An efficient dynamic reliability method for maglev vehicle-bridge systems and its application in random controller parameters analysis

Abstract

This article presents an efficient method for analyzing the dynamic reliability of maglev vehicle-bridge coupled systems by combining a theoretical model with an adaptive surrogate model and the probability density evolution method (PDEM). First, a refined theoretical model of a maglev vehicle-bridge coupling system is established. Next, an adaptive surrogate model of the equivalent extreme value of the system dynamic response is established by combining an adaptive sampling method with radial basis functions. Finally, the adaptive surrogate model and PDEM are combined to further improve the efficiency of the dynamic reliability analysis. In the numerical example, the theoretical model of the maglev vehicle-guideway system was first validated by comparing with the measured data from the Shanghai high-speed maglev line. Then, by treating the controller parameters as normally distributed random variables, the accuracy and efficiency of the proposed reliability method were verified through comparison with the Monte Carlo method and the one-stage sampling surrogate model. Additionally, the impact of the randomness of each controller parameter and the coefficient of variation of the controller parameters on the system’s dynamic reliability was discussed.

Introduction

As an innovative mode of transportation, maglev systems have attracted attention from researchers in the area of diversified rail transit due to their advantages, such as high-speed, small turning radius, little mechanical wear, safety, and ease of maintenance.^1,2 Many countries in the world, including Japan, South Korea, and China, have built their own commercial maglev lines.^3–5 The world’s first and only high-speed commercial maglev line is located in Shanghai, China, and has a maximum operating speed of 430 km/h. ⁶ To further improve the operating speed of maglev trains, China has launched the development of a high-speed maglev transport system with a speed of 600 km/h. Maglev trains have broad prospects for future applications all over the world. ¹

Due to various factors, the dynamic response of a maglev vehicle-bridge system exhibits significant randomness, meaning that reliability methods need to be used to accurately evaluate the dynamic performance of the system.^7–9 Several analysis methods have been developed for the dynamic reliability of vehicle-bridge coupled systems, including the (improved) Monte Carlo method (MCM),^10,11 the pseudo-excitation method,^12,13 the probability density evolution method (PDEM),^14–17 and the explicit time-domain method. ¹⁸ PDEM, based on the conservation of probability, has advantages such as clear mathematical principles, high efficiency, and wide applicability, meaning that it has good potential for use in this study. However, it is worth noting that due to the complexity of the vehicle-bridge system, obtaining calculation results that meet the accuracy requirements usually requires 300–600 sets of samples, ¹⁹ and the greater the number of random variables involved, the larger the sample size needs to be.

In recent years, with the development of computational science, surrogate modeling methods such as the Kriging model, radial basis function (RBF), Gaussian process regression (GPR), and artificial neural network (ANN) have been applied to improve the efficiency of structural dynamic reliability analysis.^20–22 When constructing a surrogate model, the sampling method has a significant impact on the accuracy. ²³ Sampling methods can be divided into two types: one-stage and adaptive methods. The one-stage sampling method determines the number and location of all sample points before establishing a surrogate model. Xiang et al. ²⁴ proposed an ensemble method based on the nonlinear autoregressive with exogenous input (NARX) surrogate model and the subset simulation with splitting (SS/S) method, and verified its effectiveness in predicting the vertical dynamic response of vehicle-bridge systems. Li et al. ²⁵ presented an approach to analyzing the random vibration of bridges under a coupled vehicular load using Bayesian deep learning. Zhang et al. ²⁶ proposed an efficient method based on NARX-ANN to predict the nonlinear wheel-rail force of trains on bridges caused by track irregularity. Aloisio et al. ²⁷ verified the effectiveness of the surrogate model in predicting the dynamic response of a vehicle-bridge system under braking conditions by comparing it with a physics-based model and experimental data. The studies described above^24–27 used the one-stage sampling method to construct surrogate models. Although this approach can minimize the deviation of the sample point set, it cannot reasonably allocate the position of the sample points based on the objective function, resulting in duplicate sample information and low utilization of the training set. In addition, this type of sampling method cannot determine the appropriate number of training samples in advance. ²³ In contrast, an adaptive sampling method first considers a small number of initial sample points to construct a rough surrogate model, and then uses the rough model and a criterion function to select new sample points from the candidate sample set, which are added to the training set to update the surrogate model. This process is repeated until the surrogate model meets the accuracy requirements. ²⁸ Although researchers have demonstrated in previous studies that the adaptive surrogate model (ASM) can effectively improve the efficiency of railway vehicle-bridge random analysis, ²⁹ the effectiveness of this method still needs to be verified for maglev vehicle-bridge systems.

In general, maglev trains can be divided into three categories: electrodynamic suspension (EDS), electromagnetic suspension (EMS), and high-temperature superconducting (HTS) levitation. ³⁰ Of these, EMS trains are currently the most widely used. The suspension system of an EMS train is an inherently unstable open-loop system, which requires a controller to maintain the suspension clearance within the allowable range. ³¹ Research and practice show that the controller parameters have an important influence on the formation of vehicle-guideway coupling vibration ² ; unsuitable controller parameters may result in poor passenger comfort, poor running stability and high noise levels. Li et al. ³² established a single-electromagnet proportional-derivative (PD) controller-flexible bridge coupled vibration model and analyzed the effects of the controller parameters and the delay of the signals on the self-excited vibrations of the system. In order to study the influence of track short wave length irregularity on low-speed maglev train, a vehicle model including a PD controller and five independent bogies was simulated by Yu et al. ³³ Their results showed that a reasonable adjustment to the PD controller parameters was beneficial in terms of gap fluctuation at the optimal wavelength. Wang et al. ³⁴ established a maglev vehicle with five suspension frames and studied the effect of the inner loop current gain coefficient P on the static stability. Xu et al. ³⁵ proposed a model that can be used to analyze the dynamic behaviors of high-speed maglev trains on curved viaducts. To investigate the influence of control loop faults on the dynamic responses of the vehicle and guideway, Guo et al. ³ and Xiang et al. ⁴ established a spatially coupled vibration model that includes a three-dimensional (3D) maglev train, a guideway bridge, and a PD controller. Zeng et al. ³⁶ studied the influence of controller parameters on the vibration of the vehicle-bridge coupled system and found that mismatch between the rail parameters and the controller parameters can lead to severe system vibrations. Taking into account the delay in the control system, Feng et al. ³⁷ analyzed the effects of running speed and control parameters on the vibration of the maglev vehicle-guideway system. Based on the Hopf bifurcation criterion, Sun et al. ² investigated the influence of the proportional-integral-derivative (PID) controller parameters $k_e$ , $k_c$ , and $k_v$ on the maglev vehicle-rail coupling vibrations. The results show that although $k_e$ can reduce the coupling vibration, the dynamic performance of the system deteriorates.

Several advanced controllers have also been proposed, such as a fuzzy controller ³⁸ and an intelligent controller^39,40, which can adjust the controller parameters in real time to achieve optimal control effect. However, it is important to note that the studies discussed above^2,4,32–40 focused on control algorithms or parameter analysis. Currently, few scholars have investigated the impact of the randomness and combinations of controller parameters on the system’s dynamic response. Given that controller parameters are typically determined based on experience, it is crucial to examine the influence of these parameters on the system’s dynamic response from the perspective of random vibration analysis.

In this study, we propose an efficient method for the dynamic reliability analysis of a high-speed maglev vehicle-bridge system. This method combines a theoretical model with a surrogate model and PDEM. Using the Shanghai maglev line as the engineering context, we first verify the correctness of our theoretical model for a maglev vehicle-PD controller-bridge system. Next, we assume that the control parameters are random variables, the accuracy and efficiency of the proposed method are then verified by comparison with MCM and the one-stage sampling surrogate model (OSM). Finally, we explore the impacts of the randomness in each controller parameter and the coefficients of variation of the controller parameters on the dynamic reliability results, such as the probability density function (PDF) and the probability of exceedance (POE).

Dynamic model and equation of motion for the EMS high-speed maglev vehicle-bridge coupled system

Maglev train model

In this study, we take the Shanghai EMS high-speed maglev train, as depicted in Figure 1, as an example to establish a dynamic train model. In this type of train, each car comprises 1 car body, 4 bogies, 16 rods, 14 suspension magnets, and 14 guidance magnets. Additionally, there are 2 suspension magnets and 2 guidance magnets between the front and rear bogies of adjacent cars. The bogies are composed of 2 C-shaped frames connected by a longitudinal axis. The suspension magnets have 12 magnetic poles, with a center-to-center spacing of 0.258 m, and the guidance magnets have 3 guidance magnetic poles, with a center-to-center spacing of 1.032 m. The multi-body dynamics method is employed to simulate the vibration of the vehicle model, where the car body, rockers, bogies, and magnets are modeled as rigid bodies, and the suspension system is modeled as a linear spring or damping element. Each vehicle model has 101 degrees of freedom (DOFs), and detailed information about the train model is illustrated in Figure 2.

Figure 1.

A Shanghai EMS high-speed maglev train traveling on guideway bridges.

Figure 2.

Train dynamic model: (a) front view; (b) side view; and (c) top view.

Guideway bridge model

The guideway bridge, as shown in Figure 1, consists of girders, piers, supports, and rails. The bridge has a single span of 24.768 m, and the rails on both sides are composed of eight functional units, each of which is connected to the girder via four pairs of cast iron connectors. Both the rails and girders have box-shaped cross-sections, diagrams of which are displayed in Figure 3(a). The piers are made of solid concrete, with caps and bodies featuring rectangular cross-sections of 3.0 × 1.8 m and 1.8 × 1.8 m, respectively. Two steel laminated elastic bearings are installed between the girder and pier for vibration reduction, each measuring 0.5 m in length, 0.5 m in width, and 0.04 m in height.

Figure 3.

Finite element model of a guideway bridge: (a) shape displayed and (b) shape not displayed.

The finite element method was used to establish the guideway bridge model. More specifically, the piers, girders, and rails were simulated as Timoshenko beam elements, the supports were modelled as spring damper elements, and the girder nodes and rail nodes were connected by rigid arms to simulate the eccentric loading effect of the magnetic force. When establishing the boundary conditions for the bridge model, the influence of the soil and pile vibration on the model was ignored, and the bottom of the piers was set as fixed ends. Figure 3 shows the finite element model of the maglev viaduct established using the above method.

Controller model

A maglev train features a suspension system with multiple electromagnets. Owing to the structural decoupling function of the suspension frame, the control of individual electromagnets can be considered as the fundamental unit for the overall suspension control system of the maglev train. ³⁴ The structural diagram of the single-electromagnet model is shown in Figure 4(a), and its parameters can be found in Ref. 6 Given that the EMS maglev train is an inherently open-loop unstable system, it is necessary to introduce a controller to ensure the stable suspension of the vehicle. In this study, a PD controller, as illustrated in Figure 4(b), is adopted to adjust the control current $\Delta i_e$ , and the specific calculation formula is as follows

$\Delta i_e=K_P\Delta h_e+K_D\Delta {\stackrel{.}{h}}_e=K_P\left(\Delta h_e+\tau \Delta {\stackrel{.}{h}}_e\right)$

(1)

where $K_P$ is the proportional gain, and $K_D$ is the differential gain. The control law depends on the state and is adjusted according to working conditions and related interference. The time constant $\tau$ can be determined from the root diagram of the system and $K_P$ .

Figure 4.

Model of electromagnet-rail interaction: (a) electromagnet model and (b) feedback principle for the PD controller.

Equation of motion for the coupled system

The equation of motion for the maglev vehicle-bridge coupled system can be established as follows:

$\left[\begin{array}{l}{\mathit{M}}_v\hspace{0.17em}\mathbf{0}\\ \hspace{0.17em}\mathbf{0}\hspace{0.17em}{\mathit{M}}_b\end{array}\right]\left\{\begin{array}{l}{\ddot{\mathit{U}}}_v\\ {\ddot{\mathit{U}}}_b\end{array}\right\}+\left[\begin{array}{l}{\mathit{C}}_v\mathbf{\hspace{0.17em}}{\mathit{C}}_{vb}\\ {\mathit{C}}_{bv}\mathbf{\hspace{0.17em}}{\mathit{C}}_b\end{array}\right]\left\{\begin{array}{l}{\stackrel{.}{\mathit{U}}}_v\\ {\stackrel{.}{\mathit{U}}}_b\end{array}\right\}+\left[\begin{array}{l}{\mathit{K}}_v\mathbf{\hspace{0.17em}}{\mathit{K}}_{vb}\\ {\mathit{K}}_{bv}\mathbf{\hspace{0.17em}}{\mathit{K}}_b\end{array}\right]\left\{\begin{array}{l}{\mathit{U}}_v\\ {\mathit{U}}_b\end{array}\right\}=\left\{\begin{array}{l}{\mathit{F}}_v\\ {\mathit{F}}_b\end{array}\right\}$

(2)

where $\mathit{M}$ , $\mathit{C}$ , and $\mathit{K}$ are the mass, damping, and stiffness matrices, respectively; $\mathit{U}$ , $\stackrel{.}{\mathit{U}}$ , and $\ddot{\mathit{U}}$ are the displacement, velocity, and acceleration vectors, respectively; the subscripts $v$ and $b$ represent the vehicle and guideway bridge, respectively; the subscripts “ $vb$ ” or “ $bv$ ” represent the terms caused by dynamic interactions. ${\mathit{F}}_v$ and ${\mathit{F}}_b$ represent the external load vectors for the train and guideway bridge subsystems caused by track irregularities, respectively. Equation (2) is a time-varying equation, which is calculated using the separation iteration method in this study. The specific process can refer to Ref. 6.

Dynamic reliability analysis of a vehicle-bridge system based on PDEM

In previous studies, PDEM has been successfully applied to address dynamic reliability problems in vehicle-bridge systems, which involve random excitations, random physical parameters, or both. The main calculation steps in this method are as follows:

(1) Select representative sample point sets $\mathit{\theta }={\left\{{\mathit{\theta }}_q=\left({\theta }_{q,1}{\theta }_{q,2}···{\theta }_{q,d}\right)\right\}}_{q=1}^{N_p}$ for the random variables $\mathit{\Theta }$ using a sampling strategy and calculate the corresponding assignment probability for each representative point set as:

$P_q={\int }_{{\Omega }_q}{p}_{\mathit{\Theta }}\left(\mathit{\theta }\right)\mathrm{d}\mathit{\theta }$

(3)

where $d$ represents the dimension of the random variable, $N_p$ denotes the total number of representatives, and ${\Omega }_q$ denotes the Voronoi cells that divide the probability space into $N_p$ points.

• (2) Solve the vehicle-bridge coupled vibration equation in Equation (2) at each representative sample point set ${\mathit{\theta }}_q$ and obtain the corresponding equivalent extreme value (EEV) of the system dynamic response as

$W\left({\mathit{\theta }}^q,T\right)=\{\begin{array}{l}\underset{t\in \left[0,T\right]}{\mathit{max}}\left\{Z\left(t\right)\right\},\text{dynamic}\hspace{0.17em}\text{index}\hspace{0.17em}\text{with}\hspace{0.17em}\text{one}-\text{sided}\hspace{0.17em}\text{boundary}\\ \underset{t\in \left[0,T\right]}{\mathit{max}}\left\{\left|Z\left(t\right)\right|\right\},\text{dynamic}\hspace{0.17em}\text{index}\hspace{0.17em}\text{with}\hspace{0.17em}\text{double}-\text{sided}\hspace{0.17em}\text{boundary}\end{array}$

(4)

where $T$ denotes the running time of the vehicle and $Z$ denotes the dynamic index of interest.

• (3) Construct a virtual stochastic process $\mathbb{W}$ that satisfies the following conditions:

$\mathbb{W}{\left(\tau \right)|}_{\tau =0}=0,\mathbb{W}{\left(\tau \right)|}_{\tau ={\tau }_c}=W\left(\mathit{\Theta },T\right)$

(5)

where ${\tau }_c$ is a given value. Previous studies^15,41 have shown that an accurate numerical solution can be obtained by constructing $\mathbb{W}$ as a trigonometric function:

$\mathbb{W}\left(\tau \right)=\psi \left[W\left(\mathit{\Theta },T\right),\tau \right]=\mathit{sin}\left(2.5\pi \tau \right)·W\left(\mathit{\Theta },T\right)$

(6)

Obviously, the requirements in equation (5) can be satisfied when ${\tau }_c=1$ .

(4) Solve the generalized probability density evolution equation (GPDEE) of the virtual stochastic process

$\frac{\partial p_{\mathbb{W}\mathit{\Theta }}\left(w,\mathit{\theta },\tau \right)}{\partial \tau }+\stackrel{.}{\psi }\left(\mathit{\theta },\tau \right)\frac{\partial p_{\mathbb{W}\mathit{\Theta }}\left(w,\mathit{\theta },\tau \right)}{\partial w}=0$

(7)

with the initial condition

${p_{\mathbb{W}\mathit{\Theta }}\left(w,\mathit{\theta },\tau \right)|}_{\tau =0}=\delta \left(w\right)p_{\mathit{\Theta }}\left(\mathit{\theta }\right)$

(8)

where $\delta (·)$ is the Dirac delta function and $p_{\mathit{\Theta }}$ is the joint PDF of $\mathit{\Theta }$ . In general, a numerical method is needed to solve equations (7) and (8) such as the finite-difference method with a total variation diminishing (TVD) scheme. ⁴²

• (5) Calculate the time-dependent PDF of $\mathbb{W}\left(\tau \right)$ by summing all the discretised joint PDFs as follows:

$p_{\mathbb{W}}\left(w,\tau \right)={\int }_{{\Omega }_{\mathbf{\Theta }}}{p}_{\mathbb{W}\mathit{\Theta }}\left(w,\mathit{\theta },\tau \right)\mathrm{d}\mathit{\theta }$

(9)

where ${\Omega }_{\mathit{\Theta }}$ denotes the distribution domain of $\mathit{\Theta }$ . The PDF for $W$ is obtained as

$p_W\left(w\right)={p_{\mathbb{W}}\left(w,\tau \right)|}_{\tau ={\tau }_c}$

(10)

• (6) Calculate the dynamic reliability of the maglev vehicle-bridge system by integrating $p_W$

$R\left(T\right)=\mathrm{Pr}\left\{W\left(\mathit{\Theta },T\right)\le z_B\right\}={\int }_{-\infty }^{z_B}{p}_W\left(w\right)\mathrm{d}w$

(11)

where $z_B$ is the failure threshold of the index $Z$ , which does not change with time.

Use of ASM to accelerate the dynamic reliability calculation

Outline

Based on the discussion in Dynamic reliability analysis of a vehicle-bridge system based on PDEM, we can see that the prerequisite for calculating the dynamic reliability of the maglev vehicle-bridge system is to obtain the EEVs corresponding to the representative sample point sets $\mathit{\theta }$ . This process is time-consuming and may account for 90% of the total reliability analysis time. Hence, if the EEVs of the maglev vehicle-bridge system can be obtained quickly using an ASM, the overall calculation process can be accelerated. The specific steps are as follows: first, an ASM is constructed based on a limited number of sample points to approximate the mapping relationship between the EEVs of the physical quantity of interest and the basic variables $\Theta$ in the stochastic system. Next, the EEVs ( ${\mathit{W}}_{\text{enl}}$ ) of the encrypted representative point sets ( ${\mathit{\theta }}_{\text{enl}}$ ) are predicted using the well-constructed surrogate model. Finally, the EEVs are substituted into the GPDEE of the virtual stochastic process and solved to obtain the dynamic reliability of the system. In this way, the calculation accuracy of the dynamic reliability based on PDEM can be effectively improved without an increase in the number of deterministic analyses.

RBF-based surrogate model

The surrogate model method involves the use of limited training samples to build a mathematical model $\widehat{\mathit{W}}\left({\mathit{\theta }}_{\mathit{s}}\right)$ to replace the theoretical model $\mathit{W}\left({\mathit{\theta }}_{\mathit{s}}\right)$ , as follows:

$\mathit{W}\left({\mathit{\theta }}_{\mathit{s}}\right)=\widehat{\mathit{W}}\left({\mathit{\theta }}_{\mathit{s}},\xi \right)+\mathit{\epsilon }$

(12)

where ${\mathit{\theta }}_{\mathit{s}}$ represent the sample point sets used to construct the surrogate model, $\mathit{\epsilon }$ is the approximate error, and $\xi$ is an undetermined coefficient in the surrogate model.

RBF is a commonly used surrogate model that aims to fit complex functions using linear combinations of simple radial basis functions. It has the advantages of a simple structure, fast training speed, and the capability to approximate any nonlinear function. Therefore, we utilize this approach to construct an ASM for predicting the EEVs of the vehicle-bridge system. For a d-dimensional function $f=\mathit{W}\left({\mathit{\theta }}_{\mathit{s}}\right)$ where $N_S$ sample point sets ${\mathit{\theta }}_{\mathit{s}}={\left\{{\mathit{\theta }}_{\mathit{i}}=\left({\theta }_{1,i},{\theta }_{2,i},···,{\theta }_{d,i}\right)\right\}}_{i=1}^{N_S}$ , and the corresponding EEVs $\mathit{W}=\left[W\left({\mathit{\theta }}_1\right),W\left({\mathit{\theta }}_2\right),···,W\left({\mathit{\theta }}_{N_S}\right)\right]$ are known, an RBF surrogate model $\widehat{\mathit{W}}\left({\mathit{\theta }}_{\mathit{s}}\right)$ can be established as follows:

$\widehat{\mathit{W}}\left({\mathit{\theta }}_{\mathit{s}}\right)=\sum _{i=1}^{N_S}{\lambda }_i\varphi \left(\Vert {\mathit{\theta }}_{\mathit{s}}-{\mathit{\theta }}_{\mathit{i}}\Vert \right)+P\left({\mathit{\theta }}_{\mathit{s}}\right)$

(13)

where

$\mathit{P}\left({\mathit{\theta }}_{\mathit{s}}\right)=\mathit{p}·\mathit{c}=\left(1,{\mathit{\theta }}_1,{\mathit{\theta }}_2,···,{\mathit{\theta }}_d\right){\left(c_1,c_2,···,c_d,c_l\right)}^T$

(14)

and $\mathit{\lambda }={\left({\lambda }_1,{\lambda }_2,···,{\lambda }_{N_S}\right)}^T$ represents the vector of weight coefficients to be solved, and $\Vert {\mathit{\theta }}_{\mathit{s}}-{\mathit{\theta }}_{\mathit{i}}\Vert$ represents the Euclidean distance between prediction point ${\mathit{\theta }}_{\mathit{s}}$ and the $i$ th sample point ${\mathit{\theta }}_{\mathit{i}}$ . $\varphi (·)$ is the basis function, the most commonly used forms of which are cubic, multi-quadratic, inverse multi-quadratic, and Gaussian. Following the work in Ref. 28, we adopt a cubic basis function of the form $\varphi \left(r\right)=r^3$ . $\mathit{p}=\left(1,{\mathit{\theta }}_1,{\mathit{\theta }}_2,···,{\mathit{\theta }}_d\right)$ is a $d+1$ dimensional vector, where ${\mathit{\theta }}_{\mathit{i}}$ represents the $i$ th random variable for the prediction point ${\mathit{\theta }}_{\mathit{s}}$ . $\mathit{c}={\left(c_1,c_2,···,c_d,c_l\right)}^T$ represents the coefficient vector to be evaluated. The coefficient vectors $\mathit{\lambda }$ and $\mathit{c}$ are determined by solving the following linear equations:

$\left[\begin{array}{l}\mathit{\lambda }\\ \mathit{c}\end{array}\right]={\left[\begin{array}{l}\mathit{\Phi }\hspace{0.17em}\mathit{P}\\ {\mathit{P}}^{\mathit{T}}\hspace{0.17em}\mathbf{0}\end{array}\right]}^{-\mathbf{1}}\left[\begin{array}{l}\mathit{W}\\ \hspace{0.17em}\mathbf{0}\end{array}\right]$

(15)

where ${\mathit{\Phi }}_{ij}=\varphi \left(\Vert {\mathit{\theta }}_i-{\mathit{\theta }}_j\Vert \right),i,j=1,2,···,N_S$ ; $P_{ij}=p_j\left({\mathit{\theta }}_i\right),i=1,2,···,N_S$ and $j=1,2,···,d+1$ . The process of constructing an RBF-based surrogate model is illustrated in Figure 5.

Figure 5.

Construction process for an RBF-based surrogate model.

Adaptive sampling technique

As mentioned above, this study applies an adaptive sampling technique to facilitate the efficient construction of a surrogate model. This technique comprises three main steps: generating an initial sample point set, constructing learning functions, and defining a set of termination conditions. The initial sample point set ${\mathit{\theta }}_{\text{init}}$ is used to build a prototype of the target response surface and to provide initial conditions for the implementation of adaptive sampling. The learning function is applied to actively identify the target sample points from the candidate sample point set ${\mathit{\theta }}_{\text{cand}}$ . The termination condition ensures the accuracy of the surrogate model.

In this study, we adopt a GF-discrepancy based method ⁴³ to generate the sample point sets ${\mathit{\theta }}_{\text{init}}$ and ${\mathit{\theta }}_{\text{cand}}$ . This involves generating the Sobol point sets $\mathit{u}={\left\{{\mathit{u}}_q=\left(u_{q,1},u_{q,2},···,u_{q,d}\right)\right\}}_{q=1}^{N_p}$ in a $d$ -dimensional unit hypercube and then transforming them into point sets ${\mathit{\theta }}^{″}={\left\{{\mathit{\theta }}_q^{″}=\left({\theta }_{q,1}^{″},{\theta }_{q,2}^{″},···,{\theta }_{q,d}^{″}\right)\right\}}_{q=1}^{N_p}$ , where

${\theta }_{q,i}^{″}=F_i^{-1}\left(u_{q,i}\right),i=1,2,···,d$

(16)

here, $F_i^{-1}(·)$ is the inverse function of the marginal cumulative distribution function (CDF) of the $i$ th random variable. Next, the following transformation is applied to the point sets ${\mathit{\theta }}^{″}$ to ensure that the assignment probabilities $P_q$ of the $d$ points are close to each other:

${\theta }_{m,i}^{\prime }=F_i^{-1}\left[{\displaystyle \sum }_{q=1}^{N_p}\frac{1}{N_p}·I\left({\theta }_{q,i}^{″}<{\theta }_{m,i}^{″}\right)+\frac{1}{2}·\frac{1}{N_p}\right],i=1,2,···,d$

(17)

where $I(·)$ is the indicator function.

When the assignment probabilities $P_q={\int }_{{\Omega }_q}{p}_{\Theta }\left(\mathit{\theta }\right)\mathrm{d}\mathit{\theta }$ have been obtained, the point sets $\mathit{\theta }\prime ={\left\{{\mathit{\theta }\mathbf{\prime }}_q=\left({\theta \prime }_{q,1},{\theta \prime }_{q,2},···,{\theta \prime }_{q,d}\right)\right\}}_{q=1}^{N_p}$ are transformed using the following equation to reduce the GF-discrepancy:

${\theta }_{m,i}=F_i^{-1}\left[{\displaystyle \sum }_{q=1}^{N_p}{P}_q·I\left({\theta }_{q,i}^{\prime }<{\theta }_{m,i}^{\prime }\right)+\frac{1}{2}·p_m\right],i=1,2,···,d$

(18)

Finally, $\mathit{\theta }={\left\{{\mathit{\theta }}_q=\left({\theta }_{q,1},{\theta }_{q,2},···,{\theta }_{q,n}\right)\right\}}_{q=1}^{N_p}$ are used as the target point sets for subsequent analysis.

Of the three steps involved in the adaptive sampling technique, the construction of learning functions is the most critical. A robust learning function should possess both excellent global exploration and local exploitation capabilities. Global exploration involves discovering areas with sparse sample points in probability space, while local exploitation involves finding areas with strong nonlinearity in the response function and encrypting sample points in these regions. In this study, the learning functions proposed by Mo et al. ⁴⁴ were adopted to select new sample points.

Global exploration was achieved by measuring the Euclidean distance between two points, that is,

$D_{\min }\left(\mathit{\theta }\right)=\min \left\{{\Vert \mathit{\theta }-{\mathit{\theta }}_t\Vert }_2;\mathit{\theta }\in {\mathit{\theta }}_{\text{cand}},{\mathit{\theta }}_t\in {\mathit{\theta }}_{\text{cur}}\right\}$

(19)

The aim of this step is to find the point with the smallest Euclidean distance between the candidate sample point sets ${\mathit{\theta }}_{\text{cand}}$ and the current sample point sets ${\mathit{\theta }}_{\text{cur}}$ .

To carry out local exploitation, the remainder of the first-order Taylor expansion of the function $f\left(\mathit{\theta }\right)$ was used to measure its nonlinearity at $\mathit{\theta }$ , and the expansion point was the sample point ${\mathit{\theta }}_{t,nr}$ closest to $\mathit{\theta }$ in the current sample point set ${\mathit{\theta }}_{\text{cur}}$ . In order to improve the calculation efficiency, the first derivative of the surrogate model $f\left(\mathit{\theta }\right)$ was used to approximate the first derivative of $W\left(\mathit{\theta }\right)$ . The expression is as follows:

$t\left(\mathit{\theta };{\mathit{\theta }}_{t,nr}\right)=f\left({\mathit{\theta }}_{t,nr}\right)+\stackrel{.}{f}{\left({\mathit{\theta }}_{t,nr}\right)}^T\left(\mathit{\theta }-{\mathit{\theta }}_{t,nr}\right)$

(20)

Thus, the absolute values of the remaining terms can be expressed as

$R_{\text{rem}}\left(\mathit{\theta }\right)=|f\left(\mathit{\theta }\right)-t\left(\mathit{\theta },{\mathit{\theta }}_{t,nr}\right)|$

(21)

After providing the calculation formulae for global exploration and local exploitation, the weights of both need to be considered in the search for new sample points. We therefore define the following criterion function. ⁴⁴

$J\left(\mathit{\theta }\right)=\frac{D_{\min }\left(\mathit{\theta }\right)}{\underset{\mathit{\theta }\in {\mathit{\theta }}_{\text{cand}}}{\max }{D}_{\min }\left(\mathit{\theta }\right)}+\left(1-\frac{D_{\min }\left(\mathit{\theta }\right)}{L_{\max }}\right)\frac{R_{\text{rem}}\left(\mathit{\theta }\right)}{\underset{\mathit{\theta }\in {\mathit{\theta }}_{\text{cand}}}{\max }{R}_{\text{rem}}\left(\mathit{\theta }\right)}$

(22)

and the new sample point is the index value corresponding to the maximum value of $J\left(\mathit{\theta }\right)$

${\mathit{\theta }}_{\text{new}}=\underset{\mathit{\theta }\in {\mathit{\theta }}_{\text{cand}}}{\mathit{argmax}}\left(J\left(\mathit{\theta }\right)\right)$

(23)

where $L_{\mathit{max}}$ is the maximum Euclidean distance between any two points in ${\mathit{\theta }}_{\text{cand}}$ .

To ensure the construction accuracy of the ASM, we set conditions for terminating the iterations based on the absolute error (AE):

$AE=|W\left({\mathit{\theta }}_{\text{new}}\right)-\widehat{W}\left({\mathit{\theta }}_{\text{new}}\right)|$

(24)

In view of the oscillations in $AE$ , we require that the process be terminated when $AE$ has been less than the error threshold ${\epsilon }_{\text{thr}}$ for four consecutive iterations. At the same time, in order to enable a more intuitive comparison of the training efficiency of the ASM and OSM, we introduce another evaluation indicator, the root mean square error (RMSE):

$RMSE=\sqrt{\frac{1}{m}·\sum _{i=1}^m{\left[W\left({\mathit{\theta }}_i\right)-\widehat{W}\left({\mathit{\theta }}_i\right)\right]}^2}$

(25)

where ${\mathit{\theta }}_i$ is the ith sample in the validation set, and m denotes the total size of the validation set.

Implementation process of ASM-PDEM with random controller parameters

Based on the above theoretical derivation, a corresponding program was developed using the MATLAB platform. Figure 6 shows a flowchart for the use of ASM-PDEM in a dynamic reliability analysis of a maglev vehicle-bridge system with random controller parameters.

Figure 6.

Flowchart for the use of ASM-PDEM for a dynamic reliability analysis of a maglev vehicle-bridge system with random controller parameters.

Validations

Theoretical model validation

In this section, we intend to verify the correctness of the theoretical model of the maglev vehicle-bridge coupled system by comparing it with the field measured results of Shanghai high-speed maglev line.^45,46 The vehicle is a five-car formation with a speed of 430 km/h, and the model parameters are given in Ref. 6 The bridge is a five-span simply supported beam bridge with a span of 24.768 m and a pier height of 10 m. The main parameters of the bridge model are listed in Table A1. In this section, we do not consider the randomness of the controller parameters. The suspension system controller parameters are set to $K_{P1}=3500$ and $K_{D1}=5$ , and the guidance system controller parameters are set to $K_{P2}=2600$ and $K_{D2}=50$ . The track irregularity spectrum is represented by the seven-parameter fitting formula proposed in Ref. 47, with wave length ranging from 2 to 50 m, and 4096 sampling points. Figure 7 shows the vertical and lateral track irregularity samples used in this verification, with their maximum values being 1.60 and 1.71 mm, respectively. Figure 8(a) shows the measured and simulated vertical displacement curves at the midpoint of the bridge at a vehicle speed of 430 km/h. It can be seen that the simulation results match the measured results well at the points where the train enters, crosses, and exits the bridge. The maximum values of the simulated and measured results are 1.58 mm and 1.57 mm, respectively, with a relative error of less than 1%. Figure 8(b) shows the variation in the measured and simulated dynamic coefficients of vertical displacement at the midpoint of the bridge in the speed range of 50–350 km/h. We can see that although the two sets of results show consistent behaviors when the vehicle speed changes, the values do not match. This is because the dynamic coefficient is strongly affected by track irregularities and other factors, and it is difficult to set the physical parameters of the vehicle-bridge model to match the actual model with high accuracy.

Figure 7.

Randomly generated track irregularity samples: (a) vertical and (b) lateral.

Figure 8.

Comparison between calculated and measured results for the midpoint of the bridge girder: (a) vertical deflection for v = 430 km/h and (b) dynamic coefficients for v = 50–350 km/h.

From a comparative analysis of Figure 8, we can see that the vehicle-bridge theoretical model established in Dynamic model and equation of motion for the EMS high-speed maglev vehicle-bridge coupled system simulates the dynamic characteristics of the actual model well. In the following analysis, the theoretical values of the dynamic response of the vehicle and guideway girder required for training of the surrogate model will be obtained from this model.

ASM-PDEM validation

In this section, the accuracy and efficiency of the proposed ASM-PDEM are validated by comparison with the MCM and OSM-PDEM. The distinction between OSM-PDEM and ASM-PDEM lies in their use of OSM and ASM, respectively, to predict the EEVs of the dynamic response. We assume that the controller parameters $K_{P1}$ , $K_{D1}$ , $K_{P2}$ , and $K_{D2}$ are normally distributed random variables, with mean values of 3500, 5, 2600, and 50, respectively, and their coefficients of variation are set to 0.08. The samples used for MCM are randomly generated, and the sample size is 10,000, whereas the samples used for ASM-PDEM and OSM-PDEM are generated using the GF-discrepancy-based method. In ASM-PDEM, the sizes of the initial, candidate, and validation sample sets are $5d$ , 100,000, and 10,000, respectively, where d represents the number of random variables. The vehicle operates at a speed of 430 km/h, and the other physical parameters of the vehicle-bridge system are consistent with those in Theoretical model validation.

Taking the vertical displacement and acceleration of the midpoint of the bridge girder as an example, Figure 9 shows the variation in the training accuracy evaluation indicators of OSM and ASM with the sample size. It is observed from the figure that both RMSE _OSM and RMSE _ASM gradually decrease with an increase in the sample size, but that ASM has a higher training efficiency than OSM, as the values decrease faster. For example, when the number of training samples reaches 273, the RMSE _ASM for the vertical displacement of the midpoint of the bridge girder is less than 1 × 10⁻⁶, but the RMSE _OSM corresponding to this sample size is 1.34 × 10⁻⁵, one order of magnitude larger than the former. Although AE also tends to decrease with an increase in sample size, it differs from RMSE in that it exhibits noticeable oscillation. Consequently, we set the termination condition that iterations should cease when AE remains below a certain threshold ${\epsilon }_{\text{thr}}$ for four consecutive iterations. Figure 10 shows a comparison of 10,000 EEVs for the vertical displacement and acceleration of the midpoint of the bridge girder predicted by OSM and ASM, respectively. The number of samples used in OSM to train the surrogate model is the same as the number of samples when the iterations of ASM are terminated, with 363 samples for displacement training and 132 samples for acceleration training. Reference lines are also plotted in Figure 10, where the closer the predicted results are to the reference line, the higher the prediction accuracy. It is evident from Figure 10 that the results predicted by ASM are predominantly concentrated near the reference line, while many samples predicted by OSM exhibit significant deviations from the theoretical values. This substantiates our finding that the prediction accuracy of ASM is significantly superior to that of OSM.

Figure 9.

Variation in training accuracy evaluation indicators with sample size: (a) vertical displacement and (b) vertical acceleration of the midpoint of the bridge girder.

Figure 10.

Comparison of EEVs predicted by OSM and ASM: (a) vertical displacement and (b) vertical acceleration of the midpoint of the bridge girder.

In order to illustrate the accuracy and efficiency of ASM-PDEM in the calculation of dynamic reliability, Figure 11 shows the PDFs of the EEV for the vertical displacement and acceleration at the midpoint of the bridge girder, calculated using MCM, OSM-PDEM, and ASM-PDEM. In Figure 11, “MCM₁₀₀₀₀” indicates the results obtained from the vehicle-bridge theoretical model using 10,000 randomly generated controller parameter samples, and “OSM-PDEM₁₀₀₀” and “ASM-PDEM₁₀₀₀” indicate the results obtained by OSM-PDEM and ASM-PDEM, respectively, using 1000 predicted EEVs, where the same number of training samples are used in both OSM and ASM to establish surrogate models. It can be observed from Figure 11 that the results obtained via ASM-PDEM₁₀₀₀ exhibit a higher degree of conformity with those from MCM₁₀₀₀₀ compared to those from OSM-PDEM₁₀₀₀. Taking the vertical displacement of the midpoint of the bridge girder as an example, we see that the maximum PDF values obtained from MCM₁₀₀₀₀, OSM-PDEM₁₀₀₀, and ASM-PDEM₁₀₀₀ are 6.37×10⁴, 6.01×10⁴, and 6.44×10⁴ 1/m, respectively. The relative errors for OSM-PDEM₁₀₀₀ and ASM-PDEM₁₀₀₀ are 5.65% and 1.10%, respectively. Figure 12 shows the POE curves obtained using the three methods mentioned above, and it can be seen that within the distribution range of the EEV values, the POE curves calculated by ASM-PDEM₁₀₀₀ are in good agreement with those from MCM₁₀₀₀₀. However, as the response increases, the difference between the POE curves calculated with OSM-PDEM₁₀₀₀ and MCM₁₀₀₀₀ gradually increases. Given that the failure of a vehicle-bridge system is a low-probability event, the computational precision of ASM-PDEM is significantly superior to that of OSM-PDEM for an equivalent computational effort. It is also worth noting that compared to MCM, ASM-PDEM yields an improvement in computational efficiency of an order of magnitude. For example, to obtain the vertical displacement of the midpoint of the bridge girder, ASM-PDEM only needs to solve the vehicle-bridge theoretical model 363 times, and obtaining the POE results takes a total of 5.5 h, while MCM solves the theoretical model 10,000 times, which takes a total of 124 h. The computer used in this study was an Intel(R) Core(TM) i9-9900K CPU 3.60 Hz with 32 GB RAM.

Figure 11.

Comparison of PDFs calculated with MCM, OSM-PDEM, and ASM-PDEM: (a) vertical displacement and (b) vertical acceleration of the midpoint of the bridge girder.

Figure 12.

Comparison of POEs calculated with MCM, OSM-PDEM, and ASM-PDEM: (a) vertical displacement and (b) vertical acceleration of the midpoint of the bridge girder.

To provide a more comprehensive illustration of the accuracy and efficiency of ASM-PDEM in obtaining the POEs for various dynamic responses of the maglev vehicle-bridge system, the limiting values of the dynamic responses, as determined by the three methods for different failure probabilities, are listed in Table 1. We observe that the surrogate model used to predict the vertical acceleration of the car body was GPR, rather than RBF, due to its superior predictive performance for this response. From Table 1, it can be seen that the dynamic response limits obtained by ASM-PDEM₁₀₀₀ for different failure probabilities are in good agreement with MCM₁₀₀₀₀, whereas OSM-PDEM₁₀₀₀ shows strong agreement only when the failure probability is large, and low agreement when the failure probability is small. If we take the lateral acceleration of the car body as an example, we see that the limits corresponding to a failure probability of 1×10⁻² are 0.8663, 0.8671, and 0.8669 for MCM₁₀₀₀₀, OSM-PDEM₁₀₀₀, and ASM-PDEM₁₀₀₀, respectively, which are relatively close to each other. However, when the failure probability is reduced to 1×10⁻⁴, the limits obtained by the three methods are 0.9027, 0.8969, and 0.9056, respectively. Obviously, the results obtained by ASM-PDEM₁₀₀₀ are closer to those of MCM₁₀₀₀₀. The patterns of behavior shown in Table 1 and Figure 12 are similar.

Table 1.

Limiting values of the dynamic responses determined by MCM, OSM-PDEM, and ASM-PDEM for different failure probabilities.

Item		Midpoint of bridge girder				Car body
		Displacement (mm)		Acceleration (m/s2)		Acceleration (m/s2)
		Vertical	Lateral	Vertical	Lateral	Vertical	Lateral
${\epsilon }_{\text{thr}}$		0.001	0.001	0.002	0.001	0.0001	0.002
Number of iterations		343	125	112	182	26	135
1 × 10−2	MCS10000	1.6635	0.2659	0.3358	0.2640	0.5275	0.8663
	OSM-PDEM1000	1.6660	0.2659	0.3388	0.2641	0.5276	0.8671
	ASM-PDEM1000	1.6633	0.2663	0.3344	0.2641	0.5275	0.8669
1 × 10−3	MCS10000	1.6650	0.2773	0.3421	0.2830	0.5280	0.8865
	OSM-PDEM1000	1.6722	0.2753	0.3512	0.2796	0.5278	0.8844
	ASM-PDEM1000	1.6650	0.2778	0.3404	0.2835	0.5280	0.8879
1 × 10−4	MCS10000	1.6664	0.2881	0.3470	0.3040	0.5285	0.9027
	OSM-PDEM1000	1.6778	0.2828	0.3591	0.2929	0.5279	0.8969
	ASM-PDEM1000	1.6672	0.2894	0.3448	0.3059	0.5284	0.9056

The comparison above proves that the ASM-PDEM approach proposed in this study has higher accuracy and efficiency than MCM and OSM-PDEM in terms of obtaining reliability-related results for random controller parameters. In the research described below, we therefore use ASM-PDEM for the calculations.

Numerical examples

Effects of randomness in controller parameters

The aim of this section is to investigate the effects of the randomness in each controller parameter on the dynamic reliability of the maglev vehicle-bridge system by comparing the calculation results when a single controller parameter is considered as a random variable versus when all controller parameters are considered as random variables. The vehicle-bridge system model used in this section features a five-car vehicle model and a simply supported beam bridge with 20 spans. The parameters of these models are consistent with the values provided in ASM-PDEM validation. The PDF curves for the EEV of the dynamic response under different random controller parameters are shown in Figure 13, where the legends “ $K_{P1}$ ,” “ $K_{D1}$ ,” “ $K_{P2}$ ,” and “ $K_{D2}$ ” indicate that the corresponding controller parameters are random variables, and “All” means that all controller parameters are random variables. From Figure 13(a), (c), and (e), we can see that the PDF curves corresponding to “ $K_{P1}$ ” and “All” show a high degree of agreement, meaning that the vertical responses of the system, such as the vertical displacement and acceleration of the midpoint of the bridge girder and the vertical acceleration of the car body, are mainly affected by $K_{P1}$ , while the other controller parameters have little influence. This is because the value of $K_{P1}$ significantly affects the vertical stiffness of the vehicle-bridge system, thus affecting the vertical dynamic response of the system. Figure 13(b), (d), and (f) reveal that the lateral dynamic responses of the system, such as the lateral displacement and acceleration of the midpoint of the bridge girder and the lateral acceleration of the car body, are affected by multiple control parameters; from the largest degree of influence to the smallest, the order is $K_{P2}$ , $K_{D2}$ , and $K_{D1}$ . We note that $K_{P1}$ has little effect on the lateral dynamic response. The reason for this is that the lateral constraint on the vehicle-bridge system is weak, so a change in the control parameters has a more obvious influence on the lateral dynamic response. It is also important to note that the distributions of the PDF curves marked “All” show that the PDF distribution ranges of the lateral displacement, the vertical and lateral acceleration of the midpoint of the bridge girder and the lateral acceleration of the car body are relatively wide. This implies that unsuitable values of the controller parameters can significantly amplify these responses. In contrast, under the conditions considered in this section, the controller parameters have a negligible effect on the vertical displacement at the midpoint of the bridge girder or the vertical acceleration of the car body.

Figure 13.

PDFs of the EEV for the dynamic responses of the maglev vehicle-bridge system: (a) vertical and (b) lateral displacements of the midpoint of the bridge girder; (c) vertical and (d) lateral accelerations of the midpoint of the bridge girder; (e) vertical and (f) lateral accelerations of the car body (Legend: ).

From Figure 13, it can also be seen that if we assume that the controller parameters are normally distributed random variables, the PDF curve for the EEV of the lateral vibration response approximates a normal distribution, while that for the vertical vibration response differs significantly from a normal distribution. This indicates that the relationship between the lateral vibration response and the controller parameters is approximately linear, whereas the relationship between the vertical vibration response and the controller parameters is nonlinear.

Figure 14 shows the dynamic response limits of the maglev vehicle-bridge system for different values of the failure probability. In terms of the vertical displacement and acceleration of the midpoint of the bridge girder, it can be seen that the dynamic response limits obtained by considering only $K_{P1}$ as a random variable and all controller parameters as random variables are similar for varying values of the failure probability. This indicates that the dynamic reliability of the vertical displacement and acceleration of the bridge girder is mainly affected by $K_{P1}$ . From Figure 14(e), we see that the dynamic response limits obtained by considering a controller parameter as a random variable and all parameters as random variables are almost the same for different values of the failure probability, indicating that the dynamic reliability of the vertical acceleration of the car body is little affected by the controller parameters. As can be seen from Figure 14(b), (d), and (f), the dynamic reliability of the lateral response of the maglev vehicle-bridge system is not affected equally by the different controller parameters for varying failure probabilities. Taking the lateral displacement of the midpoint of the bridge girder as an example, we see that when the failure probability is 10⁻², the order of the four controller parameters in terms of the degree of influence is $K_{P2}$ , $K_{D2}$ , $K_{P1}$ , and $K_{D1}$ . However, when the failure probability decreases to 10⁻⁴, the order becomes $K_{D2}$ , $K_{P2}$ , $K_{P1}$ , and $K_{D1}$ . This variation is attributed to the fact that the dynamic response limits corresponding to different failure probabilities are intimately connected to the shape of the PDF curve.

Figure 14.

Dynamic response limits of the maglev vehicle-bridge system corresponding to different failure probabilities: (a) vertical and (b) lateral displacements of the midpoint of the bridge girder; (c) vertical and (d) lateral accelerations of the midpoint of the bridge girder; (e) vertical and (f) lateral accelerations of the car body (Legend: ).

Effects of the coefficient of variation for controller parameters

In this section, we consider the four controller parameters as random variables that follow a normal distribution and investigate the influence of the coefficient of variation of these variables on the dynamic reliability of the maglev vehicle-bridge system. It is assumed that the coefficient of variation of the controller parameters increases from 0.04 to 0.14, with an interval of 0.02, while the remaining calculation parameters of the vehicle-bridge system are the same as those in ASM-PDEM validation. Figure 15 shows the PDF curves for the EEV of the dynamic response for a varying coefficient of variation of the controller parameters. From Figure 15, it can be seen that as the coefficient of variation increases, the distribution range of the PDF curve becomes wider, indicating that the discreteness of the dynamic response of the maglev vehicle-bridge system increases with that of the control parameters. Additionally, Figure 15 illustrates that when the coefficient of variation is small, the PDF curve approximates a normal distribution. However, as the coefficient of variation increases, the shape of the PDF deviates increasingly from the normal distribution. These findings indicate that the nonlinear relationship between the system’s dynamic response and the controller parameters becomes more pronounced as the variability of the controller parameters grows. Furthermore, it is important to note that when the coefficient of variation is low, the PDF distribution for the EEV of the car body’s vertical acceleration is very concentrated. As the coefficient of variation increases to a certain level, sporadic distributions emerge far from the mean value, indicating a significant increase in the discreteness of the EEV. These results underscore the importance of being cautious when selecting controller parameters randomly, as doing so without theoretical or empirical backing may result in abnormal dynamic responses of the maglev vehicle-bridge system.

Figure 15.

PDFs for the EEV of the dynamic responses of the maglev vehicle-bridge system for varying values of the coefficient of variation of the controller parameters: (a) vertical and (b) lateral displacements of the midpoint of the bridge girder; (c) vertical and (d) lateral accelerations of the midpoint of the bridge girder; (e) vertical and (f) lateral accelerations of the car body (Legend: ).

Figure 16 Illustrates the variation in the dynamic response limits of the maglev vehicle-bridge system with respect to the coefficient of variation, for various failure probability values. In most instances, the dynamic response limits associated with the same failure probability increase as the coefficient of variation of the controller parameters increases. This trend is attributed to the fact that an increase in the coefficient of variation leads to greater variability in the system’s dynamic response. It is also important to observe that some curves exhibit a decrease with an increase in the coefficient of variation, as seen in the limits of the vertical acceleration at the midpoint of the bridge girder under different failure probabilities. This phenomenon occurs because the tail of the PDF curve (the higher side) does not lengthen in proportion to the increase in the coefficient of variation. Additionally, Figure 16 reveals that in most cases, as the coefficient of variation increases, the disparity between the dynamic response limits for the three considered failure probabilities also gradually widens. This observation serves as an indirect indicator that the variability of the dynamic response escalates with the coefficient of variation. It underscores the importance of judicious selection of controller parameter values for the suspension system, as such selections can significantly mitigate the vibration levels within the maglev vehicle-bridge system.

Figure 16.

Variation in the dynamic response limits of the maglev vehicle-bridge system with the coefficient of variation obtained for different values of failure probability: (a) vertical and (b) lateral displacements of the midpoint of the bridge girder; (c) vertical and (d) lateral accelerations of the midpoint of the bridge girder; (e) vertical and (f) lateral accelerations of the car body (Legend: ).

Conclusions

(1) A comparison with on-site measurement results shows that the coupled vibration model of the high-speed maglev vehicle and bridge, established in this study based on multi-body dynamics and the finite element method, can effectively simulate the dynamic performance of real systems.

(2) The advantage of the adaptive surrogate model is that it can reasonably determine the positions of sample points based on the fluctuations in the response surface; that is, areas with strong nonlinearity have a denser distribution of sample points. The training efficiency and accuracy of the surrogate model can be significantly improved in this way. From a comparison with MCM and OSM-PDEM, we confirm that the proposed ASM-PDEM can significantly improve the accuracy of dynamic reliability calculation with the same number of training samples.

(3) The discreteness of the dynamic response of the maglev vehicle-bridge system increases with that of the control parameters. The vertical response of the system is mainly affected by $K_{P1}$ , while the other controller parameters have little influence. The lateral dynamic response of the system is affected by multiple control parameters, which in order from the largest to the smallest are $K_{P2}$ , $K_{D2}$ , and $K_{D1}$ . We find that $K_{P1}$ has little effect on the lateral dynamic response.

(4) As the coefficient of variation of the controller parameter increases, the nonlinear relationship between the system’s dynamic response and the control parameters becomes more apparent. A reasonable selection of values for the suspension system control parameters can effectively reduce the vibration of the maglev-bridge system.