Statistical Energy Analysis Subsystem Identification Method Based on Reduced-Order Modal Energy Cluster Analysis

Abstract

The present invention provides a statistical energy analysis subsystem identification method based on reduced-order modal energy cluster analysis. First, a modal analysis is performed on a discretized finite element model of a structure to obtain the modal frequency and modal energy of the structure within the analysis frequency band; then, the modal energy matrix is ​​reduced in order using principal component analysis to extract the main features in the modal information; and then, a cluster analysis is performed on the reduced-order modal energy matrix to identify the statistical energy analysis subsystem of the structure. The traditional statistical energy analysis subsystem identification method requires the calculation of multiple load conditions, which has a large amount of calculation, low identification accuracy, and poor calculation stability. The method proposed in the present invention only requires a finite element modal analysis of the structure, and the main modal energy information is extracted through principal component analysis, which greatly improves the identification accuracy of the statistical energy analysis subsystem.

Description

Technical Field

[0001] The present invention relates to a statistical energy analysis subsystem identification method, and in particular to a statistical energy analysis subsystem identification method based on reduced-order modal energy cluster analysis. Background Art

[0002] High-speed trains, launch vehicles and other engineering structures will face high-frequency vibration loads such as aerodynamic noise and pyrotechnic shock during their service. Statistical energy analysis theory is a common method to predict the dynamic response of structures under high-frequency loads. This method is based on the principles of statistical physics, divides the structure into several subsystems, and uses vibration energy to characterize the high-frequency dynamic response and vibration load of the structure. The high-frequency dynamic response of the structure is obtained by solving the energy balance equation between the subsystems. Therefore, the reasonable division of statistical energy analysis subsystems is the basis and key to statistical energy analysis.

[0003] At present, most researchers divide structural subsystems according to engineering experience, geometric shape, material properties, modal density, etc. This method can be applied to systems with simpler configurations, such as L-shaped plates, T-shaped plates, etc. However, for complex engineering structures, there are many problems when dividing their statistical energy analysis subsystems only through subjective judgment, such as dividing strongly coupled subsystems, merging weakly coupled subsystems, and difficulty in determining the ownership of coupling parts between structures. These problems will lead to a decrease in the accuracy of the high-frequency dynamic response prediction of the structure when conducting statistical energy analysis on complex structures in the future. Therefore, it is urgent to develop identification methods for statistical energy analysis subsystems.

[0004] Cluster analysis is an analysis process that groups a collection of physical or abstract objects into multiple classes consisting of similar objects. It is suitable for identifying statistical energy analysis subsystems. The effect of cluster analysis is directly related to the sample characteristics. The traditional analysis method is based on the virtual excitation method to obtain structural energy. This method requires the calculation of multiple load conditions. In each condition, virtual excitations are randomly applied at different locations of the structure. Cluster analysis is performed after the energy response of the structure is obtained. Therefore, the calculation amount is large when this method is used to identify statistical energy analysis subsystems. At the same time, due to the dense modal density and complex energy response of the structure in the high-frequency band, the identification accuracy of this method is low and the calculation stability is poor. Therefore, it is urgent to develop a high-precision identification method for statistical energy analysis subsystems. Summary of the invention

[0005] Purpose of the invention: The purpose of the present invention is to address the deficiencies in the prior art and provide a statistical energy analysis subsystem identification method based on reduced-order modal energy clustering analysis to solve the problems of low recognition accuracy and poor computational stability in current statistical energy analysis subsystem identification methods.

[0006] Technical solution: The present invention provides a statistical energy analysis subsystem identification method based on reduced-order modal energy cluster analysis, which is characterized by comprising the following steps:

[0007] (1) Obtain the modal energy matrix of the structure within the analysis frequency

[0008] Establish a discretized finite element analysis model of the structure, obtain the modes of each order of the structure within the analysis frequency band through modal analysis, and then obtain the modal energy of each unit of the structure in each modal order, and assemble them into the modal energy matrix of the structure;

[0009] (2) Reduce the modal energy matrix

[0010] Based on the principal component analysis method, the modal energy matrix is ​​reduced to obtain a reduced-order modal energy matrix, thereby reducing the amount of data required for subsequent cluster analysis and improving the stability of cluster analysis;

[0011] (3) Identification of statistical energy analysis subsystem based on cluster analysis method

[0012] Based on the K-Means algorithm, cluster analysis is performed on the reduced-order modal energy matrix to obtain the finite element units of the corresponding classes of each subsystem, and the identification of the statistical energy analysis subsystem is completed.

[0013] Furthermore, the step (1) of obtaining the modal energy matrix of the structure within the analysis frequency comprises the following steps:

[0014] Calculate and analyze the wavelength corresponding to the highest frequency in the frequency band [f1, f2], set the mesh size of the finite element model to a size no greater than 1 / 6 of the wavelength, and establish a finite element analysis model of the structure, with the number of units being m;

[0015] Obtain the n-order mode of the structure within the analysis frequency band through modal analysis And the modal energy of each unit in each mode, the energy corresponding to the i-th unit in the j-th mode is denoted as e ij , then we can get the modal energy matrix E with matrix dimension (m×n).

[0016] Furthermore, step (2) of reducing the order of the modal energy matrix includes the following steps:

[0017] Perform singular value decomposition on the modal energy matrix E and solve the following equation to obtain its n eigenvalues ​​s1, s2…s j …s n And the corresponding eigenvector Ψ:

[0018] E T EV=VΛ

[0019] Ψ=EV

[0020] Where: Λ=[s1,s2…s j …s n ] is a diagonal matrix of the eigenvalues ​​of the modal energy matrix E, where the eigenvalues ​​are arranged in descending order; V = [v1, v2…v j …v n ] is by E T The eigenvectors v1,v2…v of E j …v n The matrix composed of

[0021] Select n1 eigenvalues ​​whose sum of contributions to the eigenvalues ​​accounts for more than 80%, that is, the principal components in the modal energy matrix, obtain the eigenvectors corresponding to these eigenvalues, and form a reduced-order modal energy matrix with a dimension of (m×n1) Perform principal component analysis of the modal energy matrix.

[0022] Furthermore, step (3) includes the following steps of analyzing the reduced modal energy matrix based on the K-Means clustering algorithm:

[0023] 1) From the reduced modal energy matrix K samples are randomly selected as the initial K centroid vectors, and the units of the structure are randomly divided into K classes C t (t=1,2...K);

[0024] 2) For each unit e i (i=1,2...m), assuming that it is in the reduced modal energy matrix The corresponding row vector in is x i =[x i1 ,x i2 …x ij …x in ], assuming that each centroid vector is y t =[y t1 ,y t2 …y tj …y tn ], then x i With y t The standardized Euclidean distance d between them is expressed as:

[0025]

[0026] Among them, st j is the standard deviation of the component.

[0027] 3) Unit e i Classify into the class where the nearest centroid point is located;

[0028] 4) After the distance calculation of all units is completed, the new centroid vectors corresponding to each class are recalculated according to the updated class; if any centroid vector changes, repeat step 2); if all K centroid vectors do not change, go to step 5);

[0029] 5) Based on the results of cluster analysis, the constituent elements of each subsystem are obtained, thereby completing the identification of the statistical energy analysis subsystem.

[0030] Beneficial effects: The traditional analysis method obtains structural energy based on the virtual excitation method. This method needs to calculate multiple load conditions. In each condition, virtual excitations are randomly applied at different positions of the structure. Cluster analysis is performed after the energy response of the structure is obtained. Therefore, the calculation amount is large when this method is used to identify the statistical energy analysis subsystem, and the recognition efficiency is very low. At the same time, since the structure has dense modes in the high-frequency band, the amount of energy response data of the structure is huge, resulting in low recognition accuracy and poor calculation stability of the traditional analysis method. The statistical energy analysis subsystem identification method based on reduced-order modal energy cluster analysis of the present invention only needs to perform a finite element modal analysis on the structure once, and extracts the main modal energy information through principal component analysis. The calculation amount is smaller than that of the traditional analysis method, and the accuracy and stability of the statistical energy analysis subsystem identification are greatly improved, thereby quickly and accurately identifying the statistical energy analysis subsystem. BRIEF DESCRIPTION OF THE DRAWINGS

[0031] Figure 1 It is a schematic diagram of the geometric model and subsystem distribution of the I-type plate structure of the embodiment;

[0032] Figure 2 The eigenvalues ​​and contribution of the modal energy matrix of the I-type plate structure of the embodiment;

[0033] Figure 3 The result of the statistical energy analysis subsystem of the I-shaped folded plate structure of the embodiment is identified by the method of the present invention;

[0034] Figure 4 The results of the statistical energy analysis subsystem of the I-type folded plate structure of the embodiment are identified using traditional methods. DETAILED DESCRIPTION

[0035] The technical solution of the present invention is described in detail below, but the protection scope of the present invention is not limited to the embodiments.

[0036] Select as Figure 1 The I-type plate structure shown is used as an example. The thickness of each plate is set to 0.05m, and other geometric dimensions are as follows Figure 1 As shown in the marking. The material parameters of the structure are: elastic modulus is 30Gpa, density is 2400kg / m 3 , Poisson's ratio is 0.2.

[0037] The specific operations are as follows:

[0038] (1) Set the analysis frequency to [900,1200] Hz. The propagation speed of waves in metal is about 3000 m / s, and the maximum unit size calculated is 416 mm. A finite element analysis model of the I-shaped plate structure is established, with a total of 280 units. The maximum unit size of the finite element model is 267 mm, which meets the requirements of the calculation and analysis. To facilitate the subsequent observation of the cluster analysis effect, the unit numbering of subsystem 1 is defined as 1 to 70, the unit numbering of subsystem 2 is defined as 71 to 112, the unit numbering of subsystem 3 is defined as 113 to 168, the unit numbering of subsystem 4 is defined as 169 to 238, and the unit numbering of subsystem 5 is defined as 239 to 280.

[0039] The structure was subjected to modal analysis using commercial finite element software to obtain a total of 66 structural modes in the analysis frequency band [900, 1200] Hz. The modal energy of each unit in each modal order was obtained and assembled into a structural modal energy matrix with a dimension of (280×66).

[0040] (2) Perform singular value decomposition on the modal energy matrix to obtain eigenvalues ​​and eigenvectors. The contribution of the first ten eigenvalues ​​is as follows: Figure 2 The sum of the contributions of the first 7 eigenvalues ​​is greater than 80%, so the eigenvectors corresponding to the first 7 eigenvalues ​​are used to form a reduced modal energy matrix with a dimension of (280×7)

[0041] (3) K-Means clustering algorithm is used to reduce the modal energy matrix Cluster analysis was performed, and the results of cluster analysis were as follows Figure 3 As shown in Figure 1, the five subsystem classes obtained are: units 1 to 70, units 71 to 112, units 113 to 168, units 169 to 238, and units 239 to 280. The recognition results are completely consistent with the expected results. It can be seen that the method of the present invention has a good recognition effect and high calculation accuracy. The recognition results of the traditional analysis method are as follows Figure 4 As shown, the recognition error is large. It should be noted that Figure 3 and Figure 4 The number of "Subsystem Category" on the vertical axis is not the number of the statistical energy analysis subsystem, but a temporary number in the cluster analysis.

Claims

1. A statistical energy analysis subsystem identification method based on reduced-order modal energy cluster analysis, characterized in that The following steps are involved: (1) Obtain the modal energy matrix of the structure within the analysis frequency Establish a discretized finite element analysis model of the structure, obtain the modes of each order of the structure within the analysis frequency band through modal analysis, and then obtain the modal energy of each unit of the structure in each modal order, and assemble them into the modal energy matrix of the structure; (2) Reduce the modal energy matrix Based on the principal component analysis method, the modal energy matrix is ​​reduced to obtain a reduced-order modal energy matrix, thereby reducing the amount of data required for subsequent cluster analysis and improving the stability of cluster analysis; (3) Identification of statistical energy analysis subsystem based on cluster analysis method Based on the K-Means algorithm, cluster analysis is performed on the reduced-order modal energy matrix to obtain the finite element units of the corresponding classes of each subsystem, and the identification of the statistical energy analysis subsystem is completed; Step (2) of reducing the modal energy matrix includes the following steps: Perform singular value decomposition on the modal energy matrix E and solve the following equation to obtain its n eigenvalues ​​s1, s2…s j …s n And the corresponding eigenvector Ψ: E T EV=VΛ Ψ=EV Where: Λ=[s1,s2…s j …s n ] is a diagonal matrix composed of the eigenvalues ​​of the modal energy matrix E, where the eigenvalues ​​are arranged in descending order; V = [v1, v2…v j …v n ] is by E T The eigenvectors v1,v2…v of E j …v n The matrix composed of Select n1 eigenvalues ​​whose sum of contributions to the eigenvalues ​​accounts for more than 80%, that is, the principal components in the modal energy matrix, obtain the eigenvectors corresponding to these eigenvalues, and form a reduced-order modal energy matrix with a dimension of m×n1 Complete principal component analysis of the modal energy matrix; Step (3) Analyzing the reduced modal energy matrix based on the K-Means clustering algorithm includes the following steps: 1) From the reduced modal energy matrix K samples are randomly selected as the initial K centroid vectors, and the units of the structure are randomly divided into K classes C t , t=1,2...K; 2) For each unit e i , i = 1, 2...m, assuming that it is in the reduced modal energy matrix The corresponding row vector in is x i =[x i1 ,x i2 …x ij …x in ], assuming that each centroid vector is y t =[y t1 ,y t2 …y tj …y tn ], then x i With y t The standardized Euclidean distance d between them is expressed as: Among them, st j is the standard deviation of the component; 3) Unit e i Classify into the class where the nearest centroid point is located; 4) After the distance calculation of all units is completed, the new centroid vectors corresponding to each class are recalculated according to the updated class; if any centroid vector changes, repeat step 2); if all K centroid vectors do not change, go to step 5); 5) Based on the results of cluster analysis, the constituent elements of each subsystem are obtained, thereby completing the identification of the statistical energy analysis subsystem.

2. The statistical energy analysis subsystem identification method based on reduced-order modal energy cluster analysis according to claim 1 is characterized in that: The step (1) of obtaining the modal energy matrix of the structure within the analysis frequency comprises the following steps: Calculate and analyze the wavelength corresponding to the highest frequency in the frequency band [f1, f2], set the mesh size of the finite element model to a size no greater than 1 / 6 of the wavelength, and establish a finite element analysis model of the structure, with the number of units being m; Obtain the n-order mode of the structure within the analysis frequency band through modal analysis And the modal energy of each unit in each mode, the energy corresponding to the i-th unit in the j-th mode is denoted as e ij , and obtain the modal energy matrix E with a matrix dimension of m×n.

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