Dense modal response reconstruction method of condensed structure, system, device and storage medium

Abstract

The present invention discloses a method and system, equipment, and storage medium for reconstructing the dense modal response of a condensed structure. The method uses model condensation to generate a super-element model with fewer degrees of freedom based on a finite element model, which can effectively improve reconstruction efficiency. The method also divides the structural mode into dense modes and residual modes, and then reconstructs the residual modal response and dense modal response of the test point respectively. Finally, the test response is obtained by modal superposition. Compared with the existing EMD decomposition method that extracts single-frequency modal responses through a bandpass filter, the method can effectively separate dense modes and accurately reconstruct dense modal responses, which can be better applied to the response reconstruction of dense modal structures. In addition, the method can realize the reconstruction of the response of any degree of freedom of the structure based on the known structural response, improves engineering practicality, ensures accuracy, greatly improves reconstruction efficiency, saves computer memory, and has a fast analysis speed.

Description

Technical Field

[0001] The present invention relates to the technical field of structural health monitoring, and in particular to a method and system for reconstructing dense modal responses of a condensed structure, an electronic device, and a computer-readable storage medium. Background Art

[0002] The performance of civil engineering structures degrades over time during their use. To monitor the health of the structure and predict its lifespan, it is necessary to accurately obtain the system responses of critical structural areas. Structural dynamic response reconstruction has received increasing attention over the past decade. However, given the geometric complexity and component diversity of engineering structures, it is difficult to install sensors in certain locations (e.g., structural interfaces, slits, etc.), and in practice, measurements cannot always be taken at the desired locations. Therefore, predicting the dynamic responses of key points has become an important part of structural health monitoring (SHM).

[0003] Currently, time-domain response reconstruction methods based on empirical mode decomposition (EMD) are available to effectively reconstruct structural responses. However, applying this method to complex civil structures is extremely difficult for two main reasons. First, this method requires obtaining the global stiffness and mass matrices of the entire structure. Large civil engineering structures are typically large in dimension, requiring significant computational resources and time to extract these matrices. Second, due to the small difference between adjacent frequencies in dense modes, the EMD method with the intermittency criterion cannot effectively separate these dense modes, making it difficult to reconstruct the response of structures with dense modes. Summary of the Invention

[0004] The present invention provides a method and system for reconstructing the dense modal response of a condensed structure, an electronic device, and a computer-readable storage medium to solve the technical problems of the existing EMD-based response reconstruction method, which is unable to effectively separate dense modes and is difficult to achieve response reconstruction of dense modal structures.

[0005] According to one aspect of the present invention, a method for reconstructing dense modal responses of a condensation structure is provided, comprising the following steps:

[0006] Divide the finite element model into substructures and divide the degrees of freedom of each substructure;

[0007] Perform modal coordinate transformation on each substructure, and couple the substructures after modal coordinate transformation into a super-element model;

[0008] Solve the modal vibration matrix and natural frequency of the super element model;

[0009] Divide the structural modes into dense modes and residual modes based on the natural frequencies;

[0010] The residual modal response of the response to be measured is reconstructed based on the residual modal response of the acquired response using the empirical mode decomposition method.

[0011] The dense modal response of the response to be measured is reconstructed based on the mode shape matrix, the acquired response, and the residual modal response of the acquired response, and the response to be measured is obtained by using the modal superposition method.

[0012] Furthermore, the process of dividing the structural modes into dense modes and residual modes based on the natural frequencies is specifically as follows:

[0013] For the first h-order modes, a series of modes in which the difference between the natural frequencies of each pair is less than 2 Hz in the h-order modes are grouped into a group of dense modes. After all dense mode groups are divided, the remaining modes in the h-order modes are grouped into a group of residual modes.

[0014] Furthermore, the process of extracting the residual modal response from the acquired response using the empirical mode decomposition method and reconstructing the residual modal response of the response to be measured based on the residual modal response of the acquired response includes the following:

[0015] Construct a set of known responses for a superelement model:

[0016]

[0017] Among them, p m (t) represents the known response set of the superelement model, represents the modal vibration matrix of the super-element model, subscripts a and r represent the dense mode and residual mode of the structure respectively, subscripts m and u represent the degree of freedom of the response acquisition point and the degree of freedom of the response to be measured respectively, D a (t) and D r (t) are the generalized modal coordinate vectors of the dense mode and the residual mode under the modal coordinates of the super-element model. In the same model, all degrees of freedom share the same generalized modal coordinate vector. and are the dense modal response set and the residual modal response set of the collected responses, and are the dense modal response set and the residual modal response set of the responses to be measured, respectively;

[0018] The known response set p of the superelement model is extracted using the empirical mode decomposition method with the intermittency criterion. m The single-order residual modal response in (t) is used to construct the residual modal response set of the acquired response:

[0019]

[0020] in, represents the single-order residual modal response vector extracted by the empirical mode decomposition method with the intermittent criterion. It is a row vector with a length equal to the number of acquisition time points. The superscript T represents the transpose.

[0021] The residual modal response of the response to be measured is reconstructed using the following formula:

[0022]

[0023] ...

[0025]

[0026] Among them, the subscript m b Indicates the bth response in the collection response, is the corresponding known response The mode shape at the degree of freedom position.

[0027] Furthermore, the process of reconstructing the dense modal response of the response to be measured based on the mode shape matrix, the acquired response, and the residual modal response of the acquired response includes the following:

[0028] Based on formula (12), the generalized modal coordinate D is obtained a The expression of (t) is:

[0029]

[0030] And the dense modal response of the response to be measured is reconstructed by the following formula:

[0031]

[0032] The superscript + indicates the generalized inverse of the matrix, and the number of acquired responses is not less than the number of dense modes.

[0033] Furthermore, the process of obtaining the response to be measured using the modal superposition method is as follows:

[0034] Define the following matrix:

[0035]

[0036] Combining formulas (15), (17), and (18) to superimpose the dense modal response and the residual modal response, the response to be measured is given by the following formula:

[0037]

[0038] Since the response X to be measured at the corresponding position in the original model u (t) is equal to the principal mode displacement of the super-element model. Therefore, the measured response of the original model is expressed as:

[0039]

[0040] Furthermore, the process of solving the modal vibration matrix and natural frequency of the super-element model is as follows:

[0041] The undamped free vibration equation of the superelement model is:

[0042]

[0043] in, are the stiffness matrix and mass matrix of the super element model respectively, Represents the modal frequency matrix of each order of the super-element model, and the elements on its diagonal are the natural frequency values ​​of the super-element model. The modal vibration matrix of the super unit model can be obtained by solving formula (10). The modal vibration matrix Specifically, it can be expressed as:

[0044]

[0045] in, Each column represents a mode, and each element in each column represents the displacement contribution value of each degree of freedom.

[0046] Furthermore, the process of dividing the finite element model into substructures and dividing the degrees of freedom of each substructure is specifically as follows:

[0047] The finite element model is divided into multiple substructures, and the dynamic equations of the substructures can be expressed as:

[0048]

[0049] Among them, M s 、C s and K s They represent the mass matrix, damping matrix and stiffness matrix of the sth substructure in the finite element model, respectively. s (t), and Represents its displacement, velocity and acceleration respectively, f s (t) is the external force on the s-th substructure, g s (t) is the interface force of the s-th substructure;

[0050] All degrees of freedom of the substructure are divided into internal group i and boundary group j. The degrees of freedom in internal group i are not shared with any substructure, and boundary group j must contain the actual boundary, that is, the degrees of freedom shared between substructures. The degrees of freedom of the response acquisition points and the degrees of freedom of the points to be measured are added to the boundary group j. The elements of each matrix in formula (1) are reordered as follows:

[0051]

[0052] The superscript s represents the sth substructure, and the subscripts i and j represent the internal group degrees of freedom and boundary group degrees of freedom of the corresponding substructure, respectively.

[0053] In addition, the present invention also provides a dense modal response reconstruction system of a condensation structure, comprising:

[0054] Substructure division unit, used to divide the finite element model into substructures and divide the degrees of freedom of each substructure;

[0055] The coupling unit is used to perform modal coordinate transformation on each substructure and couple the substructures after modal coordinate transformation into a super-unit model;

[0056] Solving unit, used to solve the modal vibration matrix and natural frequency of the super unit model;

[0057] A modal division unit is used to divide the structural modes into dense modes and residual modes based on the natural frequencies;

[0058] A residual modal response reconstruction unit is used to extract the residual modal response from the collected response using an empirical mode decomposition method, and reconstruct the residual modal response of the response to be measured based on the residual modal response of the collected response;

[0059] The dense modal response reconstruction unit is used to reconstruct the dense modal response of the response to be measured based on the modal vibration matrix, the acquisition response, and the residual modal response of the acquisition response, and obtain the response to be measured by adopting the modal superposition method.

[0060] In addition, the present invention also provides an electronic device, including a processor and a memory, wherein the memory stores a computer program, and the processor is configured to execute the steps of the above method by calling the computer program stored in the memory.

[0061] In addition, the present invention also provides a computer-readable storage medium for storing a computer program for reconstructing the dense modal response of a condensed structure, wherein the computer program executes the steps of the method described above when running on a computer.

[0062] The present invention has the following effects:

[0063] The dense modal response reconstruction method of the condensation structure of the present invention first carries out substructure division to the finite element model according to engineering practice, then carries out degree of freedom division to each substructure, and the response acquisition degree of freedom and the degree of freedom to be measured are classified into the boundary group degree of freedom. Modal coordinate transformation is then performed on each substructure, and each substructure after the modal coordinate transformation is coupled into a super unit model. Then, the modal vibration matrix and the natural frequency of the super unit model are solved, and the structural mode is divided into dense mode and residual mode by utilizing the natural frequency, which can effectively separate the dense mode. Then, the residual modal response is extracted from the measurement data (i.e., acquisition response) by the EMD decomposition method (empirical mode decomposition method), and the residual modal response of the acquisition response is utilized to reconstruct the residual modal response of the response to be measured. Based on the modal vibration matrix, the acquisition response, and the residual modal response of the acquisition response, the dense modal response of the response to be measured is reconstructed, and finally the dense modal response and the residual modal response of the response to be measured are subjected to modal superposition to obtain the response to be measured. This method uses model reduction to generate a super-element model with fewer degrees of freedom based on the finite element model, which can effectively improve reconstruction efficiency. By dividing the structural modes into dense modes and residual modes, the residual modal response and dense modal response of the test point are reconstructed separately, and finally the test response is obtained through modal superposition. Compared with the existing EMD decomposition method that extracts single-frequency modal responses through bandpass filters, this method can effectively separate dense modes and accurately reconstruct dense modal responses, which is more suitable for the response reconstruction of dense modal structures. In addition, this method can reconstruct the response of any degree of freedom of the structure based on the known structural response, improving engineering practicality, ensuring accuracy, greatly improving reconstruction efficiency, saving computer memory, and accelerating analysis.

[0064] In addition, the dense modal response reconstruction system of the condensation structure of the present invention also has the above advantages.

[0065] In addition to the above-described objects, features and advantages, the present invention has other objects, features and advantages. The present invention will be further described in detail below with reference to the accompanying drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0066] The accompanying drawings, which constitute part of this application, are intended to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are intended to explain the present invention and do not constitute an undue limitation of the present invention. In the accompanying drawings:

[0067] Figure 1 It is a flow chart of a method for reconstructing dense modal responses of a condensation structure according to a preferred embodiment of the present invention.

[0068] Figure 2 It is a schematic diagram of a finite element model of a transmission tower in a specific simulation case of the present invention.

[0069] Figure 3a yes Figure 2 Schematic diagram of the transmission tower substructure 1 obtained by dividing the finite element model in .

[0070] Figure 3b yes Figure 2 Schematic diagram of the transmission tower substructure 2 obtained by dividing the finite element model in

[0071] Figure 3c yes Figure 2 Schematic diagram of the transmission tower substructure 3 obtained by dividing the finite element model in .

[0072] Figure 3d yes Figure 2 Schematic diagram of the transmission tower substructure 4 obtained by dividing the finite element model in .

[0073] Figure 4 yes Figure 2 Schematic diagram comparing the theoretical response value and the reconstructed response value at position Loc.R in .

[0074] Figure 5 yes Figure 2 Schematic diagram comparing the response theoretical value and the response reconstructed value using the traditional EMD reconstruction method at position Loc.R in .

[0075] Figure 6 It is a schematic diagram of the unit structure of a dense modal response reconstruction system of a condensation structure according to another embodiment of the present invention. DETAILED DESCRIPTION

[0076] The embodiments of the present invention are 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.

[0077] like Figure 1 As shown, a preferred embodiment of the present invention provides a method for reconstructing dense modal responses of a condensation structure, comprising the following steps:

[0078] Step S1: Divide the finite element model into substructures and divide the degrees of freedom of each substructure;

[0079] Step S2: performing modal coordinate transformation on each substructure, and coupling the substructures after modal coordinate transformation into a super-element model;

[0080] Step S3: solving the modal vibration matrix and natural frequency of the super unit model;

[0081] Step S4: dividing the structural modes into dense modes and residual modes based on the natural frequencies;

[0082] Step S5: extracting the residual modal response from the collected response using the empirical mode decomposition method, and reconstructing the residual modal response of the response to be measured based on the residual modal response of the collected response;

[0083] Step S6: reconstructing the dense modal response of the response to be measured based on the modal vibration matrix, the acquired response, and the residual modal response of the acquired response, and obtaining the response to be measured by adopting the modal superposition method.

[0084] It is understood that the dense modal response reconstruction method of the condensed structure of the present embodiment first divides the finite element model into substructures according to the actual project, then divides the degrees of freedom of each substructure, and classifies the response acquisition degrees of freedom and the degrees of freedom to be measured into the boundary group degrees of freedom. Then, each substructure is subjected to modal coordinate transformation, and each substructure after the modal coordinate transformation is coupled into a super unit model. Then, the modal vibration matrix and natural frequency of the super unit model are solved, and the structural mode is divided into dense modes and residual modes using the natural frequency, which can effectively separate the dense modes. Then, the residual modal response is extracted from the measurement data (i.e., the acquisition response) by the EMD decomposition method (empirical mode decomposition method), and the residual modal response of the acquisition response is used to reconstruct the residual modal response of the response to be measured. Then, based on the modal vibration matrix, the acquisition response, and the residual modal response of the acquisition response, the dense modal response of the response to be measured is reconstructed. Finally, the dense modal response and the residual modal response of the response to be measured are subjected to modal superposition to obtain the response to be measured. This method uses model reduction to generate a super-element model with fewer degrees of freedom based on the finite element model, which can effectively improve reconstruction efficiency. By dividing the structural modes into dense modes and residual modes, the residual modal response and dense modal response of the test point are reconstructed separately, and finally the test response is obtained through modal superposition. Compared with the existing EMD decomposition method that extracts single-frequency modal responses through bandpass filters, this method can effectively separate dense modes and accurately reconstruct dense modal responses, which is more suitable for the response reconstruction of dense modal structures. In addition, this method can reconstruct the response of any degree of freedom of the structure based on the known structural response, improving engineering practicality, ensuring accuracy, greatly improving reconstruction efficiency, saving computer memory, and accelerating analysis.

[0085] It can be understood that step S1 specifically includes the following contents:

[0086] First, the finite element model is divided into multiple substructures based on the structural connection characteristics, actual engineering needs, and other factors. For example, the location of the test point is used as the interface of the substructure, and the substructure of the finite element model is divided accordingly. The dynamic equation of the substructure can be expressed as:

[0087]

[0088] Among them, Ms 、C s and K s They represent the mass matrix, damping matrix and stiffness matrix of the sth substructure in the finite element model, respectively. s (t), and Represents its displacement, velocity and acceleration respectively, f s (t) is the external force on the s-th substructure, g s (t) is the interface force of the sth substructure.

[0089] Then, all the degrees of freedom of the substructure are divided into internal group i and boundary group j. The degrees of freedom in internal group i are not shared with any substructure, and boundary group j must contain the actual boundary, that is, the degrees of freedom shared between substructures. The degrees of freedom of the response acquisition points and the degrees of freedom of the points to be measured are added to the boundary group j. Rearrange the elements of each matrix in formula (1) as follows:

[0090]

[0091] The superscript s represents the sth substructure, and the subscripts i and j represent the internal group degrees of freedom and boundary group degrees of freedom of the corresponding substructure, respectively.

[0092] It can be understood that step S2 specifically includes the following contents:

[0093] The modal transformation matrix Φ extracted by the fixed interface modal synthesis method proposed by Craig-Bampton s , which is the main mode set selected by the fixed interface and the constraint mode set of all interface coordinates Composition, that is

[0094]

[0095] in, After the substructure interface is fixed, the formula Find, among them To obtain The first k columns of modes are taken, are the modal frequencies of the internal degrees of freedom of substructure s, is the identity matrix of order j, is a zero matrix with j rows and k columns.

[0096] Then, the first coordinate transformation is performed on each substructure of the finite element model, as shown in the following formula:

[0097] Among them, ΦsT is the transpose of the modal transformation matrix, are the stiffness matrix, mass matrix, damping matrix, external force and interface force of the sth substructure after modal coordinate transformation.

[0098] The response coordinate transformation is:

[0099] Among them, q s is the generalized coordinate of the structural response of substructure s after modal coordinate transformation, Respectively represent q s The interior group degree of freedom set and the boundary group degree of freedom set.

[0100] According to the second row vector of formula (4) The operation can obtain the following relationship:

[0101]

[0102] That is, after the first coordinate transformation, the response of the degrees of freedom in the boundary group in the modal coordinates is equal to the corresponding original structure response.

[0103] After the first coordinate transformation, the dynamic motion equation of the substructure s can be expressed as:

[0104]

[0105] Then the dynamic motion equation of the entire finite element model can be expressed as:

[0106]

[0107] in,

[0108]

[0109] q T =[q 1T ,...,q sT ,...,q nT ];f T =[f 1T ,...,f sT ,...,f nT ]; g T =[g 1T ,...,g sT ,...,g nT ],q T is the transpose of q, f T is the transpose of f, g T is the transpose of g, n represents the number of substructures, are the overall stiffness matrix, overall mass matrix and overall damping matrix after modal coordinate transformation, q(t) is the non-independent main modal displacement, is the non-independent main modal velocity, is the non-independent main modal acceleration.

[0110] Finally, the substructures are coupled into a super-element model by combining the Boolean matrix L. Specifically, taking the sth substructure and the s+1th substructure as an example (s>1), when performing the second coordinate transformation, the dependent generalized coordinates of the two substructures are:

[0111]

[0112] in, are the internal degree of freedom response of substructure s, the boundary degree of freedom response of substructure s, the internal degree of freedom response of substructure s+1, and the boundary degree of freedom response of substructure s+1 in generalized coordinates, respectively. The superscript T indicates transposition, and and are the corresponding equal generalized coordinates on the common boundary of the sth substructure and the s+1th substructure, that is,

[0113] When each substructure is coupled into a super unit model, the interface force is zero. According to the interface force balance condition: L T g(t)=0, the corresponding interface displacements between adjacent substructures are equal, that is, Let q = Lp, where p represents the independent coordinate. Then the independent coordinates of the two substructures after coupling after the second coordinate transformation are:

[0114]

[0115] From this, a Boolean matrix can be constructed to perform the second coordinate transformation:

[0116]

[0117] Then the dynamic motion equation of the entire super-element model can be expressed as:

[0118]

[0119] in, p(t) are the stiffness matrix, mass matrix, damping matrix and displacement mode of the super unit model, The superscript T represents the transpose of the matrix, p(t) is the independent main mode displacement, is the independent main modal velocity, is the acceleration of the independent main mode. It can be seen that the second coordinate transformation only eliminates the non-independent main modes in the coupled super-element model or changes the arrangement order of each main mode, and does not change the element size. Therefore, combined with formula (5), the following relationship can be obtained:

[0120]

[0121] in, The set of boundary groups representing the independent principal modal displacement vectors of the sth substructure in the superelement model. It can be seen that the principal modal displacements of the superelement model for the degrees of freedom in the boundary group are equal to the displacement responses of the corresponding original structure. Consequently, the principal modal velocities and accelerations of the superelement model for the degrees of freedom in the boundary group are equal to the velocity and acceleration responses of the corresponding original structure. This allows for response reconstruction on the superelement model by adding the response acquisition degrees of freedom and the degrees of freedom to be measured from the actual structure to the boundary group.

[0122] It can be understood that the step S3 is specifically as follows:

[0123] The undamped free vibration equation of the superelement model is:

[0124]

[0125] in, Represents the modal frequency matrix of each order of the entire super-element model, and the elements on its diagonal are the natural frequency values ​​of the super-element model. The modal vibration matrix of the super unit model can be obtained by solving formula (10). The modal vibration matrix Specifically, it can be expressed as:

[0126]

[0127] in, Each column represents a mode, and each element in each column represents the displacement contribution value of each degree of freedom.

[0128] It can be understood that step S4 is specifically as follows: for the first h-order modes, a series of modes in which the difference between the natural frequencies of each pair is less than 2 Hz in the h-order modes are grouped into a group of dense modes; after all dense mode groups are divided, the remaining modes in the h-order modes are grouped into a group of residual modes.

[0129] The existing empirical mode decomposition method directly extracts single-frequency modal responses from the response acquisition point signals by setting a bandpass filter. However, for dense modes, when the difference between the two-order frequencies is less than 1.5Hz, it is difficult to separate them using the bandpass filter, making it impossible to effectively separate the dense modes and, therefore, unable to reconstruct the response of structures with dense modes. The present invention, however, uses the natural frequency of the superelement model to divide the structural modes into dense modes and residual modes, and then reconstructs the dense modes and residual modes separately, effectively separating the dense modes and reconstructing the response to be measured.

[0130] It can be understood that the step S5 is specifically as follows:

[0131] Since the response acquisition degrees of freedom and the degrees of freedom to be measured are placed in the boundary group, the known response set of the superelement model can be expressed as:

[0132]

[0133] The unknown response set of the superelement model can be expressed as:

[0134]

[0135] Among them, p m (t) represents the known response set of the super-element model, p u (t) represents the unknown response set of the super-element model, subscripts a and r represent the dense mode and residual mode of the structure respectively, and subscripts m and u represent the degree of freedom of the response acquisition point and the degree of freedom of the response to be measured respectively. a (t) and D r (t) are the generalized modal coordinate vectors of the dense mode and the residual mode under the modal coordinates of the super-element model. In the same model, all degrees of freedom share the same generalized modal coordinate vector. and are the dense modal response set and the residual modal response set of the collected responses, and are the dense modal response set and the residual modal response set of the response to be measured, respectively.

[0136] Then, the empirical mode decomposition method with intermittency criterion is used to extract the known response set p of the superelement model. m (t), the residual modal response set of the acquired response can be expressed as:

[0137]

[0138] in, represents the set of residual modal responses of the acquired responses, is the single-order residual modal response vector, a row vector whose length is equal to the number of acquisition time points. The superscript T indicates the transpose. Furthermore, the process of extracting the single-order residual modal response from a known response set using the EMD method is prior art and can be referenced in the applicant's prior patent CN202011477125.5. Therefore, this description will not be repeated here.

[0139] Then, the residual modal response of the response to be measured is reconstructed based on the residual modal response of the acquired response using the following formula:

[0140]

[0141] Among them, the subscript m b Indicates the bth response in the collection response, is the corresponding known response The mode shape at the degree of freedom position.

[0142] It can be understood that the step S6 is specifically as follows:

[0143] The generalized modal coordinate D is obtained from formula (12): a The expression of (t) is:

[0144]

[0145] The superscript + represents the generalized inverse of the matrix, which is specifically expressed as The superscript - indicates the inverse of the matrix. Therefore, the number of collected responses should be no less than the number of dense modes to ensure that the generalized inverse is valid. The dense modal response of the response to be measured is reconstructed by the following formula:

[0146]

[0147] Then, define the following matrix:

[0148]

[0149] Combining formulas (15), (17), and (18) to superimpose the dense modal response and the residual modal response, the response to be measured is given by the following formula:

[0150]

[0151] Since the response acquisition degrees of freedom and the degrees of freedom to be measured are both in the boundary group of the super-element model, according to formula (9), the response to be measured X at the corresponding position in the original model is u (t) is equal to the principal mode displacement of the super-element model. Therefore, the measured response of the original model is expressed as:

[0152]

[0153] Next, if Figures 2 to 5 As shown in the figure, the implementation process of dense modal response reconstruction is described by taking the transmission tower simulation model as the research object.

[0154] Transmission tower simulation model Figure 2 As shown, the finite element model was built using ANSYS APDL. The beam element type used was beam188, the material Young's modulus was 206 GPa, and the density was 7850 kg / m 3 The transmission tower model has a total of 3132 units, 2764 nodes, 16584 degrees of freedom, six response collection locations and response test locations. Figure 2 As shown in Figure 2, the specific implementation steps of dense modal response reconstruction are as follows:

[0155] (1) The transmission tower model is divided into four substructures, such as Figure 3a to Figure 3d As shown;

[0156] (2) Generate the super-element model of the transmission tower according to equations (1) to (9). The total number of degrees of freedom of the integrated super-element model is 664, and the number of degrees of freedom of a single substructure is 166;

[0157] (3) Extract the modal vibration shapes of the super unit model according to equations (10) and (11), consider the first ten modes, and divide the 3rd to 5th order modes and the 8th to 10th order modes into two groups of dense modes according to the natural frequency, and the 1st, 2nd, 6th, and 7th order modes are regarded as a group of residual modes;

[0158] (4) The residual modal responses of each order of the six acquired responses are extracted by the EMD decomposition method with intermittent criterion, and the residual modal responses of the response to be measured are reconstructed according to formula (15);

[0159] (5) According to formulas (16) to (20), the dense modal response of the response to be measured is reconstructed, and the response to be measured is obtained by the modal superposition method.

[0160] The reconstructed value of the response to be measured reconstructed according to the above steps (1) to (5) is compared with the theoretical value of the response to be measured. Figure 4 As shown, it can be seen that the measured response reconstructed by this method is very close to the theoretical value, and the reconstruction accuracy is high.

[0161] In addition, the dense modal response reconstruction of the present invention is compared and analyzed with the traditional EMD-based response reconstruction to study the advantages of the structural response reconstruction of the present invention. The traditional method directly extracts the modal response through EMD. In this embodiment, on the super-element model, the EMD decomposition method with intermittent criterion is used to directly extract all the first ten modes of the six collected responses, and then the response to be measured is reconstructed according to formula (15). The comparison diagram of the reconstructed response under the traditional method and the theoretical value of the response to be measured is shown in Figure 2. Figure 5 As shown. Figure 5 and Figure 4 By comparison, it can be seen that the dense modal response reconstruction method of the present invention has greatly improved reconstruction accuracy in dense modal structures and has wider applicability.

[0162] In addition, the parameter matrices (stiffness and mass matrices) of large structures contain a large amount of data redundancy, that is, they contain many zero elements, which consumes a lot of time and memory during the entire reconstruction process. Table 1 below compares the amount of data required to extract the original model and the data after subdivision. This data includes the number of elements in the structural stiffness and mass matrices. It can be seen that after division into four substructures, the data involved in the calculation is greatly reduced, effectively alleviating the problem of insufficient computer memory. In addition, the order of the superelement stiffness matrix and mass matrix involved in the reconstruction process is reduced from 16,584 in the original model to 9,072 and 7,994 in the superelement model, reducing the amount of calculation.

[0163] Table 1. Comparison of the amount of data to be extracted between the original model and the substructure

[0164]

[0165] The above simulation case illustrates that the present invention can accurately reconstruct the response information of the measured point through its specific implementation steps. When applied to the dynamic response reconstruction of a large structure, the present invention can divide the substructure according to the position of the measured point, and only the response reconstruction needs to be performed under the condensed substructure, which greatly reduces the amount of calculation and improves the efficiency of response reconstruction.

[0166] In addition, if Figure 6 As shown, another embodiment of the present invention further provides a dense modal response reconstruction system for a condensation structure, preferably using the dense modal response reconstruction method for a condensation structure as described above, the system includes

[0167] Substructure division unit, used to divide the finite element model into substructures and divide the degrees of freedom of each substructure;

[0168] The coupling unit is used to perform modal coordinate transformation on each substructure and couple the substructures after modal coordinate transformation into a super-unit model;

[0169] Solving unit, used to solve the modal vibration matrix and natural frequency of the super unit model;

[0170] A modal division unit is used to divide the structural modes into dense modes and residual modes based on the natural frequencies;

[0171] A residual modal response reconstruction unit is used to extract the residual modal response from the collected response using an empirical mode decomposition method, and reconstruct the residual modal response of the response to be measured based on the residual modal response of the collected response;

[0172] The dense modal response reconstruction unit is used to reconstruct the dense modal response of the response to be measured based on the modal vibration matrix, the acquisition response, and the residual modal response of the acquisition response, and obtain the response to be measured by adopting the modal superposition method.

[0173] It can be understood that the specific working process of each unit of this system corresponds to each step of the above method embodiment, so it will not be repeated here.

[0174] It is understood that the dense modal response reconstruction system of the condensed structure of the present embodiment first divides the finite element model into substructures according to the actual project, then divides the degrees of freedom of each substructure, and classifies the response acquisition degrees of freedom and the degrees of freedom to be measured into the boundary group degrees of freedom. Then, each substructure is subjected to modal coordinate transformation, and each substructure after the modal coordinate transformation is coupled into a super unit model. Then, the modal vibration matrix and natural frequency of the super unit model are solved, and the structural mode is divided into dense modes and residual modes using the natural frequency, which can effectively separate the dense modes. Then, the residual modal response is extracted from the measurement data (i.e., the acquisition response) by the EMD decomposition method (empirical mode decomposition method), and the residual modal response of the acquisition response is used to reconstruct the residual modal response of the response to be measured. Then, based on the modal vibration matrix, the acquisition response, and the residual modal response of the acquisition response, the dense modal response of the response to be measured is reconstructed. Finally, the dense modal response and the residual modal response of the response to be measured are modally superimposed to obtain the response to be measured. This system uses model reduction to generate a super-element model with fewer degrees of freedom based on the finite element model, which can effectively improve reconstruction efficiency. By dividing the structural modes into dense modes and residual modes, the residual modal response and dense modal response of the test point are reconstructed separately, and finally the test response is obtained through modal superposition. Compared with the existing EMD decomposition method that extracts single-frequency modal responses through bandpass filters, it can effectively separate dense modes and accurately reconstruct dense modal responses, which is better suitable for the response reconstruction of dense modal structures. In addition, based on the known structural response, the system can reconstruct the response of any degree of freedom of the structure, improving engineering practicality, ensuring accuracy, greatly improving reconstruction efficiency, saving computer memory, and accelerating analysis.

[0175] In addition, the present invention also provides an electronic device, including a processor and a memory, wherein the memory stores a computer program, and the processor is configured to execute the steps of the above method by calling the computer program stored in the memory.

[0176] In addition, the present invention also provides a computer-readable storage medium for storing a computer program for reconstructing dense modal responses of condensed structures. When the computer program is run on a computer, the steps of the method described above are executed.

[0177] Common forms of computer-readable media include: floppy disks, flexible disks, hard disks, magnetic tape, any other magnetic media, CD-ROMs, any other optical media, punch cards, paper tape, any other physical medium with a pattern of holes, random access memory (RAM), programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), flash-erasable programmable read-only memory (FLASH-EPROM), any other memory chip or cartridge, or any other medium that can be read by a computer. Instructions can further be transmitted or received via a transmission medium. The term transmission medium may include any tangible or intangible medium that can be used to store, encode, or carry instructions for execution by a machine, and includes digital or analog communication signals or other intangible media that facilitate communication of such instructions. Transmission media include coaxial cables, copper wire, and fiber optics, including the wires of a bus used to transmit a computer data signal.

[0178] This invention is one of the contents of the National Natural Science Foundation of China (52078504, 51925808, U1934209).

[0179] The foregoing description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Those skilled in the art will readily appreciate that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention are intended to be within the scope of protection of the present invention.

Claims

1. A method for reconstructing dense modal responses of condensation structures, characterized in that: Includes the following: Divide the finite element model into substructures and divide the degrees of freedom of each substructure; Perform modal coordinate transformation on each substructure, and couple the substructures after modal coordinate transformation into a super-element model; Solve the modal vibration matrix and natural frequency of the super element model; Divide the structural modes into dense modes and residual modes based on the natural frequencies; The residual modal response of the response to be measured is reconstructed based on the residual modal response of the acquired response using the empirical mode decomposition method. Reconstruct the dense modal response of the response to be measured based on the mode shape matrix, the acquired response, and the residual modal response of the acquired response, and obtain the response to be measured using the modal superposition method; The process of extracting the residual modal response from the acquired response using the empirical mode decomposition method and reconstructing the residual modal response of the response to be measured based on the residual modal response of the acquired response includes the following: Construct a set of known responses for a superelement model: Among them, p m (t) represents the known response set of the superelement model, represents the modal vibration matrix of the super-element model, subscripts a and r represent the dense mode and residual mode of the structure respectively, subscripts m and u represent the degree of freedom of the response acquisition point and the degree of freedom of the response to be measured respectively, D a (t) and D r (t) are the generalized modal coordinate vectors of the dense mode and the residual mode under the modal coordinates of the super-element model. In the same model, all degrees of freedom share the same generalized modal coordinate vector. and are the dense modal response set and the residual modal response set of the collected responses, and are the dense modal response set and the residual modal response set of the responses to be measured, respectively; The known response set p of the superelement model is extracted using the empirical mode decomposition method with the intermittency criterion. m The single-order residual modal response in (t) is used to construct the residual modal response set of the acquired response: in, represents the single-order residual modal response vector extracted by the empirical mode decomposition method with the intermittent criterion. It is a row vector with a length equal to the number of acquisition time points. The superscript T represents the transpose. The residual modal response of the response to be measured is reconstructed using the following formula: Among them, the subscript m b Indicates the bth response in the collection response, is the corresponding known response The mode shape at the degree of freedom position.

2. The dense modal response reconstruction method of the condensation structure according to claim 1, characterized in that: The process of dividing the structural modes into dense modes and residual modes based on the natural frequency is specifically as follows: For the first h-order modes, a series of modes in which the difference between the natural frequencies of each pair is less than 2 Hz in the h-order modes are grouped into a group of dense modes. After all dense mode groups are divided, the remaining modes in the h-order modes are grouped into a group of residual modes.

3. The dense modal response reconstruction method of the condensation structure according to claim 1, characterized in that: The process of reconstructing the dense modal response of the response to be measured based on the mode shape matrix, the acquired response, and the residual modal response of the acquired response includes the following: Based on formula (12), the generalized modal coordinate D is obtained a The expression of (t) is: And the dense modal response of the response to be measured is reconstructed by the following formula: The superscript + indicates the generalized inverse of the matrix, and the number of acquired responses is not less than the number of dense modes.

4. The dense modal response reconstruction method of the condensation structure according to claim 3, characterized in that: The specific process of obtaining the response to be measured using the modal superposition method is as follows: Define the following matrix: Combining formulas (15), (17), and (18) to superimpose the dense modal response and the residual modal response, the response to be measured is given by the following formula: Since the response X to be measured at the corresponding position in the original model u (t) is equal to the principal mode displacement of the super-element model. Therefore, the measured response of the original model is expressed as:

5. The dense modal response reconstruction method of the condensation structure according to claim 1, characterized in that: The process of solving the modal vibration matrix and natural frequency of the super element model is as follows: The undamped free vibration equation of the superelement model is: in, are the stiffness matrix and mass matrix of the super-element model respectively. Λ represents the modal frequency matrix of each order of the super-element model. The elements on the diagonal are the natural frequency values ​​of the super-element model. The modal vibration matrix of the super unit model can be obtained by solving formula (10). The modal vibration matrix Specifically, it can be expressed as: in, Each column represents a mode, and each element in each column represents the displacement contribution value of each degree of freedom.

6. The dense modal response reconstruction method of the condensation structure according to claim 1, characterized in that: The process of dividing the finite element model into substructures and dividing the degrees of freedom of each substructure is specifically as follows: The finite element model is divided into multiple substructures, and the dynamic equations of the substructures can be expressed as: Among them, M s 、C s and K s They represent the mass matrix, damping matrix and stiffness matrix of the sth substructure in the finite element model, respectively. s (t), and Represents its displacement, velocity and acceleration respectively, f s (t) is the external force on the s-th substructure, g s (t) is the interface force of the s-th substructure; All degrees of freedom of the substructure are divided into internal group i and boundary group j. The degrees of freedom in internal group i are not shared with any substructure, and boundary group j must contain the actual boundary, that is, the degrees of freedom shared between substructures. The degrees of freedom of the response acquisition points and the degrees of freedom of the points to be measured are added to the boundary group j. The elements of each matrix in formula (1) are reordered as follows: The superscript s represents the sth substructure, and the subscripts i and j represent the internal group degrees of freedom and boundary group degrees of freedom of the corresponding substructure, respectively.

7. A system for reconstructing dense modal responses of a condensation structure, using the method for reconstructing dense modal responses of a condensation structure according to any one of claims 1 to 6, characterized in that: include: Substructure division unit, used to divide the finite element model into substructures and divide the degrees of freedom of each substructure; The coupling unit is used to perform modal coordinate transformation on each substructure and couple the substructures after modal coordinate transformation into a super-unit model; Solving unit, used to solve the modal vibration matrix and natural frequency of the super unit model; A modal division unit is used to divide the structural modes into dense modes and residual modes based on the natural frequencies; A residual modal response reconstruction unit is used to extract the residual modal response from the collected response using an empirical mode decomposition method, and reconstruct the residual modal response of the response to be measured based on the residual modal response of the collected response; The dense modal response reconstruction unit is used to reconstruct the dense modal response of the response to be measured based on the modal vibration matrix, the acquisition response, and the residual modal response of the acquisition response, and obtain the response to be measured by adopting the modal superposition method.

8. An electronic device, characterized in that: The method comprises a processor and a memory, wherein the memory stores a computer program, and the processor is configured to execute the steps of the method according to any one of claims 1 to 6 by calling the computer program stored in the memory.

9. A computer-readable storage medium for storing a computer program for reconstructing dense modal responses of a condensed structure, characterized in that: When the computer program is run on a computer, the computer program executes the steps of the method according to any one of claims 1 to 6.

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