Data-driven uncertainty modal parameter identification method based on Legendre cross section extreme value
By combining the Legendre section extremum method with LSTM neural network, the problem of uncertainty quantification and propagation in modal parameter identification of complex structures is solved, realizing efficient and accurate modal parameter identification, which is applicable to health monitoring and vibration control of key structures such as spacecraft and large bridges.
Patent Information
- Application Number
- CN202510920520.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-04
- Publication Date
- 2025-11-04
AI Technical Summary
Existing technologies struggle to effectively quantify the impact of uncertainties in modal parameter identification of complex structures, leading to biased identification results. Furthermore, they are computationally expensive, have limited generalization capabilities, and fail to meet the needs of engineering applications.
By combining the Legendre section extremum method with a long short-term memory neural network, uncertainty parameters are quantified in interval form. A multilayer perceptron and Legendre polynomial zero-point calculation framework are constructed, and an LSTM neural network is trained by combining Gaussian integral point mapping to achieve efficient identification of modal parameters and uncertainty propagation.
It significantly reduces computational costs, improves identification accuracy and stability, enhances model interpretability and generalization ability, and is suitable for intelligent health monitoring and vibration control of complex structures.
Smart Images
Figure CN120893243A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the technical field of structural health monitoring, system identification and artificial intelligence, and particularly relates to a data-driven uncertainty modal parameter identification method based on Legendre section maximum value. BACKGROUND
[0002] Structural modal parameter identification is one of the core research directions in the field of system identification, and its goal is to accurately extract the dynamic parameters such as natural frequency, damping ratio and mode shape of the structure, to provide key theoretical basis for the dynamic modeling, health monitoring and vibration control of complex structures such as spacecraft and large bridges. In the traditional method, the characteristic system realization algorithm (ERA) in the time domain identification technology is widely selected in engineering practice due to its multi-input multi-output (MIMO) processing capability and noise robustness. The algorithm extracts the minimum state space realization of the system by constructing the Hankel matrix and performing singular value decomposition (SVD), and then analyzes the modal parameters by eigenvalue decomposition. However, such methods are based on the assumption of a deterministic model, while in actual engineering, the structure parameters (such as material properties and boundary conditions) and environmental loads often have uncertainties. These unmodeled uncertainties will cause deviations in the identification results through error propagation, and if directly used for safety evaluation or control decision, it may cause serious risks.
[0003] To quantify the influence of uncertainty, probability statistical methods such as Monte Carlo simulation have been introduced. This kind of method outputs the confidence interval of modal parameters in statistical form by randomly sampling the parameter distribution on a large scale and repeatedly executing deterministic analysis. However, it has two inherent defects: first, the computational cost increases exponentially with the system degrees of freedom, which is difficult to bear for large structures or real-time monitoring scenarios; second, the accuracy depends on the pre-set probability distribution model, while the statistical properties of uncertainty sources in actual engineering are usually difficult to accurately obtain. The contradiction between model dependence and high computational cost greatly limits the application of traditional methods in complex engineering.
[0004] In recent years, deep learning technology has provided a new idea for modal parameter identification with its powerful non-linear mapping capability. The long short-term memory (LSTM) neural network solves the gradient disappearance problem of traditional recurrent neural network (RNN) through the gating mechanism, and performs well in time series modeling. However, existing researches mainly focus on deterministic identification, and fail to clearly understand the propagation mechanism of uncertainty in neural networks, which makes it difficult to evaluate the credibility of the identification results. In addition, the network design lacks deep integration with structural dynamics theory (such as modal orthogonality and superposition principle), and the generalization ability is limited. More importantly, existing methods usually separate the modal feature extraction and uncertainty quantification, and fail to establish a collaborative mechanism, not only causing computational redundancy, but also reducing the reliability of the results due to error accumulation.
[0005] Therefore, there is an urgent need for an efficient and high-precision uncertainty modal parameter identification method that can integrate data-driven technology and uncertainty quantification theory to provide reliable support for intelligent health monitoring of complex structures. SUMMARY
[0006] To solve the above technical problems, the present application provides a data-driven uncertainty modal parameter identification method based on the maximum value of Legendre section, which couples modal parameter identification and uncertainty propagation through the maximum value of Legendre section method and neural network, and improves the calculation efficiency of the maximum value of Legendre section method on this basis.
[0007] To achieve the above purpose, the present application adopts the following technical solutions:
[0008] A data-driven uncertainty modal parameter identification method based on the maximum value of Legendre section, comprising the following steps:
[0009] Step one: considering the uncertainty factors in the structure material and use scene, quantifying the uncertainty parameters of the structure model in the form of interval, obtaining the center value and interval radius of the uncertainty parameters, and constructing an interval parameter space;
[0010] Step two: according to the truncation error of the polynomial approximation model constructed based on Legendre polynomial as the basis, the number of required Gaussian integral points ;
[0011] Step three: constructing a Legendre polynomial zero point calculation framework based on multilayer perception;
[0012] Step four: calculating the zero points of each order by using the Legendre polynomial and constructing a loss function, training the Legendre polynomial zero point calculation framework, and inputting the number of Gaussian integral points into the trained zero point calculation framework to obtain Gaussian integral points, combining the interval parameter space to obtain the input sample points and sample point matrix corresponding to the uncertainty parameters;
[0013] Step five: constructing a structure finite element analysis model according to the input sample points and sample point matrix, calculating and obtaining a structure response matrix;
[0014] Step six: constructing a modal parameter identification framework based on long short-term memory neural network;
[0015] Step seven: obtaining the response data of the structure under random load excitation, training the modal parameter identification framework based on the response data, inputting the structure response matrix into the trained modal parameter identification framework, and obtaining the frequency results identified;
[0016] Step eight: according to the frequency results, an approximate equation is obtained by using Legendre polynomial approximation fitting, the maximum and minimum points of the Legendre polynomial are obtained by derivation, and the maximum and minimum sample points of the uncertainty parameters are obtained by projecting into the interval parameter space;
[0017] Step nine: according to the maximum and minimum sample points, the structural finite element analysis model is reconstructed, and the new structural response matrix is calculated and input into the trained modal parameter identification framework to obtain the uncertainty modal parameter interval.
[0018] The beneficial effects of the present application are:
[0019] Efficient uncertainty quantification and propagation: by quantifying the uncertainty of the structure parameters in the form of interval, combining the Legendre polynomial zero point calculation framework and the Gauss integral point mapping technology, the large-scale sampling requirement of the traditional Monte Carlo method is avoided, the calculation cost is significantly reduced, and the identification accuracy is guaranteed.
[0020] Deep time sequence modeling capability improvement: long short-term memory (LSTM) neural network is used as the core identification framework, and the time sequence dependence of the vibration signal is effectively captured by using the gating mechanism and memory cell structure, solving the gradient disappearance problem of the traditional recurrent neural network in long sequence modeling, and significantly improving the dynamic feature extraction capability.
[0021] Deep fusion of physical mechanism and data-driven: through the encoder-decoder architecture and the modal superposition principle constraint reconstruction process, the structure dynamics theory is embedded in the neural network design, which enhances the model's interpretability and generalization ability, and avoids the limitations of pure data-driven methods.
[0022] Multi-objective optimization improves robustness: the loss function integrates the reconstruction error, modal response independence constraint and Gaussian constraint, and through joint optimization of network parameters, noise interference is effectively suppressed, the sensitivity of the model to hyperparameters is reduced, and the stability and reliability of the identification result are significantly improved.
[0023] Strong engineering applicability: the method is suitable for modal parameter identification of complex structures in uncertain environment, and provides an efficient and accurate solution for intelligent health monitoring and vibration control of key structures such as spacecraft and large bridges. BRIEF DESCRIPTION OF DRAWINGS
[0024] Figure 1 It is a data-driven uncertainty modal parameter identification method flowchart based on Legendre section maximum value;
[0025] Figure 2 It is a large satellite truss structure schematic diagram. DETAILED DESCRIPTION
[0026] The application will be further described below in conjunction with the accompanying drawings and examples.
[0027] Figure 1 A flow chart of a data-driven uncertainty modal parameter identification method for the maximum and minimum values of Legendre section according to the application. It includes the following steps:
[0028] Step one: considering the uncertainty factors in the structure material and use scenarios, quantifying the uncertainty parameters of the structure model in interval form, obtaining the center value and interval radius of the uncertainty parameters, and constructing an interval parameter space;
[0029] Step two: according to the truncation error of the polynomial approximation model constructed based on Legendre polynomials, the number of required Gaussian integral points is obtained ;
[0030] Step three: constructing a Legendre polynomial zero point calculation framework based on a multi-layer perception machine;
[0031] Step four: calculating the zero points of each order using Legendre polynomials and constructing a loss function, training the Legendre polynomial zero point calculation framework. The number of Gaussian integral points obtained in step two is input into the trained zero point calculation framework to obtain the zero points of the Legendre polynomials of the order (i.e., the Gaussian integral points), and the Gaussian integral points are projected into the interval parameter space constructed in step one to obtain the input sample points and sample point matrix corresponding to the uncertainty parameters;
[0032] Step five: constructing a structure finite element analysis model according to the input sample points and sample point matrix obtained in step four, calculating and obtaining a structure response matrix;
[0033] Step six: constructing a modal parameter identification framework based on a long short-term memory neural network;
[0034] Step seven: obtaining the response data of the structure under random load excitation, dividing the response data into a training set and a test set, training the modal parameter identification framework using the training set, testing the identification effect of the modal parameter identification framework using the test set; inputting the structure response matrix obtained in step five into the trained modal parameter identification framework to obtain the frequency results identified;
[0035] Step eight: according to the frequency results obtained in step seven, using a Legendre polynomial to approximate and fit to obtain an approximate equation, and deriving the zero points of the derivative function. The maximum and minimum points of the Legendre polynomial are generated by the zero points of the derivative function and the endpoints of the independent variable. Projecting the maximum and minimum points into the interval parameter space constructed in step one to obtain the maximum and minimum sample points of the uncertainty parameters;
[0036] Step nine: Reconstruct the structural finite element analysis model according to the maximum sample point and the minimum sample point obtained in step eight, and calculate the new structural response matrix to input the modal parameter identification framework trained in step seven to obtain the interval of the uncertain modal parameters such as frequency.
[0037] As shown in Figure 2 , in the step one, respectively represent the lower bound and the upper bound of the interval parameter vector , and represents the interval parameter in the th of the uncertain parameters; it can be expressed in the form of the combination of the median and the radius , and the vector is the standard interval vector:
[0038] (1)
[0039] Taking a large-scale satellite truss model as a basis example, assuming that the material density and the elastic modulus of the truss beam have interval uncertainty due to the influence of processing conditions and working scenarios, they are expressed in the form of interval vector as follows:
[0040] (2)
[0041] wherein respectively represent the lower bound and the upper bound of the interval uncertain vector of the material density and the Young's modulus, respectively represent the interval of the th material density and the Young's modulus parameter, which can be expressed in the form of the combination of the median and the radius , and the number of interval uncertain parameters is .
[0042] In the step two, the number of required Gauss integral points is obtained according to the truncation error of the approximate polynomial constructed by taking the Legendre polynomial as the basis.
[0043] (3)
[0044] wherein is the rounding operator; represents the nonlinear order of the real function, that is, the number of uncertain parameters, represents the order of the selected Legendre polynomial.
[0045] In the third step, a Legendre polynomial zero point calculation framework based on a multilayer perceptron is constructed: the Legendre polynomial zero point calculation framework based on a multilayer neural network takes a multilayer perceptron neural network as a main body, the input of which is the order (i.e., the number of Gaussian integral points s) corresponding to the required Legendre polynomial zero point, and the output of which is The zero point of the order Legendre polynomial, specifically comprising:
[0046] The multilayer perceptron neural network used in the application has only one hidden layer, i.e., a three-layer structure multilayer perceptron neural network, and the three layers are input layer, hidden layer and output layer respectively. The hidden layer and the input layer are in full connection, and thus the hidden layer output is:
[0047] (4)
[0048] The output of the output layer is calculated as :
[0049] (5)
[0050] wherein, is an activation function, and the calculation formula thereof is ; and are weight matrices between the input layer and the hidden layer and between the hidden layer and the output layer respectively; is a bias vector.
[0051] The fourth step specifically comprises:
[0052] The Legendre polynomial system is an orthogonal polynomial with a weight function of 1 in the standard interval , and the order Legendre polynomial is expressed as:
[0053] (6) When
[0054] , the Legendre polynomial has the following recursive relationship:
[0055] (7)
[0056] According to formula (7), the zero point of the known order Legendre polynomial is calculated, and is denoted as :
[0057] (8) Thus, the loss function of the Legendre polynomial zero point calculation framework based on a multilayer neural network is constructed as
[0058] And train the zero-point calculation framework:
[0059] (9)
[0060] The required Legendre polynomial order Input the trained zero-point calculation framework to obtain The zeros of the Legendre polynomial are:
[0061] (10)
[0062] Project the Gaussian integral points represented by equation (10) onto the interval parameter space. The first Uncertain parameters Then the first The sample points of the approximate polynomial of the curve obtained by cutting the true response space of the structure using a 3D hyperplane can be represented as:
[0063] (11)
[0064] in, The expression for calculation is as follows:
[0065] (12)
[0066] According to formulas (11) and (12), the sample point matrix of the structural finite element analysis model can be expressed as follows: Its expression is:
[0067] (13)
[0068] in, Each row corresponds to an implementation of the input parameters for structural finite element analysis, namely:
[0069] (14)
[0070] In step five, the structural finite element analysis model is constructed based on the input sample points obtained in step four, as shown below:
[0071] (15)
[0072] in, , , and They represent the first Dimension, the Parameter vector of sample points The elements in the matrix constitute the mass matrix, damping matrix, stiffness matrix, and external load matrix.
[0073] By running Substructure finite element analysis, the structure response matrix can be calculated :
[0074] (16)
[0075] Where the node response vector at the first model input sample point (i.e. equation (12)) is:
[0076] (17)
[0077] Where, N represents the total number of degrees of freedom.
[0078] In step six, the modal parameter identification framework based on long short-term memory neural network comprises:
[0079] The modal parameter identification framework based on long short-term memory neural network comprises an encoder, a feature extractor, a decoder and a reconstructor. The first layer is a multilayer perceptron, i.e. the encoder; the second layer is a long short-term memory neural network, i.e. the feature extractor; the third layer is a multilayer perceptron, i.e. the decoder; and the fourth layer is a linear layer, i.e. the reconstructor. The basic working process of the neural network is as follows:
[0080] 1) The encoder receives the response data of the structure , and then maps it to a high-dimensional representation ;
[0081] 2) The feature extractor receives the high-dimensional representation , and obtains an intermediate representation by extracting features ;
[0082] 3) The decoder receives the intermediate representation , and then maps it to a low-dimensional space to output the modal response ;
[0083] 4) The reconstructor receives the modal response , and outputs the reconstructed response data ;
[0084] 5) Calculate the loss function .
[0085] The working process of the feature extractor is as follows:
[0086] Calculate the forgetting factor at the current time, which is used to determine how much information of the previous time to retain:
[0087] (18)
[0088] the input coefficient of the current time instant , and update the candidate memory cell state
[0089] (19)
[0090] (20)
[0091] According to the output of and , update the memory cell state of the current time instant
[0092] (21)
[0093] calculate the output coefficient of the current time instant , and generate a new hidden state
[0094] (22)
[0095] (23)
[0096] In formulas (18)-(23), , , are the forgetting coefficient, the input coefficient, and the output coefficient, respectively; , are the candidate memory cell state and the memory cell state, respectively; , are the time instant and the hidden state output by the long short-term memory neural network at the time instant ; is the input of the long short-term memory neural network at the time instant , are activation functions, and their calculation formulas are , , respectively; is element-wise multiplication, and and are the weight matrix and the bias vector, respectively.
[0097] The reconstructor is used to reconstruct the response data of the model, and its corresponding physical equation is:
[0098] (24)
[0099] wherein, This represents the structural vibration displacement data obtained from the reconstruction. The modal matrix is represented by the modal shape vectors. composition; The modal response matrix represents the modal response vectors. composition; Representing the First-order mode shape vector; Representing the The first modal response vector.
[0100] loss function Includes reconstruction error Independence constraints of modal responses and Gaussian constraint :
[0101] (25)
[0102] (26)
[0103] (27)
[0104] (28)
[0105] in, Indicates the first element in the original structural response data Line number Column elements; This indicates the first vibration displacement data obtained from reconstruction. Line number Column elements; These represent the total number of rows and columns of the vibration displacement data, respectively, and also the number of times the sensor sampled and the number of sensors, respectively. This represents the output of the third intermediate layer: the modal response of the structure; Represents the identity matrix; This represents the transfer matrix corresponding to the linear layer; ... Represents covariance operation; This indicates determinant operations.
[0106] Step seven includes: using random white noise as the structural load, and dividing the obtained structural response data into a training set and a test set in a 4:1 ratio. First, the neural network is trained using the training set, and the parameters are updated using gradient descent to minimize the loss function and bring it to convergence. Then, the test set data is input into the neural network to obtain the modal response. The modal frequencies are then obtained through power spectral density analysis. Finally, the structural response obtained in step four is input into a neural network to obtain the identified modal frequencies. :
[0107] (29)
[0108] in, Indicates the first The first uncertainty parameter corresponds to the th uncertainty parameter. The nth model input sample point First-order modal frequencies.
[0109] In step eight, the modal frequency vector of the first... The component, i.e., the first component The first modal frequency is related to the third Polynomial approximation model with interval uncertain parameters It can be represented as:
[0110] (30)
[0111] In the formula, Indicates the order of the chosen Legendre polynomial. Represents Legendre polynomials. Indicates the first One zero point.
[0112] The approximate polynomial model represented by equation (30) is for the standard interval. For a continuous function within a given range, according to the theorem for the maximum and minimum values of continuous functions, the extreme points of the function can be derived from the zeros of its derivative and the endpoints of its independent variable. The zero of the derivative function can be calculated using the following formula:
[0113] (31)
[0114] The set of extreme points can be obtained as follows:
[0115] (32)
[0116] Equation (32) represents the first The first modal frequency is related to the third The set of extreme points of an interval of uncertain parameters. Based on the polynomial approximation model. The range of values for , equation (32) should be further modified as follows:
[0117] (33)
[0118] in, and respectively represent the imaginary and real parts of the corresponding variable. The i-th element of the structural response vector is denoted as The maximum and minimum points of the i-th modal frequency with respect to the i-th uncertain parameter in the interval are calculated as:
[0119] (34)
[0120] The maximum and minimum values of the i-th modal frequency in the standard interval are obtained as:
[0121] (35)
[0122] According to equation (12), equation (35) is mapped to the interval parameter space as:
[0123] (36)
[0124] The step nine includes:
[0125] According to equation (36), equation (15) is reconstructed as:
[0126] (37)
[0127] The matrix composed of the maximum solution vector and the minimum solution vector is expressed as:
[0128] (38)
[0129] The maximum solution vector and the minimum solution vector are input into the neural network constructed in step six, respectively, to obtain the upper bound vector of modal frequency and the lower bound vector of modal frequency as:
[0130] (39)
[0131] The interval range of the i-th modal frequency is composed of the i-th element of the upper bound vector of modal frequency and the i-th element of the lower bound vector of modal frequency , and is expressed as:
[0132] (40)
[0133] The above-described specific embodiments further illustrate the purpose, technical solutions and beneficial effects of the present application, and it should be understood that the above-described is only a specific embodiment of the present application and is not intended to limit the present application, and any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.
Claims
1. A data-driven method for identifying uncertain modal parameters based on the extrema of Legendre sections, characterized in that, Includes the following steps: Step 1: Consider the uncertainties in structural materials and usage scenarios, quantify the uncertainty parameters of the structural model in interval form, obtain the center value and interval radius of the uncertainty parameters, and construct the interval parameter space; Step 2: Based on the truncation error of the polynomial approximation model constructed using Legendre polynomials as the basis, obtain the required number of Gaussian integration points. ; Step 3: Construct a Legendre polynomial zero-point calculation framework based on a multilayer perceptron; Step 4: Calculate the zeros of each order using Legendre polynomials and construct a loss function; train the Legendre polynomial zero-point calculation framework; and count the number of Gaussian integration points. The trained zero-point calculation framework is used to obtain Gaussian integration points, and combined with the interval parameter space, the input sample points and sample point matrix corresponding to the uncertainty parameters are obtained. Step 5: Construct a structural finite element analysis model based on the input sample points and sample point matrix, and calculate and obtain the structural response matrix; Step Six: Construct a modal parameter identification framework based on a long short-term memory neural network; Step 7: Obtain the response data of the structure under random load excitation, train the modal parameter identification framework based on the response data, input the structural response matrix into the trained modal parameter identification framework, and obtain the identified frequency results. Step 8: Based on the frequency results, use Legendre polynomials to approximate the equations and obtain the approximate equations. Take the derivative of the approximate equations to obtain the maximum and minimum points of the Legendre polynomials and project them onto the interval parameter space to obtain the maximum and minimum sample points of the uncertainty parameters. Step 9: Reconstruct the finite element analysis model of the structure based on the maximum and minimum sample points, and calculate the new structural response matrix. Then input the matrix into the trained modal parameter identification framework to obtain the uncertain modal parameter range.
2. The data-driven uncertainty modal parameter identification method based on Legendre section extrema as described in claim 1, characterized in that, In step one, the parameter vector in the interval parameter space is represented as: （1） in, Representing the interval parameter vectors respectively The lower and upper bounds, express The first of the uncertain parameters A range of parameters; which can be represented as the median. and radius Combination form, vector It is a standard interval vector.
3. The data-driven uncertainty modal parameter identification method based on Legendre section extrema as described in claim 2, characterized in that, The number of Gaussian integration points in step two Represented as: （3） in, This is the floor function operator; This represents the nonlinear order of the true function, i.e., the number of uncertain parameters. This indicates the order of the selected Legendre polynomial.
4. The data-driven uncertainty modal parameter identification method based on Legendre section extrema as described in claim 3, characterized in that, In step three, the Legendre polynomial zero-point calculation framework based on a multilayer perceptron is a three-layer multilayer perceptron neural network, consisting of an input layer, a hidden layer, and an output layer. The hidden layer and the input layer are fully connected, and the input is the required number of Gaussian integration points. The output is Zeros of the Legendre polynomial.
5. The data-driven uncertainty modal parameter identification method based on Legendre section extrema as described in claim 4, characterized in that, In step four, the zeros of each order are calculated using the known Legendre polynomials, and a loss function is constructed. And train the Legendre polynomial zero-point calculation framework: （9） The required number of Gaussian integration points Input the trained Legendre polynomial zero-point calculation framework to obtain The zeros of the Legendre polynomial are: （10） Will Projecting the zeros of the Legendre polynomials onto the interval parameter space The first Uncertain parameters Then the first The sample point representation of the approximate polynomial of the curve obtained by cutting the true response space of the structure using a 3D hyperplane is as follows: （11） The sample point matrix is represented as : （13）。 6. The data-driven uncertainty modal parameter identification method based on Legendre section extrema as described in claim 5, characterized in that, The structural finite element analysis model constructed in step five is shown below: （15） in, , , and They represent the first Dimension, the Parameter vector of sample points The elements in the matrix constitute the mass matrix, damping matrix, stiffness matrix, and external load matrix; By running Substructure finite element analysis to obtain the structural response matrix : （16）。 7. The data-driven uncertainty modal parameter identification method based on Legendre section extrema as described in claim 6, characterized in that, The modality parameter identification framework based on a long short-term memory neural network in step six includes an encoder, a feature extractor, a decoder, and a reconstructor. The encoder receives the response data of the structure. Then map it to a high-dimensional representation. The feature extractor receives a high-dimensional representation. Intermediate representations are obtained by extracting features. The decoder receives the intermediate representation. Then it is mapped to a low-dimensional space to output the modal response. The reconfigurator receives modal responses. Output the response data obtained from the reconstruction. ; The loss function of the modal parameter identification framework includes reconstruction error, modal response independence constraints, and Gaussianity constraints.
8. The data-driven uncertainty modal parameter identification method based on the extrema of the Legendre section according to claim 7, characterized in that, Step seven includes: Random white noise was used as the structural load. The obtained structural response data was divided into training set and test set in a 4:1 ratio. The modal parameter identification framework was trained using the training set. The neural network parameters were updated using gradient descent to minimize the loss function and make it converge. The test set data is used to input the modal parameter identification framework to obtain the weight matrix, and the modal frequencies are obtained through power spectral density analysis. The structural response matrix is input into the trained modal parameter identification framework to obtain the identified modal frequencies. : （29） in, Indicates the first The first uncertainty parameter corresponds to the th uncertainty parameter. The nth model input sample point First-order modal frequencies.
9. The data-driven uncertainty modal parameter identification method based on Legendre section extrema as described in claim 8, characterized in that, Step eight includes: The modal frequency vector of the first The component, i.e., the first component The first modal frequency is related to the third Polynomial approximation model with interval uncertain parameters Represented as: （30） In the formula, Indicates the order of the chosen Legendre polynomial. Represents Legendre polynomials. Indicates the first One zero point; function The zero of the derivative function is calculated by the following formula: （31） Get the first The first modal frequency is related to the third The set of extreme points for the uncertain parameter in each interval is: （31） The first structural response vector The first modal frequency is related to the third The maximum and minimum points of the uncertain parameters in each interval are calculated as follows: （34） No. First modal frequencies within the standard range Maximum and minimum values, i.e.: （35） (34) Mapping to the interval parameter space yields: （36）。 10. The data-driven uncertainty modal parameter identification method based on the extrema of the Legendre section according to claim 9, characterized in that, In step nine, the structural finite element analysis model is reconstructed based on the maximum and minimum value sample points respectively: （37） The maximum solution vector and minimum solution vector Input the trained modal parameter recognition framework into each parameter to obtain the modal frequency upper bound vector. and modal frequency lower bound vector for: （39） From the upper bound vector of modal frequencies and modal frequency lower bound vector The Middle The first element constitutes the First-order modal frequency The range of the interval is: （40）。