Modal method for reconstructing deformation of structural member under different boundary conditions

Abstract

The application discloses a modal method for deformation reconstruction of a complex structural member under different boundary conditions and belongs to the general engineering field. The modal method comprehensively utilizes structural strain measurement data, modal information and displacement boundary conditions to reconstruct the deformation of the structural member. The modal method avoids the problem of repeated reconstruction caused by the need to use different modal information when reconstructing the deformation of the structural member under different boundary conditions, makes the reconstruction process lengthy and high in reconstruction cost, and simultaneously solves the problem of singular equation set caused by the structural rigid body modal.

Description

Technical Field

[0001] This invention relates to the field of general engineering, specifically to a modal method for reconstructing the deformation of complex structural components under different boundary conditions. More particularly, it relates to a method for reconstructing the deformation of structural components by comprehensively utilizing structural strain measurement data, modal information, and displacement boundary conditions. Background Technology

[0002] Deformation monitoring of elastic structures under load is a key technology in engineering. One popular approach is to acquire strain measurements by placing strain sensors on or inside the structure, and then reconstruct the structural deformation using a strain-displacement transformation relationship. The most commonly used method is the modal method. When performing modal analysis using the finite element method, the displacement boundary conditions of the structure must be given; different boundary conditions yield different modal matrices. However, existing modal methods do not comprehensively consider boundary conditions. For example, the boundary conditions for an aircraft differ between its flight state and its ground taxiing or parking state. In the first state, the structure is completely free; in the second state, the aircraft has simply supported constraints at the landing gear nodes. Therefore, current classical modal methods cannot be directly used to reconstruct the deformation of aircraft under different states. Summary of the Invention

[0003] To overcome the shortcomings of existing technologies, this invention aims to propose a modal method for reconstructing the deformation of complex structural components under different boundary conditions. This method comprehensively utilizes structural strain measurement data, modal information, and displacement boundary conditions to reconstruct structural deformation, avoiding the problem of repeated reconstruction caused by using different modal information when reconstructing structural components under different boundary conditions, which makes the reconstruction process lengthy and costly. At the same time, it solves the problem of singular equations caused by rigid body modes of structures.

[0004] The objective of this invention is mainly achieved through the following technical solution: a modal method for structural component deformation reconstruction under different boundary conditions, characterized by the following specific steps:

[0005] Step 1: Divide the displacement mode matrix, strain mode matrix, and modal coordinate column vector of the structural component under free boundary conditions into blocks to obtain rigid body displacement sub-blocks and elastic deformation sub-blocks of the displacement mode matrix, strain mode matrix, and modal coordinate column vector of the structural component.

[0006] Among them, the displacement mode matrix Φ and the strain mode matrix And the sub-block expression for the modal coordinate column vector q is:

[0007]

[0008] Where r represents the sub-block corresponding to rigid body displacement, and e represents the sub-block corresponding to elastic deformation;

[0009] Step 2: Obtain the elastic deformation of the modal coordinate column vector using the strain measurement values ​​of the structural component. The expression is:

[0010]

[0011] in, These are the modal coordinates corresponding to elastic deformation; These are the strain measurements of the structural components;

[0012] Step 3: Use the displacement boundary conditions of the structural component to obtain the rigid body displacement sub-block of the modal coordinate column vector of the structural component;

[0013]

[0014] in, These are the modal coordinates corresponding to the rigid body displacement; Ξ NA A Boolean matrix consisting of 0s and 1s; For the forced displacement of the constrained degrees of freedom in the physical coordinate column vector;

[0015] Step 4: Based on the obtained displacement mode matrix, strain mode matrix, rigid body displacement sub-block and elastic deformation sub-block of modal coordinate column vector of the structural component, as well as elastic deformation and rigid body displacement sub-block of modal coordinate column vector, perform deformation reconstruction of the structural component.

[0016] Optionally, the expression for the displacement boundary conditions of the structural member in step 3 is:

[0017]

[0018] Optionally, the expression for the deformation vector of the structural component in step 4 is:

[0019]

[0020] Where, Φ i q represents the i-th displacement mode vector; i Let represent the coordinates of the i-th mode, i = 1, 2, ..., m, where m is the total number of modes used for displacement reconstruction;

[0021] Substituting equations (1), (2), and (3) into equation (5), we obtain the following expression:

[0022]

[0023] in, To reconstruct the physical coordinate column vector; To reconstruct the modal coordinate column vector.

[0024] The present invention has at least the following beneficial effects:

[0025] 1. This invention avoids the zero-strain modes corresponding to the rigid body motion of the structure from appearing in the equations for solving the modal coordinates by processing the modal matrix in blocks, thus solving the problem of singular equations that may be encountered in the classical modal method when performing deformation reconstruction.

[0026] 2. By introducing displacement boundary conditions, this invention enables the use of free-free modes when reconstructing the displacement of a structure under different boundary conditions. Compared with the classical modal method, this reduces the cost of modal analysis or modal testing and improves the efficiency of reconstruction.

[0027] 3. This invention is applicable to the deformation and reconstruction of structural components in different states, and has wide adaptability.

[0028] In this invention, the above-described technical solutions can be combined with each other to achieve more preferred combinations. Other features and advantages of this invention will be set forth in the following description, and some advantages may become apparent from the description or be learned by practicing the invention. The objectives and other advantages of this invention can be realized and obtained from the description and accompanying drawings, which are particularly pointed out. Attached Figure Description

[0029] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts.

[0030] Figure 1 This is a flowchart of the modal method of the present invention.

[0031] Figure 2 This is the finite element model of the beam.

[0032] Figure 3 This is a schematic diagram illustrating the reconstruction of beam deformation under two working conditions.

[0033] Figure 4 (a)-(b) represent the actual nodal displacements of the beam under the two working conditions.

[0034] Figure 5 (a)-(f) represent the first 6 modes of the beam in the free-free mode.

[0035] Figure 6 (a)-(f) represent the first 6 modes of the beam in mode 1.

[0036] Figure 7 (a)-(f) represent the first 6 modes of the beam in mode 2.

[0037] Figure 8(a)-(b) show the comparison of the results of beam deformation and reconstruction using different methods.

[0038] Figure 9 (a)-(b) represent the limitations of the classical modal method.

[0039] Figure 10 (a)-(b) show the reconstruction results and actual displacements of the deformation method of the present invention when the deformation includes rigid body displacement. Detailed Implementation

[0040] The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which constitute a part of the present invention and are used together with the embodiments of the present invention to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention.

[0041] A specific embodiment of the present invention, such as Figure 1-4 As shown, a modal method for reconstructing structural deformation under different boundary conditions is disclosed. The specific steps are as follows:

[0042] Step 1: Divide the displacement mode matrix, strain mode matrix, and modal coordinate column vector of the structural component under free boundary conditions into blocks to obtain rigid body displacement sub-blocks and elastic deformation sub-blocks of the displacement mode matrix, strain mode matrix, and modal coordinate column vector of the structural component.

[0043] It is understandable that free boundary conditions refer to situations where the structure is not subject to any constraints.

[0044] Among them, the displacement mode matrix Φ and the strain mode matrix And the sub-block expression for the modal coordinate column vector q is:

[0045]

[0046] Where r represents the sub-block corresponding to rigid body displacement, and e represents the sub-block corresponding to elastic deformation; let

[0047]

[0048] It is understandable that the modal coordinate column vector is the modal coordinate column vector of the main vibration mode.

[0049] Step 2: Obtain the elastic deformation of the modal coordinate column vector using the strain measurement values ​​of the structural component;

[0050] The expression is:

[0051]

[0052] in, These are the modal coordinates corresponding to elastic deformation; The strain measurement value of the structural component; Equation (2) only uses the elastic deformation sub-block of the strain mode matrix. Elastic deformation sub-block of strain mode matrix There are no all-zero columns, which avoids the singularity of the equations caused by the zero-strain modes corresponding to rigid body displacement in the classical modal method.

[0053] Step 3: Use the displacement boundary conditions of the structural component to obtain the rigid body displacement sub-block of the modal coordinate column vector of the structural component;

[0054] The expression for the displacement boundary conditions of the structural component is:

[0055]

[0056] Among them, Ξ NA A Boolean matrix consisting of 0s and 1s, used to extract the sub-blocks corresponding to the constrained degrees of freedom in the physical coordinate column vector u. It represents the forced displacement of the constrained degrees of freedom in the physical coordinate column vector.

[0057] It is understandable that the forced displacement of the constrained degrees of freedom in the physical coordinate column vector is... In this context, the forced displacement at the fixed support point is 0. For example, by using the finite element method, structural components (such as bridges, aircraft, etc.) are discretized into multiple nodes and elements. The deformation of the structural components is represented by the nodal displacements. All the nodal displacements are combined and written in the form of a column vector. This vector is called the physical coordinate column vector, which is also the column vector of displacement.

[0058] Substituting equation (2) into equation (3):

[0059]

[0060] in, These are the modal coordinates corresponding to the rigid body displacement.

[0061] Step 4: Based on the obtained displacement mode matrix, strain mode matrix, rigid body displacement sub-block and elastic deformation sub-block of modal coordinate column vector of the structural component, as well as elastic deformation and rigid body displacement sub-block of modal coordinate column vector, perform deformation reconstruction of the structural component.

[0062] The expression for the deformation vector of the structural component is:

[0063]

[0064] Where, Φ i q represents the i-th displacement mode vector; i Let represent the coordinates of the i-th mode, i = 1, 2, ..., m, where m is the total number of modes used for displacement reconstruction.

[0065] Substituting equations (1), (2), and (4) into equation (5), we obtain the following expression:

[0066]

[0067] in, To reconstruct the physical coordinate column vector; To reconstruct the modal coordinate column vector.

[0068] Compared with the classical modal method, the deformation reconstruction method proposed in this invention introduces displacement boundary conditions into the acquisition of modal coordinates. It can perform deformation reconstruction under arbitrary boundary conditions using only the modes under free boundary conditions, without having to use the modes corresponding to the boundary conditions. This avoids performing modal analysis or modal tests on the structural components under every possible boundary condition.

[0069] To verify the effectiveness of the modal method of this invention, a simulation experiment is conducted using a simple beam model as an example. The specific details are as follows:

[0070] A slender beam with a homogeneous, uniform cross-section, 1m in length, has a rectangular cross-section with a width b = 1.5mm and a height h = 35mm. Figure 2 (in the z-axis direction), the elastic modulus of the material is E = 2.1 × 10⁻⁶. 11 Pa, Poisson's ratio γ = 0.33, density ρ = 7750 kg / m³ 3 Finite element modeling of the structure: The entire beam is divided into 50 elements, with a total of 51 nodes, numbered 1 to 50 from left to right, as follows. Figure 2 As shown. Constrain the 2nd, 4th, and 6th degrees of freedom (translational degrees of freedom along the y-axis and rotational degrees of freedom about the x-axis and z-axis) of each node, and retain only the 1st, 3rd, and 5th degrees of freedom (translational degrees of freedom along the x-axis and z-axis and translational degrees of freedom about the y-axis).

[0071] The deformation of the beam under two working conditions was reconstructed:

[0072] Working condition 1: The beam is fixed at node 1 and simply supported at node 25, and a concentrated force F = 0.1 N in the z direction is applied at the rightmost end of the beam (node ​​51);

[0073] Condition 2: The beam is fixed at node 1 and simply supported at nodes 25 and 40. A concentrated force F = 0.1 N in the z-direction is applied at the rightmost end of the beam (node ​​51).

[0074] 2) Calculation process description:

[0075] The first step is to use NASTRAN to calculate the deformation state of the beam under two working conditions, including nodal displacement and element strain. The nodal displacement is used as a reference for the actual displacement, and the strain result is used as the input for deformation reconstruction.

[0076] The second step is to use NASTRAN to calculate the displacement and strain modes of the beam under free and boundary conditions corresponding to load cases 1 and 2, which are denoted as free mode, mode 1 and mode 2, respectively, as input quantities for deformation reconstruction.

[0077] The third step involves reconstructing the beam's deformation using both the classical modal method and the modal method of this invention. When using the classical modal method, modes 1 and 2 are used to reconstruct the deformation of both load cases 1 and 2, respectively. When using the modal method of this invention, free modes are used for both load cases. Furthermore, to demonstrate the limitations of the classical modal method, the results of reconstructing load case 2 using mode 1 and reconstructing load case 1 using mode 2 are also calculated.

[0078] 3) Results Analysis

[0079] The actual nodal displacements calculated in the first step under the two working conditions are as follows: Figure 4 As shown, Figure 4 (a) represents operating condition 1. Figure 4 (b) is operating condition 2.

[0080] The free modes, mode 1, and the first six modes of mode 2 calculated in the second step are as follows: Figures 1 to 6 (a)-(f) represent the beams shown.

[0081] In the third step, the results obtained using the classical modal method and the method proposed in this invention are compared, for example... Figure 4 As shown.

[0082] from Figure 4 When using the classical modal method to reconstruct the structure under different working conditions, accurate results can be obtained as long as the modes under the same boundary conditions as the calculation working conditions are used. When using the deformation reconstruction method proposed in this invention, accurate deformation reconstruction of the structure under different boundary conditions can be performed using only free-free modes.

[0083] from Figure 5 (b) As can be seen from the deformation reconstruction of load case 1 using mode 2, when using the classical modal method for deformation reconstruction, if the mode used cannot represent all possible deformations of the structure (such as mode 2 in this example, which cannot represent all possible deformations of the structure under the boundary conditions of load case 1), incorrect results will be obtained. It is important to note here that... Figure 5 (a) The case in which a mode under one boundary condition can represent all possible deformations under another boundary condition and thus correctly reconstruct the deformation using mode 1 is only a special case. In most cases, this is not satisfied. Therefore, when using the classical modal method for deformation reconstruction, different modes need to be used for different boundary conditions. This is a limitation of the classical modal method.

[0084] Furthermore, if rigid body displacement components are added to the deformations of conditions 1 and 2, the classical modal method cannot directly use free modes for deformation reconstruction (as mentioned earlier, the zero-strain mode corresponding to the rigid body displacement will lead to singular equations for solving the modal coordinates). However, using the deformation reconstruction method proposed in this paper, it is only necessary to simply add the components to equation (3). By adding rigid body displacement components, the correct result can still be obtained, such as... Figure 10 (a)-(b) are shown. This is another improvement of the method in this paper compared to the classical modal method.

[0085] The above description is only a preferred embodiment of the present invention, but the scope of protection 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 scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A modal method for reconstructing structural deformation under different boundary conditions, characterized in that, The specific steps are as follows: Step 1: Divide the displacement mode matrix, strain mode matrix, and modal coordinate column vector of the structural component under free boundary conditions into blocks to obtain rigid body displacement sub-blocks and elastic deformation sub-blocks of the displacement mode matrix, strain mode matrix, and modal coordinate column vector of the structural component. Among them, the displacement mode matrix Φ and the strain mode matrix And the sub-block expression for the modal coordinate column vector q is: Where r represents the sub-block corresponding to rigid body displacement, and e represents the sub-block corresponding to elastic deformation; Step 2: Obtain the elastic deformation of the modal coordinate column vector using the strain measurement values ​​of the structural component. The expression is: in, These are the modal coordinates corresponding to elastic deformation; These are the strain measurements of the structural components; Step 3: Use the displacement boundary conditions of the structural component to obtain the rigid body displacement sub-block of the modal coordinate column vector of the structural component; in, These are the modal coordinates corresponding to the rigid body displacement; Ξ NA A Boolean matrix consisting of 0s and 1s; For the forced displacement of the constrained degrees of freedom in the physical coordinate column vector; Step 4: Based on the obtained displacement mode matrix, strain mode matrix, rigid body displacement sub-block and elastic deformation sub-block of modal coordinate column vector of the structural component, as well as elastic deformation and rigid body displacement sub-block of modal coordinate column vector, perform deformation reconstruction of the structural component.

2. The modal method according to claim 1, characterized in that, The expression for the displacement boundary conditions of the structural member in step 3 is:

3. The modal method according to claim 1, characterized in that, Deformation vector of structural component in step 4 The expression is: Where, Φ i q represents the i-th displacement mode vector; i Let represent the coordinates of the i-th mode, i = 1, 2, ..., m, where m is the total number of modes used for displacement reconstruction; Substituting equations (1), (2), and (3) into equation (5), we obtain the following expression: in, To reconstruct the physical coordinate column vector; To reconstruct the modal coordinate column vector.

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