CN117669389B - Random vibration analysis method of earthquake-vehicle-bridge system based on deep learning - Google Patents

Abstract

The present invention relates to the technical field of seismic design of bridges, and discloses a random vibration analysis method of an earthquake-vehicle-bridge system based on deep learning. The proposed model includes three modules: a CNN module for extracting seismic data features, an attention mechanism module for enhancing information selection between time series and improving the accuracy and efficiency of final prediction, and a bidirectional gated recurrent unit (BiGRU) for predicting the response of the vehicle-bridge system. The model establishes a mapping connection between seismic excitation and train response; a train running under near-fault earthquakes and a real railway cable-stayed bridge are used to verify the proposed model. The uncertainty of the train weight and the bridge damping ratio is also considered; finally, a CNN-BiGRU model based on the attention mechanism is customized and established based on the training data. The time-varying response of the proposed model is highly consistent with the results of the verified dynamic vehicle-bridge system.

Description

Random vibration analysis method of earthquake-vehicle-bridge system based on deep learning

Technical Field

The present invention relates to the technical field of seismic design of bridges, and specifically to a random vibration analysis method of an earthquake-vehicle-bridge system based on deep learning.

Background technique

China's high-speed railway network (HSRN) has developed rapidly over the past few decades. Due to its remarkable efficiency, convenience, and punctuality, high-speed railways have become the first choice for travel and business trips. As of 2022, China's high-speed railway network has expanded to more than 42,000 kilometers, connecting various regions and promoting the development of regional economies. Therefore, the high-speed railway network must face inevitable earthquake risks during construction and operation, especially considering that China is located in the Eurasian and Pacific Rim seismic belts. At the same time, as high-speed vehicles tend to be faster and lighter, bridges are widely used in high-speed railway networks to ensure the stability of the track. When an earthquake occurs, the probability of vehicles passing through bridges increases significantly. In the past two decades, several high-speed railway vehicle accidents caused by earthquakes have occurred, so it becomes crucial to ensure the safety of vehicles on bridges under seismic excitation.

Especially considering that long-span railway bridges are often located at or across unstable boundaries of tectonic plates, near-fault earthquakes pose an inevitable and arduous challenge to the seismic design of bridges in high-speed railway networks. Compared with far-field earthquakes, near-fault earthquakes have larger velocity pulses and longer pulse periods. Researchers have achieved some results on the dynamic response characteristics of structures induced by near-fault earthquakes. Compared with the structural response under general earthquake excitation, the structural response induced by near-fault earthquake excitation is significantly different, so special attention should be paid to multi-degree-of-freedom structures under near-fault earthquakes. Due to low-frequency resonance, near-fault earthquakes have an essential impact on the dynamic response of bridges. In order to ensure the smooth operation of high-speed railway systems, the concept of "replacing railway lines with bridges" has been realized, so the dynamic response of vehicle-bridge systems has become one of the most concerned research topics in the field of civil engineering. With the revelation of the destructiveness of near-fault earthquakes, the driving safety of trains on bridges under near-fault earthquakes has become another important perspective. In general, in the research on the driving safety of vehicle-bridge systems under seismic conditions, the investigation of the driving safety of vehicle-bridge systems under near-fault earthquakes is relatively limited. At the same time, the inevitable randomness of earthquakes and the uncertainty of the vehicle-bridge system parameters should also be incorporated into the dynamic analysis of the vehicle-bridge system under near-fault earthquakes.

Currently, many studies focus on the random vibration of vehicle-bridge systems under earthquakes. In these studies, the key issue is to deal with random excitations or uncertain parameters to reduce the sampling required for dynamic responses and thus improve computational efficiency. However, hundreds of simulations are required to ensure computational accuracy. Therefore, as an alternative to vehicle-bridge models, simplified surrogate models with lower computational overhead than the rough Monte Carlo method (MCM) are gaining more and more attention. Among various types of simplified surrogate models, deep learning models have become a popular choice due to their lower computational cost.

At present, many studies have fully demonstrated the good performance and application prospects of deep learning in vehicle-bridge systems. However, the research on deep learning models for the operational safety assessment of vehicle-bridge systems under near-fault earthquakes is relatively limited. Therefore, the key challenges of operational safety analysis of uncertain systems under near-fault earthquakes include three parts: (1) complex significant random characteristics should be included in the calculation; (2) the computational cost can be further reduced through artificial intelligence methods; (3) the application of deep learning methods in the stochastic analysis of vehicle-bridge systems has not yet been confirmed.

Summary of the invention

In view of the above problems, the purpose of the present invention is to provide a random vibration analysis method for earthquake-vehicle-bridge system based on deep learning, which uses the powerful feature extraction ability of convolution to capture the features of random earthquake data, dynamically and adaptively focuses on the effective part of the features through the attention mechanism, ignores the irrelevant parts, and finally realizes the one-step prediction of the whole process of the vehicle-bridge coupled random vibration system response through BiGRU; establishes the mapping between random earthquake waves and vehicle-bridge coupled random vibration system, so as to improve the computational efficiency and reduce the computational cost, and at the same time accurately predict the response of the uncertain vehicle-bridge system. The technical solution is as follows:

The random vibration analysis method of earthquake-vehicle-bridge system based on deep learning includes the following steps:

Step 1: Establish a train model and a bridge model, extract the natural frequency and vibration mode data of the train model and the bridge model, and verify the correctness of the train-bridge model through the data calculated by analytical solution;

Step 2: According to the design technical specifications of the real long-span cable-stayed railway bridge, determine the peak acceleration value of the earthquake motion at the bridge location, the characteristic period of the earthquake motion response spectrum and the design speed of the bridge;

Step 3: Obtain seismic wave data according to the seismic design response spectrum, and perform amplitude processing on the seismic wave data according to the peak acceleration value of the ground motion at the bridge location;

Step 4: Determine the coefficient of variation Cov of the train's own weight and the bridge damping ratio, and obtain the random parameters of the train's own weight and the bridge damping ratio through Monte Carlo sampling based on the coefficient of variation and the mean;

Step 5: Based on the processed seismic wave data, the natural frequency and vibration mode data of the train-bridge model, and the fixed and random parameters of the determined train weight and bridge damping ratio, the response of the corresponding train-bridge coupling system is calculated to evaluate the operation safety and stability;

Step 6: Set a set number of seismic wave data and the determined fixed parameters of train self-weight and bridge damping ratio as input data, and form data set 1 with the corresponding vehicle-bridge coupling system response calculated in step 5; Set a set number of seismic wave data and the random parameters of train self-weight and bridge damping ratio obtained in step 4 as input data, and form data set 2 with the corresponding vehicle-bridge coupling system response calculated in step 5;

Step 7: Build a CNN-Attention-BiGRU deep learning combined network, verify its prediction accuracy, and set its hyperparameters before training based on actual data;

Step 8: Input dataset 1 into the CNN-Attention-BiGRU deep learning combined network for prediction training, obtain the prediction model of deterministic train deadweight and bridge damping ratio parameters, and verify the prediction accuracy of the network under deterministic parameter conditions;

Step 9: Input dataset 2 into the CNN-Attention-BiGRU deep learning combined network for prediction training, obtain the prediction model of uncertain train deadweight and bridge damping ratio parameters, and verify the prediction accuracy of the network under uncertain parameter conditions;

Step 10: Obtain more response samples of the vehicle-bridge coupled random vibration system through the CNN-Attention-BiGRU deep learning combined network, and calculate the mean and standard deviation based on the response samples.

Furthermore, the CNN-Attention-BiGRU deep learning combined network includes three modules: an encoder module, an attention mechanism module, and a decoder module; the details are as follows:

The encoder module is used for feature extraction of seismic data time series, including a one-dimensional convolution layer, a LeakyReLU nonlinear mapping layer and a Dropout layer; the one-dimensional convolution layer maps the one-dimensional time series to a high-dimensional space to obtain multi-dimensional features; the Dropout layer is used to prevent overfitting;

The attention mechanism module is used to weight multi-dimensional features, focus on key features, and reduce attention to other information to improve the efficiency and accuracy of task processing;

The decoder module is used for the response prediction of the vehicle-bridge coupling system and includes a BiGRU layer and a fully connected layer;

The algorithm generates the complete sequence in one step.

Furthermore, the specific calculation in the CNN-Attention-BiGRU deep learning combined network is:

The one-dimensional convolution layer in the encoder module extracts features in a one-dimensional time series. The convolution calculation is expressed in matrix form as follows:

(1);

Among them, N _cov and M are the convolution output and input respectively; C is the sparse matrix of the convolution kernel, represents the convolution operator; b is the bias term;

The attention mechanism module further weights the features output by the encoder module, dynamically and adaptively focusing on different parts of the features to provide better prediction accuracy for the target sequence; the attention mechanism module calculates the similarity between the target output and each element in the input sequence to obtain the weight of each element; then uses these weights to calculate the weighted sum of the input sequence elements to obtain a weighted sum vector, which is used as the output of the attention mechanism module; the calculation is as follows:

(2);

Where a is the output of the attention mechanism module; h is the output of the encoder and the input of the attention mechanism; Wv _, Wk and _Wq are the weight matrices of the attention module; K = hWk is _the key vector, Q = hWq is _the query _vector ; V ₌ hWv is the value vector , is the score function, d is the length of the query vector and the key vector; T is the transposition symbol;

In the CNN-Attention-BiGRU algorithm, BiGRU is used for data prediction and output; BiGRU contains two positive and negative GRU units, and the GRU unit includes two key components: update gate and reset gate. The update gate is used to determine how much past state should be retained and considered, and how to combine new input information with previous state information to help GRU maintain long-term memory; the reset gate is used to determine how much previous information should be forgotten, and can effectively discard irrelevant data; the update gate z _t and reset gate r _t are as follows:

(3);

(4);

_Where Wz and Uz represent the weight matrices learned by the update gate during training and the weight matrices determined at the _end of training, respectively; Wr and Ur represent the weight matrices learned by the reset gate during training and the weight matrices determined at the end of training _{, respectively} ; is the sigmoid function; xt is the input of the current time step, ht _{- 1 is} the hidden state of the t -1 time step; bz is the bias of the update gate, _and br _is the bias of the reset gate; the reset gate calculates the hidden state, and the update gate updates the hidden state; the detailed calculation is as follows:

(5);

(6);

in, is the hyperbolic tangent function, /> is the Hadamard product, /> represents the updated candidate state at time t; W _h and U _h are learnable weight matrices, b _h is the bias, and h _t is the hidden state at t time steps;

When z _t converges to 1, the newly calculated candidate hidden state is ignored, and the past state is not updated; when z _t converges to 0, the candidate hidden state is retained; when the sequence has short-term dependencies, the reset gate is active; when the sequence has long-term dependencies, the update gate is active;

BiGRU obtains bidirectional temporal dependency by splicing two GRUs in opposite directions. The parameterization formula of BiGRU is as follows:

(7);

(8);

(9);

in, is the hidden state of the forward propagation, /> is the hidden state of back propagation, α _t and β _t are the weights of the hidden state of forward propagation and the hidden state of back propagation respectively, and b _t is the bias vector.

Compared with the prior art, the present invention has the following beneficial effects:

The present invention mainly takes random earthquakes and uncertain train deadweight and bridge damping ratio as input data, and uncertain vehicle-bridge system response samples as output data to construct a training data set, and inputs the data set into the CNN-Attention-BiGRU deep learning combined network for training and prediction. The present invention uses the powerful feature extraction capability of convolution to capture the features of random earthquake data, dynamically and adaptively focuses on the effective part of the features through the attention mechanism, ignores the irrelevant parts, and finally realizes the one-step prediction of the whole process of the vehicle-bridge coupled random vibration system response through BiGRU (bidirectional gated recurrent unit). The deep learning algorithm establishes a mapping between random earthquake waves and vehicle-bridge coupled random vibration systems, so as to improve the computational efficiency and reduce the computational cost, and at the same time accurately predict the response of the uncertain vehicle-bridge system; the sample prediction error obtained by the present invention is within the allowable range of the project, and the effect can meet the actual needs. It provides an effective solution to solve the problems of traditional modeling difficulties, complex and time-consuming calculation processes, and will promote the application of artificial intelligence in traditional engineering.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 is a diagram of the train-bridge system dynamic analysis process.

Figure 2 is the architecture diagram of the attention module.

Figure 3 is an illustration of a GRU unit.

Figure 4 is a diagram of the BiGRU neural network structure unit.

Figure 5 is a diagram of the architecture of the CAB neural network.

Figure 6 is a diagram of the bridge structure layout.

Figure 7 (a) is a loss curve diagram for training and learning the prediction of train derailment coefficient on a bridge under near-fault earthquake: CAB model.

Figure 7(b) is a loss curve diagram for training and learning the prediction of train derailment coefficient on a bridge under near-fault earthquake: BAB model.

Figure 8 shows the derailment coefficient under a single earthquake excitation when the parameters are determined in Example 1.

Figure 9 shows the prediction of the train derailment coefficient when the parameters are uncertain in Example 1.

Figure 10 (a) shows the loss curve when training the derailment coefficient prediction with uncertain learning parameters: CAB model.

Figure 10(b) shows the loss curve when training the derailment coefficient prediction with uncertain learning parameters: BAB model.

Figure 11 shows the train derailment coefficient under an earthquake excitation in Example 2.

Detailed ways

The present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. The present invention proposes a CNN-Attention-BiGRU (CAB) neural network, which is specially designed to analyze the operational safety of trains on bridges under near-fault earthquakes. The neural network includes three modules, an encoder module, an attention mechanism module, and a decoder module. Among them, the encoder module adopts a convolutional layer (CNN) for feature extraction of seismic data time series. The attention mechanism module (Attention) is used to enhance information selection between time series to improve the accuracy and efficiency of the final prediction. The decoder module adopts a bidirectional gated recurrent unit (BiGRU) for output data prediction (operation safety assessment, stability assessment). Based on the proposed method, a real large-span cable-stayed railway bridge is applied to study its application and efficiency. The calculation involves the randomness of near-fault earthquakes and the uncertain train weight and bridge damping ratio. This method is based on Python language and uses the Pytorch library for program writing. Its computing efficiency can be increased by nearly 16 times, the computing cost is reduced by 8 times, and at the same time it can have a certain level of accuracy. The technical solution is as follows:

The random vibration analysis method of earthquake-vehicle-bridge system based on deep learning includes the following steps:

Step 1: Establish the train model and bridge model, extract the natural frequency and vibration mode data of the train model and bridge model, and verify the correctness of the train-bridge model through the data calculated by analytical solution.

(1) Dynamic model of railway vehicles

This example assumes that all components of a single-unit railway vehicle are rigid bodies and connected by springs and dampers. The vehicle model is considered as a mass-spring-damper dynamic model. The flexibility of the vehicle model is provided by spring elements, and the dampers act as energy dissipation devices such as rubber pads, shock absorbers, and absorbers.

The train is simulated as a typical Chinese CRH2 train. The train motion equation can be expressed as equation (1):

(1);

Where M _v , C _v and K _v represent the mass, damping and stiffness matrices of the train respectively. 、/> and u represent the acceleration, velocity and displacement vector of the train respectively, y _b and /> are the velocity and displacement vectors of the contact point. Organizing equation (1), we get equation (2):

(2);

The above equation is the dynamic equation of the train. The right side of the equation is the force F _bv transmitted to the train by the bridge:

(3);

The force transmitted from the bridge to the train is caused by the vibration of the bridge. Specifically, the displacement response and velocity response of the bridge are converted into the force of the bridge on the train through the dampers and springs of the train.

(2) Finite element model of bridge

The analytical finite element model for the continuous rigid frame bridge employs two-node, three-dimensional beam elements with 12 degrees of freedom (DOFs), each containing six translational and six rotational DOFs.

The equation of motion of the bridge can be expressed as:

(4);

Where M _b , C _b and K _b represent the mass, damping and stiffness matrices of the bridge, respectively. 、/> and w represent the acceleration, velocity and displacement vectors of the bridge respectively; fe _is the external force vector, which refers to the seismic excitation in the invention.

In addition, there are forces transmitted from the train to the bridge through the contact point, including the weight, inertia and damping force of the train:

(5);

Among them, g is the acceleration due to gravity; F _vb and F _bv are a pair of action and reaction forces.

(3) Vehicle-bridge system dynamic analysis process

The vehicle-bridge model consists of a vehicle (train) subsystem and a bridge subsystem. These two subsystems need to satisfy two aspects of coupling relationship:

1) To meet the displacement coordination requirements, the wheels and bridge must maintain close contact without relative movement;

2) The interaction force between the wheel and the bridge contact point needs to comply with Newton's third law.

Specifically, the forces between the train subsystem and the bridge subsystem are transmitted through the contact points. The above equations show that the coupled system of train-bridge dynamic interaction is time-varying. Therefore, it is beneficial to use a separated iteration method. In each iteration, the difference between the responses of the two subsystems is calculated until the error limit is met. The dynamic analysis process of the vehicle-bridge system is shown in Figure 1.

The wheel-rail force F _wr ( t ) consists of two parts: normal friction F _Nwr ( t ) and tangential friction F _Twr ( t ). The normal friction F _Nwr ( t ) of the wheel and rail is calculated based on the Hertz nonlinear contact theory. The tangential friction F _Twr ( t ) is calculated using the Shen-Hedrick formula. The trace method is used to determine the contact point between the wheel and the rail. The wheel-rail force F _wr ( t ) is expressed as a function of the velocity and displacement of the wheelset and the rail. The normal friction of the wheel and rail can be expressed as:

(6);

The tangential friction force can be expressed as:

(7);

Where, Irr ( t ) is _the track irregularity excitation. The present invention adopts the explicit-implicit integration algorithm of the VB system. A new explicit integration method is used to speed up the solution of the vehicle-track system; and an implicit integration method Newmark-β is used to solve the bridge system.

When a train runs on a bridge that is hit by an earthquake, the earthquake ground motion causes vibrations in the abutments through the foundations, which are then transmitted to the train through the bridge's supports and superstructure. Therefore, the earthquake force can be divided into the inertial force acting on the bridge and the inertial force acting on the train. The above equations of motion for the train and bridge under earthquake action can be expressed as:

(8);

(9);

Where, F _gv = _{and Fgb} ＝ /> They represent the seismic forces on the train and bridge, respectively, expressed as inertial forces. is the seismic acceleration vector corresponding to each degree of freedom.

Step 2: According to the design technical specifications of the real long-span cable-stayed railway bridge, determine the peak acceleration value of the seismic motion at the bridge location, the characteristic period of the seismic motion response spectrum and the design speed of the bridge.

The earthquake response spectrum is the functional relationship between the maximum response of the structure under actual earthquake motion and the natural vibration characteristics of the structure.

Step 3: Obtain a sufficient amount of seismic wave data according to the seismic design response spectrum, and perform amplitude processing on the seismic wave data according to the peak acceleration value of the ground motion at the bridge location.

The seismic design response spectrum is a reference standard for seismic design of structures. It is an idealized seismic motion response spectrum specified in a national standard. The characteristic period of the seismic motion response spectrum is required to obtain the seismic design response spectrum.

Step 4: Determine the coefficient of variation Cov of the train's own weight and the bridge damping ratio, and obtain random parameters through Monte Carlo sampling based on the coefficient of variation and the mean.

Step 5: Input the processed seismic wave data and the natural frequency and vibration mode data of the vehicle-bridge model, input the fixed parameters and random parameters of the determined train weight and bridge damping ratio, calculate the response of the corresponding vehicle-bridge coupling system, and other data used to evaluate the safety and stability of operation.

Step 6: Set a certain number of seismic wave data and the determined fixed train weight and bridge damping ratio parameters as input data, and form data set 1 with the corresponding vehicle-bridge coupling system response calculated in step 5. Set a certain number of seismic wave data and the random train weight and bridge damping ratio parameters obtained in step 4 as input data, and form data set 2 with the corresponding vehicle-bridge coupling system response calculated in step 5.

Step 7: Build a CNN-Attention-BiGRU deep learning combination network, verify its prediction accuracy, and set its hyperparameters before training based on actual data.

The CNN-Attention-BiGRU deep learning combined network consists of three modules.

The first module is the encoder module, which is used for feature extraction of seismic data time series. This module includes a one-dimensional convolution layer, a LeakyReLU nonlinear mapping layer, and a Dropout layer. The one-dimensional convolution layer maps the one-dimensional time series to a high-dimensional space to obtain multi-dimensional features. The convolution kernel is set to 5, the step size is 1, the padding is 2, and the number of channels is 64. The Dropout layer is used to prevent overfitting, and the hidden layer parameter loss rate is set to 0.5.

The second module is the attention mechanism module, which weights the multi-dimensional features, focuses on key features, reduces attention to other information, and improves the efficiency and accuracy of task processing. In this module, both K and Q values are obtained from the output of the encoder and processed by a fully connected layer using tanh as the activation function.

The third module is the decoder module, which is used for the response prediction of the vehicle-bridge coupling system. This module includes a BiGRU layer and a fully connected layer. The number of channels in the BiGRU layer is 64. Due to the long length of the final output data, the single-step loop output is prone to error accumulation. Therefore, the algorithm chooses to directly generate the complete sequence in one step.

Step 8: Input dataset 1 into the CNN-Attention-BiGRU deep learning combined network for prediction training to obtain the prediction model of deterministic train self-weight and bridge damping ratio parameters, and verify the prediction accuracy of the network under deterministic parameter conditions.

Step 9: Input dataset 2 into the CNN-Attention-BiGRU deep learning combined network for prediction training to obtain the prediction model of uncertain train self-weight and bridge damping ratio parameters, and verify the prediction accuracy of the network under uncertain parameter conditions.

Step 10: Obtain more response samples of the vehicle-bridge coupled random vibration system through the CNN-Attention-BiGRU deep learning combined network, and calculate the mean and standard deviation based on the response samples.

The specific calculation in the CNN-Attention-BiGRU deep learning combined network is:

The one-dimensional convolution layer in the CNN-Attention-BiGRU algorithm can extract features in a one-dimensional time series. The convolution calculation is expressed in matrix form as:

(10);

Among them, N _cov and M are the convolution output and input respectively; C is the sparse matrix of the convolution kernel, represents the convolution operator; b is the bias term.

The attention mechanism module in the CNN-Attention-BiGRU algorithm further weights the features output by the encoder module, dynamically and adaptively focusing on different parts of the features to provide better prediction accuracy for the prediction of the target sequence. This module calculates the similarity between the target output and each element in the input sequence to obtain the weight of each element. These weights are then used to calculate the weighted sum of the elements of the input sequence to obtain a weighted sum vector, which is used as the output of the module. The weighted sum vector represents the input sequence, with higher importance given to parts that receive higher weights. The attention module is shown in Figure 2 and is calculated as follows:

(11);

Where a is the output of the attention mechanism module; h is the output of the encoder and the input of the attention mechanism; Wv _, Wk and _Wq are the weight matrices of the attention module; K = hWk is _the key vector, Q = hWq is _the query _vector ; V ₌ hWv is the value vector , is the score function, d is the length of the query vector and the key vector; T is the transposition symbol.

In the CNN-Attention-BiGRU algorithm, BiGRU is used for data prediction and output. BiGRU contains two positive and negative GRU units to implement it. The GRU unit has two key components: the update gate and the reset gate, as shown in Figure 3. The reset gate is used to determine how much previous information should be forgotten, and can effectively discard irrelevant data. The update gate determines how much past state should be retained and considered, and how to combine new input information with previous state information to help GRU maintain long-term memory. The update gate z _t and reset gate r _t are shown below:

(12);

(13);

_Where Wz and Uz represent the weight matrices learned by the update gate during training and the weight matrices determined at the _end of training, respectively; Wr and Ur represent the weight matrices learned by the reset gate during training and the weight matrices determined at the end of training _{, respectively} ; is the sigmoid function; xt is the input of the current time step, ht _{- 1} is the hidden state of the t -1 time step; bz is the bias of the update gate, _{and br} is _the bias of the reset gate.

The reset gate calculates the hidden state, and the update gate must update the hidden state. The detailed calculation is as follows:

(14);

(15);

in, is the hyperbolic tangent function, /> is the Hadamard product, /> represents the updated candidate state at time t , W _h and U _h are learnable weight matrices, b _h is the bias, and h _t is the hidden state at time step t . The remaining parameters have the same meaning as before. When z _t converges to 1, the newly calculated candidate hidden state is ignored, which is equivalent to not updating the past state. When z _t converges to 0, the candidate hidden state is retained. The reset gate is active when the sequence has short-term dependencies. The update gate is active when the sequence has long-term dependencies.

Usually, the data at the current moment is associated with both the previous moment and the future moment. On this basis, BiGRU obtains bidirectional temporal dependency by splicing 2 GRUs in opposite directions, as shown in Figure 4. This bidirectional structure can help the recurrent neural network extract more information, thereby improving the performance of the learning process. The parameterization formula of BiGRU is as follows:

(16);

(17);

(18);

in, is the hidden state of the forward propagation; /> is the hidden state of back-propagation, α _t , β _t , and b _t are the weight and bias vectors, respectively.

As shown in Figure 5, the proposed CAB neural network architecture consists of three modules, including encoder, attention module and decoder. In the first step, the encoder is mainly composed of convolutional layers. Here, the one-dimensional time series will be mapped to a high-dimensional space by the convolutional layer to obtain multi-dimensional features. The number of feature channels output from the convolutional layer is 64, and the remaining parameters include convolution kernel 5, step size 1 and padding 2. In the second step, through the attention module, the multi-dimensional features are weighted to focus on key features, reduce attention to other information, and improve the efficiency and accuracy of task processing. In this module, both K and Q values are obtained from the output of the encoder and processed by different fully connected layers using tanh as the activation function. In the third step, the decoder consists of BiGRU layers and fully connected layers. Due to the long length of the final output data, the single-step cycle output is prone to error accumulation. Therefore, this method chooses to directly generate the complete sequence in one step.

In order to prevent overfitting and obtain better generalization ability, a dropout layer with a dropout rate of 0.5 is set in both the encoder and the decoder. In addition, this method selects ADAM as the optimizer. At the same time, this method sets L2 regularization in the optimizer to further prevent overfitting. During the training process, the initial learning rate of the optimizer is 0.01. Every 250 epochs, the learning rate is scaled by a factor of 0.8 until the learning rate drops to 5e-5, and then stops changing.

The prediction target of the present invention is train safety, including derailment coefficient, etc. Therefore, the method uses earthquake acceleration as input and train safety as output. The earthquake acceleration is normalized before input, and the data is mapped to the range of [-1,1], while the train safety value does not undergo data preprocessing.

Example verification:

A steel truss girder double-tower cable-stayed high-speed railway bridge was selected to compare the vehicle operation safety with systematic uncertainty and random near-fault earthquakes. The total length of the bridge is 864 meters, and the span arrangement is 81+135+432+135+81 meters. The overall layout of the main bridge is shown in Figure 6. The bridge deck width is 18 meters, and it is designed as two railway lanes, with the centerline of the lanes 2.2 meters away from the centerline of the bridge deck. The height of the main beam steel truss girder is 14m. The height of the main tower of the bridge is 180 meters, and the tower type is H-shaped. In the numerical model of the bridge, all main beams, piers and main towers are modeled by 3D beam elements with 6 degrees of freedom at each node, while the tension cables are simulated by 3D bar elements with 3 degrees of freedom at each node. The damping ratio of the entire bridge structure is set to 0.05.

According to the technical specifications of the actual project, the peak value of the ground vibration acceleration is set to 0.05g, the characteristic period of the ground vibration response spectrum is 0.35s, and the design speed of the vehicle on the bridge is 200km/h. The derailment coefficient, vehicle lateral and vertical acceleration are used as indicators to evaluate the operational safety of the vehicle-bridge system under near-fault earthquakes. The German low-disturbance track irregularity random excitation is added in the preparation of the data set.

(1) Example 1: Verifying the operational safety of a vehicle-bridge system under random near-fault earthquakes

This example does not consider the influence of uncertain parameters, the train weight is 26100kg, and the bridge damping ratio is 0.05. The results of the CAB model and the BiGRU-Attention-BiGRU (BAB) model are compared to show the difference in the calculation results of the two models.

By dynamically analyzing the train-bridge system affected by earthquakes, 100 prepared earthquake time series data are calculated to obtain the corresponding train safety time series data captured throughout. The calculation step size of earthquake time data and train safety time data is 0.02. In this case, a padding operation is used to make the earthquake time series a uniform length of 20 seconds, while the train safety time series is a uniform length of 15.56 seconds according to the time the train is on the bridge. In addition, a ratio of 0.8:0.1:0.1 is selected, and 80% of the data is randomly selected as a training set, 10% as a test set, and 10% as a validation set.

This method uses the mean absolute error (MAE) as the loss function for predicting the train derailment coefficient. It should be noted that there is a large gap in the derailment coefficients affected by different near-fault earthquakes. If the loss function is directly calculated during the training process to obtain the predicted data, it will cause a large fluctuation in the loss curve and reduce the prediction accuracy. In order to alleviate this problem, data processing is required before calculating the loss function to distinguish the predicted data from the real data by the absolute maximum value of the actual data. The above operation can ensure the stability of the loss curve and thus improve the prediction accuracy. Figure 7 (a) and Figure 7 (b) show the loss values of each training stage after the above data processing, and the minimum loss values are 0.0174 and 0.0192, respectively. It is worth noting that the CAB model reaches convergence faster than the BAB model.

Table 1 shows the means and variances of MAE, mean square error (MSE), and coefficient of determination (R ² ) between all predicted data and the corresponding true data in the dataset. No data processing was performed before calculating the errors. Figure 8 shows a comparison of predicted data and true data. When the uncertain parameters are not considered, the errors between the two models are not significant. For MAE, the CAB model has a smaller mean but a slightly larger variance. For MSE, the CAB model produces a slightly larger mean but a smaller variance. For R ² , the BAB model is significantly better in terms of both mean and variance.

Table 1 Derailment coefficient loss value

.

Although the two training models do not consider the uncertain parameters of the train-bridge system, they can also be used to robustly predict the data of the uncertain system. The changes in the train weight and bridge damping ratio are set in the input layer. The train weight is set to 23941.33 kg and the bridge damping ratio is 0.068. Figure 9 shows the prediction results, which are less accurate. Therefore, in order to improve the reliability and accuracy of the proposed method in predicting the response of the uncertain train-bridge system, it is crucial to integrate the uncertain parameters into the training dataset.

(2) Example 2: Verification of the operational safety of an uncertain vehicle-bridge system under random near-fault earthquakes

Train weight and bridge damping ratio are key factors affecting train safety. In order to improve the accuracy of the developed model, both parameters were given a coefficient of variation of 0.2, with an average value of 26100kg for train weight and 0.05 for bridge damping ratio. The corresponding responses of the uncertain train-bridge system were added as training data. The 30 sets of data were also divided into training set, validation set and test set in the ratio of 0.8:0.1:0.1.

MAE is selected as the loss function, and the same data processing is performed after calculation. Figure 10 (a) and Figure 10 (b) show the loss value of each epoch, and the minimum loss values are 0.042 and 0.049 respectively. As in Example 1, the CAB model can reach convergence faster than the BAB model.

Table 2 gives the mean and variance of MAE, MSE, and R2 for all predicted data in the dataset relative to the corresponding actual data. Figure 11 shows a comparison of one generated data based on the same earthquake time series in Figure 8 with the actual data. When considering the uncertain parameters, the data generated by the CAB model has a smaller mean in terms of mean absolute error and mean square error. But for the variance, the results of the BAB model are significantly smaller. In addition, the average R2 of the CAB model is significantly better than that of the BAB model, with average R2 values of 0.93866 and 0.82829, respectively. This shows that the CAB model is more advantageous, especially when considering the uncertainty of key parameters such as train weight and bridge damping ratio.

Table 2 Derailment coefficient loss values

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In summary, the CNN-Attention-BiGRU model proposed in this paper aims to predict the running safety of trains on bridges during near-fault earthquakes (NFEs). The model uses the CNN module to effectively extract features from the input data, and uses the BiGRU module to predict and output the train safety factor, and uses the Attention module to accurately locate and emphasize the key features most relevant to the output data. Through actual examples, it is found that the prediction accuracy of the CAB model and the BAB model is relatively similar when the uncertainty of the train weight and the bridge damping ratio is not considered. However, when the uncertainty of the train weight and the bridge damping ratio is considered, the CAB model shows higher prediction accuracy. In addition, compared with the BAB model, the CAB model converges faster, indicating that it is more efficient in adapting to and accurately predicting variable conditions.

Claims (3)

1. A random vibration analysis method for an earthquake-vehicle-bridge system based on deep learning, characterized in that it comprises the following steps: Step 1: Establish a train model and a bridge model, extract the natural frequency and vibration mode data of the train model and the bridge model, and verify the correctness of the train-bridge model through the data calculated by analytical solution; Step 2: According to the design technical specifications of the real long-span cable-stayed railway bridge, determine the peak acceleration value of the earthquake motion at the bridge location, the characteristic period of the earthquake motion response spectrum and the design speed of the bridge; Step 3: Obtain seismic wave data according to the seismic design response spectrum, and perform amplitude processing on the seismic wave data according to the peak acceleration value of the ground motion at the bridge location; Step 4: Determine the coefficient of variation Cov of the train's own weight and the bridge damping ratio, and obtain the random parameters of the train's own weight and the bridge damping ratio through Monte Carlo sampling based on the coefficient of variation and the mean; Step 5: Based on the processed seismic wave data, the natural frequency and vibration mode data of the train-bridge model, and the fixed and random parameters of the determined train weight and bridge damping ratio, the response of the corresponding train-bridge coupling system is calculated to evaluate the operation safety and stability; Step 6: Set a set number of seismic wave data and the determined fixed parameters of train self-weight and bridge damping ratio as input data, and form data set 1 with the corresponding vehicle-bridge coupling system response calculated in step 5; Set a set number of seismic wave data and the random parameters of train self-weight and bridge damping ratio obtained in step 4 as input data, and form data set 2 with the corresponding vehicle-bridge coupling system response calculated in step 5; Step 7: Build a CNN-Attention-BiGRU deep learning combined network, verify its prediction accuracy, and set its hyperparameters before training based on actual data; Step 8: Input dataset 1 into the CNN-Attention-BiGRU deep learning combined network for prediction training, obtain the prediction model of deterministic train deadweight and bridge damping ratio parameters, and verify the prediction accuracy of the network under deterministic parameter conditions; Step 9: Input dataset 2 into the CNN-Attention-BiGRU deep learning combined network for prediction training, obtain the prediction model of uncertain train self-weight and bridge damping ratio parameters, and verify the prediction accuracy of the network under uncertain parameter conditions; Step 10: Obtain more response samples of the vehicle-bridge coupled random vibration system through the CNN-Attention-BiGRU deep learning combined network, and calculate the mean and standard deviation based on the response samples. 2. The random vibration analysis method of earthquake-vehicle-bridge system based on deep learning according to claim 1 is characterized in that the CNN-Attention-BiGRU deep learning combined network includes three modules: an encoder module, an attention mechanism module and a decoder module; specifically as follows: The encoder module is used for feature extraction of seismic data time series, including a one-dimensional convolution layer, a LeakyReLU nonlinear mapping layer and a Dropout layer; the one-dimensional convolution layer maps the one-dimensional time series to a high-dimensional space to obtain multi-dimensional features; the Dropout layer is used to prevent overfitting; The attention mechanism module is used to weight multi-dimensional features, focus on key features, and reduce attention to other information to improve the efficiency and accuracy of task processing; The decoder module is used for the response prediction of the vehicle-bridge coupling system and includes a BiGRU layer and a fully connected layer; The algorithm generates the complete sequence in one step. 3. The random vibration analysis method of earthquake-vehicle-bridge system based on deep learning according to claim 2 is characterized in that the specific calculation in the CNN-Attention-BiGRU deep learning combined network is: The one-dimensional convolution layer in the encoder module extracts features in a one-dimensional time series. The convolution calculation is expressed in matrix form as follows: (1); Among them, N _cov and M are the convolution output and input respectively; C is the sparse matrix of the convolution kernel, represents the convolution operator; b is the bias term; The attention mechanism module further weights the features output by the encoder module, dynamically and adaptively focusing on different parts of the features to provide better prediction accuracy for the target sequence; the attention mechanism module calculates the similarity between the target output and each element in the input sequence to obtain the weight of each element; then uses these weights to calculate the weighted sum of the input sequence elements to obtain a weighted sum vector, which is used as the output of the attention mechanism module; the calculation is as follows: (2); Where a is the output of the attention mechanism module; h is the output of the encoder and the input of the attention mechanism; Wv _, Wk and _Wq are the weight matrices of the attention module; K = hWk is _the key vector, Q = hWq is _the query _vector ; V ₌ hWv is the value vector , is the score function, d is the length of the query vector and the key vector; T is the transposition symbol; In the CNN-Attention-BiGRU algorithm, BiGRU is used for data prediction and output; BiGRU contains two positive and negative GRU units, and the GRU unit includes two key components: update gate and reset gate. The update gate is used to determine how much past state should be retained and considered, and how to combine new input information with previous state information to help GRU maintain long-term memory; the reset gate is used to determine how much previous information should be forgotten, and can effectively discard irrelevant data; the update gate z _t and reset gate r _t are as follows: (3); (4); _Where Wz and Uz represent the weight matrices learned by the update gate during training and the weight matrices determined at the _end of training, respectively; Wr and Ur represent the weight matrices learned by the reset gate during training and the weight matrices determined at the end of training _{, respectively} ; is the sigmoid function; xt is the input of the current time step, ht _{- 1} is the hidden state of the t -1 time step; bz is the bias of the update gate, _{and br} is _the bias of the reset gate; The reset gate calculates the hidden state, and the update gate updates the hidden state; the detailed calculation is as follows: (5); (6); in, is the hyperbolic tangent function, /> is the Hadamard product, /> represents the updated candidate state at time t; W _h and U _h are learnable weight matrices, b _h is the bias, and h _t is the hidden state at t time steps; When z _t converges to 1, the newly calculated candidate hidden state is ignored, and the past state is not updated; when z _t converges to 0, the candidate hidden state is retained; when the sequence has short-term dependencies, the reset gate is active; when the sequence has long-term dependencies, the update gate is active; BiGRU obtains bidirectional temporal dependency by splicing two GRUs in opposite directions. The parameterization formula of BiGRU is as follows: (7); (8); (9); in, is the hidden state of the forward propagation, /> is the hidden state of back propagation, α _t and β _t are the weights of the hidden state of forward propagation and the hidden state of back propagation respectively, and b _t is the bias vector.