# Performance Analysis of Prefabricated Steel-Spring Floating-Slab Track and Its Application to Urban Express Rail Transit

Authors: Xudong Huang; Ping Li; Xinwei Luo; Liang Ling; Abdulmumin Ahmed Shuaibu; Zhiping Zeng
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The aim of the research is the design of prefabricated steel-spring floating-slab track to be applied in urban express rail transit systems. Using a developed vehicle-track dynamic-coupling equation for steel-spring floating-slab track, the effects of length, thickness, vertical damping, and use of side-mounted isolators on the floating-slab track were investigated experimentally using full-scale model and under different working conditions. The finding of the study revealed the following: (1) The prefabricated steel-spring floating-slab track can be applied to urban express rail transit, because it meets the requirements of high-speed transit while efficiently reducing noise. (2) The floating-slab track’s stability slightly increases with the increase of its length and thickness. As thickness increases, vertical displacement of the rail increases slightly, and lateral stability increases, thereby slightly improving the vehicle’s running stability. (3) When the intercity electric multiple-unit train travels along the prefabricated steel-spring floating-slab-track bed at different speeds, the wheel-axle lateral force, wheel-rail vertical force, the derailment coefficient, the wheel-weight reduction rate, and the lateral acceleration of the vehicle body are all less than the specified limits of Chinese code, thus fully meeting the safety requirements of train operation. (4) Appropriately increasing the vertical-support damping of the floating slab can improve the vehicle’s vertical dynamic performance, reduce the vertical displacement of the rail, and lower the vibration response of the floating slab. (5) Adding side-mounted vibration isolators at the joint of the floating slab could greatly improve the stability of the floating slab itself and appropriately reduce the vehicle’s vertical vibration response. Due to the optimization and establishment of relevant factors influencing the performance of prefabricated steel-string floating-slab track achieved in the study, the results obtained are particularly useful for setting safety, comfort, and stability requirements of the floating slab.


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## Body

## 1. Introduction

In recent years, the rapid development of urban express rail transit has brought great convenience to people’s travel and transportation [1]. Urban express rail transit is an emerging category of rail transit, which is a cross between railway and urban rail transit, and is mainly used to solve intercity traffic problems. The development of urban express rail transit provides a new model for urban residents to live in one city and work in a neighboring city, which is of great significance for optimizing urban patterns and alleviating traffic problems in dense urban areas. Thus, the urban express rail transit system came into being to optimize urban patterns, ease the traffic pressure in dense urban areas, and better address public-transportation challenges between city centers and suburbs or satellite communities and between key towns.Owing to the tight use of urban land, urban express rail transit lines are commonly located very close to residence areas or even in tunnels below residents’ gathering places or on viaduct structures, which are gradually adopted in large areas [2]. Because of this close proximity, a serious problem to be addressed is environmental vibration due to the high speed of urban express rail transit (such as 160 km/h, the design speed of Guangzhou Rail Transits nos. 18 and 22) [3, 4].The floating-slab track is a structure that can effectively reduce vibration and noise caused by vehicles on rails. The proposed steel-spring floating-slab track uses a prefabricated reinforced-concrete structure to form an integral track. A steel-spring isolator is used to elastically isolate the track slab from the foundation to form a mass-spring vibration-isolation system. Furthermore, a side-mounted vibration isolator is incorporated to constrain lateral displacement and vibration of the track slab. The advantages are that the steel-spring floating-slab track has better three-dimensional elasticity, less lateral displacement, good vibration-isolation performance, and easiness to maintain and replace [5, 6].Many scholars have investigated the dynamic characteristics of floating-slab track under vehicle loads [7–13]. In 2009, Zhai et al. proposed a method to investigate the dynamics of vehicle-track systems with emphasis on theoretical modeling, numerical simulation, and experimental validation. They modeled a traditional ballasted track as two continuous parallel beams supported by a discrete elastic foundation consisting of three layers, including sleepers and ballasts. They also modeled the nonballasted slab track as two continuous parallel beams supported by a series of elastic rectangular slabs on a viscoelastic foundation [7]. On the basis of the above, a coupled-dynamics computation model for metro vehicles, and steel-spring floating-slab track, was developed as the influence of factors on the coupled system was explored such as the floating-slab dimensions (thickness and length) and mass, spring rate and spatial arrangement, and running speed [8]. MFM Hussein et al. proposed a new modeling method for discontinuous floating-slab track applied to subway. The coupling relationship between two submodels of track and tunnel under train load was expressed by Fourier series. The floating-slab track studied was found to display good vibration response [9]. Lombaert established a vehicle-track-foundation coupled dynamics model through three-dimensional numerical modeling method and evaluated the vibration reduction effect of the floating-slab track under different working conditions [10]. Hunt combined a vehicle with a track model to create a method that can be used to calculate the vibration transfer between the track and building. The results obtained in the study are valuable in evaluating the vibration response between rails, fasteners, floating track, and foundation [11]. Xu et al. proposed and applied a probabilistic model to simulate the characteristic of track irregularities by employing vehicle-track and vehicle-slab coupled system [12, 13]. Wang et al. established a spatial dynamic model of train steel-spring floating-slab-track interaction and analyzed the vibration characteristics of train passing through the steel-spring floating-slab track [14]. Huang et al. built a vehicle-track coupled dynamic system and investigated how the parameters of the floating-slab track and the train’s speed influence the vehicle-track coupling system [15]. Liang et al. investigated the vibration characteristics of the damping-pad floating slab on the long-span steel-truss cable-stayed bridge in urban rail transit by developing a theoretical model of the train-track-bridge coupling interaction in the frequency domain [16]. Lu et al. established a vehicle-track dynamic interaction model to investigate the wheel-rail interaction characteristics of the steel-spring floating-slab track and carried out the dynamic analysis between the subway vehicle and the steel-spring floating-slab track under emergency braking conditions [17].At the same time, in the research of the new floating-slab track structure that has side-mounted isolator to reduce lateral vibration, Park et al. proposed a new type of vibration isolator to overcome the shortcomings of the conventional floating-slab track and achieved good results [18]. Zhu et al. effectively suppressed the low-frequency vibration of the steel-spring floating-slab track by using a dynamic vibration absorber [19]. Ding et al. obtained the low-frequency vibration performance of the floating slab by vibration testing of the floating-slab track and optimized the parameters of the floating slab [20]. The above researches have achieved certain results in the optimization and improvement of the floating-slab structural parameters, but there is no research on the effect of lateral displacement limit effect of the floating slab.The existing researches considered the interaction principles between the subway train and the steel-spring floating-slab track under normal circumstances and lower speed [21, 22]. Currently, there is still limited experience in applying steel-spring floating-slab track in urban express rail transit for higher speed worldwide. Therefore, this study focuses on the use of prefabricated steel-spring floating-slab track in urban express rail transit for higher speed. The wheel-rail dynamic performance under high-speed conditions was investigated by applying vehicle-track coupled dynamics theory and simulation technology. The analysis and evaluation of the vehicle-track dynamics, the vehicle operational safety, ride comfort, and track structural stability, were conducted according to Chinese railway dynamic performance evaluation standards. Ultimately, the study aimed to demonstrate the feasibility of using prefabricated steel-spring floating-slab track for urban express rail transit at very high running speed. In the developed model, optimization of the key structural parameters of the floating-slab track was performed to provide the theoretical basis and technical support for its engineering design and application to urban express rail transit.


## 2. Theoretical Model and Analysis of Vehicle-Track Coupled Dynamics

### 2.1. Vehicle-Track Coupled Dynamics Model of Steel-Spring Floating-Slab Track

The study proposes a model to simulate the dynamic interaction between the vehicle and the floating-slab track in order to effectively evaluate the running safety and stability of an intercity electric multiple-unit (EMU) train under conditions of different running speeds and floating-slab-track bearing stiffness. In addition, floating-slab-track vibration characteristics were investigated. The vehicle-track coupled dynamics model of the intercity EMU train was based on vehicle-track coupled-dynamics theory [23] (Figure 1). In the model, the vehicle is simulated as a rigid multicomponent system consisting of a car body, a frame, and a wheelset. The lateral, vertical, side roll, shake, and nod movements of each part were considered. The rail is simulated as a Bernoulli-Euler beam supported on a base of elastic points. The rail-support points are arranged according to the actual fastener nodes’ spacing by considering lateral, vertical, and rotational degrees of freedom. The vertical direction of the floating slab is simulated as a bidirectional curved elastic thin slab on an elastic foundation; the lateral direction is simulated as a rigid body, considering translational and rotational degrees of freedom. The concrete foundation is also simulated as a bidirectional curved elastic thin slab on the elastic foundation. The wheel-rail normal force was determined by the Hertz nonlinear elastic-contact theory, and the tangential force was determined by the nonlinear creep theory [24].Figure 1
Vehicle-track coupled dynamics model of steel-spring floating-slab track. (a) End view. (b) Front view.
(a)
(b)


#### 2.1.1. Vehicle Dynamic Equations

According to the multibody system dynamics, the vehicle subsystem is built by considering seven rigid parts involving a car body, two bogies, and four wheelsets with the primary and the secondary suspensions. Each component is, respectively, assigned with 5 degrees of freedom (DOFs) involving the vertical displacementZ, the lateral displacement Y, the roll angle Φ, the yaw angle Ψ, and the pitch angle β. Therefore, the vehicle subsystem has a total of 35 DOFs. For more details about the vehicle dynamic equations, monograph [25] can be consulted for readers.


#### 2.1.2. Track Dynamic Equations

According to the method mentioned in [26], the track model consists of rail and floating slab. The vibrations of both were considered at the same time. The equations of motion are shown in (1)–(5).(1)
Rail dynamic equationThe rail is treated as a Bernoulli-Euler beam resting on the rail pads, and the lateral, vertical, and torsional vibrations are simultaneously taken into account. By adopting the modal superposition method, the second-order ordinary differential equations of the rail vibration can be obtained:(1)q¨zk=t+ErIyρrArkπl4qzkt=−∑i=1NsFrViZkxFi+∑j=1NwFViZkxPi,k=1∼NZ,(2)q¨yk=t+ErIzρrArkπl4qykt=−∑i=1NsFrHiZkxFi+∑j=1NwPHiZkxPi,k=1∼NY,(3)q¨tk=t+GrJzρrJrkπl4qtkt=−∑i=1NsFrTiΦkxFi+∑j=1NwPTjΦkxPi,k=1∼NT,whereEr and Gr are Young’s modulus and shear modulus of the rail, respectively; Ar and ρr are the cross-sectional area and mass density of the rail, respectively; Jry, and Jrz are, respectively, the moments of inertia of the rail section to the lateral and vertical axes; Jr0 and Jrt are the polar moments of inertia and torsional moment of inertia of the rail section, respectively; Ns and Nw are the numbers of sleepers and the number of axles in the rail section; FrVi, FrHi, and FrTi are the vertical reaction force, lateral reaction force, and torsion reaction force of the i-th fulcrum of the rail, respectively; PVj, PHj, and PTj are the vertical force, lateral force, and torque of the rail acted by j-th wheel, respectively; and xFj and xPj are the x-coordinate of the i-th fulcrum of the rail and the x-coordinate of the j-th wheel set, respectively.(2)
Floating-slab dynamic equationThe floating slab is regarded as an elastic thin plate, whose governing equation is given by(4)∂4zsx,y,t∂x4+2∂4zsx,y,t∂x2∂y2+∂4zsx,y,t∂y4+CsDs+∂zsx,y,t∂t+ρshsDs∂4wx,y,t∂t2=1Ds∑i=1NpPrVitδx−xPiδy−yPi−∑j=1NFPsVjtδx−xFjδy−yFj−∑k=1NCPcVktδx−xCkδy−yCk,where PrVi is the vertical force of the i-th rail fastener point on the track slab; FsVj is the vertical reaction force of the j-th steel spring isolator under the track slab; FcVk is the vertical shear force of the k-th force hinge between the floating slabs; zs(x, y, t) is the vertical displacement or deflection of the floating slab; xPi and yPi are the positions of the i-th rail fasten points on the floating slab; xFj and yFj are the positions of the j-th steel spring isolators under the floating slab; xCk and yCk are the positions of the k-th shear joints between the floating slabs; and hs, ρs, Cs, Es, vs, and Ds are slab thickness, density, damping coefficient, modulus of elasticity, Poisson’s ratio, and bending stiffness, respectively.The generalized coordinateTmn(t) of the floating slab is introduced, and the above partial differential equation is converted into a second-order ordinary differential equation by the Ritz method, as shown in equation (7):(5)T¨mnt+CsρshsT˙mnt+DsρshsB3B2+2B4B5+B1B6B1B2Tmnt=1ρshsB1B2∑i=1NPPrVitXmxPiYnypi−∑j=1NFFsVjtXmxFjYnyFj−∑k=1NCFcVktXmxCjYnyCj,where m=1,2,…,Nx and n=1,2,…,Ny.(6)B1=∫0LsXm2xdx,B2=∫0WsYn2ydy,B3=∫0LsX‴mxXmxdx,B4=∫0LsXm″xXmxdx,B4=∫0WsYn″yYnydx,B4=∫0WsY‴nyYnydx.At momentt, the vertical displacement at point (x, y) of the track is(7)zsx,y,t=∑m=1Nx∑n=1NxXmxYnyTmnt,where Nx and Ny are the cutoff mode orders of directions for length and width of the floating slab, respectively, and Xmx and Yny are the beam-mode functions of directions for length and width of the floating slab, respectively.


### 2.2. The Wheel and Rail Interaction Principle

The vehicle-floating-slab track (FST) is a dynamic interaction system, and the wheel-rail relationship is the link between the vehicle subsystem and the track subsystem. In previous vehicle-track dynamics equations, if the wheel-rail interaction force was determined, the numerical simulation method was applied, and the dynamics simulation analysis of the vehicle-track system could be performed. In this research, the wheel-rail contact geometry was determined according to the principle of wheel-rail contact mentioned in [27]. The wheel and rail normal force and wheel and rail creep force were calculated according to the method mentioned in [26, 28]. On obtaining the wheel-rail force, the values can be substituted into the dynamic equations of the vehicle and the track as the reaction force of the wheel and the external load of the track.


### 2.3. Track Irregularities

Because the vehicle-track coupled dynamic system is very complicated and extensive, a fast explicit-integration method should be used to solve its dynamic response problem [29]. At present, China has no reliable track-irregularity data for urban express rail transit. For the purpose of analysis, considering the characteristics of the floating-slab track and the deterioration of railway-line smoothness after long-term operation, the excitation input of the vehicle-track dynamic system was based on the U.S. six-grade track spectrum that closely matches the Chinese urban express rail transit [30]. Accordingly, the power-spectral-density expressions of the track-vertical-profile, track-alignment, rail-gauge, and track-cross-level irregularities can be expressed as shown in equations (10) and (11):(1)
Track-vertical-profile irregularity:(8)SvΩ=kAvΩc2Ω2Ω2+Ωc2,whereSv(Ω) is the power-spectral density of track-vertical-profile irregularity [cm2/(rad/m)], Av is the roughness constant (cm2·rad/m), Ωc is the cutoff frequency (rad/m), and k is the safety coefficient.(2)
Track-alignment irregularity:(9)SaΩ=kAvΩc2Ω2Ω2+Ωc2,whereSv(Ω) is the power-spectral density of track-alignment irregularity [cm2/(rad/m)], Aa is the roughness constant (cm2·rad/m), and Ωc is the cutoff frequency (rad/m).(3)
Rail-gauge and track-cross-level irregularities:(10)ScΩ=SgΩ=4kAvΩc2Ω2+Ωs2Ω2+Ωc2,where Sc(Ω) and Sg(Ω) are the power-spectral densities of rail-gauge and track-cross-level irregularities [cm2/(rad/m)], respectively, and Ωs is the cutoff frequency (rad/m).According to the track power spectrum density expression, a new algorithm based on the frequency domain power spectrum equivalent was used to obtain the amplitude and stochastic phase of the spectrum. The inverse Fourier transform was used to obtain time-domain samples of the stochastic irregularity of the track (Figures2∼5), which were used as the excitation input of the vehicle-track dynamics system.Figure 2
Lateral stochastic irregularities in the left rail.Figure 3
Lateral stochastic irregularities in the right rail.Figure 4
Vertical stochastic irregularities in the left rail.Figure 5
Vertical stochastic irregularities in the right rail.


### 2.4. Numerical Integration Method

It can be seen that the developed dynamics model has large DOFs involving many nonlinear factors and time-varying parameters. Consequently, an efficient numerical integration algorithm is essential for this problem. In this paper, the Zhai method [31] is adopted to solve such a large-scale dynamic model, which has the integration form as follows:(11)Xn+1=Xn+X˙nΔt+0.5+ψX¨nΔt2−ψX¨n−1Δt2,X˙n+1=X˙n+1+φX¨nΔt−φX¨n−1Δt,where X, X˙, and X¨ are the generalized displacement, velocity, and acceleration of the system, respectively; Δt is the time step for numerical integration; the independent parameters φ and ψ are used for controlling the stability of the algorithm; the subscript n indicates the integration at the time of nΔt.


## 2.1. Vehicle-Track Coupled Dynamics Model of Steel-Spring Floating-Slab Track

The study proposes a model to simulate the dynamic interaction between the vehicle and the floating-slab track in order to effectively evaluate the running safety and stability of an intercity electric multiple-unit (EMU) train under conditions of different running speeds and floating-slab-track bearing stiffness. In addition, floating-slab-track vibration characteristics were investigated. The vehicle-track coupled dynamics model of the intercity EMU train was based on vehicle-track coupled-dynamics theory [23] (Figure 1). In the model, the vehicle is simulated as a rigid multicomponent system consisting of a car body, a frame, and a wheelset. The lateral, vertical, side roll, shake, and nod movements of each part were considered. The rail is simulated as a Bernoulli-Euler beam supported on a base of elastic points. The rail-support points are arranged according to the actual fastener nodes’ spacing by considering lateral, vertical, and rotational degrees of freedom. The vertical direction of the floating slab is simulated as a bidirectional curved elastic thin slab on an elastic foundation; the lateral direction is simulated as a rigid body, considering translational and rotational degrees of freedom. The concrete foundation is also simulated as a bidirectional curved elastic thin slab on the elastic foundation. The wheel-rail normal force was determined by the Hertz nonlinear elastic-contact theory, and the tangential force was determined by the nonlinear creep theory [24].Figure 1
Vehicle-track coupled dynamics model of steel-spring floating-slab track. (a) End view. (b) Front view.
(a)
(b)


### 2.1.1. Vehicle Dynamic Equations

According to the multibody system dynamics, the vehicle subsystem is built by considering seven rigid parts involving a car body, two bogies, and four wheelsets with the primary and the secondary suspensions. Each component is, respectively, assigned with 5 degrees of freedom (DOFs) involving the vertical displacementZ, the lateral displacement Y, the roll angle Φ, the yaw angle Ψ, and the pitch angle β. Therefore, the vehicle subsystem has a total of 35 DOFs. For more details about the vehicle dynamic equations, monograph [25] can be consulted for readers.


### 2.1.2. Track Dynamic Equations

According to the method mentioned in [26], the track model consists of rail and floating slab. The vibrations of both were considered at the same time. The equations of motion are shown in (1)–(5).(1)
Rail dynamic equationThe rail is treated as a Bernoulli-Euler beam resting on the rail pads, and the lateral, vertical, and torsional vibrations are simultaneously taken into account. By adopting the modal superposition method, the second-order ordinary differential equations of the rail vibration can be obtained:(1)q¨zk=t+ErIyρrArkπl4qzkt=−∑i=1NsFrViZkxFi+∑j=1NwFViZkxPi,k=1∼NZ,(2)q¨yk=t+ErIzρrArkπl4qykt=−∑i=1NsFrHiZkxFi+∑j=1NwPHiZkxPi,k=1∼NY,(3)q¨tk=t+GrJzρrJrkπl4qtkt=−∑i=1NsFrTiΦkxFi+∑j=1NwPTjΦkxPi,k=1∼NT,whereEr and Gr are Young’s modulus and shear modulus of the rail, respectively; Ar and ρr are the cross-sectional area and mass density of the rail, respectively; Jry, and Jrz are, respectively, the moments of inertia of the rail section to the lateral and vertical axes; Jr0 and Jrt are the polar moments of inertia and torsional moment of inertia of the rail section, respectively; Ns and Nw are the numbers of sleepers and the number of axles in the rail section; FrVi, FrHi, and FrTi are the vertical reaction force, lateral reaction force, and torsion reaction force of the i-th fulcrum of the rail, respectively; PVj, PHj, and PTj are the vertical force, lateral force, and torque of the rail acted by j-th wheel, respectively; and xFj and xPj are the x-coordinate of the i-th fulcrum of the rail and the x-coordinate of the j-th wheel set, respectively.(2)
Floating-slab dynamic equationThe floating slab is regarded as an elastic thin plate, whose governing equation is given by(4)∂4zsx,y,t∂x4+2∂4zsx,y,t∂x2∂y2+∂4zsx,y,t∂y4+CsDs+∂zsx,y,t∂t+ρshsDs∂4wx,y,t∂t2=1Ds∑i=1NpPrVitδx−xPiδy−yPi−∑j=1NFPsVjtδx−xFjδy−yFj−∑k=1NCPcVktδx−xCkδy−yCk,where PrVi is the vertical force of the i-th rail fastener point on the track slab; FsVj is the vertical reaction force of the j-th steel spring isolator under the track slab; FcVk is the vertical shear force of the k-th force hinge between the floating slabs; zs(x, y, t) is the vertical displacement or deflection of the floating slab; xPi and yPi are the positions of the i-th rail fasten points on the floating slab; xFj and yFj are the positions of the j-th steel spring isolators under the floating slab; xCk and yCk are the positions of the k-th shear joints between the floating slabs; and hs, ρs, Cs, Es, vs, and Ds are slab thickness, density, damping coefficient, modulus of elasticity, Poisson’s ratio, and bending stiffness, respectively.The generalized coordinateTmn(t) of the floating slab is introduced, and the above partial differential equation is converted into a second-order ordinary differential equation by the Ritz method, as shown in equation (7):(5)T¨mnt+CsρshsT˙mnt+DsρshsB3B2+2B4B5+B1B6B1B2Tmnt=1ρshsB1B2∑i=1NPPrVitXmxPiYnypi−∑j=1NFFsVjtXmxFjYnyFj−∑k=1NCFcVktXmxCjYnyCj,where m=1,2,…,Nx and n=1,2,…,Ny.(6)B1=∫0LsXm2xdx,B2=∫0WsYn2ydy,B3=∫0LsX‴mxXmxdx,B4=∫0LsXm″xXmxdx,B4=∫0WsYn″yYnydx,B4=∫0WsY‴nyYnydx.At momentt, the vertical displacement at point (x, y) of the track is(7)zsx,y,t=∑m=1Nx∑n=1NxXmxYnyTmnt,where Nx and Ny are the cutoff mode orders of directions for length and width of the floating slab, respectively, and Xmx and Yny are the beam-mode functions of directions for length and width of the floating slab, respectively.


## 2.1.1. Vehicle Dynamic Equations

According to the multibody system dynamics, the vehicle subsystem is built by considering seven rigid parts involving a car body, two bogies, and four wheelsets with the primary and the secondary suspensions. Each component is, respectively, assigned with 5 degrees of freedom (DOFs) involving the vertical displacementZ, the lateral displacement Y, the roll angle Φ, the yaw angle Ψ, and the pitch angle β. Therefore, the vehicle subsystem has a total of 35 DOFs. For more details about the vehicle dynamic equations, monograph [25] can be consulted for readers.


## 2.1.2. Track Dynamic Equations

According to the method mentioned in [26], the track model consists of rail and floating slab. The vibrations of both were considered at the same time. The equations of motion are shown in (1)–(5).(1)
Rail dynamic equationThe rail is treated as a Bernoulli-Euler beam resting on the rail pads, and the lateral, vertical, and torsional vibrations are simultaneously taken into account. By adopting the modal superposition method, the second-order ordinary differential equations of the rail vibration can be obtained:(1)q¨zk=t+ErIyρrArkπl4qzkt=−∑i=1NsFrViZkxFi+∑j=1NwFViZkxPi,k=1∼NZ,(2)q¨yk=t+ErIzρrArkπl4qykt=−∑i=1NsFrHiZkxFi+∑j=1NwPHiZkxPi,k=1∼NY,(3)q¨tk=t+GrJzρrJrkπl4qtkt=−∑i=1NsFrTiΦkxFi+∑j=1NwPTjΦkxPi,k=1∼NT,whereEr and Gr are Young’s modulus and shear modulus of the rail, respectively; Ar and ρr are the cross-sectional area and mass density of the rail, respectively; Jry, and Jrz are, respectively, the moments of inertia of the rail section to the lateral and vertical axes; Jr0 and Jrt are the polar moments of inertia and torsional moment of inertia of the rail section, respectively; Ns and Nw are the numbers of sleepers and the number of axles in the rail section; FrVi, FrHi, and FrTi are the vertical reaction force, lateral reaction force, and torsion reaction force of the i-th fulcrum of the rail, respectively; PVj, PHj, and PTj are the vertical force, lateral force, and torque of the rail acted by j-th wheel, respectively; and xFj and xPj are the x-coordinate of the i-th fulcrum of the rail and the x-coordinate of the j-th wheel set, respectively.(2)
Floating-slab dynamic equationThe floating slab is regarded as an elastic thin plate, whose governing equation is given by(4)∂4zsx,y,t∂x4+2∂4zsx,y,t∂x2∂y2+∂4zsx,y,t∂y4+CsDs+∂zsx,y,t∂t+ρshsDs∂4wx,y,t∂t2=1Ds∑i=1NpPrVitδx−xPiδy−yPi−∑j=1NFPsVjtδx−xFjδy−yFj−∑k=1NCPcVktδx−xCkδy−yCk,where PrVi is the vertical force of the i-th rail fastener point on the track slab; FsVj is the vertical reaction force of the j-th steel spring isolator under the track slab; FcVk is the vertical shear force of the k-th force hinge between the floating slabs; zs(x, y, t) is the vertical displacement or deflection of the floating slab; xPi and yPi are the positions of the i-th rail fasten points on the floating slab; xFj and yFj are the positions of the j-th steel spring isolators under the floating slab; xCk and yCk are the positions of the k-th shear joints between the floating slabs; and hs, ρs, Cs, Es, vs, and Ds are slab thickness, density, damping coefficient, modulus of elasticity, Poisson’s ratio, and bending stiffness, respectively.The generalized coordinateTmn(t) of the floating slab is introduced, and the above partial differential equation is converted into a second-order ordinary differential equation by the Ritz method, as shown in equation (7):(5)T¨mnt+CsρshsT˙mnt+DsρshsB3B2+2B4B5+B1B6B1B2Tmnt=1ρshsB1B2∑i=1NPPrVitXmxPiYnypi−∑j=1NFFsVjtXmxFjYnyFj−∑k=1NCFcVktXmxCjYnyCj,where m=1,2,…,Nx and n=1,2,…,Ny.(6)B1=∫0LsXm2xdx,B2=∫0WsYn2ydy,B3=∫0LsX‴mxXmxdx,B4=∫0LsXm″xXmxdx,B4=∫0WsYn″yYnydx,B4=∫0WsY‴nyYnydx.At momentt, the vertical displacement at point (x, y) of the track is(7)zsx,y,t=∑m=1Nx∑n=1NxXmxYnyTmnt,where Nx and Ny are the cutoff mode orders of directions for length and width of the floating slab, respectively, and Xmx and Yny are the beam-mode functions of directions for length and width of the floating slab, respectively.


## 2.2. The Wheel and Rail Interaction Principle

The vehicle-floating-slab track (FST) is a dynamic interaction system, and the wheel-rail relationship is the link between the vehicle subsystem and the track subsystem. In previous vehicle-track dynamics equations, if the wheel-rail interaction force was determined, the numerical simulation method was applied, and the dynamics simulation analysis of the vehicle-track system could be performed. In this research, the wheel-rail contact geometry was determined according to the principle of wheel-rail contact mentioned in [27]. The wheel and rail normal force and wheel and rail creep force were calculated according to the method mentioned in [26, 28]. On obtaining the wheel-rail force, the values can be substituted into the dynamic equations of the vehicle and the track as the reaction force of the wheel and the external load of the track.


## 2.3. Track Irregularities

Because the vehicle-track coupled dynamic system is very complicated and extensive, a fast explicit-integration method should be used to solve its dynamic response problem [29]. At present, China has no reliable track-irregularity data for urban express rail transit. For the purpose of analysis, considering the characteristics of the floating-slab track and the deterioration of railway-line smoothness after long-term operation, the excitation input of the vehicle-track dynamic system was based on the U.S. six-grade track spectrum that closely matches the Chinese urban express rail transit [30]. Accordingly, the power-spectral-density expressions of the track-vertical-profile, track-alignment, rail-gauge, and track-cross-level irregularities can be expressed as shown in equations (10) and (11):(1)
Track-vertical-profile irregularity:(8)SvΩ=kAvΩc2Ω2Ω2+Ωc2,whereSv(Ω) is the power-spectral density of track-vertical-profile irregularity [cm2/(rad/m)], Av is the roughness constant (cm2·rad/m), Ωc is the cutoff frequency (rad/m), and k is the safety coefficient.(2)
Track-alignment irregularity:(9)SaΩ=kAvΩc2Ω2Ω2+Ωc2,whereSv(Ω) is the power-spectral density of track-alignment irregularity [cm2/(rad/m)], Aa is the roughness constant (cm2·rad/m), and Ωc is the cutoff frequency (rad/m).(3)
Rail-gauge and track-cross-level irregularities:(10)ScΩ=SgΩ=4kAvΩc2Ω2+Ωs2Ω2+Ωc2,where Sc(Ω) and Sg(Ω) are the power-spectral densities of rail-gauge and track-cross-level irregularities [cm2/(rad/m)], respectively, and Ωs is the cutoff frequency (rad/m).According to the track power spectrum density expression, a new algorithm based on the frequency domain power spectrum equivalent was used to obtain the amplitude and stochastic phase of the spectrum. The inverse Fourier transform was used to obtain time-domain samples of the stochastic irregularity of the track (Figures2∼5), which were used as the excitation input of the vehicle-track dynamics system.Figure 2
Lateral stochastic irregularities in the left rail.Figure 3
Lateral stochastic irregularities in the right rail.Figure 4
Vertical stochastic irregularities in the left rail.Figure 5
Vertical stochastic irregularities in the right rail.


## 2.4. Numerical Integration Method

It can be seen that the developed dynamics model has large DOFs involving many nonlinear factors and time-varying parameters. Consequently, an efficient numerical integration algorithm is essential for this problem. In this paper, the Zhai method [31] is adopted to solve such a large-scale dynamic model, which has the integration form as follows:(11)Xn+1=Xn+X˙nΔt+0.5+ψX¨nΔt2−ψX¨n−1Δt2,X˙n+1=X˙n+1+φX¨nΔt−φX¨n−1Δt,where X, X˙, and X¨ are the generalized displacement, velocity, and acceleration of the system, respectively; Δt is the time step for numerical integration; the independent parameters φ and ψ are used for controlling the stability of the algorithm; the subscript n indicates the integration at the time of nΔt.


## 3. Basic Parameters for Dynamic Analysis

### 3.1. Vehicle Parameters

The fully loaded CRH6 intercity EMU train was considered as model vehicle when setting the vehicle parameters required for the vehicle-track-coupled dynamics simulation while keeping the overall parameters within safety limits (Table1).Table 1
Basic parameters of CRH6 intercity EMU train.
Parameter name
Value
Unit
Vehicle parameters
Bogie center distance (vehicle distance)
17,500
mm
Wheelbase
2500
mm
Wheel rolling-circle lateral span
1493
mm
Wheel rolling-circle diameter
860
mm
Wheelset inner distance
1353
mm
Wheel shape
LMA
—
Rail shape
CHN60
—
Mass/moment of inertia and center of gravity
Car-body quality
37.2
t
Body-roll moment of inertia
107.6
t·m2
Car-body nodding moment of inertia
1426.8
t·m2
Car-body shaking head moment of inertia
1402.7
t·m2
Car-weight heart distance from rail surface
1.712
m
Frame quality
2.70
t
Frame roll moment of inertia
2.19
t·m2
Frame nodding moment of inertia
1.48
t·m2
Frame moving-head moment of inertia
2.70
t·m2
Height of gravity center of frame
0.5
m
Wheelset quality
1.935
t
Wheelset rolling moment of inertia
0.95
t·m2
Wheelset nodding moment of inertia
0.15
t·m2
Wheelset shaking moment of inertia
0.95
t·m2
Structural parameters
Primary suspension
Half distance of lateral span
2.10
m
Height of upper spring-loaded point from rail surface
0.876
m
Height of lower spring-loaded point from rail surface
0.681
m
Half distance of vertical-damping lateral span
2.10
m
Longitudinal damping from vertical distance to center of bogie
1.491
m
Height of upper vertical-damping point from rail surface
0.9
m
Height of lower vertical-damping point from rail surface
0.48
m
Half distance of lateral span of axle-box boom node
2.10
m
Height of axle-box boom node from rail surface
0.46
m
Longitudinal distance of axle-box arm node from center of bogie
0.77
m
Secondary suspension
Air-spring lateral span
2.02
m
Height of upper surface of air spring from rail surface
0.884
m
Height of lower surface of air spring from rail surface
0.633
m
Lateral span of anti-movement-of-snake damper
2.6
m
Longitudinal distance between anti-movement-of-snake damper and body-connection point from center of car body (inside center pin)
9.51
m
Longitudinal distance between anti-movement-of-snake damper and frame-connection point from center of car body (inside center pin)
8.79
m
Height of anti-movement-of-snake damper and car-body-connection point from rail surface
0.556
m
Height of anti-movement-of-snake damper and frame-connection point from rail surface
0.492
m
Longitudinal distance of lateral damper from center of car body
8.9
m
Lateral distance between lateral damper and vehicle-body-connection point from center line of vehicle body
1.4
m
Lateral distance between lateral damper and frame-connection point from center line of vehicle body
0.933
m
Height of lateral damper and body-connection point from rail surface
0.764
m
Height of lateral damper and frame-connection point from rail surface
0.724
m
Vertical distance of vertical damper from center of car body
8.2
m
Lateral distance between vertical damper and vehicle-body-connection point from center line of vehicle body
1.323
m
Lateral distance between vertical damper and frame-connection point from center line of vehicle body
1.323
m
Height of vertical damper and body-connection point from rail surface
0.81
m
Height of vertical damper and frame-connection point from rail surface
0.382
m
Lateral-stop lateral span
0.211
m
Height of lateral stop from rail surface
0.49
m
Height of anti-rolling torsion bar from rail surface
0.792
m
Anti-rolling torsion-bar-mount lateral span
0.952
m
Longitudinal distance of anti-rolling torsion bar from center of bogie
0.4
m
Height of connection point between traction pin and vehicle body from rail surface
0.884
m
Suspension parameters
Primary suspension
Steel-spring longitudinal stiffness (per axle box) (including damping pad)
0.9
MN/m
Steel-spring lateral stiffness (per axle box) (including damping pad)
0.9
MN/m
Steel-spring vertical stiffness (per axle box)
1.30
MN/m
Vertical damping (per axle box)
9.8
kN·s/m
Vertical-damper series stiffness
3.5
MN/m
Axle-box boom-joint longitudinal stiffness (per axle box)
17.1
MN/m
Axle-box boom-joint lateral stiffness (per axle box)
6.1
MN/m
Secondary suspension
Air-spring longitudinal/lateral stiffness
0.20
MN/m
Air-spring vertical stiffness
0.22
MN/m
Air-spring emergency rubber stiffness
13.7
MN/m
Stiffness between emergency spring and upper cover
36
mm
Lateral-damper series stiffness
3.75
MN/m
Anti-movement-of-snake-damper series stiffness
8
MN/m
Vertical damping
9.8
kN·s/m
Lateral-stop free clearance
20
mm
Anti-rolling torsion-bar equivalent stiffness (single bogie)
3.0
MNm/rad


### 3.2. Rail Parameters

To ensure overall vehicle safety, the full-load parameters for the CRH6 intercity EMU train were considered. The basic calculation parameters for the rail are shown in Table2.Table 2
Basic parameters of rail.
Parameter name
Value
Unit
Rail elastic modulus
2.059 × 1011
Pa
Rail Poisson’s ratio
0.3
—
Rail density
7.83 × 103
kg/m3
Rail shape
CN60
—
Fastener vertical stiffness
3.0 × 107
N/m
Fastener lateral stiffness
2.0 × 107
N/m
Fastener vertical damping
7.5 × 104
N·s/m
Fastener lateral damping
5.0 × 104
N·s/m
Fastener spacing
0.600/0.625
m


### 3.3. Floating-Slab-Track Parameters and Layout Scheme

We studied and analyzed mainly 3.6 m and 4.8 m long floating-slab tracks. In order to reduce the vertical displacement at the joint of adjacent floating slabs, an increase in the stiffness transition of the floating slab by side-mounted vibration isolator was proposed. The side-mounted isolator was added to each end of the floating slab, as shown in Figures6∼8.Figure 6
Plan view of the prefabricated 4.8 m long floating-slab transition board.Figure 7
Comparison of vibration acceleration of floating slab obtained by test and simulation.Figure 8
Steel-spring floating-slab-track isolator.The basic parameters of the 3.6 m long prefabricated floating slab are shown in Table3, and layout schematics are shown in Figures 9–11.Table 3
Basic parameters of the 3.6 m long prefabricated floating slab.
Parameter name
Value
Vibration-isolator model
GSIU (double cylinder) isolator
Vibration-isolator stiffness
Vertical stiffness: 13.2 kN/mm
Lateral stiffness: 9.8 kN/mm
Vibration-isolator damping
Vertical damping: 0–100 N·s/mm
Lateral damping: 40.0 N·s/mm
Inside-isolator spacing
1800 mm
Floating-slab length
3600 mm
Floating-slab width
2900 mm
Floating-slab thicknesses
350 mm, 450 mm, 550 mm
Floating-slab and other track-bed stiffness transition modes
1 to 5 transition slabs are placed adjacent to other track beds (inside isolators are spaced 1200 mm apart)Figure 9
Prefabricated 3.6 m long floating-slab track.Figure 10
Sectional view of the prefabricated 3.6 m long floating-slab track.Figure 11
Plan view of the prefabricated 3.6 m long floating-slab-track transition-slab.The basic parameters of the prefabricated 4.8 m long floating slab are shown in Table4, and layout schematics are shown in Figures 6, 12, and 13.Table 4
Basic parameters of the prefabricated 4.8 m long floating-slab track.
Parameter name
Value
Vibration-isolator model
GSIU (double cylinder) isolator
Vibration-isolator stiffness
Vertical stiffness: 13.2 kN/mm
Lateral stiffness: 9.8 kN/mm
Vibration-isolator damping
Vertical damping: 0–100 N·s/mm
Lateral damping: 40.0 N·s/mm
Inside-isolator spacing
1800 mm
Length of floating slab
4800 mm
Width of floating slab
2900 mm
Thickness of floating slabs
350 mm, 450 mm, 550 mm
Floating-slab and other track-bed stiffness transition modes
1 to 5 transition slabs are placed adjacent to other track beds (in-slab isolators are spaced 1200 mm apart)Figure 12
Plan view of the prefabricated 4.8 m long floating slab.Figure 13
Sectional view of the prefabricated 4.8 m long floating slab.


### 3.4. Model Reliability Verification

Here only the verification results of the vibration response of the floating slab are provided. Figure7 shows comparison of the results and analysis of the vertical vibration acceleration of the floating-slab track bed in a straight section of a subway line, with train passing speed about 55 km/h. According to the vibration response results at a cross section of the floating-slab bed (Figure 7), it can be easily distinguished that the vibration acceleration responses were pronounced at measurement points when the subway vehicle is passing through. When the bogies of subway vehicle pass through the vibration measuring points, the vibration accelerations of the floating slab bed have obvious fluctuations. In the simulation calculation, the vertical acceleration of the floating-slab track bed can more clearly reflect the vibration state of each wheel set when the vehicle is passing through the measurement points.The vertical vibration acceleration test and simulation calculations of the floating-slab track bed have maximum values of 1.22 g and 1.18 g and effective values of 0.172 g and 0.163 g, respectively. The simulation calculation results are slightly smaller than the test results, but they are all within the acceptable error range. The above results show that the simulation calculation model can better reflect the vibration response process of the floating slab bed. The intercity train vehicle-floating-slab track dynamic model established in this paper can be used to evaluate the wheel-rail dynamic performance of floating-slab bed under fast driving conditions.


## 3.1. Vehicle Parameters

The fully loaded CRH6 intercity EMU train was considered as model vehicle when setting the vehicle parameters required for the vehicle-track-coupled dynamics simulation while keeping the overall parameters within safety limits (Table1).Table 1
Basic parameters of CRH6 intercity EMU train.
Parameter name
Value
Unit
Vehicle parameters
Bogie center distance (vehicle distance)
17,500
mm
Wheelbase
2500
mm
Wheel rolling-circle lateral span
1493
mm
Wheel rolling-circle diameter
860
mm
Wheelset inner distance
1353
mm
Wheel shape
LMA
—
Rail shape
CHN60
—
Mass/moment of inertia and center of gravity
Car-body quality
37.2
t
Body-roll moment of inertia
107.6
t·m2
Car-body nodding moment of inertia
1426.8
t·m2
Car-body shaking head moment of inertia
1402.7
t·m2
Car-weight heart distance from rail surface
1.712
m
Frame quality
2.70
t
Frame roll moment of inertia
2.19
t·m2
Frame nodding moment of inertia
1.48
t·m2
Frame moving-head moment of inertia
2.70
t·m2
Height of gravity center of frame
0.5
m
Wheelset quality
1.935
t
Wheelset rolling moment of inertia
0.95
t·m2
Wheelset nodding moment of inertia
0.15
t·m2
Wheelset shaking moment of inertia
0.95
t·m2
Structural parameters
Primary suspension
Half distance of lateral span
2.10
m
Height of upper spring-loaded point from rail surface
0.876
m
Height of lower spring-loaded point from rail surface
0.681
m
Half distance of vertical-damping lateral span
2.10
m
Longitudinal damping from vertical distance to center of bogie
1.491
m
Height of upper vertical-damping point from rail surface
0.9
m
Height of lower vertical-damping point from rail surface
0.48
m
Half distance of lateral span of axle-box boom node
2.10
m
Height of axle-box boom node from rail surface
0.46
m
Longitudinal distance of axle-box arm node from center of bogie
0.77
m
Secondary suspension
Air-spring lateral span
2.02
m
Height of upper surface of air spring from rail surface
0.884
m
Height of lower surface of air spring from rail surface
0.633
m
Lateral span of anti-movement-of-snake damper
2.6
m
Longitudinal distance between anti-movement-of-snake damper and body-connection point from center of car body (inside center pin)
9.51
m
Longitudinal distance between anti-movement-of-snake damper and frame-connection point from center of car body (inside center pin)
8.79
m
Height of anti-movement-of-snake damper and car-body-connection point from rail surface
0.556
m
Height of anti-movement-of-snake damper and frame-connection point from rail surface
0.492
m
Longitudinal distance of lateral damper from center of car body
8.9
m
Lateral distance between lateral damper and vehicle-body-connection point from center line of vehicle body
1.4
m
Lateral distance between lateral damper and frame-connection point from center line of vehicle body
0.933
m
Height of lateral damper and body-connection point from rail surface
0.764
m
Height of lateral damper and frame-connection point from rail surface
0.724
m
Vertical distance of vertical damper from center of car body
8.2
m
Lateral distance between vertical damper and vehicle-body-connection point from center line of vehicle body
1.323
m
Lateral distance between vertical damper and frame-connection point from center line of vehicle body
1.323
m
Height of vertical damper and body-connection point from rail surface
0.81
m
Height of vertical damper and frame-connection point from rail surface
0.382
m
Lateral-stop lateral span
0.211
m
Height of lateral stop from rail surface
0.49
m
Height of anti-rolling torsion bar from rail surface
0.792
m
Anti-rolling torsion-bar-mount lateral span
0.952
m
Longitudinal distance of anti-rolling torsion bar from center of bogie
0.4
m
Height of connection point between traction pin and vehicle body from rail surface
0.884
m
Suspension parameters
Primary suspension
Steel-spring longitudinal stiffness (per axle box) (including damping pad)
0.9
MN/m
Steel-spring lateral stiffness (per axle box) (including damping pad)
0.9
MN/m
Steel-spring vertical stiffness (per axle box)
1.30
MN/m
Vertical damping (per axle box)
9.8
kN·s/m
Vertical-damper series stiffness
3.5
MN/m
Axle-box boom-joint longitudinal stiffness (per axle box)
17.1
MN/m
Axle-box boom-joint lateral stiffness (per axle box)
6.1
MN/m
Secondary suspension
Air-spring longitudinal/lateral stiffness
0.20
MN/m
Air-spring vertical stiffness
0.22
MN/m
Air-spring emergency rubber stiffness
13.7
MN/m
Stiffness between emergency spring and upper cover
36
mm
Lateral-damper series stiffness
3.75
MN/m
Anti-movement-of-snake-damper series stiffness
8
MN/m
Vertical damping
9.8
kN·s/m
Lateral-stop free clearance
20
mm
Anti-rolling torsion-bar equivalent stiffness (single bogie)
3.0
MNm/rad


## 3.2. Rail Parameters

To ensure overall vehicle safety, the full-load parameters for the CRH6 intercity EMU train were considered. The basic calculation parameters for the rail are shown in Table2.Table 2
Basic parameters of rail.
Parameter name
Value
Unit
Rail elastic modulus
2.059 × 1011
Pa
Rail Poisson’s ratio
0.3
—
Rail density
7.83 × 103
kg/m3
Rail shape
CN60
—
Fastener vertical stiffness
3.0 × 107
N/m
Fastener lateral stiffness
2.0 × 107
N/m
Fastener vertical damping
7.5 × 104
N·s/m
Fastener lateral damping
5.0 × 104
N·s/m
Fastener spacing
0.600/0.625
m


## 3.3. Floating-Slab-Track Parameters and Layout Scheme

We studied and analyzed mainly 3.6 m and 4.8 m long floating-slab tracks. In order to reduce the vertical displacement at the joint of adjacent floating slabs, an increase in the stiffness transition of the floating slab by side-mounted vibration isolator was proposed. The side-mounted isolator was added to each end of the floating slab, as shown in Figures6∼8.Figure 6
Plan view of the prefabricated 4.8 m long floating-slab transition board.Figure 7
Comparison of vibration acceleration of floating slab obtained by test and simulation.Figure 8
Steel-spring floating-slab-track isolator.The basic parameters of the 3.6 m long prefabricated floating slab are shown in Table3, and layout schematics are shown in Figures 9–11.Table 3
Basic parameters of the 3.6 m long prefabricated floating slab.
Parameter name
Value
Vibration-isolator model
GSIU (double cylinder) isolator
Vibration-isolator stiffness
Vertical stiffness: 13.2 kN/mm
Lateral stiffness: 9.8 kN/mm
Vibration-isolator damping
Vertical damping: 0–100 N·s/mm
Lateral damping: 40.0 N·s/mm
Inside-isolator spacing
1800 mm
Floating-slab length
3600 mm
Floating-slab width
2900 mm
Floating-slab thicknesses
350 mm, 450 mm, 550 mm
Floating-slab and other track-bed stiffness transition modes
1 to 5 transition slabs are placed adjacent to other track beds (inside isolators are spaced 1200 mm apart)Figure 9
Prefabricated 3.6 m long floating-slab track.Figure 10
Sectional view of the prefabricated 3.6 m long floating-slab track.Figure 11
Plan view of the prefabricated 3.6 m long floating-slab-track transition-slab.The basic parameters of the prefabricated 4.8 m long floating slab are shown in Table4, and layout schematics are shown in Figures 6, 12, and 13.Table 4
Basic parameters of the prefabricated 4.8 m long floating-slab track.
Parameter name
Value
Vibration-isolator model
GSIU (double cylinder) isolator
Vibration-isolator stiffness
Vertical stiffness: 13.2 kN/mm
Lateral stiffness: 9.8 kN/mm
Vibration-isolator damping
Vertical damping: 0–100 N·s/mm
Lateral damping: 40.0 N·s/mm
Inside-isolator spacing
1800 mm
Length of floating slab
4800 mm
Width of floating slab
2900 mm
Thickness of floating slabs
350 mm, 450 mm, 550 mm
Floating-slab and other track-bed stiffness transition modes
1 to 5 transition slabs are placed adjacent to other track beds (in-slab isolators are spaced 1200 mm apart)Figure 12
Plan view of the prefabricated 4.8 m long floating slab.Figure 13
Sectional view of the prefabricated 4.8 m long floating slab.


## 3.4. Model Reliability Verification

Here only the verification results of the vibration response of the floating slab are provided. Figure7 shows comparison of the results and analysis of the vertical vibration acceleration of the floating-slab track bed in a straight section of a subway line, with train passing speed about 55 km/h. According to the vibration response results at a cross section of the floating-slab bed (Figure 7), it can be easily distinguished that the vibration acceleration responses were pronounced at measurement points when the subway vehicle is passing through. When the bogies of subway vehicle pass through the vibration measuring points, the vibration accelerations of the floating slab bed have obvious fluctuations. In the simulation calculation, the vertical acceleration of the floating-slab track bed can more clearly reflect the vibration state of each wheel set when the vehicle is passing through the measurement points.The vertical vibration acceleration test and simulation calculations of the floating-slab track bed have maximum values of 1.22 g and 1.18 g and effective values of 0.172 g and 0.163 g, respectively. The simulation calculation results are slightly smaller than the test results, but they are all within the acceptable error range. The above results show that the simulation calculation model can better reflect the vibration response process of the floating slab bed. The intercity train vehicle-floating-slab track dynamic model established in this paper can be used to evaluate the wheel-rail dynamic performance of floating-slab bed under fast driving conditions.


## 4. Analysis of Driving Safety and Stability Running on Prefabricated Steel-Spring Floating Slab

### 4.1. Analysis of the Influence of the Length of the Floating Slab on Vehicle-Track Dynamics Characteristics

On the basis of the Chinese railway code [32, 33], the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab tracks were chosen as focus of this research. The safety, stability, and comfort of CRH6 intercity EMU trains running at different speeds and on different radial lines were investigated. Also, the performance and the stability of the track itself were analyzed. The working conditions (Tables 5–13) considered in this study are straight line-140, straight line-160, and straight line-200, which indicates that the intercity EMU train passes through a straight line at the speeds of 140, 160, and 200 km/h, respectively, and Curve-140 and Curve-160, which indicates that the intercity EMU train travels the curve sections of R = 1100 m and R = 1500 m at the speeds of 140 and 160 km/h, respectively. L values of 3.6 and 4.8 m indicate the two lengths of the prefabricated steel-spring floating-slab track. The calculation results of wheel-rail-system dynamic response, vehicle stability and comfort, rail dynamic response, and floating-slab dynamic response under various conditions are shown in Tables 5–8. These tabulated results suggest the following:(i)
When the intercity EMU train runs on either the 3.6 m or 4.8 m long prefabricated floating track beds, its dynamic performance is basically the same, although the stability of the 3.6 m floating slab is slightly lower than that of the 4.8 m slab.(ii)
Whether the intercity EMU train runs at 140, 160, or 200 km/h on the straight sections and 140 or 160 km/h in the curved sections (R = 1100 m or R = 1500 m) of either the 3.6 m or 4.8 m prefabricated floating-slab-track bed, the wheel-axle lateral force, the wheel-rail vertical force, the derailment coefficient, and the wheel-load shedding rate are all less than the specified code limit. The lateral acceleration of the body is lower than the specified code limit, the stability index is “excellent,” and the comfort rating is “comfortable.”Table 5
Wheel-rail safety indicators for floating-slab-track bed.
Calculation working condition
Wheel-axle lateral force (kN)
Wheel-rail vertical force (kN)
Derailment coefficient
Wheel-load shedding rate
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
12.331
12.188
76.143
77.956
0.184
0.186
0.155
0.158
Straight line-160
12.913
12.890
79.518
80.343
0.182
0.187
0.187
0.179
Straight line-200
14.979
13.598
88.942
90.019
0.198
0.196
0.281
0.269
Curve-140
29.434
29.592
86.295
89.322
0.309
0.315
0.302
0.294
Curve-160
28.538
28.423
86.950
87.968
0.290
0.291
0.336
0.310
Specified code limits [32, 33]
60.92
250
1.0
0.90
L: length.Table 6
Vehicle-stability and comfort indicators for floating-slab-track bed.
Calculation working condition
Vehicle-body lateral acceleration (g)
Vehicle-body vertical acceleration (g)
Lateral-stability index
Vertical-stability index
Comfort index
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
0.032
0.034
0.047
0.042
1.707
1.708
1.549
1.558
0.733
0.733
Straight line-160
0.033
0.036
0.053
0.048
1.945
1.947
1.823
1.826
0.741
0.743
Straight line-200
0.040
0.040
0.057
0.053
2.002
2.005
1.870
1.871
0.914
0.919
Curve-140
0.058
0.057
0.045
0.041
2.117
2.115
1.619
1.626
1.102
1.105
Curve-160
0.067
0.065
0.049
0.045
2.185
2.181
1.649
1.652
1.199
1.196
Specified code limits [32, 33]
0.10
0.15
2.5 (excellent)
2.5 (excellent)
2.0 (comfortable)
L: length.Table 7
Vibration indicators of rail on floating-slab-track bed.
Calculation working condition
Rail vertical displacement (mm)
Rail vertical acceleration RMS value (g)
Rail lateral displacement (mm)
Rail lateral acceleration RMS value (g)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
2.546
2.454
5.897
5.753
0.209
0.253
0.286
0.275
Straight line-160
2.582
2.471
6.575
6.477
0.218
0.263
0.385
0.375
Straight line-200
2.683
2.540
7.613
7.408
0.375
0.356
0.635
0.639
Curve-140
2.670
2.603
5.671
5.606
0.292
0.320
0.997
1.003
Curve-160
2.698
2.601
6.483
6.329
0.286
0.303
1.068
1.077
RMS: root mean square;L: length.Table 8
Vibration indicators of floating-slab track.
Calculation working condition
Track vertical displacement (mm)
Track vertical acceleration RMS value (g)
Track lateral displacement (mm)
Track lateral acceleration RMS value (g)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
2.016
1.849
0.611
0.624
0.104
0.118
0.024
0.036
Straight line-160
2.033
1.850
0.644
0.782
0.111
0.125
0.025
0.038
Straight line-200
2.039
1.859
0.730
0.912
0.244
0.217
0.040
0.042
Curve-140
1.990
1.772
0.646
0.770
0.145
0.195
0.048
0.050
Curve-160
1.999
1.789
0.713
0.967
0.103
0.170
0.049
0.054
RMS: root mean square;L: length.Table 9
Vertical-vibration index of the rail.
Calculation working condition
Vertical displacement of the rail
RMS of lateral-vibration acceleration of the floating slab
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350∗
450∗
550∗
350∗
450∗
550∗
350∗
450∗
550∗
350∗
450∗
550∗
Straight line-140
4.794
4.860
4.919
4.045
3.980
4.025
7.754
7.701
7.682
7.572
7.577
7.671
Straight line-160
4.891
4.983
5.058
4.038
4.124
4.240
8.669
8.566
8.579
8.482
8.472
8.443
Straight line-200
5.182
5.342
5.506
4.376
4.526
4.618
10.331
10.309
10.300
9.921
9.986
9.989
Curve-140
5.258
5.256
5.291
4.299
4.236
4.218
7.657
7.610
7.586
7.578
7.579
7.607
Curve-160
5.301
5.333
5.410
4.199
4.274
4.408
9.909
8.763
8.599
8.566
8.586
8.566
Maximum per 100 mm range of change (%)
3.1
3.4
11.6
1.2
∗Thickness (mm) of steel-spring floating-slab-track bed; L: length; RMS: root mean square.Table 10
Vertical-vibration index of the floating slab.
Calculation working condition
Vertical displacement at the central part of the floating slab
RMS of vertical vibration acceleration in the middle of the floating slab
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
3.863
3.797
3.757
3.095
3.090
3.233
1.032
1.334
0.914
0.968
0.953
1.073
Straight line-160
3.832
3.765
3.760
3.192
3.217
3.189
1.311
1.139
1.551
1.147
1.249
1.084
Straight line-200
3.797
3.881
3.861
3.121
3.127
3.123
1.366
1.556
1.318
1.238
1.513
1.235
Curve-140
3.732
3.675
3.714
3.321
3.051
3.459
1.219
1.390
1.319
1.125
1.226
1.513
Curve-160
3.750
3.734
3.752
3.368
3.215
3.383
1.146
1.444
1.164
1.284
1.494
1.171
Maximum per 100 mm range of change (%)
2.2
13.4
36.2
23.3
L: length; RMS: root mean square.Table 11
Lateral-vibration index of the floating slab.
Calculation working condition
Lateral displacement of the floating slab
RMS of lateral-vibration acceleration of the floating slab
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
0.290
0.296
0.293
0.155
0.132
0.150
0.050
0.048
0.038
0.046
0.043
0.025
Straight line-160
0.339
0.374
0.383
0.155
0.201
0.224
0.058
0.051
0.053
0.040
0.043
0.030
Straight line-200
0.508
0.510
0.489
0.342
0.371
0.390
0.090
0.079
0.062
0.062
0.073
0.055
Curve-140
0.275
0.203
0.251
0.194
0.187
0.217
0.087
0.074
0.061
0.072
0.062
0.060
Curve-160
0.233
0.241
0.292
0.165
0.246
0.262
0.123
0.078
0.059
0.074
0.093
0.084
Maximum per 100 mm range of change (%)
26.2
49.3
37.0
42.3
L: length; RMS: root mean square.Table 12
Wheel-rail safety indicators of the floating-slab-track bed.
Calculation working condition
Derailment coefficient
Wheel-load shedding rate
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
0.181
0.184
0.184
0.190
0.189
0.189
0.154
0.157
0.158
0.152
0.153
0.155
Straight line-160
0.181
0.182
0.180
0.190
0.188
0.186
0.178
0.180
0.183
0.175
0.179
0.180
Straight line-200
0.186
0.189
0.195
0.182
0.189
0.194
0.267
0.284
0.280
0.283
0.284
0.278
Curve-140
0.301
0.306
0.305
0.293
0.307
0.312
0.283
0.287
0.287
0.289
0.294
0.294
Curve-160
0.271
0.272
0.278
0.268
0.272
0.273
0.283
0.286
0.295
0.302
0.305
0.306
Specified code limit [32, 33]
2.5
4.8
6.2
2.1
L: length.Table 13
Vehicle-track stationary stability and comfort indicators of the floating-slab track.
Calculation working condition
Lateral-stability index
Comfort index
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
1.701
1.700
1.698
1.708
1.708
1.709
0.737
0.736
0.734
0.741
0.746
0.749
Straight line-160
1.941
1.939
1.934
1.948
1.949
1.947
0.745
0.746
0.742
0.753
0.761
0.762
Straight line-200
2.003
2.006
2.002
2.007
2.005
2.002
0.930
0.942
0.942
0.931
0.936
0.936
Curve-140
2.120
2.117
2.112
2.122
2.121
2.120
1.105
1.099
1.092
1.119
1.119
1.119
Curve-160
2.179
2.173
2.174
2.181
2.179
2.176
1.207
1.197
1.187
1.200
1.202
1.198
Maximum per 100 mm range of change (%)
3.0
2.0
1.3
1.1
L: length.


### 4.2. Analysis of the Influence of the Track-Slab Thickness on Vehicle-Track Dynamics Characteristics

To investigate the influence of floating-slab thickness on the vehicle-track dynamics characteristics, analysis of the stability and safety of the vehicles using three different thicknesses for the steel-spring floating-slab-track beds, namely, 350, 450, and 550 mm (Tables9–13), was performed. Assuming that other parameters are unchanged, the vertical displacement of the rail, vertical-vibration acceleration of the rail, vertical displacement at the center part of the floating slab, radial acceleration of the vertical vibration of the floating slab, lateral displacement of the floating slab, root mean square of the lateral-vibration acceleration of the floating slab, derailment coefficient, wheel-load shedding rate, lateral stability of the vehicle body, and ride-comfort indicator of the floating slab were calculated. The results are shown in Tables 9–13.Calculations regarding thickness and dynamics characteristics suggest the following:(i)
In general, the influence of thickness variation of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds on the vertical displacement of rail, vertical-vibration acceleration of rail, vertical displacement of the central part of the floating slab, and the vertical-vibration acceleration of the floating slab is not apparent.(ii)
With the increase in thickness of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds, the vertical displacement of the rail slightly increased, lateral displacement and acceleration of the rail decreased to a great extent, and the lateral displacement and lateral acceleration of the rail slab decreased. In addition, as the slab thickness increases, the vertical displacement of the track bed increases, and lateral stability improves.(iii)
The thickness variation of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab tracks has little effect on the operational-safety and ride-comfort indexes of the intercity EMU trains.(iv)
With the increase of the thickness of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds, the wheel and rail safety index slightly increased, and the vehicle-stability and ride-comfort indicators were slightly reduced. The reason for the increase in the wheel-rail safety index is that the increase in the thickness of the floating slab increases the wheel-rail impact at the transition joint of the track slab.


### 4.3. Analysis of the Influence of Vertical-Support Damping of the Floating Slab on Vehicle-Track Dynamics Characteristics

To clearly investigate the influence of vertical-support damping of the floating slab (vertical damping of the vibration isolator) on the vehicle-track dynamics, no side-mounted isolator was provided during the analysis, the vertical-damping range of the isolator (Figure8) was set in the range of 0–100 kN·s/m, and other parameters were left unchanged. For the five working conditions under consideration, we calculated the dynamic response of the intercity EMU train running on the 3.6 m and 4.8 m prefabricated steel-spring floating-slab-track beds.Figures14 and 15 prove the following:(i)
The vertical acceleration of the car body (Figure12) and the vertical force between the wheel and rail are reduced with the increase of the vertical damping of the isolator, which can also improve the vertical dynamic performance of the vehicle. For the 3.6 m or 4.8 m long floating slab, when the vertical damping values are >20 and >50 kN·s/m, respectively, the vertical force between the wheel and rail and the vertical acceleration of the car body are not obvious.(ii)
An increase in vertical damping by the vibration isolator led to a decrease in the vertical displacement of the floating slab (Figure13), the vertical acceleration of the floating slab, and the vertical displacement of the rail. By appropriately increasing the isolator vertical damping, the vertical-displacement vibration response of the floating slab and the rail can be reduced to some extent. The economical and preferred vertical-damping ranges for the 3.6 m and 4.8 m long floating tracks are 10–30 and 40–60 kN·s/m, respectively, based on the vertical-vibration performance of the floating slab and the vertical deformation of the rail.(iii)
Calculation results show that the vertical damping by the vibration isolator is related to the lateral force of the axle, the derailment coefficient, the wheel-load shedding rate, the lateral acceleration of the vehicle body, the lateral stability of the vehicle body, the ride-comfort index, the vertical and lateral acceleration of the rail, the lateral displacement of the rail, the lateral displacement of the floating slab, and the lateral acceleration of the floating slab.Figure 14
Variation of the vertical acceleration of the train with vibration isolator damping. (a) 3.6 m floating-slab track. (b) 4.8 m floating-slab track.
(a)
(b)Figure 15
Variation of the vertical displacement of the train vibration isolator damping. (a) 3.6 m floating-slab track. (b) 4.8 m floating-slab track.
(a)
(b)


### 4.4. Analysis of the Influence of the Side-Mounted Isolator on Vehicle-Track Dynamics

It can be seen from the calculation results in Section4.3 that the amount of vertical displacement of the rail and the floating slab is large under various working conditions. In order to improve the local stiffness of the floating slab end and improve the dynamic performance of the floating-slab-track bed, a side-mounted isolator (Figure 16) was added at each end of the floating slab (Tables 14 and 15). We considered the situations without (Figures 17(a) and 18(a)) and with (Figures 17(b) and 18(b)) side-mounted isolators. For the five working conditions (straight lines and curves), the dynamic response of the intercity EMU train running on the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds was calculated.Figure 16
Structure of the side-mounted isolator between adjacent prefabricated steel-spring floating slabs. (a) Top view of the side-mounted isolator. (b) Side view of the side-mounted isolator.
(a)
(b)Table 14
Comparison of rail displacement of the floating-slab-track bed.
Calculation working condition
Vertical displacement of the rail (mm)
Lateral displacement of the rail (mm)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
−
+
−
+
−
+
−
+
Straight line-140
5.031
2.546
4.002
2.454
0.519
0.209
0.322
0.253
Straight line-160
5.102
2.582
4.173
2.471
0.523
0.218
0.329
0.263
Straight line-200
5.331
2.683
4.446
2.540
0.745
0.375
0.588
0.356
Curve-140
5.316
2.670
4.305
2.603
0.624
0.292
0.397
0.320
Curve-160
5.340
2.698
4.307
2.601
0.787
0.286
0.385
0.303
Maximum change (%)
49.77
40.78
63.66
39.45
L: length; −: side-mounted isolators are not installed; +: side-mounted isolators are installed.Table 15
Comparison of displacement of the floating-slab-track bed.
Calculation working condition
Vertical displacement of the floating slab (mm)
Lateral displacement of the floating slab (mm)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
−
+
−
+
−
+
−
+
Straight line-140
3.741
2.016
2.995
1.849
0.252
0.104
0.127
0.118
Straight line-160
3.759
2.033
3.006
1.850
0.347
0.111
0.196
0.125
Straight line-200
3.763
2.039
3.102
1.859
0.503
0.244
0.385
0.217
Curve-140
3.718
1.990
2.864
1.772
0.152
0.145
0.289
0.195
Curve-160
3.726
1.999
2.975
1.789
0.241
0.103
0.241
0.170
Maximum change (%)
46.48
40.07
68.01
43.64
L: length; −: side-mounted isolators are not installed; +: side-mounted isolators are installed.Figure 17
Prefabricated 3.6 m steel-spring floating-slab track. (a) Slab without side isolators. (b) Slab with side isolators.
(a)
(b)Figure 18
Prefabricated 4.8 m steel-spring floating-slab track. (a) Slab without side isolators. (b) Slab with side isolators.
(a)
(b)Results of the analysis revealed the following:(i)
For the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds, whether or not the side-mounted isolators are installed, the running-safety index and the ride-comfort index are basically similar during the intercity EMU train operation; it can be concluded that the impact of the side-mounted isolator on vehicle dynamic performance is not significant when the device is installed. Specifically, according to the analysis results, the addition of the side-mounted isolator can notably reduce the vertical force between the wheel and rail and the vertical-vibration acceleration of the vehicle body while slightly reducing the vertical stability of the vehicle body (Table16). Accordingly, the side-mounted isolator can improve the vertical dynamic performance of the vehicle.(ii)
For the 3.6 m and 4.8 m long steel-spring floating-slab-track beds, after adding the side-mounted vibration isolator, the track deformation and vibration-response index are significantly reduced. That is, side-mounted isolator improves the stability of the floating-slab bed significantly. In particular, the addition of a side-mounted isolator can greatly reduce the vertical dynamic displacement of the floating-slab track. On the basis of the current design of 3.6 m and 4.8 m long prefabricated floating slabs, four side-mounted isolators should be added. Under each working condition, the maximum vertical displacement of the rail is reduced from 5.340 to 2.698 mm and from 4.446 to 2.603 mm, respectively, indicating maximum reductions of ∼50% and ∼40%. Maximum vertical displacement of the floating slab is reduced from 3.763 to 2.039 mm and from 3.102 to 1.859 mm, respectively, indicating maximum reductions of ∼47% and ∼40%. Similarly, after the side-mounted isolator is added, the lateral displacement of the rail and floating slab was reduced within 40%–68%.Table 16
Comparison of vertical dynamic performance of the train on the floating-slab-track bed.
Calculation working condition
Vertical-vibration acceleration of the car body (g)
Vertical stability of the car body
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
−
+
−
+
−
+
−
+
Straight line-140
0.048
0.047
0.042
0.042
1.550
1.549
1.563
1.558
Straight line-160
0.055
0.053
0.050
0.048
1.825
1.823
1.829
1.826
Straight line-200
0.059
0.057
0.057
0.053
1.872
1.870
1.872
1.871
Curve-140
0.045
0.045
0.044
0.041
1.620
1.619
1.630
1.626
Curve-160
0.050
0.049
0.046
0.045
1.651
1.649
1.654
1.652
Maximum change (%)
3.64
7.01
0.12
0.32
L: length; −: side-mounted isolators are not installed; +: side-mounted isolators are installed.


## 4.1. Analysis of the Influence of the Length of the Floating Slab on Vehicle-Track Dynamics Characteristics

On the basis of the Chinese railway code [32, 33], the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab tracks were chosen as focus of this research. The safety, stability, and comfort of CRH6 intercity EMU trains running at different speeds and on different radial lines were investigated. Also, the performance and the stability of the track itself were analyzed. The working conditions (Tables 5–13) considered in this study are straight line-140, straight line-160, and straight line-200, which indicates that the intercity EMU train passes through a straight line at the speeds of 140, 160, and 200 km/h, respectively, and Curve-140 and Curve-160, which indicates that the intercity EMU train travels the curve sections of R = 1100 m and R = 1500 m at the speeds of 140 and 160 km/h, respectively. L values of 3.6 and 4.8 m indicate the two lengths of the prefabricated steel-spring floating-slab track. The calculation results of wheel-rail-system dynamic response, vehicle stability and comfort, rail dynamic response, and floating-slab dynamic response under various conditions are shown in Tables 5–8. These tabulated results suggest the following:(i)
When the intercity EMU train runs on either the 3.6 m or 4.8 m long prefabricated floating track beds, its dynamic performance is basically the same, although the stability of the 3.6 m floating slab is slightly lower than that of the 4.8 m slab.(ii)
Whether the intercity EMU train runs at 140, 160, or 200 km/h on the straight sections and 140 or 160 km/h in the curved sections (R = 1100 m or R = 1500 m) of either the 3.6 m or 4.8 m prefabricated floating-slab-track bed, the wheel-axle lateral force, the wheel-rail vertical force, the derailment coefficient, and the wheel-load shedding rate are all less than the specified code limit. The lateral acceleration of the body is lower than the specified code limit, the stability index is “excellent,” and the comfort rating is “comfortable.”Table 5
Wheel-rail safety indicators for floating-slab-track bed.
Calculation working condition
Wheel-axle lateral force (kN)
Wheel-rail vertical force (kN)
Derailment coefficient
Wheel-load shedding rate
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
12.331
12.188
76.143
77.956
0.184
0.186
0.155
0.158
Straight line-160
12.913
12.890
79.518
80.343
0.182
0.187
0.187
0.179
Straight line-200
14.979
13.598
88.942
90.019
0.198
0.196
0.281
0.269
Curve-140
29.434
29.592
86.295
89.322
0.309
0.315
0.302
0.294
Curve-160
28.538
28.423
86.950
87.968
0.290
0.291
0.336
0.310
Specified code limits [32, 33]
60.92
250
1.0
0.90
L: length.Table 6
Vehicle-stability and comfort indicators for floating-slab-track bed.
Calculation working condition
Vehicle-body lateral acceleration (g)
Vehicle-body vertical acceleration (g)
Lateral-stability index
Vertical-stability index
Comfort index
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
0.032
0.034
0.047
0.042
1.707
1.708
1.549
1.558
0.733
0.733
Straight line-160
0.033
0.036
0.053
0.048
1.945
1.947
1.823
1.826
0.741
0.743
Straight line-200
0.040
0.040
0.057
0.053
2.002
2.005
1.870
1.871
0.914
0.919
Curve-140
0.058
0.057
0.045
0.041
2.117
2.115
1.619
1.626
1.102
1.105
Curve-160
0.067
0.065
0.049
0.045
2.185
2.181
1.649
1.652
1.199
1.196
Specified code limits [32, 33]
0.10
0.15
2.5 (excellent)
2.5 (excellent)
2.0 (comfortable)
L: length.Table 7
Vibration indicators of rail on floating-slab-track bed.
Calculation working condition
Rail vertical displacement (mm)
Rail vertical acceleration RMS value (g)
Rail lateral displacement (mm)
Rail lateral acceleration RMS value (g)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
2.546
2.454
5.897
5.753
0.209
0.253
0.286
0.275
Straight line-160
2.582
2.471
6.575
6.477
0.218
0.263
0.385
0.375
Straight line-200
2.683
2.540
7.613
7.408
0.375
0.356
0.635
0.639
Curve-140
2.670
2.603
5.671
5.606
0.292
0.320
0.997
1.003
Curve-160
2.698
2.601
6.483
6.329
0.286
0.303
1.068
1.077
RMS: root mean square;L: length.Table 8
Vibration indicators of floating-slab track.
Calculation working condition
Track vertical displacement (mm)
Track vertical acceleration RMS value (g)
Track lateral displacement (mm)
Track lateral acceleration RMS value (g)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
Straight line-140
2.016
1.849
0.611
0.624
0.104
0.118
0.024
0.036
Straight line-160
2.033
1.850
0.644
0.782
0.111
0.125
0.025
0.038
Straight line-200
2.039
1.859
0.730
0.912
0.244
0.217
0.040
0.042
Curve-140
1.990
1.772
0.646
0.770
0.145
0.195
0.048
0.050
Curve-160
1.999
1.789
0.713
0.967
0.103
0.170
0.049
0.054
RMS: root mean square;L: length.Table 9
Vertical-vibration index of the rail.
Calculation working condition
Vertical displacement of the rail
RMS of lateral-vibration acceleration of the floating slab
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350∗
450∗
550∗
350∗
450∗
550∗
350∗
450∗
550∗
350∗
450∗
550∗
Straight line-140
4.794
4.860
4.919
4.045
3.980
4.025
7.754
7.701
7.682
7.572
7.577
7.671
Straight line-160
4.891
4.983
5.058
4.038
4.124
4.240
8.669
8.566
8.579
8.482
8.472
8.443
Straight line-200
5.182
5.342
5.506
4.376
4.526
4.618
10.331
10.309
10.300
9.921
9.986
9.989
Curve-140
5.258
5.256
5.291
4.299
4.236
4.218
7.657
7.610
7.586
7.578
7.579
7.607
Curve-160
5.301
5.333
5.410
4.199
4.274
4.408
9.909
8.763
8.599
8.566
8.586
8.566
Maximum per 100 mm range of change (%)
3.1
3.4
11.6
1.2
∗Thickness (mm) of steel-spring floating-slab-track bed; L: length; RMS: root mean square.Table 10
Vertical-vibration index of the floating slab.
Calculation working condition
Vertical displacement at the central part of the floating slab
RMS of vertical vibration acceleration in the middle of the floating slab
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
3.863
3.797
3.757
3.095
3.090
3.233
1.032
1.334
0.914
0.968
0.953
1.073
Straight line-160
3.832
3.765
3.760
3.192
3.217
3.189
1.311
1.139
1.551
1.147
1.249
1.084
Straight line-200
3.797
3.881
3.861
3.121
3.127
3.123
1.366
1.556
1.318
1.238
1.513
1.235
Curve-140
3.732
3.675
3.714
3.321
3.051
3.459
1.219
1.390
1.319
1.125
1.226
1.513
Curve-160
3.750
3.734
3.752
3.368
3.215
3.383
1.146
1.444
1.164
1.284
1.494
1.171
Maximum per 100 mm range of change (%)
2.2
13.4
36.2
23.3
L: length; RMS: root mean square.Table 11
Lateral-vibration index of the floating slab.
Calculation working condition
Lateral displacement of the floating slab
RMS of lateral-vibration acceleration of the floating slab
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
0.290
0.296
0.293
0.155
0.132
0.150
0.050
0.048
0.038
0.046
0.043
0.025
Straight line-160
0.339
0.374
0.383
0.155
0.201
0.224
0.058
0.051
0.053
0.040
0.043
0.030
Straight line-200
0.508
0.510
0.489
0.342
0.371
0.390
0.090
0.079
0.062
0.062
0.073
0.055
Curve-140
0.275
0.203
0.251
0.194
0.187
0.217
0.087
0.074
0.061
0.072
0.062
0.060
Curve-160
0.233
0.241
0.292
0.165
0.246
0.262
0.123
0.078
0.059
0.074
0.093
0.084
Maximum per 100 mm range of change (%)
26.2
49.3
37.0
42.3
L: length; RMS: root mean square.Table 12
Wheel-rail safety indicators of the floating-slab-track bed.
Calculation working condition
Derailment coefficient
Wheel-load shedding rate
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
0.181
0.184
0.184
0.190
0.189
0.189
0.154
0.157
0.158
0.152
0.153
0.155
Straight line-160
0.181
0.182
0.180
0.190
0.188
0.186
0.178
0.180
0.183
0.175
0.179
0.180
Straight line-200
0.186
0.189
0.195
0.182
0.189
0.194
0.267
0.284
0.280
0.283
0.284
0.278
Curve-140
0.301
0.306
0.305
0.293
0.307
0.312
0.283
0.287
0.287
0.289
0.294
0.294
Curve-160
0.271
0.272
0.278
0.268
0.272
0.273
0.283
0.286
0.295
0.302
0.305
0.306
Specified code limit [32, 33]
2.5
4.8
6.2
2.1
L: length.Table 13
Vehicle-track stationary stability and comfort indicators of the floating-slab track.
Calculation working condition
Lateral-stability index
Comfort index
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
350
450
550
350
450
550
350
450
550
350
450
550
Straight line-140
1.701
1.700
1.698
1.708
1.708
1.709
0.737
0.736
0.734
0.741
0.746
0.749
Straight line-160
1.941
1.939
1.934
1.948
1.949
1.947
0.745
0.746
0.742
0.753
0.761
0.762
Straight line-200
2.003
2.006
2.002
2.007
2.005
2.002
0.930
0.942
0.942
0.931
0.936
0.936
Curve-140
2.120
2.117
2.112
2.122
2.121
2.120
1.105
1.099
1.092
1.119
1.119
1.119
Curve-160
2.179
2.173
2.174
2.181
2.179
2.176
1.207
1.197
1.187
1.200
1.202
1.198
Maximum per 100 mm range of change (%)
3.0
2.0
1.3
1.1
L: length.


## 4.2. Analysis of the Influence of the Track-Slab Thickness on Vehicle-Track Dynamics Characteristics

To investigate the influence of floating-slab thickness on the vehicle-track dynamics characteristics, analysis of the stability and safety of the vehicles using three different thicknesses for the steel-spring floating-slab-track beds, namely, 350, 450, and 550 mm (Tables9–13), was performed. Assuming that other parameters are unchanged, the vertical displacement of the rail, vertical-vibration acceleration of the rail, vertical displacement at the center part of the floating slab, radial acceleration of the vertical vibration of the floating slab, lateral displacement of the floating slab, root mean square of the lateral-vibration acceleration of the floating slab, derailment coefficient, wheel-load shedding rate, lateral stability of the vehicle body, and ride-comfort indicator of the floating slab were calculated. The results are shown in Tables 9–13.Calculations regarding thickness and dynamics characteristics suggest the following:(i)
In general, the influence of thickness variation of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds on the vertical displacement of rail, vertical-vibration acceleration of rail, vertical displacement of the central part of the floating slab, and the vertical-vibration acceleration of the floating slab is not apparent.(ii)
With the increase in thickness of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds, the vertical displacement of the rail slightly increased, lateral displacement and acceleration of the rail decreased to a great extent, and the lateral displacement and lateral acceleration of the rail slab decreased. In addition, as the slab thickness increases, the vertical displacement of the track bed increases, and lateral stability improves.(iii)
The thickness variation of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab tracks has little effect on the operational-safety and ride-comfort indexes of the intercity EMU trains.(iv)
With the increase of the thickness of the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds, the wheel and rail safety index slightly increased, and the vehicle-stability and ride-comfort indicators were slightly reduced. The reason for the increase in the wheel-rail safety index is that the increase in the thickness of the floating slab increases the wheel-rail impact at the transition joint of the track slab.


## 4.3. Analysis of the Influence of Vertical-Support Damping of the Floating Slab on Vehicle-Track Dynamics Characteristics

To clearly investigate the influence of vertical-support damping of the floating slab (vertical damping of the vibration isolator) on the vehicle-track dynamics, no side-mounted isolator was provided during the analysis, the vertical-damping range of the isolator (Figure8) was set in the range of 0–100 kN·s/m, and other parameters were left unchanged. For the five working conditions under consideration, we calculated the dynamic response of the intercity EMU train running on the 3.6 m and 4.8 m prefabricated steel-spring floating-slab-track beds.Figures14 and 15 prove the following:(i)
The vertical acceleration of the car body (Figure12) and the vertical force between the wheel and rail are reduced with the increase of the vertical damping of the isolator, which can also improve the vertical dynamic performance of the vehicle. For the 3.6 m or 4.8 m long floating slab, when the vertical damping values are >20 and >50 kN·s/m, respectively, the vertical force between the wheel and rail and the vertical acceleration of the car body are not obvious.(ii)
An increase in vertical damping by the vibration isolator led to a decrease in the vertical displacement of the floating slab (Figure13), the vertical acceleration of the floating slab, and the vertical displacement of the rail. By appropriately increasing the isolator vertical damping, the vertical-displacement vibration response of the floating slab and the rail can be reduced to some extent. The economical and preferred vertical-damping ranges for the 3.6 m and 4.8 m long floating tracks are 10–30 and 40–60 kN·s/m, respectively, based on the vertical-vibration performance of the floating slab and the vertical deformation of the rail.(iii)
Calculation results show that the vertical damping by the vibration isolator is related to the lateral force of the axle, the derailment coefficient, the wheel-load shedding rate, the lateral acceleration of the vehicle body, the lateral stability of the vehicle body, the ride-comfort index, the vertical and lateral acceleration of the rail, the lateral displacement of the rail, the lateral displacement of the floating slab, and the lateral acceleration of the floating slab.Figure 14
Variation of the vertical acceleration of the train with vibration isolator damping. (a) 3.6 m floating-slab track. (b) 4.8 m floating-slab track.
(a)
(b)Figure 15
Variation of the vertical displacement of the train vibration isolator damping. (a) 3.6 m floating-slab track. (b) 4.8 m floating-slab track.
(a)
(b)


## 4.4. Analysis of the Influence of the Side-Mounted Isolator on Vehicle-Track Dynamics

It can be seen from the calculation results in Section4.3 that the amount of vertical displacement of the rail and the floating slab is large under various working conditions. In order to improve the local stiffness of the floating slab end and improve the dynamic performance of the floating-slab-track bed, a side-mounted isolator (Figure 16) was added at each end of the floating slab (Tables 14 and 15). We considered the situations without (Figures 17(a) and 18(a)) and with (Figures 17(b) and 18(b)) side-mounted isolators. For the five working conditions (straight lines and curves), the dynamic response of the intercity EMU train running on the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds was calculated.Figure 16
Structure of the side-mounted isolator between adjacent prefabricated steel-spring floating slabs. (a) Top view of the side-mounted isolator. (b) Side view of the side-mounted isolator.
(a)
(b)Table 14
Comparison of rail displacement of the floating-slab-track bed.
Calculation working condition
Vertical displacement of the rail (mm)
Lateral displacement of the rail (mm)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
−
+
−
+
−
+
−
+
Straight line-140
5.031
2.546
4.002
2.454
0.519
0.209
0.322
0.253
Straight line-160
5.102
2.582
4.173
2.471
0.523
0.218
0.329
0.263
Straight line-200
5.331
2.683
4.446
2.540
0.745
0.375
0.588
0.356
Curve-140
5.316
2.670
4.305
2.603
0.624
0.292
0.397
0.320
Curve-160
5.340
2.698
4.307
2.601
0.787
0.286
0.385
0.303
Maximum change (%)
49.77
40.78
63.66
39.45
L: length; −: side-mounted isolators are not installed; +: side-mounted isolators are installed.Table 15
Comparison of displacement of the floating-slab-track bed.
Calculation working condition
Vertical displacement of the floating slab (mm)
Lateral displacement of the floating slab (mm)
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
−
+
−
+
−
+
−
+
Straight line-140
3.741
2.016
2.995
1.849
0.252
0.104
0.127
0.118
Straight line-160
3.759
2.033
3.006
1.850
0.347
0.111
0.196
0.125
Straight line-200
3.763
2.039
3.102
1.859
0.503
0.244
0.385
0.217
Curve-140
3.718
1.990
2.864
1.772
0.152
0.145
0.289
0.195
Curve-160
3.726
1.999
2.975
1.789
0.241
0.103
0.241
0.170
Maximum change (%)
46.48
40.07
68.01
43.64
L: length; −: side-mounted isolators are not installed; +: side-mounted isolators are installed.Figure 17
Prefabricated 3.6 m steel-spring floating-slab track. (a) Slab without side isolators. (b) Slab with side isolators.
(a)
(b)Figure 18
Prefabricated 4.8 m steel-spring floating-slab track. (a) Slab without side isolators. (b) Slab with side isolators.
(a)
(b)Results of the analysis revealed the following:(i)
For the 3.6 m and 4.8 m long prefabricated steel-spring floating-slab-track beds, whether or not the side-mounted isolators are installed, the running-safety index and the ride-comfort index are basically similar during the intercity EMU train operation; it can be concluded that the impact of the side-mounted isolator on vehicle dynamic performance is not significant when the device is installed. Specifically, according to the analysis results, the addition of the side-mounted isolator can notably reduce the vertical force between the wheel and rail and the vertical-vibration acceleration of the vehicle body while slightly reducing the vertical stability of the vehicle body (Table16). Accordingly, the side-mounted isolator can improve the vertical dynamic performance of the vehicle.(ii)
For the 3.6 m and 4.8 m long steel-spring floating-slab-track beds, after adding the side-mounted vibration isolator, the track deformation and vibration-response index are significantly reduced. That is, side-mounted isolator improves the stability of the floating-slab bed significantly. In particular, the addition of a side-mounted isolator can greatly reduce the vertical dynamic displacement of the floating-slab track. On the basis of the current design of 3.6 m and 4.8 m long prefabricated floating slabs, four side-mounted isolators should be added. Under each working condition, the maximum vertical displacement of the rail is reduced from 5.340 to 2.698 mm and from 4.446 to 2.603 mm, respectively, indicating maximum reductions of ∼50% and ∼40%. Maximum vertical displacement of the floating slab is reduced from 3.763 to 2.039 mm and from 3.102 to 1.859 mm, respectively, indicating maximum reductions of ∼47% and ∼40%. Similarly, after the side-mounted isolator is added, the lateral displacement of the rail and floating slab was reduced within 40%–68%.Table 16
Comparison of vertical dynamic performance of the train on the floating-slab-track bed.
Calculation working condition
Vertical-vibration acceleration of the car body (g)
Vertical stability of the car body
L = 3.6 m
L = 4.8 m
L = 3.6 m
L = 4.8 m
−
+
−
+
−
+
−
+
Straight line-140
0.048
0.047
0.042
0.042
1.550
1.549
1.563
1.558
Straight line-160
0.055
0.053
0.050
0.048
1.825
1.823
1.829
1.826
Straight line-200
0.059
0.057
0.057
0.053
1.872
1.870
1.872
1.871
Curve-140
0.045
0.045
0.044
0.041
1.620
1.619
1.630
1.626
Curve-160
0.050
0.049
0.046
0.045
1.651
1.649
1.654
1.652
Maximum change (%)
3.64
7.01
0.12
0.32
L: length; −: side-mounted isolators are not installed; +: side-mounted isolators are installed.


## 5. Conclusions

Research shows that prefabricated steel-spring floating-slab track, which is traditionally used for low-speed lines, can actually be used for higher-speed lines and can also achieve significant results for the vibration and noise reduction function under conditions of driving safety and operational stability. The study is an exploratory study of the application of the new prefabricated steel-spring floating-slab track in the field of high-speed rail transportation. From the above analysis, we can draw the following conclusions:(1)
A prefabricated steel-spring floating-slab track can be applied in urban express rail transit systems and can meet the requirements of safety, comfort, and stability of high-speed vehicles while efficiently reducing noise. This is a great guiding principle for the popularization of prefabricated steel-spring floating-slab tracks on high-speed railway lines.(2)
The dynamic performance of the 3.6 m and 4.8 m long prefabricated steel-spring floating slabs is comparable, although the stability of the former is slightly lower than that of the latter. For the shorter length of the prefabricated steel-spring floating-slab track is often used in curved sections, it can be seen that when the short section of the steel-spring floating-slab track is used in curved sections, the stability of the vehicle-track coupled dynamic system will be reduced but will still remain within the acceptable range of the engineering project.(3)
For same-length prefabricated steel-spring floating-slab tracks, as slab thickness increases, the vertical displacement of the rail increases slightly, and lateral stability improves. A change of slab thickness has a little effect on the running-safety and ride-comfort indexes of the intercity EMU train, but the running stability of the vehicle can be slightly improved with increasing thickness of the floating slab. It is shown that the thickness of the track slab has a great effect on the vehicle-track coupled dynamic system; the limit of the thickness of the track slab can achieve good economic results within the acceptable range of the project.(4)
When the intercity EMU train runs at 140, 160, or 200 km/h in the straight section and at 140 or 160 km/h in the curved section (R = 1100 m and R = 1500 m) on the 3.6 m or 4.8 m long prefabricated steel-spring floating-slab track, the wheel-axle lateral force, the wheel-rail vertical force, the derailment coefficient, and the wheel-weight reduction rate are each less than the specified code limit value. The lateral and vertical accelerations of the vehicle body are each lower than the specified code limit value. In addition, the stability index is “excellent,” and the comfort rating is “comfortable.”The analysis results have subverted the previous perception that steel-spring floating-slab track can only be used for low-speed lines. The steel-spring floating-slab track in this project research has achieved very good results and can be promoted as a theoretical basis for the application of floating-slab track for high-speed railway lines.(5)
Appropriately increasing the vertical-support damping of the floating-slab track can improve the vertical dynamic performance of the vehicle, reduce the vertical displacement of the rail, and lower the vibration response of the floating-slab track. We comprehensively considered the dynamic performance of the vehicle and the stability of the track and found that the optimal ranges of vertical damping for the 3.6 m and 4.8 m long prefabricated floating-slab-track isolators are 10–30 and 40–60 kN·s/m, respectively. Choosing the vibration isolator damping in this range can obtain good vibration isolation effect and can save engineering investment.(6)
Adding side-mounted vibration isolators at the joint of the floating slabs can greatly improve the stability of the floating slab and appropriately reduce the vertical vibration response of the vehicle. The invention of the side-mounted vibration isolator is a new exploration to improve the stability of the floating-slab track. Compared with the past, simply increasing the thickness of the track slab to improve the overall quality of the track slab to achieve improved stability, the side-mounted vibrator is undoubtedly economical and effective.



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*Source: 4515319-2020-03-19.xml*