Influence of the track structure on the vertical dynamic interaction analysis of the low-to-medium-speed maglev train-bridge system

Abstract

The low-to-medium-speed maglev train is stably suspended near the rated suspension gap. The suspension force acts directly on the track and is transmitted to the bridge. The maglev track structure is novel, and the influence mechanism of the track structure on the coupled vibration of the maglev train-bridge system is unknown. Therefore, in this study, we propose vertical dynamic interaction models of the low-to-medium-speed maglev train-bridge system and the low-to-medium-speed maglev train-track-bridge system to analyse the influence mechanism of the maglev track structure on the vertical dynamic interaction of the low-to-medium-speed maglev train-bridge system. The vibration characteristics of the F-rail and the influence mechanism of the track structure on the dynamic responses of the bridge are discussed in detail. The study verifies that the local deformation of the F-rail is self-evident and cannot be ignored. In addition, the influence of the F-rail on the dynamic interaction of the maglev train-bridge system is mainly reflected in two aspects: first, the vibration of the bridge in the high-frequency band increases due to the high frequency and intensive local vibration of the F-rail itself. Second, the vibrations of the bridge and the F-rail in the low-frequency band increase due to the periodic irregularities caused by the local deformation of the F-rail. In this study, we consider the vertical dynamic interaction model of the low-to-medium-speed maglev train-track-bridge system.

Keywords

bridge system low-to-medium-speed maglev train track structure track-bridge system vertical dynamic interaction mechanism

Introduction

Maglev trains are stably operated near the rated suspension gap by regulating the suspension gap and actively controlling the electromagnetic suspension (EMS) force, so that they can run smoothly. An EMS maglev train is a type of maglev train whose rated suspension gap is generally small (∼8−10 mm) (Yim et al., 2009). When this train runs on a bridge, it causes deformation which directly affects the suspension gap and the EMS force, thus changing the dynamic interaction characteristics of the maglev train-bridge system (Wang et al., 2007; Yau, 2009a). Previous researches have shown that the EMS maglev train cannot run smoothly due to the strong coupled vibration between the maglev train and the bridge (Zhou et al., 2010, 2011), which is a large-scale complex and dynamic interaction process (Fan and Wu, 2006; Yang and Lin, 2005). Therefore, like exploring the dynamic interaction characteristics of the traditional wheel/rail train-bridge system (Biondi and Muscolino, 2003; Claus and Schiehlen, 1998; Salcher and Adam, 2015), problems involving coupled vibration of the maglev train-bridge system need to be addressed.

In recent times, Zhao and Zhai (2002) built a 10-DOF (degree of freedom) model of a TR06 maglev train and discussed the characteristics of vertical coupled vibrations due to random irregular excitation. In addition, Min et al. (2017) established two-dimensional (2D) and three-dimensional (3D) coupled maglev train-bridge models, and compared them using dynamic response analysis to study resonant vibrations and eliminate vibration problems in the train-bridge system. In the same vein, Wang et al. (2015) performed theoretical analysis and experimental verification of the dynamic interaction of the maglev train-bridge system based on state feedback theory. Furthermore, Reza et al. (2012) studied the operational stability of maglev trains with time-delay effects, parameter uncertainties, and external loads based on an adaptive robust control system.

In addition, Ju et al. (2012) established the dynamic interaction model of a maglev train-bridge-soil system using finite element method and verified the model by an analytical solution. An et al. (2012) implemented the active control of periodic vibration using maglev actuators and discovered that the proposed algorithm could efficiently compensate for the actuators’ time-varying nonlinearities and greatly attenuate the energy of periodic vibration. Zhou et al. (2017) investigated the vibration of the maglev train caused by track irregularity and found that the gap vibration of the levitation unit is larger at the rear than at the front. They also proposed an adaptive vibration control scheme to reduce the vibration of the maglev train. Moreover, Ju et al. (2014) investigated the safety of a series of maglev trains moving on multi-span bridges subject to foundation settlements and earthquakes, and discovered that EMS systems can overcome the problem of foundation settlement while electrodynamic suspension (EDS) systems can overcome both foundation settlement and earthquake problems. Song et al. (2014) presented a simple method for estimating the dynamic amplification factors (DAFs) of a double-tower multicable-stayed bridge traversed by maglev vehicles. They also suggested frequency limits for double-tower cable-stayed bridges to control the DAFs below a given reasonable value.

During operation, the low-to-medium-speed (LMS) maglev train applies suspension force directly to the track and subsequently to the bridge. However, the track structures of the LMS maglev line and the traditional wheel/rail line differ greatly. LMS maglev line is also called F-rail because the track section has an ‘F’ shape (as shown in Figure 1). The rigidity of the vertical section of F-rail is lesser in comparison with that of the track structure of traditional wheel/rail line, while the longitudinal support spacing is relatively larger (generally 1.0−1.2 m). Hence, the overall vertical rigidity is low. Therefore, the F-rail suffers from deformation and vibration under the action of maglev trains. This affects the dynamic characteristics of the suspension gap and suspension force, thereby affecting the dynamic interaction of the maglev train-bridge system. Recent researches have focused on the analysis of coupled vibration of maglev train-bridge systems or suspension control systems. However, the impact of the track structure on the dynamic interaction of the maglev train-bridge system is unknown due to the lack of sufficient relevant literatures. Thus, this study establishes the vertical dynamic interaction models of the LMS maglev train-bridge system and the LMS maglev train-track-bridge system. The vibration characteristics of the F-rail and the impact of the track structure on the dynamic responses of the bridge are also discussed in detail.

Figure 1.

Comparison of the track structures: (a) LMS maglev line and (b) traditional wheel/rail transportation.

Dynamic interaction model

LMS maglev train model

An EMS five-module LMS maglev train is considered in this study. Its main components include car body, air springs, suspension frames, suspension electromagnets, and so on. The air spring acts as a secondary suspension that connects the maglev car body to the suspension frames. There are five pairs of suspension frames for each vehicle and four pairs of suspension electromagnets for each pair of suspension frame. The EMS maglev train is shown in Figure 2(a) and (b).

Figure 2.

Theoretical simplification model of the maglev train: (a) actual maglev train, (b) cross-section of the actual maglev train and (c) theoretical simplification model.

The structure of the maglev train is more complicated than that of the traditional wheel-rail train. Li et al. (2018) and Ren et al. (2010) pointed out that, in comparison with the traditional wheel-rail train, the effect of the maglev train on the bridge is more similar to the uniform load effect. Thus, it is very important to consider the uniform load effect of the maglev train. Hence, the actual maglev train is reduced to a uniform load model, as shown in Figure 2(c). The maglev train model simplifies the five suspension frames into five masses and the maglev car body into five discrete masses corresponding to each pair of suspension frames. The arrangements of the secondary suspensions are similar to that of the actual maglev vehicle.

The vertical DOF of the car body, and the vertical and nod DOFs (Z_ca, Z_sa, rot_sy) of the suspension frame in the maglev train model are considered, as shown in Figure 2(c). The external loads, spring forces, damping forces, and inertial forces can be used to establish the force balance equations of the maglev trains using D’Alembert principle. Thereafter, the dynamic equations of motion of the maglev train model can be obtained.

Using the first suspension module as an example, the equations of motion due to the remaining suspension modules are similar, and is given by

\begin{matrix} {\begin{matrix} M_{cb} & 0 & 0 \\ 0 & M_{sb} & 0 \\ 0 & 0 & J_{sb} \end{matrix}} [\begin{matrix} {\overset{\cdot\cdot}{Z}}_{c} \\ {\overset{\cdot\cdot}{Z}}_{s} \\ {\overset{\cdot\cdot}{β}}_{s} \end{matrix}] + {\begin{matrix} 2 C_{b} & - 2 C_{b} & 0 \\ - 2 C_{b} & 2 C_{b} & 0 \\ 0 & 0 & 2 C_{b} d_{s} \end{matrix}} [\begin{matrix} {\overset{\cdot}{Z}}_{c} \\ {\overset{\cdot}{Z}}_{s} \\ {\overset{\cdot}{β}}_{s} \end{matrix}] + {\begin{matrix} 2 K_{b} & - 2 K_{b} & 0 \\ - 2 K_{b} & 2 K_{b} & 0 \\ 0 & 0 & 2 K_{b} d_{s} \end{matrix}} [\begin{matrix} Z_{c} \\ Z_{s} \\ β_{s} \end{matrix}] \\ = {\begin{matrix} 0 \\ \sum_{i = 1}^{4} f_{sbi} \\ \frac{3 l_{s}}{2} (f_{sb 4} - f_{sb 1}) + \frac{l_{s}}{2} (f_{sb 3} - f_{sb 2}) \end{matrix}} \end{matrix}

(1)

where M_cb and M_sb are the masses of the car body and the suspension frame, respectively; J_sb is the nodding moment of the suspension frame (around y-axis); C_b and K_b are the damping and stiffness of the secondary suspension, respectively; f_sbi is the i-th EMS force under a suspension frame (i = 1∼4); d_s is the half-way distance between two secondary suspensions on each suspension frame and l_s is the distance between adjacent electromagnetic forces, as shown in Figure 2(c).

The left-hand side of equation (1) represents the mass, damping, and stiffness matrix of the maglev train module respectively, while the right-hand side represents the external load acting on the maglev train.

Hence, the equation of motion of the vehicle can be defined as follows

M_{v} {\overset{\cdot\cdot}{Z}}_{v} + C_{v} {\overset{\cdot}{Z}}_{v} + K_{v} Z_{v} = F_{v}

(2)

where M_v is the mass matrix of maglev vehicle, C_v is the damping matrix, K_v is the stiffness matrix and F_v is the external load vector, which is the vector of the EMS force. The three matrices (M_v, C_v, K_v) are represented by the square matrices defined in equation (1).

Suspension control system model

Shi et al. (2007) proposed an active suspension control model based on displacement–speed–acceleration feedback using proportion–integral–derivative (PID) method, which is defined as follows

Δ i (t) = K_{sp} Δ c (t) + K_{sv} Δ \overset{\cdot}{\hat{z}} (t) + K_{sa} Δ \overset{\cdot\cdot}{z} (t)

(3)

where $Δ i (t)$ is the time-varying current, $Δ c (t)$ is the suspension gap fluctuation value and $Δ \overset{\cdot}{\hat{z}} (t)$ and $Δ \overset{\cdot\cdot}{z} (t)$ are the electromagnetic velocity and acceleration feedback variables, respectively. However, $Δ \overset{\cdot}{\hat{z}} (t)$ cannot be obtained directly and, therefore, needs to be reconstructed from the state observer. K_sp, K_sv and K_sa are the displacement, velocity and acceleration feedback coefficients of the PID active suspension control system, respectively. The time-varying current of the suspension control system can be determined using equation (3) to obtain the EMS force.

The state observer expression is defined as follows

Δ \overset{\cdot}{\hat{c}} (t) = Δ \overset{\cdot}{\hat{z}} (t) + 2 ξ_{0} ω_{0} [Δ c (t) - Δ \hat{c} (t)]

(4a)

Δ \overset{\cdot\cdot}{\hat{z}} (t) = Δ \overset{\cdot\cdot}{z} (t) + ω_{0}^{2} [Δ c (t) - Δ \hat{c} (t)]

(4b)

where $ξ_{0}$ and $ω_{0}$ are the damping and eigen frequencies of the state observer system respectively, while $Δ \hat{c} (t)$ , $Δ \overset{\cdot}{\hat{c}} (t)$ , $Δ \overset{\cdot}{\hat{z}} (t)$ and $Δ \overset{\cdot\cdot}{\hat{z}} (t)$ are the state observer variables reconstructed in the state observer.

The transfer equation of the state observer can be obtained by Laplace change, as shown in equation (5)

Δ \overset{\cdot}{\hat{z}} (s) = \frac{ω_{0}^{2} s}{ω_{0}^{2} + 2 ξ_{0} ω_{0} + s^{2}} Δ c (s) + \frac{s + 2 ξ_{0} ω_{0}}{ω_{0}^{2} + 2 ξ_{0} ω_{0} + s^{2}} Δ \overset{\cdot\cdot}{z} (s)

(5)

where $Δ \overset{\cdot}{\hat{z}} (s)$ is the constructed velocity feedback variable in the state observer and s is the Laplace variable. The system is a dual input and single output system. Hence, the velocity feedback variable can be determined using equation (5).

Thus, the suspension force can be defined as (Shi et al., 2007)

f (t) = \frac{μ_{0} A n^{2}}{4} {[\frac{i_{0} + K_{sp} Δ c (t) + K_{sv} Δ \overset{\cdot}{\hat{z}} (t) + K_{sa} Δ \overset{\cdot\cdot}{z} (t)}{c_{0} + Δ c (t)}]}^{2}

(6)

where f(t) is the suspension force, $\frac{μ_{0} A n^{2}}{4}$ is the electromagnetic scale factor and i₀ and c₀ are the rated current and the rated suspension gap, respectively.

Track-bridge system model

The track and bridge models were considered as a single system and solved using modal superposition theory (Clough and Penzien, 1993). Based on the orthogonality of mode shapes, the decoupled differential equation of the track-bridge system model is

{\overset{\cdot\cdot}{q}}_{n} + 2 ξ_{n} ω_{n} {\overset{\cdot}{q}}_{n} + ω_{n}^{2} q_{n} = {\bar{F}}_{n}^{*}

(7)

where $ω_{n}$ is the nth-order natural frequency of the track-bridge system (unit: rad), ${\bar{F}}_{n}^{*}$ is the nth-order generalised external load after the normalised mode shape given by ${\bar{F}}_{n}^{*} = {\bar{ϕ}}_{n} F_{b}$ and ${\bar{ϕ}}_{n}$ is the nth-order mode of the track-bridge system normalised with respect to the mass.

Rayleigh damping was adopted for the model to combine the mass and the stiffness matrices, as shown in equation (8a). Because the mass and stiffness matrices of the structure are orthogonal with respect to the mode shape, the damping matrix can also be decoupled by orthogonal conditions (Clough and Penzien, 1993). $ξ_{n}$ is the nth-order damping ratio of the track-bridge system, as obtained by Clough and Penzien (1993). This is shown in equation (8b)

c = a m + b k

(8a)

ξ_{n} = \frac{C_{n}}{2 ω_{n} M_{n}}

(8b)

where a and b are the combination coefficients given by Clough and Penzien (1993); c, m and k are the damping, mass and stiffness matrices of the track-bridge system, respectively; while C_n and M_n are the nth-order decoupled damping and mass matrices, respectively.

Hence, only the natural frequency of the track-bridge system and the generalised external load acting on the track or bridge need to be obtained to solve the differential equations of motion of the track-bridge system. In general, the natural frequency, mode shapes and the nth-order generalised external loads value acting on the track or bridge at any node can be easily obtained for all types of track-bridge systems by discretising the track-bridge system structure using ANSYS (a finite element analysis software). The model calculations of the track structure and bridge using ANSYS are discussed below.

Solving the dynamic interaction model

The dynamic interaction equation for a coupled maglev train-track-bridge system or maglev train-bridge system can be obtained by combining equations (2) and (7) to give equation (9). ${\bar{F_{n}}}^{*}$ and $F_{v}$ are the nth-order generalised loads and suspension force acting on the track (or bridge) and the maglev vehicle, respectively. The suspension force that connects the vehicle system and the track-bridge system (or bridge system) can be obtained by equation (5), which is related to the deformation of the track structure or bridge and the dynamic responses of the maglev vehicle. Hence, equation (9) is a coupled equation

\begin{matrix} {\begin{matrix} 1 & 0 \\ 0 & M_{v} \end{matrix}} [\begin{matrix} {\overset{\cdot\cdot}{q}}_{n} \\ {\overset{\cdot\cdot}{Z}}_{v} \end{matrix}] + {\begin{matrix} 2 ξ_{n} ω_{n} & 0 \\ 0 & C_{v} \end{matrix}} [\begin{matrix} {\overset{\cdot}{q}}_{n} \\ {\overset{\cdot}{Z}}_{v} \end{matrix}] \\ + {\begin{matrix} ω_{n}^{2} & 0 \\ 0 & K_{v} \end{matrix}} [\begin{matrix} q_{n} \\ Z_{v} \end{matrix}] = {\begin{matrix} {\bar{F}}_{n}^{*} \\ F_{v} \end{matrix}} \end{matrix}

(9)

Equation (9) is a dynamic equation of large-scale nonlinear time-varying system. In the process of solving the dynamic equation of the system, the equivalent stiffness (system characteristics) of the vehicle and track-bridge system (or bridge system) change during the operation of the vehicle. However, the system can be solved by incremental-iterative approach based on Newmark-β method which is an advanced numerical integration method, as described by Yau (2009b). The equations of motion of the track-bridge system (or bridge system) are separated from the vehicle while Newmark-β integral method is applied to solve the problem step by step. As a result, the external loads acting on the vehicle or the track-bridge system (or bridge system) are equal to the internal forces of the vehicle or track-bridge system (or bridge system), thereby achieving an equilibrium of forces. Figure 3 shows the block diagram of the solution of coupled vibration of the maglev train-track-bridge system (or train-bridge system). This study focuses on the influence of the track structure on the vertical dynamic interaction of the maglev train-bridge system. Therefore, all equations are used to obtain the vertical dynamic interaction model.

Figure 3.

The block diagram of the solution procedure of the vertical coupled vibration of maglev train-track-bridge system (or train-bridge system).

Model calculation and modal analysis

Calculation parameters

Table 1 lists the calculation parameters of the maglev train model and the suspension control system. Figure 4 shows the dimensions of the cross-sections of the bridge and track structure at mid-span for a dual-track design with 4.4 m between the track centrelines. The bridge consists of up-bound and down-bound box girders that are connected by several diaphragm plates of thickness 0.3 m and arranged in the longitudinal direction at 5 m intervals. The concrete strength and Young’s modulus of the bridge are C50 and 3.45 × 10⁴ MPa, respectively. The track structure consists of F-rails, steel sleepers, fasteners and supported rail beds. The F-rail and steel sleepers are bolted together, while the steel sleepers and concrete-supported rail beds are connected by fasteners. However, the concrete-supported rail beds are installed at a distance of 1.2 m on the bridge. The steel Young’s modulus of the F-rail is 2.1 × 10⁵ MPa, while the stiffness of the fastener is 120 kN/m.

Table 1.

Calculation parameters.

Maglev train		Suspension control system
M_cb	4 × 10⁴ (kg)	k_sp	5000
M_sb	2 × 10³ (kg)	K_sv	20
K_b	1.6 × 10⁵ (kN/m)	K_sa	0.1
C_b	1 × 10⁴ (kN/(m/s))	c₀	8 × 10^(–3) (m)
l_s	0.7 m	i₀	21 A
J_sb	3.6 × 10³ (kN·m²)	ω₀	1256.6 (Hz)
d_s	1.23 m	ξ₀	0.7 (N·m/s)

Figure 4.

Cross-sections (unit: cm): (a) bridge, (b) track structure and (c) partial enlarged view of the F-rail.

Modal analysis

The F-rail and the steel sleepers are fixed by bolts due to the large vertical rigidity of the steel sleepers. However, the impact of the steel sleepers is not considered in the model; that is, the F-rails are connected to the bridge through the fasteners directly. The F-rail is a homogeneous material and has uniform cross-section along the longitudinal direction of the bridge. Hence, the properties of the F-rail cross-section are superimposed by the left and right sides of the up-bound or down-bound box girders. The track-bridge system (model A, considering the influence of the track structure) and the bridge system (model B, not considering the influence of the track structure) were modelled using ANSYS. The bridge, F-rail and diaphragm plates were analysed using BEAM188# (an ANSYS element), while the fasteners were analysed using COMBIN14# (ANSYS spring damping element). BEAM188# has two nodes, and each node has six DOFs which are ux, uy, uz, rotx, roty and rotz. On the other hand, COMBIN14# has two nodes, and each node is only considered along the uz DOF. More information about BEAM188# and COMBIN14# can be obtained in literatures (Wang, 2007). Figure 5 shows the finite element models of models A and B. There are 744 elements for model A and 382 elements for model B.

Figure 5.

Finite element models: (a) model A and (b) model B.

Table 2 summarises the typical overall and local vertical vibration modes for bridges of model A (considering F-rail) and model B (without considering F-rail), and the corresponding vibration wavelengths of the F-rail in model A within 100 Hz. It also displays the frequencies corresponding to the three lowest mode orders for the overall vibration modes of models A and B, and six local vibration modes of the F-rail. The overall vibration modes of models A and B are the same, while the corresponding modal frequency values of model A are larger than that of model B. This is because model A considers the F-rail, which increases the overall rigidity of the structure.

Table 2.

Typical mode shapes and natural frequencies.

Description	Mode shape and natural frequency
Overall vibration modes of the bridge in model A	Frequency: 7.204 Hz First-order symmetrical vertical bend	Frequency: 27.075 Hz First-order anti-symmetrical vertical bend	Frequency: 53.593 Hz Second-order symmetrical vertical bend
Overall vibration modes of the bridge in model B	Frequency: 7.146 Hz First-order symmetrical vertical bend	Frequency: 25.491 Hz First-order anti-symmetrical vertical bend	Frequency: 50.908 Hz Second-order symmetrical vertical bend
The typical local vibration modes and corresponding wavelength of the F-rail in model A	Frequency: 58.252 Hz Wavelength: 21.4 m	Frequency: 64.090 Hz Wavelength: 13.2 m	Frequency: 65.333 Hz Wavelength: 10.8 m
	Frequency: 67.252 Hz Wavelength: 7.2 m	Frequency: 70.154 Hz Wavelength: 6.0 m	Frequency: 74.609 Hz Wavelength: 4.2 m

Figure 6 shows the distribution of all modes and the corresponding modal frequencies for model A within 100 Hz. The ordinate and abscissa are modal frequencies and have the same scale in order to properly express the frequency distribution of the model in a 2D coordinate system. The vibration of model A is due to the local vibration of the F-rail at 58 Hz, and its local vibration is more intensive and concentrated in the range of 55−75 Hz. However, the wavelength of the local vibration of the F-rail gradually decreases as the natural frequency increases. In addition, all the local vibrations of the F-rail exist in this frequency range (55−75 Hz) without any vibration mode of the bridge. Because the stiffness of the F-rail cross-section and the longitudinal support spacing are relatively large, local vibration of the F-rail easily occurs. Nevertheless, its frequency is lower than that of the track structure in traditional wheel/rail transportation lines (200 Hz; Wang et al., 2017). Therefore, the local vibration of the F-rail is very significant in the consideration of the overall vibration of the bridge structure for LMS maglev lines.

Figure 6.

Modal distribution of model A within 100 Hz.

Dynamic responses

The vehicle speed is set at 100 km/h and the dynamic response characteristics of models A and B of the maglev train, during operation, are compared. Then, the influence of the F-rail on the dynamic interaction of the system and its mechanism are explored. However, this section does not consider the random track irregularities due to the randomness of track irregularities.

Dynamic deflections

Figure 7 shows the time history curves of the dynamic deflections of the F-rail and the bridges induced by the maglev train at mid-span. Because the random irregularity is not considered, the amplitudes of the vertical deflection of the bridge and the F-rail are less. The changes of the curves are similar, except that the bridge dynamic deflection is slightly lesser in model A than in model B, with respective values of 1.85 mm and 1.87 mm. In addition, model A has greater overall rigidity due to the F-rail and lesser bridge dynamic deflection in comparison with model B.

Figure 7.

Comparison of the dynamic deflections of the F-rail and the bridges.

Moreover, the F-rail dynamic deflection of model A is significantly greater than its bridge dynamic deflection. The dynamic deflection of the F-rail consists of two parts which are caused by the deformation of the bridge and the local deformation of the F-rail, respectively. Therefore, the overall dynamic deflection of the F-rail is greater than that of the bridge.

Figure 8 presents the absolute dynamic deflection of the F-rail, that is, the dynamic deflection caused by the F-rail deformation. This deflection fluctuates more violently in the F-rail than in the bridge. The maximum value of the dynamic deflection caused by local deformation of the F-rail is 0.467 mm, while the maximum overall dynamic deflection of the F-rail is 2.290 mm. Consequently, the dynamic deflection caused by the local deformation of the F-rail is 20.4% of the overall dynamic deflection of the F-rail. Therefore, the local deformation has a considerable influence on the overall dynamic deflection of the F-rail.

Figure 8.

Absolute dynamic deflection of F-rail in model A.

Accelerations

Figure 9(a) shows the vertical acceleration time domain curves of the F-rail and the bridges at mid-span after filtering with a frequency of 20 Hz, while Figure 9(b) shows the corresponding frequency domain curves. The waveform of the bridge accelerations in models A and B is similar within 20 Hz. However, the maximum acceleration of the bridge in the model A is greater than that of model B, with respective values of 0.031 m/s² and 0.024 m/s². In addition, the maximum acceleration of the F-rail is much greater than that of the bridge in model A, having a value of 0.110 m/s².

Figure 9.

Comparison of the vertical accelerations of the F-rail and bridges: (a) time domain after 80 Hz low-pass filter and (b) frequency domain.

On the other hand, the acceleration frequency spectrum of the bridge in models A and B are similar. The acceleration amplitude of the bridge is larger in model A than in model B at the frequency of 7 Hz corresponding to the vertical fundamental frequency of models A and B (as shown in Table 2). The 7 Hz frequency is induced by the first-order symmetrical vertical bend mode of models A and B. In addition, the acceleration spectrum value of F-rail is greater than that of the bridge in model A. In addition, the dominant frequency band of acceleration is mainly concentrated in the relatively low-frequency band region (within 10 Hz).

Figure 10 shows the time domain curves after 80 Hz low-pass filtering and the frequency domain curves of the F-rail and bridges. The maximum acceleration of the bridge is larger in model A than in model B with respective values of 0.147 m/s² and 0.048 m/s², while the maximum vibration (1.225 m/s²) is greater in the F-rail than in the bridges in the time domain. Furthermore, the acceleration of the bridge in model A has distinct vibration peaks between 55 and 75 Hz in the frequency domain curves, while model B does not have such peaks at the same frequency range. In addition, the distinct vibration peaks between 55 and 75 Hz are larger for the F-rail than for the bridge and are similar to the spectrum distribution of the bridge in the model A.

Figure 10.

Comparison of the vertical accelerations of F-rail and bridges: (a) time domain after 80 Hz low-pass filter, (b) frequency domain of the bridges and (c) frequency domain of the F-rail and bridge in model A.

Suspension force and gap

Figure 11 shows the suspension force and gap curves in the time domain when the maglev train is operated using models A and B, respectively. The suspension force and gap are stable and fluctuate around the rated values. However, the fluctuation amplitudes of the suspension force and gap are both larger and more severe in model A than in model B.

Figure 11.

Comparison of the suspension force and gap of models A and B in time domain: (a) suspension force and (b) suspension gap.

The frequency spectrums of the suspension force for the two models are shown in Figure 12. The suspension forces of models A and B both have peaks at 1.7 Hz. However, the amplitude of model A is significantly larger than that of model B. Moreover, the suspension force in model A has a peak between 55 and 75 Hz, while model B has no peak in this frequency band. Thus, in comparison to the deflection of the bridge, the deflection caused by the F-rail is more similar to the short-wave irregularity, as shown in Table 2. The short-wave irregularity of the F-rail (model A) disrupts the adjustment of the suspension control system (Liang, 2015), thereby resulting in a more significant EMS force in the higher frequency range. Therefore, a lot of oscillation occurs in the acceleration of the car body in model A.

Figure 12.

Comparison of the suspension force of models A and B in the frequency domain: (a) detailed enlargement during 0−20 Hz and (b) detailed enlargement during 0−80 Hz.

Hence, the dynamic responses of the bridge are different for cases where the F-rail are considered and not considered. The vertical dynamic deflection of the bridge is less with respect to the F-rail, while its low-frequency and high-frequency vibrations are high. In addition, the fluctuations of the suspension force and the suspension gap are significantly larger and more severe with respect to the F-rail, while the suspension force is larger in the low-frequency and the high-frequency bands.

Causal analysis

In this section, the influence of the F-rail on the vertical dynamic interaction of the LMS maglev train-bridge system is discussed in detail. Figure 13 shows a part of the local vibration modes of the F-rail. The vibrational wavelengths of the F-rail for the three local vibration modes are 7.2 m, 6.0 m and 4.2 m respectively, with each mode having regular sinusoidal waveforms. Figure 14 shows the accelerations of the F-rail with respect to the local vibration modes of the F-rail with wavelengths of 7.2 m and 4.2 m in the time domain (after 20 Hz low-pass filtering) and frequency domain respectively. The sinusoidal waves of the acceleration of the F-rail are more regular in the time domain. Moreover, the accelerations of the F-rail are greater when the maglev train enters and leaves the bridge than when it covers the bridge. From equation (6), the accelerations of the F-rail are related to the generalised accelerations of each order modal while the generalised external loads acting on the F-rail are related to the modes of the F-rail and the external loads. Thus, the modal shape of the F-rail is a regular sinusoidal wave with positive and negative values. Moreover, when the maglev train covers a large area of the bridge, the external loads (EMS force) are evenly arranged on the F-rail while the positive and negative values of the mode shape are cancelled out. Hence, the generalised external load acting on the bridge has the smallest value (as shown in Figure 15). Consequently, the generalised accelerations of the F-rail are almost zero, while the accelerations at this stage are small.

Figure 13.

Different modes of the F-rail local vibration with different wavelengths.

Figure 14.

Comparison of the vertical accelerations of F-rail under different vibration modes of F-rail: (a) time domain after 20 Hz low-pass filter and (b) frequency domain between 0 and 20 Hz.

Figure 15.

Comparison of generalised loads under different vibration modes of F-rail in time domain after 20 Hz low-pass filter.

In the frequency domain, the peak accelerations of the F-rail occur at the frequencies of 3.81 Hz and 5.92 Hz respectively, with corresponding wavelengths of 7.2 m and 4.2 m. Due to the local vibration of the F-rail, an equivalent result is obtained when the periodic irregularity wavelengths of 7.2 m and 4.2 m are applied on the LMS maglev train-bridge (as shown in Table 2 and Figure 13). Furthermore, the excitation frequencies of the maglev train when the vehicle speed is 100 km/h are 3.86 Hz and 6.61 Hz, respectively, with corresponding wavelengths of 7.2 m and 4.2 m. The excitation frequency is defined as follows

f = \frac{v}{λ}

(10)

where v is the running speed of the vehicle (unit: m/s), λ is the periodic irregularity excitation wavelength (unit: m) and f is the excitation frequency caused by the periodic irregularity (unit: Hz).

The values of the excitation frequencies obtained using equation (10) are close to the peak frequency in Figure 14(b). Zhao and Zhai (2002) showed that the characteristics of the track irregularity affect the frequencies of the bridge. The effect of the maglev train on the F-rail is periodic, and the F-rail also generates periodic vibration when the deformation of the F-rail is a regular sinusoidal wave. The wavelength (λ) of the F-rail deformation affects the vibration of the car body and the spectral frequency distribution characteristics of the F-rail (Zhao and Zhai, 2002). Although the frequency of the local vibration of the F-rail is high (55−75 Hz), the periodic irregularity caused by the local deformation of the F-rail produces more obvious vibrations of the F-rail in the low-frequency band.

Figure 16 shows the suspension force in the time domain and frequency domain with respect to the local vibration of F-rail for wavelengths 7.2 m, 6.0 m and 4.2 m, respectively. When the local vibrational wavelength of the F-rail in the time domain decreases, the suspension force becomes increasingly violent in the time domain. In the frequency domain, the suspension forces caused by the local vibration of the F-rail are in the low-frequency and high-frequency bands, with respective values of 10 Hz and ∼70−75 Hz. The peak frequency in the high-frequency band corresponds to the local vibration frequency of the F-rail, with the wavelengths given as 7.2 m, 6.0 m and 4.2 m and the corresponding frequencies given as 67.196 Hz, 70.226 Hz and 74.256 Hz. Furthermore, the peak value of the suspension force decreases in the low-frequency band and increases in the high-frequency band when the perturbation wavelength decreases and the local vibration frequency of the F-rail increases. Previous study has pointed out that the suspension control system has a good regulation effect on low-frequency interference and poor real-time adjustment effect on high-frequency interference (Liang, 2015). As the wavelength decreases and the frequency of local F-rail vibration increases, the frequency of interference gradually increases, which leads to a decrease of the suspension force in the low-frequency band and an increase in the high-frequency band. Therefore, according to Table 2 and Figure 16, the local vibration of the F-rail causes an increase of the suspension force in the high-frequency band, while the increase in the low-frequency band is attributed to the periodic irregularity caused by the local vibration.

Figure 16.

Comparison of the suspension forces under different vibration modes of F-rail: (a) time domain and (b) frequency domain.

Based on the analysis above, the local vibration of the F-rail is self-evident and concentrated at the band of 55−75 Hz, thereby resulting in a relatively large value of the suspension force between 55 and 75 Hz. Thus, the vibration of the F-rail is obvious at high frequencies, thereby resulting in a significant increase in the vibration of the bridge at the high-frequency band (55−75 Hz).

The local deformation value of the F-rail cannot be ignored as it accounts for about 20.4% of the overall deformation of the F-rail. It causes periodic irregularity with different wavelengths, which is more similar to the short-wave irregularity, compared to the deformation of the bridge. The value of the suspension force increases in the low-frequency band (within 10 Hz) when the maglev train runs on a periodic irregularity track. However, the overall rigidity of the bridge is larger when the F-rail is considered, but the acceleration of the bridge is more larger in the low-frequency band due to a larger suspension force when the F-rail is not considered. Moreover, the F-rail also has obvious vibration in the low-frequency band due to the periodic track irregularities and the increase of low-frequency suspension force.

Hence, the influence of the F-rail on the dynamic interaction of the LMS maglev train-bridge system is reflected in two aspects. First, the vibration of the bridge increases in the high-frequency band due to the high frequency and dense local vibrations of the F-rail itself. Second, the vibration of the bridge increases in the low-frequency band due to the periodic irregularities caused by the local deformations and the vibration of the F-rail in the low-frequency band.

Conclusion

The major conclusions of this study are summarised as follows:

The vertical rigidity of the bridge system is slightly greater when the F-rail is considered than when the F-rail is not considered. The local deformation of the F-rail is obvious and accounts for 20.4% of the total deformation of the F-rail which is significant.

The local vibration of the F-rail occurs easily while the frequency band is relatively concentrated as it is a high-frequency vibration relative to the bridge vibration. Also, the vertical vibration of the bridge in the corresponding high-frequency band is larger when the F-rail is considered than when the F-rail is not considered.

The vibration wave formed by the local vibration of the F-rail is more similar to the short-wave irregularity in comparison with that formed by the deformation of the bridge. Due to the disturbance of short-wave irregularity, the vibrations of the bridge and the F-rail at the low-frequency band is larger when the F-rail is considered than when the F-rail is not considered.

Because the prediction of the vibration of the bridge will be too small when the F-rail is not considered, the vertical dynamic interaction model of the LMS maglev train-track-bridge system should be considered.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the National Natural Science Foundation of China (Grant Nos. U1434205 and 51878565) and Cultivation Program for the Excellent Doctoral Dissertation of Southwest Jiaotong University (Grant No. D-YB201701).

ORCID iD

Xiaozhen Li

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