The plane is deformable, and the axial strain caused by cross-sectional distortion is ignored;

[0069] Step 1.2: Based on the basic assumptions of step 1.1, the nine degrees of freedom of the wing main beam are determined by analyzing the distortion of the upper flange and the lower flange of the wing main beam, and the deformation shape function of the middle web is obtained;

[0070] The present invention regards the distortion deformation of the upper flange and the lower flange of the wing main beam as a pair of mechanical coupling deformations. Therefore, the nine degrees of freedom of the wing main beam include the The translational freedom , around the coordinate axis The angular freedom and the distortion degree of freedom, which consists of a warping degree of freedom and two cross-sectional distortion degrees of freedom, such as Figure 2a and Figure 2b As shown; the cross-sectional distortion degrees of freedom include the symmetric distortion degrees of freedom and antisymmetric distortion degrees of freedom , where the symmetric distortion degree of freedom is It is formed by the reverse twisting of the flange and the symmetrical bending of the middle web, such as Figure 3a As shown in the figure, the deformation shape function of the middle web of the wing main beam is expressed as:

[0071] (1a)

[0072] In the formula, Indicates the distance between the upper flange and the lower flange mid-surface;

[0073] Antisymmetric distortion degrees of freedom It is formed by the reverse shear deformation of the flange and the antisymmetric bending of the middle web, such as Figure 3b As shown in the figure, the deformation shape function of the middle web of the wing main beam is expressed as:

[0074] (1b)

[0075] Step 1.3: According to the deformation shape function of the middle web in step 1.2, correct the displacement of any point on the surface of the wing main beam in the direction of the coordinate axis to obtain the nine-degree-of-freedom displacement field, which is expressed as follows:

[0076] (2a)

[0077] (2b)

[0078] (2c)

[0079] In the formula, They represent the points on the surface of the wing main beam along Direction translation displacement, operator Represents the coordinate axis Derivation , the three lines in the curly brackets represent the cross-sectional deformation of the upper flange, the middle web and the lower flange from top to bottom. represents the neutral axis shear displacement, represents the area function of the cross section of the wing main beam, represents the warping degree of freedom, Indicates the distance between the upper flange and the lower flange mid-surface. represents the first-order derivative of the symmetric distortion degree of freedom, represents the first-order derivative of the antisymmetric distortion degree of freedom, The deformation shape function of the middle web of the wing main beam during symmetrical deformation, represents the deformation shape function of the middle web of the wing main beam during antisymmetric deformation, Representation function About variables The first derivative of Representation function About variables The first derivative of represents the symmetrical distortion degrees of freedom of the wing main beam, represents the antisymmetric distortion degree of freedom of the wing main beam, Respectively express about The angular degrees of freedom of the coordinate axes, They represent the displacement of any point on the neutral axis along the coordinate axis;

[0080] Among them, the area function of the cross section of the wing main beam is Defined as:

[0081] (3)

[0082] Warping freedom Defined as:

[0083] (4)

[0084] In the formula, Indicates about The first derivative of the angular freedom of the coordinate axis;

[0085] Based on the linear elastic deformation theory, the strain at any point on the surface of the wing main beam can be expressed as follows:

[0086] (5a)

[0087] (5b)

[0088] (5c)

[0089] In the formula, represents the axial strain, express The in-plane shear strain, express The in-plane shear strain, represents the theoretical neutral axis strain of the wing spar, represents the area function of the cross section of the wing main beam, represents the cross-sectional function, which is obtained by taking the partial derivative of the formula inside the curly brackets of formula (2) and is expressed as:

[0090] , , (5d)

[0091] The nine-degree-of-freedom displacement field function and its first-order derivative constitute the neutral axis strain of the wing main beam, that is, , the specific form is as follows:

[0092] , , (6a)

[0093] , , (6b)

[0094] , , (6c)

[0095] In the formula, represents the first-order derivative of the translational degree of freedom, Indicates about The first derivative of the angular freedom of the coordinate axis, Indicates about The first derivative of the angular freedom of the coordinate axis, represents the first-order derivative of the warping degree of freedom, represents the symmetric distortion degrees of freedom, represents the antisymmetric distortion degree of freedom, Indicates about The first derivative of the angular freedom of the coordinate axis, represents the first-order derivative of the symmetric distortion degree of freedom, represents the first derivative of the antisymmetric distortion degree of freedom;

[0096] At the same time, if Figure 4 As shown in the figure, for a homogeneous beam structure, the relationship between axial force, section force and moment and stress is:

[0097] , , , (7a)

[0098] , , (7b)

[0099] In the formula, represents the axial force, Indicates along The cross-sectional force in the axial direction, Indicates along The cross-sectional force in the axial direction, represents the double bending moment, Indicates winding The bending moment of the shaft rotation, Indicates winding The bending moment of the shaft rotation, represents the elastic modulus, represents the shear modulus, Indicates winding The torque of the shaft rotation, represents the Saint-Venant twist function, represents the warping torsion function, represents the cross-sectional area, represents the axial strain of the wing main spar, express The in-plane shear strain, represents the area function of the cross section of the wing spar;

[0100] Then, substituting equation (5) into equation (7), the constitutive relation is obtained as follows:

[0101] , , , (8a)

[0102] (8b)

[0103] In the formula, is the axial stiffness, and is the bending stiffness, Indicates along The moment of inertia about the axis, Indicates along Moment of inertia about the axis; is the warping constant, and It is related to the cross-sectional dimensions of the wing main beam. Indicates the thickness of the flange, Indicates the width of the flange; for a given material and structural dimensions, the parameter , , , , and are constant parameters, The neutral axis strain The first derivative of ;

[0104] Step 2: According to the constitutive relationship in step 1, the interpolation shape function order is derived and the interpolation function of the inverse unit displacement field of the wing main beam is established;

[0105] The non-uniform rational B-spline (NURBS) basis function is used as the shape function to interpolate the node degrees of freedom to form a displacement field interpolation function, which can be expressed in mathematical form as follows:

[0106] (9)

[0107] In the formula, Indicates indivual interpolation shape function, represents the node degrees of freedom;

[0108] According to the force balance equation, when the axial force and bending moment act on the end of the wing main beam, the neutral axis strain can be derived using equation (8): is a constant, is the linear neutral axis strain; furthermore, it can be deduced that the warping degree of freedom is a parabolic function. According to the constitutive equation of the Euler-Bernoulli beam, the symmetric distortion degree of freedom is set is linear, and thus the neutral axis strain is derived is a constant. Combining equation (5b) and equation (8b), the present invention proposes to convert the neutral axis strain and the neutral axis strain As a function of the same order, i.e. a quadratic function, the torque equation (8b) is guaranteed to hold; the neutral axis strain and Neutral axis strain and Determined in combination with boundary conditions. Therefore, the present invention combines the derived interpolation shape function order to express the interpolation function of the inverse unit displacement field of the wing main beam as follows:

[0109] , , (10a)

[0110] , , (10b)

[0111] , , (10c)

[0112] in, They represent the displacement of any point on the neutral axis along the coordinate axis, Respectively express about The angular degrees of freedom of the coordinate axes, They represent the antisymmetric distortion degree of freedom, the symmetric distortion degree of freedom and the warping degree of freedom respectively. denote the first-order and third-order interpolation shape functions, respectively. They represent the unknown node variables displaced along the coordinate axis, Respectively represent the The unknown nodal variables of the axis rotational degrees of freedom, the antisymmetric distortion degrees of freedom, and the symmetric distortion degrees of freedom, For the coordinate axis Derivation;

[0113] Step 3: Determine the neutral axis strain function of the wing main beam based on the measured strain detected by the FBG strain sensor, and obtain the measured neutral axis strain function of any point of the wing main beam;

[0114] Calculating the neutral axis strain of the wing main beam using the strain measurement value detected by the FBG strain sensor is a key step in the wing deformation perception modeling technology; therefore, it is necessary to establish the transformation relationship between the measured wing main beam surface strain and the neutral axis strain; the specific calculation process is as follows:

[0115] When the FBG strain sensor is installed on the outer surface of the wing main beam and aligned with the coordinate axis have When the angle is , the measured strain can be expressed by two axial strains and one shear strain, such as Figure 5a and Figure 5b As shown, Figure 5a and Figure 5b The medium orange bars represent FBG strain sensors;

[0116] Step 3.1: Calculate the neutral axis strain of discrete points on the wing main beam surface;

[0117] like Figure 5a As shown, when the FBG strain sensor is mounted on the surface of the upper flange ( When the strain is measured at any point on the plane, it can be expressed as follows:

[0118] (11a)

[0119] In the formula, represents the axial strain of the upper flange, represents Poisson's ratio, Indicates upper flange In-plane shear strain;

[0120] like Figure 5b As shown in the figure, when the FBG strain sensor is installed on the side of the middle web ( When the strain is measured at any point on the plane, it can be expressed as follows:

[0121] (11b)

[0122] In the formula, Indicates the middle web In-plane shear strain;

[0123] When the FBG strain sensor is symmetrically mounted on the surface of the upper flange and the lower flange, and along When installing the shaft, Substituting the in-plane axial strain and shear strain (as shown in Eqs. (5a) and (5b)) into Eq. (11a) yields:

[0124] (1) The position constraints of the FBG strain sensor are: , and The neutral axis strain measured at discrete points is expressed as follows:

[0125] (12a)

[0126] In the formula, Indicates the thickness of the upper flange, Indicates the distance between the upper flange and the lower flange mid-surface;

[0127] Therefore, the measured strains of the four FBG strain sensors are substituted into equation (12a) to obtain the measured neutral axis strain: and , the expression is:

[0128] , (12b)

[0129] in, and represent the surface strain measurements of the upper flange and the lower flange, respectively;

[0130] (2) Position constraints on FBG strain sensors: Indicates the installation position of the FBG strain sensor Direction coordinates, and The neutral axis strain measured at discrete points is expressed as follows:

[0131] (12c)

[0132] (12d)

[0133] Then, the measured neutral axis strain of the wing main beam is calculated using equations (12c) and (12d): and , the expression is:

[0134] , (12e)

[0135] (3) For , the angle is The FBG strain sensor position is substituted into equation (11b) by equation (5a) and (5c). The relationship between the neutral axis strain of the wing main beam and the side surface strain of the middle web is obtained as follows:

[0136] (13a)

[0137] In the formula, Indicates the measured neutral axis strain The calculated axial strain of the middle web is, Indicates the thickness of the middle web;

[0138] At the same time, considering equations (4), (6b) and (6c), the warping degrees of freedom are calculated , the expression is as follows:

[0139] (13b)

[0140] Then, combining equations (13a) and (13b), the neutral axis strain is calculated: and , the expression is:

[0141] (13c)

[0142] (13d)

[0143] In the formula, Indicates FBG strain sensors and coordinate axes The angle of Represents the axial coordinate The warping degree of freedom at

[0144] In addition, the middle web The expression for the measured strain at is:

[0145] (14a)

[0146] According to equations (13c), (13d) and (14a), the measured neutral axis strain is calculated: , the expression is:

[0147] (14b)

[0148] In the formula, Indicates FBG strain sensors installed Axis coordinates;

[0149] Step 3.2: According to the neutral axis strain of discrete points on the surface of the wing main beam and the order of the neutral axis strain function, establish the measured neutral axis strain function of the wing main beam;

[0150] According to the neutral axis strain of discrete points on the surface of the wing main beam measured by the FBG strain sensor in step 3.1, the measured neutral axis strain value of the discrete point is derived. At the same time, according to the order of the neutral axis strain function determined in step 2, the measured neutral axis function of the wing main beam can be expressed as a polynomial, as follows:

[0151] (15a)

[0152] In the formula, is an unknown parameter. At the same time, according to the parameter constraints in equation (13b) and equation (15a), it is expressed as follows:

[0153] (15b)

[0154] In the wing main beam structure, The value of is 0; then by substituting formula (15a) into formula (15b), the constraint relationship is obtained:

[0155] , , (15c)

[0156] Therefore, the unknown parameters in equation (15a) can be reduced to 11. Then, the neutral axis strain of discrete point measurement calculated by equations (12) and (13) is substituted into equation (15a), and the unknown parameters in equation (15) can be obtained, that is,

[0157] (16a)

[0158] In the formula, represents the measured neutral axis strain vector, represents a given set of FBG strain sensor position matrix, represents the unknown parameter vector;

[0159] In the formula, ,

[0160] , , , ;

[0161] in, The subscripts in the Row and The elements of the column, The subscripts in the to Row and to Elements of a column;

[0162] Then, substituting equation (16a) into equation (15a), we can obtain the measured neutral axis strain function at any point of the wing main beam, which is expressed as follows:

[0163] (16b)

[0164] In the formula, represents the position matrix of any set of FBG strain sensors;

[0165] Step 4: Based on the measured neutral axis strain function, an online evaluation model for the deformation of the wing main beam is constructed, and the unknown node degrees of freedom of the wing main beam are obtained through the online evaluation model for the deformation of the wing main beam;

[0166] In order to evaluate the deformation shape of the wing main beam in real time, the inverse finite element method is used to establish the theoretical neutral axis strain. and measure the neutral axis strain The least squares functional between

[0167] (17a)

[0168] In the formula, represents the measured neutral axis strain detected by the FBG strain sensor. Based on the finite element concept, the wing main beam is discretized into several inverse finite elements. The interpolation function of the inverse element displacement field of the wing main beam (10) can be uniformly expressed as the shape function With interpolation node degrees of freedom The product form is as shown below:

[0169] (17b)

[0170] In the formula, represents the node degree of freedom; when the wing main beam is discretized into When there are inverse units, the least squares functional is given by The least squares functional can be expanded as follows:

[0171] (18)

[0172] In the formula, represents the length of the wing main spar, represents the number of inverse units, Represents the location point to be evaluated Axial coordinates; Indicates the number of neutral axis strains actually involved in the calculation, represents the measured neutral axis strain; substituting equation (17b) into equation (6), we get the theoretical neutral axis strain function:

[0173] (19)

[0174] In the formula, the matrix is the shape function Then, substitute equation (19) into equation (18) to obtain the following quadratic equation:

[0175] (20)

[0176] In the formula, represents a constant matrix, and the matrix and matrix is defined as:

[0177] , (twenty one)

[0178] From formula (21), we can see that the matrix is a function of the coordinates of the displacement points to be evaluated on the wing main beam, and is related to the interpolation function of the inverse unit displacement field of the wing main beam; the matrix The measured neutral axis strain at any point on the wing main beam is related to the nodal degrees of freedom of equation (20). By finding the minimum variation, we can obtain the online evaluation model of the wing main beam deformation, that is, the reconstruction model, as follows:

[0179] (twenty two)

[0180] Therefore, when the measured neutral axis strain is substituted into equation (22), the unknown nodal degrees of freedom of the wing main beam are Available.

[0181] An online measurement system for dynamic deformation of a distributed wing aperture antenna main beam structure based on FBG strain sensors, comprising:

[0182] The first calculation module is used to construct a nine-degree-of-freedom displacement field, and derive the strain of any point on the surface of the wing main beam according to the nine-degree-of-freedom displacement field to obtain the constitutive relationship;

[0183] The second calculation module is used to derive the interpolation shape function order according to the constitutive relationship and establish the interpolation function of the inverse unit displacement field of the wing main beam;

[0184] The third calculation module is used to determine the neutral axis strain function of the wing main beam based on the measured strain detected by the FBG strain sensor, and obtain the measured neutral axis strain function of any point of the wing main beam;

[0185] The fourth calculation module is used to construct an online evaluation model for the deformation of the wing main beam according to the interpolation function of the inverse unit displacement field of the wing main beam and the measured neutral axis strain function of any point of the wing main beam, and obtain the unknown node degrees of freedom of the wing main beam through the online evaluation model for the deformation of the wing main beam.

Claims

1. An online measurement method for dynamic deformation of a distributed wing aperture antenna main beam structure based on FBG strain sensors, characterized in that: The steps include: Step 1: Construct a nine-degree-of-freedom displacement field, deduce the strain at any point on the surface of the wing main beam based on the nine-degree-of-freedom displacement field, and obtain the constitutive relationship; The wing main beam is an I-beam composed of an upper flange, a lower flange and an intermediate web, and the following basic assumptions are made for the wing main beam: a. Ignore the mid-surface shear strain of the upper flange, lower flange and middle web; b. Ignore the shear strain γ of the middle web xy and γ yz , shear strain of upper flange and lower flange γ xz and γ yz ; c. Assume that the upper and lower flanges are rigid in the yoz plane; d. The middle web is deformable in the yoz plane, and the axial strain caused by cross-sectional distortion is ignored; The nine degrees of freedom of the wing main beam include the translational displacement degrees of freedom (u, v, w) along the coordinate axis (x, y, z), the angular freedom (θ x ,θ y ,θ z ) and the distortion degrees of freedom, which consist of a warping degree of freedom χ and two cross-sectional distortion degrees of freedom. The cross-sectional distortion degrees of freedom include the symmetric distortion degrees of freedom θ s and the antisymmetric distortion degree of freedom θ a ; Construct a nine-degree-of-freedom displacement field, the expression is: In the formula, u x (x,y,z),u y (x,y,z),u z (x, y, z) represent the translation displacement of the wing main beam surface point along the x, y, and z directions respectively, and the operator ( ) ,i represents the derivative of coordinate axis i, i = x, z, the three lines in the curly brackets represent the cross-sectional deformation of the upper flange, the middle web and the lower flange from top to bottom, ω(y, z) represents the area function of the cross section of the wing main beam, χ represents the warping degree of freedom, h represents the distance between the middle surface of the upper flange and the lower flange, θ s,x represents the first-order derivative of the symmetric distortion degree of freedom, θ a,x represents the first-order derivative of the antisymmetric distortion degree of freedom, H s (z) represents the deformation shape function of the middle web of the wing main beam during symmetrical deformation, H a (z) represents the deformation shape function of the middle web of the wing main beam during antisymmetric deformation, H s,z (z) represents the function H s The first-order derivative with respect to variable z, H a,z (z) represents the function H a The first derivative with respect to variable z, θ s represents the symmetrical distortion degree of freedom of the wing main beam, θ a represents the antisymmetric distortion degree of freedom of the wing main beam, θ x (x), θ y (x), θ z (x) represents the angular freedom about the x, y, and z coordinate axes, respectively; u(x), v(x), and w(x) represent the displacement of any point on the neutral axis along the coordinate axis direction; Step 2: Use non-uniform rational B-spline basis functions as shape functions to interpolate node degrees of freedom and form a displacement field interpolation function; derive the order of the interpolation shape function based on the constitutive relationship in step 1, and establish the displacement field interpolation function of the wing main beam inverse unit; The interpolation shape function order is: the neutral axis strains e1 and e8 are constants, e i , i = 1, 2, 3, 4 are linear neutral axis strains, neutral axis strains e7 and e9 are functions of the same order, i.e., quadratic functions; neutral axis strains e5 and e6 are determined by neutral axis strains e8 and e9 combined with boundary conditions; the interpolation function of the inverse unit displacement field of the wing main beam obtained by the interpolation shape function order is expressed as follows: Among them, u(ξ), v(ξ), w(ξ) represent the displacement of any point on the neutral axis along the coordinate axis, θ x (ξ),θ y (ξ),θ z (ξ) represents the angular freedom about the x, y, and z axes, respectively, and θ a (ξ),θ s (ξ), χ(ξ) represent the antisymmetric distortion degree of freedom, symmetric distortion degree of freedom and warping degree of freedom respectively, C i,1 , C i,3 denote the first-order and third-order interpolation shape functions, respectively. They represent the unknown node variables displaced along the coordinate axis, They represent the unknown node variables of the rotational degrees of freedom around the x-axis, the antisymmetric distortion degrees of freedom, and the symmetric distortion degrees of freedom, respectively. ( )′ is the derivative with respect to the coordinate axis x. Step 3: Determine the measured neutral axis strain function of the wing main beam based on the measured strain detected by the FBG strain sensor, and obtain the measured neutral axis strain function of any point of the wing main beam; Step 3.1: Calculate the neutral axis strain of discrete points on the wing main beam surface. The expressions are: in, and They represent the surface strain measurement values ​​of the upper flange and the lower flange, h represents the distance between the upper flange and the lower flange mid-surface, t f Indicates the thickness of the upper flange, t w represents the thickness of the middle web, Indicates that The calculated axial strain of the middle web is, They represent the measured strains of the upper and lower flanges, y0 represents the y-coordinate of the installation position of the FBG strain sensor, and z0 = (h + t f ) / 2 represents the z-coordinate of the upper flange surface position, μ represents the Poisson’s ratio, χ(x i ) represents the axial position x i The warping degree of freedom at z i represents the z-axis coordinate of the i-th FBG strain sensor installation, β represents the angle between the FBG strain sensor and the coordinate axis x; Step 3.2: According to the neutral axis strain of discrete points on the surface of the wing main beam and the order of the neutral axis strain function, establish the measured neutral axis strain function of the wing main beam; The measured neutral axis function of the wing main beam is: The parameter constraints are: In the formula, a i , i=1, 2, 3, ···, 14 represents unknown parameters; The measured neutral axis strain function at any point of the wing main beam is: e ε (x)=POS(x)PA (16b) Where, PA=[a1,a2,a3,…,a 11 ] T represents the unknown parameter vector, POS 1,1 =POS 11,11 =1, represents the FBG strain sensor position matrix; Among them, POS i,i The subscript in POS stands for the element in the i-th row and i-th column. i-j,m-n The subscripts in represent the elements in rows i to j and columns m to n; Step 4: According to the interpolation function of the inverse unit displacement field of the wing main beam in step 2 and the measured neutral axis strain function of any point of the wing main beam in step 3, an online evaluation model of the wing main beam deformation is constructed. Through the online evaluation model of the wing main beam deformation, the unknown node degrees of freedom of the wing main beam are obtained.

2. The online measurement method for dynamic deformation of a distributed wing aperture antenna main beam structure based on a FBG strain sensor according to claim 1 is characterized in that: The unknown node degrees of freedom u of the wing main beam in step 4 e =K -1 F, the matrix K is a function of the coordinates of the displacement points to be evaluated on the wing main beam, and is related to the interpolation function of the inverse unit displacement field of the wing main beam; the matrix F is related to the measured neutral axis strain at any point on the wing main beam.

3. An online measurement system for dynamic deformation of a distributed wing aperture antenna main beam structure based on FBG strain sensors according to the method of claim 1 or 2, characterized in that: include: The first calculation module is used to construct a nine-degree-of-freedom displacement field, and derive the strain of any point on the surface of the wing main beam according to the nine-degree-of-freedom displacement field to obtain the constitutive relationship; The second calculation module is used to derive the interpolation shape function order according to the constitutive relationship and establish the interpolation function of the inverse unit displacement field of the wing main beam; The third calculation module is used to determine the neutral axis strain function of the wing main beam based on the measured strain detected by the FBG strain sensor, and obtain the measured neutral axis strain function of any point of the wing main beam; The fourth calculation module: According to the interpolation function of the inverse unit displacement field of the wing main beam and the measured neutral axis strain function of any point of the wing main beam, an online evaluation model of the wing main beam deformation is constructed, and the unknown node degrees of freedom of the wing main beam are obtained through the online evaluation model of the wing main beam deformation.

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