A method and related apparatus for response reconstruction of experimental modal and model condensation
By using experimental modal analysis and model condensation methods, the problems of difficult finite element modeling of complex structures and large computational load for response reconstruction were solved, and efficient response reconstruction was achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- CENT SOUTH UNIV
- Filing Date
- 2023-11-23
- Publication Date
- 2026-06-19
AI Technical Summary
Existing response reconstruction methods suffer from difficulties in finite element modeling and high computational costs when dealing with complex structures.
By employing the experimental modal and model condensation method, the modal information is extracted by establishing the decoupling equation of the experimental component, generating the modal expression of the experimental component, and coupling it with the finite element component to form a complete model. The response is then reconstructed using the frequency response function.
While ensuring the accuracy of modal operations, the computational degrees of freedom are reduced, the efficiency of response reconstruction is improved, and the cost of complex modeling is avoided.
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Figure CN117556669B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of structural health monitoring technology, and in particular, to a method and related apparatus for reconstructing the response of experimental modes and model condensation. Background Technology
[0002] Due to both human and natural causes, structures inevitably experience performance degradation and even partial failure during operation. Timely detection and mitigation of anomalies can often significantly reduce losses. Structural health monitoring (SHM) relies on sensor systems to provide engineers with real-time warnings to assess structural condition. However, due to factors such as monitoring budgets, limited system access, and difficulty in accessing measurement locations, sensors cannot be installed in many critical locations. Therefore, structural response reconfiguration to expand monitoring data plays a crucial role in ensuring optimal structural performance.
[0003] Model-driven response reconstruction methods are widely used due to their accuracy and efficiency. However, there are two difficulties in response reconstruction of complex structures: First, it is difficult to achieve detailed modeling of complex structures, or it requires huge costs to build finite element models with the required accuracy. Second, even if the accuracy of the finite element model is sufficient, the large number of degrees of freedom will significantly reduce the computational efficiency of the finite element model. Summary of the Invention
[0004] The purpose of this invention is to provide a response reconstruction method and related apparatus for experimental modes and model condensation, so as to solve the technical problems of difficulty in finite element modeling and large computational load in response reconstruction when dealing with complex structures in existing response reconstruction methods.
[0005] To achieve the above objectives, the present invention adopts the following technical solution:
[0006] In a first aspect, the present invention provides a response reconstruction method for experimental modes and model condensation, comprising:
[0007] Establish decoupling equations for the test components and extract modal information from test measurements;
[0008] Based on modal information, modal expressions of the test component are generated through coordinate transformation;
[0009] Generate a condensed model of the finite element component, and couple the modal expression of the test component and the condensed model of the finite element component into a complete model;
[0010] The response is reconstructed within the finite element component region by solving the complete model; and the response is reconstructed within the test component region based on the frequency response function transfer.
[0011] Optionally, decoupling equations for the test components can be established, and modal information can be extracted from the test measurements:
[0012] The superscripts E, T, and FT denote the structural matrices of the test component, finite element component, transfer component, and test assembly, where the test assembly is formed by coupling the test component and the transfer component, and has the following decoupling equation:
[0013]
[0014] Where the superscript tr indicates matrix transpose; I ET It is the identity matrix; M and K represent the mass and stiffness matrices of the structure, respectively; q ET Represents modal coordinates; x T Displacement F T ;F T and F ET Represents the load matrix in physical coordinates; It contains n ET A diagonal matrix of measurement modes; Φ ET The modal matrix is represented by the subscript m; the measurement degrees of freedom of the test assembly are denoted by the subscript m. The transmission component and the test assembly are in equal motion under the measurement degrees of freedom, as shown in the following equation:
[0015]
[0016] in This corresponds to the mass-normalized modal matrix block for the measured degrees of freedom. Based on the normalization of the measured response, the following process is performed using the p-th elastic mode shape column for degrees of freedom a and b:
[0017] 1) Using known load excitation test components, the responses of degrees of freedom a and b are recorded simultaneously to construct the frequency response function between the two degrees of freedom;
[0018] 2) After determining the p-th modal frequency of the test assembly in the Fourier response spectrum, the frequency response function is calculated based on the p-th frequency peak value. The modal constants are represented by the modal constants.
[0019] 3) Obtain the mode ratio from the measured response using the EMD method with intermittent criteria.
[0020] 4) Calculate the mass-normalized mode shape by combining the obtained mode shape ratio and modal constant.
[0021] Optionally, based on modal information, modal expressions for the test component are generated through coordinate transformation:
[0022] The degrees of freedom of the transmission component are divided into two groups: internal degrees of freedom and interface degrees of freedom, denoted by subscripts i and j, respectively; the mass and stiffness matrices of the transmission component are rearranged as follows:
[0023]
[0024] The Craig-Bampton fixed-interface modal synthesis method, CB method, is used to transform the component model to CB coordinates through the following matrix, achieving matrix dimensionality reduction:
[0025]
[0026] in and These are the actual displacement vectors corresponding to the interior and interface degrees of freedom, respectively; Construction is achieved by extracting low-order modes of the transfer components constrained across all interface degrees of freedom; ψ T Depend on calculate; It is the displacement vector in the CB coordinate system of the transmission component; therefore, the mass and stiffness matrices of the condensed model of the transmission component after CB transformation are given by the following equation:
[0027]
[0028] The equation of motion for the polycondensation transfer component is as follows:
[0029]
[0030] Replacing the matrix block corresponding to the transmission component in equation (1) with the condensed structural matrix, equation (1) is rewritten as:
[0031]
[0032] Assuming that the interface degrees of freedom are not in the set of measured degrees of freedom, equation (2) is reformulated as follows:
[0033]
[0034] Remove the transfer component from the test assembly, ψ T It is placed separately on one side of equation (8):
[0035]
[0036] in, To represent the left inverse of a matrix, it is required that... It is full column rank; based on this, the following displacement constraints are applied to perform decoupling transformation:
[0037]
[0038] Substituting equation (10) into the undecoupled system shown in equation (7), we obtain the approximate equation of motion for the test component:
[0039]
[0040] The structure matrix in equation (11) is represented as follows:
[0041]
[0042]
[0043] Optionally, generate a condensed model of the finite element component: couple the experimental component and the finite element component into a complete analytical model.
[0044] Using the CB method to process finite element components, where the superscripts or subscripts F and EF represent the matrices of the finite element component and the complete model, the complete model composed of the test component and the finite element component has the following equations of motion:
[0045]
[0046] The structure matrix in equation (13) is expressed as:
[0047]
[0048] in, It is similar to equation (4) CB transformation matrix; and These are the mass and stiffness matrices of the finite element component condensation model; It is the displacement vector in the CB coordinates of the finite element component; the compatibility condition is written in matrix form as
[0049] d EF =Lp EF (15)
[0050]
[0051] In equation (15), L is a Boolean matrix, and equation (15) ensures that the interface displacements are compatible with each other; in the initial coupling equation, a unique set of degrees of freedom d is defined. EF That is, the sum of the connecting forces acting on each interface degree of freedom is equal to zero, thus satisfying the equilibrium condition; substituting equation (15) into equation (13), the dynamic equation of the complete model is expressed as
[0052]
[0053] The structure matrix in equation (17) is represented as follows:
[0054]
[0055] Optionally, response reconstruction can be performed within the finite element component region via modal extension:
[0056] Empirical Mode Decomposition (EMD) with an intermittency criterion is used to reconstruct the response of the complete model. This method requires two types of information: one is the measured response, collected through sensor layout and processed by EMD according to the intermittency criterion to extract the modal response at each measurement location; the other is modal information extracted from the modal matrix for extrapolation of the modal response. Modal solution is then employed. The modal matrix block representation of the finite element component corresponding to the complete model is as follows:
[0057]
[0058] in and It corresponds to d EF displacement vector in and The modal factor. Equation (19) is extended from modal coordinates to physical coordinates by the following transformation:
[0059]
[0060] in It is a modal matrix block corresponding to the internal degrees of freedom of a finite element component.
[0061] Let m and r denote the measured degrees of freedom and the reconstructed degrees of freedom, respectively. The reconstructed response is given by the following equation:
[0062]
[0063] in From The extracted p-th modal response.
[0064] Optionally, response reconstruction can be performed within the test component region based on the frequency response function transfer.
[0065] For response reconstruction involving the test component region, HT, HE, HF, HET, and HEF are first defined to represent the frequency response function matrices of the transfer component, test component, finite element component, test assembly, and complete model, respectively. Degrees of freedom a and b are located inside the structure, and degree of freedom j is the interface degree of freedom. According to the transfer rules of the frequency response function between the component and the coupled structure, when the internal degrees of freedom a and b are not in the same component, the following two equations are obtained:
[0066]
[0067]
[0068] By rearranging terms in equation (2), we obtain the following equation:
[0069]
[0070] superscript The matrix right fit requires that the number of measurement degrees of freedom on the transmitted components in the test assembly be greater than the number of decoupled interface components; the first subscript of H represents the response output degree of freedom, and the second subscript represents the load input degree of freedom.
[0071] Optionally, in equations (22) to (24), The degree of freedom is constructed from the measured degrees of freedom in the test component and the finite element component. It is required that the degree of freedom information inside the test component involved in the response reconstruction be collected in advance in the test assembly. and Calculated from a given finite element model; As the interface frequency response function, it is obtained from the modal expression of the test component in equation (12); and obtained through equation (23). The mass-normalized mode shapes of the complete model in the corresponding degrees of freedom are derived, and then the response reconstruction of the test component region is achieved according to Equation (21).
[0072] In a second aspect, the present invention provides a response reconstruction system for experimental modes and model condensation, comprising:
[0073] The information extraction module is used to establish the decoupling equations of the test components and extract modal information from the test measurements.
[0074] The coordinate transformation module is used to generate the modal expression of the test component based on the modal information through coordinate transformation;
[0075] The coupling module is used to generate condensed models of finite element components, coupling the test component and the finite element component into a complete analytical model;
[0076] The response reconstruction module is used to reconstruct the response within the finite element component region by solving the complete model; and to reconstruct the response within the test component region based on the frequency response function.
[0077] Thirdly, the present invention provides a computer device including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of a response reconstruction method for experimental modes and model condensation.
[0078] Fourthly, the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of a response reconstruction method for experimental modes and model condensation.
[0079] Compared with the prior art, the present invention has the following technical effects:
[0080] The response reconstruction method based on experimental modal analysis and model simplification of this invention divides the complete model used for response reconstruction into two categories: finite element components and experimental components. Finite element components are modeled using software to obtain their analytical models, and model simplification is used for more efficient computation. Experimental components, which are difficult to model, have their modal matrix expressions derived from experimental measurements. In the experiment, a pre-designed transfer component is attached to the experimental component, and then the transfer component is analytically removed through experimental modal analysis to obtain the modal representation of the experimental component with physical interfaces, thus avoiding complex modeling. The analytical model of the complete structure is established by coupling all components in the accessible interfaces. Based on this, the structural response reconstruction within the finite element component region is achieved through modal solving and modal expansion, while the reconstruction within the experimental component region is achieved through the transfer of the frequency response function. This method uses experimental modal analysis with transfer components, which can generate component modal expressions with physical interface information, saving the cost of complex modeling. Simultaneously, model simplification is used for available finite elements, significantly reducing the computational degrees of freedom while ensuring the accuracy of modal calculations, thereby improving the efficiency of response reconstruction. Attached Figure Description
[0081] The accompanying drawings, which form part of this application, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings:
[0082] Figure 1 This is a flowchart illustrating the response reconstruction method according to a preferred embodiment of the present invention.
[0083] Figure 2 This is a schematic diagram of the actual structure of the flange-circular pipe in a specific experimental case of the present invention.
[0084] Figure 3 This refers to the dimensional information of the flange-circular pipe in a specific test case of the present invention.
[0085] Figure 4 This is a schematic diagram of the sensor arrangement.
[0086] Figure 5 This is a schematic diagram of the suspension structure used in the test measurement.
[0087] Figure 6 yes Figure 2 A schematic diagram comparing the theoretical response and the reconstructed response of the y-direction response of A5, which is reconstructed from the y-direction responses of A1 to A3.
[0088] Figure 7 yes Figure 2 A schematic diagram comparing the theoretical response and the reconstructed response of the z-direction response of A3, which is reconstructed from the y-direction and z-direction responses of A2.
[0089] Figure 8 yes Figure 2 A schematic diagram comparing the theoretical response and the reconstructed response of the x-direction response of A7, which is reconstructed from the x-direction responses of A1 to A3.
[0090] Figure 9 yes Figure 2 A schematic diagram comparing the theoretical response and the reconstructed response of the z-direction response of A7, which is reconstructed from the y-direction and z-direction responses of A6.
[0091] Table 1 compares the predicted and actual frequencies of the complete model.
[0092] Table 2 compares the amount of computational data for the original model and the super-element model of the finite element component. Detailed Implementation
[0093] The embodiments of the present invention will be described in detail below with reference to the accompanying drawings. However, the present invention can be implemented in many different ways as defined and covered below.
[0094] like Figure 1 As shown, this invention provides a response reconstruction method based on experimental modes and model condensation to solve the technical problems of existing response reconstruction methods, such as difficulties in finite element modeling and large computational load in response reconstruction when dealing with complex structures.
[0095] Step S1: Establish the decoupling equations for the test components and extract modal information from the test measurements;
[0096] Step S2: Use the modal synthesis method to construct the coordinate transformation of the transmission component and generate the modal expression of the test component;
[0097] Step S3: Generate a condensed model of the finite element component using the modal synthesis method; then couple the experimental component and the finite element component into a complete analytical model;
[0098] Step S4: Reconstruct the response within the finite element component region through modal extension;
[0099] Step S5: Based on the transfer rules of the frequency response function, reconstruct the response within the test component area;
[0100] Step S1 specifically includes the following:
[0101] Assume that the superscripts E (experiment), T (transmission), and FT represent the structural matrices of the test component, finite element component, transmission component, and test assembly, where the test assembly is formed by coupling the test component and the transmission component, and the following decoupling equations apply:
[0102]
[0103] Where the superscript tr indicates matrix transpose; I ET It is the identity matrix; M and K represent the mass and stiffness matrices of the structure, respectively; q ET Represents modal coordinates; x T Displacement F T ;F T and F ET Represents the load matrix in physical coordinates; It contains n ET A diagonal matrix of measurement modes; Φ ET Let m represent the modal matrix. Assuming the measurement degrees of freedom of the test assembly are denoted by the subscript m, theoretically, the transmission component and the test assembly are in equal motion under the measurement degrees of freedom, as shown in the following equation:
[0104]
[0105] in This corresponds to the mass-normalized modal matrix block for the measured degrees of freedom. To obtain information about this matrix, taking the p-th elastic mode shape column for degrees of freedom a and b as an example based on the normalization of the measured response, the following process is performed:
[0106] 1) Using known load excitation test components, the responses of degrees of freedom a and b are recorded simultaneously to construct the frequency response function between the two degrees of freedom;
[0107] 2) After determining the p-th modal frequency of the test assembly in the Fourier response spectrum, the frequency response function is calculated based on the p-th frequency peak value. The modal constants are represented by the modal constants.
[0108] 3) Obtain the mode ratio from the measured response using the EMD method with intermittent criteria.
[0109] 4) Calculate the mass-normalized mode shape by combining the obtained mode shape ratio and modal constant.
[0110] Step S2 specifically involves:
[0111] The degrees of freedom of the transmission component are divided into two groups: internal degrees of freedom and interface degrees of freedom, denoted by subscripts i and j, respectively. The mass and stiffness matrices of the transmission component are rearranged as follows:
[0112]
[0113] The Craig-Bampton fixed interface modal synthesis method (CB method) is adopted to transform the components to CB coordinates through the following matrix, thereby achieving matrix dimensionality reduction:
[0114]
[0115] in and These are the actual displacement vectors corresponding to the interior and interface degrees of freedom, respectively; Construction is achieved by extracting low-order modes of the transfer components constrained across all interface degrees of freedom; ψ T Depend on calculate; This is the displacement vector in the CB coordinate system of the transfer component. Therefore, the mass and stiffness matrices of the condensed model of the transfer component after the CB transformation are given by the following equation:
[0116]
[0117] The equation of motion for the polycondensation transfer component is as follows:
[0118]
[0119] Replacing the matrix block corresponding to the transmission component in equation (1) with the condensed structural matrix, equation (1) is rewritten as:
[0120]
[0121] Assuming that the interface degrees of freedom are not in the set of measured degrees of freedom, equation (2) can therefore be reformulated as
[0122]
[0123] In order to remove the transfer component from the test assembly, ψ T It is placed separately on one side of Equation 8:
[0124]
[0125] in, To represent the left inverse of a matrix, it is required that... It is full column rank. Based on this, the following displacement constraints are applied to perform decoupling transformation:
[0126]
[0127] Substituting equation (10) into the undecoupled system shown in equation (7), we obtain the approximate equation of motion for the test component:
[0128]
[0129] The structure matrix in equation (11) is represented as follows:
[0130]
[0131]
[0132] Step S3 specifically involves:
[0133] To improve the efficiency of coupled computation, the CB method is used to process finite element components. Assuming that the superscript or subscript F (finite element) and EF represent the matrices of the finite element component and the complete model, the complete model composed of the test component and the finite element component has the following equations of motion:
[0134]
[0135] The structure matrix in equation (13) is expressed as:
[0136]
[0137] in, It is similar to equation (4) CB transformation matrix; and These are the mass and stiffness matrices of the finite element component condensation model; It is the displacement vector in the CB coordinates of the finite element component. Because there is a shared interface degree of freedom between the finite element component and the test component (… and The complete model equations of motion in equation (13) are not independent. To eliminate the independent degrees of freedom, the equations of motion must satisfy both the compatibility condition and the equilibrium condition. The former stipulates that the interface displacements of the components must be compatible, while the latter ensures that the forces connecting the interface degrees of freedom are in equilibrium. The compatibility condition is written in matrix form as follows:
[0138] d EF =Lp EF (15)
[0139]
[0140] In equation (15), L is a Boolean matrix used to eliminate elements corresponding to non-independent degrees of freedom. Equation (15) ensures that the interface displacements are compatible with each other. In the initial coupling equations, a unique set of degrees of freedom d is defined. EF That is, the sum of the connecting forces acting on each interface degree of freedom is equal to zero, thus automatically satisfying the equilibrium condition. Substituting equation (15) into equation (13), the dynamic equation of the complete model can be expressed as follows:
[0141]
[0142] The structure matrix in equation (17) is represented as follows:
[0143]
[0144] Step S4 specifically involves:
[0145] Empirical Mode Decomposition (EMD) with an intermittency criterion is used to reconstruct the response of the complete model. This method requires two types of information: one is the measured response, collected through sensor layout and processed by EMD according to the intermittency criterion to extract the modal response at each measurement location; the other is modal information extracted from the modal matrix for extrapolation of the modal response. Modal solution is then employed. The modal matrix block of the finite element component corresponding to the complete model can be represented as:
[0146]
[0147] in and It corresponds to d EF displacement vector in and The modal factor. Equation (19) is extended from modal coordinates to physical coordinates by the following transformation:
[0148]
[0149] in It is a modal matrix block corresponding to the internal degrees of freedom of a finite element component.
[0150] Let m and r denote the measured degrees of freedom and the reconstructed degrees of freedom, respectively. The reconstructed response is given by the following equation:
[0151]
[0152] in From The extracted p-th modal response.
[0153] Step S5 specifically involves:
[0154] For response reconstruction involving the test component region, H is first defined. T H E H F H ET and H EF Let represent the frequency response function matrices of the transfer component, test component, finite element component, test assembly, and complete model, respectively. Considering that degrees of freedom a and b are within the structure, and degree of freedom j is the interface degree of freedom, according to the transfer rules of the frequency response function between components and coupled structures, when internal degrees of freedom a and b are not in the same component, the following two equations can be obtained:
[0155]
[0156]
[0157] By rearranging terms in equation (2), we can obtain the following equation:
[0158]
[0159] superscript The matrix right fit requires that the number of measurement degrees of freedom on the transmitted components in the test assembly be greater than the number of decoupled interface components; the first subscript of H represents the response output degree of freedom, and the second subscript represents the load input degree of freedom.
[0160] In equations (22) to (24), The degree of freedom is constructed from the measured degrees of freedom in the test component and the finite element component, which requires that the degree of freedom information inside the test component involved in the response reconstruction be collected in advance in the test assembly; and It can be calculated from a given finite element model; As the interface frequency response function, it can be obtained from the modal expression of the test component in equation (12). The calculation of the frequency response function is related to the damping ratio, which can be evaluated empirically. Since the modal frequencies and mode shapes are inherent characteristics of the model, their transmission will not be affected by inaccurate damping ratios. Once obtained through equation (23) The mass-normalized mode shape of the complete model in the corresponding degree of freedom can be derived from it, and then the response reconstruction of the region involving the test component can be realized according to Equation (21).
[0161] Next, as Figures 2 to 9 As shown, the implementation process of response reconstruction is described using the flange-circular pipe model as the research object.
[0162] Flange-circular tube test model as follows Figure 2 As shown, this includes two circular pipes connected to a flange plate. The length, outer diameter, and wall thickness of the circular pipes are 40 cm, 3 cm, and 0.1 cm, respectively. The flange has an outer diameter of 8 cm and an inner diameter of 3 cm. Figure 2 The left pipe is fixed by two sets of pipe clamps, while the right pipe is fixed to the left pipe through four bolt holes on the flange.
[0163] The ultimate goal of this experiment is to reconstruct the response on the right circular tube within the complete structure to verify the accuracy of the method. To this end, the right tube and its connected flange base plate are designated as the test component, while the remaining portion is treated as a finite element component. Furthermore, the transfer component used for experimental decoupling is designed as a ring, with eight protrusions evenly arranged along the circumference for easy sensor deployment. Eight measurement points (A8 to A15) are set on these protrusions during the experimental decoupling. The transfer component also has four bolt holes for connection to the test component. Figure 3 and Figure 4 The structural dimensions and the arrangement of the triaxial sensors (A1–A15) involved in the experiment are provided. The specific implementation steps for response reconstruction are as follows:
[0164] (1) Adopting as follows Figure 5 The suspension test assembly shown was used to conduct test measurements, collect dynamic responses under known random loads, and extract modal information, including the first six rigid body modes and thirteen elastic modes with a frequency range of 0 to 2000 Hz.
[0165] (2) Generate the condensation model of the transmission component according to equations (3) to (6), and obtain the transformation matrix through equations (8) to (10) based on the modal information measured by the experiment, and generate the modal expression of the test component after decoupling;
[0166] (3) Generate the condensation model of the finite element component according to formulas (13) to (18) and couple it with the test component to form an analytical complete flange-circular tube model;
[0167] (4) The response reconstruction of the finite element component region in the complete structure is realized by formulas (19) to (21);
[0168] (5) The response reconstruction in the test structure region of the complete structure is achieved by formulas (22) to (24) and (21).
[0169] The reconstructed response and the theoretical response of the test response obtained by following steps (1) to (5) above are compared, for example... Figure 6 As shown, the final reconstruction response at the selected reconstruction point is calculated from the average of the reconstructions at each measurement point. Table 1 also provides a comparison between the predicted frequencies calculated using the complete model matrix and the theoretical frequencies provided by measurements of the actual structure. This demonstrates the accuracy of the modal calculations using the analytical model. Figure 6 This demonstrates that the response reconstruction method can achieve accurate response reconstruction without requiring detailed modeling of complex structures. Furthermore, Table 2 compares the data volume of the original and condensed models of the finite element components. The data includes the number of elements in the structural stiffness and mass matrices. It can be seen that after model condensation, the amount of data involved in the calculation is significantly reduced, effectively alleviating the problem of insufficient computer memory and improving computational efficiency.
[0170] Table 1
[0171]
[0172] Table 2
[0173]
[0174] As illustrated by the simulation examples above, this invention, through its specific implementation steps, can reconstruct the response information of the test point relatively accurately. It has high applicability to common dense modes and significantly reduces the computational load and improves the efficiency of response reconstruction for the dynamic response reconstruction of large structures.
Claims
1. A response reconstruction method for experimental modes and model condensation, characterized in that, include: Establish decoupling equations for the test components and extract modal information from test measurements; Based on modal information, modal expressions of the test component are generated through coordinate transformation; Generate a condensed model of the finite element component, and couple the modal expression of the test component and the condensed model of the finite element component into a complete model; The response is reconstructed within the finite element component region by solving the complete model; and the response is reconstructed within the test component region based on the frequency response function transfer. Generate a condensed model of the finite element component: Couple the experimental component and the finite element component into a complete analytical model: Using the CB method to process finite element components, where the superscripts or subscripts F and EF represent the matrices of the finite element component and the complete model, the complete model composed of the test component and the finite element component has the following equations of motion: （13） The structure matrix in equation (13) is expressed as: （14） in, It is similar to equation (4) CB transformation matrix; and These are the mass and stiffness matrices of the finite element component condensation model; It is the displacement vector in the CB coordinates of the finite element component; the compatibility condition is written in matrix form as （15） （16） In equation (15), L is a Boolean matrix, and equation (15) makes the interface displacements compatible with each other; In the initial coupling equations, a unique set of degrees of freedom is defined. That is, the sum of the connecting forces acting on each interface degree of freedom is equal to zero, thus satisfying the equilibrium condition; substituting equation (15) into equation (13), the dynamic equation of the complete model is expressed as （17） The structure matrix in equation (17) is represented as follows: （18）。 2. The response reconstruction method for experimental modes and model condensation according to claim 1, characterized in that, Establish decoupling equations for the test components and extract modal information from test measurements: The superscripts E, T, and FT denote the structural matrices of the test component, finite element component, transfer component, and test assembly, where the test assembly is formed by coupling the test component and the transfer component, and has the following decoupling equation: （1） The superscript tr indicates matrix transpose; It is the identity matrix; M and K represent the mass and stiffness matrices of the structure, respectively; Represents modal coordinates; Displacement ; and Represents the load matrix in physical coordinates; It includes A diagonal matrix of measurement modes; The modal matrix is represented by the subscript m; the measurement degrees of freedom of the test assembly are denoted by the subscript m. The transmission component and the test assembly are in equal motion under the measurement degrees of freedom, as shown in the following equation: （2） in This corresponds to the mass-normalized modal matrix block for the measured degrees of freedom. Based on the normalization of the measured response, the following process is performed using the p-th elastic mode shape column for degrees of freedom a and b: 1) Using a known load to excite the test assembly, the responses of degrees of freedom a and b are recorded simultaneously, thereby constructing the frequency response function between the two degrees of freedom; 2) After determining the p-th modal frequency of the test assembly in the Fourier response spectrum, the frequency response function is calculated based on the p-th frequency peak value. The modal constants are represented by the modal constants. 3) Obtain the mode ratio from the measured response using the EMD method with intermittent criteria. ; 4) Calculate the mass-normalized mode shape by combining the obtained mode shape ratio and modal constant.
3. The response reconstruction method for experimental modes and model condensation according to claim 1, characterized in that, Based on modal information, the modal expression of the test component is generated through coordinate transformation: The degrees of freedom of the transmission component are divided into two groups: internal degrees of freedom and interface degrees of freedom, denoted by subscripts i and j, respectively; the mass and stiffness matrices of the transmission component are rearranged as follows: （3） The Craig-Bampton fixed-interface modal synthesis method, CB method, is used to transform the component model to CB coordinates through the following matrix, achieving matrix dimensionality reduction: （4） in and These are the actual displacement vectors corresponding to the interior and interface degrees of freedom, respectively; Construction is achieved by extracting low-order modes of the transfer components constrained across all interface degrees of freedom; Depend on calculate; It is the displacement vector in the CB coordinate system of the transmission component; therefore, the mass and stiffness matrices of the condensed model of the transmission component after CB transformation are given by the following equation: （5） The equation of motion for the polycondensation transfer component is as follows: （6） Replacing the matrix block corresponding to the transmission component in equation (1) with the condensed structural matrix, equation (1) is rewritten as: （7） Assuming that the interface degrees of freedom are not in the set of measured degrees of freedom, equation (2) is reformulated as follows: （8） Remove the transfer component from the test assembly. It is placed separately on one side of equation (8): （9） in, To represent the left inverse of a matrix, it is required that... It is full column rank; based on this, the following displacement constraints are applied to perform decoupling transformation: （10） Substituting equation (10) into the undecoupled system shown in equation (7), we obtain the approximate equation of motion for the test component: （11） The structure matrix in equation (11) is expressed as: （12）。 4. The response reconstruction method for experimental modes and model condensation according to claim 1, characterized in that, Response reconstruction is performed within the finite element component region by solving the complete model: Reconstructing the response of the complete model using Empirical Mode Decomposition (EMD) with an intermittency criterion requires two types of information: one is the measured response, collected through sensor placement and processed by EMD according to the intermittency criterion to extract the modal responses at each measurement location; the other is the modal information extracted from the modal matrix for extrapolating the modal responses. This is achieved through modal solving... The modal matrix block representation of the finite element component corresponding to the complete model is as follows: （19） in and It corresponds to displacement vector in and modal factors; Equation (19) is extended from modal coordinates to physical coordinates by the following transformation: （20） in It is the modal matrix block corresponding to the internal degrees of freedom of a finite element component; Let m and r denote the measured degrees of freedom and the reconstructed degrees of freedom, respectively. The reconstructed response is given by the following equation: （21） in From The extracted p-th modal response.
5. The response reconstruction method for experimental modes and model condensation according to claim 1, characterized in that, Based on the frequency response function transfer, the response is reconstructed within the test component region: For response reconstruction involving the test component region, first define , , , and Let represent the frequency response function matrices of the transfer component, test component, finite element component, test assembly, and complete model, respectively; degrees of freedom a and b are located inside the structure, and degree of freedom j is the interface degree of freedom. According to the transfer rules of the frequency response function between the component and the coupled structure, when the internal degrees of freedom a and b are not in the same component, the following two equations are obtained: （22） （23） By rearranging terms in equation (2), we obtain the following equation: （24） superscript The matrix right fit requires that the number of measurement degrees of freedom on the transmitted components in the test assembly be greater than the number of decoupled interface components; the first subscript of H represents the response output degree of freedom, and the second subscript represents the load input degree of freedom.
6. The response reconstruction method for experimental modes and model condensation according to claim 5, characterized in that, In equations (22) to (24), The degree of freedom is constructed from the measured degrees of freedom in the test component and the finite element component. It is required that the degree of freedom information inside the test component involved in the response reconstruction be collected in advance in the test assembly. , , and Calculated from a given finite element model; As the interface frequency response function, it is obtained from the modal expression of the test component in equation (12); and obtained through equation (23). The mass-normalized mode shapes of the complete model in the corresponding degrees of freedom are derived, and then the response reconstruction of the test component region is realized according to Equation (21).
7. A response reconstruction system for experimental modes and model condensation, characterized in that, include: The information extraction module is used to establish the decoupling equations of the test components and extract modal information from the test measurements. The coordinate transformation module is used to generate the modal expression of the test component based on the modal information through coordinate transformation; The coupling module is used to generate condensed models of finite element components, coupling the test component and the finite element component into a complete analytical model; The response reconstruction module is used to reconstruct the response within the finite element component region by solving the complete model; and to reconstruct the response within the test component region based on the frequency response function. Generate a condensed model of the finite element component: Couple the experimental component and the finite element component into a complete analytical model: Using the CB method to process finite element components, where the superscripts or subscripts F and EF represent the matrices of the finite element component and the complete model, the complete model composed of the test component and the finite element component has the following equations of motion: （13） The structure matrix in equation (13) is expressed as: （14） in, It is similar to equation (4) CB transformation matrix; and These are the mass and stiffness matrices of the finite element component condensation model; It is the displacement vector in the CB coordinates of the finite element component; the compatibility condition is written in matrix form as （15） （16） In equation (15), L is a Boolean matrix, and equation (15) makes the interface displacements compatible with each other; In the initial coupling equations, a unique set of degrees of freedom is defined. That is, the sum of the connecting forces acting on each interface degree of freedom is equal to zero, thus satisfying the equilibrium condition; substituting equation (15) into equation (13), the dynamic equation of the complete model is expressed as （17） The structure matrix in equation (17) is represented as follows: （18）。 8. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the response reconstruction method for experimental modes and model condensation as described in any one of claims 1 to 6.
9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the response reconstruction method for experimental modes and model condensation as described in any one of claims 1 to 6.