Hierarchical model updating for high-speed maglev vehicle/guideway coupled system based on multi-objective optimization

Abstract

The high-speed maglev vehicle/guideway coupled model is an essential simulation tool for investigating vehicle dynamics and mitigating coupled vibration. To improve its accuracy efficiently, this study investigated a hierarchical model updating method integrated with field measurements. First, a high-speed maglev vehicle/guideway coupled model, taking into account the real effect of guideway material properties and elastic restraint of bearings, was developed by integrating the finite element method, multi-body dynamics, and electromagnetic levitation control. Subsequently, simultaneous in-site measurements of the vehicle/guideway were conducted on a high-speed maglev test line to analyze the system response and structural modal parameters. During the hierarchical updating, an Elman neural network with the optimal Latin hypercube sampling method was used to substitute the FE guideway model, thus improving the computational efficiency. The multi-objective particle swarm optimization algorithm with the gray relational projection method was applied to hierarchically update the parameters of the guideway layer and magnetic force layer based on the measured modal parameters and the electromagnet vibration, respectively. Finally, the updated coupled model was compared with the field measurements, and the results demonstrated the model’s accuracy in simulating the actual dynamic response, validating the effectiveness of the updating method.

Graphical abstract

Keywords

high-speed maglev / vehicle/guideway coupled model / field measurement / model updating / neural network / multi-objective optimization

1 Introduction

The high-speed maglev transportation system overcomes the contact resistance issues of the traditional wheel-rail transportation system, through electromagnetic levitation. With the advantages of high speed, low energy consumption, and superior stability [1], the technology has become a research hotspot topic in the field of future transportation studies. To study the dynamic behavior and coupled vibration characteristics of high-speed maglev systems, it is necessary to develop a near-realistic vehicle/guideway coupled model that contributes to the analysis and evaluation of the safety and stability of vehicle operation.

In most studies to date, maglev vehicle/guideway models have comprised vehicle mechanisms and guideway structures, which are coupled by magnetic forces. For vehicles comprising car-bodies, levitation frames, and electromagnets, the simulation is generally performed by formulating equilibrium equations through multi-body dynamics. Earlier studies by Wang et al. [2] considered the vehicle as a set of uniformly distributed moving loads and explored the response of the guideway. Subsequently, to simulate a real maglev vehicle system more accurately, most researchers [3–5] treated the car-bodies and levitation frames as rigid bodies with multiple attitudes, while constructing primary and secondary suspensions using spring-damping models. Regarding the guideway, most studies modeled that using the modal decomposition method. During the process of researching the magnetic force, Lengyel and Kocsis [6] simplified it as a spring damping force and optimized the suspension parameters by calculating the system response. Later, this passive simulation was replaced by active control: magnetic forces were simulated by proportional-differential [7], proportional-integral [8], and PID [9,10] control algorithms. Kong et al. [11] proposed a sliding-mode control approach based on the Kalman filter algorithm.

However, there has typically been a discrepancy between the response calculated by the vehicle/guideway coupled model and the field measurement results, mainly because. 1) Some details in the real maglev system were ignored in the numerical modeling. Previous modeling research has mainly focused on strengthening the vehicle or magnetic force model, while the guideway was mostly assumed to be a simply supported beam. This has resulted in the disregard of the guideway’s detailed structure, thereby distorting the boundary condition simulation [12]; 2) Some real parameters were difficult to obtain, and parameter variations caused by construction errors and age-related structural degradation were often overlooked; 3) Numerous studies on high-speed maglev coupled systems relied only on theoretical approaches and lacked measured data to facilitate model updating. Therefore, to ensure the accuracy of the simulation, it is necessary to update the high-speed maglev/guideway coupled model with dynamic measurements.

The typical updating method, with measurements, has involved using either the realistic modal parameters, as a first objective, or the dynamic response of the system as a second objective. Prior to updating the model based on the first objective, test data such as the vibration and strain of the structure needs to be analyzed to identify the modal parameters. To date, common identification methods have mainly included the time-domain identification method [13], the frequency-domain identification method, and the time-frequency-domain identification method based on wavelet transform or the HHT theory [14]. For the second objective, efficient and accurate optimization of the simulation model to match the measured state is an essential issue for model updating research. Shabbir and Omenzetter [15] utilized a combination of genetic and niche algorithms to modify the finite element model of a scaled frame structure and full-scale pedestrian cable-stayed bridge. Tran-Ngoc et al. [16] proposed a model updating method for large railway bridges based on orthogonal diagonalization and improved particle swarm optimization. The results for the main vertical modes from the simulation were in close agreement with the field test results.

Nonetheless, the updating process necessitates invoking the structural model at each iteration to compute the modal information, resulting in a significant increase in the computational cost. To address this issue, surrogate models that can accurately substitute large models and enable fast computation have gained popularity. Common surrogate models have included the response surface methodology [17], radial basis function [18], Kriging model [19], etc. Li et al. [20] used the response surface function as the objective function of the particle swarm optimization algorithm, thereby effectively modifying a four-span truss model. These model updating techniques, which rely solely on frequency measurements, are single-objective optimization methods with limited correction effects. Further, Xia et al. [21] modified a bridge model based on two objective functions: principal frequency and midspan deflection. The results revealed that the mode shape obtained by the double objective function is closer to reality than the mode shape obtained through the single objective function.

In research on model updating with the dynamic response as the objective, Wang et al. [22] achieved the optimal estimation of track parameters by minimizing the error between the measured rail displacement and the modeled rail displacement for the vehicle/rail. By combining the measured rail acceleration and multi-island genetic algorithm, Jiang et al. [23] updated several track parameters in a high-speed vehicle/track model. So far, however, there has been little research on systematic model updating methods for high-speed maglev vehicle/guideway coupled models.

This paper presents a measurement-based method to hierarchically update a high-speed maglev vehicle/guideway coupled model. First, a high-speed maglev vehicle/guideway coupled model, taking into account realistic guideway constraints, is developed. Subsequently, the dynamic response of the system components is obtained through field measurements, and the modal parameters of the guideway are identified by the transfer rate function method with rational fractional orthogonal polynomials. In the guideway layer of model updating, the Elman neural network is trained by the finite element guideway model and the optimal Latin hypercube sampling method. Subsequently, the guideway parameters are identified by combining the modal parameter-based Multi-Objective Particle Swarm Optimization (MOPSO) algorithm and Grey Relational Projection (GRP) analysis. Based on this, the magnetic force layer is updated based on the vibration of the electromagnet, and the electromagnetic parameters are identified by MOPSO and GRP. Finally, the hierarchical model updating method is verified by analyzing the identified modal parameters of the guideway and the measured dynamic response of the vehicle/guideway system.

2 Simulation model of high-speed maglev vehicle/guideway system

2.1 Maglev vehicle system

The high-speed EMS maglev train system, illustrated in Fig.1, is a technologically mature and advanced high-speed transportation mode. The EMS system lifts, propels, and guides the vehicle via forces generated by electromagnets arranged along the vehicle and the functional components of the guideway. The control system precisely regulates the magnitude of these forces to ensure that the vehicles can always maintain a levitation gap of about 10 mm, even when traveling at high speeds and withstanding the disturbance of the guideway irregularity. The electromagnets are mounted on the levitation frames through metal-rubber laminated devices. To further mitigate the impact of the excitation from the lower structure on the car-body, the levitation frames are equipped with secondary suspension systems based on air springs, which serve as integral components for buffering and vibration damping.

2.2 Maglev guideway system

The guideway, which accounts for 60%−80% of the total construction cost, is critical for the safe and stable operation of the system. The focus of this study is on high-speed maglev guideways that have been implemented in Shanghai for practical use. Fig.1(b) shows the guideway model with dimensions. Each guideway consists of a prestressed concrete girder, functional components on both sides, and elastic bearings at the ends. These bearings not only prevent smoothness deterioration caused by the settlement at the end of the guideway but also protect the guideway from damage during an earthquake. The detailed structure of the bearing is illustrated in Fig.1(c). However, most of the studies mentioned above ignored the vertical elastic restraint capacity of the bearing, which directly affects the dynamic characteristics of the guideway and thus reduces the accuracy of the coupled model.

In this study, a new finite element model of the guideway was constructed using finite element software. Because the concrete girders and functional components of the guideways are not prone to plasticity during operation, the material was considered to be elastic. The elastic modulus and mass density of the steel functional components were taken as 210 GPa and 7850 kg/m³, respectively [12]. Additionally, the connections between the concrete girders and the functional components are embedded in the concrete beam on one side and secured by high-strength bolts on the other side. Thus, the connection exhibits high stiffness and could be regarded as rigid, while the elastic bearing could be viewed as a spring.

2.3 Vehicle/guideway coupled model

2.3.1 Vehicle sub-model

The high-speed maglev vehicle/guideway vertical model, shown in Fig.2, dominates the dynamic behavior of the actual vehicle/guideway interaction [24]. In the model, the suspension systems were considered to be spring-damped [25]. Moreover, the continuously distributed magnetic forces between vehicles and guideways are regulated through feedback control of the current [5]. In this study, these forces were simplified to eight concentrated forces. The vehicle system has 18 degrees of freedom, including vertical translation \left\{y_{\mathrm{c}}\right\} and pitch rotation \left\{{\theta }_{\mathrm{c}}\right\} of the car-body, vertical translation \left\{y_{\mathrm{l}i}\right\} and pitch rotation \left\{{\theta }_{\mathrm{l}i}\right\} of the levitation frame (i = 1–4), and vertical translation \left\{y_{\mathrm{m}i}\right\} of the electromagnet (i = 1–8).

Based on multi-body dynamics, the dynamic equilibrium equation of the electromagnet is expressed as follows.

(1)M_{\mathrm{m}}{\ddot{y}}_{\mathrm{m}i}+ c_{\mathrm{p}}{\dot{r}}_i+k_{\mathrm{p}}{r}_i= P_0-F_{\mathrm{m}i},

where M_{\mathrm{m}} is the mass of the electromagnet, {\ddot{y}}_{\mathrm{m}i} is the vertical acceleration of the ith electromagnet. c_{\mathrm{p}} and k_{\mathrm{p}} are the damping and stiffness of the primary suspension, respectively. By transforming the equilibrium state, p_0=(M_{\mathrm{m}}+M_{\mathrm{l}}/2+M_{\mathrm{c}}/8) g. M_{\mathrm{l}} and M_{\mathrm{c}} are the masses of a single levitation frame and a car-body, respectively. r_i represents the relative displacement between the ith electromagnet and its connected levitation frame. F_{\mathrm{m}i} is the magnetic force between the ith electromagnet and the guideway, which can be expressed as

(2)F_{\mathrm{m}i}=K_0{\left(\frac{i_0+k_i\mathrm{\Delta }{h}_i+k_{\mathrm{v}}{\dot{y}}_{\mathrm{m}i}+k_{\mathrm{a}}{\ddot{y}}_{\mathrm{m}i}}{h_i}\right)}^2,

where i_0 is the initial current. K_0 is the magnetic force coupled coefficient. h_i is the levitation gap between the ith electromagnet and the guideway, and \mathrm{\Delta } h_i is the difference between h_i and the stable levitation gap h_0. k_i, k_{\mathrm{v}}, and k_{\mathrm{a}} are the feedback control coefficients for the variation in the levitation gap, vibration speed, and vibration acceleration of the magnet force, respectively.

The dynamic equilibrium equation of the levitation frame can be expressed as (e = 1−4).

(3)\{\begin{array}{rl}{M}_{\mathrm{l}}{\ddot{y}}_{\mathrm{l}e}& +\left(k_{\mathrm{s}}{w}_i+c_{\mathrm{s}}{\dot{w}}_i-k_{\mathrm{p}}{r}_i-c_{\mathrm{p}}{\dot{r}}_i\right)+\\ & \left(k_{\mathrm{s}}{w}_{i+1}+c_{\mathrm{s}}{\dot{w}}_{i+1}-k_{\mathrm{p}}{r}_{i+1}-c_{\mathrm{p}}{\dot{r}}_{i+1}\right)=0,\\ J_{\mathrm{l}}{\ddot{\theta }}_{\mathrm{l}e}& -\left(k_{\mathrm{s}}{w}_i+c_{\mathrm{s}}{\dot{w}}_i-k_{\mathrm{p}}{r}_i-c_{\mathrm{p}}{\dot{r}}_i\right)L_{\mathrm{b}}/2+\\ & \left(k_{\mathrm{s}}{w}_{i+1}+c_{\mathrm{s}}{\dot{w}}_{i+1}-k_{\mathrm{p}}{r}_{i+1}-c_{\mathrm{p}}{\dot{r}}_{i+1}\right)L_{\mathrm{b}}/2=0,\end{array}

where M_{\mathrm{l}} and J_{\mathrm{l}} represent the mass and inertia moment of the levitation frame. c_{\mathrm{s}} and k_{\mathrm{s}} represent the damping and stiffness of the secondary suspension system, respectively. L_{\mathrm{b}} is the length of the levitation frame, and w_i represents the relative displacement between the levitation frame and car-body at the ith air spring. The dynamic equilibrium equation of the maglev car-body can be expressed as follows.

(4)\{\begin{array}{l}{M}_{\mathrm{c}}{\ddot{y}}_{\mathrm{c}}-\sum _{n=1}^8\left(k_{\mathrm{s}}{w}_i+c_{\mathrm{s}}{\dot{w}}_i\right)=0,\\ J_{\mathrm{c}}{\ddot{\theta }}_{\mathrm{c}}+\sum _{i=1}^8\left(k_{\mathrm{s}}{w}_i+c_{\mathrm{s}}{\dot{w}}_i\right)L_i=0,\end{array}

where M_{\mathrm{c}}, J_{\mathrm{c}}, and L_{\mathrm{c}} represent the mass, inertia moment, and length of the car body, respectively.

The overall equation of the vehicle system is obtained by combining the equilibrium equations for each component, as follows.

(5)\left[M_{\mathrm{v}}\right] \left\{{\ddot{u}}_{\mathrm{v}}\right\}+\left[C_{\mathrm{v}}\right]\left\{{\dot{u}}_{\mathrm{v}}\right\} +\left[K_{\mathrm{v}}\right] \left\{u_{\mathrm{v}}\right\} =\left\{P_{\mathrm{v}}\right\} ,

where \left[M_{\mathrm{v}}\right], \left[C_{\mathrm{v}}\right], and \left[K_{\mathrm{v}}\right] are the mass, damping, and stiffness matrices of the maglev vehicle, respectively. \left\{{\ddot{u}}_{\mathrm{v}}\right\}, \left\{{\dot{u}}_{\mathrm{v}}\right\}, and \left\{u_{\mathrm{v}}\right\} are the acceleration, velocity, and displacement vectors of each component of the vehicle, respectively. \left\{P_{\mathrm{v}}\right\} denotes the load vector of the maglev vehicle. The vehicle parameters are listed in Tab.1.

2.3.2 Guideway sub-model

The vibration equation of the guideway model based on the finite element method [26] is as follows.

(6)\left[M_{\mathrm{g}}\right]\left\{{\ddot{y}}_{\mathrm{g}}\right\}+\left[C_{\mathrm{g}}\right]\left\{{\dot{y}}_{\mathrm{g}}\right\}+ \left(\left[K_{\mathrm{g}}\right]+\left[K_{\mathrm{b}}\right]\right)\left\{y_{\mathrm{g}}\right\}=\left\{P_{\mathrm{g}}\right\},

where \left\{{\ddot{y}}_{\mathrm{g}}\right\}, \left\{{\dot{y}}_{\mathrm{g}}\right\}, and \left\{y_{\mathrm{g}}\right\} denote the acceleration, velocity, and displacement vectors of the guideway, respectively. \left[M_{\mathrm{g}}\right], \left[K_{\mathrm{g}}\right], and \left[C_{\mathrm{g}}\right] are the mass, stiffness, and Rayleigh damping matrices of the guideway, respectively.

(7)\left[C_{\mathrm{g}}\right]=\frac{2\left({\xi }_i{\omega }_{\mathrm{g},j}-{\xi }_j{\omega }_{\mathrm{g},i}\right)}{{\omega }_{\mathrm{g},i}^2-{\omega }_{\mathrm{g},j}^2}{\omega }_{\mathrm{g},i}{\omega }_{\mathrm{g},j}\left[M_{\mathrm{g}}\right]+\frac{2\left({\xi }_i{\omega }_{\mathrm{g},j}-{\xi }_j{\omega }_{\mathrm{g},i}\right)}{{\omega }_{\mathrm{g},i}^2-{\omega }_{\mathrm{g},j}^2}\left[K_{\mathrm{g}}\right],

where {\xi }_i and {\xi }_j are any two damping ratios of the guideway. {\omega }_{b,i} and {\omega }_{b,j} are the natural frequencies. \left[K_{\mathrm{g}}\right]\left\{y_{\mathrm{g}}\right\} represents the load reaction vector generated by the elastic bearing (boundary condition) of the guideway. The \left\{P_{\mathrm{g}}\right\} is the total load vector of the guideway.

(8)\left\{P_{\mathrm{g}}\right\} =\sum _{n=1}^8\left[{\mathrm{\Gamma }}_n\right]\left\{N_{\mathrm{c}i}\right\}\left\{F_{\mathrm{m}i}\right\},

where \left[{\mathrm{\Gamma }}_n\right] is the coordinate transmitting matrix that transforms and assembles the magnetic force vector \left\{F_{\mathrm{m}i}\right\} into a global load vector. \left\{N_{\mathrm{c}i}\right\} represents the shape function of the ith magnetic force contact unit.

2.3.3 Guideway irregularity

Guideway irregularity is an essential excitation source for coupled systems. In this model, the inertial reference method [27] was employed to generate measured samples in the spatial domain to describe vertical irregularity.

(9)r=\iint a_{\mathrm{l}}\mathrm{d}{t}^2-h_{\mathrm{l}},

where a_{\mathrm{l}} is the measured vertical acceleration of the electromagnet and h_{\mathrm{l}} is the levitation gap when the vehicle is running.

2.3.4 Numerical solution method

In this study, the vibration equations of the vehicle and guideway system were coupled by magnetic forces, and the dynamic equations were solved using the Newmark-β incremental iterative method. The equivalent stiffness equations [5] of the vehicle-bridge system are expressed as follows.

(10)\left(\left[K_{\mathrm{v}}\right]+a_1\left[M_{\mathrm{v}}\right]+a_2\left[C_{\mathrm{v}}\right]\right){\left\{{\ddot{y}}_{\mathrm{v}}\right\}}^{t+\mathrm{\Delta }t}={\left\{P_{\mathrm{v}}\right\}}^{t+\mathrm{\Delta }t}+{\left\{S_{\mathrm{v}}\right\}}^t,

(11)\left(\left[K_{\mathrm{b}}\right]+a_1\left[M_{\mathrm{b}}\right]+a_2\left[C_{\mathrm{b}}\right]\right){\left\{{\ddot{y}}_{\mathrm{b}}\right\}}^{t+\mathrm{\Delta }t}={\left\{P_{\mathrm{b}}\right\}}^{t+\mathrm{\Delta }t}+{\left\{S_{\mathrm{b}}\right\}}^t,

where {\left\{{\ddot{y}}_{\mathrm{v}}\right\}}^{t+\mathrm{\Delta }t} and {\left\{{\ddot{y}}_{\mathrm{b}}\right\}}^{t+\mathrm{\Delta }t} are the displacement vectors of the vehicle and guideway system, respectively, at time t+\mathrm{\Delta }t.

(12)\begin{array}{rl}{\left\{S_{\mathrm{v}}\right\}}^t=& \left[M_{\mathrm{v}}\right]\left(a_1{\left\{u_{\mathrm{v}}\right\}}^t+a_3{\left\{{\dot{u}}_{\mathrm{v}}\right\}}^t+a_4{\left\{{\ddot{u}}_{\mathrm{v}}\right\}}^t\right)\\ & +\left[C_{\mathrm{v}}\right]\left(a_2{\left\{u_{\mathrm{v}}\right\}}^t+a_5{\left\{{\dot{u}}_{\mathrm{v}}\right\}}^t+a_6{\left\{{\ddot{u}}_{\mathrm{v}}\right\}}^t\right),\end{array}

(13)\begin{array}{rl}{\left\{S_{\mathrm{g}}\right\}}^t=& \left[M_{\mathrm{g}}\right]\left(a_1{\left\{u_{\mathrm{g}}\right\}}^t+a_3{\left\{{\dot{u}}_{\mathrm{g}}\right\}}^t+a_4{\left\{{\ddot{u}}_{\mathrm{g}}\right\}}^t\right)\\ & +\left[C_{\mathrm{g}}\right]\left(a_2{\left\{{\ddot{u}}_{\mathrm{g}}\right\}}^t+a_5{\left\{{\dot{u}}_{\mathrm{g}}\right\}}^t+a_6{\left\{{\ddot{u}}_{\mathrm{g}}\right\}}^t\right),\end{array}

where a_i(i=1 ,2,\dots ,6) is the Newmark integration constant determined by the Newmark constants (α, β) and the time step \mathrm{\Delta }t. The numerical iterative process depicted in Fig.3 enables computation of the primary components’ dynamic response, including displacement, velocity, and acceleration.

3 Field measurements and analysis

3.1 Arrangement of field measurements

To investigate the real response of the high-speed maglev vehicle/guideway system, field dynamic measurements were conducted on a high-speed maglev test line in Shanghai. A single high-speed maglev test vehicle and a linear guide were selected as the measurement objects. The test contents are addressed as shown in Fig.4.

For the guideway, vibration tests were performed under both environmental excitation and real vehicle running conditions. To accurately identify the modal parameters (e.g., mode shapes) of the guideway, vertical acceleration sensors were arranged on both sides of the bottom of the guideway, and lateral acceleration sensors were arranged on a single side of the bottom. Fig.4 shows the sensor arrangement, which includes 15 test points in total. Each acceleration sensor was linked to the acquisition system via a shielded wire, and the acquisition system was connected to a laptop via an Ethernet line. The data was collected and stored in real time through the driver of the industrial computer.

For the vehicle system, the test contents included the vibration of the electromagnet and the real-time variation of the suspension/guide gap at the end position. The above tests were performed by the vibration sensors and laser displacement sensors, respectively. Test signals were also leveraged to synthesize the measured guideway irregularity. Each signal was input into the acquisition system in the differential mode, which prevented external interference and enabled synchronous acquisition. When the high-speed maglev vehicle passed the guideway where the test point was located, the system started recording the test data of each vehicle component and triggered the acquisition of acceleration data at the bottom of the guideway. Acquisition and recording stopped only after the vehicle had passed and the signal had stabilized.

3.2 Modal parameter identification method

Natural modal parameters, which reflect the real dynamic properties of a structure, are often used to update a structural model in practice. However, the environmental excitation that a structure like a guideway undergoes does not take the form of ideal white noise, rendering many traditional modal recognition methods ineffective. To deal with that, this section utilizes the operational modal analysis approach with the transfer function to determine the modal parameters. The transfer rate function between test points i and j [28] is expressed by Eq. (14).

(14)T_{ij}(s)=\frac{X_i(s)}{X_j(s)}=\frac{H_{il}(s)F_l(s)}{H_{jl}(s)F_l(s)}= \frac{H_{il}(s)}{H_{jl}(s)},

where s=j\omega. X_i(s) represents the response at test point i, and F_l(s) represents the excitation at test point I. H_{il}(s) denotes the frequency-response function. As s approaches the system pole S_r, the transfer rate function is converted to the ratio of {\phi }_{ir} (the amplitude of the rth mode at test point i) to {\phi }_{jr}, which is also valid for multipoint excitation.

(15)H_{il} (s)=\sum _{r=1}^m\left[\frac{K_{ilr}}{\left(s-s_r\right)}+\frac{K_{ilr}^{\ast }}{\left(s-s_r^{\ast }\right)}\right],

(16)K_{ilr}=\frac{{\phi }_{ir}{\phi }_{lr}}{\sqrt{1-{\xi }_r^2}{\omega }_r{m}_r} F_l(s),

where K_{ilr} and s_r are the residues and poles of the rth mode, ‘\text{*}’ represents the conjugate. {\omega }_r and {\xi }_r are the sth undamped natural frequency and damping ratio, respectively. To facilitate the calculation of the system poles, a rational fractional form based on the transfer rate function was constructed.

(17)\begin{array}{rl}{\mathrm{\Delta }}^{-1}{T}_{ij}^{k\text{l}}\text{(}s\text{)}=& \frac{1}{T_{ij}^k\text{(}s\text{)}-T_{ij}^l\text{(}s\text{)}}\\ =& \frac{{\displaystyle \sum _{n=0}^{2m-2}{a}_n{\mathit{\text{z}}}_n\text{(}s\text{)}}}{{\displaystyle \sum _{n=0}^{2m-1}{b}_n{p}_n\text{(}s\text{)}+b_{2m}{p}_{2m}\text{(}s\text{)}}}.\end{array}

In this section, forsythe orthogonal polynomials [29] were used to construct numerators and denominators, which facilitated the reduction of ill-conditioned properties in the solution process. {\mathit{\text{z}}}_n and p_n are the nth terms of the polynomial. Subsequently, the coefficient matrix of the denominator orthogonal polynomials was obtained based on the least squares method, and the system poles were obtained by solving the polynomials. However, ‘false poles’ often appeared in the results. To eliminate these redundant calculated poles, steady-state diagrams were created to capture the stable system poles. Finally, the natural frequency {\omega }_r and damping ratio {\xi }_r are solved by Eq. (18).

(18)\{\begin{array}{l}{s}_r=-{\xi }_r{\omega }_r+\mathrm{j}{\omega }_r\sqrt{1-{\xi }_r^2},\\ s_r^{\ast }=-{\xi }_r{\omega }_r+\mathrm{j}{\omega }_r\sqrt{1-{\xi }_r^2}.\end{array}

The mode shape corresponding to the natural frequency can be obtained by singular value decomposition of the transfer rate function matrix. The overall flowchart of the identification algorithm is presented in Fig.5.

3.3 Modal parameter identification results

The system poles for each test point were identified using the transfer rate function method based on rational fractional orthogonal polynomials. Fig.6 illustrates the corresponding steady-state diagram. When the change rate of the adjacent order frequency is less than 1% and remains low at multiple orders, the pole is deemed stable. The pole is referred to as the physical pole and is shown in the steady-state diagram, whereas the pole that does not satisfy the conditions is excluded. In the steady-state diagram, points within 0.1%–0.5% error are marked with ‘*’, and points within 0.5%–1% error are marked with o.

The system poles of each test point were solved in the same way, and the average of the identified values from all test points was taken as the result. Ultimately, the first four order frequencies identified are 23.49, 41.76, 43.77, and 79.69 Hz, respectively.

4 Model updating method

4.1 Neural network-based surrogate model

Normally, finite element models for high-speed maglev guideways are computationally expensive, particularly when heavily invoked during the updating process. Therefore, it is necessary to construct a computationally fast surrogate model to describe the complex relationship between the modal parameters and update parameters, thus improving computational efficiency.

The Elman neural network is a recurrent network with local memory and feedback capabilities; its specific structure is shown in Fig.7. It is unique in that the context layer stores the last output of the hidden layer, and the output of the hidden layer is linearized with the output of the input layer.. Thus, the Elman network not only has a powerful fitting ability, but also has stronger computational power and global stability [30]. Further, it can be utilized to simulate the relationship between guideway parameters and modal parameters, effectively reducing the computational costs caused by finite element analysis and multiple updating objectives.

(19)O_q= f_q\left({\beta }_q+\sum _{p=1}^n{w}_{pq}{x}_p+\sum _{n=1}^h{w}_{q\text{s}}{O}_{\text{c}q}\right),

(20)O_{\mathrm{c}q}(k )=O_q(k- 1),

(21)y_r=f_r\left({\beta }_r+\sum _{q=1}^h{w}_{qr},O_q\right),

where {\beta }_q is the bias from the input layer to the hidden layer and {\beta }_r is the bias from the hidden layer to the output layer. w_{pq}, w_{qs}, and w_{qr} are the weights from the input layer to the hidden layer, the context layer to the hidden layer, and the hidden layer to the output layer, respectively. O_{\mathrm{c}q} and O_q are the output values of the hidden layer and context layer, respectively. x_p is the input to the network, which represents the parameters of the model updating. y_r is the output of the network, which represents the modal information of the finite element model.

4.2 Multi-objective optimization

Vehicle-guideway coupled model updating is essentially an optimization problem, which is a process of finding the model parameters that minimize the error between the measurement and the simulation model. In this study, the hierarchical updating method included two main steps. First, the structural parameters of the guideway were updated based on its modal parameters, thereby developing a highly accurate guideway model. On this basis, a coupled model with updated guideways was developed, and the electromagnetic parameters in the model were identified based on the dynamic response of the vehicle.

4.2.1 Variables for model updating

In a maglev system, the material parameters of the vehicle, including the suspension stiffness, can be readily ascertained through experimental reports and conveniently replaced to maintain consistency. However, determining the real parameters of the guideway is challenging due to various factors such as the discreteness of the concrete, the influence of reinforcement, the degradation of the elastic bearing during extended operation, and the difficulty in replacing materials. When updating the structural model, the material properties and boundary conditions parameters are more sensitive than other parameters to the structure or vehicle [22–23,31]. Therefore, in this study, the vertical stiffness k_{\mathrm{y}} of the bearing, equivalent elastic modulus E_{\mathrm{c}}, and mass density {\rho }_{\mathrm{c}} of the concrete were taken as the updating parameters of the guideway finite element model.

In addition to the guideway parameters, the electromagnetic parameters in the vehicle system are also difficult to determine owing to the complex control and system commissioning. In previous studies [4], the displacement feedback coefficient k_{\mathrm{i}}, velocity feedback coefficient k_{\mathrm{v}}, and acceleration feedback coefficient k_{\mathrm{a}} of the magnetic force have been found to be crucial for electromagnetic levitation control, as they can seriously affect the adjustment time and stability of the controller. Hence, these three parameters were included as update parameters in this study. Tab.2 shows the range of modified parameters for the coupled model, which was designed based on relevant literature and design research information [12].

4.2.2 Multi-objective function of model updating

In addition to the natural frequencies of the finite element model, the mode shapes need to resemble the results of the field measurements to update the guideway in the coupled model more precisely. Thus, the updating of the guideway layer is a double-optimization problem, and its optimization model is detailed in Eq. (22).

(22)\{\begin{array}{l}\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{d}:E_{\mathrm{c}},{\rho }_{\mathrm{c}},k_{\mathrm{y}},\\ \mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}:G_{\mathrm{g}}=\left[G_1,G_2\right],\\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}:x_{\mathrm{g}}\text{(}j\text{)}\in \left[v_{\mathrm{g}\mathrm{l}}\text{(}j\text{)},v_{\mathrm{g}\mathrm{u}}\text{(}j\text{)}\right],j=1,2,3,\end{array}

where v_{\mathrm{g}\mathrm{l}} and v_{\mathrm{g}\mathrm{u}} are the lower and upper limits, respectively, of the updating variables of the guideway layer. G_{\mathrm{g}} is a double objective function that includes the accumulative error G_1 of the natural frequency and the accumulative error G_2 of the Modal Assurance Criterion (MAC).

(23)G_1\left(E_{\mathrm{c}},{\rho }_{\mathrm{c}},k_{\mathrm{y}}\right)=\sum _{i=1}^{N_i}{w}_i{\left(1-\frac{f_{\mathrm{s}\mathrm{i}}\left(E_{\mathrm{c}},{\rho }_{\mathrm{c}},k_{\mathrm{y}}\right)}{f_{\mathrm{m}i}}\right)}^2,

(24)G_2\left(E_{\mathrm{c}},{\rho }_{\mathrm{c}},k_{\mathrm{y}}\right)=\sum _{i=1}^{N_i}{w}_i\left(\frac{1-\sqrt{MA{C}_i\left(E_{\mathrm{c}},{\rho }_{\mathrm{c}},k_{\mathrm{y}}\right)}}{MA{C}_i\left(E_{\mathrm{c}},{\rho }_{\mathrm{c}},k_{\mathrm{y}}\right)}\right),

where w_i is the weight of the ith modal, which is taken as 1.0 in this study. f_{\mathrm{s}i} is the ith frequency value calculated using the finite element model, and f_{\mathrm{m}i} is the ith frequency value identified from the measurement data. MAC is the modal assurance criterion.

(25)MA{C}_i\left(E_{\mathrm{c}},{\rho }_{\mathrm{c}},k_{\mathrm{y}}\right)=\frac{{\left({\phi }_{\mathrm{s}i}^T{\phi }_{\mathrm{m}i}\right)}^2}{\left({\phi }_{\mathrm{s}i}^T{\phi }_{\mathrm{s}i}\right)\left({\phi }_{\mathrm{m}i}^T{\phi }_{\mathrm{m}i}\right)},

where {\phi }_{\mathrm{s}i} and {\phi }_{\mathrm{m}i} are the ith mode shape vectors identified by the simulation model and the measured data, respectively.

In the vehicle system, the electromagnet is directly subjected to magnetic force. Therefore, the vibration of the electromagnet was utilized to construct an updating function (Eq. (26)) for the electromagnetic parameters in the vehicle/guideway coupled model.

(26)\{\begin{array}{l}\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{d}:k_{\text{i}},k_{\text{v}},k_{\text{a}},\\ \mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}:G_{\text{m}}=\left[G_3,G_4\right],\\ \mathrm{S}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}:x_{\text{m}}\text{(}j\text{)}\in \left[v_{\text{ml}}\text{(}j\text{)},v_{\text{mu}}\text{(}j\text{)}\right],j=1,2,3,\end{array}

where v_{\mathrm{m}\mathrm{l}} and v_{\mathrm{m}\mathrm{u}} are the lower and upper limits of the update variables, respectively. G_{\mathrm{m}} is the double objective function of the magnetic force layer, as shown in the following equation.

(27)\begin{array}{rl}{G}_3\left(k_{\text{i}},k_{\text{v}},k_{\text{a}}\right)=& \sqrt{\sum _{k=1}^{N_k}{\left[R_{\text{s}}\left(t_k\right)-{\overline{R}}_{\text{s}}\left(t_k\right)\right]}^2\sum _{k=1}^{N_k}{\left[R_{\text{m}}\left(t_k\right)-{\overline{R}}_{\text{m}}\left(t_k\right)\right]}^2}/\\ & \left\{\sum _{k=1}^{N_k}\left[R_{\text{s}}\left(t_k\right)-{\overline{R}}_{\text{s}}\left(t_k\right)\right]\left[R_{\text{m}}\left(t_k\right)-{\overline{R}}_{\text{m}}\left(t_k\right)\right]\right\},\end{array}

(28)G_4\left(k_{\text{i}},k_{\text{v}},k_{\text{a}}\right)= \sqrt{\sum _{k=1}^{N_k}{\Vert R_{\text{s}}\left(t_k\right)-R_{\text{m}}\left(t_k\right)\Vert }^2/N_k},

where R_{\mathrm{s}}\left(t_k\right) and R_{\mathrm{m}}\left(t_k\right) are the simulated acceleration and measured acceleration of the end electromagnet at the t_k moment, respectively. {\overline{R}}_{\mathrm{s}}\left(t_k\right) and {\overline{R}}_{\mathrm{m}}\left(t_k\right) are their acceleration averages. N_k is the total number of moments.

4.2.3 Multi-objective evolutionary algorithm

With fast convergence and flexible parameter adjustment, the MOPSO algorithm [32] can be applied to solve the optimization problems in this study. This algorithm incorporates multiple fitness functions, leading to non-uniqueness of the optimal solution. However, when objectives are in conflict, non-dominant Pareto solution sets can be generated and stored in an external archive with a constant size [33].

To determine the best position of the group and archive solution set, the individual priority order was solved by the density information of the particles in the solution set. Concretely, the double objective space was equally divided into small grids, and the number of particles in each grid was regarded as particle density information. The side length of the small grid was determined by the boundaries (min{F}_1^{\mathrm{t}}, max{F}_1^{\mathrm{t}}) and (min{F}_2^{\mathrm{t}}, max{F}_2^{\mathrm{t}}) of the objective space, as shown in Eq. (29).

(29)\{\begin{array}{l}\mathrm{\Delta }{F}_1^{\mathrm{t}}={\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}{F}_1^{\mathrm{t}}-\mathrm{m}\mathrm{i}\mathrm{n}{F}_1^{\mathrm{t}}}{M}}，\\ \mathrm{\Delta }{F}_2^{\mathrm{t}}={\displaystyle \frac{\mathrm{m}\mathrm{a}\mathrm{x}{F}_2^{\mathrm{t}}-\mathrm{m}\mathrm{i}\mathrm{n}{F}_2^{\mathrm{t}}}{M}，}\end{array}

where F_1^{\mathrm{t}} and F_2^{\mathrm{t}} denote the objective function values of the particles in the tth generation, and M is the number of grids in any objective direction. Finally, the number of grids can be obtained as \{\mathrm{I}\mathrm{n}\mathrm{t}\left[\left(F_1^{\mathrm{t}}-min{F}_1^{\mathrm{t}}\right)/\mathrm{\Delta }{F}_1^{\mathrm{t}}+1\right], \mathrm{I}\mathrm{n}\mathrm{t}\left[\left(F_2^{\mathrm{t}}-min{F}_2^{\mathrm{t}}\right)/\mathrm{\Delta }{F}_2^{\mathrm{t}}+1\right]\}, and the corresponding estimated density of the particles can be calculated. The preferred principle for archive solution is that the lower the density value, the higher the probability of being selected. Thus, the group best position and the archive solution can be solved.

Subsequently, each particle is guided to converge to the final group best position according to the individual best position and the group best position of each iteration. The dth dimensional velocity v_{id}^{t+1} and position x_{id}^{t+1} of the ith particle at (t + 1) generation are as follows.

(30)v_{id}^{t+1}=w{v}_{id}^t+c_1{r}_1\left(p_{id}^t-x_{id}^t\right)+ c_2{r}_2\left(p_{gd}^t-x_{gd}^t\right),

(31)x_{id}^{t+1}=x_{id}^t +v_{id}^{t+1}

where w denotes the inertia factor. c_1 and c_2 are the individual learning factor and group learning factor of the particle, respectively. r_1 and r_2 are random numbers between (0,1). p_{id}^{\mathrm{t}} and p_{gd}^{\mathrm{t}} are the individual and group best positions of the particle, respectively.

4.3 Grey relational projection analysis

The gray relational projection method [34] was employed to select the optimal solution from the Pareto solution. This method determines the optimal objective value by comparing the superiority of each solution. First, the objectives of each solution in the archive set were normalized to avoid interference from the dimension information of the different objectives. Thus, positive and negative ideal values were obtained. Then, the gray relational coefficient between the jth objective value of the ith solution and the ideal value is calculated by Eq. (32).

(32)r_{ij}^{+(-)}=\frac{\underset{n}{\min }\underset{m}{\min }|g_{0j}^{+(-)}-g_{ij}\left|\right|+\rho \underset{n}{\max }\underset{m}{\max }\left|\right|g_{0j}^{+(-)}-g_{ij}\left|\right|}{\left|\right|g_{0j}^{+(-)}-g_{ij}\left|\right|+\rho \underset{n}{\max }\underset{m}{\max }\left|\right|g_{0j}^{+(-)}-g_{ij}\left|\right|}

where \left|g_{0j}^{+(-)}-g_{ij}\right| represents the absolute difference between the particle objective value and the positive (negative) ideal value. ρ denotes the resolution coefficient.

The projection V_i of the objective value on the ideal value and the modulus V_0 of the positive (negative) ideal values were calculated by Eq. (33).

(33)\{\begin{array}{l}{V}_i^{+\text{(}-\text{)}}={\displaystyle \sum _{j=1}^m{r}_{ij}^{+\text{(}-\text{)}}\frac{w_j^2}{\sqrt{{\displaystyle \sum _{j=1}^m{\left(w_j\right)}^2}}},}\\ V_0^{+\text{(}-\text{)}}={\displaystyle \sum _{j=1}^m\frac{w_j^2}{\sqrt{{\displaystyle \sum _{j=1}^m{\left(w_j\right)}^2}}},}\end{array}

where w_j represents the weight of the jth objective value. m is the number of solutions in the archive set. The superiority used to evaluate each particle in the archive set is defined as follows.

(34)U_i=\frac{\left(V_0-V_i^{-}\right)}{{\left(V_0-V_i^{-}\right)}^2+{\left(V_0-V_i^{+}\right)}^2},0 \le u_i\le 1.

The higher the superiority, the closer the solution is to the ideal value. The solution with the highest superiority in the archive set is the optimal solution.

4.4 Hierarchical updating

The updating process for the vehicle/guideway coupled model necessitates the joint optimization of various objectives, including the minimization of the error between the modal information of the guideway (or the dynamic response of the vehicle) and the measured results. Moreover, the guideway is the essential foundation of a system that provides traction, guidance, and support to the vehicle. Its actual properties and vibration, which are well described by modal parameters, can directly affect the state of the magnetic force and vibration of the upper vehicle. Therefore, a two-layer updating method for the high-speed maglev vehicle/guideway coupled model is proposed, and the overall flowchart is depicted in Fig.8.

5 Model updating results and analysis

5.1 Updating of guideway layer

5.1.1 Elman neural network fitting

To construct the data set for training and testing the neural network, 400 samples were randomly generated using the Optimal Latin Hypercube Sampling method [35] within the parameter ranges of the design variables (k_{\mathrm{y}}, E_{\mathrm{c}}, and {\rho }_{\mathrm{c}}). This sampling method generates data with a wide and uniform distribution, which helps cover more data features. The samples were then input into the finite element model of the guideway for modal analysis, resulting in the corresponding first four natural frequencies and mode shapes, which were used as the output of the neural network.

In the Elman network, the training and test sets were randomly generated with a 3:1 ratio from the total sample set. In terms of parameter setting, the learning rate was set to 0.01, the error goal was set to 1 × 10⁻⁶, and the epoch was set to 1000. Furthermore, the number of units in the hidden and context layers was determined by comparing the mean square errors after training at different numbers of units. The weights and biases in the Elman network were updated using the Particle Swarm Optimization algorithm to improve the approximate accuracy and generalization performance of the network. Specifically, the weight was set to 0.9, and the two learning factors were assigned values of 2.4 and 1.3, respectively. The maximum number of iterations was set to 50 and the population size was set to 50. The network ceased training upon the convergence of both training and test mean square errors to acceptably low values.

To verify the fitting and generalization performance of the trained network, statistical measures of the test set are typically utilized, including the Mean Absolute Percentage Error (MAPE), coefficient of determination (R²), and Root Mean Square Error (RMSE), as shown in Eqs. (35)–(37).

(35)R^2=1- \sum _{i=1}^k{\left(y_i-{\hat{y}}_i\right)}^2/\sum _{i=1}^k{\left(y_i-\overline{y}\right)}^2,

(36)RMSE= \sqrt{\sum _{i=1}^{n_{\text{t}}}{\left(y_i-{\hat{y}}_i\right)}^2/n_{\text{t}}} ,

(37)MAPE= \frac{100\mathrm{\%}}{n_{\text{t}}}\sum _{i=1}^{n_{\text{t}}}\left|\frac{{\hat{y}}_i-y_i}{y_i}\right|,

where {\hat{y}}_i and y_i are the predicted and simulated values of the neural network corresponding to the ith sample, respectively. \overline{y} is the average of the k simulation results. The interval of R² is [0,1]. The closer it is to unity, the more accurately the network fits. The results for these indicators are presented in Tab.3.

The R² value of the neural network in Tab.3 was greater than 0.98. The maximum MAPE was only 0.4741%, and the maximum RMSE was 0.4377, which further indicates that the Elman network has high fitting accuracy. Meanwhile, the computation time for a prediction by the Elman neural network was only 0.11s, which was 1/376 of the FE analysis. Therefore, the trained Elman network can be used as a surrogate model to assist in the subsequent optimization.

5.1.2 Multi-objective updating results

The MOPSO algorithm was also used to solve the objective function of the guide layer. A population size of 100 was employed, with an archive set size of 80 and a maximum iteration count of 200. The individual and group learning factors were assigned values of 1 and 2, respectively. The objective axis was divided into seven grids. The hardware environment used was Intel i7-9750H CPU @ 2.60GHz.

Optimization with the set parameters was performed until the iterations converged. Fig.9 shows the Pareto front for the guideway layer updating, presenting the distribution of non-dominated individuals. The evenly distributed Pareto front also indicated a negative correlation between the two objectives in the guideway layer.

The optimal solution was determined by performing GRP analysis on each solution of the Pareto front. The corresponding parameters were as follows: E_{\mathrm{c}} = 43.79 GPa, {\rho }_{\mathrm{c}} = 2534.88 kg/m³, and k_{\mathrm{y}} = 3.26 × 10¹⁰ N/m. By inputting these updated parameters into the finite element model, the first four orders of modal information after modal analysis were identified and are shown in Fig.10 and Tab.4. Compared to the modal information before updating, the modal results of the updated guideway were in better agreement with the measurement results. The maximum error between the updated and measured results was 1.14%, which corresponds to the 2nd vertical mode shape. The errors of the remaining modes were within 1%. In addition, the updated MAC for each mode shape was closer to unity, indicating that the correlation between the computed and measured mode shapes was further improved after updating.

These results reveal that the FE guideway can accurately simulate the real modal parameters after updating the guideway layer. It also indicates that the simplification of the boundary conditions and material properties of the guideway model is reasonable. In addition, the surrogate model needs to be computed a total of 2000 times during the updating process. Thus, the total time for calculating the surrogate model is 1/(376 × 2000) of the time for directly invoking the finite element model, which demonstrates the efficiency advantage of the surrogate model.

5.2 Updating of magnetic force layer

The updating of the magnetic force layer is based on a vehicle/guideway coupled model with an updated guideway model. For the magnetic force layer, a population size of 50 and an archive set size of 40 were utilized in the multi-objective particle swarm algorithm. The inertia factor was set to 0.5, and the mutation probability was set to 0.1. The algorithm continued to iterate until the pause criterion was met or the maximum iteration count (100) was reached. The Pareto front after the second-layer update is depicted in Fig.11. Using GRP analysis, a synergistic optimal solution that emanates from the most convex point of the Pareto front was found. The corresponding parameters are as follows: k_{\mathrm{i}}= 7095.21 A/m, k_{\mathrm{v}} = 40 A/(m/s), and k_{\mathrm{a}} = 0.22 A/(m/s²).

Tab.5 presents a comprehensive comparison of all parameters before and after updating. Comparing the guideway parameters before updating [4], the values obtained for elastic modulus and mass density of the concrete after updating are improved. This is mainly due to the discreteness of the actual concrete and the reinforcing effect of the steel bars inside the guideway on the concrete properties. Furthermore, the dynamic stiffness of the bearing under the influence of the moving vehicle, which was previously assumed to be infinite in the model, has been identified. Moreover, the hierarchical updating method has facilitated the variation and matching of the electromagnetic parameters.

6 Dynamic response analysis

6.1 Dynamic response analysis of the guideway

The guideway vibration resulting from the coupled model and measurements as the vehicle moves along the guideway at a constant speed of 25 km/h is depicted in Fig.12.

Fig.12(a)–Fig.12(b) illustrate the time history curves of the vertical acceleration at the mid-span of the original guideway model and the updated guideway model, respectively. These graphs demonstrate that the variation in vertical acceleration under interaction is in good agreement with the measured data, which both reflect the whole process of the vehicle gradually entering the guideway, fully located on the guideway, and gradually exiting from the guideway. However, compared to the guideway acceleration of the original model, the guideway acceleration of the updated model is more consistent with the measured results. In terms of the acceleration amplitude, the error between the maximum acceleration obtained from the updated model and the measured result (occurring at approximately 5.5 s) is 2.01%, which is significantly smaller than the error of 15.84% between the original model and the measurement. In addition, due to the vibration from the adjacent lines in reality, the measured vibration before the vehicle enters and after it leaves is greater than in the simulation.

In coupled vibration analysis [36,37], the frequency domain information of the guideway is critical. This is because it can guide the structural design by avoiding resonance. Fig.12(c)–Fig.12(d) show the frequency domain distribution of the vibration response at the mid-span of the guideway. As observed from the measured spectrum, the frequency f_{g1} (23.5 Hz) at the first prominent peak indicates the first-order natural frequency of the guideway. A comparison of the two figures reveals that the simulated frequency-domain distribution after updating is closer to the measured results, which is clearly reflected in the frequency corresponding to the first prominent peak in the spectrum. This is the first-order natural frequency identified from the coupled vibration response of the guideway [4]. The comparison results also illustrate that the spectrum distribution is significantly affected by the update parameters. Moreover, the simulation spectrum features numerous continuous small peaks around the first-order frequencies, which correspond to the densely distributed characteristic frequencies that are due to the low test speed [38]. The measured and simulated Fourier amplitudes have obvious differences around the first natural frequency of the guideway. This difference is due to the fact that some characteristic frequencies prominent in the simulated response are relatively close to the first-order natural frequency of the guideway when the maglev vehicle is running at a low speed. Due to the superposition effect of these two frequency amplitudes, the simulated response in this low-frequency range is higher than the measured response.

Fig.13 compares the measured and simulated displacement of the guideway at the mid-span. The measured and simulated displacement changes exhibit a similar trend: the rising and falling edge changes are largely symmetric, and the displacement remains generally stable when the vehicles are fully distributed on the guideway. However, compared to the peak guideway displacement of 0.491 mm obtained from the original model, the peak guideway displacement obtained from the updated model is 0.441 mm, which is closer to the measured result of 0.437 mm. The error between the updated peak guideway displacement and the measured result is only 0.86%, which further indicates that the vehicle/guideway coupled model with updated parameters is more suitable for predicting the guideway response.

In terms of the guideway, the simulation accuracy of the original model is inferior to that of the updated model, which also verifies that the simulation error is derived from the simplification of the model and the uncertainty of the design parameters.

6.2 Dynamic response analysis of vehicle

The dynamic response of vehicle components directly affects the safety and comfort of the vehicle, particularly the vibration of the electromagnet. This is because the components are directly subjected to magnetic forces and external (line) excitation.

Fig.14 depicts the simulated and measured electromagnet vibration during the movement of the vehicle on the test guideway. It is evident that the peak acceleration of the electromagnet obtained from the updated model is 5.35 m/s², which is significantly closer to the measured value than the result before updating. The above results verify the effectiveness of the multi-objective optimization algorithm.

In addition, the electromagnetic parameters can fundamentally affect the interaction between the vehicle and guideway, which is reflected in the fluctuation of the levitation gap. Similarly to the case of the vibration response of the electromagnets, the fluctuation curve of the levitation gap at the end is a fluctuation band (Fig.15) when the vehicle runs on a line consisting of guideways.

In conclusion, the coupled model with hierarchical updating can more reasonably predict the dynamic response of the vehicle and guideway, as well as the vehicle/guideway interaction.

7 Conclusions

This study proposes a hierarchical model updating method that integrates a neural network, multi-objective optimization, and gray relational projection analysis for the high-speed maglev vehicle/guideway coupled system. On the one hand, a coupled system model, which integrates the guideway and considers its elastic bearing and material effects, an 18-DOF vehicle, and a feedback current controller, was developed. On the other hand, the parameters of the coupled system were updated with the dynamic response and guideway modal information measured on the Shanghai test line during the hierarchical updating. The following conclusions are drawn.

1) The Elman network can accurately approximate the relationship between guideway parameters and modal information, with a maximum mean absolute percentage error between the predicted values and simulation results of only 0.4741%. Moreover, the Elman network can substantially improve the computational efficiency of the updating process, during which the total time for network prediction is 1/752000 of the total time calculated by directly calling the FE guideway model. Therefore, the Elman network can substitute the FE guideway model to assist in the multi-objective updating of the guideway model.

2) The coupled system parameters were updated hierarchically. For the guideway and magnetic force layer, the modal parameters and electromagnet vibration after updating are in much greater agreement with the measured results than the original results, prior to updating. Notably, the maximum error between the updated natural frequency and the measured results is only 1.14%. The results not only highlight the improved accuracy of the finite element modeling method that considers elastic bearings but also validate the effectiveness of the multi-objective optimization series algorithm.

3) The simulated and measured dynamic responses of the coupled system, including the acceleration and displacement of the guideway, electromagnet vibration, as well as levitation gap fluctuation, were analyzed to verify the updating method. The comparison results reveal that the method provides an excellent agreement between the dynamic response obtained from the updated model and the measurement results. For the guideway, the vibration peak at the midspan is reduced from 15.84% before updating to 2.01% after updating. All the comparison results further confirm the significantly improved accuracy of the updated coupling model.

The updated coupled model is capable of accurately simulating the dynamic response of the maglev vehicle/guideway system, thus facilitating further research on maglev dynamics with greater reliability, particularly regarding the mitigation of coupled vibration and system optimization. Furthermore, the hierarchical updating method with field measurements can be expanded to other vehicle/line systems with different structures, such as low-to-medium speed maglev and superconducting maglev systems with EDS.

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