CN114492112A - Satellite frame structure reliability analysis method based on sampling quantile regression - Google Patents

Satellite frame structure reliability analysis method based on sampling quantile regression Download PDF

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CN114492112A
CN114492112A CN202210004486.0A CN202210004486A CN114492112A CN 114492112 A CN114492112 A CN 114492112A CN 202210004486 A CN202210004486 A CN 202210004486A CN 114492112 A CN114492112 A CN 114492112A
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frame structure
order frequency
satellite frame
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姚雯
郑小虎
张俊
张小亚
姜廷松
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National Defense Technology Innovation Institute PLA Academy of Military Science
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Abstract

The invention discloses a satellite frame structure reliability analysis method based on sampling quantile regression, which comprises the following steps: constructing a deep neural network model for predicting the first-order frequency of the satellite frame structure; acquiring a plurality of groups of input parameter values of first-order frequency of a satellite frame structure to be analyzed based on a finite element model, determining the first-order frequency corresponding to each group of input parameter values, adding random noise in the first-order frequency, acquiring first-order frequency correction values, and acquiring a plurality of training data comprising the input parameter values and the first-order frequency correction values corresponding to the input parameter values; setting quantiles, constructing a loss function by using the quantiles, and training a deep neural network model by using training data and the loss function; and (5) carrying out reliability analysis on the satellite frame structure by using the deep neural network model. The method can avoid the problems of long calculation time and high calculation cost when finite element model analysis is adopted, and reduce the influence of computer noise on the prediction precision of the neural network model.

Description

Satellite frame structure reliability analysis method based on sampling quantile regression
Technical Field
The invention relates to the technical field of satellite structure design, in particular to a satellite frame structure reliability analysis method based on sampling quantile regression.
Background
The frame structure design of the satellite is a central link of the overall design process of the satellite. The design of the satellite frame structure needs to consider the factors of the frame structure such as strength, rigidity, modal characteristics, temperature, radiation resistance and the like, and the structure quality is reduced as much as possible on the premise of ensuring safe bearing of effective load. During launching, if the natural frequency of the satellite is close to the vibration frequency of the rocket, resonance can be generated in the launching stage, the performance of the satellite payload is influenced slightly, and the whole satellite structure is damaged seriously. The natural frequency of the satellite is influenced by factors such as the characteristics (such as density, elastic modulus and the like) of the frame structure material, the processing error and the like. Because the satellite has the characteristics of high cost and high risk, in order to ensure that the satellite frame structure is not damaged in the launching process, the reliability analysis of the satellite frame structure is required during the satellite design.
In the prior art, finite element model analysis is adopted to obtain a first-order frequency of a satellite frame structure so as to analyze the reliability of the satellite frame structure. However, in a single simulation analysis, it takes much time to analyze the first-order frequency of the satellite frame structure by using the finite element model, so that when the reliability of the satellite frame structure is analyzed, a large number of simulation experiments are required, the calculation cost and the calculation time are high, and the calculation cost and the calculation time are increased step by step along with the complexity of the satellite frame structure.
Disclosure of Invention
In order to solve part or all of the technical problems in the prior art, the invention provides a satellite frame structure reliability analysis method based on sampling quantile regression.
The technical scheme of the invention is as follows:
the method for analyzing the reliability of the satellite frame structure based on the sampling quantile regression comprises the following steps:
constructing a deep neural network model for predicting the first-order frequency of the satellite frame structure;
acquiring a plurality of groups of input parameter values of first-order frequency of a satellite frame structure to be analyzed based on a finite element model, determining the first-order frequency corresponding to each group of input parameter values by using a finite element method, adding random noise in the first-order frequency, acquiring first-order frequency correction values, and acquiring a plurality of training data comprising the input parameter values and the first-order frequency correction values corresponding to the input parameter values;
setting quantiles, constructing a loss function by using the quantiles, and training a deep neural network model by using training data and the loss function so as to fit a mapping relation between an input parameter value and a first-order frequency;
and carrying out reliability analysis on the satellite frame structure by using the trained deep neural network model.
In some possible implementations, the deep neural network model employs a deep feed-forward neural network.
In some possible implementation manners, the obtaining multiple sets of input parameter values of first-order frequencies of the satellite frame structure to be analyzed based on a finite element model, and determining the first-order frequencies corresponding to each set of input parameter values by using a finite element method includes:
according to a to-be-analyzed satellite frame structure, determining input parameters for analyzing the first-order frequency of the satellite frame structure based on a finite element model and the value ranges of the input parameters, randomly sampling once from the value range of each input parameter, acquiring a group of input parameter values for analyzing the first-order frequency of the satellite frame structure based on the finite element model, repeating the random sampling process for multiple times, acquiring multiple groups of input parameter values for analyzing the first-order frequency of the satellite frame structure based on the finite element model, and analyzing and determining the first-order frequency corresponding to each group of input parameter values by using a finite element method.
In some possible implementations, the analyzing the input parameters of the first order frequency of the satellite frame structure based on the finite element model includes: at least one of an aluminum alloy density, a spring steel density, a titanium alloy density, an aluminum alloy elastic modulus, a spring steel elastic modulus, and a titanium alloy elastic modulus of the satellite frame structure.
In some possible implementations, the first order frequency correction value is obtained using the following formula;
Figure BDA0003455024420000021
wherein, yiIndicating the first order frequency correction values corresponding to the ith set of input parameter values,
Figure BDA0003455024420000022
the first order frequency corresponding to the ith set of input parameter values determined using the finite element method is shown, and epsilon represents the random noise.
In some possible implementations, the quantile is set to τi，τiU (0,1), constructing a loss function as:
Figure BDA0003455024420000023
Figure BDA0003455024420000024
wherein, tauiDenotes the quantile, τ, corresponding to the ith training dataiU (0,1) represents the quantile τiObey [0,1 ]]The distribution of the components is uniform, and the components are uniformly distributed,
Figure BDA0003455024420000025
representing a loss function, y a first order frequency correction value,
Figure BDA0003455024420000026
representing a first-order frequency predicted value output by the deep neural network model, and x representing a first order of analyzing the satellite frame structure based on the finite element modelInput parameter values of frequency, theta represents a deep neural network model parameter, n represents a training data number, xiRepresenting the value of the ith set of input parameters,
Figure BDA0003455024420000027
i-th set of input parameter values x representing the output of the deep neural network modeliCorresponding first order frequency prediction.
In some possible implementations, the performing reliability analysis on the satellite framework structure by using the trained deep neural network model includes:
according to the satellite frame structure to be analyzed and the input parameters of the first-order frequency of the satellite frame structure analyzed based on the finite element model, random sampling is carried out once from the value range of each input parameter respectively, a group of input parameter values of the first-order frequency of the satellite frame structure analyzed based on the finite element model are obtained, the random sampling process is repeated for multiple times, and multiple groups of input parameter values { x ] of the first-order frequency of the satellite frame structure analyzed based on the finite element model are obtainedj|j＝1,2,…,M}；
Given a quantile τ, will { (x)jTau) j ═ 1,2, …, M } is input into the trained deep neural network model as input data, and a first-order frequency predicted value corresponding to the input data is obtained
Figure BDA0003455024420000031
Wherein τ is 0.5;
setting a first-order frequency critical value of a satellite frame structure, and calculating a limit state value of first-order frequency according to a first-order frequency predicted value and the first-order frequency critical value;
and counting the number of the limit state values of all the first-order frequencies which are less than 0 to determine the reliability of the satellite frame structure.
In some possible implementation modes, the extreme state value of the first-order frequency of the satellite frame structure is calculated by using the following formula;
Figure BDA0003455024420000032
wherein, Deltaj(xj) Extreme state values, y, representing first order frequencieslimRepresenting a first order frequency threshold.
In some possible implementations, the reliability of the first order frequency of the satellite frame structure is determined using the following formula;
Figure BDA0003455024420000033
where R represents the reliability score and m represents the number of less than 0 of the extreme state values for all first order frequencies.
The technical scheme of the invention has the following main advantages:
according to the satellite frame structure reliability analysis method based on the sampling quantile regression, the first-order frequency of the satellite frame structure is predicted and calculated by utilizing the deep neural network model, and the problems of long calculation time and high calculation cost in finite element model analysis can be solved; meanwhile, when training data are obtained, random noise is added to the obtained first-order frequency, the influence of computer noise on a finite element simulation analysis process can be effectively overcome, the deviation of the training data and a true value is reduced, and the prediction precision of the trained deep neural network model is improved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the embodiments or the prior art descriptions will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and other drawings can be obtained by those skilled in the art without creative efforts.
Fig. 1 is a flowchart of a satellite frame structure reliability analysis method based on sample quantile regression according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the technical solutions of the present invention will be clearly and completely described below with reference to the specific embodiments of the present invention and the accompanying drawings. It is to be understood that the described embodiments are merely a few embodiments of the invention, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.
The technical scheme provided by the embodiment of the invention is described in detail below with reference to the accompanying drawings.
Referring to fig. 1, an embodiment of the present invention provides a satellite frame structure reliability analysis method based on sample quantile regression, including the following steps:
s1, constructing a deep neural network model for predicting the first-order frequency of the satellite frame structure;
s2, acquiring a plurality of groups of input parameter values of first-order frequency of the satellite frame structure to be analyzed based on a finite element model, determining the first-order frequency corresponding to each group of input parameter values by using a finite element method, adding random noise in the first-order frequency, acquiring first-order frequency correction values, and acquiring a plurality of training data comprising the input parameter values and the first-order frequency correction values corresponding to the input parameter values;
s3, setting quantiles, constructing a loss function by using the quantiles, and training a deep neural network model by using training data and the loss function so as to fit the mapping relation between input parameter values and first-order frequency;
and S4, performing satellite frame structure reliability analysis by using the trained deep neural network model.
According to the satellite frame structure reliability analysis method based on the sampling quantile regression, the first-order frequency of the satellite frame structure is predicted and calculated by utilizing the deep neural network model, and the problems of long calculation time and high calculation cost in finite element model analysis can be solved; meanwhile, when the training data are obtained, random noise is added to the obtained first-order frequency, the influence of computer noise on a finite element simulation analysis process can be effectively overcome, the deviation of the training data and a true value is reduced, the prediction precision of the trained deep neural network model is improved, and the rapid and high-precision analysis of the reliability of the satellite frame structure is realized.
The following steps and principles of the method for analyzing the reliability of the satellite frame structure based on the sample quantile regression according to an embodiment of the present invention are specifically described.
And step S1, constructing a deep neural network model for predicting the first-order frequency of the satellite frame structure.
Specifically, in an embodiment of the present invention, the deep neural network model for predicting the first-order frequency of the satellite frame structure may adopt a deep feedforward neural network model, where the deep feedforward neural network model includes an input layer, a hidden layer, and an output layer, and each layer of neurons receives the output of the preceding layer of neurons and generates an output to the next layer.
The number of layers of the hidden layer of the depth feedforward neural network model can be determined according to the complexity of a first-order frequency analysis problem of a satellite frame structure, and the satellite frame structure is a main structure of a satellite.
Step S2, multiple groups of input parameter values of the finite element model-based analysis satellite frame structure first-order frequency of the satellite frame structure to be analyzed are obtained, the finite element method is utilized to determine the first-order frequency corresponding to each group of input parameter values, random noise is added into the first-order frequency, the first-order frequency correction value is obtained, and a plurality of training data comprising the input parameter values and the first-order frequency correction value corresponding to the input parameter values are obtained.
In order to ensure that the deep neural network model can be used for predicting the first-order frequency of the satellite frame structure, the deep neural network model needs to be trained in advance by using training data. Because real data is difficult to obtain and the number is usually small, in an embodiment of the present invention, training data is obtained through simulation analysis experiments.
Specifically, acquiring multiple sets of input parameter values of a finite element model-based first-order frequency of a satellite frame structure to be analyzed, and determining the first-order frequency corresponding to each set of input parameter values by using a finite element method, wherein the method comprises the following steps:
determining a finite element model-based component according to a satellite frame structure to be analyzedAnalyzing the input parameters of the first-order frequency of the satellite frame structure and the value ranges of all the input parameters, randomly sampling once from the value range of each input parameter respectively to obtain a group of input parameter values for analyzing the first-order frequency of the satellite frame structure based on the finite element model, repeating the random sampling process for multiple times to obtain multiple groups of input parameter values { x ] of the first-order frequency of the satellite frame structure based on the finite element modeliAnd (5) analyzing and determining the first-order frequency corresponding to each group of input parameter values by using a finite element method, wherein i is 1,2, …, n
Figure BDA0003455024420000051
In an embodiment of the present invention, the input parameter for analyzing the first-order frequency of the satellite frame structure based on the finite element model may include at least one of an aluminum alloy density, a spring steel density, a titanium alloy density, an aluminum alloy elastic modulus, a spring steel elastic modulus, and a titanium alloy elastic modulus of the satellite frame structure.
The specific value range of each input parameter can be determined according to the satellite frame structure to be analyzed.
Further, when training data are acquired by performing a simulation analysis experiment through a finite element method, due to the existence of computer noise, a certain deviation exists between the first-order frequency obtained by simulation and a true value. To this end, in one embodiment of the present invention, the first order frequency obtained by using the finite element method
Figure BDA0003455024420000052
Adding random noise epsilon to obtain first-order frequency correction value yiObtaining a plurality of training data including input parameter values and first order frequency correction values corresponding to the input parameter values (x)i,yi)|i＝1,2,…,n}。
Specifically, a first order frequency correction value is obtained using the following formula;
Figure BDA0003455024420000061
the random noise epsilon can be determined according to a specific satellite frame structure problem, the training data number n can be determined according to the prediction precision of the required deep neural network model, when the training data number is more, the prediction precision of the obtained deep neural network model is higher, and the calculation time and the calculation cost required for obtaining the training data are higher.
And step S3, setting quantiles, constructing a loss function by using the quantiles, and training a deep neural network model by using the training data and the loss function so as to fit the mapping relation between the input parameter values and the first-order frequency.
Specifically, the quantile is set to τi，τiU (0,1), constructing a loss function as:
Figure BDA0003455024420000062
Figure BDA0003455024420000063
wherein, tauiDenotes the quantile, τ, corresponding to the ith training dataiU (0,1) represents the quantile τiObey [0,1]The distribution of the components is uniform, and the components are uniformly distributed,
Figure BDA0003455024420000064
representing a loss function, y a first order frequency correction value,
Figure BDA0003455024420000065
representing a predicted value of a first-order frequency output by a deep neural network model, x representing an input parameter value of a first-order frequency of a satellite frame structure analyzed based on a finite element model, theta representing a parameter of the deep neural network model, n representing a training data number, and x representing a training data numberiRepresenting the i-th set of input parameter values, yiRepresenting the ith set of input parameter values xiThe corresponding first-order frequency correction value,
Figure BDA0003455024420000066
ith group representing deep neural network model outputInputting parameter value xiCorresponding first order frequency prediction.
According to the loss function constructed above, the training data for training the deep neural network model is actually { [ (x)i,τi),yi]|i＝1,2,…,n}。
Further, according to the obtained training data and the constructed loss function, based on a deep learning technology, a deep neural network model is trained through a minimum loss function, so that parameters of the deep neural network model are updated, and the deep neural network model capable of being used for predicting the first-order frequency of the satellite frame structure is obtained
Figure BDA0003455024420000067
Specifically, the parameters of the deep neural network model updated by training can be expressed as:
Figure BDA0003455024420000071
wherein the content of the first and second substances,
Figure BDA0003455024420000072
representing the updated deep neural network model parameters.
The deep neural network model is trained by utilizing the obtained training data and the constructed loss function, so that the deep neural network model can learn the physical law in the training data, a neural network with strong generalization capability is obtained, the first-order frequency of the satellite frame structure can be rapidly and accurately predicted, and the deep neural network is a proxy model essentially.
And step S4, performing satellite frame structure reliability analysis by using the trained deep neural network model.
Specifically, after the training of the deep neural network model is completed, when the first-order frequency prediction of the satellite frame structure is required, according to the to-be-predicted satellite frame structure, an input parameter value for analyzing the first-order frequency of the satellite frame structure based on a finite element model is given, a specific quantile is given, the determined input parameter value and the quantile are input into the trained deep neural network model, and the output of the deep neural network model, namely a first-order frequency prediction value, is obtained, wherein the given quantile can be 0.5.
Further, in an embodiment of the present invention, the analyzing reliability of the satellite frame structure by using the trained deep neural network model includes:
according to the satellite frame structure to be analyzed and the input parameters of the first-order frequency of the satellite frame structure analyzed based on the finite element model, random sampling is carried out once from the value range of each input parameter respectively, a group of input parameter values of the first-order frequency of the satellite frame structure analyzed based on the finite element model are obtained, the random sampling process is repeated for multiple times, and multiple groups of input parameter values { x ] of the first-order frequency of the satellite frame structure analyzed based on the finite element model are obtainedj|j＝1,2,…,M}；
Given a quantile τ, will { (x)jTau) j ═ 1,2, …, M } is input into the trained deep neural network model as input data, and a first-order frequency predicted value corresponding to the input data is obtained
Figure BDA0003455024420000073
Wherein τ is 0.5;
setting a first-order frequency critical value of a satellite frame structure, and calculating a limit state value of first-order frequency according to a first-order frequency predicted value and the first-order frequency critical value;
and counting the number of the limit state values of all the first-order frequencies which are less than 0 to determine the reliability of the satellite frame structure.
Specifically, when the reliability of the satellite frame structure is analyzed, the extreme state value of the first-order frequency of the satellite frame structure can be calculated by using the following formula;
Figure BDA0003455024420000074
wherein, Deltaj(xj) Extreme state values, y, representing first order frequencieslimRepresenting a first order frequency threshold value, which can be determined from the oscillation frequency of the launch vehicle at the time of the launch of the satellite to be analyzed, the limit state value Δ of the first order frequencyj(xj) And when the frequency is less than 0, the satellite frame structure generates resonance, and the structure fails.
Specifically, when the reliability of the satellite frame structure is analyzed, the reliability of the satellite frame structure can be determined by calculation by using the following formula;
Figure BDA0003455024420000081
wherein, R represents the reliability score, m represents the number which is less than 0 in the extreme state values of all the first-order frequencies, and the closer the numerical value of R is to 1, the higher the reliability of the satellite frame structure is.
It is noted that, in this document, relational terms such as "first" and "second," and the like, may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. In addition, "front", "rear", "left", "right", "upper" and "lower" in this document are referred to the placement states shown in the drawings.
Finally, it should be noted that: the above examples are only for illustrating the technical solutions of the present invention, and not for limiting the same; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.

Claims (9)

1. A satellite frame structure reliability analysis method based on sampling quantile regression is characterized by comprising the following steps:
constructing a deep neural network model for predicting the first-order frequency of the satellite frame structure;
acquiring a plurality of groups of input parameter values of first-order frequency of a satellite frame structure to be analyzed based on a finite element model, determining the first-order frequency corresponding to each group of input parameter values by using a finite element method, adding random noise in the first-order frequency, acquiring first-order frequency correction values, and acquiring a plurality of training data comprising the input parameter values and the first-order frequency correction values corresponding to the input parameter values;
setting quantiles, constructing a loss function by using the quantiles, and training a deep neural network model by using training data and the loss function so as to fit a mapping relation between an input parameter value and a first-order frequency;
and carrying out reliability analysis on the satellite frame structure by using the trained deep neural network model.
2. The method of claim 1, wherein the deep neural network model employs a deep feed-forward neural network.
3. The method for analyzing the reliability of the satellite frame structure based on the sample quantile regression as claimed in claim 1, wherein the step of obtaining a plurality of sets of input parameter values for analyzing the first-order frequency of the satellite frame structure based on the finite element model of the satellite frame structure to be analyzed, and determining the first-order frequency corresponding to each set of input parameter values by using a finite element method comprises:
according to a to-be-analyzed satellite frame structure, determining input parameters for analyzing the first-order frequency of the satellite frame structure based on a finite element model and the value ranges of the input parameters, randomly sampling once from the value range of each input parameter, acquiring a group of input parameter values for analyzing the first-order frequency of the satellite frame structure based on the finite element model, repeating the random sampling process for multiple times, acquiring multiple groups of input parameter values for analyzing the first-order frequency of the satellite frame structure based on the finite element model, and analyzing and determining the first-order frequency corresponding to each group of input parameter values by using a finite element method.
4. The method for analyzing the reliability of the satellite frame structure based on the sample quantile regression as claimed in claim 3, wherein the analyzing the input parameters of the first order frequency of the satellite frame structure based on the finite element model comprises: at least one of an aluminum alloy density, a spring steel density, a titanium alloy density, an aluminum alloy elastic modulus, a spring steel elastic modulus, and a titanium alloy elastic modulus of the satellite frame structure.
5. The method for analyzing the reliability of the satellite frame structure based on the sampling quantile regression is characterized in that a first-order frequency correction value is obtained by using the following formula;
Figure FDA0003455024410000011
wherein, yiIndicating the first order frequency correction values corresponding to the ith set of input parameter values,
Figure FDA0003455024410000012
the first order frequency corresponding to the ith set of input parameter values determined using the finite element method is shown, and epsilon represents the random noise.
6. The method of claim 5, wherein the quantile is set to τi，τiU (0,1), constructing a loss function as:
Figure FDA0003455024410000021
Figure FDA0003455024410000022
wherein, tauiDenotes the quantile, τ, corresponding to the ith training dataiU (0,1) represents the quantile τiObey [0,1]The distribution of the components is uniform, and the components are uniformly distributed,
Figure FDA0003455024410000023
representing a loss function, y a first order frequency correction value,
Figure FDA0003455024410000024
representing a predicted value of a first-order frequency output by a deep neural network model, x representing an input parameter value of a first-order frequency of a satellite frame structure analyzed based on a finite element model, theta representing a parameter of the deep neural network model, n representing a training data number, and x representing a training data numberiRepresenting the value of the ith set of input parameters,
Figure FDA0003455024410000025
i-th set of input parameter values x representing the output of the deep neural network modeliCorresponding first order frequency prediction.
7. The method for analyzing the reliability of the satellite frame structure based on the sample quantile regression of claim 6, wherein the analyzing the reliability of the satellite frame structure by using the trained deep neural network model comprises:
according to a to-be-analyzed satellite frame structure and input parameters of first-order frequency of the satellite frame structure analyzed based on the finite element model, random sampling is respectively carried out once from the value range of each input parameter, a group of input parameter values of first-order frequency of the satellite frame structure analyzed based on the finite element model are obtained, the random sampling process is repeated for multiple times, and multiple groups of input parameter values { x ] of first-order frequency of the satellite frame structure analyzed based on the finite element model are obtainedj|j＝1,2,…,M}；
Given a quantile τ, will { (x)jTau) j ═ 1,2, …, M } is input into the trained deep neural network model as input data, and a first-order frequency predicted value corresponding to the input data is obtained
Figure FDA0003455024410000026
Wherein τ is 0.5;
setting a first-order frequency critical value of a satellite frame structure, and calculating a limit state value of first-order frequency according to a first-order frequency predicted value and the first-order frequency critical value;
and counting the number of the limit state values of all the first-order frequencies which are less than 0 to determine the reliability of the satellite frame structure.
8. The method for analyzing the reliability of the satellite frame structure based on the sample quantile regression as claimed in claim 7, wherein the extreme state value of the first-order frequency of the satellite frame structure is calculated by using the following formula;
Figure FDA0003455024410000027
wherein, Deltaj(xj) Extreme state values, y, representing first order frequencieslimRepresenting a first order frequency threshold.
9. The method for analyzing the reliability of the satellite frame structure based on the sample quantile regression as claimed in claim 8, wherein the reliability of the first order frequency of the satellite frame structure is determined by using the following formula;
Figure FDA0003455024410000031
where R represents the reliability score and m represents the number of less than 0 of the extreme state values for all first order frequencies.
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CN116738553A (en) * 2023-08-14 2023-09-12 石家庄铁道大学 Uncertain parameter sensitivity analysis method for earthquake-resistant structure
CN116738553B (en) * 2023-08-14 2023-11-21 石家庄铁道大学 Uncertain parameter sensitivity analysis method for earthquake-resistant structure

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